A simple reaction–diffusion system as a possible model for the origin of chemotaxis

Figures & data

Figure 1. Local activation via positive feedback and depletion of the substrate in the cytosol generates an activator-enriched domain on the cortex [3].

Figure 2. The interconversion of Rho-GTPases between active and inactive forms can be modelled as a reaction–diffusion equation governing the dynamics of the slowly-diffusing activator u and the infinitely-diffusing substrate v.

Figure 3. Derivation of a pheromone profile generated from a heat equation.

Figure 4. Pheromone profile generated from a heat equation $f_h(x)$ and a similar linear pheromone profile $f(x)$.

Table 1. Parameters from [3].

Parameter	Description	Value/Range in simulation	Unit
k	Diffusion coefficient for u	0.01	$(\text{μ}\mathrm{m}{)}^2({\mathrm{s}}^{-1})$
α	Pheromone strength	0.5−3	${\mathrm{s}}^{-1}$
$|{\mathbf{T}}^d|$	Cell size	10	$\text{μ}\mathrm{m}$
M	Total mass	10	–
a	Strength of activation	1	$(\text{μ}\mathrm{m}{)}^2$
b	Strength of substrate	1	${\mathrm{s}}^{-1}$

Figure 5. Numeric solution to Equation (7(7) $\frac{\mathrm{\partial }u}{\mathrm{\partial }t}=(a{u}^2+\alpha f(x))\frac{1}{|{\mathbf{T}}^d|}(M-U(t))-bu+k\mathrm{\Delta }u.$(7) ) with α = 2 and other parameters given in Table 1: (a) pheromone profile given by $f_h(x)$ and (b) pheromone profile given by $f(x)$.

Figure 6. Movement speed $\frac{{\mathrm{CM}}_u(t)}{\mathrm{d}t}$ as a function of pheromone profile slope $\frac{f{(}_h{\mathrm{CM}}_u(t))}{\mathrm{d}t}$. $\frac{\mathrm{d}{f}_h(x)}{\mathrm{d}x}>0$ for $x\in (0,2.5)$.

Figure 7. Movement speed $\frac{{\mathrm{CM}}_u(t)}{\mathrm{d}t}$ as a function of pheromone profile slope $\frac{f({\mathrm{CM}}_u(t))}{\mathrm{d}t}$. $\frac{\mathrm{d}f(x)}{\mathrm{d}x}$ is a constant for $x\in (0,2.5)$.

Figure 8. Initial profile for rho-GTPase u, rho-GDPase v, and pheromone profile $f(x)$ and pheromone strength $\alpha =2$: (a) initial profile t = 0s and (b) profile when $t=10,000s$.

Figure 9. Centre of mass position ${\mathrm{CM}}_u(t)$ as a function of time with pheromone profile $f(x)$. In regions away from $x_{\mathrm{peak}}$, ${\mathrm{CM}}_u(t)$ changes linearly with time.

Figure 10. Centre of mass movement speed $\frac{d{\mathrm{CM}}_u(t)}{\mathrm{d}t}$ as a function of pheromone strength α with pheromone profile $f(x)$. In regions away from $x_{\mathrm{peak}}$, $\frac{d{\mathrm{CM}}_u(t)}{\mathrm{d}t}$ changes linearly with α.

J.-G. Chiou, S.A. Ramirez, T.C. Elston, T.P. Witelski, D.G. Schaeffer and D.J. Lew, Principles that govern competition or co-existence in Rho-GTPase driven polarization, PLoS Comput. Biol. 14(4) (2018), Article ID e1006095.

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