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Scapes - SEWALL WRIGHT, THE ADAPTIVE LANDSCAPE, AND WHAT WE DON'T KNOW (page 2)

SEWALL WRIGHT, THE ADAPTIVE LANDSCAPE, AND WHAT WE DON'T KNOW (page 2)

In 1926, Wright, his wife Louise, and their two sons (to be joined by a daughter in 1929) moved back to the Midwest, where Wright had been offered a professorship at the University of Chicago. In Chicago, Wright continued his work on guinea pigs, taught classes on zoology and genetics, and started a collection of the spiders of the region. He also updated and revised the manuscript he had written while working for the USDA, and in 1930, he submitted his whopping 62-page Evolution in Mendelian Populations was published in Genetics. This important paper was one of a few works in the early 1930’s that brought together Mendelian genetics and natural selection in a rigorous mathematical way.

Not long after this submission, Wright was sent a copy of Ronald Aylmer (RA) Fisher's The Genetical Theory of Natural Selection to review for the Journal of Heredity. This book touched on many of the same topics as Wright’s paper, combining Mendelian genetics and evolutionary theory with rigorous mathematics.

Using different techniques, Wright and Fisher derived formulas for the change of allele frequencies in populations of individuals. The two wrote letters to each other, pointing out mistakes and asking questions to clarify the differences and similarities between their formulas. In the end, they agreed on these mathematical results, producing basic formulas that relate variables like population size (N), mutation rate (u), and the selection coefficient (s) for a particular allele or mutation to results about the probably and speed of fixation of that allele (fixation meaning that the allele reaches a frequency of 100% in the population). Population genetics was born! The mathematicians could rejoice now that the whole of evolution could be described by equations! Arguments about evolution were a thing of the past

The first exclamation is true, but the second is wrong, and the third is just laughable. These formulas were an enormous step towards bringing the process of evolution into the realm of mathematical theories, but they left quite a lot to be interpreted or described. First, the variables themselves were in question; what were realistic population sizes and mutation rates for natural populations? What is the distribution of selection coefficients for random mutations for some species? These are big questions, ones that prompted debate between Fisher and Wright, and led them both to seek collaborations with researchers who could try to measure the important parameters in the field. But Provine points out the more important weakness of the formulas:

"The problem was that the available quantitative methods could easily be applied only to pairs of alleles at each locus, and frequently, as in the entire section on the distribution of gene frequencies [in Wright's 1931 paper], only one locus could be treated formally. Yet Wright's whole approach to evolution in domestic and natural populations was deeply tied to multiple alleles at interacting loci." (p. 278)

This is an often-overlooked barrier that these great minds hit: a gap between the quantitative work and the qualitative thinking that could not be crossed by exact mathematics. This is what Emanuele Serrelli calls an "epistemological gap" in his doctoral dissertation about the adaptive landscape concept (Serrelli 2011), and what Provine referred to as "the tension between the quantitative formal theory and the qualitative theory" (p. 279).

As Provine implies, this issue of multiple loci becomes problematic only if the mutations or different alleles at the loci interact. If the effects were all completely independent in their contributions to fitness (as in the special case when all the changes are neutral), the one-locus population genetics formulas could be extended to "tell the whole story" of evolution. In his Genetics paper, Wright begins to work out a two-loci case, but notes that "the frequency of A depends on the frequency and selection of B, becoming independent only if sab = sa + sb." Here sa is the selection coefficient for a change a, sb is the selection coefficient for a change b, and sab is the selection coefficient for both changes together. Wright goes on to say "It does not seem profitable to pursue this subject further for the purpose of the present paper, since in the general case, each selection coefficient is a complicated function of the entire system of gene frequencies and can only be dealt with qualitatively." This is precisely the epistemological gap that Wright, Fisher, Haldane, and others stood at the edge of in 1932. Their mathematics could describe evolution in "a given population at a given moment," but they could not describe long-term dynamics in light of the "epistatic relationships" between genes, which is exactly captured by Wright's rejection of the assumption sab = sa + sb.

Both Wright and Fisher had too much appreciation for the importance of gene interaction to quickly extend their formulas to long-term evolutionary dynamics. They recognized the issue as a formidable stumbling block, and they each came up with ways to work around it.

(Frank’s Perspective On) Fisher’s Fundamental Theorem


Fisher's introduced his fundamental theorem of natural selection modestly, comparing it to the second law of thermodynamics and claiming that it "hold[s] the supreme position among the biological sciences" (Fisher 1930, p. 37). In his book, Fisher presented the theorem like so: "The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time" (p. 35). In his review, Wright criticized Fisher's notion of "genetic variance":

"He assumes that each gene is assigned a constant value, measuring its contribution to the character of the individual (here fitness) in such a way that the sums of contributions of all genes will equal as closely as possible the actual measures of the character in the individuals of the population. Obviously there could be exact agreement in all cases only if dominance and epistatic relationships were completely lacking ... it may be safely assumed that there are always important epistatic effects. Genes favorable in one combination, are for example, extremely likely to be unfavorable in another" (Wright 1930).

According to an interesting recent article by Steven Frank (2012), this criticism of Fisher's fundamental theorem may have been misguided. Frank explains that Fisher had been aware and interested in genetic interactions for over ten years before the publication of his book. He argues that if we view the theorem as a statement about the dynamics at any one instant, it is complete and exact. Fisher’s theorem involves the "average effect" of alleles, which can be described as "the heritable contribution of each allele in the context of all of the genetic interactions in the population at any moment in time" (Frank 2012, emphasis is mine). But while Fisher's mathematics theoretically considers all of the interactions present at one specific moment in a population, it makes no comment on genetic interactions that are one or several mutational steps away from individuals in the population. This means that without an explicit model for the structure of genetic interactions, the fundamental theorem can only comment on what is happening right now, not what will happen once the first new genetic change takes place. Frank contends that Wright and many others have repeatedly failed to understand that the fundamental theorem "is clearly designed to express laws rather than to calculate long-term dynamics." More generally, Frank writes,

"Fisher realized that one cannot make a complete model of evolutionary dynamics. Too many factors come into play: changes in the physical environment, changes in competitive intensity within and between species, and changes in the complex non-additive interactions between genes that fluctuate in frequency. Given the complexity of 'open' systems in which forces flow from a variety of unknown sources, Fisher sought a way to define a 'closed' subset in which one could completely and exactly study the process of natural selection."

This is not to imply that Fisher was unconcerned with long-term evolutionary dynamics, but, according to Frank at least, he just did not think his "law" had much to say about them. In order to avoid the epistemological gap, Fisher restricted his theorem to apply in a snapshot of time and in a certain environment. In this setting, it was exact. Wright’s criticism, while perhaps misplaced, points toward the same very difficult problem that Frank is referencing above - applying an exact mathematical theorem to long-term evolutionary dynamics in a system with interactions.

Wright’s recognizes the gap, then leaps over it while people aren’t looking


In the latter part of his 1931 paper, Wright tackles the idea of long-term evolution head-on, describing the conditions that he sees as most effective for the evolution of populations. He references the mathematics from earlier in the paper, but the section is much more qualitative. He begins to describe an early version of his shifting balance theory, in which partially subdivided populations differentiate by drift, some find new beneficial genetic combinations, and those that are the most fit spread by migration to the larger populations. Both the motivation and inspiration for this theory emerges from Wright’s work at Harvard and at the USDA. The idea of beneficial combinations reachable by drift comes from Wright’s belief in the ubiquity of major genetic interactions, and the shifting balance process is intimately tied to Wright’s work on methods for livestock breeding.

In the correspondence between Fisher and Wright after Wright’s review of The Genetical Theory of Natural Selection, Wright attempts to explain how the ubiquity of gene interactions leads to his theory. This is where the first reference to what would become known as the adaptive landscape emerges. He writes:

"Some of aspects of the ideas I tried to express in pages 353 to 355 of my review [the quote above is from pg. 353] might be visualized as follows: Think of the field of visible joint frequencies of all genes as spread out in a multidimensional space. Add another dimension measuring degree of fitness. The field would be very humpy in relation to the latter because of epistatic relations, groups of mutations which were deleterious individually producing a harmonious result in combination. In the figure below this field (very imperfectly represented by a single line) is plotted against fitness." (Wright to Fisher, Feb. 3 1931, in Provine p. 272)

photo of sewall wright
Wright's two-dimensional fitness surface from his letter - redrawn by me.
I think the important thing for us to note as we sit here over 80 years later is this: Wright and Fisher did some serious, rigorous mathematics to come up with their population genetics formulas. Yet here they are both essentially taking educated guesses at what the "field of ... joint frequencies of all genes" with "another dimension measuring degree of fitness" looks like. They didn’t know, and their mathematics could not help them figure it out.

The climax of this story is in 1932, in Ithaca, New York, at the International Congress of Genetics. This was no small conference in Ithaca; it was the biggest and most important meeting of the decade for the geneticists in attendance. And the organizer, Edward Murray East, was intent on using the conference as a forum for the ideas of Wright, Fisher, and Haldane to be presented to a broad audience. Specifically, East asked the scientists to present a paper in way that a non-technical (read: don’t-want-to-look-at-all-your-complicated-equations) audience could understand. In addition, the published version of the papers in the Proceedings could fill no more than ten pages. It was in this paper and presentation that Wright fully introduced the adaptive landscape concept. He also more explicitly sketched the metaphor between his multidimensional space with a dimension for fitness and a physical landscape (where altitude is the fitness dimension). In the presentation, he displayed his now-famous diagrams:

diagram1 diagram2
wright6


Many have claimed over the years that Wright invented the landscape metaphor as a way of conveying his mathematical work for the non-mathematical audience in Ithaca. Pigliucci (2012) writes, "Wright introduced the metaphor because his advisor suggested that a biological audience at a conference would be more receptive to diagrams than toward a series of equations. But of course the diagrams are simply not necessary for the equations to do their work."

While Wright's decision to present the adaptive landscape diagrams may in part have been spurred by East’s request to present without too many equations, to claim that it was the main cause of the invention of the concept is a mistake. Remember Wright's first sketch of the concept? It was in a letter to Fisher, one of the most mathematical minds of his time. Wright didn’t introduce the concept as a substitute for his equations, he introduced it as a way of illustrating his qualitative theory, which as we have seen above, was outside of an exact mathematical treatment. The assertion that the diagrams are not needed for the equations to work is technically true, but only because the work of the equations and the work of the diagrams are fundamentally different. The equations are largely concerned with either a snapshot, one-locus, and/or additive view of evolution. The diagrams display Wright’s interpretation of how the results from his equations would lead to long-term evolutionary dynamics within a system of interacting genetic factors. The concept sits at the tense edge between quantitative analysis and qualitative interpretation, but at least in its first conception, it clearly falls on the qualitative side of the divide. As Serrelli writes,

"Wright managed to cross the epistemological gap separating his equations from the Mendelian population as a whole, for which no equations were (and are) available … he built a bridge by means of his limited laboratory experience, intuition, and heuristics" (p. 72) (by Mendelian population Serrelli means a full model of the combination space of genetic factors).

Wright didn't do the best job letting everyone know when he jumped from his quantitative work into a more qualitative realm. The images of the adaptive landscape that Wright presented looked fairly rigorous, with surfaces drawn on axes and sometimes with equations alongside them (though these were not equations for the surfaces). Provine writes,

"Since interaction between loci was integral to Wright's whole view of the process of evolution, one might guess that in turning from this section to the qualitative presentation of his actual theory of evolution in nature, he would mention the limitation directly and show how the qualitative view related to a much narrower quantitative base. Neither Wright nor Fisher did a careful job of explaining this limitation at the crucial stage of advancing beyond the strictly formal models to discussions of evolution in nature." (p. 280)
Oops.