Why is the rod only one-dimensional?

Real E. coli cells are cylindrical with three-dimensional curvature of the membrane. A 1D model collapses the rod axis as the slow direction of travel for Min waves, which is the standard teaching reduction before full 2D/3D simulations.

What does the yellow dashed line mark?

It tracks the minimum of a time-averaged MinD field, interpreted as the prospective division site where MinC-dependent inhibition of FtsZ would be weakest in the full system. The instantaneous MinD minimum often sits away from the active pole, but the time average is what should settle near x ≈ 0.5.

Why must MinE diffuse faster than MinD?

Fast MinE spreading lets the inhibitor overtake and displace a pole cluster before rebuilding it elsewhere—similar to an activator–substrate or chasing inhibitor motif. If D_v is too small, oscillations may stall or freeze at one pole.

Where is MinC in this model?

MinC is omitted for clarity. Biologically MinC rides on MinD and blocks divisome assembly; the simulator marks the time-averaged MinD minimum as a proxy for the safe midcell zone, which is how many courses first explain the geometry.

MinD / MinE Oscillation (E. coli Rod)

1D reaction–diffusion: membrane MinD u, fast MinE v; pole-to-pole oscillation; division plane at time-averaged MinD minimum.

About this model

In rod-shaped bacteria such as Escherichia coli, the MinCDE system helps place the division septum at midcell. This simulator is a phase-reduced one-dimensional reaction–diffusion caricature on a rod-shaped domain with zero-flux ends (the poles), not a full biochemical Huang–Meir–Wingreen model. The variable u represents membrane-associated MinD and v represents MinE. A MinD attachment target alternates between poles, while faster MinE (D_v > D_u) trails that zone and removes MinD through a proportional u v term. This produces the observable pole-to-pole switching pattern: a MinD polar zone forms, MinE chases it, the zone collapses, and MinD reassembles toward the opposite pole. In full biology, MinC carried by MinD inhibits FtsZ ring formation at high time-averaged MinD. Here the yellow division marker tracks the minimum of an exponentially time-averaged u field, not the instantaneous trough, because cells respond to the time-averaged Min pattern. The scheme is explicit Euler on 160 points; it is qualitative and not fitted to wild-type oscillation periods in microns.

Who it's for: Students of biophysics, cell biology, or reaction–diffusion pattern formation learning how protein waves position cytokinesis in E. coli.

Key terms

Min system
MinD
MinE
Reaction–diffusion
Pole-to-pole oscillation
Division plane
E. coli
Zero-flux boundary

How it works

Phase-reduced MinD/MinE reaction–diffusion caricature on a one-dimensional E. coli rod: a MinD attachment zone alternates between poles, while faster MinE trails and removes MinD from the membrane. The division site is marked by the time-averaged MinD minimum.

Key equations

∂u/∂t = D_u u_xx + μ(U_pole(t,x)−u) − κuv · ∂v/∂t = D_v v_xx + σ(E_trail(t,x)−v)

Frequently asked questions

Why is the rod only one-dimensional?: Real E. coli cells are cylindrical with three-dimensional curvature of the membrane. A 1D model collapses the rod axis as the slow direction of travel for Min waves, which is the standard teaching reduction before full 2D/3D simulations.
What does the yellow dashed line mark?: It tracks the minimum of a time-averaged MinD field, interpreted as the prospective division site where MinC-dependent inhibition of FtsZ would be weakest in the full system. The instantaneous MinD minimum often sits away from the active pole, but the time average is what should settle near x ≈ 0.5.
Why must MinE diffuse faster than MinD?: Fast MinE spreading lets the inhibitor overtake and displace a pole cluster before rebuilding it elsewhere—similar to an activator–substrate or chasing inhibitor motif. If D_v is too small, oscillations may stall or freeze at one pole.
Where is MinC in this model?: MinC is omitted for clarity. Biologically MinC rides on MinD and blocks divisome assembly; the simulator marks the time-averaged MinD minimum as a proxy for the safe midcell zone, which is how many courses first explain the geometry.

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