The inevitability of unconditionally deleterious substitutions during adaptation
David M. McCandlish, Charles L. Epstein, Joshua B. Plotkin
(Submitted on 4 Sep 2013)
Studies on the genetics of adaptation typically neglect the possibility that a deleterious mutation might fix. Nonetheless, here we show that, in many regimes, the first substitution is most often deleterious, even when fitness is expected to increase in the long term. In particular, we prove that this phenomenon occurs under weak mutation for any house-of-cards model with an equilibrium distribution. We find that the same qualitative results hold under Fisher’s geometric model. We also provide a simple intuition for the surprising prevalence of unconditionally deleterious substitutions during early adaptation. Importantly, the phenomenon we describe occurs on fitness landscapes without any local maxima and is therefore distinct from “valley-crossing”. Our results imply that the common practice of ignoring deleterious substitutions leads to qualitatively incorrect predictions in many regimes. Our results also have implications for the substitution process at equilibrium and for the response to a sudden decrease in population size.
Haldane's Sieve 
This paper makes a curious observation for populations in static landscapes in the weak mutation limit (monomorphic populations): There is a region in fitness in which the first mutation that fixes is more likely deleterious than beneficial. Intriguingly, this region includes the mean fitness at equilibrium.
While counter-intuitive, the explanation for this is quite simple: Time intervals between successive fixations depend on where you are in the fitness landscapes. The population moves fast at low fitness since there are many ways to adapt, but sits in high fitness genotypes for a long time since most mutations are deleterious and fix rarely. Hence even at the mean equilibrium fitness most transitions are towards lower fitness, but once at lower fitness the population quickly returns.
This is part of a more general class of phenomena that arise in stochastic ODEs when the diffusion term depends on the location. The analog of the position dependent diffusion constant in the case at hand are the different rates of fixations at different locations in the fitness distribution. A position dependent diffusion produces a spurious drift term seemingly inconsistent with the equilibrium distribution. The manifestation of this is different in the Ito and Stratonovich formulations of stochastic calculus and generates a lot of headache.
Whether or not this is important depends on how far from a fitness peak the population is: If the population is always far enough from the peak that the supply of beneficial mutations is not significantly depleted, it does not matter.
Thanks, Richard.
The basic intuition you describe is part of the intuition that we try to express in our Discussion. The weak-mutation process we study features periods of stasis punctuated by jumps between different fitnesses. The stasis periods are longer when the fitness is higher, which contributes to an equilibrium distribution that emphasizes high fitnesses. Exactly as you say, starting from the equilibrium mean, the first substitution is expected to reduce fitness, but those sample paths quickly return to higher fitnesses (and then stay there for longer).
Although they are perhaps loosely analogous, these phenomena are not manifestations of general features of Ito and Stratonovich stochastic ODEs, at least in their standard formulation. Standard stochastic ODEs always produce *continuous* sample paths, whereas our process always has *discontinuous* sample paths. We do not define the weak-mutation process as a stochastic ODE at all, but rather we specify the exact underlying Markov process — and so the issues of Ito versus Stratonovich interpretation do not arise, thankfully.
Well, I think this is semantics and I agree the analogy is not terribly tight. But you do have a position dependent mobility. In some continuum limit, this generates the terms proportional to the derivative of the diffusion constant in which Ito and Stratonovich differ. In any numerical integration which is necessarily discontinuous, you get the asymmetries you are describing. In any case, this analogy helped me understanding your paper.
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