RATE OF ADAPTIVE PEAK SHIFTS WITH PARTIAL GENETIC ROBUSTNESS

Authors


Abstract

How adaptive evolution occurs with individually deleterious but jointly beneficial mutations has been one of the major problems in population genetics theory. Adaptation in this case is commonly described as a population's escape from a local peak to a higher peak on Sewall Wright's fitness landscape. Recent molecular genetic and computational studies have suggested that genetic robustness can facilitate such peak shifts. If phenotypic expressions of new mutations are suppressed under genetic robustness, mutations that are otherwise deleterious can accumulate in the population as neutral variants. When the robustness is perturbed by an environmental change or a major mutation, these variants become exposed to natural selection. It is argued that this process promotes adaptation because allelic combinations enriched under genetic robustness can then be positively selected. Here, I propose simple two- and three-locus models of adaptation with partial genetic robustness as suggested by recent studies. The waiting time until the fixation of an adaptive haplotype was observed in stochastic simulations and compared to the expectation without robustness. It is shown that peak shifts can be delayed or accelerated depending on the conditions of genetic robustness. The evolutionary significance of these processes is discussed.

Complex phenotypes are the products of interactions among numerous genes and gene products. This interactive nature of biological organization leads to a view that the effect of a new mutation should be context dependent: its phenotypic effect depends on alleles at other loci. As a consequence of nonadditive effects of mutations, called epistasis, a large proportion of mutations may be deleterious individually but beneficial if introduced jointly. If adaptation requires joint fixation of such mutations, the population may have to suffer reduced fitness in transition to a new allelic combination. In Sewall Wright's (1932) discrete genotypic fitness landscape, the wild type is depicted as a local peak from which a higher peak can be reached through a “valley” of intermediate genotypes. He argued that adaptation involving peak shifts requires more than simple directional selection because the latter cannot make the population cross the valley of lower fitness. Genetic drift was the agent that eases the peak shift in his grand theory of adaptation, the shifting balance theory (Wright 1930, 1932). In small populations, selection against deleterious intermediate genotypes is weakened and mutant alleles may accumulate to form an advantageous allelic combination, which becomes visible to positive selection. However, whether this process is important for adaptation in real populations is not clear (Coyne et al. 1997; Wade and Goodnight 1999), whereas other mechanisms that lead to peak shifts were proposed (Kirkpatrick 1982; Whitlock 1997; Weinreich and Chao 2005).

Recent studies suggest alternative mechanisms involving genetic robustness that might facilitate peak shifts on Wright's fitness landscape. Compromising the function of heat-shock protein 90 (Hsp90) by mutation, pharmacological inhibition or environmental stress revealed an array of morphological changes in Drosophila (Rutherford and Lindquist 1998) and Arabidopsis (Queitsch et al. 2002). The phenotypic variation revealed was heritable and depended on the genetic background, thus suggesting that these mutations were “buffered” and allowed to accumulate in the population until the Hsp90 function was challenged. It was also suggested that yeast prion [PSI+] unveils silent genetic information to produce new heritable phenotypes (True and Lindquist 2000). Most phenotypic changes revealed in these experiments are quite radical and seemingly deleterious. However, these studies argued that some variation revealed in this manner might be advantageous, particularly at the time of environmental stress, and important in phenotypic evolution. They do not provide a concrete quantitative model in which a hide-and-release system yields a higher rate of adaptation than that without such a system. However, the key argument here is that mutations that are likely to be selected against otherwise accumulate under genetic robustness and, when they are exposed to selection later, they find the right allelic partners or the right environment to become beneficial (Rutherford 2000). Therefore, this hypothesis is similar to the shifting balance theory (genetic drift is replaced by genetic robustness) and possibly a promising mechanism for facilitating peak shifts.

Other studies suggest that the ability to release hidden variation does not require a special mechanism such as Hsp90. Using computer simulations of regulatory gene networks, Bergman and Siegal (2003) showed that most loss-of-function mutations at one gene reveal genetic variation at others, suggesting that genetic robustness, or canalization, arises as an emergent property of complex gene networks (Siegal and Bergman 2002). Hermisson and Wagner (2004) provided a quantitative genetic explanation for this result by deriving an analytic solution that predicted increased additive genetic variance when a change in the genetic background, possibly induced by a major mutation such as a loss-of-function mutation, redistributes the allelic effects among loci (thus in the presence of epistasis). These studies suggest that the release of hidden variation happens quite frequently in nature. Furthermore, Bergman and Siegal (2003) simulated directional selection over the population of gene networks, which were previously shown to be robust, and found out that networks allowing loss-of-function mutations evolve faster to new optimum phenotypes than those without such mutations. This supports the argument that occasional failures of genetic robustness, which release hidden variation, facilitate adaptive evolution. However, they do not provide a detailed analysis of what evolutionary genetic events happened during the simulation.

This paper aims to evaluate the importance of hide-and-release mechanisms in adaptation by computer simulations of simple population genetic models. I focus on the rate of peak shifts: I ask how quickly two individually deleterious but jointly beneficial mutations get fixed in a population. Of particular interest is how much the rate of peak shift increases or decreases relative to a system entirely missing the mechanism of genetic buffering (and thus constantly exposed to both negative and positive selection). Not all factors influencing hide-and-release adaptive processes that were (verbally) proposed in the abovementioned studies will be included in the models studied here. One argument for the evolutionary significance of Hsp90 protein is that the release of hidden variation occurs at the same time as environmental change (stress). This variation would promote directional selection (adaptation on a shifted peak). This paper explores only the adaptation by a peak shift in a constant environment. Assimilation of the revealed variation (additional mutations that enable the new phenotype to be retained even after the buffering system is restored; Waddington 1957; Rutherford and Lindquist 1998) is also not included in the model.

Constant Selection Model

First, I examine the two-locus dynamics of jointly beneficial but individually deleterious mutations in the absence of buffering mechanisms. The behavior of this system was extensively studied in previous work (Crow and Kimura 1965; Michalakis and Slatkin 1996; Phillips 1996; van Nimwegen and Crutchfield 2000; Iwasa et al. 2004; Weinreich and Chao 2005), from which many results in this section can be found. However, I highlight slightly different aspects of the system to understand the results in the following models of partial genetic robustness. I consider a haploid population of 2N individuals reproducing in discrete generations. The population is initially fixed for alleles a and b at locus A and locus B, respectively. Mutations to allele A from a and to B from b occur with probability u per generation per locus (back mutations to a and b occur at the same rate). The relative fitness of haplotypes ab, Ab, aB, and AB is 1, 1 −s1, 1 −s2, and 1 +s12, respectively. Chromosomes are assumed to randomly conjugate and recombine: recombination between the two loci occurs with probability r per generation.

The rate of adaptation is determined by the waiting time, T, until the population becomes fixed for AB chromosomes. Strong negative and positive selection is assumed, that is, Nsx >> 1; the sequential fixations of A and B by drift, as considered in Weinreich and Chao (2005), do not occur. Standard theory indicates that, with no recombination (r= 0) and weak mutation, the waiting time until the fixation of AB is approximately

image(1)

The first and second terms are the expected time until the first appearance of an AB chromosome that eventually goes to fixation and the time AB spends on the way to fixation, respectively. With strong selection, the second term becomes small enough to be ignored.

The frequencies of ab, Ab, aB, and AB chromosomes are given by x1, x2, x3, and x4, respectively. Simulation is performed by changing these frequencies in each generation using recursions for selection, mutation and recombination, and then multinomial sampling (Kim and Orr 2005). The system starts with x1= 1 and x2=x3=x4= 0. Adaptation is completed when AB reaches near-fixation (x4= 0.99). Figure 1 shows that T0 well approximates the waiting time until the fixation of AB, T, when r is small. T increases very rapidly as r approaches s12. If r is greater than s12, AB is actively blocked from reaching the fixation, with T increasing far beyond the waiting time in the absence of selection (Fig. 1). This confirms the prediction by Crow and Kimura (1965) that the fixation of AB requires r < s12/(1+s12).

Figure 1.

Waiting time until the fixation of AB chromosomes in the simulation of the constant selection model. The numbers of generations, T, scaled by population size, N= 104, are plotted for various recombination rates, r. s12= 0.04. u= 10−5. Diamonds (squares) are results for s1=s2= 0.02 (= 0.1). Horizontal dashed lines a and b show T0 from equation 1 for s1=s2= 0.02 and 0.1, respectively. Stars show the waiting time with no selection (fixation due to genetic drift).

The temporal behavior of this system can be better understood by examining the effect of selection and recombination in an infinite population. Deterministic recursions for allele frequencies and linkage disequilibrium (LD), quantified by D=x1x4x2x3, can be easily obtained, for example using the approach of Barton and Turelli (1991). Allele frequency changes predicted by these deterministic recursions are shown in Figure 2. Beginning with no allelic association between two loci (D0= 0), the marginal frequencies of alleles A and B, p1 and p2, decrease initially due to negative selection (Fig. 2A). (This decrease is seen also with s1=s2 < s12 but to a lesser degree [not shown]). In the meantime, LD between A and B builds up as haplotype AB is positively selected. With small value of r, p1 and p2 stop decreasing and start increasing. However, as r increases, the rebound of p1 and p2 weakens, and, if approximately r > s12, p1 and p2 continue to decrease (x4 thus decreases). The eventual fixation/loss of A and B also depends on the initial frequency of each allele (Fig. 2B) and initial LD (Fig. 2C): the fixation of AB is favored by higher initial frequency and larger initial D.

Figure 2.

Deterministic change of marginal allele frequency, p, in the constant selection model (s1=s2= 0.1, s12= 0.04). With identical starting frequencies (p0), the expected trajectories of both alleles A and B are given by p. (A) The effect of recombination rate (p0= 0.05 and D0= 0). From top to bottom, r= 10−5, 10−3, 0.01, 0.02, 0.03, 0.04. (B) The effect of initial frequency (D0= 0 and r= 0.045). From top to bottom, p0= 0.25, 0.2, 0.15, 0.1, and 0.05. (C). The effect of initial LD (p0= 0.05 and r= 0.04). From top to bottom, D0= 0.04, 0.03, 0.02, 0.01, 0, and −0.002.

Periodic Selection Model

I propose a two-locus model analogous to the buffering system of Hsp90 by adding steps of genetic robustness to the constant selection model above. Assume that a population alternates between buffering and selecting phase. The buffering (selecting) phase lasts for LB (LS) generations. During the buffering phase, the phenotypic expressions of alleles are suppressed. Therefore, allele frequencies change neutrally, following the standard Wright–Fisher model of reproduction. During the selecting phase, which represents the period when the Hsp90 function is challenged or [PSI+] is present, the population evolves according to the constant selection model. Simulation starts with the buffering phase, with initial haplotype frequencies x1= 1 and x2=x3=x4= 0. Again, adaptation is completed when AB reaches near-fixation (x4= 0.99). The waiting time, T, observed in the simulation is normalized as τ=T/T0 (see eq. 1). Thus, for small r, τ is approximately an indicator of the advantage of genetic robustness in adaptation because T0 approximates the waiting time under constant selection for r < s12 (Fig. 1).

Simulation was performed using two intensities of negative selection: s1=s2= 0.5s12 (“shallow valley”) and s1=s2= 2.25s12 (“deep valley”). Two recombination rates (Nr= 0.1 and 10) were used. Table 1 shows that the relative waiting time until the fixation of AB depends critically on LB and LS. τ is smaller than one (faster peak shift relative to the case of constant selection and complete linkage) only for large LB and large LS. Since the role of genetic robustness in peak shift is to allow A and B alleles to accumulate as neutral variants until haplotype AB is formed, LB needs to be sufficiently large to allow such accumulation. Too large LB (>> N), however, increases τ as expected (data not shown). When AB chromosomes do appear at the end of the buffering phase, the following phase of positive selection cannot be too brief: LS > 10 is needed to facilitate the fixation of AB (Table 1). It is also shown that the advantage of genetic buffering is greater for crossing a deep valley than a shallow valley. In the former case, τ could be as small as ∼ 0.1 (10 times faster adaptation). However, one may also notice that for some combinations of intermediate LB, short LS, large r, and deep valley, τ becomes exceedingly large. For example, τ= 23.8 when s1=s2= 0.1, LB= 100, LS= 1, and r= 10−3 (Nr= 10). In comparison, if selection is turned off in this case (LS= 0), τ= 1.83. Therefore, the fixation of AB takes even longer than it would take by genetic drift alone. Increasing recombination rates further increases τ (data not shown). This might be explained by the following argument. At the end of the buffering phase, four haplotypes (ab, Ab, aB, and AB) segregate in the population in approximate linkage equilibrium (because Nr >> 1). When selection begins, its immediate effect is to decrease the overall frequencies of A and B (Fig. 2), even though the frequency of AB chromosome may increase slightly. This creates a positive LD between alleles A and B. Recombination then acts to break this association and slows down the increase of AB. Furthermore, when the selection is turned off in the following buffering phase, the decay of the association by recombination continues, causing a net decline of AB chromosomes. Therefore, unless LS is long enough to bring a net increase of the marginal frequencies of A and B, p1 and p2, this buffering system will prohibit, not facilitate, the fixation of AB. This adverse joint effect of selection and recombination on the fixation of AB is alleviated if LB further increases (LB > 100; Table 1). This is likely due to (1) the “dilution” of the periods of negative selection with increasing LB and (2) the action of genetic drift that may generate high p1 and p2 and/or high LD by chance at the end of the buffering phase, in which case a net increase of p1 and p2 can occur at the next selecting phase (Fig. 2B,C). In summary, Hsp90-like system of genetic robustness can both facilitate and delay peak shifts depending on the duration of buffering/selective phase and recombination rate.

Table 1.  The waiting time, τ, until the fixation of AB under the periodic selection model relative to the waiting time under complete linkage with constant selection. Results are based on 1000 replicates for each parameter set. N=104.
  1. aMean ± standard error.

i) s1=s2=0.02, s12=0.04, r=10−5
  LB20100100010,000
LS   1 1.12±0.03a1.40±0.033.85±0.097.66±0.21
 2 1.04±0.031.22±0.032.80±0.066.86±0.18
   10 1.06±0.031.06±0.031.42±0.033.79±0.08
  100 1.05±0.030.91±0.030.67±0.020.95±0.02
 1000 1.02±0.031.04±0.030.62±0.020.51±0.01
ii) s1=s2=0.02, s12=0.04, r=10−3
  LB20100100010,000
LS   1 1.37±0.042.19±0.064.22±0.107.82±0.21
 2 1.18±0.042.32±0.073.01±0.067.18±0.19
   10 1.04±0.031.20±0.042.28±0.064.02±0.09
  100 1.04±0.030.91±0.030.63±0.020.84±0.02
 1000 0.99±0.030.98±0.030.56±0.020.37±0.01
iii) s1=s2=0.1, s12=0.04, r=10−5
  LB20100100010,000
LS   1 1.13±0.041.04±0.030.94±0.021.38±0.03
    2 1.08±0.041.10±0.031.07±0.031.23±0.03
   10 1.04±0.030.95±0.031.02±0.030.86±0.02
  100 0.96±0.030.49±0.020.18±0.010.21±0.005
 1000 1.13±0.030.92±0.030.23±0.010.12±0.003
iv) s1=s2=0.1, s12=0.04, r=10−3
 LB20100100010,000
LS   1 1.90±0.0623.8±0.731.13±0.031.54±0.04
    2 1.41±0.047.59±0.232.46±0.071.24±0.03
   10 1.12±0.041.24±0.0416.2±0.520.94±0.02
  100 1.02±0.030.48±0.020.17±0.0050.18±0.004
 1000 1.19±0.040.91±0.030.24±0.0080.085±0.002

Robustness-Modifier Model

Next, following Bergman and Siegal (2003), I propose a three-locus model of adaptation. Two loci affect a phenotype in the same manner as the loci in the models above and one locus determines the genetic robustness of this phenotype. This robustness-modifying locus (locus M) corresponds to the functionally important gene posited in Bergman and Siegal (2003) whose impairment reveals hidden variation at other loci. Again, I assume a population of 2N haploid individuals that undergo recombination. It is assumed that the three loci are arranged on a chromosome with order A–B–M. Recombination rates are r1 between locus A and locus B and r2 between locus B and locus M. With the “wild-type” allele, M, at locus M, the phenotypic expression of mutant alleles A and B at locus A and locus B, respectively, are suppressed. However, in the presence of the modifying allele, m, the phenotypes are expressed. Again, A and B are individually deleterious but jointly beneficial. Allele m may have its own deleterious effect, reducing fitness by sN (subscript N indicating “null”, thus implying the cost of the loss of normal function in the gene network). There are eight haplotypes in the model: abm, Abm, aBm, ABm, abM, AbM, aBM, and ABM. Relative fitness of these haplotypes are given by w1, …, w8, respectively. The fitness effect of each locus outlined above suggests a three-way epistasis: w1= 1 −sN, w2= 1 −s1sN, w3= 1 −s2sN, w4= 1+s12sN, and w5=w6=w7=w8= 1. Bidirectional mutation occurs with probability u (per locus per generation) at locus A and locus B and with probability v at locus M.

The simulations use a three-locus recursion using the formulas in Hospital et al. (1996). Again, genetic drift is incorporated by simulated multinomial sampling. It starts with a population fixed for abM or abm. Then it counts the number of generations, T, until the frequency of the AB chromosome reaches 0.99. Again, the simulation result is given by τ=T/T0 (eq. 1): for small r, this becomes the measure of how faster or slower peak shift occurs relative to the system without any mechanism of genetic robustness. Table 2 shows the results when the population is initially fixed for abM, representing the pre-existing genetic robustness in the population conferred by wild-type allele M. It is shown that this three-locus system can facilitate peak shift for various parameter sets. Similar to the case of periodic selection above, the relative advantage of robustness is larger for crossing a deeper valley (s1=s2= 0.1 vs. s1=s2= 0.02 in Table 2). For fixed v, the smallest τ (= 0.065, case 18) was obtained with no selective disadvantage of m (sN= 0) and moderate recombination rates (r1=r2= 0.01).

Table 2.  The relative waiting time until AB fixation with a robustness modifier, starting with abM. Results are based on 1000 replicates for each parameter set. (N= 104, s12=0.04, u=10−5)
Cases1=s2sNvr1r2τ
 10.02010−510−410−40.385±0.010
 20.02010−510−4 0.010.291±0.008
 30.02010−510−4 0.10.656±0.018
 40.020.03510−510−410−40.806±0.018
 50.020.03510−510−4 0.012.02±0.04
 60.020.03510−510−4 0.13.83±0.09
 70.1010−510−410−40.082±0.002
 80.1010−510−4 0.010.070±0.002
 90.1010−510−4 0.10.242±0.006
100.10.03510−510−410−40.156±0.003
110.10.03510−510−4 0.010.424±0.009
120.10.03510−510−4 0.10.845±0.020
130.1010−710−410−40.370±0.009
140.1010−610−410−40.157±0.004
150.1010−410−410−40.084±0.002
160.1010−5 0.0110−40.107±0.004
170.1010−5 0.110−40.475±0.013
180.1010−5 0.01 0.010.065±0.002
190.1010−5 0.1 0.10.662±0.016
200.10.03510−5 0.01 0.010.550±0.013
210.10.03510−5 0.1 0.11.01±0.03
220.1010−5 0.5 0.50.808±0.020

To understand how this system speeds up peak shifts, haplotype frequencies were followed until the fixation of AB occurs in a randomly chosen run with sN= 0 (Fig. 3). It is shown that haplotypes AbM, aBM, and abm—each one mutation away but with the same fitness as abM—start segregating in low frequency in mutation–drift balance. Much later, ABM begins to segregate in low frequency, after which ABm quickly enters and sweeps through the population. The fixation of AB is therefore achieved as soon as the positively selected haplotype ABm appears by mutation from ABM or by recombination between ABM and abm. The waiting time until the fixation of AB then depends on how quickly a sufficient number of ABM chromosomes appear in the population and how quickly ABm is generated thereafter. This may explain the fastest peak shifts observed with moderate recombination (Table 2; cases 8, 18), which promotes the creation of ABM from segregating AbM and aBM (r1) and ABm from ABM and abm (r2). Increasing mutation rate at locus M also reduces τ by facilitating the conversion from ABM to ABm (Table 2; cases 7, 13, 14, 15). We obtain a much shorter waiting time than T0 because this multistep process ([abM]→[AbM and/or aBM]→[ABM]→[ABm]) does not involve any deleterious intermediates. (Throughout the simulation, the frequency of two haplotypes with deleterious effects, Abm and aBm, is close to zero.)

Figure 3.

Changes of haplotype frequencies under the robustness-modifier model when the population is initially fixed for abM. No visible drift in frequency of Abm and aBm is observed at this scale.

Rapid fixation of AB shown in Figure 3 is due to positive selection conferred by robustness-breaking allele m, which goes to near-fixation along with AB. Namely, the epistatic coupling of AB and m is important for fast adaptation. Because this coupling is favored under tight linkage, τ increases as r2 increases from 0.01 to 0.1. However, even when r2 is very high (0.1 and 0.5), τ is still below one (cases 3, 9, 12, 19, and 22). This implies that the robustness-modifying locus that facilitates peak shifts can be distantly located from the trait-producing loci on the genome. The effect of recombination rate between trait-producing loci, r1, is more interesting. Figure 1 shows that, if r1 is much greater than s12, the waiting time until the fixation of AB becomes extremely large in the absence of any mechanism of genetic robustness. However, the waiting time is not so large even with r1= 0.1 or 0.5 (s12= 0.04) in Table 2 (τ < 1 in most cases; here, τ is not the reduction factor of waiting time due to robustness but simply the waiting time normalized by T0, which is calculated for r1= 0, to facilitate the comparison of absolute waiting times among cases). This suggests that the fixation of two mutations—jointly beneficial but individually deleterious—that are distantly located with each other on the genetic map is virtually impossible under constant selection but reasonably fast with a robustness-conferring allele (M). Analysis of the deterministic change of allele frequencies with free recombination (r1=r2= 0.5) provides a partial explanation for this result. The appendix shows that, if the frequency of allele m is above zero and the marginal frequencies of allele A and B become greater than sd/(s12+2sd), assuming s1=s2=sd, these alleles will tend to increase by selection (shown also in Crow and Kimura 1965). Therefore, the robustness-modifier model allows the fixation of A and B despite free recombination because allele frequencies may reach the threshold sd/(s12+ 2sd) by genetic drift when m segregates in low frequency. However, the fixation of AB in this pathway is not so fast because it takes time for A and B to reach the threshold by drift, and selection pushing p1 and p2 upward thereafter is very weak. The waiting time τ in this case is much larger than those with tight or moderate linkage but still smaller than the time obtained by genetic drift only (τ= 1.83).

It is interesting to ask how fast peak shifts occur if the system above starts with a population fixed for abm; genetic robustness is absent initially but the modifier locus can produce a robustness-conferring allele M. This model may be relevant to the discussion about the evolution of genetic robustness (see below). Table 3 shows that the fixation of AB still occurs faster than the system without locus M (τ < 1). However, τ is much larger relative to the equivalent cases in Table 2. This is because allele M (mainly in abM) first needs to increase by drift before starting the same sequence of events depicted in Figure 3. (Because M masks deleterious effects of mutations at locus A and locus B, abM has selective advantage of order 2u over abm. However, with Nu < 1 as in the case of this simulation, drift dominates the dynamics of M.) For parameters that yielded small τ (cases 4, 5, and 6 in Table 3), the frequency of M increases up to 0.8 during the process but decreases close to zero at the end of the sweep of AB. It suggests that, although genetic robustness can facilitate the peak shift, it appears in the population only transiently in this model.

Table 3.  The relative waiting time until AB fixation with a robustness modifier, starting with abm. Results are based on 1000 replicates for each parameter set. (N=104, s12=0.04, sN=0, u=v=10−5)
Caseas1=s2r1r2τpmaxbpfinalc
  1. aNumbers in parentheses point to the equivalent cases in Table 2.

  2. bMean maximum frequency of allele M during the simulation.

  3. cMean final frequency of M.

1 (1)0.0210−410−40.679±0.0180.5000.003
2 (2)0.0210−4 0.010.774±0.0200.5550.005
30.02 0.01 0.010.754±0.0200.5530.005
4 (7)0.110−410−40.226±0.0050.7120.006
5 (8)0.110−4 0.010.266±0.0080.7980.008
6 (18)0.1 0.01 0.010.286±0.0060.8390.014
70.1 0.01 0.10.705±0.0160.8780.159
80.1 0.1 0.010.774±0.0140.9920.072
9 (19)0.1 0.1 0.10.967±0.0180.9920.277

Discussion

Simulation of simple genetic models showed that robustness against new mutations may or may not facilitate the fixation of jointly beneficial alleles. When the peak shift was facilitated, the role of genetic robustness was to prevent immediate loss of alleles that may form beneficial combinations later. In addition to genetic robustness, a few other mechanisms that weaken the selection against individually deleterious mutations and facilitate peak shifts have been proposed (Wright 1930, 1932; Kirkpatrick 1982; Whitlock 1997; Hadany 2003). The periodic selection model explored here is similar to the model of peak shifts in Kirkpatrick (1982) and Whitlock (1997). They argue that environmental changes that increase genotypic variance can easily trigger peak shifts. Although their quantitative genetic model, assuming many contributing loci without genetic details, cannot be directly applicable to the current model of robustness, the key idea that temporarily relaxed stabilizing selection (due to genetic buffering in this study) around the lower peak may lead to peak shifts, is similar. Although these studies focus on temporal changes in environment, Hadany (2003) proposed a model of peak shifts in a spatially heterogeneous environment. She showed that, if subpopulations have different selective pressure against individually deleterious alleles, a subpopulation under relaxed selection would maintain increased frequencies of those alleles and supply the adaptive combination of those alleles to a subpopulation under strong selection by migration. Then, the jointly beneficial mutations can be selected and fixed in the latter subpopulation. This leads to faster peak shifts than in a homogeneous environment (Hadany 2003). This model is similar to the robustness-modifier model proposed here, because the two alleles in the robustness-modifying locus effectively create such subpopulations under relaxed and strong selection.

The models of partial genetic robustness thus look promising in facilitating adaptation in nature. However, the degree to which adaptation is promoted depends on the parameter values (e.g., depth of fitness valleys and recombination rates) and other details of the models. For the two models considered here the biological relevance of parameter values that produce significant improvements in adaptation over the case of no robustness should be evaluated. In the periodic selection model, alleles contributing to potential adaptation accumulate during the buffering phase by genetic drift. The accumulation is maximized as the period of buffering increases toward N. This condition was considered by Masel (2006), who showed by analytic calculation that jointly adaptive variants are enriched at the end of long buffering phase. (Her study included the possibility of fixing unconditionally deleterious alleles that abort adaptation. Without such alleles, the two-locus model in the current study might be more favorable for genetic robustness.) However, the final result of adaptation also depends on the length of the selecting phase following the accumulation of adaptive haplotypes. If selection lasts only one or two generations before the next buffering phase starts, the fixation of the adaptive haplotype takes longer than the case of no buffering even if the buffering phase is sufficiently long (Table 1; Fig. 2). Therefore, to facilitate phenotypic evolution by jointly beneficial alleles, the normal function of Hsp90 protein needs to be compromised for more than several generations. If it is the environmental stress that challenges Hsp90 function and reveals adaptive variation, such stress must last for many consecutive generations. We also notice that, for crossing deep valleys (s1, s2 > s12), even moderate recombination can greatly delay the fixation of adaptive haplotypes. Therefore, depending on the relative abundance of adaptive allelic combinations at tightly versus loosely linked loci in the genome, it is possible that the net effect of Hsp90-like buffering system may actually slow phenotypic evolution that crosses deep fitness valleys.

The advantage of genetic robustness in adaptation is more readily supported in the robustness-modifier model than in the periodic selection model. When the population is initially fixed for the robustness-conferring allele M, peak shifts are facilitated because hidden variation accumulates under M. Chromosomes carrying this allele are equivalent to the subpopulation with relaxed selection in the model of Hadany (2003), which serves as a “reservoir” of adaptive allelic combinations. The significance of this model can be discussed for two different recombination regimes. First, with tight or moderate linkage among loci, the shortest waiting time until the fixation of AB (T, in number of generations) is achieved. Here, after the potentially advantageous haplotype AB starts segregating by genetic drift, the robustness-breaking mutation m exposes AB to positive selection and rises to high frequency along with AB. This greatly reduces T relative to the constant selection model. Rapid invasion of m into a genetically robust population may not be surprising because in general any mutation that increases additive genetic variance is expected to increase the rate of adaptation and might thus be favored by natural selection. In this model, directional selection on AB increases the frequency of m by hitchhiking. If there is a fitness cost of fixing m (1/(2N) << sN < s12), selective pressure may exist to compensate the loss of fitness by fixing additional mutations that restore the function of allele M without reversing the phenotypic expression of AB. This next step might be considered as genetic assimilation. How likely this fast peak shift would occur in nature will depend on the availability of robustness-breaking mutations whose deleterious effect is smaller than the beneficial effect of revealed variation. Bergman and Siegal (2003) argued that loss-of-function mutations of many genes would reveal hidden variation at other loci. Recent studies found evidences for the rapid (adaptive) fixation of null mutations (Olsen and Purugganan 2002; Lamason et al. 2006; Wang et al. 2006; Xue et al. 2006). However, these mutations are more likely to have been fixed because the loss of function itself was beneficial than because they revealed hidden variation at nearby loci. It should be noted again that this model leading to fast peak shifts assumes tight linkage between loci. Although a robustness-breaking allele may reveal hidden variation at many loci across genome, only variation on the same chromosome will contribute to peak shifts that are fast enough to leave the signature of selective sweeps.

Second, with loose linkage between trait-producing loci, the robustness-modifier model with initial M does not enjoy as fast fixation of AB as it would under tight linkage, due to its dependence on long-term genetic drift and weak selection. However, under the same conditions (r > s12), the constant selection model predicts an extremely large T, as recombination in conjunction with negative selection actively eliminates A and B from the population (Figs. 1 and 2). Therefore, genetic robustness allows the fixation of these alleles, which is virtually impossible otherwise. If there are more pairs of interacting loci that are distantly located in the genome than tightly linked pairs, the advantage of genetic robustness under loose linkage might be very significant for adaptation.

The results given in Table 3, that is, τ < 1 when the population is initially fixed for abm, is also interesting. It implies that substitution from ab to AB occurs faster in the presence of a robustness-modifying locus even if the population starts with no robustness. This occurs because the robustness-conferring allele, M, can transiently increase to high frequencies by drift. However, it decreases to near-zero frequency upon the completion of the sweep of the haplotype ABm. Therefore, although tempting, these results cannot be used to support the hypothesis that adaptation on a rugged fitness landscape permanently selects for modifiers of genetic robustness.

So far, we have considered only adaptation due to individually deleterious but jointly beneficial mutations. To understand the evolutionary significance of mechanisms that aid peak shifts, such as the shifting balance theory and models of genetic robustness considered here, we need to evaluate the effect of those mechanisms on the dynamics of mutations that are unconditionally beneficial (i.e., additive among loci). Although robustness can store variation that is useful in adaptation later, it turns off selection acting immediately on new individually beneficial mutations, assuming that the phenotypic expression of additive and nonadditive mutations are equally suppressed by genetic buffering. If the population needs to respond quickly to selective pressure by fixing new beneficial mutations, genetic robustness may inhibit rather than promote adaptation in the population. In the periodic selection model above, the rate of adaptation by unconditionally beneficial mutations is expected to decrease approximately by a factor LS/(LB+LS): large LB increases the rate of peak shifts but decreases the rate of adaptation by unconditionally beneficial mutations. In the robustness-modifier model with initial robustness, a new additive beneficial mutation can be fixed only if this allele first segregates by drift and then a mutation to a robustness-breaking allele, m, occurs at a nearby site. This results in a much longer waiting time until fixation compared to simple directional selection. The same model with no initial robustness might be the only mechanism that allows rapid adaptation by both additive and epistatic beneficial mutations. However, if adaptive substitutions of additive mutations occur frequently, the frequency of allele M will be pushed to zero at each of those fixation events (allele m is coupled to the beneficial mutation) and thus kept low throughout the course of evolution. This may greatly reduce the opportunity for the adaptation by hiding-and-releasing variation.

The importance of genetic robustness in adaptation thus depends on the relative frequency of additively versus jointly beneficial mutations. It is not known how often individually deleterious but jointly beneficial mutations occur. At present we have only limited evidence of rugged fitness landscape in DNA sequence space (Kondrashov et al. 2002; Kern and Kondrashov 2004; Kulathinal et al. 2004; Poelwijk et al. 2006). Many of these studies detected compensatory changes along the phylogeny, which may not have been positively selected (adaptive).

This paper is mainly concerned about the effect of genetic robustness on the rate of adaptation, only briefly addressing the possible role of adaptive evolution in selecting/maintaining robustness. It should be noted, however, that the evolutionary significance of genetic robustness may not be solely determined by its effect on adaptive evolution, whether the effect is positive or negative, because there are other plausible causes of robustness. It has been suggested that robustness evolved by selection to reduce the effect of unconditionally deleterious mutations (Wagner 1996; Masel and Maughan 2007), by selection for environmental canalization (Wagner et al. 1997; Meiklejohn and Hartl 2002), or as a byproduct of certain forms of epistasis (Hansen and Wagner 2001; Hermisson and Wagner 2004).

In conclusion, storing variation that may be useful later in adaptation by turning off or reducing selection, the idea shared by the current models of partial genetic robustness and other models of peak shifts, can be a solution to adaptation on a rugged fitness landscape, as demonstrated by τ < 1 in the simulations reported here. However, conditions that allow the Hsp90-like system (periodic selection) to achieve fast peak shifts are quite restricted. In the robustness-modifier model, peak shifts are enhanced under much less-restrictive conditions. However, this model also cannot avoid impeding adaptation by unconditionally beneficial mutations. We thus encounter the inherent difficulty in evolutionary models that require very fine adjustments between selection and no-selection.

Associate Editor: S. Otto

ACKNOWLEDGMENTS

I thank S. Otto, J. Masel, J. Hermisson, A. Betancourt, A. Orr, D. Presgraves, and one anonymous reviewer for their comments that greatly improved the manuscript. This research was supported by National Science Foundation grant DEB-0449581 to YK.

Appendix

Deterministic changes in the robustness-modifier model with free recombination

Here, the robustness-modifier model is simplified and each generation consists of selection and recombination. With free recombination, haplotype frequencies after recombination are largely determined by marginal allele frequencies at the three loci. (Due to epistatic selection after recombination, LD does not completely disappear but remains small enough to be ignored.) Let p1, p2, and p3 be allele frequencies of A, B, and m and qi= 1 −pi. Haplotype frequencies of ABm and aBM, for example, after recombination are approximately p1p2p3 and q1p2q3, respectively. In the next generation, the allele frequency of A after selection is

image

for s1=s2=sd. If the initial frequencies of A and B are identical, p1=p2=p because of symmetry. The condition for the deterministic increase of p, using the equation above, is found to be

image

Therefore, the deterministic increase of A and B occurs if their initial frequencies are greater than sd/(s12+ 2sd). Crow and Kimura (1965) obtained the identical result (for p3= 1).

Ancillary