Fig 1.
Left: A four-color E. coli range expansion. Four strains of E. coli differing only by a heritable fluorescent marker were inoculated on an agar plate in a well-mixed droplet and expanded outwards, leaving behind a “frozen record” of their expansion. Virtually all growth occurred at the edge of the colony. The markers instilled different expansion velocities: our eCFP (blue) and eYFP (yellow) strains expanded the fastest, followed by our black strain, and finally our mCherry (red) strain. As a result of the differing expansion velocities, the yellow/blue bulges at the frontier are larger than the black bulges which are larger than the red bulges, although the significant stochastic undulations at the front mask their size. The microbes segregate into one color locally at a critical expansion radius R0 due to extreme genetic drift at the frontier [8]. After segregated domains form, genetic domain walls diffuse and collide with neighboring walls in an “annihilation” or “coalescence” event indicated by an A or C, respectively. Right: Illustration of the relevant parameters used to model range expansions. Here, a faster expanding, more fit yellow strain with expansion velocity uY is sweeping through a less fit red strain with expansion velocity uR, in a regime where the curvature of the colony can be neglected. The length expanded by the colony is L = R − R0. We characterize domain wall motion per differential length expanded dL and the wall’s differential displacement perpendicular to the expansion direction dx. is a dimensionless speed, characterizing the yellow-red (YR) domain wall’s average expansion dx per length expanded dL, i.e.,
Dw is the domain walls’ diffusion coefficient per length expanded; it controls how randomly the domain walls move. We treat the dynamics of our four strains as a one-dimensional line of annihilating and coalescing random walkers using the parameters R0, Dw, and
, where ij represents all possible domain wall types.
Table 1.
The expansion velocity ui and each strain’s selective advantage relative to the mCherry strain siR = ui/uR − 1 measured over the course of seven days for radii greater than R0 (the radius where distinguishable domain walls formed) averaged over three independent experiments conducted on separate sets of agar plates.
was the fitness of each strain relative to mCherry in liquid culture with respect to their basal growth rates gi and gR. The radial expansion velocity fitness siR did not match the well-mixed liquid-culture fitness
. However, every strain in liquid culture still grew faster than mCherry. Interestingly, the black strain grew faster than the eCFP and eYFP strains in liquid culture while on agar, the eCFP and eYFP strains expanded faster than the black strain. See the Materials and methods for additional information.
Table 2.
Parameters used in the annihilating and coalescing random-walk model.
We experimentally measured R0, Dw, and using the procedures outlined in the Materials and methods so that we could compare experimental results with our model’s predictions.
Fig 2.
Average fraction of each genotype as a function of length expanded for 20 radial expansions each when equal fractions of eCFP, eYFP, and mCherry were inoculated (left) and when 10% eCFP, 10% eYFP, and 80% mCherry were inoculated (right).
The red dot indicates the composition at the radius R0 = 3.50 mm where distinct domain walls form and the blue dot indicates the composition at the end of the experiment. The red dots are dispersed about the initial inoculated fractions due to the stochastic dynamics at the early stages of the range expansions when R < R0. The highly stochastic trajectories illustrate the importance of genetic drift at the frontier in the E. coli range expansions. The smaller ternary diagrams display the average fraction over all expansions vs. length expanded for each set of experiments. For both initial conditions, we see a small systematic drift away from the mCherry vertex indicating that the mCherry strain has a lower fitness, in agreement with the independent radial expansion velocities of each strain (see Table 1). Note that two replicates on the right resulted in the complete extinction of eCFP due to strong spatial diffusion, indicated by the trajectories pinned on the absorbing line connecting the eYFP and mCherry vertices.
Fig 3.
Two-point angular correlation functions Fij(L, ϕ) at a length expanded of L = R − R0 = 6.5 mm (R = 10 mm, R0 = 3.5 mm) in three sets of experiments where we inoculated 20 replicates with equal fractions of our eCFP, eYFP, and black strains (left), then eCFP, eYFP, and mCherry (center), and finally all four strains (right).
The shaded regions in these plots indicate standard errors of the mean. Using the measured diffusion coefficient Dw and initial radius where domain walls form R0 (see Table 2), we also plot the theoretical neutral two-point correlation functions (black dashed line; see eq. (S1.3)). The colors of each plotted correlation function were chosen to correspond to their composite strain colors; for example, two-point correlation correlation functions associated with mCherry were red or were blended with red. The subscripts correspond to the color of each strain: C = eCFP, Y = eYFP, R = mCherry, and B = Black. As judged by the magnitude of the deviation from neutral predictions, the black strain has a small selective disadvantage relative to eCFP and eYFP and the mCherry strain has an even greater disadvantage, in agreement with the independent radial expansion velocities of each strain (see Table 1).
Fig 4.
Average cumulative difference in annihilations and coalescences vs. the average cumulative number of domain wall collisions as colonies expand.
The slope of this plot gives the annihilation asymmetry ΔP. The shaded regions represent the standard error of the mean between many experiments. We use the notation C = eCFP, Y = eYFP, B = black, and R = mCherry. Despite the presence of selection, ΔP was consistent with the standard neutral theory prediction of eq (3) for q = 2, q = 3, and q = 4 (equal initial fractions of q strains), as judged by the overlap of the black dashed lines with the shaded areas in every case. We also explored an initial condition where we inoculated unequal fractions of three strains; we inoculated 10% of both eCFP and eYFP and 80% of mCherry. Our experiments agreed with the prediction of ΔP ≈ 0.51, or an effective q ≈ 2.33, from the neutral theory developed in supplementary equations (S1.8)–(S1.10).
Fig 5.
Numerical solution of eq (5) describing how LI varies as a function of κ; analytical asymptotic approximations (eq 7) are overlaid.
If κ ≳ 1, inflation does not appreciably slow selective sweeps as LI approaches the linear selection length scale Ls. In contrast, if κ ≪ 1, the inflationary selection length scale LI will be many times larger than the linear selection length scale Ls, indicating that selection will be weak compared to inflation and diffusion (but will ultimately dominate at very large lengths expanded). The three black points correspond to measurements of the κij that govern the dynamics of our competing strains; N stands for the two selectively neutral strains (eCFP and eYFP), B for black, and R for mCherry (red). See the Predicting experimental results with simulation section for more details.
Fig 6.
Simulations of the average fraction F of a less fit strain and the annihilation asymmetry ΔP collapsed onto universal curves parameterized by .
Two neutral strains swept through a less fit strain with a wall velocity vw; each strain was numerically inoculated in equal proportions and the colony’s initial radius was R0. For identical κ, despite different values of R0 and vw, both F and ΔP can be collapsed if the length traveled L is rescaled by , the linear selection length scale. Each universal curve at a fixed κ consists of six simulations with different values of vw and R0; each set of parameters has a different marker. As κ decreases, inflation slows the selective sweep of the more fit strains through the less fit strain as illustrated by the slower decrease of F. ΔP transitioned from 0 to 1 as the number of strains present in the expansion decreased from q = 3 to q = 2 (the less fit strain was squeezed out); this is expected from eq (3), ΔP = (3 − q)/(q − 1). Supplementary S3 Fig is identical to this figure except the y-axis of F(L/Ls, κ) is placed on a linear scale; this may be useful for comparison with experiments.
Fig 7.
The collapsed two-point correlation function FNR between our eCFP/eYFP strain (N) and our mCherry strain (R), which were inoculated at fractions of 2/3 and 1/3, respectively, at a length expanded of L = 6.5 mm.
The solid red line is experimental data and the shaded region is its standard error of the mean. The dashed lines are the simulated universal correlation functions corresponding to different values of . The best fitting selection length scale is
.
Table 3.
Results of fitting from the two-point correlation functions Fij and the resulting
using the two-point correlation function technique.
“CI” stands for confidence interval.
Fig 8.
Experimental average fractions and two-point correlation functions (solid lines) and their predicted dynamics (dashed lines) by using the fit Ls from Fig 7.
The shaded region is the standard error of the mean. The simulated dynamics closely match the experimental dynamics, suggesting that our fitting technique to extract Ls is robust and can be used to describe the dynamics of our strains at all L.
Fig 9.
Experimental average fractions and two-point correlation functions (solid lines) of the four strains grown together with equal initial proportions and their predicted dynamics (dashed lines) from simulations using the set of , κij, and ϕc measured in independent experiments.
No additional fitting parameters were used. The shaded region is the standard error of the mean. The simulated dynamics closely matched the experimental dynamics except at small lengths expanded (L ≲ 3 mm) where the black strain introduced significant image analysis artifacts (see Supplementary S5 Fig).
Table 4.
Quantifying the wall velocities from our fits of
by using
and an independently measured Dw which was constant between experiments.
The uncertainty in from the fit
was calculated via error propagation.
Fig 10.
A four-color E. coli range expansion (left) and the superimposed binary masks of each channel (right) [34].
Images were acquired for four overlapping quadrants and stitched together to obtain a single image with a large field of view. Overlapping regions were blended to minimize inhomogeneities. To obtain the binary masks, pixels with fluorescence above background noise were marked as “on.” A visual comparison of the raw data and the masks confirm that our binary masks accurately reflect the location and shape of individual sectors.
Fig 11.
The average colony radius versus time for each strain on one of our three independent sets of agar plates.
The error bars (comparable to symbol size at early times) are the standard errors of the mean calculated from 12 replicate expansions for each strain. The eYFP and eCFP strains had the fastest expansion velocities (data points overlap in the plot) followed by black and then mCherry. R0 is the radius at which expansions with competing strains typically demix into one color locally; R0 is approximately 1.75 times the initial inoculant radius of 2 mm (see Fig 1).
Fig 12.
The heterozygosity correlation function H(ϕ, L) (solid lines) obtained by averaging the results of 14 neutral eCFP and eYFP expansions from one set of agar plates at a variety of expansion distances L = R − R0.
The dashed lines are the theoretical fits of the heterozygosity with a constant Dw = 0.100 ± 0.005 mm. The theoretical curves track our experimental data, suggesting that a diffusive approximation to domain boundary motion is justified.
Fig 13.
Left: Single sectors of our eYFP, eCFP, and black strains sweeping through the less fit strain mCherry. The scale bar is 1 mm. The white lines are the positions of the domain walls located with our image analysis package [34]. We tracked the angular growth of sectors sweeping through a less fit strain, ϕ − ϕ0, as a function of ln(R/R0) to obtain . Right: 40 traces of each strain sweeping through mCherry from one set of agar plates. The translucent lines are the individual traces, the solid lines are the mean angular growth 〈ϕ − ϕ0〉, and the shaded area is the standard error of the mean. The slope of the mean angular growth is
.
Fig 14.
A schematic of the simulation procedure for a radial expansion.
The initial population is a circle of cells of radius R0 = N0a/2π, where N0 is the initial number of cells and a is a cell width. During each time step (generation), the expansion advances a distance a; the radius consequently grows according to R(t) = R0 + at where t is the time in generations. The dashed circle shows the population after one generation time. Each domain wall position is tracked on the inflating ring (solid lines). At each time step, domain walls (two shown) hop to the left or right with probability Pl and Pr, respectively, with an angular jump length δϕ ≡ a/R(t), and the position is updated (dashed lines). After each domain wall movement, the time in generations is incremented by 1/N where N is the number of domain walls present. For a linear simulation, the radius is simply not inflated in time, i.e. R(t) = R0.
Fig 15.
Simulations of linear (left) and inflating (right) range expansions grown for approximately 2000 generations.
The black, yellow, and blue strains all sweep through the red strain with a wall velocity of . The initial concentration of all four strains was equal and N0 = 3000 was the number of cells at the initial front of both expansions. Note that the black, yellow, and blue sectors dominate over the red sectors at the end of these expansions.