Actin filaments growing against a barrier with fluctuating shape
aa r X i v : . [ q - b i o . S C ] J un Actin filaments growing against a barrier with fluctuating shape
Raj Kumar Sadhu and Sakuntala Chatterjee
Department of Theoretical Sciences, S. N. Bose National Centre for Basic Sciences,Block JD, Sector III, Salt Lake, Kolkata 700106, India.
We study force generation by a set of parallel actin filaments growing against a non-rigid obstacle,in presence of an external load. The filaments polymerize by either moving the whole obstacle, witha large energy cost, or by causing local distortion in its shape which costs much less energy. The non-rigid obstacle also has local thermal fluctuations due to which its shape can change with time andwe describe this using fluctuations in the height profile of a one dimensional interface with Kardar-Parisi-Zhang dynamics. We find the shape fluctuations of the barrier strongly affects the forcegeneration mechanism. The qualitative nature of the force-velocity curve is crucially determined bythe relative time-scale of filament and barrier dynamics. The height profile of the barrier also showsinteresting variation with the external load. Our analytical calculations within mean-field theoryshow reasonable agreement with our simulation results.
PACS numbers: 05.40.-a, 87.16.aj, 87.16.Ka
I. INTRODUCTION
Cell motility plays an important role in a wide variety of biological processes like morphogenesis, wound healing ortumor invasion [1–4]. Actins and microtubules are cytoskeletal proteins whose polymerization and depolymerizationcan generate significant forces, without any assistance of molecular motors, and propel the cell forward. In presenceof a biological barrier, these filaments elongate and generate a pushing force against the barrier and in many in vitrostudies this force has been measured explicitly by applying an external load on the barrier in the opposite direction.With increasing load, the velocity of the barrier decreases and the functional nature of dependence of velocity on theapplied force is an important characteristic of the force generation mechanism. The maximum polymerization forcegenerated by the filaments is known as ‘stall force’ and is measured as the minimum load required in order to stallthe barrier motion completely. There has been a surge of experimental as well as theoretical research activities todetermine the stall force and the force-velocity characteristic of the cytoskeletal filaments in the last few years.Interestingly, the qualitative nature of the force-velocity curve was found to depend on the details of the experimentalset-up. A convex force-velocity characteristic was reported for actin quoted polystyrene beads [5] and magneticcolloidal particles pushed by unbranched parallel actin filaments [6, 7]. On the other hand, a concave force-velocitycurve was obtained for branched actin network [8], where velocity remains almost constant for small load and dropsrapidly at large load. An even more complex force-velocity relationship was measured for lamellipodial protrusionin a keratocyte, where velocity showed rapid decay for very small load, followed by a plateau at moderate load andanother rapid decay close to stalling [9, 10]. Although multiple filaments are expected to generate larger force thansingle filament [5, 9, 11], in [12] the stall force of approximately eight actin filaments was measured and found tobe in the piconewton range, close to a single filament stall force [13], indicating absence of co-operation among thefilaments.To investigate the force-velocity relationship theoretically, several different models have been proposed. Forcegeneration by a single actin filament growing against a barrier has been explained using a simple Brownian ratchetmechanism where thermal fluctuations of the barrier creates a gap between the barrier and the filament tip, makingit possible for the filament to grow by adding one monomer in the gap [14]. This mechanism predicts a convexforce-velocity curve. This simple model has been subsequently generalized where details of interaction between themonomers and the barrier has been considered [15] and flexibility of the filament has been included [16]. In allthese cases existence of a convex force-velocity relationship has been verified. However, when the Brownian ratchetmechanism was extended for multiple filaments, the nature of the force-velocity curve was found to crucially dependon how the details of the interaction and load-sharing among the filaments were modeled [17–20]. Certain modelseven showed a crossover from convex to concave force-velocity curve, as some model parameters are varied [21–23].Inside a cell, actin filaments grow against the plasma membrane which is not a rigid object but elastically deformable[24]. Even in vitro, when the filaments push against an obstacle as they polymerize, the obstacle may in general havelocal shape deformations. In [25] a flexible plasma membrane was explicitly modeled and it was shown that thermalfluctuation of this flexible obstacle substantially enhances the growth velocity of a filopodial protrusion. It wasargued that in the case of a flexible membrane, a filament only has to overcome the local bending energy in order topolymerize (whereas for a rigid obstacle the full load must be overcome) and this gives rise to a larger velocity for agiven load. Effect of a flexible plasma membrane on actin network growth was experimentally demonstrated in [26]when reconstituted actin networks in vitro were assembled onto synthetic lipid bilayers and it was found that themembrane elasticity causes formation of bundled filament protrusion from branched filament networks.Motivated by this, we carry out a study to probe the detailed quantitative aspects of interaction between a set ofgrowing filaments and an obstacle whose position as well as shape can fluctuate with time. To keep our descriptionsimple, we model the obstacle by a one dimensional non-rigid object whose local thermal fluctuations can alter itsshape and using a lattice gas model, we describe it by a Kardar-Parisi-Zhang (KPZ) interface [27]. In presence of anexternal load, the obstacle tends to move in the direction opposite to that of polymerization. In order to polymerize,the filaments must push against the barrier, either causing a local change in its profile (which requires less energy)or causing a global movement of the whole barrier (which involves a large energy cost). We are interested to findout how presence of the fluctuating barrier affects the dynamics of the actin filaments, and how the presence of thefilaments affects the shape of the barrier.Our numerical simulations and analytical calculations show that there is a rich interplay between the polymerizationdynamics of the filaments and the shape fluctuations of the barrier. For small and intermediate values of the externalforce, the barrier motion is governed by its global movement, and for large force, the local fluctuations becomeimportant. These local movements cost less energy and can continue even when the force is significantly large. As aresult, the stall force in our system is much higher than that for a rigid barrier [18]. Moreover, these local movementsmay be caused by filament polymerization or by independent thermal fluctuations of the barrier and hence the stallforce may also depend on the properties of the barrier. Indeed for a single filament, the stall force is found to increasewith the size of the barrier. For N filaments stall force is independent of the barrier size and scales linearly with N . The barrier shape is also affected by the growing filaments and the scaling behavior of its height profile showscontinuous variation as a function of the external load.There are two time-scales in our system, one associated with the (de)polymerization of the filaments and the otherwith the thermal fluctuations of the barrier. Our results show that the choice of these time-scales may cruciallydetermine the nature of the force-velocity curve. This is because the local movements of the barrier make increasinglyimportant contribution to its velocity as the thermal fluctuations become faster. Even for small or intermediate load,therefore, the barrier velocity is not governed by its global movement alone and this changes the qualitative natureof dependence of velocity on load. The stall force is also found to decrease for faster barrier dynamics.This paper is organized as follows. In section II, we describe our model. Our results for the single filament andmultiple filaments are presented in sections III and IV, respectively, and conclusions are in section V. II. DESCRIPTION OF THE MODEL
Our model consists of N parallel filaments growing against a barrier with a fluctuating height profile (see Fig. 1).We model the filaments as rigid polymers, made of rod-like monomers of length d , such that a (de)polymerizationevent (decreases) increases the length of the filament by an amount d . The barrier is modeled as a one dimensionalsurface. In our lattice model, the discrete surface elements are represented as lattice bonds of length λ , which canhave two possible orientations, ± π/
4. We denote these two cases by symbols / and \ and call them upslope anddownslope bonds, respectively. Height at any particular lattice site i is defined as h i = δ/ P i − j =1 tan θ j , where θ j isthe orientation of the j -th bond and δ = √ λ . The total number of such bonds is L . One / followed by a \ formsa local hill and in the reverse order \ / they form a local valley. The local height of the surface fluctuates due totransition between these hills and valleys. When a local hill (valley) at a given site flips to a valley (hill), the heightof that particular site decreases (increases) by an amount δ . We assume δ is equal to the monomer length d . Asexplained below, this assumption means that height fluctuation of the surface creates a gap which is just enough forinsertion of a monomer. Towards the end of the paper, we briefly discuss the case of δ = d .A filament whose tip is in contact with the barrier, is called a bound filament and in the absence of any suchcontact, it is called a free filament. The surface site where a bound filament can form a contact, is called a bindingsite. When a bound filament polymerizes, it creates space for insertion of another monomer by pushing the barrierup and in this process performs work against the external load (which tends to push the barrier down). When thebound filament pushes against a local valley, that valley flips to a hill and the height of the binding site increases byan amount d (Fig. 1A). However, polymerization of a bound filament, which is not in contact with a local valley,requires a global movement of the whole barrier, as shown in Fig. 1B, when height of all the L sites are increased byan amount d . Assuming F/L is the load per site, the energy cost for the first process is just
F d/L , and for the secondprocess it is
F d . Following the rule of local detailed balance, we assign rates U exp ( − βF d/L ) and U exp ( − βF d )to these two types of polymerization processes, respectively. Here, β is the inverse temperature and U is the freefilament polymerization rate that does not involve any barrier movement and hence is independent of F . We alsoassume the depolymerization rate is same for both free and bound filaments and is denoted as W . When a bound U exp(- (cid:1) Fd/L)
FF FF U exp(- (cid:0) Fd) W U R + R - (A) (B)(C) (D) FIG. 1: Schematic representation of our model. (A): Polymerization of a bound filament by causing a local change in barrierheight with rate U e − βFdL . (B): A bound filament polymerizes by causing global movement of the whole barrier with rate U e − βF d . (C): A free filament polymerizes and depolymerizes with rates U and W , respectively. Since these processes do notinvolve any barrier motion, these rates are independent of F . (D): Thermal fluctuation of the barrier: a local valley can flipto a hill with rate R + and the reverse process occurs with rate R − . We use local detailed balance, R + /R − = exp( − βF d/L ),except at the binding sites, where hill to valley transition may be blocked due to the presence of a filament. filament depolymerizes, it loses contact with the barrier and becomes a free filament. In certain configurations, whenthere is only one bound filament, its depolymerization results in an unsupported barrier.Apart from being pushed by the filaments, the barrier can also show thermal fluctuations, when local hills can flipto valleys and vice versa. However, due to presence of the filaments, these transitions can sometimes get blocked.For example, if a bound filament is in contact with a hill, then that particular hill cannot flip to a valley, until thefilament depolymerizes and a gap is created for a local downward movement of the barrier. When both forward andreverse transitions are allowed, their rates rates satisfy local detailed balance R + R − = e − βF d/L , where R + is the rate atwhich local surface height can increase (i.e. a valley flips to a hill) and R − be the reverse transition rate. Note thatin the absence of any external load F , the transition between hills and valleys become symmetric at all sites otherthan the binding sites and the surface has a local Edwards-Wilkinson dynamics [28]. For non-zero F , hill to valleytransitions are generally favored (except, possibly, at the binding site) and the barrier behaves like a KPZ surfacewith a downward bias.We assume periodic boundary condition for the surface and an equal number of upslope and downslope bonds, i.e. nooverall tilt. In one Monte Carlo step, we attempt to perform N filament updates (polymerization or depolymerization)and S independent (unaided by the filaments) surface updates. By changing the value of S we can tune the relativetime-scale between filament dynamics and barrier dynamics. For smaller (larger) S value, the barrier dynamics isslower (faster) than the filament dynamics. A relative time-scale between the surface and filament dynamics can alsobe introduced by rescaling R + and R − , but we have used R − = U and R + = U e − βF d/L throughout and controlledthe relative time-scale by S instead. We start with an initial configuration where all N filaments have unit length,containing one monomer each and the upslope and downslope bonds are placed alternatingly (a flat surface). We let V e l o c i t y ( n m / s e c ) Force (pN)(A) 0.0001 0.001 0.01 0.1 1 10 0 1 2 3 4 5 6 7 8 9 V e l o c i t y ( n m / s e c ) Force (pN) 4 5 6 7 8 20 40 60 100 200 400 F s ( p N ) L(B)
FIG. 2: Force-velocity characteristic and stall force for a single filament. (A): Force-velocity curve has a convex shape. Insetshows exponential decay of the barrier velocity for small and intermediate F , when the global motion of the barrier dominates.Close to stalling the local fluctuations become important. We have used L = 512 here. (B): Stall force increases with thebarrier size L . In both the panels, we have used S/L = 1. The free filament depolymerization rate W = 1 . s − [1, 29] and thepolymerization rate U is proportional to the free monomer concentration with a proportionality constant k = 11 . µm − s − [1, 29]. We have used a monomer concentration C = 0 . µm , which gives U = 2 . s − . The monomer size is d = 2 . nm [1, 23]. At room temperature the parameter βd = 0 . pN − . Discrete points show simulation data and continuous lines showanalytical results. the system evolve for a long time, according to above dynamical rules. All our measurements are performed in thesteady state. III. RESULTS FOR SINGLE FILAMENT
For a single filament, we first present the results for S = L and later we consider the effect of variation of S . Wedefine the velocity V of the barrier as the rate of change of the average height of the surface after the system hasreached steady state. We present the force-velocity curve in Fig. 2A. This curve has a convex shape where velocitydecays rapidly for small force, and for large force it decays slowly. In fact for small and intermediate values of force,the velocity falls off exponentially (Fig. 2A inset) and close to stalling it shows deviation from the exponential form.We explain below that the exponential dependence originates from the global movement of the barrier (as shown inFig. 1A) which dominates V for small and moderate F range. In Fig. 2B we show the variation of stall force F s with the barrier size L . Stall force increases with L , although logarithmically slowly. Note that the stall force is ofteninterpreted as the maximum polymerization force generated by the filament and therefore it is somewhat surprisingthat it depends on the size of the barrier. We show below that in our system the local fluctuations of the barrier,which depend on L , make substantial contribution towards its net velocity and this becomes particularly significantin the stalling regime.In our system there are two possible barrier movements: global and local. In a global movement, a bound filamentpolymerizes by pushing the whole barrier up, such that the average height changes by an amount d . The rate atwhich this process happens is U exp( − βF d ). Let this process contribute a velocity V to the barrier in the steadystate, which can be written as V = p dU exp ( − βF d ) . (1)Here, p is the probability that the filament is in contact with the barrier. Note that here we have ignored thepossibility that the bound filament is pushing against a valley (in that case no global movement takes place, only alocal flip is sufficient for polymerization). In fact we have verified in our simulation (data presented in Fig. A-1B )that the probability of finding a valley at the binding site is indeed small.To write V as a function of F we still need to calculate p . Define p i as the probability that the distance betweenthe filament tip and the binding site is i . Clearly, i = 0 corresponds to the contact probability. It is easy to see thatfor i >
0, the probability p i satisfies master equation for a biased random walker: dp i dt = W p i − + U p i +1 − ( W + U ) p i (2)and for i = 0 one has dp dt = U p − W p . (3)Here, we have ignored any change in p i due to height fluctuations at the binding site. For fast barrier dynamics,when height fluctuations increase, this assumption breaks down. In the steady state, these equations yield a recursionrelation p i = (cid:16) W U (cid:17) i p for positive i . This recursion relation, along with the normalization condition P i p i = 1 yieldsthe expression p = (1 − W /U ), which is independent of F . So the final expression for V becomes V = d ( U − W ) exp( − βF d ) . (4)To calculate the velocity due to local height fluctuations of the barrier, we consider a local valley (hill) flipping to ahill (valley) which increases (decreases) the average height by an amount d/L . As discussed in section 2, the transitionrates at the binding site is different from the rest of the system, since a hill to valley transition may be blocked, if afilament is in contact. Then the barrier velocity due to local height fluctuations can be written as V = dU L " (1 + p ) p v (0) + L − X i =1 p v ( i ) ! e − βF d/L − (1 − p ) p h (0) − L − X i =1 p h ( i ) (5)where p v ( i ) and p h ( i ) denote the probabilities to find a valley and a hill, respectively at a distance i from the bindingsite. In the above equation, the first term on the right-hand-side represent the situation where a valley at the bindingsite flips to a hill, due to thermal fluctuations or due to being pushed by the filament. The second term present flippingof a valley to a hill at all the other sites. The third term describe the case when there is a hill at the binding sitewhich can flip to a valley when no filament is in contact. The fourth term describe flipping of a hill to a valley in restof the system. The probabilities p v ( i ) and p h ( i ) can be calculated within a mean field approximation by consideringa KPZ surface with the binding site acting as a ‘defect site’ (see Appendix A for details), where the transition ratesare different from the rest of the system. Our calculations show that p v ( i ) and p h ( i ) have a rather weak dependenceon F and their difference [ p v ( i ) − p h ( i )] is independent of i and scales as 1 /L . For large L , the total velocity of thebarrier V = V + V can be written as V ( F ) = d ( U − W ) e − βF d + dU L " p v (0)(1 + p ) − (1 − p ) p h (0) + L − X i =1 { p v ( i )(1 − βdFL ) − p h ( i ) } (6)where we have retained terms upto order 1 /L and ignored higher order terms. In Fig. 2A we compare our calculationwith simulation results and obtain reasonably good agreement. For small F , the first term in Eq. 6 dominates thevelocity and as F increases, local fluctuations become more important. The last term in Eq. 6, within the braces,which represents the velocity due to hill-valley fluctuations at all sites, except the binding site, is the most dominantterm in the local movement. In the stalling region, the positive contribution from the global movement and thenegative contribution from the local fluctuations cancel each other, where the first and last terms of Eq. 6 determinethe major balance. The stall force F s can be obtained by graphically solving the above transcendental equation afterputting its left hand side zero. This gives stall force as a function of L and we compare this variation with simulationresults in Fig. 2B. We find good agreement for large L but as expected, for small L there are deviations. Note thatthe stall force in our system is substantially higher than that for a rigid barrier [18]. Since the local movements costmuch less energy, they can continue even when the load is high. A. Effect of faster and slower barrier dynamics
We find the nature of the force-velocity curve depends on the relative time-scale of the barrier and filament dynamics.For faster barrier dynamics, the local fluctuations of the barrier increases and as a result their contribution to thenet velocity is also higher. This means even for small force, the velocity is not dominated by the global movement(first term in Eq. 6) alone. In addition, our simple expression for the contact probability p = (1 − W /U ), whichwas derived neglecting the local fluctuations at the binding site, does not remain valid for fast barrier dynamics and V e l o c i t y ( n m / s e c ) F/F s (A) S/L=1/8S/L=1S/L=8S/L=64 4 6 8 10 12 14 16 18 0.001 0.01 0.1 1 10 F s ( p N ) S/L(B)
FIG. 3: Force-velocity characteristic for a single filament depends on the relative time-scale between the filament and thebarrier dynamics. (A): Velocity of the barrier vs scaled force for different values of
S/L . For large
S/L , the convex nature offorce-velocity characteristic is lost. As
S/L increases, the local fluctuations of the barrier become more important and even forsmall F , the barrier velocity is not governed by the global movement alone, and hence V does not decay exponentially anymore.Here, we have used L = 64. (B): Stall force decreases as a function of S/L . Since local movements of the barrier become moreimportant for large
S/L , the balance between global and local movements is reached at a smaller force. Note however, that the x -axis is plotted in a log-scale, indicating a weak dependence of stall force on the time-scale. Here we have used L = 256. Theother parameters are same as in Fig. 2. p increases with F in this case (see our data in Fig. B-1). As a result, the velocity does not decay exponentiallyfor small force, but follows a slower decay. For a given value of F, in the small or intermediate range, as the barrierdynamics becomes faster, the velocity becomes higher and the convex nature of the curve is gradually lost. Moreover,since stalling phenomenon in our system can be described as a balance between global and local velocities of thebarrier (see Eq. 6), larger contribution from local movement implies this balance is reached at a smaller value of force.Therefore, for faster barrier dynamics we have a smaller stall force. We present our data in Figs. 3A and 3B.Our data in Fig. 3B imply that in the limit of infinitely slow barrier dynamics, when the barrier can be consideredas an effectively rigid object, the stall force diverges. Note that even in this limit, our model remains different fromthe rigid barrier case studied in [18], where at least one filament is always bound to the barrier. For N = 1 thiswould mean whenever there is a depolymerization, the barrier also moves down, along with the filament tip. On thecontrary, we allow unsupported barrier in our system and when the barrier is effectively rigid, it shows only globalmovement which is always in the upward direction. The force velocity curve is perfectly exponential in this case andzero velocity is reached at F → ∞ limit. B. Variation of the shape of the barrier with load
We have seen above how the barrier fluctuations affect the growth of the filament. The barrier properties are alsoaltered in this process. As the load increases, the height profile of the barrier shows larger variation across the system.We characterize it by measuring the scaling of average height with distance from the binding site: h h ( r ) − h (0) i ∼ r α ,where h ( r ) is the height of a site at a distance r from the binding site. In Fig. 4 we plot α as a function of theexternal force, which shows that for small force α increases slowly, around the stalling force there is a sharp increaseand finally for very large force, α saturates to unity. Note that large value of α indicates presence of large hills andvalleys in the system. α = 1 corresponds to a phase separation of upslope and downslope bonds in the system whichgives rise to one single large hill, the highest point being the binding site. This situation is similar to the case of anelastic membrane, when the membrane tension is large and the membrane is stretched. IV. RESULTS FOR MULTIPLE FILAMENTS
In the case of N filaments in the system, we mainly consider the case when the ratio N/L is small. We assume thebinding sites are uniformly placed on the lattice, at a distance
L/N . Between the segment of two successive binding α F/F s FIG. 4: Variation of α as a function of external load. Close to the stalling force, α shows a sharp increase. Here, we have used S/L = 1 and L = 256 (red triangle) and 128 (blue circle). Other simulation parameters are same as in Fig. 2. sites, the same considerations as in a single filament case apply. We assume these segments are independent and applyour results for the single filament case for each segment.To start with, we consider the velocity of the barrier due to its global movement V = p N dU exp( − βF d ). Asbefore, p is the probability to find a filament in contact with the barrier and p N is the average number of boundfilaments in the system. Here, we have neglected any correlation between the binding sites. To calculate p , we writedown master equations for average number N i of filaments at a distance i from the corresponding binding sites. Thesteady state solutions of these equations can be obtained recursively for different values of i (see appendix C fordetails). For N filaments we have p = (1 − W /U )1 + ( N −
1) exp( − βF d ) . (7)For large F , the contact probability becomes same as the single filament case. For small F , the contact probabilityis approximately 1 /N times the single-filament value, indicating that for small F , at most one filament is in contactwith the barrier.For the local movement of the barrier, we need to calculate the probability to find hills and valleys. As discussedabove, for each segment between two successive binding sites, we use our results for p v ( i ) and p h ( i ) for the singlefilament case (with the modification that i in this case varies from 0 to ( L/N − V = N dU L p v (0)(1 + p ) + L/N − X i =1 p v ( i ) e − βF d/L − (1 − p ) p h (0) − L/N − X i =1 p h ( i ) (8)The total velocity to leading order in 1 /L and N/L becomes V ( F ) = d ( U − W )1 + ( N − e − βF d N e − βF d + dU NL { p v (0)(1 + p ) − p h (0)(1 − p ) } + L/N − X i =1 (cid:26) p v ( i ) (cid:18) − βF dL (cid:19) − p h ( i ) (cid:27) (9)The stall force can be obtained by solving the above transcendental equation graphically for V ( F ) = 0 and we comparethe analytical stall force with our simulation results in Fig. 5A inset. We find that the stall force is independent of L in this case and scales with N , which can be easily seen from Eq. 9. Since the value of the stall force is rather large inthis case, one can neglect global movement of the barrier close to the stalling regime. In addition, p ≈ (1 − W /U )for large force, and ( p v ( i ) − p h ( i )) is of order N/L . Using these in Eq. 9 it directly follows that the stall force for N filaments is independent of L and scales as N . We also investigate the effect of the time-scale of the barrier dynamicson the force-velocity dependence (Fig. 5B) and we find qualitatively the same effect as in N = 1 case. V e l o c i t y ( n m / s e c ) Force (pN)(A) 50 100 10 20 30 40 50 60 F s ( p N ) NL=512L=256 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 V e l o c i t y ( n m / s e c ) F/F s (B)S/L=1/64S/L=1S/L=64 20 40 60 80 100 0.01 0.1 1 10 100 F s ( p N ) S/L
FIG. 5: Force-velocity characteristic for multiple filaments. (A): Velocity shows very slow decay for large F , when globalmovement can be neglected and V can be assumed to be governed by local fluctuations alone. Here, we have used L = 512and N = 32. Inset shows stall force as a function of N for two different L values. We find stall force scales linearly with N and remains independent of L . The continuous lines show analytical results. (B): Dependence of force-velocity characteristicon the time-scale of the barrier dynamics. In this case we find same qualitative effect as in the single filament case. Here, wehave used N = 16 and L = 128. V. CONCLUSIONS
In this paper, we have studied force generation by a set of parallel filaments polymerizing against a barrier. Asimilar question has been addressed in many recent works where the barrier was modeled as a rigid wall, whichmay have a motion like a thermal ratchet [14, 15, 30, 31], or may be a passive obstacle which can move only whenpushed by the filaments [18, 20, 23, 32–34]. In this paper, we have considered a barrier with thermal fluctuationsbut instead of modeling it as a rigid wall, we allow for its shape fluctuations. In [35] a similar aspect was studiedwhere the barrier was modelled by a one dimensional Edwards-Wilkinson type membrane under tension, which wasbeing locally pushed by a set of growing filaments. The uncorrelated drive from the filaments gives rise to a KPZtype behavior in the correlated height fluctuations of the membrane, but this is associated with very slow crossover.Interestingly, the steady-state fluctuations of the driven membrane shows a non-monotonic behavior with the drivingrate, where the strongly driven and weakly driven regimes are separated by a minimum in the width of the membraneprofile. Although the filaments only impart local drive to the membrane, and no global movement of the membraneis considered in [35], the velocity still shows an exponential dependence on the membrane tension, whereas in ourmodel the exponential dependence is caused by the global movement and the local fluctuations generate a velocitythat decreases roughly linearly with the external load.One interesting result obtained in our system is the dependence of the qualitative shape of the F - V curve on therelative time-scale between the filament polymerization and barrier fluctuation. For slow barrier dynamics, the curvehas a convex shape and V shows an exponential decay for small and moderate F . But for fast barrier dynamics whenthe local fluctuations become more important, there is significant deviation from exponential dependence. A similareffect was reported in [21] for a hybrid mesoscopic model that combines the microscopic dynamics of semi-flexibleactin filaments and the viscous retrograde flow of actin network modeled as a macroscopic gel. It was shown thatthe force-velocity curve can be both convex and concave, depending on the characteristic time-scale of recoil of thegel-like network. It is remarkable that our simple lattice gas model can reproduce this same effect, which underlinesthe importance of the relative time-scale of obstacle and filament dynamics on the force generation mechanism.Throughout this paper, we have considered the case δ = d , when the local movement of the barrier occur in stepswhose size is equal to that of a monomer. We have verified (data not shown here) that many of our qualitativeconclusions remain valid for δ ≪ d . In other words, even when the shape fluctuations of the barrier occur over muchsmaller length scales, their effect cannot be ignored. We find that the stall force continues to show dependence on thebarrier properties. The relative time-scales between the filament and barrier dynamics affects the F − V curve in thesame way. However, the quantitative value of the stall force increases as smaller δ values are considered.Finally, our simple model shows that a non-rigid obstacle can produce remarkable effects on force generation ofparallel actin filaments. Our results underline the importance of the local shape distortions of an obstacle and indicatethat more research with detailed modeling of this aspect is required. Many of our conclusions are generic and canbe expected to remain valid in systems where different descriptions of a non-rigid obstacle are used. This also opensup the possibility of observing some of these effects in experiment. For example, the change of shape of the barrierwith external load can be monitored in an experiment and our prediction that the height variation across the barrierincreases with load, can be explicitly verified. The key feature of a fluctuating barrier is that one component ofvelocity comes from the local fluctuations and a direct measurement of this component will surely give insights intothe effects of barrier fluctuations. Our model shows that for multiple filaments close to stalling regime, velocity isdominated by these local movements and we also predict the scaling behavior of this velocity with filament densityand barrier size. It would be interesting to verify these predictions in experiments, which would not only shed lighton the qualitative nature of the local fluctuations but would also provide insights about their quantitative behavior. VI. ACKNOWLEDGEMENTS
The computational facility used in this work was provided through Thematic Unit of Excellence on ComputationalMaterials Science, funded by Nanomission, Department of Science and Technology, India.
Appendix A: Calculation of p v ( i ) and p h ( i ) for single filament The shape of the barrier changes due to transition between local hills and valleys. The probability to find a hill ata site s located at a distance i from the binding site is p h ( i ) and it can be written as ρ i (1 − ρ i +1 ), where ρ i is theprobability that the bond preceding the site s has π/ − ρ i +1 ) is the probability that the bondimmediately after the site s has − π/ s can similarly be written as (1 − ρ i ) ρ i +1 . The transitionrate from a hill to a valley is R − and the reverse process occurs with rate R + . For i = 0, R + /R − = exp( − βF d/L ).However, when i = 0, or, in other words, the site s is the binding site itself, then although valley to hill transitionis not affected, the reverse transition can take place only when the filament is not in contact with the binding site.We therefore make the simplifying assumption that the effect of the filament can be included by merely rescaling thehill to valley transition rate at the binding site by the probability that the filament is in contact. In section III wecalculate the contact probability p = 1 − W /U ≃ /
2. The master equations describing the time-evolution of ρ i can then be written as dρ i dt = (1 − ρ i )( R − ρ i − + R + ρ i +1 ) − ρ i [ R − (1 − ρ i +1 ) + R + (1 − ρ i − )] , for 2 ≤ i ≤ L − dρ dt = (1 − ρ )[ R − (1 − p ) ρ L + R + ρ ] − ρ [ R − (1 − ρ ) + R + ρ (1 − ρ L )] , (A-2)where we have applied periodic boundary condition, which also gives dρ L dt = (1 − ρ L )( R − ρ L − + R + ρ ) − ρ L [ R − (1 − ρ )(1 − p ) + R + (1 − ρ L − )] . (A-3)We solve the above equations in the steady state when the left hand sides vanish. To leading order in 1 /L , we find ρ i = a + bi/L , where a and b are related via the condition P Li =1 ρ i = L/ b satisfies the quadratic equation (cid:20) βF d L − p (cid:16) − L (cid:17)(cid:21) b + (cid:20) − βF d L − p (cid:16) − L (cid:17)(cid:21) b + 14 (cid:16) βF dL − p (cid:17) = 0 , (A-4)one of whose roots can be discarded from the condition that 0 ≤ ρ i ≤ i . For a given F , therefore, ρ i varies linearly with the distance from the binding site with a gradient 1 /L . For F = 0, we have a = ( √ −
1) and b = (3 − √ ≤ F ≤ F s , the range of variation of a and b are rather small and occur at third or higher decimalplaces. Therefore, ρ i does not change significantly with F . Our simulation data in Fig. A-1A show similar qualitativebehavior, although close to the binding site there is deviation of ρ i from linearity. The quantitative values of a and b however, do not match with simulations. We attribute this mismatch to the mean field theoretic assumptions usedin our calculation.We calculate p v ( i ) and p h ( i ) from ρ i and compare with simulation in Fig. A-1B. Notice that from our analyticalexpression for ρ i , it follows immediately that ( p v ( i ) − p h ( i )) is independent of i and ∼ b/L . This has importantconsequence for our calculation of V in section III. Moreover, the probability that the filament is in contact with avalley is given by p v (0) p and our numerical results in Fig. A-1B show that this probability is rather small.0 ρ i i/L(A) 0.12 0.16 0.2 0.24 0 0.1 0.2 0.3 0.4 0.5 p v ( i ) i/L(B)-6-4-2 0 2 4 6 0.2 0.4 0.6 0.8 1 [ p v ( i )- p h ( i ) ] L i/L(C) -160-120-80-40 0 0 0.01 0.02 [ p v ( i )- p h ( i ) ] L i/L(D) FIG. A-1: Average shape of the barrier for single filament. Discrete points show simulation results and continuous lines showanalytical predictions. (A): Probability ρ i to find an upslope bond as a function of scaled distance i/L from the binding site. ρ i = 1 / i = L/ i , we have ρ i = 1 − ρ i − L/ . The open symbols correspond to F = 0 and the close symbolscorrespond to F = 4 pN . Symbols ∗ and ◦ are for L = 128 and × and ✷ are for L = 256. These data show that, except closeto the binding site, ρ i increases linearly with i with a gradient ∼ /L . We also find that ρ i remains almost same for these F values. The continuous lines are analytical predictions, where green solid line is for F = 0 and blue dashed line is for F = 4 pN .(B): Probability p v ( i ) to find a valley at a distance i from the binding site. For i = 0 the probability is substantially smallercompared to the rest of the system, which means it is rather unlikely to find a valley at the binding site. The symbols ∗ and∆ represent F = 0 pN and 4 pN , respectively. We have used L = 512 here. (C) and (D): [ p v ( i ) − p h ( i )] shows a sharp jump at i = 0 and then remains constant at a value that scales as 1 /L . The open symbols correspond to F = 0 and the closed symbolscorrespond to F = 4 pN . Symbols ∗ and ◦ are for L = 256 and × are ✷ are for L = 512. Appendix B: Variation of contact probability for a single filament with load for fast and slow barrier dynamicsAppendix C: Calculation of contact probability for multiple filaments
Let N i be the average number of filaments at a distance i from the respective binding sites. By definition, N isthe average number of bound filaments and the contact probability is p = N /N . The time-evolution equations for N i can be written as dN dt = U N − { ( N − U e − βF d + W } N , (C-1) dN dt = { ( N − U e − βF d + W } N + U N − ( N U e − βF d + W + U ) N , (C-2) dN i dt = ( N U e − βF d + W ) N i − + U N i +1 − ( N U e − βF d + W + U ) N i for i ≥ . (C-3)1 p F/F s S/L=1/8S/L=1S/L=8S/L=64 [ht]FIG. B-1: Contact probability p as a function of F for single filament. Our analytical calculation yields p = (1 − W /U ) ≃ . p increases with F . The simulation parameters are as in Fig. 2. Here, we have assumed that the distance i between the filament tip and the binding site can change only due topolymerization and depolymerization dynamics and the global movement of the whole barrier due to polymerizationof bound filaments. We have neglected local height fluctuations occurring at the binding sites. As we show below,this approximation works reasonably well as long as the filament density N/L is small and the time-scale of barrierfluctuation is comparable to, or slower than the filament dynamics. For very fast motion of the barrier, the heightfluctuations at the binding sites become more frequent and this assumption breaks down.Solving the Eqs. C-1, C-2, C-3 in the steady state, we obtain the recursion relation N i +1 = (cid:18) N U e − βF d + W U (cid:19) i N ; i = 1 , , ... (C-4)and N = ( N U e − βF d + W − U e − βF d ) U N (C-5)Using the normalization relation, P N i = N we get N = N ( U − W ) U − U e − βF d + N U e − βF d (C-6)and the contact probability has the form p = ( U − W ) U +( N − U e − βFd . In Fig. C-1 we compare this result with simulationand find reasonable agreement. [1] J. Howard, Mechanics of motor proteins and the cytoskeleton, Sunderland, MA: Sinauer Associates (2001).[2] T. D. Pollard and J. A. Cooper, Actin, a central player in cell shape and movement, Science , 1208 (2009).[3] L. Blanchoin, R. B. Paterski, C. Sykes and J. Plastino, Actin dynamics, architecture and mechanics in cell motility, PhysiolRev , 235 (2014).[4] P. Friedl and D. Gilmour, Collective cell migration in morphogenesis, regeneration and cancer, Nat. Rev. Mol. Cell Biol. , 445 (2009).[5] Y. Marcy, J. Prost, M. F. Carlier, and C. Sykes, Forces generated during actin-based propulsion: A direct measurementby micromanipulation, Proc. Natl. Acad. Sci. U.S.A. , 5992 (2004).[6] C. Brangbour, O. du Roure , E. Helfer, D. D´emoulin, A. Mazurier, M. Fermigier, M. F. Carlier, J. Bibette and J. Baudry,Force-velocity measurements of a few growing actin filaments, PLoS Biol. , e1000613 (2011). N Force (pN) S/L=1/32S/L=1S/L=32
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