Pattern Formation inside Bacteria: Fluctuations due to the Low Copy Number of Proteins

We examine fluctuation effects due to the low copy number of proteins involved in pattern-forming dynamics within a bacterium. We focus on a stochastic model of the oscillating MinCDE protein system regulating accurate cell division in E. coli. We find that, for some parameter regions, the protein concentrations are low enough that fluctuations are essential for the generation of patterns. We also examine the role of fluctuations in constraining protein concentration levels.

Figure 1

Ratio of average MinD density in right-hand $30\%$ of the cell to that in left-hand $30\%$. Full line: stochastic dynamics with random initial conditions at (a) $N=200$, (b) $N=1500$; dashed line: deterministic dynamics with equivalent parameters but with inhomogeneous initial conditions.

Figure 2

(a),(b) MinD and (c),(d) MinE relative density profiles in stochastic model with (a),(c) $N=200$, (b),(d) $N=1500$. Thick lines represent averages over (a),(c) 160 and (b),(d) 110 consecutive cycles, respectively (maximum average density normalized to unity). Symbols ($+$, ◇, △, □) represent individual data sets that are averaged over only a single oscillation cycle.

Figure 3

Histogram for position of MinD density minimum in the stochastic model with (a) $N=200$, (b) $N=400$, (c) $N=800$, (d) $N=1500$. The position of each minimum is computed by averaging over a single oscillation cycle. Data is for $160$, $135$, $120$, and $110$ consecutive cycles, respectively.