Connecting gene expression to cellular movement: A transport model for cell migration

The adhesion properties and the mobility of biological cells play key roles in the propagation of cancer. These properties are expected to depend on intracellular processes and on the concentrations of chemicals inside the cell. While most existing reaction-diffusion models for cell migration consider that cell mobility and proliferation rate are constant or depend on an external diffusing species, they do not include the gene expression dynamics taking place in moving cells that affect cellular transport. In this work, we propose a multiscale model where mobility and proliferation depend explicitly on the cell's internal state. We focus more specifically on the case of cellular mobility in epithelial tissues. Wound-healing experiments have demonstrated that the loss of a key protein, E-cadherin, results in a significant increase in both mobility and invasiveness of epithelial cells, with dramatic consequences on cancer progression. We can reproduce the results of these experiments under various genetic conditions with a single set of parameters.

Figure 1

Schematic of the system. E-cadherin mediates cell-cell adhesion and is positively regulated by an external chemical factor $X$. Each site of the system can be empty or occupied by a cell. Cell hopping and cell proliferation are the two processes governing cell dynamics. A hopping process and a proliferation process are illustrated here on a 1D lattice.

Figure 2

Velocity of the front as a function of cellular E-cadherin concentration per cell $E^0$, obtained numerically (markers) and with analytical predictions (color lines) for nonlocal (NLP) and local proliferation (LP) as given by $v_{\min }^{\mathrm{NLP}}$ and $v_{\min }^{\mathrm{LP}}$, respectively. E-cadherin concentration is kept constant in space and time. A Heaviside function is used as initial condition, $a=20\mu \mathrm{m}$, ${\mathrm{\Gamma }}_0=1.5{\mathrm{h}}^{-1}$, $\alpha =0.1{\mathrm{nM}}^{-1}$, $\mathrm{\Theta }=0.1{\mathrm{h}}^{-1}$ with converged time step $dt={10}^{-3}$ h.

Figure 3

Comparison between experimental and numerical profiles. (a) Schematic of the wound-healing cell migration assay and data processing. A section of a confluent cell monolayer is removed and the healing process due to cell diffusion and proliferation is monitored. Images of the system are captured at 0, 12, and 24 h. Cell counting is used to extract cell density profiles from the images. On each image of Fig. 2 from Fan et al. [18], we drew a square grid ($80\mu \mathrm{m}\times 80\mu \mathrm{m}$) and cells were counted in each square. Averaging over the six rows of the grid and normalizing with respect to the cell occupation at the boundaries, we obtained the 1D cell occupation profiles. (b)–(e) Experimental cell occupations are obtained after wound-healing cell migration assay (plain lines and $95\%$ confidence interval reconstructed from Fan et al. [18]). The confidence intervals are due to the data processing procedure. Numerical cell occupation profiles are obtained with best fitting parameter values using the E-cadherin-dependent diffusivity and constant proliferation model (red circles) and E-cadherin-dependent proliferation rate model (blue squares). Fitted values of ${\mathrm{\Gamma }}_0$, $\alpha$, ${\mathrm{\Theta }}_0$, and $\zeta$ are shown in Table 4 and all other parameter values are shown in Table 2. Apart from the synthesis rate of E-cadherin varied to mimic the experiments, all other parameters are the same in the four panels.

Figure 4

Controlling wound-healing dynamics. (a) Cell occupation at 0 and 24 h for the values of the parameters listed in Table 2. The healed area is defined as the area beneath the cell occupation profile in the initial wound region (gray area). Time evolution of the healed area for increasing values of (b) the cell proliferation rate ${\mathrm{\Theta }}_0$ (bottom to top), (c) the hopping frequency of a cell with no E-cadherin ${\mathrm{\Gamma }}_0$ (right to left), (d) the adhesion parameter $\alpha$ (left to right), (e) the synthesis rate of E-cadherin $v_E$ (left to right), and (f) the rate constant for the secretion of the regulating factor $k_{\mathrm{out}}$ (right to left). (g) Cell occupation, hopping frequency, and cellular E-cadherin concentration profiles after 30 h for two different values of the rate of secretion $k_{\mathrm{out}}$. When not varied, the parameter values are as shown in Table 2 with a constant proliferation rate.