Distinct regimes of elastic response and deformation modes ...fcm/Papers/HeadPRE03.pdf · I. INTRODUCTION The mechanical stability, response to stress, and locomo-tion of eukaryotic

PHYSICAL REVIEW E 68, 061907 ~2003!

Distinct regimes of elastic response and deformation modes of cross-linked cytoskeletaland semiflexible polymer networks

D. A. Head,1,2 A. J. Levine,2,3 and F. C. MacKintosh1,2

1Department of Physics and Astronomy, Vrije Universiteit, Amsterdam, The Netherlands2Kavli Institute for Theoretical Physics, University of California at Santa Barbara, California 93106, USA

3Department of Physics, University of Massachusetts, Amherst, Massachusetts 01060, USA~Received 14 August 2003; published 18 December 2003!

Semiflexible polymers such as filamentous actin~F-actin! play a vital role in the mechanical behavior ofcells, yet the basic properties of cross-linked F-actin networks remain poorly understood. To address this issue,we have performed numerical studies of the linear response of homogeneous and isotropic two-dimensionalnetworks subject to an applied strain at zero temperature. The elastic moduli are found to vanish for networkdensities at a rigidity percolation threshold. For higher densities, two regimes are observed: one in which thedeformation is predominately affine and the filaments stretch and compress; and a second in which bendingmodes dominate. We identify a dimensionless scalar quantity, being a combination of the material lengthscales, that specifies to which regime a given network belongs. A scaling argument is presented that approxi-mately agrees with this crossover variable. By a direct geometric measure, we also confirm that the degree ofaffinity under strain correlates with the distinct elastic regimes. We discuss the implications of our findings andsuggest possible directions for future investigations.

DOI: 10.1103/PhysRevE.68.061907 PACS number~s!: 87.16.Ka, 62.20.Dc, 82.35.Pq

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I. INTRODUCTION

The mechanical stability, response to stress, and locotion of eukaryotic cells is largely due to networks of biopolmers that collectively form what is known as the cytosketon @1–5#. Filamentous actin~F-actin!, microtubules, andother intermediate filamentous proteins make up the cskeletal network, along with a variety of auxiliary proteinthat govern such factors as cross linking and filamgrowth. By understanding the relation of the individual filment properties and dynamically evolving gel microstructto the rheological/mechanical properties of intracellustructures, one will better understand the general framewfor cellular force generation and transduction@2,6–8#. Suchstress production and sensing underlies such fundamebiological processes as cell division, motility@9#, and adhe-sion @10–13#. Given the importance of biopolymer networkin determining the mechanical response of cells, there isobvious interest in understanding the properties of suchworks at a basic level. Understanding stress propagatiocells also has implications for the interpretation of intraclular microrheology experiments@14,15#. However, biopoly-mers also belong to the class of semiflexible polymers,called because their characteristic bending length~howeverdefined! is comparable to other length scales in the problesuch as the contour length or the network mesh size, andcannot be neglected.

Such semiflexible polymers pose interesting and funmental challenges in their own right as polymer materiaThe understanding of the properties of individual semifleible polymers is quite highly developed@16–26#; in addition,the dynamical and rheological properties of these polymin solution@24,27–33# have largely been elucidated. The rmaining problem of determining the rheology of permnently cross-linked gels of semiflexible polymers, howev

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has proved quite subtle. Related theoretical approachessidered thus far have either assumed a simplifying netwgeometry, such as a lattice@34# or a Cayley tree@35#, orassumed that the dominant deformation modes are a@36# or dominated by transverse filament fluctuations abending@37,38#.

In this paper we study the static mechanical propertiesrandom, semiflexible, cross-linked networks in the linearsponse regime with the aim of shedding light on the mcomplex, nonequilibrium cytoskeleton. Our approach isliberately minimalistic: we consider two-dimensional, athemal systems with no polydispersity in filament propertieAlthough obviously simplified, this restricts the paramespace to a manageable size and allows for a fuller charaization of the network response. The central finding of owork is the existence of qualitatively distinct regimes in telastic response and local deformation in networks, ewith characteristic signatures that should be observableperimentally.

The basic distinction between stress propagation in flible and semiflexible networks is that in the former elasdeformation energy is stored entropically in the reductionthe number of chain conformations between cross lin@39,40# while in the latter, the elastic energy is stored primrily in the mechanical bending and stretching of individuchains. In a flexible, cross-linked mesh there is only omicroscopic length scale: the mean distance between clinks or entanglements,l c . Since the actual identity of aflexible chain is immaterial on scales beyond this mecross-linking distance, chain length plays only a very smrole @41#. In a semiflexible gel, however, chains retain theidentity through a cross link because the tangent vectorsgiven chain remain correlated over distances much lonthanj, the mesh size. Thus, it is reasonable to expect thatelastic properties of the network depend on both the msize and the length of the chains.

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HEAD, LEVINE, AND MacKINTOSH PHYSICAL REVIEW E68, 061907 ~2003!

Upon increasing the density of filaments and thus the dsity of cross-links~hereafter we refer to filament density onas one quantity determines the other in two dimensions! thesystem acquires a static shear modulus via a continuphase transition at the rigidity percolation point. This corsponds to moving from left to right in the lower half of thphase diagram shown in Fig. 1. This critical point~solid linein Fig. 1! is at higher density than the connectivity percotion point. Since our model ignores the entropic elasticitythe network, below this critical density associated with rigity percolation, the material has no static shear modulusmay be considered a liquid. We will discuss the finite teperature implications of this zero-temperature phase tration in more detail. The elastic properties of the fragile g~solid! that exists just above this critical point are controllby the physics of rigidity percolation.

By increasing the cross-link density further we encouna regime over which the elastic deformation of the geldominated by filament bending and is highly nonaffine. Tnonaffine~NA! regime is consistent with prior predictions bKroy and Frey@37,38#. Within the NA regime the static sheamodulus scales linearly with the bending modulus ofindividual filaments,k, but the elastic moduli are not controlled by properties of the rigidity percolation critical poinMost remarkably, however, the deformation field under uformly applied stress is highly heterogeneous spatially olong length scales comparable even to the system sizewill quantify the degree of nonaffinity as a function of lengscale and use this nonaffinity measure to demonstrate thadegree of nonaffinity in the NA regime increases withobound as one goes to progressively smaller length scaSuch materials are then poorly described by standardtinuum elasticity theory on small length scales.

By further increasing the filament density, one approaca crossover to a regime of affine~A! deformation, in whichthe strain is uniform throughout the sample, as the velofield would be in a simple liquid under shear. This crossois shown in Fig. 1 by the dashed lines above the NA regi

FIG. 1. A sketch of the expected diagram showing the varielastic regimes in terms of molecular weightL and concentrationc;1/l c . The solid line represents the rigidity percolation transitiwhere rigidity first develops at a macroscopic level. This transitis given byL;c21. The other dashed lines indicate crossovers~notthermodynamic transitions!, as described in the text. As sketchehere, the crossovers between nonaffine and affine regimes destrate the independent nature of these crossovers from the rigpercolation transition.

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In this regime, elastic energy is primarily stored in thextension/contraction of filaments. Also, in contrast to tNA regime, the growth of the degree of nonaffinity saturaas a function of decreasing length scale. The elastic respof the network is governed primarily by the longitudincompliance of filaments, and the shear modulus can beculated from the combination of this realization and thesumption of affine, uniform deformation as shown in Re@24,30–33,36#. We will show that there is one dimensionleparameterl that controls the NA→A crossover. It is set bythe ratio of the filament length to a combination of paraeters describing the density of the network and the individfilament stiffness.

In the affine regime, the longitudinal response actuacan arise from two distinct mechanisms: there are two forof compliance of a semiflexible filament under extensiostress, one essentially entropic@36# and the other essentiallmechanical@38#. In the first case, the compliance relatesthe thermally fluctuating filament conformation, which, finstance, is straightened out under tension. A change inlength of a filament between cross links results not in simmechanical strain along that filament but rather in a redtion of the population of transverse thermal fluctuatioalong that filament thereby reducing the entropy of the fiment. This reduction results in an elastic restoring foalong the length of the filament. This is the dominant copliance for long enough filament segments~e.g., betweencross links!. In the second case, the compliance is due tchange in the contour length of the filament under tensiwhich, although small, may dominate for short segme~e.g., at high concentration!. Thus, in general, we find twodistinct affine regimes, which we refer to as entropic~AE!and mechanical~AM !.

Moving still further up and to the left in the diagramshown in Fig. 1 we would eventually reach a regime~notshown! in which the filament lengths between cross links amuch longer than their thermal persistence length and sdard rubber elasticity theory would apply. This regime isno experimental importance for the actin system. To coplete our description of the diagram, we note that by increing the filament concentration in the entropic affine regimone must find a transition from entropic to mechanical elticity (AE→AM) within a regime of affine deformation.

Here we confine our attention to two-dimensional permnently cross-linked networks and consider only enthalcontributions to the elastic moduli of the system. In effectare considering a zero-temperature system, except thaaccount for the extensional modulus of F-actin that is prcipally due to the change in the thermal population of traverse thermal fluctuations of a filament under extension.do not expect there to be a significant entropic contributto the free energy coming from longer length scale filamcontour fluctuations because of their inherent stiffness. Tjustifies our neglect of the sort of entropic contributionsthe filament free energy that are typically considered inanalysis of rubber elasticity of flexible polymers. We discuthis more below. The low dimensionality of the model sytem, on the other hand, is more significant since the estially straight chains in two dimensions will not interact ste

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DISTINCT REGIMES OF ELASTIC RESPONSE AND . . . PHYSICAL REVIEW E68, 061907 ~2003!

cally under small deformations. In our model, only olength scale is required to describe the random network,mean distance between cross links,l c . In three dimensionshowever, chains can interact sterically and two quantitiesneeded to fully describe the random network, the densitycross links and density of filaments.

In our zero-temperature analysis of the system presein this paper, we do not explicitly probe the difference btween the NA→AE and the NA→AM crossovers. Their dif-ference enters our description of the system via a chomade for the form ofm, the extension modulus of an indvidual filament. For physiological actin, we expect that trelevant transition will be the NA→AE. We discuss this fur-ther in Sec. V.

The remainder of this paper is arranged as follows. In SII we define our model system in terms of the mechaniproperties of individual filaments and the manner in whthey interconnect to form the network. An overview of thsimulation method used to find the mechanical equilibriunder an imposed strain is also given. We describe in Secthe rigidity percolation transition, at which network rigiditfirst develops. We then describe the crossover from a naffine regime above the rigidity transition to an affine regimin Sec. IV. A scaling argument is also presented for tcrossover. The macroscopic mechanical response isdemonstrated to be linked to geometric measures of thegree of affine deformation at a local level. In Sec. V, wshow how the network response in the affine regime caneither essentially thermal or mechanical in nature. In Sec.we discuss primarily the experimental implications of oresults.

II. THE MODEL

The bending of semiflexible polymers has been succfully described by the wormlike chain model, in which nozero curvatures induce an energy cost according to a benmodulusk. For small curvatures, the Hamiltonian canwritten as

Hbend51

2kE ds~¹2u!2, ~1!

where u(s) is the transverse displacement of the filameands is integrated along the total contour length of the fiment. Transverse filament dynamics can be inferred fromHamiltonian@19#. For finite temperatures, Eq.~1! can also beused to predict the longitudinal response of an isolatedment; however, asT→0 the filament buckles, preconfigurina breakdown of the linear response@38#. We also considerthe response of a filament to compression/extensional demations through the elastic Hamiltonian given by

Hstretch51

2mE dsS dl~s!

ds D 2

, ~2!

where dl/ds gives the relative change in length along tfilament. Equation~2! is just a Hookean spring responswith a stretching modulusm that is here taken to be indepe

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dent of k, although they can both be related to the crosectional radius and elastic properties of individual filame~see Sec. IV C!.

The networks are constructed by the sequential randdeposition of monodisperse filaments of lengthL into a two-dimensional shear cell with dimensionsW3W. Since theposition and orientation of filaments are uniformly distriuted over the allowed ranges, the networks are isotropichomogeneous when viewed on sufficiently large lenscales. Each intersection between filaments is identifiedcross link, the mean distance between which~as measuredalong a filament! is denotedl c , so that the mean number ocross links per rod isL/ l c21. Deposition continues until therequired cross-linking densityL/ l c has been reached. An example network geometry is given in Fig. 2.

The system Hamiltonian is found by using discrete vsions of Eqs.~1! and~2! which are linearized with respect tfilament deflection, ensuring that the macroscopic respois also linear. The detailed procedure is described in thependix. It is then minimized to find the network configurtion in mechanical equilibrium. Since entropic effects aignored, we are formally in theT[0 limit. The filaments arecoupled at cross links, which may exert arbitrary constraforces but do not apply constraint torques so that the fiments are free to rotate about their crossing points. We cment on the validity of this assumption in greater detailSec. VI below. Specifically, we find that whether or not tphysical cross links are freely rotating, the mechanical csequences of such cross links is small for dilute networprovided that the networks are isotropic. Once the displaments of the filaments under the applied strain of magnitg have been found, the energy per unit area can be calated, which within our linear approximation is equal tog2/2times the shear modulusG or the Young’s modulusY, forshear and uniaxial strain, respectively@42#. Thus the elasticmoduli can be found for a specific network. The procedurethen repeated for different network realizations and syssizes until a reliable estimate of the modulus is found. S

FIG. 2. An example of a network with a cross-link densiL/ l c'29.09 in a shear cell of dimensionsW3W and periodicboundary conditions in both directions. This example is small,W5

52 L; more typical sizes areW55L to 20L.

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Figs. 3 and 4 for examples of solved networks.The free parameters in the model are the coefficientm

andk and the length scalesL andl c . We choose to absorbkinto a third length scalel b derived from the ratio ofm andk,

l b25

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Although the simulations assume a constant angular cuture between nodes, it is possible to assignl b the physicalinterpretation of the natural length over which a free filambends when differing tangents are imposed at each end

FIG. 3. ~Color online! An example of a low-density networkwith L/ l c'8.99 in mechanical equilibrium, with filament rigiditl b /L50.006. Dangling ends have been removed, and the thickof each line is proportional to the energy density, with a minimuthickness so that all rods are visible~most lines take this minimumvalue here!. The calibration bar shows what proportion of the dformation energy in a filament segment is due to stretching or being.

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seen by a simple Euler minimization of the HamiltoniaHence we use the subscriptl b , denoting ‘‘bending length.’’Sincem now gives the only energy scale in the problem,scales out, along with one of the length scales~sayL), andthus we are left with two dimensionless control parametethe filament rigidity l b /L and the cross-link densityL/ l c .Note that there is also a fourth length scale, namely the stem sizeW, but all of the results presented below are fsufficiently large systems that theW dependence has vanished.

We now explore the various deformation regimes of tsystem beginning with the most fragile, sparse shesupporting networks that exist just above the rigidity perclation transition.

III. RIGIDITY PERCOLATION TRANSITION

For very low cross-link densitiesL/ l c the rods are eitherisolated or grouped together into small clusters, so that this no connected path between distant parts of the systemthe elastic moduli vanish. As the density of cross linksincreased, there is aconductivity percolation transition atL/ l c'5.42 when a connected cluster of infinite size first apears@43#. If there was an energy cost for rotation at crolinks, an applied shear strain would now induce a stresssponse and the elastic moduli would become nonzero@44#.This is also the case when thermal fluctuations genestresses along the filaments@45#. However, for networks withfreely rotating cross links at zero temperature, such as thunder consideration here, the network is able to defopurely by the translation and rotation of filaments. Suchfloppy mode costs zero energy and thus the elastic moremain zero. This continues to be the case until therigiditypercolation transition at a higher densityL/ l c'5.93@46–49#,when there are sufficient extra constraints that filaments mbend or stretch and the moduli become nonzero.

A full description of the network behavior just above thtransition has been given elsewhere@50#, so here we summarize the results. Just above the rigidity transition, bothG andY increase continuously from zero as a power inL/ l c , withdifferent prefactors but the same exponentf,

G, Y;S L

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same as Fig. 3 for higher densitieL/ l c'29.09 ~a! and L/ l c'46.77~b!. For calibration of the colorssee Fig. 3.

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We have found thatf 53.060.2 @50,51#, consistent with thevalue 3.1560.2 found independently@44#. It is also possibleto measure geometric properties of rigid clusters, suchtheir fractal dimension; this has been done using the pegame method, and found to give exponents that are similathat of central force~i.e., Hookean spring! percolation on alattice @46#. However, such networks cannot support bening, whereas we have found that the system Hamiltonianour model is dominated by its bending term near to the trsition. We conclude that our system is in a different univsality class to central force percolation, at least as far asideas of universality apply to rigidity percolation; indeed,casts doubt on the validity of universality for force percotion as a whole, as discussed below. Note that this discancy cannot be due to any form of long range correlationthe morphology of the system, since our random netwoare constructed in such a way as to ensure that geomcorrelations cannot extend beyond the length of a singlement. Note also that although cross links in our random nworks are connected by filament sections of varying lengthus producing a broad distribution of spring constants whcan also destroy universality according to the integralpression in Ref.@52#, our networks donot violate this con-dition. This is simply because there is a maximum lengthL,and hence a minimum spring constantm/L, between any twocross links, ensuring a low-end cutoff to the distributionspring constants. Similar considerations hold for the bendinteraction.

Of the exponents that are used to characterize the criregime, those measured by the pebble game method retopological or geometric properties of the growing rigid cluter. The exponentf is of a different class since it measures tmechanical properties of the fragile solid that appears atcritical point. Our observation of distinctf ’s for two systems~i.e., our simulations and central force lattices! that appear toshare the same geometric and topological exponents sugthat, although there are large universality classes for thepological exponents describing the interactions that prodthe appropriate number of constraints, the scaling ofshear modulus admits a larger range of relevant pertutions. We suspect that while rigidity itself is a highly nonlcal property of the network, the modulus depends criticaon how stress propagates through particular fragile, lodensity regions of the rigid cluster. Thus the moduluspends on details of how stress propagates locally throperhaps a few cross links so that the mechanical characttics of the filaments and the cross links become relevant

The possibility of experimentally observing the physiassociated with the zero-temperature, rigidity percolatcritical point @50# sensitively depends on the size of the crical region. As with other strictly zero-temperature phatransitions, there can be experimental consequences ocritical point physics only if the system can be tuned to pthrough the critical region, since the critical point itself canot be explored. At finite temperature, the network belowrigidity percolation point~but above the connectivity, or scalar percolation point! has a residual static shear modulus geerated by entropic tension in the system. One might imagthat one may crudely estimate size of the critical regi

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around the rigidity percolation transition by comparing tdecaying zero-temperature~i.e., mechanical! modulus of thenetwork above the phase transitionG;(L/ l c2L/ l cu trans)

f tothe entropic modulus below it,Gen;kBT/ l c

3 . Unfortunatelyto make such an estimate one implicitly makes assumptabout the, as yet unknown, crossover exponents. Regardwe speculate that the critical percolation point may indehave physical implications at room temperature. First, duethe significant stiffness of the filaments, this residual entrocally generated modulus should be small and thus the pics of the zero-temperature critical point may have expemental relevance. Second, one may interpret the obsedifference between the numerically extracted and calculascaling exponentsz, which describes the dependence ofl onl c / l b as evidence of corrections to mean-field scaling duethe proximity of the rigidity percolation critical point. Thisinterpretation is strengthened by the fact that additional dpoints at smaller values ofl c ~i.e., higher cross-link densityand thus farther from the rigidity percolation point! but fixedl conform more closely to our mean-field scaling exponeThus, it is reasonable to expect that one may observe pnomena associated with rigidity percolation in sparscross-linked actin systems.

IV. ELASTIC REGIMES

The coarse-grained deformation of a material is normadescribed by the strain field that is defined at all spapoints. Both the internal state of stress and the densitystored elastic energy~related by a functional derivative! arethen functions of the symmetrized deformation tensor. Inmodel, the underlying microscopic description consiststirely of the combination of translation, rotation, stretchinand bending modes of the filaments. Of these, only the latwo store elastic energy and thereby generate forces inmaterial. Thus a complete description of the energetics ofilament encompassing these two modes must lead to a mroscopic, or continuum elastic description of the material

The state of deformation itself, however, is purely a gemetric quantity; it can be discussed independently of theergetics associated with the deformation of the filamethemselves. We will characterize the deformation field asaf-fine if deformation tensor is spatially uniform under unformly applied strain at the edges of the sample. Of couthis strict affine limit is never perfectly realized within ousimulations, but, as shown below, there is a broad regionparameter space in which the deformation field is appromately affine, in the sense that quantitative measures ofnetwork response asymptotically approaches their affinedictions. This entire region shall be called ‘‘affine.’’ Sincecontinuum elastic models the stored elastic energy depeonly on the squares~and possibly higher even powers! of thespatial gradients of the symmetrized deformation tensor,clear that uniform strain is a global energy minimum of tsystem consistent with the uniformly imposed strain atboundary.

The spatial homogeneity of the strain field allows onedraw a particularly simple connection between the elaproperties of the individual elements of the network and

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collective properties. Because under affine deformationery filament experiences exactly the same deformation,collective elastic properties of the network can be calculaby determining energy stored in a single filament underaffine deformation and the averaging over all orientationsfilaments. Note this calculation constitutes a mean-fieldscription of elasticity.

If the strain field is purely affine~i.e., is uniformly distrib-uted throughout the sample! on length scales larger than thmicroscopic length scales~e.g., the distance between crolinks!, then it can be Taylor expanded to give a locally uform strain in which all elements of the strain tensor aconstants. It is then straightforward to see that the filamewould purely stretch~or compress! and the moduli would beindependent of filament bending coefficientk. Indeed, it ispossible to derive exact expressions forG andY in this case,as described below.

Conversely, nonaffine deformation on microscoplengths arises as a result of filament bending, and hendependency onk and well asm. This is what we find abovethe rigidity percolation transition, on increasing netwoconcentration or molecular weight. Surprisingly, howevthis is not restricted to the neighborhood of the transition,constitutes a broad regime of the available parameter spThis nonaffine regime is dominated by bending, as canseen by the fact thatG andY are independent ofm below. Itis important to note that continuum elasticity breaks downlength scales over which the deformation field is nonaffiIn addition, the appearance of nonaffine deformations invdates the simple, mean-field calculation of the moduli whassumes that every filament undergoes the deformation.nerically, the moduli in the nonaffine deformation regimwill be smaller than their value calculated under the assution of affine deformation. Nonaffine deformation fieldseffect introduce more degrees of freedom since the defortion field is nonuniform. Using those extra degrees of fredom, the system is able to further lower its elastic energynonaffine deformations and thereby reduce its modulus. Oupon increasing the number of mutual constraints in the stem can one constrain the system to affine deformationsthereby maximize its modulus.

A. Nonaffine, bending dominated

Starting from the most sparse networks just aboverigidity percolation transition, we first encounter the noaffine regime on increasingL/ l c . We find empirically thatthroughout this regime~until the crossover to affine deformation of one kind or the other—AM or AE! the moduli of thenetwork are controlled by the bending modes of the fiments. This nonaffine response can be distinguished fromscaling regime around the transition, which is also nonaffiby the lack of the diverging length scale associated witcontinuous phase transition; in the simulations, this cosponds to the independence ofG andY on system sizeW forrelatively smallW, as opposed to the increasingly largeWrequired for convergence close to the transition.

The dependence ofG on the system parameters in thregime can be semiquantitatively understood as follo

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Consider what happens whenk→0. In this limit, filamentscan freely bend and only stretching modes contribute todynamic response. ThisHookeancase has already been investigated by Kelloma¨ki et al. for the same random networgeometries as considered here@53#. They found the strikingresult that flexible modes exist for all densities in the lineresponse, i.e.,G[Y[0. In these flexible modes, the filaments will bend at cross links, but without costing enersincek[0. If k is now continuously increased from zerthen there should be a range of sufficiently smallk in whichthe angles remain unchanged but now incur an energyaccording to Eq.~1!, giving a total energy and henceG thatis proportional tok.

To make this idea more specific, suppose that whenk[0 the angles of filament deflection at cross links,$du%, aredistributed with zero mean and variancesdu

2 . By assump-tion, sdu

2 can only depend on the two geometric length scaL andl c , or to be more precise on their ratioL/ l c . From Eq.~1!, the energy at each cross link is

dHbend;kS sdu

l cD 2

l c . ~5!

The mean number of cross links per unit area isNL/2l c ,whereN is the number of filaments per unit area.N can beexactly related toL/ l c using the expression derived in Re@50#; however, for current purposes it is sufficient to use tapproximate relationL/ l c'(a21)/(122/a), where a52L2N/p. Thus Eq.~5! summed over the whole networgives

Gbend;ksdu

2

l c3

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Therefore plottingGL/m5sdu2 Ll b

2/ l c3 versusl b /L on log-log

axes will give a straight line of slope 2, as confirmed by tsimulations below. It is also possible to infer the variationsdu on L/ l c from either the simulations or the scaling argment presented below, but since this is not an easily meaable quantity experimentally, nothing more will be said aboit here. A bending-dominated response was assumed incalculations of Freyet al. @38# and Joly-Duhamelet al. @54#,although the two-dimensional~2D! and 3D density dependencies will of course be different.

B. Affine, stretching dominated

Under an affine strain, the network response conspurely of stretching modes and it is straightforward to calclate the corresponding modulusGaffine. Consider a rod oflengthL lying at an angleu to thex axis. Under a sheargxy ,this will undergo a relative change in lengthdL/L5gxysinu cosu, and therefore an energy cost

dHstretch51

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2 sin2u cos2u. ~7!

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rgeofhere-gth,

thegths

to

fes-taliate, itd.

di-

lyuld

con-tom.as

rre-lch

weeenich

ero


The sin2ucos2u factor reduces to 1/8 after uniformly averaing over alluP(0,p). Summing over the network as in thnonaffine case gives

Gaffine52N^dHstretch&u

gxy2

'p

16

m

L S L

l c12

l c

L23D , ~8!

where we have also corrected for dangling ends by renormizing the rod lengthsL→L22l c . In the high—density limitL/ l c→`, Eq. ~8! asymptotically approaches

Gaffine;p

16

m

l c. ~9!

Since the number of rods per unit area isN;1/(Ll c), theconcentration of protein monomers of characteristic sizea isc;NL/a;(alc)

21 and thusG;(ac)a with a51. For com-parison, theories of thermal 3D systems predicta5 7

5 forcalculations based on a tube picture@31,30,32#, a5 11

5 foraffine scaling relations@36#, and a52 for the T50 three-dimensional cellular foam@34#.

The above calculation can be repeated for an afuniaxial strain gyy to give similar expressions for thYoung’s modulusY, with the sin2ucos2u term in Eq.~7! re-placed with sin4u. Since this averages to 3/8,Yaffine differs bya factor of 3,

Yaffine53Gaffine. ~10!

Hence the Poisson ration5Y/2G21 for affinely shearednetworks isnaffine5

12 , which should be compared to the 3

lattice predictionn5 13 @34# and the 3D Cayley tree valuen

5 14 @35#.

C. Scaling argument and crossover between elastic regimes

We now attempt to identify the dominant mode governithe deviation from the affine solution. The relevant lengscales for this mode are derived, which, by comparingother lengths in the problem, allow us to estimate whencrossover between stretching-dominated and benddominated regimes should occur. This prediction correcpredicts the qualitative trends of the deviation from affinwith the lengthsL, l c , andl b . However, it is not as successful quantitatively as an empirical scaling law describedSec. IV D. Nonetheless we believe it contains the essenphysics and therefore warrants a full description.

To proceed, we note that the stretching-only solution psented above assumes that the stress is uniform along ament until reaching the dangling end. It is more realisticsuppose that it vanishes smoothly. If the rod is very long,from the ends and near the center of the rod it is stretchcompressed according to the macroscopic straing0. We as-sume that this decreases toward zero near the end, ovlengthl i , so that the reduction in stretch/compression eneis of ordermg0

2l i . The amplitude of the displacement alon

06190

l-

e

oeg-y

al

-la-

rd/

r ay

this segment which is located near the ends of the rod iorderd;g0l i . This deformation, however, clearly comesthe price of deformations of surrounding filaments, whichassume to be primarily bending in nature~the dominant con-straints on this rod will be due to filaments crossing at a laangle!. The typical amplitude of the induced curvature isorderd/ l'

2 , wherel' characterizes the range over which tcurved region of the crossing filaments extends. This repsents what can be thought of as a bending correlation lenand it will be, in general, different froml i . The latter canalso be thought of as a correlation length, specifically forstrain variations near free ends. We determine these lenself-consistently, as follows.

The corresponding total elastic energy contribution duethese coupled deformations is of order

DE152mg02l i1kS g0l i

l'2 D 2

l'l i

l c, ~11!

where the final ratio ofl i to l c gives the typical number oconstraining rods crossing this region of the filament in qution. In simple physical terms, the rod can reduce its toelastic energy by having the strain near the free ends devfrom the otherwise affine, imposed strain field. In doing soresults in a bending of other filaments to which it is coupleFrom this, we expect that the range of the typical longitunal displacementl i and transverse displacementl' are re-lated by

l'3 ; l b

2l i2/ l c . ~12!

Of course, the bending of the other filaments will onoccur because of constraints on them. Otherwise, they wosimply translate in space. We assume the transversestraints on these bent filaments to be primarily duecompression/stretch of the rods which are linked to theThese distortions will be governed by the same physicsdescribed above. In particular, the length scale of the cosponding deformations is of orderl i , and they have a typicaamplitude ofd. Thus, the combined curvature and stretenergy is of order

DE25mg02l i

l'l c

1kS g0l i

l'2 D 2

l' , ~13!

where, in a similar way to the case above,l' / l c determinesthe typical number of filaments constraining the bent onefocus on here. This determines another relationship betwthe optimal bending and stretch correlation lengths, whcan be written as

l'4 ; l b

2l il c . ~14!

Thus, the longitudinal strains of the filaments decay to zover a length of order

l i; l cS l c

l bD 2/5

, ~15!

7-7

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-sula

fene

idng

kin

e

deac

th

e

ntrmpi

r

nr-

m

t an,ly

nderin-

is-

Itnesrs,

i-

NAr in

tano

e


while the resulting bending of filaments extends over a dtance of order

l'; l cS l b

l cD 2/5

. ~16!

The physical implication of Eqs.~15! and ~16! is that alength of each filament of orderl i experiences nonaffine deformation and this nonaffine deformation causes changethe local strain field over a zone extending a perpendicdistancel' from the ends of that filament. Thus whenl ibecomes comparable to the length of the filament,L, thenetwork should deform in a nonaffine manner. We will reto this length along the filament contour over which oexpects to find nonaffine deformation asl.

These results make sense, as increased bending rigcan be expected to increase the bending correlation lel' , while decreasing the longitudinal correlation lengthl ibecause of the stiff constraints provided by the cross linBoth lengths, of course, tend to increase with decreasconcentration of cross links, i.e., with increasingl c . Thisscaling analysis assumes, however, thatl c, l i ,' . Further-more, we expect thatl b, l c in general. This is because thbending stiffness of a rodk;Yfr

4 increases with the fourthpower of its radiusr, while m;Yfr

2 increases with thesquare of the radius, whereYf is the Young’s modulus of thefilament. Thus,l b is expected to be of the order of the rodiameter, which must be smaller than the distance betwcross links, especially considering the the small volume frtionsf of less than 1% in many cases. For rods of radiusr inthree dimensions, we expect thatl c;a/Af. Thus,l b / l c maybe in the range of 0.01–0.1. This means that we expectl i. l' , although both of these lengths are of orderl c . Thus,the natural dimensionless variable determining the degreaffinity of the strain is the ratio of the filament length tol i ,the larger of the two lengths that characterize the rangenonaffine deformation; i.e. when the filaments are very locompared with the effectively stress-free ends, then mosthe rod segments experience a stretch/compression defotion determined by their orientation and the macroscostrain.

It is possible, however, thatl' above may become smallethan l c , especially for either very flexible rods or for lowconcentrations. This is unphysical, and we expect the being of constraining filaments above to extend only ovelength of orderl c when l b / l c becomes very small. This results in a different scaling ofl i , given by

l i; l c2/ l b . ~17!

Although we see no evidence for this scaling, it may becovalid for small enoughl b / l c .

D. Numerical results for elastic moduli

We now summarize our numerical results starting firslowest filament densities. Away from the rigidity transitiothe shear modulusG continues to increase monotonicalwith the cross-link densityL/ l c at a rate that only weaklydepends on the filament rigidityl b /L, as shown in Fig. 5.

06190

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Indeed, for high densitiesG approaches the affine predictioGaffine which, due to the absence of bending modes unaffine strain as discussed at the beginning of Sec. IV, isdependent ofl b . The filament rigidityl b does, however, in-fluence the crossover to the affine solution as will be dcussed below. The Young’s modulusY and the Poisson ration5Y/2G21 for a range of densities are shown in Fig. 6.is apparent thatn remains close to the affine predictionaffine5

12 , except possibly near the transition where it tak

the value 0.3560.1 @50# ~here as elsewhere, quoted erroare single standard deviations!. Note that, in two dimensionsarea-preserving deformations haven51, son50.5 does notimply incompressibility. For comparison, the thermodynamcally stable range is21<n<1 @42#. The robustness of theaffine prediction of the Poisson ratio even deep in theregime is somewhat surprising and is not accounted foour arguments.

FIG. 5. The normalized shear modulusGL/m vs the cross-linkdensityL/ l c for three different bending lengthsl b /L. The solid linegives the affine solution~8! and the dashed lines adjoining the dapoints are to guide the eye. Here and throughout, errors arelarger than the symbols.

FIG. 6. The shear and Young’s moduliG and Y for l b /L50.006 againstL/ l c . The interconnecting lines are to guide theye. ~Inset! The Poisson ration5Y/2G21 againstL/ l c for thesamel b /L.

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Varying the ratiol b /L over many orders of magnitude afixed L/ l c reveals a new regime in whichG}k, rather thanG;Gaffine}m as in the affine regime described above. Aexample is given in Fig. 7, where it can be seen thatG} l b

2

}k, suggesting that this regime is dominated bybendingmodes, a claim that is supported by the theoretical consiations presented in Sec. IV A and the work of Freyet al. @38#and Joly-Duhamelet al. @54#. We also confirm in the inset tothis figure that the regime for whichG'Gaffine is dominatedby stretching modes, and that this new regime withG}k isdominated by bending modes, as expected.

The crossover between the two regimes can be quantby the introduction of a new lengthl, being a combinationof l b and l c characterized by an exponentz,

l5 l cS l c

l bD z

. ~18!

The ratioL/l can then be used to ascertain which regimenetwork is in, in the sense thatl!L corresponds to theaffine regime, andl@L corresponds to the nonaffine regimNote that this is only possible outside the neighborhoodthe rigidity transition. For densities in the approximate ran13,L/ l c,47, a very good data collapse can be foundusingl15A@3l c

4/ l b or z5 13 , as demonstrated in Fig. 8. Th

empirical relation has already been published@51#. However,as clearly evident from the figure, it appears to fail for tsmall number of very high-density points that we have nbeen able to attain. Conversely, the scaling argument ofIV C generates the relevant length scalel25 l i5A@5l c

7/ l b2 or

z5 25 . Although the data do not collapse for this second for

as evident from Fig. 9, it appears to improve the ovecollapse for larger densities~i.e., further from the rigiditytransition point!. A possible explanation for this is thatl2 is

FIG. 7. Shear modulusG vs filament rigidity l b /L for L/ l c

'29.09, whereG has been scaled to the affine prediction for thdensity. The straight line corresponds to the bending-dominatedgime with G}k, which gives a line of slope 2 when plotted othese axes.(Inset)The proportion of stretching energy to the totenergy for the same networks, plotted against the same horizoaxis l b /L.

06190

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the correct asymptotic length, but does not apply for nworks of intermediate density, where the empirical forml1is much more successful, perhaps due to corrections to sing from the transition point as discussed in Sec. III.~This isalso the relevant range for biological applications!. However,our current computational resources cannot go to higher dsities, and so we must leave this question to be resolvedlater date, by either improved theory or increased processpeeds.

Although the goal of this paper is to characterize thehavior of semiflexible polymer networks over the wholethe parameter space, it is nonetheless instructive to alsosider parameters corresponding to physiological actin nworks. The lengthsL, l c , and l b for F-actin in physiologicalconditions can be approximated as follows. The distancetween cross links has been quoted asl c'0.1 mm @38#, which

e-

tal

FIG. 8. G/Gaffine vs L/l with l5A@3l c4/ l b for different densities

L/ l c , showing good collapse except for the highest density conered. The enlarged points forL/ l c'29.09 correspond to the samparameters as in Fig. 12.

FIG. 9. The same data as Fig. 8 plotted againstL/l2 with l2

5A@5l c7/ l b

2 , as predicted by the scaling argument in the text, shoing slight but consistent deviations from collapse for this rangeL/ l c .

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for filament lengthsL'2mm gives a cross-link densityL/ l c'20. As already argued in Sec. IV C, we can estimatel bby regarding the filament as a solid elastic cylinder wradius r, in which casel b;r'10 nm. Thus we find thal b /L;r /L;1023, which givesL/l'5. Looking at Fig. 8,this suggests that cytoskeletal networks are in the crossregion. Similarly,L/l2'4 leading to the same basic conclsion.

E. Spatial correlations

A further way of probing the degree of affinity of a nework is to consider a suitable spatial correlation functioWhatever quantity is chosen, it is clear that there can befluctuations if the strain is purely affine. This is not the cawith nonaffine strains, which will induce localized bendinmodes that couple to the local geometry of the networkthus may fluctuate from one part of the network to the neThus correlations between fluctuations should have a lonrange when the deformation is more nonaffine, qualitativspeaking. The two-point correlation function between stially varying quantitiesA(x) andB(x) can be generally defined as

CAB~r !5^A~x!B~x1r n!&2^A~x!&^B~x!&, ~19!

where the angled brackets denote averaging over all netwnodesx and direction unit vectorsn. Figure 10 shows anexample ofC«r , wherer is the local mass density of filaments and« is the energy per unit filament length, restrictto either stretching or bending energy as shown in the kThere is a clear anticorrelation between density and bforms of energy at short separations, showing that the mnitude of deformation is heterogeneously distributthroughout the network, being greater in regions of low mdensity and smaller in regions of high mass density. Quatively, the network concentrates the largest deformations

FIG. 10. The correlation functionC«r(r ) between local massdensity r and energy density«, where « is restricted to eitherstretching or bending energy as shown andr is the length of fila-ments within a radiusL/4 of the network point. The cross-linkdensity L/ l c'21.48, l b /L50.006, and both lines have been nomalized so thatuC«r(r 50)u51.

06190

er

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low mass density regions, thus reducing the macroscopicergy cost. This affects both bending and stretching moequally: there is no increased likelihood of one mode othe other for regions of given density, as demonstrated bycollapse ofC«r for both energy types after normalizationalso given in this figure.

The sizes of locally correlated regions can be inferrfrom the decay of a suitable autocorrelation function, suchthe combined energyE ~stretching plus bending! per unitlength.CEE(r ) is plotted in Fig. 11 for different density networks. The trend is forCEE(r ) to decay more slowly withrfor lower L/ l c at fixed l b /L, suggesting larger ‘‘pockets’’ ofnonuniform deformation for lower network densities, prsumably becoming infinitely large at the transition, wherecorrelation length diverges algebraically with the known eponentn'1.1760.02@46#. We have been unable to extractmeaningful length scale from ourCEE(r ) data and hence arunable to confirm the value of this exponent.

F. Measures of affinity

Intuitively, the degree to which the network deformatiois or is not affine depends on the length scale on whichlook. For length scales comparable to the system size,deformation must appear affine since we are imposingaffine strain at the periodic boundaries. Only on sosmaller length scale might deviations from affinity be oserved. If the deformation field is nonaffine on length scacorresponding to the microscopic lengthsL, l c , or l b , thenthe filaments will ‘‘feel’’ a locally nonuniform strain fieldand the assumptions leading to the prediction ofGaffine willbreak down.

To quantify the degree of affinity at a given length scaconsider the infinitesimal change in angle under an imposhear strain between two network nodes separated by atance r. Denote this angleu, and its corresponding affinepredictionuaffine. Then a suitable measure of deviation froaffinity on length scalesr is

FIG. 11. Autocorrelation of the combined~stretching plus bend-ing! energy densityE for l b /L50.006 and theL/ l c given in the key.The system sizes wereW515L (L/ l c513.92), 10L ~18.95 and

21.48!, 614 L ~31.63!, and 21

2 L ~77.42!.

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.

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atter


^Du2~r !&5^~u2uaffine!2&, ~20!

where the angled brackets denote averaging over bothwork points and different network realizations. An exampof ^Du2(r )& is given in Fig. 12, and clearly shows thatmonotonically decays with distance, as intuitively expectAlso, the deviation from affinity is uniformly higher folower l b /L at the sameL/ l c , in accord with the greater deviation of G from Gaffine observed above.

Although ^Du2(r )& decreases monotonically withr, thedecay is slow, almost power-law-like over the ranges givThis suggests that there is no single ‘‘affinity length scaabove which the deformation looks affine, and below whit does not. However, we can read off the degree of affinitythe cross-link length scaler 5 l c , which ~after normalizing tothe straing) should be!1 for an affine deformation, and@1 for a nonaffine one. This is evident when comparing F12 to theG/Gaffine for the same systems in Fig. 8; (1/g2)3^Du2( l c)&!1 does indeed correspond toG'Gaffine, andG'Gbend for (1/g2)^Du2( l c)&@1.

The monotonically increasing deviation from affinity witdecreasingL/l can also be seen using an independent afity measure, as used by Langer and Liu@55#. Consider thedisplacements$dxi% of each nodei after the strain has beeapplied, relative to their unstrained positions. Each of thhas a corresponding affine predictiondxi

affine that can be sim-ply computed given the node’s original position and the tyof strain applied. Then a scalar measure of the global detion from affinity is the root mean square of the differenbetween the measured displacements and their affine vai.e.,

m25^~dxi2dxiaffine!2&, ~21!

FIG. 12. Plot of the affinity measureDu2(r )& normalized to themagnitude of the imposed straing against distancer /L, for differ-ent l b /L. The value ofr corresponding to the mean distance btween cross linksl c is also indicated, as is the solid line (1g2)^Du2(r )&51, which separates affine from nonaffine networto with an order of magnitude~the actual crossover regime is somwhat broad!. In all cases,L/ l c'29.1 and the system size wasW5

152 L.

06190

et-

.

.’’ht

.

-

e

ea-

es,

where the angled brackets denote averaging over all nodi.In Fig. 13 we plotm/L againstL/l, and observe the expected monotonic increase of the deviation from affinity wdecreasingL/l. However, the data for differentL/ l c do notcollapse. The problem is that, unlike in Figs. 8 and 9, it is nobvious howm should be normalized to give a dimensionlequantity; we have tried using the length scalesL, l b , l c , andl, but none of these generate good collapse. It is likely tsome combination of these lengthswill collapse the data, buwe have been unable to find it empirically, and we havetheoretical prediction for this affinity measure.

V. THERMAL EFFECTS

In the above, we have considered a mechanical, puathermal model of networks governed by two microscoenergies:~i! the bending of semiflexible filaments, and~ii !the longitudinal compliance of these filaments that descritheir response to compression and stretching forces. We halready discussed the role of temperature in terms offormation of a solid via the rigidity percolation transitionNow, we examine the transition between the AM and Aregimes.

For a homogeneous filament of Young’s modulusYf , thecorresponding single-filament parametersk ~the bending ri-gidity! and m ~the one-dimensional compression/stretmodulus! are determined byYf and geometric factors:k;Yfa

4 and m;Yfa2, where 2a is the filament diameter

From here on, we refer to the later~mechanical! modulus asmM .

At finite temperature, however, there will be transverfluctuations of the filaments that give rise to an additionlongitudinal compliance. Physically, this compliance comfrom the ability to pull out the thermal fluctuations of thfilament, even without any stretching of the filament bacbone. The corresponding linear modulus for a filament sment of lengthl is @36#

-

FIG. 13. The root mean square deviation of node displacemfrom their affine prediction,m/L, plotted against the sameL/l1 asin Fig. 8. Symbol sizes are larger than errors, so the apparent scis real.

7-11

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mT5k,p

l 3;

a2,p

l 3mM , ~22!

where,p5k/(kT) is the persistence length. The full compliance of such a segment is thenl /m5 l /mM1 l /mT , corre-sponding to an effective linear modulus of

m5mMmT

mM1mT. ~23!

Thus, the thermal compliance dominates for lengths larthan A@3a2,p , while the segment behaves for all practicpurposes as a rigid rod with linear modulusmM for lengthssmaller thanA@3a2,p .

Thus, there appear to be two distinct regimes withinaffine regime: for higher concentrations, specifically forl c

&A@3a2,p , the longitudinal compliance of the filamentsgoverned by the mechanical compression/stretching ofment segments and the modulus is given by Eq.~8!; this isthe AM regime. At lower concentrations, specifically forl c

*A@3a2,p , the single-filament compliance is dominatedthermal fluctuations and

G;pk,p

16l c4

. ~24!

This is the AE regime. For the networks under discussiwhich are described by a single variablel c that both repre-sents the spacing of filaments and distance betweenstraints or cross links, the boundary between these two aregimes is simply determined by concentration, whichnaturally measured as 1/l c . For actin, we estimate the chaacteristic lengthA@3a2,p to be of order 100 nm. Thus, onlwhen the distance between cross links is less than a distof order 100 nm will the bulk response of the network dpend on the purely mechanical extension of actin filamen

Experimentally, these two affine regimes should be disguished by their scaling dependencies of the linear shmoduli G on various parameters. A clearer, qualitative dtinction, however, should be seen in their nonlinear behavSpecifically, in the thermal regime, the maximum strain~ei-ther at which the network yields or first exhibits nonlinebehavior! is expected to decrease with increasing concention of polymer or cross links@36#. This is because of thelimited extent of thermal compliance, which decreasesshorter filament segments that appear more straight. Thin contrast with mechanical networks where the nonlineties are governed by geometry~e.g., connectivity and orientation of filaments!. This would suggest a concentratioindependent maximum or characteristic strain, as is seesome colloidal gels@56#.

Moreover, the actual form of the force-extension relatifor a semiflexible polymer in the limit of segment lengthsl!,p takes on a universal form, depending only on a charteristic extensionl 2/,p and characteristic forcep2k/ l 2. Thisforce-extension relation predicts a universal strain stiffenof semiflexible gels in the affine regime, in contrast with

06190

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nonlinear modulus in the mechanical affine regime that wbe much more dependent in the type of filament@57#.

VI. IMPLICATIONS AND DISCUSSION

Starting with solutions having no~zero-frequency! elasticbehavior, as the filament concentrationcf , molecular weight~filament length! L, or density of cross links is increasethere is a point where macroscopic elastic behavior is fiobserved. This is the rigidity percolation transition, andoccurs for a fixed value ofL/ l c , where 1/l c is a particularlyconvenient measure of filament concentration, as it repsents the line density~length per volume! in our 2D system,apart from a factor of order unity.

We have shown that there are three distinct elastic behiors of semiflexible networks above the rigidity transition:~i!when either the molecular weight~filament length! or con-centration is low, a nonaffine regime is expected in whichmodulus is determined at a microscopic level by filamebending ~transverse compliance! @34,38#; ~ii ! as either themolecular weight or concentration increases, a crossoveexpected to an elastic regime in which the deformationsaffine ~uniform strain! and in which the elastic responsethe large scale is governed by the thermal/entropic longdinal compliance of filament segments;~iii ! at still higherconcentrations or cross-link densities, this single-filamcompliance becomes dominated by the mechanical comance of bare filament stretching and compression.

The crossover between~i! and ~ii ! is given by a fixedvalue of L/l of order 10, wherel is a microscopic lengthcharacterizing the range of nonaffine deformation alongfilament backbone. We expect this length to be of orderl c ,the distance between cross links. But, it should also depon the filament stiffness through the lengthl b5Ak/m. Infact, it can only depend on the two lengthsl b and l c in ournetworks. Thus, we expect thatl5 l c( l c / l b)

z. We have pre-sented a scaling argument that shows this forz52/5, whilewe find empirically thatz.1/3 for biologically relevant den-sities. The boundary between nonaffine and affine regimethus given byL;l. In the mechanically dominated regiml b5Ak/mM, while in the thermal regime,

l b5Ak/mT;A l c3k

a2,pmM

. ~25!

Thus, in the mechanical regime, the boundary is givenL; l c

11z , while in the thermal regime, the boundary is giveby L; l c

12z/2 . In either case, we see that this crossover hadifferent functional dependence on concentration, demstrating once again that the physics of this crossover istinct from the rigidity percolation transition. We showsketch of the expected diagram of the various regimespending onL and cf in Fig. 1. The boundary between mechanically dominated and thermal regimes is simply givby l c;A@3a2,p , as we have noted above.

We can make several additional observations concernthe behavior of real networks, based on our simple moFirst, we note the strong dependence of the shear modulu

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the cross-link density, as illustrated in Eq.~24! and alreadynoted for 3D affine networks in Ref.@36#. In fact, in the 2Dnetworks presented here, we have made no distinctiontween the mesh size~or typical separation of neighborinfilaments! j and the cross-link separationl c . We observefrom this strong dependence of the modulus onl c , which isindependent of filament concentration in 3D, that the molus of semiflexible gels can be varied significantly~in fact,by orders of magnitude! with changes only in cross-link density at the same filament concentration. This is very differfrom the situation for flexible polymer gels, and may wellimportant for cells, in that the mechanical properties cantuned by local variations in the densities or binding constaof various actin binding proteins.

Furthermore, as we have shown@44,51#, the modulus be-comes a very strong function of concentration~which, again,will translate to cross-link density on 3D networks! in thenonaffine regime. In the nonaffine regime, the modulusvary by several orders of magnitude with respect tostiffer affine gels. As this nonaffine regime is expectedjust a few ~specifically, of order 10 or fewer! cross linksalong a single filament, it may also be possible that thecan reduce its stiffness significantly, and even fluidize,decreasing the number of cross links per filament or thement length. In addition, by using its proximity to the N→AE crossover, the cell can tune its nonlinear mechanproperties. In the AE regime the cytoskeletal network shobe strongly strain stiffening due to the nonlinear extensioproperties of individual filaments@36#. In the NA regime,there should be a much larger linear regime since the being modes of the filaments, which dominate the deformatiin the NA regime, have a much larger linear response regiFinally, we speculate that in the affine regime, the mechacal properties of the cell should be insensitive to the detof the cytoskeletal microstructure; in the AE regime tmean-field character of the network enforced by the laratio of L to l suggests that local effects of cross-linker tyor network topology self-average. On the other hand, witthe NA regime, the cellular mechanical properties mayquite sensitive to such local network modifications.

In this model, we have assumed freely rotating crolinks. In the case of actin networks, however, it is wknown that many associated proteins can bind actin filamat either preferred or fixed angles@1#. This can have twodistinct effects: one geometric, and the other mechaniOne the one hand, the model we have described is onlyisotropic networks. Thus, if actin cross linking results inanisotropic network~e.g., with oriented bundles!, then onecannot describe such a system with the model presehere.

If, on the other hand, the networks remain isotropic, bwith rigid bond angles between filaments, then we expecsee additional rigidity of the networks as a result. The sizethis effect can be estimated for the affine regime~AE andAM, in fact!. We find that the relative contribution to thnetwork elastic modulus due to such cross links is onlyorder l b / l c , which is expected to be small for realistic neworks of actin, bothin vitro andin vivo. In simple terms, thisis simply due to the very large lever arm that a filame

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segment of lengthl c ~say, of order 1mm) between crosslinks has over a small actin binding protein of size a fenanometers. More precisely, this can be seen by notingin shearing two filaments that cross at a finite angle inshear plane, a fixed bond angle between the filamentsgive rise to a distortion~i.e., nonaffine! of the resulting fila-ment conformations within a regionl b near the cross link.~This length corresponds to the range over which finite being occurs, e.g., when a finite bending moment is imposea filament end.! The angle of this bend will be at most of thorder of the macroscopic straine in the affine regime. Thisresults in a bending elastic energy of orderke2/ l b per crosslink, compared with the longitudinal elastic energy of ordme2l c per segment between cross links. Noting thatl b

2

5k/m, we find that the latter term~which corresponds to thefreely rotating cross-link case! is larger than the former by afactor of orderl b / l c .

For simple mechanical networks atT50, as we considerin most of this paper,l b is of the order of the moleculadiameter, which is much smaller than the distance betwoverlapping filaments, let alone cross links in any real acnetwork. In thein vitro networks that have been studied, wexpect this ratio to be no larger than at most a few percenthe case of networks at finite temperature, as we discusSec. V, l b;Al c

3/,p, for which the ratio above is of ordeAl c /,p. By definition, this is smaller than 1 for semiflexiblnetworks. Thus, in any case, the corrections to the afelastic moduli due to possible fixed-angle cross links~in iso-tropic networks! are expected to be small.

In the related studies by Wilhelm and Frey@44#, who alsoconsider theT50 mechanical properties of networks suchours, the authors looked at both fixed-angle and freely roing cross links. They found that the rigidity percolation trasitions occurred at somewhat different values of concention for fixed-angle and freely rotating bonds. But, threport that very similar behavior was observed for the tcases above the critical points. Specifically, they foundstatistically significant difference in the dependence ofshear moduli with concentration in the nonaffine elasticgime. Thus, it would appear that no substantial differendue to the mechanics of cross links can be expected in eaffine or nonaffine regimes, at least for the relatively spaisotropic networks that actin forms.

ACKNOWLEDGMENTS

A.J.L. and F.C.M. would like to thank D.A. Weitz fohelpful conversations. A.J.L. would like to acknowledge thospitality of the Vrije Universiteit. D.A.H. was funded bthe European Union Marie Curie program. This work wsupported in part by the National Science Foundation unGrant Nos. DMR98-70785 and PHY99-07949.

APPENDIX

The filaments are deposited into the shear cell as alredescribed in Sec. II. This is internally represented by theof points $xi% consisting of all cross links and midpointbetween cross links~the midpoints are included so that th

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allscehe


first bending mode between any two cross links is repsented!. Relative motion between thexi contributes to thesystem Hamiltonian according to discrete versions of E~1! and~2!. A change in separation froml 0 to l 01d l betweenany two adjacently connected points incurs an energy co

dHstretch5m

2 S d l

l 0D 2

l 0 . ~A1!

In addition to this, a nonzero angledu between the vectorsxi2xi 21 and xi 112xi , wherexi 21 , xi , and xi 11 are con-secutive adjacent points on the same filament, contribute

dHbend5k

2 S du

l 8D 2

l 8, ~A2!

where l 8 is the mean of the lengths to either side of tcentral point, i.e. l 85 1

2 (uxi2xi 21u1uxi 112xi u). Cross-

.

ate

an

u

as

n-

ith

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s.

t

linked filaments are coupled by imposing the samexi at in-tersections, but there is no energy cost for relative angbetween filaments: cross links can freely rotate.

Each contribution~A1!, ~A2! is linearized with respect tochanges in thexi and summed to create the system Hamtonian H($xi%). Either a uniaxial or a shear straing is ap-plied to the system through the periodic boundaries inLees-Edwards manner@58#. The HamiltonianH($xi%) is thenminimized with respect to the$xi% by the conjugate gradienmethod@59#. Two optimizations are included. The Hessiamatrix Ai j 5]2H/]xi]xj is preconditioned byM 21, whereMhas the same diagonal 232 matrices ofA but is zero else-where. Furthermore, cross links that lie within a given smdistance, typically'1023L, are coalesced. This improvethe conditioning ofA and hence the speed of convergenconsiderably, while producing only minimal change in tmeasured quantities, except precisely at the transition.

ol.

v.

a,

ev.

nn,

y,

.

cs

vol-

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