Functional synergy of actin filament cross-linking proteins.

The organization of filamentous actin (F-actin) in resilient networks is coordinated by various F-actin cross-linking proteins. The relative tolerance of cells to null mutations of genes that code for a single actin cross-linking protein suggests that the functions of those proteins are highly redundant. This apparent functional redundancy may, however, reflect the limited resolution of available assays in assessing the mechanical role of F-actin cross-linking/bundling proteins. Using reconstituted F-actin networks and rheological methods, we demonstrate how alpha-actinin and fascin, two F-actin cross-linking/bundling proteins that co-localize along stress fibers and in lamellipodia, could synergistically enhance the resilience of F-actin networks in vitro. These two proteins can generate microfilament arrays that "yield" at a strain amplitude that is much larger than each one of the proteins separately. F-actin/alpha-actinin/fascin networks display strain-induced hardening, whereby the network "stiffens" under shear deformations, a phenomenon that is non-existent in F-actin/fascin networks and much weaker in F-actin/alpha-actinin networks. Strain-hardening is further enhanced at high rates of deformation and high concentrations of actin cross-linking proteins. A simplified model suggests that the optimum results of the competition between the increased stiffness of bundles and their decreased density of cross-links. Our studies support a re-evaluation of the notion of functional redundancy among cytoskeletal regulatory proteins.

The dynamic assembly and disassembly of filamentous actin (F-actin) 1 and its organization in ordered arrays are coordinated by various accessory proteins (1). More than 60 actinbinding proteins have already been identified in mammalian cells, among which 7 are F-actin cross-linking/bundling proteins (2). The function of actin cross-linking/bundling proteins is believed to mediate interactions between actin filaments to form both orthogonal networks and ordered bundles. Orthogonal networks and bundles appear, sometimes concurrently, in specialized subcellular complexes, including lamellipodia, filopodia, stress fibers, and focal adhesions. These organelles, which coordinate cell migration and cell spreading (3), often contain more than one actin cross-linking protein. For instance, stress fibers in adherent epithelial cells and fibroblasts contain ␣-actinin, fascin, and tensin (4), which are proteins shown to have the actin cross-linking/bundling function in vitro. Focal adhesions, which mechanically couple the actin cytoskeleton to the extracellular matrix via cell surface receptors integrins, contain the actin cross-linking proteins ␣-actinin, talin, plectin, fimbrin, and vinculin (5). Actin-filament bundles in stable microspikes and isotropic meshworks in the lamellipodia of motile cells also contain several actin cross-linking proteins, including fascin and ␣-actinin and the Arp2/3 complex, ␣-actinin, myosin, and fascin, respectively (6).
The fact that several actin cross-linking/bundling proteins co-localize to subcellular organelles suggests that these proteins complement each other and/or have overlapping functions. The relative tolerance of cells to null mutation of genes that code for a single cross-linking protein suggests that the functions of actin cross-linking proteins are highly redundant (7)(8)(9). This apparent functional redundancy may, however, reflect the limited resolution of available assays in assessing the mechanical role of F-actin cross-linking/bundling proteins in vitro and in vivo.
Recent ultrastructural work on developing Drosophila has begun to elucidate the mechanisms by which actin cross-linking proteins that co-localize in certain subcellular regions could synergistically operate. Tilney and co-workers showed that singed and forked proteins, two putative actin cross-linking proteins, were both required for proper formation of actin filament bundles in the mechanosensory bristles of Drosophila (10 -12). Forked proteins are necessary in the early development of the F-actin bundles that singed proteins subsequently stabilize via a zippering mechanism. How singed and forked proteins can complement one another to produce the propulsive forces required for the development of sensory bristles is however unknown.
Networks of actin filaments exhibit relatively poor mechanical resilience, they only resist relative shear deformations of amplitude smaller than 3% (13); moreover, the elasticity of F-actin networks in vitro is much lower than that of that found in living cells (14). Actin filaments, therefore, harness auxiliary proteins to self-organize into stiff structures that can strengthen the cytoplasm. Here, using reconstituted actin filament networks, we investigated how two types of actin crosslinking proteins could enhance the resilience of actin filaments synergistically. Resilience is measured by the shear amplitude at which the modulus of a network under stress begins to decrease, presumably due to the breakage and/or alignment of the filaments by the applied stresses. This work focuses on ␣-actinin and fascin, two F-actin cross-linking proteins that co-localize along stress fibers and in lamellipodia of epithelial cells and fibroblasts (15). Our rheological measurements show that these two proteins can generate microfilament arrays that are more resilient and more elastic than each one of these proteins separately. A finite element model of networks containing filaments and bundles show that a mechanical optimum is reached between an all-filament and all-bundle networks.

MATERIALS AND METHODS
Protein Purification-Unless specified, all reagents were purchased from Sigma. Actin was prepared from chicken breast (16) using Sephacryl S-300 for gel filtration (17). Purified actin was stored as Ca 2ϩ -actin in continuous dialysis at 4°C against buffer G (0.2 mM ATP, 0.5 mM DTT, 0.2 mM CaCl 2 , 1 mM sodium azide, and 2 mM Tris-HCl, pH 8.0). Mg 2ϩ -actin filaments were generated by adding 0.1 volume of 10ϫ KMEI (500 mM KCl, 10 mM MgCl 2 , 10 mM EGTA, 100 mM imidazole, pH 7.0) polymerizing salt buffer solution to 0.9 volume of G-actin in buffer G. ␣-Actinin was purified from chicken smooth muscle as described (18); ␣-actinin was stored by dialysis against buffer G, which was changed daily with fresh buffer. Actin and ␣-actinin were used within 5 days after purification. Human fascin was expressed as a glutathione Stransferase fusion using pGEX2T in the Escherichia coli strain JR600 as described in Ref. 19. Fascin was liberated from glutathione S-transferase by cleavage with thrombin, followed by glutathione-Sepharose chromatography. Fascin was dialyzed against 10 mM Tris, 1 mM DTT, pH 8.0, and applied to a DE52 column that was developed with a linear gradient of 0 -70 mM NaCl in 10 mM Tris, 1 mM DTT, pH 8.0. Purified fascin was dialyzed against 10 mM Tris, 10 mM NaCl, 30 mM KCl, 0.1 mM EDTA, 1 mM DTT, pH 8.0, and stored at Ϫ70°C.
Mechanical Rheometry-To probe the mechanical properties of Factin solutions, we employed a strain-controlled mechanical rheometer (ARES-100 Rheometrics, Piscataway, NJ) as described (13,20). The superior sensitivity of this rheometer is particularly well suited to probe low viscosity specimen such as F-actin networks. The G-actin solution was first mixed with either fascin, ␣-actinin, or a mixture of fascin and ␣-actinin in a test tube; 10ϫ KMEI was added to the solution (total volume 1.4 ml). We note that we did not mix one of the cross-linkers with F-actin and then added the other because strong binding to F-actin can cause the early formation of heterogeneous structures. The solution was immediately loaded onto the lower plate of the rheometer. This plate is coupled to a computer-controlled motor, which can apply steady or oscillatory shear deformations of controlled frequency and amplitude (13). The upper tool of the rheometer is a truncated cone (0.04 radian), which is connected to a sensitive torque transducer that measures the stress induced within the F-actin specimen by an applied deformation. The temperature of the specimen was maintained at T ϭ 22 Ϯ 0.1°C; evaporation was reduced using a vapor trap. The gelation process was monitored until steady state was reached as described (13).
At steady state, we determined the shear amplitude-dependent moduli, G(t, ␥ 0 ) by conducting stress-relaxation experiments where the time-dependent stress, (t, ␥ 0 ) (force per unit area), was measured after a step shear deformation of controlled amplitude ␥ 0 was applied and maintained for 1000 s. The time after application of the step deformation represents a time scale that describes the evolution of the filaments motion in the network (see "Discussion") and is equivalent to the inverse of the frequency in an oscillatory assay, t ϭ 2/. The amplitude of the deformation was varied from ␥ 0 ϭ 0.5-100%. Since large defor- mations may break actin filaments, we checked that the network had recovered before applying a new step deformation. A new stress-relaxation experiment was conducted when the overall viscoelastic modulus, ͉G*͉ ϭ ͱGЈ 2 ϩ GЉ 2 (Eq. 1) (evaluated at 2 radian/s and ␥ 0 ϭ 1%), measured during recovery after a stress relaxation measurement, was less than 10% different from the modulus G (evaluated at 1 s) measured during the 1% amplitude step relaxation experiment. From these stress-relaxation measurements, we extracted G(t, ␥ 0 ) ϭ (t)/␥ 0 as a function of ␥ 0 and time scale t. We also tested the mechanical response of F-actin networks by subjecting them to oscillatory deformations, ␥(t) ϭ ␥ 0 sint, of increasing amplitude ␥ 0 . The time-dependent stress, (t), was measured and plotted as a function of the applied time-dependent deformation ␥(t), which produced so-called Lissajoux figures (21).
Finite Element Modeling-A simple mechanical model, analyzed via the finite element method, was used to supplement the experimental investigations. The model consists of a square plane of orthogonal, equally spaced, cross-linked arrays of individual filaments and filament bundles ( Fig. 6, a-aЉ), pretensioned in-plane, and then loaded out of plane. The selected material properties and rigidities of this model network (i.e. EI, EA, GJ, see below) are based on published measurements of actin structure and rigidity; the network topology is only approximate and is designed to mimic orthogonal networks such as the actin cortex in non-muscle cells. This model does not incorporate the inherent inhomogeneities of F-actin networks (20). The simulations were conducted using ABAQUS 5.8 -16 (HKS 2000). Geometrically non-linear static analysis (large deformations, no inertia terms) was conducted; Newton's method was used for convergence of the equilibrium equations.
The filament material was modeled as linear elastic with Young's modulus of E ϭ 2.6 GPa and Poisson's ratio of ϭ 0.4 (22). The filaments and filament bundles were modeled as Euler-Bernoulli beam elements (23). The cross-section properties of these elements are the cross-sectional areas (A), moments of inertia (I), and St. Venant torsional constants (J) (23). These properties for a filament, a 7-filament bundle, and a 19-filament bundle, which were used in the calculations, are given in Table I. The out-of-plane stiffness of the filament bundle network is resisted by the bending, axial, and torsional stiffness of the filaments and filament bundles. In the linear regime, the bending stiffness of a filament is proportional to EI/L 3 , the axial stiffness is proportional to EA/L, and the torsional stiffness is proportional to GJ/L, For an individual filament, L is the length between supports; for the filament bundle network the effective L is related to the density of crossings and the rigidity of the crossings, i.e. a bundle is more rigid than a single filament. Perfect connection (no slip) was assumed between any two crossing filaments and/or bundles (24). A small in-plane pretension (1% of the length) was used on the filament and filament bundle network to engage the axial stiffness of the filaments when loaded out-of-plane. The ends of the filaments and bundles were pinned after the application of the pretension. The total out-of-plane loading was constant for all simulations and uniformly distributed on the filaments (i.e. a 7-filament bundle has seven times the load as an individual filament). The response of the center of the model to the loading was monitored.
Individual filaments are 9 nm in diameter, 10 m long, and are quite slender (L/r ϭ 4500, where r is the radius if the filament) (Fig. 6a). Filament bundles consist of seven highly packed, perfectly bonded filaments, as depicted in Fig. 6 aЈ and aЉ. Filament bundles have dramatically higher bending stiffness than a single filament, the mo- ment of inertia I is 55 times greater for the bundle, and filament bundles have lower slenderness, L/r is 1 ⁄3 that of an individual filament. Bundles containing 19 filaments were also investigated. Simulations were conducted with the number of filament bundles systematically increased, but the total number of filaments employed held constant at 10,000 (i.e. mass conservation is enforced). Typical arrays are shown in Fig. 6, b-bЉ.

Strain Hardening of F-actin/␣-Actinin/Fascin Networks-
Using rheological methods, we analyzed the mechanical response of reconstituted F-actin networks in the presence of either ␣-actinin, fascin, or both. We define a network resilience as the shear amplitude at which the network modulus begins to decrease, presumably due to the breakage and/or alignment of the filaments by the applied stresses (25). To measure the resilience and mechanical response, F-actin filament networks were sequentially subjected to step deformations of increasing amplitude ␥ 0 , which each induced a rapid stress within the network. The subsequent relaxation of the stress, (t), was monitored, from which the relaxation modulus, defined as G(t, ␥ 0 ) ϭ (t)/␥ 0 , was computed. At small shear amplitudes ␥ 0 , the curves G(t, ␥ 0 ) versus t superimposed. This describes the linear regime where the stress (t) increases linearly with the input strain amplitude, making the ratio (t)/␥ 0 independent of ␥ 0 (Figs. 1 and 2). At large shear amplitudes ␥ 0 , G(t, ␥ 0 ) versus t curves did not superimpose anymore (Figs. 1 and 2).
In the absence of cross-linking proteins, actin filament networks exhibited a slight enhancement of their stiffness for increasing strain amplitude as measured by an increase in G for increasing ␥ 0 (inset, Fig. 1b). F-actin resilience, defined as the ultimate strain ␥ c at which G evaluated at a fixed time scale t started declining, was ϳ3% and relatively independent of time scale. Despite its ability to promote extensive filament bundling, fascin was unable to enhance the resilience of F-actin networks (Fig. 1). In the linear regime of small deformation amplitudes, the modulus of F-actin/fascin networks was higher than that of F-actin. The stress relaxation modulus, G(t, ␥ 0 ), remained constant for increasingly small deformations until it monotonically decreased with the time scale (Fig. 1, a and c). The modulus G rapidly decreased with strains of amplitude ␥ 0 Ͼ ␥ c (Fig. 1, b and d). This phenomenon, shear-induced softening, occurred at all tested time scales between 0.05-100 s, i.e. shear softening was independent of the rate of deformation. Shear softening is due to either the alignment of the filaments with shear or shear-mediated filament breakage or both. The value of the strain amplitude at which the modulus of F-actin started declining, ␥ c , decreased for increasing time scales (Fig.  3a). This means that F-actin/fascin networks offered more resilience when sheared rapidly (see "Discussion"). For strain amplitudes larger than ␥ c , the rate of shear softening was relatively independent of both the fascin concentration and the rate of strain (Fig. 1, b and d).
In contrast to fascin, ␣-actinin provided F-actin with additional resistance to shear deformations (13). ␣-Actinin induced a substantial increase of elasticity particularly at high rates of shear as measured by an increase of G at small deformation amplitudes. Unlike fascin, ␣-actinin enhanced the modulus for increasing deformation amplitudes (13). This phenomenon, strain hardening, was exacerbated at short time scales, i.e. at time scales shorter than the lifetime of binding of ␣-actinin to F-actin, because sheared filaments did not have the time to slide past one another (13). This in turn prevented filaments to relax the stress. The yield strain of F-actin networks was augmented up to 8-fold by ␣-actinin and decreased with time scale (Fig. 3a), i.e. ␣-actinin enhanced the resilience of F-actin networks, a resilience that increased with the rate of shear.
The trends observed for ␣-actinin, including strain hardening and enhanced resilience, were amplified by combining ␣-actinin and fascin. The stress relaxation modulus increased for increasing strain amplitude over a wide range of time scales (Fig. 2, a and c). Accordingly, the shear modulus displayed a dramatic increase for increasing shear amplitudes, a hallmark of strain hardening, before decreasing at large shear ampli- tudes (Fig. 2, b and d). Moreover, the yield strain ␥ c of F-actin/ fascin/␣-actinin networks was increased between 2.5-and 5-fold (depending on the time scale) compared with F-actin networks cross-linked with fascin and ␣-actinin separately (at least for time scales Ͼ 0.05 s), respectively (Fig. 3a). These results indicate a synergistic enhancement of F-actin resilience by combining fascin and ␣-actinin.
To quantify the extent of strain hardening, we introduced the parameter SH ϭ (G max Ϫ G low ␥ )/G low ␥ , which is the normalized difference between the modulus measured at the maximum, G max , and the (linear) modulus measured at low strain amplitudes, G low ␥ (Fig. 3b). We (obviously) found that SH ϭ 0 for F-actin/fascin networks over a wide range of fascin concentrations since those networks display no strain-hardening. In contrast, SH increased greatly in the presence of ␣-actinin (Fig.  3B); combining ␣-actinin and fascin further enhanced that trend. Moreover, the largest modulus reached for increasing strain amplitudes, G max , was far greater in networks containing fascin and ␣-actinin than in networks containing either fascin or ␣-actinin (inset, Fig. 3b). This shows that cross-linked networks of actin filaments can strongly stiffen under shear and that combining ␣-actinin and fascin transform F-actin networks into extremely stiff and resilient structures.
Synergistic Response of Fascin and ␣-Actinin to Oscillatory Deformations-We further tested the mechanical response of F-actin networks by subjecting them to oscillatory deformations, ␥(t) ϭ ␥ 0 sint, of increasing amplitude ␥ 0 and measuring the resulting oscillatory stress (t) (inset, Fig. 4a). (t) was plotted as a function of ␥(t) to produce so-called Lissajoux figures (13) (e.g. Fig. 4a). In the absence of fascin and ␣-actinin, we confirmed that when a sinusoidal deformation was applied to F-actin networks, the stress first increased linearly with the applied strain, and then more slowly than strain (see control in Fig. 4a and Ref. 13). Therefore, an F-actin network subject to large deformations (here 10%) softened rapidly. A similar strain softening behavior, albeit at much larger stresses, was displayed by F-actin networks in the presence of fascin (Fig.  4a), at low and high levels of deformations (Fig. 4b).
The behavior of F-actin/␣-actinin/fascin networks under shear was fundamentally different from that of F-actin networks and F-actin/fascin networks. F-actin/␣-actinin/fascin networks showed a faster-than-linear increase of the time-dependent stress for increasing strain amplitude (Fig. 5a), which indicates a strain hardening behavior. This behavior persisted at very large strain amplitudes (␥ 0 Ͼ10%), but the axis of the Lissajoux figures rotated clockwise (Fig. 5a), which indicated an overall softening of the network. Strain hardening was enhanced by increasing concentrations of cross-linking proteins (Fig. 5b). When networks were subjected to repeated cycles of shear of large amplitude, the apparent modulus was decreased, an effect particularly dramatic at high concentrations of cross-linking proteins (inset, Fig. 5b).
Finite Element Model of Strain Hardening-We hypothesized that the origin of this synergistic mechanical effect of ␣-actinin and fascin was due to the presence of mixed F-actin structures formed by these proteins. ␣-Actinin forms crosslinked orthogonal networks at low concentrations and loose F-actin bundles at high concentrations (26). Fascin mostly bundles F-actin into highly ordered bundles (27). Confocal microscopy suggests that F-actin networks in the presence of both fascin and ␣-actinin display mixed structures containing bundles and orthogonal networks. 2 To better understand the origin of this synergy, we utilized a simplified model of cytoskeletal structures composed of cross-linked networks of filaments and bundles (see "Materials and Methods"). The amount of "filamentous material" employed was constant and equal to 100 ϫ 100 actin filaments. These filaments were symmetrically and orthogonally arranged on a square area of 100 m 2 , which was supported around the outer edge (see Fig. 6, a and b). The filaments spanned the 100 micrometer distance either as individual filaments or as highly packed perfectly bonded 7-filament bundles (model results were similar for 19-filament bundles). The out-of-plane stiffness of the network of filaments and filament bundles was sensitive to the total number of individual filaments and filament bundles (Fig. 7). The stiffest network consisted neither of all filaments, nor all bundles, but rather a mixture of the two structures (Fig. 7b). Of the networks of filaments and filament bundles, the network of 79 ϫ 79 filaments and 3 ϫ 3 bundles exhibited the greatest initial and tangential stiffness (for 19-filament bundles: 62 ϫ 62 filaments and 2 ϫ 2 bundles had the stiffest response). The mechanical response of the model exhibited two phenomena qualitatively similar to those observed in the experiments: (i) "strain hardening" of the modulus under deformation (Fig. 7a) and (ii) the optimum resilience consisted of a mixed system of cross-linked individual filaments and filament bundles (as opposed to purely filaments or purely bundles) (Fig. 7b). This mechanical behavior grossly corresponds to the cases of F-actin ϩ ␣-actinin, actin ϩ ␣-actinin ϩ fascin, or F-actin ϩ fascin considered above in which combinations that induced some filament bundling and some cross-linking provided an optimum solution, as opposed to individual dopants. DISCUSSION One of the central functions of cross-linked F-actin networks and F-actin bundles is to provide mechanical support to cytoplasm and reinforce cellular protrusions. We investigated potential mechanical synergism between the two cross-linking proteins, ␣-actinin and fascin, which co-localize to stress fibers and the lamelipodia of adherent cells. In several respects, the mechanical behavior of F-actin/␣-actinin/fascin is intermediate between that of F-actin/␣-actinin and F-actin/fascin systems. We previously found that the phase angle of F-actin/␣-actinin/ fascin networks is intermediate between that of the more solidlike F-actin/fascin network and the more liquid-like F-actin/␣actinin network. 3 The dynamics of F-actin cross-linking/ bundling mediated by ␣-actinin and fascin combined is also intermediate between that of fascin and ␣-actinin. 3 In several key aspects, however, the combination of ␣-actinin , and a 19-filament bundle (aЉ). A corner view of representative structures studied in the simulations is shown. These structures contain either filaments (b) or both filaments and filament bundles (bЈ and bЉ). These structures contain the same total number of filaments (100 ϫ 100). These filaments form either unbundled or compact bundles, which are symmetrically and orthogonally arranged on a square area of 100 m 2 (see also "Materials and Methods"). b, bЈ, and bЉ correspond to the case of 100 ϫ 100 filaments, 3 ϫ 3 bundles ϩ 79 ϫ 79 filaments, and 7 ϫ 7 bundles ϩ 51 ϫ 51 filaments, respectively. and fascin generate actin filament ultrastructures that synergistically exploit the functions of these proteins. Fascin and ␣-actinin generate F-actin structures that are more resilient and more elastic than produced by fascin and ␣-actinin separately. The enhancement of elasticity of F-actin provided by the combination of fascin and ␣-actinin is spectacularly increased compared with that separately provided by fascin and ␣-actinin and was about 2 orders of magnitudes higher than that of suspensions of intermediate filaments (keratin or vimentin) or actin microfilaments (25). Combining fascin and ␣-actinin also greatly enhanced strain-induced hardening and resilience of F-actin.
Structural differences among actin cross-linking proteins initially lead researchers to believe that they differed from one another by their propensity to bundle actin filaments. Compact proteins that embody a tandem of two actin binding domains, such as fascin and fimbrin, would bundle filaments via close packing. Instead, antiparallel homodimers of molecules each containing one actin binding domain separated by a long level arm, such as ␣-actinin and filamin, would induce the formation of orthogonal arrays (2). However, careful biochemical and physical characterization has now made clear that even the compact protein fascin can both cross-link and bundle F-actin at low and high concentrations, respectively (29). Vice versa, the archetypal actin cross-linking proteins ␣-actinin and fil-amin, which organize F-actin in orthogonal networks at low concentrations, can also bundle actin filaments at high concentrations (30,31). In support of this dual structural function, immunofluorescence microscopy has shown that fascin localizes not only in actin bundles that support filopodia but also in orthogonal meshworks of the lamellipodia (32,33). Similarly, ␣-actinin locates not only in the lamellipodia but also in stress fibers (28). Therefore, the roles of cross-linking proteins cannot be only distinguished by their propensity to bundle or crosslink actin filaments in vitro. Instead, we suggest that what also distinguishes cross-linking proteins from one another is their mechanical function. We note that in our experiments we mixed fascin and ␣-actinin first and then added G-actin and the polymerizing salt. We leave to future work the study of how the order of addition of the auxiliary proteins to preassembled F-actin may affect the rheological outcome.
Fascin increases F-actin elasticity at levels well below those observed in living cells (14) and does not help F-actin resist shear stresses of large magnitude. ␣-Actinin greatly enhances the elasticity, resilience, and strain hardening property of Factin, but not nearly as effectively as ␣-actinin supplemented with fascin. Our results therefore suggest that cells combine bona fide F-actin bundling proteins such as fascin and bona fide cross-linking proteins such as ␣-actinin to produce stiff and resilient cytoskeletal structures.
While the developed finite element model is not an attempt to duplicate or simulate the experiments, it provides an insight into the mechanism by which actin cross-linking proteins may work synergistically. We have recently observed that the combination of the bundling/cross-linking proteins fascin and ␣-actinin induces the formation of mixed structures. 2 Confocal microscopy reveals that these structures contain both highly ordered bundles and orthogonally oriented filaments as opposed to mostly rigid bundles (fascin alone) (29) or mostly orthogonal networks and disorganized bundles (␣-actinin alone) (20,26). Our simulations test how these mixed structures may be responsible for the synergistic enhancement of the elasticity of F-actin networks. The existence of an optimum mix of filaments and bundles was by no means obvious from the outset in the mechanical model. Consider that the bending moment of inertia (Table I) for the 7-filament bundles is nearly 8 times greater than the moment of inertia for 7 individual filaments! Thus, an optimum existing of all bundles would seem reasonable. However, as individual filaments are expended to generate the bundles, the density of crossings decreases, weakening the system, in structural terms the "unbraced length" increases as filaments are replaced by bundles. This competition leads to the optimum amount of filaments and bundles shown in Fig. 7b. The simulations suggest that mechanical modeling of networks of filaments and filament bundles can provide additional insight and explanation for the synergistic response of actin cross-linking proteins as measured in living cells.