Self-Organized Periodicity of Protein Clusters in Growing Bacteria

Chemotaxis receptors in E. coli form clusters at the cell poles and also laterally along the cell body, and this clustering plays an important role in signal transduction. Recently, experiments using fluorescence imaging have shown that, during cell growth, lateral clusters form at positions approximately periodically spaced along the cell body. In this Letter, we demonstrate within a lattice model that such spatial organization could arise spontaneously from a stochastic nucleation mechanism. The same mechanism may explain the recent observation of periodic aggregates of misfolded proteins in E. coli.

Figure 1

Schematic of the lattice model. Particles hop at random between neighboring lattice points and can join or leave an existing cluster from the boundary of the cluster. Columns of lattice points are inserted at random to mimic cell growth, and particles are inserted at random to mimic protein insertion in the membrane.

Figure 2

Snapshots of the model cell membrane at times 70, 75, 80, 85, and 90 $\times {10}^4$ Monte Carlo time steps (MCS) as defined in the text. The total size of the system in the final snapshot is 1350 columns by 50 rows, with $\approx 20\%$ of lattice sites occupied by protein particles [gray (green)]. In the inset we show a representative image of protein receptor clusters in E. coli from Victor Sourjik’s lab (courtesy: Victor Sourjik).

Figure 3

The distribution of intercluster separations. Simulations were stopped when the system reached $L_x=1900$, which corresponds to approximately eight clusters in the system. The density is ${\rho }_0=0.1$ and the growth rate is $\gamma =1\times {10}^{-5}$. The data were averaged over 70 000 simulations. The bin size is 20. If the centers of clusters were randomly distributed, the distribution of intercluster separations would be exponential as shown by the dotted curve.

Figure 4

(a) Distribution of the separation along the $x$ direction between the center of the new (second) cluster and the center of the old (first) cluster, using periodic boundary conditions in $x$ (see inset). The dotted line is a guide to the eye. (b) The particle density profile in the $x$ direction (periodic boundary conditions) with one cluster in the system. The particle density was measured when the system reached $L_x=300$ and was averaged over 10 000 simulations. The distance was normalized by the separation between the two flanking edges of the cluster as measured in the $x$ direction. The smooth line is a fit to a parabola. (c) The average intercluster separation versus the particle density ${\rho }_0$ at fixed growth rate $\gamma ={10}^{-5}$. (d) The average intercluster separation versus growth rate $\gamma$ at fixed particle density ${\rho }_0=0.1$. The smooth curves in (c) and (d) are fits to power laws ${\rho }_0^{-0.47}$ and ${\gamma }^{-0.53}$ , respectively. (e) The cluster size distribution averaged over 10 000 simulations. Cluster sizes were measured when the system reached $L_x=1900$; the solid red line is a fit to the curve $y=\exp [a_{1}x+a_{2}x\ln x+a_3\ln x+a_4]$, with $a_1\approx 0.8$, $a_2\approx -0.37$, $a_3\approx -7.6$, and $a_4\approx -0.9$, for cluster size. (f) Dependence of the standard deviation of cluster separation (normalized by the average separation) on interaction strength $J$. The parameters and the method for collecting data were the same as in  (a). In the inset we show the dependence of intercluster separation on $J$.