Long-term evolutionary conflict, Sisyphean arms races, and power in Fisher's geometric model

Abstract

Evolutionary conflict and arms races are important drivers of evolution in nature. During arms races, new abilities in one party select for counterabilities in the second party. This process can repeat and lead to successive fixations of novel mutations, without a long-term increase in fitness. Models of co-evolution rarely address successive fixations, and one of the main models that use successive fixations-Fisher's geometric model-does not address co-evolution. We address this gap by expanding Fisher's geometric model to the evolution of joint phenotypes that are affected by two parties, such as probability of infection of a host by a pathogen. The model confirms important intuitions and offers some new insights. Conflict can lead to long-term Sisyphean arms races, where parties continue to climb toward their fitness peaks, but are dragged back down by their opponents. This results in far more adaptive evolution compared to the standard geometric model. It also results in fixation of mutations of larger effect, with the important implication that the common modeling assumption of small mutations will apply less often under conflict. Even in comparison with random abiotic change of the same magnitude, evolution under conflict results in greater distances from the optimum, lower fitness, and more fixations, but surprisingly, not larger fixed mutations. We also show how asymmetries in selection strength, mutation size, and mutation input allow one party to win over another. However, winning abilities come with diminishing returns, helping to keep weaker parties in the game.

Keywords: Fisher's geometric model; adaptation; arms race; evolutionary conflict; joint phenotypes.

Figure 1

Schematic diagrams of the three versions of Fisher's geometric model studied: standard adaptation, conflict, and a moving optimum. Each panel shows one or more fitness curves as a function of a trait value z and some fixations (arrows). (a) In a standard geometric model, parties adapt a trait z to a single stable optimum by fixing beneficial mutations. (b) In models with conflict, z represents values of a joint phenotype, and there are two fitness functions, corresponding to party 1 and party 2 (we arbitrarily assign party 1 the positive optimum value). Conflict is measured by the lag load, 1−w ₀, at the point of intersection for both fitness functions. Parties can fix beneficial mutations back and forth (arrows), with party 1 fixing 2, 4, and 5 and party 2 fixing 1 and 3 in this example. (c) Abiotic environmental change is modeled by shifting a single party's optimum in a random direction. In order to compare changes of equal size to the biotic conflict scenario, in each iteration we shift the optimal value of trait z, in a random direction, by the same amount that party 1 experiences biotic environmental change in the conflict simulation—that is, by the amount that antagonistic party 2 changes z (fixations 1 and 3)

Figure 2

The effect of conflict on adaptive walks and fitness trajectories. (a) Three adaptive walks of mutations fixed with and without conflict. We include a standard model that adapts to a single optimum and models with conflict that begin at either the origin or at party 1's optimum. Horizontal dashed lines indicate the optimal phenotypes of the two parties. (b) Fitness trajectories based on averaged fitness from 1,000 simulated adaptive walks of each type. For all simulations, both parties had conflict intensity values of 0.2, average mutation sizes of 0.1, a fitness function shape parameter of 1/2, and infinite populations

Figure 3

Equilibrium properties of the geometric model under varying average mutation sizes (normally distributed) and conflict intensities (measured as lag load at the origin where the fitness functions intersect (1−w ₀) from fitness functions with a shape parameter of 1/2). Colors indicate the version of the geometric model: standard adaptation (yellow), conflict (blue), and abiotic change (pink). (a) Mean distance to the optimum. (b) Mean fitness during equilibrium. (c) Percent of mutations that are fixed during equilibrium. (d) Effect size of fixed mutations. Means of the independent variables are calculated based on data collected from iteration 500–5,500 from 1,000 replicate simulations with infinite populations. Vertices show actual mean values from simulations

Figure 4

Fitness power is usually greater for the party with higher selection strength, mutational input, or mutation size. Fitness power was calculated according to P_{w 1} = 1 − L ₁/(L ₁ + L ₂), where L_i is the average lag load for party i. Intensity of conflict was 0.2, and population sizes were infinite for all simulations. (a) Fitness power for party 1 when ω ₁ = 2 (black), 1/2 (gray), or 1/8 (light gray) and ω ₂ = ω ₁/f, where f is the relative selection strength. Each party has one mutation per iteration. (b) Fitness power for party 1, where party 1 generates r times more mutations than party 2. Colors correspond to the same ω ₁ values as shown in A, but ω is the same for both populations. (c) Fitness power for party 1 when ${\overline{m}}_1$ = κ ${\overline{m}}_2$, where κ is the relative mutation size. Each party has one mutation per iteration. Results are shown for different values of ω. The average mutation axis shows the average mutation size for party 2 (${\overline{m}}_2$). (d) Percent of mutations that are fixed for party 1 (blue) and party 2 (green) with varying relative mutation inputs and mutation sizes. Average power is calculated from 5,000 iterations as outlined in the text. Vertices show actual mean values from simulations