Symbol definitions.

<p>Symbol definitions.</p

Non-normal dynamics enables large asymmetric transients in internal state.

<p>(A) Phase space diagram of <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005429#pcbi.1005429.e031" target="_blank">Eq (8)</a> when the positive feedback dominates, <i>τ</i>_<i>E</i> = 0.1. White: streamlines without noise; magenta: the <i>r</i>-nullcline where <i>dr</i>/<i>dτ</i> = 0; black: the two <i>v</i>-nullclines where <i>dv</i>/<i>dτ</i> = 0. Heat map: noise magnitude of <i>dv</i>/<i>dτ</i> ( in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005429#pcbi.1005429.e031" target="_blank">Eq (8)</a>). (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 <i>τ</i> = 10. See also <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005429#pcbi.1005429.s006" target="_blank">S1 Movie</a>. (C,D) Same as A,B except in the negative-feedback-dominated regime, <i>τ</i>_<i>E</i> = 3. When the positive feedback dominates (<i>τ</i>_<i>E</i> = 0.1, A), the streamlines (white) are highly asymmetric around the fixed point. They tend to bring the system transiently towards <i>r</i> = 1 and <i>v</i> = 1—a result of both non-normal dynamics (non-orthogonal eigenvectors near the fixed point) and nonlinear positive feedback (growth towards <i>v</i> = 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 (<i>r</i> = 1 and <i>v</i> = 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 (<i>τ</i>_<i>E</i> = 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.</p

Different dynamical regimes of run-and-tumble gradient ascent.

<p>(A) Drift speed <i>V</i>_<i>D</i> of simulated <i>E. coli</i> cells swimming in static exponential gradients as a function of <i>τ</i>_<i>E</i> and <i>τ</i>_<i>D</i>0. Green, blue, and red: <i>τ</i>_<i>D</i>0 = 1 and <i>τ</i>_<i>E</i> = 0.1, 1, 3, respectively. Orange: <i>τ</i>_<i>E</i> = 0.1 and <i>τ</i>_<i>D</i>0 = 0.1 (dashed line: guides to the eye). White/black: sampling of a wild type population [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005429#pcbi.1005429.ref022" target="_blank">22</a>] near the bottom/top of a linear gradient. (B) Classical (red) vs. rapid climbing (green) trajectories. <i>x</i> = <i>X</i>/(<i>v</i>₀<i>t</i>_<i>M</i>) vs. time <i>τ</i> = <i>t</i>/<i>t</i>_<i>M</i> 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 <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005429#pcbi.1005429.e013" target="_blank">Eq (5)</a>; dashed: sampled distribution from agent-based simulation; colors: same parameter values as in A. Inset: zoomed view with second order analytical approximations (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005429#sec009" target="_blank">Methods</a> <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005429#pcbi.1005429.e097" target="_blank">Eq (35)</a>) in black. <i>r</i>₀ = 0.8 and <i>D</i>_<i>T</i>/<i>D</i>_<i>R</i> = 37 in all simulations. (D) Comparison of different methods to calculate <i>V</i>_<i>D</i> as a function of <i>τ</i>_<i>E</i> keeping <i>τ</i>_<i>D</i>0 = 1 fixed. Solid: numerical integration of Eqs (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005429#pcbi.1005429.e013" target="_blank">5</a>) and (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005429#pcbi.1005429.e015" target="_blank">6</a>); dashed: agent-based model simulations; dash-dot: MFT (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005429#sec009" target="_blank">Methods</a> <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005429#pcbi.1005429.e108" target="_blank">Eq (43)</a>). Details in Methods.</p

Environmental context, length scales, and receptor saturation.

<p>(A-C) Exponential gradient. (A) Schematic of a gradient of methyl-aspartate <i>C</i> = <i>C</i>₀ exp(−<i>R</i>/<i>L</i>₀) with length scale <i>L</i>₀ = 1000 <i>μm</i> and source concentration <i>C</i>₀ = 10 <i>mM</i>. Contour lines show logarithmically spaced concentration levels in units of <i>mM</i>. Contour spacing illustrates constant <i>L</i> = 1/|∂_<i>R</i> ln <i>C</i>| = <i>L</i>₀. (B) The mean trajectory over 10⁴ <i>E. coli</i> cells of the position <i>R</i> (in real units <i>μm</i>) as a function of time <i>t</i> (in <i>s</i>) when receptor saturation is taken into account. Initial values of <i>τ</i>_<i>E</i> are 0.1 (green), 1 (blue) or 3 (red). The shadings indicate standard deviations. The labels on the right axis show the concentration in <i>mM</i> at each position. (C) Corresponding time trajectories of the values of <i>τ</i>_<i>E</i> at mean positions. (D-F) Linear gradient. Similar to A-C but for <i>C</i> = <i>C</i>₁ − <i>a</i>₁<i>R</i> where the source concentration is <i>C</i>₁ = 1 <i>mM</i> and decreases linearly at rate <i>a</i>₁ = 0.0001 <i>mM</i>/<i>μm</i> with distance <i>R</i> from the source. Contour spacing decreases with distance from the source (at the top), illustrating decreasing <i>L</i> = 1/|∂_<i>R</i> ln <i>C</i>| = <i>C</i>/<i>a</i>₁ = <i>C</i>₁/<i>a</i>₁ − <i>R</i>. (G-I) Localized source. Similar to A-C but for a constant source concentration (<i>C</i>₂ = 1 <i>mM</i>) within a ball of radius <i>R</i>₀ = 100 <i>μm</i> and for <i>R</i> > <i>R</i>₀, the concentration is <i>C</i> = <i>C</i>₂<i>R</i>₀/<i>R</i> (the steady state solution to the standard diffusion equation ∂_<i>t</i><i>C</i> = ∇²<i>C</i> without decay), decreasing with radial distance as 1/<i>R</i> away from the source. Contour spacing increases away from the source (at the origin), illustrating increasing <i>L</i> = 1/|∂_<i>R</i> ln <i>C</i>| = <i>R</i>.</p

Non‐genetic diversity modulates population performance

Abstract Biological functions are typically performed by groups of cells that express predominantly the same genes, yet display a continuum of phenotypes. While it is known how one genotype can generate such non‐genetic diversity, it remains unclear how different phenotypes contribute to the performance of biological function at the population level. We developed a microfluidic device to simultaneously measure the phenotype and chemotactic performance of tens of thousands of individual, freely swimming Escherichia coli as they climbed a gradient of attractant. We discovered that spatial structure spontaneously emerged from initially well‐mixed wild‐type populations due to non‐genetic diversity. By manipulating the expression of key chemotaxis proteins, we established a causal relationship between protein expression, non‐genetic diversity, and performance that was theoretically predicted. This approach generated a complete phenotype‐to‐performance map, in which we found a nonlinear regime. We used this map to demonstrate how changing the shape of a phenotypic distribution can have as large of an effect on collective performance as changing the mean phenotype, suggesting that selection could act on both during the process of adaptation

Spatial Self-Organization Resolves Conflicts Between Individuality and Collective Migration

Files containing data for each figure of the paper in MATLAB .fig file format from which the data points can be extracted. <br