A mathematical model for axonal transport of large cargo vesicles

Abstract

In this study, we consider axonal transport of large cargo vesicles characterised by transient expansion of the axon shaft. Our goal is to formulate a mathematical model which captures the dynamic mechanical interaction of such cargo vesicles with the membrane associated periodic cytoskeletal structure (MPS). It consists of regularly spaced actin rings that are transversal to the longitudinal direction of the axon and involved in the radial contraction of the axon. A system of force balance equations is formulated by which we describe the transversal rings as visco-elastic Kelvin-Voigt elements. In a homogenisation limit, we reformulate the model as a free boundary problem for the interaction of the submembranous MPS with the large vesicle. We derive a non-linear force-velocity relation as a quasi-steady state solution. Computationally we analyse the vesicle size dependence of the transport speed and use an asymptotic approximation to formulate it as a power law that can be tested experimentally.

Appendices

A: Asymptotic expansion of transport velocity

Starting with the first equation in (22), we have the following integral equation

$$\begin{aligned} V \;&= \frac{F}{\xi }\;+\frac{1}{\xi } \;\int _A^B \left( h(y)-1-V \, h'(y) \right) h'(y) dy\;. \end{aligned}$$

Now we substitute h and its derivative given in (7) obtaining

$$\begin{aligned} 0 =V\;- \frac{F}{\xi }\;-\frac{1}{\xi } \; \int _{A}^{B} -\frac{y \left( \frac{y \, V}{\sqrt{R_v^2-y^2}}+\sqrt{R_v^2-y^2}-1\right) }{\sqrt{R_v^2-y^2}} dy. \end{aligned}$$

Coupling this to the asymptotic expansions of A and B given in (29) and (27) we conclude that

$$\begin{aligned} 0= V- \frac{F}{\xi }-\frac{1}{\xi } \left[ \frac{1}{2} V \left( \varepsilon ^2\!+\!1\right) \log \left( \frac{-y+\varepsilon ^2+1}{y+\varepsilon ^2+1}\right) \!+\! V y- \sqrt{\left( \varepsilon ^2+1\right) ^2-y^2}-y^2\right] _{A}^{B},\nonumber \\ \end{aligned}$$

(32)

where we used $R_v=1+\varepsilon ^2$. Finally, substitute the asymptotic expansion for V into (32),

$$\begin{aligned} \left( V_0-\frac{F}{\xi }\right) +V_1 \varepsilon +V_2 \varepsilon ^2+\varepsilon ^3 \left( \frac{2 \sqrt{2} F}{3 \xi }+V_3\right) +\frac{\varepsilon ^4 \left( 6 \xi V_4+4 \sqrt{2} V_1+3\right) }{6 \xi }\\+\varepsilon ^5 \left( -\frac{F}{15 \sqrt{2} \xi }+\frac{2 \sqrt{2} V_2}{3 \xi }+V_5\right) +O(\varepsilon )^6&=0 \; . \end{aligned}$$

Equating coefficients of equal powers of $\varepsilon$ we obtain

$$\begin{aligned} V_0=\frac{F}{\xi },\; V_1=0,\; V_2=0,\; V_3=-\frac{2 \sqrt{2} F}{3 \xi ^2},\; V_4=-\frac{1}{2 \xi }\; \text {and} \; V_5&=\frac{F}{15 \sqrt{2} \xi ^2}\;. \end{aligned}$$

B: Numerical test of quasi-steady state approximation

In this supplementary section we numerically test the validity of the quasi-steady state approximation (23). To this end we solve the time discrete model (8)–(9) numerically, extract the cargo velocity and compare it to the result obtained from numerically solving (23). The numerical results coincide (Fig. 12).

Fig. 12

The velocity obtained through solving the discrete system (8)–(9) for different values of $\varepsilon =\sqrt{R_v-1}$ coincides with the velocity computed numerically using the quasi-steady state approximation of the continuum model (23) for $\xi =1$, $F=2$