Filament turnover is essential for continuous long range contractile flow in a model actomyosin cortex
FFilament turnover is essential for continuous long rangecontractile flow in a model actomyosin cortex.
William M McFadden , Patrick M McCall , Edwin M Munro Abstract
Actomyosin-based cortical flow is a fundamental engine for cellular morphogenesis.Cortical flows are generated by cross-linked networks of actin filaments and myosinmotors, in which active stress produced by motor activity is opposed by passiveresistance to network deformation. Continuous flow requires local remodeling throughcrosslink unbinding and and/or filament disassembly. But how local remodeling tunesstress production and dissipation, and how this in turn shapes long range flow, remainspoorly understood. Here, we introduce a computational model for a cross-linkednetworks with active motors based on minimal requirements for production anddissipation of contractile stress, namely asymmetric filament compliance, spatialheterogeneity of motor activity, reversible cross-links and filament turnover. Wecharacterize how the production and dissipation of network stress depend, individually,on cross-link dynamics and filament turnover, and how these dependencies combine todetermine overall rates of cortical flow. Our analysis predicts that filament turnover isrequired to maintain active stress against external resistance and steady state flow inresponse to external stress. Steady state stress increases with filament lifetime up to acharacteristic time τ m , then decreases with lifetime above τ m . Effective viscosity PLOS a r X i v : . [ q - b i o . S C ] D ec ncreases with filament lifetime up to a characteristic time τ c , and then becomesindependent of filament lifetime and sharply dependent on crosslink dynamics. Theseindividual dependencies of active stress and effective viscosity define multiple regimes ofsteady state flow. In particular our model predicts the existence of a regime, whenfilament lifetimes are shorter than both τ c and τ m , in which dependencies of effectiveviscosity and steady state stress cancel one another, such that flow speed is insensitiveto filament turnover, and shows simple dependence on motor activity and crosslinkdynamics. These results provide a framework for understanding how animal cells tunecortical flow through local control of network remodeling. Author Summary
In this paper, we develop and analyze a minimal model for a 2D network of cross-linkedactin filaments and myosin motors, representing the cortical cytoskeleton of eukaryoticcells. We implement coarse-grained representations of force production by myosinmotors and stress dissipation through an effective cross-link friction and filamentturnover. We use this model to characterize how the sustained production of activestress, and the steady dissipation of elastic stress, depend individually on motor activity,effective cross-link friction and filament turnover. Then we combine these results to gaininsights into how microscopic network parameters control steady state flow produced byasymmetric distributions of motor activity. Our results provide a framework forunderstanding how local modulation of microscopic interactions within contractilenetworks control macroscopic quantities like active stress and effective viscosity tocontrol cortical deformation and flow at cellular scales.
Introduction
Cortical flow is a fundamental and ubiquitous form of cellular deformation thatunderlies cell polarization, cell division, cell crawling and multicellular tissuemorphogenesis [1–6]. These flows originate within a thin layer of cross-linked actinfilaments and myosin motors, called the actomyosin cortex, that lies just beneath theplasma membrane [7]. Local forces produced by bipolar myosin filaments are integrated
PLOS in vitro , it is unlikely toaccount for the rapid cortical deformation and flow observed in living cells [26, 28–31].Experimental studies in living cells reveal rapid turnover of cortical actin filaments ontimescales comparable to stress relaxation (10-100s) [32–36]. Perturbing turnover canlead to changes in cortical mechanics and in the rates and patterns of corticalflow [34, 37]. However, the specific contributions of actin turnover to stress relaxationand how these depend on network architecture remain unclear.
PLOS in vitro [40], and the kinematics ofcontraction observed in these studies support a mechanism based on asymmetricalfilament compliance and filament buckling. However, in these studies, the filamentswere preassembled and network contraction was transient, because of irreversiblenetwork collapse [41], or buildup of elastic resistance [42], or because networkrearrangements (polarity sorting) dissipate the potential to generate contractileforce [43–46]. This suggests that network turnover may play essential role(s) in allowingsustained production of contractile force. Recent theoretical and modeling studies havebegun to explore how this might work [47–49], and to explore dynamic behaviors thatcan emerge when contractile material undergoes turnover [15, 50]. However, it remains achallenge to understand how force production and dissipation depend individually onthe local interplay of network architecture, motor activity and filament turnover, andhow these dependencies combine to mediate tunable control of long range cortical flow.Here, we construct and analyse a simple computational model that bridges betweenthe microscopic description of cross-linked actomyosin networks and the coarse graineddescription of an active fluid. We represent actin filaments as simple springs withasymmetric compliance; we represent dynamic binding/unbinding of elastic cross-linksas molecular friction [51–53] at filament crossover points; we represent motor activity asforce coupling on a subset of filament cross-over points with a simple linearforce/velocity relationship [54]. Finally, we model filament turnover by allowing entirefilaments to appear with a fixed probability per unit area and disappear with fixedprobabilities per unit time. We use this model first to characterize the passive responseof a cross-linked network to externally applied stress, then the buildup and maintenanceof active stress against an external resistance, and finally the steady state flowsproduced by an asymmetric distribution of active motors in which active stress and
PLOS
Models
Our goal was to construct a minimal model that is detailed enough to capture essentialmicroscopic features of cross-linked actomyosin networks (actin filaments withasymmetric compliance, dynamic cross-links, active motors and and continuous filamentturnover), but simple enough to explore, systematically, how these microscopic featurescontrol macroscopic deformation and flow. We focus on 2D networks because theycapture a reasonable approximation of the quasi-2D cortical actomyosin networks thatgovern flow and deformation in many eukaryotic cells [11, 55], or the quasi-2D networksstudied recently in vitro [40, 56].Fig. 1 provides a schematic overview of our model’s assumptions. We model eachfilament as an oriented elastic spring with relaxed length l s . The state of a filament isdefined by the positions of its endpoints x i and x i + marking its (-) and (+) endsrespectively. The index i enumerates over all endpoints of all filaments. We refer to thefilament connecting endpoint i and i+1 as filament i, and we define ˆu i to be the unitvector oriented along filament i from endpoint i to endpoint i+1. Asymmetric filament compliance
We assume (Fig. 1A) that local deformation of filament i gives rise to an elastic force: F µ i , i + = µγ i ˆu i (1)where γ i = ( | x i − − x i | − l s ) /l s is the strain on filament i, and the elastic modulus µ is a composite quantity that represents both filament and cross-linker compliance as inthe effective medium theory of Broederz and colleagues [57]. To model asymmetricfilament compliance, we set µ = µ e if the strain is positive (extension), and µ = µ c ifthe strain is negative (compression). The total elastic force on a filament endpoint i canbe written as: PLOS elasi = F µ i , i + − F µ i − , i (2)In the limit of highly rigid cross-links and flexible filaments, our model approachesthe pure semi-flexible filament models of [58, 59]. In the opposite limit (nearly rigidfilaments and highly flexible cross links), our model approaches that of [57] in smallstrain regimes before any nonlinear cross link stiffening. Drag-like coupling between overlapping filaments
Previous models represent cross-linkers as elastic connections between pairs of points onneighboring filaments that appear and disappear with either fixed or force-dependentprobabilities [25, 57]. Here, we introduce a coarse-grained representation of crosslinkdynamics by introducing an effective drag force that couples every pair of overlappingfilaments, and which represents a molecular friction arising from the time-averagedcontributions of many individual transient crosslinks (Fig. 1B). This coarse-grainedapproximation has been shown to be adequate in the case of ionic cross-linking ofactin [60, 61], and has been used to justify simple force-velocity curves for myosin boundfilaments in other contexts [54].To implement coupling through effective drag, for any pair of overlapping filaments jand k, we write the drag force on filament j as: F ξ j , k = − ξ ( v j − v k ) (3)where ξ is the drag coefficient and v j , v k are the average velocities of filaments j andk. We apportion this drag force to the two endpoints ( j, j+1) of filament j as follows: If x j , k is the position of the filament overlap, then we assign (1 − λ j , k ) F ξ j , k to endpoint jand λ j , k F ξ j , k to endpoint j+1, where λ j , k = | x j , k − x j | / | x j + − x j | .The total crosslink coupling force on endpoint i due to overlaps along filament i andi-1 can then be written: F xli = (cid:88) j (1 − λ i , j ) F ξ i , j + (cid:88) k λ i − , k F ξ i − , k (4)where the sums are taken over all filaments j and k that overlap with filaments i and PLOS
Active coupling for motor driven filament interactions
To add motor activity at the point of overlap between two filaments j and k ; for eachfilament in the pair, we impose an additional force of magnitude υ , directed towards its(-) end (Fig. 1C): F υ i = − υ ˆu i (5)and we impose an equal and opposite force on its overlapping partner. We distributethese forces to filament endpoints as described above for crosslink coupling forces. Thus,the total force on endpoint i due to motor activity can be written as: F motori = υ (cid:88) j (1 − λ i , j ) ( ˆu i − ˆu j ) q i,j + υ (cid:88) k ( λ i − , k ) ( ˆu i − − ˆu k ) q i − ,k (6)where j and k enumerate over all filaments that overlap with filaments i and i-1respectively, and q j,k equals 0 or 1 depending on whether there is an “active” motor atthis location. To model dispersion of motor activity, we set q i,j = 1 on a randomlyselected subset of filament overlaps, such that ¯ q = φ , where ¯ q indicates the mean of q (Fig. 1C). Equations of motion
To write the full equation of motion for a network of actively and passively coupledelastic filaments, we assume the low Reynold’s number limit in which inertial forces canbe neglected, and we equate the sum of all forces acting on each filament endpoint tozero to obtain: 0 = − l s ζ v i − F xli + F elasi + F motori (7) PLOS ζ is the dragcoefficient.
2D network formation
We used a mikado model approach [62] to initialize a minimal network of overlappingunstressed linear filaments in a rectangular 2D domain. We generate individualfilaments by laying down straight lines, of length L, with random position andorientation. We define the density using the average distance between cross-links alonga filament, l c . A simple geometrical argument can then be used to derive the number offilaments filling a domain as a function of L and l c [58]. Here, we use the approximationthat the number of filaments needed to tile a rectangular domain of size D x × D y is2 D x D y /Ll c , and that the length density is therefore simply, 2 /l c . Modeling filament turnover
In living cells, actin filament assembly is governed by multiple factors that controlfilament nucleation, branching and elongation. Likewise filament disassembly isgoverned by multiple factors that promote filament severing and monomer dissociationat filament ends. Here, we implement a very simple model for filament turnover inwhich entire filaments appear with a fixed rate per unit area, k app and disappear at arate k diss ρ , where ρ is a filament density (Fig. 1D). With this assumption, in theabsence of network deformation, the density of filaments will equilibrate to a steadystate density, k app /k diss , with time constant τ r = 1 /k diss . In deforming networks, thedensity will be set by a competition between strain thinning ( γ >
0) or thickening( γ < τ s < . · τ r (i.e. 1% of the equilibration time), we selected a fraction, τ s /τ r ,of existing filaments (i.e. less than 1% of the total filaments) for degradation. We thengenerated a fixed number of new unstrained filaments k app τ s D x D y at random positionsand orientations within the original domain. We refer to k diss = 1 /τ r as the turnoverrate, and to τ r as the turnover time. PLOS imulation methods
Further details regarding our simulation approach and references to our code can befound in the Supplementary Information (S1 Appendix A.1). Briefly, equations 1-7define a coupled system of ordinary differential equations that can be written in theform: A · ˙x = f ( x ) (8)where x is a vector of filament endpoint positions, ˙x the endpoint velocities, A is amatrix with constant coefficients that represent crosslink coupling forces betweenoverlapping filaments, and f ( x ) represents the active (motor) and elastic forces onfilament endpoints. We smoothed all filament interactions, force fields, and constraintslinearly over small regions such that the equations contained no sharp discontinuities.We numerically integrate this system of equations to find the time evolution of thepositions of all filament endpoints. We generate a network of filaments with randompositions and orientations as described above within a domain of size D x by D y . For allsimulations, we imposed periodic boundaries in the y-dimension. To impose anextensional stress, we constrained all filament endpoints within a fixed distance0 . · D x from the left edge of the domain to be non-moving, then we imposed arightwards force on all endpoints within a distance 0 . · D x from the right edge of thedomain. To simulate free contraction, we removed all constraints at domain boundaries;to assess buildup and maintenance of contractile stress under isometric conditions, weused periodic boundary conditions in both x and y dimensions.We measured the local velocity of the network at different positions along the axis ofdeformation as the mean velocity of all filaments intersecting that position; wemeasured the internal network stress at each axial position by summing the axialcomponent of the tensions on all filaments intersecting that position, and dividing bynetwork height; finally, we measured network strain rate as the average of all filamentvelocities divided by their positions.We assigned biological plausible reference values for all parameters (See Table 1).For individual analyses, we sampled the ranges of parameter values around thesereference values shown in S1 Table. PLOS able 1.
Simulation parameters with reference values
Parameter Symbol Reference Value extensional modulus µ e nN compressional modulus µ c . nN cross-link drag coefficient ξ unknown solvent drag coefficient ζ . nNsµm filament length L 5 µm cross-link spacing l c . µm active filament force υ . nN active cross-link fraction φ . < . D x × D y × µm Results
The goal of this study is to understand how cortical flow is shaped by the simultaneousdependencies of active stress and effective viscosity on filament turnover, crosslink dragand on “network parameters” that control filament density, elasticity and motor activity.We approach this in three steps: First, we analyze the passive deformation of across-linked network in response to an externally applied stress; we identify regimes inwhich the network response is effectively viscous and characterize the dependence ofeffective viscosity on network parameters and filament turnover. Second, we analyze thebuildup and dissipation of active stress in cross-linked networks with active motors, asthey contract against an external resistance; we identify conditions under which thenetwork can produce sustained stress at steady state, and characterize how steady statestress depends on network parameters and filament turnover. Finally, we confirm thatthe dependencies of active stress and effective viscosity on network parameters andfilament turnover are sufficient to predict the dynamics of networks undergoing steadystate flow in response to spatial gradients of motor activity.
Filament turnover allows and tunes effectively viscous steadystate flow.
Networks with passive cross-links and no filament turnover undergo threestages of deformation in response to an extensional force.
To characterizethe passive response of a cross-linked filament network without filament recycling, wesimulated a simple uniaxial strain experiment in which we pinned the network at oneend, imposed an external stress at the opposite end, and then quantified network strain
PLOS ∼ exponential approachto a fixed strain (S1 FigA), which represents the elastic limit in the absence of cross-linkslip predicted by [58]. At intermediate times, the local stress and strain rate wereapproximately constant across the network (Fig. 2B), and the response was effectivelyviscous; internal stress remained constant while the network continued to deform slowlyand continuously with nearly constant strain rate (shown as dashed line in Fig. 2C) asfilaments slip past one another against the effective cross-link drag. In this regime, wecan quantify effective viscosity, η c , as the ratio of applied stress to the measured strainrate. Finally, as the network strain approached a critical value ( ∼
30% for thesimulation in Fig. 2), strain thinning lead to decreased network connectivity, localtearing, and rapid acceleration of the network deformation (see inset in Fig. 2C).
Network architecture sets the rate and timescales of deformation.
Tocharacterize how effective viscosity and the timescale for transition to effectively viscousbehavior depend on network architecture and cross-link dynamics, we simulated auniaxial stress test, holding the applied stress constant, while varying filament length L ,density l c , elastic modulus µ e and cross link drag ξ (see S1 Table). We measured theelastic modulus, G , the effective viscosity, η c , and the timescale τ c for transition fromviscoelastic to effectively viscous behavior, and compared these to theoretical predictions.We observed a transition from viscoelastic to effectively viscous deformation for theentire range of parameter values that we sampled. Our estimate of G from simulationagreed well with the closed form solution G ∼ µ/l c predicted by a previous theoreticalmodel [58] for networks of semi-flexible filaments with irreversible cross-links (Fig. 3B).A simple theoretical analysis of filament networks with frictional cross link slip,operating in the intermediate viscous regime (see S1 Appendix A.2), predicted that theeffective viscosity η c should be proportional to the cross-link drag coefficient and to thesquare of the number of cross-links per filament: η c = 4 πξ (cid:18) Ll c − (cid:19) (9) PLOS η c /G sets the timescale fortransition from elastic to viscous behavior [63]. Combining our approximations for G and η c , we predict a transition time, τ c ≈ L ξ/l c µ . Measuring the time at which thestrain rate became nearly constant (i.e. γ ∼ t n with n > .
8) yields an estimate of τ c that agrees well with this prediction over the entire range of sampled parameters (Fig.3C). Thus the passive response of filament networks with frictional cross link drag iswell-described on short (viscoelastic) to intermediate (viscous) timescales by an elasticmodulus G , an effective viscosity η c , and a transition timescale τ c , with well-defineddependencies on network parameters. However, without filament turnover, strainthinning and network tearing limits the extent of viscous deformation to small strains. Filament turnover allows sustained large-scale viscous flow and defines twodistinct flow regimes.
To characterize how filament turnover shapes the passivenetwork response to an applied force, we introduced a simple form of turnover in whichentire filaments disappear at a rate k diss ρ , where ρ is the filament density, and newunstrained filaments appear with a fixed rate per unit area, k app . In a non-deformingnetwork, filament density will equilibrate to a steady state value, ρ = k ass /k diss , withtime constant τ r = 1 /k diss . However, in networks deforming under extensional stress,the density will be set by a competition between strain thinning and densityequilibration via turnover.We simulated a uniaxial stress test for different values of τ r , while holding all otherparameters fixed (Fig. 4A-C). For large τ r , as described above, the network undergoesstrain thinning and ultimately tears. Decreasing τ r increases the rate at which thenetwork equilibrates towards a steady state density ρ . However, it also increases therate of deformation and thus the rate of strain thinning (Fig. 4B). We found that theformer effect dominates, such that below a critical value τ r = τ crit , the network canachieve a steady state characterized by a fixed density and a constant strain rate (S2Fig). Simple calculations (S1 Appendix A.3) show that the critical value of τ r isapproximately: PLOS crit = ξ (cid:16)(cid:113) Ll c − (cid:17) σ . (10)where σ is the applied stress, L/l c the linear cross link density, and ξ is the effectivecrosslink drag.For τ r < τ crit , we observed two distinct steady state flow regimes (Fig. 4B,C). Forintermediate values of τ r , effective viscosity remains constant with decreasing τ r .However, below a certain value of τ r ( ≈ for the parameters used in Fig. 4C),effective viscosity decreased monotonically with further decreases in τ r . To understandwhat sets the timescale for transition between these two regimes, we measured effectiveviscosity at steady steady for a wide range of network parameters ( L, µ, l c ), crosslinkdrags ( ξ ) and filament turnover times (Fig. 4D). Strikingly, when we plotted thenormalized effective viscosity η r /η c vs a normalized recycling rate τ r /τ c for allparameter values, the data collapsed onto a single curve, with a transition at τ r ≈ τ c between an intermediate turnover regime in which effective viscosity is independent of τ r and an high turnover regime in which effective viscosity falls monotonically withdecreasing τ r /τ c (Fig. 4D).This biphasic dependence of effective viscosity on filament turnover can beunderstood intuitively as follows: As new filaments are born, they become progressivelystressed as they stretch and reorient under local influence of surrounding filaments,eventually reaching an elastic limit where their contribution to resisting networkdeformation is determined by effective crosslink drag. The time to reach this limit isabout the same as the time, τ c , for an entire network of initially unstrained filaments toreach an elastic limit during the initial viscoelastic response to uniaxial stress, as shownin Fig. 2b. For τ r < τ c , individual filaments do not have time, on average, to reach theelastic limit before turning over; thus the deformation rate is determined by the elasticresistance of partially strained filaments, which increases with lifetime up to τ r = τ c .For τ r > τ c , the deformation rate is largely determined by cross-link resistance to slidingof maximally strained filaments, and the effective viscosity is insensitive to furtherincrease in τ r .These results complement and extend a previous computational study of irreversiblycross-linked networks of treadmilling filaments deforming under extensional stress [64]. PLOS τ r < τ crit , this behavior can besummarized by an equation of the form: η = η c τ c /τ r ) m (11)For τ r (cid:29) τ c , η ≈ η c : effective viscosity depends on crosslink density and effectivecrosslink drag, independent of changes in recycling rate. For τ r (cid:28) τ c , effective viscosityis governed by the level of elastic stress on network filaments, and becomes stronglydependent on filament lifetime: η ∼ η c ( τ r /τ c ) m . The origins of the m = 3 / Filament turnover allows persistent stress buildup in activenetworks
In the absence of filament turnover, active networks with free boundariescontract and then stall against passive resistance to network compression.
Previous work [38, 40, 65] identifies asymmetric filament compliance and spatial
PLOS ∼ exponential approachto stall (Fig. 5). The time to stall, τ s , scaled as Lξ/υ (S3 FigA). On even longertimescales, polarity sorting of individual filaments, as previously described [42, 44–46]lead to network expansion (see S2 Video).During the rapid initial contraction, the increase in network strain closely matchedthe increase in mean compressive strain on individual filaments Fig. 5B, as predictedtheoretically [38, 39] and observed experimentally [40]. Contraction required asymmetricfilament compliance and spatial heterogeneity of motor activity ( µ e /µ c > φ <
1, S3FigB). Thus our model captures a minimal mechanism for bulk contractility indisordered networks through asymmetric filament compliance and dispersion of motoractivity. However, in the absence of turnover, contraction is limited by internal buildupof compressive resistance and the dissipative effects of polarity sorting.
Active networks cannot sustain stress against a fixed boundary in theabsence of filament turnover.
During cortical flow, regions with high motoractivity contract against a passive resistance from neighboring regions with lower motoractivity. To understand how the active stresses that drive cortical flow are shaped bymotor activity and network remodeling, we analyzed the buildup and maintenance ofcontractile stress in active networks contracting against a rigid boundary. We simulatedactive networks contracting from an initially unstressed state against a fixed boundary(Fig. 6A,B), and monitored the time evolution of mean extensional (blue),compressional (red) and total (black) stress on network filaments (Fig. 6C,D). Wefocused initially on the scenario in which there is no, or very slow, filament turnover,sampling a range of parameter values controlling filament length and density, motoractivity, and crosslink drag.
PLOS σ m , and then decayed towards zero (Fig. 6C,D). The rapid initial increase in totalstress was determined largely by the rapid buildup of extensional stress (Fig. 6C,D) ona subset of network filaments (Fig. 6A,B t = 10 s ). The subsequent decay involved twodifferent forms of local remodeling: under some conditions, e.g. for higher motoractivity (e.g. Fig. 6A,C), the decay was associated with rapid local tearing andfragmentation, leading to global loss of network connectivity as described previouslyboth in simulations [48] and in vitro experiments [41]. However, for many parameters,(e.g. for higher motor activity as in Fig. 6B,D), the decay in stress occurred with littleor no loss of global connectivity. Instead, local filament rearrangements changed thebalance of extensile vs compressive forces along individual filaments, leading to a slowdecrease in the average extensional stress, and a correspondingly slow increase in thecompressional stress, on individual filaments (see Fig. 6D).Combining dimensional analysis with trial and error, we were able to find empiricalscaling relationships describing the dependence of maximum stress σ m and the time toreach maximum stress τ m on network parameters and effective crosslink drag( σ m ∼ √ µ e υ/l c , τ m ∼ Lξ/ √ µ e υ , S3 FigC,D). Although these relationships should betaken with a grain of salt, they are reasonably consistent with our simple intuition thatthe peak stress should increase with motor force ( υ ), extensional modulus ( µ e ) andfilament density (1 /l c ), and the time to reach peak stress should increase with crosslinkdrag ( ξ ) and decrease with motor force ( υ ) and extensional modulus ( µ e ). Filament turnover allows active networks to exert sustained stress on afixed boundary.
Regardless of the exact scaling dependencies of σ m and τ m onnetwork parameters, these results reveal a fundamental limit on the ability of activenetworks to sustain force against an external resistance in the absence of filamentturnover. To understand how this limit can be overcome by filament turnover, wesimulated networks contracting against a fixed boundary from an initially unstressedstate, for increasing rates of filament turnover (decreasing τ r ), while holding all otherparameter values fixed (Fig. 7A-C). While the peak stress decreased monotonically withdecreasing τ r , the steady state stress showed a biphasic response, increasing initiallywith decreasing τ r , and then falling off as τ r →
0. We observed a biphasic response
PLOS σ/σ m ) vs normalized recycling time ( τ r / τ m ) for a wide range of network parameters,the data collapse onto a single biphasic response curve, with a peak near τ r /τ m = 1(Fig. 7D). In particular, for τ r < τ m , the scaled data collapsed tightly onto a singlecurve representing a linear increase in steady state stress with increasing τ r . For τ r > τ m , the scaling was less consistent, although the trend towards a monotonicdecrease with increasing τ r was clear. These results reveal that filament turnover can“rescue” the dissipation of active stress during isometric contraction due to networkremodeling, and they show that, for a given choice of network parameters, there is anoptimal choice of filament lifetime that maximizes steady state stress.We can understand the biphasic dependence of steady state stress on filamentlifetime using the same reasoning applied to the case of passive flow: During steadystate contraction, the average filament should build and dissipate active stress onapproximately the same schedule as an entire network contracting from an initiallyunstressed state (Fig. 7B). Therefore for τ r < τ m , increasing lifetime should increase themean stress contributed by each filament. For τ r > τ m , further increases in lifetimeshould begin to reduce the mean stress contribution. Directly comparing thetime-dependent buildup and dissipation of stress in the absence of turnover, with thedependence of steady state stress on τ r , supports this interpretation (S5 Fig)As for the passive response (i.e. Equation 11), we can describe this biphasicdependence phenomenologically with an equation of the form: σ ss = σ m ( τ r /τ m ) n + τ m /τ r (12)where the origins of the exponent n remain unclear. Filament turnover tunes the balance between active stressbuildup and viscous stress relaxation to generate flows
Thus far, we have considered independently how network remodeling controls thepassive response to an external stress, or the steady state stress produced by active
PLOS υ , with ψ = 0 . γ = σ ss η (13)where σ ss is the active stress generated by the right half-network (less the internalresistance to filament compression), η is the effective viscosity of the left half networkand strain rate ˙ γ is measured in the left half-network. Note that strain rate can berelated to the steady state flow velocity v at the boundary between right and left halvesthrough v = ˙ γDx . Therefore, we can understand the dependence of flow speed onfilament turnover and other parameters using the approximate relationships summarizedby equations 11 and 12 for η and σ ss . As shown in Fig. 8, there are two qualitativelydistinct possibilities for the dependence of strain rate on τ r , depending on the relativemagnitudes of τ m and τ c . In both cases, for fast enough turnover ( τ r < min ( τ m , τ c )), weexpect weak dependence of strain rate on τ r ( ˙ γ ∼ τ / r ). For all parameter values thatwe sampled in this study (which were chosen to lie in a physiological range), τ m > τ c .Therefore we predict the dependence of steady state strain rate on τ r shown in Fig. 8A.To confirm this prediction, we simulated the simple scenario described above for arange of values of τ r , with all other parameter values initially fixed. As expected, weobserved a sharp dependence of steady state flow speeds on filament recycling rate (Fig.9B,C). For very long recycling times, ( τ r = 1000 s , dark blue line), there was a rapidinitial deformation (contraction of the active domain and dilation of the passivedomain), followed by a slow approach to a steady state flow characterized by slowcontraction of the right half-domain and a matching dilation of the left half-domain (see PLOS τ r , steady state flow speeds increasedsteadily, before reaching an approximate plateau on which flow speeds varied by lessthan 15 % over more than two decades of variation in τ r (Fig. 9C).We repeated these simulations for a wider range of parameter values, and saw similardependence of ˙ γ on τ r in all cases. Using equation 11 with τ r < τ c and equation 12 with τ r < τ m , and the theoretical or empirical scaling relationships found above for η c , τ c , σ m and τ m , we predict a simple scaling relationship for ˙ γ (for small τ r ):˙ γ = υξL ( τ r ) / (14)Indeed, when we plot the steady state measurements of ˙ γ , normalized by υ/ξL , forall parameter values, the data collapse onto a single curve for small τ r . Thus. oursimulations identify a flow regime, characterized by sufficiently fast filament turnover, inwhich the steady state flow speed is buffered against variation in turnover, and has arelatively simple dependence on other network parameters. Discussion
Cortical flows are shaped by the dynamic interplay of force production anddissipation within cross-linked actomyosin networks. Here we combined computationalmodels with simple theoretical analyses to explore how this interplay depends onfilament turnover, crosslink dynamics and network architecture. Our results reveal anessential requirement for filament turnover during cortical flow, both to sustain activestress and to continuously relax elastic resistance without catastrophic loss of networkconnectivity. Moreover, we find that biphasic dependencies of active stress and passiverelaxation on filament lifetime define multiple regimes for steady state flow with distinctdependencies on network parameters and filament turnover.We identify two regimes of passive response to external stress: a low turnover regimein which filaments strain to an elastic limit before turning over, and effective viscositydepends on crosslink density and effective crosslink friction, and a high turnover regimein which filaments turn over before reaching an elastic limit and effective viscosity isproportional to elastic resistance and ∼ proportional to filament lifetime. Thus our PLOS τ r < τ m ), steady state stress increases linearly withfilament lifetime because filaments have more time to build towards peak extensionalstress before turning over. For longer loved filaments ( τ r > τ m ), steady state stressdecreases monotonically with filament lifetime because local rearrangements decreasethe mean contributions of longer lived filaments. These findings imply that for cortical PLOS ∼ linear dependencies of steady state stress and effectiveviscosity on filament lifetime for short-lived filaments define a fast turnover regime inwhich steady state flow speeds are buffered against variations in filament lifetime, andare predicted to depend in a simple way on motor activity and crosslink resistance.Measurements of F-actin turnover times in cells that undergo cortical flow [32, 67–71]suggests that they may indeed operate in this fast turnover regime, and recent studiesin C. elegans embryos suggests that cortical flow speeds are surprisingly insensitive todepletion of factors (ADF/Cofilin) that govern filament turnover [11], consistent withour model’s predictions. Stronger tests of our model’s predictions will require moresystematic analyses of how flow speeds vary with filament and crosslink densities, motoractivities, and filament lifetimes.
Supporting Information
S1 Appendix. Code Reference and Supplementary Methods A.1)
Referenceto simulation and analysis code.
A.2)
Derivation of effective viscosity.
A.3)
Derivationof critical turnover timescale for steady state flow
S1 Table. Parameter values.
List of parameter values used for each set ofsimulations.
PLOS
Plots of normalizedstrain vs time during the elastic phase of deformation in passive networks underextensional stress. Measured strain is normalized by the equilibrium strain predicted fora network of elastic filaments without crosslink slip γ eq = σ/G = σ/ (2 µ/l c ). S2 Fig. Filament turnover rescues strain thinning. A)
Plots of strain vs timefor different turnover times (see inset in (b)). Note the increase in strain rates withdecreasing turnover time. B) Plots of filament density vs strain for different turnovertimes τ r . For intermediate τ r , simulations predict progressive strain thinning, but at alower rate than in the complete absence of recycling. For higher τ r , densities approachsteady state values at longer times. S3 Fig. Mechanical properties of active networks. A)
Time for freelycontracting networks to reach maximum strain, τ s , scales with Lξ/υ . B) Freecontraction requires asymmetric filament compliance, and total network strain increaseswith the applied myosin force υ . Note that the maximum contraction approaches anasymptotic limit as the stiffness asymmetry approaches a ratio of ∼ C) Maximumstress achieved during isometric contraction, σ m , scales approximately with √ µ e υ/l c . D) Time to reach max stress during isometric contraction scales approximately with
Lξ/ √ µ e υ . Scalings for τ s , σ m and τ m were determined empirically by trial and error,guided by dimensional analysis. S4 Fig. Filament turnover prevents tearing of active networks.
Plots ofnormalized strain vs time during the elastic phase of deformation in passive networksunder extensional stress. Measured strain is normalized by the equilibrium strainpredicted for a network of elastic filaments without crosslink slip γ eq = σ/G = σ/ (2 µ/l c ). S5 Fig. Bimodal dependence on turnover time matches bimodal buildupand dissipation of stress in the absence of turnover. A)
Bimodal buildup ofstress in a network with very slow turnover ( τ r = 1000 s ). B) Steady state stress fornetworks with same parameters as in (a), but for a range of filament turnover times.
PLOS
Plots of stress and strain vs position fornetworks in which motor activity is limited to the right-half domain and filamentturnover time is either A) τ r = 10000 or B) τ r = 10 s . Blue indicates velocity whileorange represents total stress, measured as described in the main text. S1 Video. Extensional strain in passive networks.
Movie of simulation setupshown in Fig. 2. Colors are the same as in figure.
S2 Video. Active networks contracting with free boundaries.
Movie ofsimulation setup shown in Fig. 5. Colors are the same as in figure.
Acknowledgments
We would like to thank Shiladitya Banerjee for stimulating discussions.
References
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A.1 Simulation and Analysis Code Available Online
All of the simulation and analysis code for generating the figures in this paper isavailable online. To find the source code please visit our Github repository athttps://github.com/wmcfadden/activnet
A.2 Steady-state Approximation of Effective Viscosity
We begin with a calculation of a strain rate estimate of the effective viscosity for anetwork described by our model in the limit of highly rigid filaments. We carry this outby assuming we have applied a constant stress along a transect of the network. Withmoderate stresses, we assume the network reaches a steady state affine creep. In thissituation, we would find that the stress in the network exactly balances the sum of thedrag-like forces from cross-link slip. So for any transect of length D, we have a forcebalance equation.
PLOS = 1 D (cid:88) filaments (cid:88) crosslinks ξ · ( v i ( x ) − v j ( x )) (15)where v i ( x ) − v j ( x ) is the difference between the velocity of a filament, i , and thevelocity of the filament, j , to which it is attached at the cross-link location, x . We canconvert the sum over cross-links to an integral over the length using the average densityof cross-links, 1 /l c and invoking the assumption of (linear order) affine strain rate, v i ( x ) − v j ( x ) = ˙ γx . This results in σ = 1 D (cid:88) filaments (cid:90) L ξ · ( v i ( s ) − v j ( s )) ds cos θl c = (cid:88) filaments ξ ˙ γLl c cos θ · ( x l + L θ ) (16)Here we have introduced the variables x l , and θ to describe the leftmost endpointand the angular orientation of a given filament respectively. Next, to perform the sumover all filaments we convert this to an integral over all orientations and endpoints thatintersect our line of stress. We assume for simplicity that filament stretch and filamentalignment are negligible in this low strain approximation. Therefore, the max distancefor the leftmost endpoint is the length of a filament, L, and the maximum angle as afunction of endpoint is arccos( x l /L ). The linear density of endpoints is the constant D/l c L so our integrals can be rewritten as this density over x l and θ between ourmaximum and minimum allowed bounds. σ = 1 D (cid:90) L dx l (cid:90) arccos( xlL ) − arccos( xlL ) πdθ ξ ˙ γLl c · DLl c · ( x l cos θ + L cos θ ) (17)Carrying out the integrals and correcting for dangling filament ends leaves us with arelation between stress and strain rate. σ = 4 π (cid:18) Ll c − (cid:19) ξ ˙ γ (18)We recognize the constant of proportionality between stress and strain rate as aviscosity (in 2 dimensions). Therefore, our approximation for the effective viscosity, η c , PLOS σ = 4 π (cid:18) Ll c − (cid:19) ξ (19) A.3 Critical filament lifetime for steady state filament extension
We seek to determine a critical filament lifetime, τ crit , below which the density offilaments approaches a stable steady state under constant extensional strain. To thisend, let ρ be the filament density (i.e. number of filaments per unit area). We considera simple coarse grained model for how ρ changes as a function of filament assembly k ass ,filament disassembly k diss , ρ and strain thinning ˙ γρ . Using ρ = k ass k diss , τ r = k diss , and˙ γ = ση c . dρdt = 1 τ r (cid:18) ρ − ρ − στ r η c ( ρ ) ρ (cid:19) (20)where η c = η c ( ρ ) on the right hand side reflects the dependence of effective viscosityon network density. The strength of this dependence determines whether there exists astable steady state, representing continuous flow. Using η c ( ρ ) ∼ ξ (cid:16) Ll c ( ρ ) − (cid:17) fromabove (ignoring the numerical prefactor) and ρ ∼ Ll c ( ρ ) , we obtain: dρdt = 1 τ r (cid:18) ρ − ρ − στ r ξ ( ρL / − ρ (cid:19) (21)Fig. A.1 sketches the positive ( ρ ) and negative ( ρ + στ r η c ( ρ ) ρ ) contributions to theright hand side of Equation 6 for different values of τ r . For sufficiently large τ r , there isno stable state, i.e. strain thinning will occur. However, as τ r decreases below a criticalvalue τ crit , a stable steady state appears. Note that when τ r = τ crit , ρ + στ r η c ( ρ ) ρ passesthrough a minimum value ρ at ρ = ρ ∗ . Accordingly, to determine τ crit , we solve:0 = ddρ (cid:18) ρ + στ r η c ( ρ ) ρ (cid:19) = 1 − στ r ξ ( ρL / − (22)From this, with some algebra, we infer that ρ ∗ = 2 L (cid:32) (cid:18) στ r ξ (cid:19) / (cid:33) (23) PLOS στ r η c ( ρ ∗ ) = (cid:18) στ r ξ (cid:19) / (24)We seek a value for τ r = τ crit at which ρ ∗ + στ crit η c ( ρ ∗ ) ρ ∗ = ρ (25)Substituting from above, and using ρ = Ll c , we have:2 L (cid:32) (cid:18) στ crit ξ (cid:19) / (cid:33) (cid:32) (cid:18) στ crit ξ (cid:19) / (cid:33) = 2 Ll c (26)Finally, rearranging terms, we obtain τ crit = ξσ (cid:32)(cid:114) Ll c − (cid:33) (27) Table 2.
Parameter values sampled for individual figuresParameter
Figure 3 Figure 4 Figure S3a,b Figure S3c,d Figure 7 Figure 9 L , , , ,
10 3 3 , , , , l c . , . , . , . . , . . . , . , . , . . , . . , . , . , . µ e /µ c
100 100 3 −
300 100 100 100 µ c .
01 0 .
01 0 . − . . − .
03 0 .
01 0 . ξ . , . , . , . , . , . , . , , . . , υ . , . , . , . , , . φ .
25 0 . . , .
75 0 . τ r . − . − . − σ . − .
01 0 . − . PLOS undamental Parameters: L = filament length (µm) e = extensional modulus (nN) c = extensional modulus (nN) l c = cross-link spacing (µm) a pp li e d f o r ce l c r L s li d i ng r a t e pulling force ec f il a m e n t s li p time F/ F o l d f il a m e n t s time r e -t/ D) = active filament force (nN) = cross-link active fraction r = filament recycling time (s) ) E)A) B) C) l s F) G) filament strain (L - l s /l s ) Fig 1.
Schematic overview of modeling framework and assumptions. A) Filaments areoriented linear springs that are stiffer in extension than in compression. B) Cross-linking occurs at all filament crossings; we represent cross link resistance as aneffective drag, proportional to the relative velocity of the overlapping filaments. C) Werepresent motor activity as a linear force-velocity relationship with a fixed force at zerovelocity directed towards a filament’s (-) end. We implement spatial heterogeneity byimposing motor activity at a fixed fraction of filament crossover points, resulting invariation in the magnitudes of compressive vs extensile vs translational forces alongindividual filament segments. D) Whole filaments disappear at a constant rate; newfilaments appear with random positions and orientations at the constant rate per unitarea, such that entire network refreshes on a characteristic timescale τ r . e-g) Threedifferent simulation scenarios: E) Passive response to uniaxial stress, F) Freecontraction of an active network and G) Isometric contraction against a fixed boundary.
PLOS = . s t = s Position (µm) S t r e ss ( n N ) × 10 -3 V e l o c it y ( µ m / s ) × 10 -3 t = s Time (s) S t r e ss ( n N ) × 10 -3 S t r a i n S t r a i n A) B)C) -0.060.000.06strain
Fig 2.
Networks with passive cross-links and no filament turnover undergo threestages of deformation in response to an extensional stress. A) Three successive timepoints from a simulation of a 4 × . µm network deforming under an applied stress of0.005 nN/µm . Stress (tan arrows) is applied to filaments in the region indicated by thetan bar. In this and all subsequent figures, filaments are color-coded with respect tostate of strain (blue = tension, red = compression). Network parameters: L = 1 µm , l c = 0 . µm , ξ = 100 nN · s/µm . B) Mean filament stress and velocity profiles for thenetwork in (a) at t=88s. Note that the stress is nearly constant and the velocity isnearly linear as predicted for a viscous fluid under extension. C) Plots of the meanstress and strain vs time for the simulation in (a), illustrating the three stages ofdeformation: (i) A fast initial deformation accompanies rapid buildup of internalnetwork stress; (ii) after a characteristic time τ c (indicated by vertical dotted line) thenetwork deforms at a constant rate, i.e. with a constant effective viscosity, η c , given bythe slope of the dashed line; (iii) at long times, the network undergoes strain thinningand tearing (see inset) Predicted G (2µ/l c ) Predicted τ c (G /η c ) A) B) C) E s ti m a t e d G E s ti m a t e d τ c E s ti m a t e d η c Predicted η c (ξ (L/l c -1) ) Fig 3.
Network architecture sets the rate and timescales of deformation. (a-c)
Comparison of predicted and simulated values for: A) the bulk elastic modulus G , B) the effective viscosity η c and C) the timescale for transition from viscoelastic to viscousbehavior τ c , given by the ratio of the bulk elastic modulus G to effective viscosity, η c .Dotted lines indicates the relationships predicted by theory. PLOS γ = 0.005 τ r = s γ = 0.04 Time (s) S t r a i n τ r = τ r = 10000τ r = 1000τ r = 100τ r = 10τ r = 1 Turnover time ( τ r ) E ff ec ti v e v i s c o s it y ( n N s / µ m ) Normalized turnover time ( τ r /τ c ) -5 N o r m a li ze d v i s c o s it y ( η / η c ) -3 -2 -1 A) B)C) strain D) τ r = ∞ s Fig 4.
Filament recycling defines two regimes of effectively viscous flow. A) Comparison of 20 × µm networks under 0.001 nN/µm extensional stress without(top) and with (bottom) filament turnover. Both images are taken when the networkshad reached a net strain of 0.04. For clarity, filaments that leave the domain of appliedstress are greyed out. B) Plots of strain vs time for identical networks with differentrates of filament turnover. Network parameters: L = 5 µm , l c = 0 . µm , ξ = 10 nN · s/µm . C) Plot of effective viscosity vs turnover time derived from thesimulations shown in panel b. Square dot is the τ r = ∞ condition. D) Plot ofnormalized effective viscosity ( η/η c ) vs normalized turnover time ( τ r /τ c ) for a largerange of network parameters and turnover times. For tau r (cid:28) τ c , the viscosity of thenetwork becomes dependent on recycling time. Red dashed line indicates theapproximation given in equation 11 for m = 3 / PLOS = 0 s t = 49 s t = 99 s t = 149 s -0.50.000.05
Time (s) S t r a i n -0.4-0.3-0.2-0.100.1 Extensional strainCompressional strainNetwork Strain
Time (s) S t r e ss ( n N / µ m ) Extensional StressCompressional StressTotal Stress
A)B) C) strain
Fig 5.
In the absence of filament turnover, active networks with free boundariescontract and then stall against passive resistance to network compression. A) Simulation of an active network with free boundaries. Colors represent strain onindividual filaments as in previous figures. Note the buildup of compressive strain ascontraction approaches stall between 100 s and 150 s. Network parameters: L = 5 µm , l c = 0 . µm , ξ = 1 nN · s/µm , υ = 0 . nN . B) Plots showing time evolution of totalnetwork strain (black) and the average extensional (blue) or compressive (red) strain onindividual filaments. C) Plots showing time evolution of total (black) extensional (blue)or compressive (red) stress. Note that extensional and compressive stress remainbalanced as compressive resistance builds during network contraction.
PLOS = 1 s = . n N t = 10 s t = 100 s t = 1000 s -0.20.000.02 Time (s)
200 400 600 800 1000
Extensional StressCompressional StressTotal Stress S t r e ss ( n N / µ m ) t = 1 s υ = n N t = 10 s t = 25 s t = 57.5 s -0.60.000.3 Time (s) S t r e ss ( n N / µ m ) Extensional StressCompressional StressTotal Stress
A)B)C) D) strainstrain υ Fig 6.
In the absence of filament turnover, active networks cannot sustain continuousstress against a fixed boundary. A) Simulation of an active network with fixedboundaries. Rearrangement of network filaments by motor activity leads to rapid loss ofnetwork connectivity. Network parameters: L = 5 µm , l c = 0 . µm , ξ = 1 nN · s/µm , υ = 1 nN . B) Simulation of the same network, with the same parameter values, exceptwith ten-fold lower motor activity υ = 0 . nN . In this case, connectivity is preserved,but there is a progressive buildup of compressive strain on individual filaments. C) Plots of total network stress and the average extensional (blue) and compressive (red)stress on individual filaments for the simulation shown in (a). Rapid buildup ofextensional stress allows the network transiently to exert force on its boundary, but thisforce is rapidly dissipated as network connectivity breaks down. D) Plots of totalnetwork stress and the average extensional (blue) and compressive (red) stress onindividual filaments for the simulation shown in (b). Rapid buildup of extensional stressallows the network transiently to exert force on its boundary, but this force is dissipatedat longer times as decreasing extensional stress and increasing compressive stressapproach balance. Note the different time scales used for plots and subplots in C) and D) to emphasize the similar timescales for force buildup, but very different timescalesfor force dissipation. PLOS = s r = s -0.5 0.00 0.1 r = s r = s Time (s) S t r e ss ( n N ) τ r = 1000τ r = 100τ r = 10τ r = 1τ r = 0.1 Turnover time ( τ r ) s t ea dy s t a t e s t r e ss ( n N ) -5 -4 -3 -2 -1 Normalized turnover time (τ r /τ m ) -4 -2 N o r m a li ze d s t ea dy s t a t e s t r e ss ( σ / σ m ) -4 -3 -2 -1 A) B)C) D) τ ττ τ
Fig 7.
Filament turnover allows active networks to exert sustained stress on a fixedboundary. A) Snapshots from simulations of active networks with fixed boundaries anddifferent rates of filament turnover. All other parameter values are the same as in Fig.6A. Note the significant buildup of compressive strain and significant remodeling forlonger, but not shorter, turnover times. B) Plots of net stress exerted by the networkon its boundaries for different recycling times; for long-lived filaments, stress is builtrapidly, but then dissipates. Decreasing filament lifetimes reduces stress dissipation byreplacing compressed with uncompressed filaments, allowing higher levels of steady statestress; for very short lifetimes, stress is reduced, because individual filaments do nothave time to build stress before turning over. C) Plots of ≈ steady state stress estimatedfrom the simulations in B) vs turnover time. D) Plot of normalized steady state stressvs normalized recycling time for a wide range of network parameters and turnover times.Steady state stress is normalized by the predicted maximum stress σ m achieved in theabsence of filament turnover. Turnover time is normalized by the predicted time toachieve maximum stress τ m , in the absence of filament turnover. Predictions for σ m and τ m were obtained from the phenomenological scaling relations shown in (Fig. C,D).Dashed blue line indicates the approximation given in equation 12 for n = 1. PLOS e l a ti v e un it s R e l a ti v e un it s -2 -3 -2 -1 S t r a i n r a t e A)B) C)
StressViscosityStrain Rate
Turnover time ( τ r ) τ c τ m τ m τ c τ r /τ m τ c / τ m Fig 8.
Filament recycling tunes the magnitudes of both effective viscosity and steadystate stress. A) Dependence of steady state stress, effective viscosity, and resultingstrain rate on recycling time τ r under the condition τ m < τ c . B) Same as a) but for τ c < τ m . C) State space of flow rate dependence relative to the two relaxationtimescales, τ r and τ c normalized by the stress buildup timescale, τ m . PLOS
Time (s) S t r a i n τ r = 1000τ r = 333.3τ r = 100τ r = 33.33τ r = 10τ r = 3.333 Turnover time ( τr) -2 -5 -4 -3 Turnover time ( τr) -2 (˙ γ L ξ / υ ) -4 -3 -2 -1 A) B)C) D) S t r a i n r a t e ( / s ) N o r m a li ze d s t r a i n r a t e τ r = . s τ r = s Fig 9.
Filament recycling allows sustained flows in networks with non-isotropicactivity. A) Example simulations of non-isotropic networks with long ( τ r = 1000) andshort ( τ r = 33) recycling timescales. In these networks the left half of the network ispassive while the right half is active. Network parameters are same as in Fig.s 6 and 7.Importantly, in all simulations τ m < τ c . B) Graph of strain for identical networks withvarying recycling timescales. With long recycling times, the network stalls; reducing therecycling timescale allows the network to persist in its deformation. However, for theshortest recycling timescales, the steady state strain begins to slowly return to 0 netmotion. Measurements are based on the passive side of the network. C) Steady statestrain rates for the networks in (b). D) Graph of network’s long-term strain rate as afunction of recycling timescale. Dashed line is form of dependence predicted by thetheoretical arguments shown in Fig. 8.
PLOS rc ( ) = L r decrease Fig. A.1
Flux balance analysis of network density. Qualitative plots of ρ + στ r η c ( ρ ) ρ (redcurves) vs ρ (green line) for different values of τ r . For sufficiently large τ r , there are nocrossings. For τ r < τ crit , there are two crossings: The rightmost crossing represents astable steady state. PLOS
50 100 150 200
Time (s) N o r m a li z ed S t r a i n ( / / l c ) S1 Fig.
Fast viscoelastic response to extensional stress. Plots of normalized strain vstime during the elastic phase of deformation in passive networks under extensionalstress. Measured strain is normalized by the equilibrium strain predicted for a networkof elastic filaments without crosslink slip γ eq = σ/G = σ/ (2 µ/l c ). Time (s) S t r a i n Strain F il a m e n t l e ng t h d e n s it y ( µ m - ) τ r = τ r = 1000 τ r = 100 τ r = 10 A) B) ∞ S2 Fig.
Filament turnover rescues strain thinning. A) Plots of strain vs time fordifferent turnover times (see inset in (B)). Note the increase in strain rates withdecreasing turnover time. B) Plots of filament density vs strain for different turnovertimes τ r . For intermediate τ r , simulations predict progressive strain thinning, but at alower rate than in the complete absence of recycling. For higher τ r , densities approachsteady state values at longer times. PLOS
Lξ/υ) T i m e t o s t a ll ( τ s ) Stiffness Asymmetry (µ e /µ c ) M a x i m u m s t r a i n υ/µ c -1 T i m e t o m a x s t r e ss ( τ m ) -1 -2 -1 M a x s t r e ss ( σ m ) -2 -1 A) B)C) D) /l c √µ e υ Lξ/√µ e υ S3 Fig.
Mechanical properties of active networks. A) Time for freely contractingnetworks to reach maximum strain, τ s , scales with Lξ/υ . B) Free contraction requiresasymmetric filament compliance, and total network strain increases with the appliedmyosin force υ . Note that the maximum contraction approaches an asymptotic limit asthe stiffness asymmetry approaches a ratio of ∼ C) Maximum stress achievedduring isometric contraction, σ m , scales approximately with √ µ e υ/l c . D) Time toreach max stress during isometric contraction scales approximately with
Lξ/ √ µ e υ .Scalings for τ s , σ m and τ m were determined empirically by trial and error, guided bydimensional analysis. PLOS = 0.1 s t = 12 s t = 25 s t = 57.5 s -0.60.000.3
Time (s) S t r e ss ( n N / µ m ) CompressionalExtensionalTotal Stress t = 0.1 s t = 12 s t = 25 s t = 57.5 sTime (s) S t r e ss ( n N / µ m ) A)B)C) D) strain
S4 Fig.
Filament turnover prevents tearing of active networks. A) An active networkundergoing large scale deformations due to active filament rearrangements. B) Thesame network as in (A) but with a shorter filament turnover time. C) Plots of internalstress vs time for the network in (A). D) Plots of internal stress vs time for the networkin (B).
Turnover time (τ r ) -2 S t ea dy s t a t e s t r e ss ( n N / µ m ) Time (s) -2 S t r e ss ( n N / µ m ) A) B)
CompressionalExtensionalTotal Stress
S5 Fig.
Bimodal dependence on turnover time matches bimodal buildup anddissipation of stress in the absence of turnover. A) Bimodal buildup of stress in anetwork with very slow turnover ( τ r = 1000 s ). B) Steady state stress for networks withsame parameters as in (Aa), but for a range of filament turnover times.
PLOS osition (µm) v e l o c it y ( µ m / s ) -0.003 -0.00150 0.0015 0.003 s t r e ss ( n N / µ m )
0 0.0010.0020.0030.0040.005 τ r =10000 s position (µm) v e l o c it y ( µ m / s ) -0.003 -0.00150 0.0015 0.003 s t r e ss ( n N / µ m )
0 0.0010.0020.0030.0040.005 τ r =10 s A) B)
S6 Fig.