Fig 1.
Reaction-diffusion model of chiral growth accurately describes the behavior of sector boundaries in compact microbial colonies.
Population dynamics are visualized by the spatial pattern produced during the growth of two neutral strains expressing different fluorescent proteins. The growth is largely limited to the colony edge, so the patterns behind the front do not change over time. Although initially the strains are well-mixed, strong genetic drift leads to local extinctions of one of the strains, which manifests as a characteristic pattern of sectors in both experiments (A) and simulations (B). The boundaries between the sectors fluctuate due to genetic drift and twist counterclockwise due to a chiral bias in cell motion. This bias is quantified in (C) for experiments and in (D) for simulations by plotting the polar angle θ, averaged over many sector boundaries, vs. the radius r. A constant boundary velocity along the colony edge should result in a linear increase of θ with lnr [35]. Consistent with this expectation, both plots show that sector boundaries are logarithmic spirals. The excellent agreement between experiments and simulations indicates that our reaction-diffusion model is suitable for the study of competition between chiral strains in compact microbial colonies. The experimental data was obtained from the Dryad digital data repository associated with Ref. [18]. Here, m0 = ms = mb = md = 0, g = 0.03, N = 100, ml = 0.045, mr = 0.005 for both strains. Radius of initial circle was 30 on a lattice of 700x700 sites.
Fig 2.
Chirality provides a fitness advantage in competition, but not in overall growth.
The first two panels show that a non-chiral (A) and a chiral strain (B) expand with the same velocity. To facilitate this comparison, we started the simulations with a linear instead of a circular front. Panel (C) demonstrates the chiral strain displaces the non-chiral strain when they compete within the same colony: A small initial population of the right-handed strain (shown in red) expands over time and eventually takes over. Because the simulations respect the mirror symmetry, the results were the same irrespective of whether the chirality was due to a left-handed or a right-handed bias in motility. Note that the fate of a strain is not determined solely by its expansion velocity because the colonies expand as pushed waves [47, 49–51]. The expansion velocity of a pushed wave depends not only on the growth and migration rates at the leading edge, but also on the non-linear population dynamics within the wave front. As a result, the outcome of the competition is affected by how each of the strains responds to the presence of the other strain. In our simulations, expansions are pushed because of the density-dependent motility; see Methods and SI. Here, m0 = ms = mb = md = 0, g = 0.1, N = 200 for both strains. ,
,
on a lattice of 600x3300 sites. All distances were measured in 100 units where Δx = Δy = 1 unit.
Fig 3.
Selection for coexistence between strains with opposite handedness.
Panel (A) shows that a left-handed mutant (shown in green) can invade a right-handed population (shown in red). The reverse invasion also occurs and is shown in panel (B). This negative frequency-dependent selection is further illustrated in panel (C), which shows how , the spatially averaged relative abundance of the first strain, changes over time starting from different initial conditions. At t = 0, the strains are spatially separated in (A) and (B), but well-mixed in (C). In this figure, the strains have exactly opposite chiralities, but coexistence occurs more generally; see Fig 7. Note that the selection for coexistence relies on the presence of boundaries between the strains. When strains intermix (as shown in this figure), we observe a strong and time-invariant selection for coexistence. When strains do not intermix (see Figs 5 and 6), the number of boundaries slowly declines over time due to neutral coarsening [18, 41]. In such cases, robust coexistence relies on occasional external re-mixing events, e.g. during the establishment of a new colony. Here, m0 = ms = mb = md = 0, g = 0.1, N = 200 for both strains.
,
,
,
on a lattice of 600x3000 sites. All distances were measured in 100 units where Δx = Δy = 1 unit.
Fig 4.
Effects of chirality persist despite growth rate differences.
We competed strains with different growth rates and chirality starting from well-mixed initial conditions. The abundances of the strains were equal at the beginning of the simulations, but changed over time leading either to stable coexistence or to the exclusion of one of the strains. The outcome of the competition depends on the relative growth rates of the strains quantified by Δg. The competition between a left-handed strain and a faster growing non-chiral strain is shown in (A). The non-chiral strain is outcompeted even if it has a growth advantage as high as to 2%. The chiral strain becomes extinct only when its growth penalty exceeds 7%. For intermediate Δg, the two strains stably coexist. Similar dynamics occur during the competition between the strains with equal, but opposite chirality shown in panel (B). The stable coexistence between the strains is destroyed only by growth rate differences higher than about 7%. Here, m0 = ms = mb = md = 0 for both strains and N = 200. In (A), ,
,
,
on a lattice of 1000x3600 sites. We fixed g = 0.01 for the left handed-strain and varied the growth rate of the non-chiral strain according to g(1 + Δg/100%). In (B),
,
,
,
on a lattice of 500x3000 sites. We set g = 0.1 for the left-handed strain, and varied the growth rate of the right-handed strain according to g(1 + Δg/100%). The distances on the x-axis were measured in 100 units where Δx = Δy = 1 unit. We verified stable coexistence by starting the simulation above and below the observed steady-state relative abundances and checking that they return to the same steady-state values.
Fig 5.
Boundaries between strains with different chiralities create front undulations.
(A) A magnified view of the colony front shows bulges and dips near in-flow and out-flow boundaries. The analytical solutions for the shape of bulges and dips are shown in panels (B) and (C) respectively. Note that both the theory and the simulations predict an approximately triangular bulge shape. Here, m0 = ms = mb = md = 0, g = 0.1, N = 200 for both strains, and ,
,
,
on a lattice of 2400x2100 sites in panel(A). The chirality of the strains is shown with thick white arrows. All distances were measured in 100 units where Δx = Δy = 1 unit.
Fig 6.
Bulges drive selection by “pushing” out-flow boundaries.
The panels show the motion of an out-flow boundary due to the expansion of the bulges formed at the surrounding in-flow boundaries. Initially, the bulges are small (AB). Bulge growth has no effect, until one of the bulges comes in contact with the boundary of interest (C). After that, the expansion of the bulge displaces the out-flow boundary (D). The movement of the boundary stops when it is locked between two nearest bulges (EF). The relationship between the location of bulge and the motion of the out-flow boundary is further clarified in schematics above each of the panels. The black lines in the last panel show the locations (but not the actual size) of the bulges at earlier times. To illustrate the dynamics most clearly, we chose the model parameters in the no-mixing regime (see Fig 8). As a result, both in-flow and out-flow boundaries appear almost equally sharp. The chirality of the strains is shown with thick white arrows. Here, m0 = ms = mb = md = 0, g = 0.1, N = 200 for both strains. ,
,
,
on a lattice of 2400x6400 sites.
Fig 7.
Equilibrium fractions change with relative chirality.
(A) shows the steady-state spatial structure for two strains with opposite handedness, but unequal magnitudes of chirality (f* ≠ 1/2). (B) shows the relative abundance of the strains as a function of their relative chiralities. This relationship is approximately linear in agreement with Eq (5); see also S1 and S2 Figs. Here, m0 = ms = mb = md = 0, g = 0.1 for both strains. The chirality of the strains is shown with thick white arrows. In (A) ,
,
,
, N = 100. In (B), the chirality of the strains was varied with the difference in their chiralities
and ml + mr = 0.1 at N = 400. Length of lattice was varied to ensure steady state was reached.
Fig 8.
Transition from segregation to strain intermixing.
The intermixing of the strains was quantified by heterozygosity H = 〈2f(1 − f)〉, which is nonzero only when both strains are present at the same spatial location. (A,B) show that there is a phase transition between an intermixed regime, where H has a nonzero value at steady state, and a regime, where the strains spatially segregate with H vanishing in the long-time limit. The transition is controlled by the relative magnitude of strain chirality and the strength of genetic drift. The latter depends on the number of the organisms at the growing edge. (C) shows the spatial patterns in the demixed regime starting either from a single boundary or from well-mixed initial conditions. (D) shows how the relative abundance of the strains change starting from well-mixed initial conditions and different initial fractions. Note that, even in the demixed regime, there is negative frequency-dependent selection towards coexistence. (E,F) are the same as (C,D) but in the intermixed regime. All data in this figure are for strains with opposite handedness, but equal magnitude of chirality. Similar results were obtained for unequal magnitudes of chirality; see S4 Fig. Here, m0 = ms = mb = md = 0, g = 0.1 for both strains. In panels (A), (C), (D), (E), (F), ,
,
,
on a lattice of 1000x8000 sites with N = 40 and N = 320 for no intermixing and intermixing respectively. In (B)
, N = 200 was held fixed as chirality was varied. All distances were measured in 100 units where Δx = Δy = 1 unit. The right-handed strain is shown in red, and the left-handed strain is shown in green. Note that the equilibration of H requires that f is in steady state; therefore, H equilibrates more slowly.
Fig 9.
Isotropic, but chiral migration in a lattice-based model.
The panels illustrate the computation of migration fluxes given by Eq (9) for different orientations of the migration direction relative to the lattice. Note that the choice of the migration coefficients is always made relative to the direction of migration and not relative to the coordinate system.