Stochastic dynamics and mechanosensitivity of myosin II minifilaments

Stochastic dynamics and mechanosensitivity of myosin II minifilaments

Abstract

Tissue cells are in a state of permanent mechanical tension that is maintained mainly by myosin II minifilaments, which are bipolar assemblies of tens of myosin II molecular motors contracting actin networks and bundles. Here we introduce a stochastic model for myosin II minifilaments as two small myosin II motor ensembles engaging in a stochastic tug-of-war. Each of the two ensembles is described by the parallel cluster model that allows us to use exact stochastic simulations and at the same time to keep important molecular details of the myosin II cross-bridge cycle. Our simulation and analytical results reveal a strong dependence of myosin II minifilament dynamics on environmental stiffness that is reminiscent of the cellular response to substrate stiffness. For small stiffness, minifilaments form transient crosslinks exerting short spikes of force with negligible mean. For large stiffness, minifilaments form near permanent crosslinks exerting a mean force which hardly depends on environmental elasticity. This functional switch arises because dissociation after the power stroke is suppressed by force (catch bonding) and because ensembles can no longer perform the power stroke at large forces. Symmetric myosin II minifilaments perform a random walk with an effective diffusion constant which decreases with increasing ensemble size, as demonstrated for rigid substrates with an analytical treatment.
Correspondence: Ulrich.Schwarz@bioquant.uni-heidelberg.de

I Introduction

Cytoskeletal molecular motors are a large class of proteins that generate movement and force in biological cells by cycling between states bound and unbound from a cytoskeletal filament Howard (1997); Schliwa and Woehlke (2003). In general, they can be classified as processive or non-processive motors. Processive motors like kinesin, dynein or myosin V have a duty ratio (fraction of time of the motor cycle spent on the filament) close to unity and therefore are particularly suited for persistent transport of cellular cargo, such as vesicles, small organelles or viruses. Using small groups of processive motors increases the walk length and the efficiency of transport compared to the single motor Holzbaur and Goldman (2010). A theoretical treatment with a one-step master equation showed that the effective unbinding rate decreases exponentially with the size of the motor ensemble Klumpp and Lipowsky (2005). Moreover groups of motors can also work against larger load than the single motor Koster et al. (2003). If motors of different directionality on the substrate are attached to the same cargo, bidirectional movement can ensue Müller et al. (2008), as often observed in cargo transport. A similar tug-of-war setup has been used earlier to explain mitotic spindle oscillations Grill et al. (2005). Non-processive motors such as myosin II have a duty ratio significantly smaller than unity. Therefore, non-processive motors have to operate in groups in order to generate appreciable levels of force. Similar to processive motors, the duty ratio of a group of non-processive motors increases with the size of the group and can become large enough that the group effectively behaves like a processive motor. This is certainly true for the sarcomeres in skeletal muscle, where typically hundreds of myosin II work together as one group. Combining structural investigations of skeletal muscle with modeling has led to the swinging cross-bridge model for single myosin II Huxley (1957); Huxley and Simmons (1971). A statistical treatment then has allowed to accurately model the dynamics of the motor ensemble in muscle sarcomeres Duke (1999); Vilfan and Duke (2003).

Groups of myosin II motors also play a crucial role for the mechanics and adhesion of non-muscle tissue cells. Cytoskeletal myosin II assembles into bipolar minifilaments consisting of - motors Verkhovsky and Borisy (1993). They interact with an actin cytoskeleton which is much less ordered than in muscle, mainly in the actin cortex as well as in the contractile actin networks and bundles associated with cell adhesion and migration Vicente-Manzanares et al. (2009). Recently it has been shown that the activity of myosin II minifilaments contributes to the sorting of actin filament orientation because of the asymmetric elasticity of actin filaments Lenz et al. (2012); Murrell and Gardel (2012). The myosin II based forces generated in the actin cytoskeleton are transmitted to the extracellular environment via adhesion sites, which have been shown to harbor different mechanosensitive processes Geiger et al. (2009); Schwarz and Gardel (2012). In particular, mature focal adhesions are often connected to actin stress fibers consisting of parallel bundles of actin filaments with alternating polarity enabling myosin II minifilaments to contract the bundles and thus mechanically load the adhesion sites. To apply these forces effectively, the extracellular matrix underlying the adhesion sites must not be too soft. Therefore, cells are sensitive to the elasticity of the substrate and adhere preferentially to stiffer substrates Discher et al. (2005). If the environment is much stiffer than the cell, it essentially deforms itself and becomes insensitive to the environmental stiffness Schwarz et al. (2006). Therefore cellular stiffness sets the scale for the sensitivity of rigidity sensing Solon et al. (2007). Due to the complex interplay of many components in a cell, it is difficult to identify the exact contribution of myosin II to the rigidity response of cells. One promising experimental route is the reconstruction of in vitro systems of motors and filaments Surrey et al. (2001); Mizuno et al. (2007); Bendix et al. (2008); Soares e Silva et al. (2011); Thoresen et al. (2011, 2013); Murrell and Gardel (2012); Shah and Keren (2014), which in the future might allow us to probe these relations in more quantitative detail.

With the focus on the description of large assemblies of myosin II motors in the muscle sarcomere, theoretical progress has been made mainly through mean-field models Huxley (1957); Leibler and Huse (1993); Vilfan et al. (1999) or computer simulations Duke (1999, 2000); Günther and Kruse (2007). For ensembles consisting of a large number of motors, details about internal motor states are less important and experimentally accessible. Instead, collective quantities such as velocity, walk length and number of bound motor are of large interest. For example, generic two-state ratchet models have been used to study the behavior of mechanically coupled motors Jülicher et al. (1997); Plaçais et al. (2009); Guérin et al. (2010). Here we aim at understanding minifilaments with few myosin II molecules for which molecular details and stochastic effects are expected to be more important. In this context, cross-bridge models are appropriate, which have been studied before mainly with computer simulations Duke (1999, 2000); Walcott et al. (2012). However, this approach is numerically costly, in particular for extensions to systems with multiple minifilaments. Recently the parallel cluster model (PCM) based on the cross-bridge cycle has been introduced as an efficient yet detailed model for stochastic effects in small myosin II ensembles Erdmann and Schwarz (2012); Erdmann et al. (2013).

In this manuscript, we extend the PCM to myosin II minifilaments by modeling them as two ensembles of myosin II motors working against each other by walking along two actin tracks with opposing polarity. This situation can be considered as a tug-of-war of the two ensembles of non-processive motors, in analogy to a tug-of-war of processive motors Müller et al. (2008); Grill et al. (2005). In contrast to those studies, however, we do not use a phenomenological force-velocity relation, but rather a cross-bridge model to explicitly include the molecular details of the motor cycle of myosin II. In particular, we account for the catch bond character of myosin II unbinding (dissociation rate decreases under load, in contrast to the classical case of a slip bond) and for the detailed kinetics of the power stroke. From our model definition, it becomes clear that the mechanical situation in bipolar myosin II minifilaments is very complex, with an effective spring constant that depends on internal mechanics, external mechanics and the exact state of the motor ensembles. Our main result is that myosin II minifilaments show a kind of mechanosensitivity that is reminiscent of the way cells respond to environmental stiffness. We show that this effect not only results from the molecular catch-bonding property, but also from the inability to perform the power stroke in a stiff environment with sufficiently large force. We also find that catch-bonding of myosin II on stiff substrates leads to frequent switches of direction of the ensemble movement and therefore to an effective diffusion constant which decreases with increasing ensemble size, in marked contrast to a tug-of-war of processive motors with slip bonds.

Ii Model

In the parallel cluster model (PCM), individual myosin II motors are described by a cross-bridge model with three discrete states and stochastic transitions between them Erdmann and Schwarz (2012); Erdmann et al. (2013). Here we generalize this model for myosin II minifilaments and discuss it with parameter values originally introduced for modeling skeletal muscle Duke (2000); Vilfan and Duke (2003). For cytoskeletal myosin II, these values depend on the exact isoform one is considering. The parameter values used here result in a duty ratio of , which lies in the range of duty ratios reported for cytoskeletal myosin II B Kovács et al. (2003); Rosenfeld et al. (2003). As shown schematically in Fig. 1 (a), a motor comprises three mechanical elements. The motor head binds to the substrate and contains the ATP hydrolysis site, which binds ATP or its hydrolysis products ADP and P. The rigid lever arm is hinged to the motor head and alternates between stretched and primed conformation, thus amplifying conformational changes in the motor head. The linear elastic neck linker with spring constant anchors the lever arm to the rigid motor filament. In the unbound (ub) state, the motor head is loaded with ADP and P and the lever arm is primed. The motor reversibly transitions to the weakly-bound (wb) state with on-rate and off-rate . With release of P, the lever arm swings to the stretched conformation and the motor enters the post-power-stroke (pps) state. The power stroke is reversible with forward rate and reverse rate but is driven towards the pps state by the free energy bias . This energy is stored in the primed conformation of the lever arm as part of the energy released in ATP hydrolysis. Replacing ADP by ATP, unbinding and ATP hydrolysis brings myosin II from the pps to the ub state, thus completing the motor cycle. These events are subsumed in a single reaction with unloaded rate which is assumed to be irreversible due of the hydrolysis of ATP. Myosin II dynamics is characterized by the load dependence of power stroke and unbinding from pps state: the power stroke moves the lever arm by the power-stroke length ; unbinding from the pps state requires an additional movement of the lever arm by a distance . Thus, both reactions become slower under load. The reduced rate of unbinding under load makes the pps state of myosin II a catch bond rather than a slip bond in the range of forces considered here.

Figure 1: (a) Mechanical setup and hydrolysis cycle of myosin II. In the unbound (ub) state, the lever arm is primed and the neck linker has vanishing strain. In the weakly-bound (wb) state the motor head is bound to actin. The lever arm is primed but the neck linker generally has non-zero strain. In the post-power-stroke (pps) state the lever arm is stretched. Unbinding from pps state is the only irreversible transition because of the hydrolysis of ATP. (b) Bipolar minifilament with motors in the ensemble moving to the right and motors in the ensemble moving to the left. The state of the ensemble is described by the number of bound motors and the number of motors in the post-power-stroke state. The configuration in (b) corresponds to in the ensemble and in the ensemble. The displacement of the ensembles on the actin filaments is denoted by and . Bound motors within each ensemble are arranged in parallel; the two ensembles are arranged in series. (c) The parallel cluster model (PCM) applied to a bipolar minifilament treats the motor ensembles as two adhesions clusters of parallel bonds coupled in series with external springs. All bound motors in equivalent mechano-chemical states have the same strain. The strain of weakly-bound (wb) motors is denoted by and . The power stroke shortens the rest length of the neck linker by the power-stroke length so that post-power-stroke (pps) motors have the strain .

The arrangement of myosin II motors in a bipolar minifilament is depicted schematically in Fig. 1 (b). The minifilament consists of two ensembles of motors working in opposite direction. The motors are anchored to the rigid motor filament joining the two ensembles. Within each ensemble, motors are arranged in parallel whereas the two ensembles operate in series. The total number of motors in the ensemble working in direction (towards the right in Fig. 1 (b)) is denoted by ; the number of motors in the ensemble working in direction (towards the left in Fig. 1 (b)) is denoted by . The ensembles move on actin filaments of opposite polarity which are attached to linear elastic elements with spring constant and to represent the effective elasticity of the environment. The opposite polarity allows the minifilament to slide the actin filaments relative to each other and thereby to stretch the springs. The first approximation of the PCM is to assume that motors in equivalent mechano-chemical states exert equal forces (equal load sharing). This mean-field approximation is justified by the small duty ratio of non-processive motors. The elongation or strain of wb motors in and ensemble is denoted by and , respectively, so that wb motors exert the force . The strain is positive, when the neck linker is stretched against the moving direction of the ensemble (inwards in Fig. 1 (b)). With the assumption of equal load sharing, motor ensembles are mechanically equivalent to adhesion clusters of parallel bonds as depicted in Fig. 1 (c) Erdmann and Schwarz (2004). In contrast to the adhesion cluster, however, the rest length of the bond is not fixed, but is reduced by a length due to the power stroke. Thus, motors in the pps state have the strain and exert the force . The strain of wb as well as pps motors is determined by the offset between the bound motor head on the substrate and the anchor in the motor filament. The state of a minifilament is described by the numbers and of bound motors and and of pps motors in both ensembles. The number of ub motors is and that of wb motors . With pps motors and wb motors, the ensemble exerts the force . To determine the offset of wb and pps motors, we assume that a minifilament is always in mechanical equilibrium. For the arrangement of motor ensembles and external springs in series as in Fig. 1 (c) this requires that the ensemble forces balance the force exerted by the external springs. The latter is where is the effective external spring constant and the displacement of the ensemble from its origin on the actin filament. The extension of the external springs is assuming they are relaxed when and all bound motors are in the wb state with . Solving the balance of forces, , yields as function of minifilament state and contraction ,

(1)

The force as function of minifilament state and contraction then reads

(2)

where the total spring constant is defined as

(3)

and therefore varies dynamically, in contrast to . When all bound motors in the ensemble are in the wb state (), the offset is positive, , and the neck linkers are stretched against the direction. For growing contraction , the offset increases because increases. A growing number of pps motors in the ensemble increases further, because transitions to the pps state shorten the minifilament and increase . On the other hand, decreases and can become negative for a growing number of pps motors in the ensemble, although increases further. For , wb motors contribute to the external load which is carried by the pps motors whose strain is always positive, .

The assumption of equal load sharing of equivalent motors defines a four dimensional network of minifilament states . The second approximation of the PCM reduces this network further by assuming that bound states are in local thermal equilibrium (LTE). This is justified by the strong separation of time scales between fast power-stroke and slow binding kinetics Vilfan and Duke (2003); Erdmann and Schwarz (2012); Erdmann et al. (2013). In LTE, the conditional probability that motors are in the pps state when motors are bound is the Boltzmann distribution

(4)

where is the partition sum. The energy of a minifilament is the sum of the elastic energy in the external springs, , the elastic energy in the neck linkers, where , and the free energy bias towards the pps state. Inserting Eq. (1) for yields as function of minifilament state and contraction ,

(5)

The elastic energy is split into two contributions: the first is due to overall stretching of external springs and neck linkers. For , it increases with increasing contraction and number of bound motors in pps state. The second contribution is due to internal tension caused by motors in different bound states. It vanishes when all bound motors in an ensemble are either in wb state or in pps state, that is, for or , and is positive for intermediate states with .

The LTE assumption for the bound states leaves the numbers and of bound motors as the only remaining variables. Binding and unbinding changes the state by and . Binding proceeds only to the wb state and with constant on-rate . Because motors can bind independently the effective forward rate for the transition in the ensemble is

(6)

The forward rate is independent of and depends only on in the respective ensemble. Unbinding of motors proceeds either with constant off-rate from the wb state or with load dependent off-rate from the pps state. Thus, the reverse rate for the transition in state is

(7)

The effective reverse rate for transitions in state is obtained by averaging over with the LTE distribution from Eq. (4),

(8)

We use a Kramers type load dependence for the off-rate from the pps state, . The off-rate decreases exponentially with increasing load on a motor to describe the catch bond character of the pps state, where sets the unbinding force scale. Inserting effective forward and reverse rate, a two dimensional master equation for the probability that motors are bound can be formulated as

(9)

The probability for a specific state is the product of the coarse-grained probability distribution with the conditional LTE probability distribution . The master equation cannot be separated in two one-dimensional equations because the reverse rates and depend on and in both ensembles.

Because the effective reverse rates depend on the contraction , the master equation for binding dynamics has to be solved together with rules for the displacement of the ensembles upon binding and unbinding Erdmann et al. (2013). We define the position of an ensemble as the average position of bound motor heads on the substrate. The position of the motor filament is then given by , where is the offset of the motors. New motors are assumed to bind with vanishing offset, thus shifting the ensemble position by . Note that the offset is negative for ensembles subjected to small forces and with the majority of motors in the pps state. To implement the rules for ensemble movement with Eq. (9) the offset needs to be averaged with the LTE distribution. This gives in state and the position change upon binding of a motor. Unbinding of a motor does not change ensemble position because all bound motors are assumed to be at the same position . Complete detachment of one ensemble relaxes the external springs and the still attached ensemble moves freely with offset . Reattachment therefore places the ensemble at . Because and are defined in opposite directions, the sum describes the contraction of the actin substrates. With these definitions, the PCM for myosin II minifilaments is completely defined.

Iii Results

iii.1 Transitions in the power-stroke probability

The LTE distribution is shaped by the three parts of the minifilament energy in Eq. (5). The second part—elastic energy due to internal tension of opposing motors in both ensembles—is symmetric against exchanging and . It vanishes when all bound motors are either in pps state () or in wb state (). Between and (or to ) the energy increases by so that the relative occupancy is . This implies that intermediate states with are hardly occupied and is close to a binary distribution, in which either none or all of the bound motors in an ensemble perform the power stroke. Only the four states , , and , which are local minima of , can be appreciably occupied. The third part of in Eq. (5) decreases by the gain of conformational energy for each of the pps motors. This conformational energy bias is opposed by the first contribution from the elastic energy to in Eq. (5) which increases with . This elastic energy bias increases with contraction (if and ) and total spring constant and eventually exceeds the conformational bias towards the pps state.

Figure 2: (a) Power-stroke probability (see Eq. (4)) for a minifilament with bound motors as function of contraction for external spring stiffness , that is, and . The power-stroke probability is shown for , and ; is smaller than . (b) Effective reverse rates and (see Eq. (8)) as function of for the minifilament from (a).

Fig. 2 (a) shows the power-stroke probability for a minifilament with bound motors as function of contraction . For small , the gain of conformational energy in the power stroke exceeds the increase of elastic energy and the minifilament is in state with probability . At an intermediate value of , the minifilament switches to in a sharp transition. This means that above this threshold, only the ensemble with larger number of bound motors () is able to perform the power stroke. Above a second threshold, neither ensemble performs the power stroke and the minifilament switches to . The plot confirms that the LTE distribution can be almost neglected for intermediate states with and ensembles are said to be either in pps or wb state according to the dominant state of the bound motors.

The thresholds for the transitions can be determined by comparing the energy in the four states with or . For the transition from to occurs at and from to at . In the symmetric case , the states and are degenerate and the minifilament occupies both with equal probability. It is important to note that the thresholds contain an additional dependence on via . For a given value of , fluctuations of will therefore induce transitions to the wb state for small and to the pps state for large .

Fig. 2 (b) plots the effective reverse rates in and ensemble as function of for the same minifilament setup as in Fig. 2 (a). For small the minifilament is in state and both rates decrease exponentially with due to catch bonding of pps motors. Note, however, that the relation between and force depends on power-stroke probability: it is for , for and for . At the transition to , the bound motors in the ensemble can no longer perform the power stroke and the effective reverse rate of the ensemble drops to the value determined by the small off-rate from of the wb state. Because the transition to the wb state decreases the force on the bound motors in the ensemble, the effective reverse rate increases during the transition. With the transition to the bound motors in the ensemble enter the wb state and the effective reverse rate drops to .

iii.2 Stochastic trajectories

The two dimensional master equation Eq. (9) describes a non-equilibrium process without detailed balance. Moreover, there is a nonlinear feedback between minifilament displacement and binding dynamics so that the master equation cannot be solved analytically. Instead, we analyze minifilament dynamics numerically using the direct method of the Gillespie algorithm. Fig. 3 shows typical stochastic trajectories of symmetric minifilaments with varying size and external spring stiffness. The lower panel of each plot shows the number of bound motors in the two ensembles. The upper panel plots the minifilament force from Eq. (2) for a given weighted with the appropriate conditional probability over . Due to the binary nature of the power-stroke probability, is dominated by the states with or . Fig. 3 (a) shows a trajectory of a small minifilament with motors for , that is, . Due to the soft external springs, the power stroke in both ensembles does not increase the load on the motors appreciably so that the effective reverse rate remains close to its large intrinsic value. Moreover, the threshold value of for the transition from pps to wb state is large and at least one of the ensembles typically detaches before reaching the threshold. Therefore, trajectories are characterized by frequent detachment of ensembles.

Figure 3: Stochastic trajectories of symmetric minifilaments ( and ) with varying ensemble size and varying external spring stiffnesses. In each plot, the bottom panel shows the number of bound motors in ensemble (, red) and ensemble (, blue). The top panel shows the force on the ensembles, that is, (see Eq. (2)) weighted with . (a) and (), (b) and () and (c) and ().

Fig. 3 (b) shows a trajectory of a minifilament with the same size as in Fig. 3 (a) but for stiffer external springs with and . Detachment is much less frequent than in Fig. 3 (a) and the series of short force peaks is no longer observed. Instead, initial attachment is followed by a gradual increase of force towards a state with strongly fluctuating but on average constant force. This is the result of two effects. First, the force generated by the power stroke at decreases the off-rate of pps motors appreciably so that the time to detachment of the minifilament is increased. Second, the threshold for the transition from pps to wb state is lowered and—in combination with smaller off-rate—is more likely to be reached before detachment. To stabilize attachment, it is sufficient that the power stroke cannot be performed for so that the last motor in each ensemble unbinds slowly from the wb state. For the transition from to occurs at a force below and above the transition. The fast fluctuations of force upon binding and unbinding in Fig. 3 (b) thus indicate transitions between pps and wb state, while the slower variations are due to variations of following multiple binding events. Fluctuations to small values of allow both ensembles to perform the power stroke and the increased reverse rate can lead to detachment of the minifilament.

Fig. 3 (c) shows a trajectory of a larger minifilament with motors for soft external springs as in Fig. 3 (a). Initially, the miniflament detaches repeatedly as observed for the smaller ensemble. Due to its larger size and detachment time, the minifilament eventually reaches the threshold above which the last motors in both ensembles unbind from the wb state. Although the minifilament is stalled for , it continues to builds up larger force because the typical number of bound motors is larger. For forces above , the trajectory of the number of bound motors shows that it becomes unlikely to find a single bound motor. This force is large enough to keep the last two bound motors in the wb state with low unbinding rate and to make detachment of any of the two ensembles unlikely.

The dynamics of force can be understood considering the sequence of pps to wb transitions in Fig. 2. For small , both ensembles perform the power stroke and increases quickly through the activity of both ensembles. The increase of is terminated by the detachment of one of the ensembles. If the minifilament remains attached sufficiently long, reaches the threshold above which the ensemble with smaller number of bound motors enters the wb state. In this case, the pps ensemble moves forward upon binding of motors as long as . The wb ensemble, on the other hand, will step backward upon binding because without pps motors. The contraction continues to increase as long as the forward movement of the pps ensemble is faster than the backward movement of the wb ensemble. As reaches the threshold above which both ensembles enter the wb state, can only decrease because for so that both ensembles step backwards upon binding of motors. Since the threshold is reached first for small , detachment of the minifilament becomes very unlikely. On the other hand, for an increasing number of such states the minifilament enters an isometric state in which contraction and force fluctuate with a constant average, because forward movement at large is balanced by backward stepping at small .

The trajectories in Fig. 3 are for symmetric minifilaments with the same number of motors in and ensemble and equal external spring constants, . Differences of do not affect results, because enters the dynamic description only via the effective external spring constant . Differences of ensemble sizes, , do affect minifilament dynamics but trajectories are qualitatively similar as long as the difference is not too large. Most importantly, asymmetric minifilaments display a net movement in the direction of the larger ensemble. When both ensembles are attached and perform the power stroke, fewer motors are bound on average in the smaller ensemble. These are subject to larger force so that position steps are smaller or even negative; catch bonding of pps motors reduces this difference. Moreover, the smaller ensemble detaches more frequently allowing the larger ensemble to move freely. Finally, the smaller ensemble is more likely to transition from pps to wb state and form a passive anchor for the larger ensemble. The frequency of detachment of asymmetric minifilaments and the probability to reach large forces, though, is determined by the smaller ensemble.

iii.3 Mechanosensitivity

The stochastic trajectories reveal a switch in minifilament dynamics from transient attachment without sustained force (see Fig. 3 (a)) to near permanent attachment (see Fig. 3 (b-c)) in response to increasing external spring stiffness. To elucidate this mechanosensitivity further, Fig. 4 (a) plots the mean force obtained by averaging in the steady state generated by symmetric minifilaments with varying ensemble size as function of external spring stiffness . Fig. 4 (a) reveals a strongly nonlinear increase of with , which is caused by catch bonding of pps motors in combination with pps to wb transitions in the power-stroke probability.

Figure 4: Mean force generated by symmetric minifilaments with varying number of motors as function of external spring constant . is obtained by time averaging over stochastic trajectories. (a) Mean force for , , and . (b) Mean force for for several model variants in comparison with the full model: no wb state, that is, fixed power-stroke probability , no catch bonding of pps motors which unbind with constant off-rate , increased off-rate from the wb state and single myosin II ensemble working against an external spring with and as for the minifilament.

For small stiffness, increases linearly with . Here, the contraction reached before one of the ensembles detaches is on the order of the power-stroke length . For small values of , the corresponding force is too small to increase the time to detachment significantly, so that the typical contraction hardly increases with . In the limit of small , it is so that the force is proportional to . The mean force along a trajectory is proportional to the duty ratio of a minifilament, that is, the fraction of time both ensembles are attached. The duty ratio is hardly affected by small forces but increases with minifilament size so that the slope in the linear regime increases with . Catch bonding of pps motors eventually increases detachment time and duty ratio so that increases beyond the force free case. The mutual positive feedback between force and duty ratio leads to a rapid increase of which is reinforced by transitions from pps to wb state. On the other hand, these transitions limit the increase of force, because pps motors are required for forward movement of the ensembles and an increase of . Once minifilaments are almost permanently attached, increases slowly with further increasing . This reflects the increase of the average number of bound motors in the isometric state, which is caused by increased force fluctuations for large and the nonlinear dependence of the off-rate on force. The increasing number of bound motors allows the minifilament to reach larger values of although the pps to wb transition occurs for increasing values of . The linear regime is most prominent for small where the intermediate, super linear regime cannot be discerned. With increasing , the linear regime shrinks and the super linear growth in the intermediate regime approaches a step increase of , because detachment time depends sensitively on changes of the off-rate for large ensembles Erdmann et al. (2013).

Fig. 4 (b) plots the mean force generated by minifilaments with as function of for model variants in which components of the myosin II motor cycle are omitted in order to elucidate their contribution to force generation. Without the transition from pps to wb state (no wb state), that is, for fixed power-stroke probability , the super linear increase of at intermediate is comparable in steepness to the full model but is shifted to larger values of . Without stabilization through the pps to wb transition, the increase of duty ratio and mean force is due to catch bonding of pps motors alone. On the other hand, reaches significantly larger values and continues to increase over the whole range of because is no longer limited by the transition to the wb state of both ensembles. Instead, is limited by the stall force at which the offset in both pps ensembles vanishes. Thus the transition of the power-stroke probability is required to increase the sensitivity of the response and to generate a switch-like behavior with a plateau at large forces. Without this transition, the model would become unphysical because then the power stroke would require more energy than provided by the ATP hydrolysis. Without catch bonding, that is, with constant off-rate from the pps state, the super linear increase of at intermediate is present but occurs for larger and is much weaker than in the full model. Also the overall level of is strongly reduced. Trajectories show that minifilaments do reach the transition from pps to wb state, but continue to detach frequently. Thus, catch bonding of pps motors does not only provide the feedback between force and duty ratio needed for the steep increase of at intermediate , but also stabilizes the isometric state by increasing the average number of bound motors. With a large off-rate of wb motors, transitions from pps to wb state increase the effective reverse rates (see Fig. 2). Hence, pps to wb transitions induce detachment and neutralize the effect of catch bonding. The level of is smaller than in the case without catch bonding and a steep intermediate regime is not observed. Thus a low off-rate from the wb state is required to conserve the large level of force in the isometric state and to generate the observed switch-like response. A single myosin II ensemble working against an external spring (representing one half of a minifilament), generates slightly larger than minifilaments because the frequency of detachment is reduced. Due to the larger total spring constant, this difference increases as approaches . For very large , on the other hand, the mean force generated by a single ensemble collapses because the power stroke can no longer be performed at . This does not occur for minifilaments because is limited by . Thus the interplay between external and internal mechanics is essential for the functioning of minifilaments.

iii.4 Ensemble movement on rigid substrates

The contraction is confined within a narrow range around a stable fixed point for attached minifilaments. Fluctuations to large are limited by the transition of both ensembles to the wb state. Fluctuations to small induce transitions to the pps state in both ensembles so that forward movement of both ensembles increases rapidly. The range of narrows with increasing . To facilitate analysis of this situation, we replace the external springs by rigid anchorage. The total spring constant reduces to and the contraction is identical to the sum of the offset in the two ensembles, . Due to the large stiffness of the neck linkers, at most one ensembles can be in pps state at , while the other is in wb state. The threshold for the transition to state , at which the minifilament is stalled, is small and reached within few binding steps. As a consequence a stationary state is established quickly and the time dependent solution of the master equation Eq. (9) can be replaced by the stationary limit . Results of stochastic simulations for reveal two clearly separated peaks at a negative and a positive value of (data not shown). The peak at negative corresponds to the ensemble in pps state with force per motor and the peak at positive to the ensemble in wb state with force per motor. The separation of the peaks equals the power-stroke length, , which is the expected difference of for (see Eq. (1)). Finite width and asymmetry of the observed peaks are due to fluctuation of and for the case . The same results applies to the offset in the ensemble.

Stochastic simulations confirm that the ensemble with larger number of bound motors usually performs the power stroke while the other ensemble is in wb state (see Fig. 2). For , the minifilament is in state or with equal probability. Therefore, the power-stroke probability from Eq. (4) can be replaced by

(10)

where is the Heaviside step function with and the Kronecker delta. As a consequence of the mechanical coupling of ensembles and the mechanosensitivity of myosin II, the numbers and of bound motors in and ensemble are synchronized. The correlation increases with but is reduced for minifilaments with soft external springs. Synchronization is due to the transition to the wb state of the ensemble with fewer bound motors. The small off-rate of wb relative to pps motors (see Fig. 2 (b)) tends to equalize the number of bound motors. Soft external springs weaken the mechanical coupling of ensembles so that motors in one ensemble are less sensitive to variations of the number of bound motors in the other. Because minifilaments move in the direction of the pps ensemble, synchronization of the number of bound motors tends to reverse the direction of motion of minifilaments and prevents long, persistent runs. This is different from a tug-of-war of processive motors, which are usually described as slip bonds which favor large differences of the number of bound motors.

To derive an approximation for the stationary distribution we replace the continuous distribution of by a discrete one with two -peaks. Assuming constant contraction during a typical binding and unbinding cycle through states allows to estimate the negative offset in the pps ensemble as . The offset in the wb ensemble is . For , excellent agreement of this constant strain approximation with stochastic simulations is observed. The load dependent off-rate of pps motors becomes independent of . The strain of pps motors in the leading ensemble is and their off-rate . Together with the binary approximation of Eq. (10) for the power-stroke probability, the off-rate of motors in the ensemble (see Eq. (8)) reduces to

(11)

The first term describes unbinding from the pps state, the second from the wb state and the third is for vanishing force in the case of a minifilament with detached ensemble. An analogous expression holds for .

Figure 5: Steady state distribution for symmetric minifilaments with . (a) Results of stochastic simulations for as function of and . Solid and dashed line indicate slices of for constant and . (b) Comparison of simulation results (lines) and analytical results (symbols) from the constant offset approximation for as function of for constant and .

The constant offset approximation yields a two dimensional network of states with constant transition rates per motor. Analytical solutions for the stationary distribution are found by solving the corresponding linear system of equations. Fig. 5 (a) shows numerical results for from the exact model for a minifilament in a tug-of-war with . The distribution is symmetric with respect to exchanging and and strongly peaked at . It is centered along the diagonal with which expresses the effect of synchronization of the ensembles. Fig. 5 (b) shows the stationary probability for fixed as function of and compares the numerical solution to the analytical solution obtained via the constant offset approximation. Considering the approximations entering the analytical solution the excellent agreement is quite remarkable.

Figure 6: Diffusion constant of a minifilament as function of ensemble size for symmetric minifilaments. Numerical results (symbols) are compared to results from the constant strain approximation (see Eq. (12)). The inset shows a logarithmic plot of the data for a wider range of .

The absolute position of the minifilament can be defined as the mean position of and ensemble. Symmetric minifilaments perform an unbounded random walk with vanishing mean which is characterized by the diffusion coefficient . Due to the limited range of the contraction , the diffusion coefficient of will be identical to that of and in the long time limit, . We calculate via the limit of the second jump moment per time in the limit for a vanishing time step as van Kampen (2007). For , binding in the ensemble yields where . Detachment of the ensemble releases the strain of both ensembles and yields for . Movement of the detached ensemble through stepping of the ensemble contributes by for and . The average diffusion constant of the minifilament position is

(12)

Within the constant offset approximation, the offset is and when both ensembles are attached and if one of the ensembles is detached. Using the approximate expression for from Eq. (10), the contributions to the diffusion constant reduce to

(13)
(14)
(15)

Fig. 6 plots the diffusion constant