Nonlinearities of biopolymer gels increase the range of force transmission

We present a model of biopolymer gels that includes two types of elastic nonlinearities, stiffening under extension and softening (due to buckling) under compression, to predict the elastic anisotropy induced by both external as well as internal (e.g., due to cell contractility) stresses in biopolymer gels. We show how the stretch-induced anisotropy and the strain-stiffening nonlinearity increase both the amplitude and power-law range of transmission of internal, contractile, cellular forces, and relate this to recent experiments.

Figure 1

Stretch-induced anisotropy in Young's moduli for semiflexible polymer gels under external, uniaxial tensile stress ${\sigma }_1$ in the limit of ${\epsilon }_m^{\prime }\ll {\epsilon }_m$. Here ${\epsilon }_m=0.1$, ${\epsilon }_m^{\prime }=0$, ${\rho }_0=0.1$; ${\nu }_0=0.49$ and $E_0=2{\mu }_0(1+{\nu }_0)$. The Young's modulus $E_1\equiv \partial {\sigma }_1/\partial {\epsilon }_1$ is constant ($=E_0$) for small stress and increases asymptotically to $\sim {\sigma }_1^{3/2}$ with increasing stress. $E_2$ is the transverse Young's modulus of the gel under given longitudinal prestress ${\sigma }_1$ is constant ($\approx {\rho }_0{E}_0\ll E_1$) in the entire range of stress calculated. Inset: Uniaxial stress vs strain diverges near ${\epsilon }_m$.

Figure 2

(a) Schematic illustration of a spherical cell exerting contractile force on the extracellular biopolymer gel in the quasilinear limit of ${\epsilon }_m\to \infty$ and ${\epsilon }_m^{\prime }\ll 1$. Two regimes separated at $\tilde{R}={\tilde{R}}_T$ are identified. (b) Numerical solutions of the equilibrium equation (5): normalized displacement vs normalized radius for ${\rho }_0=0.1$. Case 1: ${\nu }_0=0$ (blue dashed line). The decay of displacement can be well approximated by composite materials: a near-field, linear anisotropic material for $1<\tilde{R}\ll {\tilde{R}}_T$ with a far-field, linear isotropic material for $\tilde{R}\gg {\tilde{R}}_T$. The near-field displacement decays as a power law with exponent $m\approx 1.2$ and the far-field displacement as ${\tilde{R}}^{-2}$. The red dash-dotted line is the analytical solution (7) and (9) of linear composite materials. Case 2 – weak compressibility: ${\nu }_0=0.49$ (black dotted line). Its solution is similar to that of linear isotropic materials as shown for comparison (green solid line). Inset: Normalized effective far-field displacement ${\tilde{u}}_{\mathrm{eff}}$. It shows that the stretch-induced anisotropy in the gel can increase the range of force transmission.

Figure 3

(a) Schematic illustration of a spherical cell exerting contractile force on the extracellular biopolymer gel in the nonlinear limit of ${\epsilon }_m^{\prime }\sim {\epsilon }_m\ll 1$ or ${\epsilon }_m^{\prime }\gg {\epsilon }_m$. Two regimes separated by $\tilde{R}={\tilde{R}}_{*}$ are identified. (b) Numerical solutions of the equilibrium equation (4) in the highly nonlinear limit of $A=-u_0/R_0{\epsilon }_m\gg 1$: normalized displacement vs normalized radius for $A=50$ and ${\rho }_0=0.1$. Case 1: ${\nu }_0=0$ (red dotted line and blue dashed line). The near-field displacement decays linearly with $\tilde{R}$ (to first order of $A^{-1}$, gray dotted line). The far-field displacement decays as ${\tilde{R}}^{-2}$, i.e., as in linear isotropic materials (as shown in the green solid line). Note that the two cases of ${\epsilon }_m^{\prime }={\epsilon }_m$ and ${\epsilon }_m^{\prime }\gg {\epsilon }_m$ lie on the same line. Case 2 – weak compressibility: ${\nu }_0=0.49$ (black solid line). The solution is qualitatively similar to Case 1 of ${\nu }_0=0$.

Figure 4

(a) Schematic illustration of a spherical cell exerting contractile force on the extracellular biopolymer gel in the nonlinear limit of ${\epsilon }_m^{\prime }\ll {\epsilon }_m\ll 1$. Three regimes separated by $\tilde{R}={\tilde{R}}_{*}$ and $\tilde{R}={\tilde{R}}_T$ are identified. (b) Numerical solutions of the equilibrium equation (4) in the highly nonlinear limit of $A=-u_0/R_0{\epsilon }_m\gg 1$: normalized displacement vs normalized radius for $A=50$ and ${\rho }_0=0.1$. Case 1: ${\nu }_0=0$ (blue dashed line). The near-field displacement decays linearly with $\tilde{R}$ (to first order of $A^{-1}$, gray dotted line). In the intermediate quasilinear regime, the displacement decays as in a linear, anisotropic material as ${\tilde{R}}^{-m}$ with $m\approx 1.2$. The far-field displacement decays as ${\tilde{R}}^{-2}$, i.e., as in linear isotropic materials (as shown in green solid line). Case 2 – weak compressibility: ${\nu }_0=0.49$ (black solid line). There are only two regimes, as shown in Fig. 3. That is, the effects of stretch-induced anisotropy are important only for compressible gels.

Figure 5

Normalized effective far-field displacement ${\tilde{u}}_{\mathrm{eff}}$ vs nonlinearity parameter $A\equiv -u_0/R_0{\epsilon }_m$ in the nonlinear limit of ${\epsilon }_m^{\prime }\sim {\epsilon }_m\ll 1$. Here lines are plotted only for guidance.