Figure 1

Determination of the maximum time step, $\Delta t_{\max }$. The maximum size of the time step is set by the requirement that each particle can interact with at most one other particle during that time step; it can thus travel a distance of at most $\Delta r_{\max ,i}$ during a time step, as indicated by the circles. We have used the operational criterion $\Delta r_{\max ,i}=H\sqrt{6D_i\Delta t_{\max ,i}}$, with $D_i$ being the diffusion constant of particle $i$. A value of $H=3$ was found to yield a good conservation of the spatial distribution of particles. In this example, each particle is assumed to have the same diffusion constant; the time step is limited by the constraint that particles $i$ and $k$ should not interact as particle $i$ can already interact with particle $j$. Note that with this maximum time step the many-body problem of propagating the $N$ particles is reduced to that of propagating single particles and pairs of particles.Reuse & Permissions

Figure 2

The mean protein number $\overline{N_P}$ as a function of the protein production rate $k_{\mathrm{prod}}$ as obtained from the GFRD simulations for the reaction scheme shown in Eqs. (9, 10, 11). The solid line denotes the mean-field prediction given by Eq. (12).Reuse & Permissions

Figure 3

The noise in protein level ${\eta }_P$ as a function of synthesis rate $k_{\mathrm{prod}}$ for the reaction scheme shown in Eqs. (9, 10, 11). To elucidate the effect of spatial fluctuations, we fix the mean number of proteins at $\overline{N_P}=1000$ by changing the protein degradation rate $k_{\mathrm{decay}}$.Reuse & Permissions

Figure 4

The distribution of propagation times $\Delta t$ in GFRD for a system consisting of a single particle $A$ and $N_B$ particles $B$ that can react according to the scheme in Eq. (9). In brute-force Brownian dynamics, the time step is $\Delta t\approx 1\times {10}^{-10}\mathrm{s}$.Reuse & Permissions