evolutionary theory

Chapter 2 of R. A. Fisher's Genetical Theory of Natural Selection is remarkable for many reasons. In it, he presents a model of selection in an age-structured population, the concept of reproductive value, and the Fundamental Theorem. Toward the end of the chapter, he discusses "The Nature of Adaptation," presenting a geometric model to justify the assertion that the probability of favorable genetic changes declines as the effect size of those changes increases.

In order to consider in outline the consequences to the organic world of the progressive increase of fitness of each species of organism, it is necessary to consider the abstract nature of the relationship which we term 'adaptation.' This is the more necessary since any simple example of adaptation, such as the lengthened neck and legs of the giraffe as an adaptation to browsing on high levels of foliage, or the conformity in average tint of an animal to its natural background, lose, by the very simplicity of statement, a great part of the meaning which the world really conveys. For the more complex the adaptation, the more numerous the different features of conformity, the more essentially adaptive the situation is recognized to be. An organism is regarded as adapted to a particular situation, or to the totality of situations which constitute its environment, only in so far as we can imagine an assemblage of slightly different situations, or environments, to which the animal would on the whole be less well adapted; and equally only in so far as we can imagine an assemblage of slightly different organic forms, which would be less well adapted to that environment (38).

I've highlighted that last sentence, which is saying that organisms fit their environments in possibly many different ways, so that their fitness is not actually tied in any single feature, such as "tint," but is instead an optimum of many features, with respect to any single factor the organism may be more or less well adapted. The rest of that paragraph continues on to make the same point.

Then:

The statistical requirements of the situation, in which one thing [the organism] is made to conform to another [the environment] in a large number of different respects, may be illustrated geometrically. The degree of conformity may be represented by the closeness with which a point A approaches a fixed point O. In space of three dimensions we can only represent conformity in three different respects, but even with only these the general character of the situation may be represented. The possible positions representing adaptations superior to that represented by A will be enclosed by a sphere passing through A and centered at O.

This is really very simple, and the geometric model reveals an interesting switch. Suppose that we imagine an organism as a set of a traits, each of which lies at some distance d₁, d₂, d₃, ..., d_a from the optimum value for that trait, O. We could imagine adaptation as a stepwise process, in which a any one of the a traits may change, and only those changes that reduce d_a will potentially be selected.

But there's no reason at all why we should consider every trait as an independent entity. Suppose that a single genetic change could improve two traits, or three, or even all the traits at the same time.

Or, more interesting, a change might greatly improve trait 1, while making all the rest of the traits marginally worse. Without a word, Fisher has removed the issue of adaptation from the fit of many separate variables, to a single distance in multidimensional space -- a change from cartesian to polar coordinates, as it were.

If A is shifted through a fixed distance, r, in any direction its translation will improve the adaptation if it is carried to a point within this sphere, but will impair it if the new position is outside.

The geometric model really assumes very little. It tells us nothing at all about the relationship between fitness and any particular phenotype, aside from assuming that (1) the relation is continuous within the sphere centered on O with radius A, and (2) there are no "holes" of low fitness within that sphere. This is not a fitness landscape. Fisher's view, as will become clear, was that species are well-adapted in nature. He assumes that the distance between A and O is always rather small in comparison to the kinds of phenotypic effects that mutations might cause. So in assuming that the fitness function is continuous, and that the area is small, he more or less automatically arrived at the assumption that there's only one peak at O.

The next part is the famous one that people remember:

If r is very small it may be perceived that the chances of these two events are approximately equal, and the chance of an improvement tends to the limit 1/2 as r tends to zero; but if r is as great as the diameter of the sphere or greater, there is no longer any chance whatever of improvement, for all points within the sphere are less than this distance from A.

In this model, small changes are roughly 50-50 beneficial versus deleterious, but since their effect is very small, it hardly matters. Big changes are much less likely to be beneficial -- and if they exceed twice the distance from A to O, they can never be beneficial.

After this quote, he gives an exact expression for the probability that a given change r is beneficial ((1/2)(1-(r/diameter))), but this is limited to the three-dimensional model. Then, there's another interesting one-sentence switch:

The chance of improvement thus decreases steadily from its limiting value 1/2 when r is near zero, to zero when r equals d. Since A in our representation may signify either the organism or its environment, we should conclude that a change on either side has, when this change is extremely minute, an almost equal chance of effective improvement or the reverse; while for greater changes the chance of improvement diminishes progressively, becoming zero, or at least negligible, for changes of a sufficiently pronounced character.

Remember that A represents the current "degree of conformity" of the organism to its "particular situation." This degree of conformity might be changed either by changing the organism or its environment -- the implication fo the last confusing sentence from the first paragraph, above.

The point O is not an objective location (as it would be if we assumed it is the optimum within the current environment). Instead, it is a geometric abstraction that exists only in so far as it bears a relation to A in the present multidimensional space of "adaptation". We may change the distance from A to O either by changing the organism or by changing its environment. Fisher refers explicitly to this "assemblage of slightly different situations" with respect to both -- and again, the concept of "slightly different" underlies the assumption that the fitness function within the sphere is continuous.

It seems to me that the idea of "niche construction" falls very easily within Fisher's model. An organism that systematically alters its environment may thereby change its level of adaptation. So even though mutation is an obvious candidate for a process described by the model, I've continued to refer to "change" as a nonspecific term for the unit of adaptation.

The representation in three dimensions is evidently inadequate; for even a single organ, in cases in which we know enough to appreciate the relation between structure and function, as is, broadly speaking the case with the eye in vertebrates, often shows this conformity in many more than three respects. It is of interest therefore, that if in our geometrical problem the number of dimensions be increased, the form of the relationship between the magnitude of the change r. and the probability of improvement, tends to a limit which is represented in Fig. 3. The primary facts of the three dimensional problem are conserved in that the chance of improvement, for very small displacements tends to the limiting value 1/2, while it falls off rapidly for increasing displacements, attaining exceedingly small values, however, when the number of dimesnions is large, even while r is still small compared to d.

Here we see the problem of universal pleiotropy emerging. As the number of dimensions of adaptability increases, the probability that one change will be a net increase in fitness, considering all dimensions, must decrease. This probability remains larger when the distance between A and O is larger, but declines as the number of dimensions increases.

However, what we would call "universal pleiotropy" is intrinsic to Fisher's assumption that the direction of changes in this multidimensional space is random. If changes may occur in any direction, this is the same as asserting that a mutation may induce correlated changes between any set of phenotypes. With thousands of genes, and therefore thousands of dimensions of genetic "adaptedness", we might guess that the dimensionality is high enough to make this assumption useful.

If on the other hand, the direction of changes is constrained in some way, then the dimensionality of the space accordingly is smaller by some degree. This is the rough equivalent of modularity in Fisher's model: if we say that some genetic correlations cannot be changed, we are saying that the phenotypic structure of the organism is modularized.

The remainder of the section discusses the general case in spaces with higher than three dimensions. The basic point is that the probability that a change of a given size will be adaptive increases with the distance from A to O, decreases with the effect size r, and also decreases with the number of dimensions.

Fisher finishes with a paragraph that, had he begun the section with it, might have made everything much clearer:

The conformity of these statistical requirements with common experience will be perceived by comparison with the mechanical adaptation of an instrument, such as the microscope, when adjusted for distinct vision. If we imagine a derangement of the system by moving a little each of the lenses, either longitudinally or transversely, or by twisting through an angle, by altering the refractive index and transparency of the different components, or the curvature, or the polish of the interfaces, it is sufficiently obvious that any large derangement will have a very small probability of improving the adjustment, while in the case of alterations much less than the smallest of those intentionally effected by the maker or the operator, the chance of improvement should be almost exactly half.

But there is an obvious objection: if you twist a knob on a microscope, it may move one of the lenses, but it isn't going to polish it. It's not possible to effect simultaneous changes on distinct elements of the microscope, because a microscope is in fact modular in its construction and controls. That doesn't disprove Fisher's model, but at least in the microscope case, the possible changes are not random in direction, but are constrained to "Cartesian"-like independent axes.

Fisher's general idea is different from the fitness landscape, in the sense used by Sewall Wright. Fisher assumes a single adaptive optimum; Wright espoused the possibility of a "rugged" landscape in which large genetic changes might diverge from the nearest local optimum yet place the population near an alternate (possibly higher) local optimum. But the geometric model shares many properties with the fitness landscape, including its assumptions about hill-climbing toward the local optimum.

I pointed to Sergey Gavrilets work a couple of weeks ago. Fisher's geometric model is most similar to the second meaning of fitness landscape, which pertains to the mean fitness of a population given a discrete genotype -- or in this case, a discrete phenotype-environment combination. Fisher's model is entirely about the relationship of two equilibria: the population mean fitness before and after a given change. It does not deal with the dynamics of the transition from one state to another.

References:

Fisher RA. 1930. The Genetical Theory of Natural Selection. Clarendon Press, Oxford.

I'd like to point readers to a recent essay in Evolution, by Scott V. Edwards, titled, "Is a new and general theory of molecular systematics emerging?"

Edwards covers some of the recent progress and problems encountered when using molecular evidence to test phylogenetic hypotheses. A sampling of the issues: How do we combine information from different sets of molecular data? Can we just compile sequences from many gene loci together into one analysis ("concatenation"), or do we need to make allowances for genealogical diversity among loci? How do prior assumptions affect the outcomes of analyses, like the presence or absence of polytomies (branching points where three or more species emerge simultaneously)?

I try to think of things that students should read as they get up to speed with evolutionary genetics. Edwards' essay raises many important points, and as I read through it, I reflected on the ways that paleoanthropologists increasingly need to be aware of the inner workings of molecular studies of phylogeny.

If we're interested in the phylogeny of species, we need to know how the "tree" of relationships of species may be manifested in the genealogical relationships among genes. Discordances between genes result from the fact that gene trees are not species trees. Species are genetically variable, and the living descendants of an ancient species may have inherited different parts of the variation of ancient species. Depending on the demography of that ancient population, gene trees representing the evolution of two distinct genetic loci may have different topological properties.

From Edwards:

John Avise encapsulated the relationship between gene and species trees well in 1994: “Gene trees and species trees are equally “real” phenomena, merely reflecting different aspects of the same phylogenetic process. Thus, occasional discrepancies between the two need not be viewed with consternation as sources of “error” in phylogeny estimation. When a species tree is of primary interest, gene trees can assist in understanding the population demographies underlying the speciation process” (pp. 133 and 138 in Avise 1994). This essay is in part meant to reemphasize Avise' perspective and to remind readers that species trees are in fact the “primary interest” of systematics.

Genealogies involve some unknown parameters. Applying the fossil and archaeological record may let us constrain those parameters, just as applying molecular biology and pedigree comparisons may let us constrain the parameters describing the mutational process.

To my mind, this is where paleoanthropologists need to be most attentive: Molecular methods are not in conflict with fossil approaches, they implicitly depend upon them. Yet, communication between the two fields rarely involves actual numbers, so a frequent occurrence is that a "bottleneck" in paleoanthropology with a 10 percent reduction in population becomes a "bottleneck" in genetics with a 1000-fold reduction in population.

Testing of demographic hypotheses moved on to genome-wide polymorphism data several years ago. The logical equivalent for species divergences is lineage sorting -- a model that's been applied since the mid-1990's. The hominoids are extremely well studied from the standpoint of molecular systematics, and remain the central example in most theoretical papers incorporating multiple loci. This year I have noticed several interesting implementations of whole-genome polymorphism comparisons among species embedded in phylogenetic trees. The higher mutation rate of CpG sites has long been known, but we now know that a 50-bp or longer flanking region may influence local mutation rate. As we move from genes to gene networks, our comparisons will not be the same nucleotide, but classes of mutations across classes of genes.

This is another of those cases where the future lies in better algorithms. Edwards seems a man after my own heart -- the computer programs lend a superficial veneer of rigor, when the underlying assumptions are in need of challenge:

Producing phylogenies directly from gene sequences essentially in one step, without additional transformations, is now the dominant mode of phylogenetic analysis and indeed it has advanced the field enormously. Nonetheless, I suggest that the very success of this paradigm and the ease with which phylogenies could be produced directly from DNA matrices led to a comfort zone in phylogenetics. If we can imagine systematic methods themselves as a likelihood surface, I suggest that the current paradigm is a local optimum in that surface, an optimum that is useful but ultimately incomplete in so far as it has failed to model the potential for gene tree/species tree discordance even cursorily (Fig. 3) (Edwards 2009:6).

His theme is an old one -- how do we use "total evidence" methods in phylogenetics. Variance among loci gives the problem a newish twist, one that may add information that other techniques have left on the table. But we have to wring it out of the data.

References:

Edwards SV. 2009. Is a new and general theory of molecular systematics emerging? Evolution 63:1-19. doi:10.1111/j.1558-5646.2008.00549.x

I'd like to point readers to James Crow's article in the open access Journal of Biology. Titled, "Mayr, mathematics and the study of evolution," it's a brief summary of some of the important results from mathematical genetics -- sort of a follow-on to Haldane's "A defense of beanbag genetics".

Coming fifty years after Haldane's effort, Crow has been able to include a lot more stuff -- in particular the consequences of the mathematical development of neutral theory, and the effects of computers, permutation tests, and molecular clock models in phylogenetics.

I cannot help but quote this passage, which is direct:

Ironically, Mayr himself unwittingly provided an especially compelling argument for mathematical analysis. His theory of “genetic revolutions” assumed that from a well integrated population, genetic drift in a small founder offshoot will sometimes produce a population with a new set of genotypes integrated in a new way. Intuitively, a small founder population seemed a particularly unlikely place to find a new favorable gene combination, and this was indeed shown to be the case in a very detailed mathematical analysis by Barton and Charlesworth [5]. If Mayr had had more respect for mathematical population genetics, he never would have made what most theorists regard as the mistake of proposing that small founder populations are a likely source of major evolutionary changes by genetic drift (Crow 2009:13.2).

Lest you think this is an argument against the role of chance, Crow later describes the more au courant view of speciation:

Until recently, mathematical theory had contributed little to the study of speciation. Mayr emphasized allopatric speciation and the prevailing model, due to Dobzhansky and Muller [9], prevailed. Recent mathematical studies [10] support it and favor the view that speciation genes correspond to normal genes, selected for their effects within the species. Furthermore, there is evidence that these genes evolve rapidly. Thus, hybrid incompatibility is a by-product of ordinary selection in geographically isolated populations (Crow 2009:13.4).

This model of speciation recognizes chance and contingency, but not mainly from stochastic fluctuations in allele frequency (drift). Instead we have the stochastic processes of mutation and environmental change and the (possibly complex) epistatic interactions among selected alleles.

There's more in the essay. Crow does refer to human evolution -- the out of Africa scenario and Neandertal genetics make appearances not entirely to my taste, but he notes that selective sweeps -- dear to my heart -- are an important feature of the recent landscape of mathematical genetics as well.

Crow could have included a number of other mathematical developments, particularly the Price equation, Hamilton's contributions, and Maynard Smith's "evolutionarily stable strategies", all of which share his theme of the mathematical derivation coming first, and the non-mathematical descriptive formulations only coming later.

References:

Crow JF. 2009. Mayr, mathematics and the study of evolution. J Biol 8:13. doi:10.1186/jbiol117