Efficient simulations of tubulin-driven axonal growth

Abstract

This work concerns efficient and reliable numerical simulations of the dynamic behaviour of a moving-boundary model for tubulin-driven axonal growth. The model is nonlinear and consists of a coupled set of a partial differential equation (PDE) and two ordinary differential equations. The PDE is defined on a computational domain with a moving boundary, which is part of the solution. Numerical simulations based on standard explicit time-stepping methods are too time consuming due to the small time steps required for numerical stability. On the other hand standard implicit schemes are too complex due to the nonlinear equations that needs to be solved in each step. Instead, we propose to use the Peaceman–Rachford splitting scheme combined with temporal and spatial scalings of the model. Simulations based on this scheme have shown to be efficient, accurate, and reliable which makes it possible to evaluate the model, e.g. its dependency on biological and physical model parameters. These evaluations show among other things that the initial axon growth is very fast, that the active transport is the dominant reason over diffusion for the growth velocity, and that the polymerization rate in the growth cone does not affect the final axon length.

Appendices

Appendix A: Proof of Lemma 1

We begin by noting that any positive definite matrix is invertible and that any square matrix is positive definite when its symmetric part has only positive eigenvalues. That is, the linear system of Eqs. (23) has a unique solution when the eigenvalues of

$$\mathbf{S}(\mathbf{U}^{n+\frac{1}{2}},\mathbf{y}) := \frac{\left( \mathbf{I} - \frac{\Delta\tau}{2}\mathbf{A}(\mathbf{U}^{n+\frac{1}{2}},\mathbf{y})\right) + \left( \mathbf{I} - \frac{\Delta\tau}{2} \mathbf{A}(\mathbf{U}^{n+\frac{1}{2}},\mathbf{y})\right)^{\mathrm{T}}}{2} $$

are all positive. The entries of the main diagonal, respectively the super and sub diagonals are given as follows

$$\begin{array}{@{}rcl@{}} &&s_{j,j}(\mathbf{U}^{n+\frac{1}{2}},\mathbf{y}) = 1 + \frac{\Delta\tau}{2} \left( \frac{2D}{a({\Delta} y)^{2}}\frac{1}{L^{n+\frac{1}{2}}} + \frac{g}{a}L^{n+\frac{1}{2}}\right),\\[-2pt] && j = 1, \dots, M-1, \\[-2pt] &&s_{j,j+1}(\mathbf{U}^{n+\frac{1}{2}},\mathbf{y}) = s_{j+1,j}(\mathbf{U}^{n+\frac{1}{2}},\mathbf{y})\\[-2pt] &&\quad = - \frac{\Delta\tau}{2} \left( \frac{r_{\mathrm{g}}}{4a}(C_{\mathrm{c}}^{n+\frac{1}{2}}-c_{\mathrm{c}}^{\infty}) + \frac{D}{a({\Delta} y)^{2}}\frac{1}{L^{n+\frac{1}{2}}}\right),\\[-2pt] &&j = 1, \dots, M-2. \end{array} $$

Thus, S(U ^n+1/2,y) is a symmetric, tridiagonal Toeplitz matrix meaning that the eigenvalues are given by the following formula

$$\begin{array}{@{}rcl@{}} \lambda_{k} &=& s_{j,j}(\mathbf{U}^{n+\frac{1}{2}},\mathbf{y}) + 2s_{j,j+1}(\mathbf{U}^{n+\frac{1}{2}},\mathbf{y}) \cos(k\pi{\Delta} y), \quad \\ k &=& 1, \dots, M-1. \end{array} $$

Then, any λ _k fulfils the following inequality

$$\begin{array}{@{}rcl@{}} \lambda_{k} &\geq& s_{j,j}(\mathbf{U}^{n+\frac{1}{2}},\mathbf{y}) - |2s_{j,j+1}(\mathbf{U}^{n+\frac{1}{2}},\mathbf{y}) \cos(k\pi{\Delta} y)| \\ &\geq& s_{j,j}(\mathbf{U}^{n+\frac{1}{2}},\mathbf{y}) - |2s_{j,j+1}(\mathbf{U}^{n+\frac{1}{2}},\mathbf{y})|. \end{array} $$

By plugging in the entries and using the triangle inequality we get

$$\begin{array}{@{}rcl@{}} \lambda_{k} &\geq& 1 + \frac{\Delta\tau}{2} \left( \frac{2D}{a({\Delta} y)^{2}}\frac{1}{L^{n+\frac{1}{2}}} + \frac{g}{a}L^{n+\frac{1}{2}}\right) \\ &&- {\Delta}\tau \left\lvert\frac{r_{\mathrm{g}}}{4a}(C_{\mathrm{c}}^{n+\frac{1}{2}}-c_{\mathrm{c}}^{\infty}) + \frac{D}{a({\Delta} y)^{2}}\frac{1}{L^{n+\frac{1}{2}}}\right\rvert \\ &\geq& 1 + \frac{\Delta\tau}{2} \left( \frac{2D}{a({\Delta} y)^{2}}\frac{1}{L^{n+\frac{1}{2}}} + \frac{g}{a}L^{n+\frac{1}{2}}\right) \\ &&- {\Delta}\tau \left( \left\lvert\frac{r_{\mathrm{g}}}{4a}(C_{\mathrm{c}}^{n+\frac{1}{2}}-c_{\mathrm{c}}^{\infty})\right\rvert + \left\lvert\frac{D}{a({\Delta} y)^{2}}\frac{1}{L^{n+\frac{1}{2}}}\right\rvert\right) \\ &=& 1 + {\Delta}\tau \left( \frac{g}{2a}L^{n+\frac{1}{2}} - \frac{r_{\mathrm{g}}}{4a}|C_{\mathrm{c}}^{n+\frac{1}{2}}-c_{\mathrm{c}}^{\infty}| \right) \\ &\geq& 1 - {\Delta}\tau \frac{r_{\mathrm{g}}}{4a}|C_{\mathrm{c}}^{n+\frac{1}{2}}-c_{\mathrm{c}}^{\infty}| \ \ \geq \ \ 1 - {\Delta}\tau \frac{r_{\mathrm{g}}}{4a}\gamma, \end{array} $$

where we have used the definition of γ given by Eq. (24). We conclude that the eigenvalues of S(U ^n+1/2,y) are positive when

$${\Delta}\tau < \frac{4a}{r_{\mathrm{g}}} \cdot \frac{1}{\gamma}, $$

and therefore, for these values of Δτ, the system (23) has a unique solution.

Appendix B: Two-dimensional slices of Figs. 2b and 8

We complement the three-dimensional plots (Figs. 2b and 8) with two-dimensional slices at different x and t values. Recall that in Fig. 2b the tubulin concentration along the axon is plotted for nominal values on the biological and physical parameters. For Fig. 8 a three times larger advection velocity a is used. The slices of Figs. 2b and 8 are presented next to each other in Figs. 13, 14, 15 and 16 for easy comparison.

Fig. 13

Slices of Fig. 2b at three different x values and at the growth cone. Note that the spatial position of the latter changes with time. The spikes in the second and third subfigures correspond to times where the respective slice is close to the cone. Further, note that the concentration in the growth cone is largely unaffected by the soma concentration, instead a higher value in the latter results in a longer axon

Fig. 14

Slices of Fig. 8 at three different x values and at the growth cone. As the three times larger advection velocity a gives a far longer axon, different x values are chosen compared to those in Fig. 13. Note how the higher advection velocity gives smaller concentration values in the inner of the axons; the tubulin is concentrated close to the soma and the growth cone

Fig. 15

Slices of Fig. 2b at four different t values. See also (26) for the times of decline and increase in soma concentration. Note the characteristic profile of the concentration along the axon, cf. Diehl et al. (2014, Section 4)

Fig. 16

Slices of Fig. 8 at four different t values. Note the increased length of the axon as an effect of the three times larger advection velocity a. Note also the sharper gradient close to the growth cone, cf. Diehl et al. (2014, Section 4)