Figure 2.

Fluctuations in mRNA level in the case of diffusion in a dilute environment, leading to noncompact exploration. The nucleus volume is $V={10}^3\mu m^3$, and we take $D=1\mu m^2/s$, and $\langle m\rangle =5$. (A) Typical evolution of the mRNA copy number, $n_M\left(t\right)$ (red line), obtained by numerical simulations of the model for n = 10, ${\lambda }_d=0.02{\text{min}}^{-1}$. Here, the nucleus is modeled as a 3-dimensional cubic lattice and $\langle T\rangle =648\text{min}$. Green bars mark the activation of the gene. (B) Waiting-time distribution, $f_n\left(T\right)$, between activation events for n = 1 (violet circles) and n = 10 (magenta triangles), compared to the theoretical prediction, derived in the Supporting Material (plain and dashed lines, respectively). (C) Normalized autocorrelation function $R_M\left(t\right)/n$ for diffusion on a 3-dimensional cubic lattice. Numerical simulations (symbols) for different values of ${\lambda }_d$ and n are compared to the theoretical prediction (plain and dashed lines). ${\lambda }_d=0.1{\text{min}}^{-1}$ (n = 1 (green circles) and n = 10 (light green crosses)); ${\lambda }_d=0.02{\text{min}}^{-1}$ (n = 1 (dark blue triangles) and n = 10 (light blue inverted triangles)); ${\lambda }_d=0.005{\text{min}}^{-1}$ (n = 1 (red triangles) and n = 10 (orange triangles)). The dotted line is obtained by assuming a homogeneous concentration $c_0=1/V$ of TFs in infinite space (case ${\lambda }_d=0.02{\text{min}}^{-1}$). (D) Normalized autocorrelation function $R_M\left(t\right)/n$ for diffusion with dilute obstacles (here modeled by a 3-dimensional supercritical percolation cluster with $p=0.8$). ${\lambda }_d=0.02{\text{min}}^{-1}$ (n = 1 (dark blue triangles) and n = 10 (light blue inverted triangles)); ${\lambda }_d=0.005{\text{min}}^{-1}$ (n = 1 (red triangles) and n = 10 (orange triangles)).