# Deterministic models.

Somatic incompatibility is known to exist in animals, protists, and fungi (Buss 1982; Grosberg 1988). It is expressed by an incompatibility reaction following tissue contact between genetically different conspecifics and is characterized by a high degree of polymorphism. Well-known examples are vertebrate immune systems, like HLA in man (Klitz et al. 1992), and allorecognition systems in invertebrates, like the colonial ascidian Botryllus schlosseri (Oka and Watanabe 1957; Scofield et al. 1982). In fungi, somatic incompatibility is also very common, and is most often called vegetative incompatibility, although the terms somatic, protoplasmic, or heterokaryon incompatibility are also used.The model described attempts to explain the evolution of polymorphism in vegetative incompatibility, as found in a large group of fungi, the filamentous ascomycetes. We model those species that are obligately asexual, the so-called imperfect fungi (e.g., Aspergillus niger). These are haploid fungi, growing as individual mycelia that develop from spores, the conidia. A mycelium consists of hyphae, vegetative filaments that are constructed of separated segments, which usually contain several nuclei each. During growth, the mycelia may encounter conspecifics. If the two conspecifics are vegetatively compatible, they will fuse by a process called anastomosis, and form a heterokaryon.

Heterokaryon formation, the formation of a coenocytic state where two (or even more) genetically different haploid nuclei are present, is typical for the fungi. Its occurrence has been considered as evolutionarily beneficial: because two different haploid nuclei are combined, the organism can gain from some kind of heterosis. Also, the parasexual cycle, in which recombination occurs without meiosis, can function as an alternative for sex in imperfect species (Pontecorvo 1946; Snyder 1961; Davis 1966; Caten 1987). However, heterokaryons are rarely isolated from nature (Caten and Jinks 1966), and the formation of heterokaryons between different wild isolates of a species is normally restricted. This is caused by vegetative incompatibility, which prevents the formation of heterokaryons in nature.

Many examples of vegetative incompatibility in filamentous ascomycetes are known. In many species, a large number of vegetative compatibility groups (VCGs) occur ([ILLUSTRATION OMITTED]; also see Carlile 1987). Within such groups heterokaryon formation and the exchange of genetic material are possible, but not between groups. In the species studied so far, the incompatibility reactions are always mediated by many nuclear loci (Puhalla and Spieth 1985), mostly with two and occasionally with multiple alleles. Since one allelic difference can cause incompatibility, large numbers of VCGs can exist.

If the function of vegetative incompatibility is [TABULAR DATA OMITTED] to prevent the formation of heterokaryons, the widespread occurrence of vegetative incompatibility suggests that the disadvantages of heterokaryon formation will, on the average, be greater than the advantages. The precise selective mechanisms that favor vegetative incompatibility, however, are still unknown. Protection of the genetic integrity of the individual seems the most plausible (Todd and Rayner 1980). This can be interpreted as the prevention of a conflict between two different nuclear genomes in a heterokaryon (Hartl et al. 1975), and the prevention of an invasion by harmful cytoplasmic elements, such as plasmids, viruses, mitochondria (Day 1968; Caten 1972). Alternatively, Davis (1966) and Bernet (1992) proposed that the incompatibility reaction is an unselected by-product of genes regulating cell death and a consequence of genetic divergence of natural isolates. Finally, Esser and Blaich (1973) and Boucherie and Bernet (1980) consider the restriction of outbreeding as an explanation for the evolution of vegetative incompatibility.

Hartl et al. (1975) first modeled the evolution of vegetative incompatibility in fungi. They concluded that a parasitic nuclear gene can favor the evolution of vegetative incompatibility in Neurospora crassa. However, they limited their analysis to the coexistence of only two VCGs, far fewer than probably exist in most natural populations. Here we extend their model in two key ways. First, we increase the number of possible VCGs and vary the number of potential fusing partners. Second, we analyze whether harmful cytoplasmic elements (e.g., plasmids, viruses, mitochondria), instead of parasitic nuclear genes, can promote the evolution of vegetative incompatibility.

In these deterministic models, assuming infinite population sizes, we analyze the question of why so many incompatibility types have evolved and persist. In particular, we investigate whether parasitic nuclear genes or harmful cytoplasmic elements can lead to the evolution of many VCGs, as found in nature. Furthermore, we explore the fate of mutants that are Heterokaryon Self-Incompatible or Omnicompatible. Self-incompatible strains are found in a few ascomycete species. They are incompatible with every other strain, even if they are clonally related. Omnicompatible strains are assumed to be compatible to all other strains. No such strains are known to exist, and their existence is questionable. Still, their dynamics are studied for heuristic reasons.

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The organism whose biology we model would be like an imperfect soil fungus such as Aspergillus niger. In this case, recombination between incompatibility genes is impossible, because heterokaryon formation between VCGs is impossible by definition. This frees the model from dealing with the number of loci, number of alleles per locus, and linkage. It also means that new VCGs arise by mutation only. During its life cycle this organism grows from a germinating spore, forms a mycelium, interacts with conspecifics that it encounters, and sporulates.

We compare the results of this analysis with the evolution of VCG diversity in a sexual species in the discussion.

The Model of Hartl et al. 1975

Hartl et al. (1975) considered a nuclear gene in Neurospora crassa, with a pair of alleles, H and h. Homokaryons with only h nuclei have a fitness disadvantage compared to homokaryons with only H nuclei. In a heterokaryon, however, h nuclei have a proliferative advantage at the cost of H nuclei. The h allele exploits H in a heterokaryon and can therefore be regarded as parasitic. (It differs essentially from a recessive deleterious allele with heterozygotic advantage, because the advantage applies only to the [haploid] h genotype and not the [diploid, heterozygous] Hh genotype). An actual gene pair in Neurospora crassa with properties similar to those postulated here has been reported by Pittenger and Brawner (1961).

Hartl et al. (1975) assume only vegetative reproduction in discrete nonoverlapping generations and random dispersion of conidia in infinite populations. The conidia germinate and develop into new mycelia. The mycelia meet in pairs, and if they are compatible, they fuse to form a heterokaryon. If they are incompatible, they survive normally and grow side by side as two homokaryons. Then they produce conidia again.

In the model, a homokaryon with only (parasitic) h nuclei is assumed to have a fitness 1 - s (0 [less than] s [less than] 1), whereas h + H heterokaryons and H homokaryons have a fitness I. In a H + h heterokaryon, the h nucleus has a proliferative advantage over the H nucleus, so that the ratio of h:H conidia from such heterokaryons is [Theta]:1 - [Theta], where 0.5 [less than] [Theta] [less than] 1.

Let x be the frequency of h, and y the frequency of H (thus x + y = 1) at the moment of germination of the conidia that grow out to form mycelia. Analysis of the recursion relations describing the dynamics of the model - which can easily be recovered from system (1) given below, by putting i = 1, 2 - shows that there is a stable equilibrium x = (2[Theta] - 1)/s if s [greater than] 20 - 1, or x = 1 if s [less than] 20 - 1. This means h can always invade the population. Consequently, the presence of h will lower the mean fitness of that population.

Next, suppose that there is a gene for vegetative incompatibility with two alleles, [c.sub.1] and [c.sub.2], creating two VCGs which are incompatible. Then, in a population with either H or h nuclei (but not both), vegetative incompatibility is neutral. But in a population with four types H[c.sub.1], H[c.sub.2], h[c.sub.1], and h[c.sub.2], selection will operate. Such a population will result from an initial monomorphic state after (repeated) mutations to all types. Depending on the values of the parameters s and [Theta], a population polymorphic for all four types will evolve toward one of three different compositions.

(1) 20 - 1 [less than] s. - The population will end up polymorphic for H[c.sub.1] and H[c.sub.2]. Because of the effects of vegetative incompatibility, the disadvantageous parasitic h gene disappears.

(2) s [less than] 2[Theta] - 1 [less than] 2s. - The population will become polymorphic for [c.sub.1], [c.sub.2], h, and H. The gene frequencies keep changing indefinitely.

(3) 2s [less than] 2[Theta] - 1. - The population will become polymorphic for h[c.sub.1] and h[c.sub.2]. The nuclear exploitation of h is too strong to be overcome by the effects of vegetative incompatibility.

Although Hartl et al. (1975) could not find any limit cycle in situation (2), our iterations show that limit cycles do occur. In some cases, this limit cycle passes through values very close to zero, which implies extinction in a finite population. A new mutation to the extinct type, however, will be able to invade the population again.

The main conclusion of Hartl et al. (1975) was that if s [greater than] 2[Theta] - 1 the disadvantageous allele h will be driven out of the population by vegetative incompatibility. Polymorphism in VCGs could be explained by the action of a parasitic nuclear gene, like h.

Some of the assumptions of Hartl et al.'s (1975) study require further discussion. First, fusion of the mycelia in pairs is highly unlikely; heterokaryon formation is not a sexual process. The occurrence of mycelial contact will depend on the density of the population. Some mycelia will never encounter a partner, others will meet one, two, three, or even more partners. The model we develop here explicitly considers variation in the probability of encounters between mycelia.

Second, their model assumed that some recombination occurs between the h/H and the c loci. In Neurospora crassa (for which the model was made) this can take place after a (rare) sexual cross, but in imperfect species this can never happen. In the latter case, a polymorphism with four types - situation (2) above - can only occur after two independent mutations, both to a new VCG and H. These mutations have to occur nearly simultaneously, because a situation with only three types is always unstable.

Finally, the Hartl et al. (1975) model only considers situations with one or two VCGs. Since there are always many more VCGs in natural populations, questions remain as to the dynamics of these VCGs and the number of VCGs that can be maintained. Our models, as described in the next sections, will address these questions.

An Extended Model for a Parasitic Nuclear Gene

If vegetative incompatibility has any adaptive significance, contact between mycelia must occur regularly. This contact will take place when two or more mycelia develop and meet. We incorporate encounter possibilities into the model by assuming that each individual mycelium can potentially have n neighbors, that is, there are n neighboring sites for each individual (n = 1, 2, 3 . . .; the Hartl et al. model has n = 1 [meeting in pairs]; n = 4 means each individual can have four neighbors, as if it lived on a chessboard.) Since not every site will be occupied (by a conspecific), a fraction p of all these sites is empty (i.e., each site has a probability P to be unoccupied; P = 0 in the Hartl et al model). For simplicity, the model further assumes that all individuals and empty sites are randomly distributed in space.

Since many VCGs (denoted [c.sub.1]) coexist in nature, the model allows mutations to new VCGs. Each mutation is considered a mutation to a novel VCG.

In some species, Heterokaryon Self-Incompatible strains exist. These strains are incompatible with every other strain they meet, even with clonally related (and thus genetically identical) strains. The fate of this type of individual ([c.sub.SI]) is also considered. It is assumed to have a lowered fitness value [f.sub.SI]([f.sub.SI] [less than] 1), because the mycelium of a Heterokaryon Self-Incompatible strain will not be able to form anastomoses (i.e., to fuse with itself within the mycelium), potentially reducing growth (Correll et al. 1989; A. J. M. Debets pers. comm. 1992).

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Finally, the model analyzes an Omnicompatible type in which individuals are compatible with all others (except Heterokaryon Self-Incompatibles). Such a type has not been reported in the literature. Its existence is doubtful since the underlying assumption is that it is physiologically possible for one genotype to determine the course of events in every interaction with other genotypes. (To some extent the Omnicompatible type can be compared with the "belligerent" type, as described by Grosberg and Quinn [1989], in a model on allorecognition specificity in sessile cnidarians. Like the Omnicompatible type, the aggressive behavior of the belligerent type is unaffected by the genotype it encounters.) Nonetheless, for heuristic reasons, we examine whether and when such a type could invade a population, were it to arise by mutation.

Based on these assumptions, we derive a set of recursion relations that relate the frequencies of all genotypes between consecutive (discrete, nonoverlapping) generations. The derivations and the general form of these equations are given in Appendix B.

We study the evolutionary dynamics of this set of equations under different parameter values, and we analyze the fate of mutations to new VCGs by introducing new VCGs in low frequency. We assess if and how a population will evolve towards a stationary state and how many VCGs will exist at equilibrium.

The Simplest Case. - The first case with n = 1 and P = 0, without Heterokaryon Self-Incompatibles and Omnicompatibles, differs from the Hartl et al. (1975) model only in that more than two VCGs are allowed.

System (B2) becomes

W[x[prime].sub.i] = [x.sub.i][(1 - s)(1 - [y.sub.i]) + 2[Theta][y.sub.i]]

W[y[prime].sub.i] = [y.sub.i][1 + [x.sub.i](1 - 2[Theta])]

with

W = 1 - s [summation over j] [x.sub.j](1 - [y.sub.j]), (1)

with [x.sub.i] representing the frequency of VCG [c.sub.i] individuals with h nuclei, and [y.sub.i] those with H nuclei (thus, the frequency of [c.sub.i] equals [x.sub.i] + [y.sub.i] = [z.sub.i], and [summation of] [z.sub.i] = 1).

From this, it follows that in a population without h nuclei ([x.sub.i] = 0, [for every]i), h cannot invade VCG [c.sub.i] if

[z.sub.i] [less than] s/2[Theta] + s - 1, (2)

and H cannot invade VCG [c.sub.i] in a population with only h nuclei ([y.sub.i] = 0, [for every]i) if

[z.sub.i] [greater than] s/2[Theta] - 1. (3)

This means that h will spread in a common VCG, but not in a rare one, because it can only profit from heterokaryon formation when it encounters enough compatible conspecifics. Only if neither of the inequalities (2) and (3) hold will polymorphism of H and h be found.

The dynamics of the process are illustrated in figure 1. It shows the course of events in populations with initially only (nonparasitic) H nuclei, where mutation(s) to (parasitic) h occur. In this example, s = 0.055 and [Theta] = 0.6, thus, the critical value for invasion of h is [z.sub.i] = 0.2157, and for H it is 0.275, according to inequalities (2) and (3).

Figure 1A shows the invasion of a single h mutant in a VCG with a relatively high frequency, where inequality (2) does not hold. The frequency of h increases. However, homokaryotic h individuals have a lower fitness (1 - s) than homo- and heterokaryotic individuals with H nuclei that have fitness 1. Therefore, the mean fitness of individuals belonging to the invaded VCG is lowered, and thus the frequency of that VCG decreases. Because of this decrease, the h mycelia will encounter fewer compatible H mycelia, and consequently form fewer heterokaryons. Consequently, a low frequency of a VCG is disadvantageous for the h allele, which then profits little from heterokaryon formation with H. If all h nuclei are eliminated by selection, the population reaches a new equilibrium.

Figure 1B-D show the dynamics if mutations to h occur in all VCGs. The h type can invade in those VCGs where inequality (2) does not hold (those with a frequency higher than 0.2157). In those VCGs, the mean fitness will be lowered. Still, the h nuclei become fixed in a population with three VCGs, because all VCGs then suffer from this lower fitness. With three VCGs, each VCG can have a frequency higher than 0.275; inequality (3) may hold for all VCGs in equilibrium.

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A chaotic polymorphism is maintained in a population with four VCGs. Now no VCG can remain at a high frequency due to the invasion of h. In contrast, the frequency of a VCG will increase once its frequency is low; if inequality (2) holds, h cannot invade, and the VCG will not suffer from a lower mean fitness (and thus have a relatively high fitness).

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The H nuclei become fixed in a population with five VCGs. In this case, all VCGS can simultaneously have a frequency lower than 0.2157; inequality (2) will hold for all VCGs in equilibrium.

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The fate of a new, and presumably rare, VCG depends on its own genotype (h or H), and the composition of the population. Only a VCG associated with H can invade, and only if the parasitic allele h is present in the population (i.e., if W [less than] 1 in eqs. 1). A new VCG associated with h can never invade a population, as it cannot parasitize an H of the same VCG.

The total frequency of h ([summation of][x.sub.i], the shaded areas) decreases when the number of VCGs increases. New VCGs can invade until all h alleles disappear ([summation of][x.sub.i] = 0). When the critical number of VCGs for which all h alleles disappear is reached, the frequencies z, of all VCGs tend to be equal. The number of VCGs at equilibrium is thus the lowest number that allows inequality (2) to hold for all VCGs, that is, the smallest integer N that satisfies

N [greater than or equal to] 2[Theta] + s - 1/s. (4)

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In an asexual species without any recombination, however, this need not be the actual number of VCGs at equilibrium. When inequality (3) holds for all VCGs in the population, the whole population carries only h nuclei ([summation of][x.sub.i] = 1). H will be able to invade the population only when there is a double mutation to both H and a new VCG. This ought to be extremely rare. If such a double mutation were to occur, the frequency of the new VCG could increase. It would reach a value for which inequality (2) no longer held, and h could invade this VCG after mutation. Its frequency might increase even further to a value that satisfied inequality (3). This might lead to a population monomorphic for h again, but with one more VCG. A subsequent double mutation both to H and to a new VCG would be necessary for a new VCG to invade. Thus, without this necessity of double mutations, there appears to be a threshold in the number of VCGs that must exist before selection to new VCGs could act.

This threshold can be calculated with inequality (3), assuming equal frequencies of all VCGs, as the largest integer [N.sub.Thr] that satisfies

[N.sub.Thr] [less than or equal to] 2[Theta] - 1/s. (5)

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Figure 2 shows the values of the critical numbers N and [N.sub.Thr] as given by equations (4) and (5). From these equations it can easily be calculated that [N.sub.Thr] = N - 2. Because selection will only operate if the number of VCGs lies between [N.sub.Thr] and N, this means that selection for more VCGs will only occur for two possible numbers of VCGs.

Figure 2 also shows that N and [N.sub.Thr] are very high for values of s close to zero (i.e., if the fitness disadvantage of h is low). Simulations of the dynamics in a population with mutations to new VCGs, and from H to h and vice versa, showed that the selection process decelerates rapidly when s is very small, and that it takes a very long time to reach the equilibria.

The General Case. - If the number of neighboring sites n is greater than one and the fraction of unoccupied sites P is greater than zero, the model becomes much more complicated to analyze. Qualitatively, however, the results do not differ much from those presented above. Again, there are three kinds of populations possible: those with few VCGs and without H nuclei, those showing chaotic behavior with more VCGs and both h and H nuclei, and those with many VCGs and without h nuclei. A typical example of the dynamics in a population with the regular introduction of a new VCG with genotype H is given in figure 3. Critical values for the number of VCGs can be calculated numerically with the formulas (B6) and (B8) given in Appendix B.

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The impact of the parameters n and P on the results can be evaluated as follows: in general, higher values of n and lower values of P (i.e., a higher population density) cause a higher threshold [N.sub.Thr] and allow selection for more VCGs as soon as this threshold is passed. When the expected number of neighbors n(1 - P) increases, both [N.sub.Thr], N and the interval N - [N.sub.Thr] where selection can operate increase.

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A Heterokaryon Self-Incompatible (HSI) Type. - The fate of a Heterokaryon Self-Incompatible (HSI) mutant, which is even incompatible with itself, depends primarily on its fitness [f.sub.SI]. If [f.sub.SI] = 1, the HSI type will become fixed if mutations to h occur. If [f.sub.SI] [less than] 1 - s, the HSI type will always have the lowest fitness, and it will be unable to invade any population. More precisely, as shown in Appendix B, an HSI type can invade the population if

[f.sub.SI] [greater than] 1 - s[summation of][x.sub.i][(1 - (1 - P)[y.sub.i]).sup.n]. (6)

With only one VCG, a stable equilibrium exists, as in the example given in figure 11 in Appendix A. As the number of VCGs grows, the frequency of h nuclei in the population decreases, as does the frequency of HSI. Finally, the HSI type will disappear.

The fate of an HSI mutant with [f.sub.SI] = 0.99 in a population with the same parameter values as in figure 3 is given in figure 4. It shows that an initially invading HSI mutant disappears when the number of VCGs increases.

An Omnicompatible Type. - A hypothetical Omnicompatible (OC) type, capable of forming heterokaryons with all other types, will invade a population if the OC type carries the parasitic h allele. Because it is compatible with all VCGs, this h-OC type will behave as an h type in a population with only one VCG. Therefore, it can always invade the population, independent of the existence of any number of other VCGs. Because the OC type encounters more compatible conspecifics than the other types, many heterokaryons will be formed but only a few (low-fitness) homokaryons with only h nuclei. So the disadvantage of h-OC is very low compared to h nuclei in other groups.

As a consequence the OC type will become fixed and can eventually cause the total disappearance of all instances of vegetative incompatibility in the population. So the model suggests that this type can destabilize the system of incompatibility.

Thus, if an omnicompatible type were feasible, it would probably be common. The fact that the Omnicompatible type is not known to exist may imply that a mutation to an OC type is physiologically impossible.

A Model of the Evolution of Vegetative Incompatibility Driven by Harmful Cytoplasmic Elements

Harmful cytoplasmic elements (i.e., plasmids, viruses, mitochondria) may also contribute to selection for vegetative incompatibility (Day 1968; Caten 1972). In this section, we analyze a model which assumes the action of a harmful cytoplasmic element, instead of a nuclear gene like h. This element can be regarded as an infection. Infected individuals, developing from spores carrying the element, have a lowered fitness, 1 - c (0 [less than] c [less than] 1). The infection is passed on to a fraction [Alpha] of the (asexual) spores, so the transmission rate is [Alpha]. Furthermore, all compatible neighbors of an infected individual are infected. Newly infected individuals are assumed to have no lowered fitness but will transmit the element in a fraction, [Alpha], of their spores. The genetics of VCG determination are the same as in the parasitic nuclear-gene model.

Under these assumptions, without considering the Heterokaryon Self-Incompatibles and Omnicompatibles, with [x.sub.i] equaling the frequency of the infected individuals of VCG [c.sub.i] and [y.sub.i] equaling the frequency of the uninfected individuals of VCG [c.sub.i]. then

W[x[prime].sub.i] = [Alpha][[x.sub.i](1 - c) + [y.sub.i](1 - [(1 - P)[x.sub.i]).sup.n])]

W[y[prime].sub.i] = (1 - [Alpha])[[y.sub.i] + [x.sub.i](1 - c)] + [Alpha][[y.sub.i][1 - (1 - P)[x.sub.i]].sup.n]

with

W = 1 - c [summation over i] [x.sub.i]. (7)

If [x.sub.i] = 0, [for every]i, a stable equilibrium [x.sub.i] = 0 occurs if

[y.sub.i] [less than] 1 - [Alpha](1 - c)/[Alpha]n(1 - P), (8)

and [y.sub.i] = 0 is only stable if [Alpha] = 1 and

[(1 - (1 - P)[x.sub.i]).sup.n] [greater than] 1 - c. (9)

If [Alpha] [less than] 1, [y.sub.i] = 0 is never stable, because a fraction 1 - [Alpha] of the spores produced by an infected individual will not be infected and belong to the same VCG. This means that the population can only become fixed for infected individuals if the transmission rate [Alpha] = 1. This condition is improbable, since it implies that the cytoplasmic element is present in every spore that is produced. Therefore, the infected type can never become fixed, as did the parasitic nuclear type h in the previous model. Consequently, no threshold in the number of VCGs ([N.sub.Thr]) necessary for the operation of selection exists. Whenever a cytoplasmic element as described in this model occurs, and inequality (8) is not true, there will be selection for more VCGs.

Figure 6 illustrates the dynamics of the process of invasion of new VCGs. It shows that the frequencies of all types tend to equalize, thus, [z.sub.i] = [x.sub.i] + [y.sub.i] = 1/N if there are N VCGs. Then for n = 1, the expected value of [Sigma][x.sub.i] in equilibrium can be calculated as

[Mathematical Expression Omitted].

The expected number of VCGs selected for is the smallest integer N that satisfies

[Mathematical Expression Omitted].

Equation (11) shows that the expected number of VCGs N increases for large n and small P (i.e., with high population density), small c (the fitness disadvantage for infected individuals), and high a (transmission rate). Figure 7 shows the solution to inequality (11) for different values of a and c if n(1 - P) = 2.4 (e.g., n = 4 and P = 0.4). By rescaling the y-axis according to equation (11), expected numbers of VCGs can be calculated for other values of n(1 - P). As in the case of a parasitic nuclear eerie, the selection process slows for small values of c. [For c = 0, inequality (11) would predict a large number of groups, but no selection on VCGs can act in that case.]

If the HSI and OC types are considered, the system of equations can be derived as given in Appendix C.

Introduction of an HSI type has the same consequences as in the previous (nuclear-parasite) model. As shown in Appendix C (inequality C4), HSI can invade the population if its fitness [f.sub.SI] [greater than] 1 - c[Sigma][x.sub.i]. Thus, only if 1 - c [less than] [f.sub.SI] [less than] 1 can an equilibrium with the HSI type be stable. When the number of VCGs rises, [Sigma][x.sub.i] will decrease and it will be less likely that the HSI type can be maintained in the population, For example, the HSI type invades if the population consists of only two VCGs, but its frequency decreases when more VCGs invade. With five VCGs, the frequency of the HSI type drops to zero.

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Introduction of an OC mutant has the opposite effect, compared to the nuclear-parasite model. Here, the OC type can never invade the population. In the previous model, the fixation of the OC type results from the coupling of the h and [c.sub.oc] genotype in the nucleus, as the linked [c.sub.oc] genotype can profit from the proliferative advantage of h. In the present cytoplasmic model, no proliferative advantage and no linkage exist; the OC type cannot profit from the infectious behavior of the cytoplasmic element, as it could from the parasitic behavior of h.

DISCUSSION

This paper explores whether vegetative incompatibility in filamentous ascomycetes could have evolved as a defense mechanism against the invasion of disadvantageous genetic elements. Hartl el al. (1975) first developed a population-genetic model examining the stability of a nuclear gene in Neurospora crassa with a parasitic and a non-parasitic allele. They assumed that the parasitic allele was competitively superior in heterokaryons but lowered the fitness of homokaryons. They concluded that vegetative incompatibility could evolve as a consequence of the existence of such a parasitic gene, but whereas their model considers only two vegetative compatibility groups (VCGs), in reality (many) more VCGs are found in ascomycete species.

The extended model presented in this study assumes a similar parasitic nuclear gene, in an infinitely large population of asexual ascomycetes. The analysis shows that there are two critical numbers of VCGs, [N.sub.Thr] and N, whose values depend on the fitness parameters of the parasite and population density. If the number of VCGs is less than [N.sub.Thr] the parasitic allele will be fixed in all VCGs, and a new VCG can only invade if it possesses the nonparasitic allele. If the number of VCGs is greater than N, then the nonparasitic allele will be fixed in all VCGs, and no new VCG can invade on account of the low population density of each VCG. Only if the number of VCGs lies between [N.sub.Thr] and N will the population be polymorphic for the parasitic and the nonparasitic alleles; only in that case will a new VCG increase in frequency. Once one or more new VCGs invade a polymorphic population, the parasitic allele will not be able to maintain itself, and the polymorphism (and, consequently, the selective pressure for more VCGs) will disappear.

The analysis further shows that the conditions for coexistence of parasitic and nonparasitic alleles will be harder to fulfill for purely asexual species than for sexual species. If the number of VCGs lies below [N.sub.Thr], a new VCG also has to differ at the parasitic gene locus to invade in a population. To generate the suitable combination of alleles at the two different loci (one for VCG type and one for the parasitic gene) requires a double mutation in an asexual species, whereas recombination of two independent single mutations can be sufficient in a sexual species.

Selection for numerous VCGs (as found in nature) requires rather extreme parameter values in our model. Population density [expressed by the mean number of neighbors n(1 - P)] must be high, and the selective disadvantage of the parasitic types must be low. To our knowledge, there are no empirical estimates of these parameter values. However, the values necessary to maintain high levels of VCG polymorphism seem so restrictive that the model may be unrealistic.

One modification would be the occurrence of many different parasitic nuclear genes. Then the threshold number of VCGs ([N.sub.Thr]) that must be exceeded before selection can operate may be less relevant, as the different thresholds of different parasitic genes can successively be passed. The upper limit in the number of VCGs (N), however, will remain restricted: If the number of VCGs is such that the number of compatible interactions is sufficiently limited (i.e., to prevent invasion of the most severe parasitic gene until then), more (different) parasites will not be able to cause selection for more VCGs unless one of them has even more extreme parameter values.

Alternatively, a highly skewed VCG frequency distribution, as in a subdivided population with many subpopulations, could probably maintain numerous VCGs. Our model predicts that, at equilibrium, all VCGs will have equal frequencies, none of them carrying parasitic nuclei. If a new subpopulation is founded with few VCGs, a parasite may invade, leading to parasite-non-parasite polymorphism. Consequently, selection will favor additional VCGs in such a subpopulation.

Both possibilities require rather special conditions. Without information concerning the nature and occurrence of parasitic nuclei, and the structure of natural populations of filamentous fungi, the pertinent assumptions will be difficult to evaluate.

Another result of our study is that when VI is selected as defense against parasitic nuclear genes, incompatibility will break down after introduction of an OC mutant. This is a (hypothetical) type showing vegetative compatibility with all individuals in the population. Such OC types am not known to exist. Perhaps their occurrence is highly unlikely because several independent mutations in the same individual are required, or because their phenotype is physiologically impossible. This can be judged only when detailed information becomes available on the mechanisms and genetics of VI.

The results from the model of cytoplasmic parasitic element differ from those of the nuclear gene in two respects. First, them is no critical threshold number of VCGs ([N.sub.Thr]) that has to be passed before selection for more VCGs can operate. The key assumption causing this difference is the imperfect transmission ([Alpha] [less than] 1) of the cytoplasmic element during reproduction by spores. Second, an OC mutant cannot invade the population. Cytoplasmic parasitic genetic elements may therefore be more likely candidates for causing selection for VI than nuclear elements. However, vegetative incompatibility does not always provide an absolute barrier to cytoplasmic elements (Caten 1972; Gordon and Okamoto$1991). Further analysis is needed to explore possibilities for selection for VCGs in situations where vegetative incompatibility can be leaky, allowing harmful cytoplasmic elements occasionally to slip through.

Our model can account for HSI types (see table 1 for documented cases) if self-incompatibility causes no fitness loss. In that case, a self-incompatible type will invade and subsequently become fixed. Stable coexistence of self-incompatible and (normal) self-compatible types is not possible in the model. If the fitness loss caused by self-incompatibility is less than a specified amount (given by inequality 6), such a type can invade the population, but will disappear again when the number of VCGs increases.

The models presented in this paper mainly focus on asexual fungi. To some extent, they can be applied to sexual fungi as well. In sexual species, however, recombination can generate large numbers of new VCGs under the realistic assumption of multilocus genetic determination. But this raises the question of why so many incompatibility loci do exist. A smaller number of loci would suffice to generate the number of VCGs necessary to prevent the invasion of disadvantageous genetic elements, as suggested here.

[ILLUSTRATION OMITTED]

Perhaps the explanation for the evolution of vegetative incompatibility is outside the scope of our model. It may be that vegetative incompatibility is important in interactions with other species (e.g., resistance against pathogens); it may be that it is necessary to occupy and preserve a territory for living (Rayner 1991); or it may be a pleiotropic effect of genes having their main effect in other functions. For example, in Podospora anserina, incompatibility genes also regulate the formation of fruiting bodies (Bernet 1992). However, in the same species, Turcq et al. (1991) found no other function for a different VI gene.

Vegetative incompatibility in fungi can be compared with other highly polymorphic incompatibility systems, like the vertebrate immune system (Klitz et al. 1992), the self-incompatibility system in plants (Emerson 1939; Wright 1969) and the allorecognition systems in invertebrates (Grosberg 1988). The vertebrate immune system, however, is different, as it apparently is not a direct adaption for rejection of nonself con-specific tissue. Here, the maintenance of a high degree of polymorphism is explained by parasite-driven selection, selective abortion, and mating preferences (Potts et al. 1991).

One of the consequences of the evolutionary dynamics of the process is that all frequencies of the different VCGs tend to be equal. This is analogous to the case of evolution of self-incompatibility alleles in plants (Wright 1969). The dynamics of these two incompatibility systems show much similarity. However, they operate in an opposite direction, by promoting (self-incompatibility) and preventing (vegetative incompatibility) self-nonself fusions. There is one major difference between the nature of frequency-dependent selection acting on VCGs and self-in-compatibility alleles: in the case of self-incompatibility alleles, a difference between a self-self and a self-nonself reaction directly implies a difference in fitness; in an infinitely sized population, selection favors an infinite number of alleles. In our model on VCGs, however, fitness differences are determined by additional genetic elements that may profit from self-self reactions; in an infinitely sized population, selection favors only a restricted number of alleles.

The present models do not consider the specific role of population size and mutation rate. Mutation to new types is essential in the model (and is assumed to occur occasionally at low rates) and random drift can cause the loss of VCGs. A specific study on these aspects will be described in a subsequent paper (Nauta and Hoekstra MS).

In his review, Grosberg (1988) mentioned three hypotheses invoking natural selection for the evolution of numerous "VCGs," as they have also been found in several sessile, clonal invertebrates: frequency-dependent selection, over-dominance, and spatial or temporal variation in selective pressures. He argues that the first hypothesis offers the only viable mechanism, without, however, presenting a quantitative analysis. Since our models clearly are examples of frequency-dependent selection, this study confirms its importance. But the very high number of VCGs, as found in many species, remains difficult to explain. In haploid organisms overdominance cannot play a role, but the potential importance of spatiotemporal variation in selective pressures requires further analysis.

It appears that the fungal incompatibility systems are highly comparable to the incompatibility systems of higher organisms, and haploid sexual and asexual fungi are well suited for experiments. However, there is a conspicuous lack of information concerning the occurrence, genetics, and physiology of vegetative incompatibility in ascomycetes. Differences in ecology and population structure are probably important, and await further empirical studies.

ACKNOWLEDGMENTS

The authors thank two anonymous reviewers for helpful comments. These investigations were supported by the Foundation for Biological Research (BION), which is subsidized by the Netherlands Organization for Scientific Research (NWO; grant number 811-439-085).

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APPENDIX A

Hartl et al. (1975, p. 560) "performed extensive simulations looking for limit cycles and other kinds of attractors" in a population with two VCGs, but "no such instances were found." Our simulations of the same model however show that such instances can easily be found.

Figure A1 shows simulation results of a case where 2s [greater than] 2[Theta] - 1 [greater than] s for two VCGs, thus, there is polymorphism of h and H as described in the main text. Figure A2 shows the same simulation in another way, with [Sigma][x.sub.i], the sum in frequency of all h nuclei plotted against the frequency of one VCG. This shows a clear limit cycle, which is the "stable state," independent of starting frequencies.

A Heterokaryon Self-Incompatible (HSI) type (as described in the third section of the nuclear-parasite model) can invade the population if its fitness ([f.sub.SI]) is not too small (see inequality 6), and if selection does not remove h-type nuclei from the population. Figure A3 shows that a stable equilibrium point exists if HSI is introduced in a population with only one VCG. From equations (B2) this point can be calculated for n = 1 as the point at which

[Mathematical Expression Omitted],

[Mathematical Expression Omitted],

and

[Mathematical Expression Omitted].

In a population with two VCGs, where a limit cycle occurs, this cycle will be shifted by the introduction of an HSI type, as shown in figure A4. After the introduction of more (and more) VCGs, the HSI type will finally disappear.

APPENDIX B

The general system of equations for the nuclear-parasite model can be derived as follows.

In the model (with n neighboring sites, which are empty with probability P) a VCG that can form a heterokaryon with a fraction x of the strains, has a probability,

[Phi](x) = [(1 - (1 - P)x).sup.n] (B1)

of not forming a heterokaryon with any neighbor, and consequently staying homokaryotic.

Provided that heterokaryons form only with direct neighbors, and there are n potential neighbors, an individual remains a homokaryon (consisting of one haploid genotype) when it has no compatible neighbors, or forms a heterokaryon consisting of 2, 3, . . . n + 1 individuals when it has 1, 2, . . . n compatible neighbors. If such a heterokaryon consists of X h-individuals and Y H-individuals, the spores will produce a fraction X[Theta]/[X[Theta] + Y(1 - [Theta]] h spores and a fraction Y(1 - [Theta])/[X[Theta] + Y(1 - [Theta])] H spores.

To calculate the frequency of the different VCGs in the next generation, the frequencies of spore genotypes from all possible homo- and heterokaryons, weighted by the frequency of occurrence of the latter, must be added. Each VCG [c.sub.i] only forms heterokaryons with members of the same VCG and with the OC type. The HSI type forms no heterokaryons at all. This leads to the following system of equations, with x representing the frequencies of individuals with h nuclei, and y those of individuals with H nuclei, and the subscripts i, OC, and SI standing for VCG [c.sub.i], the omnicompatible and the self-incompatible individuals:

W[x[prime].sub.i] = [x.sub.i](1 - s)[Phi]([y.sub.i] + [y.sub.OC]) + [x.sub.i][[Phi].sub.i], (B2a)

W[x[prime].sub.OC] = [x.sub.OC](1 - s)[Phi]([Sigma][y.sub.i] + [y.sub.OC]) + [x.sub.OC][[Phi].sub.OC], (B2b)

W[y[prime].sub.i] = [y.sub.i][Phi]([x.sub.i] + [x.sub.OC]) + [y.sub.i][T.sub.i], (B2c)

W[y[prime].sub.OC] = [y.sub.OC][Phi]([Sigma][x.sub.i] + [x.sub.OC]) + [y.sub.OC][T.sub.OC], (B2d)

and

W[y[prime].sub.SI] = [f.sub.SI][y.sub.SI], (B2e)

with

[Mathematical Expression Omitted],

[Mathematical Expression Omitted],

[Mathematical Expression Omitted],

[Mathematical Expression Omitted],

and with mean fitness W standing for the sum of the right-hand sides of the five equations of (B2).

In formulas (B2a) to (B2d), the first part of the equation to the right of the equals sign is the fraction of that genotype produced by homokaryons. The second part is the fraction produced by the heterokaryons. The parameters [Theta] and T, that are given more explicitly in (B3a) to (B3d), can be derived from the multinomial distribution, leaving out the nonheterokaryotic types.

For [x.sub.OC] = [y.sub.OC] = 0, the mean fitness W becomes

W = 1 - s[Sigma][x.sub.i][Phi]([y.sub.i]) - [y.sub.SI](1 - [f.sub.SI]). (B4)

We could not solve the system of equations (B2) analytically. It can be shown, however, under which conditions [x.sub.i] = 0, [for every]i or [y.sub.i] = 0, [for every]i are stable (i.e., when h or H are unable to invade).

If [x.sub.OC] = [y.sub.OC] = [y.sub.SI] = 0, for([x.sub.i]) = [x[prime].sub.i], equation (B2a) can be rewritten as

f([x.sub.i] = [x.sub.i](1 - s)[[1 - (1 - p)[y.sub.i]].sup.n] + [[Theta].sub.i]/1 - s [summation over j] [x.sub.j][[1 - (1 - p)[y.sub.j]].sup.n]. (B5)

Consequently, f(0) = 0 is an equilibrium.

This is stable if [absolute value of f[prime](0)] [less than] 1, i.e., for [x.sub.i] = 0, [for every]i if

[Mathematical Expression Omitted]

Therefore, if (B6) is true, an h type will not be able to invade VCG [c.sub.i] if there are no h nuclei in the population.

Also, for g([y.sub.i]) = [y.sub.i], equation (B2c) becomes

g([y.sub.i]) = [y.sub.i][(1 - (1 - p)[x.sub.i]).sup.n] + [T.sub.i]/1 - s [summation over j] [x.sub.j][(1 - (1 - p)[y.sub.j]).sup.n]. (B7)

Here again g(0) = 0 is a stable equilibrium if [absolute value of g[prime](0)] [less than] 1 for [y.sub.i] = 0, for every i if

[Mathematical Expression Omitted].

If this condition is satisfied, H will not be able to invade VCG [c.sub.i].

It is easy to see that equation (B6) solved for n = 1, p = 0 gives (2), and solving (B8) gives (3). For larger n, (B6) and (BS) can only be solved numerically. As an example, the results of these calculations with n = 4 and p = 0.4 are given in figure B1.

The HSI type will increase in frequency if

[f.sub.SI] [greater than] 1 - s [Sigma] [x.sub.i][Phi]([y.sub.i])/1 - [y.sub.SI], (B9)

and decrease otherwise.

APPENDIX C

The general system of equations for the dynamics of a cytoplasmic element can be derived in the same way as the system above.

With [Phi] defined as in appendix B, [x.sub.i] the frequency of the infected individuals of VCG [c.sub.i], and [y.sub.i] that of the uninfected ones, the system of recurrence relations for all types in case of a harmful cytoplasmic element becomes

W[x[prime].sub.i] = [Alpha]{(1 - c)[x.sub.i] + [y.sub.i][1 - [Phi]([x.sub.i] + [x.sub.OC])]},

W[x[prime].sub.OC] = [Alpha]{(1 - c)[x.sub.OC] + [y.sub.OC][1 - [Phi]([Sigma] [x.sub.i] + [x.sub.OC])]},

W[y[prime].sub.i] = (1 - [Alpha])[[y.sub.OC] + [x.sub.OC](1 - c)] + [Alpha][y.sub.i][Phi][x.sub.i] + [x.sub.OC]),

W[y[prime].sub.OC] = (1 - [Alpha])[[y.sub.OC] + [x.sub.OC](1 - c)] + [Alpha][y.sub.OC][Phi]([Sigma][x.sub.i] + [x.sub.OC]), C1

and

W[y[prime].sub.SI] = [f.sub.SI][y.sub.SI],

with

W = 1 - c([Sigma] [x.sub.i] + [x.sub.OC]) - [y.sub.SI](1 - [f.sub.SI]). (C2)

For the Heterokaryon Self-Incompatible type, [y.sub.SI] will increase if

[f.sub.SI] [greater than] 1 - c([Sigma] [x.sub.i] + [x.sub.OC])/1 - [y.sub.SI], (C3)

so that it cannot invade the population if

[f.sub.SI] [less than] 1 - c([Sigma] [x.sub.i] + [x.sub.OC]). (C4)

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Title Annotation: | Evolution of Vegetative Incompatibility in Filamentous Ascomycetes, Part 1; includes appendices |
---|---|

Author: | Nauta, Maarten J.; Hoekstra, Rolf F. |

Publication: | Evolution |

Date: | Aug 1, 1994 |

Words: | 9050 |

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