Founder effects and peak shifts without genetic drift: adaptive peak shifts occur easily when environments fluctuate slightly.
Recent investigations into the quantitative genetic version of the SBT have demonstrated that drift-mediated peak shifts are possible but extremely improbable events (Barton and Rouhani 1987; Whitlock 1995). This has caused many researchers to doubt the importance of peak shift and founder events in evolution. However, Wright's perspective had at least two components: the existence of rugged adaptive landscapes and the mechanism (SBT), which he proposed as a solution to the problems posed by those landscapes. There is increasing evidence that adaptive landscapes are in fact rugged (Whitlock et al. 1995). This raises the question: Can peak shifts occur in nature? Here it is shown that subtle transitions in the function describing the relationship between an individual's phenotype and its fitness deterministically cause transitions between adaptive peaks much more readily than drift.
Many authors have suggested that environmental change is also capable of causing peak shifts (e.g., Wright 1931a; Simpson 1953; Dodson and Hallam 1977; Kirkpatrick 1982; Milligan 1986). I show here that this is a much more effective mechanism for transitions between locally stable states than the drift-dependent mechanisms proposed by Wright. I focus here on the landscapes generated by the relationship between individual fitness and the phenotype, with an assumed additive genetic basis; similar arguments can be made about the landscape created by the relationship between genotype and fitness.
THE ADAPTIVE LANDSCAPE
The adaptive landscape is a function of the relationship between the form of an organism and its environment. Changes in the environment, as measured by changes in the intensity and nature of selection, can generate changes in the adaptive landscape that allow the further evolution of the species. Not only can the local optima change, but the strength of stabilizing selection around these optima and their relative heights can change, causing transitions between peaks. The idea that changes in the fitness function can cause shifts is not new; in fact this was suggested by Wright himself (Wright 1931b). What has been unrecognized is that the changes needed can be very small relative to the likely amount of variation of fitness in nature. This point is illustrated here using a simple fitness function, the sum of two Gaussian functions having different means (see Kirkpatrick 1982 and Fig. 1a). The appropriate measure of fitness when predicting the evolutionary trajectory of a population is mean fitness [ILLUSTRATION FOR FIGURE 1B OMITTED], which can be either a unimodal or bimodal function of phenotype, even when the individual fitness function is bimodal [ILLUSTRATION FOR FIGURE 1A OMITTED].
Following Kirkpatrick (1982), the relationship between fitness (W) and some character z, such as body size, for an individual is assumed to be the weighted sum of Gaussian functions, such that
W = C(exp[- s[(z - [[Mu].sub.1]).sup.2]/2] + H exp[- s[(z - [[Mu].sub.2]).sup.2]/2]), (1)
where C is a constant, [[Mu].sub.1] and [[Mu].sub.2] are the optima, H is the relative height of the two peaks, and s is proportional to the strength of stabilizing selection around each optimum [ILLUSTRATION FOR FIGURE 1A OMITTED]. If the distribution of phenotypes is normal, with phenotypic variance [V.sub.P], the relationship between mean phenotype [Mathematical Expression Omitted] and mean fitness [Mathematical Expression Omitted] of a population is
[Mathematical Expression Omitted]. (2)
This function [ILLUSTRATION FOR FIGURE 1B OMITTED] is much less extreme and is often unimodal even when the individual fitness function is not.
The various parameters which describe the individual fitness functions were examined numerically to find the values which cause a bimodal mean fitness function to become unimodal, which therefore allows deterministic evolution to the highest peak. The populations were initially assumed to be at equilibrium, with a mean phenotype equal to the value of the lower peak of the mean fitness function and a variance determined by [V.sub.G] = [V.sub.M] + [square root of 2[V.sub.M]([V.sub.E] + 1/s + [V.sub.M]/2)], where [V.sub.G] is the genetic variance, [V.sub.E] is the environmental variance, and [V.sub.M] is the variance due to mutation per generation (Bulmer 1980). All of the examples reported in this paper used [V.sub.M] = [10.sup.-3] [V.sub.E]. Since [V.sub.G] is a function of the strength of selection, with more variance maintained with weaker selection, the equilibrium genetic variance is calculated for the s after the environmental change. This difference in [V.sub.G] contributes very slightly to the changes in the topology of the new fitness surface.
To investigate the time scale of environmental change required for transitions to occur, the recursion equations of Kirkpatrick (1982) were used instead to study the changes in the mean, genetic variance, and linkage disequilibrium due to a short-term environmental perturbation. Again, recursive numerical methods allowed critical values of change in the selective parameters s and H, which allow permanent transition to the higher peak to be identified.
Transitions between Peaks by Environmental Change
Calculations of the probability of transitions from the one adaptive peak to another usually assume that the mean fitness function is static (Barton and Charlesworth 1984; Barton and Rouhani 1987; Barton and Hewitt 1989). This mathematical convenience does not allow for the easy accounting of the effects of environmental change. Explicit consideration of the variation in individual fitness functions shows that relatively minor changes in fitness functions can be responsible for peak transitions.
Figure 2 demonstrates the effects of changes in the selection pressure resulting from environmental change. Small changes in the strength of selection can significantly alter the topology of the fitness function. For a valley 90% as fit as the starting peak [ILLUSTRATION FOR FIGURE 1B OMITTED], a decrease in the strength of stabilizing selection of only 16-30% is sufficient to cause the landscape to become unimodal and allow the population to evolve to the higher peak, regardless of the size of the population. By comparison, the probability of transition by a population bottleneck for a 90% valley for the parameters used in Figure 1 is approximately 4 x [10.sup.-11], using the equations given by Barton and Rouhani (1987). It seems clear that, with the relatively large changes in most environments over short time scales and over small spatial scales, the odds of the strength of selection varying by merely 25% are greater by several orders of magnitude than 4 x [10.sup.-11]. Even with the effects of variance-induced shifts accounted for in the drift models (Whitlock 1995), environmental fluctuations must account for orders of magnitude more phenotypic peak shifts than drift-mediated processes.
This is not an isolated example. There are several combinations of the relative height of peaks and the strength of selection that can cause the same depth of adaptive valley (expressed relative to the height of the lower peak). Figure 3 shows that, for a given depth of the valley, the range of change needed in either of the two environmental variables needed to allow a shift is small. Thus the mean fitness at an adaptive valley (as a ratio of the mean fitness at the lower peak) largely predicts the magnitude of change in the environment that would be needed for a transition from one to another peak. The transitions that are easiest with drift-mediated mechanisms are trivially common by means of environmental fluctuations; those that may be considered unlikely by environmental change are nearly impossible by drift mechanisms [ILLUSTRATION FOR FIGURE 4 OMITTED]. The duration of these changes need not be very long [ILLUSTRATION FOR FIGURE 5 OMITTED].
The implications for founder effect speciation are profound: by far the most likely way for a transition for an isolated population from one type to another is not by genetic drift, but rather by the differences in the environment from one habitat to another. It is clear (although there is little explicit data) that islands (in the broad sense of a section of habitat surrounded by another different habitat) have different selective regimes from the mainland or from other islands (Schluter and Grant 1984; Schluter, in press). For example, islands with different sizes have different species compositions (MacArthur and Wilson 1967), and therefore different selection due to other species, in the form of competition, predation, parasitism, and food. These differences in environments result in differing selection coefficients, and, as has been shown in this paper, only slightly varying selection coefficients can translate into topologically different fitness functions. Founder effects in nature are likely to be substantially unaffected by the size of the founder population (unlike the predictions of drift-mediated theories), except in a negative sense that smaller populations are for a variety of reasons less likely to persist.
Transitions on adaptive landscapes may be responsible for some of the rapid evolution and stasis seen in the paleontological record (Newman et al. 1985; Lande 1986; Milligan 1986). While these transitions have been linked to changing environments (Milligan 1986), the results shown here demonstrate that the changes in the environments may be subtle indeed, allowing a much larger change in phenotype than in the environment. The existence of multiple peaks allows phenotypic transitions to be large when they happen; small changes in the environment allow these transitions to occur.
Unfortunately, the observations contained in this paper have one very pessimistic implication: it will be impossible to ever design an experiment that unambiguously tests the shifting balance theory sensu strictu. The reason is simple: for an experiment to be able to allow a peak shift to occur in an experimental time frame, the fitness function must be such that the depth of the fitness valley is not great (otherwise no peak shifts will occur within even a very long experiment). However, given such a fitness function, minor perturbations in fitness, caused either by slight changes in the experimental conditions or the selection imposed, can be enough to cause the mean fitness function to be temporarily unimodal. Furthermore, the pleiotropy and frequency dependence of natural selection on the genes involved can change the fitness relationships in subtle, almost unmeasureable, ways. If this happens, and it can be argued that no experimental system can be well enough controlled to prevent this kind of noise, then it can not be asserted with confidence that any peak shifts that are observed happen by Wright's method.
The results in this paper refer to the selection on phenotypes, not directly on genotypes. The selection on the interaction of alleles affecting metabolic pathways may not be as subject to changes due to external environmental change, and therefore drift may play an important role in the navigation of adaptive landscapes on a genotypic scale. However, an important analog to the processes reported here may be that as genotype frequencies change due to selection at some loci, the "genotypic environment" of other loci can change, thereby instigating a deterministic change in allele frequencies to a combination that was previously inaccessible (Price et al. 1993).
We have few direct measures of the fluctuations in selection intensity, but many indications that the intensity of change is great. Environments vary in both space (Schulter and Grant 1984; Weis et al. 1992; Dudley 1996) and time (Gibbs and Grant 1987; Weis et al. 1994). (These studies have shown that the environment changes are enough to reverse even the sign of directional selection; the types of changes required for transitions as shown by this paper are far less extreme.) The simple fact that species diversity varies greatly implies that many of the biotic interactions that affect the optimum phenotype must change substantially, by changing the food available, competition, and predation (Schluter and Grant 1984; Gibbs and Grant 1987, 1993). Furthermore, the properties of other species constantly change (Lively 1993). The climate changes in short and long-term scales. It is clear that to understand the evolution of and on complex landscapes we must have more information about the variability in space and time of the strength of selection.
I am very grateful for the very useful comments of N. Barton, M. Kirkpatrick, A. Mooers, S. Otto, P. Phillips, T. Price, D. Schluter, B. Walsh, and an anonymous reviewer. This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada.
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|Author:||Whitlock, Michael C.|
|Date:||Aug 1, 1997|
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