Field of Science

Crossing valleys in fitness landscapes

ResearchBlogging.orgWith his "holey adaptive landscapes", Sergey Gavrilets (e.g. 1997) solved the problem of crossing valleys of low fitness in the fitness landscape* by positing that for high-dimensional landscapes (which is realistic - typically the genotype consists of thousands of genes and many more DNA nucleotides) there is always a ridge between fitness "peaks" (which are then not really peaks). The only rationale for that idea is that the more neighbors a genotype has, the higher the chance that the fitness of one of them is about the same. However, there are indications that this is generally not true. Gavrilets himself says that if all the high fitness genotypes are over in one "corner" of the fitness landscapes, then there could not always be ridges. One way to formulate this is Kauffman's Massif Central hypothesis, which just says that the chance of finding a fitness peak of high fitness is higher closer to another high peak; peaks cluster in genotype space, and there is a correlation between peak fitness of neighbors. This has already been shown to be true in Kauffman's NK landscape (Østman et al., 2010), and is under investigation in other models as well. Stay tuned.

Ridges are irrelevant
That is not to say that it couldn't be true that there are ridges in real biological fitness landscapes of extremely high dimensionality. After all, the numerical landscapes investigated have few genes/loci in comparison. But, looking at real biological fitness landscapes empirically, there is every reason to believe that they are rugged, containing many peaks varying in height/fitness. I could state it like this: no one (that I have heard of) have shown that there is always a path of about constant fitness between any two genotypes. In other words, there are generally (at least) not any accessible paths between genotypes separated by more than one mutation that does not vary enough in fitness that selection can distinguish between them**. However, even if there is, it doesn't matter! If there were always paths of neutral fitness - ridges - between any two genotypes, it would be extremely unlikely that the population would find them. The ridges, supposedly, appear in fitness landscapes of very high dimensionality, meaning that the number of neighboring genotypes is going to be huge, so when increasing dimensionality and the first ridge appears, there are already a fantastic number of mutational neighbors, making it very improbable that the ridge will be discovered by stochastic processes (as evolutionary processes inherently are).

Fitness landscapes are not static
Not only has it been shown that valleys can be crossed when the mutation rate is not prohibitively low (Østman et al., 2012), as in the strong-selection weak-mutation regime (SSWM), where each new mutation is lost or goes to fixation alone, so no two mutations segregate in the population at the same time. But Gavrilets assumed that fitness landscapes are static in both space and time. Static in space means that they are the same in different geographical locations (wet vs. dry conditions, for example). Static in time means that for one location it stays constant and fitness does not change from one point in time to another. And I really should not have to explain how not true this is, right? I mean, not only is it obvious that a genotype adapted to a wet environment will not have the same fitness in a dry environment, it should also make immediate sense that the environment at one location can change over time, for example from a wet climate to a dry one. Fitness landscapes are clearly not static functions (references, you lazy bastard!).

The fitness landscape changes when 1) environmental conditions change and 2) when the population changes. Changes in population size can reduce the strength of selection; the larger the population is, the better able selection is to distinguish small fitness effects. When the population size is low, stochastic effects dominate, and genetic drift rules. In this case, valleys may be crossed in small populations just because the decrease in fitness while crossing the valley matters less (but see Weissman et al., 2009).

The effect of changing the environment is to change the fitness landscape. This could result in peaks shifting position in genotype-space, peaks appearing and disappearing, and deep valleys becoming shallow or vice versa. In two dimensions, instead of thinking of a rigid landscape, think of a seascape of water (e.g., Mustonen and Lässig, 2009). In this case, where fitness as a function of genotype is forever changing, evolutionary dynamics (i.e., moving around in the fitness landscape, crossing valleys, and locating new peaks) may be reduced to moving only uphill, rather than having to actually tolerate deleterious mutations at all. Some landscapes may indeed be rather static over longer periods of time, and then the dynamics of populations crossing valleys may be relevant. But it is totally possible that all the important evolutionary changes occur when the fitness landscape changes, rendering theories of valley-crossing somewhat immaterial.

* There are these two terms in use among researchers: adaptive landscapes and fitness landscapes. The only thing the former term has going for it is that Sewall Wright (1931) - accredited as the inventor of the idea - called it "adaptive landscapes". However, most people actually call it "fitness landscapes", but in addition to that important fact, it also makes a lot more sense. A fitness landscape is a function where fitness is given by genotype or phenotype values (rather than frequencies - incidentally, Gavrilets and I agree that fitness as a function of population allele frequencies makes no sense), so it makes a lot more sense to call it that. On top of that, fitness as a function of genotype/phenotype does not have to have anything to do with adaptation. The fitness landscapes can be flat, in which case there will be no adaptation going on. To my exasperation I just discovered a new book here at the Evolution 2012 conference in Ottawa by the title of The Adaptive Landscape in Evolutionary Biology (Oxford University Press). Nearly all the chapters, written by more or less famous people in the field, have 'adaptive landscape' in the title. Piss me off, it does.

 ** If two genotypes differ in fitness by a small amount, it may be too small for selection to distinguish between them. Since evolution is an inherently stochastic process, in which genetic drift is always present, and selection only chooses who gets to reproduce based on probabilities (fitness can be thought of as this probability), having higher fitness than your neighbor does not guarantee that you will have more offspring; it only makes it so on average. Generally, selection can distinguish fitness effects that are greater than one divided by the population size (s>1/N). If the selection coefficient (the measure of the fitness effect of a mutation, s=w'/w-1, where w' is the fitness with mutation, and w without) is less than one over the population size, then that mutation/genotype will drift, and selection makes no difference. The smaller the population is, the larger a mutation's fitness effect has to be for selection to see it, and therefore selection is weaker in small populations. This is the basis of Sewall Wright's Shifting Balance Theory (Wright, 1982), which explains how crossing valleys in a rugged fitness landscape can be done by breaking the population up into smaller groups (demes), which are then able to drift across the valleys, because selection is now weaker.

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

Mustonen V and Lässig M (2009). From fitness landscapes to seascapes: non-equilibrium dynamics of selection and adaptation. Trends in genetics : TIG, 25 (3), 111-9 PMID: 19232770

Weissman DB, Desai MM, Fisher DS, and Feldman MW (2009). The rate at which asexual populations cross fitness valleysTheoretical population biology, 75 (4), 286-300 PMID: 19285994

Wright S (1931). Evolution in Mendelian populations Genetics (16), pp. 97–159

Wright S (1982). The shifting balance theory and macroevolution. Annual review of genetics, 16, 1-19 PMID: 6760797

Østman B, Hintze A, and Adami C (2010). Critical properties of complex fitness landscapes Proc. 12th Intern. Conf. on Artificial Life, H. Fellerman et al., eds. (MIT Press, 2010), pp. 126-132 arXiv: 1006.2908v1

Østman B, Hintze A, and Adami C (2012). Impact of epistasis and pleiotropy on evolutionary adaptation. Proceedings. Biological sciences / The Royal Society, 279 (1727), 247-56 PMID: 21697174


  1. Okay, so there may not be ridges of constant fitness between peaks. What about bridges that make the valley less deep and the peaks less high? Anyway, Gavrilets has pointed out one way in which the image of a static, 3-dimensional, rugged fitness landscape is false (by adding more dimensions) and you another (by adding dynamic changes of the landscape). That's scientific inquiry: there's a problem, let's look for a solution. Michael Behe, on the other hand, has seized on the image of a static, rugged fitness landscape and thrown up his hands claiming another Edge beyond which evolution is impossible.

  2. Yes, I have seen your post on Behe and fitness landscapes.

    On top of what you say, I don't think there is any evidence for ridges, and Gavrilets' mathematical argument for it may not at all be valid. And, as I pointed out, even if there are ridges, those trajectories are highly unlike to ever be found, and therefore are not solutions to the evolutionary dynamics problem of valleys.

  3. Philosophically speaking, an argument will be valid, if the conclusion is a matter of course given the premises. In this sense Gavrilets's argument is mathematically valid, I guess, but some of his premises may not hold in nature.

    P.S.: I predict that the day Behe or some other creationist realizes the static-rugged-landscape image is not correct he'll turn around 180° and claim that therefore evolution cannot work, where beforehand that very image has been used for the same argument.

  4. Right, I should say that Gavrilet's mathematical argument does not mean that there are ridges in biological fitness landscapes.

  5. And while I'm not an expert and have not followed the history of fitness landscape research, my hunch is that Gavrilets deserves some share in bringing back an issue into rigorous scientific inquiry that has been degenerated to a mere metaphor.

    1. Fitness landscapes were introduced as a metaphor by Wright in 1932. They were a visual aid that metaphorically expressed what he mathematically demonstrated in his 1931 paper. Many authors other authors also used them as a metaphor.

  6. Sure, he deserves credit for that. And for lots of other great research.

  7. Dare I say that the various claims of ridges in fitness space (or on fitness landscapes) is a little reminiscent of the widespread invocation of land bridges to explain biogeographical distributions before plate tectonics? :)


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