Stress relaxation in F-actin solutions by severing
S. Arzash, P.M. McCall, J. Feng, M.L. Gardel, F.C. MacKintosh
SStress relaxation in F-actin solutions by severing
S. Arzash,
1, 2
P.M. McCall,
3, 4, 5, 6, 7
J. Feng, M.L. Gardel,
3, 4, 8 and F.C. MacKintosh
1, 2, 9, 10 Department of Chemical & Biomolecular Engineering, Rice University, Houston, TX 77005 Center for Theoretical Biological Physics, Rice University, Houston, TX 77030 Department of Physics, University of Chicago, Chicago, IL 60637 James Franck Institute, University of Chicago, Chicago, IL 60637 Max Planck Institute of Molecular Cell Biology and Genetics,Pfotenhauerstraße 108, 01307 Dresden, Germany Max Planck Institute for the Physics of Complex Systems, N¨othnitzerstraße 38, 01187 Dresden, Germany Center for Systems Biology Dresden, Pfotenhauerstraße 108, 01307 Dresden, Germany Institute for Biophysical Dynamics, University of Chicago, IL 60637 Department of Chemistry, Rice University, Houston, TX 77005 Department of Physics & Astronomy, Rice University, Houston, TX 77005
Networks of filamentous actin (F-actin) are important for the mechanics of most animal cells.These cytoskeletal networks are highly dynamic, with a variety of actin-associated proteins thatcontrol cross-linking, polymerization and force generation in the cytoskeleton. Inspired by recentrheological experiments on reconstituted solutions of dynamic actin filaments, we report a theoreticalmodel that describes stress relaxation behavior of these solutions in the presence of severing proteins.We show that depending on the kinetic rates of assembly, disassembly, and severing, one can observeboth length-dependent and length-independent relaxation behavior.
INTRODUCTION
Networks of actin filaments (F-actin) constitute a keycomponent of the cytoskeleton of most animal cells.This cytoskeleton governs the organization and mechan-ics of cells, as well as a variety of transport properties.Actin filaments are double helical chains of globular actinmonomers (G-actin). These filaments exhibit molecularpolarity by their head-tail arrangement. Their two endsare referred to as barbed and pointed. This polarity is akey feature of filamentous actin in the cytoskeleton andis essential for a variety of cellular processes such as cellmotility [1, 2]. Actin filaments show dynamic associationand dissociation from both their barbed and pointed ends[3]. Under physiological conditions, there is net polymer-ization of the barbed end and net depolymerization ofthe pointed end, resulting in steady-state filament tread-milling [4], which we assume throughout this paper. Thepolymerization, cross-linking, branching and dynamics ofthe actin cytoskeleton are governed by a variety of asso-ciated proteins. Among these are severing proteins suchas ADF/cofilin, which play an important role in the re-cycling and turn-over of actin monomers [5, 6]. Figure1 shows a simplified sketch of an actin filament with thekey reactions.These polymerization, depolymerization and severingreactions result in a steady-state described by a time-independent distribution of filament length or molecularweight. This steady-state is necessarily dynamic; thelength distribution is set by the steady-state reactionrates, which are themselves tuned by the concentrationsof different components. Interestingly, the steady-stateis also driven away from equilibrium. Conformationaldifferences between actin monomers in filaments F-actin
FIG. 1. Sketch of an actin filament and its key molecu-lar reactions. The notation P and Q are used to track totalfilament length distribution and hence finding the stress re-laxation behavior. In our model, we assume a constant netpolymerization rate r of P filaments and a constant net de-polymerization rate γ of Q fragments. ATP-actin is convertedto ADP-actin at the same rate r, such that only a single ATP-actin subunit is present per filament and located at the fila-ment barbed end. We assume a uniform severing rate of α along the length of the filament. Using the tube model of en-tangled polymeric systems, we claim that polymerizing newand stress-free subunits (The green section of tube) have noeffect on relaxation of initial stress. As we will show, severingreaction has a large effect on changing the initial tube andrelaxing the initial stress. and actin monomers in solution G-actin result in a morethan 10 -fold increase in the hydrolysis rate of adeno-sine triphosphate (ATP) bound to F-actin vs G-actin [7].ATP hydrolysis on filaments introduces chemically dis-tinct actin species into the system, which participate inthe polymerization, depolymerization, and severing reac-tions with distinct rate constants. Crucially, the effec-tively irreversible nature of ATP hydrolysis breaks de-tailed balance, resulting in a net flux of ATP-actin intofilaments and thus non-equilibrium steady-state dynam-ics. While this non-equilibrium flux, measured experi- a r X i v : . [ c ond - m a t . s o f t ] J u l mentally as the actin turnover rate, is typically very smallfor purified actin in the absence of regulatory proteins,the presence of ADF/cofilin has been shown to increasethe steady-state flux more than 20-fold [8, 9].Recent experimental studies on reconstituted actin so-lutions have shed light on various aspects including themechanical behavior of cytoskeletal systems undergo-ing non-equilibrium turnover [9]. Specifically, rheologi-cal measurements of F-actin networks and solutions inthe presence of various actin-associated proteins haverevealed regimes with both elasticity and stress relax-ation. Stress relaxation in solutions of high molecular-weight polymers typically depends on reptation, in whichpolymers diffuse along their contour, subject to the con-straints provided by neighboring polymers [10]. Stressrelaxation due to reptation is typically very slow athigh molecular weight or polymer length L , with acharacteristic relaxation time τ r ∼ L . Polymeriza-tion/depolymerization reactions can also lead to stressrelaxation. Since the resulting treadmilling is directed,the corresponding relaxation time is expected to vary as τ r ∼ L , as previously shown [11].By adding cofilin, however, a length-independent re-laxation time is observed [9]. In order to explain thisexperimental observation, we develop a minimal theoret-ical model of the actin length distribution depending onsevering and (de)polymerization. We then extend this todetermine the time-dependent stress relaxation from thedynamic filament length distribution. We find that oursimple model predicts three distinct relaxation regimes,including two regimes in which the relaxation rate is ex-pected to be independent of average filament length ormolecular weight. These regimes are summarized in Fig.2. A natural characteristic length scale in a polymericnetwork is the entanglement length L e where polymerchains shorter than this length move easily through thenetwork without being constrained by neighboring chains[10]. Another characteristic length scale arises from thecompetition of the depolymerization reaction of Q frag-ments (Fig. 1) and the severing reaction of filaments:we define this depolymerization length scale L d = (cid:112) γα ,where γ is the net depolymerization rate (in units oflength per time) of Q fragments and α is the rate ofsevering per length. This is a length for which the de-polymerization time is comparable to the time betweenconsecutive severing events. Likewise, a characteristicpolymer length can be identified as (cid:112) rα , where r is thenet polymerization rate (in units of length per time) of P filaments (Fig. 1). For this length, the time between twoconsecutive severing events is comparable to the time topolymerize the filament.We find that the stress relaxation behavior of actin so-lutions depends on the relative magnitudes of three char-acteristic length scales: the depolymerization length L d ,the entanglement length L e , and the initial average fil-ament length (cid:104) L (cid:105) . In the limit of instant disassembly of fragments, the stress relaxation is length-dependentwith a characteristic timescale inversely proportional tothe initial average length (Regime I in Fig. 2). On theother hand, for very slow rates of fragment disassembly γ , the characteristic timescale during stress relaxation isinversely proportional to L e which is shown as regime IIIin Fig. 2. Moreover, for intermediate rates of fragmentdepolymerization where L e < L d < (cid:104) L (cid:105) , the relaxationtime behaves as τ ∼ /αL d (Regime II in Fig 2). Asthe average filament length becomes comparable to orsmaller than the entanglement length, the actin networkbehaves as a viscous fluid. This regime is denoted as asolution in Fig. 2a and b. Moreover, for large depolymer-ization length L d , i.e., for very small severing rate α → (cid:104) L (cid:105) > L e , the solution’s behavior is dominated by reptation [10]. This regime is better understood by usingthe severing rate α directly in the phase diagram (seeFig. 2b). We estimate the boundaries between regimesI & II and regimes II & III by equating the relaxationtime scaling relationships (displayed in Fig 2a) for eachregime pair, and solving for α as a function of (cid:104) L (cid:105) . Sim-ilarly, we estimate the boundaries between the reptationregime and each of regimes I, II, and III by equating therelaxation time scaling relationship for each regime withthe reptation time τ r = (cid:104) L (cid:105) /D r where D r = k B T /ζ (cid:104) L (cid:105) , k B is the Boltzmann constant, T is temperature, and ζ is the drag coefficient per unit length.In the following sections, we study both the steady-state length distribution, as well as the correspondingdynamics of stress relaxation. In both cases, we considertwo limits: (1) very rapid fragment disassembly, corre-sponding to the limit γ → ∞ and (2) finite disassem-bly. The steady-state length distribution of F-actin withsevering has been considered previously in Refs. [12–14].Refs. [12, 13] introduced a model for severing by Gel-solin, in which the two fragments ( P and Q in Fig. 1)were equivalent, corresponding to γ = 0 in our modelbelow. The limit of instantaneous disassembly of frag-ments without an ATP-cap (fragment Q ), correspondingto γ → ∞ in our model, has recently been examined inRef. [14]. In this limit, the average filament length (cid:104) L (cid:105) is proportional to the characteristic length (cid:112) rα . We ex-tend the approach introduced in Refs. [12, 13] to accountfor finite disassembly rates γ of unstable fragments. Theprior models, however, only considered the steady-statelength distribution and not the dynamics of stress re-laxation. A simplified model for stress relaxation wasrecently introduced in Ref. [9] for the limit of no disas-sembly ( γ = 0). In the presence of disassembly, the twofragment species must be considered: those with ( P ) andwithout ( Q ) ATP-actin at the barbed ends. (b)(a) FIG. 2. Phase diagram of stress relaxation in actin solutions.(a) Schematic phase boundaries of stress relaxation behaviorin terms of the solution’s characteristic length scales, ignor-ing reptation. When the initial average length (cid:104) L (cid:105) is less thanthe entanglement L e , the system is in the solution state whichis understood by hydrodynamic laws. In the case of instantevaporation of short fragments, the stress relaxation stronglydepends on the initial average length (cid:104) L (cid:105) (Region I). RegionIII shows a length-independent relaxation behavior where en-tanglement length is less than the initial average length butlarger than the depolymerization length L d < L e < (cid:104) L (cid:105) , i.e.,very slow disassembly rate γ of fragments. In this regime,our model predicts a relaxation time which is inversely pro-portional to the entanglement length. By increasing the dis-assembly rate γ to a point where L e < L d < (cid:104) L (cid:105) , we findthat the relaxation time is determined by L d as sketched inregion II. (b) Same phase diagram as in (a) but accountingfor reptation and now in terms of experimentally-measurablesevering rate α and initial average length (cid:104) L (cid:105) in dimensionalunits. We used the entanglement length L e = 0 . µ m and netdepolymerization rate of γ = 0 . µ ms − . The regime bound-aries in (b) are estimated by equating the relaxation timesfor each pair of regimes and using the scaling relationshipsfrom (a) to determine the functional dependence of α on (cid:104) L (cid:105) for each boundary. The reptation timescale is estimated as τ r = ζ (cid:104) L (cid:105) /k B T with ζ = 3 π × − pN/nm and k B T = 4 . STEADY STATE LENGTH DISTRIBUTION
Assuming a constant pool of monomers, each of unitlength, we calculate the steady-state length distributionof actin filaments resulting from the addition and sub-traction of monomers by polymerization, depolymeriza-tion, and severing reactions (see Fig. 1). Two distinctlimits of depolymerization rate γ are studied here. In thecase of very large depolymerization rate, the ADP-richfragments formed by severing reactions dissolve rapidlyand do not contribute to the filament length distribution.On the other hand, for finite γ we obtain the distribu-tions for both P (stable filaments with ATP barbed end)and Q (less stable fragments with ADP barbed end) asshown in Fig. 1.In order to remain analytically tractable in the face ofthe large number of distinct reactions, our model makes anumber of simplifying approximations. Specifically, actinbinding proteins (e.g. cofilin, profilin and formin) aretreated implicitly via corresponding reaction rates, whichare treated in a mean-field manner. The monomer poolis assumed to be exclusively ATP-bound G-actin and tobe constant in time. Filaments are assumed to be com-posed of ADP-bound actin subunits, except for a singleATP-bound subunit located at the barbed end of eachP filament. The rate of filament severing is assumed tobe uniform along the chain and equal for P and Q fila-ments. Filament annealing is neglected and nucleation isassumed to occur in steady-state at a rate proportionalto the monomer concentration. Many of these approx-imations are motivated by the conditions of recent ex-periments [9] containing high concentrations of the pro-teins profilin and formin, which regulate actin assemblyat barbed ends. Unstable Fragments: γ → ∞ By assuming rapid depolymerization of unstable frag-ments after severing, we are able to write the masterequation for filament length distribution P L . One of thekey assumption in our model is a uniform rate of severingreaction along every fiber, i.e., we assume equal proba-bility of severing event happening on any site betweenadjacent monomer units. Hence, the master equation inpresence of severing reaction is as following˙ P L = − α ( L − P L + α ∞ (cid:88) m =1 P L + m − rP L + rP L − , (1)where P L represents the number of filaments of length L and α and r are severing and polymerization rates,respectively. Here, for L = 1 the final term in Eq. (1) isabsent. The number P L of filaments of length L decreasesby severing, which can occur at any of L − L + 1. This number can also increase by severingof longer filaments, or by the addition of single monomersto a filament of length L −
1. This master equation hasbeen solved for the steady state condition (each ˙ P L = 0)using a recursive method [14]. Here, we solve this usinga continuous approach similar to Ref. [12]. In addition tothe steady-state solution, this method enables us to findthe dynamic solution needed for the relaxation behaviorin the subsequent section. The continuous form of Eq. (1)using F ( (cid:96), t ) as the continuous probability distribution isgiven by ∂F ( (cid:96), t ) ∂t = − α(cid:96)F ( (cid:96), t ) + α (cid:90) ∞ (cid:96) F ( s, t ) ds − r ∂F ( (cid:96), t ) ∂(cid:96) (2)By defining a new variable, V ( (cid:96), t ) = (cid:82) ∞ (cid:96) F ( s, t ) ds , Eq.(2) becomes − ∂ V ( (cid:96), t ) ∂t∂(cid:96) = α(cid:96) ∂V ( (cid:96), t ) ∂(cid:96) + αV ( (cid:96), t ) + r ∂ V ( (cid:96), t ) ∂(cid:96) (3)The steady state solution of this equation is obtainedusing the normalization condition for the probabilities V ( (cid:96) = 0 , t ) = 1 and also using the fact that the probabil-ity distribution is a bounded function V ( (cid:96) ) = exp (cid:0) − α(cid:96) r (cid:1) (4)Thus, the continuous distribution is F ( (cid:96) ) = αr (cid:96) exp (cid:0) − α(cid:96) r (cid:1) (5)This is indeed a Rayleigh distribution with the scaleparameter as (cid:112) rα . Therefore, the average steady-statefilament length is calculated as (cid:104) L (cid:105) = (cid:90) ∞ (cid:96)F ( (cid:96) ) d(cid:96) = (cid:114) π rα (6)Higher polymerization rates or smaller severing rates re-sults in a larger average length. This natural length scaleis a key parameter for determining the overall stress re-laxation behavior, as shown below. Role of fragments: finite γ At finite depolymerization rate, the fragments Q con-tribute to the overall length distribution, which affectsboth steady state and dynamic length distributions. Al-though previous models have ignored these fragments[12–14], we show that including these fragments canstrongly affect both steady state distributions and stressrelaxation. In order to find the total length distributionof actin filaments, we track filaments P and fragments Q separately. In addition to Eq. (1), which is unchanged,we also consider the master equation for Q L :˙ Q L = − α ( L − Q L + α ∞ (cid:88) m =1 (2 Q L + m + P L + m ) (7) − γQ L + γQ L +1 In contrast to P L , the distribution Q L is affected by dis-assembly ( γ ), rather than assembly ( r ). Moreover, al-though stable filaments P can only come from severingof stable filaments, fragments ( Q ) can arise from the sev-ering of either stable filaments or fragments. The factorof 2 in the severing term in Eq. (7) is due to the fact that,unlike stable filaments, there are two sites on a fragmentlonger than L which result in a fragment of length L aftersevering. The two sets of coupled master equations areneeded for a complete model. By subtracting two con-secutive terms of P in Eq. (1) and also Q in Eq. (7), weare able to establish the following recursive relations P L +1 = (cid:18) α ( L −
1) + 2 rα ( L + 1) + r (cid:19) P L − (cid:18) rα ( L + 1) + r (cid:19) P L − (8) Q L +2 = (cid:18) α ( L + 2) + 2 γγ (cid:19) Q L +1 − (cid:18) α ( L −
1) + γγ (cid:19) Q L + (cid:18) αγ (cid:19) P L +1 (9)These recursion relations provide the steady-statelength distribution of filaments. Each recursion relationrequires two boundary conditions to fully specify the dis-tributions. We generate the P filament distribution byforward recursion of Eq. (8), and thus require boundaryconditions on P L for two sequential and small values L .Rather than finding conditions on P and P , we takeadvantage of the fact that P is not physically meaning-ful and use P = 0 as one boundary condition in Eq.(8). The second boundary condition is on P , which wespecify below. We note that the steady-state length dis-tribution of P filaments is a function of P . To solve theequation for Q , we use backward recursion since we knowthat the tail of the Q distribution goes to zero. As withsimilar recursion relations arising from second order lin-ear differential equations, we can expect two solutions.Since only the growing solution under backward recur-sion (i.e., the decaying solution under forward recursion)is physical, the result should be insensitive to the initialchoice apart from an overall prefactor, provided that therecursion is started sufficiently far into the tail. In par-ticular, we use the two boundary conditions Q N = 0 and Q N − = 0 for large N = 5000. Since the Q distributionis coupled to the P distribution through to the presenceof the P L +1 term in Eq. (9), and since the P distributionis a function of P as mentioned above, the steady state (a) (b) FIG. 3. Steady state distributions of both filaments P and fragments Q . (a) Comparing the corresponding distri-butions for finite depolymerization rate γ = 1 . Q L for the same γ and decreases rapidly for largelengths. (b) The total length distribution for different γ val-ues are shown. By increasing the depolymerization rate, weclearly see that the total distribution shifts to the instant de-polymerization limit shown by thick black curve. We used apolymerization rate of r = 1 . α = 10 − event/monomer/s. length distribution of Q filaments is therefore a functionof P as well. Finally, P is obtained by using the factthat the number of filaments and monomers is constantat steady-state, i.e., (cid:80) ∞ L =1 ( P L + Q L ) = constant. Wenote that the normalization constant has no effect on thestress relaxation behavior due to the fact that the stressis measured relative to its initial value.The steady state distributions are shown in Fig. 3 forboth infinite and finite values of depolymerization rate γ . Figure 3 a shows that the fragment distribution Q L decays rapidly with the length, since long fragments aresubjected to both severing and disassembly. The effect offragments on the total length distribution ( P L + Q L ) canbe clearly seen by comparing both limits of infinite andfinite depolymerization rates (see Fig. 3 a). Fig. 3 b illus-trates that by increasing γ , the total length distributionconverges to the limit of immediate disassembly. STRESS RELAXATION
In order to characterize the relaxation of stress, weuse the well-established model of entangled solutions ofsemiflexible polymers [15–18], based on the tube conceptof topological entanglements that constrain the lateralmotion of a polymer chain [10, 19]. This model predictsa linear plateau modulus given by G ∼ ρkT /L e , (10)where ρ is the total length of (entangled) polymer pervolume in the solution and L e is the characteristic lengthbetween entanglement points along a polymer that is as-sumed to be longer than this length. We consider the time evolution of stress for such a solution that is sub-ject to a step-strain experiment. In general, this stresscan relax by three mechanisms: (1) reptation or longitu-dinal diffusion of chains along their confining tube [10] (2)treadmilling by combined polymerization at the barbedend and depolymerization at the pointed end and (3) thecombination of severing and fragment dissolution. Thefirst of these is known to lead to a relaxation time τ r thatgrows approximately with the third power of the molec-ular weight or filament length (cid:104) L (cid:105) [17, 18]. Rheology inthe presence of motile polar polymers, e.g., due to mo-tors or active treadmilling, has been studied before andthe resulting relaxation time is expected to grow linear in (cid:104) L (cid:105) , as previously shown [11]. In both of these cases, theresidual stress is determined by the total polymer length, ρ , per volume remaining in the original tube, since thepolymer in newly explored regions, either by the diffus-ing or actively driven ends, can be expected to be stress-free on average. In particular, newly added monomer bypolymerization will not contribute to the stress. Thus,for severing (3), we consider the time evolution of theoriginal polymer at the instant of the applied step strain.As sketched in Fig. 1, severing and depolymerization re-actions have large effects on changing the original tubeand enhancing relaxation of the initial stress. Therefore,to find the dynamic length distribution of load-bearingfilaments, we remove the assembly reaction from the dy-namic master equation. Using our derived steady statesolutions in the previous section as the initial condition,we are able to solve the dynamic equations and relatethe remaining initial stress to the amount of load-bearingfilaments. As above, we discuss the dynamics for bothinfinite and finite γ . Unstable Fragments: γ → ∞ The dynamic master equation of load-bearing fila-ments in the case of infinite depolymerization of frag-ments is given by Eq. (1) for r = 0. We solve this inits continuous form by using Eq. (5) as the initial con-dition, i.e., we assume the actin network is in its steadystate before applying a step strain. The dynamic lengthdistribution of load-bearing filaments is given by F ( (cid:96), t ) = (cid:0) αt + α(cid:96)r (cid:1) exp (cid:0) − α(cid:96) t − α(cid:96) r (cid:1) (11)where F ( (cid:96), t ) is the continuous form of the discrete lengthdistribution P ( L, t ).Filaments shorter than L e diffuse easily through thenetwork and do not contribute to the stress relaxation.Thus, we relate the residual stress in the system to theportion of the distribution with L > L e : σ ( t ) ∼ ∞ (cid:88) L = L e LP ( L, t ) (12)or in continuous form σ ( t ) ∼ (cid:90) ∞ (cid:96) = L e (cid:96)F ( (cid:96), t ) (13)Thus, we find the following relation for the stress in limitof infinite γσ ( t ) = exp (cid:18) − αL e ( L e + 2 rt )2 r (cid:19) (cid:20) L e + (14)erfc (cid:18)(cid:114) α r ( L e + rt ) (cid:19)(cid:114) πr α exp (cid:18) α ( L e + rt ) r (cid:19)(cid:21) where erfc( x ) is the complementary error function, .Fig. 4 shows the length distributions calculated fromEq. (11) at different times scaled by severing rate (˜ t ≡ αt ). As time increases, the length distribution of load-bearing filaments shifts toward shorter filaments due tosevering events, which leads to a stress relaxation asshown in the inset of Fig. 4. The initial average filamentlength (cid:104) L (cid:105) , which is obtained in Eq. (6), is a natural char-acteristic length scale relating polymerization to severingrate and governs the network relaxation behavior in thelimit of instant depolymerization. We find that the ini-tial stress relaxation is approximately single-exponentialwith relaxation time τ ∼ √ πα (cid:104) L (cid:105) . At longer times, however,we find an additional single-exponential relaxation time τ ∼ αL e in this regime. The relaxation times are derivedin Appendix A. This counter-intuitive, inverse length de-pendence of the relaxation time can be understood interms of severing, the rate of which increases with length,due to the increased number of potential severing sites.The rapid dissolution of fragments means that each sever-ing event results in an order of unity fractional reductionof stress per polymer. Thus, this instantaneous dissolu-tion limit, as considered in Refs. [12–14], cannot accountfor the observed length-independent stress relaxation [9].With finite depolymerization of fragments, however, weobserve qualitatively different relaxation regimes, as de-scribed in the following section. Role of fragments: finite γ By introducing a finite rate of depolymerization, weproceed solving coupled master equations for initially-stressed filaments. As we argued before, disassembly ofactin filaments changes the hypothetical tube that con-strains the filament’s motion and affects the relaxationprocess. Therefore, the dynamic master equation for P L is again given by Eq. (1) with r = 0, since polymerizationresults in unstressed filament segments. The equation (7)for Q L is unchanged. Using the derived steady state so-lution of Eq. (8) and (9) as the initial condition, we solvethese coupled systems of linear differential equations nu-merically. The remaining initial stress is calculated using FIG. 4. Dynamic length distribution in the limit of instantdisassembly of fragments. Using Eq. (11) in the text, lengthdistributions of load-bearing filaments at infinite γ and dif-ferent scaled time are shown. For longer times, filaments getshorter due to the severing process. The red dashed line in-dicates the entanglement length L e = 100 which is used tocalculate stress. Also we used (cid:104) L (cid:105) = (cid:112) π rα = 1253. Inset:Showing the residual stress calculated from Eq. (14) in thetext normalized by the initial stress for three different val-ues of (cid:104) L (cid:105) which are shown in the legend. The superpositionof the curves during the first 90% of the stress decay whentime is rescaled by length indicates that the relaxation timeis length-dependent. the total length distribution as following σ ( t ) ∼ ∞ (cid:88) L = L e L (cid:0) P ( L, t ) + Q ( L, t ) (cid:1) . (15)As mentioned earlier, we define the depolymerizationlength scale as L d = (cid:112) γα . This length scale togetherwith the network’s entanglement length L e provides twodifferent regimes, (cid:104) L (cid:105) > L d > L e (II) and (cid:104) L (cid:105) > L e > L d (III). If the entanglement length L e is larger than (cid:104) L (cid:105) , thesystem should exhibit simple viscous behavior. Thus, wefocus on the limit (cid:104) L (cid:105) > L e . It is noted that the regimewhere L d > (cid:104) L (cid:105) > L e (I) has been investigated in theprevious section where γ → ∞ .Fig. 5 illustrates the effect of depolymerization length L d on the stress relaxation in the regime (II) where L e < L d < (cid:104) L (cid:105) . The inset of Fig. 5 shows that this regimeis characterized by an approximate single-exponential re-laxation, in this case with relaxation time τ ∼ αL d . Wealso find that the stress relaxation in this regime is inde-pendent of the initial average filament length (cid:104) L (cid:105) priorto applying a step strain (see Appendix B). This strik-ing length-independent relaxation behavior can be un-derstood by noting that, the time for significant stressrelaxation is determined by the time at which the typi-cal length of initial load-bearing filaments is reduced bysevering to L d , since the dissolution becomes very rapidfor filaments of this length and shorter. Increasing de-polymerization rate γ (increasing L d ) shifts the lengthdistribution Q L toward monomeric units and hence thestress relaxation becomes faster. FIG. 5. Relaxation curves for different depolymerizationlength. Showing normalized stress versus time scaled by sev-ering rate (˜ t = αt ) for different values of depolymerizationlength scale L d which are specified in the legend. We usedentanglement length of L e = 20 and initial average length of (cid:104) L (cid:105) = (cid:112) π rα = 1253. Inset: Showing the collapse of stresscurves versus ˜ tL d , which indicates that the stress relaxationis determined by L d in this regime. The approximate straightline in this semi-log plot shows a single-exponential behavior. As the effects of fragment dissolution become less im-portant, L e can exceed L d . Here, we also find that thestress relaxation has no dependence on the initial averagelength (cid:104) L (cid:105) (see Appendix B). The preceding argumentsconcerning L d apply in this limit for L e . In the limit ofslow or absent dissolution of fragments (small γ ), to afirst approximation severing simply reduces the averagelength of load-bearing filaments, while conserving the to-tal length of these. Only when a significant portion of theinitial length distribution shifts from longer filaments tofilaments shorter than L e will the stress begin to relax sig-nificantly. This will occur when filaments of length ∼ L e are severed with significant probability, i.e., for times t ∼ ( αL e ) − (see inset of Fig. 6). Both of the regimes IIand III are consistent with the recent experiments on re-constituted actin solutions in the presence of cofilin show-ing a length-independent relaxation process. Combiningour results in different regimes of length scales, we areable to construct a phase diagram for stress relaxationbehavior of F-actin networks (see Fig. 2). These regimesare, in principle, experimentally accessible by varying re-action rates via actin-binding proteins such as profilin,cofilin, and formin [3, 5]. By increasing concentrationof profilin, as a nucleation inhibitor, the initial averagelength of actin filaments (cid:104) L (cid:105) decreases. On the otherhand, adding formin promotes nucleation and increases (cid:104) L (cid:105) . Cofilin concentration also controls the rate of sev-ering reaction [9]. However, one important caveat whencomparing the model with experiments is that reachinga true steady state of actin solutions during the experi- FIG. 6. Relaxation curves for different entanglement length.Normalized stress versus time scaled by severing rate (˜ t = αt )for three different L e as shown in the legend for depoly-merization length L d = 20 and initial average length of (cid:104) L (cid:105) = (cid:112) π rα = 1253. The inset shows a collapse of the re-laxation curves versus ˜ tL e , which implies that L e determinesthe relaxation behavior in this regime. Also the approximatestraight line in this semi-log plot shows a single-exponentialstress relaxation. ments may be slow, particularly if diffusive length fluc-tuations are relevant [20], making it likely that the ex-perimental filament length distributions are collected ina quasi-steady state. LIMITATIONS OF THE MODEL
Due to the multiple molecular reactions occurring inF-actin solutions, it has been a challenge to model eventheir (dis)assembly, let alone the consequences of this forstress relaxation. We present above a minimal model ofstress relaxation based on the temporal evolution of thelength distribution of load-bearing filaments. In order tomake the model tractable, we make a number of simpli-fying assumptions. In particular our model is a coarse-grained one, appropriate for sufficiently high molecularweight. The model considers all filaments to be com-posed of ADP-bound actin subunits, with the exceptionof a single ATP-bound terminal monomer at the barbedend of each P filament. Thus, we do not resolve thefinite size of an ATP-cap. This simple nucleotide distri-bution ensures that exactly one P and one Q filamentare formed as a result of severing of P filaments, con-sistent with experiments [9, 24, 25]. Similarly, filamentnucleation is not treated in detail in our model, althoughthe final term in Eq. (1) for L = 2, i.e., rP representsthe nucleation rate, with P being an implicit additionalparameter to account for nucleation. Changing P has atrivial multiplicative effect on the amplitude of the lengthdistribution and does not affect the time dependence ofstress relaxation.Furthermore, we neglect filament annealing [26], aswas deemed appropriate in recent experimental studiesof actin solutions in presence of formin and profilin [9].The presence of formin at barbed ends is sufficient tosuppress annealing of elongating filaments [27], and thebinding of profilin to ADP-bound barbed ends of depoly-merizing filaments generates a steric clash we expect toinhibit annealing at barbed ends exposed by severing.We note that by including filament annealing at zero de-polymerization rate γ = 0, our model becomes similar tothe viscoelastic model for worm-like micelles [28].Rather than an explicit treatment, the activities ofactin binding proteins are implicitly included in themodel through reaction rates. Although the reactionrates depend on the concentrations of different compo-nents in the solution [21, 22], we simplify our model byassuming constant reaction rates. In particular we as-sume a uniform and equivalent severing rate along bothfilament types P and Q . The possible non-uniform sever-ing reaction in the vicinity of an ATP-cap (on filament P )should be characterized by a local interaction on the scaleof monomers, which can be neglected for high molecularweight. We also note that various reaction rates in actinsolutions can depend on each other, e.g., in the observedsynergy effect of cofilin and Arp2/3 in actin solutions [29–31], which is not incorporated in our simplified model.Moreover, we assume that the monomer pool consistsonly of ATP-bound G-actin in complex with profilin andis constant in time. This is indeed the major species inreconstituted actin solutions in presence of profilin andcofilin at steady state [9, 23]. CONCLUSION
Considering all of these assumptions and limitations,our model takes into account polymerization, depolymer-ization, and also severing reactions with constant ratesand phenomenologically relates the magnitude of remain-ing stress after applying a step strain to the amount ofinitially-stressed large filaments. Depending on the rel-ative values of different reaction rates, we observe bothlength-dependent and length-independent relaxation pro-cess.Assuming instantaneous disassembly of unstable frag-ments ( Q in Fig. 1) after severing events gives a Rayleighdistribution for filament length in steady state. Thispeaked distribution was indeed investigated in previousworks [12–14]. Moreover, using the dynamic length dis-tributions, we find that the stress relaxation has a strongand surprisingly inverse dependence on the initial averagefilament length (cid:104) L (cid:105) .By including finite disassembly of fragments in ourmodel, we find a significant change in both steady stateand dynamic length distributions and hence the resultingrelaxation behavior. For finite fragment disassembly rate γ , there is an enhancement of short filaments, comparedto the limit of instant disassembly γ → ∞ . This is due tothe presence of fragments with ADP barbed ends (Fig.1). As we increase γ , this distribution tends to the lengthdistribution without fragments. In the limit of very slowrate of disassembly γ where L d < L e < (cid:104) L (cid:105) (regime IIIin Fig. 2), stress relaxation of F-actin solutions is in-dependent of initial filaments length. Interestingly, thecharacteristic timescale in this regime is inversely pro-portional to the entanglement length of the network L e .For the intermediate γ values in which L d becomes largerthan L e but still smaller than (cid:104) L (cid:105) (regime II in Fig. 2),we also find a length-independent stress relaxation witha characteristic timescale as ( αL d ) − .Recent rheological experiments on reconstituted actinsolutions show a length-independent relaxation behav-ior [9], consistent with regimes II and III in the presentmodel. Further experiments will be needed to determinewhich, if either of these regimes is observed. One wayto explore this, for instance, would be to vary the con-centration of actin and, thereby the entanglement length L e . CONFLICTS OF INTEREST
There are no conflicts to declare.
ACKNOWLEDGMENTS
S. A., J. F. and F. C. M. were supported, in part,by The Center for Theoretical Biological Physics (NSFPHY-1427654). S. A. and F. C. M. were also supportedin part by NSF (DMR-1826623). P. M. M. and M. L.G. were supported by University of Chicago MaterialsResearch Science and Engineering Center (NSF DMR-1420709). P. M. M was also supported in part by anELBE postdoctoral fellowship.
APPENDICESAppendix A: Timescales in the case of unstablefragments ( γ → ∞ ) By rewriting Eq. (14) in terms of the two length scales,i.e., the entanglement length L e and the initial averagelength (cid:104) L (cid:105) , we obtain σ ( t ) = L e exp (cid:0) − R − t/τ (cid:1) + (cid:104) L (cid:105) erfc (cid:0) R + t/τ (cid:1) exp (cid:0) ( t/τ ) (cid:1) (16)where R = √ π L e (cid:104) L (cid:105) , τ = αL e , and τ = √ πα (cid:104) L (cid:105) . Thisexpression gives two different timescales τ and τ indi-cating that in the regime where L e < (cid:104) L (cid:105) < L d , stressinitially decays as (cid:104) L (cid:105) (see inset of Fig. 4 in the maintext) and then relaxes as L e . Appendix B: Length-independent stress relaxationfor finite γ Figure 7 shows the stress relaxation for two differentinitial average length (cid:104) L (cid:105) in the regime where L e < L d < (cid:104) L (cid:105) (regime II in Fig. 2a and b in the main text). Asexpected, the stress relaxation is length-independent inthis regime. The deviation of the curve corresponding tothe smaller length is due to numerical errors. Likewise,by plotting stress relaxation curves for two different (cid:104) L (cid:105) in the regime where L d < L e < (cid:104) L (cid:105) (regime III in Fig.2a and b in the main text), which is shown in Fig. 8, weclearly see a length-independent relaxation. FIG. 7. Stress relaxation for two different (cid:104) L (cid:105) as shown inthe legend in the regime where L e < L d < (cid:104) L (cid:105) for L e = 20and L d = 150.FIG. 8. Stress relaxation for two different (cid:104) L (cid:105) as shown inthe legend in the regime where L d < L e < (cid:104) L (cid:105) for L e = 150and L d = 20. [1] J. A. Cooper, Annual Review of Physiology , 585(1991).[2] T. D. Pollard and G. G. Borisy, Cell , 453 (2003).[3] T. D. Pollard, The Journal of Cell Biology , 2747(1986).[4] B. Bugyi and M.-F. Carlier, Annual Review of Biophysics , 449 (2010).[5] L. Blanchoin, R. Boujemaa-Paterski, C. Sykes, andJ. Plastino, Physiological Reviews , 235 (2014).[6] E. M. De La Cruz, Biophysical Reviews , 51 (2009).[7] M. McCullagh, M. G. Saunders, and G. A. Voth, Journalof the American Chemical Society , 13053 (2014).[8] M.-F. Carlier, V. Laurent, J. Santolini, R. Melki,D. Didry, G.-X. Xia, Y. Hong, N.-H. Chua, and D. Pan-taloni, The Journal of Cell Biology , 1307 (1997).[9] P. M. McCall, F. C. MacKintosh, D. R. Kovar, and M. L.Gardel, Proceedings of the National Academy of Sciences, 201818808 (2019).[10] P. de Gennes, Scaling concepts in polymer physics (Cor-nell Univ. Pr., Ithaca, 1979).[11] T. B. Liverpool, A. C. Maggs, and A. Ajdari, PhysicalReview Letters , 4171 (2001).[12] L. Edelstein-Keshet and G. B. Ermentrout, Bulletin ofmathematical biology , 449 (1998).[13] G. B. Ermentrout and L. Edelstein-Keshet, Bulletin ofmathematical biology , 477 (1998).[14] L. Mohapatra, B. L. Goode, P. Jelenkovic, R. Phillips,and J. Kondev, Annual review of biophysics , 85(2016).[15] H. Isambert and A. C. Maggs, Macromolecules , 1036(1996).[16] D. C. Morse, Macromolecules , 7030 (1998).[17] D. C. Morse, Macromolecules , 7044 (1998).[18] P. Lang and E. Frey, Nature Communications (2018),10.1038/s41467-018-02837-5.[19] M. Doi and S. F. Edwards, The Theory of Polymer Dy-namics , International Series of Monographs on Physics(Clarendon Press, 1988).[20] L. Mohapatra, T. J. Lagny, D. Harbage, P. R. Jelenkovic,and J. Kondev, Cell Systems , 559 (2017).[21] J. Roland, J. Berro, A. Michelot, L. Blanchoin, and J.-L.Martiel, Biophysical Journal , 2082 (2008).[22] E. M. De La Cruz and D. Sept, Biophysical Journal ,1893 (2010).[23] D. Didry, M.-F. Carlier, and D. Pantaloni, Journal ofBiological Chemistry , 25602 (1998).[24] H. Wioland, B. Guichard, Y. Senju, S. Myram, P. Lap-palainen, A. J´egou, and G. Romet-Lemonne, CurrentBiology , 1956 (2017).[25] C. Suarez, J. Roland, R. Boujemaa-Paterski, H. Kang,B. R. McCullough, A.-C. Reymann, C. Gu´erin, J.-L.Martiel, E. M. De La Cruz, and L. Blanchoin, CurrentBiology , 862 (2011).[26] B. McCullough, E. Grintsevich, C. Chen, H. Kang,A. Hutchison, A. Henn, W. Cao, C. Suarez, J.-L. Mar-tiel, L. Blanchoin, E. Reisler, and E. M. De La Cruz,Biophysical Journal , 151 (2011).[27] D. R. Kovar, J. R. Kuhn, A. L. Tichy, and T. D. Pollard,The Journal of Cell Biology , 875 (2003).[28] M. E. Cates and S. J. Candau, Journal of Physics: Con-densed Matter , 6869 (1990). [29] I. Ichetovkin, W. Grant, and J. Condeelis, Current Bi-ology , 79 (2002).[30] V. DesMarais, F. Macaluso, J. Condeelis, and M. Bailly,Journal of Cell Science , 3499 (2004). [31] N. Tania, J. Condeelis, and L. Edelstein-Keshet, Bio-physical Journal105