Min-protein oscillations in Escherichia coli with spontaneous formation of two-stranded filaments in a three-dimensional stochastic reaction-diffusion model

We introduce a three-dimensional stochastic reaction-diffusion model to describe MinD/MinE dynamical structures in Escherichia coli. This model spontaneously generates pole-to-pole oscillations of the membrane-associated MinD proteins, MinE ring, as well as filaments of the membrane-associated MinD proteins. Experimental data suggest MinD filaments are two-stranded. In order to model them we assume that each membrane-associated MinD protein can form up to three bonds with adjacent membrane-associated MinD molecules and that MinE induced hydrolysis strongly depends on the number of bonds MinD has established.

Figure 1

Geometric shape used to model the bacterium E. coli: cylinder of length $H$ and radius $R$ with two hemispheres of radius $R$ at either end. The origin of the coordinate system $(x,y,z)$ is placed in the center of the bacterium. Two distances from the membrane, $d_{\mathit{min}}$ and $d_{\mathit{max}}$, are used to define the region near the membrane $(d<d_{\mathit{min}})$ and the region far from the membrane $(d>d_{\mathit{max}})$, respectively. For details see text. All parameters shown in the figure are scaled equally, except the parameter $d_{\mathit{min}}$.

Figure 2

Schematic representation of four stages in MinD/MinE proteins dynamics. (1.) MinD:ATP binds to the inner layer of the cytoplasmic membrane; (2.) MinE binds to the membrane-associated MinD:ATP; (3.) MinE induces ATP hydrolysis, releasing subsequently MinD:ADP, MinE, and phosphate into the cytoplasm; (4) MinD:ADP is converted back into MinD:ATP by nucleotide exchange.

Figure 3

(a) Orthogonal projection of the membrane-associated MinD proteins onto a plane parallel to the line connecting two poles of the bacterium. Each projected protein is represented with a dot. Five time frames are shown. They refer to times $0,\frac{1}{4}T,\frac{2}{4}T,\frac{3}{4}T,T$; where $T\approx 80\sec$ is the period of oscillation obtained with parameters specified in the text. (b) A portion of (a) is enlarged to clearly show two-stranded filaments.

Figure 4

(a) Histogram for the number of the membrane-associated MinD (MinE) proteins versus x-coordinate, for the same time frames as in Fig. 3. Size of the bin used is $0.08\mu \mathrm{m}$. (b) Time average of the number of the membrane-associated MinD (MinE) proteins over three periods.

Figure 5

Dependence of the oscillation period on the total number of MinD and MinE molecules, and the cell length. Points marked by ◻ are obtained with parameters used to generate Fig. 3. In (a) and (b) parameters are varied one at a time while keeping the others fixed. In (c), instead of keeping the number of MinD and MinE fixed, their concentrations are fixed at values used in Fig. 3 while cell length is varied.

Figure 6

Space-time plot of the membrane-associated MinD proteins in the filamentous cell $(L=15\mu \mathrm{m})$. Each MinD is represented with a dot. For the sake of clearness, we have reduced the number of dots by a factor of 100. Concentrations of MinD and MinE, plus all the remaining model parameters are fixed at values used to obtain results in Fig. 3.

Figure 7

Membrane-associated MinD proteins oscillations; only a single time frame is given. Parameter values are the same as those used in Fig. 3, except ${\sigma }_{Dd}=0.02\mu {\mathrm{m}}^3∕\sec$, ${\sigma }_D=0.0005\mu \mathrm{m}∕\sec$, and ${\sigma }_{\mathit{de}}^{\left(0\right)}=11.0{\sec }^{-1}$ [parameters ${\sigma }_{\mathit{de}}^{\left(1\right)}$, ${\sigma }_{\mathit{de}}^{\left(2\right)}$, ${\sigma }_{\mathit{de}}^{\left(3\right)}$ were modified to obey (12)]. Two- stranded MinD filaments are longer then those obtained in Fig. 3.