Figure 3.

Fluctuations in mRNA levels in the case of diffusion with fractal crowding leading to compact exploration. (A) Typical evolution of the mRNA copy number, $n_M\left(t\right)$ (red line), obtained by numerical simulations of the model. The nucleus is modeled as a fractal environment (here, a 3-dimensional critical percolation cluster embedded in a 12³ cube) embedded in a volume $V=1.7{10}^3\mu {\text{m}}^3$, such that $\langle T\rangle =500\text{min}$. Other parameter values are $\langle m\rangle =5$, n = 10, ${\lambda }_d=0.02{\text{min}}^{-1}$. 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) for the same critical percolation cluster. (C) Normalized autocorrelation function $R_M\left(t\right)/n$ for diffusion on the same 3-dimensional critical percolation cluster. Numerical simulations (symbols) for different values of ${\lambda }_d$ and n are compared to the theoretical prediction (plain and dashed lines, respectively). ${\lambda }_d=0.1{\text{min}}^{-1}$ (n = 1 (dark 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)). (D) Normalized autocorrelation function $R_M\left(t\right)/n$ for diffusion on a Sierpinski gasket (example of deterministic fractal) such that $\langle T\rangle =729\text{min}$. ${\lambda }_d=0.1{\text{min}}^{-1}$ (n = 1 (dark green circles) and n = 10 (green crosses)), ${\lambda }_d=0.02{\text{min}}^{-1}$ (n = 1 (dark blue triangles) and n = 10 (light blue inverted triangles)).