Field of Science

Evolutionary dynamics in holey fitness landscapes

ResearchBlogging.orgWhat do real fitness landscapes look like? Do they look more like the image on the left, a nearly-neutral holey fitness landscape, or the one on the right, a rugged fitness landscape with many distinct peaks?

Those are only in two dimensions, so the question is also if depicting anything in two dimensions conveys intuitions that are at all correct.

Holey fitness landscapes (Gavrilets and Gravner, 1997, Gavrilets 1997) are approximations of real fitness landscapes where all genotypes are assigned a fitness value of either zero or one. After normalizing fitnesses to be between zero and one, those that are lower than one are assigned a fitness of zero1. Because real fitness landscapes are of extremely high dimensionality2, and assuming that genotypes have fitnesses that are randomly distributed3, it follows that there exist a nearly-neutral network of genotypes connected by single mutations that has fitness (effectively) equal to one.

The proposition is then that this holey landscape model is a good approximation of real fitness landscapes. It hypothesizes that the evolutionary dynamics on real fitness landscapes is similar to that on holey landscapes, and that distinct peaks like in the image on the right do not really exist. And this is a testable prediction.

Take a look at these videos. They depict populations evolving in two-dimensional fitness landscapes at a very high mutation rate. (You can also download the videos from my research website.)


In all three cases the population size is 2304 (that's (3*16)2, in case you're wondering), mutation rate is 0.5, the grid is 200x200 pixels (i.e. genotypes), and mutations cause organisms to move to a neighboring pixel. Ten percent of the population is killed every computational update (which gives an approximate generation time of 10 updates), and those dead individuals are replaced by offspring from the survivors selected with a probability proportional to fitness (asexual reproduction). Top: neutral landscape where all genotypes have the same fitness. Middle: Half-holey landscape with square holes of 10% lower fitness (size of holes is 14x14 pixels). Bottom: Holey landscape where the genotypes in the holes have fitness zero.

The proposition is that the dynamics of the populations should be the same no matter how deep the holes are. The populations in the half-holey and in the holey landscapes should evolve in comparable ways if the holey landscape is a good approximation.

So what do you think?

What I think is that the evolving population in the top (neutral) and middle (half-holey) landscapes resemble each other, whereas they look nothing like the bottom (holey) landscape. In the half-holey landscape the population takes advantage of the holes all the time, meaning that many individuals who are in them reproduce, even though they have a clear fitness disadvantage. The lesson is that being disadvantaged is just okay, and populations can easily cross quite deep valleys in the fitness landscape. But obviously not when the valleys consist of genotype with zero fitness; evolution in holey landscapes is much impeded compared to rugged landscapes, which is why I think they are not a good approximation.

Caveats: These populations are evolving at a very high mutation rate. When I redid it with a much lower mutation rate (0.05), the neutral and half-holey landscapes stop resembling each other, and the half-holey and holey landscapes look more alike. However, evolution happens so slowly in this case that it is difficult to distinguish the dynamics, so the matter is unresolved so far (however, I have other evidence that lower and more realistic mutation rates do not change this conclusion - some preliminary data in Østman and Adami (2013)). A second caveat is that the whole holey landscape idea relies on the fitness landscape being multidimensional, and so how can I even allow myself to compare evolution of populations in half-holey and holey landscapes in just two dimensions? That is valid question: the intuitions we get from these animations may lead us to think we know something about evolution in multi-dimensional landscapes, while the original premise of Gavrilets' idea was that we exactly cannot. Unfortunately, while this is an empirical question - meaning that it could be tested - the holey landscape model posits that the neutral network appears at very high dimensionality. What this dimensionality is is unclear, so even if I were to evolve populations in 2,000 dimensions (which is not computationally feasible - the limit is a little over 30 binary loci), one could always claim that not even that many are enough. Sighs.

1 Genotypes with fitness greater than 1 divided by the population size, N, are effectively the same, because selection cannot "see" differences smaller than 1/N.
2 High dimensionality means a large number of genes (loci) or number of nucleotides.
3 We already know that this is not a very good assumption, as there are indications that fitness landscapes are non-randomly structured with high fitness genotypes clustered with other fit genotypes (Østman et al, 2010), but we don't know if it is enough to render the holey landscape model useless.

Gavrilets S, and Gravner J (1997). Percolation on the fitness hypercube and the evolution of reproductive isolation. Journal of theoretical biology, 184 (1), 51-64 PMID: 9039400

Gavrilets S (1997). Evolution and speciation on holey adaptive landscapes. Trends in ecology & evolution, 12 (8), 307-12 PMID: 21238086

Østman B and Adami C (2013). Predicting evolution and visualizing high-dimensional fitness landscapes, in Recent Advances in the Theory and Application of Fitness Landscapes" (A. Engelbrecht and H. Richter, eds.). Springer Series in Emergence, Complexity, and Computation DOI: 10.1007/978-3-642-41888-4_18


  1. Nice post. One thing I wonder about is that all my work with small population sizes and selection suggests that one effect of drift and selection is to effectively lower the dimensionality of landscapes. This suggests that perhaps the holey landscape model is appropriate early in the process of adaptation, but less appropriate late in the process. Sadly, I have no clue how to go about investigating that speculation.

  2. I see what you mean. But the effect would depend on how much dimensionality is lowered. Can you quantify that? However, either way, I have other reasons to believe that real landscapes do not resemble holey landscapes even at high dimensions. Gavrilets' mathematical argument depends on genotypes having random fitnesses, such that there is no particular structure to the landscape, and that is probably not true at all (genotype have correlated fitnesses).

    Also, if the population is maladapted initially, then there is not much to gain from a holey landscape model, since all the population does at first is to climb the closest peak(s).

  3. I don't think fitness is a scalar, I think it is a vector having components like lifespan and number of offspring. (Just as utility is not a single scalar like money. see Theory of games and economic behavior by von Neumann and Morgenstern, pg 29.)

  4. Robert, fitness is as a matter of fact a scalar as used in evolutionary biology (despite how utility is defined in economics). You can definitely say that there are different aspects of an organism's traits that contribute to fitness, such as fecundity, offspring viability, lifespan, and others, but fitness itself is defined as a scalar. If you don't want to take this only from me, take a look at Joe Felsenstein's comments in this post on where we discussed the exact same thing contra another economist.

  5. Interesting post!
    Why do you say that the computational limit is 30 binary loci?


  6. Right, I do actually know that more has been done, but I can't personally do more than 31 or 32 and still record line-of-descent data, which for my purposes is crucial.


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