Table 1.
Symbol definitions.
Fig 1.
Different dynamical regimes of run-and-tumble gradient ascent.
(A) Drift speed VD of simulated E. coli cells swimming in static exponential gradients as a function of τE and τD0. Green, blue, and red: τD0 = 1 and τE = 0.1, 1, 3, respectively. Orange: τE = 0.1 and τD0 = 0.1 (dashed line: guides to the eye). White/black: sampling of a wild type population [22] near the bottom/top of a linear gradient. (B) Classical (red) vs. rapid climbing (green) trajectories. x = X/(v0tM) vs. time τ = t/tM for cells in the positive-feedback- (green) and negative-feedback-dominated (red) regime (thin: 5 samples; thick: mean over 104 samples). (C) Marginal probability distribution of the internal variable at steady state ; solid: numerical solution of Eq (5); dashed: sampled distribution from agent-based simulation; colors: same parameter values as in A. Inset: zoomed view with second order analytical approximations (Methods Eq (35)) in black. r0 = 0.8 and DT/DR = 37 in all simulations. (D) Comparison of different methods to calculate VD as a function of τE keeping τD0 = 1 fixed. Solid: numerical integration of Eqs (5) and (6); dashed: agent-based model simulations; dash-dot: MFT (Methods Eq (43)). Details in Methods.
Fig 2.
Non-normal dynamics enables large asymmetric transients in internal state.
(A) Phase space diagram of Eq (8) when the positive feedback dominates, τE = 0.1. White: streamlines without noise; magenta: the r-nullcline where dr/dτ = 0; black: the two v-nullclines where dv/dτ = 0. Heat map: noise magnitude of dv/dτ ( in Eq (8)). (B) Two example trajectories starting in positive (cyan) or negative (magenta) direction. Each trajectory starts from black and lasts over the same time period of τ = 10. See also S1 Movie. (C,D) Same as A,B except in the negative-feedback-dominated regime, τE = 3. When the positive feedback dominates (τE = 0.1, A), the streamlines (white) are highly asymmetric around the fixed point. They tend to bring the system transiently towards r = 1 and v = 1—a result of both non-normal dynamics (non-orthogonal eigenvectors near the fixed point) and nonlinear positive feedback (growth towards v = 1 away from the fixed point)—before eventually falling back to the fixed point. High noise near the fixed point causes the system to quickly move away from it (magenta in B). Low noise in the upper right corner (r = 1 and v = 1) facilitates longer runs in the correct direction (cyan in B). Taken together, these effects result in a fast “ratchet-like” gradient climbing behavior. In contrast, when the negative feedback dominates (τE = 0.1, C) the streamlines all point back directly to the fixed point and small deviations do not grow (cyan and magenta in D). Details in Methods.
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.