Alignment and nonlinear elasticity in biopolymer gels

We present a Landau-type theory for the nonlinear elasticity of biopolymer gels with a part of the order parameter describing induced nematic order of fibers in the gel. We attribute the nonlinear elastic behavior of these materials to fiber alignment induced by strain. We suggest an application to contact guidance of cell motility in tissue. We compare our theory to simulation of a disordered lattice model for biopolymers. We treat homogeneous deformations such as simple shear, hydrostatic expansion, and simple extension, and obtain good agreement between theory and simulation. We also consider a localized perturbation which is a simple model for a contracting cell in a medium.

Figure 1

Small area of a network before (left) and after (right) simple shear. Red dashed lines: the part of the fiber segments that align with the direction of strain after easy bending modes are exhausted. Solid lines: other fiber segments before (gray) and after (blue) deformation.

Figure 2

(a) Simulation (data points) and model fit (dashed curves) based on Eqs. (3) and (5) for shear stress as a function of strain, for lattices with $p=0.6$ (lower blue), and $p=0.8$ (higher red). (b) Alignment order $q$ as function of shear strain for the same deformation and color coding as in (a). (c) Stress as a function of strain for hydrostatic deformations for lattices with $p=0.6$ (lower blue), and $p=0.8$ (higher red). (d) Stress strain curves for shear deformation at more values of $p$ (from top to bottom, $p=1.00,0.80,0.70,0.65,0.60,0.55$). Inset: fitted parameter $C\left(p\right)$, which controls the nonlinearity in $V$. For (a)–(d) we used $\kappa /\left(k{a}^2\right)={10}^{-3}$.

Figure 3

Simulation results (data points) and theory prediction (dashed lines) for simple extension at $p=0.6$ (lower blue), and $p=0.8$ (higher red) and $\kappa /\left(k{a}^2\right)={10}^{-3}$ using parameters fitted from shear and the hydrostatic expansion.

Figure 4

Simulation results (points) and theory predictions (solid lines) for a model for a cell in ECM using the parameters from the homogeneous deformation case.

Figure 5

Comparison of $q_{\mathrm{affine}}$ (top red), $q$ (middle blue), and $q-q_{\mathrm{affine}}$ (bottom green) for a simple extension at $p=0.6$ and $\kappa /\left(k{a}^2\right)={10}^{-3}$.