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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 @xcite . 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 @xcite . a theoretical treatment with a one - step master equation showed that the effective unbinding rate decreases exponentially with the size of the motor ensemble @xcite . moreover groups of motors can also work against larger load than the single motor @xcite . if motors of different directionality on the substrate are attached to the same cargo , bidirectional movement can ensue @xcite , as often observed in cargo transport . a similar tug - of - war setup has been used earlier to explain mitotic spindle oscillations @xcite . non - processive motors such as 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 work together as one group . combining structural investigations of skeletal muscle with modeling has led to the swinging cross - bridge model for single @xcite . a statistical treatment then has allowed to accurately model the dynamics of the motor ensemble in muscle sarcomeres @xcite . groups of motors also play a crucial role for the mechanics and adhesion of non - muscle tissue cells . cytoskeletal assembles into bipolar minifilaments consisting of @xmath1-@xmath2 motors @xcite . 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 @xcite . recently it has been shown that the activity of minifilaments contributes to the sorting of actin filament orientation because of the asymmetric elasticity of actin filaments @xcite . the 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 @xcite . in particular , mature focal adhesions are often connected to actin stress fibers consisting of parallel bundles of actin filaments with alternating polarity enabling 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 @xcite . if the environment is much stiffer than the cell , it essentially deforms itself and becomes insensitive to the environmental stiffness @xcite . therefore cellular stiffness sets the scale for the sensitivity of rigidity sensing @xcite . due to the complex interplay of many components in a cell , it is difficult to identify the exact contribution of to the rigidity response of cells . one promising experimental route is the reconstruction of _ in vitro _ systems of motors and filaments @xcite , 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 motors in the muscle sarcomere , theoretical progress has been made mainly through mean - field models @xcite or computer simulations @xcite . 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 @xcite . here we aim at understanding minifilaments with few 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 @xcite . however , this approach is numerically costly , in particular for extensions to systems with multiple minifilaments . recently the parallel cluster model ( ) based on the cross - bridge cycle has been introduced as an efficient yet detailed model for stochastic effects in small ensembles @xcite .    in this manuscript , we extend the to minifilaments by modeling them as two ensembles of 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 @xcite . 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 . in particular , we account for the catch bond character of 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 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 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 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 . in the parallel cluster model ( ) , individual motors are described by a cross - bridge model with three discrete states and stochastic transitions between them @xcite . here we generalize this model for minifilaments and discuss it with parameter values originally introduced for modeling skeletal muscle @xcite . for cytoskeletal , these values depend on the exact isoform one is considering . the parameter values used here result in a duty ratio of @xmath3 , which lies in the range of duty ratios reported for cytoskeletal b @xcite . as shown schematically in ( a ) , a motor comprises three mechanical elements . the motor head binds to the substrate and contains the hydrolysis site , which binds or its hydrolysis products and . 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 @xmath4 anchors the lever arm to the rigid motor filament . in the unbound ( ) state , the motor head is loaded with and and the lever arm is primed . the motor reversibly transitions to the weakly - bound ( ) state with on - rate @xmath5 and off - rate @xmath6 . with release of , the lever arm swings to the stretched conformation and the motor enters the post - power - stroke ( ) state . the power stroke is reversible with forward rate @xmath7 and reverse rate @xmath8 but is driven towards the state by the free energy bias @xmath9 . this energy is stored in the primed conformation of the lever arm as part of the energy released in hydrolysis . replacing by , unbinding and hydrolysis brings from the to the state , thus completing the motor cycle . these events are subsumed in a single reaction with unloaded rate @xmath10 which is assumed to be irreversible due of the hydrolysis of . dynamics is characterized by the load dependence of power stroke and unbinding from state : the power stroke moves the lever arm by the power - stroke length @xmath11 ; unbinding from the state requires an additional movement of the lever arm by a distance @xmath12 . thus , both reactions become slower under load . the reduced rate of unbinding under load makes the state of a catch bond rather than a slip bond in the range of forces considered here . motors in the ensemble moving to the right and @xmath13 motors in the ensemble moving to the left . the state of the ensemble is described by the number @xmath14 of bound motors and the number @xmath15 of motors in the post - power - stroke state . the configuration in ( b ) corresponds to @xmath16 in the ensemble and @xmath17 in the ensemble . the displacement of the ensembles on the actin filaments is denoted by @xmath18 and @xmath19 . bound motors within each ensemble are arranged in parallel ; the two ensembles are arranged in series . ( c ) the parallel cluster model ( ) 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 ( ) motors is denoted by @xmath20 and @xmath21 . the power stroke shortens the rest length of the neck linker by the power - stroke length @xmath22 so that post - power - stroke ( ) motors have the strain @xmath23.[fig : setup ] ]    the arrangement of motors in a bipolar minifilament is depicted schematically in ( 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 ( b ) ) is denoted by @xmath24 ; the number of motors in the ensemble working in direction ( towards the left in ( b ) ) is denoted by @xmath25 . the ensembles move on actin filaments of opposite polarity which are attached to linear elastic elements with spring constant @xmath26 and @xmath27 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 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 motors in and ensemble is denoted by @xmath20 and @xmath21 , respectively , so that motors exert the force @xmath28 . the strain is positive , when the neck linker is stretched against the moving direction of the ensemble ( inwards in ( b ) ) . with the assumption of equal load sharing , motor ensembles are mechanically equivalent to adhesion clusters of parallel bonds as depicted in ( c ) @xcite . in contrast to the adhesion cluster , however , the rest length of the bond is not fixed , but is reduced by a length @xmath22 due to the power stroke . thus , motors in the state have the strain @xmath23 and exert the force @xmath29 . the strain of as well as motors is determined by the offset @xmath30 between the bound motor head on the substrate and the anchor in the motor filament . the state @xmath31 of a minifilament is described by the numbers @xmath32 and @xmath33 of bound motors and @xmath34 and @xmath35 of motors in both ensembles . the number of motors is @xmath36 and that of motors @xmath37 . with @xmath15 motors and @xmath37 motors , the ensemble exerts the force @xmath38 $ ] . to determine the offset of and motors , we assume that a minifilament is always in mechanical equilibrium . for the arrangement of motor ensembles and external springs in series as in ( c ) this requires that the ensemble forces @xmath39 balance the force @xmath40 exerted by the external springs . the latter is @xmath41 $ ] where @xmath42 is the effective external spring constant and @xmath43 the displacement of the ensemble from its origin on the actin filament . the extension of the external springs is @xmath44 assuming they are relaxed when @xmath45 and all bound motors are in the state with @xmath46 . solving the balance of forces , @xmath47 , yields @xmath30 as function of minifilament state @xmath31 and contraction @xmath48 , @xmath49 the force @xmath50 as function of minifilament state and contraction then reads @xmath51\,,\ ] ] where the total spring constant is defined as @xmath52 and therefore varies dynamically , in contrast to @xmath53 . when all bound motors in the ensemble are in the state ( @xmath54 ) , the offset is positive , @xmath55 , and the neck linkers are stretched against the direction . for growing contraction @xmath56 , the offset @xmath20 increases because @xmath40 increases . a growing number @xmath57 of motors in the ensemble increases @xmath20 further , because transitions to the state shorten the minifilament and increase @xmath40 . on the other hand , @xmath20 decreases and can become negative for a growing number @xmath58 of motors in the ensemble , although @xmath40 increases further . for @xmath59 , motors contribute to the external load which is carried by the motors whose strain is always positive , @xmath60 . the assumption of equal load sharing of equivalent motors defines a four dimensional network of minifilament states @xmath31 . the second approximation of the reduces this network further by assuming that bound states are in local thermal equilibrium ( ) . this is justified by the strong separation of time scales between fast power - stroke and slow binding kinetics @xcite . in , the conditional probability that @xmath61 motors are in the state when @xmath62 motors are bound is the boltzmann distribution @xmath63 where @xmath64 is the partition sum . the energy @xmath65 of a minifilament is the sum of the elastic energy in the external springs , @xmath66 ^ 2 / 2 $ ] , the elastic energy @xmath67 in the neck linkers , where @xmath68 / 2 $ ] , and the free energy bias @xmath69 towards the state . inserting for @xmath30 yields @xmath70 as function of minifilament state @xmath31 and contraction @xmath71 , @xmath72 ^ 2     + \frac{\kl d^2}{2 } \left[\frac{\jp\left(\ip-\jp\right)}{\ip } + \frac{\jm\left(\im-\jm\right)}{\im } \right ]    + \left(\jp + \jm\right ) \epp\,.\ ] ] the elastic energy is split into two contributions : the first is due to overall stretching of external springs and neck linkers . for @xmath56 , it increases with increasing contraction and number of bound motors in 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 state or in state , that is , for @xmath73 or @xmath74 , and is positive for intermediate states with @xmath75 . the assumption for the bound states leaves the numbers @xmath76 and @xmath77 of bound motors as the only remaining variables . binding and unbinding changes the state @xmath62 by @xmath78 and @xmath79 . binding proceeds only to the state and with constant on - rate @xmath80 . because @xmath36 motors can bind independently the effective forward rate for the transition @xmath81 in the ensemble is @xmath82 the forward rate is independent of @xmath15 and depends only on @xmath14 in the respective ensemble . unbinding of motors proceeds either with constant off - rate @xmath83 from the state or with load dependent off - rate @xmath84 from the state . thus , the reverse rate for the transition @xmath85 in state @xmath31 is @xmath86 the effective reverse rate for transitions in state @xmath62 is obtained by averaging over @xmath15 with the distribution from , @xmath87 we use a kramers type load dependence for the off - rate from the state , @xmath88 . the off - rate decreases exponentially with increasing load @xmath89 on a motor to describe the catch bond character of the state , where @xmath90 sets the unbinding force scale . inserting effective forward and reverse rate , a two dimensional master equation for the probability @xmath91 that @xmath62 motors are bound can be formulated as @xmath92 { p_{\ip,\im}\xspace}\ , . \end{split}\ ] ] the probability for a specific state @xmath31 is the product of the coarse - grained probability distribution @xmath91 with the conditional probability distribution @xmath93 . the master equation can not be separated in two one - dimensional equations because the reverse rates @xmath94 and @xmath95 depend on @xmath76 and @xmath77 in both ensembles .    because the effective reverse rates @xmath96 depend on the contraction @xmath71 , the master equation for binding dynamics has to be solved together with rules for the displacement of the ensembles upon binding and unbinding @xcite . we define the position @xmath43 of an ensemble as the average position of bound motor heads on the substrate . the position of the motor filament is then given by @xmath97 , where @xmath30 is the offset of the motors . new motors are assumed to bind with vanishing offset , thus shifting the ensemble position by @xmath98 . note that the offset @xmath30 is negative for ensembles subjected to small forces and with the majority of motors in the state . to implement the rules for ensemble movement with the offset needs to be averaged with the distribution . this gives @xmath99 in state @xmath62 and the position change @xmath100 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 @xmath43 . complete detachment of one ensemble relaxes the external springs and the still attached ensemble moves freely with offset @xmath101 . reattachment therefore places the ensemble at @xmath102 . because @xmath18 and @xmath19 are defined in opposite directions , the sum @xmath103 describes the contraction of the actin substrates . with these definitions , the for minifilaments is completely defined . the distribution @xmath93 is shaped by the three parts of the minifilament energy @xmath70 in . the second part elastic energy due to internal tension of opposing motors in both ensembles  is symmetric against exchanging @xmath15 and @xmath37 . it vanishes when all bound motors are either in state ( @xmath74 ) or in state ( @xmath73 ) . between @xmath73 and @xmath104 ( or @xmath74 to @xmath105 ) the energy increases by @xmath106 so that the relative occupancy is @xmath107 . this implies that intermediate states with @xmath75 are hardly occupied and @xmath93 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 @xmath108 , @xmath109 , @xmath110 and @xmath62 , which are local minima of @xmath70 , can be appreciably occupied . the third part of @xmath70 in decreases by the gain @xmath69 of conformational energy for each of the @xmath111 motors . this conformational energy bias is opposed by the first contribution from the elastic energy to @xmath70 in which increases with @xmath15 . this elastic energy bias increases with contraction @xmath71 ( if @xmath112 and @xmath113 ) and total spring constant @xmath114 and eventually exceeds the conformational bias towards the state . ( see ) for a minifilament with @xmath115 bound motors as function of contraction @xmath103 for external spring stiffness @xmath116 , that is , @xmath117 and @xmath118 . the power - stroke probability is shown for @xmath119 , @xmath110 and @xmath120 ; @xmath121 is smaller than @xmath122 . ( b ) effective reverse rates @xmath94 and @xmath95 ( see ) as function of @xmath71 for the minifilament from ( a).[fig : psprob ] ]    \(a ) shows the power - stroke probability @xmath93 for a minifilament with @xmath115 bound motors as function of contraction @xmath71 . for small @xmath71 , the gain of conformational energy in the power stroke exceeds the increase of elastic energy and the minifilament is in state @xmath123 with probability @xmath124 . at an intermediate value of @xmath71 , the minifilament switches to @xmath125 in a sharp transition . this means that above this threshold , only the ensemble with larger number of bound motors ( @xmath126 ) is able to perform the power stroke . above a second threshold , neither ensemble performs the power stroke and the minifilament switches to @xmath108 . the plot confirms that the distribution can be almost neglected for intermediate states with @xmath127 and ensembles are said to be either in or 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 @xmath73 or @xmath74 . for @xmath128 the transition from @xmath119 to @xmath110 occurs at @xmath129 and from @xmath130 to @xmath120 at @xmath131 . in the symmetric case @xmath132 , the states @xmath133 and @xmath109 are degenerate and the minifilament occupies both with equal probability . it is important to note that the thresholds contain an additional dependence on @xmath62 via @xmath114 . for a given value of @xmath71 , fluctuations of @xmath62 will therefore induce transitions to the state for small @xmath14 and to the state for large @xmath14 . \(b ) plots the effective reverse rates @xmath96 in and ensemble as function of @xmath71 for the same minifilament setup as in ( a ) . for small @xmath71 the minifilament is in state @xmath134 and both rates decrease exponentially with @xmath71 due to catch bonding of motors . note , however , that the relation between @xmath71 and force depends on power - stroke probability : it is @xmath135 for @xmath119 , @xmath136 for @xmath133 and @xmath137 for @xmath108 . at the transition to @xmath138 , 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 @xmath139 determined by the small off - rate @xmath140 from of the state . because the transition to the state decreases the force on the @xmath141 bound motors in the ensemble , the effective reverse rate @xmath142 increases during the transition . with the transition to @xmath108 the bound motors in the ensemble enter the state and the effective reverse rate drops to @xmath143 . the two dimensional master equation 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 can not be solved analytically . instead , we analyze minifilament dynamics numerically using the direct method of the gillespie algorithm . 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 @xmath62 in the two ensembles . the upper panel plots the minifilament force @xmath50 from for a given @xmath62 weighted with the appropriate conditional probability over @xmath61 . due to the binary nature of the power - stroke probability , @xmath61 is dominated by the states with @xmath144 or @xmath74 . ( a ) shows a trajectory of a small minifilament with @xmath145 motors for @xmath146 , that is , @xmath147 . 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 @xmath71 for the transition from to 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 . and @xmath148 ) with varying ensemble size and varying external spring stiffnesses . in each plot , the bottom panel shows the number of bound motors in ensemble ( @xmath76 , red ) and ensemble ( @xmath77 , blue ) . the top panel shows the force on the ensembles , that is , @xmath50 ( see ) weighted with @xmath93 . ( a ) @xmath145 and @xmath149 ( @xmath147 ) , ( b ) @xmath145 and @xmath150 ( @xmath151 ) and ( c ) @xmath152 and @xmath153 ( @xmath154).[fig : trajectories ] ]    \(b ) shows a trajectory of a minifilament with the same size as in ( a ) but for stiffer external springs with @xmath155 and @xmath151 . detachment is much less frequent than in ( 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 @xmath156 decreases the off - rate of motors appreciably so that the time to detachment of the minifilament is increased . second , the threshold for the transition from to 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 can not be performed for @xmath157 so that the last motor in each ensemble unbinds slowly from the state . for @xmath157 the transition from @xmath133 to @xmath120 occurs at a force @xmath158 below and @xmath159 above the transition . the fast fluctuations of force upon binding and unbinding in ( b ) thus indicate transitions between and state , while the slower variations are due to variations of @xmath71 following multiple binding events . fluctuations to small values of @xmath71 allow both ensembles to perform the power stroke and the increased reverse rate can lead to detachment of the minifilament . \(c ) shows a trajectory of a larger minifilament with @xmath152 motors for soft external springs @xmath154 as in ( 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 @xmath160 above which the last motors in both ensembles unbind from the state . although the minifilament is stalled for @xmath161 , it continues to builds up larger force because the typical number of bound motors is larger . for forces above @xmath162 , 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 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 to transitions in . for small @xmath71 , both ensembles perform the power stroke and @xmath71 increases quickly through the activity of both ensembles . the increase of @xmath71 is terminated by the detachment of one of the ensembles . if the minifilament remains attached sufficiently long , @xmath71 reaches the threshold above which the ensemble with smaller number of bound motors enters the state . in this case , the ensemble moves forward upon binding of motors as long as @xmath163 . the ensemble , on the other hand , will step backward upon binding because @xmath164 without motors . the contraction @xmath71 continues to increase as long as the forward movement of the ensemble is faster than the backward movement of the ensemble . as @xmath71 reaches the threshold above which both ensembles enter the state , @xmath71 can only decrease because @xmath165 for @xmath166 so that both ensembles step backwards upon binding of motors . since the threshold is reached first for small @xmath14 , 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 @xmath71 and force @xmath40 fluctuate with a constant average , because forward movement at large @xmath14 is balanced by backward stepping at small @xmath14 . the trajectories in are for symmetric minifilaments with the same number @xmath167 of motors in and ensemble and equal external spring constants , @xmath148 . differences of @xmath168 do not affect results , because @xmath169 enters the dynamic description only via the effective external spring constant @xmath53 . differences of ensemble sizes , @xmath170 , 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 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 to 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 . the stochastic trajectories reveal a switch in minifilament dynamics from transient attachment without sustained force ( see ( a ) ) to near permanent attachment ( see ( b - c ) ) in response to increasing external spring stiffness . to elucidate this mechanosensitivity further , ( a ) plots the mean force @xmath171 obtained by averaging in the steady state generated by symmetric minifilaments with varying ensemble size @xmath167 as function of external spring stiffness @xmath172 . ( a ) reveals a strongly nonlinear increase of @xmath171 with @xmath169 , which is caused by catch bonding of motors in combination with to transitions in the power - stroke probability .     generated by symmetric minifilaments with varying number @xmath167 of motors as function of external spring constant @xmath173 . @xmath171 is obtained by time averaging over stochastic trajectories . ( a ) mean force @xmath171 for @xmath174 , @xmath175 , @xmath176 and @xmath177 . ( b ) mean force @xmath171 for @xmath178 for several model variants in comparison with the full model : @xmath179 no state , that is , fixed power - stroke probability @xmath180 , @xmath181 no catch bonding of motors which unbind with constant off - rate @xmath182 , @xmath183 increased off - rate @xmath184 from the state and @xmath185 single ensemble working against an external spring with @xmath24 and @xmath26 as for the minifilament.[fig : meanforce ] ]    for small stiffness , @xmath171 increases linearly with @xmath169 . here , the contraction @xmath71 reached before one of the ensembles detaches is on the order of the power - stroke length @xmath22 . for small values of @xmath53 , the corresponding force is too small to increase the time to detachment significantly , so that the typical contraction @xmath71 hardly increases with @xmath53 . in the limit of small @xmath186 , it is @xmath187 so that the force @xmath188 is proportional to @xmath53 . 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 @xmath189 . catch bonding of motors eventually increases detachment time and duty ratio so that @xmath71 increases beyond the force free case . the mutual positive feedback between force and duty ratio leads to a rapid increase of @xmath171 which is reinforced by transitions from to state . on the other hand , these transitions limit the increase of force , because motors are required for forward movement of the ensembles and an increase of @xmath71 . once minifilaments are almost permanently attached , @xmath171 increases slowly with further increasing @xmath169 . this reflects the increase of the average number of bound motors in the isometric state , which is caused by increased force fluctuations for large @xmath53 and the nonlinear dependence of the off - rate on force . the increasing number of bound motors allows the minifilament to reach larger values of @xmath71 although the to transition occurs for increasing values of @xmath14 . the linear regime is most prominent for small @xmath189 where the intermediate , super linear regime can not be discerned . with increasing @xmath189 , the linear regime shrinks and the super linear growth in the intermediate regime approaches a step increase of @xmath171 , because detachment time depends sensitively on changes of the off - rate for large ensembles @xcite . \(b ) plots the mean force generated by minifilaments with @xmath190 as function of @xmath169 for model variants in which components of the motor cycle are omitted in order to elucidate their contribution to force generation . @xmath179 without the transition from to state ( no state ) , that is , for fixed power - stroke probability @xmath180 , the super linear increase of @xmath171 at intermediate @xmath169 is comparable in steepness to the full model but is shifted to larger values of @xmath169 . without stabilization through the to transition , the increase of duty ratio and mean force is due to catch bonding of motors alone . on the other hand , @xmath171 reaches significantly larger values and continues to increase over the whole range of @xmath169 because @xmath71 is no longer limited by the transition to the state of both ensembles . instead , @xmath171 is limited by the stall force at which the offset @xmath30 in both 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 hydrolysis . @xmath181 without catch bonding , that is , with constant off - rate @xmath191 from the state , the super linear increase of @xmath171 at intermediate @xmath169 is present but occurs for larger @xmath169 and is much weaker than in the full model . also the overall level of @xmath171 is strongly reduced . trajectories show that minifilaments do reach the transition from to state , but continue to detach frequently . thus , catch bonding of motors does not only provide the feedback between force and duty ratio needed for the steep increase of @xmath171 at intermediate @xmath169 , but also stabilizes the isometric state by increasing the average number of bound motors . @xmath183 with a large off - rate @xmath192 of motors , transitions from to state increase the effective reverse rates ( see ) . hence , to transitions induce detachment and neutralize the effect of catch bonding . the level of @xmath171 is smaller than in the case without catch bonding and a steep intermediate regime is not observed . thus a low off - rate from the state is required to conserve the large level of force in the isometric state and to generate the observed switch - like response . @xmath185 a single ensemble working against an external spring ( representing one half of a minifilament ) , generates slightly larger @xmath171 than minifilaments because the frequency of detachment is reduced . due to the larger total spring constant , this difference increases as @xmath169 approaches @xmath193 . for very large @xmath169 , on the other hand , the mean force generated by a single ensemble collapses because the power stroke can no longer be performed at @xmath156 . this does not occur for minifilaments because @xmath114 is limited by @xmath193 . thus the interplay between external and internal mechanics is essential for the functioning of minifilaments .      the contraction @xmath71 is confined within a narrow range around a stable fixed point for attached minifilaments . fluctuations to large @xmath71 are limited by the transition of both ensembles to the state . fluctuations to small @xmath71 induce transitions to the state in both ensembles so that forward movement of both ensembles increases @xmath71 rapidly . the range of @xmath71 narrows with increasing @xmath53 . to facilitate analysis of this situation , we replace the external springs by rigid anchorage . the total spring constant reduces to @xmath194 and the contraction @xmath71 is identical to the sum of the offset @xmath30 in the two ensembles , @xmath195 . due to the large stiffness of the neck linkers , at most one ensembles can be in state at @xmath156 , while the other is in state . the threshold for the transition to state @xmath196 , 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 @xmath91 of the master equation can be replaced by the stationary limit @xmath197 . results of stochastic simulations for @xmath30 reveal two clearly separated peaks at a negative and a positive value of @xmath20 ( data not shown ) . the peak at negative @xmath20 corresponds to the ensemble in state with force @xmath198 per motor and the peak at positive @xmath20 to the ensemble in state with force @xmath199 per motor . the separation of the peaks equals the power - stroke length , @xmath200 , which is the expected difference of @xmath30 for @xmath201 ( see ) . finite width and asymmetry of the observed peaks are due to fluctuation of @xmath14 and @xmath71 for the case @xmath202 . the same results applies to the offset @xmath21 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 state ( see ) . for @xmath203 , the minifilament is in state @xmath133 or @xmath109 with equal probability . therefore , the power - stroke probability from can be replaced by @xmath204 where @xmath205 is the heaviside step function with @xmath206 and @xmath207 the kronecker delta . as a consequence of the mechanical coupling of ensembles and the mechanosensitivity of , the numbers @xmath76 and @xmath77 of bound motors in and ensemble are synchronized . the correlation increases with @xmath189 but is reduced for minifilaments with soft external springs . synchronization is due to the transition to the state of the ensemble with fewer bound motors . the small off - rate of relative to motors ( see ( 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 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 @xmath208 we replace the continuous distribution of @xmath30 by a discrete one with two @xmath209-peaks . assuming constant contraction @xmath71 during a typical binding and unbinding cycle through states @xmath210 allows to estimate the negative offset in the ensemble as @xmath211 . the offset in the ensemble is @xmath212 . for @xmath213 , excellent agreement of this constant strain approximation with stochastic simulations is observed . the load dependent off - rate of motors becomes independent of @xmath62 . the strain of motors in the leading ensemble is @xmath214 and their off - rate @xmath215 . together with the binary approximation of for the power - stroke probability , the off - rate of motors in the ensemble ( see ) reduces to @xmath216\,.\ ] ] the first term describes unbinding from the state , the second from the state and the third is for vanishing force in the case of a minifilament with detached ensemble . an analogous expression holds for @xmath95 .     for symmetric minifilaments with @xmath190 . ( a ) results of stochastic simulations for @xmath208 as function of @xmath76 and @xmath77 . solid and dashed line indicate slices of @xmath208 for constant @xmath217 and @xmath218 . ( b ) comparison of simulation results ( lines ) and analytical results ( symbols ) from the constant offset approximation for @xmath208 as function of @xmath77 for constant @xmath217 and @xmath218.[fig : probability ] ]    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 . ( a ) shows numerical results for @xmath208 from the exact model for a minifilament in a tug - of - war with @xmath190 . the distribution is symmetric with respect to exchanging @xmath76 and @xmath77 and strongly peaked at @xmath219 . it is centered along the diagonal with @xmath220 which expresses the effect of synchronization of the ensembles . ( b ) shows the stationary probability for fixed @xmath76 as function of @xmath77 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 .     of a minifilament as function of ensemble size @xmath167 for symmetric minifilaments . numerical results ( symbols ) are compared to results from the constant strain approximation ( see ) . the inset shows a logarithmic plot of the data for a wider range of @xmath24.[fig : diffusion ] ]    the absolute position of the minifilament can be defined as the mean position @xmath221 of and ensemble . symmetric minifilaments perform an unbounded random walk with vanishing mean which is characterized by the diffusion coefficient @xmath222 . due to the limited range of the contraction @xmath71 , the diffusion coefficient of @xmath223 will be identical to that of @xmath18 and @xmath19 in the long time limit , @xmath224 . we calculate @xmath225 via the limit of the second jump moment per time in the limit for a vanishing time step as @xmath226 @xcite . for @xmath227 , binding in the ensemble yields @xmath228 where @xmath229 . detachment of the ensemble releases the strain of both ensembles and yields @xmath230 for @xmath231 . movement of the detached ensemble through stepping of the ensemble contributes by @xmath232 for @xmath233 and @xmath234 . the average diffusion constant of the minifilament position @xmath223 is @xmath235 { p_{\ip,\im}(\infty)\xspace}\,.\ ] ] within the constant offset approximation , the offset is @xmath236 and @xmath237 when both ensembles are attached and @xmath238 if one of the ensembles is detached . using the approximate expression for @xmath208 from , the contributions to the diffusion constant reduce to @xmath239 \quad\text{for}\quad \ip > 0\\   d_{+}^{\rm detach } & = ( \xneg+\xpos)^2 \left[\kzns \theta(\ip-\im ) + \ken \theta(\im-\ip ) \right]\quad\text{for}\quad \ip = 1\\   d_{+}^{\rm drag }    & = \gmin(\im ) \frac{d^2}{(\im+1)^2 } \quad\text{for}\quad \ip = 0 , \im > 0\,.\end{aligned}\ ] ] plots the diffusion constant @xmath222 of a minifilament as function of ensemble size @xmath167 . the diffusion coefficient decreases with increasing ensemble size . simulation results show that for large @xmath240 , where detachment of ensembles is negligible , the diffusion coefficient scales as the inverse of ensemble size , @xmath241 . for smaller @xmath189 , ensemble detachment leads to a stronger dependence on @xmath189 . in the approximation of , only @xmath242 contributes to @xmath222 for @xmath240 . for a given @xmath243 , the squared step @xmath244 scales as @xmath245 and the forward rate as @xmath246 . using @xmath247 , this yields the scaling expression @xmath248 which fits well to the observed dependence @xmath249 . the deviations between simulation and analytical results are probably due to correlations in the fluctuations of the offset @xmath250 . the decrease of @xmath222 is in contrast to the dependence of @xmath222 on ensemble size in a tug - of - war of processive motors . processive motors are usually described as slip bonds so that the time to switch between directions increases exponentially with ensembles size , which induces an exponential increase of the diffusion coefficient @xcite . the synchronization of the number of bound motors as a consequence of the catch bond character of leads to a weak dependence of the switching time on ensemble size and prevents the increase of the diffusion coefficient . here we have introduced a new model for the stochastic dynamics of minifilaments . analyzing the stochastic trajectories and the mean force generation predicted by our model revealed a switch of the dynamic behavior of the minifilaments in response to changes of the environmental elasticity as a consequence of two independent mechano - sensitive processes . first catch bonding of motors depends directly on the load on the motors . a gradual increase of load decreases the reverse rate and increases the number of bound motors to stabilize ensemble attachment . second the transition from to state provides another type of catch bonding because unbinding from state is slower than from state , @xmath140 . there are two important differences : @xmath179 to transitions reduce the reverse rates @xmath96 abruptly and @xmath181 the transition is sensitive not only to minifilament load but also to the elasticity of the environment . for single ensembles , to transitions only occur in elastic environments , but not for ensembles working against constant external load @xcite . in a miniflament , the elastic neck linkers provide an elastic environment for the ensembles even with rigid anchorage . therefore our model demonstrates the importance of considering both the internal and external mechanics when investigating the function of a minifilament . both mechano - sensitive processes cooperate in stabilizing minifilament attachment . while permanent attachment can be reached with catch bonding of motors alone , the transition from to state shifts the switch to smaller external stiffness and steepens the transition . on the other hand , the transition to the state alone is not sufficient for stable attachment . the two mechano - sensitive processes have contrasting effects on minifilament dynamics . catch bonding of motors allows to distribute load on a growing number of bound motors . this increases the velocity of an ensemble at a given load and increases the stall force . in the context of sarcomeric and cytoskeletal ensembles working against a constant external load , it was shown that catch bonding of motors leads to the characteristic upward convex force - velocity relation @xcite . for minifilaments , catch bonding of motors increases the mean force exerted in an elastic environment . transitions from to state , on the other hand , limit the increase of force because motors are required for an increase of contraction , so that minifilaments adjust themselves to an isometric state . in single ensembles working against a linear external load , the force generated by the ensembles breaks down for very stiff external load because the power stroke can no longer be performed at @xmath156 . in minifilaments , the effective stiffness is limited by the elasticity of the neck linkers and this breakdown is prevented . therefore , minifilaments in the isometric state generate a mean level of force , which depends mainly on the number of motors available for binding , but is robust to variations of elastic properties of the environment . transitions to the state limit the mean force as long as they occur before the stall force of an ensemble in state is reached . this is the case for the parameters in our model ; for significantly smaller neck linker stiffness as used in ref.@xcite with comparable power - stroke length , the stall force is reduced strongly and could be reached before the transition to the state . for the smaller stall force , however , catch bonding of motors would not be sufficient to decrease the reverse rate sufficiently to achieve stable attachment . the mechanosensitivity of minifilaments has important implications for the structure and function of acto - myosin networks . in the actin cortex or reconstituted mixtures of actin with minifilaments , the apparent stiffness of an actin filament depends largely on the level of crosslinking . due to their mechanosensitivity , minifilaments will most efficiently form crosslinks with and exert force to filaments which are already firmly linked to the network . loose filaments , on the other hand , which do not contribute to overall network tension , could not be integrated by minifilaments . this could help to create densely linked networks in which minifilaments provide active crosslinks maintaining a constant level of tension , which is restored after deformations . stretching a network would initially reinforce attachment and prevent rupture of the network . on longer time scales , the increased tension would be released by the flow of the network as a consequence of backward stepping of minifilaments . compression , on the other hand , would reduce the load on the minifilament crosslinks . subsequent forward movement of ensembles then helps to restore the initial tension in the network . persistent compression , however , could lead to detachment and network disintegration . in an actin network of random polarity , mechanosensitivity of minifilaments in combination with the asymmetric elasticity of actin filaments could lead to a selection of filaments according to their orientation , because large forces can be exerted when filaments are stretched , but small forces on filaments buckling under compression @xcite . this could also contribute to the observed contraction rather than expansion of actin networks of random orientation . unbalanced preferential attachment to strongly linked filaments could cause aggregation of minifilaments and actin , as it is observed in reconstituted assays @xcite . in adherent cells elasticity of the extracellular matrix contributes to the effective stiffness experienced by minifilaments . actin stress fibers tend to form and reinforce at focal contacts on sufficiently stiff substrates @xcite . mechanosensitivity of minifilaments could contribute to discrimination of substrate stiffness because they are not able to crosslink and to build up forces in soft environments . although this possible relation is rather speculative at this stage and the exact role of myosin minifilaments in the rigidity response of cells is hard to pin down , it is worth mentioning that activity is an integral part of mechanosensing , which is completely disrupted by inhibiting activity , e.g.  with the pharmacological inhibitor blebbistatin @xcite .    according to our model , the dynamics of minifilaments is strongly determined by the existence of at least two load bearing bound states of motors . for small load , the free energy bias allows the motors to perform the power stroke and complete the motor cycle ; for large load , however , the increase of the elastic energy exceeds the free energy bias and the power stroke can no longer be performed . because the free energy bias is limited by the energy gained in hydrolysis , this transition is inevitable , independent of the parameters . the exact value of the transition thresholds , however , depends on the stiffness @xmath193 of the neck linkers . this also means that the value of @xmath193 determines the range of external spring stiffness @xmath169 to which minifilaments are sensitive . the values for @xmath193 used in the literature range over an order of magnitude , @xmath251 @xcite . although the effect of varying @xmath193 is reduced by combining small @xmath193 with large values of the power - stroke length @xmath22 , small values as in ref . @xcite could lead to stalling of minifilaments before the transition of the power - stroke probability is reached . unless , however , the unbinding force scale @xmath252 is decreased as well , this stall force would not be enough to achieve stable attachment . another set of parameters determining the dynamics of minifilaments in our model are the on- and off - rates of motors , which determine the duty ratio . the load - free duty ratio ( @xmath253 ) of with the transition rates used here is comparable to values reported for non - muscle isoform b but significantly larger than for non - muscle isoform a or smooth muscle . changes of the transition rates will not change the outcome of the model qualitatively as long as the off - rate from the state is small compared to the off - rate from the state under load . smaller values of the duty ratio could be compensated by larger ensemble sizes in order to achieve the stabilization of minifilament attachment observed here . we have further shown with an analytical treatment for the case of minifilaments on rigid substrates that the effective diffusion constant decreases with ensemble size , in marked contrast to the case of the tug - of - war of processive motors , where the diffusion coefficient increases exponentially with the number of motors @xcite . processive motors are usually described as slip bonds favoring strongly asymmetric states , in which one of the ensembles is completely detached . with increasing number of motors , a tug - of - war of processive motors generates persistent movement in either direction , with exceedingly rare reversal of the direction of motion . this dependence of the diffusion coefficient on ensemble size is in line with the function of processive motors for transport , which becomes more efficient with increasing number of motors . non - processive motors , on the other hand , serve as active , force generating crosslinks which are not required to move through the cytoskeleton . an experimental realization of a tug - of - war of two ensembles of non - processive motors , e.g.  using a solid substrate covered by actin filaments with nematic order but random polarity or bipolar actin constructs on a surface covered with motors , could be used to investigate the characteristic difference between processive , slip - bond motors and non - processive , catch - bond motors . in general , we envision that biomimetic assays provide rewarding avenues to experimentally test our theoretical predictions .
tissue cells are in a state of permanent mechanical tension that is maintained mainly by minifilaments , which are bipolar assemblies of tens of molecular motors contracting actin networks and bundles . here 
 we introduce a stochastic model for minifilaments as two small 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 cross - bridge cycle . 
 our simulation and analytical results reveal a strong dependence of 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 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 . 
 + @xmath0correspondence : ulrich.schwarz@bioquant.uni-heidelberg.de