Redundancy and cooperativity in the mechanics of compositely crosslinked filamentous networks

# Redundancy and cooperativity in the mechanics of compositely crosslinked filamentous networks

Moumita Das Department of Physics and Astronomy, Vrije Universiteit, Amsterdam, The Netherlands    D. A. Quint Department of Physics, Syracuse University, New York, United States    J. M. Schwarz Department of Physics, Syracuse University, New York, United States
###### Abstract

111All authors contributed equally to this work.

## I Introduction

Here, we address this redundancy versus cooperativity issue by studying a model network of semiflexible filaments crosslinked with two types of crosslinkers. We first study the mechanical properties of the model network with one type of crosslinker and then add the second type of crosslinker and look for mechanical similarities and differences with the original model network. In addition, we also address the redundancy versus cooperativity issue of two types of crosslinkers for networks made of flexible filaments.

As for the two types of crosslinkers, we consider crosslinkers that allow the crossing filaments to rotate freely (freely-rotating crosslinks) and crosslinkers that constrain the angle between two filaments. The ABP -actinin is a candidate for the former type of crosslinking mechanics: optical trapping studies demonstrate that two filaments bound by -actinin can rotate easily courson . As an example of the latter, we consider filamin A (FLNa), which binds two actin filaments at a reasonably regular angle of ninety degrees, suggesting that FLNa constrains the angular degrees of freedom between two filaments nakamura . Here, we do not take into account the possible unfolding of FLNa since the energy to unfold filamin A is large nakamura ; didonna ; gardel.filamin , nor do we take into account the kinetics of FLNa since we seek to understand fully the mechanics in the static regime first. There exist other possible examples of angle-constraining crosslinkers such as Arp2/3 that serves a dual role as an F-actin nucleator and a crosslinker blanchoin . While its role as a nucleator has been emphasized in lamellipodia formation pollard ; svitkina , its role constraining the angle between the mother and daughter filaments is presumably also important for lamellipodia mechanics. Better understanding of the mechanical role of Arp2/3 in lamellipodia may also help to distinguish between the dendritic nucleation model for lamellipodia formation and a new model where Arp2/3 only nucleates new filaments but does not produce branches urban .

In studying the mechanical properties of compositely crosslinked filamentous networks, we focus on the onset of mechanical rigidity as the filament concentration is increased above some critical threshold. This onset is otherwise known as rigidity percolation degennes ; feng.sen ; feng ; thorpe ; thorpe.cai ; sahimi ; latva-kokko . Above this critical threshold, both experiments and theoretical studies of F-actin networks have observed distinct mechanical regimes. For dense, stiff networks the mechanical response is uniform or affine and the strain energy is stored predominantly in filament stretching modes. While for sparse, floppy networks one finds a non-affine response dominated by filament bending where the observed mechanical response of the network is inhomogeneous and highly sensitive to the lengthscale being probed head ; heussinger ; wilhelm ; gardel ; das . It has been recently reported that there exists a bend-stretch coupled regime for intermediate crosslinking densities and filament stiffnesses broedersz .

While considerable progress has been made in understanding the mechanics of cytoskeletal networks that are crosslinked by one type of crosslinkers, compositely crosslinked networks are only beginning to be explored experimentally schmoller ; esue as are composite filament networks with one type of crosslinker theoretically huisman ; das2 .

Here we investigate the mechanics of such networks as a function of the concentration and elasticity of the crosslinkers and the filaments.

## Ii Model and Methods

We arrange infinitely long filaments in the plane of a two-dimensional triangular lattice. The filaments are given an extensional spring constant , and a filament bending modulus . We introduce finite filament length into the system by cutting bonds with probability , where , with no spatial correlations between these cutting points. The cutting generates a disordered network with a broad distribution of filament lengths. When two filaments intersect, there exists a freely-rotating crosslink preventing the two filaments from sliding with respect to one another. Next, we introduce angular springs with strength between filaments crossing at angles with a probability , where denotes non-collinear. These angular springs model the second type of crosslinker. See Fig.1 for a schematic.

We study the mechanical response of this disordered network under an externally applied strain in the linear response regime. For simplicity we set the rest length of the bonds to unity. Let be the unit vector along bonds and the strain on the bond . For small deformation , the deformation energy is

 E = α2∑⟨ij⟩pij(%\boldmath$u$\unboldmathij.\boldmathr\unboldmathij)2+κ2∑⟨ˆijk=π⟩pijpjk((\boldmathu\unboldmathji+\boldmathu\unboldmathjk)×\boldmathr\unboldmathji)2 (1) + κnc2∑⟨ˆijk=π/3⟩pijpjkpncΔθijk2

where is the probability that a bond is occupied, represents sum over all bonds and represents sum over pairs of bonds sharing a node. The first term in the deformation energy corresponds to the cost of extension or compression of the bonds, the second term to the penalty for the bending of filament segments made of pairs of adjacent collinear bonds, and the last term to the energy cost of change in the angles between crossing filaments that meet at angle. Furthermore, for small deformations . It is straightforward to see that the angular spring between and will contribute to an effective spring in parallel with , giving rise to an enhanced effective spring constant .

### ii.1 Effective medium theory

We study the effective medium mechanical response for such disordered networks following the mean field theory developed in feng ; thorpe for central force networks and das for filament bending networks. The aim of the theory is to construct an effective medium, or ordered network, that has the same mechanical response to a given deformation field as the depleted network under consideration. The effective elastic constants are determined by requiring that strain fluctuations produced in the original, ordered network by randomly cutting filaments and removing angular springs vanish when averaged over the entire network.

Let us consider an ordered network with each bond having a spring constant , a filament bending constant for adjacent collinear bond pairs , and an angular bending constant between bonds making angles. Under small applied strain, the filament stretching and filament bending modes are orthogonal, with stretching forces contributing only to deformations along filaments () and bending forces contributing only to deformations perpendicular to filaments (), and hence we can treat them separately. The angular forces due to the angular (non-collinear) springs, when present, contribute to stretching of filaments as discussed earlier, where we only consider three body interactions. For these springs to contribute to bending one needs to consider four-body interactions which is outside the scope of this paper and will be addressed in future work.

We start with the deformed network and replace a pair of adjacent collinear bonds with bending rigidity by one with a rigidity , and a bond spring with extensional elastic constant by a spring with an elastic constant and the facing angular spring by . This will lead to additional deformation of the above filament segments and the angle which we calculate as follows. The virtual force that needs to be applied to restore the nodes to their original positions before the replacement of the bonds will have a stretching, a bending and an angular contribution: , , and . The virtual stretching force is given by , the virtual filament bending force is , while the virtual force to restore the angle is , where , and are the corresponding deformations in the ordered network under the applied deformation field. By the superposition principle, the strain fluctuations introduced by replacing the above bending hinges and bonds in the strained network are the same as the extra deformations that result when we apply the above virtual forces on respective hinges and segments in the unstrained network. The components of this “fluctuation” are, therefore, given by:

 dℓ∥ = Fsμm/a∗−μm+α+(3/2)κnc dℓ⊥ = Fbκm/b∗−κm+κ dθ = Fθκnc,m/c∗−κnc,m+κnc (2)

The effective medium spring and bending constants, , and , respectively, can be calculated by demanding that the disordered-averaged deformations , , and vanish, i.e. , , and . To perform the disorder averaging, since the stretching of filaments is defined in terms of spring elasticity of single bonds , the disorder in filament stretching is given by . Filament bending, however, is defined on pairs of adjacent collinear bonds with the normalized probability distribution . Similarly, for the angular springs, the normalized probability distribution is given by . This disorder averaging gives the effective medium elastic constants as a function of and as

 p3parp(μm−α−3κarp/2μm/a∗−μm+α+3κarp/2)+(1−p)p2parp(μm−3κarp/2μm/a∗−μm+3κarp/2) (3) + p(1−p2parp)(μm−αμm/a∗−μm+α)+(1−p)(1−p2parp)(μmμm/a∗−μm)=0

The constants , and for the network contribution to the effective spring constant of bonds, to the filament bending rigidity , and the bending rigidity of angular springs making angles respectively, are given by . The sum is over the first Brillouin zone and is the coordination number. The stretching, filament bending and non-collinear bending contributions, respectively, to the full dynamical matrix , are given by:

 \boldmathD\unboldmaths(q) = μm∑⟨ij⟩[1−e−i\boldmathq% \unboldmath.\boldmathr\unboldmathij]\boldmathr% \unboldmathij\boldmathr\unboldmathij \boldmathD\unboldmathb(q) = κm∑⟨ij⟩[4(1−cos(\boldmathq\unboldmath.\boldmathr\unboldmathij)) −(1−cos(2\boldmathq\unboldmath.\boldmathr% \unboldmathij))](\boldmathI\unboldmath−\boldmath% r\unboldmathij\boldmathr\unboldmathij) \boldmathD\unboldmathnc(q) = 32κnc,m∑[2(1−cos(\boldmathq% \unboldmath.\boldmathr\unboldmathij))+2(1−cos(\boldmathq% \unboldmath.\boldmathr\unboldmathik)) (4) −2(1−cos(\boldmathq\unboldmath.\boldmathr% \unboldmathjk))]\boldmathr\unboldmathij\boldmathr\unboldmathik

with the unit tensor and the sums are over nearest neighbors  feng . Note that for small , and have the expected wavenumber dependencies for bending and stretching.

By definition, , where is the dimensionality of the system. At the rigidity percolation threshold , , and vanish, giving , and . For semiflexible filament networks with only freely-rotating crosslinks i.e. filament stretching and bending interactions only, the rigidity percolation threshold is given by . For networks with angle-constraining crosslinks, at , we obtain rigidity percolation thresholds for the case of flexible filament networks, and for semiflexible filament networks. We also calculate how changes on continuously increasing from to .

### ii.2 Numerical Simulations

Simulations were carried out on a triangular lattice with half periodic boundary conditions along the shear direction for the energetic terms whose small deformation limit is given in Eq. (1). Networks were constructed by adding bonds between lattice sites with probability . Next, a shear deformation was applied to the two fixed boundaries of magnitude . The lattice was then relaxed by minimizing its energy using the conjugate gradient method  numrecipes allowing the deformation to propagate into the bulk of the lattice. Once the minimized energetic state was found within the tolerance specified, in this case the square root of the machine precision , the shear modulus was then measured using the relation, , using small strains , with denoting the system length and denoting the area of the unit cell for a triangular lattice which is equal to in our units. System size was studied, unless otherwise specified, and sufficient averaging was performed.

## Iii Results

Mechanical integrity as measured by the shear modulus: On a triangular lattice, networks made solely of Hookean springs lose rigidity at a bond occupation probability around  maxwell ; alexander ; feng . This result corresponds to the central force isostatic point at which the number of constraints is equal to the number of degrees of freedom on average. In contrast, networks made of semiflexible filaments become rigid at a smaller due to extra constraints placed on the system via filament bending. For semiflexible networks with freely-rotating crosslinks, our effective medium theory shows that the shear modulus, , approaches zero at as shown in Fig.2 . This result is in good agreement with our simulation results yielding and previous numerical results broedersz . See Fig.2 . A different formulation of the EMT yields  broedersz . By introducing additional crosslinks that constrain angles between filaments at , the rigidity percolation threshold is lowered. Our EMT yields and our simulations yield for (Fig.2 and ). The cooperative mechanical interplay between these crosslinks and their interaction with filaments allows the network to form a rigid stress-bearing structure at remarkably low crosslinking densities, almost immediately after it attains geometric percolation, , which agrees with a calculation by Kantor and Webman kantor . For flexible filament networks, introducing angle-constraining crosslinkers also lowers the rigidity percolation threshold as compared to the isostatic point with the network attaining rigidity at for our EMT and in the simulations ((Fig.2 and ). Incidentally, our result agrees very well with a previous simulation hughes . We also compute analytically and numerically how changes with . See Fig.3, and . Note that is lowered continuously as the concentration of angle-constraining crosslinks is increased.

Just above the rigidity percolation threshold, for a semiflexible network with freely-rotating crosslinks, we find a bending-dominated regime for sparse networks with the shear modulus eventually crossing over to a stretch dominated affine regime at higher filament densities. The purely stretch dominated regime is represented by the macroscopic shear modulus staying almost constant with increasing , while in the purely bend dominated regime the network is highly floppy and is a sensitive function of , decreasing rapidly as is lowered. This behavior has been observed previously in  head ; heussinger ; wilhelm ; das ; broedersz . For , both the effective medium theory and the simulations yield a bend-stretch coupled regime, which is characterized by an inflection in as a function of as observed most clearly for (with ).

We find a similar non-affine to affine crossover for the compositely crosslinked flexible filament networks and semflexible filament networks as is increased. For the flexible filament networks, however, the bend-stretch coupling regime occurs for , i.e. replaces . For semiflexible filament networks, as long as , the bend-stretch coupled regime is robust (for fixed ). In contrast, for , the angle-constraining crosslinker suppresses the bend-stretch coupled regime and enhances the shear modulus to that of an affinely deforming network (for fixed ). The mechanics of the network has been altered with the introduction of the second type of crosslinker.

Non-affinity parameter: To further investigate how the interaction of the crosslinkers affects the affine and non-affine mechanical regimes, we numerically study a measure for the degree of non-affinity in the mechanical response, , defined in Ref.broedersz as:

 Γ=1L2γ2N∑i(ui−uaff)2. (5)

The non-affinity parameter can be interpreted as a measure of the proximity to criticality, diverging at a critical point as we approach infinite system size. We find that develops a peak at the rigidity percolation threshold, which progressively moves to smaller values of as the concentration of angular crosslinkers is increased (Fig.4 ). A second peak develops near the isostatic point for as seen in Fig.4 . As both the collinear and non-collinear bending stiffnesses tend to zero, the network mechanics approaches that of a central force network, and the second peak in at the isostatic point becomes increasingly more pronounced.

On the other hand, this second peak can be suppressed by increasing (Fig.4 ), or by increasing the concentration (Fig.4 ) even for very small values of . This further corroborates that adding angle-constraining crosslinkers to non-affine networks can suppress non-affine fluctuations, provided they energetically dominate over filament bending. The reason for this suppression can be understood by considering the effect of adding a constraint which prohibits the free rotation of crossing filaments. As the concentration of these non-collinear crosslinks is increased (at fixed avg. filament length) microscopic deformations will become correlated. The lengthscale associated with this correlation will increase on increasing either or , and will eventually reach a lengthscale comparable to system size even at at large enough concentration and/or stiffness of the angular springs. As a result the mechanical response of the network will approach that of an affinely deforming network. Upon decreasing the value of relative to we again recover the second peak because energetically the system can afford to bend collectively near the isostatic point.

Scaling near the isostatic point: Finally, using scaling analysis we quantify the similarity in mechanics between freely-rotating crosslinked semiflexible networks and compositely crosslinked flexible networks. To do this, we examine the scaling of the shear modulus near the isostatic point with . For (or ), the shear modulus scales as (or broedersz ; wyart . For both , and , , the EMT predicts and as shown in Fig.5(a) and (b), indicating that both types of networks demonstrate redundant, or generic, mechanics. To compare the EMT results with the simulations, we use the position in the second peak in to determine the central force percolation threshold, , and then vary and to obtain the best scaling collapse. For case (a), , and . For case (b), , and . Both sets of exponents are reasonably consistent with those found in Ref. broedersz for a semiflexible network with freely-rotating crosslinks only. Preliminary simulations for compositely crosslinked semiflexible networks indicate that the shear modulus scales as also with a similar and a similar with .

## Iv Discussion

In the limit of small strain, we conclude that the presence of multiple crosslinkers in living cells can be simultaneously cooperative and redundant in response to mechanical cues, with important implications for cell mechanics. Redundant functionality helps the cytokeleton be robust to a wide range of mechanical cues. On the other hand, different crosslinkers can also act cooperatively allowing the system to vary the critical filament concentration above which the cytoskeleton can transmit mechanical forces. This may enable the cytoskeleton to easily remodel in response to mechanical cues via the binding/unbinding of crosslinkers (tuning concentration) or their folding/unfolding (tuning stiffness and type of crosslinker). Since the cytoskeleton consists of a finite amount of material, the ability to alter mechanics without introducing major morphological changes or motifs may play important role in processes such as cell motility and shape change.

The second cooperative interplay between the two crosslinkers depends on the energy scale of the angle-constraining crosslinker to the filament bending energy. For , the freely-rotating semiflexible filament system exhibits large non-affine fluctuations near the isostatic point. Upon addition of the angle-constraining crosslinkers, for , the non-affine fluctuations near this point become suppressed and the mechanics of the angle-constraining crosslinker dominates the system. Once again, with a small change in concentration of the second crosslinker, the mechanical response of the network is changed dramatically.

Redundancy: We observe two redundant effects in these compositely crosslinked networks, the first of which depends on energy scales. For with , the non-affine fluctuations near the isostatic point in the freely-rotating crosslinker semiflexible filament network remain large even with the addition of the angle-constraining crosslinker. In other words, the angle-constraining crosslinkers are redundant near the isostatic point. Their purpose is to decrease the amount of material needed for mechanical rigidity as opposed to alter mechanical properties at higher filament concentrations.

Redundancy is also evident in the mechanics of these networks sharing some important, generic properties. All three networks studied here (free-rotating crosslinked semibflexible networks and compositely crosslinked semiflexible and flexible networks) have three distinct mechanical regimes: a regime dominated by the stretching elasticity of filaments, a regime dominated by the bending elasticity of filaments and/or stiffness of angle-constraining crosslinkers, and an intermediate regime which depends on the interplay between these interactions. The extent of these regimes can be controlled by tuning the relative strength of the above mechanical interactions. In particular, the ratio of bending rigidity to extensional modulus of an individual actin filament is head . Since the bend-stretch coupled regime has not been observed in prior experiments on in-vitro actin networks crosslinked with FLNa only, we conjecture that the energy cost of deformation of angles between filaments crosslinked with FLNa is larger than the bending energy of filaments. The qualitative redundancy becomes quantitative, for example, near the isostatic point where we obtain the same scaling exponents for as a function of and (or ) for the free-rotating crosslinked semiflexible network and the compositely crosslinked flexible network. Preliminary data suggests the same scaling extends to compositely crosslinked semiflexible networks. This result is an indication of the robustness of these networks and should not be considered as a weakness. Whether or not this robustness extends to systems experiencing higher strains such that nonlinearities emerge is not yet known.

Lamellipodia mechanics: The interplay between cooperative and redundant mechanical properties may be particularly important for the mechanics of branched F-actin networks in lamellipodia. Within lamellipodia, there exist some filament branches occuring at an angle of around with respect to the plus end of the mother filament (referred to as junctions). These branches are due to the ABP Arp2/3 blanchoin . During lamellipodia formation, these branches are presumed to be the dominant channel for filament nucleation. The mechanics of Arp2/3 can be modeled as an angular spring between the mother and daughter filament with an angular spring constant of approximately  blanchoin . In other words, Arp2/3 is an angle-constraining crosslinker for junctions (as opposed to junctions), and thereby plays an important role in lamellipodia mechanics as demonstrated in this work. The mechanical role of Arp2/3 in lamellipodia has not been investigated previously and may help to discriminate between the dendritic nucleation model pollard ; svitkina and a new model urban by predicting the force transmitted in lamellipodia as a function of the Arp2/3 concentration.

In addition to Arp2/3, FLNa localizes at junctions in the lamellipodia and is thought to stabilize the dendritic network revenu . Both angle-constraining crosslinkers lower the filament concentration threshold required for mechanical rigidity in the system. Depending on the energy scale of FLNa as compared to the energy scale of Arp2/3, addition of the FLNa may or may not modulate, for example, the bend-stretch coupling regime at intermediate filament concentrations. Again, at times mechanical redundancy is needed and at times not. With three crosslinkers, the system can maximize the redundancy and the cooperativity. Of course, lamellipodia are dynamic in nature and are anisotropic since the Arp2/3 is activated from the leading edge of a cell. Both attributes will modulate the mechanical response.

###### Acknowledgements.
DAQ would like to thank Silke Henkes and Xavier Illa for useful discussions regarding lattice simulations. MD would like to thank Alex J. Levine, F. C. MacKintosh, C. Broedersz, T.C. Lubensky, C. Heussinger and A Zippelius for discussions on the mechanics of semiflexible networks. MD and JMS also acknowledge the hospitality of the Aspen Center for Physics where some of the early discussions took place. JMS is supported by NSF-DMR-0654373. MD is supported by a VENI fellowship from NWO, the Netherlands.

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