Stochastic force generation by small ensembles of myosinII motors
Forces in the actin cytoskeleton are generated by small groups of non-processive myosinII motors for which stochastic effects are highly relevant. Using a crossbridge model with the assumptions of fast powerstroke kinetics and equal load sharing between equivalent states, we derive a one-step master equation for the activity of a finite-sized ensemble of mechanically coupled myosinII motors. For constant external load, this approach yields analytical results for duty ratio and force-velocity relation as a function of ensemble size. We find that stochastic effects cannot be neglected for ensemble sizes below . The one-step master equation can be used also for efficient computer simulations with linear elastic external load and reveals the sequence of build-up of force and ensemble rupture that is characteristic for reconstituted actomyosin contractility.
Generation of motion and force by ATP-powered molecular motors is a hallmark of living systems Howard (1997). In their cellular environment, molecular motors usually operate in groups Guerin et al. (2010). A striking example is force generation in skeletal muscle, where hundreds of non-processive myosinII motors are assembled into the thick filaments of the sarcomeres. Since the pioneering work of Huxley Huxley (1957), the statistical physics of large ensembles of myosinII motors has been studied in great detail. It has been shown that in order to describe the response of skeletal muscle to varying loading conditions, it is essential that the unbinding rate of myosinII from actin is strain-dependent and decreases under load Duke (1999); Vilfan and Duke (2003). In contrast to e.g. the processive motor kinesin, this makes myosinII a catch rather than a slip bond Veigel et al. (2003); Guo and Guilford (2006) and leads to recruitment of additional crossbridges under load Piazzesi et al. (2007).
The collective activity of myosinII motor ensembles is also essential for the generation of motion and force in the actin cytoskeleton of non-muscle cells. In this case, the actin structures are far more disordered than in muscle and non-muscle myosinII is usually organized in mini-filaments comprising motors Verkhovsky et al. (1995). For such small numbers of motors, stochastic effects will become important and are indeed observed in experiments. Measurements of tension generated by myosinII motors in reconstituted assays, e.g. in three bead assays Finer et al. (1994); Veigel et al. (2003); Debold et al. (2005), active gels Mizuno et al. (2007); Silva et al. (2011) or motility assays Placais et al. (2009), reveal noisy trajectories, typically with a gradual increase of tension followed by an abrupt release, which is likely due to detachment of the whole ensemble (slip). However, a detailed and analytically tractable description for this biologically important situation is still missing.
The collective activity of mechanically coupled molecular motors has been investigated before in the framework of a generic two-state Fokker-Planck equation in which ensemble size enters into the noise intensity Jülicher and Prost (1997); Placais et al. (2009). In order to study effects of molecular details for ensembles of myosinII motors, crossbridge models originally developed for skeletal muscle can be used as a starting point Duke (1999); Vilfan and Duke (2003). Due to their complexity, these models are usually studied by computer simulations. Analytical progress has been made with a mean field approximation for large system size Hexner and Kafri (2009). Exploiting a separation of time scales in the myosinII cycle and using the assumption of equal load sharing between motors in equivalent states, here we derive a one-step master equation which explicitly includes the effects of catch bonding and small system size. A one-step master equation has been introduced before for transport by finite-sized ensembles of processive motors with slip bond behavior Klumpp and Lipowsky (2005), but not for non-processive motors with catch bond behavior. Our results suggest that stochastic effects are particularly important for ensemble sizes below , which corresponds to the typical size of cytoskeletal mini-filaments.
Model. We model the myosinII cycle by three discrete mechano-chemical states. The cycle is shown schematically in Fig. 1a. To allow for comparison with earlier work, transition rates and most other molecular parameters are taken from Refs. Duke (1999); Vilfan and Duke (2003). In practice, they will depend on ATP concentration and the exact type of myosinII Veigel et al. (2003); Silva et al. (2011). In the unbound state , the motor-head is loaded with ADP and P and the lever-arm is in its primed conformation. The motor then reversibly transitions to the weakly bound state with forward rate and reverse rate . After release of P, the lever-arm swings to the stretched conformation and the motor enters the post-powerstroke state . The transition rates between the two bound states are relatively high, with . Replacing ADP by ATP, unbinding from the substrate and hydrolysis of ATP brings the motor back to the unbound state . This last step is irreversible, with rate . Most important in our context, both powerstroke and unbinding depend on load. The powerstroke moves the lever-arm forward by and strains the elastic neck-linker. Unbinding from requires further movement of the lever-arm, thus making unbinding slower under load (catch bonding).
As shown schematically in Fig. 1b, the (upper) motor-filament mechanically couples the different motors in an ensemble to each other. Due to the strain-dependence of the rates, they are also dynamically coupled. Hence, a complete description of the ensemble dynamics has to include conformational state and strain of every motor. To arrive at a tractable model, we first note that the motors pull in parallel. We next assume that all motors in the same mechano-chemical state exert the same force and hence have the same neck-linker strain. We thus arrive at the parallel cluster model (PCM) depicted in Fig. 1c, in which the state of an ensemble with motors is characterized by the number of bound motors and the number of motors in the post-powerstroke state. The number of motors in the weakly bound state follows as . In the PCM, each motor in the weakly bound state has the same strain , where the indices indicate the dependence of the motor strain on the ensemble state . The powerstroke stretches the elastic neck-linker by , so that motors in the post-powerstroke state have the strain . The strain of the weakly bound motors follows from the balance of the external load and the elastic motor forces: . Here is the spring constant of the neck-linkers. For , the force balance leads to
Thus the strain of the weakly bound motors is a state variable determined by external load and both binding and powerstroke dynamics. If all motors are in the weakly bond state (), it is positive. It can become negative if sufficiently many motors have gone through the powerstroke and if the external load is not too large. The strain of the post-powerstroke motors always stays positive and eventually drives force generation and motion.
In the PCM, the network of reactions between states is two-dimensional (see Fig. 1d). Due to slow binding and unbinding, local thermal equilibrium (LTE) is maintained for the bound states Vilfan and Duke (2003). When motors are bound, the probability that motors are in the post-powerstroke state follows the Boltzmann distribution , where is the appropriate partition sum. The energy in state is the sum of elastic energy stored in the neck-linkers, free energy bias towards the post-powerstroke state, and a possible external energy contribution . For , we have .
LTE of the bound states allows us to project the -axis onto the -axis, thus arriving at a one-dimensional reaction scheme with index as shown in Fig. 1e. Then the probability that motors are bound at time obeys the one-step master equation
The probability to find an ensemble in state is . The forward rate is because free motors can bind. Unbinding is possible from states and so that the reverse rate for given and is . Averaging over gives . The off-rate from state depends on the applied load as , where . The strain-dependence of makes myosinII a catch bond. With these prescriptions, the one-step master equation Eq. (2) is fully specified for the case of constant external load, . If the external load depends on the position of the ensemble, like in the case of linear elastic loading, Eq. (2) has to be solved together with additional prescriptions for ensemble movement (see below).
Binding dynamics for constant load. Mathematically, the reduction to Eq. (2) is a dramatic simplification, because many general results are known for one-step master equations van Kampen (1992). The stationary distribution is
Fig. 2a plots the average number of bound motors, , as function of for different ensemble sizes (lines). The increase of is due to the catch bond character of the post-powerstroke state. With increasing load, decreases to , so that is small because is strongly biased towards large and . For skeletal muscle, the recruitment of additional crossbridges under load has been observed experimentally Piazzesi et al. (2007) as predicted by computer simulations Duke (1999). Here it follows in a relatively simple way from analytical considerations. In order to validate the PCM leading to Eq. (2), in Fig. 2a we also show results of computer simulations which incorporate an individual strain value for each motor (symbols). The agreement is very good, except at very small load, where differences in the strain values between different motors reduces unbinding, an effect which is less relevant under larger load.
Next we discuss the effect of system size . In general, smaller ensembles are more likely to detach as a whole. The mean first passage time for ensemble detachment after binding of the first motor is
It is a polynomial of order in the ratio of binding to unbinding rate and increases exponentially with ensemble size. Once the ensemble has detached, on average it takes the time to rebind. We define the duty ratio of an ensemble as
Fig. 2b plots as function of for different (lines). Because increases and decreases with , the duty ratio increases quickly with and reaches unity for ensemble sizes around . With increasing force, increases faster because of the increasing . Again the agreement with the simulation of the crossbridge model with individual motor strains (symbols) is rather good except at very small force. Stochastic effects are expected to be important for duty ratios below unity, i.e. below ensemble sizes around . This implies that myosin mini-filaments in the cytoskeleton are typically at the verge of stochastic instability.
Ensemble movement. We now consider the spatial coordination schematically depicted in Fig. 1b, that is, we assume an immobile substrate over which an ensemble moves to the right. The PCM assumes that all bound motor-heads are at the same position, which we denote by the coordinate . The anchors of the motors in the (upper) motor-filament are located at the common position . Note that whereas increases to the right in Fig. 1b, external load and strain of the motors are defined in the opposite direction. For the case of constant external load, , is a variable which is slaved to the binding dynamics. When the ensemble works against a linear external load, , the value of enters the force balance and hence feeds back into the system state. In addition, here one has to include an external elastic energy , where is the external spring constant.
Although more complicated assumptions might be possible, here we make the following simple assumptions for the dynamics of within the PCM. Starting with a state , we assume that a motor binds to the substrate with vanishing strain at the position just below its anchor point. Binding of a new motor thus changes the average position of the bound motor-heads by . Unbinding does not change , because all bound motor-heads are at the same position. The powerstroke does not change either, because it does not affect the positions of the motor-heads. However, is affected by binding and unbinding as well as by the powerstroke via the force balance, so that the position of the motor-filament is affected by all these transitions. When an ensemble detaches completely from the substrate, the motor-heads relax to the position of the anchors. The detached ensemble then moves backwards with velocity (slip), where is the effective mobility of the motor-filament.
With these additional prescriptions, the rates defined for Eq. (2) can now be used to investigate the details of the stochastic movement of the motor ensemble for arbitrary laws for the external load. To simulate stochastic trajectories, we use the Gillespie algorithm Gillespie (1976). After every change of , and are updated to calculate the average strain of the weakly bound motors, , and the transition rates and . In case of binding, we change by . Fig. 3a shows a stochastic trajectory of an ensemble working against constant load. The lower panel shows the number of bound motors , the upper panel the average head position as function of time. When bound, the ensemble moves forward with fluctuations around a steady state velocity. A slip leads to backsteps of average size . Fig. 3b shows a trajectory for an ensemble working against a linear elastic load. The ensemble is slowed down by the load building up by the forward motion. An increasing load stabilizes the ensemble because increases. However, the very small ensemble frequently detaches before reaching the stall force. Detachment leads to a noisy trajectory in which the load fluctuates around an effective stall force. This type of trajectories, with gradual buildup and quick release of tension, resembles those experimentally observed in three bead assays Finer et al. (1994); Veigel et al. (2003); Debold et al. (2005), active gels Mizuno et al. (2007); Silva et al. (2011) and motility assays Placais et al. (2009).
Force-velocity relation for constant load. In state , one can identify the ensemble velocity with (the ensemble only moves to the right when the strain defined to the left is negative). The average stationary velocity of a bound ensemble is
with from Eq. (3). This is the force-velocity relation of the bound ensemble at constant load. Fig. 4a plots as function of the external load per motor for different . With increasing load, the velocity decreases. The upward convex shape of is due to the increase of with , which allows the ensemble to resist larger forces. For small ensembles for , bound velocity and also the stall force increase with increasing . Above , the force-velocity curve is independent of . This confirms our conclusion from the duty ratio that stochastic effects cannot be neglected up to a system size of (compare Fig. 2b).
Assuming that the stationary velocity is established quickly after binding to a substrate, the walk-length of a motor-filament in one attachment event is given by . Although the bound velocity decreases, increases with because the detachment time increases strongly. Only upon passing the stall force, the walk-length drops to negative values. Comparison with numerical solutions of the master equation (not shown) reveals that is a good approximation except for very small values of . Because the sliding velocity is negative, , the effective velocity is reduced by the occurrence of slip events:
Fig. 4b plots the effective velocity as function of the external load per motor. Because the duty-ratio increases and the rebinding time decreases with , i.e., detachment is less frequent and backsteps are smaller, the velocity at small now increases with . In addition, detachment of small ensembles leads to a faster decrease of under load and a smaller stall force. Moreover, detachment leads to large fluctuations of at the effective stall force: instead of being stationary, the ensemble alternates between slow forward motion when bound and fast backward slipping when detached. Above the threshold of , where the duty ratio is close to unity, the effective velocity is identical to the bound velocity.
The insets in Fig. 4 compare the analytical results using the PCM to the computer simulations without PCM for and as function of . The agreement is rather good. Due to the molecular friction resulting from differences in strain, the bound velocity at vanishing load now decreases with and the curvature of the force-velocity relation is less pronounced. The stall force and the role of ensemble size for stochastic effects is predicted well.
Discussion. In this Letter, we have derived a mathematically tractable model for the collective behavior of small ensembles of myosinII motors as a function of system size. Our main assumption, the parallel cluster model (PCM) for the load sharing, was validated by computer simulations of a cross-bridge model with individual motor strains. These assumptions decrease the disorder in the motor strains, so that the model cannot describe powerstroke synchronization through load as it has been done before with a detailed model for skeletal muscle Duke (1999). However, our model makes accurate predictions for central quantities such as duty ratio and force-velocity relation as a function of ensemble size. For processive motors, the strains of the motors are homogenized because fast moving motors are slowed down by the increasing load. For the non-processive motors studied here, this mechanism cannot operate. However, here the differences in the strain of the bound motors are reduced by the small duty ratio, thereby making the PCM a reasonable assumption for our purposes. Due to its computational simplicity, in the future the approach introduced here can be used for studies of the intriguing interplay between actin filaments and small ensembles of myosin II motors in the actin cytoskeleton of non-muscle cells and reconstituted actomyosin systems.
Acknowledgements.We thank Philipp Albert for helpful discussion. TE and USS were supported by a Frontier-grant from Heidelberg University. USS is a member of the Heidelberg cluster of excellence CellNetworks and was supported through the MechanoSys-grant from the BMBF.
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