POPULATION STRUCTURE, LEVELS OF SELECTION, AND THE EVOLUTION OF INTRACELLULAR SYMBIONTS

Authors


Abstract

Many obligately intracellular symbionts exhibit a characteristic set of genetic changes that include an increase in substitution rates, loss of many genes, and apparent destabilization of many proteins and structural RNAs. Authors have suggested that these changes are due to increased mutation rates, or, more commonly, decreased effective population size due to population bottlenecks at the symbiont or, perhaps, host level. I propose that the increase in substitution rates and accumulation of deleterious mutations is a consequence of the population structure imposed on the endosymbionts by strict host association, loss of horizontal transmission and potentially conflicting levels of selection. I analyze a population genetic model of endosymbiont evolution, and demonstrate that substitution rates will increase, and the effect of those substitutions on endosymbiont fitness will become more deleterious as horizontal transmission among hosts decreases. Additionally, I find that there is a critical level of horizontal transmission below which natural selection cannot effectively purge deleterious mutations, leading to an expected loss of fitness over time. This critical level varies across loci with the degree of correlation between host and endosymbiont fitness, and may help explain differential retention and loss of certain genes.

Many insect species harbor intracellular bacterial symbionts (Buchner 1965). Although little is known about many of the associations, several well-studied cases exist. Among these, obligate, mutualistic symbionts (primary endosymbionts) often play a role in host nutrition, such as providing essential amino acids (Douglas 1998) or recycling nitrogen compounds (Gil et al. 2003). These symbionts often display strict maternal transmission, cospeciation with their insect hosts, and their removal by antibiotic treatment is generally fatal to the host (Moran et al. 1993; Douglas 1996; Sauer et al. 2000). In contrast, “secondary endosymbionts” are facultative (Sandstrom et al. 2001; Dale and Moran 2006) and, although vertical transmission from mother to offspring is the most common mode of transmission, they may also be horizontally transmitted (Dale and Moran 2006).

Vertically transmitted bacterial endosymbionts often share several characteristics of genome evolution, together referred to as “reductive evolution.” Typically the genome of the symbiont is smaller, more A+T biased, has an increase in overall substitution rate, and a greater nonsynonymous/synonymous substitution rate relative to related free living hosts (Moran 1996; Clark et al. 1999; Shigenobu et al. 2000; Akman et al. 2002; van Ham et al. 2003; Gil et al. 2003; Woolfit and Bromham 2003; Degnan et al. 2005). Some primary endosymbionts also display a lack of adaptive codon bias (Wernegreen and Moran 1999), decreased protein folding efficiency (van Ham et al. 2003), and destabilization of ribosomal RNAs (Lambert and Moran 1998). To date, the bacterium with the smallest known genome (160 kb) and the greatest degree of A+T bias (83.5%) is a primary endosymbiont (Carsonella ruddii, the endosymbiont of the gall psyllid Pachypsylla venusta, Nakabachi et al. 2006). Secondary endosymbionts, although less well studied, appear to display an intermediate level of genome reduction (Moran et al. 2005; Toh et al. 2006; Dale and Moran 2006). For instance, Sodalis glossinidius, a facultative endosymbiont of tsetse flies, exhibits a modestly reduced genome composed of nearly 50% pseudogenes, a finding which appears consistent with elevated substitution rates and future reductions in genome size (Toh et al. 2006).

The causes of reductive evolution in endosymbiotic bacteria are not well understood, and current hypotheses may be grouped into three broad categories. First, the loss of mutational repair pathways may have increased mutation rates, which in turn may cause increased substitution rates (Itoh et al. 2002). Second, the transition to a permissive intracellular environment may have relaxed selection across many loci. Relaxed selection increases the probability of fixation of deleterious mutations, which in turn may result in increased sequence evolution rates and the accumulation of deletions (e.g., Lambert and Moran 1998; Dagan et al. 2006). Third, the observed patterns may result from a decrease in the effective size of the endosymbiont population. A reduction in effective population size (Ne) increases the importance of genetic drift, which reduces both the strength of purifying selection and the probability that beneficial mutations will fix. Several studies of intrapopulational variation in Buchnera aphidicola, the primary endosymbiont of aphids, have found low levels of polymorphism but an increase in the number of relatively rare alleles, a finding consistent with a decrease in effective size (Funk et al. 2000; Abbot and Moran 2002; Herbeck et al. 2003). Many authors now cite reduced effective population size as the primary cause of increased substitution rates and the accumulation of deleterious mutations in endosymbionts (Lambert and Moran 1998; Clark et al. 1999; Wernegreen and Moran 1999; Fares et al. 2002b; Woolfit and Bromham 2003; Wernegreen and Funk 2004; Fry and Wernegreen 2005; Dale and Moran 2006).

Although changes in effective size are sufficient to explain some features of endosymbiont evolution, other characteristics are more difficult to interpret in the context of effective size change alone. In particular, selection at the host level may influence the evolutionary rate of some (or many) endosymbiont genes. Canbäck et al. (2004) noted that, in B. aphidicola and W. glossinidia, genes in functional categories thought to be associated with host fitness (amino acid, fatty acid and cofactor biosynthesis) evolved more slowly and were more numerous than genes in other functional categories. Additionally, Oliver et al. (2005) found that different clones of the pea aphid endosymbiont Hamiltonella defensa conferred different degrees of resistance to the parasitoid wasp Aphidius ervi, indicating that heritable variation exists in endosymbionts for host-level traits. Nonetheless, how changes in the effective size of the endosymbiont population interact with natural selection at the host and symbiont levels is not well understood. In particular, the question of whether selection at the host level can slow or stop the accumulation of deleterious mutations, and if so, at which loci this may occur, is unanswered.

Two studies have sought to quantify the relative effects of selection at the host and symbiont levels on the rate of accumulation of deleterious mutations (Rispe and Moran 2000; Pettersson and Berg 2007). Both studies examined evolution by Muller's Ratchet, which posits that small, asexual populations will decline in fitness due to the repeated, random loss of the least mutated class of individuals (Muller 1964). The studies involved computer simulation of a population of primary endosymbionts within hosts (horizontal transmission was not considered), and found that increasing the number of hosts, as well as the number of symbionts transmitted from mother to offspring, significantly slowed the rate of accumulation of substitutions. Pettersson and Berg (2007) found that, if there are more than about 105 hosts, then purifying selection at the host level is so effective that very few mutations deleterious to hosts are likely to accumulate. Thus, if most mutations deleterious to hosts are also deleterious to endosymbionts, then endosymbionts may be protected from complete degradation by selection at the host level. Both models assumed that, within a given simulation run, all mutations had identical effects on fitness. This assumption may lead to inaccurate conclusions regarding Muller's Ratchet (Butcher 1995), and complicates interpretation of the models because the net effect of various types of mutations is not explicitly considered. Below, I generalize these studies by examining a model that includes the full distribution of mutational effects at the host and endosymbiont levels (see Fig. 1). Models of this sort include the possibility that all types of mutations arise (beneficial or deleterious at both levels, “selfish” or “altruistic” from the endosymbiont perspective) and specify their associated probability. Including the full range of mutational effects then allows a more thorough investigation of the extent to which selection at the host level may protect endosymbionts from complete genetic degradation.

Figure 1.

Samples drawn from three different distributions of fitness effects, each point represents a single mutation with effect on endosymbiont (s) given by the x-coordinate, and effect on host (v) given by the y-coordinate. All distributions are bivariate Gaussian with (A) no correlation between host and endosymbiont fitness. (B) Positive correlation, (C) Negative correlation.

The evolution of endosymbionts is similar to that of organelles and has been investigated mathematically by a number of authors (Takahata and Slatkin 1983; Walsh 1992, 1993; Bergstrom and Pritchard 1998; Roze et al. 2005). These models examine the effect of transmission bottlenecks and biparental transmission on the probability of fixation of new mutations and the equilibrium number of deleterious mutations. The studies have established that reduced numbers of endosymbionts transmitted from mother to offspring favors selection at the host level, and that occasional paternal “leakage” of organelles to offspring can increase the efficiency of selection at the organellar level. Because biparental transmission of organelles is a process similar to horizontal transmission of endosymbionts among hosts, the results suggest that horizontal transfer may have important consequences for the types of mutations that fix as well as for the overall substitution rate. In particular, because secondary endosymbionts are sometimes horizontally transmitted and display a level of genome reduction less dramatic than that of primary endosymbionts, investigating the role of horizontal transmission may provide insight into the genetic differences between primary and secondary endosymbionts.

In this article I propose that several important features of endosymbiont evolution, including increased substitution rates, accumulation of deleterious mutations, and heterogeneous evolutionary rates across loci, are best understood as products of the population structure imposed by vertical transmission and resulting conflicts between levels of selection. Using analytic models, I investigate a population in the latter stages of progression toward primary endosymbiosis; prevalence of the endosymbiont in hosts is assumed to be high and vertical transmission is perfect (no uninfected offspring are born). I demonstrate that as horizontal transmission of endosymbionts among hosts decreases, substitution rates increase, and the expected effect of each substitution becomes more deleterious to the endosymbionts. I then expand the model to include correlations in the distribution of selective effects between hosts and symbionts, and show that this degree of correlation strongly influences evolutionary rate and, ultimately, whether or not the locus under consideration is eliminated or retained.

Methods and Results

To facilitate analysis I make several assumptions regarding the structure of endosymbiont populations. First, I assume that the host population has no significant spatial structure and follows Wright–Fisher dynamics with constant population size Nh. Little is known about the demography of insect host populations in general, and for analytic simplicity I assume the host population is well mixed. Second, I also assume that the endosymbiont reproduction within hosts follows Wright–Fisher dynamics with constant size Ns. Although the partitioning of the endosymbionts into bacteriocytes may render this assumption inaccurate, the role that subdivision at this level might play is unclear, and I limit my analysis to a single level of subdivision.

PROBABILITY OF FIXATION

To begin, I consider the simplest possible configuration of hosts and endosymbionts, where each host is infected with a single endosymbiont acquired from the host's mother. Below, I generalize the calculations to allow for an arbitrary number of endosymbionts per host, and horizontal transmission of endosymbionts among hosts. In the case in which each host is infected with a single endosymbiont, there is no intrahost competition, and the dynamics are equivalent to those of familiar asexual populations. Each endosymbiont behaves in the same manner as a host gene, and the probability of fixation of a new mutation initially present as a single copy with selective effect v on host reproduction is given by Kimura's (1964) formula

image(1)

The behavior of equation (1) is fairly simple. As long as the number of hosts is relatively high (that is, greater than the reciprocal of v) purifying selection is effective and the probability of fixation of beneficial mutations is approximately 2v.

Equation (1) may be generalized to allow for the possibility of multiple endosymbionts per host, say Ns. The endosymbionts reproduce within hosts according to Wright–Fisher dynamics for τ generations before each round of host reproduction. Furthermore, a new mutation segregating among the endosymbionts has selective effect s on endosymbiont fitness. If τ is relatively large and Ns fairly small, then the probability that a new mutation initially present as a single copy will either fix or be lost before the first round of host reproduction is close to one. In this limit, the probability that the new mutation reaches global fixation is the product of the probability that it reaches fixation in the first host and the probability that a host mutation with selective effect v reaches global fixation

image(2)

Some endosymbionts may be transferred horizontally within the host population, and this process might affect the probability of fixation of new endosymbiont mutations. To allow for the possibility of horizontal transfer I introduce a new parameter m, which describes the probability that a single endosymbiont is transferred from one host to another during one host generation. I assume that the state of the donor host is unaltered by endosymbiont emigration. If x reflects the global frequency of mutant endosymbionts, then each host receives mx mutant and m(1 −x) wild-type endosymbionts each host generation. If τ is large enough so that a newly introduced endosymbiont either is fixed or lost before the next round of host reproduction, then the probability that mutant endosymbionts “take over” a host with wild-type endosymbionts is

image(3)

Furthermore, the probability that wild-type endosymbionts, introduced in a group of colonizers, “take over” a mutant-occupied host is equal to the probability that the mutants are lost from the host. Because a colonizing group of endosymbionts of total size m contains fraction (1 −x) of wild-types, the frequency of mutants within a mutant-occupied host immediately after colonization is 1 −m(1 −x). The probability that these mutants do not reach fixation is then

image(4)

Using these two expressions, the probability of fixation of a mutation that has reached fixation in one host is approximately

image(5)

Here Nve=Nh/(2Nssm+ 1) is an approximate variance effective population size, se=v+ 2Nssm is an approximate effective selection coefficient (see Appendix A), and p is the initial frequency of hosts fixed for the mutant endosymbiont (assumed to be 1/Nh for all calculations below). A similar expression was derived by Walsh (1993), but in the case of biparental inheritance of organelles (see Appendix A). To calculate the probability of fixation of a mutation initially present as a single copy, the above expression must be multiplied by the probability that the mutant fixes in the first host.

Figure 2 compares the probability of fixation for a single endosymbiont per host, 100 endosymbionts per host, and 100 endosymbionts per host with a small amount of horizontal transfer among hosts, for a range of v encompassing mutations strongly deleterious to hosts (v < − 0.01) to mutations very beneficial to hosts (v > 0.01). In the absence of horizontal transfer, the probability of fixation (relative to a neutral mutation) is not much altered by increasing the number of endosymbionts per host from 1 to 100. This is because intrahost selection can only play a role during the fixation of the initial host, and if |Nss| < 1, the new allele will behave in a manner similar to a neutral allele during this phase. In contrast, small amounts of horizontal transfer may increase the probability of fixation for mutations beneficial to the endosymbionts by several orders of magnitude (dashed line on Fig. 1), indicating substantial increases in the efficiency of natural selection at lower levels for a modest degree of horizontal transfer.

Figure 2.

Comparison of the probability of fixation of a new endosymbiont mutation with selective effect on hosts v, relative to a neutral mutation, for a single endosymbiont per host (equivalent to a gene in an asexual host population), many endosymbionts per host, and many endosymbionts per host with moderate horizontal transfer. Nh= 1000, s= 0.005 where applicable.

Figure 3 compares equation (5), multiplied by the probability of fixation in the first phase, to simulation results for several different combinations of s and v. Although the mathematics assume that all mutations reach fixation or loss in each host before host reproduction, the simulations demonstrate that this approximation is valid even if as few as five endosymbiont generations pass for each host generation. The simulations also indicate that the model slightly overestimates the probability of fixation when s and v are jointly near N−1s, a possible consequence of the assumption that migration does not influence the probability of fixation in the first host.

Figure 3.

The probability of fixation of a new allele initially present as a single copy, with different fitness effects on host (v) and endosymbiont (s), as a function of horizontal transfer rate (m). Solid lines: analytic formulas, Markers: simulation results. Ns= 100, Nh= 100 in all cases.

Figure 4 examines the effect of horizontal transfer and selection at the endosymbiont level on probability of fixation. Increasing the transfer rate increases the effectiveness of selection at the symbiont level in a manner similar to that found by Walsh (1993) and Roze et al. (2005), and decreases the effectiveness at the host level. For example, in the absence of horizontal transfer (m= 0), a new mutation beneficial to hosts (v= 0.005) and deleterious to endosymbionts (s=−0.01) will fix with a probability greater then 10−5. However, small amounts of horizontal transfer favor selection at the endosymbiont level, and decrease the probability of fixation by over four orders of magnitude. Similarly, mutations deleterious to hosts and beneficial to endosymbionts have a very small probability of fixation in the absence of transfer, but increasing m from 0 to 0.005 may increase this probability by more than 1000-fold.

Figure 4.

The probability of fixation of a new allele as a function of its effect on endosymbiont fitness (s), for several horizontal transfer rates (m), and fitness effects on host (v). Ns= 100, Nh= 1000 in all cases.

These results indicate that the outcome of conflicts between levels of selection strongly depends on the horizontal transfer rate among hosts. At very low levels of horizontal transfer, natural selection is more efficient at the host level. Populations of hosts with little horizontal transfer more efficiently purge mutations in the endosymbionts deleterious to hosts, and fix mutations beneficial to hosts. The endosymbionts in such populations fare worse; their ability to purge deleterious and fix beneficial mutations is greatly impaired. In populations with intermediate degrees of horizontal transfer selection at the two levels is more comparable, and both hosts and endosymbionts may enjoy relatively efficient natural selection.

PROBABILITY OF SUBSTITUTION

Although the probability of fixation is a tool useful for understanding the fate of particular mutations, to understand long-term evolutionary rate we must investigate the probability that any newly arising mutation reaches fixation. This may be accomplished by combining the probability of fixation with the probability that a mutation with a particular effect on fitness occurs. This latter function, sometimes called a distribution of mutational (or selective) effects, in this case specifies the probability with which a mutation with a given effect on both endosymbiont (s) and host (v) arises. Letting φ(s, v) describe this distribution, then the probability that a randomly drawn mutation reaches fixation is given by

image(6)

where u(s, v, m) is the probability that a new allele, initially present as a single copy, reaches fixation. Little empirical data exist to inform the choice for φ. One recent study using vesicular stomatitis virus found a roughly exponential distribution of deleterious mutations (Sanjuan et al. 2004). Another study, using random transponson insertions in E. coli, found a distribution best fit by a combination of the gamma and uniform distributions (Elena et al. 1998). For analytic simplicity, I assume that s and v are independent, and s is drawn from an exponential distribution that extends from the origin in the negative direction with parameter λ. I also assume that λ > Ns such that most new mutations will have s < 1/Ns. In Appendix B, I demonstrate that an approximate probability of substitution is given by

image(7)

where I have made the additional assumption that Nh− 1 ≈Nh and v < − 1/Nh (relaxing the restriction on v leads to a somewhat more complicated solution, see Appendix B). psub(m) is decreasing in m, indicating that as horizontal transmission shrinks to zero substitution rates increase, regardless of the value of v. As expected, psub(m) increases with λ, which indicates that reducing the expected deleteriousness of mutations, as well as the variance of their distribution, increases the substitution rate.

This analysis indicates that decreasing horizontal transfer among hosts leads to an increase in the probability that a newly arising mutation will fix. Over time, this increase in the probability of fixation of each new mutation will lead to an increase in the number of substitutions, that is, an increase in the number of substitutions per unit time relative to populations with more horizontal transfer. If the progression to primary endosymbiosis involves a decrease in transmission of endosymbionts among hosts, then these calculations imply that an increase in substitution rates is an expected consequence of primary endosymbiosis.

THE EXPECTED EFFECT OF SUBSTITUTIONS

Although the above section demonstrates that decreasing horizontal transfer may lead to increased evolutionary rate, it is so far unclear what effect these substitutions may have on endosymbiont or host fitness. If the probability of a substitution is given by equation (6), then the expected effect of a substitution on endosymbiont fitness may be found by multiplying the integrand by s,

image(8)

likewise the mean fitness effect on host fitness is

image(9)

In general performing the above double integrations is difficult, but analytic formulas may be found in certain cases. For equation (8), I begin by assuming that the distribution of fitness effects is independent on the host and symbiont levels. I also assume a large number of hosts as well as negative v, such that inline image is a large positive number. For analytic simplicity I take s to be distributed as back-to-back exponentials that meet at zero, with parameters λ1 and λ2 in the negative and positive directions, respectively, and probability of deleterious mutation p. Additionally, I assume that λ2 > 4NsNhm. Under these restrictions integration over s yields

image(10)

where γ= 4NhNsm, α= 4NhNsm− 4Nsm, and β= exp[2Nhv](see Appendix C). If deleterious mutations are more common than beneficial mutations and have, on average, a greater effect on fitness (λ−11 > λ−12), then this function will be negative at m= 0. If hosts are numerous enough such that NhNh− 1, then the critical migration rate, mcrit, at which the net effect of evolution changes sign, exists at

image(11)

This value describes the (approximate) level of horizontal transmission necessary for natural selection to ensure that the mean fitness of endosymbionts is constant or increasing. Below this level natural selection is so compromised that deleterious substitutions outweigh beneficial substitutions, and the fitness of the endosymbiont population is predicted to decrease over time. As expected, this function is increasing in p, which implies that an increase in the fraction of mutations that are deleterious will accelerate degeneration. In addition, mcrit is also increased by decreasing the variance of the distribution. By letting λ2=xλ1, where x denotes the ratio of the mean effect of deleterious mutations relative to the mean effect of beneficial mutations, then mcrit is increasing in λ1 as long as x > 1 and p > 1/2. Because increasing λ1 will decrease the spread of the distribution, this result implies that distributions with smaller variance may facilitate degeneration (the restrictions on x and p indicate that this is only true if most mutations are deleterious and have a greater mean than beneficial mutations). Hence, factors that mitigate the effects of deleterious mutations, such as overexpression of the heat shock protein GroEL (Fares et al. 2002a), may in the long run promote decay.

Figure 5 compares numeric solutions of (8) (assuming that v is constant) to the analytic formula in (10). At low levels of horizontal transfer the expected effect of a new substitution is negative, but increases with m and becomes positive at some intermediate m (given by 11). In accordance with the analysis above, decreasing the variance of the distribution of selective effects decreases the mean effect of substitutions (compare parameter combinations in (a) to (b)), and thus makes the endosymbiont population more likely to experience degenerative evolution.

Figure 5.

The expected effect of a new substitution on endosymbiont fitness, comparison of numeric solution of equation 8 (solid lines) to the analytic approximation in 10 (dashed lines, visible only near left side of figure). (a) λ1= 200, λ2= 500, Ns= 100, Nh= 500, p= 0.9. v=−0.05 (b) λ1= 300, λ2= 600, Ns= 100, Nh= 500, p= 0.9, v=−0.05, and (c) λ1= 500, λ2= 2000, Ns= 400, Nh= 800, p= 0.9, v=−0.01

Calculations for the mean effect of a substitution on host fitness are more difficult, and I have been unable to find a closed-form approximation that is accurate for a variety of parameter combinations. Figure 6 presents numerical solutions to (8) and (9) for several parameter combinations, using Gaussian distributions of fitness effects for both s and v. Figure 6 demonstrates that, contrary to the effect on endosymbionts, decreasing horizontal transfer benefits the host population by making the expected effect of a substitution more beneficial, although this effect is small compared to the effect on endosymbionts. For no degree of horizontal transmission is the expected effect negative, and natural selection remains efficient at the host level.

Figure 6.

The expected fitness effect of a substitution on hosts (black lines) and endosymbionts (gray lines). Distributions of selective effects are independent, identical and Gaussian. (i) μ=−0.001, σ= 0.0075. (ii) μ=−0.001, σ= 0.005. (iii) μ=−0.0005, σ= 0.0075. Ns= 200, Nh= 1000 in all cases.

CORRELATION IN THE DISTRIBUTIONS OF SELECTIVE EFFECTS

The analysis above suggests that all endosymbiont populations that lack horizontal transfer among hosts are bound for extinction (because setting m= 0 causes the expected effect of a substitution to be negative). However, these analyses have assumed that there is no correlation in the distribution of selective effects at the host and endosymbiont levels. It is likely that at some loci s and v are correlated, either positively or negatively. Positive correlations, where mutations deleterious to the host are also likely to be deleterious to the endosymbiont (and similarly for beneficial mutations), may be likely to occur in genes essential to endosymbiont metabolism. A deleterious mutation here is likely to compromise the bacterium's competitive ability, and may also affect, for instance, the quantity of amino acids provided to the host. On the other hand, some loci may be more likely to generate mutations of conflicting effect. Mutations that decrease the quantity of amino acids supplied to the host may increase endosymbiont intrahost competitive ability, but decrease the fitness of the host. At these loci the opportunity for “selfish” or altruistic behavior is present, depending on whether a mutation has a positive effect on endosymbiont or host fitness, respectively.

Figure 7 examines the role of correlations between levels of selection by assuming that the selective effects of new mutations are drawn from a bivariate Gaussian distribution with correlation ρ. For a given level of horizontal transmission m, positive correlations improve the plight of endosymbionts and increase the expected “beneficiality” of each substitution. If the correlation is strong enough, a locus may be rescued from degeneration by raising the expected effect of a substitution above zero, even in the absence of horizontal transfer. Negative correlations facilitate degenerative evolution by decreasing the expected fitness effect of substitutions. Additionally, the degree of correlation required for maintenance of a locus increases as horizontal transmission decreases, such that at a moderate level of transfer even loci with a correlation close to zero may be maintained. As horizontal transfer decreases however, a greater degree of correlation is required for natural selection to purge mutations at the locus.

Figure 7.

Expected effect of a substitution on host fitness (black lines) and endosymbiont fitness (gray lines). Distribution of selective effects is bivariate Gaussian with correlation ρ, and μ=−0.001, σ= 0.0075. Ns= 200, Nh= 1000 in all cases.

Evolution at the host level is not greatly affected by alterations in the correlation coefficient, although the effect of horizontal transmission may be reversed. For instance, if ρ≤ 0, the effect of substitutions on hosts increases as m decreases. However, if the correlation is strong and positive the dependence of the host fitness effect on m may become positive, such that decreasing horizontal transfer makes more deleterious the expected effect of substitutions on hosts. However, this effect appears to be small for most parameter combinations, and the expectation remains positive.

Discussion

Several conclusions may be drawn from the above analysis. First, decreasing horizontal transmission increases the probability that a given new mutation will fix in the endosymbiont population. Assuming a constant mutation rate, an increased probability of substitution will result in an increase in evolutionary rate. Thus, when compared to free-living relatives or to ancestors with more horizontal transfer among hosts, endosymbionts are expected to accumulate substitutions more rapidly. This process is qualitatively similar to Muller's Ratchet (Muller 1964), but here includes both beneficial and deleterious mutations (at both levels), and thus does not necessarily imply a decrease in endosymbiont fitness.

Second, decreasing horizontal transmission leads to increasing deleteriousness of substitutions on endosymbionts, although the fitness effect on hosts is not greatly altered. Thus, as horizontal transmission decreases and the association progresses toward primary endosymbiosis, endosymbionts are expected to accumulate greater numbers of deleterious mutations. Although all loci are predicted to accumulate some deleterious substitutions, in some cases these negative fitness effects may overwhelm the positive effects of beneficial substitutions, leading to an expected decrease in fitness over time. Such a process may lead to the more rapid fixation of a stop codon or frameshifting indel at the locus, causing nonfunctionalization of the gene and facilitating its eventual deletion. The predicted decrease in fitness is qualitatively similar to the “mutational meltdown” (Lynch and Gabriel 1990; Lynch et al. 1993), but here relates to the loss of a particular locus from a population, and does not necessarily involve a synergistic feedback with decreases in population size. However, if the accumulation of mutations in some loci leads to a reduction in horizontal transmission, then a positive feedback cycle may occur, leading to eventual loss of the loci involved as well as an overall reduction in interhost transfer.

Whether natural selection can effectively maintain a given locus depends on several factors. The critical horizontal transmission rate, below which a locus is expected to degenerate, is inversely proportional to the total number of endosymbionts in the population (the product of Nh and Ns), meaning that either larger populations of hosts or greater numbers of endosymbionts per host may slow the accumulation of deleterious substitutions and allow retention of more loci. The critical transmission rate is increased by decreasing the variance of the distribution of fitness effects, which suggests that mutations of small effect are most responsible for the loss of fitness. These effects may all be offset to some extent if the distributions of fitness effects between host and endosymbiont are positively correlated, such that a mutation deleterious to the endosymbiont is also likely to be deleterious to the host. If the correlation is strong enough, a given locus may be rescued from degeneration, even if the horizontal transmission rate is zero. Because this correlation is likely to vary across loci, some loci are predicted to be retained permanently, whereas others may be quickly lost. Variance in this parameter across loci may be partly responsible for the heterogeneity in dN/dS ratios observed in various studies (Clark et al. 1999; Fry and Wernegreen 2005).

These findings do not imply that increased mutation rates, relaxed selection, and transmission bottlenecks are not important factors affecting evolutionary rate and trajectory in endosymbionts. Indeed, loss of DNA repair mechanisms must to some extent increase mutation rates (unless offset by decreased exposure to mutagens), and this will likely influence the rate of sequence divergence. The fact that most (but not all) endosymbionts are AT biased provides further support for an important role for altered mutational patterns, although whether loss of repair pathways is a cause or consequence of genome reduction is not well understood. In addition to alterations of mutation rate and pattern, selection is likely to be quite relaxed at a number of loci whose functions are no longer needed in a stable, intracellular environment, and these loci will degenerate quickly. Nonetheless, it remains unclear how this might contribute to the destabilization of, for instance, ribosomal RNAs, whose function remains critical despite host association.

Transmission bottlenecks will further increase the importance of stochastic effects within host lineages and also are likely to contribute to increased sequence evolution rates and decreased efficiency of purifying selection. Previous investigations of the role of bottlenecks (e.g., Bergstrom and Pritchard 1998; Rispe and Moran 2000; Roze et al. 2005) have indicated that they increase variance in allele frequency among hosts and reduce variance within-hosts. This in turn increases the efficiency of selection at the host level and reduces it at the endosymbiont level. The relative importance of bottlenecks depends to some extent on the level of within-host variability maintained in the absence of bottlenecks. If m= 0 and τ is quite large, within-host variability may be so low that bottlenecks will not substantially decrease it. However, for a moderate degree of horizontal transmission bottlenecks may play a more important role, thus bottlenecks may affect the evolution of secondary endosymbionts more significantly than primary endosymbionts.

The recent study by Pettersson and Berg (2007) suggested that purifying selection at the level of the host may be sufficient to slow or stop the accumulation of deleterious mutations in endosymbionts. The analysis here supports the conclusion that few mutations deleterious to hosts are likely to fix, and thus many of the observed substitutions are expected to be neutral at the host level. However, by considering the full range of mutational effects at both host and symbiont levels, the model here includes the possibility that some mutations may have very little effect on host fitness but a negative impact on endosymbiont fitness. Because purifying selection at the endosymbiont level is compromised, these mutations are likely to fix and will continue to drive increased evolutionary rate and, possibly, loss of some loci. The ultimate fate of endosymbionts may depend on the extent to which host fitness depends on symbiont loci. If this correlation is particularly strong some endosymbiont loci may be preserved indefinitely. However, if correlation is weakened (perhaps by shift in host ecology, e.g., Moran and Wernegreen 2000), the endosymbiont is predicted to degrade completely.

The theory outlined above appears consistent with current observations regarding the differences between “primary” and “secondary” endosymbionts. Most primary endosymbionts, such as B. aphidicola, Blochmannia floridanus, and B. pennsylvanicus, and Wigglesworthia glossinidia, display no evidence of horizontal transfer, are relatively ancient, and contain genomes substantially smaller than those of secondary endosymbionts. In contrast, secondary endosymbionts such as H. defensa and S. glossinidius, although they are most often transmitted vertically, also display some level of horizontal transfer, contain genomes of intermediate size (1.0–4.0 Mbp), and the associations appear to be more recent (Dale and Moran 2006). Secondary endosymbionts may also differ from primary endosymbionts in characteristics not accounted for by this model. For instance, their prevalence among hosts is likely to be less than 100% and the presence of phage may allow recombination in both H. defensa and S. glossinidius. It is possible that these factors, in addition to horizontal transfer, have influenced the rate of genome reduction, although their expected effect is unclear.

Although I have made the assumption that progression toward primary endosymbiosis involves a decrease in horizontal transfer, I have intentionally avoided the question of the cause of this change. The interesting question as to whether endosymbiosis is an adaptation from the perspective of the symbiont, or a condition that has been “forced upon them” by the host, has been treated in other theoretical papers (see Lipsitch et al. 1995, 1996, and Yamamura 1993, 1996), and is closely related to the evolution of virulence. Yamamura (1993, 1996), who used a deterministic Evolutionary Stable Strategy analysis, found that, when the efficiency of vertical transmission increases above a critical level, further increases in vertical transmission may be favored by both endosymbiont and host. This finding, combined with the results here, suggests that the ancestors of current endosymbionts may have “chosen” a lifestyle of strict vertical transmission, which later resulted in large-scale genome degradation. In addition, a positive feedback may exist between reductions in horizontal transfer and loss of genes that facilitate such transfer. For instance, a moderate decrease in horizontal transfer may decrease purifying selection enough to result in the loss of some genes required for interhost transmission and establishment, which may further decrease the possibility for interhost exchange.

In bacteria, genome content is to a large extent shaped by the opposing forces of gene gain and loss (Ochman et al. 2000, Lawrence and Hendricks 2005). Understanding the construction of bacterial genomes depends critically on understanding why some genes are retained where others are eliminated. This work demonstrates that in some circumstances retention of a gene depends on more than the selective benefit conferred. Ecological factors such as the population structure imposed by host association and the degree and mode of transmission among hosts may play an important role.

Associate Editor: T. Hansen

ACKNOWLEDGMENTS

I would like to thank F. Adler, C. Dale, and J. Seger for thoughtful discussion of these topics, and two anonymous reviewers for insightful comments on the manuscript. The work was financially supported by a National Institutes of Health Genetics Training Grant.

Appendices

Appendix A

I use Kimura's (1964) general formula for the probability of fixation of a new allele,

image((A1))

where

image((A2))

where the allele under consideration has initial frequency p, expected change per generation Mδx, and variance of that change Vδx.

The above formula assumes that the mean change and its variance are relatively small each generation, and that the other moments of the distribution of change may be safely neglected. For practical purposes this means that selection coefficients and migration rates should not be too large.

With the above assumptions outlined in Models and Results, the expected change in frequency of a mutation with current frequency x is given by

image((A3))

where α(x) is the probability that a host fixed for the wild-type endosymbiont is “taken over” by mutant endosymbionts, β(x) is the probability that a mutant host remains fixed for the mutant endosymbiont. The first term reflects the change in mutant endosymbiont frequency due to selection between hosts, the second term the fraction of wild-type hosts that become dominated by mutants due to horizontal transfer, and the third term the fraction of mutant hosts that remain mutant despite horizontal transfer.

As noted in the text, each host receives xm mutant immigrants and (1 −x)m wild-type immigrant symbionts each host generation. Using equation (3), the probability of a wild-type host that receives xm mutant endosymbionts becoming fixed for the mutants is

image((A4))

Likewise, using equation (4), the probability that a mutant host remains fixed for mutants despite receiving (1 −x)m wild-type immigrants, is

image((A5))

where the approximations are valid for small m. Substituting the approximate α and β into (A3) and further assuming v is small such that 1 +vx≈ 1, the mean change becomes

image((A6))

This derivation is similar to that given in Walsh (1993), but with at least one difference. In the biparental inheritance case the probability that a wild-type female/mutant male mating gives rise to mutant offspring is (in Walsh's notation) Uc(α)x(1 −x) where x is the frequency of the mutant and Uc(α) is a function that describes the proportion of mutant offspring born given degree of “paternal leakage”α. Walsh's Uc(α) is analogous to my α(x) or β (x), but is independent of x. In my model each host experiences (via migration) the average state of the population, and thus the conversion probabilities are functions of x. Given a small m, second-order Taylor expansion leads to a probability of fixation similar to Walsh's equation (A6).

To calculate the variance in the expected change first note that in a standard Wright–Fisher population the variance in allele frequency change is the variance of the binomial distribution, x(1 −x)/N, which must be the variance due to sampling of hosts during reproduction. Horizontal transmission and intrahost selection increase the variance, and because each mutant host has an equal probability of becoming a wild-type host, and similarly for wild-type hosts, these two probabilities are also binomial random variables, with parameters α(x) and β (x). Thus the total variance in allele frequency change in one host generation is approximately

image((A7))

where Nh is the number of hosts in the population. Once again expanding the exponentials in the numerator and dropping terms of m2 and higher leaves

image((A8))

In a simple Wright–Fisher population with allelic selection (no dominance), the expected change in gene frequency is approximately sx(1 −x) (Ewens 1979). Because the equation for the expected change is proportional to x(1 −x), the object v+ 2Nssm is an effective selection coefficient, such that the population behaves in some regards as if it was a Wright–Fisher population with selection coefficient v+ 2Nssm. Similarly, if the variance in allele frequency change in a Wright–Fisher model is x(1 −x)/Nve, then a variance effective population size can be found by equating this expression with the equation for the variance given in equation (8). Solving for Nve yields

image((A9))

Because both Mδx and Vδx are proportional to x(1 −x), their ratio is constant in x, and the integrations in equations (A1) and (A2) are easy to perform. The probability of fixation is

image((A10))

where se and Nve are the effective selection coefficient and effective population size calculated above. Here p expresses the initial frequency of hosts fixed for the endosymbiont bearing the new allele; once again this calculation is conditional on fixation in the first host. To calculate the probability of fixation of a new allele present in one copy in a single endosymbiont u(p) must be multiplied by the probability that the allele fixes within a single host. If |s| < 1/Ns, then the probability of fixation in the first phase (the probability that the original host will become fixed for the mutant endosymbiont) is ≈1/Ns. In this case the probability that a single mutant endosymbiont will reach global fixation becomes

image((A11))

Appendix B

The probability of substitution given in equation (7) may be found in the following manner. Ignoring the integration with respect to v, and assuming that the selective effect of a new mutation on endosymbionts is drawn from a negative exponential distribution, equation (6) becomes

image((B1))

Setting β= exp[− 2Nhs], γ= 4NhNsm, and α=λ− 4Nsm the above equation can be written as

image((B2))

Carrying out the integration leads to a solution in terms of the hypergeometric function

image((B3))

If the number of hosts is large enough so that Nh− 1 ≈Nh, the above equation reduces to

image((B4))

In the case in which purifying selection at the level of the host is relatively strong, such that exp (−2Nhv) ≫ 1, terms of order β−2 and greater may be dropped, leading to

image((B5))

Appendix C

The mean fitness effect of a substitution on the endosymbiont population can be calculated beginning with equation (8). If the distribution of fitness effects of new mutations is given by back-to-back exponential distributions that meet at zero with parameters λ1 and λ2 in the negative and positive directions, respectively, and probability of deleterious mutation p, then (8) becomes

image((C1))

If purifying selection at the level of the host is relatively strong, which implies that exp [− 2Nhv]≫ 1, then 1 − exp [− 2Nhv] exp [− 4NhNsms]≈− exp [− 2Nhv] exp [− 4NhNsms], which leads to

image((C2))

Performing the integrations and simplifying leads to

image((C3))

where γ= 4NhNsm, α= 4NhNsm− 4Nsm and β= exp[2Nhv]. If the number of hosts is large enough so that Nh− 1 ≈Nh, then γ≈α. Further simplification followed by solution for m yields equation (11).

Ancillary