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The science
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 kBT 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.