Acta Biomaterialia 7 (2011) 2109–2118

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Acta Biomaterialia

journal homepage: www.elsevier .com/locate /actabiomat

On the role of the filament length distribution in the mechanicsof semiflexible networks

Mo Bai a, Andrew R. Missel b, Alex J. Levine b,c, William S. Klug a,c,⇑a Department of Mechanical and Aerospace Engineering, UCLA, Los Angeles, CA 90095, USAb Department of Chemistry and Biochemistry, UCLA, Los Angeles, CA 90095, USAc California Nanosystems Institute, UCLA, Los Angeles, CA 90095, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 29 September 2010Received in revised form 14 December 2010Accepted 20 December 2010Available online 25 December 2010

Keywords:ActinCytoskeletonElasticityMechanicsPolydispersity

1742-7061/$ - see front matter � 2011 Acta Materialdoi:10.1016/j.actbio.2010.12.025

⇑ Corresponding author at: Department of Mechaning, UCLA, Los Angeles, CA 90095, USA. Tel.: +1 310 79

E-mail addresses: klug@ucla.edu, klug@seas.ucla.e

This paper explores the effects of filament length polydispersity on the mechanical properties of semi-flexible crosslinked polymer networks. Extending previous studies on monodisperse networks, we com-pute numerically the response of crosslinked networks of elastic filaments of bimodal and exponentiallength distributions. These polydisperse networks are subject to the same affine to nonaffine (A/NA) tran-sition observed previously for monodisperse networks, wherein the decreases in either crosslink densityor bending stiffness lead to a shift from affine, stretching-dominated deformations to nonaffine, bending-dominated deformations. We find that the onset of this transition is generally more sensitive to changesin the density of longer filaments than shorter filaments, meaning that longer filaments have greatermechanical efficiency. Moreover, in polydisperse networks, mixtures of long and short filaments interactcooperatively to generally produce a nonaffine mechanical response closer to the affine prediction thancomparable monodisperse networks of either long or short filaments. Accordingly, the mechanical affin-ity of polydisperse networks is dependent on the filament length composition. Overall, length polydisper-sity has the effect of sharpening and shifting the A/NA transition to lower network densities. We discussthe implications of these results on experimental observation of the A/NA transition, and on the design ofadvanced materials.

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1. Introduction

Semiflexible networks refer to elastic structures composed ofcrosslinked semiflexible filaments, which can store elastic energyboth in bending and stretching deformations. Such systems derivetheir name from their relation to semiflexible polymers, macro-molecules having a long thermal persistence length as a result oftheir large bending rigidity. For the purpose of designing advancedstructural materials, stiff synthetic polymers and even carbonnanotubes are attractive examples of filaments that may be usedto form semiflexible networks [1,2]. However, the most ubiquitousexamples are all found in molecular and cell biology, where thebiopolymer filament bending rigidity, j, is typically reported interms of a thermal persistence length ‘P � j=kBT , i.e. the distancealong the polymer backbone over which the local tangent vectorsbecome decorrelated in thermal equilibrium, typically at roomtemperature. Biological examples include double-stranded DNAð‘P ¼ 50 nmÞ, filamentous actin (F-actin, ‘P ¼ 1:7� 104 nmÞ andmicrotubules ð‘P � 5� 106 nm ¼ 5 mmÞ. The first of these is a

ia Inc. Published by Elsevier Ltd. A

ical and Aerospace Engineer-4 7347; fax: +1 310 206 4830.du (W.S. Klug).

covalently bonded polymer coupled noncovalently to its so-calledconjugate pair, while the latter two are actually noncovalentlybonded aggregates of globular proteins that, in the appropriatesolution conditions, self-assemble into filaments having diametersof 7 and 25 nm, respectively, but lengths of many microns [3]. Forour purposes, we neglect hereafter both the complex microstruc-ture of these objects and the dynamics of their assembly and disas-sembly. These macromolecular aggregates can then be treated aselastic beams when considering the collective mechanics of a gelor network formed by crosslinking them.

Particularly in the case of F-actin it is possible to produce per-manently [4–6] or otherwise [5,7–9] crosslinked networks inwhich the mean distance between the crosslinks is significantlyshorter than the thermal persistence length of the constituent fila-ments. Such networks are termed semiflexible to distinguish themfrom traditional polymeric solids for which the opposite orderingof length scales typically applies. For example, in the work of Miz-uno et al. [5], F-actin filaments were crosslinked into a networkhaving a mean spacing between crosslinks on the order of 2 lm,while the ‘P of these filaments is about an order of magnitude lar-ger. Such systems can be thought of as highly simplified in vitromodels of the cytoskeleton, a chemically heterogeneous networkof F-actin, crosslinking proteins, microtubules, molecular motors

ll rights reserved.

2110 M. Bai et al. / Acta Biomaterialia 7 (2011) 2109–2118

and other constituents that pervades the ex-nuclear interior ofeukaryotic cells, providing them with their mechanical rigidityand allowing them to generate and respond to forces in the extra-cellular matrix [10–14].

Understanding the mechanics of even simple semiflexible net-works has proven to be difficult. For random statistically isotropicnetworks of this type, their collective elastic properties should de-pend only on the mechanics of the individual filaments and thenetwork density. In more complex systems, e.g. those exhibitingbroken rotational symmetries [15], composites having multiple fil-ament types [16,17] and elastically compliant crosslinks [18–20],other relevant variables are present. We do not consider such com-plexities here, but instead focus on the role of length polydispersityin random and isotropic networks of mechanically identical semi-flexible filaments. Furthermore, while the frequency-dependentrheology of these systems [21–25], and especially of living cells[26–32], is interesting and the focus of much attention, in the pres-ent article we confine our interest to the static or zero frequencymechanical properties.

The mechanical response of these networks deviates from thepredictions of traditional rubber elasticity theory applicable to gelsof flexible polymers. The underlying reasons are clear. Flexiblepolymer networks are composed of filaments whose thermal per-sistence length is typically much shorter than the mean distancebetween crosslinks on a given filament. Each polymer acts as a ran-dom walk and, under imposed strain, stores elastic free energy(entropically) in response to only the change in the separation be-tween consecutive crosslinks along the chain. There is no bendingenergy cost. Semiflexible filaments, on the other hand, can storeelastic energy by bending on scales comparable to or larger thanthe mean distance between crosslinks. Under uniformly appliedshear strain, the sample, when deforming affinely (as in the ex-pected minimum energy configuration required by stress balancefor uniform density), should not store any energy in these bendingmodes. This means that, with the appropriate modification of thesingle filament force extension curve, one should be able to under-stand the linear elastic properties of a semiflexible network [33].For sufficiently sparse networks, however, the assumption of affinedeformation breaks down at small and intermediate length scalesthat are typically longer than both the mean distance betweencrosslinks and even the length of the filaments. In such nonaffinelydeforming networks, elastic energy is stored almost entirely in thebending of filaments. A new theoretical approach to the mechanicsof the system is needed.

Previous experimental and theoretical work has shown thatdecreasing crosslink density in monodisperse semiflexible net-works leads to a sharp affine to nonaffine (NA/A) crossover froma high-density affine regime to a nonaffine one, characterized bybending-dominant, nonaffine deformation [6,7,34–45]. Mechani-cally, nonaffine networks are several orders of magnitude morecompliant than is predicted by continuum linear elasticity for auniform, affine strain field; in contrast, affine networks can achievemore than 90% of the stiffness predicted by continuum elasticity.Geometrically, when subjected to uniform strains along theirboundaries, nonaffine networks have a heterogeneous deformationfield, where denser regions in the network tend to undergo local ri-gid body motions compelling sparser ‘‘connector’’ regions to expe-rience large bending deformations, whereas affine networks followa homogeneous strain field compatible with the boundary condi-tions and consistent with continuum elasticity theory. Energeti-cally, because of the sparsely connected regions with largerdistance between crosslinks, filaments in nonaffine network tendto bend more than stretch; however, in affine networks, stretchingenergy accounts for as much as 99% of the total energy.

In this article we examine the linear elastic response of semi-flexible networks with certain classes of length polydispersity.

The purpose of doing so is twofold. First, we want to better connectthe current state of the theory, chiefly focused on monodispersenetworks, to both in vitro experiments on typically polydisperseF-actin [46–48] and eventually to studies of the elastic propertiesof living cells [49]. These experimental systems are generallyhighly polydisperse; we seek to understand the implications ofthat type of disorder on the affine to nonaffine crossover. Thereis a clear reason for so doing: in previous work it has been shownthat the crossover can be understood in terms of a single controlparameter, the ratio of the length of a filament to the so-callednonaffinity length, defined below. While this provides a universalframework for understanding monodisperse systems, it remainsto be seen whether the crossover in polydisperse networks canbe characterized in an analogous manner. Second, we propose a ba-sic question related to the engineering of optimal semiflexible net-works. One desirable feature of these systems is that they makefairly rigid solids at low volume fractions. For example, the cyto-skeleton occupies less than one percent of the cell’s volume butconfers mechanical rigidity. Assuming that one could control thelength distribution of the semiflexible filaments, how can onechoose the optimal length distribution to maximize the rigidityof the network at fixed mass density? We term this contributionto the network’s collective rigidity per unit mass density as itsmechanical efficiency and discuss what network design principlesmaximize this quantity.

We address these questions about the mechanics of polydis-perse semiflexible networks by numerical simulation using two-dimensional model networks of linear elastic beams, introducedin Section 2. We start our investigation of the length polydispersitywith bidisperse networks consisting of only two classes of filamentlength: long and short. We subsequently consider model networkswith exponentially distributed lengths, which more faithfully de-scribe experimental F-actin systems. In Section 3, we present oursimulation results: first a set of complementary phase diagramsdescribing the mechanical response of bidisperse networks withvaried compositions of long and short filaments, and second, com-parisons among monodisperse, bimodal and exponential lengthdistributions of the effects on mechanical modulus, deformationaffinity, and energy storage. For the polydisperse networks thatwe examine, our first general finding is that while there remainsa well-defined affine to nonaffine crossover, the network densityfor the cross-over can be significantly depressed relative to the pre-dictions determined from monodisperse networks having filamentlengths equal to mean length of the polydisperse system. Specifi-cally, a small fraction of longer filaments can have a disproportion-ately strong effect of increasing the mechanical stiffness of anotherwise nonaffine network of shorter filaments, and therebydelaying the network softening transition to lower network densi-ties. Secondly, when we examine directly the geometry of thestrain field or the partitioning of energy between bending andstretching modes, we see little difference between monodisperseand polydisperse networks. Thus it appears to be important to di-rectly measure the geometry of the deformation field or the parti-tioning of elastic energy between stretching and bending modes toobserve affine to nonaffine crossover in polydisperse networks. Ourthird key finding is that, deep in the nonaffine regime where evenpolydisperse networks are highly compliant relative to the stan-dard prediction based on affine deformation, broad distributionsof filament lengths provide for the most rigid networks per unitof filament length density. This appears to be a useful design prin-ciple for creating low density but rigid filament networks when inthe nonaffine regime. In Section 4 we discuss the implications ofthese results for the mechanics of the semiflexible F-actin net-works that are the frequent subject of in vitro experimental modelsof the cytoskeleton, and consider the bearing of our results on thestrategies for design of advanced materials.

M. Bai et al. / Acta Biomaterialia 7 (2011) 2109–2118 2111

2. Model

We consider two-dimensional networks composed of semiflex-ible filaments having uniform elastic properties but lengths drawnrandomly from a distribution PðLÞ. Each filament is placed withrandom position and orientation in a square box of width W andarea A ¼W2. Wherever two filaments cross with each other, a rigidpin (crosslink) is placed at their intersection. Each crosslink con-strains the relative displacements of two filaments while allowingfree rotation, and does not contribute to the network’s total energy.An example network demonstrating the model for a bimodallength distribution PðLÞ is shown in Fig. 1.

2.1. Single filament mechanics

We model the mechanics of individual filaments using classical,Bernoulli–Euler (B–E) linear elastic continuum beam theory andconsider both bending and stretching of the neutral axis. For a fil-ament with undeformed length L, its energy after deformation canbe represented as

E ¼Z

Lds

l2

d~rds

���� ����2 þ j2

dhds

� �2" #

ð1Þ

where s is the arc-length along undeformed filament,~r is the posi-tion of filament’s cross-section after deformation and h is the angleof rotation of the cross-section between the undeformed and de-formed configurations. All filaments have the same stretching mod-ulus l and bending modulus j. To compute the strain energynumerically, we discretize each filament into several piecewisestraight segments. Nodes between segments are placed at connect-ing crosslinks and the midpoints between each pair of adjacentcrosslinks. Our mesh refinement studies (not shown here) haveestablished that this level of filament refinement is sufficient fornumerical convergence. Adding more nodes between crosslinkshas no significant impact on the mechanical response. Approximat-ing derivatives by finite-differences, we then write a discrete ver-sion of the energy as a summation of bending and stretchingenergies over segments and adjacent segment pairs as

eE ¼ l2

Xsegments

d� d0ð Þ2

d0þ j

2

Xangles

b2

‘0ð2Þ

Fig. 1. A typical simulated bimodal length polydisperse network, with shortfilaments shown in blue/grey and long filaments in red/black, with Lh i=k ¼ 12 andlong/short length ratio r ¼ 5. Arrows show the shearing directions imposed on thetop and bottom periodic boundaries.

where the d0 and d parameters represent the undeformed/restlength and deformed length of the segments, b represents the anglebetween two adjacent segments and the ‘0 parameters are the aver-age of the rest lengths of two segments adjoined at an angle spring.Consistent with the small strains explored herein, we employ thesmall angle approximation b2=2 � 1� cos b. We investigate themechanical and geometric response of the network while applyingshearing deformation through Lees–Edwards sheared periodicboundary conditions [50]. Static equilibrium of the network underimposed boundary shear is obtained by minimizing the total energyusing a iterative quasi-Newton optimization code [51,52].

The relative stiffness of single filament bending and stretchingmodes is described by the bending length ‘b ¼

ffiffiffiffiffiffiffiffiffij=l

p. For a zero-

temperature prismatic rod having constant cross-sectional andelastic properties along its length, B–E continuum beam theorygives the familiar results l ¼ EA and j ¼ EI, where E is thethree-dimensional Youngs modulus, and A and I are the areaand area-moment of inertia of the cross-section, such that‘b ¼

ffiffiffiffiffiffiffiffiffiI=A

pis of the order of filament diameter. We note that for

semiflexible filaments such at F-actin at room temperature theeffective longitudinal compliance can be significantly larger thanthe zero-temperature value. In this case, tensile loading extendsthe filament primarily through the pulling out of thermally gener-ated undulations [33]. This leads to an effective l that is smallerthan the stretching rigidity EA of the underlying proteins, and thusa larger value of ‘b.

2.2. Network geometry

Throughout this paper, we employ simulation length units of mi-crons, with nominal filament length L0 ¼ 2 for monodisperse net-works and bending length ‘b ¼ 0:012 representing typical valuesfor F-actin. The dangling ends on both ends of filament do not con-tribute to the mechanics of the network [43], and are removed fromour simulation network. The overall mechanical response of the fil-ament network is therefore determined by ‘b along with the meandistance between crosslinks ‘c , which serves as an effective mechan-ical description of network density. For monodisperse semiflexiblegels, it has been shown that the mechanical and geometric nonaffin-ities depend on the combination of ‘b and ‘c through a single lengthscale, the so-called nonaffinity length,

k ¼ ‘cð‘c=‘bÞz

where previous numerical simulations [42] have empirically identi-fied z ¼ 1=3.

Extending the approach of Head et al. [42–44], we calculate ‘c

for a generic polydisperse network as follows. Considering a fila-ment of length L lying along the x-axis, the probability of crossingwith another filament placed with random position and orienta-tion can be calculated as

PcðLÞ ¼2LhLipA

ð3Þ

where the angled brackets here and throughout this article denotethe average over filament lengths for the given polydispersity PðLÞ.Thus, hLi is the average length taken over all filaments in the network.For a large number of filaments, N � 1, the probability pnðLÞ that thisfilament of length L is crosslinked n times is given by a binomial dis-tribution with N trials and success probability PcðLÞ, which can beapproximated well by an exponential distribution

pnðLÞ ¼e�NPcðLÞ NPcðLÞ½ �n

n!ð4Þ

The mean distance between crosslinks is then calculated as the ratio‘c ¼ K=nc of the average total distance between crosslinks pairs (fil-ament length less the dangling ends),

2112 M. Bai et al. / Acta Biomaterialia 7 (2011) 2109–2118

K ¼Z 1

0dLPðLÞ

X1n¼2

pnðLÞLn� 1nþ 1

¼ hLi þ hLe�NPcðLÞi � 2hLihNPcðLÞi

1� he�NPcðLÞi� �

ð5Þ

to the mean number of crosslink pairs per filament

nc ¼Z 1

0dLPðLÞ

X1n¼2

pnðLÞðn� 1Þ ¼ hNPcðLÞi þ e�NPcðLÞ�

� 1 ð6Þ

2.3. Affine response

In comparing the mechanical responses of different networks,we report throughout the computed shear modulus relative tothe value it would have under a uniform (affine) shear deforma-tion, Gaffine, which can be calculated analytically in a similar man-ner by averaging the shear stiffness of each filament of length Lat angle h as

Gaffine ¼lN2pA

Z 1

0dLPðLÞ

Z 2p

0dh sin2 h cos2 h

Xn ¼ 21pnðLÞL

n� 1nþ 1

which simplifies to

Gaffine ¼lN8A

K ð7Þ

There are two good reasons to report the numerically computedshear modulus G relative to the value that would be expected underan affine shear strain Gaffine. First, the primary mechanical signatureof the affine to nonaffine transition is the deviation of network’sshear modulus G from its affine prediction Gaffine, as given in Eq.(7). Second, the calculation of the affine shear modulus Gaffine auto-matically accounts for the mechanical effect of dangling ends.Clearly, in the strained network the dangling ends – i.e. the lengthsof the filaments extending past the crosslinks closest to the ends ofthe filaments – cannot store elastic energy. They are irrelevant tothe elasticity of the system. If one were to compare two monodis-perse networks having the same total length density but differentfilament lengths, the system with the shorter filaments (but ingreater number) would have more of these dangling ends andwould thus be more elasticity compliant. This dangling end effectis small enough to be neglected in most cases; nonetheless, by nor-malizing the observed shear modulus by the affine prediction, wecan take the effect into account completely, allowing us to concen-trate on the less trivial changes in the mechanics. Finally, we note inpassing that, as a consequence of dangling ends, a network ofmonodisperse filament lengths has, in fact, a polydisperse distribu-tion of active, load-bearing filament lengths, as discussed further inAppendix A. Hence, in the sense of mechanically relevant filamentlengths, all random semiflexible networks are polydisperse. To beclear, then, we use the term ‘‘monodisperse’’ in the commonlyunderstood sense, referring to the lengths of filaments in isolation.

3. Results

3.1. Bimodal length distribution

We began the numerical studies considering bidisperse net-works, composed of filaments of only two lengths: Ns ‘‘short’’ fila-ments of length Ls and Nl ¼ N � Ns ‘‘long’’ filaments of lengthLl ¼ rLs, with the length ratio r > 1. In anticipation of the applica-tion of our work to polydisperse F-actin networks, we fix themechanical properties of all the filaments to be identical. All fila-ments are assigned the same bending and stretching moduli land j. This makes the bending length ‘b a constant. For studies

of mechanically heterogeneous networks, e.g. F-actin and microtu-bule systems, see the work by Lin et al. [16] and Bai et al. [17].

For such bimodal distributions the composition of the networkcan be described by three scalar quantities: (i) the total filamentlength density; (ii) the fraction of that density stored in the longer(or equivalently the shorter) filaments; and (iii) the ratio of thelength of the long filaments to the short ones. Other parameteriza-tions of this information are clearly possible; we discuss belowwhich parameterizations are most directly obtainable from exper-imental parameters and which are most useful in understandingthe mechanics of the network. Although such bidisperse filamentdistributions are not easily realizable in experiment or commonlyfound in biological materials, understanding this system is usefulsince it is the most easily characterized form of length polydisper-sity due to the simplicity of the distribution.

In the following we present a mechanical phase diagram thatpresents the observed shear modulus of the network G as a func-tion of two of the parameters characterizing the density and lengthdistribution of the network. To do this we must fix one of the threeparameters discussed above. We fix r ¼ Ll=Ls the ratio of the lengthof the long filaments to the short ones and vary independently thelength density of the short and long filaments. For the bimodal-length networks considered here, the expression for the averagelength

hLi ¼ Ns

NLs þ

Nl

NLl ¼

Ns þ rNl

NLs ð8Þ

leads to a natural decomposition of the length density (i.e. the totallength of filaments per unit area in our two-dimensional system) as

q ¼ NhLiA¼ NsLs

Aþ NlLl

Að9Þ

where we denote partial densities of long and short filaments as

ql ¼NlLl

Aand qs ¼

NsLs

Að10Þ

These two densities reflect quantities that have the most directmeaning in experiment, representing the total amount of material(or length) stored in the forms of long and short filaments. Lastly,we note that, although we separate the filament density into longand short densities, the shear moduli G and Gaffine are computedfor the entire bimodal network as a whole. Furthermore, while fora given network composition Gaffine is defined uniquely by Eq. (7),G is subject to some variation from one randomly generated net-work instance to another. Although the points plotted in the phasediagram and elsewhere below each represent results for a singlenetwork instance, we have compared multiple instances for a fewcompositions, measuring variations in G as much as on the orderof 5% for networks in the NA regime. Networks in the A regimeare more self-averaging, showing sample-to-sample variation ofthe modulus of <1%. Most important, however, is the observationthat these sample-to-sample variations are everywhere dramati-cally smaller than the (average) differences between affine and non-affine samples. Based on these considerations, and because in the Aregime G=Gaffine approaches one independent of network composi-tion, we attempted to curtail computational costs by generally clus-tering simulation data points in the NA region.

Fig. 2 shows a mechanical phase diagram for bimodal networkswith long/short length ratio r ¼ 5. The color contours report thesimulated network shear modulus G normalized by Gaffine over arange of long and short filament densities ql and qs, rendereddimensionless by multiplication with short filament length Ls. Justas for monodisperse networks [43], G=Gaffine generally increaseswith increasing density q (or decreasing ‘c). However, the plotmakes it clear that the mechanics of polydisperse networks is alsostrongly dependent on the composition of filament lengths, as

Fig. 2. The phase diagram of G=Gaffine vs. filament densities ql and qs normalized byshort filament length Ls , for a series of bimodal-length networks with r ¼ Ll=Ls ¼ 5.Blue dots indicate actual simulation data points. Contours were generated bypiecewise cubic interpolation of the simulation data using the subroutine grid-

data from MATLAB (The MathWorks, Inc.). Dashed lines of constant total filamentdensity q ¼ ql þ qs are shown as a visual guide to distinguish the compositiondependence of G=Gaffine.

Fig. 3. Phase diagram of hLi=k as computed from Eqs. (5) and (6) for a bimodalnetwork with long/short length ratio r ¼ Ll=Ls ¼ 5, as a function of partial densitiesof long and short filaments ql and qs defined in Eq. (10), and normalized by shortfilament length Ls .

M. Bai et al. / Acta Biomaterialia 7 (2011) 2109–2118 2113

indicated by ql=qs. Specifically, the contours of constant modulusðG=GaffineÞ are generally curved ‘‘downward’’ toward the ql axis, rel-ative to the straight (dashed) lines of constant total density q withslope �1. In other words, as the composition is shifted from short-dominated ðql < qsÞ toward long-dominated ðql > qsÞ, the gradientdirection of G=Gaffine turns increasingly toward the ql axis. Conse-quently, the phase diagram suggests an enhanced stiffening effectfrom redistributing the sum of filament lengths into long filaments.That is, as the concentration of long filaments increases, addition ofa unit of length density in the form of long filaments stiffens thenetwork increasingly more than addition of the same unit in theform of short filaments.

Conversely, one can consider changes in the densities of thelong and short filaments along contours of constant G=Gaffine. Forshort-dominated networks with ql � qs, contours of constantmodulus track closely with lines of constant total density. Thus,simply exchanging a unit of short-filament density for a unit oflong-filament density, say by splicing together r short filamentsto create one new long filament, will have little effect on themechanical response of the network. However, as the concentra-tion of long filaments increases, the contours of constant modulusturn continuously downward, such that many units of short-fila-ment density can be exchanged for a single unit of long-filamentdensity while leaving the mechanical response unchanged. In sum-mary, long filaments are more efficient than short filaments instiffening the network. We refer to this property as a greatermechanical efficiency of the long filaments relative to that of theshorter ones.

To understand the greater mechanical efficiency of the longerfilaments distinct from the dangling ends effect, it is useful to re-view the results of the mechanics of monodisperse filament net-works and the nonaffine to affine crossover. There, as notedabove, it was found that the shear modulus G and the so-calledmechanical affinity measure G=Gaffine depends on the filamentmechanics and network density through only one parameter L=k,the ratio of the length of the filaments to the nonaffinity lengthk ¼ ‘cð‘c=‘bÞ

13. If we have two monodisperse networks of long and

short filaments having the same value of G=Gaffine (such pairs ofnetworks can be read off of the vertical and horizontal axes ofFig. 2), then it must be that Ll ¼ rLs and Ls=ks ¼ Ll=kl since the

function G=GaffineðL=kÞ is one-to-one. Recalling that the mechanicsof the individual filaments in each network are identical, it thenfollows that

ð‘cÞlð‘cÞs

¼ r34 ð11Þ

For monodisperse networks at all but the smallest densities, ‘c isproportional to the inverse of the length density [53]. Thus, fromEq. (11) and r ¼ 5, we find that networks of long and short filamentswith the same G=Gaffine will have a significant difference in thelength density ql ¼ r�

34qs � 0:3qs. This is consistent with the values

of G=Gaffine along the axes of Fig. 2. The longer filament network isequally rigid (compared to the affine prediction) as a shorter fila-ment network at about three times the length density. In this exam-ple, we may say that the longer filaments are three times moremechanically efficient.

One might imagine that this concept of the mechanical effi-ciency of a filament accounts for the entire structure of the linesof constant modulus and not just for their endpoints on the twoaxes of the mechanical phase diagram, and thus explains why linesof equal mechanics deviate significantly from lines of constant to-tal length density (shown as black dashed lines in Fig. 2) in bidis-perse networks. These ideas, however, have only been validated formonodisperse networks. To generalize these results to polydis-perse ones, we propose replacing the filament length L in the ratioL=k by the mean filament length in the network hLi. The determina-tion of k proceeds as before. There is no a priori reason to demandthat the functional dependence of G=Gaffine on hLi=k is identical tothat of the same quantities for monodisperse networks, but we ex-pect such an analogous relation between modulus and hLi=k forfixed composition to be at least monotonically increasing andone-to-one. If this be the case, contours of constant hLi=k corre-spond to contours of constant modulus and larger values of hLi=kcorrespond to more affine and mechanically incompliant networks.In Fig. 3 we plot the contours of hLi=k vs. the partial densities. Thelines of constant G=Gaffine in Fig. 2 appear to agree to much higherprecision with the lines of constant hLi=k in Fig. 3 than with the(dashed) lines of constant length density.

To make a quantitative comparison between Fig. 2 and 3, it ishelpful to redefine the two axes of the mechanical phase diagramin a manner that directly incorporates hLi=k. For bidisperse net-works we may use the following decomposition:

0 5 10 15 20 2510−3

10−2

10−1

100

Fig. 5. Normalized shear modulus G=Gaffine vs. hLi=k for the whole network. Thedangling ends which do not contribute to the network’s mechanics have beenremoved. Curves are grouped by long filament length fraction, ql=q, and long/shortlength ratio r. The inset shows a significant increase in G=Gaffine betweenmonodisperse and bimodal polydisperse networks at hLi=k ¼ 3.

2114 M. Bai et al. / Acta Biomaterialia 7 (2011) 2109–2118

hLik¼ NlLl

Nkþ NsLs

Nkð12Þ

Fig. 4 shows the mechanical phase-diagram replotted as G=Gaffine vs.NlLlNk and NsLs

Nk . Dashed lines indicate the straight contours of hLi=k withslope �1, revealing that contours of constant G=Gaffine tend to curvetoward lower hLi=k for mixed compositions of long and short fila-ments. In other words, for each value of hLi=k there is a mechani-cally optimal composition NlLl=NsLs ¼ ql=qs at which G=Gaffine

attains a maximum value. It appears that this maximum is attainednear to the line of equal composition, ql ¼ qs, although a preciselocation is difficult to identify due to the noise in the contour plotgenerated by interpolation over a rather sparse set of data points.Throughout the phase diagram, and most clearly at lower densities(see the inset of Fig. 4), bidisperse networks, i.e. those for whichql;qs–0 are always stiffer than monodisperse networks (whereeither ql or qs ¼ 0) of the same hLi=k. Thus, the proposal thatG=Gaffine depends only on hLi=k is not precisely satisfied, but linesof constant hLi=k correspond more closely to lines of constantmechanics. Interestingly, the residual corrections, especially atsmall values of hLi=k, where the network is expected to deform ina highly nonaffine manner, show that maximal mechanical effi-ciency occurs generically for compositions in which the two contri-butions to hLi=k are roughly equal: NlLl

Nk �NsLsNk .

The stiffening effect of length polydispersity is more dramati-cally demonstrated by comparing fixed composition cuts throughthe phase diagram, as shown in Fig. 5, which plots G=Gaffine as afunction of hLi=k for a selection of different long/short filamentlength ratios r ¼ Ll=Ls and compositions ql=q. As pointed out inFig. 4, we find that the ratio of the mean filament length to thenonaffinity length does not determine uniquely the mechanics ofthe network. G=Gaffine is dependent on both the length ratio r andcomposition ql=q at fixed hLi=k. Consistent with the phase diagramin Fig. 2 and 4, we see that polydispersity in filament lengths in-creases a network’s stiffness, or more precisely, the mechanicalmeasure of affinity, G=Gaffine, and that this effect is most significantfor sparse networks that are in the nonaffine regime wherehLi=kK 15. For example, at hLi=k � 2:5, a polydisperse networkwith about 10% of its density ðql=q ¼ 0:11Þ stored in the form oflong filaments ðr ¼ 5Þ is an order of magnitude stiffer than a mono-disperse network of the same density. Likewise, distributing one

Fig. 4. The phase diagram of G=Gaffine vs. Nl LlNk and Ns Ls

Nk for the same series of bimodal-length networks with r ¼ Ll=Ls ¼ 5 plotted in Fig. 2. This alternative parameteri-zation has the advantage of rendering contours of constant hLi=k as straight(dashed) lines of slope �1 (see text). Contours of G=Gaffine bend toward the originaway from lines of constant hLi=k, demonstrating that polydisperse networks arestiffer, or mechanically more affine, than monodisperse networks of equal hLi=k.Inset: magnified view of the highly nonaffine region of the phase diagram.

third of the same density ðql=q ¼ 0:35Þ to filaments that are 10times longer ðr ¼ 10Þ than the rest stiffens the network a hundred-fold. This demonstrates that the network stiffening effect is sensi-tive to both the long filament length fraction ql=q and the long/short length ratio r. The latter dependence is clearly demonstratedby comparison of the ql=q ¼ 0:36; r ¼ 5 (magenta) andql=q ¼ 0:35; r ¼ 10 (green) curves. These have the same totallength density of long filaments but differ by a factor of two inthe long/short length ratio, illustrating that simply splicing to-gether pairs of long filaments stiffens the network roughly fivefold,at least deep in the nonaffine regime. As expected for affine net-works (i.e. those with hLi=k > 20), there is essentially no mechani-cal effect of the filament length distribution once the dangling endseffect has been removed.

3.2. Exponential length distribution

Bidisperse networks provide a restricted class of polydispersesystems that can be easily characterized and thus systematicallyexplored, as in the previous section. It is well known, however, thatthe most ubiquitous example of semiflexible networks is found inthe predominantly F-actin-based cytoskeleton and its simplifiedin vitro models. Previous theoretical and in vitro experimentalstudies have predicted and shown that the polymerization processresults in polydisperse filament networks with filaments of varyinglength. In vivo, dynamically growing and remodeling actin net-works are also known to have a rather broad length distribution.Theoretically, the lengths of actin filaments have been predictedto be exponentially distributed [54]. Viamontes et al. [48] mea-sured the length distribution of the filaments used in their exper-iment and found a peaked distribution, which appearedexponential at larger lengths. They claimed that the polydispersityis responsible for affecting orientational ordering and perhaps alsoother properties of the network. In recent theoretical and experi-mental studies on composite F-actin/microtubule networks, Baiet al. [17] and Lin et al. [16] found that polydispersity in themechanical stiffness of filaments can have a large effect on theoverall mechanical response of a gel, while affecting the geometricnonaffinity to a lesser degree. Those studies revealed that the addi-tion of just a small fraction of stiff microtubules to a soft, nonaffine,actin gel can effect an increase in the effective shear of severalorders of magnitude. Here we consider whether an exponential

M. Bai et al. / Acta Biomaterialia 7 (2011) 2109–2118 2115

distribution of filament lengths can produce similar cooperativechanges to the mechanical response of networks.

Based on our intuition gleaned from the bidisperse case, we ex-pect the rare long filaments to have a disproportionately large ef-fect on the mechanics of the network deep in the nonaffineregime. To explore this, we consider an exponential lengthdistribution

PðLÞ ¼1‘e�

L�‘0‘ ; for L P ‘0

0; otherwise

(ð13Þ

cutoff at ‘0 ¼ 0:25L0. Here L0 is the filament length of our monodis-perse networks to which we compare our data. The details of thedistribution at short lengths, smaller than that of the mean distancebetween crosslinks, are mechanically irrelevant. The exponentialdistribution is characterized by a single decay length ‘, which weadjust to set the desired mean filament length

hLi ¼Z 1

‘0

PðLÞLdL ð14Þ

Using this measure, in Fig. 6 we plot G=Gaffine for networks having anexponential length distribution as a function of hLi=k, along with theanalogous mechanical data from the previously discussed mono-and bidisperse networks.

Comparing the mechanics of networks having exponential dis-tributions (red triangles and squares) to those of a monodisperse(black filled circles) and two bidisperse (blue squares and trian-gles) networks, we note two points. First, while the mechanicsare relatively insensitive to the mean length (compare the redsquares and red triangles), at least over the range explored, overallin the nonaffine regime the exponential filament distribution leadsto the stiffest networks for a given value of hLi=k. Second, thecurves are essentially identical in shape. Hence, it appears thatmeasurement of the G=Gaffine vs. hLi=k alone does not provide ameans to determine the form of the filament length distribution.The principal effect of polydispersity is simply to push the cross-over to nonaffine mechanics to even lower values of hLi=k. We alsonote that exponential length distributions generate the greatestmechanical efficiency within the set of distributions studied. Theseresults are consistent with the observations that within the bidis-perse networks larger r and roughly equal length densities (i.e.broader distributions) exhibit greater mechanical efficiency. It is

Fig. 6. Normalized shear modulus G=Gaffine vs. hLi=k, for monodisperse, bimodalpolydisperse and exponential polydisperse networks. Bimodal networks weredefined with Ls ¼ Lmono and r ¼ 5. Exponential length distributions were truncatedat ‘0 ¼ L0=4 as indicated in Eq. (13), so as to neglect filaments too short to bemechanically active.

tempting to speculate that all of these results, when taken to-gether, point to the principle that broader filament length distribu-tions generally lead to higher mechanical efficiency in thenonaffine regime. We return to this point in the discussion.

3.3. Deformation field and elastic energy storage

Previous studies on monodisperse gels have shown that thechanges in the mechanical properties across the affine–nonaffinetransition are driven, as semantically implied, by changes in thenature of the deformation field from affine at high crosslink densi-ties to nonaffine at low densities. Under purely affine strains, all fil-aments in a network are subjected to pure stretching (lengthchange) deformation along their lengths. However, in sparse andnonaffinely deforming networks, filaments undergo a combinationof stretching and bending deformations. To explore whether thesame mechanisms apply for polydisperse networks, we plot inFig. 7 the fraction of energy stored in stretching as a function ofhLi=k for representative monodisperse, bimodal and exponentialnetworks. It is clear from this plot that, just as in the monodispersecase, polydisperse networks shift their energy from stretching tobending as the density of filaments/crosslinks is decreased. A clo-ser comparison to the monodisperse case reveals that the polydis-perse networks undergo a slightly delayed, and sharper shift fromstretching to bending. Decrease in the stretching fraction sets inmore abruptly, and at lower values of hLi=k. Thus, the enhancedstiffening seen in polydisperse networks at low densities is corre-lated with and likely derived from a suppression of appearanceof bending deformations as the network is made more sparse.

To test the intuition from monodisperse studies that the in-crease in bending energy should also be a signature of increasinglynonaffine strains, we also examined the network deformationfields directly. Fig. 8 shows a vector plot of the displacement fieldfor a random sampling of nodes in three typical bidisperse net-works ðr ¼ 5; ql=q ¼ 0:11Þ above, below and just at the onset ofthe A/NA transition. At and above the transition ðhLi=k ¼ 10; 25Þ,the strain field is visually indistinguishable from a pure (affine)shear; below the transition ðhLi=k ¼ 3Þ, the presence of nonaffinedeformation is clearly manifested in the vortex-like patternsexhibiting significant vertical (and even horizontally retrograde)displacements. To quantitatively describe the degree of nonaffinedeformation, we consider the geometric nonaffinity measure intro-duced previously [42–44]:

Fig. 7. The stretching energy fraction in total energy is plotted against hLi=k. Twocomparisons are made: the effect of long filament length fraction in bimodalpolydispersity and the effects of bimodal and exponential polydispersities.

Fig. 8. Vector field plots of the displacement field in bidisperse networks ðr ¼ 5; ql=q ¼ 0:11Þ well above ðhLi=k ¼ 25Þ, well below ðhLi=k ¼ 3Þ and just at the onsetðhLi=k ¼ 10Þ of the A/NA transition. Network filaments are shown superimposed in gray. Color contours represent the value of horizontal displacement normalized by themaximum affine displacement at the top boundary. Vector glyphs are plotted at a randomly sampled subset of nodes, with lengths uniformly rescaled for better visibility.

2116 M. Bai et al. / Acta Biomaterialia 7 (2011) 2109–2118

CðrÞ ¼ hðh� haffineðrÞÞ2i=c2 ð15Þ

where h is the rotation angle of the line connecting two networknodes separated by distance r, haffineðrÞ is the corresponding rotationangle under an affine deformation and c is the shear strain. In Fig. 9we plot C vs. separation distance r normalized by ‘c , for monodis-perse, bimodal and exponential networks above ðhLi=‘c ¼ 25Þ, nearðhLi=‘c ¼ 10Þ and below ðhLi=‘c ¼ 3Þ the A/NA transition. The plotshows that polydispersity has little overall impact on the qualitativetrends of geometric nonaffinity: for affine networks C increases butplateaus as r ! 0, while for nonaffine networks it continues to growas r is reduced. Quantitatively, for very sparse networks ðhLi=k ¼ 3Þ,polydispersity leads to some increase – albeit slight – in the overallnonaffinity as compared with monodisperse networks. For higherhLi=k, the overall quantitative differences in nonaffinity betweenmonodisperse and polydisperse are more ambiguous. The discrep-ancy between the impact of polydispersity on mechanical affinity(e.g. G=Gaffine and Estretch=E) and geometrical nonaffinity (C) suggeststhat the correlation between network mechanical properties andthe deformation field that is apparent for monodisperse networksis not a general feature of networks of polydisperse length distribu-tions. A similar observation of independence between mechanicaland geometrical measures of non-affinity was also found in our re-cent work on composite networks of soft F-actin-like filaments andstiff microtubule-like filaments [17].

Fig. 9. Nonaffinity C plotted against normalized measuring distance r=‘c . Mono-disperse and polydisperse networks with different filament densities:hLi=k ¼ 3; hLi=k ¼ 10 and hLi=k ¼ 25 are tested and plotted.

4. Discussion

The principal conclusion of this work is that for polydispersenetworks the mean filament length alone is not sufficient to deter-mine its mechanics when that network is in the nonaffine regime.In order to quantify this property, we examined bidisperse net-works and introduced the mechanical efficiency of a filamentlength distribution – the ratio of the network’s shear modulus toits affine value as a function of total length density. In other words,given a fixed mass of filament making material (e.g. monomeric ac-tin) to span a given region, one may ask which partitioning of thatmass into filaments (i.e. which length distribution) provides for themost rigid network. Deep in the nonaffine regime, it appears thatthe broader the distribution of lengths, the stiffer the network.Such broad length distributions proved the most efficient use ofthe material to create a stiff random structure. In the affine regime,on the other hand, the filament length distribution is irrelevant ex-cept for the rather trivial dangling end effect.

There are two main consequences of this result. First, experi-ments that seek to observe the nonaffine regime (primarily in F-ac-tin gels) must be concerned about the role of polydispersity andparticularly the presence of long ‘‘impurity’’ filaments in the sam-ple. These rare long filaments suppress the mechanical effect ofnonaffine deformation, requiring one to create even more sparsenetworks to observe a regime in which G=Gaffine � 1. One mayspeculate that for a sufficiently broad filament length distributionit may be possible to suppress the highly compliant nonaffinemechanical regime down to network densities approaching therigidity percolation transition. In this case there may be nomechanically distinct nonaffine regime intermediate between thecritical regime associate with stress percolation and theG=Gaffine � 1 regime normally associated with an affinely deform-ing elastic solid. Regardless of the suppression of the dramatic net-work softening in the nonaffine regime seen in monodispersenetworks, the geometric signature of nonaffine deformation andthe change in the partitioning of elastic energy between filamentbending and stretching associated with the entry into the nonaf-fine regime are still found in polydisperse networks. It appears thatthese latter two signatures of the nonaffine regime originally ob-served in simulations of monodisperse networks are more robustto network heterogeneity than the mechanical signature, i.e.G=Gaffine � 1. This observation is also supported by simulations[17] and experiments [16] on F-actin and microtubule compositenetworks. We suggest that measures of either energy storage inbending degrees of freedom or the direct geometric measure ofnonaffine deformation are more reliable experimental measuresof nonaffinity than the mechanical effect. The analytical and

M. Bai et al. / Acta Biomaterialia 7 (2011) 2109–2118 2117

numerical prediction of the nonaffine/affine transition by Wilhelmand Frey [36], Heussinger and Frey [37,38], Heussinger et al. [39],Heussinger and Frey [40,41], Head et al. [42–44], Das et al. [45],Heussinger et al. [55] has been found to be qualitatively consistentwith in vitro experiments; however, quantitative validation is sofar not definitive. Wagner et al. [34] performed a series of experi-ments measuring overall stiffness of actin networks under varyingcrosslink densities with three different kinds of crosslinking pro-teins. While this study found that the overall shear modulus ofthe network increased with crosslink density, the measured stiff-ness of networks predicted to be in the nonaffine region as definedin the paper by Head et al. [43] did not deviate noticeably from theaffine shear modulus prediction. Liu et al. [35] experimentallyprobed the geometric nonaffinity in F-actin networks at various ac-tin/crosslink densities, finding that the measured nonaffinities forboth dense and sparse networks lie between the theoretical pureaffine and pure nonaffine limits. However, the differences in mea-sured nonaffinity among networks at various densities were foundto be rather small. Experimentalists have not yet observed the pre-dicted abrupt change in modulus associated with the affine to non-affine crossover in both theory and simulation. Based on ourresults regarding the effect of polydispersity on the affine to non-affine transition, it appears that one factor contributing to this dis-crepancy is polydispersity, or, more specifically, the presence of atleast a low density of longer filaments. We suggest that these long-er filaments push the mechanical softening to even smaller net-work densities, but that more direct measures of geometricnonaffinity are a better probe of the phenomenon.

Secondly, if one were to engineer biomimetic semiflexible net-works that are sparse and yet wish to maximize the linear modulusfor a fixed amount of material, developing a random network witha broad filament distribution appears to be ideal. This suggests adesign principle for the construction of simultaneously lightweight(due to the low volume fraction of the network) and rigid materialsfrom filament networks. This principle may be exploited using F-actin, microtubules or even carbon nanotubes. By analogy to tradi-tional fiber-based composites, semiflexible networks have the dis-tinguishing feature that filaments play the roles of both fiberreinforcement and the matrix. Accordingly, by pushing the A/NAtransition toward the percolation threshold, length polydispersitygreatly enhances the strength-to-weight ratio of the network atlow fiber densities. Furthermore, in contrast to traditional fibercomposites, for which design strategies are needed to control local(nonaffine) strain gradients, which serve as the main source fiberdebonding failure, semiflexible networks are free of bimaterial, fi-ber–matrix interfaces, and can therefore perform robustly despitenonaffine strains. These latter observations raise new questionsabout the optimal design for such strong and light materials cre-ated out of crosslinked stiff filaments. Is there a particular lengthdistribution, set of impurities (e.g. stiffer filaments) or even net-work organization (e.g. local nematic order [15]) that maximizesthe mechanical efficiency of the filaments? In this study we notethat polydispersity enhances mechanical efficiency. It remains tobe seen how length polydispersity interacts with the addition ofmechanically stiffer filaments to enhance (or not) the mechanicalefficiency of the composite network. Moreover, it is an open ques-tion whether the structure of the cytoskeleton is so organized as totake advantage of the enhanced mechanical efficiency of highlypolydisperse networks.

We close by pointing out some of the limitations of the presentanalysis. First, in contrast to monodisperse networks, for which thenetwork mechanical response is universally described in terms of asingle parameter, L=k, polydisperse networks have a more complexresponse depending in a non-trivial way on the density,mechanical properties and length distribution of filaments. As isclear in Fig. 4, the network stiffness when plotted against hLi=k is

not independent of length distribution. In other words, there ap-pears to be no simple way to completely deconvolve the effectsof length distribution and density (as described by either ‘c or k).Furthermore, we have confined our attention to polydisperse net-works of filaments with fixed mechanical properties, as describedby the bending length ‘b. As noted above, previous studies ofmonodisperse networks found that scaling by the nonaffinitylength k universally captures the effects of changing crosslink den-sity ‘c and the relative stiffness of filaments in bending and stretch-ing. While herein we have reported modulus results only forsimulations where ‘c is varied with ‘b fixed, we have also examinedthe modulus as a function of ‘b for a few select length compositions(not shown). In these few cases, G=Gaffine still collapses as a functionof hLi=k along with the cases where ‘c was varied.

Secondly, the present analysis focuses purely on the linear shearresponse of networks. This leaves open the question of the effect oflength polydispersity on the nonlinear network response. In partic-ular, previous theoretical studies by Onck et al. [56,57] and Mahad-evan et al. [58,59] have shown that changes in the local geometryof filaments brought about by large network deformations areimportant in defining the nonlinear response. By defining the net-work geometry with initially straight filaments and restricting ouranalysis to small shear strains, we neglect these nonlinear effects.Likewise we neglect the role of initial prestress, which is likelypresent in real semiflexible networks cross-linked in the presenceof thermal fluctuations. Exhaustive studies of the interaction oflength polydispersity with each of these effects would be interest-ing to pursue as future research.

Acknowledgement

The authors gratefully acknowledge support for this work fromNSF-CMMI-0800533.

Appendix A

We here derive the probability distribution for the dangling endlength mentioned at the end of Section 2. Imagine a filament oflength L with n P 2 crosslinks. We begin by noting that the prob-ability that no crosslinks exist within a distance s1 of one end of thefilament and s2 of the other end of the filament is given by1� ðs1 þ s2Þ=L½ �n. Taking the derivative with respect to s1 and again

with respect to s2 gives the joint probability density for having theoutermost crosslinks on the filament at distances s1 and s2 from theends. This probability is given by:

pðs1; s2; L;nÞ ¼nðn� 1Þ

L2 1� s1 þ s2

L

h in�2

Integrating across s1 and s2 subject to the constraints L� ðs1 þ s2Þ ¼Lmech and s1 þ s2 6 L results in a probability density for the mechani-cally relevant filament length Lmech as a function of L and n:

pðLmech; L; nÞ ¼nðn� 1Þ

L2

Lmech

L

� �n�2

L� Lmechð Þ

To arrive at a probability density that is independent of the numberof crosslinks, we must sum over n with the weight pnðLÞ given in Eq.(4). This results in the probability distribution for the mechanicallength of a filament of length L that is crosslinked at least twice;from this, it is trivial to derive the probability distribution for hav-ing a dangling end length of Lend given a filament length of L:

pðLend; LÞ ¼x2Lende�xLend=L

L2½1� e�xð1þxÞ�

where xðLÞ ¼ 2qLhLi=p, with q the total number density of fila-ments and hLi the mean filament length.

2118 M. Bai et al. / Acta Biomaterialia 7 (2011) 2109–2118

Appendix B. Figures with essential colour discrimination

Certain figures in this article, particularly Figs. 1–9, may be dif-ficult to interpret in black and white. The full colour images can befound in the on-line version, at doi:10.1016/j.actbio.2010.12.025.

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