Biophysical model of self-organized spindle formation patterns without centrosomes and kinetochores Stuart C. Schaffner* and Jorge V. Jose´*†‡ *Center for Interdisciplinary Research on Complex Systems and Physics Department, Northeastern University, 360 Huntington Avenue, Boston, MA 02115; and †Physics Department and Department of Physiology and Biophysics, 516 Capen Hall, University at Buffalo, State University of New York, Buffalo, NY 14260-1611 Communicated by Morrel H. Cohen, Rutgers, The State University of New Jersey, Piscataway, NJ, June 9, 2006 (received for review January 24, 2006)
Eukaryotic cell division and chromosome segregation depend crucially on the mitotic spindle pattern formation. The usual pathway for spindle production involves microtubule polymerization from two centrosomes. However, experiments using Xenopus extracts with micrometer-sized chromatin-coated beads found, remarkably, that spindle patterns can form in the absence of centrosomes, kinetochores, and duplicated chromosomes. Here we introduce a previously undescribed biophysical model inspired by the heuristic interpretations of the experiments that provides a quantitative explanation and constraints for this type of experiment. The model involves plus-directed (chromokinesin and Eg5) and minus-directed (cytoplasmic dynein oligomers) motors walking on microtubules and the boundary conditions caused by the chromatin-coated spheres. This model combines the effects of the plus-directed cross-linking motor Eg5 and any chromokinesin on the chromatincovered beads, reflecting current uncertainties in the division of function between the two kinds of motors. The model can nucleate dynamically a variety of self-organized spindle patterns over a wide range of biological parameter values. Our calculations show that spindles will form over a wide range of parameter values. Some parameter values cause a monaster to form instead of a bipolar spindle. Varying the processivity and the dynein microtubule attachment and detachment rates, we find stability parameters for spindle formations. These results not only constrain the possible parameter values, but they point toward the proper division of function between Eg5 and chromokinesin in this spindle formation pathway. The model results suggest experiments that would further enhance our understanding of the basic elements needed for spindle pattern formation in this pathway. mitosis 兩 modeling 兩 molecular motors 兩 microtubule 兩 Xenopus
U
nderstanding how a complex set of proteins and polymers can replicate faithfully is a central problem in cell biology (1, 2). In eukaryotic cell division, the mitotic spindle plays an important role in chromosome segregation (3), and there are alternate pathways for its formation (4). The most common pathway for animal eukaryotic cells is based on centrosomes and kinetochores (5, 6). These specialized structures directly force the pattern to form. However, there is an alternative chromosome-directed spindle pattern formation pathway. In this pathway, spindle patterns self-assemble without centrosomes and chromosome pairs attached by kinetochores. The details of this alternative pathway are not well understood. The pathway for Drosophila oocytes (7, 8) involves chromosome pairs attached by kinetochores but does not involve complete centrosomes. Heald et al. (9) performed a series of experiments to explore the chromosome-directed pathway. These experiments used Xenopus meiotic extracts with magnetic micrometer-sized chromatin-coated beads. They showed that self-organized spindle patterns can form in the absence of chromosomes, kinetochores, and centrosomes. Here we introduce a previously undescribed biophysical model that explains the dynamics involved in these experiments and introduces constraints on
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the possible mechanisms involved. The model can dynamically produce self-organized spindle patterns over a biologically reasonable range of parameter values. Our quantitative model is physically accurate but conceptually simple and contains only a small number of functional elements. The model results can guide further experimentation and limit hypotheses for microtubule and motor dynamic behavior. In particular, our results severely constrain models of microtubule nucleation and subsequent growth (1, 10, 11). They also limit motor force and processivity values needed to form the spindles. Spindle formation pathways feature certain characteristics: The microtubules must grow, the poles must form, the equator must form, and the chromosomes must attach to the spindle. These events occur at different times in different pathways, and this time may suggest different mechanisms for spindle formation in different pathways. In the centrosome-directed pathway, the centrosomes first move to opposite sides of the nucleus, and in this way, fix the position of the spindle poles. The centrosomes also nucleate microtubules, which grow to form two asters. The growing asters intersect, forming a spindle equator. The growing asters also contact and capture the chromosomes. In the chromosome-directed pathway, however, asters do not form. Instead, the microtubules grow randomly over the bundle of chromosomes. After the microtubules attach to the chromosomes, the microtubule minus ends are pushed away from the chromosomes. Finally, the minus ends assemble into two opposite poles. Different biophysical models can examine particular mechanisms in isolation and, thereby, can shed light on which mechanisms are operative in a given biological environment. Ne´de´lec and colleagues (12, 13) have done calculations based on a model that gives asters a central role. In one version of their model, asters formed in bulk and combined until there were only two. In another version, their model was initialized with two asters, as if they had been nucleated by two centrosomes. These models were successful at forming spindles. They always involved the initial creation of at least two independent asters, followed by their coalescence into a spindle. This approach is consistent with the sequence of events in the centrosome-directed pathway but not in the chromosome-directed pathway. Heald et al. (9) observed that in vitro a cluster of microtubules first appeared over the chromatin-covered beads and only then did the spindle poles form (14). Our model describes the sequence of events that takes place in this spindle formation pathway. It is likely, however, that a full treatment of in vivo cellular spindle formation mechanisms may involve elements of more than one model. To explain the results of their experiments, Heald et al. proposed a heuristic interpretation involving microtubules and three types of molecular motors (9, 15). One type was a simple kinesin-like motor (assumed in our model to be chromokinesin) with a plus-directed head and a foot that bound to the surface of chromatin. Another type was a dynein-like motor complex Conflict of interest statement: No conflicts declared. ‡To
whom correspondence should be addressed. E-mail:
[email protected].
© 2006 by The National Academy of Sciences of the USA
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(assumed in our model to be an oligomer of cytoplasmic dynein) with two minus-directed heads. This motor is the simplest dynein complex that provided the required microtubule cross-linking functionality. A third type was a kinesin-like complex with two plus-directed heads (assumed in our model to be Eg5). All of these motors are assumed to be processive, so by ‘‘head’’ we mean a closely coupled pair of motor domains. The first type of motor served to move the minus ends of the microtubules away from the chromosomes. The second type pulled the minus ends together into poles. The third type formed antiparallel bundles of microtubules and also pushed the minus ends away. Although this heuristic interpretation is plausible, Heald et al. (9) did not develop a biophysical quantitative model to test their hypothesis.
Schaffner and Jose´
Fig. 1. Motors apply forces to microtubules. The kinesin motors are attached to the column of chromatin-covered beads. They act as pivots for the microtubules and apply axial forces tending to move the plus ends of the microtubules toward the column. They can be overpowered by forces generated by the dynein motors. A dynein motor applies tensile force between two microtubules. When a dynein attaches to two microtubules, there are three distinct geometrical cases depending on whether each head is on the same side of the pivot as the minus end or on the opposite side. (a and b) a shows the case in which both heads are on the same side, and b shows the case in which both heads are on opposite sides. The dynein heads tend to move toward the minus ends of the microtubules, with a speed that varies with the axial component of the tensile force. The mixed case should be clear and is not shown. As shown in c, differences in head speed quickly align the dynein motor so that the axial components of the tensile force are equal in both microtubules.
kinesin motors. The statistical distribution of these positions reflects the distribution of chromatin-covered beads in the Heald et al. (9, 15) experiments. A schematic picture of our mechanical force model is shown in Fig. 1. In Table 1 we list the biophysical parameters used in the model. The definition of the equations of motion and the calculational approach is described below. In our calculations we used experimental generic properties for kinesins and dyneins (19–21). The cytoplasm is assumed to extend well beyond the chromatin-covered beads. An attached dynein motor head can become detached either because it is overstretched or because it is not completely processive. The kinesin motors are assumed to have a sufficiently high density that a kinesin pivot exists wherever a microtubule crosses the beads’ axis. Kinesin heads are assumed to be processive (22, 23). Even if individual kinesin motors are not completely processive, if the density of kinesin motors on the beads’ surfaces is high enough, then it is likely that more than one kinesin motor will interact with each microtubule. This extra interaction would raise the effective kinesin processivity. At the same time, the small moment arm between the PNAS 兩 July 25, 2006 兩 vol. 103 兩 no. 30 兩 11167
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Results and Discussion Our biophysical model derives from this heuristic proposal but is simpler in that it contains only one kinesin-like motor that lumps together the combined effects of the Eg5 and chromokinesin motors. This lumped component reflects current uncertainties regarding the proper division of function between the two (16, 17). There are three known functions associated with the two kinds of motor. First, both the Eg5 and the chromokinesins push the minus ends of the microtubules away from the chromosomes. Second, the chromokinesins bind the plus ends of the microtubules to the chromosomes or chromatin-covered beads. Third, the Eg5 motors help sort the microtubules into antiparallel bundles. The first function is required because the minus ends of the microtubules must be pushed through the viscous cytoplasm away from the chromosomes, and the dynein motors cannot do this alone. The second function is required because the spindle must be geometrically related to the bundle of chromosomes. We do not model the third function. The fact that spindles form readily without it shows that the dynein motors alone are sufficient to sort the microtubules into poles, provided that the minus ends have been pushed away from the chromosomes and that the plus ends can exert forces on the chromosomes. This fact implies that the antiparallel sorting by the Eg5 is a redundant mechanism. Most biological systems contain such redundant mechanisms, providing greater robustness for critical processes. One of the more notable components among different biophysical models of spindle formation is their sensitivity to boundary conditions. A mitotic spindle by definition is not spherically symmetric, so there must be either a broken geometric symmetry or there must be asymmetries in the boundary conditions. In the models considered by Ne´de´lec and colleagues (12, 13), the boundary conditions are either at the outer physical boundaries of the cytoplasm or they are determined by the initial positions of the two centrosomes. Our model has central boundary conditions: The microtubules form over and are captured by a spherically asymmetric bundle of chromatin-covered beads, which we assumed to be along a line. The orientation of the spindle is then determined by this asymmetry. We use these boundary conditions to mirror the experimental constraints found in the Heald et al. (9, 15) experiments. Our biophysical model includes one kind of microtubule, two kinds of motors that walk on those microtubules, plus the cytoplasm. The first kind of motor has one foot and one plus-directed head with properties appropriate to the kinesin family of motors. The foot binds permanently to the chromatin on the beads. The second kind of motor is a complex with two heads and no feet. Each head has properties appropriate to cytoplasmic dynein (18). For brevity, we call the two motors by their family names when no confusion is likely. The microtubules differ from each other only in length, and all lengths are specified at the start of the computations. The cytoplasm is treated as a uniform viscous medium and also as a reservoir of dynein motors. The boundary conditions are the fixed positions of the
Table 1. Parameters that drive the model Parameter Microtubule length, m No. of microtubules Time increment per simulation step, sec Center bar height, m Temperature, K Microtubule angular diffusion coefficient Microtubule translation diffusion coefficient Stall force for kinesin motors, pN Stall force for dynein motors, pN Spring constant for dynein, pN兾nm Dynein relaxed length, nm Dynein head speed, m兾sec Dynein dissociation rate New dynein rate Microtubule length distribution types Parameter for length distribution
Value Varies, usually 10 Varies, usually 400 5 5 300 0.02 2.5 ⫻ 10⫺13 5 10 1 100 1–10 Varies Varies Uniform, Gaussian, exponential Varies
closely spaced motors would not generate enough torque to interfere with the pivoting action. Heald et al. (9) reported that the spindles formed in two stages. In the first stage, microtubules grew over the chromatin-covered beads and formed antiparallel bundles that extended from the bead area. These extended microtubules then aggregated into two opposite spindle poles. Our model is initialized between the two stages. Thus, we work with microtubules that are already at least partially grown and that are already anchored on the chromatin. The total number of microtubules does not vary during a model calculation, and all are placed initially with both random positive兾negative polarity and random angular orientations distributed uniformly between 0 and 2. Microtubule lengths can vary initially but are fixed during a simulation run. We do not have enough quantitative experimental data to model the initial aggregation into bundles nor the dynamic changes of microtubule lengths during the second stage. In our simulations, we found some aggregation into bundles due to motor forces, but it is not yet clear whether this is the mechanism that causes the aggregation as described by Heald et al. (9). The kinesin motors are anchored permanently with a random distribution over the surfaces of the chromatin-covered spheres at a density high enough to not be a limiting factor for model behavior. The number of dynein motors walking on the microtubules varies, and it is assumed to involve diffusive exchange with the reservoir of motors in the cytoplasm. The cytoplasm maintains this dynein motor reservoir so that the exchange rates do not vary greatly over the time period being modeled. Initially there are no dynein motors walking on the microtubules. They are dynamically attached to the microtubules by a Poisson process with a constant time rate. Another Poisson process allows attached dynein motors to detach because of having finite processivity and be returned to the cytoplasmic reservoir. The microtubules are embedded in a viscous medium subject to motor forces and thermal fluctuations, with their angular and longitudinal diffusion coefficients. The dynein head velocity is denoted by vdh. The motor stall forces are fd and fk, for dynein and kinesin, respectively. The dynein spring constant is k, its relaxed length L0, its dissociation rate ddr, and its new dynein rate ndr. The maximum number of dynein motors is nd, and the number of microtubules is nm (3, 9, 24–26). In discussing our results, it is useful to use the derived processivity parameter p ⫽ 1 ⫺ ((ddr ⫻ dx)兾vdh), with the dynein head step distance dx ⫽ 16 nm. Taking a different step distance will lead to a 11168 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0604721103
Fig. 2. Dynein processivity values affect spindle pattern morphology. The images show simulation runs after 2.5 sec with different dynein head processivities: All runs are made with 400 microtubules, all of the same length, and 1,500 new dynein motors per sec. The microtubules are shown in yellow with green minus ends and red plus ends.
different processivity. Recent results suggest that the step size may vary with load (27). We ran the program coding this model a number of times with different parameter values, and we found that a spindle pattern readily forms over a wide range of biophysical conditions. However, we found some parameter values slightly outside of those limits that prevent proper spindles from forming. In particular, we found that the processivity of the dynein motors must be high, that the dynein motors must be added to the system at a controlled rate, and the length distribution of the microtubules must all vary slightly about an average value. All of these limits and constraints have important biological ramifications. Fig. 2 shows the important dependence of the spindle pattern morphological properties as a function of the dynein processivity parameter. We find that to form geometrically ideal spindle patterns, we need to have high dynein processivity. As the processivity parameter decreases, the two spindle poles break into sets of subpoles, but the overall shape is still that of a spindle. For lower processivity values, the spindle poles become more fragmented, and the overall resemblance to a spindle pattern decreases. Even when a spindle does not form, however, there is substantial microtubule aggregation into bundles. This latter pattern also was seen by Heald et al. (9) as a first step in the spindle formation pattern, although they did not connect it to the dynein’s processivity as we found in our calculations. At the spindle poles, where the microtubule density is large, motor reattachments can be to a different microtubule. These reattachments contribute to the formation of subpoles. In real in vivo cells, there however may be additional agents that stabilize the poles (28). Next we varied the new dynein attachment rate to determine its effect on the spindle pattern formation. When ndr was small, it took a long time before there were enough motors in the system to form a spindle pattern. For high rates, sudden large forces were generated that disrupted spindle formation. We found that there was a range of ndr values that produced good spindle patterns in a reasonable amount of time. When the Schaffner and Jose´
microtubule lengths were 10 m long, and the unloaded dynein head speed was 5 m䡠sec⫺1, the mean traversal time for an unloaded dynein motor was ⬇1 sec. We would expect then that the minimum time to form a spindle would be on the order of 1 sec. Indeed this formation happens, because Fig. 3 shows an ndr value of 1,650 per sec. However, Fig. 3 also shows that such a rapid introduction of dynein motors puts severe strains on the kinesin anchors on the chromatin column. A previously undescribed instability, due to the force competition between the plus and minus motors, develops. This instability puts an upper limit on the dynein rate values to form a spindle pattern. This result leads to the important conclusion that by changing the dynein production rate one can stop spindle formation completely. Fig. 1 shows how such forces imbalances can be generated. The ndr parameter controls how quickly minus-directed motors are added to the system. For a large range of other Schaffner and Jose´
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Fig. 3. Attaching dynein motors to microtubules too fast causes a monaster formation. Results are from two simulation runs in which the only difference is a 5% variation in the new dynein microtubule attachment rate. Processivity is high (0.985). (a–d) The new dynein rate is 1,650 per sec. (e–h) The new dynein rate is 1,700 per sec. t ⫽ 0 (a and e), 0.833 (b and f ), 1.667 (c and g), and 2.5 (d and h) sec.
biophysical parameter values, there is a band of ndr values above which an instability occurs and below which it does not. As the processivity increases, this band narrows. For example, Fig. 3 shows two identical systems that differed only in their ndr values by 5% but that lead to entirely different resulting morphologies. The stability boundary depends also on rdk, the ratio of dynein and kinesin stall forces. An experimental value for rdk is not yet precisely known (22, 29, 30). When rdk is large, the boundary occurs for relatively low values of the new dynein rate. When rdk is 2, as we have used here, the boundary is found at a substantially higher ndr value. Note that the boundary still exists when the dynein and kinesin stall forces are equal but only for high ndr values and, consequently, for a large number of active dynein motors. In the in vitro experiments of Heald et al. (9), as well as in actual eukaryotic cells, spindle formation time is on the order of 10 min rather than 1 sec. Thus, there may be another ratelimiting mechanism, such as a lower ndr or dynein head speed that we have not yet included in our model. We have varied dynein head speeds in the range 1–10 m兾sec and found that spindle formation time varies but morphology does not. Similarly, lower ndr slows spindle formation but does not prevent its formation. The simulation results shown in Figs. 2 and 3 involve sets of microtubules that are all of the same length that do not vary during the course of the simulations. This restriction is greatly at variance with the dynamic instability displayed in natural spindle formation, where each microtubule passes through a series of catastrophes and rescues states. Whether a particular microtubule grows or shrinks over the entire course of the spindle formation depends on the detailed balance between catastrophe and rescue. This balance is not known specifically for the Heald et al. (9) experiments, so we did not include full dynamic instability in the model, which also would have made it more computationally involved. Further, trying to include dynamic instability in the model would have critically depended on parameters that have not yet been measured. Instead, we allowed the microtubules to have a distribution of fixed lengths. To see why we think this is a good first modeling step, it is convenient to closely look at dynamic instability modeling. Consider an ensemble of a system containing microtubules, which each at any given time are either growing or shrinking. The probability density at time t to have growing microtubules with length x is p⫹ (x, t). Similarly, p⫺ (x, t) will be the probability density at time t of shrinking microtubules that have length x. Then the integral N(t) ⫽ 兰⬁0 dx{p⫹ (x, t) ⫹ p⫺ (x, t), gives the ensemble averaged mean number N(t) of microtubules. A microtubule is assumed to grow at a rate v⫹ and to shrinks at a rate v⫺. A growing microtubule has a probability rate f⫹⫺ of converting to a shrinking microtubule, and a shrinking microtubule has a probability rate f⫺⫹ of converting to a growing microtubule. The time evolution of the probability distributions is governed by the balance equations ⭸tp⫹ ⫽ ⫺⭸x(v⫹p⫹) ⫺ f⫹⫺p⫹ ⫹ f⫺⫹p⫺ and ⭸tp⫺ ⫽ ⭸x(v⫺p⫺) ⫹ f⫹⫺p⫹ ⫹ f⫺⫹p⫺. These equations were first applied to microtubules by Dogterom and Leibler (10). Shrinking microtubules that are very short will disappear completely before having a chance to convert to growing microtubules. Very short growing microtubules have to come from nucleation, and the equation in the limit of extremely small microtubules becomes ⭸tN ⫽ (p⫹v⫹ ⫺ p⫺v⫺)兩x30, where N is the total number of microtubules. The distance between spindle poles during mitosis is found experimentally to change only very slowly, despite the rapid changes of length changes of individual microtubules. For this reason we look for stationary solutions to this system of equations. As a first approximation, we can assume that the four parameters in the equations are constant. Although these assumptions are unlikely to be true in a real biological system (10), the results from this simple case shows some of the
salient features of the dynamics of the system. In particular, if the quantity g ⫽ v⫹f⫺⫹ ⫺ v⫺f⫹⫺ is positive, growth dominates over shrinkage. When g is zero, growth and shrinkage balance each other on the average. Because the pole-to-pole distance does not change greatly after a spindle forms, the effects of growth and shrinkage must almost exactly balance. The only nonzero stationary solution with bounded length occurs when g is negative, that is, shrinkage dominates over growth everywhere. The stationary solution is an exponential distribution of microtubule lengths, and it requires continuous nucleation of new growing microtubules to maintain it. To get other kinds of stationary distributions, it is required that the equation parameters not be constant. One simple way to have this situation is to have g varying smoothly with x, the distance from the spindle equator toward the poles. If g(x) ⫽ (⫺a(x ⫺ x0), for a positive constant a, and a characteristic length x0, then long microtubules longer than x0 will experience net shrinkage and microtubules shorter than x0 will experience net growth. The constant a is the strength of the negative feedback. Numerical simulations indicate that if the feedback is large enough, then there will be a stationary distribution with a Gaussian distribution of lengths ⬇x0. This kind of feedback could be generated, for example, by having concentration gradients of growth-promoting elements regarding the chromosomes. Another way would be to have microtubule growth be stimulated by tension or suppressed by force compression. Microtubules that were too long to meet at a pole would be placed under compression and microtubules that were too short would be placed under tension. Because there is inadequate data to accurately model dynamic instability in a system as complex as a mitotic spindle and, in particular, as it applies to the Heald et al. (9) experiment, we decided to model a probabilistic distribution of fixed microtubule lengths. Our model allows both exponential and Gaussian length distributions, to match the two major classes of stationary distributions that are derived by using the balance equations. Fig. 4 shows the effect on spindle formation patterns when we have a distribution of microtubule lengths. A spindle will still form, even with small variations in microtubule lengths, but large variations produce distorted spindle poles or may even prevent spindle formation entirely. In particular, having a large number of short microtubules seems to prevent spindle formation in our biophysical model. This phenomenon admits two primary possibilities. First, if our model is correct, distributions featuring many small microtubules are ruled out. Second, our model could be missing a functional element that would counteract the disruptive effects of having short microtubules. Which of these two cases is true can be determined and tested by experiment. If future experiments show that there are only a few short microtubules, then our model makes the further prediction that there must be some kind of negative feedback mechanism regarding a characteristic microtubule length. The calculations we have performed so far do not favor any particular kind of negative feedback over any other. On the other hand, if short microtubules are found to be common in a developing spindle, then we must add another functional element to our model. There are many candidates if one is required. The most obvious candidate is to replace our single plus-directed motor by two motors, chromokinesin and Eg5. The cross-linking function of the Eg5 can be used to make the connection between the plus ends of the microtubules and the chromosomes more robust. There is already evidence for this action in that when Heald et al. (9) deactivated Eg5, spindles did not form. We also see evidence for this action in the performance of our model. When our model fails to produce a spindle, it is usually because the plus ends of the microtubules have been driven back from the kinesin pivot points. This effect can be seen in Figs. 3h 11170 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0604721103
Fig. 4. Having different microtubule length distributions affects spindle pattern morphology. Results of three simulation runs after 5 sec. In all cases, there were 400 microtubules, and dynein motors were added at a rate of 400 per sec. Processivity was 0.970. (a, c, and e) Images show the microtubule distribution. (b, d, and f) Images show the dynein motor distribution. For a and b, microtubule lengths had a Gaussian distribution with a mean value of 10 m and variance ⫽ 1 m. In c and d, ⫽ 2 m. In e and f, microtubule lengths had an exponential distribution, with a mean length of 2.5 m.
and 4e. The resulting pattern is a monaster shape rather than a bipolar spindle. Our model would then predict that one vital function of the plus-directed motors is to prevent the microtubule plus ends from being pushed far back from the kinesin pivot points on the chromatin-covered beads. Methods In our present model, microtubules are rigid cylinders that move in the x–y plane through a viscous medium. Microtubules do not interact directly but interact via the applied forces from the molecular motors in the presence of thermal fluctuations. Each mechanical degree of freedom ‘‘x’’ satisfies the finite temperature equation of motion described by the Langevin equation: ˙x⫽ DF ⫹ 冑2D. Here, is a Gaussian random variable that represents the temperature fluctuations in the degree of freedom ‘‘x,’’ with mean value 具(t)典 ⫽ 0 and covariance 具(t)(t ⫹ ␣)典 ⫽ ␦(␣). D ⫽ (1兾) denotes the diffusion coefficient of degree of freedom ‘‘x,’’ F is the sum of all of the applied forces acting on x, and is the cytoplasm viscosity.  ⫽ 1兾kT, with T the temperature in Kelvin and k Boltzmann’s constant. To solve this large number of equations, we do numerical computations where we approximate the differential Langevin equations by a discrete time Schaffner and Jose´
冉
␣2 P共␣, t兲 ⫽ exp ⫺ 22 冑2 1
冊
冘冘 M
and
⫽
冑2Dh.
The microtubule motion is constrained to the x–y plane and described by two degrees of freedom (s, ), where is the microtubule angle about the pivot and s is the signed distance between the center of the microtubule and the pivot point. The model contains M microtubules, equal to the number of kinesin pivots, and N dynein motors. There are 2M discrete Langevin stochastic equations, one for s and another for for each i motor:
␦i ⫽ D共si兲Fi h ⫹ 冑2D共si兲␣i1 共si兲
and
␦si ⫽ DsFi h ⫹ 冑2Ds␣i2. To integrate these discrete equations, we used a second-order stochastic Runge–Kutta approximation (31). The angular diffusion coefficient of pivoting the ith microtubule with length L is: D共si兲 ⫽
of indices of all motors attached between microtubule i and j. Then,
Dr . 12si2 1⫹ 2 D
F i ⫽
j⫽1 k僆Ij
冘冘 M
Fki
and
F si ⫽
F ski,
j⫽1 k僆I ij
where Fki is the torque of motor k on microtubule i and Fski is the longitudinal force of motor k on microtubule i. To calculate these forces, define nជ ik to be the vector from the pivot point for microtubule i to the attachment point for motor k. Also, let m ជ kji be the vector from the point where motor k attaches to microtubule j to the point where it attaches to microtubule i. Finally, ជ (vជ ) ⫽ 0ជ if 兩vជ 兩 ⱕ L0, and K ជ (vជ ) ⫽ ⫺(vជ 兾兩vជ 兩)kD for any vector vជ , set K (兩vជ 兩 ⫺ L0) otherwise. Here, kD is the spring constant for a dynein motor, and L0 is the relaxed length of the motor. Then Fki ⫽ (nជ ik ជ (m ជ (m ⫻K ជ kji))䡠zˆ, and Fski ⫽ (nជ ik ⫻ K ជ kji))兾兩nជ ik兩. We programmed the model by building two computer programs: motor and vizmotor. motor is written in Fortran 90 and performs the simulations. vizmotor is written in C⫹⫹. It reads output files generated by motor, performs data analysis, and generates graphical displays of the results. motor can be run on either Unix or Microsoft (Redmond, WA) Windows. vizmotor depends on the user interface of Microsoft Windows. Some further data analyses were performed by using Mathematica.
Here, Dr is the rotational diffusion coefficient, and Ds is the longitudinal diffusion coefficient for any microtubule. D(si) depends on the pivot position because the microtubule does not always rotate about the center of pressure. The forces Fi and Fsi represent the sums of all motor forces on the ith microtubule. Fi is the torque. Consider Iij to be the set
We thank J.-F. Chauwin and F. Gibbons for collaboration in the initial stages of the development of the work described in this article. J. Banavar, M. Desposito, R. Heald, R. Li, D. Sharp, F. Ne´de´lec, and K. Burbank provided helpful comments and contributions to the work presented here. Partial research support was provided by the National Science Foundation and the Center for the Interdisciplinary Research on Complex Systems.
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PNAS 兩 July 25, 2006 兩 vol. 103 兩 no. 30 兩 11171
CELL BIOLOGY
representation with a finite time step h, ␦x ⫽ DFh ⫹ 公2D␣, where ␣ is a Gaussian random variable with probability distribution:
