Selective sweeps in growing microbial colonies

Abstract

Evolutionary experiments with microbes are a powerful tool to study mutations and natural selection. These experiments, however, are often limited to the well-mixed environments of a test tube or a chemostat. Since spatial organization can significantly affect evolutionary dynamics, the need is growing for evolutionary experiments in spatially structured environments. The surface of a Petri dish provides such an environment, but a more detailed understanding of microbial growth on Petri dishes is necessary to interpret such experiments. We formulate a simple deterministic reaction-diffusion model, which successfully predicts the spatial patterns created by two competing species during colony expansion. We also derive the shape of these patterns analytically without relying on microscopic details of the model. In particular, we find that the relative fitness of two microbial strains can be estimated from the logarithmic spirals created by selective sweeps. The theory is tested with strains of the budding yeast Saccharomyces cerevisiae for spatial competitions with different initial conditions and for a range of relative fitnesses. The reaction-diffusion model also connects the microscopic parameters like growth rates and diffusion constants with macroscopic spatial patterns and predicts the relationship between fitness in liquid cultures and on Petri dishes, which we confirmed experimentally. Spatial sector patterns therefore provide an alternative fitness assay to the commonly used liquid culture fitness assays.

Figure 1

(Colour online) Comparison of spatial segregation during a range expansion of Baker’s yeast, Saccharomyces cerevisiae with (a) equal and (b) different growth rates of the two competing strains. The Petri dishes were inoculated with a well-mixed population occupying a narrow horizontal linear region at the bottom of the images from which the sectors appear. As the populations expand, they segregate into well defined domains. Different colours label different genotypes. In (a), the two strains (yellow and blue) have the same fitness and the demixing is driven primarily by number fluctuations (genetic drift) [15, 6]. It is likely that the small variations with horizontal position in boundary slopes are related to undulations of linear fronts, which are hard to suppress when the front is very long [26]. In (b), the sector is formed by the fitter strain (black), and the sector expansion is caused by the difference in growth rates of the strains, or, in other words, by natural selection. In both (a) and (b), the scale bars are 500 µm.

Figure 2

(Colour online) Colony edge at single cell resolution (mature yeast cells are 5 µm in diameter). (a), (b), (c), and (d) are successive images (at two hour intervals) of the same region near the edge of a growing S. cerevisiae colony inoculated with a razor blade. Note the formation of a green (light gray) sector on the lower right. The two strains have approximately the same fitness in this experiment.

Figure 3

Schematic representation of four different behaviors of equation (2) in phase space. (a) ε₁ > 0 and ε₂ > 0. (b) ε₁ > 0 and ε₂ < 0. (c) ε₁ < 0 and ε₂ > 0. (d) ε₁ < 0 and ε₂ < 0. The arrows represent the direction of trajectories in the phase space, and the dots represent the fixed points.

Figure 4

(Colour online) Solutions of equation (2) plotted in phase space (c₁, c₂) for negative ε₁ and positive ε₂. The arrows indicate the direction of time. For the initial conditions, we chose positive c₁(0) and c₂(0) close to zero with each trajectory having a different value of c₁(0)/c₂(0). In this plot, g₁ = 1.5, g₂ = 1, ε₁ = −0.1, ε₂ = 0.6. Note that, initially, all trajectories bend toward the c₁-axis (increasing c₁/c₂), but the first strain is nevertheless eliminated at long times, as the system approaches the stable fixed point (0, 1).

Figure 5

(Colour online) Both (a) and (b) show the solution of equation (4) plotted for g₁ = 11.8, g₂ = 10, ε₁ = ε₂ = 0, α₁ = α₂ = 1, D₀₁ = 1.18 · 10⁻³ and D₀₂ = 10⁻³; note that we chose g₁/g₂ = D₀₁/D₀₂ = 1.18 to facilitate comparison with figure 1b (see discussion in text). In these units, the habitat is a 1 × 2 rectangle and is initially empty; only the bottom sixty percent of the habitat is shown because the top part remains empty throughout the expansion. The origin of the expansion is a line at the bottom edge of the images, where we impose the boundary conditions that c₂(t, x, 0) = 1 and c₁(t, x, 0) = 0 except in the width 2⁻⁸ region near the center of this boundary, where c₂(t, x, 0) = 0 and c₁(t, x, 0) = 1. No-flux boundary conditions are imposed along all other edges. Equation (4) is solved on a square grid of 256 × 512 points. (a) The concentration of the first strain is shown in red (dark gray) and of the second strain in green (light gray). The maximal colour intensity corresponds to the concentration of 1, and the lowest to the concentration of 0. This colour scheme is chosen to facilitate the comparison with the experimental data shown in figure 1. (b) The same solution as in (a), but only the concentration of the first strain is shown to highlight its establishment as a sector early in the expansion. Brighter regions correspond to higher concentration of the first (red) strain.

Figure 6

(Colour online) The solution of equation (4) plotted for g₁ = 12.5, g₂ = 10, ε₁ = ε₂ = 0, α₁ = α₂ = 1, and D₀₁ = D₀₂ = 0.01. The habitat is a 10 × 10 square inoculated with a circular drop of radius 2 shown in gray. We assign c₁ and c₂ in the initial circular boundary to be either 0 or 1 (in blocks) to mimic the sectoring pattern produced by a short period of genetic drift with a relatively small selective advantage. No-flux boundary conditions are imposed along all edges. Equation (4) is solved on a square grid of 2560 × 2560 points. The concentration of the first strain is shown in red and of the second strain in green. The maximal intensity (of red or green) corresponds to the concentration of 1, and the lowest to the concentration of 0.

Figure 7

(Colour online) The length of the initial stage of interacting sector boundaries from equation (4) as a function of the ratio of expansion velocities v₁/v₂. The quantity y_1/2 is the distance from the origin of the sector to the closest point where c₁ = 1/2. Here, we vary g₁ while keeping D₀ = 10⁻³ and g₂ = 10 fixed. For small fitness differences, we expect y_1/2 ~ (g₁ − g₂)⁻¹ from equation (6). The data from numerical solutions of equation (4) is shown as dots, and the solid line is a fit to A/(g₁ − g₂) + B, where A and B are fitting parameters.

Figure 8

Schematic illustration of transverse boundary motion behind a front advancing in the y-direction for g₁ > g₂ and ε₁ < 0 < ε₂. The plots show concentration profiles along a linear cut along the x-direction, parallel to the front and inoculant. (a) The concentration profiles a short distance behind the population frontier. There is an overlap region, where both c₁(x) (dashed line) and c₂(x) (solid line) are nonzero. This region has a finite width (from A to B) because the discreetness of the number of cells is inconsistent with infinitesimally small values of the concentrations. (See Ref. [35] for a more detailed discussion of this issue in a related model.) (b) The concentration profile at the same spatial location as in (a), but after a very long time. The interval between A and B is now occupied exclusively by the cells of the second strain, which wins out under crowded conditions.

Figure 9

(Colour online) The comparison between selective advantage in liquid culture and on Petri dishes within the reaction-diffusion model. The red circles show the results of the numerical solution of equation (4) for single strain expansions. The black line shows the theoretically predicted linear dependence. We varied g₁ to mimic different growth rates in the experiments and used g₂ = 10, ε₁ = ε₂ = 0, α₁ = α₂ = 1, D₀₁ = 10⁻⁴g₁, and D₀₂ = 10⁻³. In these units, the habitat was a 1 × 10 rectangle and was initially empty. Each expansion was started at the shorter edge of the habitat, where we imposed the Dirichlet boundary condition forcing strain density to be 1. No-flux (Neumann) boundary conditions were imposed along all other edges. The grid size used was 128 × 1280 points.

Figure 10

(Colour online) Equal-time argument and sector shape in a linear geometry. The wiggles in the sector boundaries represent genetic drift, neglected in most of this paper.

Figure 11

(Colour online) Equal-time argument and sector shape for a circular inoculant of radius R₀.

Figure 12

(Colour online) Illustration of the equal-time argument for the shape of the bulge induced by a faster growing strain (strain 1) in the circular geometry; see also figure 11. The central part of the bulge (between C and D) is an arc of a circle bounded by two tangents CK and DK (black dashed lines) to the sector boundaries at their origin. The rest of the bulge (between F and C, and between D and E) is described parametrically by equation (12), where, for any point A on this part of the bulge, the parameter ρ̃ is the distance between the center of the homeland and the intersection point B between the closest sector boundary and its tangent AB passing through A. Equating the total expansion time of the fitter strain, first, along the sector boundary KB and, then, along the tangent AB to the expansion time of the other strain along the radius of the green segment immediately yields equation (12).

Figure 13

The enclosure of a slower growing strain (initially in the majority) by the faster growing strain during a competition experiment. The fitter strain is shown in gray, and the other strain is shown in black. Four consecutive snapshots of the numerical solution of equation (4) are shown in (a), (b), (c), and (d). The enclosure leading to the heart-shape occurs shortly before the snapshot shown in (d).

Figure 14

(Colour online) Schematic picture of a colony collision.

Figure 15

(Colour online) The radii of yeast colonies as a function of time. Yellow squares and black circles correspond to colonies of the wild-type and the advantageous sterile mutant, respectively. After an initial transient (time < 90h), described in more detail in the text, the radii are well fitted by a straight line (red), in accordance with a constant expansion velocity. Inset: Instantaneous velocity ratio as function of time. The instantaneous velocities at a specific time are determined from linear fits to the radii of the five surrounding time points. The black vertical line indicates the starting time for the fit in the main figure.

Figure 16

(Colour online) Fitness estimation from linear expansion sectors. (a) S. cerevisiae colony grown from a linear inoculation at the bottom of the picture. A sector of the advantageous sterile mutant (black) emerges in the predominantly wild-type colony (yellow). The sector boundaries inferred from the image are shown with red lines. The scale bar is 500 µm. (b) Sector boundaries (blue dots) extracted from the image shown in (a), and fits (lines, r² > 0.995) to equation (8). Note that the early part of sector growth differs from the later part. One possible explanation for this difference is the sector establishment process discussed in Sec. 4 when sector boundaries are not fully separated and interact with each other. The equal-time argument does not apply in this case.

Figure 17

(Colour online) Fitness estimation from radial expansion sectors. The same as figure 16, but with a circular geometry. In (a), only the top half of a circular colony is shown. The smaller red circle shows the inoculum, and the larger red circle marks the colony radius. The scale bar is 1 mm.

Figure 18

(Colour online) Fitness estimation from colony collisions. The wild-type (yellow) colony meets the colony of the advantageous sterile mutant (black). The red lines are the fits of colony boundaries by circles. The relative fitness of the colonies can be measured from the radius and center of the circle fitted to the interface between the colonies; see equations (18) and (19). The scale bar is 1 mm.

Figure 19

(Colour online) Comparison of the selective advantage in well-mixed liquid culture and in spatial expansions. We varied the relative fitness using the drug cycloheximide for competitions of the cycloheximide-sensitive wild-type with a cycloheximide-resistant mutant. The fitness s measured with radial expansion sectors (red circles) and colony collisions (green squares) agrees well with the fitness s_wm from the liquid competition assay, since all points lie close to the diagonal (black line, not a fit). The agreement of the spatial fitness s with the liquid fitness s_wm is predicted by our theoretical model when migration is driven by cell growth, see the corresponding figure 9.