Abstract
Motivated by the structure of networks of cross-linked cytoskeletal biopolymers, we study orientationally ordered phases in two-dimensional networks of randomly cross-linked semiflexible polymers. We consider permanent cross-links which prescribe a finite angle and treat them as quenched disorder in a semimicroscopic replica field theory. Starting from a fluid of un-cross-linked polymers and small polymer clusters (sol) and increasing the cross-link density, a continuous gelation transition occurs. In the resulting gel, the semiflexible chains either display long-range orientational order or are frozen in random directions depending on the value of the crossing angle, the cross-link concentration, and the stiffness of the polymers. A crossing angle θ~2π/M leads to long-range M-fold orientational order, for example, "hexatic" or "tetratic" for θ=60° or 90°, respectively. The transition to the orientationally ordered state is discontinuous and the critical cross-link density, which is higher than that of the gelation transition, depends on the bending stiffness of the polymers and the cross-link angle: The higher the stiffness and the lower the M, the lower is the critical number of cross-links. In between the sol and the long-range ordered state, we always expect a gel which is a statistically isotropic amorphous solid with random positional and random orientational localization of the participating polymers.
Highlights
The cytoskeleton is a network of linked protein fibers which plays an important role in several functions of eukaryotic cells, such as maintenance of morphology, mechanics, and intracellular transport [1]
The function of the actin cytoskeleton is modulated by a large number of actin-binding proteins (ABPs) [2,3]
The organization of actin filaments into networks is regulated by ABPs which can be classified into two broad categories: Cross-linkers promote binding of the filaments at finite crossing angles, whereas bundlers promote formation of bundles consisting of parallel or antiparallel filaments
Summary
The cytoskeleton is a network of linked protein fibers which plays an important role in several functions of eukaryotic cells, such as maintenance of morphology, mechanics, and intracellular transport [1]. In vitro structural studies by Wong et al [17] of F-actin in the presence of counterions have shown the formation of sheets (“rafts”) with tetratic order without any cross-linking proteins It appears that electrostatic interactions are the main mechanism behind the effective “π/2” cross-linking of the actin filaments in this experiment. We are led to study the equilibration of the thermal degrees of freedom {ri(s)} in the presence of quenched disorder represented by a given cross-link configuration C = {ie,je; se,se}Me=1 which is characterized by the set of pairs of polymer segments that are involved in a cross-link. Given the Hamiltonian, the constraints due to cross-linking and the distribution of cross-link realizations, the specification of the model is complete and we proceed to calculate the disorder-averaged free energy [F ]
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