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Population genetics, Fisher's theorem, fitness landscapes, drift, and draft

The foundations for the critically important synthesis of Darwinism and genetics were set in the late 1920s and early 1930s by the trio of outstanding theoretical geneticists: Ronald Fisher, Sewall Wright, and J. B. S. Haldane. They applied rigorous mathematics and statistics to develop an idealized description of the evolution of biological populations. The great statistician Fisher apparently was the first to see that, far from damning Darwinism, genetics provided a natural, solid foundation for Darwinian evolution. Fisher summarized his conclusions in the seminal 1930 book The Genetical Theory of Natural Selection (Fisher, 1930), a tome second perhaps only to Darwin's Origin in its importance for evolutionary biology.5 This was the beginning of a spectacular revival of Darwinism that later became known as Modern Synthesis (a term mostly used in the United States) or neo-Darwinism (in the British and European traditions).

It is neither necessary nor practically feasible to present here the basics of population genetics.6 However, several generalizations that are germane to the rest of the discussion of today's evolutionary biology can be presented succinctly. Such a summary, even if superficial, is essential here. Basically, the founders of population genetics realized the plain fact that evolution does not affect isolated organisms or abstract species, but rather affects concrete groups of interbreeding individuals, termed populations. The size and structure of the evolving population largely determines the trajectory and outcome of evolution. In particular, Fisher formulated and proved the fundamental theorem of natural selection (commonly known as Fisher's theorem), which states that the intensity of selection (and, hence, the rate of evolution due to selection) is proportional to the magnitude of the standing genetic variation in an evolving population, which, in turn, is proportional to the effective population size.

Box 1-1 gives the basic definitions and equations that determine the effects of mutation and selection on the elimination or fixation of mutant alleles, depending on the effective population size. The qualitative bottom line is that, given the same mutation rate, in a population with a large effective size, selection is intense. In this case, even mutations with a small positive selection coefficient ("slightly" beneficial mutations) quickly come to fixation. On the other hand, mutations with even a small negative selection coefficient (slightly deleterious mutations) are rapidly eliminated. This effect found its rigorous realization in Fisher's theorem.

A corollary of Fisher's theorem is that, assuming that natural selection drives all evolution, the mean fitness of a population cannot decrease during evolution (if the population is to survive, that is). This is probably best envisaged using the imagery of a fitness landscape, which was first introduced by another founding father of population genetics, Sewall Wright. When asked by his mentor to present the results of his mathematical analysis of selection in a form accessible to biologists, Wright came up with this extremely lucky image. The appeal and simplicity of the landscape representation of fitness evolution survive to this day and have stimulated numerous subsequent studies that have yielded much more sophisticated and less intuitive theories and versions of fitness landscapes, including multidimensional ones (Gavrilets, 2004).7 According to Fisher's theorem, a population that evolves by selection only (technically, a population of an infinite size—infinite populations certainly do not actually exist, but this is convenient abstraction routinely used in population genetics) can never move downhill on the fitness landscape (see Figure 1-1). It is easy to realize that a fitness landscape, like a real one, can have many different shapes. Under certain special circumstances, the landscape might be extremely smooth, with a single peak corresponding to the global fitness maximum (sometimes this is poetically called the Mount Fujiyama landscape; see Figure 1-1A). More realistically, however, the landscape is rugged, with multiple peaks of different heights separated by valleys (see Figure 1-1B). As formally captured in Fisher's theorem (and much in line with Darwin), a population evolving by selection can move only uphill and so can reach only the local peak, even if its height is much less than the height of the global peak (see Figure 1-1B). According to Darwin and Modern Synthesis, movement across valleys is forbidden because it would involve a downhill component. However, the development of population genetics and its implications for the evolutionary process changed this placid picture because of genetic drift, a key concept in evolutionary biology that Wright also introduced.

Figure 1-1

Figure 1-1 Fitness landscapes: the Mount Fujiyama landscape with a single (global) fitness peak and a rugged fitness landscape.

As emphasized earlier, Darwin recognized a crucial role of chance in evolution, but that role was limited to one part of the evolutionary process only: the emergence of changes (mutations, in the modern parlance). The rest of evolution was envisaged as a deterministic domain of necessity, with selection fixing advantageous mutations and the rest of mutations being eliminated without any long-term consequence. However, when population dynamics entered the picture, the situation changed dramatically. The founders of quantitative population genetics encapsulated in simple formulas the dependence of the intensity of selection on population size and mutation rate (see Box 1-1 and Figure 1-2). In a large population with a high mutation rate, selection is effective, and even a slightly advantageous mutation is fixed with near certainty (in an infinite population, a mutation with an infinitesimally small positive selection coefficient is fixed deterministically). Wright realized that a small population, especially one with a low mutation rate, is quite different. Here random genetic drift plays a crucial role in evolution through which neutral or even deleterious (but, of course, nonlethal) mutations are often fixed by sheer chance. Clearly, through drift, an evolving population can violate the principle of upward-only movement in the fitness landscape and might slip down (see Figure 1-2).8 Most of the time, this results in a downward movement and subsequent extinction, but if the valley separating the local peak from another, perhaps taller one is narrow, then crossing the valley and starting a climb to a new, perhaps taller summit becomes possible (see Figure 1-2). The introduction of the notion of drift into the evolutionary narrative is central to my story. Here chance enters the picture at a new level: Although Darwin and his immediate successors saw the role of chance in the emergence of heritable change (mutations), drift introduces chance into the next phase—namely, the fixation of these changes—and takes away some of the responsibility from selection. I explore just how important the role of drift is in different situations during evolution throughout this book.

Figure 1-2

Figure 1-2 Trajectories on a rugged fitness landscape. The dotted line is an evolutionary trajectory at a high effective population size. The solid line is an evolutionary trajectory at a low effective population size.

John Maynard Smith and, later, John Gillespie developed the theory and computer models to demonstrate the existence of a distinct mode of neutral evolution that is only weakly dependent on the effective population size and that is relevant even in infinite populations with strong selection. This form of neutral fixation of mutations became known as genetic draft and refers to situations in which one or more neutral or even moderately deleterious mutations spread in a population and are eventually fixed because of the linkage with a beneficial mutation: The neutral or deleterious alleles spread by hitchhiking with the linked advantageous allele (Barton, 2000). Some population-genetic data and models seem to suggest that genetic draft is even more important for the evolution in sexual populations than drift. Clearly, genetic draft is caused by combined effects of natural selection and neutral variation at different genomic sites and, unlike drift, can occur even in effectively infinite populations (Gillespie, 2000). Genetic draft may allow even large populations to fix slightly deleterious mutations and, hence, provides them with the potential to cross valleys on the fitness landscape.

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