Nonlinear elasticity of stiff biopolymers connected by flexible linkers

Abstract

Networks of the biopolymer actin, cross-linked by the compliant protein filamin, form soft gels. They can, however, withstand large shear stresses due to their pronounced nonlinear elastic behavior. The nonlinear elasticity can be controlled by varying the number of cross-links per actin filament. We propose and test a model of rigid filaments decorated by multiple flexible linkers that is in quantitative agreement with experiment. This allows us to estimate loads on individual cross-links, which we find to be less than 10 pN. DOI: 10.1103/PhysRevE.79.041928 PACS number͑s͒: 87.16.Ka, 83.60.Df, 87.15.La Cells interact mechanically with their environment largely through their cytoskeleton, a mechanical framework consisting of filamentous protein polymers and associated proteins that regulate cytoskeletal microstructure and connectivity ͓1,2͔. The in vivo cytoskeleton is remarkably complex, making in vitro studies of purified cytoskeletal networks useful for elucidating basic physical principles governing cytoskeletal mechanics ͓3,4͔ and mechanosensing ͓5͔. Networks of filamentous actin ͑F-actin͒, a major component of the cytoskeleton, exhibit unusual material properties ͓6-14͔. Among the most striking properties is a strongly nonlinear response to shear ͓6-10͔; this depends sensitively on the cross-linking protein. For noncompliant cross-links, the network response arises from the compliance of the actin filaments themselves ͓7-10,15͔. By contrast, for compliant cross-links, such as those commonly found in cells, actin networks exhibit dramatically different elasticity ͓16-19͔. One such cross-link is human filamin, a large protein with a contour length of 150 nm ͓20,21͔, ͓Fig. 1͑b͔͒. For small forces, filamin behaves like a flexible wormlike chain; whereas for forces larger than 50-100 pN, unfolding of individual Ig-like domains occurs ͓22͔. Human filamin forms a weak gel with F-actin, with linear shear moduli of ϳ1 Pa. However, these gels exhibit a large mechanical stiffening under strain ͓16,17͔, and can withstand very large stresses, as high as 100 Pa, at shear strains of order of one. A theoretical understanding of the molecular origins of this unusual behavior will help elucidate the basic design principles of cytoskeletal mechanics.Here, we show that the nonlinear elastic behavior of filamin-F-actin gels is controlled by the number of crosslinks per actin filament, and we account for this unusual behavior by modeling the networks as rigid rods connected with multiple flexible linkers ͓23͔. The model quantitatively explains the dramatic nonlinear stiffening of filamin-F-actin networks, providing fundamental insight into its origins. It also provides an estimate of the maximum load experienced by individual cross-links, which is less than 10 pN, too small to result in significant unfolding of filamin Ig-like domains.We reconstitute networks of F-actin cross-linked with filamin A using purified monomeric actin, filamin A ͓24͔, and gelsolin ͓25͔. We control network microstructure by varying the actin concent...

Strain-driven criticality underlies nonlinear mechanics of fibrous networks

“…We show that the equation accurately describes the stiffness of collagen networks with a single fit parameter. The excellent agreement of model predictions with the experiments provides strong evidence for criticality as the underlying mechanism of the well-known nonlinear mechanics of athermal fibrous networks such as collagen [23,33,40,41] and bundled actin [56][57][58].…”

Cyclic hardening in bundled actin networks

“…Reconstituted actin networks are ideal candidates for investigating the effect of cyclic loads at the microscopic scale using confocal fluorescence microscopy due to their large mesh sizes, and their propensity to form large bundles 11,12 . Interestingly, bundled actin networks also exhibit strain stiffening 11,13 one hallmark of materials which show the Mullins effect. From a biological perspective, insight into the response of in vitro biopolymer networks to cyclic loads may be important, as a passive reorganization induced by external mechanical stimuli should prove to be a powerful tool to facilitate the high adaptability of living materials.…”

The filamins

“…They behave as weak elastic solids under low shear stress due to the flexible nature of actin-FLNa crosslinks, yet can support large shear stresses and have pronounced nonlinear strain-stiffening behaviors. [73][74][75] These mechanical properties are attributed to FLNa's unique structure and how it interacts with F-actin to form orthogonal branches. High avidity binding to F-actin due to dimerization and multiple binding to F-actin through FLNa ABD and rod 1 confers strain-stiffening on actin networks.…”