Bacterial Chemotaxis 2D

How a bacterium climbs a hill it cannot see

E. coli is about 2 µm long and swims at roughly 20 µm/s, but it cannot compare concentrations between its head and tail (the difference is far below the noise). Instead it compares now with the recent past: a run of about one second is long enough to notice whether the ligand level has risen or fallen. Straight runs alternate with short tumbles (about 0.15 s) when the flagellar bundle flies apart and the cell reorients by roughly 68°. Lowering the tumble rate when things get better, and keeping it when they do not, turns an unbiased random walk into a biased one.

Receptors: a cooperative MWC cluster

A cluster of N receptors switches together between an active and an inactive state. The free-energy difference is F = N [ α(m0 − m) + ln((1 + L/KI)/(1 + L/KA)) ] and the probability of being active is a = 1/(1 + eF). Here L is the ligand concentration, KI = 18.2 µM and KA = 3000 µM are the dissociation constants of the inactive and active states, m is the methylation level and α = 1.7 k_BT per methyl group. Attractant binds the inactive state more tightly, so more ligand lowers a. Cooperativity (N) amplifies a small fractional change in ligand into a large change in activity.

Methylation feedback gives perfect adaptation

Two enzymes adjust m slowly: dm/dt = kR(1 − a) − kB a. The only steady state is a* = kR/(kR + kB), which does not contain L at all. After a step in ligand the activity jumps, then relaxes back to a* with the time constant τa = 1/[(kR + kB) N α a*(1 − a*)] found by linearising around the steady state. This memory of the last few seconds is what the cell compares against.

From activity to tumbling

Activity sets the motor's clockwise probability through a steep Hill function p = aH/(aH + KH) (H ≈ 10, Cluzel 2000). With mean tumble time τt the run→tumble rate is p/[(1 − p) τt], so the steady tumble fraction equals p. Receptor activity therefore changes the tumble rate exponentially fast for small changes of a.

Random-walk diffusion (an independent check)

In a uniform field the cell is a persistent random walker. Runs have length ℓ, successive directions have mean cosine αt = ⟨cos θ⟩ and the diffusion constant in d dimensions is D = [⟨ℓ2⟩ + 2⟨ℓ⟩2 αt/(1 − αt)] / (2 d T) where T is the mean run-plus-tumble time. The page evaluates this expression with the exact discrete-time (geometric) run and tumble statistics of the stepping code and compares it with the mean-square displacement of 3000 simulated cells.

Three in-page numerical checks

1. Perfect adaptation. A deterministic single-cell integration at L = 0.1, 10 and 1000 µM must end at a* whatever L is. 2. Adaptation time. The integrated e-folding time after a 5 % step must match the linear-theory τa. 3. Diffusion. The measured mean-square displacement must match the formula above within four standard errors.

Honest simplifications

The ligand field is static (no consumption or diffusion in time); receptor noise is a simple slow Gaussian process; the motor responds instantly to activity (the real CheY-P delay is about 0.1 s); the tumble angle is drawn from a truncated normal distribution rather than the measured skewed one; cells do not interact, do not feel Brownian rotation (which would add a few degrees of drift per run) and the arena has reflecting walls. Cell and flagellum sizes are drawn larger than life. Parameter values follow the textbook ranges (Berg 2004; Tu, Shimizu & Berg 2008; Cluzel et al. 2000) but are illustrative.

• Start with Wild type, exponential gradient: watch the green centroid line in the second graph move right, the tumbling fraction fall below the baseline and the Up-gradient index (mean cosine of the swimming direction with the gradient) rise to about 0.3.
• Switch to Step stimulus (uniform field), press Pulse x10: activity dives, tumbling stops, then methylation brings it back to a* in about 6 s. Press Clear step for the opposite transient.
• Choose Slow adaptation (40 s) and compare the Up-gradient index (about 0.29 for wild type, about 0.12 here): cells with a sluggish memory compare against stale information and climb worse.
• Reduce the number of receptors N to 1: the gain drops and so does the index (about 0.12); more receptors help, with diminishing returns.
• Set the mean tumble angle to 10 degrees (tumble-blind): tumbling no longer randomises direction, so the gradient bias disappears (index near 0).
• Try the Capillary assay: the cloud collects around the tip as in Adler's 1966 experiment.