Abstract
Interspecific genetic interactions in hostsymbiont systems raise intriguing coevolutionary questions and may influence the effectiveness of public health and management policies. Here we present an analytical and numerical investigation of the effects of host genetic heterogeneity in the rate of vertical transmission of a symbiont. We consider the baseline case with a monomorphic symbiont and a single diallelic locus in its diploid host, where vertical transmission is the sole force. Our analysis introduces interspecific disequilibria to quantify nonrandom associations between host genotypes and alleles and symbiont presence/absence. The transient and equilibrium behavior is examined in simulations with randomly generated initial conditions and transmission parameters. Compared to the case where vertical transmission rates are uniform across host genotypes, differential transmission (i) increases average symbiont survival from 50% to almost 60%, (ii) dramatically reduces the minimum average transmission rate for symbiont survival from 0.5 to 0.008, and (iii) readily creates permanent hostsymbiont disequilibria de novo, whereas uniform transmission can neither create nor maintain such associations. On average, heterozygotes are slightly more likely to carry and maintain the symbiont in the population and are more randomly associated with the symbiont. Results show that simple evolutionary forces can create substantial nonrandom associations between two species.
GENETIC variation at loci involved in interspecific interactions can have profound coevolutionary implications. The qualitative and quantitative characteristics of symbiotic interactions, be they commensalist, mutualist, or parasitic, are especially likely to vary with the genotypes of the two interacting individuals (Flor 1956; Wakelin 1978; Reisseret al. 1982; Van transgene are thought to Looet al. 1987; Slade 1992; Frank 1993; Rautianet al. 1993; Antonovics and Thrall 1994; Guptaet al. 1994; Anderson 1995; Kover and Clay 1998; Burdon and Thrall 1999). These interspecific genetic interactions arise because of the close physical, and in many cases physiological (Arilloet al. 1993; Heddiet al. 1993; Dunnet al. 1995; Wilkinson and Douglas 1995), genetics (Weis and Levine 1996; Charleset al. 1997; Chunget al. 1997; Escalanteet al. 1998), and phylogenetic (Yanet al. 1994; Escalante and Ayala 1995; Díaz 1997; Lutzoni and Pagel 1997; Morzunovet al. 1998) relationships between host and symbiont (Spencer 1988; Mulveyet al. 1991; Graf and Ruby 1998). For example, the strong host genotypebystrain interactions between legumes and their symbiotic bacteria may permit the host's genetic diversity to maintain genetic polymorphism in the symbiont (Spoerkeet al. 1996; LievenAntoniou and Whittam 1997). Similarly, human genetic variability in disease resistance has had major historical and demographic consequences (May 1984; Cadavid and Watkins 1997), by rendering us more or less vulnerable to infection by parasites such as the vertically transmitted human papillomavirus, HPV (Favreet al. 1998; Ramozet al. 1999), and the human immunodeficiency virus, HIV (Fauci 1996; RowlandJones 1998).
These interspecific interactions raise intriguing evolutionary questions (Price 1980; Hamilton and Zuk 1982; Futuyma and Slatkin 1983; Futuyma and Keese 1992; Hamilton 1993; Herre 1993; Frank 1994; Grenfell and Dobson 1995; Henter and Via 1995; Ebert and Hamilton 1996; Thompson 1999). They also have important practical implications for management policies in public health, wildlife, forestry, and agriculture dealing with such critical issues as infectious diseases, crop productivity, or pest control, where the fitness of the plant or pest is dependent on symbionts (Allard 1990; Schardl and Tsai 1992; Douglas 1996; Grenieret al. 1997; Sáez Nieto and Vázquez 1997). These policies may be more effective if population heterogeneity is considered (Tibayrencet al. 1990; Anderson and May 1991; Chabora and Koepfer 1991; Ewald 1994; Gupta and Hill 1995; Riley 1996; B. R. Levinet al. 1997; S. A. Levinet al. 1997; Lipsitch 1997; Weatherallet al. 1997; Schorket al. 1998).
Here we provide an extensive theoretical investigation of two new issues regarding hostsymbiont systems. The first involves the effects of host genetic heterogeneity in the rate of vertical transmission of the symbiont from parents to offspring (Fine 1975; Yamamura 1993; Lipsitchet al. 1996). This addresses an important theoretical gap that has been pointed out by previous authors in the hostparasite literature (Busenberg and Cooke 1993). The second innovation is the use of interspecific disequilibria for studying systems of two interacting species. Here we use disequilibria to quantify nonrandom associations between host genotypes and alleles and symbiont presence (symbiotic host) or absence (aposymbiotic host). Similar measures have proven useful in a theoretical study of hostparasite systems with multiple parasite types (Sánchezet al. 1997) and resemble the geneculture disequilibrium used to measure associations between genes and vertically inherited cultural traits in studies of geneculture coevolution (Feldman and CavalliSforza 1984; Feldman and Zhivotovsky 1992). These new interspecific statistics are analogous to cytonuclear disequilibria between nuclear markers and cytoplasmic markers in either host organellar DNA (Asmussenet al. 1987; Asmussen and Basten 1994) or cytoplasmically inherited microorganisms (Turelliet al. 1992; Dean and Arnold 1998).
As with the cytonuclear disequilibria on which they are based, hostsymbiont associations may allow us to draw evolutionary inferences that are difficult to obtain by other means (Asmussenet al. 1989; Siteset al. 1996; Van Der Looet al. 1997; Goodismanet al. 1998). For example, in hostparasite systems they may indicate which are the more susceptible and higher transmitting genotypes, because we expect such individuals to be infected more often than would be expected by chance. One such potential application is provided by the vertically transmitted sigma virus of Drosophila melanogaster (Wayneet al. 1996; Yampolski et al. 1999), where the varying degrees of association of the virus with host genotypes could be used to infer the degree of resistance of the different host genotypes.
Here we use hostsymbiont disequilibria to help us delimit how differential vertical transmission rates of a symbiont across host genotypes affect the dynamics and equilibria of the two species. We focus on the baseline situation where vertical transmission is the sole selective factor determining symbiont survival and distribution; there is no symbiontinduced selection or other evolutionary forces acting on the system. This case is of theoretical and practical interest, because the mode and rate of transmission of a symbiont can be a determining factor in the biology of both the symbiont and the host (Fenner 1968; Ewald 1987; Maynard Smith, pp. 676–677 in Saffo 1991; Garnick 1992; Michael 1993; Clayton and Tompkins 1994; De Leo and Dobson 1996; Lipsitchet al. 1996). Moreover, transmission rates may be representative of the overall fitness of the symbiont, with a higher transmission rate reflecting a higher fitness; this can occur when only the symbiont is affected by the interaction (as in commensalism), or in hostparasite systems when pathogen virulence and transmissibility are independent (Frank 1996; Lipsitch and Moxon 1997). The survival and dynamics of nonpathogenic parasite strains (Capucciet al. 1996; Chasey 1997) may also be determined essentially by their transmission rates. To facilitate the genetic analysis (Leonard 1997), we assume that the vertical transmission rates are constant, and we use discretetime difference equations based on discrete generations in the host to mimic the natural discontinuity between host generations that occurs with each mating event.
We first derive the explicit dynamic equations that describe the change through time of the different host and symbiont classes and use these to obtain precise analytic conditions for symbiont survival. We then explore numerically the symbiont's prevalence and distribution across host genotypes at equilibrium and how these depend on the absolute and relative values of the vertical transmission rates and the host allele frequencies. This is followed by the definition of formal measures of nonrandom association (disequilibria) between the host and symbiont and an extensive analytical and numerical investigation of their dynamical and equilibrium behavior in the case of (1) arbitrary initial conditions (general disequilibrium analysis) and (2) no initial disequilibria between the symbiont and host genotypes and alleles (de novo disequilibrium analysis). In addition, we discuss how the expected behavior of hostsymbiont frequencies and disequilibria under differential vertical transmission can improve our understanding of the evolution of hostsymbiont systems, for both theoretical and practical reasons.
THE HOSTSYMBIONT MODEL
We assume no stochastic forces, mutation, or migration affect the dynamics of either species. The host population is diploid, sexual, and panmictic and is characterized by a diallelic autosomal locus (alleles A_{1} and A_{2}) with Mendelian segregation. The symbiont is haploid (or equivalently a clonal diploid), genetically monomorphic, and exclusively vertically transmitted. Generations of both species are discrete and coincide in time. Selection affects only the symbiont's survival and distribution in the population and is caused by variation across host genotypes in the rate of transmission of the symbiont to the next host generation. This is the sole evolutionary force acting on the system. The genotype and allele frequencies of the host across symbiont classes (S^{+} is the symbiotic host class and S^{−} is the aposymbiotic host class) are given in Tables 1 and 2, respectively. Adding down columns provides the corresponding marginal frequencies of the host genotypes (g_{11}, g_{12}, g_{22}) and alleles (p, q), while the row sums provide the frequencies of the symbiont classes (u, y).
Vertical transmission: The symbiont's survival and distribution in the population are governed by the three vertical transmission rates β_{11}, β_{12}, β_{22} (after Anderson and May 1979), where β_{ij} is the probability that an A_{i}A_{j}/S^{+} individual transmits the symbiont to its offspring. Under uniform vertical transmission β_{11} = β_{12} = β_{22} = β, while under differential vertical transmission at least one host genotype has a rate different from the others.
The frequency of the symbiotic and aposymbiotic host classes in the next generation and the overall dynamics of the system depend on the proportion of offspring from each mating that fall in each hostsymbiont class, given the vertical transmission rates of the symbiotic host parent(s) (Table 3). We assume that an S^{+} parent transmits the symbiont independently of the Mendelian segregation at its nuclear marker and independently of symbiont transmission by the other parent, should both be S^{+}. For example, in a mating between an S^{−} and an A_{i}A_{j}/S^{+} individual, the offspring is S^{+} with probability β_{ij} and S^{−} with probability 1 − β_{ij}. When both parents are S^{+}, the progeny is S^{+} if and only if at least one parent transmits the symbiont. This occurs with probability
Hostsymbiont dynamics: Our analytical results show that the frequency of the hostsymbiont and symbiont classes change after each round of random mating and symbiont transmission. The dynamics of the system are governed by those of the two quantities,
SYMBIONT MAINTENANCE
We first obtained explicit analytical formulas for the equilibria of the system (appendix a). Because these involve a complex cubic equation, further analytical insight into symbiont survival was gained by analyzing the stability of the boundary equilibrium corresponding to loss of the symbiont (that is, to
Due to the analytical complexity of these inequalities, further information regarding symbiont maintenance was obtained via a numerical analysis (a summary of our numerical methods is given in appendix b). In these simulations the discriminant of (4) was always positive, indicating that both local stability eigenvalues are real numbers for all host allele frequencies (p, q) and vertical transmission rates (β_{ij}). Further numerical investigations revealed that the single condition A − B > 1, corresponding to
Prevalence and distribution of the symbiont at equilibrium: On average, 60% of host individuals are symbiotic at equilibrium, which in the hostparasite literature corresponds to parasite prevalence (Anderson and May 1991). Heterozygotes are slightly more likely to be symbiotic at equilibrium than homozygotes (the average of the ratio
Further indication of the importance of host genetic heterogeneity is given by its effect on the maintenance of the symbiont: differential vertical transmission increases the chances of symbiont survival by a factor of almost 18% over uniform vertical transmission, from 50 to 58.9% (Fine 1975; Sánchezet al. 1997). The conditions on the average vertical transmission rate in the population, once HardyWeinberg equilibrium is reached in the first generation,
Factors determining symbiont survival: We next explored how the relative values of the three vertical transmission rates affect the likelihood of maintaining the symbiont. The probability of symbiont survival is slightly higher when heterozygotes have the maximum vertical transmission rate (61.2%), rather than one of the homozygotes (57.7%). Likewise, by adding the corresponding entries in column 3 of Table 4, we see that when the symbiont survives, heterozygotes are slightly more likely to have the highest transmission (34.7%) than one of the homozygotes (32.7%); β_{12} is intermediate 33.4% and lowest 31.9% of the time.
While in the uniform case, symbiont survival is determined exclusively by the single vertical transmission rate (β > 0.5); in the differential case, it involves a complicated interplay between host allele frequencies and the three vertical transmission rates, shown in (5). With higher frequencies of allele A_{1} (p), the transmission rate of A_{1}A_{1} homozygotes (β_{11}) is more likely to be maximal when the symbiont is retained, and likewise for A_{2} and β_{22} (Table 4). For instance, the proportion of time that β_{11} is maximal increases from 32.6% when p is in [0,1] to 43.1% when p is in [0.5,1] and to 49.3% when p is in [0.8,1]. Although the corresponding values for β_{12} decrease from 34.7 to 34.6 to 30.9%, the heterozygote
transmission rate still plays a role. For example, when p is in [0.5,1] the case β_{12} > β_{22} > β_{11}, where β_{12} is maximal and β_{11} is minimal, has a higher frequency (15.6%) than the case β_{22} > β_{11} > β_{12} (11.3%), where β_{11} is intermediate and β_{12} is minimal. The probability β_{12} is maximal when the symbiont survives is greater the smaller the average transmission rate (e.g., β_{12} is maximal 70% of the time when
Further insight into the effects of host allele frequencies (p, q) on symbiont fate is gained by partitioning their values into the subintervals 0.0–0.1, 0.1–0.2, …, 0.4–0.5. Although the differences are fairly small, the symbiont is most likely to be lost from the population when the host is near fixation (45% of the time for p or q in 0.0–0.1 vs. 40–41% in the other intervals). In those cases where the symbiont is maintained, the host allele frequencies are almost uniformly distributed across all intervals (19% fall in 0.0–0.1 vs. 20–21% in the other intervals). As a general rule, however, a smaller total difference in the frequency of homozygotes and heterozygotes p^{2} − 2pq + 2pq − q^{2} appears to give the symbiont a marginal survival advantage.
HOSTSYMBIONT DISEQUILIBRIA
Nonrandom associations in a hostsymbiont system can arise at the level of both host genotypes and alleles. By analogy to cytonuclear systems (Asmussenet al. 1987), we define three hostsymbiont genotypic disequilibria
The frequency of each hostsymbiont class can be decomposed in terms of the appropriate hostsymbiont disequilibrium and marginal frequencies as shown in Tables 5 and 6. For genotypic disequilibria, a zero value represents completely random association between the symbiont and that host genotype; a positive disequilibrium represents an excess of that particular symbiotic class and a deficit of its aposymbiotic counterpart with respect to that expected when the symbiont is randomly distributed, whereas a negative value represents the reverse. In the same way, a positive allelic disequilibrium implies the symbiont is associated more often with the A_{1} allele, and a negative value indicates that it is associated more often with the A_{2} allele, as compared to random expectation.
The four hostsymbiont disequilibria are related by
Disequilibrium recursions: The disequilibrium recursions are best derived by substituting those for the frequencies of the different host and symbiont categories
from (2) and (3) into the disequilibrium relationships for the aposymbiotic class in Tables 5 and 6. This yields the compact equations
De novo disequilibria: The most important result from (10) is that differential vertical transmission can by itself generate hostsymbiont disequilibria de novo starting from a state with no initial disequilibria between host and symbiont (
Further insight is possible when the host is initially at HardyWeinberg equilibrium (p^{2}, 2pq, q^{2}), which in this model is reached within one generation. The initial de novo disequilibria in (11) then reduce to
Ignoring the trivial case where β_{ij} ≡ y = 1, we see that allelic and homozygote disequilibria will be generated de novo under (12)–(14) if and only if the two alleles have different marginal transmission rates
The initial dynamics of the symbiont are also especially straightforward if the host is in HardyWeinberg equilibrium and nonrandom associations between host
and symbiont are not present in the system. In such cases, the symbiont can increase in frequency in the first generation if and only if the symbiont's initial frequency y^{(0)} is below the critical value
DETECTABILITY OF DISEQUILIBRIA
One of our primary goals is to show that hostsymbiont disequilibria provide a valuable new tool for coevolutionary studies. We have investigated the practical utility of these new types of nonrandom associations through an extensive numerical analysis of their magnitude, duration, sign, and dynamical and equilibrium behavior under our model. Results are given both for runs with arbitrary initial frequencies and disequilibria (general case) and for runs with no initial disequilibria (de novo case). We consider a disequilibrium measurable if it exceeds 0.01 in magnitude, which is the minimum level at which disequilibria can usually be detected at a 0.05 significance level given reasonable sample sizes and marginal frequencies (Asmussen and Basten 1994; Basten and Asmussen 1997; Sánchezet al. 1997). Trajectories with disequilibria >0.01 in magnitude in at least one generation are called measurable and are classified as permanent if they have measurable associations at equilibrium and as transient otherwise (Babcock and Asmussen 1996, 1998). Because nonzero disequilibria require the presence of the symbiont (Table 7), necessarily all measurable trajectories that lose the symbiont are transient.
Although interrelated, for clarity we have partitioned our results into four main sections (general case, de novo case, normalized disequilibria, and sign statistics). Unless noted otherwise, these share three common features. First, we have calculated results separately for (i) all trajectories, (ii) trajectories where the symbiont is maintained at equilibrium, and (iii) trajectories where the symbiont is lost. Tables include the breakdown for these three cases only when there are substantial differences among them, and, for simplicity, we primarily focus in the text on the results when all trajectories are included in the analysis. When differences do exist, on average the disequilibria have a greater magnitude and persist longer in trajectories where the symbiont is maintained than in trajectories where it is ultimately lost; as expected, the results based on all trajectories have intermediate values. The second common feature is that the heterozygote disequilibrium usually has approximately half the magnitude of homozygote and allelic disequilibria and, when transient, is measurable for a smaller average number of generations. In general, the statistics for the homozygote and allelic disequilibria usually are very similar, although there are notable exceptions. Last, in all cases the average final magnitudes of the disequilibria are calculated considering only trajectories where the symbiont is maintained. Disequilibrium statistics regarding the maximum and minimum values across all simulations are reported in appendix c.
General case—arbitrary initial disequilibria: In this first set of simulations, the initial frequencies were generated randomly, with no constraints on the symbiont's initial distribution in the host population. To reduce the confounding effects of the arbitrarily generated initial disequilibria, which are in no way related to the transmission dynamics of the model, the analysis here excludes the values in the initial generation. The effects of these initial associations on the relevant statistics are discussed in appendix d.
Likelihood and duration of measurable associations: How often hostsymbiont disequilibria reach experimentally detectable levels and how long transient associations remain measurable are data critical to the practical importance of these measures. The results in Table 8 show that, on average, all three types of disequilibria are apt to be measurable, with allelic (D) and homozygote (D_{ii}) disequilibria being so much more frequent than the heterozygote (D_{12}) association (82–85 vs. 51% of the time). Moreover, a substantial fraction of measurable trajectories have permanent disequilibria (47% for D_{ii} and D, 41% for D_{12}), and particularly so when the symbiont is maintained (72% for D_{ii} and D, 60% for D_{12}), because all trajectories that lose the symbiont are necessarily transient.
Transient disequilibria tend to be fairly short lived, being measurable for an average of 5 (D_{12}) to 8 (D_{ii}, D) generations when the symbiont is maintained and 4 (D_{12}) to 5 (D_{ii}, D) generations when the symbiont is lost. The maximum number of generations transient disequilibria are measurable can be quite high, however,
and is substantially affected by symbiont fate. These values are 315 for D_{12}, 368 for D, and 520 for D_{ii} when
Magnitude of disequilibria: As shown in Table 9, the average maximum magnitude along a disequilibrium trajectory while the symbiont is maintained (y^{(t)} > 10^{−8}) is 0.016 for D_{12} and 0.030 for the other associations when all trajectories are considered. The values are slightly higher (0.018 and 0.034) for trajectories where the symbiont is maintained at equilibrium, but still measurable (0.013 and 0.024) for those where the symbiont is ultimately lost. When there is permanent symbiosis, the final magnitudes of the disequilibria are also, on average, all measurable, with the average heterozygote association (0.010) again being approximately half that for the alleles and homozygotes (0.021). These average final values are 30–37% smaller than the average maximum disequilibria along a complete trajectory, which indicates that there tends to be an overall decrease in the magnitude of the disequilibria as equilibrium is approached. The decrease does not have to be monotonic (Figure 1), because it involves a complicated interaction between the transmission rates and the frequencies of the hostsymbiont categories.
Sign changes: Differential vertical transmission alone can also bring about a change in sign of the disequilibria (that is, the system can go from an excess to a deficit of a particular hostsymbiont class, and vice versa, with respect to expectations under random association). Heterozygote disequilibria have the highest proportion of trajectories with at least one sign change (61%), followed by homozygote (53%) and allelic disequilibria (50%). The average number of sign changes in a trajectory is also highest for D_{12} (0.74), intermediate for the D_{ii} (0.56), and lowest for D (0.51), but is always <1. These differences across genotypes may reflect the fact that heterozygotes are generated from a higher number of matings and therefore are more sensitive to the symbiont's distribution across host genotypes (see discussion and appendix c).
De novo case—no initial disequilibria: The bearing of disequilibria on the study of hostsymbiont systems is perhaps best seen when the two species are initially randomly associated, because now all disequilibrium values and dynamics will be generated exclusively by the differential vertical transmission of the symbiont. The sections below highlight the ways in which the statistics for the de novo case differ from the general case. Note that there are no substantial differences in the statistics for the final disequilibria, since there is a unique equilibrium for each set of vertical transmission rates and host allele frequencies (See symbiont maintenance).
Likelihood, duration, and magnitude of measurable de novo associations: Now that we do not have any arbitrarily generated initial disequilibria, the proportion of measurable trajectories is naturally lower than in the general case, with the greatest reduction occurring when the symbiont is lost (Table 8). The differences between the two cases are caused solely by the transient trajectories, because, as mentioned above, the initial associations do not affect the final disequilibria. The most important discovery, however, is that measurable transient and permanent associations are both readily generated de novo by differential vertical transmission alone: this occurs 67% of the time for D, 63% of the time for D_{ii}, and 47% of the time for D_{12}.
Furthermore, de novo transient associations are measurable, on average, for a slightly higher number of generations than those generated by arbitrary initial conditions (9 generations for D_{12}, 14 for D, D_{ii} when
Firstgeneration de novo disequilibria: To more directly assess the power of differential vertical transmission to create hostsymbiont associations, we investigated the disequilibria generated de novo by a single generation of random mating and symbiont transmission. A substantial proportion of the initial de novo associations is measurable, with the fraction when the symbiont is maintained (47.6% for D_{ii}, 31.3% for D_{12}, and 50.4% for D) being slightly higher than when it is lost (45% for D_{ii}, 31.1% for D_{12}, and 48.2% for D). Moreover, the average initial magnitudes are measurable in all cases: 0.010 for D_{12} and 0.015 for the other disequilibria, with no appreciable differences when trajectories are partitioned according to symbiont fate. Interestingly, the maximum values of the first generation de novo disequilibria across all runs are equivalent to those along complete trajectories. This indicates that, in terms of magnitude, most of the de novo disequilibria are in fact created in the very first generation.
Sign changes: One of the greatest contrasts to the general case is that de novo disequilibria very rarely change sign, presumably because there is no need to counteract arbitrary initial conditions; ~90% of the time the system goes directly to an excess or deficit of a particular hostsymbiont combination and remains there. Across all trajectories, the proportion of de novo runs with a sign change is only 13.2% for D_{12}, and 8.4% for D_{ii} and D. The corresponding decimal values represent the average number of sign changes within a run.
Normalized disequilibria: Because the disequilibria are constrained by the marginal frequencies (Table 7), normalized values that take these constraints into account can provide further insight into the practical interpretation of observed hostsymbiont associations. The normalized disequilibrium d′ is obtained by dividing the observed disequilibrium d by the maximum possible magnitude for a disequilibrium of that sign (Lewontin 1964; Asmussen and Basten 1996), so that
Magnitude of normalized disequilibria: The average maximum magnitudes of normalized disequilibria along a trajectory (Table 9) show the same basic features as the actual disequilibria: (i) on average, these are slightly greater than the final normalized disequilibria, which indicates that normalized disequilibria also tend to decrease slightly in magnitude during their approach to equilibrium; (ii) trajectories where the symbiont survives have a greater average maximum magnitude than those where it is lost; and (iii) the normalized de novo magnitudes are 10–14% lower than the general ones, except for the de novo heterozygote association, which is 7–13% higher (this may stem from the different probability distributions of the initial variables in the two cases, which may have a greater effect on the heterozygote association because of its overall smaller magnitude).
Normalization accentuates the difference in the degree of association of the symbiont with the different host genotypes: the average maximum and average final magnitudes of the normalized heterozygote disequilibrium are only onethird those of the homozygotes vs. half for the actual disequilibria. For the general case, the average normalized values of equilibrium are 0.301 for
In Figure 3 we have plotted the distribution of the magnitude of the final normalized disequilibria when the symbiont is maintained. In >55% of such runs the normalized heterozygote values are in the lowest interval (0.0, 0.1); successively higher values are increasingly rarer. The normalized homozygote and allelic disequilibria also tend to have low values, although on average they are higher than the heterozygote association. In accordance with previous results, the allelic association is slightly more apt to be low and consequently less apt to be high relative to homozygote associations. Note that all four associations can be at or very near their maximum possible values for the corresponding host and symbiont frequencies in the population.
Normalized firstgeneration de novo disequilibria: The average magnitudes of the first generation normalized de novo disequilibria confirm that differential vertical transmission can generate substantial hostsymbiont associations in a single generation. Values are very similar whether the symbiont is ultimately lost or maintained, and the magnitude of
Sign statistics: Last, because the sign of a disequilibrium tells us if the symbiont is associated more or less often than expected with a particular host genotype or allele, we investigated whether the relevant statistics above differed according to the sign of the final disequilibrium. We do not make a distinction here between the general and the de novo cases because their results are very similar.
Likelihood of measurably positive and negative disequilibria: At equilibrium, measurable homozygote and allelic disequilibria are equally likely to be positive or negative; the heterozygote disequilibrium, however, is much more apt to be positive (61%) than negative (39%), which is consistent with heterozygotes having a greater probability of being symbiotic at equilibrium (see Prevapplence and distribution of the symbiont at equilibrium).
Magnitudes of positive and negative disequilibria: The average final magnitudes of the disequilibria (Table 10) do not present substantial differences by sign (0.02 for D^_{ii} and D^ and 0.01 for D^_{12} in both cases). On the other
hand, the maximum positive value of
Disequilibrium sign and relative transmission rates: Further insight is obtained by partitioning permanent measurable disequilibria according to sign, conditioned on symbiont survival and the relative transmission rate of the heterozygotes (Table 11). When β_{12} is maximal,
The allelic disequilibrium shows a roughly equal likelihood of being measurably positive or negative at equilibrium, whatever the transmission rate relationships. Permanent positive and negative associations between symbiont and host alleles are both most apt to be created when β_{12} is intermediate (38%), most probably because the symbiont is then systematically transmitted more often with the allele corresponding to the homozygote with the highest transmission rate. (For the same reason, measurable disequilibria occur much more often for homozygotes than for heterozygotes when β_{12} is intermediate, because the symbiont will then accumulate in the highest transmitting homozygote). When β_{12} is minimal the probability of having measurably positive or negative
Joint sign patterns: Only four joint sign patterns are possible for the final disequilibria under this model, and these are almost equally likely (Table 12). Within each sign pattern there are substantial differences in the proportion of runs in which the heterozygote trans
mission rate (β_{12}) is maximal, intermediate, or minimal. As expected, when
DISCUSSION
We have conducted an extensive analytical and numerical investigation of the effects of host genetic heterogeneity in the rate of symbiont transmission from parents to offspring. We have focused here on the baseline, deterministic formulation in which the differential vertical transmission rates are constant and are the sole force affecting the symbiont's survival, prevalence, and distribution across host genotypes. Our analysis has introduced the use of hostsymbiont disequilibria, which provides a convenient way to quantify nonrandom associations between two interacting species. To better understand the bearing of nonrandom hostsymbiont associations on the behavior of the system, we performed our disequilibrium analyses both with arbitrary initial conditions (general case) and with no initial disequilibria (i.e., D_{ij} ≡ D = 0) between the symbiont and host genotypes and alleles (de novo case). Because of the analogy to cytonuclear systems, estimates of observed hostsymbiont associations and their statistical significance can be calculated by following existing procedures for cytonuclear disequilibria (Asmussenet al. 1987; Asmussen and Basten 1994, 1996; Dean and Arnold 1996; Basten and Asmussen 1997).
Comparison between uniform and differential vertical transmission: Under our baseline model, we find critical differences between hostsymbiont systems in which vertical transmission rates vary or are equal across host genotypes. As compared to the uniform case, differential vertical transmission (i) increases the overall chances of symbiont survival from 50% to almost 60%, (ii) dramatically reduces the minimum average vertical transmission rate at which the symbiont can survive (from 0.5 to 0.008), and (iii) can create permanent hostsymbiont disequilibria de novo that are, practically speaking, the maximum values they can take, whereas uniform transmission can neither create nor maintain such associations.
A key consequence of points (i) and (ii) is that particular combinations of the transmission rates and host allele frequencies may permit the symbiont's survival in the differential case, even when the average vertical transmission rate across the host population (β) is very low. The symbiont can survive when one or two of the host genotypes do not effectively transmit the symbiont, as long as at least one other genotype transmits it at a high enough rate and is sufficiently common in the population. However, when there is a single uniform transmission rate for all host genotypes, this compensatory effect among the different host genotypes is not possible. If we view the host as “symbiont habitat” (May and Nowak 1994), habitat diversity therefore decreases the symbiont's overall chances of extinction. Accordingly, symbiont loss is more likely in the differential transmission case when the host population is near fixation (45% for host allele frequencies in 0.0–0.1 vs. 40–41% otherwise). Other theoretical studies have also found enhanced species survivorship in relation to habitat heterogeneity, which ties into diversity as a stabilizing ecological factor that can promote species diversity (MacArthur and MacArthur 1961; Yu 1972; Anderson and May 1978; Levinet al. 1984; Gupta and Hill 1995).
Hostsymbiont disequilibria: In relation to point (iii) above, hostsymbiont disequilibria provide a valuable indication of how diverse (or uniform) the host population is for the symbiont, because these measures reflect the extent to which the symbiont is associated more or less often with a given host genotype or allele than under random expectation. For both actual and normalized disequilibria, the average magnitudes along a trajectory tend to be greater, and the transient associations are measurable for a larger number of generations when the symbiont survives than when it is ultimately lost. These interspecific measures therefore have practical implications for the coevolution of the two species, because, on average, the symbiont is more unevenly distributed across host genotypes during trajectories where it is maintained (that is, there is greater habitat diversity for the symbiont). The maximum magnitudes of the disequilibria along a trajectory do not always follow this pattern, because they can be similar or even slightly higher when the symbiont is lost. This may be because very high disequilibria may occur under conditions that lead to symbiont extinction or simply because maximum values do not always reflect the overall pattern due to stochastic effects in the simulations.
The practical utility of hostsymbiont disequilibria ultimately depends upon the rate and magnitude at which they are generated. Our analytical investigation shows that differential vertical transmission virtually always creates disequilibria de novo. The true potential usefulness of these measures in empirical studies, however, is confirmed by our numerical demonstration that experimentally detectable hostsymbiont associations are readily generated and maintained. In simulations initialized with arbitrary conditions, measurable homozygote (D_{ii}) and allelic (D) disequilibria are produced >80% of the time, while measurable heterozygote disequilibria (D_{12}) are found approximately half the time. Even in the de novo case, where the symbiont is initially distributed at random in the host population, the proportion of measurable disequilibria is still high (67% for D, 63% for D_{ii}, and 47% for D_{12}). Furthermore, when the symbiont is maintained, approximately twothirds of the measurable homozygote and allelic trajectories and more than onethird of the measurable heterozygote trajectories have permanent, nonzero associations. The average final magnitudes are all measurable (0.021 for
With regard to the dynamic behavior of the disequilibria, the average maximum magnitude along a trajectory is higher than the average final magnitude for actual and normalized disequilibria for all initial conditions. This indicates that, on average, the symbiont gradually becomes more uniformly distributed across the host population, even when measurable associations are retained at equilibrium. In addition, disequilibrium trajectories frequently are not monotonic in this baseline model. This is of critical importance for empirical studies: disequilibrium trends (increasing or decreasing) are easily reversed even when only one force, differential vertical transmission, is driving the system. Most importantly, our study shows that substantial levels of permanent hostsymbiont disequilibria can be generated by differential vertical transmission alone in the absence of any symbiontinduced selection.
Role of heterozygotes vs. homozygotes: Symbiont prevalence and distribution across the different host genotypes is a consequence not only of the transmission rates but also of the differential impact of homozygotes and heterozygotes upon the transmission process. Heterozygotes can receive the symbiont from any of the three host genotypes, and they likewise can distribute it to all three, because they are produced by and can produce offspring with any of the three host genotypes. Homozygotes on the other hand are generated by and can produce only like homozygotes and heterozygotes; they cannot receive or give the symbiont to the other homozygote class. As a result, on average, (i) heterozygotes are slightly more likely to carry the symbiont at equilibrium (61%) than homozygotes (58.5%); (ii) the symbiont has a slightly greater probability of surviving when the vertical transmission rate is higher for heterozygotes (61.2%) than for homozygotes (57.7%); (iii) the heterozygote disequilibrium (D_{12}) has a lower magnitude than the other disequilibria, implying the symbiont is distributed closer to random expectation with regard to heterozygotes than to homozygotes or alleles; (iv) measurable heterozygote disequilibria are 56% more likely to be positive than negative at equilibrium, while homozygote and allelic disequilibria are equally likely to be of either sign.
Although heterozygotes facilitate symbiont maintenance more than homozygotes, symbiont survival involves a complicated interplay between host allele (and thus genotype) frequencies and the differential vertical transmission rates. For example, when the symbiont survives, the transmission rate of a homozygote is more likely to be maximal the higher the frequency of the corresponding allele (e.g., β_{11} and p for allele A_{1}). In the context of hostparasite systems, the relative frequencies of the host genotypes may therefore be important not only because of possible immunological differences in susceptibility to infection of homozygotes and heterozygotes (Hedrick and Thomson 1983; Hughes and Nei 1988; Van Der Looet al. 1991; McGuireet al. 1994) but also because of the inherent mathematical dynamics of the system.
Implications: The results from this baseline study of differential vertical transmission in hostsymbiont systems have both theoretical and practical implications and open the door to further research. For example, on the theoretical side our simulations suggest that substantial nonrandom associations between two different interacting species can occur in sympatry, due exclusively to simple evolutionary forces such as differential vertical transmission. Different coevolutionary tracks can ensue because of the different degree of association between host genotypes and the symbiont. This could ultimately lead to significant differentiation among sympatric groups of the same species, and possibly even to a speciation event in one or both species, as has been proposed in the case of the vertically transmitted endosymbiont Wolbachia (Price 1977; Rice 1984; Somersonet al. 1984; Breeuwer and Werren 1990; Jaenike 1993; CarlssonGraner 1997; Craiget al. 1997; Mauricio and Rausher 1997).
Interspecific disequilibrium measures such as the ones used here may prove useful in studies of many systems of interacting species. Similar disequilibria have already provided insight into the effects of Wolbachia infection on the dynamics of the nuclear genome of their Drosophila host (Turelliet al. 1992). In this same context, the nonrandom associations between Wolbachia and the different mitochondrial DNA haplotypes of D. simulans contain important information regarding the inheritance and spread of the bacteria in natural populations (Hale and Hoffman 1990; Turelliet al. 1992). Such nonrandom associations between two cytoplasmic elements could also be studied with disequilibrium measures analogous to the allelic disequilibrium described here (Schnabel and Asmussen 1989).
Other practical applications stem from our demonstration that vertical transmission can maintain the symbiont at a substantial frequency in the host population without the aid of other selective forces. This key finding suggests nonpathogenic parasite strains have the potential to be used in human, animal, and plant populations as attenuated and/or handicapped vaccines (Bertagnoliet al. 1996; Sánchezet al. 1997; Ewald 2000). This apparently beneficial application should be interpreted with caution, however, because our results also indicate that these strains could remain in the host population long enough for harmful pathogens to arise through mutations from these initially innocuous strains.
To get a more complete picture of hostsymbiont coevolution, we will need to include other factors at work in these systems, particularly symbiontinduced selection. We expect interspecific disequilibria will prove to be a valuable new tool in these future models, as well as in empirical studies where such nonrandom associations may play a key role in elucidating the coevolutionary processes at work.
Acknowledgments
We thank J. C. Avise, R. Dean, R. E. Fundyga, T. C. Gard, W. D. Hamilton, S. KresicJuric, D. E. L. Promislow, M. H. Smith, two anonymous reviewers, and members of the J. Arnold and M. A. Asmussen laboratories for helpful comments and discussion. We also acknowledge the rabbits of Navarra for providing the inspiration for this work. This work was funded in part by a Government of Navarra research grant (M.S.S.), a National Institutes of Health (NIH) training grant 5T32GM0710324 (M.S.S.), and NIH Grant GM48528 (M.A.A.).
APPENDIX A: HOSTSYMBIONT EQUILIBRIA
The equilibrium frequencies of the six hostsymbiont genotypic classes are
APPENDIX B: NUMERICAL METHODS
Each of the simulations in the numerical phase of our investigation comprises 10^{5} − 10^{6} runs, where each run has a different set of initial conditions and parameter values. We considered that equilibrium was reached when the sum of the absolute changes in the frequencies of the six hostsymbiont classes (Table 1) between two consecutive generations was <10^{−8}. The system reached an equilibrium in all simulations conducted, which occurred anywhere between 7 generations and >10^{5} generations. On average, equilibrium is attained faster when the symbiont ultimately goes extinct (273 generations for symbiont maintenance vs. 197 for symbiont loss). No cycling was observed. Vertical transmission rates, and in certain simulations the host allele (p) or symbiotic class (y) frequencies, were generated randomly from uniform distributions on [0,1]. In simulations started from arbitrary initial conditions, the hostsymbiont classes were generated via the brokenstick method, by selecting five random numbers to divide the interval [0,1] into six segments, whose lengths were then taken as the frequency of the six hostsymbiont classes (Karlin 1969, pp. 241–242). A similar brokenstick method was used to generate the marginal host genotype frequencies in simulations initiated with random hostsymbiont associations. All numerical simulations were performed in MATLAB.
APPENDIX C: MAXIMUM AND MINIMUM DISEQUILIBRIUM STATISTICS
To further characterize the dynamic and equilibrium behavior of the disequilibria, we calculated several maximum and minimum statistics across all runs. When appropriate, the values discussed below are given in Table C1.
Duration of transient disequilibria: The maximum number of generations disequilibria are measurable in transient trajectories is similar for the general and the de novo analyses. Even though homozygote and allelic disequilibria have similar average durations, surprisingly the maximum number of generations with nonrandom associations is consistently smaller for allelic disequilibrium vs. genotypic disequilibrium when the symbiont is lost. When the symbiont is maintained, the maximum number of generations allelic disequilibrium is measurable is intermediate between the values obtained for the heterozygote and homozygote disequilibria. The minimum number of generations with measurable disequilibria, calculated over all measurable trajectories, is 1 for all cases.
Magnitude of disequilibria: The maximum magnitudes of the disequilibria across all trajectories (
Sign changes: The heterozygote disequilibrium D_{12} is also distinctive in having twice the maximum number of sign changes in a trajectory as the other disequilibria (four vs. two in the general case and two vs. one in the de novo case).
Normalized disequilibria: No noticeable differences exist between trajectories classified according to their initial values (general or de novo) or symbiont fate (lost or maintained) except for
Normalized firstgeneration de novo disequilibria: The maximum magnitudes of the normalized disequilibria in the first generation are almost 1, except for
APPENDIX D: EFFECTS OF INITIAL DISEQUILIBRIA IN THE GENERAL CASE
The effects of the initial (generation zero) disequilibria on the frequency of measurable, transient, and permanent disequilibria are seen by repeating our analyses of the general case with the random initial generation included. Both measurable and transient trajectories are now more frequent, and measurable heterozygote associations occur nearly (rather than half) as often as allelic and homozygote associations (Table D1). The effects of the initial disequilibria are more pronounced in the case of transient D_{12} trajectories, which occur at a substantially higher frequency than for the other disequilibria. The average maximum magnitudes of the disequilibria along a complete trajectory are also two to four times higher when we include the initial generation, although the most striking observation is that the heterozygote association is virtually the same as the other three (0.06 over all trajectories), whereas in the previous analysis its average magnitude statistics are approximately half those of the other disequilibria.
In consequence, on average, the magnitudes of the disequilibria are greater and more similar to each other when they are created arbitrarily than when they are generated solely through the vertical transmission process. In addition, D_{12} is more susceptible to this initial effect, because under the dynamics of the model this association tends to have a smaller magnitude than the others. Moreover, results show that the pattern in which D_{12} has half the magnitude of the other disequilibria is becoming established in only one generation of random mating and symbiont transmission. Last, there is an interesting common feature that holds whether or not the initial generation is included in the analysis: disequilibrium trajectories with permanent symbiosis have a slightly larger average maximum magnitude than those where the symbiont is lost.
Footnotes

Communicating editor: M. W. Feldman
 Received November 12, 1998.
 Accepted November 2, 1999.
 Copyright © 2000 by the Genetics Society of America