Wright meets AD: not all landscapes are adaptive
Mark Kirkpatrick, Section of Integrative Biology C-0930, University of Texas, Austin, TX 78712, USA.
Waxman & Gavrilets (2005) have provided a concise and accessible introduction to adaptive dynamics. The paper is a great resource for students and colleagues who want to know what the excitement over this rapidly developing field is all about.
Early on, the review refers briefly to connections between the famous ‘adaptive landscape’ of Sewall Wright and the ‘pairwise invasibility plots’ used in adaptive dynamics. The image of an evolutionary landscape has such visual power that it has become part of the conceptual foundation of evolutionary biology. Unfortunately, this representation sometimes takes precedence over the actual equations it is meant to describe, even among evolutionary cognoscenti. In fact, the confusion began with Wright himself: his figures representing the landscape are inconsistent (Provine, 1986).
As the landscape concept has such a visceral appeal, we will try to straighten out its tortured legacy. We discuss how different conceptions of evolutionary landscapes relate to each other and to the equations that describe evolutionary change, placing them in a historical context. These views are grounded in various approaches that make different assumptions about the genetics underlying the evolutionary process. We begin by reviewing the landscapes described by Wright, then relate them to the landscapes developed in different branches of evolutionary theory, including adaptive dynamics.
Individual fitness, population mean fitness, and Wright's adaptive landscape
All landscapes begin with an individual fitness function, which is simply the function that relates an individual's phenotype, Z, to its fitness, W. With a single trait that varies continuously and fitnesses constant in time, we can visualize the function as a curve whose altitude varies along a single dimension. The idea applies equally to a locus with two alleles; now the fitness function consists of just three points. If fitnesses depend on more than one trait or locus, then the number of horizontal dimensions (that is, arguments of the fitness function) increases accordingly. When relative fitnesses are frequency-dependent, then the horizontal dimensions also include things that depend on the state of the population. In typical situations, fitnesses depend on the trait mean (or allele frequency), and so the individual fitness function can be written as .
Wright's earliest studies showed that the evolutionary dynamics for one locus with two alleles (say A and a) can be visualized as an adaptive landscape. He began with the key assumptions that mating is random and that genotypic relative fitnesses are constant in time. This rules out the possibility of frequency-dependent selection, the situation in which the relative fitnesses of genotypes vary as functions of their frequencies.
Wright's algebra showed that the dynamics can be described in terms of the population's mean fitness, . To relate Wright's work to other approaches, it is helpful to think of individual's phenotype Z as the number of A alleles that it carries. The mean for this trait, , is simply twice the value of p, the frequency of allele A, and the genetic variance for Z is 2pq. Table 1 shows the equation that describes the rate of evolution of the population. That rate depends on two things: the genetic variance for Z, and the selection gradient (or force of directional selection) acting on that trait.
Table 1. Fitness landscapes and dynamic equations under different approaches.
|1 Individual fitness function||–||–||–|
|2 Wright's adaptive landscape|| |
|3 Wright's frequency-dependent ‘fitness function’||F()|| || |
|4 Phenotypic|| || |
|5 Invasion landscape|| || |
|6 Pairwise invasibility plot|| || ||(As above)|
Wright (1932, 1935) discovered that selection gradient is determined by the population mean fitness function, . This is Wright's famous adaptive landscape, a plot whose horizontal axis is and whose altitude is . Selection drives populations to local peaks on this landscape, and the selection gradient for a given value of is equal to the steepness of the landscape there. Peaks in the adaptive landscape do not generally correspond to peaks in the individual fitness function. Consider an overdominant locus with fitnesses (1 − s) :: 1 :: (1 − t). The peak in the individual fitness function corresponds to the heterozygote whose phenotype is at Z = 1. The equilibrium is at , however, which differs from 1 if the homozygotes have unequal fitnesses.
The notion of the adaptive landscape can be extended in several ways. Wright (Wright, 1935, 1969) focused a lot of attention on the case of multiple loci. Here he assumed that linkage disequilibrium between the loci is negligible so that the population's state is described entirely by its allele frequencies, and the evolution of each locus can be treated independently. The landscape now has one horizontal dimension for each locus, and again the altitude is given by the population's mean fitness.
The landscape can be applied equally to quantitative traits (Lande, 1976, 1979). Now the horizontal axes are the trait means, and the altitude is again the population's mean fitness. Assume we have a quantitative trait Z that is measured on scale in which the phenotypic variance is independent of the mean. Then, just as for a single locus, the adaptive landscape is a plot that shows how varies with . If breeding values are normally distributed and the fitness function also has a Gaussian shape, then the trait mean evolves at a rate determined by the additive genetic variance and the selection gradient (Table 1). And once again, peaks in the adaptive landscape do not necessarily coincide with peaks in the individual fitness function. Phenotypic variation averages over the individual fitness function, smoothing it out. This can cause peaks in the individual fitness function to disappear entirely from the adaptive landscape, with interesting evolutionary consequences (Kirkpatrick, 1982).
Ever since its inception, Wright's adaptive landscape has had a bipolar personality as a mathematical description of evolutionary dynamics for a certain group of models and as a metaphor for a much broader range of evolutionary problems. Both uses are valuable. Even technical discussions of models for peak shifts often bring the metaphor into play.
Landscapes in frequency-dependent selection
Wright emphasized that the adaptive landscape does not apply when relative fitnesses change in time, including the important case of frequency-dependent genotypic selection coefficients. It is not difficult to see why. With interference competition, for example, a nasty phenotype can spread if the fitness it gains outweighs the cost of the behaviour, even if it decreases the fitness of the population as a whole (Wright, 1949). In the extreme, a population can evolve to extinction. Selection is certainly not driving populations towards the peaks of the adaptive landscape described by the population's mean fitness function.
In some cases, however, it is still possible to find a landscape that describes the force of selection. Imagine that fitnesses depend on just one variable, for example an allele frequency. Then selection can always be described in terms of a function F that Wright (1969) referred to (a bit confusingly) as a ‘fitness function’, which plays the same role as the mean fitness function when selection is frequency-independent. That is, the selection gradient is again given by the slope of this fitness function, and equilibria again correspond to peaks in it. This landscape, however, has not seen much use in evolutionary theory. What is more, it does not even exist in some cases. Wright (1949) showed this can happen when fitness depends on more than one variable that changes in time. He discussed an example in which the frequencies of three alleles at a locus are under a kind of selection analogous to the classic rock-scissors-paper game. The allele frequencies cycle indefinitely. Here there clearly can be no function whose peaks determine the equilibria, as there simply are no equilibria.
The force of selection can always be described in algebraic terms, of course, regardless of whether there is a landscape to represent it graphically. Whether or not selection is frequency dependent, the rate of evolution can be written as the product of the genetic variance, G, and the selection gradient, β:
For a single locus under random mating, the selection gradient is equal to the difference in the mean (or marginal) relative fitness of allele A and that of all the other alleles segregating at the locus: β =. For quantitative traits with normally distributed breeding values, equation (1) is equivalent to the famous breeder's equation (Falconer & Mackay, 1996). The selection gradient β is equal to the coefficient of the linear regression of relative fitness onto the trait value, which is also equal to the derivative of the individual fitness function with respect to the trait, averaged over the population's phenotypic distribution (Lande & Arnold, 1983). In many frequency-dependent models, the selection gradient depends only on the population mean, as shown in equation (1). More generally, however, it can depend on any aspect of the population's distribution.
In brief, describing the response to selection in algebraic terms is easy, at least for these simple models. But when fitnesses are frequency-dependent, it is not always possible to translate the dynamic equations into a landscape that represents the force of selection. Although equation (1) is deterministic, for a single locus it is easy to add genetic drift by way of diffusion approximations. These are described by two quantities, the effective size of the population, and a weak selection approximation for the expected change in allele frequency, which is effectively given by the selection gradient.
Game theory and phenotypic models
Game theory has been a major development in the study of frequency-dependent selection (Maynard Smith, 1982). This approach seeks to find the ‘evolutionary stable strategies’ or ESS, defined as states of a population that cannot be invaded by mutations with alternative phenotypes. Key assumptions that greatly simplify the analysis of complex problems are that the evolutionary dynamics are not of interest, that the resident population initially has no genetic variation for the phenotype of interest, and that alternative mutations are very rare. Although these assumptions can be restrictive, they allow us to work on problems where we may be unwilling to make a more detailed genetic model.
The notion of a fitness landscape is useful in game theory when phenotypes vary continuously (Brown & Vincent, 1987). The object that we need to study this kind of problem is the individual fitness function, . For a state of the population to be an ESS, no possible phenotype can have a fitness higher than that of the population as a whole. Although the individual fitness function predates game theory by many years, focusing on the landscape that depicts is a conceptual shift from the usual population genetics perspective. Wright's approach was to take the phenotypes (i.e. the fitnesses) of the mutations as given, and work out the evolutionary consequences. Game theory, on the other hand, asks what are the consequences of all possible mutations. Again, analysis of that question is made tractable using the assumption that there is no genetic variation at an ESS.
In its original form, game theory simplifies finding the evolutionary equilibria, but at the cost of giving up all understanding of the evolutionary dynamics. This gap motivated several workers to fuse concepts from classical quantitative genetics with those from game theory. The idea behind these ‘phenotypic’ models of evolution is to assume that there standing genetic variation for the trait, but that the variation is very small relative to the curvature of the individual fitness function in the vicinity of the population's mean. Then the selection gradient β is well approximated by the slope of the individual fitness function at that point (Iwasa et al., 1991; Taper & Case, 1992; Abrams et al., 1993), which makes it easy to calculate.
To find the rate of evolution, we need to also assume something about inheritance. If breeding values are normally distributed or if selection is sufficiently weak that allele frequencies change slowly, then the breeder's equation again applies. That is, the rate of change in the population's mean is simply the product of the selection gradient and the additive genetic variance (equation 1; Table 1).
The landscape relevant to phenotypic models is simply the individual fitness function. Typically, there are two horizontal dimensions: the population's current mean phenotype, and the phenotype of individuals. The altitude of landscape is just the fitness of an individual with that phenotype, given the mean of the population. Brown & Vincent (1987) dubbed this function the ‘frequency-dependent adaptive landscape,’ a term that is perhaps unfortunate as this function is not a new concept and as the outcome of evolution with frequency-dependent selection may not be adaptive.
Fitness landscapes and adaptive dynamics
This finally brings us to the fitness landscapes seen in adaptive dynamics (AD). This research program borrows key simplifying assumptions from previous approaches. As in the phenotypic models, new mutations are assumed to have only small phenotypic effects, which simplifies calculations involving the fitness function. But adaptive dynamics departs from the phenotypic models in two important ways. First, the invading mutant is inherited asexually, just as in game theory. Under some situations, this allows the trait's distribution to split into two or more modes (‘evolutionary branching’). That cannot happen in quantitative genetic and phenotypic models, which start with the assumption that the population has a unimodal distribution. When the problem of interest involves sexual reproduction (for example, speciation), conclusions from an adaptive dynamics analysis must be checked using traditional population genetics models (e.g. Dieckmann & Doebeli, 1999).
Secondly, adaptive dynamics assumes that the time needed for a favourable mutations to spread through a population is much shorter than the waiting time between the appearance of new successful mutations. Hence the genetic control on the rate of evolution comes from the rate of supply of new mutations rather than standing additive genetic variance. In population genetics, this situation is called the ‘strong selection-weak mutation’ (or SSWM) limit (Gillespie, 1991). An unresolved issue is whether results that use this assumption apply to many of the traits that motivate AD models, which often have substantial amounts of standing genetic variation.
Adaptive dynamics simplifies thinking about selection by the assumption that the effect of the mutation on the phenotype, ΔZ, is small. The mutation's fate is then determined by the shape (first and second derivatives) of the individual fitness function near the ancestral phenotype, (see Table 1). If the mutation has higher fitness than its ancestor, adaptive dynamics assumes it will spread. In the absence of ‘branching’, adaptive dynamics assumes the mutation will become fixed, and so the change in the phenotype of the lineage in which the mutant occurs is ΔZ (see Table 1). As noted by Waxman and Gavrilets, adaptive dynamics neglects the consequences of genetic drift. However, the first derivative of the individual fitness function that it considers is the selection gradient, from which fixation probabilities and other consequences of drift can be computed (Rousset, 2004).
Thus the landscape relevant to adaptive dynamics is yet again the individual fitness function, , which Waxman and Gavrilets call the ‘invasion landscape’. The horizontal axes are the phenotype of a new mutation, Z, and the population's (or lineage's) current state, , whereas the vertical axis is the fitness of the individual.
Representing the landscape is simplified further in the ‘pairwise invasibility plots’ of adaptive dynamics (Christiansen & Loeschcke, 1980). Here the only information that is kept about the fitness of a new mutation is whether it is greater or smaller than that of its ancestor. A practical advantage is that the landscape can be pictured easily in two dimensions. With the horizontal axes again representing Z and , the regions in which a new mutation will invade [that is, where ] can be identified by shading.
Despite the semantic cascade that Wright's term has triggered, the different visions of a landscape used by evolutionary biologists are minor variations on the two conceptions that Wright proposed. The first is the individual fitness function. This is the most general and least abstract of the variants, and the one that has found use in adaptive dynamics. The second vision is a function that depends on the state of the population. The most celebrated of these is the population's mean fitness function, which gives rise to Wright's adaptive landscape when relative fitnesses are constant in time. Under some special conditions, a similar landscape can sometimes be constructed when fitnesses are frequency-dependent. But here there is no guarantee that evolution is adaptive, and we court confusion by using that word in this context.
We dedicate this paper to John Maynard Smith. His clear thinking made profound contributions to our thinking about of frequency-dependent selection, and his joy in discussing the subject was an inspiration in itself.
We thank Troy Day, Kate Lessells, Monty Slatkin, and an anonymous reviewer for helpful comments. This work was supported by NSF grants DEB-9973221and EF-0328594 to M.K.