Division accuracy in a stochastic model of Min oscillations in Escherichia coli

Abstract

Accurate cell division in Escherichia coli requires the Min proteins MinC, MinD, and MinE as well as the presence of nucleoids. MinD and MinE exhibit spatial oscillations, moving from pole to pole of the bacterium, resulting in an average MinD concentration that is low at the center of the cell and high at the poles. This concentration minimum is thought to signal the site of cell division. Deterministic models of the Min oscillations reproduce many observed features of the system, including the concentration minimum of MinD. However, there are only a few thousand Min proteins in a bacterium, so stochastic effects are likely to play an important role. Here, we show that Monte Carlo simulations with a large number of proteins agree well with the results from a deterministic treatment of the equations. The location of minimum local MinD concentration is too variable to account for cell division accuracy in wild-type, but is consistent with the accuracy of cell division in cells without nucleoids. This finding confirms the need to include additional mechanisms, such as reciprocal interactions with the cell division ring or positioning of the nucleoids, to explain wild-type accuracy.

Fig. 1.

Reactions and geometry of the stochastic Min oscillation model. (A) Reaction cycle. Cytosolic MinD in its ADP-bound form converts to an ATP-bound form with rate k₁. MinD-ATP binds to the membrane alone with rate k₂, and membrane-bound MinD (with or without MinE) catalyzes its own addition to the membrane at rate k₃. Cytosolic MinD binds membrane-bound MinD with rate k₄. Finally, the MinE/MinD complex dephosphorylates and dissociates into cytosolic MinE and MinD-ADP at rate k₅. (B) Snapshot of a simulation running inside a 4-μm-long, 0.5-μm-radius triangulated cylinder (transparent surface) after 3 min of simulated time. Each colored dot is a single molecule. Colors for each state are from A. There are 5,400 proteins and a MinD/MinE ratio of 4.0.

Fig. 2.

Validation of the stochastic simulations. (A) Qualitative agreement between models. MinE concentration along the long axis of the cell (vertical axis) is plotted over time (horizontal axis). MinE moves from pole to pole of the cell in the stochastic simulations with cylindrical geometry (Cyl). Similar patterns are seen in a stochastic simulation run in a box (Box) and in a deterministic solution in a box (Det). (B) Stochastic fluctuations in oscillation period. The instantaneous oscillation periods for the cylindrical (thin line), box (thick line), and deterministic models (gray line) are shown. The data are from A. (C) Quantitative agreement between models. Oscillation periods are shown (diamonds) along with the means (large bars) and standard errors (small bars) for three simulations: cylindrical (n = 10 different random number streams), box (n = 82), and deterministic box.

Fig. 3.

Stochastic effects on oscillation period. (A) Oscillations are disrupted by decreasing protein number. MinE waves are plotted for the deterministic case (∞) and for stochastic models with decreasing numbers of proteins (indicated on left). In all cases, the total MinD/MinE ratio is D/E = 4.0, and the reaction rates have been altered to match the deterministic case (see Methods). (B) Rapid transition from stable to unstable oscillations. The coefficient of variation of the period of oscillation was determined for D/E ranging from 2.7 to 4.0, and protein numbers ranging from 540 to 10,800. Stable oscillations produce a low coefficient of variation in the period. The deterministic model has no dynamic solution for D/E ≤ 2.6. (C) Dependence of oscillation period on MinD/MinE. Shown is the mean oscillation period for a range of D/E values (indicated on right) and protein number (horizontal axis). Error bars indicate standard deviation of the period. Only stable oscillations are shown (CV < 0.15 from B).

Fig. 4.

Midpoint determination accuracy. (A) Membrane-bound MinD averaged over 20 min of oscillations in a rectangular box with MinD/MinE = 4.0 and 5,400 proteins. Average MinD concentration (black line) can be approximated as a fourth-order polynomial (gray line). (B) Depth of MinD concentration minimum at true midpoint of cell, measured as central value divided by mean value. The central value was taken from the polynomial fit to the stochastic data. (C) Accuracy of MinD minimum. Interaction distances were approximated as a Gaussian blur of the concentration profile, and the position of minimum MinD was measured on each smoothed profile. Midpoint error (solid line) is the mean rms distance between the true midpoint and the MinD minimum, measured as a fraction of cell length. Experimentally determined errors (from ref. 1) for wild-type (dashed line) and anucleate cells (dotted line) are included for reference. (D) Distribution of errors. Cumulative probability distribution of the midpoint error is shown for 82 full simulations (black line) and for 1,000 mimicked data sets (gray line) generated from 5 of the 82 simulations. The interaction distance was set to 50 nm. (E) Comparison with experimental data. A total of 1,000 mimicked data sets were generated from five simulations for each value of the parameters shown. Shown is the rms distance between the true cell midpoint and the MinD minima in the mimicked data sets.