Symbiont Survival and Host-Symbiont Disequilibria Under Differential Vertical Transmission

Corresponding author: María S. Sánchez, Department of Environmental Science and Policy, University of California, Davis, CA 95616., mssanchez{at}ucdavis.edu (E-mail)

Communicating editor: M. W. FELDMAN

	ABSTRACT

TOP ABSTRACT THE HOST-SYMBIONT MODEL SYMBIONT MAINTENANCE HOST-SYMBIONT DISEQUILIBRIA DETECTABILITY OF DISEQUILIBRIA DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D LITERATURE CITED

Interspecific genetic interactions in host-symbiont 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 host-symbiont 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 ; REISSER et al. 1982 ; VAN DER LOO et al. 1987 ; SLADE 1992 ; FRANK 1993 ; RAUTIAN et al. 1993 ; ANTONOVICS and THRALL 1994 ; GUPTA et 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 (ARILLO et al. 1993 ; HEDDI et al. 1993 ; DUNN et al. 1995 ; WILKINSON and DOUGLAS 1995 ), genetics (WEIS and LEVINE 1996 ; CHARLES et al. 1997 ; CHUNG et al. 1997 ; ESCALANTE et al. 1998 ), and phylogenetic (YAN et al. 1994 ; ESCALANTE and AYALA 1995 ; DIAZ 1997 ; LUTZONI and PAGEL 1997 ; MORZUNOV et al. 1998 ) relationships between host and symbiont (SPENCER 1988 ; MULVEY et al. 1991 ; GRAF and RUBY 1998 ). For example, the strong host genotype-by-strain interactions between legumes and their symbiotic bacteria may permit the host's genetic diversity to maintain genetic polymorphism in the symbiont (SPOERKE et al. 1996 ; LIEVEN-ANTONIOU 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 (FAVRE et al. 1998 ; RAMOZ et al. 1999 ), and the human immunodeficiency virus, HIV (FAUCI 1996 ; ROWLAND-JONES 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 ; GRENIER et al. 1997 ; SAEZ NIETO and VAZQUEZ 1997 ). These policies may be more effective if population heterogeneity is considered (TIBAYRENC et al. 1990 ; ANDERSON and MAY 1991 ; CHABORA and KOEPFER 1991 ; EWALD 1994 ; GUPTA and HILL 1995 ; RILEY 1996 ; B. R. LEVIN et al. 1997 ; S. A. LEVIN et al. 1997 ; LIPSITCH 1997 ; WEATHERALL et al. 1997 ; SCHORK et al. 1998 ).

Here we provide an extensive theoretical investigation of two new issues regarding host-symbiont 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 ; LIPSITCH et al. 1996 ). This addresses an important theoretical gap that has been pointed out by previous authors in the host-parasite 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 host-parasite systems with multiple parasite types (SANCHEZ et al. 1997 ) and resemble the gene-culture disequilibrium used to measure associations between genes and vertically inherited cultural traits in studies of gene-culture coevolution (FELDMAN and CAVALLI-SFORZA 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 (ASMUSSEN et al. 1987 ; ASMUSSEN and BASTEN 1994 ) or cytoplasmically inherited microorganisms (TURELLI et al. 1992 ; DEAN and ARNOLD 1998 ).

As with the cytonuclear disequilibria on which they are based, host-symbiont associations may allow us to draw evolutionary inferences that are difficult to obtain by other means (ASMUSSEN et al. 1989 ; SITES et al. 1996 ; VAN DER LOO et al. 1997 ; GOODISMAN et al. 1998 ). For example, in host-parasite 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 (WAYNE et 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 host-symbiont 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 symbiont-induced 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 ; LIPSITCH et 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 host-parasite systems when pathogen virulence and transmissibility are independent (FRANK 1996 ; LIPSITCH and MOXON 1997 ). The survival and dynamics of nonpathogenic parasite strains (CAPUCCI et 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 discrete-time 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 host-symbiont frequencies and disequilibria under differential vertical transmission can improve our understanding of the evolution of host-symbiont systems, for both theoretical and practical reasons.

	THE HOST-SYMBIONT MODEL

TOP ABSTRACT THE HOST-SYMBIONT MODEL SYMBIONT MAINTENANCE HOST-SYMBIONT DISEQUILIBRIA DETECTABILITY OF DISEQUILIBRIA DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D LITERATURE CITED

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 A₁ and A₂) 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 Table 1 and Table 2, respectively. Adding down columns provides the corresponding marginal frequencies of the host genotypes (g₁₁, g₁₂, g₂₂) 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 ß₁₁, ß₁₂, ß₂₂ (after ANDERSON and MAY 1979 ), where ß_ij is the probability that an A_iA_j/S⁺ individual transmits the symbiont to its offspring. Under uniform vertical transmission ß₁₁ = ß₁₂ = ß₂₂ = ß, 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 host-symbiont 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_iA_j/S⁺ individual, the offspring is S⁺ with probability ß_ij and S^- with probability 1 - ß_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 ß_ij(1 - ß_ij) + (1 - ß_ij) ß_ij + ß²_ij = ß_ij(2 - ß_ij) when both parents have the same genotype (A_iA_j) and with probability ß_ij(1 - ß_km) + (1 - ß_ij) ß_km + ß_ijß_km = ß_ij + ß_km - ß_ijß_km when the parental hosts have different genotypes (A_iA_j and A_kA_m). In each case, the probability of an S^- progeny is simply the product of the probabilities that neither parent transmits the symbiont.

Host-symbiont dynamics:
Our analytical results show that the frequency of the host-symbiont 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,

(1)

where F_i is the probability that a host transmits an A_i allele together with the symbiont. The exact change per generation in the frequency of the host genotype-symbiont classes is given by the recursions

(2)

where a prime (') denotes the value of the variable in the next generation. Note that the behavior of the system depends on both the absolute and the relative values of the vertical transmission rates. The corresponding recursions for the change in frequency of the two symbiont classes are

(3)

where F = F₁ + F₂. Because the host is not under any kind of selection pressure, host allele frequencies do not change (p' = p, q' = q), and host genotype frequencies reach Hardy-Weinberg equilibrium (g^'₁₁ = p², g^'₁₂ = 2pq, g^'₂₂ = q²) in one generation.

	SYMBIONT MAINTENANCE

TOP ABSTRACT THE HOST-SYMBIONT MODEL SYMBIONT MAINTENANCE HOST-SYMBIONT DISEQUILIBRIA DETECTABILITY OF DISEQUILIBRIA DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D LITERATURE CITED

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 ₁₁ = ₁₂ = ₂₂ = 0). This approach is analogous to analyzing the maintenance of alleles or species via a protected polymorphism (LEVENE 1953 ; PROUT 1968 ; LEON 1974 ; MAYNARD SMITH 1974 ). If at least one eigenvalue of the local stability matrix at this boundary equilibrium is >1 in magnitude, the equilibrium is locally unstable and the symbiont will never be lost, because whenever the system is close to this equilibrium it will move away from it. These two eigenvalues (₁, ₂) satisfy the characteristic equation

(4)

which is a complicated quadratic of the form ² - A + B = 0. Because A > 0, the roots of this equation are both <1 in magnitude if and only if A < 1 + B < 2 (GOLDBERG 1958 ; MAY et al. 1974 ). The symbiont will thus be maintained if either inequality is reversed (A - B > 1 or B > 1).

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 (ß_ij). Further numerical investigations revealed that the single condition A - B > 1, corresponding to

(5)

is both necessary and sufficient for maintenance of the symbiont, where we considered the symbiont to be lost when its frequency was <10^-8. The symbiont's survival is thus determined exclusively by its vertical transmission rates in the three host genotypes and the allele frequencies in the host. These same factors uniquely determine the equilibrium of the full system (Appendix A), which in consequence is independent of the symbiont's initial frequency and distribution across host genotypes.

Prevalence and distribution of the symbiont at equilibrium:
On average, 60% of host individuals are symbiotic at equilibrium, which in the host-parasite 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 ₁₂/₁₂ is 0.610 vs. 0.585 for _ii/_ii in trajectories where the symbiont is maintained). In addition, all members of a host genotype class can eventually carry the symbiont, because the maximum equilibrium frequency observed for each symbiotic class (_ij) is very close to the maximum possible frequency for the corresponding host genotype (1.0 for homozygotes and 0.5 for heterozygotes).

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 ; SANCHEZ et al. 1997 ). The conditions on the average vertical transmission rate in the population, once Hardy-Weinberg equilibrium is reached in the first generation,

(6)

can help us understand why. The symbiont will be maintained under uniform transmission if and only if = ß > 0.5, whereas under differential vertical transmission > 0.5 is sufficient but not necessary for symbiont survival. In fact, when the symbiont was maintained, >16% of the time < 0.5, and the minimum observed was 0.008 (this occurred when ß₁₁ = 0.224, ß₁₂ = 0.999, ß₂₂ = 0.002, p = 0.003, and q = 0.997). Symbiont survival places no restriction on the cross products of the vertical transmission rates in (5), because the symbiont was maintained for values of ß_ijß_km ranging from 0.0 to 0.99.

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%); ß₁₂ 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 (ß > 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 A₁ (p), the transmission rate of A₁A₁ homozygotes (ß₁₁) is more likely to be maximal when the symbiont is retained, and likewise for A₂ and ß₂₂ (Table 4). For instance, the proportion of time that ß₁₁ 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 ß₁₂ 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 ß₁₂ > ß₂₂ > ß₁₁, where ß₁₂ is maximal and ß₁₁ is minimal, has a higher frequency (15.6%) than the case ß₂₂ > ß₁₁ > ß₁₂ (11.3%), where ß₁₁ is intermediate and ß₁₂ is minimal. The probability ß₁₂ is maximal when the symbiont survives is greater the smaller the average transmission rate (e.g., ß₁₂ is maximal 70% of the time when 0.2 < < 0.3 and 90% of the time when < 0.2); this probably occurs because at lower the chance of symbiont extinction is higher, so the advantage heterozygotes have in maintaining the symbiont now plays a greater role.

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 (|p² - 2pq| + |2pq - q²|) appears to give the symbiont a marginal survival advantage.

	HOST-SYMBIONT DISEQUILIBRIA

TOP ABSTRACT THE HOST-SYMBIONT MODEL SYMBIONT MAINTENANCE HOST-SYMBIONT DISEQUILIBRIA DETECTABILITY OF DISEQUILIBRIA DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D LITERATURE CITED

Nonrandom associations in a host-symbiont system can arise at the level of both host genotypes and alleles. By analogy to cytonuclear systems (ASMUSSEN et al. 1987 ), we define three host-symbiont genotypic disequilibria,

(7)

which measure the departure from random association between the symbiont and each of the host genotypes. The host-symbiont allelic disequilibrium,

(8)

similarly quantifies nonrandom associations between the symbiont and the host alleles.

The frequency of each host-symbiont class can be decomposed in terms of the appropriate host-symbiont disequilibrium and marginal frequencies as shown in Table 5 and Table 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 A₁ allele, and a negative value indicates that it is associated more often with the A₂ allele, as compared to random expectation.

The four host-symbiont disequilibria are related by

(9)

Although these interrelationships reduce the four disequilibria to two independent measures, we analyze all four because, as for cytonuclear associations (ASMUSSEN et al. 1989 ; SITES et al. 1996 ; GOODISMAN and ASMUSSEN 1997 ; BABCOCK and ASMUSSEN 1998 ) the joint sign patterns may provide valuable biological inferences. All the host-symbiont disequilibria are constrained by maximum and minimum values determined by the marginal frequencies of the host and symbiont classes (Table 7). These bounds are derived from the parameterizations in Table 5 and Table 6 and are analogous to those obtained by ASMUSSEN and BASTEN 1996 for cytonuclear disequilibria.

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 Table 5 and Table 6. This yields the compact equations

(10)

where F = F₁ + F₂, and F₁ and F₂ are defined in (1). The final values of the disequilibria (₁₁, ₁₂, ₂₂, ) are obtained by substituting the equilibrium values for ₁ and ₂ from Appendix A into the right-hand side of (10).

De novo disequilibria:
The most important result from (10) is that differential vertical transmission can by itself generate host-symbiont disequilibria de novo starting from a state with no initial disequilibria between host and symbiont (D⁽⁰⁾₁₁ = D⁽⁰⁾₁₂ =D⁽⁰⁾₂₂ = D⁽⁰⁾ = 0) in the initial generation). Using the relations in Table 5, we find from (1) and (10) that the de novo disequilibria after one generation are

(11)

Here

is the average vertical transmission rate of the symbiont in generation 0; this equals the sum of the two quantities

where g⁽⁰⁾_ij is the initial frequency of A_iA_j. These equations apply whatever the initial host frequencies and show that disequilibria cannot be generated de novo by uniform vertical transmission (where all ß_ij ß), because then = ß, ₁ = pß, and ₂ = qß, yielding D⁽¹⁾_ij D⁽¹⁾ = 0.

Further insight is possible when the host is initially at Hardy-Weinberg equilibrium (p², 2pq, q²), which in this model is reached within one generation. The initial de novo disequilibria in (11) then reduce to

(12)

where

(13)

are the marginal transmission rates of the two alleles. These satisfy

(14)

where is the average vertical transmission rate when the host is at Hardy-Weinberg frequencies, as defined in (6).

Ignoring the trivial case where ß_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 (_{1 2}). This is a necessary but not sufficient condition for the generation of heterozygote disequilibrium de novo, because D⁽¹⁾₁₂ could be 0 when _{1 2} in the special case where p(1 - ₁py) = q(1 - ₂qy) or, equivalently, (1 - ₁y)(1 - ₂y) = (1 - y)². Together these results show that if the host is a Hardy-Weinberg equilibrium, differential vertical transmission will always generate host-symbiont disequilibria de novo, except in the unlikely event that the host allele frequency has the value

and either ß₁₂ > ß₁₁, ß₂₂ or ß₁₂ < ß₁₁, ß₂₂.

The initial dynamics of the symbiont are also especially straightforward if the host is in Hardy-Weinberg 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 y* = ; this is possible only when the average transmission rate () exceeds 0.5. This threshold frequency of the symbiont class corresponds to the equilibrium frequency of the symbiont under uniform vertical transmission (SANCHEZ et al. 1997 ).

	DETECTABILITY OF DISEQUILIBRIA

TOP ABSTRACT THE HOST-SYMBIONT MODEL SYMBIONT MAINTENANCE HOST-SYMBIONT DISEQUILIBRIA DETECTABILITY OF DISEQUILIBRIA DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D LITERATURE CITED

One of our primary goals is to show that host-symbiont 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 ; SANCHEZ et 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 , BABCOCK and ASMUSSEN 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 host-symbiont 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 (D₁₂) 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 D₁₂), and particularly so when the symbiont is maintained (72% for D_ii and D, 60% for D₁₂), because all trajectories that lose the symbiont are necessarily transient.

View this table: In this window In a new window   Table 8. Frequency of measurable, transient and permanent disequilibria over all trajectories and those with ( > 0) or without ( = 0) the symbiont at equilibrium

Transient disequilibria tend to be fairly short lived, being measurable for an average of 5 (D₁₂) to 8 (D_ii, D) generations when the symbiont is maintained and 4 (D₁₂) 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 D₁₂, 368 for D, and 520 for D_ii when > 0 and go down to 78, 42, and 84, respectively, when = 0 (Appendix C).

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 D₁₂ 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 (Fig 1), because it involves a complicated interaction between the transmission rates and the frequencies of the host-symbiont categories.

View larger version (14K): In this window In a new window Download PPT slide   Figure 1. Transient host-symbiont disequilibria for a population with initially nonrandom associations, where the symbiont is ultimately lost. Vertical transmission rates: ß11 = 0.609, ß12 = 0.016, ß22 = 0.016. Initial host-symbiont frequencies: u11 = 0.058, u12 = 0.133, u22 = 0.178, y11 = 0.219, y12 = 0.045, y22 = 0.369.

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 host-symbiont 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 D₁₂ (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 host-symbiont 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 D₁₂.

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 D₁₂, 14 for D, D_ii when > 0, and 5 and 6 generations, respectively, when = 0). The maximum number of measurable generations under de novo conditions is fairly similar to the general case for all disequilibria (391 for D₁₂, 430 for D, and 497 for D_ii when > 0, and 98, 45, and 106, respectively, when = 0; Appendix C). The magnitudes of the disequilibria along de novo trajectories are also surprisingly close to the general case (Table 9 and Appendix C). For instance, the average maximum magnitudes are all measurable and only slightly smaller than before. An example illustrating the substantial disequilibria that can be generated de novo is provided in Fig 2.

View larger version (15K): In this window In a new window Download PPT slide   Figure 2. Permanent host-symbiont disequilibria generated de novo from an initial state of no disequilibrium between host and symbiont. Vertical transmission rates: ß11 = 0.846, ß12 = 0.525, ß22 = 0.203. Initial host-symbiont frequencies: u11 = 0.659, u12 = 0.163, u22 = 0.159, y11 = 0.013, y12 = 0.003, y22 = 0.003.

First-generation de novo disequilibria: To more directly assess the power of differential vertical transmission to create host-symbiont 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 D₁₂, and 50.4% for D) being slightly higher than when it is lost (45% for D_ii, 31.1% for D₁₂, and 48.2% for D). Moreover, the average initial magnitudes are measurable in all cases: 0.010 for D₁₂ 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 host-symbiont combination and remains there. Across all trajectories, the proportion of de novo runs with a sign change is only 13.2% for D₁₂, 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 host-symbiont 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

where the minimum and maximum disequilibria are defined in Table 7. Despite minor differences, the same conclusions are derived from both the general and the de novo analyses.

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 one-third those of the homozygotes vs. half for the actual disequilibria. For the general case, the average normalized values of equilibrium are 0.301 for ^'_ii, 0.105 for ^'₁₂, and 0.253 for '. Interestingly, the normalized de novo values are all slightly higher (0.312 for ^'_ii, 0.127 for ^'₁₂, and 0.261 for '); the different initial probability distributions are again the most likely cause.

In Fig 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.

View larger version (26K): In this window In a new window Download PPT slide   Figure 3. Distribution of the magnitude of the final normalized disequilibria when the symbiont is maintained.

Normalized first-generation de novo disequilibria: The average magnitudes of the first generation normalized de novo disequilibria confirm that differential vertical transmission can generate substantial host-symbiont associations in a single generation. Values are very similar whether the symbiont is ultimately lost or maintained, and the magnitude of D^'₁₂ is again approximately half that of the other disequilibria (0.21 for D^'_ii, 0.08–0.09 for D^'₁₂, and 0.17 for D').

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 _ii and and 0.01 for ₁₂ in both cases). On the other hand, the maximum positive value of _ii across all runs is 37% smaller in magnitude than its minimum negative value (0.086 vs. -0.136), while the minimum negative ₁₂ is 63% smaller than its maximum positive value (0.123 vs. -0.045); shows no sign effect (0.078 vs. -0.078). It is interesting that even though ₁₂ has the smallest average magnitude of the three disequilibria, it has the highest maximum positive magnitude.

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 ß₁₂ is maximal, ₁₂ is measurably negative only 0.2% of the time and is measurably positive 34.6% of the time; on the other hand, when ß₁₂ is minimal these values reverse to 41.2 and 0.7%, respectively. Differences between positive and negative values for the homozygote associations are not as pronounced as for ₁₂. When ß₁₂ is minimal, _ii is more apt to be measurably positive (34.8%) than negative (27.3%); the opposite occurs when ß₁₂ is maximal (20.5 vs. 28.8%, respectively). When ß₁₂ is intermediate, measurably positive and negative _ii occur at similar high frequencies (37 and 35.9%, respectively). The significance of the sign of the disequilibria is apparent here because, even though the final homozygote disequilibria are, on average, more apt to be measurable than the final heterozygote associations, this is not always the case when partitioned according to sign.

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 ß₁₂ 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 ß₁₂ is intermediate, because the symbiont will then accumulate in the highest transmitting homozygote). When ß₁₂ is minimal the probability of having measurably positive or negative decreases slightly (33%) and is smallest when ß₁₂ is maximal (27%).

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 transmission rate (ß₁₂) is maximal, intermediate, or minimal. As expected, when ₁₂ is positive, ß₁₂ is most apt to be maximal (55% of the time) and least apt to be minimal (10% of the time); when ₁₂ is negative this relationship is reversed. For all sign patterns, the frequency of cases with an intermediate heterozygote rate (ß₁₁ > ß₁₂ > ß₂₂ or ß₁₁ < ß₁₂ < ß₂₂) is intermediate.

	DISCUSSION

TOP ABSTRACT THE HOST-SYMBIONT MODEL SYMBIONT MAINTENANCE HOST-SYMBIONT DISEQUILIBRIA DETECTABILITY OF DISEQUILIBRIA DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D LITERATURE CITED

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 host-symbiont disequilibria, which provides a convenient way to quantify nonrandom associations between two interacting species. To better understand the bearing of nonrandom host-symbiont 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 host-symbiont associations and their statistical significance can be calculated by following existing procedures for cytonuclear disequilibria (ASMUSSEN et al. 1987 ; ASMUSSEN and BASTEN 1994 , ASMUSSEN and BASTEN 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 host-symbiont 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 host-symbiont 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 () is very low. The symbiont can survive when