Persistence of direction increases the drift velocity of run and tumble chemotaxis

Escherichia coli is a motile bacterium that moves up a chemoattractant gradient by performing a biased random walk composed of alternating runs and tumbles. Previous models of run and tumble chemotaxis neglect one or more features of the motion, namely (i) a cell cannot directly detect a chemoattractant gradient but rather makes temporal comparisons of chemoattractant concentration, (ii) rather than being entirely random, tumbles exhibit persistence of direction, meaning that the new direction after a tumble is more likely to be in the forward hemisphere, and (iii) rotational Brownian motion makes it impossible for an E. coli cell to swim in a straight line during a run. This paper presents an analytic calculation of the chemotactic drift velocity taking account of (i), (ii) and (iii), for weak chemotaxis. The analytic results are verified by Monte Carlo simulation. The results reveal a synergy between temporal comparisons and persistence that enhances the drift velocity, while rotational Brownian motion reduces the drift velocity.

💡 Research Summary

The paper tackles a long‑standing gap in the theoretical description of Escherichia coli chemotaxis by constructing a model that simultaneously incorporates three biologically realistic features that previous run‑and‑tumble frameworks have typically omitted. First, E. coli does not sense spatial gradients directly; instead it makes temporal comparisons of chemoattractant concentration, evaluating the change Δc over a short memory interval Δt and adjusting its tumble frequency accordingly. Second, tumbles are not completely random reorientations; experimental observations show a forward bias, meaning the post‑tumble direction tends to lie in the forward hemisphere. This “persistence of direction” is quantified by a parameter α (0 ≤ α ≤ 1), with α = 0 corresponding to isotropic reorientation and α → 1 to almost straight continuation. Third, during a run the cell experiences rotational Brownian motion, characterized by a rotational diffusion coefficient D_r, which prevents perfectly straight swimming.

The authors formulate a stochastic master equation that couples the run‑time distribution ψ(τ) = λ e^{‑λτ} (λ is the baseline tumble rate) with a directional kernel f(θ) = (1 + α cosθ)/2π for the new heading after a tumble, and with an angular diffusion term ⟨Δθ²⟩ = 2D_r τ for rotational Brownian motion. Temporal comparison enters through a modulation of the tumble rate: λ(t) = λ₀