Pulled, pushed or failed: the demographic impact of a gene drive can change the nature of its spatial spread

Technical Summary: "Pulled, pushed or failed: the demographic impact of a gene drive can change the nature of its spatial spread"

Problem Statement
The paper addresses the critical need to understand the temporal and spatial spread of gene drive alleles before field experiments are conducted. While early models utilized classical population genetics frameworks assuming discrete, non-overlapping generations in well-mixed populations, these simplifications fail to capture the spatial heterogeneity and continuous demographic dynamics of real-world scenarios. Specifically, the introduction of maladapted transgenic individuals can reduce population size, creating a demographic flux from dense to sparse areas that opposes the spread of the drive allele. The central problem is to clarify how variations in population density (demographic dynamics) affect the propagation speed and nature of gene drive waves across space, particularly under extreme growth regimes.

Methodology
The authors employ a deterministic reaction-diffusion modeling approach in a one-dimensional spatial context. The study focuses on traveling wave solutions, which represent stationary profiles moving at a constant speed $v$.

1. Model Formulation:

   • The models describe the dynamics of three genotypes: wild-type homozygotes ($WW$), drive homozygotes ($DD$), and heterozygotes ($DW$).
   • Fitness is assigned as $f_{WW}=1$, $f_{DD}=1-s$ (where $s$ is the fitness cost), and $f_{DW}=1-sh$ (where $h$ is the dominance coefficient).
   • Gene conversion (homing) is modeled with a probability $c \in [0, 1]$, occurring either in the zygote or the germline.
   • The system incorporates a density-dependent growth term parameterized by the intrinsic growth rate $r$ and a diffusion term (normalized to 1).

2. Analytical and Numerical Approaches:

   • Extreme Regimes: The authors analytically investigate two limiting cases for the intrinsic growth rate $r$:
     • $r \to +\infty$: Represents a scenario where the population instantaneously restores carrying capacity, minimizing demographic flux.
     • $r = 0$: Represents a scenario where births exactly balance deaths in a wild-type population, maximizing the demographic impact of fitness costs. This regime is analyzed by establishing a mathematical connection to epidemiological SI (Susceptible-Infected) models.
   • Partial Conversion: Unlike previous works assuming 100% conversion, this study explicitly models partial conversion ($c < 1$) and the resulting presence of heterozygotes.
   • Numerical Simulations: Extensive numerical simulations of the Cauchy problem bridge the gap between the $r=0$ and $r=+\infty$ regimes, generating heatmaps of invasion speeds across parameter spaces ($s, h, c, r$).

Key Contributions and Results

1. Characterization of Wave Types:
   The paper distinguishes between pulled waves (where the speed is determined by the dynamics at low density/frequency) and pushed waves (where the speed is strictly larger than the linearized minimal speed, driven by the bulk of the population).

   • In the $r \to +\infty$ regime: The results align with and extend previous studies. Depending on the fitness cost $s$ and conversion rate $c$, the drive can fail, invade as a pulled wave, invade as a pushed wave, or lead to wild-type invasion.
   • In the $r = 0$ regime: A significant finding is that waves cannot be pushed when the intrinsic growth rate vanishes. The system reduces to a form analogous to SI models, where the drive either invades as a pulled wave (if the fitness cost is low enough) or fails completely (gene drive clearance).

2. Impact of Demographic Flux:
   The study demonstrates that demographic effects are most pronounced when $r$ is low. In this regime, the demographic flux (movement from high to low density) acts as a barrier to the spread of costly drive alleles. Consequently, a drive that successfully invades in a high-growth regime ($r \to +\infty$) may fail to invade or clear entirely in a low-growth regime ($r=0$).

3. The "Always Pulled" Conjecture:
   Through numerical simulations bridging the two extreme regimes, the authors observe that if a wave is pulled in both the $r=0$ and $r=+\infty$ limits, it remains pulled for all intermediate values of $r$. In these cases, the wave speed appears independent of the demographic parameter $r$. This occurs when:

   • The fitness cost $s$ is sufficiently small.
   • There is a stable coexistence of drive and wild-type alleles after invasion.

4. Thresholds and Stability:
   The paper provides analytical thresholds for the transition between drive invasion, wild-type invasion, and coexistence. For instance, in the $r=0$ limit with zygotic conversion, the drive invades if $s < s_{2,z} = \frac{c}{2c + h(1-c)}$. If $s$ exceeds this threshold, the drive clears.

Significance and Claims
The paper claims to delineate the specific conditions under which demographic dynamics fundamentally alter the spread of gene drives. By connecting the $r=0$ limit to SI epidemiological models, the authors provide analytical tools to predict drive failure in populations with low intrinsic growth rates.

The study emphasizes that assuming a constant population density (the $r \to +\infty$ limit) may lead to overly optimistic predictions of gene drive success. Specifically, the nature of the wave (pulled vs. pushed) and the success of the invasion can change drastically depending on the population's ability to recover from the fitness costs imposed by the drive. The authors conclude that demographic dynamics are not merely a background factor but can determine whether a gene drive successfully spreads, fails, or changes the genetic composition of the population in a spatially structured environment.