Statistical Mechanics of Semiflexible Bundles of Wormlike Polymer Chains

Abstract

We demonstrate that a semiflexible bundle of wormlike chains exhibits a state-dependent bending stiffness that alters fundamentally its scaling behavior with respect to the standard wormlike chain. We explore the equilibrium conformational and mechanical behavior of wormlike bundles in isolation, in cross-linked networks, and in solution. DOI: 10.1103/PhysRevLett.99.048101 PACS numbers: 87.16.Ka, 82.35.Lr, 83.10.ÿy, 87.15.La In recent decades, the wormlike chain (WLC) has emerged as the standard model for the description of semiflexible polymers [1]. The defining property of a WLC is a mechanical bending stiffness f that is an intrinsic material constant of the polymer. Within this framework, numerous correlation and response functions have been calculated, providing a comprehensive picture of the equilibrium and dynamical properties of WLCs [2 -4]. A number of experimental studies have demonstrated the applicability of the WLC model to DNA [5] and F-actin [6], among other biological and synthetic polymers. Significant progress has also been made towards the description of the collective properties of WLCs, for example, in the form of entangled solutions. One of the hallmarks of this development is the scaling of the plateau shear modulus with concentration G c 7=5 [7][8][9], which is well established experimentally [10,11].Another important emerging class of semiflexible polymers consists of bundles of WLCs [12,13]. Semiflexible polymer bundles consisting of F-actin or microtubules are ubiquitous in biology [14] and have unique mechanical properties that may well be exploited in the design of nanomaterials [13]. As shown by Bathe et al. [15,16], wormlike bundles (WLBs) have a state-dependent bending stiffness B that derives from a generic interplay between the high stiffness of individual filaments and their rather soft relative sliding motion. In this Letter, we demonstrate that this state dependence gives rise to fundamentally new behavior that cannot be reproduced trivially using existing relations for WLCs. We explore the consequences of a state-dependent bending stiffness on the statistical mechanics of isolated WLBs, as well as on the scaling behavior of their entangled solutions and cross-linked networks.We consider the bending of ordered bundles with an isotropic cross section. A bundle consists of N filaments of length L and bending stiffness f . Filaments are irreversibly cross-linked to their nearest neighbors by discrete cross-links with mean axial spacing . Cross-links are modeled to be compliant in shear along the bundle axis with finite shear stiffness k and to be inextensible transverse to the bundle axis, thus fixing the interfilament distance b [17]. Bundle deformations are characterized by the transverse deflection r ? s of the bundle neutral surface at axial position s along the backbone and by the stretching deformation u i s of filament i. The torsional stiffness of the bundle is assumed to be of the same order as the bending stiffness. Thus, as long as transverse deflections remain sm...

The role of structure in the nonlinear mechanics of cross-linked semiflexible polymer networks

“…For example, the interplay between the monomer size and the cross-linker properties may determine the fiber dispersity and propensity to bundling, which in turn will affect the strain transmission mechanism and the overall mechanical behavior of the network. 23,46 By choosing the effective potential energies described in the Methods section and variable values listed in Table I, we have in the present study focused our attention to the mechanics of a subset of biopolymer systems. However, we expect our approach to be robust and applicable to understanding the structural origin of the mechanics of a wide range of biophysical networks.…”

Mechanics of Bundled Semiflexible Polymer Networks

“…The bundle stretching is related to the transverse displacement na by simple geometric considerations as l 2 c 2 na l c 2 . The floppy mode description thus applies as long as this stretching is small compared to the available thermal excess length l c b=Nl p [20]. This defines a critical strain c l c b=N 1=2 R ÿy x b ÿ1=6 R ÿ1:0 ÿ1=12 for the onset of nonlinear effects.…”

“…On the other hand, if one were to apply an affine stretching model, a different picture emerges, where c f has to be tuned to obtain a reasonable data fit. In such a model one would assume the modulus to be given by G aff k k aff , where k k 3=2 N f 2 =l 4 c is the stretching stiffness of the bundle [20]. The deformations are assumed to be affine, implying aff l c .…”

Anisotropic Random Networks of Semiflexible Polymers

“…A first step towards understanding the behavior of such complex aggregates is to study in vitro solutions of many filaments whose interactions are controled by crosslinkers [2,3,4]. There is currently significant activity on the mechanics and elasticity of crosslinked bundles [5,6]. Previous attempts to theoretically describe the formation of ordered structures from disordered solutions use a generalized Onsager approach [7] or a Flory-Huggins theory [8].…”