## Abstract

Estimating the fixation probability of a beneficial mutation has a rich history in theoretical population genetics. Typically, to attain mathematical tractability, we assume that generation times are fixed, while the number of offspring per individual is stochastic. However, fixation probabilities are extremely sensitive to these assumptions regarding life history. In this article, we compute the fixation probability for a “burst–death” life-history model. The model assumes that generation times are exponentially distributed, but the number of offspring per individual is constant. We estimate the fixation probability for populations of constant size and for populations that grow exponentially between periodic population bottlenecks. We find that the fixation probability is, in general, substantially lower in the burst–death model than in classical models. We also note striking qualitative differences between the fates of beneficial mutations that increase burst size and mutations that increase the burst rate. In particular, once the burst size is sufficiently large relative to the wild type, the burst–death model predicts that fixation probability depends only on burst rate.

ESTIMATING the fixation probability of an initially rare beneficial mutation is fundamental to our understanding of adaptation. Such estimates are critical to studies of evolution under controlled laboratory conditions and are also essential for predicting the rate of adaptation of natural populations—for example, the rate of adaptation in response to environmental change or the rate of emergence of novel, or drug-resistant, pathogens.

To date, a considerable body of theoretical literature has been devoted to this question, beginning with classic articles such as Fisher (1922) and Haldane (1927). To attain mathematical tractability, these approaches have necessarily imposed a number of simplifying assumptions. In particular, the life-history models of classical population genetics typically assume that generation times are fixed, with no death between generations. However, recent work in our group (*e.g*., Wahl and DeHaan 2004) has emphasized that fixation probabilities are extremely sensitive to the underlying life-history model.

The work described here is motivated, in part, by the extensive and important body of literature regarding the experimental evolution of lytic viruses (*e.g*., Bull *et al*. 1997, 2000; Burch and Chao 1999, 2000; Wichman *et al*. 1999; Sanjuán *et al*. 2005; Manrubia *et al*. 2005). For lytic bacteriophages in particular, each “generation” time consists of an *attachment time*, during which the viral particle “searches” for a cell to infect, and a *lysis time*, the time between attachment and lysis, during which phage accumulate within the infected cell. Although the lysis (or “phage-accumulation”) time may be tightly regulated, attachment times are well modeled by a first-order process; *i.e*., they are exponentially distributed (Abedon *et al*. 2001).

When attachment times are very short, the standard assumption of fixed generation times is clearly valid. In this article, however, we develop a life-history model in which generation times vary. In particular, we treat the case when “death” occurs at a constant rate, and thus generation times (or, more correctly, lifetimes) are exponentially distributed. This may be an appropriate model for estimating fixation probabilities for lytic viruses, particularly when the attachment time is long compared to the phage-accumulation time, and may also be applied to other natural and experimental populations where generation times vary widely.

## A BURST–DEATH LIFE-HISTORY MODEL

We have developed the “burst–death” life-history model in analogy to the well-studied birth–death process. There are two ways, intuitively, to derive this model. First, we can consider an individual with a constant probability of death, *M*Δ*t*, in any small interval of time, Δ*t*. An example of such a process is the clearance of free virus or bacteria from a chemostat, or, more generally, any failure of a virion to infect a new cell. We also note that *M* may be zero, a case we treat explicitly in the sections that follow. While alive, the individual has a constant probability of reproducing, *L*Δ*t*, in time interval Δ*t*. These assumptions yield an exponential distribution of lifetimes. Note, however, that the latter assumption also implies that the time until reproduction does *not* depend on the age of the organism. The assumption that the probability of reproducing is constant for any age is the main limitation of the burst–death model and is clearly an avenue for future work. We return to this idea in the discussion.

If the individual reproduces, a “burst” of a fixed number of offspring is produced. In analogy to a linear birth–death process, we can either assume that the parent organism survives when *B* new offspring enter the population or assume that the parent dies when the offspring are produced, but *B* + 1 offspring are produced. These are mathematically equivalent since the age of the organism does not matter.

A second way to derive the same model is to imagine that each individual has a lifetime drawn from an exponential distribution. At the end of its life, the individual reproduces, producing either zero or *B* + 1 offspring. Although this second formulation may seem somewhat different from the first, for the purposes of computing fixation probabilities they are precisely equivalent. In particular, we have equivalence when lifetimes (generation times) in the second model are drawn from an exponential distribution with mean 1/(*M* + *L*), and the probability of having zero offspring is given by *M*/(*M* + *L*). We note that while *M* and *L* are positive real numbers, *B* is constrained to be a positive integer.

In the results and discussion to follow, we often compare the predictions of the burst–death model to the predictions of what we refer to as the “classic” model (Fisher 1922; Haldane 1927), which has the following assumptions: (1) generation times are fixed, (2) there is no death between generations, (3) the number of offspring produced by each individual follows a Poisson distribution, and (4) beneficial mutations confer an increased average number of offspring. Thus in the classic model, lifetimes are fixed but stochasticity occurs via the Poisson distribution of offspring, whereas in the burst–death model, lifetimes are stochastic but the number of offspring is fixed.

#### Selective advantage:

Suppose that the wild-type population consists of individuals as described above, with constant burst rate *L*, constant death rate *M*, and burst size *B*. A population of size *N*_{0} at time zero would then have total expected size *N*_{0}*e*^{(BL−M)t} at time *t*. We initially restrict our attention to the case when *BL* = *M*; otherwise the population is on average growing or decaying exponentially, and the fixation probability is of little interest. In the next section, we relax this assumption and introduce population bottlenecks.

Analogously, an individual with a beneficial mutation has burst rate λ, death rate μ, and burst size β. Since the mutant reproduces more effectively than the wild type, we might have, for example, β = *B* + Δ*B*, where β, *B*, and Δ*B* are all integers. To find the selective advantage, *s*, of such a mutant, recall that in standard population genetics models, a beneficial mutation increases the expected number of offspring in a generation from *W* to *W*(1 + *s*). In this case, however, the expected number of offspring in a mean wild-type generation time of 1/(*L* + *M*) increases fromtoSolving, we find that the classically defined selective advantage, *s*, is given by(1)Note that Equation 1 applies only when the beneficial mutation confers an increased burst size, β = *B* + Δ*B*. A similar expression can likewise be derived for *s* when the mutation confers an increased burst rate, λ > *L*, or decreased death rate, μ < *M*. For a mutation with burst rate λ = *L*(1 + δ), for example, we find that(2)The selective advantage *s* thus depends on the wild-type burst size, as well as on the wild-type burst and death rates (this factor becomes important in interpreting Figure 2 in the results).

#### Fixation and extinction probabilities:

We use *X* to denote the extinction probability, the probability that a lineage carrying a beneficial mutant is ultimately lost due to stochastic fluctuations in the size of the lineage. Throughout this article, we also use π to denote the complement of this probability, π = 1 − *X*. In reality, a beneficial mutation could also be driven to extinction through clonal interference (Gerrish and Lenski 1998) or could fix through quasispecies interactions (Wilke 2003); we ignore both of these possibilities. Thus technically, π does not give the fixation probability; we nonetheless use this parameter extensively (for example, in the figures) as shorthand for 1 − *X*.

#### Linear birth–death process:

When β = 1, the burst–death model yields the classic linear birth–death process, which has been studied in depth (Taylor and Karlin 1998). It is well known that the extinction probability for a birth–death process is one if μ ≥ λ, which can occur only if *s* is negative. The more interesting case occurs for positive *s*; in this case we have μ < λ and the extinction probability is given by *X* = μ/λ (Grimmett and Stirzaker 2001).

#### Burst–death process:

When β > 1, the extinction probability can also be obtained in a straightforward way. There are two ways for a lineage to become extinct through stochastic fluctuations. Either the progenitor dies before bursting or the progenitor has a burst of β + 1 offspring, but each of these lineages goes extinct. This yields(3)where μ/(μ + λ) gives the probability that an individual dies before bursting. Solving this equation for *X*, we find that as long as β > 4 or 5, *X* ≈ μ/(μ + λ). This result is quite straightforward from a mathematical point of view, but is somewhat surprising from a biological perspective. In particular, we would typically expect that as *s* grows very large, the fixation probability should approach one. Here, when either μ decreases or λ increases, this remains true. However, when *s* increases through an increase in burst size, β, the fixation probability approaches a plateau, λ/(μ + λ), which may be much less than one. In fact, this maximum value of π is equal to the probability of bursting before dying. This feature comes up again in the results (Figures 1 and 2).

## POPULATION BOTTLENECKS

A limitation of the burst–death model is that unless the overall rate at which offspring are produced exactly balances the death rate, *BL* = *M*, the population will grow exponentially without bound or decay exponentially to extinction. We extend this model, relaxing the assumption that *BL* is precisely balanced by *M*. In particular, we explore the case when *BL* > *M*, such that the population on average experiences sustained intervals of growth. We then assume that these growth intervals are balanced by population bottlenecks that reduce the population to its initial size.

Once again, we use the simplest possible model to allow tractability. We assume, as in previous work (Wahl and Gerrish 2001; Heffernan and Wahl 2002), that the wild-type population grows for τ time units, at which point a population bottleneck occurs. Each individual in the population independently survives the bottleneck with probability *D*, and the process repeats. Assuming that, on average, the bottleneck restores the population to its size at the start of the growth phase, we find that *D* must be given by *D* = *e*^{−(BL−M)τ}.

To solve for the fixation or extinction probability in this case is somewhat more difficult. Our approach is as follows. We first consider a beneficial mutation that exists as a single copy at the beginning of a growth phase. We then derive an implicit equation for *G*(*x*, τ), the probability generating function (pgf) that describes the number of offspring in this mutant lineage at the end of the growth phase. Each individual in the population survives the bottleneck with probability *D* and does not survive with probability 1 − *D*; thus the pgf for the bottleneck process is given by *H*(*x*) = 1 − *D* + *Dx*. Overall, then, the pgf for the number of individuals in the mutant lineage after one cycle of growth and sampling is given by *F*(*x*) = *G*(*H*(*x*), τ). Finally, the extinction probability for a single lineage present at the start of a growth phase, which we denote *X*_{0}, is given by the fixed point of this pgf, *i.e*., by the solution to(4)The extinction probability for mutations that first occur at other times during the growth phase can be computed very simply from *X*_{0}, as described in a later section. The difficult step in this approach is deriving, and then solving, an implicit equation for *G*(*x*, *t*).

#### Birth–death model with bottlenecks:

For the linear birth–death model (β = 1), the probability generating function *G*(*x*, *t*) is well known and is typically found as the solution to the following partial differential equation (PDE), which can be derived from first principles in a straightforward way (Grimmett and Stirzaker 2001):This is a PDE of the Lagrange type and can be solved using the partial fraction expansion of the auxiliary equation and the usual boundary conditions (*G*(*x*, 0) = *x*, *G*(1, *t*) = 1), to give(Grimmett and Stirzaker 2001). The fixed point of *G*(1 − *D* + *Dx*, τ) gives the extinction probability, *X*_{0}, and can be found exactly:While *X*_{0} explicitly depends on the life-history parameters of the mutant (λ and μ), the analogous parameters for the wild type (*L* and *M*) also affect *X*_{0} through the constant *D*.

#### Burst–death model with bottlenecks:

For β > 1, we can derive the following PDE from first principles, as described in detail in the appendix:(5)Once again this is a PDE of the Lagrange type and can be solved in an analogous way to the β = 1 case. The details of this solution are provided in the appendix, which demonstrates that *G*(*x*, *t*) is given by the value of *y* on [0, 1] that satisfieswhere the *x _{i}* are the β roots of the polynomial λ

*x*

^{β}+ λ

*x*

^{β−1}+ . . . + λ

*x*− μ, and the formulas for

*w*and for the

*L*are provided in the appendix.

_{i}This solution is inelegant, but numerically tractable even for very large values of β, for example, β = 100.

#### Pure burst model with bottlenecks:

A more elegant description of *G*(*x*, *t*) can be found for the case when μ = 0, which we call a *pure burst process* with periodic bottlenecks. In this simpler case, we can use a result from the study of continuous-time branching processes; for a very clear derivation, see Allen (2003, p. 240). This result holds when individuals in the population have exponentially distributed lifetimes with mean 1/λ, and at the end of a lifetime produce a random number of offspring according to the probability generating function *f*(*x*). In this case, if a lineage begins with a single individual at time zero, the pgf describing the total number of individuals in a lineage at time *t*, *P*(*x*, *t*), must satisfy(6)This formulation yields the pure burst process when *f*(*x*) is given by *f*(*x*) = *x*^{β+1}; that is, lifetimes are exponentially distributed with mean 1/λ, and each individual has exactly β + 1 offspring.

Substituting *f*(*x*) into Equation 6, we obtain the solutionOnce again, if regular bottlenecks are imposed every τ-time units, the extinction probability for a novel beneficial mutation that first occurs at time zero, just after a bottleneck, is given by the solution to(7)

#### Mutations that occur at time *t* during growth:

We now consider the case of mutations that occur at time *t* during a growth phase, with 0 < *t* ≤ τ. In this case, the lineage has τ − *t* time units to grow before facing the first bottleneck. Thus *G*(*x*, τ − *t*) gives the pgf describing the number of individuals in the lineage just before the first bottleneck, and *G*(1 − *D* + *Dx*, τ − *t*) gives the pgf for the number of individuals in the lineage after the first bottleneck. Thus, if *p _{i}* denotes the probability that there are

*i*individuals in the lineage just after the first bottleneck, we could writeNow consider the extinction probability,

*X*, for a lineage that first appears as a single copy at time

_{t}*t*. If this lineage has no survivors after the first bottleneck, the lineage is extinct; this occurs with probability

*p*

_{0}. Or, a single individual could survive the first bottleneck, with probability

*p*

_{1}, but that lineage could go extinct, with probability

*X*

_{0}, as derived in Equation 4. Or, two individuals could survive the first bottleneck (probability

*p*

_{2}), but both of those lineages independently go extinct, with probability (

*X*

_{0})

^{2}. This line of reasoning leads us to the following result:(8)where in the most general case

*X*

_{0}is the solution to Equation 4 and

*G*(

*x*,

*t*) is the solution to Equation 5.

## RESULTS

Here we illustrate the fixation probability as obtained through the numerical solution of the equations presented in the previous section. To verify this analytical and numerical work, we also performed extensive Monte Carlo (individual-based) simulation. In the simulations, we start with a single individual with a beneficial mutation, whose burst time is drawn from an exponential distribution, and who faces a constant probability of death, at rate μ. If the individual survives until the burst time, β + 1 offspring are produced. The offspring each behave independently, dying or bursting as described above, with independently drawn burst times. At time τ, each individual in the population survives the bottleneck with probability *D*. This process is repeated until the mutant population either becomes extinct or grows to a size sufficient that the probability of future extinction is negligible. This procedure constitutes a single “run” of the Monte Carlo simulation; we typically use 100,000 runs to estimate fixation probabilities. In all cases, we found that the simulation results were indistinguishable from the numerical results reported below.

Figure 1 illustrates the fixation probability *vs*. the selective advantage, *s*, as defined in Equations 1 and 2, for a burst–death model with a constant population size. In this case there are no population bottlenecks, and the rate at which new individuals are produced, *BL*, must be exactly balanced by the death rate, *M*. In Figure 1, we have set *L* = 1 and increased the wild-type burst size, *B*, from 1 to 100 (labels on the right). The fixation probability is given for beneficial mutations that increase the burst size to give the selective advantage as described in Equation 1. We see that for populations with larger wild-type burst sizes, and thus correspondingly large death rates, the fixation probability is dramatically reduced, even for large values of *s*.

The interesting result here is that in the burst–death model, the fixation probability may reach a plateau (which is far less than one) for high values of *s*. This implies that once a burst size is sufficiently large compared to the wild type, further increases no longer increase the fixation probability. The fundamental difference between the burst–death model illustrated here and classical models is that unless μ = 0 (pure-burst model), there is always a chance, given by μ/(μ + λ), that the individual carrying the beneficial mutation will die before bursting. For mutations that increase the burst size, β, this probability remains fixed, and results in an upper bound on π.

For comparison, the thick line in Figure 1 shows the classic fixation probability obtained in a constant population size, under the assumptions for the classic case that we enumerated earlier. Here π approaches one as *s* grows large, as expected. This is because as *s* grows large, the mean of the Poisson distribution of offspring grows large, and the probability of having zero offspring becomes negligible. We also note, as an aside, that the classic approach yields the same values of π for any value of *B* offspring per generation. This is because a Poisson distribution with mean *B*, randomly sampled with probability 1/*B* (such that only one offspring survives on average, maintaining a constant population size) gives the same Poisson distribution, with mean one, for any value of *B*.

In Figure 1b, we again explore the case without population bottlenecks (*M* = *BL*) and plot the fixation probability for beneficial mutations that increase the burst rate, λ = *L*(1 + δ). In this case, the probability of dying before bursting, μ/(μ + λ), approaches zero as λ increases, and thus π approaches one, albeit only when *s* is very large (not shown). We also note that π is very different, at the same value of *s*, between Figure 1a and 1b; in the burst–death model, the fixation probability is extremely sensitive to the *mechanism* of the selective advantage. In a constant population, increases in burst rate are more likely to fix than increases in burst size.

In Figures 2–4⇓, we explore fixation probabilities for cases when *BL* ≥ *M*, and thus population bottlenecks must be imposed, every τ-time units, to restrict population growth. For all of these figures, we have assumed that the wild-type population has burst rate *L* = 1 and burst size *B* = 100. The beneficial mutation first occurs in a single copy at time zero (we vary this in Figure 4). The death rate, *M*, varies between zero, a pure burst process, and *M* = *BL*, a constant population size.

We note that despite these differences in *M*, the overall probability that a single individual will die, over one cycle of growth and sampling, is constant. At *M* = 0, however, all death occurs at the bottleneck, whereas at *M* = *BL*, all death occurs *between* bottlenecks. This is because the latter case represents the limit at which *D* = 1, and effectively bottlenecks no longer occur. Thus decreasing *M* from *BL* to zero can be thought of as gradually imposing more severe bottlenecks, while concomitantly reducing the death rate between bottlenecks.

Also in Figures 2–4⇑, we typically show results for τ = 0.3. Since the mean lifetime is given by 1/*L* = 1, this bottleneck time may appear short compared to experimental practice. However, when the burst size is 100, and there is no delay between attachment and lysis (as our model assumes), the wild-type population size grows extremely quickly. For example, at the highest death rate, *M* = *BL*, the bottleneck ratio, *D*, is one. However consider the next highest death rate, *M* = *BL*/2. Here the virus is cleared 50 times faster than the burst rate, yet at τ = 0.3 the bottleneck ratio is given by *D* = exp(−(*BL* − *BL*/2)τ) ≈ 10^{−7}, and this becomes even more severe for lower death rates. Thus τ = 0.3 represents a relatively “long” bottleneck time, and we explore shorter times (less severe bottleneck ratios) in Figures 3 and 4c.

In Figure 2, we illustrate the behavior of π with increases in burst size (Figure 2a) and burst rate (Figure 2b). We find that as the wild-type death rate, *M*, increases, the fixation probability decreases. Thus π decreases even though the bottlenecks are becoming *less* severe. This result critically depends on the time at which the beneficial mutation first occurs, however; in Figure 2 we consider a mutation that occurs at the beginning of a growth phase. Thus, for mutations that occur early during growth, the death rate between bottlenecks is a more important factor than the severity of the bottleneck itself (but see Figure 4).

In Figure 2a, we again observe an upper bound imposed on π when the beneficial mutation confers an increase in burst size, due to the fixed probability of dying before bursting. No such upper bound occurs when the burst rate is increased; although not illustrated in the figure, each of the curves in Figure 2b approaches one when the burst rate is sufficiently high. For Figure 2, a and b, we have plotted π *vs*. the relevant life-history parameter, rather than *s*. This is in part because *s* varies with *B*, *L*, and *M* as described previously, and this additional transformation of the data does not improve clarity. We do note, however, that changes in β and λ may confer extremely large *s*-values, particularly when *B* is large. For example, when *B* = 100, μ = *BL*/5, and *L* = 1, doubling the burst rate to λ = 2 yields a selective advantage *s* = 116. Thus the mutant population is predicted to grow 117 times as quickly as the wild type in a single wild-type generation.

Figure 3 explores the effect of varying the time between bottlenecks, τ. Similar to the case when population bottlenecks are added to classic life-history models (Wahl and Gerrish 2001; Heffernan and Wahl 2002), we find that longer periods of growth between bottlenecks increase the chance that a beneficial mutation will survive. Once again, this effect is most pronounced when growth between bottlenecks is rapid, and bottlenecks are severe (*M* = 0). We note that for the parameter sets illustrated here, the bottleneck fraction *D* = *e*^{−(BL−M)τ} ranges from 1 (at *M* = *BL*) to 10^{−13} (at *M* = 0) when τ = 0.3. Thus values of τ on the right side of Figure 3 may not be experimentally feasible when *M* is small.

The results illustrated in Figures 2 and 3 sensitively depend on the time at which the mutation first occurs; we explore this effect in Figure 4. In Figure 4a, we find, not surprisingly, that π is much higher for mutations that occur early during the growth phase and that this effect is more pronounced when the bottlenecks are more severe. In the limit, for a constant population size (*M* = *BL*), the time at which the mutation occurs does not affect the fixation probability.

In Figure 4b, we repeat the results shown in Figure 2b, but use a mutation that first occurs near the end of the growth phase, at *t*_{0} = 0.25 when τ = 0.3. We see that when *s* is small, the relative influence of death between or during bottlenecks is reversed: for mutations that occur just before the bottleneck, severe bottlenecks are a more important factor than the death rate between bottlenecks. This effect is mitigated when *s* is very large, presumably because the burst rate is sufficiently high that a burst may occur before the bottleneck.

Finally, in Figure 4c, we repeat the results shown in Figure 3, but for a mutation that occurs near the end of the growth phase, at *t*_{0} = 0.85τ. Here again we observe a reversal of previous observations: for mutations that occur late in the growth phase, a long period of growth does not increase the fixation probability; rather, a less severe bottleneck increases π. We present and discuss a summary of the key results from these figures in the following section.

## DISCUSSION

The fixation probability of a beneficial mutation is sensitive both to the life history of the organism and to the means by which the mutation changes that life history, which we call the *mechanism* of the selective advantage. The burst–death model examined here proposes a life-history model that is mathematically tractable, but may be more appropriate, for many organisms, than classic models that assume fixed generation times and no death between generations.

For populations that maintain a constant size, we find that the fixation probability, π, predicted in a burst-death model is substantially lower than that predicted in the classic case. This is because the death rate has an important effect on π, and this death rate must be quite high to maintain a constant population size when the burst size is large. Overall, we see that the classic 2*s* approximation is a very poor predictor of fixation probability when death occurs between generations.

We also note that the mechanism of the selective advantage plays a crucial role in predicting π. For populations that maintain a constant size, the fixation probability is higher for mutations that increase burst rate, rather than increase burst size, for the same selective advantage. The fundamental insight here is that extinction is almost entirely determined by the ways in which the mutant lineage may die, be cleared, or fail to produce offspring during the first few generations of its existence. In constant-size populations, there is no death via bottlenecks, and new mutations are lost only through the death rate μ, during the period before the first burst. Since the probability of dying before bursting is μ/(μ + λ), increases in λ thus have a profound effect on extinction probabilities in the burst–death model.

In contrast, in the classic model (generation times are fixed but offspring numbers are stochastic, with no death between generations), we find the reverse effect: increases in offspring number are predicted to confer a larger π, for the same value of *s*, than reductions in generation time (Wahl and DeHaan 2004). This is because, in the classic model, mutant lineages are lost only by having zero offspring. Since increasing offspring number (the mean of the stochastic offspring distribution) reduces the probability of having zero offspring, increasing burst size has a profound effect on extinction.

When population bottlenecks are imposed, the predictions of the burst–death model depend on when, during the growth phase, the mutation first occurs. For mutations that occur early during growth, π is increased by reducing the death rate between bottlenecks and increasing the time between bottlenecks. For mutations that occur late in the growth phase, π is increased by reducing the severity of the bottleneck, which occurs when the time between bottlenecks is short. We also see that π may be orders of magnitude larger for mutations that occur early during the growth phase. However, since many more mutations will occur late during growth when the wild-type population is large, the “best” time at which to impose bottlenecks is unclear. Thus a goal for future work is to predict, using the burst–death model, the time between bottlenecks that will maximize the overall rate of adaptation (see Wahl *et al*. 2002).

The dichotomy between burst rate and burst size mutations described above for constant populations may or may not hold, in general, once population bottlenecks are introduced. When new mutant lineages may also be lost in the bottleneck, we expect that the effect of burst size or burst rate mutations will be sensitive to the time at which the mutation first appears or to the balance between death rate and bottleneck severity. Elucidating this dependence on the other model parameters, in detail, is beyond the scope of this article. We are, however, able to draw some conclusions in the limiting case when *s* is large, even in the presence of bottlenecks. In this situation, the burst–death model predicts that π has an upper bound, which can be much less than one, for mutations that increase burst size. In other words, once the burst size is sufficiently large relative to the wild type, the fixation probability no longer depends on burst size, but depends only on the probability of bursting before dying. In contrast, the fixation probability does approach one in the burst–death model when the burst rate, λ, is increased and *s* is very large. Recent work by Bull *et al*. (2006) has elucidated the dynamics of adaptation when changes in both burst size and lysis time are possible.

A necessary extension of our approach is to relax the assumption that the probability of reproducing is constant, irrespective of the age of the organism. In particular, a model that is appropriate for lytic viruses when both attachment and lysis times are nonnegligible would clearly extend the experimental relevance of the work. Such a model introduces a delay term in the partial differential equation (Equation 5) and thus presents both analytical and numerical challenges.

## APPENDIX

To derive Equation 5, from first principles, we let *G*(*x*, *t*) be the probability generating function for the number of individuals in the mutant lineage at time t,where *p _{i}*(

*t*) is the probability that there are

*i*individuals in the lineage at time

*t*. For convenience, we use

*p*to denote

_{i}*p*(

_{i}*t*) in the following. Note that(A1)Now consider the possible changes to

*G*(

*x*,

*t*) in a small interval of time, Δ

*t*. At time

*t*, the lineage consists of a single individual with probability

*p*

_{1}, and a burst occurs with probability λΔ

*t*. If such an event occurs, the probability mass must be subtracted from the coefficient of

*x*and added to the coefficient of

*x*

^{1+B}. There are two individuals in the lineage at time

*t*with probability

*p*

_{2}, and in this case a burst occurs with probability 2λΔ

*t*. Using the same logic for deaths, we deriveIn the limit as , we therefore haveUsing Equation A1, we have thus derived Equation 5:This equation is a Lagrange-type partial differential equation, which takes the formThis can be solved using the auxiliary equationwhich in our case yields the following two equations:(A2)(A3)If we are able to solve for two independent solutions of the auxiliary equation,where

*c*

_{1}and

*c*

_{2}are constants, then the general solution of the Lagrange-type PDE can be given aswhere

*F*is an arbitrary analytical function and can be solved explicitly for

*G*.

From Equation A3, we find that *dG* = 0, or *G*(*x*, *t*) = *c*_{1}, and so we take *f*_{1} to be simply *G*. Integrating Equation A2, we findThe previous step is obtained by partial fraction expansion, where *x _{i}* are the β roots of the polynomial λ

*x*

^{β}+ λ

*x*

^{β−1}+… + λ

*x*− μ, and the

*L*are given byIn general, these roots cannot be found analytically, especially as β increases, but can be easily found numerically. We thus take the arbitrary analytical function . We rearrange this to find that

_{i}*G*(

*x*,

*t*) is given by some analytical function

*H*(

*w*), where(A4)Our solution must satisfy the boundary condition

*G*(

*x*, 0) =

*x*, so we set

*t*= 0 in Equation A4 and solve for

*x*. The solution to this procedure yields

*G*(

*x*,

*t*), which must be on the interval [0, 1]. In simple cases, such as where β = 1, this can be solved analytically. In the general case, we find that

*G*(

*x*,

*t*) is given by the value of

*y*on [0, 1] that is a root of the equation(A5)

We note that for any *t*, *x* = 1 implies that *w* = 0, and in this case the root of Equation A5 is clearly *y* = 1. Thus our solution also satisfies the second boundary condition, *G*(1, *t*) = 1.

## Acknowledgments

We thank two anonymous referees whose insightful comments improved the manuscript. This work was supported by the Natural Sciences and Engineering Research Council of Canada and by the Ontario Ministry of Science, Technology and Industry.

## Footnotes

Communicating editor: M. W. Feldman

- Received February 12, 2007.
- Accepted May 1, 2007.

- Copyright © 2007 by the Genetics Society of America