Variance-induced peak shifts.
This stochastic process, or shifting balance, proposed by Wright (1932, 1940, 1977, 1982), has ranked as one of the most compelling theories within evolutionary biology, because the shifting-balance process seems to overcome some of the obvious difficulties in making transitions between fit forms. Wright proposed that a population could cross from one adaptive form to another through a phase of decreased mean fitness if the population experienced a phase of small size, allowing drift to bring the mean of the phenotypic distribution far enough toward the new peak to allow a transition from one peak to another to be completed by deterministic processes.
This model has been controversial from the beginning. Other workers have preferred other mechanisms for this kind of shift. Fisher (1958) believed that changes in the fitness function would cause a phenotype to track a changing optimum and that the amount of drift possible in even a very small population would be insufficient to drive Wright's process. Others have reinforced this idea of the changing fitness function (Kirkpatrick 1982; Barton and Charlesworth 1984; Milligan 1986). Another perspective, that of Kirkpatrick (1982), is that changes in the environment can cause changes in the phenotypic variance, which also can change the evolutionary dynamics of a population.
Peak shifts have been widely recognized as microevolutionary processes that would generate the macroevolutionary pattern of punctuated evolution (Charlesworth et al. 1982; Kirkpatrick 1982; Lande 1985, 1986; Newman et al. 1985). By any of the proposed modes of shifts between stable equilibria, the expected time during the transition is expected to be much smaller than the amount of time between phenotypic changes (Kirkpatrick 1982; Lande 1985). This is exactly the pattern of rapid change and relative stasis observed in much of the fossil record (Eldridge and Gould 1972).
The controversy over the involvement of drift-mediated processes in mechanisms of peak shifts has received a great deal of attention (Barton and Charlesworth 1984; Carson and Templeton 1984; Lande 1986; Barton and Rouhani 1987, 1993a,b; Rouhani and Barton 1987a,b; Charlesworth and Rouhani 1988; Crow et al. 1990; Barton 1992; Phillips 1993; Whitlock and Wade 1995). These analyses have varied greatly in their biological assumptions, and the many authors involved have nearly as many interpretations of the plausibility and importance of peak shifts.
These papers do share at least one attribute; they all assume that the amount of genetic variance in a population after a bottleneck or founder event is slightly less than in the outbred source population. If this assumption is made explicit, the amount of genetic variance in a founder population becomes (1 - F) of what it was before the bottleneck, where F is Wright's coefficient of inbreeding. This is the expectation of genetic variance for a bottleneck population, but this is only the expectation, and any given population can have a very different amount of genetic variance. In other words, there can be quite a large variance among populations in the amount of genetic variance within populations (Avery and Hill 1977). I will show that this assumption, that variance itself does not vary, can lead to extremely inaccurate ideas of the probabilities of peak shifts as a result of small population size.
The probability of shifts from one stable state to another depends quite directly on the amount of phenotypic variance within populations. In fact, if the amount of phenotypic variance is large enough, what was once a bimodal distribution of mean fitness becomes unimodal, and selection can deterministically take a population from one state to another. With reasonable amounts of variance in variance, we might expect that the probability of peak shifts could be very different from the prediction made using only the expectation of the variance.
Here I present a new model of morphological evolution as a result of peak shifts. This model, referred to as the variance-induced peak-shifts model (VIPS), accounts for the possible changes of phenotypic variance in small populations. A small fraction of founder populations will have higher genetic variance than the outbred population from which they came. In those populations, the relationship of the character mean and mean fitness becomes flatter, thus, the difference between the mean fitness at the adaptive valley and the fitness peaks becomes less. In some cases, in fact, the valley disappears altogether. As a result of these changes, shifting from one peak to another becomes easier than in a model that does not consider changes in variance. Some populations will change from one state to another even with no genetic drift in the phenotypic mean. The total probability of transitions can be much higher than in models of shifting balance, as it is generally interpreted.
Felsenstein (1979) and Kirkpatrick (1982) devised a fitness function that allows for multiple fit states for individual organisms, based on the sum of two Gaussian functions. This model is extremely useful for two reasons: it is very similar to models of stabilizing selection, and it makes explicit the relationship between the fitness function for individuals and the distribution of the trait in the population. Imagine a character z that has two optima, one [[Mu].sub.1] and the other at [[Mu].sub.2]. Let there be Gaussian stabilizing selection around each optimum, such that the intensity of selection can be inversely measured by [Gamma] around [[Mu].sub.1] and around [[Mu].sub.2], with the peak at [[Mu].sub.2] H times higher than the peak at [[Mu].sub.1]. The fitness of an individual is therefore defined as
W [infinity] (e - [(z - [[Mu].sub.1]).sup.2]/2[Gamma] + He - [(z - [[Mu].sub.2]).sup.2]/2[Gamma])/[square root of [Gamma]]. (1)
An individual fitness function does not, by itself, say anything about the course of evolution; we must consider instead the shape of the mean fitness function. When characters are determined additively, the mean fitness of a population is increased by natural selection. With weak selection, the mean fitness remains a good predictor for the outcome of selection, even with dominance and epistasis (Nagylaki 1993). If [Mathematical Expression Omitted] is normally distributed in a population with mean [Mathematical Expression Omitted] and variance [V.sub.p], the mean fitness of the population ([Mathematical Expression Omitted]) is
[Mathematical Expression Omitted].
One of the disadvantages of this sort of fitness function is that the derivative with respect to [Mathematical Expression Omitted] is a transcendental function, thus, most maxima and minima cannot be analytically located. In the case where H = 1, there is a maximum or minimum at ([[Mu].sub.2] - [[Mu].sub.1])/2 (Felsenstein 1979). Kirkpatrick (1982) has presented recursion equations for the mean and variance of a trait evolving on this adaptive landscape.
An example of a bi-Gaussian fitness function is constructed in figure 1. Note that as the amount of phenotypic variance increases, the [Mathematical Expression Omitted] function becomes more and more smooth, until what was a bimodal function with small phenotypic variance becomes unimodal when the variance gets larger. This is because as the phenotypic variance increases, the population is experiencing a broader range of fitness values, so the mean fitness therefore changes less with small changes in [Mathematical Expression Omitted]. In other words, as the variance in phenotypes increases, the relationship between the phenotypic mean and the mean fitness becomes "noisier," and therefore more uniform. With increasing phenotypic variance, differences in the phenotypic mean give smaller and smaller differences in mean fitness, because the correlation between the mean fitness and the mean phenotype is less and less precise.
Peak Shifts by Drift in the Mean Phenotype
The neutral effects of founder events on the mean phenotype of a population have been well studied. If the character in question is determined genetically by loci that interact additively, the distribution of the phenotypic mean of populations derived by bottlenecks should have variance equal to [V.sub.A]/N, where N is the effective population size of the bottlenecked population. Depending on genetic assumptions, this distribution is approximately or exactly Gaussian, such that the probability that the phenotypic mean of a bottlenecked population being greater than some critical value is determined by the probability density of the tail of the normal, which is given by the error function (erf). Rouhani and Barton (1987b) have shown that, even under selection, the neutral model of this distribution determines the dynamics following a severe bottleneck. Therefore, the probability that the mean of a bottlenecked population has crossed into the domain of attraction of a new selective peak is simply
[Mathematical Expression Omitted],
where [Mathematical Expression Omitted] is the difference between the mean phenotype before the bottleneck and the edge of the domain of attraction (Rouhani and Barton 1987b).
The domain of attraction for this one-dimensional model must be carefully defined. When the asymmetry term (H) is not equal to 1 (that is, one peak is higher than the other), the amount of phenotypic variance can change the location of the minimum of the mean fitness function. The expected change in [V.sub.A] is -F[V.sub.A], thus, the amount of variance that determines the landscape (and therefore the direction of selection) after a bottleneck is, on the average, (1 - F)[V.sub.A], which means that the valley moves slightly away from the lower peak. This effect has been taken into account in simulation studies of the mean drift phenomenon (Charlesworth and Rouhani 1988).
The Variance Threshold
We have seen that phenotypic variance can increase sufficiently to allow shifts from one fitness peak to another [ILLUSTRATION FOR FIGURE 1 OMITTED]. However, these changes in the phenotypic variance can subsequently be reversed by natural selection. After a bottleneck, as selection becomes more effective with increasing population size, stabilizing selection tends to return the amount of genetic variance to an equilibrium between mutation and selection. The directional component of selection must therefore change the mean of the phenotypic distribution sufficiently to change the sign of the selection on the second moment, or else have changed the mean so far that there is now stabilizing selection around the new fitness peak. In other words, the mean must change toward intermediate values sufficiently to cause stabilizing selection to become disruptive selection, which would cause the amount of genetic variance to increase as a result of selection. If this doesn't happen, stabilizing selection eventually reduces the variance to such a point that the fitness function of the mean again becomes bimodal, and selection then returns the mean to its previous value.
Kirkpatrick (1982) has described the changes in mean, variance, and linkage disequilibrium for populations evolving on the bi-Gaussian fitness function used here. Explicit solutions for these equations are not possible, but the results can be numerically determined for any starting conditions [ILLUSTRATION FOR FIGURE 2 OMITTED]. Using these equations, the domain of attraction of each peak can be determined, as illustrated in figure 3, where the domain of attraction does not correspond exactly to the saddle of the bivariate fitness function. The reason is straightforward. For drift in the variance without changes in the mean, the variance must stay above a critical value long enough to allow the mean to move past the saddle point, as described above. Conversely, if the mean drifts sufficiently close to the saddle point, past the transition between stabilizing and disruptive selection, then increases in variance resulting from disruptive selection can change the mean fitness function enough to move the saddle point past the mean. This means that the domain of attraction of a higher peak is somewhat closer to the lower peak than is the saddle point, making adaptive transitions easier than expected in models with static variance.
The variance effect is therefore enhanced by anything that slows the rate of change of variance resulting from stabilizing selection, such as high mutation rates and weak selection. In fact, mutation can be so high relative to selection that mutation alone makes the transition from bimodal to unimodal fitness functions.
The Distribution of Genetic Variance
To estimate the distribution of genetic variance following peak shifts requires making certain assumptions about the nature of genetic variation. Here we assume that the distribution of allelic effects is Gaussian, because of the mathematical simplicity this assumption allows: We also neglect the effects of linkage disequilibria, even though linkage disequilibrium can contribute significantly to the variance among populations in the variance within populations (Avery and Hill 1977). Linkage disequilibria decays very rapidly, and as such does not allow enough time for selection to change the mean to the static domain of attraction of the new peak. We can therefore conservatively understand the relevant changes in the mean fitness landscapes by considering only the changes in genetic variance resulting from drift in gene frequencies.
To find the distribution of variance within a population for a species where the genetic variance is determined by a Gaussian distribution of allelic effects x, let [Mathematical Expression Omitted] be the variance in allelic effects for alleles at each locus in the outbred population. Assume that there is no dominance or epistasis. The variance among individuals at one locus is therefore [Mathematical Expression Omitted]. Let [V.sub.g] be the total variance among individuals, and [n.sub.L] be the number of loci that affect a trait, so that [Mathematical Expression Omitted].
Let N be the number of diploid individuals in the bottleneck. We can define a value
[Mathematical Expression Omitted],
where [x.sub.i] is the allelic effect of allele i, and [Mathematical Expression Omitted] is the mean of the [x.sub.i]. Q is [x.sup.2] distributed with 2N - 1 degrees of freedom. If [Mathematical Expression Omitted] is the variance among individuals in a sample for one locus, then
[Mathematical Expression Omitted],
so that the frequency of populations with variance [s.sup.2] is
[[Psi] ([s.sup.2] [where] N) = [(Q).sup.N - 3/2][e.sup.-Q]/[2.sup.N - 1/2][Gamma](N - 1/2). (5)
Because the distribution of the sum of [x.sup.2]-distributed variables is another [X.sup.2] distribution (Abramowitz and Stegun 1972), the distribution for [n.sub.1] equivalent loci is simply the [X.sup.2] distribution with [n.sub.L](2N - 1) degrees of freedom, or for [Mathematical Expression Omitted],
[Psi] (V [where] N) = [(2N[n.sub.L]R).sup.[n.sub.L](N - 1/2) - 1] [e.sup.-(2N[n.sub.L]R)]/[2.sup.[n.sub.L](N - 1/2)] [Gamma]([n.sub.L](N - 1/2)), (6)
where R is the ratio of the variance after the bottleneck over the variance before. R has a mean of [n.sub.L] (2N - 1) and a variance of 2[n.sub.L] (2N - 1); thus, the mean of V is [1 - (1/2N)], and variance of V is (1/2N [1 - (1/2N)]([Mathematical Expression Omitted]). Figure 4 plots an example of the distribution of genetic variance within populations after a bottleneck.
The Probability of Variance-Induced Peak Shifts
The above calculations include the information necessary to determine the probability of a shift from one peak to another by this variance effect. Unfortunately, because the domain of attraction is impossible to calculate explicitly, only numerical results are possible.
One attribute of the Gaussian model simplifies analysis; the mean and variance of the samples are independent. The joint distribution of the mean and variance of populations following bottlenecks of size N is therefore
[f.sub.N] [m, V] = [Phi](m [where] N) [Psi](V [where] N), (7)
where [Phi](m [where] N) is the distribution of the mean following a bottleneck. [Phi](m [where] N) is known to be Gaussian, with mean [Mathematical Expression Omitted] and variance [V.sub.A]/N. This distribution is plotted for N = 4 in figure 5.
For any given value of the strength of stabilizing selection relative to the distance between two individual fitness peaks ([Gamma]), the equilibrium mean and genetic variance were calculated by the recursions in Kirkpatrick (1982), assuming a Gaussian distribution of allelic effects after the bottleneck, which is more accurate as the bottleneck population size increases. This numerically obtained value of the mutation-selection balance is preferable to the available analytic solutions for two reasons: the same recursions are used to calculate the trajectories of populations, and other models of calculating the genetic variance at the mutation-selection balance assume a single fitness peak, rather than two or more. The recursions from Kirkpatrick (1982) were used to calculate the domains of attraction of the two peaks with regard to both mean phenotypes and phenotypic variance. Forty points on the domain were calculated, spaced from zero genetic variance to the genetic variance sufficiently large to allow peak shifts to occur without an initial change in the phenotypic mean. The domains were then estimated with a regression of critical variance on the log of the critical mean, which yielded a description of the domain curve with an [r.sup.2] greater than 0.99. The probability of populations falling within the domain of attraction of the other peak was calculated with numerical integration over the distribution of genotypic means and variances following a bottleneck of a given size.
It is most informative to plot the probability of peak shifts against the strength of stabilizing selection around one of the peaks on the individual fitness function. As selection becomes more intense, the equilibrium value of the phenotypic mean more closely approaches a single peak on the individual fitness function, and the equilibrium amount of genetic variance maintained in a balance between selection and mutation is less. Thus, the valley on the mean fitness function becomes farther away from the equilibrium mean in both absolute terms and in terms of the number of genetic standard deviations. Also, as the strength of selection increases, the genetic variance converges more rapidly to the equilibrium value; therefore, the bimodality of the fitness function lasts a shorter time, and the mean has less time to move beyond the valley in mean fitness at equilibrium.
Figure 6 shows the ratio of the probability of peak shifts resulting from the two-dimensional stochastic changes in the phenotypic mean and genetic variance, as compared to the shift probability caused only by drift in the mean. As can be seen, the possibility of peak shifts by either mechanism is strongly dependent on the strength of stabilizing selection. Most importantly, when stabilizing selection is sufficiently weak that there is any real chance of peak shifts, the VIPS mechanism is a far more likely cause than simple drift in the mean. Figure 7 shows the absolute probability of peak shifts as a function of the strength of stabilizing selection; only when the strength of stabilizing selection is close to the critical value, when the two peaks will merge into one, will peak shifts occur often enough to be an important force in morphological evolution. The ratio of the probabilities of peak shifts by the different mechanisms approaches unity only in the circumstances when the absolute value of the probability of change is very small (compare [ILLUSTRATION FOR FIGURES 6 AND 7 OMITTED]). Thus, in the circumstances when drift-mediated peak shifts are most likely to occur, the importance of variance effects are most extreme.
The probability of a shift within a population from one fit type to another, through a set of intermediates with lower fitness, has been a persistent concern in evolutionary biology. Many mechanisms have been proposed to explain these shifts, most prominently Wright's (1977) shifting balance; others have included Fisher's (1958) idea that individual fitnesses change frequently and Goldschmidt's (1940) macromutations. Here I suggest a modification of Wright's idea that applies when adaptive landscapes are defined with respect to phenotypes - that changes in the phenotypic variance resulting from drift allow transition to occur. The idea is superficially a hybrid of Wright's and Fisher's ideas on the subject; mean fitness functions change in response to drift-driven changes in phenotypic variance.
Under many circumstances, peak shifts are more likely to occur via changes in phenotypic variance (VIPS) than by drift in the mean alone. Many evolutionary properties affect the probability of shifts by either mechanism; we need to consider the likelihood that these properties are capable of giving sufficient opportunity for peak shifts in natural populations.
The two most important factors that control the probability of shifts are the strength of stabilizing selection relative to the distance between optima and the effective population size of populations. As the strength of selection increases, the amount of genetic variance decreases in populations at equilibrium between mutation and selection. Lower genetic variance decreases the probability of peak shifts by either mean drift or VIPS. For mean drift, an important parameter is the distance from the lower peak to the domain of attraction, measured in genetic standard deviations; thus, lower genetic variance translates into less drift on an absolute scale. For VIPS, lower equilibrium genetic variance means that R, the ratio of the critical variance to the standing variance, is much increased; lower variance means that it is more difficult for the phenotypic variance to drift to sufficiently high levels. These constraints imposed on the shapes of mean fitness functions by the strength of stabilizing selection allow only a narrow band of selection intensities in which the mean fitness function is bimodal, yet allows a reasonable chance of peak shifts. In other words, rarely is drift both necessary and sufficient to cause peak shifts. If the strength of selection is slightly weaker than this critical set of values, the mean fitness function becomes unimodal. If the strength of selection becomes too strong, then peak shifts are unlikely, even over long periods.
How rare is too rare depends on the second important parameter - population size. Extremely small population sizes are required to allow peak shifts by either mechanism, by mean drift or by VIPS. Population bottlenecks or founder events probably occur in almost all species at some times; however, there is a wide range of local population-size variation among species, and very little data about population sizes during bottlenecks (for references, see Whitlock 1992). Furthermore, population bottlenecks resulting from local fluctuations in population size are more likely than a small constant effective population size. We have seen above that variance-induced peak shifts are less sensitive to population size than is mean drift, although in both cases the highest probability of shifts occur with smaller population sizes [ILLUSTRATION FOR FIGURE 7 OMITTED].
Other parameters affect the probability of peak shifts by variance-induced mechanisms. In particular, the number of loci affecting traits is critical to VIPS, but not to mean drift mechanisms. As the number of loci increases, the likelihood of a peak shift occurring through drift in genetic variance becomes extremely small. Unfortunately, little data for evaluating the number of loci that contribute much to the genetic variance within populations exists (Barton and Turelli 1989). From interspecific studies we know that many loci can separate two species (for examples, see Coyne 1992). However, for most of the traits analyzed, the observed response to selection is attributable to changes at a very small number of loci (Orr and Coyne 1992). Neither of these lines of evidence tells much about the architecture of genetic variance. The first steps towards these kinds of analyses report that less than 20 loci make up 95% of the genetic variance among selected lines (Zeng 1992).
There is another plausible variance-effect mechanism that has not been explored here, but that does not depend on the number of loci affecting a trait. Studies of genotype-by-environment interactions commonly find that there is genetic variation for the degree to which phenotypes are affected by environmental differences (Stearns 1992). Any characteristic under genetic control can drift in finite populations. Some bottlenecked populations may drift to increased sensitivity to environmental differences, and the environmental-variance component of phenotypic variance would be increased. Nothing in the mechanism of variance-induced peak shifts requires that the increased phenotypic variance be due to changes in the genetic variance per se; as long as there is some additive genetic variance for the character, the population can evolve along the revised mean fitness landscape to a new peak.
The idea that variance may increase following a population bottleneck may seem counterintuitive. After all, the expected amount of additive variance following a bottleneck should be (1 - F) times the additive variance of the outbred source population. The effects of nonadditivity aside, this expectation must be remembered to be just that, an expectation. On the average, the additive genetic variance will decrease by a factor F, but this is only one attribute of an entire distribution of possible outcomes. As shown above, there can be a substantial tail of this distribution where the additive variance increases.
With nonadditive interactions among alleles and loci, the distribution of changes in variance becomes much more complicated. In fact, under some circumstances, the total genetic variance may be expected to increase (Robertson 1952; Goodnight 1988; Whitlock et al. 1993). Nonadditivity also inflates the variance in variance across populations (Whitlock unpubl. data). Genotype-by-environment interactions imply that the environmental variance will increase in some subpopulations (see above), and genotype-by-environment interactions are common.
There are some empirical data that pertain to this problem (see Wright 1977, ch. 4). Studies of genetic homeostasis have often compared the morphological variance of inbred lines to that of the [F.sub.1] between these lines and found that there was less variability in the [F.sub.1] (Mather 1950; Rasmusen 1952; Robertson and Reeve 1952; Lewis 1953; Gruneberg 1954; McLaren and Michie 1954; Thoday 1955). The levels of in-breeding in these studies are too extreme for our purposes, and the genetic structure of [F.sub.1]'s is different from outbred individuals. Many studies have reported a significant relationship between enzyme heterozygosity and morphological variability (for a review, see Mitton and Grant 1984; Eanes 1978; Mitton 1978; Leamy 1982; Fleisher et al. 1983; but also see negative results by Handford 1980; McAndrew et al. 1982; Zink et al. 1985; Houle 1989; Booth et al. 1990). Other studies have reported the phenotypic variance of multiple inbred lines relative to that in a control. In these cases where data is reported, there is evidence that the phenotypic variance can sometimes increase as a result of inbreeding (Lopez-Fanjul and Villaverde 1989; Lopez-Fanjul et al. 1989). In each of these studies, the median change in phenotypic variance is a decrease, but there are some lines in which phenotypic variance goes up, sometimes to an extent that makes the process presented here feasible.
It is difficult to interpret these changes at face value. As Avery and Hill (1977; see also Zeng 1992) have shown, the majority of the variance in variance for characters determined by many loci is caused by drift in linkage disequilibrium. Increased variance resulting from linkage disequilibrium does change the mean fitness landscape, but linkage disequilibrium is likely to disappear relatively quickly; so fast, in fact, that the mean phenotype may not have evolved to within the "static" domain of attraction of a new peak. Only one study has investigated this expected decline resulting from segregation, and it has not found evidence for the expected decrease through time (Lopez-Fanjul et al. 1989). Finer-scale measures of the changes in and cause of variance in variance are needed.
The details of the model, as presented here, depend on the genetic variance being maintained within populations as a result of the balance between mutation and stabilizing selection. However, we do not know what ultimately maintains genetic variance in populations. This lack of information leaves a problem for the model of peak shifts presented here. If [V.sub.g] depends exactly on [Gamma], then there is a small range of the strength of stabilizing selection in which the VIPS mechanism can function, bounded on one side by values of [V.sub.g] too small relative to [Gamma], and on the other by equilibrium unimodal mean fitness functions that do not require any form of phase transition. Furthermore, if variance is maintained by other mechanisms than a pure mutation-selection balance, then those other mechanisms would presumably also affect the probability of peak shifts in as yet unspecified ways. Pleiotropy can significantly change the level of variance (Lande 1980; Barton 1990) and can also significantly affect the direction of evolution on a multivariate surface (and therefore change the probability of shifts from one state to another; Price et al. 1993). There is no reason why other mechanisms for the maintenance of variation would not allow variance-induced peak shifts, but, obviously, the relationship between the strength of stabilizing selection and the probability of peak shifts would be substantially changed.
It is clear that the changes in phenotypic variance associated with (among other things) periods of small population size can be a significant factor in the evolution of phenotypic characters. Random increases in the phenotypic variance of populations translate into smoother mean fitness landscapes, which in turn allow selection to deterministically move a population from one stable state to another. Including changes in variance in our analyses of the adaptive exploration of fitness landscapes significantly increases the power of this model of evolution to explain rapid and adaptive evolution.
This paper has benefited from the kind criticism and discussion of N. Barton, K. Fowler, P. Phillips, and M. Wade. This work was funded by the Science and Engineering Research Council.
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MICHAEL C. WHITLOCK Institute of Cell, Animal and Population Biology, Ashworth Laboratory, King's Buildings, University of Edinburgh, Edinburgh EH9 3JT, Scotland
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|Author:||Whitlock, Michael C.|
|Date:||Apr 1, 1995|
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