A multistranded polymer model explains MinDE dynamics in E. coli cell division

Abstract

In Escherichia coli, the location of the site for cell division is regulated by the action of the Min proteins. These proteins undergo a periodic pole-to-pole oscillation that involves polymerization and ATPase activity of MinD under the controlling influence of MinE. This oscillation suppresses division near the poles while permitting division at midcell. Here, we propose a multistranded polymer model for MinD and MinE dynamics that quantitatively agrees with the experimentally observed dynamics in wild-type cells and in several well-studied mutant phenotypes. The model also provides new explanations for several phenotypes that have never been addressed by previous modeling attempts. In doing so, the model bridges a theoretical gap between protein structure, biochemistry, and mutant phenotypes. Finally, the model emphasizes the importance of nonequilibrium polymer dynamics in cell function by demonstrating how behavior analogous to the dynamic instability of microtubules is used by E. coli to achieve a sufficiently rapid timescale in controlling division site selection.

FIGURE 1

Proposed dimer model. (A) MinE undergoes dimerization. The N-terminus anti-MinD domain is involved in inducing hydrolysis by MinD and the C-terminus is involved in dimerization and E-ring formation. Residues 45 and 49, involved in E-ring formation, are shown as light patches on the underside of the MinE dimer. (B) MinD cycles from the cytosol where it binds ATP (i), dimerizes (ii), attaches to the membrane (iii), recruits MinE (iv), hydrolyzes ATP, and is released from the membrane (v). MinE is proposed to undergo a conformation change in which the anti-MinD domain moves to the MinD dimerization face (v) simultaneously blocking polymerization of dimers and inducing MinD ATPase activity. (C) If MinD attaches at the poles, hydrolysis and release is assumed to be slower leading to an accumulation of dimers at the pole.

FIGURE 2

Progression of vesicle tubulation. (A) MinD binds to a slack vesicle, (B) begins to polymerize thereby pulling out any extra membrane, and (C) eventually stretches the membrane as the membrane/polymer tube grows. In panel B, the force F ≈ 0–3 pN is opposed mostly by membrane bending. In panel C, the force F ≈ 10 pN is opposed by the membrane tension (T), which is assumed to be in mechanical equilibrium with the osmotic pressure (π) induced by volume change.

FIGURE 3

E-ring formation and function. (A) E-ring nucleation by two pathways: sequential monomer binding or dimer binding. The monomer pathway is less preferred than the dimer pathway due to the extra intermediate step during which hydrolysis could lead to a loss of MinD subunits and MinE monomer at the tip. As a result, with a cytosolic dimerization dissociation constant of K_E = 0.6 μM, nucleation is unlikely below that concentration and more likely above it. Finally, once MinE is attached at the tip, MinD is prevented from binding by the anti-MinD domain halting further polymerization. (B) Top view of the MinD dimer structure proposed in Lutkenhaus and Sundaramoorthy (39) superimposed on the cartoon shape of MinD. ATP are in dark shading. The α-helices α-4 and α-7, known to influence the anti-MinD activity of MinE, are in green and yellow, respectively (51,52). Note how the α-4 and α-7 domains from opposite monomers come together when the dimer forms. The blue α-helix is present in the structure of the analogous ATPase NifH but is missing from MinD (39). We propose that the anti-MinD domain of MinE takes its place.

FIGURE 4

Approximate solution in the limit of rapid polymer growth. The exact solution differs only in the growth phase through which linear growth is replaced by an exponential approach to linear growth (dashed curve). The polymer at the lower pole is capped at t = 0 and is only seen disassembling under the influence of the E-ring (dark shading). The value t_nuc denotes the time at which the length of the lower polymer drops below the nucleation threshold (long dashes), meaning that the cytosolic MinD concentration is sufficient to nucleate the upper polymer. The value t_dis is the time at which the lower E-ring begins net disassembly. The value t_cap is the time at which the tip of the lower polymer crosses the capping threshold meaning the cytosolic MinE concentration is high enough to form an E-ring on the upper polymer. This process repeats with a period T = 2t_cap. Light shading represents MinD polymer without MinE and hence the region in which Z-ring formation is inhibited.

FIGURE 5

Numerical solution to stochastic implementation. (A) A set of traces from one run of the stochastic simulation. Blue curves represent MinD polymers tips; blue shading represents MinD polymers; red dashed curves denote the growing end of the E-rings; red shading represents E-rings. (B) A sequence of images showing approximately one half-period, generated from the traces in panel A. Each frame corresponds to a dashed line in panel A (from 57 to 94.5 in 7.5-s intervals). MinD polymer (blue circles, outlined), MinE ring (red, outlined), cytosolic MinD (blue, no outline), cytosolic MinE (red, no outline). For clarity, only half of the cytosolic MinD monomers, all cytosolic MinE dimers (but no monomers) and one out of every eight monomers in polymer form are shown. Note that in the model, polymer lengths and diffusively well-mixed cytosolic concentrations are tracked as scalar quantities; for visualization only, spatial distribution in the cytosol is by uniform random placement and the helical shape is prescribed (consistent with measurements from the images of (24)). Frame 1: A preexisting polymer is almost entirely disassembled (bottom). A new polymer (top) is growing. Note that cytosolic MinE dimer concentration is high, and as a result E-ring formation will occur soon on the growing polymer. Frame 2: An E-ring has formed; MinE dimer concentration is low and remains low until Frame 5; E-ring treadmilling is driven by monomer addition. Cytosolic MinD concentration is also low. Frame 3: Cytosolic MinD concentration rises as the upper polymer disassembles. Frame 4: The same trend continues. Frame 5: A MinD polymer has formed at the bottom and cytosolic MinE dimer concentration has begun to rise. Frame 6: Cytosolic MinE concentration has risen sufficiently to allow an E-ring to form on the lower polymer (equivalent to Frame 2, one half-period later).

FIGURE 6

Polymerization dynamics in three different types of theoretical cells. One cell with nonlinear nucleation of MinD and nonlinear capping by MinE (top), one with linear nucleation of MinD and nonlinear capping by MinE (middle), and one with nonlinear nucleation of MinD and linear capping by MinE (bottom). Notice the loss of regularity in the lower two panels.

FIGURE 7

The MinE^D45A/V49A mutant model. Each pair of solid (shaded) curves correspond to the positions of the ends of a single polymer through time. Arrows denote nucleation. Nucleation alternated from one pole to the other provided the preceding polymer began disassembly at the nonnucleating pole. Occasionally, disassembly began at the same pole as nucleation. Circles highlight stuttering.