New sector morphologies emerge from anisotropic colony growth

Competition during range expansions is of great interest from both practical and theoretical view points. Experimentally, range expansions are often studied in homogeneous Petri dishes, which lack spatial anisotropy that might be present in realistic populations. Here, we analyze a model of anisotropic growth, based on coupled Kardar-Parisi-Zhang and Fisher-Kolmogorov-Petrovsky-Piskunov equations that describe surface growth and lateral competition. Compared to a previous study of isotropic growth, anisotropy relaxes a constraint between parameters of the model. We completely characterize spatial patterns and invasion velocities in this generalized model. In particular, we find that strong anisotropy results in a distinct morphology of spatial invasion with a kink in the displaced strain ahead of the boundary between the strains. This morphology of the out-competed strain is similar to a shock wave and serves as a signature of anisotropic growth.

💡 Research Summary

In this paper the authors investigate how spatial anisotropy influences the morphology of competing strains during a two‑dimensional range expansion. Building on their earlier “thin‑edge” framework, they describe the colony front by a height field h(x,t) and the relative abundance of a mutant strain by a scalar field f(x,t). The dynamics of h obey a modified Kardar‑Parisi‑Zhang (KPZ) equation, while f follows a modified Fisher‑Kolmogorov‑Petrovsky‑Piskunov (FKPP) equation. The two equations are coupled through a term β ∂ₓh ∂ₓf, which captures the fact that a tilted front advects the genetic composition along the front.

Anisotropy is introduced by allowing the local expansion speed v to depend on the heading angle θ of the front. Expanding v(θ) around the slowest‑growth direction (θ = 0) yields v(θ) ≈ v₀ + ½ v″(0)θ². This modifies the KPZ non‑linearity λ from v₀ to v₀ + v″(0) and changes the FKPP coupling β from v₀ to v₀(1 + ε′(0)), where ε(θ)≈ε′(0)θ describes the deviation of a neutral sector boundary from the front normal. Consequently, in the isotropic limit λ = β = v₀, whereas in the anisotropic case λ and β become independent parameters. The ratio β/λ therefore quantifies the strength of anisotropy; β/λ = 1 corresponds to trivial anisotropy that can be removed by simple rescaling, while β/λ ≠ 1 signals a genuinely anisotropic system.

After nondimensionalisation the model is governed by three independent dimensionless groups: the diffusion‑ratio D_h/D_f, the anisotropy measure λ/β, and the combined fitness‑anisotropy parameter αβ/(s₀D_f) (α is the difference in expansion speeds between the two strains, s₀ the selective advantage of the mutant). By varying these groups the authors explore the full phase space of possible sector morphologies.

The central theoretical finding is that when β/λ exceeds a critical threshold, the usual sector shape—where the boundary remains perpendicular to the expanding front—breaks down. Instead, a “shock‑wave” morphology emerges: the out‑competed strain develops a pronounced kink that protrudes ahead of the main front, resembling a shock front in fluid dynamics. This kink is a direct consequence of the anisotropic advection term β∂ₓh∂ₓf; a tilted front pushes the mutant fraction forward faster than the bulk, creating a thin, fast‑moving front of the disadvantaged strain. In the limit of strong anisotropy (β/λ ≫ 1) the shock becomes almost planar and propagates ahead of the fitter strain, effectively allowing the weaker strain to “lead” the expansion locally.

To validate the analytical predictions, the authors perform two sets of simulations. First, they discretise the coupled KPZ‑FKPP equations on a lattice, confirming that increasing β/λ produces the predicted kinked sectors. Second, they implement a more mechanistic reaction‑diffusion model of cell growth in two dimensions, which includes nutrient diffusion and local logistic growth. This more realistic model reproduces the same transition, demonstrating that the shock‑wave morphology is robust to the details of the underlying biology.

The paper also discusses experimental relevance. In microbial colonies, anisotropy can arise from collagen fibers, microfluidic channels, chemical gradients, or external magnetic fields, all of which bias the local expansion speed. The presence of a forward‑projecting kink in a sector of the out‑competed strain would therefore serve as a clear signature of such anisotropic conditions. Moreover, because β/λ can be measured from the geometry of sector boundaries, the framework offers a quantitative tool for inferring hidden directional biases in real biological expansions, including biofilms and tumor growth.

Overall, the work extends the previously isotropic theory of range expansions to include directional growth, provides a concise parameter (β/λ) to quantify anisotropy, and predicts a novel, experimentally observable sector morphology—a shock‑wave‑like kink—that distinguishes anisotropic from isotropic expansion dynamics. This advances our theoretical understanding of spatial competition and opens new avenues for probing environmental anisotropy in microbial and cellular systems.