## Abstract

Muller's ratchet operates in asexual populations without intergenomic recombination. In this case, deleterious mutations will accumulate and population fitness will decline over time, possibly endangering the survival of the species. Mutator mutations, *i.e*., mutations that lead to an increased mutation rate, will play a special role for the behavior of the ratchet. First, they are part of the ratchet and can come to dominance through accumulation in the ratchet. Second, the fitness-loss rate of the ratchet is very sensitive to changes in the mutation rate and even a modest increase can easily set the ratchet in motion. In this article we simulate the interplay between fitness loss from Muller's ratchet and the evolution of the mutation rate from the fixation of mutator mutations. As long as the mutation rate is increased in sufficiently small steps, an accelerating ratchet and eventual extinction are inevitable. If this can be countered by antimutators, *i.e*., mutations that reduce the mutation rate, an equilibrium can be established for the mutation rate at some level that may allow survival. However, the presence of the ratchet amplifies fluctuations in the mutation rate and, even at equilibrium, these fluctuations can lead to dangerous bursts in the ratchet. We investigate the timescales of these processes and discuss the results with reference to the genome degradation of the aphid endosymbiont *Buchnera aphidicola*.

IN many bacterial populations—both experimental and natural—one finds subpopulations with a significantly (10- to 1000-fold) higher mutation rate than the rest (Sniegowski *et al.* 1997; Björkholm *et al.* 2001). These so-called mutator strains are usually present at relatively low frequencies because they accumulate more deleterious mutations than the others. Also antimutator strains, with a significantly lower mutation rate, can be found (Fijalkowska *et al.* 1993). This suggests that mutation rates are adjustable and subject to selection (Taddei *et al.* 1997; Sniegowski *et al.* 2000; De Visser 2002). Mutation rates can be modified in many different ways, for instance, through changes in the accuracy of DNA polymerase or of one of the many enzymes in the various repair pathways that normally keep mutations at bay. Purifying selection does not act primarily on the mutator alleles directly, but on their effects on other loci in the genome; this is referred to as indirect selection. Thus, at mutation–selection balance there will be a continuous flux where new mutators appear in the population through mutation from the wild-type while older mutators that have picked up too many deleterious mutations are lost through indirect selection. There is also direct selection: if an increased mutation rate is due to less stringent replication or repair, there could be savings in time and free-energy consumption that translate to a selective advantage for the mutator allele. A direct negative selection could be the result if the increased mutation rate is coupled to the disruption of other cellular functions. There is also a negative selection due to the increased level of lethal mutations in a mutator. If there is frequent recombination, however, the linkage between the mutator allele and its consequences, the higher mutational load, will be lost. Thus, in a sexual species (Johnson 1999), the indirect selection on mutator alleles will be much weaker than in an asexual one.

Mutator alleles can be carried to fixation by hitchhiking with adaptive mutations (Taddei *et al.* 1997; Shaver *et al.* 2002; Tanaka *et al.* 2003; André and Godelle 2006; Palmer and Lipsitch 2006; Gerrish *et al.* 2007; Wylie *et al.* 2009). If environmental conditions change, adaptive mutations may be most likely to occur first in a mutator subpopulation, which can then be carried to dominance through the positive selection on the adaptation. However, the extra load of deleterious mutations will be carried along with it. If there is some recombination, linkage can be broken and the adaptive mutation can become associated instead with a genome without the mutator allele and the extra load. Thus, in total, there are a plethora of effects that can influence the fate of mutator alleles—and therefore the evolution of mutation rates—in populations under various conditions.

In this article we study the fate of mutators in an asexual population where intergenomic recombination and adaptive mutations are rare, *i.e.*, where Muller's ratchet operates (Muller 1964; Felsenstein 1974; Haigh 1978). The ratchet-like behavior derives from the fact that, once all members of the population by chance have picked up some deleterious mutation, an unloaded genome cannot be recreated, and the ratchet will have “clicked” one step. The best, *i.e.*, most fit, genome will now carry at least one mutation and once all members by chance have picked up at least one more mutation the ratchet will click again. Without recombination or back mutation, this process will lead to a relentless fitness loss in the population. A prime example of where the ratchet is expected to operate is in endosymbionts, where clones are isolated from each other and live in a highly controlled environment inside cells of another species (Lynch 1996; Moran 1996; Bergstrom and Pritchard 1998). However, if the conditions are right, the ratchet may be moving so slowly even for these organisms that the fitness loss becomes insignificant in an evolutionary perspective. Such a slowdown of the ratchet serves as an important protection against fitness loss for an asexual species and has been suggested as a reason behind the genetic stability of some endosymbiont species (Pettersson and Berg 2007). The question then arises whether this situation really is stable. Since the ratchet allows the accumulation of deleterious mutations, could one of these be a mutator allele? Fixation of a mutator allele would lead to an increase in the ratchet rate, which in turn could promote fixation of stronger mutators and a further increase in the ratchet rate. Such a cascading effect would inevitably end in a mutational catastrophe. Relentless fitness deterioration is in the nature of the ratchet; it is the timescale that will determine survival.

In this article, the appearance and fate of mutator alleles were studied through simulation of an asexual population where the ratchet is operating. This assumes fairly constant environmental conditions where selective sweeps are unlikely to occur, and a mutator allele consequently is unlikely to hitchhike with adaptive mutations. Thus, the main selective force acting on a mutator allele will come from the extra load of deleterious mutations it causes, both as indirect selection on the deleterious mutations that accumulate in the ratchet and as direct selection on the lethal mutations. In a previous model (Söderberg and Berg 2007) we studied the behavior of Muller's ratchet on the basis of the assumption that the effects of deleterious mutations have an exponential distribution. Here we extend this model to include new variants with altered mutation rate and study their fates under various conditions. It is found that the fixation of mutators can be facile, in particular if the mutation rate can increase in a series of relatively small changes. When back mutations generating antimutators are introduced, the mutation rate may come to an equilibrium. However, even then, fluctuations leading to a temporary high mutation rate can be sufficiently frequent and long lasting to bring on a precipitous fitness loss. We also investigate properties of the ratchet that may curtail mutator fixation and thereby delay the ultimate mutational catastrophe.

## MODEL

#### Muller's ratchet:

Computer simulations were used to propagate a population of *N* haploid individuals. Deleterious mutations occur in the model with rate *U* per genome replication and each mutation alters the fitness with a factor of (1 − *s*). Lethal mutations occur with rate *Ut*_{let} and mutator mutations with rate *Ut*_{mut}, giving the total rate of all nonneutral mutations as *U*(1 + *t*_{mut} + *t*_{let}). In a general case, the deleterious effect, *s*, of each mutation may be very different. Here we assume an exponential distribution over 0 < *s* < 1 with average and in addition some that are directly lethal (*s* = 1). As each effect is unique, the population model needs to account separately for each individual, which is represented by its fitness and its mutation rate. Here we use a Moran model, as described previously (Söderberg and Berg 2007). First in the algorithm, one individual is chosen for propagation randomly in proportion to its fitness. Thereafter, the progeny is subject to deleterious mutations, the number of which is Poisson distributed with a frequency of *U*, and the effect of each is chosen from an exponential distribution with average . Lethal mutations and mutator mutations are introduced in the same way with the frequencies *Ut*_{let} and *Ut*_{mut}, respectively. Each mutator mutation increases the mutation rate by a given factor *f*, and several can be present simultaneously such that an individual with *n* mutator mutations would have the mutation rate . The new progeny, unless killed by a lethal mutation, is then inserted into the population and a random individual is killed to keep the population size constant. Strictly lethal mutations do not contribute to the fitness loss in the ratchet since they do not accumulate. However, they do contribute to different survival probabilities for the progeny of individuals with different mutation rate. The state of the population is recorded as its mean fitness per individual and the rate of fitness loss, , is calculated from the slope of the natural logarithm of fitness *vs.* time. The calculations with the Moran model were carried out with a population size 2*N* to mimic the results of a conventional Wright–Fisher model with discrete generations and population size *N* (Moran 1958; Ewens 1979; Söderberg and Berg 2007). Here and below, *N* always refers to the effective population size of a Wright–Fisher model.

With distributed *s*-values, the dominant effect of Muller's ratchet can be explained by the reduction in effective population size due to background selection (Charlesworth 1994); individuals that have picked up strongly deleterious mutations will not contribute to later generations and should not be considered as part of the propagating population. Previously (Söderberg and Berg 2007), we found that, for an exponential distribution of *s*-values, the effective population size due to background selection, *N*_{eb}, can be determined from the implicit relation(1)Furthermore, we found that the expected rate of fitness loss for the ratchet under a broad range of parameter values (if and ) can conveniently be expressed as(2)As described by these equations, mutations with *sN*_{eb} > 4 are not fixed in the population but contribute to background selection, while those with *sN*_{eb} < 1 behave as near neutral. With distributed *s*-values there are no distinct clicks of the ratchet. Instead fitness loss is a continuous process with as the most relevant measure.

#### Parameter settings and population-size scaling:

As expected from theory (Ewens 1979) and shown in a number of simulations (*e.g.*, Söderberg and Berg 2007), fixation times, *T*_{fix}, and rates of mutation accumulation, *R*, scale with population size such that *T*_{fix}/*N* or *RN* depend only on the parameter combinations *UN* and *N*. Equations 1 and 2 clearly satisfy such a scaling law. We tested the scaling also in the present model (results not shown) for *N* = 160, 500, and 1600 with *UN* = 12.5 and *N* = 50; as expected it holds very well also for the fixation time of mutators. Thus, to save computer time, simulations can be carried out for much smaller values of *N* and proportionally larger values for *U* and . Most calculations reported here were carried out with *N* = 500, *U* = 0.025, and = 0.1; this corresponds to *N*_{eb} = 330 from Equation 1 and gives = 2.5 × 10^{−6} from the simulations. If this system is scaled up by a factor of 30, one will get the—in our view—more realistic parameter values *N*_{eb} = 10^{4}, *U* = 8 × 10^{−4}, and = 3 × 10^{−3} (see discussion). The resulting calculated values for fixation times (and other timescales) should be increased by a factor of 30 and the calculated fitness loss rate should be decreased by a factor of 1/30^{2} as it involves both a rate and a fitness factor. Thus after such a rescaling, = 3 × 10^{−9} and it would take 3 billion generations to reduce fitness by a factor of 0.37; with these parameter values, the ratchet would be very efficiently stalled.

#### Annealing/burn-in:

To get comparable results for the fixation time of a mutator, the system was allowed to run for 10^{4} generations without mutator mutations just to allow the relative fitness distribution of the individuals to reach a stationary state. When this is achieved, the distribution is saved. Thereafter, the system is restarted using the obtained fitness distribution of individuals and the mutator mutations are allowed to spread and data are collected.

## RESULTS

#### Ratchet rate *vs.* mutation rate:

An increase in the mutation rate can have a much larger than proportional effect on the fitness-loss rate in the ratchet. The main effect comes from the fact that an increase in *U* will lead to a decrease in the background-selection effective population size, *N*_{eb}. For very small *U*-values, such that *U*/ < 0.1, *N*_{eb} ≈ *N*, mutation accumulation is dominated by genetic drift, and is proportional to *U* (Söderberg and Berg 2007). However, in the broad parameter region where the ratchet effects are important, varies much more strongly with *U*; the slope in the log–log diagram of Figure 1 shows that an order of magnitude increase in *U* will increase by three to five orders of magnitude. Thus, a relatively modest increase in mutation rate can easily set a stalled ratchet in motion. The question is how fast such a mutator allele can actually spread in the population.

#### Fixation of mutator alleles in the presence of a ratchet:

How much and how fast increases in *U* can be fixed in the population depends on the parameters. One obvious effect is that when starting at a very small *U*, even large increments reach fixation, albeit slowly. Figure 2 (black curve) shows an example where the mutation rate increases by a factor *f* = 10, from a very low initial value (*U*_{0} = 5 × 10^{−5}). The indirect selection is weak since neither the initial mutation rate nor that of the first mutator is sufficiently large to set the ratchet in motion. In fact, the fixation time for the first steps in Figure 2 is close to the neutral expectation, 1/*Ut*_{mut} = 2 × 10^{6} generations. With a low mutation rate, the fixation of mutator mutations is facile, although like other mutations, their rate of appearance is small. In the parameter range where the ratchet starts to become effective, fixation of large increments in mutation rate becomes much more difficult; *e.g.*, when starting from *U*_{0} = 0.0025, no jump by a factor of 10 was observed over 2 × 10^{7} generations. Figure 2 (red and blue curves) shows that the mutation rate increases much faster and can reach a much higher level when the mutators increase the rates by a smaller factor. Here a rate *U* = 0.02 is stable against increases by a factor *f* = 2 (red curves), but not against *f* = 1.5 (blue curves).

The simulations show that the fixation time increases nearly exponentially with *f* (see Figure 3). Thus, mutators with a large increment factor in *U* cannot be fixed easily and will be present only in small fractions in the population. Small mutator mutations, on the other hand, can accumulate almost freely in the population. The exponential increase in fixation time with increasing *f* in Figure 3 can be interpreted as an effective counterselection. Thus, we can set , where is the neutral result and *s*_{m} is the effective counterselection against a mutator. The best-fit curves shown in Figure 3 extrapolate well to the neutral expectation, , when the increment factor in mutation rate approaches *f* = 1. The direct counterselection against an increased mutation rate is expected to be equal to the change in lethal mutation rate, *s*_{let} = (*f* − 1)*U*_{0}*t*_{let}. This corresponds quite well to the difference between the results with *t*_{let} = 0.1 and *t*_{let} = 0 in the two sets of curves in Figure 3. Even though the fraction of sites that are lethal, *t*_{let} = 0.1, is chosen fairly large in these calculations, the direct selection is of minor consequence; decreasing *t*_{let} has a relatively small effect on the fixation time of a mutator. Instead, the main counterselection, as determined by *s*_{m} *− s*_{let}, comes from the indirect selection due to the increased genetic load in the mutator. Thus, increasing the mutation rate from a low initial value (red curves in Figure 3) is under much weaker indirect selection than increasing it from a higher initial value (blue curves).

Thus, a large increase in the mutation rate is very unlikely to occur in a single event. On the other hand, increasing the mutation rate in a series of small steps is very facile and the mutation rate would increase exponentially with time, at least in the beginning of the process (Figure 4). The fixation time for a mutator that increases the mutation rate from *U*_{0} = 0.025 in two steps of a factor of 1.3, *i.e.*, a factor of 1.69, takes only 8 × 10^{4} generations (Figure 4B), while an increase by a factor of 1.7 in a single step takes on average 10^{6} generations (Figure 3, blue squares). Similarly, increasing *U* by 30% 12 times (corresponding to a factor of 23.3) takes 3 × 10^{5} generations (Figure 4B), while the same increase in a single step is virtually impossible on any realistic timescale. As long as the mutation rate can increase in small steps through changes in modifier genes, the mutation rate can be expected to increase faster than exponentially with time. The fitness-loss rate, in turn, increases much faster than proportionally to *U* (Figure 1), and fitness crash can be rapid (Figure 4 and Table 1). Even on the log-scale of Figure 4, the mutation rate shows an accelerating increase with time. An exponential behavior is expected from the fact that the occurrence rate of mutators increases in proportion to the overall mutation rate. The acceleration in the rate (the upward bend in Figure 4) comes from the fact that an increased mutation rate decreases the effective population size, *N*_{eb} (Equation 1), and thereby decreases the effective counterselection on all deleterious mutations, including the mutators.

Figure 4 shows that the mutation rate increases much faster when it takes place through the accumulation of small-effect mutators. But as *f* becomes even smaller, the rate increase will again decline and disappear when . In our simulations, the fastest rate increase occurs for . In general, mutators are present in the population only at a small fraction (on average <5%; results not shown) until one by chance goes to fixation. This implies that the mutators have very little influence on the background effective population size and on the fitness loss rate as long as they do not become fixed.

#### Effects of antimutators:

Since the accumulation of small mutators can lead to an accelerating fitness loss, unless stopped in some way, we did numerous simulations where also antimutators could appear. Thus, in the system where mutators increasing the mutation rate by a factor *f* appear with rate *Ut*_{mut}, antimutators that reduce the mutation rate by the factor 1/*f* were allowed to appear with rate *Ut*_{mutb}. When the ratio *Q* = *t*_{mutb}/*t*_{mut} is not sufficiently large, the results look very much like those of Figure 4 with a rapid fitness loss. As long as *Q* > 0, mutator accumulation is expected to stop somewhere. We have chosen to consider the cases where *U* approaches *U* = 1 as effectively crashed. At this point, the fitness-loss rate is faster than 10^{−3} and fitness declines very rapidly. These simulations were repeated 30 times for each case and the results are summarized in Table 1. The time it takes to reach fitness = 0.01 was chosen as an estimate for comparisons of the time required for fitness to crash. At this point, fitness loss is very rapid and thus the exact choice of cutoff is not crucial. In fact, for the results in Table 1, the difference in the times it takes to reach fitness 10^{−1} and 10^{−3} is at most a factor of 1.7. The crash time in Table 1 is on the order of 10^{5} generations using the initial rate *U*_{0} = 0.025. For small values of *f*, it is dominated by the time it takes to reach a dangerous *U*-value. For the larger value (*f* = 1.5), it is dominated by the rate of the ratchet at *U*_{0}.

A fairly low frequency of antimutator occurrence can efficiently stop the accumulation of mutator mutations, particularly for large values of *f*. Figure 5 shows some examples where the resulting mutation rate fluctuates around a low mean value. There are two striking features in Figure 5. First, there are large fluctuations in mutation rate that appear to occur randomly. Second, these fluctuations are linked to even larger fluctuations in the rate of fitness loss. With a small increment factor, the fluctuations in mutation rate become more rapid and *t*_{mutb} must be chosen much larger to constrain the rates of mutation and fitness loss. Thus with *f* = 1.1 and *Q* = 0.2, the fitness crashes after ∼10^{5} generations (Table 1), while with *Q* = 0.5, the mutation rate fluctuates around the mean *U*_{mean} = 0.038 and fitness crash is expected on a timescale of ∼10^{6} generations (Figure 5A and Table 2). Similar results hold for *f* = 1.3, where *Q* = 0.2 leads to *U*_{mean} ∼ 0.02 (Figure 5B and Table 2), while *Q* = 0.05 leads to fitness crash after 4 × 10^{5} generations (Table 1). The results for parameter choices where the mutation rate appears to equilibrate as in Figure 5 are summarized in Table 2. Here we have quantified the properties of the fluctuations through the correlation time of the autocorrelation function for the variations in *U*-value and defined *T*_{corr} as the time after which the autocorrelation has reached the value exp(−1); this gives a measure of the memory—or average lifetime—of individual fluctuations. There is also an estimate for the appearance time, *T*_{wait}, of fluctuations, here calculated as the number of generations in each simulation divided by the number of occurrences of a fluctuation for which the *U*-value is equal to or greater than twice its median value. If these fluctuations are purely random, this would be an estimate of the average waiting time in the corresponding Poisson process. Naturally, when the number of such fluctuations is small, this estimate is very uncertain.

The lethal target size, *t*_{let}, has a minor influence on the fixation of mutators (Figure 3 and Table 1). This can be seen also in Table 2, although there is a general trend that a larger *t*_{let} increases the waiting time for large fluctuations and decreases the average rate of fitness loss (Table 2). These simulations have been run for up to 2 × 10^{7} generations and in some cases to unrealistically low absolute fitness (*cf.* the left *y*-scale of Figure 5). However, what is of primary interest here is the rate of change, *i.e*., the slope of the fitness curve. The dynamics of the system are determined by the relative fitness among the individuals of the population and are independent of their average fitness relative to an arbitrary starting point. Thus, the fitness curve illustrates how quickly and how far the average fitness will fall for the duration of a mutation-rate fluctuation. The fitness declines at a relatively modest rate—corresponding roughly to the median value of —for long periods of time and then suffers a precipitous loss whenever a fluctuation occurs. Thus, due to the randomness of the fluctuations, the average rate of fitness loss is not sufficient to predict the survival time even at mutator equilibrium. The important timescale over which the mutation rate fluctuates in Figure 5 (the correlation time) is on the order of 10^{5} generations (Table 2); this is also an estimate of the time required to reach equilibrium. It should also be noted that if the population is rescaled, *e.g.*, by increasing *N* a factor of 10 while keeping *UN* and *N* constant, the graphs in Figure 5 would look the same except for the units on the *x*-axis (time) and the left *y*-axis (log of fitness). In this case, 10^{6} generations would become 10^{7} and fitness 10^{−40} would become 10^{−4}, thereby moving also the absolute fitness scale into a more realistic range.

## DISCUSSION

These calculations describe the behavior of Muller's ratchet in the presence of mutators and, conversely, the evolution of mutation rate in the presence of the ratchet. Various processes that can invalidate the ratchet were not considered, *e.g.*, occasional recombination events and back mutations or other adaptive mutations. In the same vein, positive selection for an increased mutation rate was not considered either. Rather than including all possible effects in the same model, here we have focused on the ratchet and the effects of the deleterious mutations including the lethal ones. As the calculations assume constant population size, extinction is not directly modeled here. The main result of the simulations is given by the rate of fitness loss, and it is assumed that when this rate is high, the population is headed for extinction on a timescale determined by the inverse of this rate (*cf.* Gerrish *et al.* 2007).

The effect of adaptive mutations in the ratchet has been modeled by Bachtrog and Gordo (2004) and their results show that fixation of adaptive mutations is slowed down by the ratchet. However, adaptive mutations have little effect on the ratchet as long as their rate of occurrence is low and their beneficial effect on fitness is not larger than the deleterious effects of the ratchet. Back mutations are a special class of adaptive mutations, where each deleterious mutation accumulated in the ratchet can be reversed. They are mostly of small effect as most deleterious mutations that can be fixed in the ratchet are of small effect. Furthermore, the target for back mutation will be appreciable only when the ratchet has accumulated many mutations. We studied the effects of back mutations in a simplified model where all deleterious mutations have the same effect (Haigh 1978). The results show that mutator accumulation is nearly unaffected by back mutations as long as the mutation rate is low. When *U* increases, a point is reached where back mutations start to interfere with the accumulation of both ratchet and mutator mutations. However, at this point the fitness-loss rate is already so large that extinction seems imminent (R. J. Söderberg and O. G. Berg, unpublished results). Large-effect adaptive mutations can occur as a way of compensating several deleterious mutations or in response to changes in the environment. Most previous modeling of the fixation of mutators in asexual populations has focused on the effects of hitchhiking with adaptive mutations (Taddei *et al.* 1997; Tanaka *et al.* 2003; André and Godelle 2006; Palmer and Lipsitch 2006; Wylie *et al.* 2009). Gerrish *et al*. (2007) studied a model where both deleterious/adaptive and mutator/antimutator mutations occur in an asexual population. They found that the complete linkage between adaptive mutations and mutator mutations could drive the mutation rate to intolerable levels. In contrast, our calculations focus on the deleterious mutations in the ratchet, showing that random fluctuations in the mutation rate can produce devastating spurts of fitness loss (Figure 5) even in the absence of hitchhiking with adaptive mutations. If fixation of rare adaptive mutations plays a major role, the present calculations would show the behavior of mutators in the ratchet between such events.

#### Effects of modifier mutations, mutators, and antimutators:

Mutation rates can be modified in many different ways. The destruction of an entire repair pathway could easily bring about a 10-fold or 100-fold increase in the rate. If caused by a genomic deletion, such a change would also be virtually impossible to reverse, although it could be partially alleviated by increasing the accuracy in other pathways. Mutators of small effect would not be easily observed experimentally and their existence is a conjecture based on the large number of enzymes that contribute to replication accuracy. We assume that small increases in the mutation rate can be achieved by mutations that reduce the stability or activity of individual enzymes. Such changes could easily be reversed by back mutations or by compensatory mutations at other loci. Thus, mutators could have a very broad range of effects, from small increases to very large, while most antimutators, on the other hand, probably have relatively small effects.

In general, one expects that mutations—including mutators—that have very small deleterious effects will be fixed in a population with some significant probability. Considering the large number of sites where such mutations can occur, at any one time there should always be a significant number of weakly deleterious changes that can be reversed or compensated. In this picture, the optimal situation would leave the system with many sites where mutations could either increase or decrease the mutation rate. Although we expect there to be more ways of reducing accuracy rather than increasing it, this difference may become smaller if the system has already suffered substantial degradation. Thus, the mutation rate could be tunable both up and down.

#### Fixation of mutators in the ratchet:

Although it is stochastic in nature, Muller's ratchet induces a relentless and steady fitness loss that appears almost deterministic. When mutators are introduced, the behavior of the ratchet changes dramatically; fitness loss is still relentless but much more erratic. Although the mutator mutations are part of the ratchet, they do not accumulate in the same way as other deleterious mutations. This is because the main selection is the indirect selection that derives from the increase in the ratchet rate and the concomitant fitness loss that they cause. When the ratchet is very slow, mutators accumulate more easily because the relative increase in fitness-loss rate is small and the indirect selection is weak. If instead the mutator causes a large increase in the fitness-loss rate, indirect selection is strong and mutator accumulation becomes much more difficult. However, a runaway ratchet will result if the change in mutation rate can occur in a series of sufficiently small steps. Increasing the mutation rate leads to a decrease in effective population size, thereby alleviating further mutator fixations and accelerating the ratchet.

Introducing antimutators is akin to introducing back mutations in the ratchet, which is contrary to the basic assumptions. However, antimutators have properties that are very different from ordinary back mutations due to their effect on the ratchet. By reducing the ratchet rate, they can be strongly positively selected. Even if antimutators occur at a frequency that is much lower than that of mutators, they can easily become fixed in the population, thereby establishing a kind of equilibrium for the mutation rate. However, even if equilibrium can be established around a point where the mutation rate is not too high, fluctuations in the rate may still lead to extinction over a long time period. The ratchet is particularly sensitive to these kinds of fluctuations because they are amplified by the background selection: when the mutation rate increases, *N*_{eb} decreases and the effective selection is weakened, both against a further increase and for a decrease. Still, the mutational catastrophe from an accelerating Muller's ratchet may be thwarted or delayed by the fact that enzymes are tunable—both up and down—and not always set for maximum activity or accuracy.

#### Parameter choices and fitness loss:

The choice of parameter values used in these calculations was guided in part by estimated values for an endosymbiotic bacterium like *Buchnera aphidicola*. This bacterium has been stably associated with its host, an aphid, for at least 50 MY (Tamas *et al.* 2002). It has a genome of 6 × 10^{5} bp and spends its entire life cycle inside its aphid host, passed on directly from mother to offspring. The population dynamics are most easily described as a ratchet on the host level (Pettersson and Berg 2007). Assuming that there are 4 × 10^{5} sites in the genome that can pick up deleterious mutations (0 < *s* < 1), *t*_{let} = 0.1 implies that ∼4 × 10^{4} sites are lethal (*s* = 1) while *t*_{mut} = 0.01 and *t*_{mutb} = 2 × 10^{−4} imply that there would be 4 ×10^{3} sites and 80 sites for mutator and antimutator mutations, respectively. The mutation rate has recently been estimated as 2.2 × 10^{−7} per site per year (Moran *et al.* 2009). Counted per host generation—assuming ∼10 generations per year (Clark *et al.* 1999)—this gives the rate *u* ≈ 2.2 × 10^{−8} per site per host generation and the deleterious mutation rate *U* ≈ 0.009. On the basis of the neutral diversity θ = 0.0036 (Funk *et al.* 2001) for Buchnera, the effective population size (on the host level) can be estimated as *N*_{eb} = θ/2*u* ≈ 8 × 10^{4}. If, for instance, it is assumed that the average deleterious effect is = 3 × 10^{−4}, we estimate the average fitness-loss rate from Equation 2 as = 2 × 10^{−8} per host generation. With this rate it would take only 6 MY to get a significant reduction in fitness; the genome of Buchnera would therefore be expected to be quite unstable, if these parameter values prevail. However, the fitness-loss estimate is very sensitive to the average deleterious effect, (*cf.* Figure 1), which is not adequately known. For other organisms, estimates vary between 10^{−2} for *Escherichia coli* (Kibota and Lynch 1996; Trindade *et al.* 2010) and 10^{−4} for Drosophila (Loewe *et al.* 2006). If the average effect in Buchnera is more like that of its close relative *E. coli*, the ratchet would be much slower; *e.g.*, with = 3 × 10^{−3} and *U* and *N*_{eb} as above, the ratchet would be running with = 10^{−9} and significant fitness loss would occur on a timescale of 100 MY. On the other hand, in this system, the deleterious effects that matter act mostly on the host level, and *s*-values may therefore be more like those of Drosophila with its smaller . Not only the average , but also the distribution around the average is poorly known. If the distribution is broader than exponential, *e.g.*, gamma distributed with a shape factor <1, we expect the fitness-loss rate to be larger. Furthermore, the observed mutation rate could actually be on its way to an exponential increase through the accumulation of small-effect mutators (*cf.* Figure 4), in which case the ratchet would be accelerating and unstoppable. However, the system has already survived so long that it may be more likely to have reached equilibrium between mutator and antimutator (*cf.* Figure 5). Nevertheless, even if the equilibrium mutation level would allow a tolerable fitness-loss rate on average, large fluctuations in the rate are inevitable.

Most of the simulations reported above were carried out with *N* = 500, *U*_{0} = 0.025, and = 0.1, giving *N*_{eb} = 330. Scaled up to a factor of 150, this corresponds to *N*_{eb} = 5 × 10^{4}, *U*_{0} = 1.7 × 10^{−4}, and = 6.7 × 10^{−4}; thus, this mutation rate is ∼50-fold lower than the recent estimate for Buchnera (Moran *et al.* 2009) and more in line with older estimates (Tamas *et al.* 2002). The starting value *U*_{0} in the simulations is therefore closer to that of a free-living bacterium. The chosen parameter values allow the simulations to run from regions where the ratchet is effectively standing still to regions where it would quickly lead to extinction. The simulations could therefore mimic the process after the initial stages when a free-living bacterium with low mutation rate has been “captured” to become an obligate intracellular symbiont. Initially, there would be both adaptation to the new life style and genome degradation as properties that did not contribute sufficiently to the fitness of the host would easily be lost. During this early process, when the ratchet is still inactive and selection weak, strong mutators could also be fixed fairly easily. After that, the fixation of strong mutators becomes virtually impossible and the mutation rate can continue to increase only in small steps. The calculations above show how the ratchet may respond under different assumptions. However, the relevant parameter values for any real system, like Buchnera, are still too uncertain to allow an estimate of the timescale of present-day genome degradation. And, even if these parameter values were adequately known, the system may be so inherently erratic to make the long-term behavior unpredictable.

## Acknowledgments

This work was supported by The Swedish Research Council.

## Footnotes

Communicating editor: H. G. Spencer

- Received November 1, 2010.
- Accepted January 24, 2011.

- Copyright © 2011 by the Genetics Society of America