chanical transitions of the single motors, which determine the free energy .... the chemical reaction as a Markov process, as in [87] for kinesin, the motor attains .... Inserting Eq.(2.7) into the latter expression yields a relation between the rate constants ..... Kinesin transforms the energy of ATP hydrolysis into mechanical work, ...
Max-Planck-Institut für Kolloid- und Grenzflächenforschung Abteilung Theorie und Biosysteme
Coupled Molecular Motors: Network Representation & Dynamics of Kinesin Motor Pairs
Dissertation zur Erlangung des akademischen Grades “doctor rerum naturalium” (Dr.rer.nat.) in der Wissenschaftsdisziplin “Theoretische Physik”
eingereicht an der
Mathematisch-Naturwissenschaftlichen Fakultät der Universität Potsdam
von
Corina Keller
Potsdam, den 26. Oktober 2012
Statement of authenticity I declare that this thesis is my own work and has not been submitted in any form for another degree at any university or other institute of tertiary education. Information derived from the published and unpublished work of others has been acknowledged in the text and a list of references is given. Hiermit versichere ich, dass ich die vorliegende Arbeit selbstständig verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel benutzt habe.
Corina Keller
Abstract Cargo transport by two molecular motors is studied by constructing a chemomechanical network for the whole transport system and analyzing the cargo and motor trajectories generated by this network. The theoretical description starts from the different nucleotide states of a single motor supplemented by chemical and mechanical transitions between these states. As an instructive example, we focus on kinesin-1, for which a detailed chemomechanical network has been developed previously. This network incorporates the chemical transitions arising from ATP hydrolysis on both motor heads. In addition, both the chemical and the mechanical transition rates of a single kinesin motor were found to depend on the load force experienced by the motor. When two such motors are attached via their stalks to a cargo particle, they become elastically coupled. This coupling can be effectively described by an elastic spring between the two motors. The deflection of this spring determines the mutual interaction force between the motors and, thus, affects all chemical and mechanical transition rates of both motors. As a result, cargo transport by two motors leads to a combined chemomechanical network, which is quite complex and contains a large number of motor cycles. However, apart from the single motor parameters, this complex network involves only two additional parameters, the coupling parameter which correponds to the spring constant of the elastic coupling between the motors and the rebinding rate for an unbound motor. We show that these two parameters can be determined directly from cargo trajectories as well as from trajectories of individual motors. Both types of trajectories are accessible to experiment and, thus, can be used to obtain a complete set of parameters for cargo transport by two motors. From statistical analysis of the trajectories we establish the activity state diagram which separates the parameter regime, in which the cargo is mainly pulled by one motor, from the regime, in which it is more likely that the cargo is pulled by two motors. The coarse grained activity state network then implies that the average 1-motor run time depends on the rebinding rate but not on the coupling parameter whereas the 2-motor run time depends on the coupling parameter but not on the rebinding rate. In general, this parameter separation is reflected in the different transport properties. The average cargo run properties of the motor pair are combinations of the activity state properties, and thus, depend on both motor pair parameters. We find that, in general, a motor pair is able to cover a larger run length than the single motor. The dependence of the single motor dynamics on the nucleotide concentrations is also reflected in the dynamics of the motor pair. In agreement with recent experiments, we find that for small product concentrations the cargo run length of the motor pair decreases with increasing ATP concentration and that it is strongly increased compared to the single motor run length for small ATP concentrations. This is quite counterintuitive since the 1motor runlength increases with increasing ATP concentration and the 2-motor run length is rather independent of the ATP concentration for small product concentrations. Even though unbinding is rapid under these conditions, rebinding of the inactive motor dominates over the unbinding of the active motor. We also incorporate external load forces both into the force balance relations and into the chemomechanical network for the motor pair. We find that the force velocity relations of the acitivity states as well as of the cargo run are in good agreement with experimental data for substall forces.
Contents 1 Introduction 1.1 Molecular motors . . 1.2 Kinesin motors . . . . 1.3 Cooperative transport 1.4 Overview . . . . . . .
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2 Methods 2.1 Enzymatic networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Description of single motors 3.1 Chemomechanical coupling . . . 3.2 Network description . . . . . . . 3.3 Motor dynamics and observables 3.4 Specification of transition rates . 3.5 Physically meaningfull parameter 3.6 Summary and discussion . . . .
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4 Chemomechanical network of a motor pair 4.1 Theoretical description . . . . . . . . . 4.2 Structure of the motor pair network . . 4.2.1 Combination of motor cycles . . 4.2.2 Pathways on the network . . . . 4.2.3 Activity states of the motor pair 4.3 Simulation details and observables . . . 4.4 Summary and discussion . . . . . . . .
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5 Properties of cargo transport by motor pairs 5.1 Coupling parameter and rebinding rate . . . . . . . . . . . . . . 5.1.1 Motor pair trajectories . . . . . . . . . . . . . . . . . . . 5.1.2 Activity state properties . . . . . . . . . . . . . . . . . . 5.1.3 Summary and discussion . . . . . . . . . . . . . . . . . . 5.2 Dependence of transport properties on nucleotide concentrations 5.2.1 Influence on the motor pair properties . . . . . . . . . . . 5.2.2 Cargo run length in the limit of small ADP and P . . . . 5.2.3 Velocity of 2-motor runs and cargo runs . . . . . . . . . . 5.2.4 Summary and discussion . . . . . . . . . . . . . . . . . . 5.3 Dependence of transport properties on external load force . . . . 5.3.1 Force velocity relation . . . . . . . . . . . . . . . . . . . . 5.3.2 Relevance of cargo rotation for small cargo . . . . . . . . 5.3.3 Operation modes of 2-motor runs . . . . . . . . . . . . . 5.3.4 Summary and discussion . . . . . . . . . . . . . . . . . . 6 Conclusion
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7 Bibliography
101
Appendix
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A Details of the 7 state model B Supplementary plots
i ix
C Application of external load force
xvii
D Glossary
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1 Introduction 1.1 Molecular motors Molecular motors are biological machines that perform mechanical work on the nanometer scale within living organism. Motor proteins are the driving force behind most of the active transport of proteins and vesicles in the cytoplasm [1]. Intracellular transport proceeds along the cytoskeleton, a complex cell-spanning network that is composed of different types of filaments linked by molecular motors [2, 3]. Today it is known that most transport processes in cells are performed by molecular motors [4]. A particular class of motors are cytoskeletal motors that move along cytoskeletal filaments and transport intracellular cargo such as vesicles or whole organelles [5, 6, 7]. Powered by the hydrolysis of adenosine triphosphate (ATP) into adenosine diphosphate (ADP) and inorganic phosphate (P), these motors are able to transport large vesicles along filaments within cells. For large particles such as vesicles and organelles, the intracellular fluid is too crowded and they are too large to diffuse to their destinations [8]. Motor proteins act as transporters of such large cargo and, in contrast to diffusion, provide directed transport to a given destination. Fig. 1.1 shows a sketch of the cytoskeleton with cytoskeletal motors and outlines experimental methods to study the transport properties of such motors.
Families of cytoskeletal motors Cytoskeletal motors which walk along cytoskeletal filaments and transport intracellular cargo such as vesicles, as shown in Fig. 1.1(a) and (b), are distinguished in three superfamilies: myosin, dynein and kinesin motors [14, 9]. The most prominent example of a motor protein is the muscle protein myosin II which fuels the contraction of muscle fibers [15]. Quantitative in vitro motility experiements for myosin motors also led to the discovery of new classes of motors, dynein and kinesin superfamilies [16]. The prototypes of the three superfamilies of cytoskeletal motors are shown in Fig. 1.1(c). These are the microtubule-based motor conventional kinesin, the actin-based motor skeletal muscle myosin and the microtubule motor cytoplasmic dynein. These motors consist of a protein dimer of two heavy chains which form motor domains and a stalk and several light chains that consist of associated polypeptides [10]. In general, a molecular motor is of the size of about one hundred nanometers and is able to pull vesicles which are large compared to its own size [2]. Motor proteins travel in a specific direction along a specific filament. Myosin motors walk on actin filaments whereas kinesin and dynein are microtubule motors [9, 16] as shown in the close-up in Fig. 1.1(a). Microtubules are protofilaments of tubulin molecules, and kinesin and dynein move along through interaction with tubulin [14]. Actin filaments are linear polymers of actin subunits, they are the thinnest filaments of the cytoskeleton [2]. An important difference between molecular motors and macroscopic engines is that molecular motors operate in an environment where the fluctuations arising from thermal noise are significant. Due to the noisy environment of the cell interior, the distance a motor molecule can move on a filament is finite. On average, it makes about 100 steps [11] before it detaches from the filament. Hence, it has been found, that motors often cooperate in cargo transport in order to increase this distance [17].
1
1.1 Molecular motors
(a) Actin filament
Vesicle carried
Vesicle carried
by kinesins
by dynein
Microtubule Vesicle carried by myosin
Centrosome
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(c) Motor domain
Associated polypeptides Motor domain
Stalk
Stalk Associated polypeptides
Kinesin
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Associated polypeptides
Myosin
Dynein
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Figure 1.1: (a) Sketch of the cytoskeleton of an eucaryotic cell: microtubules are oriented from the centrosome to the cell periphery, actin filaments are distributed within the cell. The close-up shows transport by cytoskeletal motors: Kinesin transports vesicles towards the cell periphery, i.e., the plus end of a microtubule, whereas dynein walks towards the centrosome, i.e., the minus end of a microtubule. Myosin motors walk on actin filaments. Based on a picture adapted from [9]. (b) Aspects of transport by cytoskeletal motors: Transport of (1) centrosomal components, (2) of intermediate filaments, (3) of ribonucleoprotein complexes. Interactions of (4) myosin, kinesin and dynein with the microtubule plus-end complex, (5) dynein with actin, (6) kinesin with actin and (7) dynein with catenin. Picture adapted from [6]. (c) Prototypes of the three superfamilies of cytoskeletal motors: the microtubule motor conventional kinesin, the actin-based motor skeletal muscle myosin and the microtubule motor cytoplasmic dynein. The motors consist of a protein dimer of two heavy chains, wound around each other in a coiled-coil structure, which form motor domains and a stalk and several light chains which consist of associated polypeptides. Picture adapted from [10]. There are two different basic experimental setups in vitro to adapt intracellular movement and transport of molecular motors: (d) Bead assay: kinesin-coated polymer beads move along immobilized microtubules, used e.g. in [11]. (e) Gliding assay: taxol-stabilized microtubules glide across kinesin-coated glass surfaces, used e.g. in [12]. Pictures (d) and (e) taken from [13].
2
1 Introduction The mechanism of movement is thought to be on principle the same for all stepping motors [18]. Most kinesin motors walk with a stepsize of 8 nm [19] towards the plus end of a microtubule which entails transporting cargo from the centre of the cell towards its periphery [2]. This motion is a processive movement [16], i.e., the motor is firmly bound to the microtubule and executes many steps before it detaches. Dynein motors also transport cargo along microtubules, but towards their minus end. Dyneins are much larger and more complex than kinesin or myosin motors. Their step size is variable and varies between 8 and 32 nm, depending on the load [20]. Myosin motors are responsible for muscle contraction and walk along actin filaments. Myosin V, for example, has a step size of about 36 nm [21, 22], according to the helical repeat length of actin filaments [23]. Myosin V spirals left-handed [23] around the filament towards the plus end, whereas myosin VI, for instance, walks in opposite direction, towards the minus end [1] of the actin filament. The role that cytoskeletal motors play beyond membrane transport is also included in the sketch in Fig. 1.1(b), for example transport of centrosomal components or intermediate filaments, or interactions of motor proteins with the different filaments or components of the filaments. For more details about these phenomena see [6].
Motors in motility assays During the past 40 years, molecular motors have been studied in great detail [24]. The structure of molecular motors has been resolved by X-ray crystallography, e.g. in [25], revealing that kinesins and myosins share a similar fold structure whereas dyneins have a very different structure. Typically, intracellular movement and transport processes by stepping motors are mimicked in in vitro motility assays, as reviewed in [26]. The basic experimental setups as shown in Fig. 1.1(d) and (e) are bead assays, where the translocation of motor-coated polymer beads along immobilized microtubules are examined, as used e.g. in [11, 19] or gliding assays, where microtubules are gliding across motor-coated glass surfaces, as used e.g. in [12]. In both cases, the motion of either the microtubule or of the bead can then be monitored in a light microscope. Since a gliding assay typically involves several motors, the bead assay is more useful to study the properties of a single motor. Furthermore, the bead can be captured in an optical trap which allows to apply external loads to the motor.
1.2 Kinesin motors In 1985, the Vale lab [12] published their discovery of a large protein complex, that they found to be distinct from myosin and dynein motors. They concluded that these translocators represent a novel class of motility proteins and called them ’kinesin’, from the Greek word kinein, to move. Kinesins are, like dyneins, microtubule-associated motor proteins [16]. Members of the kinesin family vary in shape. The prominent conventional kinesin or kinesin-1 motor consists of two identical heavy chains, which fold into two enzymatic motor domains or heads, and two identical light chains, which mediate the binding to cargo particles in vivo [7]. Each motor head contains a nucleotide binding pocket for the binding and hydrolysis of ATP and a microtubule binding site, by which the head attaches to the filament. A rather flexible necklinker [14, 2] combines the two chains into dimeric or two-headed kinesin-1. Single kinesin motors have been typically studied using bead assays, in which the motor is attached to a bead that can be manipulated by an optical trap, see, e.g. [27, 28, 29]. It
3
1.2 Kinesin motors is known that kinesin is a processive motor [16] which walks in discrete steps of 8 nm [19] towards the plus end of microtubules [2]. On average, it makes about 100 mechanical steps [11] in a hand-over-hand fashion [30, 31] before it detaches from the filament. This stepping mechanism displaces the unbound kinesin head by 16 nm and the center of mass by 8 nm and this step size corresponds to the lattice constant of the microtubule [4]. During each mechanical step, kinesin hydrolizes one ATP molecule [32, 33]. The mechanical steps are fast and completed within 15 µs [27] whereas the chemical transitions take several ms [34, 35]. In the presence of a sufficiently large load force, kinesin performs frequent backward steps [36, 27]. Single kinesin motors have been studied in much detail, both experimentally and theoretically: Since the motion of molecular motors is stochastic, an appropriate theoretical description for their behavior is given by driven Brownian ratchets and networks [37]. Theoretical applications to molecular motor systems are reviewed in [4, 38, 39] and, particularly regarding two particles in [40, 41, 42]. In this study, we will focus on conventional kinesin or kinesin-1, which has been studied by many single molecule experiments, see, eg., [27, 43, 28, 29, 44], and for which we have a good theoretical description on the single motor level [45, 46]. The latter showed that observations, which have been obtained by different experimental groups, can be described quantitatively in the framework of a network theory for single kinesin-1 motors [45]. This theory is based on discrete motor states as defined by the nucleotide occupancy of the two motor heads as well as chemical and mechanical transitions between these states. One important property of these networks is that they involve several motor cycles, which provide the free energy transduction between ATP hydrolysis and mechanical work. As one varies the nucleotide concentrations and the external load force, the fluxes on these cycles change and different cycle fluxes dominate for different parameter regimes. In this sense, the chemomechanical networks of a single motor as introduced in [45, 46] contain several competing motor cycles. Kinesin motor properties The motor velocity is thought to be an important property, which characterizes the performance of a motor [4]. Accessible control parameters for the motor velocity are the nucleotide concentration [28, 29] of ATP, ADP or P and the external load force [27, 47]. The motor velocity increases with the ATP concentration and exhibits a saturation behaviour. In the limit of small external load forces the motor velocity as a function of the ATP concentration obeys Michaelis-Menten kinetics [28]. Furthermore, the motor velocity decreases with increasing load forces applied on the motor against the direction of its motion [27]. In experimental biophysics, the activity of molecular motors is observed with many different experimental approaches. Various motor properties such as the motor velocity [43, 27, 28], the bound state diffusion coefficient and the run length [28] or ratio of forward to backward steps [27, 36] were measured as a function of the ATP concentration and the external load force.
4
1 Introduction
1.3 Cooperative transport In vivo, cargo transport is usually performed by small teams of motors that may belong to the same or to different motor species, see, e.g., [48, 49, 50]. Each motor of such a team has a finite run length, after which it unbinds from the filament. Furthermore, such an unbound motor is likely to rebind to the filament as long as the cargo is still connected to the filament by the other motors. As a consequence, the number of actively pulling motors is not fixed but fluctuates. The associated binding and unbinding processes are stochastic and have been shown, in previous theoretical studies, to play a crucial role for the overall transport properties of cargos that are transported by teams of identical motors [17, 51], by two teams of antagonistic motors that perform a stochastic tug-of-war [52, 53], and by an actively pulling team supported by a passively diffusing team [54, 55]. Experimental observations, both in vitro [56, 57, 58, 59, 60, 61] and in vivo [62, 63] confirm the importance of stochastic motor binding and unbinding as predicted theoretically.
Experimental approaches Experimentally, the field of cooperative motors has been explored using various approaches: Focusing on the question of fractional stepping in cargo trajectories by assemblies of several kinesin-1 motors [59], in vitro gliding motility assays have been used where microtubules coated with quantum dots are driven over a glass surface by a known number of kinesin-1 motors. The authors of [59] find successive 4 nm steps for motor pairs and distinct jumps on the order of 10 nm for three motors and conclude that there is no coordination between the motors. Examination of bead assays covered with several kinesin-1 motors by [56] reveal that the velocity does not dependent on the number of motors pulling the beads, whereas the run length is strongly increased if more motors are involved. Similar results have been found in another study with bead assays in [61], where multiple kinesin-1 motor proteins are attached to giant vesicles. Here, long traveling distances up to the millimeter range appear, whereas the velocity of the assembly is unchanged compared to the velocity of a ’cargo-free’ motor. The authors of [61] estimate the number of actively pulling motors to typically 5–10 motors. The experimental findings discussed in [60] support the multimotor model by [17]. Here, in vitro gliding assays in which kinesin-1 motors move the microtubule against an external load provided by an magnetic field gradient, are used. The authors of [60] find a force-velocity relation for multiple motors that is similar to that of a single motor. A method of organizing exactly two kinesin-1 motors on linear scaffolds is presented in [58]. They find that such motor pairs produce longer average run lengths than single kinesins. Examination of the load dependent properties of such motor assemblies [64] reveals that motor pairs can produce forces and move with velocities beyond the abilities of single kinesin molecules. On the other hand, they observe that such large forces occur only rarely and thus conclude that the arrangement of motors on the filament dictates how loads are distributed within the motor pair. Furthermore, they find that cargo is mainly pulled by one kinesin motor which then provides an alternative explanation for previous observations which reveal that intracellular transport depends weakly on kinesin number. The same lab recently studied also two identical coupled myosin V motors [65]. Recently, another method to ensure that exactly two motors attached to a bead has been presented in [66]: two kinesin motors bind to an antibody which has two distinct binding sites. The authors of [66] consider the single motor velocity as a tuning parameter for multiple motor
5
1.3 Cooperative transport transport. They find that the cargo run length of a motor pair is strongly increased compared to the single motor run length if the velocity of the individual motors in the assembly is reduced. The reduction of the velocity is controlled by the ATP concentration. Bi-directional transport by the processive motors kinesin and myosin V has been studied in vitro in [57]. For such kinesin-myosin assemblies, the diffusive behavior of myosin V on microtubule leads to an elongation of the processive run length of the kinesin motor and vice versa on actin filaments. The authors propose that ’one motor acts as a tether for the other’ which prevents its diffusion. In vivo studies of bi-directional transport by kinesin and dynein motors are presented in [62] and [63]. Inside living Dictyostelium cells the authors of [62] find many weak dyneins in a tug-of-war against one strong kinesin. In [63] they study the fungus Ustilago maydis and find one dynein motor opposing several kinesin-3 motors. Both studies reveal reversals during the motion of the complexes.
Theoretical studies There are several theoretical studies on cooperation of multiple motors which use different approaches: A stochastical model for motor molecules that cooperate in large groups, for instance, has been presented in [67] applying for a nonprocessive linear type of myosin in muscles and for motility assays with a high concentration of motor molecules. A decade later, a theoretical study [17] about transport of cargo particles that are pulled by several molecular motors was published. This model covers various numbers of motors on the cargo and shows that the properties depend on the maximal number of motors, i.e., increasing this maximal number leads to a strong increase of the average run length of the cargo particle. Since unbinding of individual motors in the assembly is incorporated in [17], the actual number of motors attached to the bead may vary. Discussion of the cooperative cargo transport by several motors with focus to the question how motor teams can drag cargos through a viscous environment is studied in [8]. Moreover, in [68] the collective dynamics of interacting molecular motors experiencing an external force is presented, using a lattice model. The authors of [68] find a strong relation between effective dynamic interactions between the motors and their performance. The multiple kinesin motors model in [69] includes stochastic fluctuations and predicts average velocities that differ from preceding results. In [69], an unevenly shared force leads to an increase of the assembly run length and for small loads to slightly slower velocities than for single motors. With the incorporation of nonlinear force– velocity relations and stochastic load sharing to this model [70], the model exhibits robustness of the mechanical properties of the transport if only two or three motors transport the cargo. For larger numbers of motors, the authors of [70] predict that the effective diffusion of the cargo driven by multiple motors under load increases by an order of magnitude compared to the single motor. A theoretical framework for the analysis of intracellular transport processes by several molecular motors, which may belong to one or several motor species, is described in [55]. Here, the the performance of cargos is explained by the transport properties of the individual motors and their interactions. Distinct transport regimes for two identical, elastically coupled motors are calculated in [51] which explain the differences in results by previous studies on the same topic. The elastic coupling may reduce the motor velocity and/or enhance the unbinding of the motors from the filament. The resulting regimes, derived by
6
1 Introduction explicit calculations and time scale arguments, can be explored experimentally by varying the elastic coupling strength. Concerning bi-directional transport aspects, i.e., transport by different types of motors which walk in opposing directions, a symmetric two-state model has been introduced in [71] which is based on numerical simulations of the two-state model which contains a large number of motors and which reveals bimodal velocity distributions. The tug-of-war mechanism has been studied in [52] incorporating transport properties of individual motors derived form single motor experiments. In contrast to previous expectations, the authors of [52] find that such a tug-of-war is highly cooperative.
1.4 Overview In the previous theoretical studies on cooperative cargo transport, the chemical and mechanical transitions of the single motors, which determine the free energy transduction of these motors, were not taken into account explicitly but only implicitly via the resulting force–velocity relations. In the present study, we will introduce and study a more detailed theoretical description, in which we take the chemomechanical motor cycles of the individual motors into account. In this refined theory, each of the four motor heads of the two dimeric motors can bind ATP, hydrolyze it into ADP and P, and subsequently release the latter nucleotides, first P and then ADP. The reverse chemical reactions corresponding, e.g., to ADP binding and ATP synthesis, are also included as well as the coupling of these chemical reactions to the forward and backward mechanical steps. It is important to note that all of these transitions are stochastic as well. Furthermore, we impose cyclic balance conditions [72, 46] on all motor cycles, which ensures that the network description satisfies both the first and second the law of thermodynamics. We focus on a pair of kinesin-1 motors and start from the chemomechanical network for a single motor as developed in [45, 46]. We attach the two motors by their flexible stalks to a cargo particle and describe these two stalks as elastic springs. Since these two springs are only coupled via the cargo, which is taken to be rigid, we can effectively describe the system by one linear spring with an effective spring constant, the coupling parameter K. This coupling generates an elastic force between the two motors, as soon as one of these motors performs a mechanical step. It is important to note that the force arising from a mechanical step of one motor affects the chemical and mechanical transition rates of both motors. As a result, we obtain a uniquely defined chemomechanical network for the motor pair. Even though this network is rather complex and contains a large number of motor cycles, it involves only two additional parameters: the coupling parameter K as well as the rebinding rate πsi of a single motor. A systematic analysis of the cargo and motor trajectories allows the separation of these two parameters: The properties of a 1-motor run during which the cargo is pulled by one active motor, depend on the rebinding rate but not on the coupling parameter, whereas 2motor runs during which the cargo is pulled by two active motors, depend on the coupling parameter but not on the rebinding rate. We study the influence of these intrinsic motor pair parameters as well as the influence of the external control parameters such as the nucleotide concentrations and the load force, on the motor pair dynamics.
7
1.4 Overview
Structure This thesis is organized as follows. Chapter 2 explains the theoretical framework of enzymatic networks, the approach from graph theory to the single motor network and the numerical method used for the coupled motor pair system. Chapter 3 gives an overview of the description of the single motor model as introduced in [45, 46] and of the modifications to this model. Chapter 4 addresses the incorporation of this single motor description into a model for motor pairs. We describe how the networks of the individual motors are assembled to a complex motor pair network, which then is studied by simulations. In Chapter 5 we focus on the properties of motor pairs strating from trajectories and focusing in particular on distinct activity state properties and their dependence on the different parameters. In the first part, we focus on the internal parameters such as the coupling parameter and the rebinding rate and in the last parts on the external control parameters like the concentration of nucleotides and the influence of the external load force. We adapt our system to various experimental data sets for the single motor and compare the simulation results of the motor pair system to experimental motor pair data. Finally, we conclude with a summary and outlook in Chap. 6
List of publications • C. Keller, F. Berger, S. Liepelt and R. Lipowsky. Network complexity and parametric simplicity for cargo transport by two molecular motors. Submitted to Journal of Statistical Physics (2012). • F. Berger, C. Keller, R. Lipowsky and S. Klumpp. Elastic coupling effects in cooperative transport by a pair of molecular motors. Submitted to Cellular and Molecular Bioengineering (2012). • F. Berger, C. Keller, S. Klumpp, and R. Lipowsky. Distinct transport regimes for two elastically coupled molecular motors. Physical Review Letters 108, 208101 (2012). • F. Berger, C. Keller, M. J. I. Müller, S. Klumpp, and R. Lipowsky Co-operative transport of molecular motors. Biochemical Society Transactions 39, 1211-1215 (2011).
8
2 Methods The dynamics and activity of molecular motors are current topics of study in theoretical and experimental biophysics. As mentioned above, a lot of researchers have been and are presently engaged in understanding how such motors operate in environments with nonnegligible thermal noise. Considering the direct movement of a molecular motor in such an environement intuitively leads to a ratchet model. On the other hand, focusing on the conversion of chemical energy into mechanical work implies a network approach. In this chapter, we will focus on enzymatic networks and computational methods. For the single motor network as depicted below in Fig. 3.3 one can calculate the steady state solution of the master equation as introduced in [73]. Alternatively, one can approach this network with a graph-theoretic method [74, 75]. With regard to the relatively small size of the single motor network graph, it is more convenient to use this latter method which will be outlined below. In the second part, we discuss numerical methods which can be used to approach molecular motors. In particular, we describe a specific Monte Carlo method, the Gillespie algorithm [76]. The state space for the motor pair system, which will be introduced in Chap. 4, is rather complex and hardly amenable to analytical calculations, which is why we use a numerical method to generate random walks on this motor pair network. The advantage of the Gillespie algorithm is that it is exact apart from numerical round-off errors.
2.1 Enzymatic networks Molecular motors are relatively small objects which operate in a noisy environment, which on a first glance, implies overdamped Brownian motion. It is not obvious, that they exhibit directed movement which seems to be in conflict with the second law of thermodynamics, which states that a perpetuum mobile of the second kind is impossible. First in 1912, Smoluchowski [77] and later Feynman [78] proposed a thought experiment on this conflict, the so-called Brownian ratchet or Feynman-Smoluchowski ratchet. The basic idea is a simple machine consisting of a paddle wheel and a ratchet which should be able to convert thermal energy from the environment into mechanical work. Feynman stated, that the system must be out of the thermodynamical equilibrium in order to convert fluctuations into directed motion. More recently, in 1993 Magnasco [79] stated that if the fluctuations were correlated, macroscopic movement may appear, and introduced an assymetric potential which governs the motion of the object and a correlated fluctuating force against which the object performs work. This example is governed by either the Langevin Equation [80], describing the position of the object, or alternatively, by a Fokker-Planck Equation [81, 82, 73], focusing on the probability to find the object at a certain time at a certain position. Ratchets and ratchet models of molecular motors described in detail in, e.g., [83, 84, 80, 40]. The link between ratchet models and the network method, which will be introduced next, is given by the definition of localized transitions [38, 85]. Discretization of the continuous potentials then leads “from ratchets to networks” [86]. Concerning molecular motors, the argument for discretization of the energy landscape of a ratchet model is mainly based on the separation of time scales in the motor system. The chemical reactions are rather long compared to the mechanical stepping. Additionally, the chemistry of the system is explicitely accessible to experiments. Thus, experimentally obtained reaction rates can be implemented as the transition rates of the network model [45].
9
2.1 Enzymatic networks
Kinetic diagrams The movement of a molecular motor on the filament is determined by the chemical reaction taking place within the catalytic domains of the motor which is coupled to a conformational change of the motor domains that causes translational motion of the motor. Considering the chemical reaction as a Markov process, as in [87] for kinesin, the motor attains different chemical states and the transitions between these states can be interpreted as a random walk on the network constituted by the chemical states. In 1977, Hill [75] introduced a method based on network theory in enzyme kinetics and emphazised the importance of a cycle representation for enzymatic networks in the steady state. The basic idea is, to interpret the enzymatic activity of a molecular motor as different chemical states the motor can attain, and the change in the chemical conformation of the motor as transitions between these states. The states of such a network are governed by certain dwell times which are exponentially distributed in a continuous–time Markov process [73]. The average dwell time in a state is affiliated with all transitions rates out of this state. When we perform a random walk on such a network, the probability to find the motor in a certain state at a certain time is determined by the Master equation [73]. a)
b)
ADP E
E . ADP
ATP
1
4
2
3
P E . ATP
E . ADP . P
Figure 2.1: (a) Four state cycle C ± of an ATPase E. (b) Kinetic diagram of an ATPase. Figures adapted from [75].
As an example of enzymatic networks [75], the enzymatic cycle of an ATPase is shown in Fig. 2.1(a), the ATPase is indicated by E and the cycle will be denoted by C ± . In the first step, the ATPase binds ATP, then hydrolyzes the ATP to the products ADP and phosphat P, followed by the subsequential release of the products. The direction of the hydrolysis is indicated by the arrow and is refered to as ’positive’ direction of the cycle C + and synthesis of ATP takes place in the clockwise or ’negative’ direction of the cycle C − . The different confirmations of the ATPase then are refered to as the discrete states i =1, 2, 3, 4 of the ATPase in the kinetic diagram in Fig. 2.1(b). The transitions between the states i are stochastic and governed by transition rates ωij . The rate law for an elementary reaction which is first order with respect to a reactant X =ATP, ADP or P of the example in Fig. 2.1 is given by ωij = −
d[X] =κ ˆ ij [X] dt
(2.1)
with the concentration of the nucleotides denoted as [X] and the first order rate constant κ ˆ ij in the units of s−1 µM−1 . Transitions without reactants involved, follow the zero-order
10
2 Methods reaction law with ωij = κij in the units of s−1 . The average time which is spent in state i before a transition occures is defined as
hτi i =
X
−1
ωij
(2.2)
Πij = ωij hτi i .
(2.3)
j
and with this, the probability for a transition from state i to j can be written as
The chemical potentials µX of the example in Fig. 2.1 are given by µX = µ0X + kB T ln[X]
(2.4)
with the standard chemical potential µ0X . In the following, when refered to the nucleotides in a subscript, we will denote ATP by T, ADP by D and phosphate by P. Then, the thermodynamical force which drives the cycle can be written as the chemical potential differnce ∆µ(C) in the cycle C ∆µ(C + ) = µT − µD − µP
(2.5)
= µ0T − µ0D − µ0P + kB T ln
[ATP] [ADP][P]
(2.6)
At equilibrium, the chemical potential difference vanishes, ∆µ = 0, and thus, with the equilibrium constant ! [ADP][P] µ0D + µ0P − µ0T Keq ≡ = exp − , (2.7) [ATP] eq kB T the expression in Eq. (2.6) reads as ∆µ(C + ) = kB T ln
[ATP] Keq [ADP][P]
.
(2.8)
At equilibrium, each elementary process should be equilibrated by its reverse process, i.e., each transition pair of forward and backward rates must obey the detailed balance Pjeq ωij = eq ωji Pi
(2.9)
with this probability Pieq to be in state i at full equilibrium, i.e., in thermal, chemical, and mechanical equilibrium. Applying this to each transition of the network in Fig. 2.1, multiplication leads to the relation κ ˆ 12 κ23 κ34 κ41 κ21 κ32 κ ˆ 43 κ ˆ 14
[AT P ] [ADP ][P ]
eq
=
P1eq P2eq P3eq P4eq = 1. P2eq P3eq P4eq P1eq
(2.10)
Inserting Eq. (2.7) into the latter expression yields a relation between the rate constants κ ˆ12 κ23 κ34 κ41 = κ21 κ32 κ ˆ43 κ ˆ14 Keq
(2.11)
11
2.1 Enzymatic networks which implies that not all the rate constants of a cycle can be chosen independently, they must obey this relationship. A more general expression, the balance condition on cycles at any arbitrary steady state, is given by Ω(C + ) ∆µ = exp (2.12) Ω(C − ) kB T where the relation in Eq. (2.8) is used and the transition rate product of cycle Cνd with the number ν and the direction d = ± of the cycle as given by d
d
Ω(C ) =
C Y
(2.13)
ωij .
|iji
The rate constants of each cycle in the network must obey the balance condition in Eq. (2.12). Since the network example in Fig. 2.1 consists of a single cycle C ± , the subscript ν has been neglected here. The local excess fluxes along the edges |iji of a kinetic diagram such as in Fig. 2.1(b) are defined as ∆Jij ≡ Jij − Jji = − (Jji − Jij ) = −∆Jji (2.14) with the local fluxes Jij ≡ ωij Pist
and Jji ≡ ωji Pjst .
(2.15)
In steady state, one can define a cyclic flux J(C d ) in each direction d = ± as the average rate at which cycles C d are compled for the network in Fig. 2.1(b). Then, the net cycle flux ∆J(C) for completion of any cycle C of the network can be calulated via
∆J(C) = J C + − J C − with the simple relation
∆µ Ω C+ J C+ = = exp − J (C ) Ω (C − ) kB T
(2.16)
(2.17)
where Eq. (2.12) has been used. Inspection of this equation shows, that ∆J(C) and ∆µ have the same sign in agreement with the second law of thermodynamics. Note, that at full + equilibrium ∆µ = 0 and the net cycle flux around any closed cycle vanishs J C = J (C − ) and ∆J(C) = 0 and ∆Jij = 0. Then, the transition rates ωij must obey the detailed balance condition for all edges in the network. Note, that here, the cycle fluxes are equal to the two local transition fluxes since the network example in Fig. 2.1 consists of a single cycle C ± . This is not necessarily true for networks with more than one cycle. Details of the transition rates and the actual balance conditions of the single motor network of kinesin will be introduced in chapter 3.
Graph-theoretic method In 1847, Kirchhoff [74] proposed a method to solve the equations which arise in electrical circuits with regard to the conservation of charge and energy. The nodal rule says that at any node in an electrical circuit, the sum of currents flowing into that node is equal to the sum of currents flowing out of that node. The mesh rule says that the directed sum of the electrical
12
2 Methods potential differences around any closed network is zero. Adapted to graph theory [88, 89] this method enables to derive steady state solutions of network graphs, as will be outlined here. For instance, the network graph shown in Fig. 2.2(a) consists of three different states, displayed by the three vertices i = 1, 2, 3 and three edges hiji refering to the transition rates ωij between the states i and j. A spanning tree Tm of a network graph contains all vertices of the network graph but no cycles, i.e., not every edge is a part of a spanning tree. The subscript m = 1, 2, ..., M distinguishes the different spanning trees of a network graph. The three spanning trees, T1 , T2 and T3 of the example network in Fig. 2.2(a) are shown in Fig. 2.2(b). In a next step, one establishes for each vertex j so-called arborescences Am,j of each spanning tree Tm . All edges in an arborescence Am,j of the root vertex j are directed towards this root vertex, as shown in Fig. 2.2(c). For each arborescence of a root vertex one defines a transition rate product which is given by Ω(Am,j ) ≡
Y
ωij
(2.18)
|iji
where the product includes all directed edges |iji to the root vertex j which are included in the arborescence Am,j . Summation over all spanning trees for each root vertex j separately then yields XY Ωj ≡ ωij (2.19) m |iji
which is a multilinear polynomial in the transition rates ωij , as the example in Fig. 2.2(c) illustrates, and corresponds to unnormalized steady state probabilities to be in vertex j Pjst = Ωj /Ω with the normalization factor Ω = a)
P
j
(2.20)
Ωj .
c) 1 2
3
1: ω 21ω 31 + ω 23ω 31 + ω 21ω 32 = Ω1
b) 2: ω 12ω 31 + ω 13ω 32 + ω 12ω 32 = Ω2
3: ω 13ω 21 + ω 13ω 23 + ω 12ω 23 = Ω3
Figure 2.2: (a) Network graph with three vertices i and three edges hiji, with the transition rates ωij . The colors are a guide for the eye. (b) The three spanning trees Tm for the network in (a). (c) The arborescences Am,j for the root vertex j = 1 (top row), j = 2 (middle row) and j = 3 (bottom row) with the transition rate products Ωj . Arrows indicate the directed edges.
13
2.2 Numerical method This approach via the graph-theoretic method to obtain the steady state solution of a network is effectively and elegant. However, this method has its limits, it is practicable only for rather simple networks with not to much cycles, since the number of spanning trees increases strongly with the complexity of the network.
2.2 Numerical method Several decades of studying molecular motors gave rise to a rich pool of models and approaches in this field. Starting from an atomistic view and looking into enzymatic reactions, molecular dynamics simulations are often used, see e.g. in [90]. The hierarchy of biomolecular simulations, as drafted in Fig. 2.3 in adaption from [91], shows the different approaches on different time and/or length scales. At each scale, the parameters are determined by averaging over the underlying scale. The region of our interest lies in length scale inbetween 8 nm, corresponding to the stepsize of a kinesin motor and several micrometer which corresponds to the run length of a kinesin motor and in time scales inbetween microseconds, corresponding to the transition time for a mechanical step, and several seconds, which is the run time of a kinesin motor. In Fig. 2.3 this length scale reaches down to the region of atoms, covered by molecular dynamics simulations, and the time scale corresponds clearly to the region of segments in mesoscale dynamics. On the scale of micrometers, we are interested in the interplay of motors with filaments and/or the environement, which is for example covered by ratchet models [86]. A more coarse grained view on the transport in cells, focusing on motion, allows analytical approaches as in [51], at the loss of information internal processes. Motor dynamics on filaments including the chemical configurational changes in the motor domains can be studied using Monte Carlo techniques, e.g. as in [92, 93]. The Gillespie algorithm [76] is of Monte Carlo type and its advantage is that it generates a statistically correct solution of a stochastic equation.
Molecular Dynamics (MD) simulations Molecular dynamics (MD) is a computer simulation of physical movements of molecules and atoms, the fundamental unit. Simulation techniques are used to mimic the behavior of molecules and allow insight into molecular motion on an atomic scale. The resulting forces of interactions between atoms or molecules and the potential energy are defined by molecular mechanics force fields. One then solves the Newton’s equations to describe the motion of the particles. MD simulations cover large systems of usually about ten thousands of atoms and up to a million atoms or hundreds of angstroms in a coarse grained model which summarizes groups of atoms, covering time scales of several nanoseconds. The result of MD simulations is a set of parameters that describe forces between assemblies of atoms which determines macroscopic thermodynamic properties of the system. This method is applied for instance to studies of enzyme reaction mechanisms as done by [90, 94] or calculations of binding energies which is essential, e.g., for drug design as done by [95, 96]. Concerning molecular motors this method may be usefull for example to identify confirmational changes of the nucleotide binding pockets of a motor domain during or after hydrolysis of ATP. However, the focus of the present motor pair system on the aspect of motion of the entire protein or even the entire assembly which is out of scale for MD methods to sample all the underlying structures. As indicated in Fig. 2.3 by the highlighted lines, the current problem can be located in the mesoscale region, since it ranges from microseconds and several
14
2 Methods electrons
atoms
segments
grids ORGAN BEHAVIOUR
TIME years
hours CELLULAR LEVEL SIMULATIONS
minutes
CHEMOTAXIS seconds MESOSCALE DYNAMICS
microsec
proteins & membranes
nanosec MOLECULAR DYNAMICS F=MA
Binding Energies Tertiary structure Group properties QSAR
picosec QUANTUM MECHANICS H Ψ =EΨ femtosec 1A
Force Field Charges
1 nm 10 nm micron
mm
cm DISTANCE
Figure 2.3: The hierarchy of biomolecular simulations adapted from [91]. Red lines indicates the length and time scales of interest in the present thesis.
nanometer, which refers to mechanical stepping, to seconds and micrometer, refering to cargo run length and run times.
Monte Carlo (MC) simulations A Monte Carlo simulation is a statistical sampling technique using a repeated random search algorithm, e.g., Metropolis-Hastings algorithm [97, 98]. It is a numerical method to study dynamic behavior of molecules, such as diffusion or chemical reactions. There are various MC methods applied to many problems, e.g. [99, 100, 101]. The protocol typically starts from a given molecular conformation with a given set of transition rates. The next step generates a new conformation by random displacement of one or more atoms. Calculation of the change in the potential energy of the current step and comparsion with the previous step triggers either acceptance of the current step or rejection. After acceptance, the new sample is used as the starting conformation for the next random displacement. Rejection leads back to the start and a different random displacement is generated on the old sample. This protocol results in a Boltzmann ensemble of conformations. Note, that in chemistry, this method is referred to as “dynamic” MC method whereas in physics the terminology is “kinetic” MC method. A particular variant is provided by the Gillespie algorithm [76], which generates a statistically correct trajectory of a stochastic equation, see next subsection.
15
2.2 Numerical method
Gillespie algorithm In 1977, Gillespie proposed an stochastic simulation algorithm to analyze chemical reactions involving large numbers of species with complex reaction kinetics [76]. The algorithm is a Monte Carlo type method and it is an exact procedure for numerically simulating the time evolution of coupled chemical reactions, i.e., it generates a statistically correct trajectory of a stochastic equation, e.g. the master equation. As computers have become faster, the algorithm has been used to simulate increasingly complex systems. The algorithm is particularly useful for simulating chemical reactions within cells where the number of reagents is typically large. Such systems are typically governed by distinct transition rates ωij . The average dwell time P hτi i = 1/ j ωij in state i is determined by all outgoing rates of this state as in Eq. (2.2) and the transition probability Πij from state i to j can be written as Πij = ωij hτi i the product P of the dwell time and the outgoing transition rate as in Eq. (2.3). The condition j Πij = 1 ensures a transition for each event. The following procedure generates a random walk on a network graph: (i) Start at time t = 0 in state i with a given set of transition rates ωij . (ii) Choose an exponentially distributed dwell time τi for state i: a random number 0 < rτ < 1 is chosen which determines the dwell time via τi = −hτi i ln(rτ ). Choose a second random number 0 ≤ rΠ ≤ 1 in order to determine the transition event e. Assign intervals of size Πij to each state j which is connected to the current state i and compare the random number rΠ successively to these intervals. If rΠ is within an interval, accept the transition to the new state j, otherwise reject it. (iii) Update the clock time te = te−1 + τi and use this new state as a starting point for the next transition. (iv) Last step: Stop the simulation at time te ≥ tmax . Note that tmax should be sufficiently large compared to the time required to complete a cycle Cν of the network. A detailed description of the adaption of this algorithm to the motor pair network is given in section 4.3.
16
3 Description of single motors This chapter provides information about the description of the single motor model as introduced and extensively analyzed in [45, 46], including the chemomechanical coupling of ATP hydrolysis to the spatial displacement on a filament. We introduce the network description of the kinesin motor governed by transition rates as well as the aspects of macroscopic observables such as motor velocity. The second part of this chapter focuses on the modifications to this basic description such as the specification of transition rates with regard to the force dependence and arising time scaling constrains.
3.1 Chemomechanical coupling The dynamics of a single motor can be described by a continuous-time Markov process on a discrete state space or network [102, 45], the vertices of which represent different chemical states of the two motor heads and the edges describe transitions between these states. Following in general the description in [102], the main idea of kinesin motor networks is basically outlined here.
Catalytic domains of kinesin motor heads Kinesin transforms the energy of ATP hydrolysis into mechanical work, moving unidirectionally along a microtubule. Each motor head has two seperate binding sites, one for binding to the microtubule and a catalytical domain, which adsorbs and hydrolyses ATP. The catalytic domain can be empty (E) or can be occupied by ATP (T ) or the cleavage products (θ) of the hydrolysis or contain ADP (D), which is indicated by the four different states in Fig. 3.1(a). Summarizing the cleavage transition |T θi and the P release transition |θDi into the single transition |T Di leads to a reduced network with the three states E, T , and D which a single motor head can attain and six transitions, where each represents binding or release of nucleotides in Fig. 3.1(b). The chemical network for a kinesin motor with two identical heads is shown in Fig. 3.1(c). Each motor head can attain three states: empty (E), contain ATP (T ) or ADP (D). Two motor heads can then occupy 32 = 9 states, each of which is connected to four other states, indicated by the solid lines, via chemical transitions that describe the binding or release of ATP or ADP as well as the combined process of ATP cleavage and the release of P. Per convention, the right head represents the leading head in respect of the direction of motion. If a motor head is empty or contains ATP, it is strongly bound to the filament whereas it is only loosely bound to the filament if it contains ADP [103, 104], which is indicated by the vertical displacement of the blue bullets labeled with D in respect to the bullets labeled with E or T . Thus, the two states (EE) and (T T ) are characterized by two strongly bound heads and should be irrelevant for the processive motion of kinesin. Focusing on states for which the two heads have different chemical composition leads to the 6-state network in Fig. 3.1(d). Then, in Fig. 3.1(e), the six chemical states are rearranged and enumerated, each state is connected to two neighbours.
Spatial displacement The spatial displacement of the motor along the filament is shown in Fig. 3.2 as the network with the six chemical states in Fig. 3.1(e), which a motor can attain on each binding site of the
17
3.1 Chemomechanical coupling filament. The network at motor position x on the filament is connected to the neighbouring binding site x+ℓ via the mechanical transition indicated by the dashed line between the states (D, T ) and (T, D). The motion of a kinesin motor is in a hand-over-hand fashion which implies that a mechanical step requires the interchange of position of leading and trailing head of the motor. a)
b)
c) EE
D
E
TE
P θ
D
D
P E
ADP
D
DD
T
D
ADP T
E
T
E
ATP
ET
ATP
d)
D
TT
T
e)
D
E
E
D
ET
TE
T
D
T
D
5 T
4 E
3
D
D
ET
6 TE
D
D
1 E
T
2
Figure 3.1: (a) Chemical network of a kinesin motor head domain describing the metabolic motor cycle of ATP hydrolysis. The motor head can be empty (E), occupied by ATP (T ), the cleavage products (θ) or occupied by ADP (D). (b) Summarizing the cleavage transition |T θi and the P release transition |θDi into the single transition |T Di leads to a reduced network with the three states E, T , and D and six transitions, where each represents binding or release of nucleotides. (c) Chemical network for a kinesin motor with two heads. Each motor head is described by the three states E, T , and D, which implies a chemical network with 32 = 9 states with the 18 transitions hiji indicated by the lines. (d) Reduced 6-state network omitting states where both motor heads are in the same chemical state (T T ), (DD)and (EE) and the respectiv transitions. Then, rearranging and enumeration of the chemical states leads in (e) to a 6-state network of a kinesin motor with two heads. Figures are adapted from [102].
In principle, there are several possibilities to fullfill these condition for mechanical stepping, see Fig. 3.1(c). Previous studies, e.g., [72, 45], show that “the processive motion of kinesin is governed, to a large extent, by those motor states for which the two heads differ in their chemical composition”. It is shown in [102] that the only mechanical transition which is compatible with single motor data is the transition from (DT ) to (T D). Therefore, in this
18
3 Description of single motors work, we consider the the latter transition as the mechanical stepping transition of the kinesin motor, as shown in Fig. 3.2.
T
E
D
TE
D
D
D
ET
T
D
TE
D
E
E
T
ET
0
D
D
T
D
TE
D
E
E
T
ET
D
D
T
D
TE
D
E
E
T
ET
D
D
E
T
x l
2l
3l
Figure 3.2: Spatial displacement of the motor along the filament. The x-axes at the bottom represents the filament, the four digits on the axes represent four filament binding sites at intervals of the stepsize ℓ. The motor can attain six chemical states at each binding site, indicated by the black frame. The solid lines of the networks correspond to chemical transitions. The broken lines between a (DT ) and a (T D) state of two neighbouring binding sites represents a spatial displacement of the motor along the filament, the motor mechanically steps with the stepsize ℓ.
An equivalent representation for the kinesin network in Fig. 3.2 is provided by a network graph in a more compact form as in [45]. However, studying run length of a motor or motor pair system requires a finite motor run and thus unbinding events must be incorporate into the motor model which leads to the 7-state model of kinesin as in [45]. Since a motor head is only loosely bound to the filament if it contains ADP [103, 104], the (DD) state seems most likely the state from which unbinding events happen. Neglecting the transitions from state ED and T D to the state DD in the following, since it has been found that they are rare [105], results in the 7-state model as depicted in Fig. 3.3. This network graph consists of seven chemical states and one unbound state, it has 16 chemical and two mechanical transitions and one unbinding transition.
Motor states and transitions The seven states of the chemomechanical network in Fig. 3.3 will be labeled by i = 1, ..., 7 and the transitions |iji from state i to j are described by the transition rates ωij . In the network, the transition |iji corresponds to a directed edge whereas a non-directed edge will be denoted by hiji. In Fig. 3.3, the direction of the ATP hydrolysis is indicated by filled arrows and takes place during the transitions |61i, |34i and |57i. The mechanical step is marked by the dashed line, which represents both the mechanical forward step |25i and the mechanical backward step |52i. The motor can dissociate with unbinding rate ω70 from state i = 7, in which both heads contain ADP and the motor is loosely bound to the filament [103], to the unbound motor state i = 0, which represents an absorbing state as far as the directed stepping of the motor along the filament is concerned. The open arrow indicates an unbinding event from the (DD) state.
19
3.1 Chemomechanical coupling
Motor cycles and dicycles The network graph in Fig. 3.3 has six cycles Cν , three of which are fundamental cycles in the sense of mathematical graph theory [106]: The forward cycle F = h12561i, the backward cycle B = h45234i and the thermal slip cycle T = h16571i. The directed cycles F + = |12561i and B + = |45234i both involve the hydrolysis of a single ATP molecule coupled to a single mechanical step whereas the dicycles T + and T − involve both the hydrolysis of one ATP molecule and the synthesis of such a molecule. The alternative forward cycle FDD = h12571i as well as the two enzymatic slip cycles E = h1234561i and EDD = h1234571i during which two ATP molecules are hydrolysed without mechanical stepping, can be constructed as linear combinations of the fundamental cycles. By convention, a directed cycle Cν+ is completed in the counter-clockwise direction.
DD
DD
7
0
T D
TE
E
6
1 D
2
F T
D
5
T
B ET
3
E
D
4
Figure 3.3: Chemomechanical network of kinesin: The nucleotide binding pocket of each motor head can be empty (E), occupied by ATP (T ), or occupied by ADP (D). In general, the two-headed motor can then attain 32 = 9 states but two of these states, namely (EE) and (T T ), should not play any prominent role for the processive motion of kinesin, since, for these two states, both motor heads are strongly bound to the microtubule. Neglecting the states (EE) and (T T ), one arrives at the 7-state network shown here. The dashed line represents the mechanical stepping and the filled arrows indicate the direction of the ATP hydrolysis that takes place during the transitions |61i, |34i and |57i. The transition |25i corresponds to the mechanical forward step which implies the convention that the leading head of the dimeric motor is the one on the right, whereas the trailing head is the one on the left. Motor dissociation from the filament is only possible if the motor is in the (DD) state. The unbound motor state will be denoted by i = 0 and represents an absorbing state as far as the directed walks of a single motor are concerned. The forward cycle F contains the mechanical forward step |25i and the backward cycle B the mechanical backward step |52i. The thermal slip cycle cycle T includes hydrolysis |57i and synthesis |16i of one ATP molecule without mechanical stepping. Figure in the style of [45].
The transition |25i corresponds to the mechanical forward step. Thus, we use the implicit convention in Fig. 3.3 that the leading head of the dimeric motor is the head on the right, whereas the trailing head is the one on the left. Therefore, the network contains pairs of states that can be transformed into each other by swapping the positions of the leading and the trailing head. It seems plausible to assume the chemical transition rates between the different motor states do not depend on the spatial ordering of the two heads. However, because each cycle is characterized by a certain balance condition as introduced in section 2.1
20
3 Description of single motors and because the rates for forward and backward mechanical stepping are very different, at least one chemical transition of the cycle B must be different from the corresponding transition of the cycle F. As in Ref. [45] , we choose this transition to be |54i, the rate of which is then determined by the balance conditions rather than by the rate of the transition |21i.
Energy conservation Following the concept of energy transduction in [72, 102], we find that the changes in internal energy ∆Uij during the transition from state i to state j are quantified by the balance of chemical energy ∆µij from the reservoirs for ATP, ADP, and P, the mechanical work Wij which the motor performs against the load force F and the released heat Qij . Then, the local energy balance relation is given by ∆Uij = Uj − Ui = ∆µij − Wij − Qij .
(3.1)
The chemical potential difference ∆µ in Eq. (2.5) depends on the chemical potentials µX of ATP, ADP and P. The chemical potential difference ∆µij during the transition |iji is positive if the transition contains the binding of a nucleotide, negative for the release of a nucleotide an zero elsewise. The mechanical work is affiliated with the mechanical stepping of a motor, which is the transition h25i in Fig. 3.3 and can therefor be defined as W25 = ℓF and W52 = −ℓF with the stepsize ℓ and Wij = 0 for all chemical transitions. As in [102] we use the convention that a load force F > 0 pulls against the direction of motion of the motor and the convention that Qij > 0 if it increases the internal energy of the heat reservoir. Using the transition entropies ∆Sij as defined by [107] during the transition |iji ωij ∆Sij ≡ kB ln ωji
= −∆Sji .
(3.2)
and the relation Q = T ∆S, the local energy balance can be rewritten as ∆Uij = ∆µij − ℓij F − kB T ln
ωij ωji
!
(3.3)
.
or, more convinient as ωij exp(−Ui /kB T ) = exp(−∆µij + ℓij F )/kB T ωji exp(−Uj /kB T )
(3.4)
for each edge of the network in Fig. 3.3. For a directed cycle Cνd in Fig. 3.3 the sum the local energy balance yields ∆U
Cνd
ν,d X
ν,d X
ωij = (Uj − Ui ) = ∆µij − Wij − kB T ln ωji |iji |iji
= 0.
(3.5)
21
3.2 Network description
3.2 Network description The 7-state model of the kinesin motor in Fig. 3.3 is determined by the transition rates between the different motor states. In this section, we explain the parametrization of these transition rates established in [45]. For the description of the force dependence of the chemical rates we follow another procedure, according to the generalization in [93].
Parametrization of transition rates The transition rates ωij between two states i and j depend on the molar nucleotide concentrations [X], with X = ATP, ADP or P, and on the load force F . It will be convenient to express the force dependence in terms of the dimensionless force ℓF F¯ ≡ kB T
(3.6)
which involves the step size ℓ = 8 nm of the kinesin motor and depends on the thermal energy given by Boltzmann constant kB times temperature T . We use the convention that positive and negative load forces F or F¯ correspond to resisting and assisting forces, respectively. In general, these rates can be parametrized in the factorized form ωij ≡ ωij,0 Φij (F )
with Φij (F = 0) ≡ 1
(3.7)
and ωij,0 ≡ κ ˆij [X] ≡ κij
for X-binding
(3.8)
for X-release
where the units of the rate constants κ ˆ ij and κij are given by s−1 µM−1 and s−1 , respectively. The force dependence of the transition rates is described by the factor Φij (F ). For chemical transitions, these factors are taken to have the form Φij (F ) =
1 + exp[−χij F¯ij′ ] 1 + exp[χij (F¯ − F¯ij′ )]
(3.9)
which involves the dimensionless parameter χij and the characteristic force F¯ij′ = ℓ Fij′ /(kB T ). For F¯ij′ = 0, the expression in Eq. (3.9) reduces to the force-dependent factors Φij (F ) as used in [45]. The parameter F¯ij′ represents a force threshold for the influence of the external force onto the corresponding chemical transition as in [93]. For the mechanical transitions, we take the exponential dependence Φ25 (F ) = exp(−θ F¯ ) Φ52 (F ) = exp((1 − θ)F¯ )
forward transition |25i
backward transition |52i
(3.10) (3.11)
The force dependence of the chemical transition rates can be adjusted by the load distribution factors χij , which are taken to satisfy χij = χji with 0 ≤ χij ≤ 1, and the dependence of the mechanical rates is governed by θ with 0 ≤ θ ≤ 1. A good description of the experimental single motor data is obtained, if one chooses two different values, F¯1′ and F¯2′ , for the force ′ =F ¯ ′ = F¯ ′ = F¯ ′ ≡ F¯ ′ for the ATP binding and ATP release transition thresholds with F¯12 1 54 45 21
22
3 Description of single motors and F¯ij′ ≡ F¯2′ for all other chemical rates. Likewise, two different values, χ1 and χ2 , were chosen for the load distribution factors with χ12 = χ21 = χ45 = χ54 ≡ χ1 for the ATP binding and ATP release transition and χij ≡ χ2 for all other chemical rates [45]. An important issue is the dissociation of the motor from the filament. First, it is plausible that the motor unbinds primarily from state i = 7, in which both heads contain ADP and are therefore loosely bound to the filament [103]. The state of the unbound motor will be labeled by i = 0. Thus, motor dissociation is described by the transition from state i = 7 to state i = 0 in Fig. 3.3 [45]. Second, the unbinding rate ω70 should depend on force since the motor will unbind faster if it experiences a load force as can be concluded from the load force dependence of the run length as observed experimentally [29]. The latter force dependence is roughly exponential in agreement with transition state or Kramers theory. Thus, we will use the parameterization ω70 = κ70 exp(|F |/FD ) (3.12) which depends on the detachment force FD = 3 pN [46]. In the steady state, the overall unbinding rate εsi of the single motor from the filament is then given by [46] εsi = P7st ω70 = P7st κ70 exp(|F |/FD )
(3.13)
where P7st describes the load-dependent steady state probability for the motor to occupy state i = 7.
Balance conditions of the fundamental cycles As mentioned above, not all transition rate constants can be chosen independently, they must obey the balance condition in Eq. (2.9). The network graph in Fig. 3.3 consists of three fundamental cycles, F, B and T . As a consequence, three transition rate constants are determined by balance conditions. As established in [45], we choose κ ˆ65 of the forward cycle, κ54 of the backward cycle and κ ˆ 75 of the thermal slip cycle according to κ ˆ 65 ≃
κ ˆ12 κ25 κ56 κ61 κ21 κ52 Keq κ ˆ 16
κ54 ≃ κ21 κ ˆ 75 ≃
κ52 κ25
κ ˆ 65 κ ˆ16 4κ ˆ17
2
(3.14)
(3.15)
(3.16)
with the equilibrium constant Keq = 4.9 × 1011 µM [108] and kB T = 4 pN nm at room temperature.
23
3.3 Motor dynamics and observables
3.3 Motor dynamics and observables The motor dynamics is described by the probabilities Pi = Pi (t) to find the motor in state i at time t, corresponding to a continuous-time Markov process on the chemomechanical network in Fig. 3.3. The time evolution of these probabilities is governed by the master equation X d [Pj ωji − Pi ωji ] for Pi = dt j6=i
i = 1, 2, . . . , 7 .
(3.17)
d st The steady state is characterized by time-independent probabilities Pi = Pist with dt Pi = 0. P One then has to solve a system of linear equations as provided by j6=i [Pj ωji − Pi ωji ] = 0 for i = 1, 2, . . . , 7, either by linear algebra or by graph-theoretical methods. As a result, one finds that the steady state probabilities Pist can be expressed as ratios of two polynomials, which are multilinear in the transition rates ωij as defined by Eq. (3.7) and depend on the external load forces F ≡ Fext and the nucleotide concentrations [72]. The explicit solution of the 7-state modelis given in appendix A. The steady state probabilities Pist to find the motor in state i is shown in Fig. 3.4(a) as a function of the external force for saturating ATP as well as low ADP and P concentrations. For small forces below the stall force F = Fst = 7.2 pN, the states 5, 6 and 7 dominate, for large forces above the stall forces, the motor primarily visits the states 2 or 3 of the backward cycle. Because of the large ATP concentration, the states 1 and 4 are rarely occupied for all values of the external load force.
(a)
(b) 1
60
800 2,3
-1
4
-3
M data; [ATP]=4.2
200
[ATP]=4.2
20
M
M
data; [ATP]=1.6 mM [ATP]=1.6 mM
si
10
400
-1
-2
10
40
[nm s ], [ATP]=4.2
1
600
si
10
0
0
v
[nm s ], [ATP]=1.6 mM
7
v
st
steady state probability P
i
5,6 -1
-4
10
-5
0
5 F
ext
[pN]
10
15
-10
-5
0 F
ext
5
10
[pN]
Figure 3.4: (a) Steady state probabilities Pist to find a single motor in state i as a function of external force Fext for saturating ATP concentration [ATP] = 1 mM and low ADP and P concentrations [ADP] = [P] = 10 µM (semi-logarithmic plot). For small Fext , the motor prefers to visit the states + 5, 6 and 7 and to complete the forward dicycles F + and FDD . The probabilities P2st and P3st increase with increasing force Fext and the motor becomes more likely to complete the backward dicycle B + . Because of the large ATP concentration, the motor stays in the states 1 and 4 for a relatively short time, and the corresponding probabilities P1st and P4st are small for all values of Fext . (b) Motor velocity as a function of the external load force: Comparsion between network calculations (full lines) and experimental data as reported in [43] (circles).
24
3 Description of single motors (a)
(b) 103
1000
data [ADP]=[P]=0.1
M
[nm/s] [ADP]=[P] = 1
M
v
v
[ADP]=[P] = 0.1
102
si
100
si
[nm/s]
[ADP]=[P]=0
v =v
M
[ADP]=[P] = 10 [ADP]=[P] = 100
si
v
M
102
103
m
= (672 +/- 30) nm/s
sat
m
si
10
[ATP]/(K +[ATP])
K = (68 +/- 10)
data; v
10
sat
fit parameter:
M
10
104
1
10
100
M
1000
[ATP] [ M]
[ATP] [ M]
Figure 3.5: Single motor velocity vsi as a function of the ATP concentration in a log-log plot for different values of ADP concentrations and at zero external load force. (a) Different values of ADP concentrations with [ADP]=[P]=0.1 µM (dotted line), [ADP]=[P]=1 µM (dashed line), [ADP]=[P]=10 µM (solid line), [ADP]=[P]=100 µM (dash-dotted line). Experimental data as reported in [43] displayed by a red circle. (b) Small product concentrations with [ADP]=[P]=0.1 µM (black line) and [ADP]=[P]=0 (dashed cyan line) compared to experimental data as reported in [32], displayed by black circles and data fit (red line) via Eq. (3.19) with the fit parameters vsat and Km .
Motor velocity Using the steady state probabilities Pist , we can calculate motor properties such as the motor velocity which is given by the excess flux along the |25i transition [45] and, thus, by
vsi = ℓ ∆J25 = ℓ P2st ω25 − P5st ω52
(3.18)
with the step size ℓ = 8 nm as before. Note that both, the mechanical transition rates ω25 and ω52 as well as the steady state probabilities P2st and P5st depend on the load force Fext . Fig. 3.4(a) shows that P2st strongly increases with increasing Fext up to the stall force and saturates for superstall forces. In contrast, the probability P5st is essentially constant for substall forces and strongly decays for superstall forces. The motor velocity as a function of the external load force is shown in Fig. 3.4(b) for both large and small ATP concentration in good agreement with the experimental data in [43]. We use the transition rate constants derived in [45] except for the ATP binding rate κ ˆ 12 , the ADP release rate κ56 and the rate of the alternative pathway κ57 = κ56 /2 [105]. These latter rates were adjusted to the experimental data in [43]. Specification of the transition rates are given below in section 3.4. For saturating ATP concentration, the motor velocity is proportional to ω56 whereas for small ATP concentration the limiting rate is the ATP binding rate ω12 . As a result of this procedure, we found the parameter values as given in Tab. A.1. Both force velocity curves decrease with resisting forces and vanish at the stall force Fst = 7.2 pN. Inspection of Fig. 3.4(b) shows that, for saturating ATP concentration the motor velocity is hardly affected by assisting forces, i.e., by pulling the motor in the direction of motion, while for small ATP concentration the velocity already starts to decrease at small assisting forces.
25
3.3 Motor dynamics and observables For negligible loads, the motor velocity vsi ([ATP]) =
vsat [ATP] Km + [ATP]
(3.19)
obeys Michaelis-Menten kinetics [38] with the saturation velocity vsat and the MichaelisMenten constant, Km . Note, that in general, the saturation velocity and the Michaelisconstant should depend on the external load force and as well on the concentrations of the hydrolysis products. The motor velocity as a function of the ATP concentration is shown in Fig. 3.5(a) for zero external load force and for different ADP and P concentrations and for vanishing product concentration in Fig. 3.5(b), the latter is in good agreement with the experimental data as reported in [32]. The motor velocity obeys Michaelis-Menten kinetics, using vsat and Km as fitting parameters, the results in Fig. 3.5 can be fitted by Michaelis-Menten curves. For a large ATP concentration with [AT P ] ≥ 1 mM the motor velocity approaches the same saturation level with vsi ∼ 670 nm/s for all four product concentrations in good agreement with the experimental data vsi = (668 ± 8) nm/s [43]. For small ATP concentrations the ratio between ATP and product concentrations is significant: for small product concentrations [ADP]=[P]≤ 10 µM, the effect of a variation in the ADP and P concentration is rather small, the velocity is shifted from 60 nm/s to 75 nm/s for small ATP concentration. In contrast, a relatively large ADP and P concentration decreases in the motor velocity to vsi ∼ 20 nm/s for small ATP-concentrations. This effect is reflected in the Michaelis-Menten constant Km as shown in Tab. 3.1. [ADP]=[P] [µM] Km [µM]
0 78 ± 5
0.1 80 ± 4
1 83 ± 5
10 110 ± 10
100 375 ± 30
Table 3.1: The motor velocity as a function of the ATP concentration obeys the relation in Eq. (3.19). For different values of ADP and P concentration in Fig. 3.5 we determine the Michaelis-Menten constant Km with vsat ≈ 670 nm/s.
Since the ratio of the motor velocity vsi and the stepsize ℓ of the motor determines the stepping rate of a motor, the average dwell time between two steps is given by h∆τsi i =
ℓ . vsi
(3.20)
The single motor velocity vsi can experimentally be determined by measuring the average runlength hxsi i and the average run time htsi i of a single motor run. Note, that piecewise analysis of motor trajectories would lead to an averaging of local velocities and thus yield a different average velocity vsi = h∆xsi /∆tsi i.
Run time and unbinding events The time a single motor travels on the filament before it detaches is called the motor run time. The single motor run time is affiliated with the overall unbinding rate htsi i = 1/εsi as defined in Eq. (3.13) which depends on the external load force and within P7st also on the nucleotide concentrations. As introduced above, the steady state probabilities Pist can be expressed as ratios of two polynomials, which are multilinear in the transition rates ωij as
26
3 Description of single motors defined by Eq. (3.7) which are both, force and concentration dependend. The steady state probability P7st is shown in Fig. 3.6(a) as a function of the ATP and the product concentrations for zero load force. Small product concentrations provide for a small ATP concentration a small unbinding rate εsi ∼ k7 0 P7st which increases with the ATP concentration. Large ADP and P concentrations lead for small ATP to a large unbinding rate which decreases with increasing ATP. As a result, the single motor run time htsi i shown in Fig. 3.6(b) varies more than one order of magnitude with the product concentrations [ADP]=[P] for small ATP concentrations. Very small [ADP]=[P]=0.1 µM lead to a large single motor run time of htsi i ≈ 12.6 s, whereas for large [ADP]=[P]=100 µM the single motor run time decreases to htsi i ≈ 0.66 s for [ATP]=10 µM. For large ATP concentrations, the single motor run time htsi i ≈ 1.54 s, which is in good agreement with the experimental data [43], is independent of the product concentrations. Thus, the influence of a variation in the ATP concentration is significant for very small and for large product concentrations, but if we choose a relatively low [ADP]=[P]= 10 µM for instance, a change in the ATP concentration has only a minor effect on the run time. The different experiments in [43, 27, 29, 58] which we study in the next section, use different ATP concentrations. In order to compare these different experimental results we choose in general the product concentrations with [ADP]=[P]≈ 10 µM, unless noted otherwise. (b)
(a) 2
10
st
P
7
(F
ext
=0)
[ADP]=[P] = 0.1
0.50
10
[ADP]=[P] = 10
M
0.40
[ADP]=[P] = 100
0.35
data; = / v
0.25 0.20 0.15 0.10
si
si
M si
si
0.30
1
M
M
[s]
[ADP]=[P]
[ M]
0.45 10
[ADP]=[P] = 1
0.05 0.00
1
-1
10
10
2
10
3
10
[ATP] [ M]
4
10
10
102
103
104
[ATP] [ M]
Figure 3.6: a) Contour plot of the steady state probability P7st to be in state 7 as a function of the nucleotide concentrations [ATP] and [ADP]=[P] in a log-log plot. The gray-scale map starts with small values of P7st in black to large values in white. b) Average run time htsi i of a single motor as a function of the ATP concentration in a log-log plot for different values of ADP concentrations, [ADP]=[P]=0.1 µM (dotted line), [ADP]=[P]=1 µM (dashed line), [ADP]=[P]=10 µM (solid line), [ADP]=[P]=100 µM (dash-dotted line), at zero external load force. The red circle represents the average value htsi i = (1.54 ± 0.03) s, calculated from experimental data in [43] via htsi i = hxsi i/vsi with hxsi i = (1.03 ± 0.17) µm and vsi = (668 ± 8) nm/s .
Motor run length The distance a single motor travels on the filament before it detaches is called the motor run length. As mentioned above, the probability of detachment is given by the unbinding rate
27
3.3 Motor dynamics and observables
(a)
(b)
100 [ADP]=[P] = 0.1 [ADP]=[P] = 1
10
10 -4
-2
0
2
F
ext
4
6
8
M
M
[ADP]=[P] = 10 [ADP]=[P] = 100
si
100
1000
si
100
[nm]
M
M
[ATP]=4.2
[nm]; [ATP]= 4.2
data
si
[nm]; [ATP]=1.6 mM
[ATP]=1.6 mM
1000
M M
data;
10
10
si
10
[pN]
102
103
104
[ATP] [ M]
Figure 3.7: a) Semi-logarithmic plot of the average run length hxsi i of a single motor calculated via Eq. (3.21) as a function of the external load force Fext for two different values of ATP concentration, [ATP] = 1.6 mM (black line) and [ATP]=4.2 µM (red line). The product concentrations are chosen to [ADP] = [P] = 10 µM. The motor becomes unprocessive for a run length hxsi i < ℓ = 8 nm smaller than the stepsize. b) Average run length hxsi i as a function of the ATP concentration in a log-log plot for different values of ADP concentrations, [ADP]=[P]=0.1 µM (dotted line), [ADP]=[P]=1 µM (dashed line), [ADP]=[P]=10 µM (solid line), [ADP]=[P]=100 µM (dash-dotted line), at zero load force. Experimental data hxsi i = (1.03 ± 0.17) µm as reported in [43] displayed by circles in both plots.
ω70 and the probability P7 to be in the (DD) state, where both motor heads are only loosely bound to the filament. Thus, the average run length of the motor is given by hxsi i =
vsi st P7 ω70
=
vsi = vsi htsi i . εsi
(3.21)
The average run length hxsi i of a single motor calculated via Eq. (3.21) is shown in Fig. 3.7 as a function of the external load force Fext and as a function of the ATP concentration. The motor run length in Fig. 3.7(a) decreases with increasing resisting forces and the motor becomes unprocessive for a run length hxsi i < ℓ which is smaller than the stepsize. Note, that the decline in the run length is less pronounced for assisting forces, since the velocity is almost constant for assisting forces. In the limit of vanishing product concentrations, the single motor run length in Fig. 3.7(b) is independent of the ATP concentration [45]. Thus, for vanishing ADP and P concentrations, the relation in Eq. (3.21) implies that the overall unbinding rate is proportional to the average single motor velocity which leads with Eq. (3.19) to the expression εsi ([ATP]) ≈
28
[ATP] vsat . hxsi i Km + [ATP]
(3.22)
3 Description of single motors
3.4 Specification of transition rates In general, we use the etablished method for kinesin motors to calculate the transition rate constants for our model. Since not every set of experimental data provides all necessary details, we have to assemble and merge the information of several data sets, summarized in Tab. A.1. Based on the fact that the experiments by [27] and [29] provide most of the needed data, we start as in [45] and then find our way to the data sets of [43] with force–velocity relations for different ATP concentrations, [29] runlength as a function of external load force and [64] force–velocity relations. In the following, we call the rate constant sets by a capital corresponding to the experimental data as follows: parameter set B refers to [43], parameter set C refers to [27], parameter set D to [64] and parameter set E to [29]. The different results of the different experimental groups are mainly caused by the different kind of kinesin which they use. In the study by [27] they use pig brain microtubules and full length drosophila kinesin and polystyrene beads of diameters of 500 nm, 560 nm and 800 nm. The group of [29] use borine brain microtublues and kinesin of squid optic lobes and glas beads of 500 nm diameter. In [58, 64, 109] truncated human kinesin and bovine brain microtubules and axonemes purified from sea urchin sperm are used and eventually attached to streptavidin-functionalized beads with a diameter of 500 nm. In [43] 500 nm diameter silica beads were mixed with native kinesin which is purified from Loligo pealei and microtubules which are polymerized from purified bovine brain tubulin. In general, our calculation and simulation in the following chapters are accomplished by using the parameter set B, unless noted otherwise, which is based on the experimental data by [43]. Since other experimental data for kinesin, and especially on motor pairs is available, we introduce here the parameter settings for our model to these data, see Tab. 3.2 and Tab. A.1. For small external forces, the motor velocity as a function of ATP concentration obeys Michaelis-Menten kinetics [38] as in Eq. (3.19). The saturation velocity vsat as well as the Michaelis-Menten constant Km depend on the external load force and on the concentration of the hydrolysis products as in Tab. 3.1. For small load forces and in the limit of vanishing product concentrations the explicit solution of Eq. (3.18) can be approximated by v ≈ ℓω56 (1 + α)/3
v≈
ℓω12
for saturating [ATP]
(3.23)
for small [ATP]
(3.24)
with the transition rates ωij = ωij,0 Φij as in Eq. (3.7) and α = ω57 /ω56
where Φ56 = Φ57
leads to
α = α0 = κ57 /κ56 .
(3.25)
The latter ratio α0 = 1/2 has been determined in [105]. For zero external load force the motor velocity v(F = 0) = v0 can then be written as v0 ≈ ℓω56,0 /2 = ℓκ56 /2
v0 ≈
ℓω12,0
= ℓˆ κ12 [ATP]
for saturating [ATP]
(3.26)
for small [ATP] .
(3.27)
In general, the motor velocity then has the form v(F ) ≈ v0 Φij (F )
(3.28)
29
3.4 Specification of transition rates
exp. data by B [43] C [27] D [64] E [29]
1 2
χ1
F¯1′
χ2
F¯2′
0.4 0.4 (0.4)∗ 0.3
4 0.6 (4)∗ 4
0.6 0.3 0.3 0.2
8 6 9 10.2
Table 3.2: Dimensionless force factors χij and characteristic forces F¯ij′ = ℓFij′ /kB T determined for experimental data [43], [29], [27] and [64]. The superscript ()∗ indicates a value adopted from another column (no data available). Parameter settings are named in capitals as introduced above.
with the force dependent factors Φij (F ) of the chemical transition rates as introduced above in Eq. (3.9) 1 + exp(−χij F¯ij′ ) (3.29) Φij (F ) = 1 + exp(χij (F¯ − F¯ ′ )) ij
which involves the dimensionless parameter χij and the characteristic force F¯ij′ = ℓ Fij′ /(kB T ) and the condition Φij (F = 0) ≡ 1. Inserting this relation into the expression for the motor velocity in Eq. (3.28) leads to 1 + exp(−χ2 F¯2′ ) 1 + exp(χ2 (F¯ − F¯2′ )) 1 + exp(−χ1 F¯1′ ) v(F ) ≈ v0,small 1 + exp(χ1 (F¯ − F¯ ′ ))
v(F ) ≈
v0,sat
for saturating [ATP]
(3.30)
for small [ATP]
(3.31)
1
′ ≡ F ¯ ′ and χ12 ≡ χ1 for the ATP binding transition and F¯ ′ ≡ F¯ ′ and χ56 ≡ χ2 for with F¯12 2 56 1 the ADP release transition as introduced in section 3.2.
Since the velocity is assumed to be constant for large assisting forces as in [43], we can formulate the condition lim [v(F )] = vmax ≈ v0 (1 + exp(−χij F¯ij′ ))
F → −∞
(3.32)
where Eq. (3.28) and Eq. (3.29) have been used. The right side in Eq. (3.32) can be transposed to a conditional equation for the the force factor χij and the characteristic force F¯ij′ χij F¯ij′ = ln
v0 vmax − v0
(3.33)
which involves the zero force velocity v0 and the saturation velocity vmax for large assisting forces. Both velocities are accessible to experiments as in [43]. The force threshold Fij′ can be estimated with the force, at which v(F ) ≈ vmax /2. Since some of the experimental data [29, 64] covers force velocity relations only for resisting forces, vmax is not accesable for these data sets. Here we find the force, at which v(F ) ≈ v0 /2 as an upper limit for the force threshold. The dimensionless parameter χij is determined by the slope mij of the force velocity curve at the characteristic force F¯ij , which one obtains by carfully fitting the experimental data near the characteristic force Fij′ . Derivation of Eq. (3.28) leads to the relation ∂Φij ∂v = v ≡ mij . (3.34) 0 ∂F F =Fij′ ∂F F =Fij′ 30
3 Description of single motors Together with the derivation of the force dependent factors Φij in Eq. (3.9) at Fij′ ∂Φij ℓχij =− (1 + exp(−χij F¯ij′ )) ∂F F =Fij′ 4 kB T
(3.35)
and Eq. (3.32) this leads to a conditional equation for the dimensionless parameters χij χij = −
4m ˜ ij vmax
(3.36)
where m ˜ ij = mij kB T /ℓ has been used. The results of this procedure are summarized in Tab. 3.2 for the different data sets. (a)
(b)
si
[ATP]=1.6 mM
0
0
parameter set E 20
data; [ATP]=2 mM
200
[ATP]=2 mM data [ATP]=5 [ATP]=5
-5
0 F
ext
5
0 -5
10
0
5 F
[pN]
(c)
M
M
M
0 -10
[nm/s]; [ATP]=5
M
data; [ATP]=1.6 mM
si
[ATP]=4.2
40 400
si
200
20
M
-1
data; [ATP]=4.2
600
v
parameter set B
v
400
60
[nm/s]; [ATP]=2 mM
M
800
[nm s ], [ATP]=4.2
40
80
v
-1
[nm s ], [ATP]=1.6 mM
600
si
v
1000
60
800
ext
10
[pN]
(d) 500
200
v
v
si
si
100
0
-15
0
-10
-5
0 F
ext
[pN]
5
10
15
[ATP]=2mM
parameter set D
400
100
M
M
[ATP]=10
120
80
300
60
200
40
100
20
0
[nm/s]; [ATP]=10
200
[nm/s]; [ATP]=1 mM
400
300
[ATP]=2mM
(i) : data (ii);
[ATP]=2 mM
500
si
parameter set C
data (ii) (i);:
si
400
v
M
M
M
[ATP]=10
600
[nm/s]; [ATP]=10
data [ATP]=10
[nm/s]; [ATP]=1 mM
600
[ATP]=1mM
v
data; [ATP]=1mM
800
0
-5
0
5 F
ext
10
15
[pN]
Figure 3.8: Calculated force velocity curves via Eq. (3.18) with parameter sets B, E, C and D compared to experimental data in (a) [43], (b) [29], (c) [27], (d) (ii) [64] and (i) [58] (blue square). For saturating ATP, data points and lines are drawn in black on the left axis, for small ATP concentration in red on the right axis.
Including the specification of the force dependence to the transition rates, our single motor system is fully determined. This allows us to calculate the force velocity relation from Eq. 3.18 which is shown in Fig. 3.8 compared to the experimental data in [43], [29], [27] and [58, 64]. The simplifications in Eq. (3.30) and Eq. (3.31) seem to be adequat at least up to the stall force. Our parametrization of the transition rates nicely agrees with the data set in (a),
31
3.5 Physically meaningfull parameter regimes whereas in (b) the results match for saturating ATP-concentration but slightly differ for small ATP concentrations. Since the data in (c) is widely distributed for assisting forces, the constant velocity vmax apparently is considered as an average velocity at assisting forces. For resisting forces up to the stall force, the results in (c) and in (d) are good approximations of the force–velocity curves for these data sets.
3.5 Physically meaningfull parameter regimes The transition rates of the chemomechanical network in Fig. 3.3 depend on the nucleotide concentration as well as on the external load force. For limiting cases, e.g., for a very small ATP-concentration or for a large external load force, the motor may be caught in a state, where it will dwell for a long time. In this section, we establish the limiting constraints and physically meaningfull parameter regimes for our model, with special regard to the simulation method.
Nucleotide concentration as a limiting factor The chemomechanical network in Fig. 3.3 has three cycles which contain a mechanical transition as well as the hydrolysis of an ATP molecule, these are the forward cycles F and FDD and the backward cycle B. The supplied chemical energy of the motor system can be expressed in terms of the chemical potential [102] ∆¯ µ=
[ATP] ∆µ = ln Keq kB T [ADP][P]
(3.37)
with the equilibrium constant Keq = 4.9×1011 µM [108]. Chemical equilibrium between ATP hydrolysis and ATP synthesis corresponds to ∆¯ µ = 0, which implies [ATP]eq = [ADP][P]/Keq . Since Keq is large, the balance between ATP hydrolysis and ATP synthesis requires very small ATP concentrations. The transition rate ωij of a transition that includes nucleotide binding depends linearly on the corresponding nucleotide concentration, as in Eq. (3.9). Hence, in the forward cycles F and FDD , a very small ATP concentration leads to a large average transition time t12 = 1/ω12 ∼ 1/[ATP] . Thus, the rate limiting transition for small ATP concentration is the ATP-binding transition and the average time it takes to complete the forward cycle tF = t12 + t25 + t56 + t61 is proportional to t12 for small ATP concentrations. The transition time t12 for the ATP binding transition compared to the time tF to complete the forward cyle F is shown in Fig. 3.9(a) as a function of the chemical potential ∆µ for different values of ADP and P concentrations. The transition time t12 is proportional to the inverse ATP concentration and decays exponentially with the chemical potential, whereas the other transition times t25 , t56 and t61 of the forward cycle F + are independent of the nucleotide concentrations. Thus, one obtains a certain ∆µ at which tF & t12 . For low ADP and P concentration, solid lines in Fig. 3.9(a), the chemical potential ∆µ should exeed ∆µ = 25 kB T whereas we can reach lower ∆µ values only for very large ADP and P concentrations. At ∆µ = 0 the concentrations are equilibrated [ATP]eq = [ADP][P]/Keq . For small ADP and P concentrations this leads to a transition time up to several years to undergo the ATP binding transition at equilibrium whereas for very large ADP and P concentrations it takes only some seconds. Note, that if we apply an additional external force t12 is increased by the according factor 1/Φ12 . For a random walk on the network, the [ATP]-dependent transition should be
32
3 Description of single motors fast compared to the cycle time to assure a sufficient number of completed cycles during a motor or motor pair run to reach a steady state of the system. (a)
(b) 10
10
t 8
10
t
12
F
+
1.0
[ADP]=[P]=10 6
10
M
[ADP]=[P]=1 mM
F
/ t
12
t [s]
[ADP]=[P]=100 mM 4
10
2
t
10
0.5
1
-2
10
[ADP]=[P]=0.1
M
[ADP]=[P]=
1
M
[ADP]=[P]= 10
M
0.0
0
5
10
15
20
[k T] B
25
30
0
10
20
30
40
[k T] B
Figure 3.9: (a) Semi-logarithmic plot of transition time t12 (black lines) compared to the time to complete the forward cycle F + (red lines) as a function of the chemical potential ∆µ in zero force regime and for three different ADP and P concentrations, low [ADP] = [P] = 10 µM (straight lines), large [ADP] = [P] = 1 mM (dashed lines) and very large [ADP] = 100 mM (dotted lines). Except ω12 all rates in F + are independent of the nucleotide concentrations, thus t12 decays exponentially with increasing ∆µ. At ∆µ = 0 the concentrations are equilibrated. For low ADP concentrations this leads to a cycle time up to several years to undergo this transition at equilibrium whereas for the very large ADP concentration it takes only some seconds. (b) Ratio of transition time t12 to cycle time tF as a function of the chemical potential ∆µ in zero force regime and for three different values of small [ADP]=[P] concentrations: 0.1 µM (dashed line), 1 µM (dotted line) and 10 µM (solid line).
The ratio of transition time t12 to the cycle time tF is shown in Fig. 3.9(b) as a function of the chemical potential ∆µ in zero force regime and for three different values of small product concentrations. With decreasing ADP and P concentration, the time scale at which tF & t12 is shifted to larger values of the chemical potential. As a consequence, we have to deliberate about whether we choose a very small product concentration which is in agreement with the most experimental setups but leads to huge dwell times for small ATP concentrations or if we choose a sufficiently small but not vanishing product concentration which ensures even for small [ATP] a sufficiently good ratio t12 /tF . With regard to experiments, large timescales are not convenient. In the same way, very large or very small concentrations are not suitable. Our simulations do not provide timescales of days for a single transition rate, this exceeds the computational time. Note, that very large ADP concentrations lead in our model even at saturating ATP concentrations to an oscillation between state 1 and 7 in Fig. 3.3. On the other hand, very small ADP concentrations cause for sufficiently small ATP concentrations huge dwell times in state 1 and 4. In general, we choose the ADP and P concentration with [ADP]=[P]=10 µM on the one hand as sufficiently small compared to the saturating ATP concentration which is in the order of mM. On the other hand choosing these values for the concentrations provides reasonable dwell times of the order of ms to s regarding experimental setups.
33
3.5 Physically meaningfull parameter regimes
Large external load force as a limiting factor Note, that if we apply a small resisting force to the motor, the transition time t12 in Fig. 3.9 is increased by the factor 1/Φ12 what then leads for small ADP concentrations to a transition time of several hundred of years and for large ADP concentrations to about 30 hours. In order to detect reasonable properties of our system, we have to pay attention to this transition times first, i.e. the [ATP] dependent transition should be fast, especially under large external load force, compared to the cycle time to assure a sufficient number of cycle completions during a motor run to reach a steady state of the system. Since all transition rates in our model, not only the mechanical stepping rates, but also the chemical transition rates depend on the external load force, this may comprise some problems for large forces, as will be depicted in the following. The overall rate for leaving state i is given by ωi =
X
ωij =
j
1 hτi i
(3.38)
i.e., by the inverse of the average dwell time hτi i in state i. (b)
(a) 6
4
10
10
3
10
5
10
5 2
7
i
10
< > [s]
-1
> [s ]
3
10 1,4
3
10
Fst , but as mentioned before, we neglect the rare transition |27i [105]. Since the overall rate ω2 out of state 2 and ω3 out of state 3 are very small for superstall forces, i.e., the motor dwells rather long in these states as shown in Fig. 3.10, it will take a long time to reach state 7, from which unbinding finally is possible. Addionally, the chemical transition rates ω75 and ω71 from state 7 are small compared to the unbinding rate ω70 for large forces. Thus, the first time, the motor arrives in state 7 it is most likely, that it will immediately unbind from filament. Since the rebinding of an inactive motor occures to state 7, large external forces may lead to enhanced unbinding of the motors and the motor pair.
(b)
(a)
-1
10
-1
7
-2
P
P
7
10
10
[ADP]=[P]= 10
M
[ATP]= 10
M
-3
10
-3
10
-5
10
[ATP]=0.1 mM [ATP]=
M
[ATP]= 10
M
[ATP]=0.1 mM [ATP]=
1 mM
[ATP]= 10 mM
1 mM
10
-1
[s ]
[ATP]= 10 mM
-1
[s ]
2
10
[ADP]=[P]=0.1
si
si
10 1
-1
10
-1
10
-3
10 -10
-5
0
5
F
ext
[pN]
10
15
-10
-5
0
5
F
ext
10
15
[pN]
Figure 3.11: Semi-logarithmic plot of the steady state probability P7st and the overall unbinding rate εsi as functions of the external load force Fext for different values of the ATP concentration, [ATP]=10 µM (red lines), [ATP]=0.1 mM (green lines), [ATP]=1 mM (black lines) and [ATP]=10 mM (blue lines) for two different product concentrations, a)[ADP]=[P]=10 µM and b)[ADP]=[P]=0.1 µM.
The probability to be in state 7 is very sensitive to a change in the nucleotide concentrations for superstall forces, as shown in Fig. 3.11 for two different product concentrations and in detailed contour plots in the appendix in Fig. B.1. Therefore, the overall unbinding rate calculated via Eq. (3.13) in Fig. 3.11 varies about two orders of magnitude, depending on the nucleotide concentrations, and has a local minimum for superstall forces which deepens with increasing ATP-concentrations and also with decreasing ADP and P concentration. Due to the large force threshold F¯2′ , this minimum is shifted towards larger forces for increasing ATP concentrations.
35
3.6 Summary and discussion A motor run is artificially elongated or shortcutted depending on the starting state for superstall forces. In order to ensure steady state conditions the applicable load force on a single motor should not exceed the stall force. Generally, one should consider as well the rare transitions h27i and h47i or even incorporate tearing-off rates, i.e., unbinding from more than one state, in order to avoid constraints on the applicable force. Additionally, an alternatively stepping transition between the (DE) and (ED) state as studied in [110, 111] should be considered. In particular, these considerations would lead to a vast increase in network cycles for the motor pair system.
3.6 Summary and discussion In this chapter we display the general description of the single motor model established in [45, 46]. First the concept of the chemomechanical coupling of the ATP hydrolysis to the spatial displacement on a filament is introduced. Second, the network description of the kinesin motor which is governed by transition rates as well as the aspects of macroscopic observables such as motor velocity is outlined. This generic model can be adapted to various single motor setups and reliably reproduces experimental findings as shown in [45]. However, we choose a more general description of the force dependence of the chemical transition rates as in [93] which allows invariant motor velocity for assisting forces in agreement with experimental data [43]. We show the specification of the transition rates and the application to several experimental data for different kinds of kinesin: purified from Longfin Inshore Squid [43], drosophila [27], pig [32] and human [58] kinesin. There are several time scaling constraints to the motor pair simulations which dictate certain limits to the applicable chemical potential and the external load force. Arising from the nucleotide and force dependence of the single motor transition rates, we find that very small ATP concentrations as well as large external load forces lead to very long transition and dwell times, which are not convenient in computer simulations. Generally, one should consider the rare transitions h27i and h47i as well or even incorporate tearing-off rates, i.e., the motor unbinds from more than one state with a certain rate for a sufficiently large force, in order to avoid constraints on the applicable force. Also, an alternatively stepping transition between the (DE) and (ED) state as studied in [110, 111] should be considered. In particular, these considerations would lead to a vast increase in network cycles for the motor pair system and is not convenient either. Therefor, in this study, external load forces should not exceed the single motor stall force. The single motor model, with all modifications and constraints which are introduced in this chapter, is the starting point for our motor pair model, which will be explained in the following chapter.
36
4 Chemomechanical network of a motor pair In this chapter we introduce our basic concept for coupled molecular motors [112]. First, we consider theoretically the idea of how to transform two individual motors into a motor pair. In order to obtain a detailed description for a pair of coupled motors, we describe each motor by its chemomechanical network and couple these two motors and, thus, the two networks by an elastic spring. Because of the elastic strain forces mediated by this spring, mechanical steps by one motor influence the transition rates of both motors if they are both connected to the filament. The elastic coupling may lead to interference effects: the two motors may stall each other or one motor may pull the other motor from the filament. However, we also expect a motor pair to cover larger distances compared to a single motor since a detached motor is still close to the filament and hence is able to rebind to the filament before the other motor detaches as well. Second, geometry aspects and the structure of the motor pair network are explained. The single motor model in Fig.3.3 consists of 7 chemical states and an unbound state, which provides 64 possible state combinations for two such motors. Taking into account, that the interaction force each of these motors experiences is not constant but changes with each mechanical step, leads to a huge and complex network. In the last part of this chapter we explain how to find pathways on this complex motor pair network and give detailed information about the simulation setup and the characteristic observables of a motor pair system.
4.1 Theoretical description The motor system considered here consists of two kinesin motors which are attached to the same cargo and walk on the same microtubule as indicated in the top row of Fig. 4.1. Because the stalk of the kinesin molecule is flexible, we consider this stalk to behave as a harmonic spring with spring constant κ and focus on the deflections of this spring parallel to the filament. The corresponding rest length Lk will be of the order of the stalk length, which is about 80 nm, if the separation between the cargo and the filament is about (17 ± 2) nm as observed in [113],see Fig. 4.2. The mutual interaction forces that the two motors and the cargo experience parallel to the filament are then described by two linear, harmonic springs attached to the cargo as indicated in the middle row of Fig. 4.1. The leading motor, which walks in front of the other motor, has position xle on the filament and its states are labeled by i = ile . Likewise, the trailing motor is located at position xtr with states labeled by i = itr . The stalks of the two motors are anchored at the surface of the rigid cargo. The distance between the two anchor points is assumed to be fixed in order to ensure that the motors do not interact sterically. As long as both motors are attached to the filament, we also preserve the ordering of the two motors with respect to the filament, i.e., the motors are not allowed to pass each other. In general, a reordering of motors is possible during 1-motor runs. Thus, rebinding of the detached motor may occur either in front or behind the active motor if we include the possibility that the cargo rotates during the 1-motor runs and the trailing and leading motors are interchanged. The rotational diffusion constant for a spherical particle with radius R in water is given by Drot ≃
µm R
3
0.2 s
(4.1)
37
4.1 Theoretical description at room temperature. Then, a rotation of the bead by π or 180◦ leads to a typical time scale trot with hβ 2 i = π 2 ≃ 2Drot trot (4.2) which implies trot
5π 2 ≃ 2
R µm
3
s.
(4.3)
which has to be compared to the run time of 1-motor runs. Typically, a single motor run takes only a few seconds [11]. As we will see below, the run time of a 1-motor run is always short compared to the run time of a single motor. For bead radii that are larger than 350 nm the expression in Eq. (4.3) exceeds the single motor run time. In the following, we will focus on relatively large beads, for which appreciable cargo rotation and, thus, interchange of the trailing and leading motor can be neglected. We will return to the aspect of cargo rotation in section 5.3. Force balance during 2-motor runs Since each motor can unbind from and rebind to the filament, the cargo can be actively pulled by one, two or no motors corresponding to three different activity states. If both motors provide a connection between the cargo and the filament, the cargo performs a 2motor run. During such a run, the cargo with center-of-mass position xca is subject to two forces arising from the two motors. The force that the leading motor exerts onto the cargo is given by Fle,ca = κ(xle − xca − Lk ) (4.4) which is positive if xle − xca > Lk , i.e., if the leading spring pulls in the positive x-direction. Likewise, the force that the trailing motor exerts onto the cargo is Ftr,ca = −κ(xca − xtr − Lk ) .
(4.5)
We now assume that, for given positions xle and xtr of the two motors, the elastic forces balance each other on time scales that are small compared to the time scales of the single motor transitions. This assumption allows us to eliminate the cargo position from the theory, as indicated in the bottom row of Fig. 4.1 . Elastic force balance implies Fle,ca + Ftr,ca = 0 or the average cargo position (4.6) x ¯ca = 21 (xle + xtr ) . Thus, after the forces have balanced, both springs have the same deflection as given by 1 2
L≡
1 2
(xle − xtr )
(4.7)
and the elastic force acting on the leading motor is equal to Fca,le = −κ(xle − x ¯ca − Lk ) = −K(L − L0 )
(4.8)
with effective spring parameters K ≡ κ/2
38
and
L0 ≡ 2Lk .
(4.9)
4 Chemomechanical network of a motor pair
...
...
1−motor run ca
...
2−motor run
1−motor run ca
ca
tr
ca
tr le
tr
le
∆t1 ,∆x1
tr
le
tr
le
offside
tr
le
le
tr
le
tr
le
∆t2 , ∆x2 ∆tca ,∆xca
Figure 4.1: (Top row) Overall cargo run for two kinesin motors (blue), each of which has two motor heads. Both motors are attached to the same cargo (light gray) and walk along the same filament (black line); (Middle row) Reduced representation of the overall cargo run in terms of three ‘particles’ corresponding to the leading motor at position xle , the trailing motor at position xtr , and the cargo with center-of-mass position xca . These three ‘particles’ are connected via two linear springs with spring constant κ and rest length Lk . (Bottom row) Reduced representation of the overall cargo run in terms of only two motor ‘particles’ connected by a single spring. As long as both motors are attached to the filament and, thus, active as indicated by the blue ‘balls’ they perform a 2-motor run, during which each mechanical step of one motor affects, via the elastic spring, the force experienced by both motors. After unbinding from the filament, an active motor becomes inactive as indicated by the white ‘balls’. If the cargo is pulled by only one active motor, the cargo performs a 1-motor run until (i) it either unbinds as well, leading to an unbound motor pair, or (ii) the inactive motor rebinds to the filament and the cargo starts another 2-motor run. We will denote the distances and times traveled during 2-motor runs by ∆x2 and ∆t2 , those during 1-motor runs by ∆x1 and ∆t1 and those during the overall cargo run by ∆xca and ∆tca . Here and below, all quantities that refer to 1-motor and 2-motor runs will be labeled by subscript 1 and 2, and those refering to the cargo run by the subscript ca, respectively. Unbound cargo states, in which both motors are inactive, will be indicated by the subscript 0. We do not take diffusion of the unbound cargo state into account.
39
4.1 Theoretical description Thus, a stretched spring with xle − x ¯ca > Lk generates a force Fca,le < 0 acting in the negative x-direction. Likewise, the elastic force acting on the trailing motor is given by Fca,tr = κ(¯ xca − xtr − Lk ) = K(L − L0 ) .
(4.10)
Since the elastic forces depend only on the deflection ∆L ≡ L − L0 = xle − xtr − L0 ,
(4.11)
of the motor-motor separation, the leading motor exerts the interaction force Fle,tr = Fca,tr = K∆L
(4.12)
onto the trailing motor, which exerts the force Ftr,le = Fca,le = −K∆L = −Fle,tr
(4.13)
on the leading motor as required by Newton’s third law. In this way, we obtain a reduced description of the motor pair in terms of two kinesin motors connected via a single linear spring with the effective rest length L0 = 2Lk and the effective spring constant K = κ/2 which will be denoted as the coupling parameter of the motor pair in the following. If we focus on one of the two motors during a 2-motor run, the elastic force in Eq. (4.13) may be regarded as a load force acting on this motor, which thus enters its force-dependent transition rates in Eq. (3.9) and its force-dependent unbinding rate in Eq. (3.13). In these latter relations, we used the convention that resisting forces are positive, whereas assisting forces are negative. Therefore, the forces that enter these relations are F = Fle with Fle ≡ −Ftr,le = −Fca,le = K∆L for the leading motor
(4.14)
Ftr ≡ −Fle,tr = −Fca,tr = −K∆L for the trailing motor .
(4.15)
and F = Ftr with
Thus, the leading motor experiences a resisting force Fle > 0 if the trailing motor is subject to an assisting force Ftr < 0 and vice versa, as required by Newton’s third law. Transitions between 2-motor and 1-motor runs Now, consider a 2-motor run and assume that one of the two motors visits its single motor state 7, which is loosely bound to the filament, see Fig. 3.3. This motor then unbinds from the filament with transition rate ω70 , which transforms the 2-motor run into a 1-motor run by the other motor, which is still bound to the filament, see Fig. 4.1. The latter motor now pulls both the cargo and the detached motor along with it. During such a 1-motor run, mechanical equilibrium between the elastic forces acting on the cargo implies that the two linear springs are relaxed and that ∆L = 0. The detached motor is still connected to the cargo and, thus, remains close to the filament. When this motor rebinds to the filament, the 1-motor run is terminated and another 2-motor run begins. We will ignore chemical transitions of the unbound motor, which implies that motor rebinding to the filament only occurs back to the single motor state i = 7. The corresponding rebinding rate will be denoted by ω07 ≡ πsi . (4.16)
40
4 Chemomechanical network of a motor pair Furthermore, since the 1-motor runs are characterized by ∆L = 0, rebinding of a detached motor initially leads to a state with ∆L = 0 as well, corresponding to no elastic force between the two bound motors. Such a force is generated as soon as one of the motors performs a mechanical step which leads to a stretching or compression of the effective spring between the two motors. When bound to the filament, the motors are active and undergo their chemomechanical cycles. During a 1-motor run, the actively pulling motor exhibits the same properties as a single motor. The latter properties have been studied in great detail and are taken into account here via the detailed chemomechanical network description. The average run length and run time of a 1-motor run are, however, shorter than the corresponding quantities of a single motor since the 1-motor runs may be terminated by the rebinding of the second motor. During a 2-motor run, on the other hand, both motors undergo their individual chemomechanical cycles but, at the same time, experience the mutual interaction force as in Eq. (4.13), which depends on the deflection ∆L of the motor-motor separation from its rest length L0 . Whenever one of the two motor performs a mechanical step, this step changes the deflection ∆L and, thus, the mutual interaction force. Since this force enters all transitions of both motors, a single mechanical step affects all subsequent transitions of both motors. This feedback mechanism between mechanics and chemistry represents the most important aspect of the system considered here. As explained in the following, we will construct a network representation for the motor pair that is based on a combination of the chemomechanical networks for the two individual motors. In this way, we obtain a unique representation of the state space for the motor pair. Furthermore, we take all transitions between these motor pair states into account that arise from transitions of individual motors. Simultaneous transitions of both motors, on the other hand, will not be considered since the motor dynamics will be described as a continuous-time Markov process. For such a process, the probability that a single motor transition occurs during a small time interval of size dt is proportional to dt. Therefore, the probability for simultaneous transitions of both motors is proportional to dt2 and, thus, of higher order in dt. As explained before, a motor performing a mechanical step exerts an interaction force onto the other motor and then feels the corresponding reactive force as in Eq. (4.13). As long as these forces are small, each motor will essentially behave as a single, non-interacting motor, and the probability distribution for its single motor states will then be close to the one displayed in Fig. 3.4(a). On the other hand, if these forces become sufficiently large compared to the motors’ detachement force, either by a large deflection ∆L or by a strong elastic coupling K, the interactions are no longer negligible and will lead to a variety of interference effects as recently categorized in [51].
Geometry aspects As indicated in Fig. 4.1 and emphasized in in Fig. 4.2 we reduce the geometry in the motormotor complex to a one dimensional description. The flexible stalks of each kinesin are considered to behave as harmonic springs. First, we consider the cargo position with regard to the filament. The separation between the cargo and the filament is about (17 ± 2) nm as observed in [113]. Therefore the rest length Lk of each spring parallel to the filament will be of the order of the stalk length, which is about 80 nm. Next, the cargo position can be eliminated from the description, since we assume the cargo to be rigid and the anker points of
41
4.2 Structure of the motor pair network the motors at this cargo to be fixed in distance. The elastic forces of the two identical springs, that represent the individual motor stalks, are assumed to balance each other on time scales which are small compared to any of the single motors transitions which allows to define the cargo position as the center-of-mass position. R>350 nm
~ 80 nm
fixed
17 nm
ca L = 80 nm
tr
le
Figure 4.2: Geometry aspects of the motor pair model: (Top row) Both motors are attached to the same cargo which has a radius of R > 350 nm and the stalks of each kinesin are considered to behave as harmonic springs with a rest length of about 80 nm, anker points at the cargo are fixed in distance. The separation between the cargo and the filament is about (17 ± 2) nm. From this, the parallel part of the spring can be estimated. (Bottom row) Reduced representation, two motor ‘particles’ connected by a single spring parallel to the filament with a restlength of 2 · Lk . Cargo position at the center of mass is indicated by the black arrow
The perpendicular force component is balanced by repulsive interactions between filament and cargo [114] and need not be considered here since we want to describe the mechanical motion parallel to the filament. Thus, we will focus on the force component Fext parallel to the filament. Then, the motor pair consists of two identical particels, that are connected via a single spring orientated parallel to the filament, which is determined by the coupling parameter K and the deflection ∆L of the motor-motor separation. The coupling parameter can be determined by different approaches, as introduced in section 5.1, whereas the deflection ∆L is a prominent state variable of the motor pair system, since it determines the forces that occur between the motors during a 2-motor run.
4.2 Structure of the motor pair network A motor pair system contains two individual single motors, that are determined by their chemomechanical networks as introduced in section 3.1 and shown in Fig. 3.3. A motor pair walk as indicated in Fig. 4.1 consists of three possible activity states, one motor bound to the filament, two motors bound to the filament or no motor bound to the filament. After unbinding from the filament, an active motor becomes inactive. As long as one of the motors is bound to the filament, the motor pair performs a 1-motor run, if both motors are attached the motor pair performs a 2-motor run and a cargo run ends, if both motors are inactive.
4.2.1 Combination of motor cycles Starting a cargo run with a 1-motor run, the state space of this situation is governed by both, the individual motor states of the active motor including the possibility to unbind, i.e., state 0, and the rebinding possibility of the inactive motor. As shown in Fig. 4.3, depending on which motor is inactive, we project the single motor network of the leading motor onto the
42
4 Chemomechanical network of a motor pair horizontal axis and that of the trailing motor onto the vertical axis. Nevertheless, we have either of the two possibilities, never both for 1-motor runs. 1−motor runs tr
1
2
3
le
4
5
le
tr
6
7
0
ile
7 0
0
ile
0
7
7
itr
6 5 4 3 2 1
itr
Figure 4.3: State space of motor pair during a 1-motor run as described by two coordinates: the motor states ile of the leading motor (horizontal axis), the motor states itr of the trailing motor (vertical axis). Left: State space of the motor pair during a 1-motor run performed by the leading motor. Open circles represent motor pair states, black lines represent the chemical transitions between these states. The green line describes the mechanical stepping transition of the leading motor, emanating from a state ile = 2 and reaching a state ile = 5. Red lines represent the rebinding events of the inactive trailing motor from state itr = 0 to itr = 7 and the unbinding transition of the leading motor from ile = 7 to ile = 0, describing the unbound motor pair. Right: State space of the motor pair during a 1-motor run performed by the trailing motor, description likewise the 1-motor run of the leading motor, but the mechanical stepping transition of the trailing indicated by a blue line.
After rebinding of an inactive motor, the motor pair performs a 2-motor run. The state space of the motor pair during such a 2-motor run requires the combination of all motor states of the individual motors of this motor pair. In general, the 7-state model of the single motor as in Fig. 3.3, implies 72 = 49 motor pair states and additionally the possibility to unbind, which holds true as long as the spring is equilibrated with L = L0 as shown in Fig. 4.4 (left). Moreover, a mechanical step of one of the motors modifies the elastic force between the two motors, which alternates all transition rates of both motors. Since this elastic force that occur between the motors during a 2-motor run is determined by the deflection ∆L of the motor-motor separation, the deflection ∆L is a state variable of the motor pair system, displayed in Fig. 4.4 (right) as a third axes in the state space. Combination of all these consideration leads to a complex quasi-3-dimensional network graph which will be introduced next.
43
4.2 Structure of the motor pair network L0 le
tr
1
2
L 0 ∆L
2−motor runs
3
4
5
le
tr
6
7
0
ile
1
2
3
4
5
6
7
0
ile
0
0 7
0 7
1
6
2 6
5 ∆L
5
4 4 3 3 2 2 1 1 itr itr
Figure 4.4: State space of the motor pair during a 2-motor run as described by three coordinates: the motor states ile of the leading motor, the motor states itr of the trailing motor, and the deflection ∆L of the elastic spring. The motor pair states form a stack of layers, each of which corresponds to a fixed value of ∆L. Open circles represent motor pair states, black lines represent the chemical transitions between these states. Full green stubs describe mechanical foward steps of the leading motor reaching a state in the overlying ∆L layer, broken green stubs describe mechanical backward steps of the leading motor reaching a state in the underlying ∆L layer. Likewise, the full and broken blue stubs describe forward and backward steps of the trailing motor. Red lines represent unbinding events between the individual motor states i = 7 and i = 0. Left: Single layer of state space of motor pair during a 2-motor run with ∆L = 0. Right: Stack of two layers of state space of motor pair during a 2-motor run: the layer with ∆L = 0 is colored in dark gray, the layer with ∆L = 1 in light grey with all transitions included. The unbinding transitions from the layer ∆L = 1 lead to the 1-motor run states on the axes within the layer ∆L = 0.
4.2.2 Pathways on the network ... In order to address the stochastic behavior of a motor pair consisting of two identical kinesin motors, we describe each motor by a single motor network as depicted in Fig. 3.3. A motor pair state a is now defined by the single motor states i = ile and i = itr of the leading and the trailing motors as well as by the deflection ∆L of the motor-motor separation L from its rest length L0 . In this way, we generate a new state space for the motor pair as shown in Fig. 4.5(a). As explained before, 1-motor runs are characterized by ∆L = 0 corresponding to a relaxed elastic coupling between the two motors. The motor pair in state a = (ile , itr ; ∆L) may undergo transitions to all neighbouring states b according to a = (ile , itr ; ∆L) → b = (jle , itr ; ∆L′ ) ′
= (ile , jtr ; ∆L )
transition by leading motor
(4.17)
transition by trailing motor.
(4.18)
The corresponding transition rates will be denoted by ωab . In general, ∆L′ = ∆L for all chemical transitions, but ∆L′ 6= ∆L for all mechanical transitions during 2-motor runs.
44
4 Chemomechanical network of a motor pair It is important to realize that, for the continuous-time Markov process considered here, each transition rate ωab can be identified with a single motor rate as given by ωab = ωij,le = ωij,0 Φij (Fle ) = ωij,tr = ωij,0 Φij (Ftr )
transition by leading motor
(4.19)
transition by trailing motor.
(4.20)
which has the same form as the expression in Eq. (3.7) where Fle and Ftr represent the forces experienced by the leading and the trailing motors as introduced in Eq. (4.14) and Eq. (4.15). (a)
1
2
3
4
5
6
7
(b)
0
ile
0
1
2
3
4
7
6
5
0
ile 0 −2
7
−1
5
7
0
6
1 ∆L
6
2
5
4
4
3
3
2
2
1
1
∆L itr
itr
Figure 4.5: State space of motor pair as described by three coordinates: the motor states ile of the leading motor (horizontal axis), the motor states itr of the trailing motor (vertical axis), and the deflection ∆L of the elastic spring (axis perpendicular to the plane of the figure). The motor pair states form a stack of layers, each of which corresponds to a fixed value of ∆L. (a) Single layer of state space with ∆L = 0. Open circles represent motor pair states with ∆L = 0, black lines represent the chemical transitions between these states. Full green stubs describe mechanical foward steps of the leading motor, emanating from a state (2, itr ; 0) and reaching a state (5, itr ; 1) in the overlying layer with ∆L = 1. Broken green stubs describe mechanical backward steps of the leading motor, emanating from a state (5, itr ; 0) and reaching a state (2, itr ; −1) in the underlying layer with ∆L = −1. Likewise, the full and broken blue stubs describe forward and backward steps of the trailing motor. Red lines represent binding and unbinding events between the single motor states i = 7 and i = 0. When a motor unbinds from a state (7, itr ; ∆L) with ∆L 6= 0, it reaches a state (0, itr ; 0) with ∆L = 0. The unbound motor pair is described by the pair state (0, 0; 0) in the upper right corner. (b) Stack of five layers: the layer with ∆L = 0 is colored in red, the two layers with ∆L = ±1 in dark grey, and the two layers with ∆L = ±2 in light grey. Each layer is connected to two boundary lines, which are defined by the motor pair states (ile , 0; 0) along the ile -axis and by the states (0, itr ; 0) along the itr -axis. All of these latter states represent 1-motor runs whereas all other motor pair states within the different layers represent 2-motor runs. The unbinding transitions from 2-motor run states with ∆L > 0 and ∆L < 0 to 1-motor run states with ∆L = 0 are described by the full and broken black lines, respectively. The red lines represent both the unbinding transitions from 2-motor run states with ∆L = 0 to 1-motor run states as well as the rebinding transitions from 1-motor run states to 2-motor run states with ∆L = 0. Red arrows represent transitions to the unbound motor pair state.
As shown in Fig. 4.5(b), the motor pair states form a stack of layers, each of which is characterized by a constant value of ∆L. The layer with ∆L = 0, see Fig. 4.5(a), is special since it contains the unbound motor pair state (ile , itr ; ∆L) = (0, 0; 0) as well as two boundary lines. The first boundary line is defined by motor pair states
45
4.2 Structure of the motor pair network
(ile , itr ; ∆L) = (i, 0; 0)
with i 6= 0
(4.21)
and represents 1-motor runs of the leading motor. The second boundary line is defined by the states (ile , itr ; ∆L) = (0, i; 0) with i 6= 0 (4.22) and represents 1-motor runs of the trailing motor. Each of these boundary lines represents a copy of the single motor network depicted in Fig. 3.3. Any 1-motor run may be terminated in two ways: (i) by unbinding of the active motor, which leads to the unbound motor pair state (ile , itr ; ∆L) = (0, 0; 0), or (ii) by rebinding of the inactive motor back to the filament. Rebinding of the inactive trailing motor leads from state (ile , itr ; ∆L) = (i, 0; 0) to state (ile , itr ; ∆L) = (i, 7; 0) whereas rebinding of the inactive leading motor leads from state (ile , itr ; ∆L) = (0, i; 0) to state (ile , itr ; ∆L) = (7, i; 0), see Fig. 4.5. All of these rebinding transitions are governed by the rebinding rate πsi as in Eq. (4.16). The unbound motor pair is described by the pair state (ile , itr ; ∆L) = (0, 0; 0) in the upper right corner of Fig. 4.5(a). In general, a motor pair run is terminated after arriving in this state. However, as far as the unbound state is concerned, the cargo is assumed to stay at the position, at which it became completely detached from the filament and the position of the instantly rebinding motor is calculated with respect to this cargo position. This procedure provides long trajectories of the motor pair. Mechanical steps during 2-motor runs lead to transitions between neighbouring ∆L-layers as indicated by the blue and green stubs in Fig. 4.5(a). These stubs emanate from motor pair states, for which one of the two motors dwells in the single motor state i = 2 or i = 5, compare Fig. 3.3. The transitions between 2-motor and 1-motor runs are provided by binding and unbinding events that are indicated by red lines and black stubs in Fig. 4.5(a) and by red and black lines in Fig. 4.5(b). Unbinding of one motor from a pair state with ∆L 6= 0 corresponds to a transition back to a pair state with ∆L = 0 and either ile = 0 or itr = 0 since the deflection ∆L is taken to vanish in a 1-motor run. As explained before, the deflection ∆L continues to vanish directly after a rebinding event, i.e., directly after a transition that emanates from the two boundary lines. Since the state space for a motor pair as described above is hardly amenable to analytical calculations, we will study it by stochastic simulations as described below in the following section. It is important to note, however, that our description of the motor pair system, which is based on the motor cycles of a single motor, involves only two additional parameter, coupling parameter K, and the rebinding rate πsi .
4.2.3 Activity states of the motor pair As previously mentioned, the network in Fig. 4.5 can be decomposed into three parts, (i) the unbound state with (ile , itr ; ∆L) = (0, 0; 0), in which both motors are inactive, (ii) the two boundary lines as defined by Eq. (4.21) and Eq. (4.22), corresponding to 1-motor runs with one active and one inactive motor; and (iii) the remaining stacked layers of states, which represent 2-motor runs with two active motors. These distinct parts of the motor pair network define the activity states α of the motor pair. The unbound state corresponds to activity state α = 0, 1-motor runs correspond to activity state α = 1 and 2-motor runs to activity state α = 3, respectively. As shown in Fig. 4.5, each layer is characterized by a fixed value of the deflection ∆L of the motor-motor separation and consists of 72 = 49 states for
46
4 Chemomechanical network of a motor pair 2-motor runs. The deflection ∆L determines the elastic interaction force between the two motors. This interaction force changes as soon as one of the motors performs a mechanical step, which leads to a transition to the neighboring ∆L layer. Mechanical forward steps of the leading motor and mechanical backward steps of the trailing motor increase the deflection from ∆L to ∆L + 1 wheras backward steps of the leading motor and forward steps of the trailing motor descrease the deflection from ∆L to ∆L − 1, as indicated by the blue and green stubs in Fig. 4.5(a) Thus, mechanical steps during 2-motor runs lead to transitions parallel to the ∆L axis of the state space while chemical transitions connect two motor pair states within the same ∆L layer. The three different activity states are connected by the overall unbinding and rebinding transitions in Fig. 4.5. The transition |10i corresponds to the overall unbinding of the last active motor, the transition |12i corresponds to the rebinding of the inactive motor, and the transition |21i terminates a 2-motor run by the unbinding of one of the two active motors. We will use this to establish the coarse grained activity state network in Fig. 5.2.
4.3 Simulation details and observables The state space for the motor pair as introduced above is rather complex. The network of the motor pair as displayed in Fig. 4.5, is hardly amenable to analytical calculations, which is why we use the Gillespie algorithm [76], as introduced in section 2.2, to generate random walks on this motor pair network. In the following, the adaption of Gillespie algorithm on the motor pair network will be explained in detail. Adaption of Gillespie algorithm As previously mentioned in section 4.2, transitions between the motor pair state a = (ile , itr ; ∆L) and the motor pair state b = (i′le , itr ; ∆L′ ) for the leading motor or b = (ile , i′tr ; ∆L′ ) for the trailing motor, respectively, are governed by single motor transition rates ωab = ωij,le or ωab = ωij,tr respectively as in expression Eq. (4.19) and Eq. (4.20). P Then we can define the average dwell time hτa i = 1/ b ωab for the pair state a = (ile , itr ; ∆L) and transition probabilities Πab = ωab hτa i from pair state a = (ile , itr ; ∆L) to pair state b = (i′le , itr ; ∆L′ ) or b = (ile , i′tr ; ∆L′ ) respectively. These transition probabilities P are normalized with b Πab = 1 for all motor pair states a. Note that the average dwell times hτa i are related to the average dwell times of the individual motors via hτa i−1 = hτile i−1 + hτitr i−1
(4.23)
with hτile i = 1/ j ωij,le and hτitr i = 1/ j ωij,tr respectively, depending on the single motor transition rates of each motor. Thus, for a motor pair state a that belongs to a 1-motor run, the average dwell time hτa i is affected by the inactive motor, which may rebind to the filament. P
P
Procedure for generating motor pair walks In order to generate random walks on the motor pair network in Fig. 4.5, we use the following procedure:
47
4.3 Simulation details and observables (i) Start at time t = 0 in state a = (ile , 0; 0) corresponding to an inactive trailing motor and an active leading motor in motor state ile = 7. As explained in subsection 4.1, the motor pair attains this state directly after the rebinding of the leading motor to the filament. (ii) Choose a random dwell time τa for state a. Since the dwell times τa are taken to be exponentially distributed, a random number rτ is chosen with uniform probability density over the interval 0 < rτ < 1, from which the random dwell time τa is calculated via τa = −hτa i ln(rτ ). Choose a random number 0 ≤ rΠ ≤ 1 to determine the transition event e. Assign intervals of size Πab to each state b which is connected to a and compare rΠ successively to these intervals. If rΠ lies within an interval, accept the transition to the new state b, otherwise reject it. (iii) Update the clock time te = te−1 + τa , calculate the position of the cargo x ¯ca , the deflection ∆L and the effective forces Fle and Ftr . Update the transition rates and the average dwell time in the new state b and transition probabilities out of this state, and record required quantities. Note, that the position of the cargo could be a floating point number, whereas the position of the motors on the filament has to be an integer, since the binding sites on the filament are discrete with the lattice constant of ℓ = 8 nm. The position of the inactive motor during a 1-motor run is identified with the position of the cargo. When the transition event e is an rebinding event of the inactive motor, calculate the new position of this motor on the filament in respect to the cargo as described in Eq. (4.6). If the calculated new position on the filament lies between two binding sites, choose one of these binding sites with probability 1/2. Repeat step (ii) and (iii). Last step: Stop the walk at time tmax . Note that t = tmax should be sufficiently large compared to the time required to complete a cycle Cν of the single motor network. In principle, a motor pair walk is terminated after it arrives in state a = (0, 0; 0), i.e., when no motor is bound to the filament. It is, however, more convenient to continue the motor pair walk by immediately rebinding one of the motors where each motor is chosen with probability 1/2. The cargo is assumed to stay at the position, at which it became completely detached and the position of the instantly rebinding motor is calculated with respect to this cargo position. In this way, we replace an ensemble average over many relatively short walks by a time average over one relatively long walk. For computational reasons, each of these long walks is stopped after tmax ≃ 104 s; the simulation time has been chosen in such a way that the cargo performs approximately 106 steps in the force free case. Specification of parameters The overall unbinding rate εsi for zero external load force as given by Eq. (3.13) with F = 0 has been determined from the experimental data in [43] for the single motor runlength hxsi i = (1.03 ± 0.17) µm and velocity vsi = (668 ± 8) nm/s at saturating ATP-concentration. As a result, we found εsi ≃ vsi /hxsi i ≃ 0.65/s. Inserting this value of εsi together with the calculated probability P7st into expression Eq. (3.13) with F = 0 we determine the rate constant κ70 . We choose the single motor rebinding rate πsi = 5 /s as found in [56] and independent of the external load force, since it has been found that the effect of a force-dependent rebinding rate is negligible for kinesin motors [115]. The system is typically set up at zero external load force and at saturating ATP concentration [ATP] = 1.6 mM and low ADP and P concentrations [ADP] = [P] = 10 µM, unless noted otherwise.
48
4 Chemomechanical network of a motor pair We will study the coupling parameter K = κ/2 of the assembly in the following range: We refer to the largest elastic modulus found in [116] for AMP-PNP governed kinesin heads with κ ≃ 0.92 pN/nm and therefore limit our calculations to K ≤ 0.5 pN/nm. We choose K ≥ 0.02 pN/nm, since smaller values of K are not convenient: for K < 0.02 pN/nm, the deflection of the motor-motor separation ∆L > 160 nm gets large compared to the size of the assembly. These arguments lead to the range 0.02 ≤ K [pN/nm] ≤ 0.5 for the coupling parameter K with a weak coupling regime K & 0.02 pN/nm and a strong coupling regime with K . 0.5 pN/nm. The transport properties of single motors can be reproduced by disabling rebinding transitions with κ07 ≈ 0 in order to check the algorithm. The starting state in step (i) is then chosen according to the probability distribution as shown in Fig. 3.4(a). On the other hand, by disabling unbinding transitions with κ70 ≈ 0, we obtain long 2-motor runs and therefor better statistics for certain properties of these runs such as the distribution for ∆L or force– velocity relations. In the latter case, we start with two active motors with ∆L = 0 in step (i) in order to avoid intrinsic effects, such as force induced elongation of the spring during a preceding 1-motor run. The individual initial states of the two motors are chosen according to the single motor probability distribution as shown in Fig. 3.4(a).
Characteristic observables In principle, the characteristic observables of single motor runs as introduced in section 3.3, are the same for motor pair walks: Run length, run time and velocity. In particular, there are different levels which define a motor pair system: First, the properties of the activity states, i.e., the distance and time travelled during 1-motor runs and 2-motor runs, respectively. Second, the properties of a cargo run of a motor pair, which are combinations of the activity state properties. Hence, the observables of the single motor run triple for motor pair walks. The distances and times the cargo travels during 2-motor runs with two active motors are denoted by the relative position ∆x2 and the relative time ∆t2 and those during 1-motor runs with one active and one inactive motor likewise by ∆x1 and ∆t1 . Each of these quantities represents a stochastic variable that is governed by a certain probability distribution Pα,x and Pα,t that can be determined by the stochastic simulation. The average run length h∆xα i and the average run time h∆tα i during a activity state run determine the corresponding activity state velocity vα , i.e., the 2-motor run velocity v2 and the 1-motor run velocity v1 . The dwell time ∆τα between to mechanical steps during 1- and 2-motor runs, is governed by a certain probability distribution Pα,∆τ that can be determined by stochastic simulations. As the single motor run time is affiliated with the overall unbinding rate εsi , likewise the average 2-motor run time provides the termination rate ε2 of 2-motor runs. The average distance and time, the cargo travels during a cargo run is denoted by h∆xca i and h∆tca i. Each cargo run consists of alternating 1- and 2-motor runs. Hence, the cargo run length h∆xca i and the cargo run time h∆tca i are given by the average run length and run time during the 1- and 2-motor runs and the average number hnα i of 1- and 2-motor runs. The average number of of 1- and 2-motor runs are simply related via hn1 i = hn2 i + 1, since each cargo run starts and ends with a 1-motor run. The corresponding cargo velocity vca is determined by the ratio of average cargo run length and cargo run time of a motor pair. All of these observables depend on external control parameters like the nucleotide concentration and the external load force. Concerning the intrinsic motor pair parameters, the
49
4.4 Summary and discussion coupling parameter K and the rebinding rate πsi , the properties of the activity states allow the separation of these two parameter: 1-motor runs depend on the rebinding rate but not on the coupling parameter and 2-motor runs depend on the coupling parameter but not on the rebinding rate. Cargo run properties on the other hand are affected by both parameters, since cargo runs consist of alternating 1- and 2-motor runs. Motor pair walks provide some properties which have no equivalent for single motor runs, such as the average number hnα i of 1- and 2-motor runs and the probability Pα to be in the activity state α. The latter are simply related via P1 = 1 − P2 , since we assume that the cargo instantly rebinds to the filament after termination, i.e., P0 = 0, as mentioned before.
4.4 Summary and discussion The motor pair system displayed in Fig. 4.1 consists of two kinesin motors which are attached to the same cargo and walk on the same filament and which are denoted by the leading and the trailing motor. Each motor can unbind from and rebind to the filament separatly. As a consequence, the cargo is actively pulled by either one or two motors or is occupied by two inactive motors. These different activity states are denoted by 1-motor run, 2-motor run and unbound motor pair. We consider the flexible stalk of a kinesin motor as a harmonic spring. A mechanical step of one of the motors during a 2-motor run then leads to an mutual interaction force between the two motors and the cargo. We assume that the elastic forces balance each other on time scales which are small compared to a motor transition and hence we can eliminate the cargo position from the motor pair description. In this way, we obtain a reduced description of the motor pair in terms of an effective spring constant and an effective restlength. In order to adress the stochastic behavior of a motor pair, we describe each motor by the single motor network in Fig. 3.3. We define a motor pair state by the single motor states ile and itr of the leading and the trailing motor and by the the deflection ∆L of the motor-motor separation from the restlength. This new state space for the motor pair results in a complex network as depicted in Fig. 4.5 which involves a huge number of cycles. Since we consider a motor pair walk as continuous-time Markov process, each of the transitions in the motor pair network can be identified with a single motor transition. 1-motor runs occure on one of the boundary lines in Fig. 4.5 which represent the single motor network. Any 1-motor run may be terminated either by unbinding of the active motor which leads to the unbound motor pair state, or by the rebinding of the inactive motor which results in a 2-motor run. While 1-motor runs and rebinding events are characterized by ∆L = 0, 2-motor runs occure on a stack of layers which are characterized by a constant value of ∆L. Mechanical steps during 2-motor runs lead to transitions between neighbouring ∆Llayers. Termination of a 2-motor run by unbinding of one of the active motors corresponds to a transition back to the ∆L = 0 layer, in particular to one of the boundary lines. Even though this chemomechanical motor pair network has a complex structure it involves, apart from the single motor parameters, only two additional parameters, the coupling parameter K and the single motor rebinding rate πsi [112]. Nevertheless, the network is hardly amenable to analytical calculations, which is why we use stochastic simulations on the motor pair network.
50
5 Properties of cargo transport by motor pairs In this chapter we study the dynamics of two elastically coupled kinesins. The elastic coupling leads to mutual interaction forces between the two motors, which each motor effectively experiences as an external load force. With each mechanical step of one of the motors, the deflection of the motor-motor separation and with this the interaction force changes, which then affects all transition rates of both motors, since all chemical and mechanical transitions of both motors depend on the load force. This feedback mechanism between mechanics and chemistry is an important aspect of the system considered here. In the first section 5.1 we focus on the influence of the intrinsic motor pair parameters, the coupling parameter and the rebinding rate, on motor pair trajectories and the characteristic observables of motor pair walks such as run length, run time and velocity. First, we will show how inspection of the trajectories allow to determine the motor pair parameters in experiments [112]. Then, we establish the activity state diagram for this parameter regime and study the influence on the motor pair properties. Comparison to experimental motor pair data leads to distinct combinations of the coupling parameter and the rebinding rate, which will be used in the following parts of this chapter. In the sections 5.2 and 5.3 we study the dependence of the motor pair properties on the external control parameters, first the nucleotide concentration and second the external load force in comparison to the single motor performance and to experimental data.
5.1 Coupling parameter and rebinding rate In the first part of this section we show motor pair trajectories of the cargo run and of the individual motors of a motor pair. We will see how inspection of these trajectories allow to determine the unknown motor pair parameters, the coupling parameter K and the rebinding rate πsi , experimentally. The cargo trajectory can be classified in alternating sequences which are defined by different displacements of the cargo arsing from the number of active motors pulling the cargo. Measuring the average run times of the 1- and 2-motor runs can be used to determine the motor pair parameters. From individual motor trajectories we can deduced the probability distribution of the deflection of the motor-motor separation, which is a property of the 2-motor runs and depends on the coupling parameter K. The relative frequency of 1- and 2-motor runs during a cargo run depends on the motor pair parameters. The termination rate ε2 of 2-motor runs is determined by the rebinding rate if the two probabilities are equal. Thus, we show the activity state diagram which separates dominant 1-motor run and dominant 2-motor run regimes in dependence of the motor pair parameters, the coupling parameter K and the rebinding rate πsi . In the second part of this section we show how these motor pair parameters influence the motor pair properties and we study the activity state diagrams for the different experimental data sets which are introduced in section 3.4. On the basis of experimental motor pair data in [58] we determine distinct combinations of the coupling parameter K and the rebinding rate πsi .
51
5.1 Coupling parameter and rebinding rate
5.1.1 Motor pair trajectories For gliding assays, in which the motors are immobilized on a substrate surface, it is possible to distinguish 4 nm and 8 nm steps of the filaments which permits to distinguish between 1-motor run and 2-motor run regions as shown in [59]. For bead assays as considered here, one could perform analogous experiments, in particular if one uses two different fluorescent labels for the two motors, a method that has been recently applied to the two heads of a single dynein motor [117]. A time resolution in the range of ms would then allow to measure the different run times during a motor pair walk and to detect the dwell times between individual steps even at saturating ATP concentrations. Dwell times between individual motor steps The overall cargo trajectory in Fig. 5.1(a) consists of two segments. During the first 55 ms, the cargo performs a 1-motor run with 8 nm displacements. At 55 ms, the other motor rebinds and the cargo undergoes a 2-motor run with 4 nm displacements, in close analogy to the experimental results in [59]. Indeed, a cargo transported by two motors is displaced by 4 nm when one of the motors makes a 8 nm step. The cargo trajectory in Fig. 5.1(a) is labeled by the dwell times between two steps. The unlabeled part of the trajectory in the middle of the plot represents the switch region between the 1-motor and the 2-motor run, i.e., the dwell time between the last mechanical step during the 1-motor run and the rebinding event. For the histogram in Fig. 5.1(b), we count only dwell times between two mechanical steps. (a)
(b) 72 64
x
ca
[nm]
56 48
2
40
4
5
4
3
4
1 10
8
1-motor runs 0.2
Relative Frequency
80
7
32
9
24
3
16
17
8 0
20
40
60 t [ms]
80
100
2-motor runs
0.1
0.0 0
6
12
18
24
30
36
42
[ms]
Figure 5.1: (a) Cargo trajectory of a motor pair system with a coupling parameter of K = 0.02 pN/nm for zero external force. Cargo steps during a 2-motor run are marked by red color. During the 1-motor run the cargo moves by 8 nm steps whereas it performs 4 nm steps during a 2motor run. The trajectory is labeled by the dwell times (in units of ms) between two successive cargo steps. The unlabeled segment of the trajectory represents the transition from the 1-motor run to the 2-motor run via a rebinding event. (b) Relative frequency count of the dwell times between two steps of the cargo trajectory. Red bars describe the dwell time distribution of 2-motor runs, black bars the dwell time distribution of 1-motor runs. The average dwell times between two cargo steps are found to be h∆τ1 i = (11.6 ± 0.1) ms for 1-motor runs and h∆τ2 i = (6.02 ± 0.02) ms for 2-motor runs.
In general, a transition from a 1-motor to a 2-motor run is defined by a 8 nm step followed by a 4 nm step in the cargo trajectory or vice versa. Inspection of Fig. 5.1(a) indicates that the dwell times ∆τ2 between two cargo steps during a 2-motor run are smaller than the dwell
52
5 Properties of cargo transport by motor pairs times ∆τ1 during a 1-motor run. The dwell time distributions in Fig. 5.1(b) which are derived by stochastic simulations of rather long trajectories, confirm this inequality. If the interactions between the two motors can be neglected, one has a 2-motor run stepping rate that is twice the single motor stepping rate [17]. The average stepping rate 1/h∆τ2 i = (166.1±0.6) /s of the 2-motor run for weak coupling with K = 0.02 pN/nm in Fig. 5.1 is almost twice the 1-motor run stepping rate 1/h∆τ1 i = (86.2 ± 0.7) /s. The latter value correponds to the inverse of the completion time for the forward cycle F and is in good agreement with the experimental value vsi /(8 nm) = (83.5 ± 1.0) /s for single kinesin motors with vsi = (668 ± 8) nm/s as found in [43]. Run times for 1- and 2-motor runs As previously mentioned, the distances and times traveled during 2-motor runs are denoted by ∆x2 and ∆t2 and those during 1-motor runs by ∆x1 and ∆t1 . Each of these quantities represents a stochastic variable that is governed by a certain probability distribution P that can be determined by stochastic simulations. We then define the average run length of 1motor and 2-motor runs by h∆x1 i ≡
X
∆x1
P1,x (∆x1 ) ∆x1
and h∆x2 i ≡
X
∆x2
P2,x (∆x2 ) ∆x2
(5.1)
and the average run times for these runs by h∆t1 i ≡
X ∆t1
P1,t (∆t1 ) ∆t1
and h∆t2 i ≡
X ∆t2
P2,t (∆t2 ) ∆t2 .
(5.2)
The average velocities of the 1-motor and 2-motor runs are given by v1 = h∆x1 i/h∆t1 i
and v2 = h∆x2 i/h∆t2 i .
(5.3)
In general, 1-motor runs may be terminated in two ways: First, the active motor may unbind from the filament as well, which implies a transition from the state a = (7, 0; 0) to the unbound state b = (0, 0; 0). The corresponding transition rate ωab is equal to the single motor unbinding rate ω70 . Second, the nonactive motor may rebind to the filament, which leads to transitions from the motor pair states a′ = (i, 0; 0) to the states b′ = (i, 7; 0), see Fig. 4.5. These rebinding events are all governed by the same transition rate ωa′ b′ = πsi . Therefore, the average run length h∆x1 i as well as the average run time h∆t1 i of 1-motor runs is smaller than the average run length hxsi i and run time htsi i of a single motor. The average velocity v1 , on the other hand, is equal to the average velocity vsi of a single motor. Inspection of Fig. 5.2 shows that the average run time h∆t1 i of 1-motor runs is given by 1/h∆t1 i = εsi + πsi .
(5.4)
with the overall unbinding rate εsi = 1/htsi i and the rebinding rate πsi of a single motor. This relation implies h∆t1 i < 1/(εsi ) as long as εsi < πsi . A 2-motor run is terminated by the unbinding of one of the two active motors. The overall rate for this process will be denoted by ε2 ≡ 1/h∆t2 i
(5.5)
53
5.1 Coupling parameter and rebinding rate
0
εsi
πsi 1
2 ε2
Figure 5.2: Three activity states α = 0, 1 and 2 of a cargo corresponding to its unbound state, 1-motor runs, and 2-motor runs, respectively. The transition |10i with the rate εsi corresponds to the unbinding of the active motor, the transition |12i with rate πsi to the rebinding of the inactive motor, and the transition |21i with rate ε2 to the unbinding of one of the two active motors.
as indicated in Fig. 5.2 . If the interactions between the two motors can be neglected, one has [8, 17] ε2 = 2εsi (5.6) On the other hand, if the motors exert the elastic forces ±K∆L onto each other, the termination rate ε2 of the 2-motor runs will be affected by interference effects arising from these motor-motor interactions [51]. The cargo trajectories of two motor pairs in two different transport regimes, a weakly coupled motor pair with the K = 0.02 pN/nm and a strongly coupled one with K = 0.5 pN/nm, are compared in Fig. 5.3(a), 2-motor runs are marked in red and blue. The different slopes of the two trajectories depicted in Fig. 5.3(a) reflect the general property that the average cargo velocity decreases with an increasing coupling parameter [51]. The run time distributions P1,t (∆t1 ) during 1-motor runs and P2,t (∆t2 ) during 2-motor runs for the two different motor pairs are shown in Fig. 5.3(b). The broad run time distribution for 2-motor runs in the weak coupling regime implies an average value h∆t2 i that is more than twice the corresponding value in the strong coupling regime. Thus, weak coupled motor pairs spend more time in a 2-motor run than strong coupled motor pairs. The run time distributions for the 1-motor runs are essentially identical for both coupling parameter. The average value h∆t1 i as deduced from the trajectories is in good agreement with the value obtained via the expression in Eq. (5.4). As mentioned before, the average run time h∆t2 i determines the termination rate ε2 = 1/h∆t2 i of the 2-motor runs. The average values for the 2-motor run times in Fig. 5.3(b) imply ε2 = (1.62 ± 0.01) /s for weak coupling and ε2 = (4.18 ± 0.03) /s for strong coupling and lead to the ratios in Table 5.1. Hence, the termination rate ε2 of 2-motor runs increases with increasing coupling parameters K as experimentally observed in a recent experimental study [58]. K [pN/nm] ε2 /εsi
0 2
0.02 2.50 ± 0.02
0.1 3.54 ± 0.03
0.5 6.44 ± 0.06
Table 5.1: Ratio of the termination rate ε2 of the 2-motor runs to the overall unbinding rate εsi of single motors for different coupling parameter K. The termination rate ε2 of the 2-motor runs is determined by Eq. (5.6) for the non interacting motors with K = 0 and by Eq. (5.5) for the 2-motor run times in Fig. 5.3(b) as well as for K = 0.1 pN/nm.
Motor pair parameters from cargo trajectories As mentioned before, even though the chemomechanical network of a motor pair is rather complex, it involves only two new parameters, the single motor rebinding rate πsi and the
54
5 Properties of cargo transport by motor pairs
(a)
(b) 2,t
3.2
P
1-motor run
t
0.4
t
and
2-motor run (K=0.02 pN/nm) 2-motor run (K=0.5 pN/nm)
t 0.3
t
Relative Frequency
1.6
x
ca
[ m]
P
1,t
2.4
0.8
0.0 0
1
2
3
4
5
1
1
2
2
(K=0.02 pN/nm) (K=0.5 pN/nm) (K=0.02 pN/nm) (K=0.5 pN/nm)
0.2
0.1
0.0 0.0
t [s]
0.2
0.4
0.6
0.8
1.0
1.2
1.4
t [s]
Figure 5.3: (a) Cargo trajectories of motor pairs in two different transport regimes: weak coupling with K = 0.02 pN/nm and strong coupling with K = 0.5 pN/nm. Cargo steps during 2-motor runs are marked in red for weak coupling and in blue for strong coupling. 1-motor runs are in black. (b) Run time distributions P1,t (∆t1 ) of 1-motor runs and P2,t (∆t2 ) of 2-motor runs for the two different coupling parameters K. The distributions of 1-motor runs should not depend on the coupling parameter as confirmed by the unshaded and shaded bars that represent the data for K = 0.02 pN/nm (weak coupling) and K = 0.5 pN/nm (strong coupling) respectively. Lines with symbols describe the distribution of the 2-motor run times for weak coupling (red squares) and strong coupling (blue circles). The broad run time distribution for 2-motor runs in the weak coupling regime leads to the average run time h∆t2 i = (651 ± 6) ms, which is more than twice the corresponding value h∆t2 i = (249 ± 2) ms in the strong coupling regime. The run time distributions for the 1-motor runs as obtained from both trajectories are in good agreement with each other and imply h∆t1 i = (176 ± 2) ms and h∆t1 i = (179 ± 1) ms, respectively, which should be compared with h∆t1 i = 177 ms as derived from Eq. (5.4).
coupling parameter K. In the simulations, we can choose certain values for these parameters. Such a choice is not possible in experimental studies, where those two parameters depend on molecular architecture of the cargo/motor complex. Therefore, we will now describe procedures, by which one can deduce the values of πsi and K from the observation and analysis of cargo trajectories. In fact, the properties of the 1-motor and 2-motor runs allow us to determine these two parameters separately: 1-motor run properties depend on the rebinding rate πsi but not on the coupling parameter K whereas 2-motor run properties depend on the coupling parameter K but not on the rebinding rate πsi . Measuring the 1-motor run time h∆t1 i in the cargo trajectories of the motor pair as in Fig. 5.3, the rebinding rate πsi can be calculated via the relation in Eq. (5.4) or πsi =
1 − εsi . h∆t1 i
(5.7)
Further investigation of the cargo trajectories in Fig. 5.3 provide the average 2-motor run time h∆t2 i of a cargo run, which determines the coupling parameter K. The 2-motor run time h∆t2 i, rescaled by the single motor run time, is shown in Fig. 5.4 as a function of the coupling parameter K. If the interactions between the two motors can be neglected, i.e., for K = 0, one has h∆t2 i = htsi i/2 as in Eq. (5.6). The 2-motor run time shown in Fig. 5.4(a) decreases with increasing coupling parameters K in a roughly double exponential manner. There is a sharp decline to 30% of the single motor run time at small coupling parameters K < 0.1 pN/nm. For larger coupling parameters, the 2-motor runtime decreases more slowly.
55
5.1 Coupling parameter and rebinding rate
(a)
(b)
0.5
0.5
[ATP]=
[ADP]= 0.1
1 mM
[ATP]= 0.1 mM
[ADP]=
10
[ATP]= 10
[ADP]=
1
M
M mM
si
t > /
0.4
si
t > /
0.4
M
2
0.3
P2 are dominant for relatively small values of πsi /εsi , whereas dominant 2-motor runs with P2 > P1 require large values of πsi /εsi .
The ratio P2 /P1 is shown in Fig. 5.7(a) as a function of the rebinding rate πsi . The simulation results for three different values of the coupling parameter K, corresponding to the three transport regimes for kinesin as introduced in [51], are compared to the calculated relation in Eq. (5.10), where we have used ε2 as a fit parameter, see inset in Fig. 5.7(a). These fit parameter correspond to the average values in Table 5.1. When the ratio P2 /P1 is less than 1, the motor pair walk is dominated by the 1-motor runs whereas it is dominated by the 2-motor runs for P2 /P1 > 1 . As shown in Fig. 5.7(a), the ratio P2 /P1 depends linearely on the rebinding rate πsi . Inspection of Fig. 5.7(a) also reveals that the ratio P2 /P1 is strongly reduced for increasing
60
5 Properties of cargo transport by motor pairs coupling parameter K, which implies that the termination rate ε2 of the 2-motor runs is strongly increased by interference effects arising from the motor-motor interactions. Observing dominant 2-motor runs of a strongly coupled motor pair requires a large rebinding rate with πsi ≥ 4 /s, whereas weakly coupled motor pairs have as many 2-motor runs as 1-motor runs for a relatively small rebinding rate πsi ≃ 1.5 /s. The expression in Eq. (5.10) implies that the characteristic rebinding rate πc , at which P1 = P2 , of a coupling parameter, determines the 2-motor run termination rate ε2 (K) = πc which is consistent with the average values in Table 5.1. The crossover line, at which P1 = P2 , is shown in Fig. 5.7(b) as a function of the coupling parameter and the rebinding rate. The characteristic rebinding rate πc is rescaled by the single motor overall unbinding rate εsi = 0.65/s. This crossover line seperates the parameter regime, in which 1-motor runs dominate the cargo run, from the regime, in which 2-motor runs are more likely. The motor pair spends most of its time in 1-motor runs for all coupling parameters if we choose a relatively small rebinding rate, whereas a dominance of 2-motor runs requires relatively large rebinding rates. (b)
(a)
8
6
7
P
1
1 mM
M
[ATP]=
M
1
6
2
[ATP]=10
2
[ATP]=0.1 mM
P=
[ATP]= 7
5
[ADP]=[P]= 10
M
3
2
1
3
P =2P
2
P=P
2
1
2
1
1
2
1 0.0
P =P
4
/
4
si
c
/
si
si
5
0 0.1
0.2
K
0.3
[pN/nm]
0.4
0.5
0.0
0.1
0.2
K
0.3
0.4
0.5
[pN/nm]
Figure 5.8: (a) The crossover line P2 = P1 as a function of the coupling parameter K and the characteristic rebinding rate πc for different ATP concentrations, with [ATP]=1 mM (solid line), [ATP]=0.1 mM (dashed line), [ATP]=10 µM (dotted line) and [ATP]=1 µM (dash-dotted line). The characteristic rebinding rate πc is rescaled by the corresponding single motor overall unbinding rate εsi . Note, that εsi depends on the nucleotide concentrations, as shown in Fig. 3.6(a). (b) Activity state diagram at saturating ATP concentration: Crossover lines of the ratio P2 /P1 as a function of the coupling parameter K and the characteristic rebinding rate πc rescaled by the single motor overall unbinding rate εsi = 0.65/s. The solid line corresponds to P2 = P1 as in (a) and in Fig. 5.7(b). The dashed and the dotted black lines are the crossover lines, at which P1 = 2 P2 and P2 = 2 P1 , respectively.
The ratio P2 /P1 in Eq. (5.10) depends on the average 2-motor run time h∆t2 i. As shown in Fig. 5.4 the 2-motor run time depends on the ATP-concentration. Hence, the ratio P2 /P1 is affected by a change in the ATP-concentration. Fig. 5.8(a) shows the crossover line of the ratio P2 = P1 for several values of ATP-concentrations. The region of dominant 2-motor runs enlarges with a decreasing ATP-concentration. The dominance of 2-motor runs for strongly coupled motor pairs requires relatively large rebinding rates πsi ≥ 6.4εsi for saturating ATP
61
5.1 Coupling parameter and rebinding rate concentration, whereas for very low ATP-concentrations a rebinding rate of πsi ≃ 3.2εsi is sufficient. Which activity state is dominant depends on the rebinding rate πsi and the coupling parameter K as summarized in Fig. 5.8(b). The crossover line P2 = P1 seperates the parameter regime, in which 1-motor runs dominate the cargo run, from the regime, in which 2-motor runs are more likely. The motor pair spends most of its time in 1-motor runs for all coupling parameters if we choose a relatively small rebinding rate, whereas a dominance of 2-motor runs requires relatively large rebinding rates. The crossover lines at which P1 = 2 P2 and at which P2 = 2 P1 , indicated by the broken and dotted black lines, give more information about dominance of 1-motor and 2-motor runs. A small rebinding rate with πsi ≤ εsi provides for all coupling parameters a clear dominance of 1-motor runs with P1 ≥ 2 P2 . Here, 2-motor runs presumably serve as anchors, so that the motor pair does not diffuse away from the filament. In contrast, a clear dominance of 2-motor runs with P2 ≥ 2 P1 can only be accomplished by motor pairs with a small coupling parameter and only for relatively large rebinding rates with πsi ≥ 4εsi .
Run length, run time and velocity In general, a motor pair walk consists of several cargo runs, separated by the activity state α = 0 in Fig. 5.2. A cargo run consists of alternating 1- and 2-motor runs, which represent the activity states α = 1 and α = 2 and a final 1-motor run ending in activity state α = 0 . As previously mentioned in Sec. 4.2, the distances and times traveled in activity state α are denoted by ∆xα and ∆tα . Each of these quantities represents a stochastic variable that is governed by a certain probability distribution P that can be determined by stochastic simulations and which are defined in Eq. (5.1), Eq. (5.2) and Eq. (5.3) as h∆xα i ≡ h∆tα i ≡
X
∆xα
X
∆tα
Pα,x (∆xα ) ∆xα
(5.18)
Pα,t (∆tα ) ∆tα
(5.19)
vα = h∆xα i/h∆tα i .
(5.20)
In principle, 1-motor runs may be terminated in two ways: First, the active motor may unbind from the filament as well and the corresponding transition rate ωab is equal to the single motor unbinding rate ω70 . Second, the inactive motor may rebind to the filament, these rebinding events are all governed by the same transition rate πsi . Therefore, the average run length h∆x1 i as well as the average run time h∆t1 i of 1-motor runs is smaller than the average run length hxsi i and run time htsi i of a single motor. The average velocity v1 , on the other hand, v1 = vsi (5.21) is equal to the average velocity vsi of a single motor. The 1-motor run time depends on the rebinding rate, as in Eq. (5.4) 1 . (5.22) h∆t1 i = εsi + πsi With these expressions, the 1-motor run length is given by vsi . (5.23) h∆x1 i = v1 h∆t1 i = εsi + πsi
62
5 Properties of cargo transport by motor pairs Rebinding of an inactive motor to the filament leads to a 2-motor run. In principle, 2-motor runs are terminated by the unbinding of one of the active motors. Therefore, as mentioned above, the 2-motor run time can be expressed by this termination rate h∆t2 i =
1 . ε2
(5.24)
Inspection of the motor pair trajectories above exhibit, that the average value of this termination rate increases if the coupling parameter increases, as shown in Tab. 5.1. This leads with Eq. (5.6) to the following relation for the 2-motor run time 1 . 2εsi
h∆t2 i ≤
(5.25)
If the interactions between the two motors can be neglected, one has v2 = vsi [17]. On the other hand, if the motors exert the elastic forces ±K∆L onto each other, the 2-motor run velocity will be affected by interference effects arising from these motor-motor interactions [51] in terms of v2 ≤ vsi (5.26) and with this, the average 2-motor run length can be written as h∆x2 i ≤
xsi . 2
(5.27)
Note that the 2-motor run properties in Eq. (5.24), (5.25) and (5.27) are functions of the coupling parameter K. If the interactions between the two motors can be neglected, i.e., K = 0, one has h∆t2 i = 1/(2εsi ) and v2 = vsi as in [17] and with this h∆x2 i = hxsi i/2. Note also that these relations are considered for the force free case. As explained in section 4.3, once the motor pair reaches the activity state α = 0, it is immediately returned to α = 1 by rebinding one of the two motors back to the filament. This protocol is used in order to replace an ensemble average over many short cargo runs by a time average over one long motor pair walk. Each cargo run starts with a 1-motor run and ends with a 1-motor run, which implies that the average number hn1 i of 1-motor runs and the average number hn2 i of 2-motor runs in a cargo run are simply related via hn1 i = hn2 i + 1. We then define the average cargo run length and run time, as before in Eq. (5.12) by h∆xca i ≡ h∆tca i ≡
X α
hnα ih∆xα i = (hn2 i + 1) h∆x1 i + hn2 ih∆x2 i
X α
hnα ih∆tα i = (hn2 i + 1) h∆t1 i + hn2 ih∆t2 i
(5.28) (5.29)
and the cargo velocity by (5.30)
vca ≡ h∆xca i/h∆tca i .
Inserting in these latter expressions the average number of 2-motor runs hn2 i = πsi /εsi as in Eq. (5.15), the 1-motor run length as in Eq. (5.23) as well as the run times as in Eq. (5.4) and (5.5) one obtains h∆xca i =
πsi v1 + h∆x2 i εsi εsi
and
h∆tca i =
ε2 + πsi ε2 εsi
(5.31)
63
5.1 Coupling parameter and rebinding rate
(a)
(b)
/ 1
0.8
si
/
/
1
1
si
si
0.6
0.4
0.2
0.0
2-motor run properties
1-motor run properties
1.0 1.0
0.8
/ 2
0.6
si
/ (/2)
/ (/2)
2
2
si
si
0.4
0.2 0
1
2
3
4
si
/
5
6
7
0.0
0.1
0.2
K
si
0.3
0.4
0.5
[pN/nm]
Figure 5.9: Activity state properties run length h∆xα i (red dashed line), run time h∆tα i (green dotted line) and velocity vα (black line) (a) of 1-motor runs as a function of the rebinding rate πsi divided by the unbinding rate εsi = 0.65 /s at saturating ATP-concentration. All properties of the 1-motor run are divided by the corresponding single motor properties hxsi i, htsi i and vsi . The relations in Eq. (5.4) and (5.23) show that the 1-motor run time and the 1-motor run length decrease with increasing z = πsi /εsi like 1/(1 + z) . (b) of 2-motor runs as a function of the coupling parameter K. All properties of the 2-motor run are divided by the single motor properties hxsi i/2, htsi i/2 and vsi .
which is in accordance with the results in [17] for K = 0. Then, the average cargo velocity is given by ε2 v1 + πsi v2 vca = (5.32) ε2 + πsi which implies that the cargo velocity is determined by the acitivity state velocities and the probabilities to be in the corresponding activity state vca = P1 v1 + P2 v2
(5.33)
where Eq. (5.9) and Eq. (5.3) have been used. Since P1 + P2 = 1 and v1 = vsi, as in Eq. (5.21), the latter relation can be rewritten as vca = vsi + P2 (v2 − vsi ) .
(5.34)
The cargo velocity now only depends on the probability to be in a 2-motor run and the velocity difference between 2-motor run and single motor, since the the single motor velocity is a measured property. As summarized in Fig. 5.8(b), which activity state is dominant depends on the rebinding rate πsi and the coupling parameter K. The crossover line P2 = P1 seperates the parameter regime, in which 1-motor runs dominate the cargo run, from the regime, in which 2-motor runs are more likely. In the limit of small probability to be in a 2-motor run P2 ≃ 0, the cargo run will presumably consist of a single motor run, and the cargo velocity then is given by the single motor velocity vca ≃ vsi
for small P2 ≃ 0 .
(5.35)
As shown in Fig. 5.8(b), the probability to be in a 2-motor run is rather small for a relatively small rebinding rate and even smaller for low ATP-concentrations, as shown in Fig. 5.8(a).
64
5 Properties of cargo transport by motor pairs In the limit of large probability to be in a 2-motor run P2 ≃ 1, the cargo run consists of long 2-motor run and short 1-motor runs, and the cargo velocity then is given by the 2-motor run velocity vca ≃ v2 for large P2 ≃ 1 . (5.36) As shown in Fig. 5.8(b), the dominance of 2-motor runs requires relatively large rebinding rates. The velocity during a 2-motor run does not depend on the rebinding rate πsi , but on the coupling parameter K. Therefore, the velocity difference between 2-motor run and single motor, determines the cargo velocity vca = vsi
if
v2 = vsi
(5.37)
vca < vsi
if
v2 < vsi
(5.38)
vca > vsi
if
v2 > vsi
(5.39)
For a weakly coupled motor pair, with a small coupling parameter, the 2-motor run velocity is comparebale to the single motor velocity, whereas for a strongly coupled motor pair, the 2-motor run velocity is essentially reduced [51]. We will see in section 5.3, that the last case in Eq. (5.39) can be obtained by relatively large external load forces. Summarized, these arguments determine the cargo velocity as follows for small πsi or small K
(5.40)
vca < vsi
for large K
(5.41)
vca > vsi
for large Fext .
(5.42)
vca ≃ vsi
Coupling parameter and rebinding rate As mentioned before, the properties of a 1-motor run do not depend on the coupling parameter whereas the properties of a 2-motor run do not depend on the rebinding rate. Fig. 5.9 shows the runlength, the run time and the velocity of a 1-motor run as a function of the rebinding rate and the properties of the 2-motor runs as a function of the coupling parameter. The properties of a 1 motor run are shown in Fig.. 5.9(a) as a function of the rebinding rate πsi . The 1-motor run velocity does not depend on the rebinding rate v1 = vsi as in Eq. (5.21). The relations for the 1-motor run length in Eq. (5.4) and 1-motor run time in Eq. (5.23) both decrease with increasing z = πsi /εsi like 1/(1 + z) . Note, that for a large rebinding rate of πsi = 5 /s, as typically used in this work, if not mentioned otherwise, both, the 1-motor run time and the 1-motor runlength are significantly smaller than the single motor properties. In the limit of a small rebinding rate πsi ≃ 0, the motor pair in a 1-motor run behaves as a single motor. Therefore, the 1-motor run properties in Fig. 5.9(a) are rescaled by the correponding single motor properties. The properties of a 2 motor run are shown in Fig. 5.9(b) as a function of the coupling parameter K. If the interactions between the two motors can be neglected, i.e., for K = 0, the 2-motor run velocity v2 ≃ vsi complies with the single motor velocity. Both, the 2-motor run length h∆x2 i ≃ hxsi i/2 and the 2-motor run time h∆t2 i ≃ htsi i/2 correspond to half of the single motor properties, since there are two individual motors involved now. Therefore, the 2-motor run properties in Fig. 5.9(b) are rescaled by the correponding fraction of the single motor properties. All 2-motor run properties decrease with increasing values of the coupling parameter. For a strongly coupled motor pair, the 2-motor run velocity drops to about 80 % of the single motor runlength and the 2-motor run time drops to about 30% of
65
5.1 Coupling parameter and rebinding rate
(a)
(b)
/ si
7
>/ si
3.5
2
1
2.0
P = 2P
1.5
si
2.5
2
P
4.0 3.5
2
1
3.0 2.5 2.0
3
P = 2P
2
1
=
4
1.0
2
1
2
P
5
3.0
4
3
4.5
/
si
P
si
P
=
/ si
6
4.0
P=
2
4.5
5
1.5 1.0
2
1
1
0 0.0
ca
2
P 2
P=
6
t
1
ca
P
x
1
< 7
0 0.1
0.2
K
0.3
[pN/nm]
0.4
0.5
0.0
0.1
0.2
K
0.3
0.4
0.5
[pN/nm]
Figure 5.10: Contour plots of (a) the cargo run length h∆xca i, divided by the single motor run length hxsi i, as a function of the coupling parameter K and the rebinding rate πsi divided by the unbinding rate εsi = 0.65 /s at saturating ATP-concentration. (b) the cargo run time h∆tca i, divided by the single motor run time htsi i, as a function of the coupling parameter K and the rebinding rate πsi divided by the unbinding rate εsi = 0.65 /s at saturating ATP-concentration. The gray-scale map in both plots starts with small values in black to large values in white. The red lines represent the three crossover lines in Fig. 5.8(b).
htsi i/2, which is in accordance to the strong coupling regime in [51]. Note that the 2-motor run times in Fig. 5.4 are rescaled by htsi i, therefore, the inital values correpond to 0.5 htsi i. Since the cargo properties are composed of the activity state properties, both, the cargo run length and the cargo run time, and with these the cargo velocity depend on both parameters, the rebinding rate and the coupling parameter. The cargo run length h∆xca i rescaled by the single motor run length hxsi i is shown in Fig. 5.10(a) as a function of the coupling parameter K and the rebinding rate πsi , the latter is rescaled by the unbinding rate εsi . The crossover lines from Fig. 5.8(b) for dominating 2-motor or 1-motor runs are drawn in red here. The solid line represents P2 = P1 which corresponds to the contour line of h∆xca i = 2hxsi i, i.e. if we choose the parameters πsi and K thus the motor pair performs 1-motor runs and 2-motor runs with equal probability, then the cargo run length is twice the single motor run length. The dotted line is the crossover line, at which P2 = 2 P1 , which corresponds to the contour line of h∆xca i = 3hxsi i. The dashed line is the crossover line, at which P1 = 2 P2 , which corresponds to the contour line of h∆xca i = 1.5hxsi i. Although the motor pair walk consists mainly of 1-motor runs, the cargo run length exceeds the single motor run length. An inactive motor in a motor pair is sterically still near the filament since it is bound to the cargo and the other motor remains active. Without any forces involved, it is likely that an inactive motor rebinds before the active motor unbinds as long as we choose πsi > εsi . Therefore, a motor pair is able to cover a larger run length than the single motor. Likewise, the contour lines of the cargo run time h∆tca i in Fig. 5.10(b) correspond to the crossover lines.
66
5 Properties of cargo transport by motor pairs
(a)
(b)
0.6
data
data
0.8
0.5
=0.5/s
Relative Frequency
Relative Frequency
si
=0.61/s
si
0.6
=0.65/s
si
=1/s
si
0.4
0.2
0.0
=1 /s
K=0.02 pN/nm; K=0.1 pN/nm;
0.4
K=0.5 pN/nm;
si
=1.25 /s
si
=3 /s
si
0.3
0.2
0.1
0.0 0
1
2
3
x
si
4
5
6
0
1
2
[ m]
3
4
x
ca
5
6
7
8
[ m]
Figure 5.11: (a) Histogram of the single motor run length ∆xsi for parameter set D as introduced in section 3.4 for different values of the overall unbinding rate, εsi = 0.5/s (black bars), εsi = 0.61/s (red bars), εsi = 0.65/s (green bars) and εsi = 1/s (blue bars), compared to experimental data (gray bars) from [58]. (b) Histogram of the cargo run length ∆xca : Experimental data (gray bars) from [58] compared to simulation results of parameter set D for different values of the rebinding rate πsi and the coupling parameter K. Weak coupling with K = 0.02 pN/nm and a small rebinding rate of πsi = 1/s displayed by the black bars, K = 0.1 pN/nm with πsi = 1.25/s displayed by the red bars and strong coupling with K = 0.5 pN/nm and a rebinding rate of πsi = 3/s displayed by the blue bars.
Comparison with experimental motor pair data The experimental findings in [58] provide run length histograms for both, the single kinesin motor and a coupled motor pair system. Using the parameter set D as introduced above in section 3.4, we produce single motor run length xsi histograms which are displayed in Fig. 5.11(a) for several values of the overall unbindingrate εsi . In agreement with the experimental data, we find εsi = 0.61/s. For the motor pair system, we can find various combinations of the motor pair parameter πsi and K which provide cargo run length ∆xca histograms in accordance with the experimental data, see appendix B in Fig. B.6. The examples are shown in Fig. 5.11(b) for a weakly coupled motor pair and a small rebinding rate and a strongly coupled motor pair with a rather large rebinding rate. Both combinations are located slightly below the characteristic line P2 = P1 in the 1-motor run dominance region in Fig. 5.12(b). The average values of the velocity and the runlength of the single motor and of the motor pair are specified in [58]. Inserting these values into Eq. 5.31 leads to a conditional equation for the 2-motor run time depending on the cargo properties of the motor pair h∆t2 i =
1 πsi
vsi h∆xca i −1 vca hxsi i
(5.43)
where the relations vca = h∆xca i/h∆tca i and vsi = hxsi i/htsi i have been used. Note that the 2-motor run time itself does not depend on the rebinding rate πsi , as mentioned above. The dependence on πsi in Eq. (5.43) is due to the cargo properties of the motor pair. Comparison between the run time h∆t2 i calculated via Eq. (5.43) to the simulation results leads to the
67
5.1 Coupling parameter and rebinding rate same parameter combinations as derived by the analysis of the run length distributions in Fig. 5.11(b). The detailed analysis and comparison is given in appendix B in Fig. B.7. Adaption to experimental data By now, we have used the data set B in Tab. A.1 corresponding to the experimental data in [43] for the basic single motor parameters, except for the previous subsection where we introduced results for the parameter set D. In this subsection, we additonally use the single motor parameter sets C and E in Tab. A.1, corresponding to the experimental data in [27] and [29] to our motor pair system and compare the results. We perform cargo trajectories and trajectories of individual motors for the parameter sets C, D and E and analyse them in in the same way as introduced for the parameter set B. (b)
(a)
8
0.5
7
0.2
D 2
E
E
5 si
C
D
/
B
6
c
3
B C
L=0)
0.3 4
amplitude P (
5
width
7
0.4
6
4
P=P
3
1
2
0.1
[ADP]=[P]= 10
1
0.0
2
0.1
0.2
K
0.3
[pN/nm]
0.4
0.5
0.0
M
[ATP]= 1.6 mM 0.1
0.2
K
0.3
0.4
0.5
[pN/nm]
Figure 5.12: Parameter sets B (solid line), C (dashed line), D (dotted line) and E (dash-dotted line): (a) Width and amplitude of P(∆L) as a function of the coupling parameter K at saturating ATP-concentration. (b) Activity state diagram: Crossover lines of the ratio P2 /P1 as a function of the coupling parameter K and the characteristic rebinding rate πc rescaled by the corresponding single motor overall unbinding rate εsi .
The width σ and the peak amplitude P(∆L = 0) of the probability distribution P(∆L) of the deflection of the motor-motor separation are shown in Fig. 5.12(a) as functions of the coupling parameter K for the different parameter sets B, C, D and E at saturating ATP-concentration. While the amplitude P(∆L = 0) increases exponentially, the width σ decreases in an double exponential manner with increasing K. The parameter sets D and E yield similar results to the parameter set B whereas the width of the distribution is significantly reduced for the parameter set C. Likewise, the the the probability for ∆L = 0 is increased for the parameter set C. As listed in Tab. 3.2, the force threshold F¯ij′ is quite similar for the parameter sets B, D and E, but is reduced for set C. Since the distribution of ∆L is related to the distribution of interaction forces, see Eq. (4.13), we expect to find this difference reflected in the results for this parameter set. The activity state diagram in Fig. 5.12(b) compares the different parameter sets C, D and E, to the results of the parameter set B. Which activity state is dominant depends on the rebinding rate πsi and the coupling parameter K. The crossover line P2 = P1 seperates the
68
5 Properties of cargo transport by motor pairs parameter regime, in which 1-motor runs dominate the cargo run, from the regime, in which 2-motor runs are more likely. The characteristic rebinding rate πc for which P2 = P1 , is rescaled by the single motor overall unbinding rate εsi for the corresponding parameter set, via κ70 as in Tab. A.1. Again, parameter set D and E yield similar results compared to the parameter set B. The characteristic line of the parameter set C is shifted towards smaller values of πc , which implies that 2-motor runs are more likely for a smaller rebinding rate and/or coupling parameter, compared to the other parameter sets. The differences in the characteristic lines of the different parameter sets is again explained by their different the force threshhold F¯ij′ .
5.1.3 Summary and discussion The complex motor pair network introduced in chapter 4 consists of a huge number of cycles and the dependence of the transitions on force is highly nonlinear. Therefore, the network is not amenable to analytic calculations. Instead, we use Gillespie algorithm [76] in order to simulate the motor pair dynamics and to create trajectories, where we distinguish between cargo trajectories from individual motor trajectories. Both types of trajectories are accessible to experiments and, thus can be used to obtain a complete set of parameters of cargo transport by motor pairs. These trajectories can always be decomposed into 1-motor run sections, where the cargo is displaced by 8 nm and 2-motor run sections, during which the cargo performs 4 nm steps. The statistics of these runs can be used to extract reliable estimates of the two intrinsic motor pair parameters [112], namely the coupling parameter K and the rebinding rate πsi . The rebinding rate can be directly obtained from the average 1-motor run time. The average 2-motor run time is related to the termination rate of these runs which can be used to determine the coupling parameter. Alternatively, the coupling parameter can be determined by statistical analysis of the individual motor trajectories. During 2-motor runs, the individual motor trajectories reveal the time dependent deflection ∆L of the motor-motor separation. The distribution of these deflections can then be used to determine the coupling parameter. Since the trajectories can be decomposed into 1-motor and 2-motor runs, we introduce the simple 3-state network in Fig. 5.2 which describes the three activity states of the motor pair and the transitions between these states. Which activity state is dominant during a motor pair walk depends on the rebinding rate and via the 2-motor run termination rate on the coupling parameter. The corresponding activity state diagram in Fig. 5.8 shows the crossover line which separate the parameter regime, in which 1-motor runs dominate the cargo run from the regime, in which 2-motor runs are more likely. A small rebinding rate leads to a clear dominance of 1-motor runs for all values of the coupling parameter, whereas clear dominance of 2-motor runs is only found for a relatively large rebinding rate and small coupling parameters. The activity state network in Fig. 5.2 implies that the average 1-motor run time depends on the rebinding rate but not on the coupling parameter whereas the 2motor run time depends on the coupling parameter but not on the rebinding rate. In general, this parameter separation is reflected in the transport properties. The run length and the run time of 1-motor runs decay as predicted in Eq. (5.4) and Eq. (5.23) for increasing values of the rebinding rate. The resulting average 1-motor run velocity corresponds to the single motor velocity, as expected. There is a sharp decline in the runlength and the run time of 2-motor runs as functions of the coupling parameter for weakly coupled motor pairs. For larger coupling parameters, the run length and run time decrase more slowly. In contrast to
69
5.1 Coupling parameter and rebinding rate the 1-motor run properties, the 2-motor run length decays faster than the 2-motor run time dependent on the coupling parameter, which leads to the decrease in the average 2-motor run velocity as a function of the coupling parameter, as predicted in [51]. The average cargo run properties of a motor pair are combinations of the activity state properties, hence depend on both motor pair parameters. Comparison between the activity state diagram and the contour plots of the cargo properties as functions of the motor pair parameters, reveals, that the cargo run length is twice the single motor run length if we choose the parameter K and πsi that the motor pair performs 1-motor runs and 2-motor runs with equal probability, which corresponds to the crossover line P1 = P2 . The crossover line that marks the region of clear dominance of 2-motor runs correponds to a cargo run length thrice the single motor run length. Although the motor pair walk consists of mainly 1-motor runs for small rebinding rates, the cargo run length exeeds the single motor run length. Without any external strain involved, it is likely, that an inactive motor rebinds before the active motor detaches. Therefore, in general, a motor pair is able to cover a larger run length than the single motor. Note, that in the limit of non-interacting motors, our model reproduces the results in [17]. The experimental findings in [58] provide run length histograms of the single motor and the motor pair. In agreement with these data we determine different combinations of the coupling parameter and the rebinding rate. The average cargo run length h∆xca i = (1.7 ± 0.2)hxsi i in [58] corresponds to a region with mainly 1-motor runs in the activity state diagram, which is in good agreement with their subsequent findings in [64]. Comparison of the corresponding activity state diagrams and analysis of the distributions of the deflection show little difference for the parameter sets B, D and E whereas the results for the parameter set C are significally reduced. The force thresholds F¯ij′ of the chemical transition rates are quite in the same range for the parameter sets B, D and E whereas the estimated force thresholds in parameter set C are significantly smaller. With regard to the application of external load force on the motor pair system, it is important to note, that already a slight change in the force dependence of the chemical transition rates may lead to such differences.
70
5 Properties of cargo transport by motor pairs
5.2 Dependence of transport properties on nucleotide concentrations The dynamics of a single motor depends on the nuclotide concentrations as introduced in section 3.3, i.e., a small ATP concentration slows the motion of the motor whereas sufficiently large ATP concentration leads to a saturation velocity. A change in the product concentrations of ADP and P has little effect on the single motor properties for large ATP concentrations whereas the influence is significant for small ATP concentrations. The dependence of the single motor dynamics on the nucleotide concentrations is also reflected in the motor pair dynamics.
5.2.1 Influence on the motor pair properties As explained above in section 5.1, the motor pair properties can be expressed by the unbinding and rebinding rates between the activity states and the single motor properties. The average run time h∆t1 i of a 1-motor run in Eq. (5.22) depends on the rebinding rate πsi and the single motor overall unbinding rate εsi . The latter corresponds to the inverse of the average single motor run time htsi i = 1/ε1 . As mentioned before, we choose a constant rebinding rate πsi = κ07 . The overall unbinding rate in Eq. (3.13) on the other hand, depends on the steady state probability P7st for the motor to occupy state i = 7. The steady state probabilities Pist as introduced in section 3.3 can be expressed as ratios of two polynomials, which are multilinear in the transition rates ωij as defined in Eq. (3.7) and depend on the nucleotide concentrations [72]. The influence of the concentration-dependent steady state probability P7st on the overall unbinding rate is shown in Fig. 3.6(a). For a small ADP and P concentration, P7st increases with an increasing ATP concentration whereas for a large ADP and P concentration, P7st decreases with increasing ATP. For a product concentration of [ADP]=[P]=10 µM, P7st is almost independent of the ATP concentration. Concerning the 1-motor run time h∆t1 i in Eq. (5.22), the effect of a change in the nucleotide concentrations is small. As long as εsi ≪ πsi the 1-motor run time yields h∆t1 i ≈ 1/πsi . (5.44) For small nucleotide concentrations, the single motor unbinding rate is very small, εsi ≪ πsi .Thus, on average an inactive motor rebinds before the active motor detaches which implies long motor pair run times. Additionally, a small single motor unbinding rate leads to a large average number of 2-motor runs hn2 i = πsi /εsi as in Eq. (5.15). For non-interacting motors with K = 0, the average 2-motor run time h∆t2 i in Eq. (5.24) is proportional to the average single motor run time htsi i and thus, likewise reflects the dependence on the nucleotide concentrations. As shown in Fig. 5.4, the strong decrease in the 2-motor run time with an increasing coupling parameter is extenuated for a small ATP-concentration. The average cargo run time h∆tca i of the motor pair is a combination of 1- and 2-motor run time. Using h∆t2 i = 1/ε2 and htsi i = 1/εsi the expression in Eq. (5.31) can be written as h∆tca i = htsi i (1 + πsi h∆t2 i) .
(5.45)
The average cargo run time h∆tca i and the average activity state run times h∆t2 i and h∆t1 i compared to single motor run time htsi i are shown in Fig. 5.13 as functions of the ATP concentration for two different ADP and P concentrations. The different values of the
71
5.2 Dependence of transport properties on nucleotide concentrations
(a)
(b) [ADP]=[P]= 10
M
[ADP]=[P]= 0.1
M
100
K=0.02 pN/nm
1
t
ca
2
t > 1
si
[s]
si
[s] 10
run time
run time
ca
2
x > 1
[ m]
si
10
runlength
[ m]
x
10 pN before detachement of the motor pair are rare and they conclude that specific conditions such as the orientation of the assembly may be required for a motor pair to produce such large forces. We will consider this in terms of cargo rotation in the next subsection. On account of this experimental observation, that large forces are rare, the focus of the comparison to the experimental data in Fig. 5.21 and Fig. 5.22, is on resisting forces Fext < 10 pN, where we find good agreement with the data. In [64] the authors conclude, that the shape of the cargo force velocity relation indicates P1 > P2 ,
87
5.3 Dependence of transport properties on external load force
(a)
(b) 600
600
500
v
K=0
= 1/s
si
ca
500
K=0.02 pN/nm
= 5/s
K=0.1 pN/nm
si
K=0.1 pN/nm;
K=0.5 pN/nm
400 [nm/s]
300
v
ca
200
K=0.5 pN/nm;
si
si
si
=1/s
=1.25/s =3/s
300 200
v
[nm/s]
400
ca
data (iii)
K=0.02 pN/nm;
100
v
data (ii)
v
data (iii)
ca
ca
100 0
0
-2
0
2
4
6
8
F
ext
10
12
14
16
18
0
20
[pN]
2
4
6
F
ext
8
10
12
[pN]
Figure 5.22: Experimental data (ii) from [64] (squares) and (iii) from [109](circles) for the cargo velocity as a function of the external load force Fext (a) compared to the simulation results of the cargo velocity of parameter set D as defined in section 3.4 for two different values of the rebinding rate πsi = 1/s (solid lines) and πsi = 5/s (dashed lines) and different values of the coupling parameter: no coupling K = 0 (black line), weak coupling K = 0.02 pN/nm (red line), K = 0.1 pN/nm (blue line) and strong coupling with K = 0.5 pN/nm and (b) compared to the simulation results of the cargo velocity of parameter set D for the according parameters K and πsi as determined in Fig. 5.11(b): K = 0.02 pN/nm with πsi = 1/s displayed by the red line, K = 0.1 pN/nm with πsi = 1.25/s by the blue line and K = 0.5 pN/nm with πsi = 3/s displayed by the green line. Note, that the applicable external load force is restricted by 2KLk.
in agreement with our results for small rebinding rates πsi and small coupling parameters K, as shown in Fig. 5.20(b). The motor pair model, as studied here, does not predict stall forces for the 2-motor and cargo velocity. However, neither do the experiments in [64] and [109] provide any evidence for such forces.
5.3.2 Relevance of cargo rotation for small cargo In general, a reordering of motors is possible during 1-motor runs as indicated in section 4.1. Thus, rebinding of the detached motor may occur either in front or behind the active motor if we include the possibility that the cargo rotates during the 1-motor runs and the trailing and leading motors are interchanged. The rotational diffusion time of the cargo is calculated in appendix in section C.5. The rotational diffusion of a sphere is described by the EinsteinSmoluchowski relation Drot = kB T /frot as in Eq. (C.1) which relates the rotational diffusion constant Drot to the rotational frictional drag coefficient frot . The mean square deviation in time of the diffusion of a sphere by an angle β about any axis is related to the diffusion constant Drot , see Eq. (4.3), which implies for the typical time scale of the diffusion
trot ≃
88
4πη 2 3 hβ iR . kB T
(5.68)
5 Properties of cargo transport by motor pairs Then, a rotation of the bead by π in water at room temperature leads for bead radii that are larger than 350 nm to a rotation time trot > htsi i which exceeds the single motor run time. Next, we take another cargo construct, a hinge-like connection which represents the DNAlinker in [58] as depicted in in the appendix C in Fig. C.5(a), into account. The shape of the latter can be approximated by a streched ellipsoid of revolution, i.e., an extreme prolate as shown in Fig. C.5(b), with the semiaxes p > q ≡ q1 = q2 . The rotational diffusion time of the ellipsoid is calculated in appendix C in section C.5. The 50 nm DNA-linker in [58] then has p ≃ 25 nm q ≃ 1 nm, which implies and axial ratio of ρ = q/p = 0.04 ≪ 1 which is much smaller than 1. Thus, higher orders of ρ can be neglected in Eq. (C.10), which leads to an approximate time scale of the rotational diffusion about one of the equatorial axis of the hinge-like connection of trot = tq ≃
hβq2 ip3 4πη . 3kB T 2 ln (2/ρ) − 1
(5.69)
Then, a rotation by βq = π about one of the equatorial axis, in water and at room temperature, leads to a rotation time of tq ≃ 40 µs ≪ htsi i which is very small compared to the single motor run time. Hence, such a rotation experimentally looks like flipping. In the previous sections, we focused on relatively large beads, for which appreciable cargo rotation and, thus, interchange of the trailing and leading motor can be neglected. For sufficiently small beads or hinge-like assemblies on the other hand we have to consider this phenomena. Cargo rotation adds another random shift to the cargo trajectories of the motor pair walk. The rotational shift is constant has the magnitude of ∆¯ xrot = ±L0 . If we assume, that the probability for cargo rotation depends on the applied external load force and that it is likely that the cargo will rotate in the direction of the force, this will lead on average to backward shifts of the cargo for resisting forces and to forward shifts of the cargo for assisiting forces. This rotational shift will not affect the activity state run lengths, since these are defined by relative positions, but leads to an additive shift h∆¯ xrot i in Eq. (5.62) if one considers an experimental overall cargo run in a low resolution.
5.3.3 Operation modes of 2-motor runs As introduced in section 3.3 for the single motor and above for the 2-motor runs, the velocity depends on the nucleotide concentration and on the external load force. One can define a characteristic line Fst (∆¯ µ) which separates the region of prevailing forward and prevailing backward stepping, the operation modes of a motor [118] or a motor pair respectively. The force velocity relations in Fig. 5.21(b) have no stall force, the velocity reaches a constant positive value for large forces. This is mainly caused by the assumption of force-free rebinding events and force induced elongation of the spring during a preceding 1-motor run which causes rebinding to non zero ∆L-layers. In order to avoid such intrinsic effects we use a variation to the simulations as introduced in section 4.3 which provides a random walk on the different ∆L-layers in the network graph in Fig. 4.5 without the boundary lines ile and itr . Here, we start with two active motors with ∆L = 0. By disabling unbinding transitions, κ70 ≈ 0, we obtain one long 2-motor run and therefore a better statistics for the average velocity of such runs. Another advantage of this method is that the we do not have to consider the limit of the applicable external force due to overstreched springs since we eliminate 1-motor runs. Thus,
89
5.3 Dependence of transport properties on external load force this method should provide stall forces for the 2-motor assembly in contrast to the curves in Fig. 5.21(b). The average velocity v2,tot is then given by the ratio of the total distance xtot the cargo travels in the fixed simulation time t v2,tot =
xtot . t
(5.70)
(b)
(a)
700
10 v (F si
600
/2)
v (F
ext
K=0.5
pN/nm
[ADP]=[P]= 0.1
K=0.02 pN/nm
6
[nm/s]
pN/nm
M
[ATP]= 1.6 mM
2,tot
300
v
200
v
[nm/s]
400
K=0.1
/2)
ext
K=0
K=0.02 pN/nm
500
2,tot
si
8
K=0
K=0.1
pN/nm
K=0.5
pN/nm
4
2
0
100 -2 0 -4 -5
0
5
10
F
ext
[pN]
15
20
14.0
14.5
15.0
F
ext
15.5
16.0
[pN]
Figure 5.23: (a) The velocity v2,tot of a long 2-motor run with an unbinding rate κ70 ≈ 0, as a function of the external load force for different values of the coupling parameter K: K = 0 (black line), weak coupling with K = 0.02 pN/nm (red line), K = 0.1 pN/nm (blue line) and strong coupling with K = 0.5 pN/nm (green line). The approximation v2 (Fext ) ≈ vsi (Fext /2) (gray line) is covered by the black line. (b) Close-up of (a), displaying the stall force Fext ≈ Fst of the different curves, indicated by the corresponding drop lines. The concentrations correspond to ∆µ ≈ 39 kB T .
Fig. 5.23 shows the force velocity relation of a long 2-motor run for different values of the coupling parameter K. The shape of the force velocity curves in Fig. 5.23(a) is similar to the single motor foce velocity curve in Fig.3.4(b): almost constant velocity for assisting forces, decreasing velocity for increasing resisting forces and a region of small negative velocity for very large forces. The approximation v2 (Fext ) ≈ vsi (Fext /2) for non-interacting motors corresponds to the simulation results for K = 0. For small external forces, the velocity decreases for increasing coupling parameters. For large external forces the velocity of the strongly coupled motor pair is still positive whereas the K = 0 case and the weakly coupled motor pairs have a negative velocity. The close-up of the force velocity relations in Fig. 5.23(b) shows the forces for which the velocity vanishs. This stall force depends on the coupling parameter, for non-interacting motors, the stall force is twice the single motor stall force Fst,2 ≈ 14.46 pN ≈ 2 Fst and even larger for the strongly coupled motor pair Fst,2 ≈ 15.9 pN. The stall force also depends on the ADP and P concentration, a product concentration of [ADP]=[P]=0.1 µM has been used in Fig. 5.23. Due to the time scaling constrains for small ATP concentrations as introduced in section 3.5, it is not convenient to calculate stall forces at small ∆µ via the simulation method. Therefore we solve Eq. (3.18) for vsi = 0 which defines the single motor stall force and in general follow the procedure in [118] for the single motor. Simulations of the long 2-motor
90
5 Properties of cargo transport by motor pairs run yield reliable results corresponding to the time constrains, i.e., the larger the ADP and P concentration the better the results of the force velocity relation for small ∆µ. Reliable results of the simulation method for the 2-motor runs can only be achieved when t12 = 1/ω12 is small compared to the cycle time tF . We use the approximation v2 (Fext ) ≈ vsi (Fext /2) to gain the stall force of the 2-motor run velocity for non-interacting motors with K = 0. The stall forces of the 2-motor assemblies with K > 0 are calculated via v2,tot (Fext ) = 0.
single motor
40
K=0.02 pN/nm K=0.5 pN/nm v (F si
ext
/2)=0
B
[k T]
30 20 10 [ADP]=10
M
[ADP]=100 mM
0
0
5
10 F
ext
15
[pN]
Figure 5.24: Operation mode diagram for kinesin motor pairs: Stall force of long 2-motor runs compared to the single motor stall force (black lines) depending on the nucleotide concentrations by ∆µ in Eq. (3.37) for two different product concentrations, [ADP] = [P] = 10 µM (dotted lines) and [ADP] = [P] = 100 mM (solid lines). The boundaries between the region of forward and backward stepping are given by the stall force lines ∆¯ µ(Fst ). The single motor stall force is calculated via Eq. (3.18). The gray lines represent the approximation v2 (Fext ) ≈ vsi (Fext /2) for non-interacting motors. The stall force Fst,2 of the 2-motor run velocity v2,tot = 0 is displayed by red lines for weak coupling K = 0.02 pN/nm and by green lines for strong coupling K = 0.5 pN/nm. The gray lines are partially coverd by the red lines.
Fig. 5.24 shows the stall force of long 2-motor runs, for a weakly and for a strongly coupled motor pair compared to the single motor stall force depending on the nucleotide concentrations expressed by ∆µ in Eq. (3.37) ∆µ [ATP] Keq ∆¯ µ= = ln kB T [ADP][P]
.
(5.71)
The characteristic lines ∆¯ µ(Fst ) and ∆¯ µ(Fst,2 ) separate the regions of prevailing forward and prevailing backward stepping, the operation modes of a motor [118] and here of a motor pair, respectively. Two different values of ADP and P concentration are used in Fig. 5.24, a rather low concentration with [ADP] = [P] = 10 µM displayed by the dotted lines and a very large concentration with [ADP] = [P] = 100 mM displayed by the solid lines. Reliable results of the simulation method for the 2-motor runs can only be achieved when t12 = 1/ω12 is small compared to the cycle time tF . The characteristic lines for a large ADP and P concentration
91
5.3 Dependence of transport properties on external load force in Fig. 5.24 increases linearly with the 2-motor stall force beyond ∆¯ µ ≈ 16.5. In this region, the influence of the coupling parameter is rather small for ∆¯ µ < 15. For both product concentrations the characteristic lines are independ of chemical potential for large ∆¯ µ ≥ 20, similar to the single motor lines. For non-interacting motors with K = 0, the 2-motor stall force corresponds to two times the single motor stall force. The stall force of a weakly coupled motor pair is only slightly larger compared to the stall force with K = 0 whereas the stall force of a strongly coupled motor is about 1 pN larger. Note, that the small product concentration used here is larger than the product concentration used in Fig. 5.23. Compared to the single motor operation mode diagram, which correspnds to the results in [118], the region of forward stepping is enlarged for 2-motor runs. A kinesin motor pair is able to perform a processive run with a non zero velocity even for large resisting forces. Although the actual stall force during a 2-motor run depends on the coupling parameter, the 2-motor run stall force is considerably larger than the single motor stall force.
5.3.4 Summary and discussion The chemomechanical motor pair network in Fig. 4.5 is governed by the single motor transition rates which depend on the external load force as introduced in chapter 3. In the present section, we develop a method to incorporate the external load force into our motor pair model. During 2-motor runs, the cargo experiences an external load force in addition to the elastic forces arising from the elastic coupling. Mechanical equilibrium now leads to a cargo position which is closer to the trailing motor if a resisting force is applied, and and closer to the leading motor for assisting forces. We assume, that the external force is equally shared between the leading and the trailing motor and the effective forces experienced by each motor are obtained by simple superposition of the shared external force and the mutual interaction force. Thus the single motor forces which enter the force-dependent transition rates are these effective forces acting on the leading and on the trailing motor, respectively. During a 1-motor run, the active motor carries the total external load force. Since the individual spring of the active motor during a 1-motor run is determined by the spring constant κ = 2 K, the external load force induces a deflection of this spring according to Fext /κ. Comparison to the cargo position of the preceeding 2-motor run reveals, that the cargo undergoes a shift as one of the motors unbinds. In general, resisting forces shift the cargo backwards and assisting forces shift the cargo forwards. Still, a sufficiently large deflection ∆L of the motor-motor separation immediately before the unbinding of one of the motors during the preceeding 2-motor run, may lead to a cargo shift in the opposite direction. However, the distributions of the cargo shift in Fig. 5.18 show, that the average cargo shift corresponds to the half of the force induced deflection on the individual motor spring during a 1-motor run, i.e., h¯ xca i = Fext /2 κ. Note that the 1/2 arises from the equal force sharing. The width of the cargo shift distribution depends on the coupling parameter, in agreement with the distribution of the deflection ∆L of the motor-motor separation in Fig. 5.6 and Fig. C.3. The transition from a 1-motor run to a 2-motor run is governed by the force-independent rebinding rate πsi . Note that the effect of a force-dependent rebinding rate is negligible for kinesin motors [115]. We assume, that the individual spring of an inactiv motor is relaxed. As a consequence, the elastic coupling between the rebinding motor and the cargo should be relaxed. Thus, immediately after rebinding, the effective force acting on the rebinding motor is zero whereas the other motor experiences effectively the total external load force.
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5 Properties of cargo transport by motor pairs These forces change during the subsequent 2-motor run whenever one of the motors performs a mechanical step. On the other hand, the force induced deflection of the individual motor spring during the preceeding 1-motor run leads to a non-zero deflection ∆L in the beginning of each 2-motor run. So far, there is a need for more sustained discussion about the influence of our assumptions. For instance, an interesting aspect is the time scale on which the motors attain equal force sharing after rebinding of an inactive motor, which should depend on the coupling parameter. In the second part of this section we study the force velocity relations of the activity states and the cargo run in comparison to the experimental findings in [64] and [109]. The force dependence of the single motor velocity and the velocity of the activity states is shown in Fig. 5.21 and of the cargo velocity in Fig. 5.22 compared to the experimental data. Slightly different to our definition of 1- and 2-motor runs, the authors distinguish between two classes of ’assembly microstates’ where either one or two motors bear the applied external force corresponding to slow and fast segments of the trajectory. Nevertheless, for substall forces, the results of our motor pair model are in good agreement with this experimental data. For superstall forces, our results for the average 2-motor and cargo velocity are qualitatively in agreement with the data in [109] and not consistent with the experimental findings in [64]. The deviations are on one hand caused by the different definitions of activity states and microstates. On the other hand, large resisting forces in our network model may lead to large dwell times in a state without unbinding possibility as mentioned in section 3.5. Thus, a final concluding interpretation of the simulation results for superstall forces, especially regarding all assumptions, is still owing. For completion, one should consider the fact, that in the experiments large forces Fext > 10 pN before detachement, are rarely observed [64]. The authors assume that special conditions may be required for a motor pair to produce such large forces. Accounting this, we consider the influence of cargo rotation during 1-motor runs. The possibility of cargo rotation results in an rotational backward or forward shift which depends on the direction of the external load force, and has on average the magnitude of the restlength L0 . The rotational shift does not affect the activity state properties, since the interchange of leading and trailing motor makes no difference for two identical motors. The motor pair model, as studied here, does not predict any stall forces for the 2-motor or the cargo velocity. However, neither do the experiments in [64] and [109] provide any evidence of such forces. In order to avoid the effects which cause a non-zero velocity for superstall forces, we study the 2-motor run properties in a closed 2-motor network, i.e., we start the simulations with two active motors in the limit of a vanishing unbinding rate κ70 ≈ 0. In this way, we obtain long 2-motor runs and better statistics for the force–velocity relation which now exhibits stall forces. The force dependence of this 2-motor run velocity v2,tot is displayed in Fig. 5.23 for different coupling parameters. We find that the stall force depends on the coupling parameter. For non-interacting motors with K = 0, the 2-motor stall force corresponds to two times the single motor stall force. The stall force of a weakly coupled motor pair is only slightly larger compared to the stall force with K = 0 whereas the stall force of a strongly coupled motor pair is about 1.5 pN larger. In agreement with the results for the single motor which correspond to those in [118], the stall force also depends on the ADP and P concentration. In general, a large ADP and P concentration provides a slightly larger stall force at saturating ATP concentrations as shown in the operation mode diagram in Fig. 5.24 for the single motor as well as for the motor pair.
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5.3 Dependence of transport properties on external load force
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6 Conclusion In this thesis, we present a network representation of two elastically coupled kinesin motors and study the dynamics of the system. The starting point of our model is the general description of the single motor model as introduced and extensively analyzed in [45]. When a single motor is bound to the filament at a certain spatial position, it can attain different chemical states depending on the nucleotides bound to the two motor domains. These chemical states are embodied in the chemomechanical network for a single motor as displayed in Fig. 3.3. Specifying the force dependence and the transition rate constants of these transition rates allows the quantitative description of various experimental single motor data. The theoretical approach for coupled motors is introduced in chapter 3. The motor system considered here consists of two kinesin motors, a leading and a trailing motor, which are attached to the same cargo and walk on the same filament as indicated in Fig. 4.1. Each motor can unbind from and rebind to the filament separatly. As a consequence, the cargo is actively pulled by either one or two motors or is attached to two inactive motors. These different activity states are denoted by 1-motor run, 2-motor run and unbound motor pair. We consider the flexible stalk of a kinesin motor as a harmonic spring. A mechanical step of one of the motors during a 2-motor run then leads to a mutual interaction force between the two motors and the cargo. We assume that the elastic forces balance each other on time scales which are small compared to a single motor transition and hence can eliminate the cargo position from the motor pair description. In this way, we obtain a reduced description of the motor pair in terms of an effective spring constant and an effective restlength. The state of the elastically coupled motor pair is characterized by three variables, the chemical states ile and itr of the leading and trailing motor as well as the deflection ∆L of the motor-motor separation. The resulting chemomechanical network has a layer structure as shown in Fig. 4.5, where each layer corresponds to a constant value of ∆L. 1-motor runs occur on one of the boundary lines of the network with ∆L = 0, which represent the single motor network. Any 1-motor run may be terminated either by unbinding of the active motor which leads to the unbound motor pair state, or by the rebinding of the inactive motor which results in a 2-motor run. Mechanical steps during 2-motor runs lead to transitions between neighbouring ∆L-layers. Even though this chemomechanical motor pair network has a complex structure it involves, apart from the single motor parameters, only two additional parameters, the coupling parameter K and the single motor rebinding rate πsi . Because the resulting network consists of a huge number of cycles and the dependence of these single motor rates on force is highly nonlinear, reflecting the complex molecular structure of the motor proteins, the network dynamics is not amenable to analytical methods. Therefore, we use the Gillespie algorithm in order to study this dynamics and to generate trajectories of the motor pair. In the stochastic simulations, we studied the trajectories as generated from the chemomechanical network of the motor pair for certain values of the single motor rebinding rate πsi and the elastic coupling parameter K. In experimental studies, the values of these two parameters are not known but have to be determined in a consistent manner. Therefore, we show how one can deduce the values of these two parameters from the statistical properties of the trajectories. The single motor rebinding rate πsi can be directly obtained from the average 1-motor run time. The coupling parameter K, on the other hand, can be deduced in
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two different ways. First, one can determine K by measuring the average value of the 2-motor run time and using the functional relationship between K and h∆t2 i as displayed in Fig. 5.4. Second, the coupling parameter K can be determined from the trajectories of the individual motors by analyzing the probability distribution of the deflection of the motor-motor separation during 2-motor runs. The width and the peak amplitude of the probability distribution can be used to deduce K by using the functional relationship displayed in Fig. 5.6. Since the trajectories can be decomposed into 1-motor and 2-motor runs, we introduce the coarse grained 3-state network in Fig. 5.2 which describes the three activity states of the motor pair and the transitions between these states. Which activity state is dominant during a motor pair walk depends again on the rebinding rate and, via the 2-motor run termination rate, on the elastic coupling parameter. The corresponding activity state diagram in Fig. 5.8 shows the crossover line that separates the parameter regime, in which 1-motor runs dominate the cargo run, from the regime, in which 2-motor runs are more likely. A small rebinding rate leads to a dominance of 1-motor runs for all values of the coupling parameter, whereas a larger probability of 2-motor runs is only found for a relatively large rebinding rate and small coupling parameters. The activity state network implies that the average 1-motor run time depends on the rebinding rate but not on the coupling parameter whereas the 2-motor run time depends on the coupling parameter but not on the rebinding rate. In general, this parameter separation is reflected in different transport properties. The average cargo run properties of a motor pair are combinations of the activity state properties, hence depend on both motor pair parameters. Comparison between the activity state diagram and the cargo properties as functions of the motor pair parameters reveals that the cargo run length is twice the single motor run length if the motor pair performs 1-motor runs and 2-motor runs with equal probability, which corresponds to the crossover line P1 = P2 . The crossover line that marks the region of clear dominance of 2-motor runs correponds to a cargo run length thrice the single motor run length. Although the motor pair walk consists of mainly 1-motor runs for small rebinding rates, the cargo run length exceeds the single motor run length. Without any external strain involved, it is likely that an inactive motor rebinds before the active motor detaches. Therefore, in general, a motor pair is able to cover a larger run length than a single motor. The binding transitions of nucleotides to a single motor head are governed by transition rates that are proportional to the nucleotide concentrations. Thus, the dynamics of the single motor depends on the nuclotide concentrations as introduced in section 3.3. The dependence of the run time and the run length of a single motor on the ATP concentration is displayed in Fig. 3.6(b) and Fig. 3.7(b) for different product concentrations. A variation in the hydrolysis product concentrations of ADP and P has little effect on the single motor properties as long as the ATP concentration is sufficiently large whereas such a variation has a significant influence for small ATP concentrations: For small product concentrations [ADP]=[P] up to a few µM, an increase in the ATP concentration from 10 µM to 1 mM leads to a strong decrease of the run time of a single motor whereas its run length remains essentially constant. In contrast, for a product concentration of [ADP]=[P]=10 µM such an increase in the ATP concentration leads to a strong increase of the run length without affecting the run time. For larger product concentrations increasing the ATP concentration increases both, the run time and run length of a single motor. The dependence of the single motor dynamics on the nucleotide concentrations is also reflected in the motor pair dynamics:
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6 Conclusion The dependence on the ATP concentration of the run time and the run length of the activity states and of the cargo run are displayed in Fig. 5.13 and Fig. 5.14. The 1-motor run time does not depend on any nucleotide concentration, since a 1-motor run is limited by the constant rebinding rate πsi . The 1-motor run length on the other hand, is related to the single motor velocity vsi which decreases for decreasing ATP concentrations as shown in Fig. 3.5(a). The 2-motor run time and run length are almost proportional to the single motor run time and run length, respectively. The coupling parameter has in both cases only a scaling effect, i.e., the actual value of the 2-motor run length and run time is larger for weak coupling, which is reflected in the smaller 2-motor run velocity for strong coupling as shown in Fig. 5.15. As a consequence, for small product concentrations, the cargo run time is significantly increased for small compared to large ATP concentrations and almost constant for all ATP concentrations for [ADP]=[P]=10 µM. The influence of the variation of the product concentrations at small [ATP] on the cargo run length on the other hand, is not intuitiv. First, since the 1-motor and the 2-motor run length are small for [ADP]=[P]=10 µM and increase with increasing ATP concentration, the cargo run length also increases with increasing ATP concentrations, as expected. Surprisingly, for small product concentrations the cargo run length is strongly enhanced for small compared to large ATP concentrations and decreases with increasing ATP concentration. Here, the 2-motor run length is almost independent of the ATP concentration and the 1-motor run length increases with increasing ATP concentration. The average number hn2 i of 2-motor runs on the other hand, which is proportional to the single motor run time, see Eq.(5.15), is quite large for small nucleotide concentrations. Therefore, vanishing ADP and P concentrations cause an enhanced cargo run length for small ATP concentrations which decreases with an increasing ATP concentration. Note, that this effect is not reflected in the average cargo velocity in Fig. 5.16. Here, the influence of the different ADP and P concentrations correspond to the effect on the single motor velocity. The strong increase of the average cargo run length of a motor pair, in comparison to the single motor run length, for small nucleotide concentrations is in agreement with recent experiments [66]. The authors of [66] consider the single motor velocity as a potential tuning parameter of the multiple motor transport and implement the reduction of the single motor velocity by variations in the ATP concentration. These experiments were done in the absence of ADP and P. Our model predicts that this property of the average cargo run length of a motor pair can be observed only in the limit of very small product concentrations. For a sufficiently large ADP and P concentration of [ADP]=[P]& 10 µM, the effect disappears and the cargo run length increases proportionally to the single motor run length with an increasing ATP concentration. The chemical and mechanical single motor transition rates depend on the external load force as introduced in chapter 3. We develop a method to incorporate the external load force to our motor pair model: During 2-motor runs, the cargo experiences an external load force in addition to the elastic forces arising from the elastic coupling. We assume, that the external force is equally shared between the leading and the trailing motor and the effective forces experienced by each motor are obtained by simple superposition of the shared external force and the mutual interaction force. Thus the single motor forces which enter the forcedependent transition rates are these effective forces acting on the leading and on the trailing motor, respectively. During a 1-motor run, the active motor carries the total external load force. Since the individual spring of the active motor during a 1-motor run is governed by the spring constant
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κ, the external load force induces a deflection of this spring according to Fext /κ. The transition from a 1-motor run to a 2-motor run is governed by the force-independent rebinding rate πsi . We assume, that the individual spring of an inactive motor is relaxed. As a consequence, the elastic coupling between the rebinding motor and the cargo should be relaxed. Thus, immediately after rebinding, the effective force acting on the rebinding motor is zero whereas the other motor experiences effectively the total external load force. These forces change during the subsequent 2-motor run whenever one of the motors performs a mechanical step. Note that the force-induced deflection of the individual motor spring during the preceeding 1-motor run leads to a non-zero deflection ∆L in the beginning of each 2-motor run. In Fig. 5.21 and Fig. 5.22 we compare the force–velocity relations of the activity states and of the cargo run to the experimental findings in [64] and [109]. Slightly different to our definition of 1- and 2-motor runs, the authors distinguish in [64] between two classes of “assembly microstates” where either one or two motors bear the applied external force corresponding to slow and fast segments of the trajectory. Nevertheless, for substall forces, the results of our motor pair model are in good agreement with this experimental data. For large forces, our results for the average 2-motor and cargo velocity are qualitatively in agreement with the data in [109] and not consistent with the experimental findings in [64]. This deviations are on one hand caused by the different definitions of activity states and microstates. On the other hand, large resisting forces in our network model may lead to large dwell times in a state without unbinding possibility as discussed for the single motor in section 3.5. Thus, a final concluding interpretation of our simulation results for superstall forces with special regard to the assumptions is still owing. For completion, one should consider the fact, that in the experiments large forces Fext > 10 pN before detachement are rarely observed [64], which implies that unbinding occurs from more than one state of the single motor network in Fig. 3.3. The motor pair model, as studied here, does not predict any stall forces for the 2-motor or the cargo velocity. However, neither do the experiments in [64] and [109] provide any evidence of such forces. In order to avoid the effects which cause a non-zero velocity for superstall forces, we study the 2-motor run properties in a closed 2-motor network with two active motors in the limit of a vanishing unbinding rate κ70 ≈ 0. This way, we obtain long 2-motor runs and better statistics for the force–velocity relation which then provides stall forces. In agreement with the results for the single motor which correspond to those in [118], the stall force depends on the ADP and P concentration. In general, a large ADP and P concentration provides a slightly larger stall force at saturating ATP concentrations as shown in the operation mode diagram in Fig. 5.24, for the single motor as well as for the motor pair. Additionally, we find that the stall force depends on the coupling parameter. The stall force of a weakly coupled motor pair is only slightly larger compared to the stall force of non-interacting motors with K = 0, which is twice the single motor stall force, whereas the stall force of a strongly coupled motor is more than one pN larger.
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6 Conclusion
Outlook In this thesis we present a network representation for coupled molecular motors which incorporates a detailed description of the single motor model. Thus, not only the overlaying properties of the cargo can be studied with this model, but also the behavior of the individual motors of a motor pair can be examined independently. Predictions from this model are in agreement with a large body of experimental data. For large external forces, the comparison to experimental data reveals qualitative deviations. Therefore, in the future one should focus on the influence of our assumptions - a single unbinding state, equal force sharing and relaxed individual spring of inactive motors - on the motor pair dynamics. For instance, an interesting aspect is the time scale on which the motors attain equal force sharing after rebinding of an inactive motor, which should depend on the coupling parameter. Furthermore, it stands to reason to define the hydrolysis rate for the motor pair system and study the force dependence of balanced activities in 2-motor and cargo runs and in consequence of the 2-motor and cargo run efficiency as established for the single motor in [118] and [110]. In the future, this model can be extended in various ways, e.g., including geometry effects and/or external friction or considering a rather flexible or soft cargo which would lead to an additional spring constant to the coupling parameter. Concerning the geometry, one can also think of two motors that walk on parallel filaments. Here, the motors would be able to pass each other without steric interaction. Another interesting point is, to extend the model to several cooperating motors. A motor triple for instance would provide 3-motor runs, in addition to the 1-motor and 2-motor, during which the deflection of the motor-motor separation would consist of two parts. So far, we studied two identical motors, in particular kinesin-1 motors since this motor is extensively analyzed. It is straightforward to extend this model to other pairs of motors, e.g., two myosin V motors as in recent experiments [65], or even pairs of different motors, e.g., kinesin-myosin V assemblies [57]. In the latter case our model should reveal the tugof-war behavior as found in [52] in the limit of two non-interacting motors. In principle, the experimental realization of two coupled dynein motors should not be too challenging, if one uses two different fluorescent labels for the two motors, a method that has been recently applied to the two heads of a single dynein motor [117].
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Software LATEX Kile Version 2.0.85. http://kile.sourceforge.net Mathematica 8 by Stephen Wolfram. www.wolfram.com OriginPro 8.5 by OriginLab Corporation. www.originLab.com Xfig 3.2.5a by Tom Sato and Brian V. Smith. http://www.xfig.org GCC 4.4.3 GNU Compiler Collection. http://gcc.gnu.org mtt.c Random number generator coded by Takuji Nishimura and Makoto Matsumoto. Copyright 1997 - 2002. www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt.html
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Appendix
A Details of the 7 state model
i
A.1 Spanning trees of the 7 state network graph A.2 Explicit solution of the 7 state model A.3 Transition rate constants of the 7 state model A.4 Influence of the force threshold on the single motor
B Supplementary plots
ix
B.1 Nucleotide concentration and force dependent unbinding B.2 Motor pair properties depend on [ATP], [ADP] and [P] B.3 Comparsion with experimental motor pair data
C Application of external load force
xvii
C.1 Cargo shift with and without force C.2 Rebinding under external load force C.3 Probability distribution of the deflection under force C.4 Comparsion to experimental motor pair data C.5 Rotational diffusion time of the cargo
D Glossary
xxiii
D.1 List of symbols D.2 List of abbreviations D.3 List of figures D.4 List of tables
109
this is...
A Details of the 7 state model A.1 Spanning trees of the 7 state network graph Fig. A.1 shows on the left side the 7-state network of the single motor model as established in [45] and below, the kinetic diagram of this network containing the states and the transitions. The nucleotide binding and release transitions are displayed in red for ATP and in black for ADP and P. The mechanical transition is displayed in blue. On the right side of Fig. A.1 the 44 spanning trees corresponding to the 7-state network graph are shown. ...............
DD
T 7 D
TE
E
6
1 D
2
F T
D
5
T
B ET
E
3
1 2 3
6
D
4
7 5 4
ATP binding/release ADP and P binding/release mechanical transition
Figure A.1: Left: Detailed 7-state network as shown in Fig.3.3 and the kinetic diagram with 7 vertices and 9 edges. Red lines indicate the ATP binding and release transitions, black lines represent the ADP and P binding and release transitions and the blue line displays the mechanical stepping transition. Right: m = 44 spanning trees for a 7-state network.
i
A.2 Explicit solution of the 7 state model
A.2 Explicit solution of the 7 state model Explicit solution for the steady state configuration of the 7-state model according to Eq. 2.19, where the rates of the backward cycle have been identified with those of the forward cycle, namely ω23 = ω56 , ω45 = ω12 , ω34 = ω61 , ω43 = ω16 , ω32 = ω65 . Ω1 =Υ(F)[ω56 ω61 (ω21 + ω25 ) + ω52 ω21 (ω65 + ω61 )] + Υ(T )ω57 ω71 (ω61 + ω65 ) + ω54 ω16 ω65 (ω71 + ω75 )(ω21 (ω61 + ω65 ) + ω56 ω61 ) Ω2 =Υ(F)[ω61 ω21 (ω56 + ω52 ) + ω52 ω65 (ω16 + ω12 )] + Υ(B)ω54 ω43 ω32 + γ(ω65 + ω61 )(ω17 ω75 ω52 + ω57 ω71 ω12 )
(A.1)
(A.2)
Ω3 =(ω12 + ω61 )ω12 ω56 [ω56 ω61 (ω71 + ω75 ) + ω57 ω71 (ω65 + ω61 )] + ω21 ω54 ω56 [ω16 ω65 (ω71 + ω75 ) + ω17 ω75 (ω65 + ω61 )]
(A.3)
+ Υ(B)[ω54 ω16 (ω25 + ω56 ) + ω52 ω56 (ω16 + ω12 )] Ω4 =ω21 ω54 (ω65 + ω61 )[ω16 ω65 (ω71 + ω75 ) + ω17 ω75 (ω65 + ω61 )] + ω12 ω56 ω61 [ω56 ω61 (ω71 + ω75 ) + ω71 ω57 (ω65 + ω61 )]
(A.4)
+ Υ(B)[ω56 ω61 (ω52 + ω54 ) + ω25 ω54 (ω65 + ω61 )] Ω5 =ω12 ω21 (ω65 + ω61 )Υ(F) + Υ(T )Υ(B)
(A.5)
Ω6 =(ω71 + ω75 )[ω56 ω61 ω12 ω56 (ω12 + ω16 ) + ω21 ω16 ω54 ω16 ω65 ] + Υ(F)[ω21 ω16 (ω56 + ω52 ) + ω25 ω56 (ω12 + ω16 )]
(A.6)
+ Υ(T )(ω57 ω71 ω61 + ω17 ω75 ω56 ) Ω7 =ω21 (ω65 + ω61 )[γ(ω17 (ω52 + ω12 ω57 ) + ω54 ω16 ω65 ] + Υ(T )[ω17 ω61 (ω57 + ω56 ) + ω57 (ω65 ω17 + γ)]
(A.7)
with γ = ω12 ω65 + ω61 ω12 + ω16 ω65 and the influent factors of the 7 state network: Υ(F) =(ω71 + ω75 )γ
(A.8)
Υ(B) =(ω71 + ω75 )γ + ω17 ω75 (ω61 + ω65 )
(A.9)
Υ(T ) =(ω12 + ω25 )γ + ω12 ω56 ω65 .
(A.10)
With this, one finds the steady state probabilities in Eq. 2.20 for state i with Pist = Ωi /Ω P and the normalization factor Ω = i Ωi .
ii
A Details of the 7 state model
A.3 Transition rate constants of the 7 state model The transition rate constants κ and κ ˆ and the force scaling factors χij and F¯ij′ of the 7-state network are summarized in Tab. A.1 for the different parameter sets B, C, D and E derived from experimental data in B [43], C [27], D [58, 64] and E [29] as introduced in section 3.4. B; [43]
C; [27]
E; [29]
D; [58, 64]
notes
κ ˆ 12
1.1
2.0
1.8
(2.0)‡
κ56
175
130
220
130
≡κ ˆ 45
κ61
175
130
220
130
κ57
87.5
65
110
65
κ25
(2.9 × 105 )‡
(2.9 × 105 )†
(2.9 × 105 )†
(2.9 × 105 )‡
κ21
(100)‡
(100)†
(100)†
(100)‡
κ ˆ 17
(3.23)‡
(3.23)†
(3.23)†
(3.23)‡
κ ˆ 16
(0.02)‡
(0.02)†
(0.02)†
(0.02)‡
κ54
(6.8 × 10−11 )∗
(6.8 × 10−11 )∗
(6.8 × 10−11 )∗
(6.8 × 10−11 )∗
(6.4 × 10−5 )∗
(6.5 × 10−5 )∗
(1.7 × 10−4 )∗
(6.5 × 10−5 )∗
χ1
0.4
0.4
0.3
0.4
χ2 F¯ ′
0.6
0.3
0.2
0.3
1
4
0.6
4
4
F¯2′
8
6
10.2
9
θ
(0.65)‡
(0.65)†
(0.3)†
(0.65)‡
κ52
κ ˆ 65 κ ˆ 75 κ70
(0.24)‡
(0.04)∗ 2.05
(0.24)†
(0.04)∗ 1.86
(0.24)†
(0.1)∗ 3.10
≡ κ23 ≡ κ34 ≡ κ71
(0.24)‡
(0.04)∗
≡κ ˆ 43 ≡κ ˆ 32
1.86
Tab. A.1: Transition rate constants κ in the units of 1/s and κ ˆ in the units of 1/(µM s) for the network in Fig. 3.3 adjusted to experimental data: parameter set B [43], C [27], D [58, 64] and E [29]. The chemical rate constants of the backward cycle B are identified with the corresponding ones of the forward cycle F , except k54 . Force factors χij and F¯ij′ = ℓFij′ /kB T as introduced in section 3.4. Superscripts: (†) value adopted from [45]; (‡) value adopted from another column; (∗) value via balance conditions. The unbinding rate constant κ70 ≡ vsi /(hxsi iP7st ) is calculated via the steady state probability P7st and the corresponding experimental data of single motor velocity and run length for zero load force and saturating ATP concentration (1.6 mM in B, 1 mM in C and 2 mM in D and E) and small product concentrations [ADP]=[P]=0.1 µM.
iii
A.4 Influence of the force threshold on the single motor
A.4 Influence of the force threshold on the single motor Beside the mechanical properties of the single motor, the chemical activity of a motor can be expressed by the hydrolysis rate hsi which is given by the local excess fluxes ∆Jij through those transitions of the network that involve ATP hydrolysis, namely |61i, |34i and |57i as introduced in [118]: hsi = ∆J61 + ∆J34 + ∆J57 = P6st ω61 − P1st ω16 + P3st ω34 − P4st ω43 + P5st ω57 − P7st ω75 .
(A.11)
The balanced activities ∆µb are then defined by those concentrations of ATP, ADP and P, for which the hydrolysis rate hsi = 0 vanishes in the steady state. Fig. A.2(b) shows the hydrolysis rate hsi as a function of the external load force and the chemical potential ∆µ. The red line displays the balanced activities ∆µb with hsi = 0, which is independent of the external load force for superstall forces. Large ∆µ and small Fext lead to a large hydrolysis rate whereas for small ∆µ the hydrolysis rate is negative. Thus, for small ∆µ the synthesis of ATP is dominant. Note, that the linear part of hsi = 0 corresponds to the linear part of vsi = 0 in Fig. A.2(a). Thus, these lines in the ∆µ-Fext diagram separate different operation modes [118] of the single motor. The region of dominant forward stepping is coupled to ATP hydrolysis whereas backward stepping occures for ATP hydrolysis if ∆µ is large, and as well for synthesis of ATP if ∆µ is small. This corresponds to the first quadrant of the operation mode diagram of kinesin motors in [118]. (b)
(a)
45
v
si
/ l
45
[1/s]
h
si
40
80
80
70
35
70
35
60 30
60 30
25
30 20
20
10 0
15
-10
40 30 20
20
10 0
15
10
10
5
5
0
50
25
B
40
[k T]
50
B
[k T]
[1/s]
40
-10
0 0
2
4
F
ext
6
8
10
[pN]
0
2
4
F
ext
6
8
10
[pN]
Figure A.2: a) Contour plot of the single motor velocity vsi calculated via Eq. (3.18) as a function of the external load force Fext and and the chemical potential ∆µ for [ADP]=[P]=1 µM. The characteristic stall force line at vsi = 0 is emphasized by the red line. The gray-scale map starts with small values of vsi in black to large values of vsi in white. b) Contour plot of the single motor hydrolysis rate hsi calculated via Eq. (A.11) as a function of the external load force Fext and and the chemical potential ∆µ for [ADP]=[P]=1 µM. The balanced activities hsi = 0 are emphasized by the red line. The gray-scale map starts with small values of hsi in black to large values of hsi in white.
The chemomechanical coupling parameter λsi =
iv
hsi P6st ω61 − P1st ω16 + P3st ω34 − P4st ω43 + P5st ω57 − P7st ω75 = nf +b P2st ω25 + P5st ω52
(A.12)
A Details of the 7 state model relates the hydrolysis rate hsi to the mobility of the motor which is given by the total number nf +b of steps per unit time, nf +b = nf + nb = P2st ω25 + P5st ω52
(A.13)
with the number nf of forward steps and nb of backward steps per unit time. When every hydrolysis of an ATP molecule is followed by a mechanical step, obviously λsi = 1. Another relation between mechanical and the chemical performance of a motor is given by ηsi =
Fext vsi . ∆µ hsi
(A.14)
which can be considered as the motor efficiency [110]. For stall forces, the velocity is zero by definition. Hence the motor efficiency is zero for the stall force and zero external load.
Comparison between the parameter sets The force dependence of the transition rates is described by the factor Φij (F ). For the chemical transitions, these factors are taken to have the form Φij (F ) =
1 + exp[−χij F¯ij′ ] 1 + exp[χij (F¯ − F¯ij′ )]
(A.15)
as in Eq. (3.9), which involves the dimensionless parameter χij and the characteristic force F¯ij′ = ℓ Fij′ /(kB T ). As introduced in section 3.2, a good description of the experimental single motor data is obtained, if one chooses two different values, F¯1′ and F¯2′ , for the force thresholds with ′ = F ¯ ′ = F¯ ′ = F¯ ′ ≡ F¯ ′ for the ATP binding and ATP release transition and F¯ ′ ≡ F¯ ′ F¯12 2 1 54 45 21 ij for all other chemical rates. Likewise, two different values, χ1 and χ2 , were chosen for the load distribution factors with χ12 = χ21 = χ45 = χ54 ≡ χ1 for the ATP binding and ATP release transition and χij ≡ χ2 for all other chemical rates [45]. In section 3.4, we specify these force factors according to different experimental data. The specific force factors are summarized in Tab. 3.2 and repeated here in Tab. A.2. parameter set B [43] C [27] D [64] E [29]
χ1 0.4 0.4 (0.4)∗ 0.3
F¯1′ 4 0.6 (4)∗ 4
χ2 0.6 0.3 0.3 0.2
F¯2′ 8 6 9 10.2
Table A.2: Dimensionless force factors χij and characteristic forces F¯ij′ = ℓFij′ /kB T determined for experimental data [43], [29], [27] and [64]. The superscript ()∗ indicates a value adopted from another column. Parameter settings are named in capitals as introduced above.
Fig. A.3 shows the single motor velocity vsi calculated via Eq. (3.18) and the single motor hydrolysis rate hsi calculated via Eq. (A.11) for the different parameter sets B, C, D and E. Both properties are rescaled by their inital values at zero force in order to compare the influence of the different force factors on the shape of the force-velocity and the force-hydrolysis rate curves. Both, the velocity and the hydrolysis rate decrease with an increasing external
v
A.4 Influence of the force threshold on the single motor
(a)
(b)
1.0
~30 kBT; [ADP]=10
B
1.0
C
M
D
[ATP]=2.15 mM
0.8
E
0.8
si,0
/ h
0.6
si
0.4
v
h
si
/ v
si,0
0.6
0.4
0.2
0.2
0.0
-0.2
0.0
0
2
4
F
ext
[pN]
6
8
0
2
4
F
ext
6
8
[pN]
Figure A.3: a) Single motor velocity vsi calculated via Eq. (3.18), rescaled by vsi (Fext = 0), and b) single motor hydrolysis rate hsi calculated via Eq. (A.11), rescaled by hsi (Fext = 0), as functions of the external load force Fext for ∆µ ≈ 30 kB T for the different parameter sets B (black lines), C (red lines), D (green lines) and E (blue lines).
load force. The influence of the force in Fig. A.3(a) is strong for the parameter set C, here the velocity decreases fast and almost linearly with the external load force until stall force is reached. Here, the force threshold is small compared to the thresholds of the other parameter sets. The force dependence of the velocity of parameter sets B and E is highly nonlinear which is caused by the large force threshold F¯2′ . The velocity of parameter set B decreases fast with the force whereas the velocity of parameter set E decreases slowly, since the force scaling factor χ2 is small compared to the force scaling factors of the other parameter sets. The velocity of parameter set D decreases similar to the velocity of parameter set E for small forces, since F¯1′ and χ1 are equal for these sets, and decreases almost linearly with the force Fext > 3 pN until stall force is reached. A similar influence of the force factors is shown by the force-dependence of the hydrolysis rate in Fig. A.3(b). Fig. A.4 shows the comparison of several single motor properties as functions of the external load force for the different parameter sets B, C, D and E which are introduced in section 3.4. For each parameter set, the ratio between the single motor velocity vsi and the hydrolysis rate hsi in Fig. A.4(a) is equal to one for small forces and vanishes with the velocity vsi for the stall force. For substall forces, the ratio for the parameter set E is large compared to the other parameter sets, since the velocity of this parameter set is large, almost twice the velocity of the other sets for saturating ATP concentrations, compare Fig. 3.8. The influence of the force fators on the ratio vsi /hsi corresponds to the effect shown in Fig. A.3. This influence is likewise reflected in the force-dependence of the motor efficiency in Fig. A.4(c). The force dependence of the chemomechanical coupling parameter λsi of the different parameter sets is shown in Fig. A.4(b). For the parameter set C, the results correspond to the previous results in [105]. The chemomechanical coupling parameter is equal to one for small forces and for large superstall forces, which implies that the motor consumes one ATP molecule per step. The chemomechanical coupling parameter has a local minimum at the stall force, here the motor makes a lot of steps without hydrolyzing ATP. The chemomechanical
vi
A Details of the 7 state model
1.0
v
si
si
/ (l h )
0.8
si
0.6
0.4
0.2
0.0 0.5 ~30 kBT; [ADP]=10
0.4
M
[ATP]=2.15 mM B
0.3
si
si
C D
0.2
si
E
0.1
0.0 0
2
4
F
ext
[pN]
6
8
0
5
10
F
ext
15
20
[pN]
Figure A.4: Single motor properties as functions of the external load force Fext for ∆µ ≈ 30 kB T for the different parameter sets B (black lines), C (red lines), D (green lines) and E (blue lines): a) Ratio of average velocity vsi , rescaled by the step size ell, and the hydrolysis rate hsi . b) Chemomechanical coupling parameter λsi as in Eq. (A.12). c) Motor efficiency ηsi as in Eq. (A.4) d) Product of chemomechanical coupling parameter λsi and motor efficiency ηsi corresponding to the ’stepping efficiency’ of the motor.
coupling parameter of parameter set D reveals qualitatively a similar relation, since the force scaling factors χij of these parameter sets are the same. But for small forces, the chemomechanical coupling parameter of parameter set D additionally yields a local maximum with λsi > 1 which corresponds to ATP hydrolysis without mechanical stepping. There is also a local maximum for the parameter set B for small forces. The parameter sets B and D have the same χ1 and F¯1′ which are responsible for the large λsi at small forces. In contrast to the parameter sets C and D, the chemomechanical coupling parameter of parameter set B and E strongly decreases with increasing force and does not return to λsi = 1 in the given force range, which is caused by the rather large force threshold F¯2′ of these sets. Although the maximal motor efficiency of parameter set E in Fig. A.4(c) is large, its stepping efficiency in Fig. A.4(d) is small compared to the other parameter sets, which reflects the highly nonlinear force-velocity relation. The stepping efficiency of the parameter sets B, C and D is quite similar.
vii
A.4 Influence of the force threshold on the single motor
viii
B Supplementary plots B.1 Nucleotide concentration and force dependent unbinding Fig. B.1 shows the single motor overall unbinding rate εsi as a function of the nucleotide concentrations [ATP] and [ADP]=[P] for different external load forces Fext . Compared to the force free case in Fig. 3.6(a), with εsi = k70 P7st , the overall unbinding rate for Fext ≈ 7 pN is significantly increased: for [ATP]> 40 µM and [ADP]=[P]> 2 µM, the unbinding rate is εsi > 1/s large. For superstall forces Fext ≈ 8 pN on the other hand, the unbinding rate decreases again compared to the stall force plot. A large unbinding rate εsi > 1/s requires [ATP]> 100 µM. For Fext ≈ 9 pN, the contour plot reveals again the patterns as for the force free case, very small unbinding rates εsi < 0.1/s are possible for small [ADP]=[P]< 0.2 µM and relatively small [ATP]< 50 µM. For a larger force Fext ≈ 10 pN, the unbinding rate is very small εsi < 0.1/s for large [ATP]≈ 10 mM and relatively small [ADP]=[P]< 2 µM on the one hand but very large εsi > 1/s for small [ATP]< 100 µM and small [ADP]=[P]≈ 1 µM. These plots supplement the plots in Fig. 3.11 where the unbinding rate as a function of the external load force for different ATP and ADP, P concentrations is shown.
2
10
F
[ADP] [ M]
ext
=7 pN
F
ext
=8 pN
10
-1
si
1
[s ]
1 0.9 -1
10
0.8 0.7
2
10
F
[ADP] [ M]
ext
F
=9 pN
ext
0.6
=10 pN
0.5 0.4
10
0.3 0.2 0.1
1
0
-1
10
10
2
10
3
10
[ATP] [ M]
4
10
10
2
10
3
10
4
10
[ATP] [ M]
Figure B.1: Contour plots of the single motor overall unbinding rate εsi as a function of the nucleotide concentrations [ATP] and [ADP]=[P] in a log-log plot for different external load forces Fext . Color code from black to white increasing values of εsi .
ix
B.2 Motor pair properties depend on [ATP], [ADP] and [P]
B.2 Motor pair properties depend on [ATP], [ADP] and [P] Fig. B.2 shows contour plots of the cargo properties of a motor pair as functions of the nucleotide concentrations for weak and strong coupling in the middle and the bottom row, respectively, compared to the single motor properties in the first row. Additionally, Fig. B.3 shows the cargo run length of a motor pair for two different rebinding rates. Fig. B.4 and Fig. B.5 show the contour plots of the 1- and 2-motor run properties as functions of the nucleotide concentrations for weak and strong coupling. The left column in Fig. B.2, Fig. B.4 and Fig. B.5 shows the velocity, the middle column displays the run length and the right column the run times of the cargo run and the activity state runs, respectively.
velocity
runlength
2
[ADP] [ M]
si
1
3
10
10
4
10
2
10
2
10
3
10
4
10
2
10
ca
120
-1 2
3
10
4
10
2
240 300
10
[ADP] [ M]
420
3
10
4
10
540
[s]
ca
1
480 1
2
10
0
360 10
> [ m]
ca
10
-1
10
10
K=0.02 pN/nm =
t
2
2
P
2,t
< t > = [v < x 2
si
> / (v
ca
) - 1] /
ca
si
0.4
4.5 si
4.0
2.5 0.2
2.0
[1/s]
3.0
si
0.3
K
[pN/nm]
3.5
1.5 0.1
1.0 0.5
0.0 0.0
0.0 0.2
0.4
0.6
[s] 2
Figure B.7: Determination of the coupling parameter K via the 2-moto run time h∆t2 i. The black line displays the simulation results for the 2-motor run time as a function of the coupling parameter (left axis). The red circles correspond to the h∆t2 i calculated via Eq. (5.43) based on experimental data in Tab. B.1 as a function of the rebinding rate πsi (right axis).
Moreover, the run length data in [58] has been fitted by a distribution function using the rebinding rate πsi as a floating parameter, which results in πsi ≈ 1.03/s for the motor pair [58]. Inserting this value in Eq. (5.43) leads to the 2-motor run time of h∆t2 i ≈ (0.7 ± 0.3) s which corresponds to a coupling parameter of K ≤ 0.17 pN/nm which is in agreement with the estimation in [58].
xv
B.3 Comparison with experimental motor pair data
xvi
C Application of external load force C.1 Cargo shift with and without force Fig. C.1 shows the distribution of the cargo shift of a motor pair walk for weak and strong coupling in the force free case. For both coupling parameters, the distribution of the cargo shift is symmetrically distributed around ∆¯ xca = 0 and the average cargo shift h∆¯ xca i = 0 is zero. Thus, one can observe local cargo shifts in motor pair trajectories for the force free case but these shifts do not affect the transport properties of the motor pair in the force free case. On the other hand, the run lengths of the acitivity states are defined as relative positions as shown in Fig. 4.1, thus, independent of the external force, the cargo shift affects neither the average run length of the activity states nor the cargo run length of the motor pair which is determined by the activity state run lengths as in Eq. 5.28. However, if one can not distinguish between the individual motors, i.e., a experimental setup with a low resolution, the average cargo run length is determined by the total distance the cargo travels on the filament, i.e., one measures how far the cargo travels on the filament before detachement. This total distance h∆xca itot in Eq. (5.62) then includes the cargo shift.
K=0.02 pN/nm
8
K=0.5 pN/nm
F
ext
= 0
Count
x10
3
6
4
2
0 -80
-60
-40
-20
0
x
ca
20
40
60
80
[nm]
Figure C.1: Histograms of the cargo shift ∆¯ xca for two different values of the coupling parameter K, weak coupling with K = 0.02 pN/nm (black columns) and strong coupling with K = 0.5 pN/nm (red columns) for the force free case Fext = 0.
C.2 Rebinding under external load force Each time an inactive motor rebinds to the filament, the force induced deflection of the single motor spring of the preceeding 1-motor run induces a non-zero deflection ∆L of the motormotor separation in the beginning of each 2-motor run. Hence, the rebinding now occurs to a non-zero ∆L-layer as indicated in Fig. C.2. Note, that the deflection of the single motor spring Fext /κ usually is a floating-point number, whereas the position of the motors on the filament must be an integer. Thus, the actual ∆L-layer to which the rebinding occurs is determined by rounding Fext /(κℓ) to an integer.
xvii
C.3 Probability distribution of the deflection under force tr
le
Fle = Fext
1
2
3
4
7
6
5
0
ile
0
7
0 Fext κ
κ ∆L
itr
Figure C.2: Rebinding under external load force. The deflection of the individual motor spring Fext /κ determines the ∆L-layer to which the inactive motor will rebind.
C.3 Probability distribution of the deflection under force The probability distribution P(∆L) is shown in Fig. C.3 for different external load forces, Fext = 3 pN in (a) and Fext = 6 pN in (b). For both forces, there are deviations in the symmetry, the width and the amplitude compared to the distribution P(∆L) for the force free case in Fig. 5.6(a). Nevertheless, P(∆L) is similarly distributed around ∆L = 0. The number of accessible ∆L-layers decreases not only for increasing the coupling parameter K as in Fig. 5.6(a), but also for increasing the external load force. The red lines in Fig. C.3 indicate the maximal values of the deflection ∆L observed in the simulations.
20
20
P( L) F
ext
15
P( L) F
= 3 pN
0.00
ext
15
= 6 pN
0.00
0.05 10
0.05 10
0.10
0.10
0.15 5 0
0.30
L / l
0.25
L / l
0.15 5
0.20
0.20 0.25
0
0.30
0.35
0.35
-5
-5
-10
-10
-15
-15
-20
-20 0.1
0.2
K
0.3
[pN/nm]
0.4
0.5
0.1
0.2
K
0.3
0.4
0.5
[pN/nm]
Figure C.3: Contour plot of the probability distribution P(∆L) for the deflection ∆L as a function of the coupling parameter K for different values of the external load force (a) Fext = 3 pN and (b)Fext = 6 pN. The deflection is given in the units of the step size ℓ. The red line indicates maximal values of the deflection ∆L observed in the simulations. The gray-scale map starts with small values of P(∆L) in white to large values of P(∆L) in black.
xviii
C Application of external load force
C.4 Comparsion to experimental motor pair data In section 5.3 we show that the cargo velocity of the motor pair is for substall forces in good agreement with and for large forces qualitatively accords to the experimental data from [109]. For completion, we show in Fig. C.4 the comparison to the experimental data from [109] of the run length and the run time of the single motor run and the motor pair cargo run as a function of the external load force Fext . In both plots, the experimental data of the single motor run is displayed by gray circles and of the motor pair complex by red circles. The cargo properties of the motor pair are calculated for the according parameters K and πsi as determined in Fig. 5.11(b): K = 0.02 pN/nm with πsi = 1/s displayed by the red line, K = 0.1 pN/nm with πsi = 1.25/s by the blue line and K = 0.5 pN/nm with πsi = 3/s displayed by the green line.
10
si
< x
>: K=0.02 pN/nm;
< x
>: K=0.1 pN/nm;
< x
>: K=0.5 pN/nm;
ca
ca
1
>: K=0.02 pN/nm;
=1.25/s
< t
>: K=0.1 pN/nm;
=3/s
< t
>: K=0.5 pN/nm;
ca
ca
si
ca
si
=1/s
si
=1.25/s
si
=3/s
si
[s] run time
run length
si
< t
si
[ m]
ca
=1/s
0.1
1
; data (iii)
; data (iii)
; data (iii)
ca
2
4
6
F
ext
8
10
12
[pN]
Figure C.4: Experimental data (iii) from [109] compared to the simulation results of the run length in (a) and the run time in (b) of the single motor run and the cargo run as functions of the external load force Fext . In both plots, the experimental data of the single motor run is displayed by gray circles and of the two kinesin run by red circles. The cargo properties are simulated for the according parameters K and πsi as determined in Fig. 5.11(b): K = 0.02 pN/nm with πsi = 1/s displayed by the red line, K = 0.1 pN/nm with πsi = 1.25/s by the blue line and K = 0.5 pN/nm with πsi = 3/s displayed by the green line. Note, that the applicable external load force is restricted by 2KLk .
We find that as well the run lengths as the run times in Fig. C.4(a) and (b) are in good agreement to the experimental data for substall forces. For large forces, the results of the cargo run length of the motor pair qualitatively agrees with the experimental findings, since all curves increase again for superstall forces, as the data does. The cargo run time of the motor pair strongly increases for superstall forces whereas the data decreases for superstall forces. On the other hand, large forces are rarely observed in the experiments, which implies, that the unbinding may occur from more than one state of the single motor network in Fig. 3.3. In order to find better accordance with the experimental data for large forces, one should consider unbinding events also from other states of the single motor network, which then may reduce the run times for large forces which will also lead to better accordance for the velocities.
xix
C.5 Rotational diffusion time of the cargo
C.5 Rotational diffusion time of the cargo Technically, the reordering of motors on the filament is possible during 1-motor runs as indicated in section 4.1. Thus, rebinding of the detached motor may occur either in front or behind the active motor if we include the possibility that the cargo rotates during the 1-motor runs and the trailing and leading motors are interchanged. The rotational diffusion of a sphere is described by the Einstein-Smoluchowski relation Drot =
kB T frot
(C.1)
which relates the rotational diffusion constant Drot to the rotational frictional drag coefficient frot . This frictional coefficient is determined by Stokes’ law for a sphere with volume V and radius R by frot = 6ηV = 8πηR3 (C.2) with the dynamic viscosity η. Then, the rotational diffusion constant in Eq. (C.1) for a spherical particle can be written as Drot =
kB T . 8πηR3
(C.3)
The mean square deviation in time of the diffusion of a sphere by an angle β about any axis, as introduced above in Eq. 4.3, is given by hβ 2 i = 2Drot trot
(C.4)
which implies with Eq. (C.3) for the typical time scale of the diffusion trot ≃
(a)
4πη 2 3 hβ iR . kB T
(C.5)
βq
(b)
βq q βq
1
p
βp
q2
βq Figure C.5: (a) Hinge-like connection between two motors in adaption from the DNA-linker in [58]. Concerning the exchange of the leading and trailing motor via ’cargo’ rotation, this assembly has two rotational axes displayed by the red dashed lines and the angle βq . (b) Prolate ellipsoide with the semiaxes p > q ≡ q1 = q2 and the rotation angles βq and βp .
xx
C Application of external load force Second, we will also take into account another construct of a motor pair, a hinge-like connection, representing the DNA-linker in [58] as depicted in Fig. C.5(a). For the rotational diffusion time calculations, the cylindric shape of the latter can be approximated by a streched ellipsoid of revolution, i.e., an extreme prolate as shown in Fig. C.5(b), with the semiaxes p > q ≡ q1 = q2 and the axial ratio q (C.6) ρ ≡ < 1. p The corresponding frictional coefficients fp and fq ≡ fq1 = fq2 associated with the rotation about the axis p and q1 or q2 , respectively, with q ≡ q1 = q2 , are given by [120] fp = Cp f0
and
fq = Cq f0
(C.7)
with the frictional coefficient f0 = 6ηV0 = 8πηa3 ρ2 of a sphere of equivalent volume V0 and the Perrin frictional factors Cp and Cq . Perrin [120] defines a parameter S, an elliptic integral, which results for prolate ellipsoids in 2 S= p ln p 1 − ρ2
1+
p
1 − ρ2 ρ
!
(C.8)
and defines the Perrin frictional factors Cp and Cq as Cp =
4 1 − ρ2 3 2 − ρ2 pS
and
Cq =
4 1 − ρ4 . 3ρ2 (2 − ρ2 ) pS − 2
(C.9)
Twisting about the axial axis p of the motor pair system coupled via DNA-linker, as depicted in Fig. C.5(a) does not displace the cargo position on the filament, thus we focus on the rotation about one of the equatorial axis q1 or q2 which leads to a reordering of motors on the filament. Using the relations in Eq. (C.8) and (C.9) the frictional coefficient fq of the rotation about one of the equatorial axis q in Eq. (C.7) yields fq =
16πηp3 1 − ρ4 √ . 2 2 3 √2−ρ ln 1+ ρ1−ρ − 1 2
(C.10)
1−ρ
The mean square deviation in time of the diffusion of a prolate by an angle βq about an equatorial axis is given analoguos to Eq. (C.4) by hβq2 i = 2Dq tq
(C.11)
which implies with Dq = kB T /fq as in Eq. (C.3) for the typical time scale of the diffusion tq =
hβq2 ifq . 2kB T
(C.12)
For the 50 nm DNA-linker in [58] one has p ≃ 25nm, q ≃ 1nm, which implies an axial ratio ρ = 0.04 ≪ 1 which is small compared to one. Hence, higher orders of ρ can be neglected in Eq. (C.10), which leads to an approximate time scale of trot = tq ≃
hβq2 ip3 4πη . 3kB T 2 ln (2/ρ) − 1
(C.13)
xxi
C.5 Rotational diffusion time of the cargo With the given values of p and q, a rotation by βq = π, in water and at room temperature, leads to a rotation time of trot ≈ 40 µs which is very small compared to the single motor run time and even to the average dwell time between two cargo steps during a 1-motor run. The latter is about h∆τ1 i ≈ 10 ms, see Fig. 5.1(b). Since such a motor pair construct may rotate several hundred times between two cargo steps during a 1-motor run, we refer to its rotation as flipping of the cargo.
xxii
D Glossary D.1 List of symbols Roman symbols Symbol a, b Cp , Cq d dt Drot Dq e fp , fq frot F F¯ FD Fext Fij′ Fle Fle,ca Fle,tr Fst Ftr Ftr,ca hsi i, j ile itr |iji hiji Jij J(C d ) ∆Jij ∆J(C) kB K ¯ K Keq Km ℓ ℓij L Lk L0
motor pair states consisting of motor states ile and itr and ∆L Perrin frictional factors direction of the network cycle small time intervall rotational diffusion constant, sphere rotational diffusion constant, ellipsoid simulation event rotational drag friction coefficients, ellipsoid rotational drag friction coefficient, sphere force dimensionless force detachement force external load force scaling force threshold for chemical transitions force acting on the leading motor of a motor pair force that the leading motor exerts onto the cargo interaction force stall force force acting on the trailing motor of a motor pair force that the trailing motor exerts onto the cargo hydrolysis rate of a single motor motor states motor state of the leading motor motor state of the trailing motor directed transition form i to j non-directed transition form i to j local flux from i to j cyclic flux of the directed cycle C d local excess flux along the edge |iji net cycle flux for completion of cycle C Boltzmann constant coupling parameter dimensionless coupling parameter equilibrium constant in the unit of µM Michaelis-Menten constant stepsize of a motor displacement of a motor during the transition |iji motor-motor separation equilibrated length of a kinesin motor equilibrated spring length
xxiii
D.1 List of symbols ∆L m hn1 i hn2 i nb nf nf +b nα p Pi Pist q, q1 , q2 Qij R rτ , rΠ S ∆Sij t tq ∆tα h∆tα i tmax trot ∆tca h∆tca i T Ui ∆Uij v v0 vα vα,sat v2,tot vca vca,sat vca,tot vsat vsi V Wij x xca xle x ¯le ∆xα h∆xα i
xxiv
deflection of the motor-motor separation number of spanning trees of a network graph average number of 1-motor runs average number of 2-motor runs number of backward steps per unit time number of forward steps per unit time total number of steps per unit time number of acitivity runs axial semiaxis of a prolate ellipsoid probability to be in state i steady state probability to be in state i equatorial semiaxes of a prolate ellipsoid heat release during the transition |iji radius of a sphere random numbers elliptic integral for prolate ellipsoids transition entropy time rotation time about axes q of an ellipsoid time spent in activity state α = 0, 1, 2 average run time in activity stateα maximal simulation time rotation time cargo run time of a motor pair average cargo run time of a motor pair temperature internal energy in state i difference of the internal energy from state i to state j velocity velocity for zero load force average velocity in the activity states saturation velocity in the activity states total 2-motor run velocity average cargo velocity saturation cargo velocity total cargo velocity single motor saturation velocity average single motor velocity volume of a sphere mechanical work performed against load force during the transition |iji motor position on the filament position of the cargo transported by a motor pair position of the leading motor on the filament xle − L0 in the units of the stepsize ℓ run length in activity stateα average run length in activity stateα
D Glossary xsi xtr x ¯tr ∆xca h∆xca i ∆¯ xca ∆¯ xrot xtot [X]
single motor run length position of the trailing motor on the filament xtr in the units of the stepsize ℓ cargo run length of a motor pair average cargo run length of a motor pair force induced shift in cargo position rotational shift in cargo position total distance the cargo travels on the filament nucleotide concentration X = ATP, ADP or P
Greek symbols Symbol α β, βp , βq ∆L ∆µ ∆¯ µ ∆µb ∆τα ∆τsi ε2 εsi η ηsi θ κ κij κ ˆ ij λsi µ ν πsi πc Πab Πij ρ σ τa τi Υ(C) Φij (F ) χij ωab ωi ωij
subscript of activity runs α = 0, 1, 2 rotation angles deflection of the motor-motor separation chemical potential difference dimensionless chemical potential difference balanced activities for Hsi = 0 dwell time between two mechanical steps during activity state α dwell time between two mechanical steps by the single motor termination rate of 2-motor runs overall unbinding rate of a single motor dynamic viscosity single motor efficiency load distribution factor for mechanical transition rates spring constant of the individual motor spring transition rate constant in the unit of s−1 transition rate constant in the unit of s−1 µM−1 chemomechanical coupling parameter of the single motor chemical potential subscript of a cycle of a network graph single motor rebinding rate characteristic rebinding rate for which P1 = P2 transition probability between motor pair state a and b transition probability between motor state i and j axial ratio between the axes of an ellipsoid width of the probability distribution P(∆L) dwell time in motor pair state a dwell time in motor state i influent factor for cycle C force depending factor of the transition rates load distribution factor of chemical transitions transition rate between two motor pair states a and b overall rate for leaving state i transition rate between two motor state i and j
xxv
D.2 List of abbreviations ωij,le ωij,tr Ω Ω(Am,j ) Ω(C) Ωj
transition rate of transitions by the leading motor transition rate of transitions by the trailing motor normalization factor transition rate product of the aborescences Am,ji of vertice j transition rate product of the cycle C unnormalized probabilities to be in state j
Caligraphy symbols Symbol An,i B C, Cνd E EDD F FDD Pα Pα,t Pα,x T Tm
aborescences of the spanning tree Tm of a network graph backward cycle cycle of a network enzymatic slip cycle enzymatic slip cycle including the (DD) state forward cycle forward cycle including the (DD) state probability to be in activity state α probability distribution of run times in activity state α probability distribution of run lengths in activity state α thermal slip cycle spanning trees of a network graph
D.2 List of abbreviations ATP ADP (D) (E) P (P) (T)
adenosine triphosphate adenosine diphosphate motor head occupied by ADP empty motor head phosphate motor head occupied by phosphate motor head occupied by ATP
p. Fig. Eq. Tab. i.e. e.g. cp. et al. s.e.m.
page figure equation table id est (lat. ’that is’) exempli gratia (lat. ’for example’) compare et alii (lat. ’and others’) standard error of a mean
ca eq
cargo equilibrium
xxvi
D Glossary ext le rot st st si tot tr
external leading rotation subscript: stall superscript: steady state single total trailing
Units µm nm s ms µs mM µM pN J K
micrometer nanometer second millisecond microsecond millimolar micromolar piconewton joule kelvin
Parameter sets B C D E
based based based based
on on on on
experimental experimental experimental experimental
data data data data
by by by by
Block et.al. [43] Carter et.al. [27] Diehl et.al. [58, 64] Schnitzer et.al. [29]
xxvii
D.3 List of figures
D.3 List of figures 1.1
Introduction to molecular motors . . . . . . . . . . . . . . . . . . . . . . . . .
2
2.1 2.2 2.3
Kinetic diagram of an ATPase . . . . . . . . . . . . . . . . . . . . . . . . . . . Spanning trees for a 3-state network . . . . . . . . . . . . . . . . . . . . . . . The hierarchy of biomolecular simulations . . . . . . . . . . . . . . . . . . . .
10 13 15
3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11
Network description of kinesin motors . . . . . . . . . . . . . . . . . . . Translocation of the chemical network of kinesin motors . . . . . . . . . 7 state network of the single motor . . . . . . . . . . . . . . . . . . . . . Single motor steady state probabilities and velocity as a function of Fext Single motor velocity as a function of ATP,ADP and P . . . . . . . . . . Single motor run time and P7st as a function of ATP, ADP and P . . . . Single motor run length as a function of Fext and ATP, ADP and P . . Force velocity curves compared to different experimental data . . . . . . Cycle time and ATP-transition time as a function of ∆µ . . . . . . . . . Leaving rates and dwell times in state i as a function of Fext . . . . . . Unbinding under external force for several values of ATP, ADP and P .
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18 19 20 24 25 27 28 31 33 34 35
4.1 4.2 4.3 4.4 4.5
Coupled motor pair model . . . . . . . Geometry aspect of motor pair model Assembling the 1-motor run network . Assembling the 2-motor run network . Coupled motor pair network . . . . . .
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39 42 43 44 45
5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 5.21 5.22
Cargo trajectory and distributions of dwell times between steps . . . . . Coarse-grained description: Activity state network . . . . . . . . . . . . . Cargo trajectories and distributions of run times in the activity states . . 2-motor run time as a function of the coupling parameter K and ATP . . Individual motor trajectories and distribution of the deflection ∆L . . . . Distribution of the deflection ∆L of motor-motor separation . . . . . . . . Activity state probabilities as a function of K and πsi . . . . . . . . . . . Crossover line for different ATP-concentrations and activity state diagram Properties of the activity states as a function of K and πsi . . . . . . . . . Cargo properties compared to the activity state diagram . . . . . . . . . . Run length distributions and comparison to experimental data . . . . . . Distribution P(∆L) and activity state diagram of sets B, C, D and E . . Motor pair run times as functions of nucleotide concentrations . . . . . . Motor pair run lengths as functions of nucleotide concentrations . . . . . 2-motor run velocity as a function of K, ATP and ADP, P . . . . . . . . . Cargo velocity as a function of K and πsi , ATP and ADP, P . . . . . . . . Coupled motor pair model under external load force . . . . . . . . . . . . Histograms of the local cargo shifts under external load force . . . . . . . Trajectories of the motor pair under external load force . . . . . . . . . . Trajectory for a large force and activity state diagram under force . . . . Activity state velocities: Comparison to experimental motor pair data . . Cargo velocity: comparsion to experimental motor pair data . . . . . . . .
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52 54 55 56 57 58 60 61 64 66 67 68 72 73 74 75 79 81 82 84 87 88
xxviii
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List of tables 5.23 Velocity v2,tot of a long 2-motor run as a function of Fext . . . . . . . . . . . . 5.24 Operation mode diagram of a coupled motor pair . . . . . . . . . . . . . . . .
90 91
A.1 A.2 A.3 A.4
Spanning trees for 7-state network . . . . . . . . . . . . . . . . . . . . . Single motor velocity and hydrolysis rate as a function of Fext and ∆µ . Single motor velocity and hydrolysis rate for the different parameter sets Single motor properties as a function of Fext . . . . . . . . . . . . . . . .
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i iv vi vii
B.1 B.2 B.3 B.4 B.5 B.6 B.7
Unbinding under superstall depending on nucleotide concentrations Cargo properties as functions of nucleotide concentrations . . . . . Cargo run length as functions of nucleotide concentrations and πsi 1-motor run properties as functions of nucleotide concentrations . . 2-motor run properties as functions of nucleotide concentrations . . Comparison to experimental motor pair data . . . . . . . . . . . . Determination of K via h∆t2 i based on experimental data . . . .
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ix x xi xii xiii xiv xv
C.1 C.2 C.3 C.4 C.5
Histogram of the local cargo shifts for Fext = 0 . . . . . . . . . . . . . . . . . xvii Rebinding under external force . . . . . . . . . . . . . . . . . . . . . . . . . . xviii Distribution of the deflection of motor-motor separation under force . . . . . xviii Cargo run length and run time: comparsion to experimental motor pair data xix Hinge-like connection and prolate ellipsoide . . . . . . . . . . . . . . . . . . . xx
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D.4 List of tables 3.1 3.2
Michaelis-Menten constant Km for the single motor velocity . . . . . . . . . . Force factor parameters of the different parameter sets . . . . . . . . . . . . .
26 30
5.1
Termination rate ε2 as a function of coupling parameter K
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54
A.1 Transition rate constants for the different parameter sets . . . . . . . . . . . . A.2 Copy of Tab. 3.2: Force factor parameters of the different parameter sets . . .
iii v
B.1 Average velocity and run length data from [58] . . . . . . . . . . . . . . . . .
xiv
xxix
D.4 List of tables
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this is...
All my thanks to... ... my supervisor Prof. Lipowsky for introducing me to Biophysics and especially to the topic of molecular motors, for inspiring discussions and constant support ... the theory group for all these nice years at the institute and a bunch of good questions on my talks ... Vroni, Stefan and Tobias for proofreading parts of this work ... Jonas for various discussions on physics and maths, on politics and philosophy and life in general; for providing me with all the necessary things I recently forgot, like food ... Vroni for scientific and other discussions, for help in programming and mathematica, entertainment, teaching and finding creative solutions to various kind of problems ... Florian and Stefan for all the discussions on motors and motor pairs and the scientific collaboration ... Steffen for all the groundwork on which this work based and for the training in scientific working ... Angelo and Jörg for helpful discussions about dwell times and networks ... Nico and Marko for relaxing and pleasurable coffee breaks and mutual encouraging ... Oksana and Yu for continuously providing me with birds milk, sweets and green tea and encouraging me whenever I felt like I’ll never get this work done ... my family in Bretten as well as in Berlin for supporting me and being patient with me ... Meriam and Susi for helping me solving all the non-scientific stuff ... René and Marco for IT support ... Peter for the insight, that not every problem is analytically solveable... ... and Peter Westermeier for the insight that there actually is a solution for any problem (and thanks for plenty of apple pie!) ... my lunchgroup for (alsmost always) avoiding work-talk during lunch ... all my friends, who supported me on my way through these years and therefore met me rarely
