GENERAL METHODS FOR CALCULATING FIXATION PROBABILITIES

The branching process model as originally developed (Fisher 1922, 1930; Haldane 1927) assumes that W is the same for all copies of the beneficial allele and is constant over time. Building on work by Pollak (1966a, 1972), Barton (1995 and references therein) has applied the theory of multitype branching processes (see, e.g., Harris 1963) to study how fixation probabilities are influenced when W varies according to the “site” in which a given copy of the beneficial allele finds itself. This is a flexible approach in which sites can represent demes or microhabitats (Pollak 1966a, 1972; Barton 1987, 1993), genetic backgrounds (Barton 1994, 1995), age classes, or any other aspect of population structure. Relatively complex models can be analyzed in a straightforward way by considering the probability Q_i,t that a single copy of the beneficial allele present in site i in generation t is ultimately lost from the population. An expression for Q_i,t can be found by summing over all possible numbers of offspring and all possible movements of offspring between sites. If each copy of the allele present in site i at time t independently gives rise to n offspring, where n is drawn from a Poisson distribution with mean W_i,t, and the independent probability that each of these offspring will be in site j at time t + 1 is m_i,j, and each offspring in site j has independent probability Q_j,t+1 of being lost, then Qi,t=∑n=0∞e−Wi,t(Wi,t)nn!(Qi,t∗)n≡exp[Wi,t(Qi,t∗−1)], (3) where Qi,t∗=∑jmi,j(Qj,t+1) (4) is the probability of loss given that the allele at time t has exactly one offspring (Barton 1995). These equations extend in a straightforward way to give the generating function for the distribution of the number of copies of the beneficial allele.

The assumption that separate copies of the allele are independent means that the branching process is an appropriate model only when the number of copies of the beneficial allele is small relative to the total population size. When s_b ⪢ 1/N it is reasonable to assume that deterministic forces will prevail when the branching process model breaks down in this way, and in this case an allele that arises in background i at time t and that is never lost is said to become established. This occurs with probability P_i,t = 1 - Q_i,t. The probability of establishment for a beneficial mutation that arises in a random genetic background is denoted P_fix,t and is calculated as an average of the P_i,t, weighted by the probability of the beneficial mutation initially occurring in each background i. If an allele is established, it is not actually guaranteed fixation, but its frequency might approach a polymorphic equilibrium or it might become fixed only in some sites.

Because the branching process model assumes that the fate of an allele of interest is determined while it is rare, it cannot be used to calculate fixation probabilities for slightly deleterious mutations or for beneficial mutations that confer an advantage that is weak relative to the effects of genetic drift. For the same reason, branching process models cannot be used to determine the distribution of times taken until ultimate fixation of an allele. To address these types of questions Kimura (1957) used a diffusion approximation, which is valid when change is approximately continuous in time (large population sizes and weak selection). This approach has been used extensively to study the expected time to fixation (Kimura and Ohta 1969) and how fixation probabilities are influenced by population subdivision (Maruyama 1970, 1971; Nagylaki 1982; Barton 1987, 1993; Barton and Rouhani 1987), by “background selection” due to segregating deleterious alleles (Charlesworth 1994), or by changing population size (Otto and Whitlock 1997). Despite the advantages of the diffusion approximation, the branching process model remains a useful tool because it can often be analyzed much more easily.

MODEL

The model used is identical to the one studied by Manning and Thompson (1984) and Peck (1994). The notation used is summarized in Table 1. We consider the fate of a beneficial allele within a haploid population of fixed size N. Individuals without the beneficial allele are called “wild type.” We assume that N is sufficiently large for deterministic results to apply to the wild-type subpopulation and for the effect of Muller’s ratchet (loss of the most fit genotype by genetic drift; see Muller 1964; Haigh 1978; Gordo and Charlesworth 2000) within the wild-type subpopulation to be negligible over the timescale of interest. (A sufficient condition for this is exp[-U/s_d] ⪢ 1/N.) We assume that deleterious mutations arise at a constant rate U per genome per generation, that there are an effectively infinite number of equivalent biallelic loci, and that at each locus the deleterious allele reduces fitness by a factor (1 - s_d) with multiplicative effects across loci. Genetic backgrounds (sites) can therefore be enumerated by i, where i is the number of deleterious alleles and takes discrete values i = 0, 1, 2,.... Back mutations are ignored. We first consider a wild-type population at mutation-selection equilibrium. We then consider the more complex scenario of a wild-type population that is initially free of deleterious alleles and is approaching equilibrium. The correspondingly more complex calculations have been placed in appendices a and b.

We assume that a single beneficial allele arises in a randomly chosen wild-type individual, and its presence increases relative fitness by a factor (1 + s_b) regardless of the genetic background on which it is expressed; that is, we assume no epistasis for fitness. Except for its small size, the subpopulation carrying the beneficial allele is identical to the large wild-type (sub)population. That is, deleterious mutations arise at the same rate U and have the same effect on fitness (1 - s_d). Because the number of copies of the beneficial allele is initially small, we consider the progress of Muller’s ratchet within this subpopulation. We calculate the fixation probability for the beneficial allele, P_fix, by considering its copy number in different genetic backgrounds as a multitype branching process. We assume a Poisson distribution of offspring number.

To estimate the long-term average rate of adaptation, measured as the rate of fitness improvement, we embed our model for fixation probabilities within a more complex model. This is a generalization of the model of Orr (2000b). We make the standard assumption that the beneficial mutation origination process is Poisson and occurs at rate kU per individual per generation, where k ⪡ 1 is the ratio of beneficial to deleterious mutations. Because the population size is N the total number of new beneficial alleles arising per generation is NkU. We then make the questionable assumption that each such mutation is fixed with equal and independent probability P_fix, so that the mutation fixation process is also Poisson and occurs at rate NkUP_fix. This assumption requires that the perturbations due to beneficial substitutions are small and transient enough that the fixation probabilities of most beneficial mutations are well approximated by the equilibrium population results. In making this assumption we also ignore interference between multiple beneficial alleles (the Hill-Robertson effect; Fisher 1930; Muller 1932; Hill and Robertson 1966; Felsenstein 1974; Barton 1995; Gerrish and Lenski 1998; Gerrish 2001). We assume that successively fixed beneficial alleles have multiplicative effects on fitness.

ANALYSIS

Fixation probabilities: Fitness relative to the fittest wild-type individual is denoted w. Hence, when the beneficial allele is present in the fittest possible individual it has relative fitness w = w₀ = (1 + s_b). A beneficial allele present in genetic background i has relative fitness wi=(1+sb)(1−sd)i. (5) At this stage, a minor technical point should be made about two factors that have not been made very explicit in some previous analyses, although they were discussed in the Appendix of Peck (1994). In the branching-process model the expected number of offspring of a given genotype is its absolute fitness W_i, which with a constant population size is its fitness relative to the population mean fitness, W_i = w_i/w. Because the beneficial allele is rare by assumption, w is equal to the mean fitness of the wild-type subpopulation, and because we assume here that the wild-type population is at equilibrium w = e^-U (Kimura and Maruyama 1966) and therefore Wi=wieU. (6) The importance of this difference between absolute and relative fitness is not necessarily apparent when both the wild-type population is at equilibrium and only unmutated offspring are of interest. A fraction e^-U of the offspring of a given individual carrying the beneficial allele are free from additional deleterious mutations, and so in this special case these two factors cancel out and correct results can be derived by assuming that the “effective absolute fitness” is e^-Uw e^{U i} = w_i. This simplification does not apply generally, however.

In Equations 3 and 4, Q_i,t is the probability of loss of a single copy of the beneficial allele present after selection followed by movement between sites. Since here “movement between sites” represents deleterious mutation, the probability of the beneficial allele arising in site i is calculated using the frequencies of the different sites after deleterious mutation. Because the number of deleterious alleles, i, carried by a randomly chosen wild-type individual is Poisson distributed with mean λ= U/s_d (Haigh 1978), we have that fi=λtie−λi! (7) is the probability of the beneficial mutation arising with i deleterious alleles (i.e., the frequency of background i). Equation 7 holds only when exp[-U/s_d] ⪢ 1/N.

It is more convenient to rewrite Equation 4 in terms of the probability P^{* i} that an allele in background i is never lost (where P^{* i} = 1 - Q^{* i}). The probability that a given copy of the beneficial allele in site i is moved to site i + j by deleterious mutation is simply mi,i+j=e−UUjj!. (8) When we substitute (4) and then (8) into (3) we obtain Pi,t=1−exp[−Wi,t∑j=0∞e−UUjj!Pi+j,t+1]. (9)

As mentioned briefly earlier in this article and discussed at more length by Peck (1994), there may be backgrounds in which the beneficial allele confers a net advantage in the short term but that will ultimately give rise to a lineage with a lower mean fitness than the wild-type population; such beneficial alleles are assumed not to ultimately fix. Consider a beneficial allele subpopulation that has become large and that includes individuals with i, i + 1, i + 2,... deleterious alleles. The most fit individual in this subpopulation has fitness max{w_B} = (1 + s_b)(1 - s_d)ⁱ and so the subpopulation will ultimately approach mutation-selection balance with w¯_B = (1 + s_b)(1 - s_d)ⁱ e^-U. The wild-type subpopulation ultimately approaches w¯_W = e^-U and if w¯_B > w¯_W, equivalently (1 + s_b)(1 - s_d)ⁱ > 1 or i≤imax=⌊ln[1+sb]−ln[1−sd]⌋, (10) then the beneficial allele subpopulation will ultimately replace the wild-type subpopulation. In condition (10) ⌊.⌋ denotes the integer part and i_max is the largest value of i where the condition is satisfied, noting that it is always satisfied for i = 0. Here we have assumed that the wild-type population is sufficiently large that the beneficial allele subpopulation does not have time to fix before reaching approximate mutation-selection balance, and we have ignored back mutation of deleterious alleles. (The validity of these assumptions is discussed below.) To find the probability that the beneficial allele ultimately fixes we need to follow only the number of copies in backgrounds 0 ≤ i ≤ i_max, because if it is lost from all of these backgrounds then it can never ultimately fix. Therefore we can replace the ∞ in the upper limit of the sum in Equation 9 with (i_max - i).

As t → ∞ with N constant the state of the wild-type population becomes constant over time, and at equilibrium both P_i,t+1 and P_i,t converge to the single value P_i (an abbreviation for P_i,∞).

By making the simplifications described in the previous two paragraphs, we obtain a set of simultaneous equations in P_i for i = 0, 1, 2,..., i_max. In general, we can start by solving Pimax=1−exp[−e−UWimaxPimax]=1−exp[−wimaxPimax]=pimax, (11) where p_i is an abbreviation for p[w_i - 1] and p[s] is the fixation probability of a Poisson branching process with mean 1 + s (see Equation 1). Each P_i can then be calculated numerically in descending order of i by solving Pi=1−exp[−wi∑j=0imax−iUjj!Pi+j]. (12) Once the P_i are known, the net fixation probability P_fix can be calculated by averaging over all of the different genetic backgrounds Pfix=∑i=0imaxfiPi, (13) where the f_i are given by setting λ= U/s_d in Equation 7. There does not appear to be a more concise general expression for P_fix, except for the special case where i_max = 0, which was solved by Peck (1994; see Equation 2 above). The calculations described here have been automated in a Mathematica (Wolfram 1996) notebook available from T. Johnson on request.

The rate of adaptation: Our model for long-term adaptation assumes that beneficial mutations are rare enough that they can be considered, to a good approximation, to arise in close-to-equilibrium populations, so that P_fix is relevant. We wish to find an approximation for C, the expected increase in mean log-fitness per generation. To do this we first find an approximation for ΔC, the expected increase in log-fitness per beneficial substitution, which in turn requires an approximation for ΔC_i, the expected increase in log-fitness per beneficial substitution conditional on the beneficial allele arising in background i.

If the beneficial allele arises on background i and is ultimately fixed, then some number h_i of deleterious alleles will be fixed by hitchhiking (allowing the possibility of h₀ = 0). Clearly h_i ≥ i, but the problem is that h_i conditional on fixation is a random variable and may exceed i in the event that the beneficial mutation is lost from the background on which it originally arose but ultimately does becomes fixed. We make use of the decomposition P_i = p_i + x_i, which is detailed in appendix c. Here p_i = p[w_i - 1] is the probability of fixation given that the beneficial allele fixes in the background in which it arose (i), and x_i is the probability of fixation given that the beneficial allele is lost from the background in which it arose. x_i can be calculated directly by numerical solution of Equation C2 or simply by taking the difference between P_i and p_i. This partitioning of fixation events into two mutually exclusive possibilities suggests that for U ⪡ 1 it might be reasonable to suppose that the fate of a beneficial allele lost by mutation from background i, conditional on eventual fixation, is identical to the fate of a beneficial allele that arises in background i + 1, again conditional on ultimate fixation. This leads to the approximation E(hi)≃1Pi(pii+xiE(hi+1)), (14) which can be calculated in decreasing order of i because x_imax = 0. The expected increase in population mean fitness given that a beneficial allele arising on background i becomes fixed can be similarly approximated, and because for small s_b the mean increases in fitness and log-fitness are roughly equal we obtain ΔCi≃1Pi(pi((1+sb)(1−sd)i−1)+xiΔCi+1). (15) Both of these quantities can be averaged over the distribution of f_iP_i. Then we obtain the expected number of deleterious alleles that will hitchhike with a beneficial allele arising on a random background, conditional on its ultimate fixation E(h)≃1Pfix∑i=0imaxfiPiE(hi) (16) and the expected increase in mean log-fitness per fixation ΔC≃1Pfix∑i=0imaxfiPiΔCi. (17)

As explained above, the assumption of our model for continued adaptation is that the mutation fixation process is Poisson and occurs at rate NkUP_fix. Because we assume multiplicative effects of multiple beneficial alleles, the long-term average rate of mean log-fitness increase is given approximately by C≃NkUPfixΔC. (18)

When U is varied and the other model parameters are held constant, C is maximized at a particular value of U, which we call the optimum mutation rate U_opt. Orr (2000b) proved that when s_b < s_d (so that i_max = 0) the optimum mutation rate is given simply by U_opt = s_d. This result can be seen by substituting (2) into (18) and differentiating to obtain dCdU≃Nke−U∕sd(1−Usd)2sbΔC (19) (because ΔC ≃ s_b does not depend on U for i_max = 0) and noting that dC/dU = 0 and d²C/dU ² < 0 when U = s_d (Orr 2000b). For i_max > 0 no such simple derivation is possible, but U_opt can be found by numerical calculation reasonably efficiently using a golden section search, automated in a Mathematica (Wolfram 1996) notebook available from T. Johnson on request. It is notable that because both P_fix and ΔC are only functions of U, s_b, and s_d and do not depend on N or k, then U_opt must be a function of s_b and s_d only and also will not depend on N or k.

NUMERICAL RESULTS

Fixation probabilities when many genetic backgrounds are relevant: As we described in the Introduction (see also Fisher 1930; Manning and Thompson 1984; Charlesworth 1994; Peck 1994; Barton 1995; Orr 2000b), when s_b ≤ s_d a beneficial allele must arise in the single most fit genetic background to have any chance of fixation, and the net fixation probability is given by Equation 2. However, when s_b ⪢ s_d there are many genetic backgrounds in which a beneficial allele can arise and have some probability of fixation and so the situation is more complex. This is illustrated in Figure 1, which shows P_i (top, line), the fixation probability of a beneficial allele arising in a background with i deleterious alleles, assuming a wild-type population at equilibrium and parameter values s_b = 5 × 10^-3, s_d = 5 × 10^-4, and U = 3 × 10^-3. The probability of arising in each background, f_i (top, dots), and the way that P_i and f_i combine (bottom) to determine the net fixation probability P_fix are also shown. For beneficial alleles arising in the least fit background of relevance, with i = i_max = 9 deleterious alleles, the fixation probability is very small (w_imax < ∼1 + s_d and hence P_imax < ∼2s_d for s_d ⪡ 1) because the advantage of the beneficial allele is almost totally eliminated by the deleterious alleles it is linked to and because any new deleterious mutation will eliminate that advantage altogether. As i decreases P_i increases and approaches an asymptote, which is P₀ ≃ 2(U + s_b) = 1.6 × 10^-2 for U ⪡ 1, s_b ⪡ 1. This asymptote occurs because the beneficial allele is in a genotype with absolute fitness approaching (1 + s_b)exp[U] and because many new deleterious mutations must occur to reduce that advantage. A mathematical description of this behavior, which is a good approximation when U ⪢ s_b, is detailed in appendix c. In this example the main contribution to the net fixation probability P_fix is from moderate fitness backgrounds with i intermediate between zero and i_max. For the parameter values used in Figure 1 a beneficial allele that fixes is most likely to be one that arose on a background of i = 5 deleterious alleles, and therefore at least one-half its selective advantage will be negated by the deleterious alleles that hitchhike to fixation with it.

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Figure 1.

—A beneficial allele has probability f_i (top, dots) of arising in a background of i deleterious alleles. Given that it does, it has fixation probability P_i (top, line). The sum of f_iP_i (bottom) over all backgrounds gives the net fixation probability P_fix ≃ 7.8 × 10^-3 for the parameter values used here (s_b = 5 × 10^-3, s_d = 5 × 10^-4, U = 3 × 10^-3).

Fixation probabilities in an equilibrium population: Figure 2 shows results for the situation where the beneficial allele arises in a population at equilibrium under selection and deleterious mutations of fixed effect. It explores the region of the parameter space where U/s_d ≤ 10, so that f₀ ≥ e^-10 ≃ 4.5 × 10^-5 will represent a large number of individuals for population sizes that are realistic, at least for bacteria. Shown is the relative fixation probability when there is interference, R = P_fix/ p[s_b], where the fixation probability of the beneficial allele is P_fix and the value it would take in the absence of any interference is p[s_b]. A built-in Mathematica algorithm (Wolfram 1996) was used to fit contours to a lattice of values calculated by numerical solution of Equations 11 and 12. The irregularities in the contour lines are not caused by numerical inaccuracies but occur near where the number of relevant genetic backgrounds i_max changes from one integer value to the next, which causes the gradient of P_fix to change abruptly.

For the region of the parameter space considered in Figure 2 the relative fixation probability spans almost five orders of magnitude. A logarithmic scale of fixation probabilities is appropriate if one wishes to understand in which regions of the parameter space evolution essentially cannot proceed. However, for other questions a linear scale is more appropriate. In the case of competing subpopulations, a small difference in rates of adaptation is critical, and the rate of adaptation is approximately linear in the fixation probability. In a log-log-linear space the (s_b, s_d, R) surfaces show regions with roughly constant either high (R ≃ 1) or low (R ≃ 0) fixation probability. To help visualize this we show additional contour lines at R = 0.9 (dotted lines) and R = 0.5 (dashed lines), which together with the first solid contour at R = 0.1 roughly define the transition from no effect to a severe effect of interference from deleterious alleles.

In all of Figure 2, the area above and to the left of the diagonal s_b = s_d represents the region of the parameter space where Equation 2 applies, and the area below and to the right of this diagonal represents the region of the parameter space where the methods developed above are necessary to calculate the fixation probability.

Several trends worthy of comment are evident in Figure 2. First, by comparing across all three panels we can see that fixation probabilities are reduced more severely by interference as the rate of deleterious mutation U increases. This result is expected. Second, the severity of the reduction in fixation probability often but not always increases monotonically as the strength of selection against deleterious alleles s_d decreases. An example of nonmonotonicity in s_d can be seen when U = 10^-3 and s_b = 10^-2.5 (Figure 2, top). This perhaps counterintuitive result occurs because there are two opposing forces at work here. As s_d decreases the frequency of deleterious alleles increases, and hence they are more likely to be present in the background on which the beneficial allele arises, but the effect of any one deleterious allele in reducing the advantage of the beneficial allele is less. The graphs show that, for the parameter space explored, the former force tends to dominate the latter. For s_b < s_d examination of Equation 2 shows that R increases as U increases and s_d decreases, but it was not at all clear that this dependency would be true for most but not all s_b.

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Figure 2.

—The fixation probability P_fix of a beneficial allele conferring a selective benefit s_b is reduced by interference from segregating deleterious alleles causing disadvantage s_d that arise at completely linked loci at rate U. The relative fixation probability in an equilibrium population (R = P_fix/ p[s_b], measured relative to the fixation probability without interference) is shown as a function of s_b and s_d. From top to bottom U = 10^-3, U = 10^-2, and U = 10^-1.

A third visible trend is that beneficial alleles of larger effect have fixation probabilities that are less influenced by segregating deleterious alleles. This is an intuitively reasonable result, but one that is not true when s_b < s_d. Equation 2 shows that R is independent of s_b, which can be seen to be true in the parts of Figure 2 where this equation applies, i.e., everything above and left of the line s_b = s_d. To the right of and below the line we see that the situation is more complex. The final trend worth noting is that in some parts of the parameter space there is a catastrophic reduction in fixation probability caused by interference from segregating deleterious alleles. To a coarse approximation, it can be said that R ≃ 0.5 when s_b = max{s_d,U} and R will be very small when both s_b ⪡ s_d and s_b ⪡ U.

The three parts of Figure 2 are almost perfect replicas of each other, offset by the value of U, suggesting that R depends only on two compound parameters s_b/U and s_d/U. Equation 2 shows that this is true when s_b < s_d. In appendix c we show that this is also true when s_d ⪡ s_b ⪡ 1 and U ⪡ 1, that is, when selection and mutation are both weak and many genetic backgrounds are relevant.

Nonequilibrium populations: In appendices a and b we derive results for fixation probabilities when the population of interest was free of segregating deleterious alleles at some time t = 0 in the past. This initial condition approximates the effect of a rapid selective sweep, a severe population bottleneck with rapid recovery, or the founding of a laboratory evolution experiment from a single clone. The fixation probability of a beneficial mutation that arises at some subsequent time t =τ is denoted P_fix,τ. The results above for an equilibrium population are a special limiting case of this scenario, where τ → ∞.

Consider first the simplest case τ= 0. At that time a beneficial mutation is guaranteed to arise in a background free of deleterious alleles (f₀ = 1), which would increase its net fixation probability. However, at the same time the population mean fitness w¯ would be unity, causing the absolute fitness of genotypes containing the beneficial allele to be lower than in an equilibrium population (where w¯ = e^-U; see Equation 6), which would cause a decrease in its net fixation probability. There are therefore two opposing forces at work, and their combined effect on the fixation probability of a beneficial allele is not obvious. It is not necessarily sufficient to argue that the net fixation probability is reduced by variance in fitness across backgrounds and is therefore higher when there is zero variance at τ= 0, because this argument does not take into account the changing mean fitness of the wild-type population. (Indeed a variance-based argument fails to predict how P_fix depends on s_d in an equilibrium population.)

In appendix b we prove that, for the special case where s_b < s_d, the fixation probability for a beneficial mutation occurring at time τ< ∞ is always greater than in an equilibrium population (τ= ∞). To see the importance of this result, consider further the case of a weakly selected beneficial allele with s_b ⪡ U, s_b ⪡ s_d arising at time τ= 0. Genotypes containing such a beneficial allele all give rise to, on average, less than one offspring of the same genotype at t = 0 and for some period of time thereafter (because e^-UW_i,0 < 1, see appendix a). To have any probability of fixation at least one copy of the allele must persist until the wild-type mean fitness has decayed significantly to have absolute fitness greater than one. It is therefore quite surprising to find that such beneficial alleles always have greater fixation probability than they would if they arose in an equilibrium population. We offer the following explanation for our perhaps counterintuitive result. As we go backward in time there is a decrease in fixation probability due to increasing w¯_t and an increase in fixation probability due to increasing f_0,t. Because f_0,t+1/f_0,t = e^-U/w¯_t these two forces are coupled, and the nature of this coupling ensures that, working backward in time from an equilibrium fixation probability, the fixation probability will always increase. On the basis of our numerical results for s_b > s_d we speculate that this is true for all values of s_b and s_d.

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Figure 3.

—The fixation probability P_fix of a beneficial allele conferring a selective benefit s_b is inflated if it arises some short time τ after segregating deleterious alleles have been purged from an asexual population. The degree of inflation (I = P_fix,τ/P_fix,∞, measured relative to the fixation probability in a population at mutation-selection equilibrium) is shown as a function of τ and s_b. Left, U = 10^-2, s_d = 10^-2, and the maximum inflation is I ≃ 1.58. Right, U = 10^-1, and s_d = 10^-2, and the maximum inflation is I ≃ 37.7. In both sides the same shading scheme is used, solid contours are at geometric intervals of ∼1.44, and dashed contours are at geometric intervals of ∼1.048.

We measure the effect of a nonequilibrium population by the inflation in fixation probability, I = P_fix,τ/ P_fix,∞, measured relative to an equilibrium population. Figure 3 shows I as a function of s_b and of τ. In these calculations we assumed that after 1000 generations the population would be close to equilibrium. The choice of s_d = 10^-2 for these plots was influenced by the computer time required for the calculations (see appendix a), and the choices of U = 10^-2 (left plot) and U = 10^-1 (right plot) represent situations where there is a moderate and a large reduction in fixation probability at equilibrium. For these parameter values, any inflation in fixation probability is a relatively short-lived effect and is negligible [in the sense that (I - 1)/(max{I} - 1) < 0.05] after 300 generations. This is to be expected because a population perturbed from mutation-selection balance decays toward its equilibrium state on a timescale proportional to 1/s_d = 100 generations (Johnson 1999). What is more remarkable is that the inflation in fixation probability occurs only for beneficial alleles with certain selection coefficients. For U = 10^-2 only beneficial alleles with s_b ≃ 10^-2 have inflated fixation probabilities and the effect is moderate (I ≤ 1.6). For U = 10^-1 beneficial alleles with selection coefficients s_b ≃ 10^-2 or greater have inflated fixation probabilities and the effect is substantial (I ≃ 40). To understand why this is so, it is necessary to consider these fixation probabilities relative to the case with no interference, measured by R = P_fix,τ/p[s_b], which are shown in Figure 4. The abrupt changes in gradient visible in the graph are caused when the number of relevant genetic backgrounds i_max changes from one integer value to the next.

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Figure 4.

—Relative fixation probabilities, R, in a nonequilibrium population. Top, s_d = 10^-2, U = 10^-2; bottom, s_d = 10^-2, U = 10^-1. Note the very different scales of the y-axes. Curves show τ= 0 (dotted), τ= 100 (dashed), and τ= 1000 ≃ ∞ (solid).

Assuming that the fixation probability is always less than it would be in the complete absence of deleterious mutation (we prove this for s_b < s_d in appendix b and speculate that it is true always), then if fixation probabilities are only moderately reduced, as in the case s_d = 10^-2, U = 10^-2 (Figure 4, top), then the inflation in fixation probability I can be at most moderate. On the other hand when fixation probabilities are substantially reduced, as in the case s_d = 10^-2, U = 10^-1 (Figure 4, bottom), then the inflation in fixation probability I can also be substantial. The slightly mysterious peak of inflation I at s_b ≃ 10^-2 for U = 10^-2 can be partly explained because, for s_b ⪢ 10^-2, there is no reduction in fixation probability at equilibrium and hence there can be no transient inflation.

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Figure 5.

—Under the model assumptions (see text and Figure 6), the optimum deleterious mutation rate U_opt maximizes the long-term average rate of fitness improvement in an asexual population. U_opt depends only on s_b and s_d, against which it is plotted here.

Figures 3 and 4 show that, for the specific departure from equilibrium that we have studied, there is little effect on fixation probability for weakly selected beneficial alleles. It is possible that this is because the fate of such alleles takes a long time to be determined, and the transient nonequilibrium state of the wild-type population is therefore of little relevance.

The rate of adaptation and the optimum mutation rate: Figure 5 shows numerical calculations of the optimum mutation rate U_opt as a function of the two parameters on which it depends, s_b and s_d. The optimum mutation rate is the rate that maximizes the long-term average rate of fitness increase, as estimated using Equation 18. These results are in agreement with Orr’s (2000b) finding that U_opt = s_d when s_b ≤ s_d (Figure 5, left). However, when s_b > s_d the dependence on s_d is much weaker, and to a very coarse approximation U_opt ≃ max{s_b, s_d} for the whole parameter space examined here. This result can be explained in terms of our results for fixation probabilities. For any chosen combination of s_b and s_d imagine the appropriate points in Figure 2. Consider first a very small value of U. The fixation probability is high. Now imagine increasing U so that a “hole” starts to appear in the corner of the plane. At first, there is a less-than-linear decline in fixation probability with U and the rate of beneficial mutations increases linearly with U, so the rate of fitness improvement C increases. Suddenly, when U exceeds max{s_b, s_d} there is a catastrophic decline in fixation probability and an associated decline in C. Hence the rate of adaptation is maximized just before this catastrophe, at U_opt ≃ max{s_b, s_d}. This is illustrated in Figure 6.

APPENDIX A: NONEQUILIBRIUM WILD-TYPE POPULATION

There are many ways in which the wild-type population could be out of equilibrium. For simplicity, we study only one possibility. We assume that the population of interest is descended from one that was free of segregating deleterious alleles at time t = 0, where t is measured in generations, and that any deleterious alleles that were fixed at t = 0 do not revert to wild-type alleles by back mutation. This initial condition approximates the effect of a rapid selective sweep, a severe population bottle-neck with rapid recovery, or the founding of a laboratory evolution experiment from a single clone. As time increases, the frequencies of deleterious alleles will increase and the population will approach an equilibrium at mutation-selection balance. We assume that a single beneficial allele arises at some time t =τ> 0. The analysis in the main part of the article is therefore a special case with τ= ∞. Since the fate of the beneficial allele depends on the time of its origin, we write P_fix,τ for its fixation probability.

Because we assume that there are no segregating deleterious alleles in the wild-type population at time zero, the mean fitness and distribution of number of deleterious alleles per wild-type individual will change over time. Johnson (1999) gives equations describing these dynamics, in terms of the Laplace transforms (see, e.g., Feller 1971; Pinkus and Zafrany 1997) of the distributions involved. When deleterious alleles have a fixed effect (1 - s_d), the distribution of effects on log fitness has Laplace transform d~_(z) = (1 - s_d)^z, where z is a dummy variable. Equation 12 of Johnson (1999) shows that the Laplace transform of negative log fitness at time t is g∼(z,t)=exp[U∑i=0t−1(1−sd)z+i−(1−sd)i] (A1) (where g here corresponds to f in the notation of Johnson 1999). This simplifies to g∼(z,t)=exp[U(1−(1−sd)t)sd((1−sd)z−1)]. (A2) The population mean fitness at time t, relative to a wild-type mutation-free (i = 0) individual, is given by w‒(t)=g∼(z=1,t)=exp[U((1−sd)t−1)]. (A3) Equation A2 also shows that the distribution of negative log fitness is a scaled Poisson distribution and therefore that the number of deleterious alleles, i, carried by a randomly chosen wild-type individual at time t is Poisson distributed with mean λt=U(1−(1−sd)t)sd, (A4) where the quantities in Equations A3 and A4 are given after mutation and before selection. Equation A4 allows us to calculate the probability of the beneficial mutation arising in a given background at time t =τ, and Equation A3 allows us to calculate the absolute fitnesses of individuals carrying the beneficial allele for all subsequent times t ≥τ. As expected, as t → ∞ these quantities approach their equilibrium values w¯ → e^-U (Kimura and Maruyama 1966) and λ → U/s_d (Haigh 1978).

Equations A3 and A4 can also be derived using factorial cumulant-generating functions, as described by Dawson (1999).

The fixation probability of a beneficial mutation arising at time t =τ can be obtained by choosing some large time t = T, setting P_i,T = P_i,∞, and then finding the P_i,t recursively in decreasing order of t using Equation 9 with the upper limit in the sum set to i_max. Results obtained in this way, using T = 1000, are shown in Figures 3 and 4. The choice of T = 1000 is justified by noting that w¯_t and λ_t approach their equilibria over a timescale proportional to 1/s_d = 100 generations. Similar calculations for smaller s_d would be time-consuming at present because both the appropriate value for t and the number of genetic backgrounds to be considered for a given s_b (i_max) scale with 1/s_d and hence the number of calculations scales with 1∕sd2 .

Given the computationally intensive nature of these calculations, an approximation is desirable. It is tempting to assume that w¯_t changes slowly relative to the time taken for the fate of the branching process to be determined (which is typically “rapid”; see, e.g., Feller 1968, p. 297), which would give P_i,t+1 ≃ P_i,t ≃ P_i,τ. Then the method used for τ= ∞ could be followed with appropriately modified values for the W_i and f_i. Numerical results (not shown) show that this approximation is highly unsatisfactory for many combinations of parameter values. Specifically, this approximation often gives P_fix,τ < P_fix,∞ when more accurate calculations show that P_fix,τ > P_fix,∞. Because this approximation assumes that the fate of the beneficial allele is determined rapidly relative to the mean fitness dynamics in the wild-type subpopulation, it is most accurate for large s_b and/or large τ. However, sufficiently large s_b are so large that there is negligible effect from interfering deleterious alleles and sufficiently large τ are so large that solutions are indistinguishable from solutions for equilibrium wild-type populations.

APPENDIX B: NONEQUILIBRIUM POPULATION AND BENEFICIAL ALLELE OF SMALL EFFECT

For the special case s_b ≤ s_d we can prove an important result. This is that the net fixation probability of a beneficial allele arising in a nonequilibrium population (for the specific departure from equilibrium considered in appendix a) is greater than in an equilibrium population. More specifically, going backward in time from a close-to-equilibrium population, the net fixation probability increases. We speculate that this result is probably true in general but do not see an easy way to prove it. We can also show that for s_b ≤ s_d the net fixation probability increases by at most a factor (1 + s_b) per generation backward in time. This may explain why beneficial alleles of small effect do not have inflated fixation probabilities in nonequilibrium populations.

When s_b ≤ s_d it is necessary only to consider the fate of beneficial alleles in a background of zero deleterious alleles; that is, i_max = 0. The recursion for the fixation probability of a beneficial allele on such a background is P0,t=1−exp[−(1+sb)e−Uw‒tP0,t+1]. (B1) The net fixation probability of a beneficial allele that arises at time t =τ is Pfix,τ=f0,τP0,τ, (B2) where f_0t = exp[-λ_t] is the probability of arising in a background free of deleterious alleles, at time t. Equation A4 shows that f0,t+1f0,t=e−Uw‒t. (B3) By substituting Equations B2 and B3 into Equation B1 we can show that Pfix,τ=f0,τ(1−exp[−(1+sb)f0,τPfix,τ+1]). (B4) The series expansion Pfix,τ=(1+sb)Pfix,τ+1−(1+sb)22f0,τPfix,τ+12+(1+sb)36f0,τ2Pfix,τ+13−… (B5) converges for all f_0,τ > 0 and shows that Pfix,τ≤(1+sb)Pfix,τ+1 (B6) as claimed.

Our proof that net fixation probabilities increase as we go backward in time seems a little contorted, but we have been unable to find a more transparent proof. We first assume that a beneficial allele that arises in a nonequilibrium population at some large time τ= T has net fixation probability equal to that for an equilibrium population Pfix,T=Pfix,∞ (B7) and then prove by induction that Pfix,0<Pfix,τ<Pfix,τ+1<Pfix,T (B8) for all 0 <τ < T - 2. As T → ∞ condition (B7) is necessarily satisfied and therefore Pfix,0<Pfix,τ<Pfix,τ+1<Pfix,∞ (B9) for all 0 < τ.

To prove Equation B8, we note first that, from (B4), the equation Pfix,τ=Pfix,τ+1 (B10) is satisfied only when P_fix,τ = 0 or P_fix,τ = f_0,tp[s_b]. Therefore a graph of P_fix,τ against P_fix,τ+1 does not touch the line P_fix,τ = P_fix,τ+1 between these two points, and since it has gradient (1 + s_b) at the origin (see Equation B5) it must in fact lie above the line. Therefore Pfix,τ>Pfix,τ+1when0<Pfix,τ+1<f0,τp[sb]. (B11) Noting that P_fix,t is a monotonically increasing function of P_fix,t+1 (see Equation B4) and since f_0,τ-1 > f_0,τ we have Pfix,τ<f0,τp[sb]<f0,τ−1p[sb]when0<Pfix,τ+1<f0,τp[sb]. (B12) Observing that the condition in Equations B11 and B12 is satisfied when τ+ 1 = T because by assumption P_fix,T = P_fix,∞ = f_0,∞p[s_b], Equations B11 and B12 constitute a proof by induction for Equation B8.

APPENDIX C: APPROXIMATIONS

In the analysis section of the article we described a method that allows us to calculate P_fix exactly (given the model assumptions) for any parameter values. However, this analysis makes no real improvement to our understanding of how fixation probabilities are influenced by segregating deleterious alleles. There does not appear to be a more concise exact expression for P_fix, except for the special case where s_b < s_d and hence i_max = 0, which was solved by Peck (1994).

In this section we assume an equilibrium wild-type population, but the methods used could in principle be applied to nonequilibrium wild-type populations.

It is useful to write P_i = p_i + x_i. Here the first term gives the probability that the beneficial allele fixes within the genetic background in which it arose, and the second term gives the probability, conditional on it being lost from that background and that it was lost by mutation, that it fixes in some lower fitness background where it still confers a net advantage. x_imax = 0 but Manning and Thompson’s (1984) analysis erroneously assumed x_i = 0 for all i. By making this decomposition of P_i, separating out the j = 0 term from the sum in Equation 12 and then rearranging we can obtain pi+xi=(1−exp[−e−UWipi])+exp[−e−UWipi]×(1−exp[−e−UWi(xi+∑j=1imax−iUjj!Pi+j)]), (C1) which by using the identity p_i = 1 - exp[-e^-UW_ip_i] can be simplified to xi=(1−pi)×(1−exp[−e−UWi(xi+∑j=1imax−iUjj!Pi+j)]). (C2)

When selection and mutation are weak, we can assume 1 - p_i ≃ 1, e^-UW_i ≃ 1, and ignore terms in U ² and higher powers in the summation to obtain xi≃1−exp[−xi−UPi+1], (C3) which can be solved by making a series expansion to obtain the quadratic equation xi≃xi+UPi+1−12(xi2+2xiUPi+1+U2Pi+12)0≃xi2+xi2UPi+1+U2Pi+12+2UPi+1 (C4) and hence xi≃−UPi+1+2UPi+1−(UPi+1)2, (C5) which for UP_i+1 ⪡ 1 gives xi≃2UPi+1 and so Pi≃pi+2UPi+1. (C6) Unfortunately, even this simple expression does not seem to lead to an approximation for P_i in terms of the model parameters alone (i.e., which does not depend on P_i+1). Such an approximation can be found only when selection is weak relative to mutation, so that pi≪2UPi+1 , and then we have Pi≃PimaxA(2U)1−A, (C7) where A = 1/(2ⁱ_max-ⁱ). This shows that P_imax-_i → 2U as i increases (see Figure 1). This approximation may be useful more generally, so long as pi≪2UPi+1 for the backgrounds that contribute significantly to P_fix. However, we do not see a way to sum this approximation for the P_i over all backgrounds and hence do not see a way toward an approximate formula for P_fix.

We can use Equation C6 to show that when s_b ⪢ s_d the relative fixation probability R = P_fix/p[s_b] depends only on the compound parameters σ_b = s_b/U and σ_d = s_d/U. First note that under these conditions i_max ≃ s_b/s_d =σ_b/σ_d and that the f_i depend only on λ= U/s_d = 1/σ_d. Let the contribution to R from background i be R_i = P_i/p[s_b] ≃ P_i/2s_b. Then from Equation C6, which assumes weak selection and weak mutation, we have Ri≃2(sb−isd)2sb+2U2sbPi+12sb, (C8) which rearranges to give Ri≃1−iσdσb+1σbRi+1. (C9) Although R_imax cannot be written in terms of the compound parameters σ_b and σ_d alone, the R_i rapidly become independent of R_imax as i decreases. Therefore, to a good approximation R depends only on σ_b and σ_d.