More on high-dimensional fitness landscapes

In a post from a few days ago about a paper I just got published, John Wilkins asked how my work ties in with Gavrilets' stuff on high dimensional landscapes. Here's my answer:

In terms of the discussion about the strength of the metaphor of the fitness landscape, my view is that the people who argue that it's not useful or even misleading are wrong. Yes, some people may take the analogy with a geographic landscape too far. However, a fitness landscape is simply a fitness function, and mathematically functions can of course be of as many variables as one likes. Thus, the high-dimensionality of real fitness landscapes does nothing to diminish the value of thinking in terms of fitness landscapes. Also, while high-dimensional fitness landscapes cannot be nicely visualized, there are other ways to get information about its structure (which is crucial for evolutionary dynamics). Here's is a paper from last year I wrote on that: Critical properties of complex fitness landscapes.

You can read Gavrilet's contribution to the Altenberg conference here: High-dimensional fitness landscapes and the origins of biodiversity.

I agree with Gavrilets' conclusion that much of empirical fitness landscapes will contain neutral networks, akin to his 'holey landscapes'. He cites papers showing this is the case in RNA, proteins, bacteria, viruses, and artificial life. However, there is ample evidence from those as well that fitness landscapes are rugged, i.e., contains multiple peaks and valleys. The paper by Chou and Khan cited in the OP are direct evidence that there is at least one peak, for example. The particular implementation of NK that I am using here does not result in neutral networks, but even if it was set up to contain many neighboring genotypes of effectively equal fitness, it wouldn't change the results much, provided there are still peaks and valleys.

Also in the comments to the OP, Nofutur876 mentions Kryazhimskiya et al. (2009). That paper makes one assumption that is very unrealistic in bacterial species, such as E. coli (which they compare with): "We assume that the mutation rate is sufficiently small that, at most, one mutant segregates in the population at any time". But the mutation-supply rate can be as much as 20,000, which results in multiple mutations segregating at the same time, so I am not comfortable accepting their results. They look at three different types of landscapes, two of which are highly unrealistic (an uncorrelated fitness landscape in which there is no correlation in fitness between neighboring genotypes, and the "stairway to heaven", in which all mutations have the same effect), and then one where there is no epistasis, which is only somewhat unrealistic, in that there could be parts of biological fitness landscapes that are non-epistatic. However, evolutionary dynamics in these single-peak landscapes bears little resemblance to that in rugged - and more realistic - landscapes.

Nofutur876 also asks how I measure landscape ruggedness. Ruggedness is simply a measure of how many peaks a fitness landscape contains. The more peaks the more rugged it is. One may also describe a landscape with the same density of peaks but with a greater range in fitness (global peak minus deepest dip) as more rugged. However, there aren't any formal definitions, as far as I know. Since knowing the structure of real fitness landscape is very difficult, it is not a measure that has much utility. In the NK model ruggedness increase both as a function of N and K. That is, the more loci the model contains (N), the more peaks is has, and the more loci interact to produce the fitness of each trait (K+1), the more peaks there are and the greater the fitness range will be.