Eur. Phys. J. E (2012) 35: 119 DOI 10.1140/epje/i2012-12119-5
THE EUROPEAN PHYSICAL JOURNAL E
Regular Article
Nano-swimmers in biological membranes and propulsion hydrodynamics in two dimensions Mu-Jie Huang1 , Hsuan-Yi Chen1,2,3 , and Alexander S. Mikhailov4,a 1 2 3 4
Department of Physics, National Central University, Jhongli 32001, Taiwan Institute of Physics, Academia Sinica, Taipei 11520, Taiwan Physics Division, National Center for Theoretical Sciences, Hsinchu, 30013, Taiwan Abteilung Physikalische Chemie, Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany Received 3 May 2012 and Received in final form 9 September 2012 c EDP Sciences / Societ` Published online: 23 November 2012 – a Italiana di Fisica / Springer-Verlag 2012 Abstract. Active protein inclusions in biological membranes can represent nano-swimmers and propel themselves in lipid bilayers. A simple model of an active inclusion with three particles (domains) connected by variable elastic links is considered. First, the membrane is modeled as a two-dimensional viscous fluid and propulsion behavior in two dimensions is examined. After that, an example of a microscopic dynamical simulation is presented, where the lipid bilayer structure of the membrane is resolved and the solvent effects are included by multiparticle collision dynamics. Statistical analysis of data reveals ballistic motion of the swimmer, in contrast to the classical diffusion behavior found in the absence of active transitions between the states.
1 Introduction It is well known that macroscopic objects can actively move through fluids, or swim, by periodically changing their spatial configuration. The swimming behavior is furthermore characteristic for many biological microorganisms and often underlies the capacity of biological cells, such as bacteria, to move themselves towards nutrition sources or away from danger. As shown in the classical study by P.M. Purcell [1], swimming at microscales should have special properties since it takes place under the conditions of small Reynolds numbers. Particularly, self-propulsion by cyclic configuration changes is only possible if the reverse motion in a cycle does not represent a simple reciprocal of the forward motion. General theoretical analysis of the propulsion phenomena at low Reynolds numbers has been performed [2] and various examples of micro-swimmers have been proposed and investigated [1,3–5]. The question, which is currently being debated [6– 9], is whether active swimming is also possible on the nanoscales, at the level of single biomolecules. Many important functions in a biological cell are executed by protein machines that act as enzymes or motors, process other macromolecules, such as RNA or DNA, and serve Supplementary material in the form of a .mov file available from the Journal web page at http://dx.doi.org/10.1140/epje/i2012-12119-5 a e-mail:
[email protected]
as active ion channels. Their operation is based on conformational changes, typically induced by binding of an ATP molecule, its hydrolysis and products release [10]. The turnover cycles are continued as long as fresh ATP molecules are supplied and reaction products are removed from the solution. In each cycle, mechanochemical motions inside a protein take place, so that the physical shape of this macromolecule is repeatedly varied. Recently, cyclic conformational motions have been directly observed in single-molecule experiments by using fast atomic force microscopy [11,12]. They could also be seen in structurally resolved coarse-grained dynamical simulations of the molecular motor hepatitis C virus (HCV) helicase [13] and of the enzyme adenylate kinase [14]. An important aspect of mechanochemical motions in molecular machines is that such cyclic motions are not generally reciprocal. Indeed, the forward and reverse conformational changes are taking place in the presence or absence of ligands and, hence, in the systems whose composition and dynamical properties are different. Therefore, each individual machine essentially represents a (nano)swimmer and active propulsion motion of the machines should be observed when energy, in the form of ATP molecules, is continuously supplied. Previously, it has been shown how the propulsion velocity and the stall force of a machine can be determined if internal conformational motions in a protein are known [7]. There are many proteins which stay bound to biological membranes formed by lipid bilayers. Biomembranes play a fundamental role in the living cell; they build up
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the wall separating a cell from its environment and make up the vesicles used in intracellular transport. Typically, up to 40% of the molecular weight of a vesicle falls to its protein content and similar numbers are characteristic for cell membranes. While some of the membrane proteins are passive and represent, e.g., nanopores through which ions can pass, there is also a large number of active proteins, such as ion pumps, inside the membranes. Referring to them, one usually talks about active protein inclusions. The presence of actively operating molecular machines brings the entire biomembrane into a state far from thermal equilibrium. Membrane instabilities and fluctuation phenomena caused by active inclusions have been extensively investigated (see, e.g., [15–17]). Recently, we have shown that stationary Turing-like patterns and traveling waves can spontaneously develop in such systems [18]. In contrast to previous publications [15–18] where collective effects leading to membrane deformations were considered, our attention in the present study is focused on the properties of a single inclusion in a biological membrane. Our aim is to demonstrate that an inclusion, immersed into a lipid bilayer and cyclically changing its conformation, can behave as a swimmer and actively propel itself through the membrane. Under physiological conditions, biological membranes are in the liquid phase. Because their viscosity is much higher then that of the surrounding solvent (predominantly water), hydrodynamic coupling to the solvent can be neglected on the scales shorter than a micrometer and, on such length scales, the membrane can be approximately described as a two-dimensional (2D) fluid [19,20]. Generally, reduction from three to two dimensions is known to have a profound effect on fluid mechanics (see e.g. [21]). While hydrodynamic interactions in three dimensions are fading away as the inverse of the distance at large separations between the particles, such interactions in 2D fluids have a logarithmic dependence on the distance between the particles and, therefore, do not disappear even at long separations. Since hydrodynamic interactions play a principal role in propulsion phenomena, the behavior of micro-swimmers in 2D systems should be significantly different. The detailed numerical modeling of a single active protein in lipid bilayers is a computationally demanding problem and we do not yet address it in the present study. Instead, we consider a simple model of an active membrane inclusion which represents a chain of three beads linked by elastic links. Our 2D membrane swimmer is similar to that considered by A. Najafi and R. Golestanian in three dimensions [4] (see also [8]), M. Leoni and T.B. Liverpool also considered the hydrodynamic dipole and quadrupole in two dimensions [22]. The difference is that we do not assume that the motions of the beads are predefined. In our model (similar to ref. [7]), natural lengths of the elastic links connecting the beads are repeatedly changed (increased and decreased). Such cyclical state changes induce internal conformational relaxation motions which lead to propulsion.
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It should be noted that this simple mechanical model grasps nonetheless some essential aspects of the behavior of real protein machines. As an example, the operation of the enzyme adenylate kinase (ADK) can be considered [14]. This protein consists of three stiff domains connected by two flexible hinges. Being an enzyme, it catalyzes a reaction where adenylate triphosphate (ATP) reacts with adenylate monophosphate (AMP) to produce two adenylate diphosphate (ADP) molecules. First, ATP binds to the left protein domain and this leads to shortening of the distance between the left and the central domains. Second, AMP binds to the right domain and, as a result, the latter also moves closer to the central domain. Now, when two ligands (ATP and ADP) are in contact, the reaction between them takes place and the products leave the enzyme molecule. After that, both side domains move away from the middle one, and the initial protein configuration is recovered. Our model can be viewed as a toy version of such an enzyme (note, however, that ADK itself is not a membrane protein). In the next section, the behavior of the model is analytically treated in the framework of the far-field Oseen description for 2D fluids. By numerical integration of dynamical equations, propulsion of the membrane swimmer is investigated. The third section provides an example of a more realistic simulation. Now, each lipid molecule in the bilayer is modeled as a short polymer with three hydrophilic and one hydrophobic particle connected by elastic strings. For the solvent, the multipartcile collision (MPC) dynamics is used. Potential interactions between lipid molecules and also between such molecules and the solvent particles are explicitly considered. It has been recently shown [23] that this coarse-grained model yields a good description of biological membranes, including also their hydrodynamic properties. In the present study, such an approximate description is used to simulate swimming of the machine in a realistic membrane environment. Simulations both for an active swimmer and for the same object in a passive regime are performed and their statistical analysis is undertaken. The paper ends with conclusions and the discussion of obtained results.
2 The two-dimensional swimmer We consider a model swimmer consisting of three beads (i = 1, 2, 3) which are connected by two elastic links a and b and thus form a short string (fig. 1). It is assumed that the string has a high bending stiffness and always remains straight. If the x-axis is chosen along the string direction, the elastic energy of this object is 2 2 1 k |x1 − x2 | − la (s) + |x2 − x3 | − lb (s) . 2 (1) Here x1 , x2 and x3 are the coordinates of the beads, k is the stiffness constant, and la (s) and lb (s) are the natural lengths of the links a and b that depend on the discrete state variable s. We assume that otherwise all the particles and the links are identical. E=
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to the solvent needs to be taken into account and the system effectively becomes three-dimensional [19,20]. It is convenient to introduce dimensionless variables ξ and τ by measuring lengths in units of the natural (long) length l0 of the links and time in units of τ0 = (μk)−1 . For the dimensionless variables, the explicit dynamical equations of the model are
Fig. 1. The three-state swimmer. The swimmer consists of three particles (1, 2, and 3) connected by two elastic links (a and b) with variable natural lengths. In the state s = 1, the natural lengths of both links are long. In the state s = 2, the natural length of the link a is short, while the natural length of the link b is still long. In the state s = 3, both links have short natural lengths. The states are cyclically switched. The swimmer dynamics consists of the relaxation processes which follow after each switching event.
There are three states of the swimmer, with different natural lengths of the links. We have la (s = 1) = l0 , la (s = 2) = la (s = 3) = l0 (1 − ), lb (s = 1) = lb (s = 2) = l0 , lb (s = 3) = l0 (1 − ).
∂E ∂E ∂E − L12 − L13 , ∂x1 ∂x2 ∂x3
(3)
and similar equations hold for the other two particles. In eq. (3), μ is the mobility of the beads inside the membrane. The coefficients L12 and L13 are the longitudinal components of the mobility tensor that determine how the velocity of particle 1 along the x-direction will be affected by the application of forces in the same direction to the particles 2 and 3. For the membranes, the longitudinal components are approximately given by [20] Lij (xij ) = −
1 ln(κxij ), 4πηh
(5)
Here, the following notations have been used: (2)
Thus, both links are long in the first state. In the second state, the link a is short, while the link b is long. In the third state, both links are short. The swimmer is immersed into a 2D fluid. In the Oseen approximation for the regime of low Reynolds numbers, the motion of the first bead is governed by the equation x˙1 = −μ
dξ1 = −[ξ12 − da (s)] dτ ν − ln(σξ12 )[ξ23 − ξ12 + da (s) − db (s)] 4π ν ln(σξ13 )[ξ23 − db (s)], + 4π dξ2 = −[ξ23 − ξ12 + da (s) − db (s)] dτ ν ln(σξ12 )[ξ12 − da (s)] + 4π ν − ln(σξ13 )[ξ23 − db (s)], 4π dξ3 = −[ξ23 − db (s)] dτ ν ln(σξ23 )[ξ23 − ξ12 + da (s) − db (s)] − 4π ν + ln(σξ13 )[ξ12 − da (s)]. 4π
(4)
where xij = |xi −xj | is the distance between beads i and j, the parameter η specifies the (three-dimensional) viscosity of the membrane, and h is the membrane thickness. The coefficient κ can be estimated [20] as κ = ηf (ηh)−1 , where ηf is the solvent viscosity. The description in terms of the mobility tensor (4) is applicable for the distances between the particles which are much larger than their own sizes, but still smaller than the characteristic distance κ−1 . At larger distances, hydrodynamic coupling of the membrane
ξij = |ξi − ξj |, ν = (μηh)−1 ,
da,b (s) = la,b (s)l0−1 , σ = κl0 .
(6)
Before proceeding to numerical simulations of the model, characteristic values of the dimensionless parameters need to be discussed. It is known [20] that the membrane viscosity is approximately 1000 times higher than that of water, i.e. we can take η/ηf = 103 . It can be furthermore assumed that the natural lengths of the links in the swimmer are ten times larger than the membrane thickness, so that we have l0 /h = 10. With these choices, the dimensionless parameter σ is equal to 0.01. To estimate the dimensionless parameter ν in eqs. (5), the membrane mobility μ of an individual bead needs to be known. Using dimensionality arguments, we can notice that the mobility should have the form μ = (ηReff )−1 where Reff is some effective size of the particle. Substitution of this expression leads to the estimate ν = Reff /h. Below, we shall assume that the effective size of the particles is equal to the membrane thickness and therefore take ν = 1. If the state s = 1, 2, 3 of the swimmer is maintained fixed, eqs. (5) describe the relaxation of the elastic chain to the equilibrium with the lengths ξ12 = da (s) and ξ23 = db (s), as affected by hydrodynamic interactions between the particles. To make this object active, switching . . . → 1 → 2 → 3 → 1 . . . between the states is introduced. When the equilibrium configuration, corresponding to a certain dynamical state s, is approached, the state variable s is switched to the next state s in the sequence and relaxation to this state begins. Thus, the swimmer is cyclically changing its configuration, never settling into the rest (see fig. 1).
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0.05
s=2
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s=1
Position
1
0.8
0.02 0.01
0.6
s=3 0.4 0.4
0.03
0
0.6
0.8
0
1
Fig. 2. Periodic phase trajectory of the swimmer in the plane of pair distances ξ12 and ξ23 between the particles for = 0.5. For comparison, dashed curves show the same orbit when hydrodynamic interactions between the particles are switched out.
20
40
60
80
100
Fig. 3. Time dependence of the mass center position Ξ(t) of the swimmer for = 0.5.
-3
The following explicit switching conditions, defined in terms of the time-dependent distances ξ12 (τ ) and ξ23 (τ ) between the particles, have been employed in our simulations: If |ξ12 − 1| < ρ and |ξ23 − 1| < ρ, transition to the state s = 2 takes place. If |ξ12 − 1 + | < ρ and |ξ23 − 1| < ρ, transition to the state s = 3 occurs. Finally, if |ξ12 − 1 + | < ρ and |ξ23 − 1 + | < ρ, transition to the state s = 1 is implemented. A small value ρ = 0.025 of the switching control parameter was chosen. Although switching conditions are only formally introduced in the present study, their qualitative physical interpretation can also be given. If beads 1, 2 and 3 correspond to three different protein domains, we can say that the state s = 1 is the free state of the protein, in the absence of any ligands. In the state s = 2, the ligand is attached to the first domain, modifying its interaction with the domain 2, whereas the domain 3 still stays free. The state s = 3 corresponds then to a situation when the ligands are bound to both domains 1 and 3. Hence, transitions from the state s = 1 to s = 2 and from s = 2 to s = 3 are caused by ligand binding events. In contrast, the transition from the state s = 3 to the free state s = 1 is a consequence of the reaction between the two ligands and of the instantaneous product release. Since the dynamics is dissipative, energy must be supplied to the swimmer within each its cycle. In the absence of thermal fluctuations, sufficient energy should be brought by the ligands, so that the dissipation of energy within a cycle is balanced. Numerical integration of eqs. (5) with the parameters ν = 1, σ = 0.01 and ρ = 0.025 was performed for different values of the parameter , determining the extent of change of the link lengths within a cycle. Figure 2 shows phase trajectories of the membrane swimmer in the plane of distances ξ12 and ξ23 between the particles (the initial transient is not displayed here). For comparison, we also show the cycle of the same active object when hydrodynamic interactions are neglected, i.e. if ν = 0 in eqs. (5). In addition to pair distances, coordinates of all three particles have been traced in the simulations and instantaneous
Velocity
210
-3
110
0 0
0.2
0.4
0.6
0.8
1
Fig. 4. Dimensionless propulsion velocity of the swimmer as a function of the parameter . The dashed line shows the quadratic fitting approximation.
mass-center positions Ξ(τ ) = (1/3)(ξ1 (τ ) + ξ2 (τ ) + ξ3 (τ )) have been determined. Figure 3 displays the motion of the mass center. As we see, after each cycle a shift by a certain distance takes place and the propulsion behavior is indeed observed. Repeating simulations for different values of the parameter , the dependence of the dimensionless propulsion velocity V on this parameter was determined (fig. 4). For small , the velocity is proportional to 2 , in agreement with previous estimates for three-dimensional machine swimmers [7].
3 Microscopic membrane simulations In the above description, the presence of a membrane was taken into account only indirectly, in terms of a twodimensional Oseen tensor. Now, we want to make our description more detailed. The all-atom molecular dynamics (MD) simulation would have treated the problem at an atomic resolution, so that the molecular structures of the protein and the lipids, as well as that of the surrounding water, were completely resolved. Such simulations are possible (see, e.g., [24]), but computationally expensive, so that only the behavior on very short time scales may be traced. There are, however, efficient coarse-grained dy-
Eur. Phys. J. E (2012) 35: 119
namical descriptions of proteins. They picture a protein as an elastic network (EN) formed by point particles, each of which corresponds to a certain residue. Such networks are constructed by using the experimental structural data for specific proteins. Using EN descriptions, conformational cycles of the first protein machines could already be successfully reproduced in a structurally resolved way [13, 14]. For the solvent, an efficient coarse-grained simulation method is provided by the multiparticle collision (MPC) dynamics (see review [25]). Here, the fluid is divided into a grid of cells and, at regular time intervals, all particles that currently find themselves in a cell are undergoing a multi-collision in which their velocities are randomly exchanged. The MPC simulations are known to correctly reproduce the motions of the fluid, including hydrodynamic fluctuations, on the scales larger than the size of a cell. They have been used to treat polymers in solutions and have also been applied to describe micro-swimmers in three-dimensional fluids (see, e.g., [26]). It has been shown how EN models for proteins can be combined with the MPC methods for fluids in order to obtain simulations of entire protein machine cycles, with hydrodynamic effects included [14,27]. Recently, a method for coarse-grained structurally resolved numerical simulations of biomembranes, which employs the MPC dynamics for the solvent, was developed [23]. It has been demonstrated that, by using this method, self-assembly of biomembranes from a uniform mixture of lipids can be observed. It has been found [23] that the method correctly reproduces the gel and liquid phases of the membranes at different temperatures. Moreover, it yields realistic predictions for such averaged membrane properties, as the surface tension coefficient or the bending rigidity modulus; it is also correctly reproducing hydrodynamic fluctuations of membrane flows. Membrane-bound proteins can be treated in numerical simulations, if an EN description for a protein is combined with the coarse-grained dynamical description for biomembranes in the presence of solvent. In this way, membrane swimmers, representing active protein machines, can be simulated. Such full realistic simulations are, however, still a challenge for the future studies. While they remain to be performed, below we present an example of a numerical simulation where the membrane and the solvent are modeled in a structurally resolved manner, but the swimmer is described in a simple way, with only three its domains being explicitly considered. The detailed description of the coarse-grain model for the membrane and the solvent can be found in a separate publication [23]. Each lipid was modeled as a string of four beads, one corresponding to the hydrophilic head and three forming the hydrophobic tail of the molecule. The neighboring particles interacted via elastic FENE bonds; the string was straightened by the second-neighbor repulsive elastic forces which made it stiff. The interactions between particles belonging to different lipids were described by the truncated Lennard-Jones potentials. The solvent was formed by a large number of particles interactions between which were accounted through the MPC
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Fig. 5. The microscopic membrane swimmer. Three cylindric domains are immersed into the lipid bilayer. The top and the bottom parts of each domain are hydrophilic, whereas the middle parts are hydrophobic. The domains are linked through elastic interactions applied to their mass centers. The natural lengths of these links are cyclically switched.
dynamics approximation. Attractive interactions between hydrophilic heads and the solvent were present, whereas repulsive interactions between the hydrophobic tails and the solvent particles were assumed. In our study, the set of model parameters which was found [23] to correspond to the liquid phase of the membrane is used. Below, the characteristic length of the Lennard-Jones interaction, σ0 , was chosen as the length unit, the mass of a lipid bead and a solvent particle, m0 , was the mass unit, and the strength of lipid head-tail interaction, 0 , was the energy unit. Time is reported in MD time steps δt. The parameter values and further details of the membrane model were given in ref. [23]. To incorporate the swimmer into this description, we have modeled it as consisting of three cylindric domains each spanning the membrane (fig. 5). An individual domain was formed by 42 beads, where the mass of a swimmer bead was 10 times larger than the mass of a solvent particle and a lipid bead. For simplicity, we have assumed that these beads were organized into the same four-bead strings as lipid molecules. Thus, each domain possessed the hydrophilic top and bottom parts, whereas its middle part had hydrophobic properties. The seven strings, constituting a cylindrical domain, were kept together through a stiff interaction potential. However, their interactions with other membrane particles and with the solvent were the same as for the lipid strings in the considered model. The internal mechanics of the swimmer was described by the elastic energy equation (1) where the coordinates x1,2,3 were replaced by two-dimensional vectors R1,2,3 specifying instantaneous positions of the mass centers of the domains 1, 2 and 3 in the membrane plane. To prevent bending of the machine in the membrane plane, high bending stiffness was introduced. We have defined θ as the angle between vectors R12 = R1 −R2 and R23 = R2 −R3 and introduced the bending potential U (θ) = β(1 − cos θ) with a high stiffness constant β. Planar distances between the mass centers were used to formulate the switching conditions, similar to what is described in sect. 2. To introduce interactions affecting only the motion of the mass centers, forces with the same magnitude and direction were applied to all particles in the chosen domain. The natural lengths of the links, connecting three domains, were equal to 8 when the swimmer was in state s = 1; they were shortened by 50% in every cycle. These elastic links had the stiffness which was 5 times higher
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Fig. 6. A series of six consecutive snapshots from a simulation of the membrane swimmer; the video of the entire simulation is available in the Supplementary Material. Time moments (in MD steps) are indicated for each snapshot. The solvent is not displayed and, for clarity, only half of the membrane, with the cross-section along the initial swimmer position, is shown. The elastic links, connecting the cylindric domains, are not visualized here. Periodic boundary conditions are employed and, therefore, in the last snapshot the swimmer is seen re-entering the membrane from the opposite direction.
swimming length (
18 16 14 12 10 8 6
8.010
5
8.510
5
9.010
MD steps
5
5
9.510
Fig. 7. Total length of the swimmer as a function of time for a relatively short time interval including 23 swimmer cycles.
than the stiffness of the elastic FENE bonds in the lipids. The switching parameter was ρ = 0.2. Periodic boundary conditions were used. The simulations box size was 40×40×25. The simulated system included 2592 lipids and 137,322 solvent particles. On average, the solvent number density ns in the bulk was equal to five. The initial orientation of the swimmer was in the direction along the x-axis and the center of mass of the swimmer was in the center of the membrane. The system was equilibrated for 50000 δt during which time the swimmers were kept passive and freely diffused in the membrane. After the equilibration, the swimmers were made active and performing cyclic self-propelled motions. Figure 6 displays several snapshots from the membrane-resolved simulation of a swimmer. The video of the entire simulation
is available in the Supplementary Material. For clarity, only half of the membrane is displayed, with the crosssection through the initial position of the swimmer. In the simulation shown, about 100 cycles have taken place and the swimmer has moved over a distance comparable to its total length. Figure 7 shows 23 consequent cycles monitoring the total length of the swimmer (i.e., of the distance between the mass centers of the domains 1 and 3), each peak corresponds to the state s = 1 of the swimmer. As revealed by this figure, the durations of the cycles are strongly fluctuating and, sometimes, the swimmer stays for a while in one of its states before the motion can continue. This behavior is due to the fact that the dynamics of the swimmer takes place inside a densely packed medium and, when one of its domains tries to move, it needs to push aside the surrounding lipids. In contrast to the Oseen description for 2D fluids, thermal noises from the fluctuating environment, solvent plus lipids, were present in the simulation. An important question is whether the effects of propulsion prevail for an active swimmer or its dynamics is still dominated by thermal diffusion. To analyze it, a series of long simulations for the active swimmer and also, for comparison, for the passive swimmer has been performed. For the passive swimmer, active switching between the states has been excluded, so that it was always found in the equilibrium conformation (s = 1). The results of such simulations and their statistical analysis are presented below. In fig. 8(a), five different trajectories of the active swimmer obtained in independent numerical simulations are shown. Each trajectory corresponds to 5 × 106 δt. A single cycle of the swimmer takes, on average, about 7288 ± 75δt, so that these trajectories cover about 675 to 700 swimmer cycles. For comparison, we show in fig. 8(b) five trajectories of the passive swimmer under the same conditions. Examining the trajectories, one can already notice that the trajectory of an active swimmer is more extended than that of its passive analog. Statistical analysis of the simulation data for active and passive swimmers has been undertaken. Figure 9 shows the dependences of the mean-square displacement (MSD) on time for the two objects. For the active swimmer (circles), the observed dependence is well approximated by the quadratic law, whereas the classical diffusion law with the MSD linearly proportional to time is found for the passive swimmer under the same conditions (squares). Hence, the motion of the active swimmer is essentially ballistic on the considered time scales, whereas the passive swimmer performs the classical diffusive motion on the same scales. From fig. 9, the mean propulsion velocity V of the active swimmer can be estimated, yielding V = 1.2 × 10−5 . The diffusion constant of the passive swimmer is estimated as D = 6.3 × 10−6 . This implies that, for the system under consideration, the propulsion effects dominate over the effects of thermal noise. In addition to the translational motion, characterized by the position of the mass center, we have also analyzed the orientational motions of the active swimmer. The ori-
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(a)
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1
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Y
C(t)
0
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0.4 0
(b)
0
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40
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-20
-40 0
X
20
40
Fig. 8. Five trajectories of the active (a) and passive (b) swimmers. Each trajectory corresponds to 5 × 106 MD steps. In the passive swimmer, active transitions between the conformational states are prohibited, so that it is always in the equilibrium state s = 1.
3
MSD
10
2
10
1
10
6
110
t (MD steps)
6
310
Fig. 10. The plot of the orientational correlation function, C(t), for the active swimmer, where the solid line is the exponential fit.
0
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6
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t (MD steps)
-40 -20 20
6
110
6
510
Fig. 9. Log-log plots of the mean square displacement of the swimmer mass center showing ballistic motion for active swimmer (circles), MSD(t) ∼ t2 (solid line), and diffusive motion for passive swimmer (squares), MSD(t) ∼ t (dashed line).
entation was characterized by the instantaneous direction ˆ 31 (t), which is pointing from domain of the unit vector R 1 to domain 3, and the orientational correlation was comˆ 31 (t) · R ˆ 31 (0). The averputed as the average, C(t) = R ages were taken over simulation snapshots every 1000δt from five realizations. The computed orientational correlation function for the active swimmer is displayed in fig. 10. As we see, it is well approximated by an exponential law with C(t) proportional to exp(−t/τc ) with the correlation time τc 4 × 107 δt. Note that this correlation time is about ten times longer than the simulation time. This explains why only the ballistic motion of the active swimmer was observed in our simulations. It should be expected that, on the time scale much longer than orientational correlation time τc , the ballistic motion would be replaced by diffusive motion with an enhanced diffusion constant [28,29]. Our simulation times are not yet long enough to resolve this asymptotic behavior.
4 Discussion On submicrometer scales, biomembranes behave as viscous two-dimensional fluids. The inclusions, immersed into a membrane and cyclically changing their conformation, can propel themselves through the membrane and thus act as micro-swimmers. Such active inclusions can represent protein machines, changing their conformation in response to binding of ATP and other ligands [30]. Instead of considering real proteins, we have treated in our study a model swimmer that consists of three domains connected by elastic links which can change their natural lengths depending on the internal state of the swimmer. By using the far-field expressions for the mobility tensor in two-dimensional fluids, we could directly investigate the propulsion behavior of such model swimmer and numerically determine its propulsion velocity. We have also provided examples of microscopic membrane simulations where the three-domain swimmer was placed into an actual lipid bilayer. Energetic interactions between lipid molecules were explicitly included into such
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simulations, as well as the interactions between the lipids and the solvent particles. The solvent hydrodynamics was approximately treated within the multiparticle collision dynamics approach. Our simulations have shown that propulsion behavior of a swimmer in a low-Reynoldsnumber environment could indeed be observed. The Reynolds numbers of the swimmer with respect to the solvent and the membrane can be estimated. According to refs. [31,32], the kinematic viscosity of the solvent is νs = 0.75. Using the linear size of a swimmer L = 16, the mean velocity of the swimmer V = 1.2×10−5 , and solvent kinematic viscosity νs , the Reynolds number for swimming in the solvent is estimated as Re = vL/νs 2.6 × 10−4 . Because the viscosity of the membrane is 1000 times larger than the solvent [20], the Reynolds number for the swimmers with respect to the membrane is smaller by a factor of 103 . This indicates that, consistent with the model described in sect. 2, the considered swimmer indeed operated in the low Reynolds number regime. In our previous computational study of biomembranes [23], the length unit in the simulations has been identified as σ0 = 0.8 nm and the MD time step as δt = 10 ps. By using these values, physical properties of the considered model swimmer can be accessed. For the passive swimmer, we find the diffusion constant D = 6.3 × 10−6 σ02 /δt 0.4 μm2 /s, which is comparable to the typical diffusion constant of membrane proteins [33]. For the active swimmer, we find that the propulsion velocity is V = 1.2 × 10−5 σ0 /δt = 960 μm/s. The mean cycle time of the swimmer is 7300δt 0.1 μs. As we have already noted in the Introduction, our simple model swimmer can be considered as an idealization of real protein machines and, specifically, of enzymes where mechanochemical conformational motions inside the cycles are often involved. The turnover times of such enzymes vary from a microsecond to milliseconds and even seconds. Thus, our active membrane swimmers would correspond to rapid enzymes with very short turnover cycles. It is possible in our simulations to include stochastic waiting times for the transitions between the internal swimmer states (cf. [8,9]), and also to consider swimmers with slower internal conformational motions. When the swimmer is cycling less frequently, its propulsion velocity would be reduced. Note that, even then, propulsive motions may be visible on the background of thermal diffusion. For example, if the cycle time of the swimmer is 1000 times longer than in the present simulations, the propulsion velocity would be about V 1 μm/s. A particle with such a velocity would move ballistically over a micrometer in a second. Since the diffusion constant of the swimmer is D 0.4 μm2 /s, it would also move diffusively, on the average, over a distance about a micrometer within the time of a second. Hence, propulsion effects would still be significant for the slower membrane swimmers on such time scales. In the reported study, only the dynamics of an individual membrane swimmer has been considered. It would be interesting to investigate interactions between two or more swimmers that move through a membrane. In 2D fluids, hydrodynamic interactions depend logarithmically
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on the distance between the particles, in contrast to the power law characteristic for three dimensions. As a result, strong interactions over long distances should be observed. One can expect that membrane swimmers would feel each other even at large separations, leading to orientation alignment and possible dynamical synchronization. Groups of membrane swimmers, representing active protein inclusions, can furthermore form rafts traveling in definite directions. Such effects, both for model swimmers and for real active protein inclusions, should be a subject of future investigations. Partial financial support from the DFG Research Training Group (GRK 1558) “Nonequilibrium collective dynamics in condensed matter and biological systems” and the National Science Council of the Republic of China (Taiwan) is gratefully acknowledged. We thank R. Kapral for many stimulating discussions.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.
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