Molecular discreteness in reaction-diffusion systems yields steady states not seen in the continuum limit

We investigate the effects of the spatial discreteness of molecules in reaction-diffusion systems. It is found that discreteness within the so-called Kuramoto length can lead to a localization of molecules, resulting in novel steady states that do not exist in the continuous case. These states are analyzed theoretically as the fixed points of accelerated localized reactions, an approach that was verified to be in good agreement with stochastic particle simulations. The relevance of this discreteness-induced state to biological intracellular processes is discussed.

Figure 1

(Color online) Time series of $N_1$ and $N_2$. $r=1$, $a=4$, $N=1000$, $L_x=1000$. (a) $D=10$, (b) $D=100$, (c) $D=1000$. Initially, $(N_1,N_2,N_3)=(250,250,500)$. For $D=10$, $X_3$ reaches 0, which corresponds to the unstable fixed point $(2c∕3,c∕3,0)$.

Figure 2

(Color online) Average concentration of $X_2$, for different $r$ and $D$ ($a=4$, $N=1000$, $L_x=1000$, sampled over $5000<t<\mathrm{10\hspace{0.17em}000}$, and ten trials. The error bars show the standard deviation between the trials). The dotted lines correspond to $0.1$ molecule per the Kuramoto length $l_1=\sqrt{D∕50r}$ for each $r$.

Figure 3

(Color online) The acceleration factor $\alpha$, plotted against ${\lambda }_2∕l_1$. We measure the relation from simulations with different $r$, $D$, and $a$ ($N=1000$, $L_x=1000$, sampled over $5000<t<\mathrm{10\hspace{0.17em}000}$, and ten trials. The error bars show the standard deviation of $c_2$ between the trials). This is very close to the theoretical estimation $\alpha =1+1∕\left(2\sqrt{\pi }\right)·{\lambda }_2∕l_1$.