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A M . ZOOLOCIST, 11:381-398 (1971). The Genetic Structure and Evolutionary Fate of Parthenogenetic Amphibian Populations as Determined by Markovian Analysis JAMES H. ASHER, JR. AND GEORCE W. NACE The Department of Zoology and The Center for Human Growth and Development, The University of Michigan, Ann Arbor, Michigan 48104 SYNOPSIS. One-locus, two-allele models are presented which describe the genetic consequences of naturally occurring and experimentally induced parthenogesis in triploid and diploid amphibians. The models may in general be used to investigate genetic change resulting from apomictic (ameiotic) and automictic (meiotic) parthenogenetic reproduction. These models quantify the influence of mutation, segregation, and selection upon genetic variability in parthenogenetic populations. They also allow an estimate of the relative importance of stochastic forces in altering this variability. They thus provide a basis for understanding evolution in these populations. Some of the conclusions derived from this study contradict previous predictions regarding genetic variability in parthenogenetic populations. First, if mutation is the sole source of genetic change (i.e., strict apomixis), parthenogenetic populations should not become completely heterozygous. Second, small amounts of segregation occurring in apomictic populations have enormous effects upon the genetic variability of these populations, i.e., they should lose much of their heterozygosity. In addition to these conclusions, the results of this study suggest that studies of protein variability in parthenogenetic species should contribute toward answering the question: How much of the genetic variability observed in nature is evolutionarily relevant? Many reproductive mechanisms are em- "populations," and (3) considering the ployed by animals in their adaptation to evolutionary implications o£ these models, Polyploidy and parthenogenesis are relaconstant and changing environments. Parthenogenesis as one of these mechanisms is tively rare in the amphibia. Naturally ocexamined here by: (1) reviewing natural curring polyploidy has been reported in and experimental parthenogenesis in am- three genera of anurans, Ceratophrys, phibians, (2) presenting mathematical Odontophrynus, and Hyla (Be^ak, Bec,ak, models which describe changes in the gen- and Rabello, 1966, 1967; Saez and Brumetic structures of some parthenogenetic Zorrilla, 1966; Bogart, 1967; Wasserman, 1970) and in three genera of urodeles, We thank Henry Wilbur for his helpful sugges- Ambystoma, Eurycea, and Notophthalmus tions and discussions made during the preparation ( F a n k h a u s e v of this manuscript. We also wish to extend our lyb4 1 9 3 8 ' 1938& 1 9 3 9 ,nAn U z z d l , appreciation to George Estabrook for suggesting the use of Markovian Theory in modeling the parthenogenetic systems presented here and for his patience in explaining this and other mathematical concepts. Finally, we wish to thank Dr. Dan L. Hartl for his very valuable suggestions made during and after the symposium presentation of this 5 Book, 1940; Fankhauser and Humphrey, 1942). Parthenogenesis, in the form of natural gynogenetic reproduction, j , , o ( x u r s { f u r o d d e s ; ° ° (Ambystoma) and involves a complex composed of four species (Uzzell, 1963, paper. These investigations were supported by a Horrace H. Rackham Postdoctoral Fellowship to one of us (Asher) and by a grant from the National UzzeU a n d Goldblatt 196?.Wilbur> „, . . , ' . ' . T h e fro u r s e c i e s Science Foundation, GB-8187. plex are: A. jeffersonanium (2n) , A. later- 1964, 1969; Macgregor and Uzzell, 1964; 1971). ' P involved in this COm- SSI 382 JAMES H. ASHER, JR. AND GEORGE W. NACE Ancestral Form GLACIATION WISCONSIN A. laterale (2n) A. jeffersonianum (2 n) X 2n Hybrid (LJ) A. laterale X UNREDUCED EGG J 3n Hybrid (2LJ) A. tremblayi (3n) X A. jeffersonianum I 3n Hybrid (L2J) A. platineum (3n) FIG. 1. Diagrammatic representation o£ the origin of triploid parthenogenetic Ambystoma estimated at 10,000 years or 5000 generations ago (proposed by Uzzell, 1963, 1964; and Uzzell and Goldblatt, 1967). ale (2n) , A. platineum (3n), and A. tremblayi (3n). The parthenogenetic triploids probably arose immediately following the Wisconsin glaciation (10,000 years ago) as a result of hybridization between the two diploid bisexual species (U/zell, 1963, 1964). If we assume the length of each generation in the triploids is two years (Wilbur, personal communication), the Lriploid populations have had at most 5,000 generations in which to diverge. Figure 1 presents a summary of this proposed evolutionary relationship. The genomic compositions of the triploids were deduced from a consideration of morphological and serum protein phenotypes (Uzzell, GENETICS OF PARTHENOGENETIC POPULATIONS 1964; Uzzell and Goldblatt, 1967). Based on the differences in electrophoretic mobilities and relative concentrations of serum proteins of the "modern" forms, it was hypothesized that A. jeffersonianum contributed two chromosomal sets and A. laterale one set to A. platineum (L2J) while A. jeffersonianum contributed one set and A. laterale two sets to A. tremblayi (2LJ). Cytological studies of oocytes from the triploids (Macgregor and Uzzell, 1964) revealed a 6n ploidy prior to the completion of meiotic events. This suggested the occurrence of an endomitotic duplication of the somatic 3n ploidy. Activation of such eggs with sperm from the diploid species— A. jeffersonianum males in the case of A. platineum, and A. laterale males in the case of A. tremblayi—is accompanied by a normal meiosis. Thus, Ambystoma triploids are technically automictic parthenogens.1 The cytological analysis further suggested that only sister chromosomes produced by the pre-meiotic endomitosis pair during the prophase of meiosis I. Hence, the regular meiosis should result in the production of progeny that are genetically identical to the mother and to each other. Figure 2 summarizes these cytological observations. Two events can alter the genetic consequences of this mode of reproduction: (1) segregation resulting from the infrequent formation of quadrivalents which does occur (See Fig. 2), and (2) mutation. The first objective of this paper is to present and evaluate several models which predict the genetic consequences of these two "rare" events. These models are also applicable to any apoomictic (ameiotic) parthenogenetic reproduction. Experimentally induced parthenogenesis is well documented in both the anurans 383 and urodeles (Guyer, 1907; Loeb, 1913; Parmenter, 1933; Kawamura, 1939; Tyler, 1941, 1955; Rostand, 1950; Beatty, 1967; Kawamura and Nishioka, 1967). These reproductions have been used to investigate a variety of problems including those of fertilization, cleavage, sex differentiation, and genetic mapping (Fankhauser, 1938a; Kawamura, 1939; Tyler, 1941; Lindsley, Fankhauser, and Humphrey, 1956; Kawamura and Nishioka, 1967; Nace, Richards, and Asher, 1970; Volpe, 1970). To employ experimental parthenogenesis in the mapping of genekinetochore distances and in the development of "inbred" strains, a mathematical model was developed (Nace et ah, 1970; Asher, 1970) which described the genetic consequences of parthenogenetic reproduction by inhibition of meiosis I (or central fusion) or meiosis II (or terminal fusion). A second objective of this paper is to present an extension of this earlier model and to consider the genetic consequences of both experimentally induced and naturally occurring automictic parthenogenetic reproduction. Genetic studies of parthenogenetic species (both apomictic and automictic forms) are important from the standpoint of understanding evolution and development in these organisms. They may, however, provide information with far broader consequences. A third objective of this paper, then, is to suggest genetic studies which may provide solutions to general problems in evolution and development. THEORETICAL CONSIDERATIONS AND RESULTS Natural populations For the purposes of this discussion, we consider only one-locus, two-allele models. As a consequence of this limitation, triploid Ambystoma may exist in only one of four genotypic states: (I) AAA, (II) i Throughout the text, we use two terms as they are defined and used by White (1948) and A'A'A', (III) AA'A', and (IV) AAA'. The Suomalainen (1950). By apomictic parthenogenetic Roman numerals are used throughout the reproduction, we mean parthenogenesis in the ab- text as synonyms for these four states. We sence of meiosis. By automictic reproduction, we symbolize the frequency of mutation from mean parthenogenesis employing some form of A to A' by p and the reverse frequency by meiosis. 384 JAMES H. ASHER, JR. AND GEORGE W. NACE 3N OOGONIUM PRE-MEIOTIC MITOShS WITHOUT CYTOKINESIS 6N CHROMATID REPLICATION OOCYTE FOR QUADRIVALENT FORMATION 3/8 OF ZYGOIDS SURVIVE 5/8 DIE 1/3/ \2/3 METAPHASE MEIOSIS I METAPHASE AAA (P/24) A (4P/24) (P/24) MEIOSIS E Y= 2/3 AAA (27P/56O) (86P/56O) (27P/56O) ZYGOID AAA1 (I-P) FIG. 2. Meiosis and quadrivalent formation obser\ed in Ambystoma triploids (based on the cytological analyses of Macgregor and Uzzell, 1964). The genetic consequences of quadrivalent formation are given assuming that: P is the probability that a quadrivalent forms; Y is the probability that (3P/24) recombination occurs where Y = 0 implies absolute linkage and Y z= 2/3 implies independent assortment; quadrivalent formation is independent of chromosomal origin; and /v^oids with deviations in the 3n kinetochore number are lethal. GENETICS OF PARTHENOGENETIC POPULATIONS GENERATION GENOTYPIC 385 STATES N -3-J N+l AAA AAA FIG. 3. Diagrammatic representation of mutational changes in triploid apomictic populations. The frequency of mutation from A to A' is given by /i. The frequency of the reverse mutation is given by v. Only first order terms are considered in this model. v. We also make three assumptions regarding the ancestral triploid Ambystoma populations: (1) ancestral A. tremblayi were AAA', (2) ancestral A. platineum were AA'A', and (3) ancestral forms arose 10,000 years ago (Uzzell, 1964; Uzzell and Goldblatt, 1967). Finally, we assume that generations of reproduction do not overlap and that population size remains constant. Mutation: Figure 3 is a schematic representation of genetic change in the triploid populations caused by mutation alone. This model assumes that P (the probability of quadrivalent formation given in Fig. 2) is zero, and considers only first-order mutation effects. Hence, the values fi, v, 2/i, 2v, etc. of Figure 3 represent the probabilities that an individual in any genotypic state produces individuals in other states in the next generation. These values represent transition probabilities (probabilities of making the transition from one state to another) and can be arranged to form a transition matrix T (below). The Roman numeral in the column at the left of the matrix represents the i th starting genotypic state at generation n, while a Roman numeral labeling the column represents the j t h transition state at generation n -f- 1. The ij th entry in the matrix represents the probability of passing from the ith state to the j t h state in a single generation. Hence, the probability that an AAA (I) individual gives rise to an AAA' (IV) is 3/x (three times the mutation frequency from A to A') while the probability that an A'A'A' (II) individual gives rise to an AA'A' (III) is 3i/ (three times the mutation frequency from A' to A). To obtain the frequencies of each genotypic state in the n -|- 1 generation, a row vector FW whose elements are the genotypic fre- I II I "l - $>X 0 II 0 1 - 3v T = III 0 M 1 IV V 0 IV III 0 3v 3M 0 2v 2M 2v 1 - 2M'- v 386 JAMES H. ASHER, JR. AND GEORGE W. NACE quencies /("> in generation (n) is postmultiplied by the transition matrix T. Hence, 0) where (2) and TABLE 1. Genotypic equilibria for mutation fi/v AAA Genotypes AAA' AA'A' 1 2 5 10 .1250 .0370 .0046 .0008 .3750 .2222 .0694 .0225 .3750 .4444 .3472 .2254 alone* A'A'A' .1250 .2963 5787 .7513 * Computed from equations 5 through 8, recall that the Roman numerals I, II, III, and IV found in these equations refer to genotypes AAA, A'A'A', AA'A', and AAA' respectively. 3 represents an independent trials process, then the expected distribution of genotypic frequencies for this model at generation n, considering a finite population size N, is given by (/i^n'-)-/ii(n) + /m ( n > +/iv ( n ) ) N - Hence, the probability that all individuals in a population will retain the ancestral genotype is given by Tn represents the product of the transition (/iv*n')N f° r t n e triploid A. trimblayi. Some matrix multiplied by itself n times. The numerical values for this expectation are process described by Figure 3 and equa- given in Table 2. tions 1 through 4 represents a Markov Segregation: Two variables can affect chain. From the theory of Markovian pro- the proportion of segregants produced by cesses (Kemeny et ah, 1959), we determine triploid Ambystoma: (1) frequency of the equilibrium state, F^), for the model synapsis of unlike chromosomes, and (2) represented by Figure 3 to be: probability of recombination between the locus in question and its kinetochore. 1 Co) Accepting the cytological mechanism (5) (Fig. 2) attributed to these triploids, the 1 +3 (£ first variable is a function of quadrivalent formation. Based on data from Macgregor (6) and Uzzell (1964), an estimate of the frequency of quadrivalent formation in these 1+ triploids is P = 1 quadrivalent/208 bivalent equivalents (approximately .005). To (7) obtain an estimate of the proportion of fin™ = The values fjW, /„<">, fllIW> a n d fIYW, are the frequencies of each designated genotype at generation (n). The genotypic frequencies occurring n generations following some ancestral state 2""<0> are given by: WWW 1 +M- 1 £)+(")+KfT TABLE 2. Probability* that all members of a small apomictic population are of genotype AAA'. (8) Thus, given the starting genotypic frequencies and the mutation frequencies, /x and v, the population structure at each generation is described by equation 4, and the equilibrium conditions are described by equations 5 through 8. Some numerical values obtained from these equations are given in Table 1. If we assume that Figure Generation * n 1000 2000 3000 4000 5000 .9708 .9431 .9168 .8919 .8682 20 .5527 .3097 .1760 .1014 .0593 Population Size (N) 30 50 100 .4109 .1724 .0739 .0323 .0144 .2271 .0534 .0130 .0033 .0009 .0516 .0029 .0002 .0000 .0000 • Probability that all individuals are AAA' — (I AAA') • ** Expected frequency of genotjpe AAA' at generation (n) for any population size where: u. = v — 10-*;/<»!.= l:/«2>, =.3750. 387 GENETICS OF PARTHENOGENETIC POPULATIONS GENERATION GENOTYPIC N AAA N+ l AAA AAA STATES AA'A' A'A'A' AAA A'A'A' FIG. 4. Diagrammatic representation of the segregational changes in triploid apomictic populations. P is the probability of quadrivalent formation while the segregation ratio expected from heterozygous females (AAA') is g± (AAA): g2 (AAA'): gt (AA'A'). segregants produced by this mode of reproduction (Fig. 2), we must make three additional assumptions: (1) The frequency of quadrivalent types is independent of the ancestral chromosome type (i.e., independent of whether the chromosome was derived from A. laterale or A. jeffersonianum); (2) Kinetochore segregation from quadrivalents at anaphase of meiosis I is an independent trials process (i.e., the probability that a cell gets 0, 1, 2, 3, or 4 kinetochores is binomially distributed); and (3) Zygoids with an aneuploid number of kinetochores do not survive. From assumption one, we observe that 2/3 of the quadrivalents should result in segregation while 1/3 should not. From assumptions two and three, we observe that 3/8 of the eggs with quadrivalents should yield viable zygoids while 5/8 should yield inviable ones. Thus, the frequency of quadrivalents leading to viable segregants and nonsegregants should be 6P/24 and 3P/24 respectively. and (2) independent assortment of the locus and its kinetochore (y = 2/3). If the ratio of segregants from a heterozygous female (AAA') is given by gt (AAA) : e» (AAA') : gl (AA'A') where 2gl + g2 = " l , then the value of gr (non-parental segregants) for these two extreme cases of linkage are given by: Once a synaptic configuration occurs which can lead to segregation (occurring with a frequency of 6P/24), the proportion of segregants will depend upon the probability of recombination between the locus and its kinetochore. For the purposes of this discussion, we consider two extreme values of recombination: (1) absolute linkage to the kinetochore (y = 0), gl = (P/24)/(l - 5P/8) fory = 0 (9) gi = (27P/560)/(l - 5P/8) for^v = 2/3 (10) Substituting P =: .005 (the value observed for triploid Ambystoma) into equations 9 and 10, we find that 2.0 X 10- 4 <gx<2.5 X 10~4. Thus, we expect approximately two non-jsarental segregants among every 5,000 progeny produced by a heterozygous female. The genetic change resulting from this segregation is represented schematically in Figure 4. As in the case of mutation alone, the process modeled by Figure 4 is a Markov chain with a transition matrix Z given by: I II III IV I 0 0 0 0 II 0 0 1 7 = III gl gl gl IV gl 0 The Markov chain described by Z is an 388 JAMES H. ASHER, JR. AND GEORGE W. NACE absorbing chain, i.e., it is impossible to leave states I and II. In terms of the Ambystoma triploids, this means that for segregation alone, the population should eventually become completely homozygous at the ^4-locus. We may ask two questions regarding this process: (1) What is the average number of generations to complete homozygosity, and (2) What is the proportion of each homozygous class observed at this equilibrium point? Using procedures outlined by Kemeny et al. (1959), both of these questions may be answered. The average number of generations to complete homozygosity, i.e., fixation, were obtained for various values of P (probability of quadrivalent formation) and y (probability of recombination) and are presented in Table 3. The final frequency of homozygotes depends only upon the starting (ancestral) genotypic frequencies. For the A. tremblayi population with the ancestral form of AAA', the final genotypic proportion is expected to be 2/3 (AAA): 1/3 (A'A'A'). Thus, the time required to reach this final equilibrium state depends upon the values of P and y (Table 3); however, the final state is itself independent of P and y provided P > 0 . Mutation and segregation: The two models presented to this point describe the independent influences of mutation and segregation upon the genetic structure of the triploid Ambystoma populations. To determine the composite effect of these two events (mutation and segregation), the two transition matrices T and Z are multiplied to give a new transition matrix. This multiplication, TZ, produces a transition TABLE 3. Generations to fixation for alone*. P y= 0 y = 2/3 .0001 .0005 .0010 .0050 239,984 47,985 23,985 4,785 207,394 41,469 20,728 4,135 P .0100 .0500 .1000 .5000 segregation y = o ,i = 2/3 2,385 465 225 33 2,061 402 194 29 matrix for the case of pre-meiotic mutation while the reverse order, ZT, gives a transition matrix for the case of postmeiotic mutation. For the purposes of this paper, we will present only the latter, for which the transition matrix is given by ZT = M where: II 0 I I "l - 3M 0 II M = III gi" IV _ fe III 0 3» & & 1 — 3v gllx IV 3M 0 h where - 3M - 2v) & = v + 4M - 2) - 2v - /i + 4» - 2) - 2n - v - 2M - 5v) 2v - 2 v - 2M 5M) As in the case of mutation alone, this process is a Markov chain with an equilibrium state, F^i, given by: feai - i where + k) (gin 3M") Sample values obtained from equations 11 through 14 are given in Table 4 for this model which incorporates both post• P — probability of quadrivalent formation; y = meiotic mutation (modeled in Fig. 3) and probability of recombination between the locus and segregation (modeled in Fig. 4). its kinetochore. GENETICS OF PARTHENOGENETIC POPULATIONS 389 TABLE 4. Genotypic equilibria* for mutation and segregation. 1/10 1/5 1/2 1 2 5 10 AAA AAA' AA'A' A'A'A' .8395 .7536 .5913 .4398 .2918 .1455 .0086 .0815 .0873 .0775 .0602 .0409 .0207 .0012 .0456 .0647 .0712 .0602 .0428 .0222 .0014 .0333 .0944 .2600 .4398 .6245 .8116 .9888 ""b £i/ni(n'Win,} WH, (17) (18) • Equilibria computed from equations 11 through 14 where: p. — 10"5; P = .005; and y = 0. Mutation, segregation, and selection: Two additional factors may influence the genetic structure of any parthenogenetic population: migration and selection. If we assume that current triploid populations arose as a consequence of a single or a few hybridization events (Fig. 1) and that migration is proportional to the genotypic frequencies, then migration would have no influence upon the genetic structure of this population. Selection, on the other hand, could play a very important role in altering the genetic structure. Since selection alters the transition probabilities from generation to generation, we were unable to use fully the theories of Markovian processes to answer questions concerning equilibrium conditions where mutation, segregation, and selection are acting simultaneously. The transition matrix M previously given may be used, however, to obtain a set of recurrent equations describing the genetic structure of such a population where two fitness components are considered: (1) fecundity, and (2) zygoid survival. Multiplying M on the left by a matrix whose principal diagonal elements are the fitness components with respect to fecundity and on the right by a matrix whose principal diagonal elements are the fitness components with respect to zygoid survival gives a new matrix S. The recurrent equation which considers post-meiotic mutation, segregation and selection thus takes the form: F<.n+i) (15) Expanding this equation produces a set of simultaneous equations for the frequency of each genotype: (19) The symbol W represents the component of fitness considered where subscript 1 refers to fecundity while 3 refers to zygoid survival, and the Roman numerals refer to the genotypic states as previously defined. Because of the non-linearity of these four simultaneous equations, iterative procedures were used to solve for the equilibrium values for various values of the fitnesses considered. Under all conditions investigated, the equilibrium values appeared to be stable and non-oscillatory. Sample iterative solutions which describe triploid populations where segregation, selection, and post-meiotic mutation occur are presented in Table 5. Analytical solutions for these equilibrium states are at present being developed using the eigenvalues of S as the proportionality constants of F(eq) SccFti). Summarizing the results of this section, we observed that in apomictic populations mutation, segregation, and selection all play roles in maintaining heterozygosity. In this process, very small amounts of segregation appear to dominate mutational forces. In the presence of both mutation and segregation, intense selective forces are needed to maintain the ancestral genotypic state. Experimental Populations Experimentally induced parthenogenesis, or reproduction involving the manipulation of normal meiotic mechanisms, in amphibians has been used for genetic investigations in two genera: Rana and Ambystoma. These studies have been 390 JAMES H. ASHER, J R . AND GEORGE W. NACE TABLE 5. Genotypic frequencies for mutation*, segregation**, and selection* Genotype Fitness 5000 Equl Fitness 5000 Equl Fitness 5000 Equl AAA 1.000 1.000 1.000 1.000 .456 .210 .164 .170 .440 .060 .060 .440 1.000 .999 .999 .999 .952 .029 .010 .008 .975 .021 .003 .001 .999 1.000 .999 .999 .204 .620 .134 .042 .209 .612 .133 .045 .999 .999 .999 1.000 .109 .052 .054 .786 .001 .003 .021 .975 .000 1.000 .999 .000 .000 .822 .178 .000 .000 .821 .179 .000 .000 1.000 .970 .000 .000 .993 .007 .000 .000 .993 .007 .000 AAA' AA'A' A'A'A' AAA AAA' AA'A' A'A'A' *n=v=10r>. * * P = . O O 5 ; y = 0. *** Differences in fitness are attributed to zygoid survival alone and the frequencies are given at generations 5000 (Fig. 1) and at equilibrium. concerned with mapping gene-kinetochore distances for previously identified loci, uncovering new mutants at other loci, and producing homozygous strains (Lindsley et STAGE ADULTS al., 1956; Nace et al., 1970; Volpe, 1970). The development off theory needed l d d to predict the genetic consequences of these meiotic manipulations was presented by STATES GENOTYPIC AA AA' A'A' WAA. "A'A1, I ' EGGS A A' AA GENERATION A'A' MEIOSIS N N I K2 { ZYGOIDS AA A ' A> AA' \ 2M AA N+ I ZYGOIDS AA AA' A'A' N+ 1 ADULTS AA AA' A'A1 N +1 FIG. 5. Diagrammatic representation of genotypic change in a diploid automictic parthenogenetic population uhere segregation, post-mciotic muta- tion, and selection occurs. Fitness (W) in this model has two components: (1) fecundity, stage 1, and (2) z>goid survival, stage 3. 391 GENETICS OF PARTHENOGENETIC POPULATIONS Nace et al. (1970) and Asher (1970). When meiosis I is inhibited, heterozygous (AA') females are expected to produce three genotypes in the ratio y/i (AA): 1 _ y/z (AA'): y/4 (A'A') where y is the probability of recombination between the locus and its kinetochore. In the case of inhibition of meiosis II, segregation ratios are (1 — y)/2 (AA): y (AA'): (1 — y)/2 (A'A'). Map distances may be computed from these ratios with the aid of a mapping function (Barratt et al., 1954; Nace et al., 1970) which describes the relationship between y (the probability of recombination) and x, the corrected map distance between the locus and its kinetochore. Both mechanisms of reproduction (inhibition of meiosis I or II) should produce homozygosity at a very rapid rate in the absence of selection (Nace et al., 1970; Asher, 1970). Since these mechanisms of reproduction have naturally occurring counterparts, they warrant further discussion. Inhibition of meiosis I has been shown to be genetically equivalent to central fusion (observed in Drosophila mangabeirai, Murdy and Carson, 1959) while inhibition of meiosis II is equivalent to terminal fusion (also observed in D. mangabeirai, Murdy and Carson, 1959), or fusion of the second polar body with the egg nucleus (suggested for Lacerta saxicola, Darevsky, 1966). The mathematical models presented by Nace et al. (1970) and Asher (1970) predict that in these natural automictic parthenogenetic populations as well as in the experimental populations, in the absence of selection for heterozygosity, the populations should become genetically uniform. However, if selection does occur, heterozygosity can be maintained indefinitely. The models presented by Asher (1970) considered frequent segregation (a consequence of automictic reproduction) and selection acting only upon zygoid survival. In this paper, we wish to extend this model in two ways: (1) to consider both fecundity and zygoid survival as components of fitness, and (2) to include mutation as another source of genetic change. Figure 5 diagrams the genetic change between generations of automictic parthenogenetic organisms reproducing by either or both inhibition of meiosis I and meiosis II. In this model, many of whose components are more completely described in Nace et al. (1970) and Asher (1970), only a single locus with two allelic states in diploid populations is considered. The mutation frequencies fx, and v are as previously defined. If selection, segregation, and first-order, post-meiotic mutation are considered, the following non-normalized transition matrix is derived from the process modeled by Figure 5: AA' AA B = A'A'\ 0 AA'ly* where 7 i = (1 72 = 2ix)WAAlWAA, 2nWAAlWAA,z 7 3 = (1 -2v)WA.A,,WA,A.3 74 = 2VWA.A.1WAA,3 7s = (1 2ti)K1WAA,lWAA, + vKtWAA-JVAA, 76 = (1 + 2»)K1WAA'JVA>A>, IIKJVAA'JVA'A^ 77 = 20* + ti ( VIWAA'I K2 = Ep + EtQ. -y/2) Ei = the probability of inhibition of meiosis II or terminal fusion £2 = the probability of inhibition of meiosis I or central fusion E2 = 1 - Ei y = the probability of recombination between the gene and its kinetochore 392 JAMES H. ASHER, JR. AND GEORGE W. NACE and the number subscripts on the selection terms indicate the stage in the life cycle at which selection occurs (Fig. 5). An understanding of this complete model can be attained most directly by simplifications that consider first, mutation and segregation alone, and then segregation and selection. Thus, the joint influence of segregation and post-meiotic mutation upon genetic change in these diploid automictic populations, is determined by setting the fitness components all equal to one thus reducing matrix B to: AA AA' A'A' 0 AAr 1 - 2 = A'A'l 0 AA'\_ : j. 2M 2, X •] where Xi = (1 - 2M)A"I + vKt X2 = (1 - 2v)K~i + M/G X3 ^ 1 — Xi — X2 Again, t h e process is M a r k o v i a n a n d F(ei"> is given by: JAA — \*V) n (21) 2v 1 (22) AI 271 + 2v + l Assuming that //. — v, equation 22 reduces to: 1 (23) +2 The maximum value thai equilibrium heterozygosity may attain for equation 22 occurs when K1 = 0. This result obtains when reproduction is by central fusion (£2 := 1) and when the gene is absolutely linked to the kinetochore (y = 0). Under these conditions, / / i./ ( c q ) " 1/2. Rephrased, this states that in the case of reproduction by inhibition of meiosis I, the frequency of heterozygotes at equilibrium for genes absolutely linked to the kinetochore will be 1/2 provided all genotypes are equally fit and the frequencies of mutation from one allelic state to another are equal. If we assume that the smallest value that y actually attains is y = .001, which is the probability of recombination between adjacent nucleotides (Guest and Yanofsky, 1966), we may obtain another estimate of heterozygosity for this mode of reproduction. With E2 = 1 and y = .001, Kx = .00025. If we further assume that ^ = v = IO-5, then /4/(«i> = .037. The two values calculated above for fAA(e(l) (-500 and .037) represent a wide range, but they do not indicate the extremes of the heterozygosity maintained by a balance of mutation and segregation in populations reproducing by central fusion or by inhibition of meiosis I. This is appreciated by recalling that the value of /^/< e i) depends upon the value of K±. For reproduction by central fusion (E2 = 1), O^K^l/6 where Kx = y/4 (y is the probability of recombination between the gene and kinetochore). Substituting these extreme values into equation 23, 6/A < /^/(eq) ^ 1/2. The frequency of equilibrium heterozygotes decreases rapidly, however, as y increases from zero. This rate of decrease can be observed by substituting x/t = .Ki into equation 23. Hence, f.i/^il = l / ( x + 2). For a gene located 2 map units from the kinetochore, y ss .04 (Nace et al, 1970) and K = .01. If /x = 10- 5 , x = 1000 and fAA' =3 10- 3 . Thus populations reproducing by means of central fusion should be homozygous for all loci having neutral allelic variation except for those loci which are very closely linked to the kinetochore. These results thus verify and quantify the prediction made by Carson (1967) that heterozygosity can be retained in automictic parthenogenetic populations reproducing by central tusion or inhibition of meiosis I. The limits of heterozygosity maintained in populations reproducing by terminal fu- 393 GENETICS OF PARTHENOGENETIC POPULATIONS sion (£i = 1) or inhibition of meiosis II can be obtained in a similar manner. For this mode of reproduction, l/Q^K^l/2 where K1 =: (1 - y)/2. Substituting these extreme values into equation 23 gives 2/A</AA'(eq)<6/x. Hence, these automictic populations are expected to be homozygous for all neutral alleles regardless of linkage relationships. These results indicate that neutral genetic variability, i.e., heterozygosity, though not retained in automictic parthenogenetic populations reproducing by terminal fusion, is retained by central fusion; however, since equilibrium heterozygosity decreases rapidly as y increases, this variability is restricted to an area very closely linked to the kinetochore. Turning now to the joint influence of segregation and selection, we obtain from the transition matrix B a set of recurrent equations which take these two factors into consideration by setting ^ = v = 0. These recurrent equations are given by: WAMWAM = the fitness of the AA genotype = the fitness of the AA' genotype ^ A J A A , = the fitness of the A'A' genotype In order for this equation to give a nonzero equilibrium, the following inequalities must be true: JVAA', > WA'A-xWA'A't Since 0^K2^l, non-zero equilibria can exist only if the fitness of the heterozygote is greater than the fitness of either homozygote. To determine the conditions for the maximum value of the frequency of heterozygosity with respect to genotypic fitness, the following partial differential equation was solved: (28) = 0 + /AA' M WAA'.K,} WAA-JK (24) WA.A.JK (25) WAA^/K (26) where where WK represents the fitness of genotype X. Substituting equation 27 into equation 28, we obtain: _r dWx'XIK.WAA^ - WAAlWAA "1 __ Q w.AA'J^A'A's dWxlKiWAA'iWAA', WA.A^WA-A',} K= This set of equations can be solved for the equilibrium conditions where heterozygosity (/j/ (e(l) ) is maintained. The derivation is parallel to that presented by Hayman (1953), Workman and Jain (1966) and Asher (1970) and yields the equation: Solving equation 29 with respect to the various fitness components where WAAl = and WAM = k,WA.A.a (kt and k2 being arbitrary constants), we find that /A1,(eq) c a n have a maximum only when the fitness of both homozygotes is equiva(27) (eq) — . - where ] WAA3 WAAl AAlW + +1 394 JAMES H. ASHER, JR. AND GEORGE W. NACE TABLE 6. Summary of equilibrium conditions for diploid parthenogenetic reproduction by either or both central and terminal fusion. Genotype Fitness Component Fecundity W Zygoid Survival W IV n AA\ W AAa ^AAwAAi Fitness /AA,(eq> > o -> AA\ W AA' 1A'3 > WAA If WAA — kJF A ,, and W , , = •A\ W A'A' W AA> WAA and w w A'A\ A'A'a W,A'S A A3 1 fAA,(°'l) to be a maximum T h e n for V I A'A' AA' AA AAr A (b) k 1 = l / k 2 lent or when the ratio of fitness components of the homozygotes at one stage is equal to the reciprocal of the fitness components for the homozygotes at another stage (i.e., kx = 1/^s) • Table 6 presents a summary of these statements. Numerical examples for au tomictic parthenogenetic reproduction are presented in Nace et al. (1970) and Asher (1970) and are not repeated here. Evaluation of the complete model indicated by matrix B which considers segregation, selection, and post-meiotic mutation will not be presented at this time. However, the comparisons of mutation with segregation, and segregation with selection made in this presentation suggest that segregation and selection have the greatest influence upon the genetic structure of automictic parthenogenetic populations with respect to maintenance of heterozygosity. Mutation can affect this structure; however, this influence is restricted to loci closely linked to the kinetochore. four factors can influence this structure: (1) mutation, (2) segregation, (3) selection, and (4) migration. Among these factors, mutation, segregation, and selection should play major roles. In addition, these factors may have a directional, or deterministic, component as well as a nondirectional, or stochastic, component which must be distinguished when evaluating models. Hence, we discuss briefly the significance of the stochastic component when considering the triploid Ambystoma population. If, for parthenogenetic species that have had a single origin, we assume that migration into and out of the population is proportional to the frequencies of each genotype, migration should play no role in altering their genotypic structure. If, on the other hand, migration is not proportional to genotypic frequency, then, migration becomes a component of selection. Thus, the influence of migration is either considered in the model, i.e., as selection, or it is of no consequence. DISCUSSION One intent of this paper is to present mathematical models which describe the genetic structure of apomictic and automictir parthenogenetic populations, \t least Mutation The influence of mutation upon genetic structure is markedl) different ior apomicfic and. automictic populations, Dof.crmin- GENETICS OF PARTHENOGENETIC POPULATIONS isLic predictions state that in the absence of segregation and selection mutation should lead to polymorphisms of all loci in a triploid apomictic population (See equations 10 through 13, and. Table 1). Stochastic predictions, while confirming this conclusion, add the prediction that this polymorphic state will not necessarily be the ancestral state (Table 2) even in cases of small population size (N ^ 15 to 250) as observed for the triploid Ambystoma (Wilbur, 1971). This deviation from the ancestral state is equally true for so-called "neutral" variability. Evidence presented by Uzzell and Goldblatt (1967) suggest to the contrary that serum protein genotypes (AAA' and A A'A') postulated for the two ancestral triploids have been maintained in the "modern" triploids. The absence of this divergence from the ancestral genotype with respect to a serum protein and various enzymes (Uzzell, personal communication) suggests that, while mutations probably do occur, other factors with homeostatic influence act upon and dominate the composition of the genome of the triploids. Hence, the protein phenotypes of these triploids are not neutral. Within automictic populations, heterozygosity is maintained by mutation at loci closely linked to the kinetochore in populations reproducing by central fusion or inhibition of meiosis I. The frequency of equilibrium heterozygosity decreases rapidly, however, as y increases from zero. These results support and quantify the predictions made by Carson (1967). Considerations of equilibrium heterozygosity in populations reproducing by terminal fusion or inhibition of meiosis II, on the other hand, indicate that these automictic populations should be completely homozygous for all loci regardless of linkage relationships. These observations lead to the general conclusion that mutation plays a relatively small role in maintaining heterozygosity in automictic species and, therefore, that selection is absolutely essential to the retention of this variability. In the absence of selection, very few loci should exist in a polymorphic state. 395 Segregation Within apomictic species, segregation occurs infrequently because of a lack of synapsis and genetic exchange. With a relatively small probability of synapsis, apomictic populations, in the absence of mutation and selection, should become homozygous (Table 3). Using values of the probability of synapsis (.005) suggested by the data of Macgregor and Uzzell (1964), triploid Ambystoma should become completely homozygous within 5,000 generations (Table 3). By contrast, complete homozygosity should be produced within 30 generations for many loci in automictic populations (Asher, 1970). Thus, any mode of parthenogenetic reproduction which allows even small proportions of auto-segregation should, in the absence of mutation or selection, become completely homozygous. The importance of small amounts of auto-segregation in apomictic species was suggested both by Gustafsson (1942) and Suomalainen (1961). The data presented in Table 3 quantify the importance of this small amount of segregation and contradicts the predictions made by Darlington (1937), White (1948), and others that this mode of reproduction must lead to complete heterozygosity. Mutation, Segregation, and Selection Values for equilibrium states predicted by models which incorporate the simultaneous occurrence of mutation and segregation for triploid apomictic populations are presented in Table 4. The equilibrium states are markedly different from those expected with mutation alone. (Compare the values in Tables 1 and 4, and the upper left values of Table 5). Since the data of Uzzell and Goldblatt (1967) which indicate that triploid Ambystoma maintain their ancestral heterozygosity contradict the predictions made in Table 4 and the case of equal fitness in Table 5, two alternative hypotheses are available: (1) segregation and mutation are not affecting the genome, or (2) selection is maintaining the constancy of the genome. 396 JAMES H. ASHER, JR. AND GEORGE W. NACE Table 5 illustrates the influence of selec- obtained, in part, for Ambystoma triploids. tion upon the genetic structure of triploid Using this information and the models apomiclic populations where segregation, presented in this paper, we have remutation and selection occur simultane- examined predictions made with respect to ously. These data suggest that for triploid the role of parthenogenetic reproduction Ambystoma where both segregation and in the evolution of these species. mutation are expected to be rare events It has been stated that species employing heavy selective pressures are required to the parthenogenetic mode of reproducretain the ancestral genotype through tion represent evolutionary "dead ends," time. In this case, it appears that the de- i.e., the probability of parthenogenetic spependence of the triploids, e.g. A. plat- cies becoming extinct is one, while for ineum, upon the diploids, e.g. A. jefjer- bisexual species this probability is less than sonianum, for egg activation has allowed one. This evolutionary prediction was little or no genetic deviation from the based upon two general types of speculatriploid ancestral genome. tion. First, parthenogenetic populations must lack genetic variability: i.e., apomicSince mutation plays such a small role in maintaining heterozygosity in automictic tic species become completely heterozygous populations, we have only considered the while automictic species become comjoint effects of selection with segregation in pletely homozygous (Darlington, 1937; this paper. Equation 27 represents the White, 1948; Suomalainen, 1950; and othequilibrium state for this mode of repro- ers). Second, newly arising benefical mutaduction and represents a generalization of tions cannot be incorporated into the equation 8 presented by Asher (1970). genomes of asexual species as rapidly as The conditions for maintaining stable they can in bisexual species (Muller, 1932, non-zero polymorphisms in these popula- 1964; Crow and Kimura, 1965). The tions are given in Table 6. In this case, the present paper deals specifically with the fitness of the heterozygote must be greater first set of speculations. Although the modthan either homozygote. els are severely limited in their predictive power because they are restricted to onelocus, two-allele systems, they do provide a Evolutionary Considerations series of illuminating predictions with reOne of the goals of population biology is spect to genetic variability in parthenogento develop theories which describe and etic populations. predict evolutionary change. The miniBased upon these models, it appears that mum information needed to make such previous speculations made about genetic predictions is: (1) the mechanism of variability in parthenogenetic populations reproduction, (2) the number of gener- are not correct. First, apomictic populaations of reproduction, (3) the initial tions should not become completely composition of the genome, and (4) the heterozygous as a consequence of mutation fitness of a particular genomic configura- alone but should attain some equilibrium tion at each generation. The function of state in which there exist both homozypopulation genetics is to develop models gotes and heterozygotes. Selection and small which use this information to predict the amounts of segregation should drastically change in the genome from an initial alter this equilibrium state. Second, autostate to some future state, thus permitting mictic populations need not be completely an understanding of both the past and fu- homozygous provided selection favors hetture evolution of a species and providing eio/.)gOiity. The magnitude of the selecthe data base needed to anticipate the tion which is needed is rigorously determanner in which imposed changes may mined by the mode ot automictic reproinfluence this evolution. It is recognized duction and linkage relationships (Table that acquisition of this information is a 6, see also Asher, 1970). These two predicmajor undertaking; however, it has been tions thus represent a new starting point GENETICS OF PARTHENOGENETIC POPULATIONS for evaluating evolutionary change in parthenogenetic populations. While analyzing the models developed to describe apomictic reproduction, we were led to the conclusion that apomictic species should not retain their ancestral genotype. 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