Chase P. Broedersz

Modeling semiflexible polymer networks

This is an overview of theoretical approaches to semiflexible polymers and their networks. Such semiflexible polymers have large bending rigidities that can compete with the entropic tendency of a chain to crumple up into a random coil. Many studies on semiflexible polymers and their assemblies have been motivated by their importance in biology. Indeed, cross-linked networks of semiflexible polymers form a major structural component of tissue and living cells. Reconstituted networks of such biopolymers have emerged as a new class of biological soft matter systems with remarkable material properties, which have spurred many of the theoretical developments discussed here. Starting from the mechanics and dynamics of individual semiflexible polymers, the physics of semiflexible bundles, entangled solutions, and disordered cross-linked networks are reviewed. Finally, recent developments on marginally stable fibrous networks, which exhibit critical behavior similar to other marginal systems such as jammed soft matter, are discussed.

Criticality and isostaticity in fibre networks

In fibre networks, mechanical stability relies on the fibres’ bending resistance—in contrast to rubbers, where entropic stretching is the key. The extent to which the mechanics of fibre networks is controlled by bending is, however, an open question. The study of a general lattice-based model of fibrous networks now reveals two rigidity critical points, one of which controls a rich crossover from stretching-dominated to bending-dominated behaviour.

Divalent Cations Crosslink Vimentin Intermediate Filament Tail Domains to Regulate Network Mechanics

Intermediate filament networks in the cytoplasm and nucleus are critical for the mechanical integrity of metazoan cells. However, the mechanism of crosslinking in these networks and the origins of their mechanical properties are not understood. Here, we study the elastic behavior of in vitro networks of the intermediate filament protein vimentin. Rheological experiments reveal that vimentin networks stiffen with increasing concentrations of Ca(2+) and Mg(2+), showing that divalent cations act as crosslinkers. We quantitatively describe the elastic response of vimentin networks over five decades of applied stress using a theory that treats the divalent cations as crosslinkers: at low stress, the behavior is entropic in origin, and increasing stress pulls out thermal fluctuations from single filaments, giving rise to a nonlinear response; at high stress, enthalpic stretching of individual filaments significantly modifies the nonlinearity. We investigate the elastic properties of networks formed by a series of protein variants with stepwise tail truncations and find that the last 11 amino acids of the C-terminal tail domain mediate crosslinking by divalent ions. We determined the single-filament persistence length, l(P) approximately 0.5 mum, and Youngs modulus, Y approximately 9 MPa; both are consistent with literature values. Our results provide insight into a crosslinking mechanism for vimentin networks and suggest that divalent ions may help regulate the cytoskeletal structure and mechanical properties of cells.

Nonlinear elasticity of composite networks of stiff biopolymers with flexible linkers.

Motivated by recent experiments showing nonlinear elasticity of in vitro networks of the biopolymer actin cross-linked with filamin, we present an effective medium theory of flexibly cross-linked stiff polymer networks. We model such networks by randomly oriented elastic rods connected by flexible connectors to a surrounding elastic continuum, which self-consistently represents the behavior of the rest of the network. This model yields a crossover from a linear elastic regime to a highly nonlinear elastic regime that stiffens in a way quantitatively consistent with experiment.

Actin Filament Length Tunes Elasticity of Flexibly Cross-Linked Actin Networks

Networks of the cytoskeletal biopolymer actin cross-linked by the compliant protein filamin form soft gels that stiffen dramatically under shear stress. We demonstrate that the elasticity of these networks shows a strong dependence on the mean length of the actin polymers, unlike networks with small, rigid cross-links. This behavior is in agreement with a model of rigid filaments connected by multiple flexible linkers.

Filament-Length-Controlled Elasticity in 3D Fiber Networks

We present a model for disordered 3D fiber networks to study their linear and nonlinear elasticity. In contrast to previous 2D models, these 3D networks with binary crosslinks are underconstrained with respect to fiber stretching elasticity, suggesting that bending may dominate their response. We find that such networks exhibit a bending-dominated elastic regime controlled by fiber length, as well as a crossover to a stretch-dominated regime for long fibers. Finally, by extending the model to the nonlinear regime, we show that these networks become intrinsically nonlinear with a vanishing linear response regime in the limit of flexible or long filaments.

Broken detailed balance at mesoscopic scales in active biological systems

Identifying nonequilibrium dynamics Living systems clearly operate out of thermodynamic equilibrium at the molecular scale. How these activities are manifest at the cellular scale, however, has been unclear. Battle et al. use video microscopy together with statistical thermodynamics to unambiguously identify which random fluctuations at the cellular scale are out of equilibrium (see the Perspective by Rupprecht and Prost). Transitions between states obey a detailed balance in equilibrium, whereas imbalanced transitions point to nonequilibrium dynamics. For instance, nonequilibrium dynamics can be identified in the periodic beating of a flagellum and in the nonperiodic fluctuations of primary cilia. Science, this issue p. 604; see also p. 514 Nonequilibrium dynamics can be identified in randomly fluctuating mesoscopic systems. Systems in thermodynamic equilibrium are not only characterized by time-independent macroscopic properties, but also satisfy the principle of detailed balance in the transitions between microscopic configurations. Living systems function out of equilibrium and are characterized by directed fluxes through chemical states, which violate detailed balance at the molecular scale. Here we introduce a method to probe for broken detailed balance and demonstrate how such nonequilibrium dynamics are manifest at the mesosopic scale. The periodic beating of an isolated flagellum from Chlamydomonas reinhardtii exhibits probability flux in the phase space of shapes. With a model, we show how the breaking of detailed balance can also be quantified in stationary, nonequilibrium stochastic systems in the absence of periodic motion. We further demonstrate such broken detailed balance in the nonperiodic fluctuations of primary cilia of epithelial cells. Our analysis provides a general tool to identify nonequilibrium dynamics in cells and tissues.

Condensation and localization of the partitioning protein ParB on the bacterial chromosome

Significance The ParABS system is responsible for chromosome and plasmid segregation in many bacteria. A large, coherent ParB–DNA complex forms the partitioning module at the heart of this segregation machinery. Here we provide a simple theoretical model for interacting proteins on DNA to elucidate the structure of the ParB–DNA complex. We show that that both 3D bridging and 1D spreading interactions between DNA-bound ParB proteins are required to ensure the formation of a coherent protein–DNA complex. This combination of protein–protein interactions implies a surface tension that drives the condensation of ParB proteins on the DNA. The formation of such a condensed protein complex is essential for understanding how a single centromeric parS site can localize ParB on the DNA. The ParABS system mediates chromosome segregation and plasmid partitioning in many bacteria. As part of the partitioning mechanism, ParB proteins form a nucleoprotein complex at parS sites. The biophysical basis underlying ParB–DNA complex formation and localization remains elusive. Specifically, it is unclear whether ParB spreads in 1D along DNA or assembles into a 3D protein–DNA complex. We show that a combination of 1D spreading bonds and a single 3D bridging bond between ParB proteins constitutes a minimal model for a condensed ParB–DNA complex. This model implies a scaling behavior for ParB-mediated silencing of parS-flanking genes, which we confirm to be satisfied by experimental data from P1 plasmids. Furthermore, this model is consistent with experiments on the effects of DNA roadblocks on ParB localization. Finally, we show experimentally that a single parS site is necessary and sufficient for ParB–DNA complex formation in vivo. Together with our model, this suggests that ParB binding to parS triggers a conformational switch in ParB that overcomes a nucleation barrier. Conceptually, the combination of spreading and bridging bonds in our model provides a surface tension ensuring the condensation of the ParB–DNA complex, with analogies to liquid-like compartments such as nucleoli in eukaryotes.

Nonlinear effective-medium theory of disordered spring networks

Disordered soft materials, such as fibrous networks in biological contexts, exhibit a nonlinear elastic response. We study such nonlinear behavior with a minimal model for networks on lattice geometries with simple Hookian elements with disordered spring constant. By developing a mean-field approach to calculate the differential elastic bulk modulus for the macroscopic network response of such networks under large isotropic deformations, we provide insight into the origins of the strain stiffening and softening behavior of these systems. We find that the nonlinear mechanics depends only weakly on the lattice geometry and is governed by the average network connectivity. In particular, the nonlinear response is controlled by the isostatic connectivity, which depends strongly on the applied strain. Our predictions for the strain dependence of the isostatic point as well as the strain-dependent differential bulk modulus agree well with numerical results in both two and three dimensions. In addition, by using a mapping between the disordered network and a regular network with random forces, we calculate the nonaffine fluctuations of the deformation field and compare them to the numerical results. Finally, we discuss the limitations and implications of the developed theory.

Effective-medium approach for stiff polymer networks with flexible cross-links

Recent experiments have demonstrated that the nonlinear elasticity of in vitro networks of the biopolymer actin is dramatically altered in the presence of a flexible cross-linker such as the abundant cytoskeletal protein filamin. The basic principles of such networks remain poorly understood. Here we describe an effective-medium theory of flexibly cross-linked stiff polymer networks. We argue that the response of the cross-links can be fully attributed to entropic stiffening, while softening due to domain unfolding can be ignored. The network is modeled as a collection of randomly oriented rods connected by flexible cross-links to an elastic continuum. This effective medium is treated in a linear elastic limit as well as in a more general framework, in which the medium self-consistently represents the nonlinear network behavior. This model predicts that the nonlinear elastic response sets in at strains proportional to cross-linker length and inversely proportional to filament length. Furthermore, we find that the differential modulus scales linearly with the stress in the stiffening regime. These results are in excellent agreement with bulk rheology data.