This author post is by Joshua B. Plotkin and David McCandlish on their preprint “The inevitability of unconditionally deleterious substitutions during adaptation”, arXived here.

The idea for this paper came to us while we were re-reading an earlier study by Sergey Kryazhimskiy and others, on the dynamics of adapting populations (Kryazhimskiy et al. 2009). Kryazhimskiy et al. studied the “fitness trajectory” — that is, the mean fitness across an ensemble of populations, as a function of time. The basic idea of their study was to infer the structure of an underlying fitness landscape by observing the fitness trajectory in experimental populations of evolving organisms (such as the ones from Lenski’s long-term experiments).

In re-reading Sergey’s paper, we noticed that the fitness trajectories were always monotonic, that is, the expected fitness would either always decrease or always increase. Indeed Kryazhimskiy et al. 2009 had presented a detailed analytical theory for how the fitness trajectory should behave (at least for a large class of models), and according to this theory the fitness trajectories should always be monotonic. However, when we looked more carefully at how this analytical theory was derived, we saw that the apparent impossibility of non-monotonic fitness trajectories was actually an unintended consequence of a seemingly innocuous technical assumption. The theory had been thoroughly tested against simulations for the examples explored in the paper, and it had performed quite well. But still, we wondered, could we construct fitness landscapes with non-monotonic fitness trajectories?

The answer was yes. In fact, we found conditions that produced non-monotonic fitness trajectories in one of the simplest and widely used models of a fitness landscape: the house of cards model, where the fitness of each new mutation is drawn from some fixed probability distribution. We also noticed an interesting pattern. If the population starts at a very low fitness then the fitness trajectory is be monotonically increasing. But if the starting fitness of the population is closer to the equilibrium mean fitness (that is, the value that the fitness trajectory would eventually tend to) the fitness trajectories will become non-monotonic: fitness will initially decrease, and then, eventually, increase to
its asymptotic value.

After much coffee, we eventually proved that this basic pattern must occur for any house of cards model whose equilibrium fitness distribution has a finite mean (at least under a Moran process in the limit of weak mutation). That result was the germ that eventually developed into our paper, which includes further results on the house of cards model, and on Fisher’s geometric model.

Why are non-monotonic fitness trajectories interesting? On the one hand, this is a population-genetic curiosity in a vein similar to McVean and Charlesworth (1999)’s observation that increasing the strength of purifying selection can sometimes increase the nucleotide site diversity. It’s somewhat counter-intuitive that the expected selection coefficient of the first mutation to fix in an adapting population can be negative, even on a fitness landscapes that contains no local maxima!

On the other hand, we think that this result has important implications studying adaptive evolution. It is common in such studies to assume that deleterious mutations can never fix (e.g. by approximating the probability of fixation for a new mutation as 2s). Our results on the surprising prevalence of deleterious substitutions during adapation should hopefully spur others to consider carefully the circumstances under which ignoring deleterious fixations is justified.

Joshua B. Plotkin and David McCandlish

Works cited:

Kryazhimskiy S, Tkacik G, Plotkin JB. The dynamics of adaptation on correlated fitness landscapes. PNAS 106: 18638-18643 (2009)

McVean, G. A., and Charlesworth, B. (1999). A population genetic model for the evolution of synonymous codon usage: patterns and predictions. Genetical research, 74:145-158.