A two-phase thin-film model for cell-induced gel contraction incorporating osmotic effects

Abstract

We present a mathematical model of an experiment in which cells are cultured within a gel, which in turn floats freely within a liquid nutrient medium. Traction forces exerted by the cells on the gel cause it to contract over time, giving a measure of the strength of these forces. Building upon our previous work (Reoch et al. in J Math Biol 84(5):31, 2022), we exploit the fact that the gels used frequently have a thin geometry to obtain a reduced model for the behaviour of a thin, two-dimensional cell-seeded gel. We find that steady-state solutions of the reduced model require the cell density and volume fraction of polymer in the gel to be spatially uniform, while the gel height may vary spatially. If we further assume that all three of these variables are initially spatially uniform, this continues for all time and the thin film model can be further reduced to solving a single, non-linear ODE for gel height as a function of time. The thin film model is further investigated for both spatially-uniform and varying initial conditions, using a combination of analytical techniques and numerical simulations. We show that a number of qualitatively different behaviours are possible, depending on the composition of the gel (i.e., the chemical potentials) and the strength of the cell traction forces. However, unlike in the earlier one-dimensional model, we do not observe cases where the gel oscillates between swelling and contraction. For the case of initially uniform cell and gel density, our model predicts that the relative change in the gels' height and length are equal, which justifies an assumption previously used in the work of Stevenson et al. (Biophys J 99(1):19-28, 2010). Conversely, however, even for non-uniform initial conditions, we do not observe cases where the length of the gel changes whilst its height remains constant, which have been reported in another model of osmotic swelling by Trinschek et al. (AIMS Mater Sci 3(3):1138-1159, 2016; Phys Rev Lett 119:078003, 2017).

Keywords: Cell-extracellular matrix interactions; Fluid mechanics; Gel; Mathematical model; Multiphase model; Osmosis; Thin films; Tissue engineering.

Fig. 1

Thin film domain $\mathrm{\Omega }={\mathrm{\Omega }}_g\cup {\mathrm{\Omega }}_s$. ${\mathrm{\Omega }}_g$ is the gel region with ${\theta }_p>0$, ${\theta }_s>0$ and cell density $n\ge 0$. ${\mathrm{\Omega }}_s$ is the region of pure solvent surrounding the gel wherein ${\theta }_p=n=0$ and ${\theta }_s=1$. The gel is symmetric about the x-axis and y-axis, and the ratio of gel height to length is small

Fig. 2

Time evolution of a cell-free thin gel with uniform initial conditions; ${\theta }_p$ is shown in blue, and h in maroon, with $h=L$. Common parameters ${\theta }_I=0.6$, $n_I=0$, $h_I=1$, $N=100$, $\mathcal{R}=1$. With $\chi =0.75$ (Fig. 2a) the gel swells uniformly across its domain to the equilibrium (${\theta }^{\ast },h^{\ast },L^{\ast })=(0.45,1.16,1.16)$. With $\chi =1.5$ (Fig. 2b) the gel contracts to the equilibrium (${\theta }^{\ast },h^{\ast },L^{\ast })=(0.86,0.84,0.84)$

Fig. 3

Time evolution of a thin cell-gel system with uniform initial conditions. In Fig. 3a and b, ${\theta }_p$ is shown by the solid blue curve, $h=L$ is the solid maroon line, and n is the dotted purple line, for common parameter values ${\theta }_I=0.6$, $n_I=1$, $h_I=1$, $\chi =0.75$, $N=100$, $\mathcal{R}=1$, $\lambda =1$. With ${\tau }_0=1$ (Fig. 3a) the gel contracts to the equilibrium (${\theta }^{\ast },n^{\ast },h^{\ast },L^{\ast })=(0.86,1.44,0.83,0.83)$ due to the presence of cells. With ${\tau }_0=0.1$ (Fig. 3b) the gel swells to a steady state due to osmoticpressure counteracting weak cell traction, (${\theta }^{\ast },n^{\ast },h^{\ast },L^{\ast })=(0.54,0.91,1.05,1.05)$. Figure 3c compares the effect of the size of the resistance parameter on the evolution of the gel height; $\mathcal{R}=1$ is shown by the light blue dotted line, $\mathcal{R}=5$ by the maroon dashed line, parameter values otherwise as for Fig. 3a

Fig. 4

Time evolution of a cell-free gel with non-uniform initial polymer fraction ${\theta }_I=0.6+0.02cos(\pi X)$, and parameters $n_I=0$, $h_I=1$, $\chi =0.75$, $\xi =1$, $\mathcal{R}=1$, expanding to equilibrium (${\theta }^{\ast },{\overline{h}}^{\ast },L^{\ast })=(0.45,1.16,1.16)$. Figure 4a shows ${\theta }_p$ versus time T at $X=0$ (solid blue curve) and $X=1$ (dashed red curve); the curves are identical to graphical accuracy. Figure 4b shows h versus T at $X=0$ (dashed light blue curve) and $X=1$ (dotted maroon curve), along with L(T) as the solid gold curve. Figure 4c and d show curves ${\theta }_p$ versus X and h versus X, respectively, at times $T=0,0.1,0.2,0.5,1,2,8,120$, with time increasing in the direction of the black arrow. The polymer fraction evens out to a uniform equilibrium as the gel swells (Fig. 4a and c). Spatial variations develop in the height in response to the initial non-uniform polymer distribution and persist to equilibrium (Fig. 4b and d)

Fig. 5

Time evolution of a cell-gel system with non-uniform initial polymer fraction ${\theta }_I=0.6+0.02cos(\pi X)$, and parameters $n_I=1$, $h_I=1$, $\chi =0.75$, $\xi =1$, $\mathcal{R}=1$, ${\tau }_0=1$, $D=1$, contracting to equilibrium (${\theta }^{\ast },n^{\ast },{\overline{h}}^{\ast },L^{\ast })=(0.86,1.44,0.83,0.83)$. Figure 5a shows that spatial variations in the polymer profile decay quickly over time, whilst in Fig. 5b small spatial variations briefly emerge in the cell density, but dissipate before the gel equilibrates. By contrast, in Fig. 5c spatial variations which emerge in the gel height increase in magnitude throughout to equilibrium. Profiles are plotted at $T=0,0.1,0.2,0.5,0.8,1.2,1.6,3$, with time increasing in the direction of the black arrow

Fig. 6

Time evolution of a cell-gel system with non-uniform initial cell density $n_I=1+0.02cos(\pi X)$, and parameters ${\theta }_I=0.6$, $h_I=1$, $\chi =0.75$, $\xi =1$, $\mathcal{R}=1$, ${\tau }_0=1$, $D=1$, contracting to equilibrium (${\theta }^{\ast },n^{\ast },{\overline{h}}^{\ast },L^{\ast })=(0.86,1.44,0.83,0.83)$. Spatial variations briefly emerge in the polymer fraction (Fig. 6a) and the gel height (Fig. 6c), which dissipate before the gel equilibrates, whilst spatial variations in the cell density decay quickly over time (Fig. 6b). Profiles are plotted at $T=0,0.06,0.1,0.2,0.5,0.8,1.2,3$, with time increasing in the direction of the black arrow

Fig. 7

The effect of varying the drag parameter, $\xi$. For large drag ($\xi =4$, Fig. 7a) spatial variations in polymer density slowly recede as the gel contracts over time. By contrast, in the case of low drag ($\xi =0.2$, Fig. 7b), the spatial variations quickly smooth out as the gel contracts. Profiles are plotted at $T=0,0.02,0.05,0.1,0.2,0.4,0.6,0.8$, with time increasing in the direction of the black arrow. Initial conditions and parameter values otherwise as given in Fig. 5

Fig. 8

The effect of varying the resistance parameter, $\mathcal{R}$. When resistance is high ($\mathcal{R}=4$, Fig. 8a) the polymer fraction decreases very slowly due to the impermeability of the boundary, whilst when the boundary is more permeable ($\mathcal{R}=0.4$, Fig. 8b), contraction is much faster. Profiles are plotted at $T=0,0.02,0.05,0.1,0.2,0.4,0.6,0.8$, with time increasing in the direction of the black arrow. Initial conditions and parameter values otherwise as given in Fig. 5

Fig. 9

Time evolution of a cell-gel system with zero cell diffusion, a non-uniform initial cell density $n_I=1+0.02cos(\pi X)$, and parameters ${\theta }_I=0.6$, $h_I=1$, $\chi =0.4$, $\xi =1$, $\mathcal{R}=1$, ${\tau }_0=1$, $D=0$, contracting to equilibrium (${\overline{\theta }}^{\ast },{\overline{n}}^{\ast },{\overline{h}}^{\ast },L^{\ast })=(0.78,1.29,0.88,0.88)$. Spatial variations in the cell profile grow and persist at equilibrium (Fig. 9a), whilst similar spatial variations in the polymer density also emerge and persist (Fig. 9b); profiles are plotted at $T=0,0.05,0.1,0.2,0.4,0.8,1.6,8$, with time increasing in the direction of the black arrow. Figure 9c shows curves of the velocity $v_p$, versus time T, at $X=0$ (blue), $X=0.25$ (red), $X=0.5$ (yellow), $X=0.75$ (purple), $X=1$ (green); $v_p$ goes to zero across the spatial domain as the gel reaches its spatially varying equilibrium