Crossing a fitness valley in a changing environment: With and without pit stop

TL;DR

We analyze crossing of fitness valleys in a changing environment using a stochastic, individual-based adaptive dynamics model for an asexually reproducing population. With carrying capacity $K\to\infty$ and mutation probability $\mu_K=K^{-1/\alpha}$, environmental parameters vary on an intermediate time scale $\lambda_K$ via piecewise-constant phases. We prove two main results: (i) in a strict valley all intermediate mutants remain unfit in every phase and the successful valley crossing occurs on the time scale $\mathcal{O}\left(1/(K\mu_K^L)\right)$ with rate $R^{\mathrm{eff}}_L$, the phase-averaged crossing rate, and (ii) in a valley with a pit stop a single intermediate trait $w$ attains positive fitness during a phase and accelerates crossing to a macroscopic size, yielding a rate roughly $\mathcal{O}\left(K\mu_K^L e^{\lambda_K T_1 f^1_{w,0}}/\lambda_K\right)$. The paper provides heuristics, precise assumptions, and rigorous proofs, and discusses extensions to more phases, non-constant residents, multiple pit stops, and other valley configurations. The findings quantify how environmental periodicity and brief growth spurts can dramatically affect evolutionary trajectories and valley traversal.

Abstract

We consider a stochastic individual-based model of adaptive dynamics for an asexually reproducing population with mutation. Biologically motivated by the influence of seasons or the variation of drug concentration during medical treatment, the model parameters vary over time as piecewise constant and periodic functions. We study the typical evolutionary behavior of the population by looking at limits of large populations and rare mutations. An analysis of the crossing of valleys in the fitness landscape in a changing environment leads to various interesting phenomena on different time scales, which depend on the length of the valley. By carefully examining the influence of the changing environment on each time scale, we are able to determine the crossing rates of fit mutants and their ability to invade the resident population. In addition, we investigate the special scenario of pit stops, where single intermediate mutants within the valley have phases of positive fitness and can thus grow to a diverging size before going extinct again. This significantly accelerates the traversal of the valley and leads to an interesting new time scale.

Figures (2)

• Figure 1: Exemplary plot of $g_w(0,s)$ for $\ell=4$ parameter phases. The blue dot marks the optimal/typical occurrence time $t^*$ of a $w$ mutant to initiate a population reaching its highest possible size. The blue line marks this growth phase, at the end of which (at time $t^*+s^*$), $L$-mutants are produced with the highest possible rate.
• Figure 2: Exemplary plot of $g_w(0,s)$ and $g_L(0,s)$ for $\ell=2$ parameter phases, temporarily unfit trait $L$ and A) asynchronous or B) synchronous fit phases. Blue dots and trajectories mark the optimal/typical occurrence time $w$ mutants and their successive growth phase. Dashed arrows mark the typical transition time to trait $L$ and red dots and trajectories mark the occurrence and growth of $L$-mutants. The set $A$ of possible arrival times of successful $L$-mutants is marked in orange.

Theorems & Definitions (34)

• Definition 2.1: Invasion fitness
• Theorem 2.2
• Remark 1
• Theorem 2.3
• Lemma 4.1
• proof
• Lemma 4.2: Equilibrium size of mesoscopic traits
• Remark 2
• proof
• Lemma 4.3