Cell Repolarization: A Bifurcation Study of Spatio-Temporal Perturbations of Polar Cells

Abstract

The intrinsic polarity of migrating cells is regulated by spatial distributions of protein activity. Those proteins (Rho-family GTPases, such as Rac and Rho) redistribute in response to stimuli, determining the cell front and back. Reaction-diffusion equations with mass conservation and positive feedback have been used to explain initial polarization of a cell. However, the sensitivity of a polar cell to a reversal stimulus has not yet been fully understood. We carry out a PDE bifurcation analysis of two polarity models to investigate routes to repolarization: (1) a single-GTPase (“wave-pinning”) model and (2) a mutually antagonistic Rac-Rho model. We find distinct routes to reversal in (1) vs. (2). We show numerical simulations of full PDE solutions for the RD equations, demonstrating agreement with predictions of the bifurcation results. Finally, we show that simulations of the polarity models in deforming 1D model cells are consistent with biological experiments.

Appendix

1.1 Computational Methods for Bifurcation Diagrams

Bifurcation diagrams are computed using numerical continuation. At its core, numerical continuation uses a predictor (Euler step) and a corrector (Newton solver) to compute steady state solution branches.

1.1.1 Function Approximation and Collocation

As we are dealing with systems of one dimensional partial differential equations, their solutions are found in the function spaces $H^2$ and the related $L^2$. Elements of these spaces are approximated using truncated Chebyshev series. Throughout this work we truncate the series at machine precision (using the truncation algorithm presented in Platte and Trefethen (2010)). For mathematical details on Chebyshev polynomials and their use in function approximation we refer the reader to the book by Trefethen (2019).

Motivated by the use of Chebyshev polynomials for function approximation, we choose a closely related collocation approach to discretize linear operators on function spaces (e.g. the Fréchet derivative required in the Newton iteration). We choose the well-conditioned collocation method introduced in Olver and Townsend (2013) based on ultraspherical polynomials, which yields sparse matrices.

When solving a linear operator the optimal truncation of the solution’s Chebyshev series is computed using the “naive” method outlined in Olver and Townsend (2013). The allowable discretization sizes are $N = 2^n + 1$, with $n = 4$ being the minimum. Each linear system is consecutively solved for increasing values of n, until the $L^2$ norm between consecutive solutions is below a user defined tolerance (we use $10^{-12}$). Boundary conditions are applied using the rectangular collocation ideas discussed in Driscoll and Hale (2016).

Our implementation is in Python, and builds on the infrastructure developed in numpy (Harris et al. 2020) and scipy (Virtanen et al. 2020). In particular, we overload many of the existing numpy ufuncs to seamlessly work for our function objects e.g. numpy.sum(function) will compute the integral of function, or numpy.diff(function) will compute the derivative of function.

Both linear and nonlinear operators are defined by a user provided string, which is translated into sympy (Meurer et al. 2017) expressions. Nonlinear operators are symbolically linearized using sympy. Using sympy’s code printing abilities, the sympy expressions are translated into callable “numpy code”, which is used by the collocation method to generate the discretized approximations to the linear operators. This approach has several advantages: (1) Numerical approximations of the Fréchet derivatives are not required; and (2) rapid adaptation of the existing code to new problems.

1.1.2 Nonlinear Solver

Nonlinear operators are solved using Newton’s method. We employ two related Newton methods: (1) NLEQ-ERR, and (2) the quasi-Newton method QNERR. For more details on each of these methods we refer the reader to Deuflhard (2011). These damped Newton methods have been shown to perform well for nonlinear operators of elliptic partial differential equations (Deuflhard 2011). Our implementation of NLEQ-ERR switches to QNERR whenever possible.

1.1.3 Numerical Continuation Algorithm

We use a predictor-corrector scheme for continuation. Here we give a brief overview of the method implemented for this work. For more details, we refer the reader to (Deuflhard et al. 1987; Deuflhard 2011). Given the nonlinear function $F(x, \mu ) = 0$, where $x \in {\mathbb {R}}^n$ is the solution (here x is a vector of the Chebyshev coefficients) and $\mu \in {\mathbb {R}}$ is a parameter, the goal is to trace out how the solution x changes with respect to the parameter $\mu$. To simplify notation, we denote a point along a solution branch by $y:=(x, \mu )$. Starting from a known solution $y_0$ our goal is to consecutively compute adjacent points on the solution branch eventually tracing out the entire solution branch $(x, \mu )$.

• Step 1 Euler-step prediction: First, the unit tangent vector at the known solution branch point $y_0$, denoted $t(y_0)$ (satisfying $F'(y_0)t(y_0) = 0$) is computed. The prediction $y_p$ of the next point along the solution branch is then computed using Euler’s method:

  $$\begin{aligned} y_p(s) = y_0 + s t(y_0) \end{aligned}$$

  where s is the step-size.

• Step 2 Correction: Since the predicted point $y_p(s)$ is not on the solution branch, we use a nonlinear solver to refine the prediction. This guarantees that the newly computed solution point satisfies $F(y) = 0$. In more detail, the nonlinear system $F(x, \mu ) = 0$ is solved using the following iterative scheme.

  $$\begin{aligned} \varDelta y^k&= -F'(y^k)^{+} F(y^k) \\ y^{k+1}&= y^k + \varDelta y^k \end{aligned}$$

  where $F'(x, \mu )$ is the Fréchet derivative of $F(x, \mu )$, which is given by $F'(x, \mu ) = \begin{pmatrix} F_x(x, \mu )&F_\mu (x, \mu ) \end{pmatrix}$. Since this is a $(n, n+1)$ sized matrix, we use the Moore-Penrose inverse denoted by $()^{+}$. The initial guess is $y^0 = y_p$. The iteration stops when the $L^2$ norm of $\varDelta y^k$ is below $10^{-12}$. To avoid consecutive expensive evaluations of the Fréchet derivative, we use the rank-1 update strategy described in Deuflhard et al. (1987).

An advantage of this method is the availability of rigorous convergence estimates from which step size adaptation algorithms can be derived. Here the continuation step-size is controlled using the algorithm derived in Deuflhard et al. (1987). Furthermore, we place two additional constraints on the step size:

• 1. We provide an upper bound to the step size $s_{\text {max}}$.

• 2. We require that the angle between subsequent tangent vectors be below a user provided tolerance. Denote by $y_n$, and $t_n$ the n-th solution point and normalized tangent respectively. A newly computed solution point $y_{n+1}$ is only accepted when $(t_n, t_{n+1})_{L^2} \ge \alpha$ i.e. the angle between tangents at adjacent points on the solution branch are not too large. Here we choose $\alpha = 0.3$. If this check fails, the newly computed point $y_{n+1}$ is not accepted, and the computation of the next point is repeated with a halved step-size.

Turning points are detected using the standard monitoring function which triggers when detecting a sign change in the parameter component of the tangent vector of the solution branch. Upon detection of a turning point its estimate is corrected Hermite interpolation and a Newton-Gauss iteration. (For more details see Pönisch and Schwetlick 1981).

1.1.4 Discovery of New Solution Branches

Since spatial heterogeneous perturbations will introduce disconnected solution branches, we employ deflation to discover new solution branches. During deflation the nonlinear operator is modified so that known solutions are removed from its solution set. An overview of deflation techniques in the computation of solutions of partial differential equations can be found in Farrell et al. (2015). Here we use a standard deflation operator using the $H^1$ norm with $\sigma =1,\ p=2$. In a slight variation of the algorithm outlined in Farrell et al. (2016), we completely compute each branch following its discovery.

1.1.5 Linear Stability of Solution Branches

We ascertain the linear stability of the solution branches by computing the eigenvalues of the Fréchet derivative along the solution branch. When $\max _{i} {\text {Re}}[\lambda _i] < 0$, a point is said to be linearly stable. The eigenvalues of the Fréchet derivative can then be calculated from a generalized eigenvalue problem using a QZ decomposition. For more details we refer the reader to Driscoll and Hale (2016).

1.2 Computational Methods for Time Dependent Equations

A modular Python modelling toolkit for one dimensional cells was introduced in Buttenschön et al. (2020). Using a change of variables the concentrations inside the moving domain are mapped to a constant sized domain. The resulting constant sized domain is discretized into uniformly sized intervals ($N = 512$ per unit length). The fluxes of the reaction-diffusion equation are computed using a finite volume approach. The resulting ordinary differential equations together with the equations for the cell’s mechanics are integrated using the ROWMAP integrator (Weiner et al. 1997), with absolute and relative tolerances of $10^{-6}$.

1.3 Mechanical Model Equations

For the single-GTPase with a deforming 1D cell (domain $\varGamma =\{x|x_1(t)\le x \le x_2(t)\}$ of length $L(t)=x_2(t)-x_1(t)$), we modified the original equations to include length dependence.

$$\begin{aligned} \begin{aligned} \frac{\partial G}{\partial t}&= D \varDelta G + \left( {\tilde{b}} + {\tilde{\gamma }} \frac{G^n}{1 + G^n}\right) G_i - {\tilde{I}} G - G \left( \frac{{\dot{L}}(t)}{L(t)}\right) , \quad x \in \varGamma , t \ge 0 \\ \frac{\partial G_i}{\partial t}&= D_i \varDelta G_i - \left( {\tilde{b}} + {\tilde{\gamma }} \frac{G^n}{1 + G^n}\right) G_i + {\tilde{I}} G- G_i \left( \frac{{\dot{L}}(t)}{L(t)}\right) , \quad x \in \varGamma , t \ge 0. \end{aligned}\nonumber \\ \end{aligned}$$

(6)

For the Rac-Rho model, the same idea leads to the PDEs

$$\begin{aligned} \varepsilon \frac{\partial R}{\partial t}=&{} \varepsilon ^2 \varDelta R + \left( {\tilde{b}}_R+ {\tilde{\gamma }}_R \frac{1}{1+ \rho ^n}\right) R_i - {\tilde{I}} R - R \left( \frac{{\dot{L}}(t)}{L(t)}\right) , \quad x \in \varGamma , t \ge 0, \nonumber \\ \varepsilon \frac{\partial R_i}{\partial t}=&{} D \varDelta R_i - \left( {\tilde{b}}_R+ {\tilde{\gamma }}_R \frac{1}{1+ \rho ^n}\right) R_i + {\tilde{I}} R - R_i \left( \frac{{\dot{L}}(t)}{L(t)}\right) , \quad x \in \varGamma , t \ge 0,\nonumber \\ \varepsilon \frac{\partial \rho }{\partial t}=&{} \varepsilon ^2 \varDelta \rho + \left( {\tilde{b}}_\rho + {\tilde{\gamma }}_\rho \frac{1}{1+ R^n}\right) \rho _i - {\tilde{I}} \rho - \rho \left( \frac{{\dot{L}}(t)}{L(t)}\right) , \quad x \in \varGamma , t \ge 0, \nonumber \\ \varepsilon \frac{\partial \rho _i}{\partial t}=&{} D \varDelta \rho _i - \left( {\tilde{b}}_\rho + {\tilde{\gamma }}_\rho \frac{1}{1+ R^n}\right) \rho _i + {\tilde{I}} \rho - \rho _i \left( \frac{{\dot{L}}(t)}{L(t)}\right) , \quad x \in \varGamma , t \ge 0.\nonumber \\ \end{aligned}$$

(7)

Table 4 Cell length and mechanical parameters used for the deforming cell simulations using Eqs. (8)

The ends of the cell satisfy a set of ODEs

$$\begin{aligned} \mu \frac{dx_1}{dt}&= {\hat{F}}_1(x_1) + E(x_2 - x_1 - L_0) + \eta \left( \frac{dx_2}{dt} - \frac{dx_1}{dt} \right) , \end{aligned}$$

(8a)

$$\begin{aligned} \mu \frac{dx_2}{dt}&= {\hat{F}}_2(x_2) - E(x_2 - x_1 - L_0) - \eta \left( \frac{dx_2}{dt} - \frac{dx_1}{dt} \right) , \end{aligned}$$

(8b)

where the forces are GTPase-dependent with the forms:

$$\begin{aligned} {\hat{F}}_1(x_1)&=F(R(x_1,t),\rho (x_1,t))= \alpha _\rho \rho (x_1,t)-\alpha _R R(x_1,t), \end{aligned}$$

(8c)

$$\begin{aligned} {\hat{F}}_2(x_2)&=F(R(x_2,t),\rho (x_2,t))= \alpha _R R(x_2,t)-\alpha _\rho \rho (x_2,t). \end{aligned}$$

(8d)

In Eq. (8), the ${\hat{F}}_j$ are forces on cell ends due to the local activities of Rac and Rho. Rac causes the cell edge to protrude outwards, and Rho causes it to retract inwards. For the single-GTPase model, we have the two choices: either $G=R, \alpha _\rho =0$ or $G=\rho , \alpha _R=0$ (used in Fig. 11). E is an elastic coefficient governing the cell’s tendency to return to a rest-length $L_0$, and $\eta$ is a drag coefficient of the frictional forces that resist cell motion. Values of the parameters for the mechanical part of the model, (8) are given in Table 4. The system of equations represents a common linearly elastic spring-mass system operating in the overdamped regime, where the spring creeps back to its rest-state in the absence of external forces. In this version of the model, the length of the cell can changes, and is given by

$$\begin{aligned} \mu \frac{dL}{dt} = (F_2-F_1) -2 E(L - L_0) -2 \eta \left( \frac{dL}{dt} \right) . \end{aligned}$$

(9)