A 2D fluid-based model for individual cell crawling

Abstract

This paper presents the analysis and numerical simulations of a two-dimensional model of single-cell crawling migration, focusing on intracellular dynamics. The fluid-based model relies on the interaction between the actin cytoskeleton and a generic molecular regulator of its activity. The model is formulated as a Poisson problem with Darcy’s law, coupled with a nonlocal advection–diffusion equation for the molecular concentration, where the non-local term involves boundary values. Existence and uniqueness of resting and travelling stationary states are established in certain parameter domains, along with linear stability in the resting case. Numerical simulations illustrate both cell regimes. This demonstrates the model’s ability to capture key features of cell migration.

Appendix A Numerical framework and cell polarity indices

In this part, we detail the finite volume numerical framework used to perform the numerical simulations. Then, we detail and illustrate the cell polarity indices used to characterise the internal cell dynamics.

1.1 A.1 Numerical framework

The radial discretization is given by $\{r_j=j \, \Delta r, j =1 \dots N_r\}$, where $r_{N_r + \frac{1}{2}} = 1$.

Similarly, the angular discretization is given by $\{\theta _k=k \, \Delta \theta , k \in 1\dots N_{\theta } \}$, with $\theta _{N_{\theta }}=2\pi$. For $j\in \{ 1 \dots N_r \}$ and $k \in \{1 \dots N_{\theta } \}$, the control volumes $W_{(j,k)} \subset \mathbb {R}^2$ are given by

$$\begin{aligned} W_{(j,k)} = \left( r_{j-\frac{1}{2}},r_{j+\frac{1}{2}}\right) \times \left( \theta _{k-\frac{1}{2}},\theta _{k+\frac{1}{2}}\right) \,, \end{aligned}$$

such that $\left| W_{(j,k)} \right| = r_j \Delta r \Delta \theta$. The central control volume is given by $W_0 = B(0,r_{\frac{1}{2}})$ such that $\left| W_0 \right| = \pi r_{\frac{1}{2}}^2$. We use a finite volume scheme in space. More precisely, keeping the same notation for simplicity, the discretised values of the molecular concentration at time $t>0$ are given by

$$\begin{aligned} {\left\{ \begin{array}{ll} c_{0}(t):= \frac{1}{\pi r_{\frac{1}{2}}^2} \iint _{B(0,r_{\frac{1}{2}})} c(t,r,\theta ) r \, {\mathrm d}r \, {\mathrm d}\theta \,,& \\ c_{j,k}(t):= \frac{1}{\left| W_{(j,k)} \right| } \iint _{W(j,k)} c(t,r,\theta ) r \, {\mathrm d}r \, {\mathrm d}\theta \,,& j\in \{ 1 \dots N_r \},\, k \in \{1 \dots N_{\theta } \}\,, \end{array}\right. } \end{aligned}$$

and their time dynamics are obtained using Eq. (1). The advection term based on Eqs. (2)–(4) is approximated using a first-order upwind scheme. Finally, let $\{t^n = n\, \Delta t,\, n=0 \dots N_f\}$ be the time discretization, with $T_f:= t^{N_f}$ the final time of the simulation. Then, $(c^n_{j,k}:= c_{j,k}(t^n))_{1\le j \le N_r, \,1 \le k \le N_{\theta }}$ and $c^n_{0}:= c_{0}(t^n)$ are the fully discretised values of the solution at time $t^n$ obtained using an Euler scheme. Unless indicated otherwise, we take $\Delta t = 0.01$, and $N_{\theta } = 200$, so that $\Delta r = \Delta \theta = \frac{2\pi }{N_{\theta }} \simeq 0.03$.

1.2 A.2 Cell polarity indices

Two cell polarity indices are considered in this paper.

• the Relative Standard Deviation (RSD) or Coefficient of Variation (CV) is given by the perimeter-normalized ratio between the standard deviation of $\{c^n_{N_r,k}\}_{1 \le k \le N_y}$ and its mean value $\bar{c}^n:= \frac{1}{N_{\theta }} \sum _{k=1}^{N_{\theta }} c^n_{N_r,k}$. More precisely,

  $$\begin{aligned} \text {RSD}^n := \frac{1}{2\pi \bar{c}^n} \sqrt{ \frac{1}{N_y} \sum _{k=1}^{N_y} | c^n_{N_r,k} - \bar{c}^n |^2 }\,. \end{aligned}$$

• the normalized Mean Resultant Vector is given by

  $$\begin{aligned} RV^n:= \frac{1}{\sum _{i=1}^{N_{\theta }} c^n_{N_r,k} } \sum _{i=1}^{N_{\theta }} c^n_{N_r,k} \varvec{\vec {e}}_{k} \end{aligned}$$

  as defined in circular statistics Jammalamadaka and Sengupta (2001). Its norm, the Resultant Vector Length (RVL) provides another polarisation index.

Fig. 6

Several boundary concentration profiles with the corresponding polarity indices. a Homogenous boundary concentration associated with a resting state, with null Relative Standard Deviation and Resultant Vector Length. b Unipolar boundary concentration given by $\theta \mapsto 1-\cos (\theta )$, associated with a travelling state, characterised by nonzero Relative Standard Deviation and Resultant Vector Length. c Boundary concentration given by $\theta \mapsto \cos (\theta )\left( \mathbbm {1}_{[-\pi /2,\pi /2]} - \mathbbm {1}_{[\pi /2,3\pi /2]}\right)$, showing two opposite poles and associated with a resting state. The Relative Standard Deviation is nonzero while the Resultant Vector Length is zero. d Boundary concentration given by $\theta \mapsto 2\mathbbm {1}_{[\pi -5\Delta \theta ,\pi +5\Delta \theta ]}$, with a steep polarisation associated with a travelling state. Both Relative Standard Deviation and Resultant Vector Length are nonzero, although smaller than case (b)

Figure 6 shows some boundary concentration profiles together with the corresponding RSD and RVL values. In Fig. 6a, the boundary concentration is constant, both indices are null and the cell is in a resting state. In Fig. 6b, the boundary concentration shows a single pole, both the RSD and the RVL are nonzero, and the cell is in a travelling state. In Fig. 6c, the boundary concentration shows two symmetric poles, such that the cell is in a resting state and the RVL is zero while the RSD is nonzero. Finally, in Fig. 6d, the boundary concentration shows a steep polarisation associated with a travelling state. However, both polarity indices are smaller than in Fig. 6b, although nonzero.