Pattern formation in Escherichia coli: a model for the pole-to-pole oscillations of Min proteins and the localization of the division site

Abstract

Proper cell division requires an accurate definition of the division plane. In bacteria, this plane is determined by a polymeric ring of the FtsZ protein. The site of Z ring assembly in turn is controlled by the Min system, which suppresses FtsZ polymerization at noncentral membrane sites. The Min proteins in Escherichia coli undergo a highly dynamic localization cycle, during which they oscillate between the membrane of both cell halves. By using computer simulations we show that Min protein dynamics can be described accurately by using the following assumptions: (i) the MinD ATPase self-assembles on the membrane and recruits both MinC, an inhibitor of Z ring formation, and MinE, a protein required for MinC/MinD oscillation, (ii) a local accumulation of MinE is generated by a pattern formation reaction that is based on local self-enhancement and a long range antagonistic effect, and (iii) it displaces MinD from the membrane causing its own local destabilization and shift toward higher MinD concentrations. This local destabilization results in a wave of high MinE concentration traveling from the cell center to a pole, where it disappears. MinD reassembles on the membrane of the other cell half and attracts a new accumulation of MinE, causing a wave-like disassembly of MinD again. The result is a pole-to-pole oscillation of MinC/D. On time average, MinC concentration is highest at the poles, forcing FtsZ assembly to the center. The mechanism is self-organizing and does not require any other hypothetical topological determinant.

Figure 1

Components of the center-finding system in E. coli. MinC/D, MinE, and FtsZ are assumed to be pattern-forming systems that assemble in a self-enhancing way on the membrane. In this simulation, the elements are introduced one at a time to show the interplay of the subsystems. Shown is a one-dimensional simulation of patterning at the membrane along the long axis of the cell. Local concentrations are plotted as function of time; concentrations are indicated by the densities of pixels. (A) FtsZ (blue) alone can make a pattern, but the location of the maximum need not be central (σ_d = 0; σ_e = 0; μ_DE = 0). (B) MinD precursor production switched on (σ_d = 0.0035). MinD (green) on its own does not make a pattern but suppresses FtsZ patterning. (C) MinE precursor production switched on (σ_e = 0.002). MinE (red) on its own would make a stable pattern. (D) However, because MinE removes MinD from the membrane (μ_DE = 0.0004) and MinE association depends on MinD, the MinE maximum destabilizes itself and shifts toward a region of higher MinD concentration. Shortly before the MinE wave reaches the pole, MinD and MinE concentrations collapse. On its way, the MinE wave removes MinD from the membrane. Meanwhile, a new plateau of membrane-bound MinD is rising in the other part of the cell. A new high MinE concentration is triggered at its flank, causing this peak to disappear also, leading to a polar MinD oscillation in counter phase. Because of the low MinC/D level at the center, the FtsZ signal for septum formation appears there. (E) FtsZ remains in place there even after switching off of MinD (σ_d = 0). MinE disappears from the membrane, although the precursor is still produced. Eight hundred iterations are calculated between each pixel row; 80,000 time steps (iterations) are required for one full cycle. The total region has been subdivided into 15 spatial units. Assuming a length of E. coli of 3 μm and a full cycle of 50 s, the spatial unit size equals ≈0.15 μm, and one iteration corresponds to 0.6 ms.

Figure 2

Center finding during growth and division. (A) Only to demonstrate the correct center detection, a unilateral enlargement of the field is assumed. For that, the rightmost spatial element is doubled (after each 100,000 iterations = 50 pixel lines). Both daughter elements initially have identical concentrations. The FtsZ signal remains at the actual central position. (B) After separation of the large field into two parts, traveling MinE waves and the pole-to-pole oscillation of MinD are re-established quickly (shown is the left half). (C) The FtsZ ring becomes relocalized rapidly to the new center. For animated simulations, see the supporting information.

Figure 3

Oscillation in counterphase in long extended filaments. (A) In a cell surpassing the critical size (≈23 space units with the parameters chosen), a transition occurs from one to two zones in which MinE sweeps back and forth. Localization of the FtsZ ring follows the corresponding change in MinD distribution. (B and C) In longer filaments, several sites of MinE “sweeping,” and thus several potential division sites emerge. MinD oscillates in counter phase. Each pixel row corresponds to the distribution after 800 iterations; 32 (B) and 42 (C) space units are used. For animated simulations, see the supporting information.

Figure 4

Inverse relationship of the MinD and MinE concentrations on oscillation frequency. (A) “Normal” pattern. (B) An increase of the MinE precursor production (σ_e from 0.002 to 0.004) leads to a higher oscillation frequency, because less time is required to remove MinD. (C) Conversely, a decrease (σ_e = 0.001) leads to a lower frequency (note that the MinE wave does not have to reach a pole before MinD can trigger at the opposite pole). (D) The lowering of MinD precursor synthesis (σ_d from 0.0035 to 0.002) leads to more rapid oscillations, because less MinD has to be removed from the membrane. All simulations start with identical initial situations.

Figure 5

Simulation on a cylinder and the problem of ring formation. (A) Assuming a more realistic cylindrical geometry for the bacterium, a simple activator-depletion mechanism can lead to unpredictable patterns. Several maxima may emerge, preferentially at opposite positions of the cylinder. The simulation corresponds to the static MinE pattern formation as shown in Fig. 1C. (B–F) In the mechanism proposed, the diffusion of MinE leads to a synchronization of the wave and to ring-shaped bands. Shown are the MinD (green) and MinE (red) distributions in one full MinD cycle. (G–I) Patterning of the FtsZ ring. (G) The FtsZ ring also would decay into individual patches. (H) By a saturation of the F autocatalysis (κ_F = 0.2), this decay can be avoided. Nevertheless, the position of the ring(s) would be unpredictable. (I) The elaborate mechanism proposed is able to generate one central band as required. Simulations are made of the surface of a cylinder; the diffusion within the cylinder is not considered (cell length = 19 space elements, circumference = 9 space elements). Except for a diffusion term generalized for two dimensions, the same equations and parameters as in Figs. 1–4 are used. For animated simulations, see the supporting information.