Motile curved bacteria are Pareto-optimal

Abstract

Curved rods are a ubiquitous bacterial phenotype, but the fundamental question of why they are shaped this way remains unanswered. Through in silico experiments, we assessed freely swimming straight- and curved-rod bacteria of a wide diversity of equal-volume shapes parameterized by elongation and curvature, and predicted their performances in tasks likely to strongly influence overall fitness. Performance trade-offs between these tasks lead to a variety of shapes that are Pareto-optimal, including coccoids, all straight rods, and a range of curvatures. Comparison with an extensive morphological survey of motile curved-rod bacteria indicates that the vast majority of species fall within the Pareto-optimal region of morphospace. This result is consistent with evolutionary trade-offs between just three tasks: efficient swimming, chemotaxis, and low cell construction cost. We thus reveal the underlying selective pressures driving morphological diversity in a widespread component of microbial ecosystems.

Keywords: evolution; morphology; motility; shape; swimming.

Fig. 1.

Survey of extant curved-rod morphologies, with geometric parameters and definitions of dimensionless shape parameters $\mathrm{ℒ}$ and $\mathrm{𝒦}$ (Inset). The circles represent species median shapes based on segmented images, with circle radius proportional to equivalent spherical diameter (ESD) when scale data available (filled) or overall median ESD when not (open). The smallest ellipse containing 50% of all species is shown in red, and overall median species shape (and ESD) is depicted by a red filled circle, highlighted by a triangle. Silhouetted individuals (green), rescaled to constant volume, are shown for selected species (green filled circles).

Fig. 2.

(A) Simulated 2D morphospace of equal-volume body shapes. Empty region at upper left consists of nonphysical self-intersecting shapes. (B) Example body + flagellum with simulated swimming trajectory traced by the body midpoint, which appears to lack any symmetry viewed off-axis but reveals long-range rotational symmetry viewed axially (Inset). (C) The random Brownian rotation that would be superimposed onto the swimming trajectory can be quantified by three anisotropic rotational diffusivities, depicted as circles with diameter proportional to diffusivity. These rotations occur around principal axes (light gray lines) passing through the center of diffusion (black circle) (30). The largest of these (dashed) corresponds to rotations around an axis close to the flagellar axis, but it is the other two (solid) that determine how long the cell can maintain its course. (D) Comparison of optimal flagellar shapes for a sphere ($\mathrm{ℒ}=1$, $\mathcal{K}=0$) and highly elongated curved rod ($\mathrm{ℒ}=10$, $\mathcal{K}=0.5$), for Swimming Efficiency (red) and Chemotactic SNR (blue). Example second-order triangular surface meshes are shown in B–D; the flagella were similarly fully meshed (SI Appendix, section 5).

Fig. 3.

Performance landscapes for putative tasks critical to curved-rod bacteria: Swimming Efficiency ${\mathrm{\Psi }}_{swim}^n$ (A), Chemotactic SNR ${\mathrm{\Psi }}_{chemo}^n$ (B), and body shape Construction Ease ${\mathrm{\Psi }}_{constr}^n$ (C). In each case, performance (pseudocolor and contours) is normalized relative to that of a spherical body, i.e., ${\mathrm{\Psi }}^n=\mathrm{\Psi }/{\mathrm{\Psi }}_{\left(\mathrm{ℒ}=1,\mathcal{K}=0\right)}$, and is thus dimensionless and scale-invariant (SI Appendix, section 10.2). In addition to selected shapes for reference (gray), performance maxima within our morphospace are shown (black); in A, these include both the global and a local maximum (see text).

Fig. 4.

Ratio of performance of curved ($\mathcal{K}=0.14$, i.e., the species median) to straight ($\mathcal{K}=0$) rods at different tasks, calculated by evaluating the interpolants for each task (Fig. 3 and SI Appendix, Figs. S10 and S12 and section 10.1) along these transects. Some tasks are improved by curvature (solid lines), while others become worse (dashed lines).

Fig. 5.

Pareto-optimal curved-rod morphologies. Colored regions represent the set of shapes that are Pareto-optimal resulting from trade-offs between Swimming Efficiency, Chemotactic SNR, and Construction Ease; white region represents suboptimal shapes; and dots represent observed species medians. Selected simulated morphologies are plotted along the main optimal/suboptimal Pareto front as well as at the centroid of the disconnected Pareto-optimal “island.” RGB color values were assigned by normalizing performances in each task within the optimal region between 0 and 1. Color thus signifies relative, not absolute, trade-offs between tasks as indicated by the color triangle (Inset). Not all colors in the triangle are realized because not all possible trade-offs are realized (e.g., the shapes that excel at Construction Ease also excel at Swimming Efficiency).