# Is local equilibrium a reasonable assumption for evolutionary processes?

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Whenever I look at discussions of fitness landscapes (in particular, Kauffman's NK model) the questions tend to resemble:

The population is at a local equilibrium, but another equilibrium of higher fitness exists, how will the population cross the fitness valley between these equilibria?

These sort of statements assume that the population has reached a local equilibrium. Although, the local equilibria must exist, why do the people working in this field believe that they can be found before environmental (or other external events) change the fitness function? Are the timescales required to go from a random initial population to one that is at a local equilibrium compatible with the typical time-scales on which a fixed fitness landscape is an appropriate approximation?

If we switch to the polar opposite model of complete frequency-dependent selection (say replicator dynamics in evolutionary game theory) then limit-cycles (think rock-paper scissors game) and chaotic-attractors are common and it is possible for the population genetics to be constantly changing and never at equilibrium.

In an experimental setting, it also seems like although beneficial point-mutations are much more rare than deleterious, they do exist. This would suggest that experimentally, organisms are not at a local equilibrium. Do model organisms tend to be at local fitness equilibria?

In general, is the local equilibrium assumption in fitness landscapes research a reasonable assumption?

Nothing is at a genome-wide local equilibrium. Graham Bell wrote fairly extensively on this (IIRC).

Some loci

• will be at what are likely global optimums (e.g. Cytochrome oxidase)
• will be at local but not global optimums (e.g. low-fitness malaria resistance vs. high fitness malaria resistance: for the extremely cool story check out this page)
• will not be at any optima because the environment shifts too quickly (e.g. loci clearly under frequency-dependent selection, such as MHC).

So, in general, it depends what you are trying to model. If you want to model a specific kind of evolutionary change, then model that kind of evolutionary change. If you want to model a different kind of change, then model that kind of evolutionary change. Natural populations will likely follow any model some of the time, but will probably never follow any model all of the time.

I think the jury is out on this one, there are examples of evidence both for and against reachability of local equilibrium and even these examples can be interpreted in many ways. I present three pieces of evidence and some interpretations. In general, my feeling after reading about this is that the assumption of equilibrium is ingrained in mathematical models for convenience and historic reasons not because there is solid evidence for it. (I expand more on this in this blog post).

### Evidence against: vestigial features and macroevolutionary change

From a genome-wide perspective, it seems that equilibrium is at odds with the intuition of naturalists. Consider for example vestigial features of your own body like your appendix, goose bumps, tonsils, wisdom teeth, third eyelid, or the second joint in the middle of your foot made immobile by a tightened ligament (see video below). Wouldn't it be more efficient (and thus produce marginally higher fitness) if you didn't spend the energy to construct these features? Of course, this naturalist argument is not convincing since we don't know if there are any small mutations that could remove these vestigial features from our development, I could just be describing a different local optimum that lays on the other side of a fitness valley from my current vertex.

The other tempting naturalist example of macroevolutionary changes like speciation is also not convincing for static fitness landscapes. The usual retort is that on these timescales the environment is not constant and depends on the organisms through mechanisms like niche-construction or frequency-dependence. This defense of local equilibria is actually a central part of the punctuated equilibrium theory of evolution; the environment changes (either through an external effect like meteor or internal effect like migration or niche-construction) and the wild-type becomes not locally optimal, but adaptation quickly carries the species to a new nearby local optimum where it remains for a long period of time until the next environmental change. Naturalistic observations are insufficient to settle this question, so we need to turn to experiments.

### Evidence for: affinity maturation

The length of evolutionary process leading to affinity maturation is very short, typically a local equilibrium is found after only 6-8 nucleotide changes in CDR (Crews et al., 1981; Tonegawa, 1983; Clark et al., 1985), so you need only a few point mutations to quickly develop a drastically better tuned antibody -- an adaptive process that happens on the order of days. There are two reservations to keep in mind:

First, the adapted B-cells were not experimentally isolated and all of their point-mutations were not checked to guarantee that a fitness peak was reached. In both theoretical and experimental treatments of evolution, it is known that fitness increases tend to show a pattern of geometrically diminishing returns (Lenski & Travisano 1994; Orr, 1998; Cooper & Lenski, 2000) which means that after a few generations the fitness change will be so small that the fixation time in the large population of B-cells will be longer than the presence of the pathogen causing the immune response. We might not be seeing more steps because the next steps might have a fitness increase too small to fixate before the environment (and thus, fitness landscape) changes again.

Second, as can be seen from the AID protein (and other mechanisms) increasing the rate of mutation by a factor of $$10^6$$ along the gene encoding antibody proteins, this is a fitness landscape that has been shaped by previous evolution of the human immune system to find fit mutants as quickly as possible. This biases the phenomenon towards landscapes where local maxima would be easier to find than usual, and thus makes it not a good candidate for considering evolution under more typical conditions.

### Evidence against: long-term evolution experiment

A more typical setting might be the evolution of E. coli in a static fitness landscape. Here, biologists have run long-term experiments tracking a population for over 50,000 generations (Lenski & Travisano 1994; Cooper & Lenski, 2000; Blount et al. 2012) and continue to still find adaptations and marginal increases in fitness. This suggests that a local optimum is not quickly found, even though the environment is static. However, it is difficult to estimate the number of adaptive mutations that fixed in this population, and Lenski (2003) estimated that as few as 100 adaptive point-mutations fixated in the first 20,000 generations. It is also hard to argue that the population doesn't traverse small fitness valleys between measurements, which could be used to suggest that the colony is hopping from one easy-to-find local equilibrium to the next.

### References

Blount, Z. D., J. E. Barrick, C. J. Davidson, and R. E. Lenski. (2012). Genomic analysis of a key innovation in an experimental Escherichia coli population. Nature 489: 513-518.

Clark, S.H., Huppi, K., Ruezinsky, D., Staudt, L., Gerhard, W., & Weigert, M. (1985). Inter- and intraclonal diversity in the antibody response to influenza hemagglutin. J. Exp. Med. 161: 687.

Cooper, V.S., & Lenski, R.E. (2000). The population genetics of ecological specialization in evolving Escherichia coli populations. Nature 407: 736-739.

Crews, S., Griffin, J., Huang, H., Calame, K., & Hood, L. (1981). A single V gene segment encodes the immune response to phosphorylcholine: somatic mutation is correlated with the class of the antibody. Cell 25: 59.

Kauffman, S.A., & Weinberger E.D. (1989). The NK model of rugged fitness landscapes and its application to maturation of the immune response. Journal of Theoretical Biology, 141(2): 211-245.

Lenski, Richard E. (2003). "Phenotypic and Genomic Evolution during a 20,000-Generation Experiment with the Bacterium Escherichia coli". In Janick, Jules. Plant Breeding Reviews (New York: Wiley) 24(2): 225-65.

Orr, H.A. (1998). The population genetics of adaptation: the distribution of factors fixed during adaptive evolution. Evolution 52: 935-949.

Tonegawa, S. (1983). Somatic generation of antibody diversity. Nature 302: 575.