Emergent evolutionary forces in spatial models of luminal growth and their application to the human gut microbiota

Abstract

The genetic composition of the gut microbiota is constantly reshaped by ecological and evolutionary forces. These strain-level dynamics are challenging to understand because they depend on complex spatial growth processes that take place within a host. Here we introduce a population genetic framework to predict how stochastic evolutionary forces emerge from simple models of microbial growth in spatially extended environments like the intestinal lumen. Our framework shows how fluid flow and longitudinal variation in growth rate combine to shape the frequencies of genetic variants in simulated fecal samples, yielding analytical expressions for the effective generation times, selection coefficients, and rates of genetic drift. We find that over longer timescales, the emergent evolutionary dynamics can often be captured by well-mixed models that lack explicit spatial structure, even when there is substantial spatial variation in species-level composition. By applying these results to the human colon, we find that continuous fluid flow and simple forms of wall growth alone are unlikely to create sufficient bottlenecks to allow large fluctuations in mutant frequencies within a host. We also find that the effective generation times may be significantly shorter than expected from traditional average growth rate estimates. Our results provide a starting point for quantifying genetic turnover in spatially extended settings like the gut microbiota and may be relevant for other microbial ecosystems where unidirectional fluid flow plays an important role.

Keywords: genetic drift; natural selection; separation of timescales; spatial structure.

Fig. 1.

Dynamics of a genetic variant in a spatially extended population in the intestinal lumen. (A) Schematic of spatial growth model. Nutrients enter the colon from the small intestine (on the left), and fecal samples are generated at the distal end (on the right). Intestinal wall contractions mix contents of lumen (Bottom Right), producing effective diffusion in the axial direction (26, 28). Continuous fluid flow through the colon generates longitudinal gradients in steady-state bacterial growth rates [$\lambda (x)$, orange line] and population densities [$\rho (x)$, shaded blue region]. Mutation events (red star) find new lineages that begin to grow and spread out in space [${\rho }_m(x,t)$, shaded red region]. After ${\tau }_{\text{mix}}$ generations, the spatial distribution of the mutant lineage approaches that of the wild-type population but with a smaller overall size. (B) Mutation frequencies observed in fecal samples over time. Two hypothetical trajectories are shown (red lines), illustrating the stochastic fluctuations that arise due to genetic drift.

Fig. 2.

Analytical solutions for a minimal model of bacterial growth. (A and B) Schematic of bacterial growth rates. A step-like growth response curve (A) leads to a sharp transition between high growth ($\lambda (x)\approx \lambda$) and no growth ($\lambda (x)\approx 0$) at a critical position $\ell$ (B). (C and D) Spatial profiles of population density, $\rho (x)$ (blue line), neutral fixation probability, u(x) (red line), and distribution of future common ancestors, $g(x)=u(x)\rho (x)$ (green shaded region), for examples of (C) diffusion-dominated growth ($4D\lambda /v^2\approx 10$) and (D) flow-dominated growth ($4D\lambda /v^2\approx 1.04$). In both cases, the dashed black lines indicate the boundary of the growth region in B. (E) Scaled effective growth rate, ${\lambda }_e/\lambda$, and (F) scaled effective population size, $(N_e/N)·(vL/2D)$, as a function of the washout parameter $\alpha \equiv 4D\lambda /v^2$. Black lines denote the analytical predictions in SI Appendix, section 2, while the red lines illustrate the asymptotic behavior in the $\alpha \gg 1$ and $\alpha -1\ll 1$ limits.

Fig. 3.

Dynamics of selected mutations. (A) Schematic trajectory of a beneficial mutation, which either establishes and grows exponentially at rate s_e (dashed line) or else goes extinct (light red line). (B) The spatially resolved fixation probability $w_s(x)$ of a beneficial mutation that originates at position x. Red lines denote the numerical solutions of Eq. 13 for the minimal model in Fig. 2 with a constant growth rate advantage ${\lambda }_m(x)=(1+s)\lambda (x)$ (SI Appendix, section 4). The dashed lines indicate the transition from $\lambda (x)\approx \lambda$ to $\lambda (x)\approx 0$. (Top) A population in the diffusion-dominated growth regime ($4D\lambda /v^2\approx 1.9$) and (Bottom) a population in the flow-dominated growth regime ($4D\lambda /v^2\approx 1.04$). In both cases, fixation profiles were calculated for $s={10}^{-3},\hspace{0.17em}{10}^{-2}$, and ${10}^{-1}$. (C) The effective fixation probability $p_{\text{fix},e}$ obtained by averaging $w_s(x)$, the range of initial positions. (Top and Bottom) The same populations as in B. Dashed lines indicate the separation-of-timescales prediction, $p_{\text{fix},e}=2s_e$. Deviations from this prediction emerge in the flow-dominated growth regime, due to the enhanced fixation probability of mutations that originate at larger x values (illustrated in B).

Fig. 4.

Applications to bacteria in the human colon. (A) Estimates of the flow velocity along the first 80 cm of the human colon (28). Colors indicate the positions of different colonic segments. (B and C) Steady-state growth rate and density profiles estimated for two representative species using the results of ref. . Growth rate profiles were extracted from the figures in ref. , and the corresponding density profiles were calculated from Eq. 2 using the methods described in SI Appendix, section 5. (D and E) Predictions for the neutral fixation profile u(x) (D) and distribution of common ancestors g(x) (E) obtained from the estimated parameters in A–C (SI Appendix, section 5). In both cases, the predicted ancestral distributions contain a large contribution from regions with low growth rates. (F–H) Predictions for the effective growth rate, effective population size, and the net fixation probability of beneficial mutations (SI Appendix, section 5).

Fig. 5.

Mutation frequencies within a Bacteroides massiliensis population from a single human subject. (Top) Relative abundance of B. massiliensis and (Bottom) the frequencies of single nucleotide variants in this species inferred from longitudinally sequenced fecal samples from a previous study (15) (SI Appendix, section 5). Blue trajectories highlight a cluster of mutations that experienced a large shift in frequency during the study ($\mathrm{\Delta }f\approx 50\%$), despite relatively small changes in the overall relative abundance of B. massiliensis (approximately twofold; Top). The timescale of this genetic shift ($\mathrm{\Delta }t\sim 25$ d) is many orders of magnitude faster than our predicted timescale of genetic drift ($\mathrm{\Delta }t\sim 1/\mathrm{\Lambda }$; Fig. 4 and Eq. 16), even after accounting for the observed fluctuations in overall relative abundance.