Selection of the fittest or selection of the luckiest: the emergence of Goodhart's law in evolution

TL;DR

It is shown through simulations and mathematical analysis, that the speed of adaptation increases with increasing selection pressure only up to a threshold, and this intricate response of evolution to natural selection can be mathematically explained by a novel phase transition for pulled traveling waves.

Abstract

Biological evolution depends on the passing down to subsequent generations of genetic information encoding beneficial traits, and on the removal of unfit individuals by a selection mechanism. However, selection acts on phenotypes, and is affected by random contingencies. Thus, a combination of fitness and luck determines which individuals will successfully reproduce and give rise to the next generation. To understand how randomness in the selection mechanism affects the long-term patterns of evolution, we studied an idealized evolution model. We show through simulations and mathematical analysis, that the speed of adaptation increases with increasing selection pressure only up to a threshold. Beyond the threshold, any increase of the selection pressure results in more weight given to random effects rather than on genetic fitness in determining which individuals will successfully reproduce. This severely reduces the speed of adaptation and the diversity in the gene pool. Our findings may be considered as a biological instance of Goodhart's law: "When a measure becomes a target, it ceases to be a good measure". Finally, we show that this intricate response of evolution to natural selection can be mathematically explained by a novel phase transition for pulled traveling waves.

Figures (5)

• Figure 1: Rates adaptations and effective population sizes in a branching-selection particle system with a population of size $10^5$, plotted as a function of the genotypic to phenotypic standard deviation ratio $\mu$ and the selection pressure $\gamma$. The Top Panels where simulated using $\alpha=1$ (Laplace distribution for the phenotypic and genotypic distributions), whereas the Bottom Panels were made using $\alpha =2$ (Gaussian distribution for phenotypic and genotypic distributions). The phenomenological picture appears identical for different values of $\alpha \geq 1$. Left Panels: Estimated rate of adaptation of the genotype profile. The dotted line corresponds to the critical line $\gamma_{c}(\mu)$ segregating the strong and the weak regime (respectively lower and upper part of the figure). For each $\mu$, $\gamma_c(\mu)$ is the selection pressure that maximises the rate of adaptation. Right Panels: The effective population size $N_e$. The same function $\mu \mapsto \gamma_c(\mu)$ computed from the corresponding left pannel is reproduced.
• Figure 2: Top left panel. Distribution of genotypes in a large population, plotted on a logarithmic scale. We observe for large populations the apparition of a deterministic genotypic profile. The profile is well approximated by a quadratic profile (in red). Top right panel. Convergence to the traveling wave solution of the deterministic dynamic, started from an initial Gaussian distribution (i.e. a parabolic log-profile). Blue curves show the evolution of the first 8 steps of the genotypic profile, the red curves steps 16 to 24. We observe that by that time, the phenotypic profile has converged to a traveling wave solution. Bottom panels. Schematic description of the iteration of the traveling wave over one step. The blue line represents the initial traveling wave profile, the orange line the genotypic profile of the children (that is after reproduction). The green line correspond to the genotypic profile of the selected chidlren, and is a translation of the blue profile. The vertical red dotted line corresponds to the position of the selection threshold --in particular, note that the green and orange lines are identical to the right of that threshold, indicating that genotypes to the right of the threshold will be selected. The bottom purple dots correspond to iterations of the ancestral function in the stationary setting indicating the successive ancestors in the population.
• Figure 3: Top panels. Rate of adaptation for the deterministic limiting model and its finite-size population correction, for the original stochastic model on the left pannel, and the noisy BRW right pannel (believed to be its scaling limit). In both cases, the rate of convergence is notably slow. Bottom left panel. Comparison between the critical value $\gamma_c$ estimated for each $\mu$ as the value of $\gamma$ maximizing $\gamma \mapsto v^{(N)}_{\gamma,\mu}$. Bottom right panel. Effective population size for $N=10^7$ (thick brown curve, average over 100 realizations) and $N=10^2$ (blue curve, average over 1000 realizations). The shaded areas show the 10th to 90th percentiles of the distribution. The solid black line is the theoretical approximation \ref{['eq:Ne-below-gammac']} in the strong regime; the thin brown and blue lines are the theoretical approximation \ref{['eq:Ne-above-gammac']} in the weak selection regime. For large populations ($N=10^7$), a change of monotonicity occurs close to the closed predicted value which corresponds to the end point of the curve $1/\mu$.
• Figure 4: Left panel. Speed for the asexual model as predicted by the explicit solution of (\ref{['eq: deterministicDynamic2']}). Right panel. Speed for the sexual model as predicted by the iterating the modified equation (\ref{['eq: deterministicDynamic2sex']}). For both models, if $\mu\in(.5,1)$, the speed is maximized at intermediary values. However, in the sexual case, the speed is always $0$ when the phenotypic noise is too large $\mu<1/2$. Bottom panel. Comparison of the speed of evolution in the sexual and asexual models.
• Figure 5: Fitness wave for the sexual model. In blue : generation $0$ to $8$. In red: generation $16$ to $24$. From left to right: static regime ($\mu=.48, \gamma= .43$); close to static ($\mu=.51$, $\gamma=.14$); moving ($\mu=.61$, $\gamma=.61$)

Theorems & Definitions (33)

• Theorem S.1.1
• Remark
• Theorem S.1.2
• Remark
• Theorem S.1.3
• Lemma S.2.1
• proof
• Proposition S.2.2
• Remark
• proof