Feedback between motion and sensation provides nonlinear boost in run-and-tumble navigation
Fig 3
Environmental context, length scales, and receptor saturation.
(A-C) Exponential gradient. (A) Schematic of a gradient of methyl-aspartate C = C0 exp(−R/L0) with length scale L0 = 1000 μm and source concentration C0 = 10 mM. Contour lines show logarithmically spaced concentration levels in units of mM. Contour spacing illustrates constant L = 1/|∂R ln C| = L0. (B) The mean trajectory over 104 E. coli cells of the position R (in real units μm) as a function of time t (in s) when receptor saturation is taken into account. Initial values of τE are 0.1 (green), 1 (blue) or 3 (red). The shadings indicate standard deviations. The labels on the right axis show the concentration in mM at each position. (C) Corresponding time trajectories of the values of τE at mean positions. (D-F) Linear gradient. Similar to A-C but for C = C1 − a1R where the source concentration is C1 = 1 mM and decreases linearly at rate a1 = 0.0001 mM/μm with distance R from the source. Contour spacing decreases with distance from the source (at the top), illustrating decreasing L = 1/|∂R ln C| = C/a1 = C1/a1 − R. (G-I) Localized source. Similar to A-C but for a constant source concentration (C2 = 1 mM) within a ball of radius R0 = 100 μm and for R > R0, the concentration is C = C2R0/R (the steady state solution to the standard diffusion equation ∂tC = ∇2C without decay), decreasing with radial distance as 1/R away from the source. Contour spacing increases away from the source (at the origin), illustrating increasing L = 1/|∂R ln C| = R.