Mahsa Vahabi

Uncoupling shear and uniaxial elastic moduli of semiflexible biopolymer networks: compression-softening and stretch-stiffening

Gels formed by semiflexible filaments such as most biopolymers exhibit non-linear behavior in their response to shear deformation, e.g., with a pronounced strain stiffening and negative normal stress. These negative normal stresses suggest that networks would collapse axially when subject to shear stress. This coupling of axial and shear deformations can have particularly important consequences for extracellular matrices and collagenous tissues. Although measurements of uniaxial moduli have been made on biopolymer gels, these have not directly been related to the shear response. Here, we report measurements and simulations of axial and shear stresses exerted by a range of hydrogels subjected to simultaneous uniaxial and shear strains. These studies show that, in contrast to volume-conserving linearly elastic hydrogels, the Young’s moduli of networks formed by the biopolymers are not proportional to their shear moduli and both shear and uniaxial moduli are strongly affected by even modest degrees of uniaxial strain.

Porosity governs normal stresses in polymer gels

When sheared, most elastic solids including metals, rubbers, and polymer gels dilate perpendicularly to the shear plane. This behavior, known as the Poynting effect, is characterized by a positive normal stress. Surprisingly, fibrous biopolymer gels exhibit a negative normal stress under shear. Here we show that this anomalous behavior originates from the open-network structure of biopolymer gels. Using fibrin networks with a controllable pore size as a model system, we show that the normal-stress response to an applied shear is positive at short times, but decreases to negative values with a characteristic time scale set by pore size. Using a two-fluid model, we develop a quantitative theory that unifies the opposite behaviors encountered in synthetic and biopolymer gels.

Hydrodynamic and subdiffusive motion of tracers in a viscoelastic medium

We investigate the diffusive motion of micron-sized spherical tracers in a viscoelastic actin filament network over the time span of 8 orders of magnitude using optical-tweezers single-particle tracking. The hydrodynamic interactions of a tracer with the surrounding fluid are shown to dominate at microsecond time scales, while subdiffusive scaling due to viscoelastic properties of the medium emerges at millisecond time scales. The transition between these two regimes is analyzed in the frame of a minimal phenomenological model which combines the Basset force and the generalized Stokes force. The resulting Langevin equation accounts for various dynamical features of the thermal motion of endogenous or exogenous tracers in viscoelastic media such as inertial and hydrodynamic effects at short times, subdiffusive scaling at intermediate times, and eventual optical trapping at long times. Simple analytical formulas for the mean-square displacement and velocity autocorrelation function are derived.

Driven diffusive systems with mutually interactive Langmuir kinetics

We investigate the simple one-dimensional driven model, the totally asymmetric exclusion process, coupled to mutually interactive Langmuir kinetics. This model is motivated by recent studies on clustering of motor proteins on microtubules. In the proposed model, the attachment and detachment rates of a particle are modified depending upon the occupancy of neighboring sites. We first obtain continuum mean-field equations and in certain limiting cases obtain analytic solutions. We show how mutual interactions increase (decrease) the effects of boundaries on the phase behavior of the model. We perform Monte Carlo simulations and demonstrate that our analytical approximations are in good agreement with the numerics over a wide range of model parameters. We present phase diagrams over a selective range of parameters.

Area coverage of radial Levy flights with periodic boundary conditions

We consider the area coverage of radial Lévy flights in a finite square area with periodic boundary conditions. From simulations we show how the fractal path dimension d(f) and thus the degree of area coverage depends on the number of steps of the trajectory, the size of the area, and the resolution of the applied box counting algorithm. For sufficiently long trajectories and not too high resolution, the fractal dimension returned by the box counting method equals two, and in that sense the Lévy flight fully covers the area. Otherwise, the determined fractal dimension equals the stable index of the distribution of jump lengths of the Lévy flight. We provide mathematical expressions for the turnover between these two scaling regimes. As complementary methods to analyze confined Lévy flights we investigate fractional order moments of the position for which we also provide scaling arguments. Finally, we study the time evolution of the probability density function and the first passage time density of Lévy flights in a square area. Our findings are of interest for a general understanding of Lévy flights as well as for the analysis of recorded trajectories of animals searching for food or for human motion patterns.

Nonlinear Mechanics of Athermal Branched Biopolymer Networks

Naturally occurring biopolymers such as collagen and actin form branched fibrous networks. The average connectivity in branched networks is generally below the isostatic threshold at which central force interactions marginally stabilize the network. In the submarginal regime, for connectivity below this threshold, such networks are unstable toward small deformations unless stabilized by additional interactions such as bending. Here we perform a numerical study on the elastic behavior of such networks. We show that the nonlinear mechanics of branched networks is qualitatively similar to that of filamentous networks with freely hinged cross-links. In agreement with a recent theoretical study,1 we find that branched networks also exhibit nonlinear mechanics consistent with athermal critical phenomena controlled by strain. We obtain the critical exponents capturing the nonlinear elastic behavior near the critical point by performing scaling analysis of the stiffening curves. We find that the exponents evolve with the connectivity in the network. We show that the nonlinear mechanics of disordered networks, independent of the detailed microstructure, can be characterized by a strain-driven second-order phase transition, and that the primary quantitative differences among different architectures are in the critical exponents describing the transition.

On-site residence time in a driven diffusive system: Violation and recovery of a mean-field description

We investigate simple one-dimensional driven diffusive systems with open boundaries. We are interested in the average on-site residence time defined as the time a particle spends on a given site before moving on to the next site. Using mean-field theory, we obtain an analytical expression for the on-site residence times. By comparing the analytic predictions with numerics, we demonstrate that the mean-field significantly underestimates the residence time due to the neglect of time correlations in the local density of particles. The temporal correlations are particularly long-lived near the average shock position, where the density changes abruptly from low to high. By using domain wall theory, we obtain highly accurate estimates of the residence time for different boundary conditions. We apply our analytical approach to residence times in a totally asymmetric exclusion process (TASEP), TASEP coupled to Langmuir kinetics (TASEP+LK), and TASEP coupled to mutually interactive LK (TASEP+MILK). The high accuracy of our predictions is verified by comparing these with detailed Monte Carlo simulations.