Heteroplasmy and Organelle Gene Dynamics

Corresponding author: Ronald K. Chesser, Savannah River Ecology Laboratory, P.O. Drawer E, Aiken, SC 29802., chesser{at}srel.edu (E-mail).

Communicating editor: B. S. WEIR

	ABSTRACT

TOP ABSTRACT ORGANELLE DIVERSITY IN ONTOGENY SPATIAL AND TEMPORAL DYNAMICS DISCUSSION LITERATURE CITED

This study assesses factors that influence the rates of change of organelle gene diversity and the maintenance of heteroplasmy. Losses of organelle gene diversity within individuals via vegetative segregation during ontogeny are paramount to resultant spatial and temporal patterns. Steady-state losses of organelle variation from the zygote to the gametes are determined by the effective number of organelles, which will be approximately equal to the number of intracellular organelles if random segregation prevails. Both rapid increases in organelle number after zygote formation and reductions at germ lines will reduce variation within individuals. Terminal reductions in organelles must be to very low copy numbers (<5) for substantial losses in variation to occur rapidly. Nonrandom clonal expansion and vegetative segregation during gametogenesis may be effective in reducing genetic variation in gametes. If organelles are uniparentally inherited, the asymptotic expectations for effective numbers of gametes and spatial differentiation will be identical for homoplasmic and heteroplasmic conditions. The rate of attainment of asymptote for heteroplasmic organelles, however, is governed by the rate of loss of variation during ontogeny. With sex-biased dispersal, the effective number of gametes is maximized when the proportional contributions of the sex having the higher dispersal rate are low.

GENETIC studies are seldom conducted solely to ascertain the status of genetic characters existing in a population. Typically, genetic observations together with spatial data are used to infer important biological information regarding behaviors, movements, and dynamics of ecological processes that influence genetic characters. Genetic information is rapidly becoming a tool that is useful for unprecedented probes for toxicological impacts of man's activities and serves as a sensitive biomarker of individual exposures to mutagens and clastogens (BICKHAM and SMOLEN 1994 ; BAKER et al. 1996 ). As such, genetic data may provide vital details of population histories and probable trajectories over space and time, as well as signatures of chemical and elemental stressors affecting individuals (SHUGART and THEODORAKIS 1994 ). The effectiveness of conclusions based on observed genetic patterns depends on an understanding of which patterns may be expected under natural conditions given the behavioral ecology of the species and on the nature and dynamics of genetic material.

Accurate inference of parameters that produce observed genetic patterns is dependent on the availability of realistic theory. Interpretation of behavioral influences and historical factors on the placement of gene diversity and the rate of its change depends on (1) the mode of inheritance of specific gene characters, (2) tactics of genetic transfer (mating strategies), (3) relative sizes of families and breeding groups, and (4) the rate of genetic exchange (or lack thereof) among other breeding groups within a population. Proliferation of methods of isolation and identification of genetic characters for organelles such as mtDNA, chloroplasts, and paternal markers, as well as nuclear genes, has enhanced the power of population biologists to visualize suites of genes that may have different modes of inheritance (AVISE 1994 ). If these techniques are used in concert, results of genetic studies may convey great power for scientists to predict the factors that give rise to observed patterns. This is particularly true if the parameters necessary to generate the various models are congruent (CHESSER and BAKER 1996 ).

CHESSER and BAKER 1996 showed that spatial differentiation and temporal rates of change for uniparentally inherited organelle genes may be substantially different from those for diparentally inherited nuclear genes. The only difference between the two models was the mode of inheritance of the genetic material. In addition to uniparental inheritance, their models assumed that organelles were homoplasmic; that is, specific genes for organelles were invariant within individuals and in the gametes they produced. Some organelles such as chloroplasts and mitochondria in some taxa are known to be transmitted, at least partially, by both sexes (OHBA et al. 1971 ; KIRK and TILNEY-BASSETT 1978 ; NEALE et al. 1986 ; KONDO et al. 1990 ; AVISE 1991 ; GYLLENSTEN et al. 1991 ) allowing heteroplasmy to arise as a function of organelle variation between mates. Heteroplasmy may also arise when organelles are uniparentally inherited if the contributing gamete contains variable forms of genes (BIRKY 1975 , BIRKY 1994 ; BERMINGHAM et al. 1986 ; LAIPIS et al. 1988 ; Rand AND HARRISON 1989 ).

BIRKY et al. 1983 , BIRKY et al. 1989 considered heteroplasmy in their assessment of loss of genetic diversity of organelles. They speculated that diversity present within the zygote would diminish during cytokinesis because of random assortment of organelles in vegetative segregation. Diversity would be reduced as a function of the number of cell divisions that had transpired since zygote formation. Likewise, diversity among cells would be a function of the number of cell divisions since divergence of the cell lines. BIRKY et al. 1983 , BIRKY et al. 1989 , however, did not consider that there may be variation of organelle genes among progeny produced from different gametogenic tissues in the same organism. Gametes may not covary for all genetic variance lost during their ontogeny. Losses of diversity within gametes during ontogeny may be partially offset by variation among gametes, particularly among those produced by different gametogenic tissues. Therefore, consideration of heteroplasmy of organelle genes presents some unique problems, and accurate accounting for potential losses of variation during ontogeny is of paramount importance for estimating dynamics of heteroplasmic organelle genes.

This work presents an extension of several earlier models, each of which invoked various assumptions regarding development and transmission of organelle genes (BIRKY et al. 1983 , 1988; TAKAHATA and PALUMBI 1985 ; CLARK 1988 ; CHESSER and BAKER 1996 ). The works of BIRKY et al. 1983 , BIRKY et al. 1989 (also see BACKER and BIRKY 1985 ; BIRKY 1994 ) set the stage for applications of theoretical and empirical research into patterns of organelle variation. Prior models have documented the negative relationship between spatial differentiation and rate of loss of genetic variation (CHESSER et al. 1993 , CHESSER et al. 1996 ). In this article I develop models to estimate the spatial patterns and temporal dynamics of organelle genes that may be heteroplasmic and that may be conveyed to progeny via maternal, paternal, or diparental contributions. Because heteroplasmy may arise by variation among uniparentally contributed genes, variation between gametes of diparentally inherited genes, or by mutation of organelle genes subsequent to zygote formation (Figure 1), there is considerable attention given to the mode of inheritance and differentiation during ontogeny. The models are applicable to heteroplasmy in zygotes, potential losses of variation in the zygote during ontogeny and gametogenesis, and finally, to the genetic characterization of gametes contributed to the subsequent generation. Finally, models predicting the spatial differentiation and temporal dynamics of organelle genes are presented. The models will apply either to subdivided populations in which subunits may exchange genetic material at specified rates or to single, isolated populations with no emigration or immigration. Also, when possible, the parameter definitions and symbol denotations will follow those presented in earlier models (CHESSER 1991A , CHESSER 1991B ; CHESSER et al. 1993 ; CHESSER and BAKER 1996 ).

View larger version (21K): In this window In a new window Download PPT slide   Figure 1. Diagram depicting heteroplasmy as a function of variation between mates for diparentally inherited organelles (A) and via variation within the gamete of the contributing sex when organelles are inherited from a single parent (e.g., B). Of course mutation can cause heteroplasmy regardless of the means of organelle inheritance, and variation can be present in the gametes of either or both parents when organelle genes are diparentally transmitted.

	ORGANELLE DIVERSITY IN ONTOGENY

TOP ABSTRACT ORGANELLE DIVERSITY IN ONTOGENY SPATIAL AND TEMPORAL DYNAMICS DISCUSSION LITERATURE CITED

Determination of the dynamics of organelle gene diversity is fundamentally different from that for diploid, segregating genes. Unlike nuclear genes, variation of organelle genes within cells may be lost in the process of vegetative segregation during mitosis or meiosis (BIRKY et al. 1989 ). As a result, tissues may exhibit quite different organelle genotypes (e.g., MARCHINGTON et al. 1997 ) depending on the number of cell divisions that have transpired since they shared a common parental cell. Somatic variations may occur such that particular organs are homoplasmic while others are either heteroplasmic or homoplasmic for different organelle types (KOEHLER et al. 1991 ; CIAFALONI et al. 1992 ; MATTHEWS et al. 1994 ; SHITARA et al. 1998 ). If organelle diversity is lost via a random, steady-state process during cell divisions, then organs derived early in ontogeny would be expected to possess greater proportions of the original variation present in the zygote. Other evidence, however, has shown that tissue-specific heteroplasmy may be a function of mitotic activity. Brain cells and oocytes have high incidence of heteroplasmy relative to tissues with greater mitotic activity (JAZIN et al. 1996 ; MARCHINGTON et al. 1997 ). Heteroplasmy is also more prevalent in older individuals, showing that accumulation of mutations over time affects the diversity of organelle genes (JAZIN et al. 1996 ).

To understand the progressive changes of organelle genetic variation, it is important to determine the fraction of the diversity in the zygote that is passed on to progeny. Therefore, the variations within and among the gametes are pivotal to the determination of spatial and temporal dynamics of organelle genetic diversity. Figure 2 depicts three examples of the ontogeny of gonadal tissues subsequent to zygote formation. Clearly, each progeny is expected to have lower variance of organelles due to losses in cell divisions (BIRKY et al. 1983 , BIRKY et al. 1989 ). However, some portion of the original variation of the zygote potentially may be conserved among progeny borne from gametes contributed by the same ovary or testis or among different gonads unless there exists some concerted mechanism for the maintenance of organelle homogeneity within organisms (e.g., BIRKY 1994 ). The determination of the rate of loss of organelle diversity during ontogeny is replete with difficulties. BIRKY et al. 1983 , BIRKY et al. 1989 assumed a steady-state loss dependent on pairing of organelles in daughter cells or relaxed partitioning (BIRKY 1994 ). Empirical studies, however, have concluded that the process of loss may be rather abrupt with bottlenecks limiting the number of organelle types in gametes (KOEHLER et al. 1991 ; AVISE 1994 ). Reductions in organelle numbers are often cited as the primary mechanism for relative homogeneity of gene variation within individuals and the general rarity of heteroplasmy (BROWN 1985 ; TAKAHATA 1985 ; CLARK 1988 ; LAIPIS et al. 1988 ; Rand AND HARRISON 1989 ). Others, however, have suggested that rapid clonal expansion of a limited number of organelles may reduce the heteroplasmy of gametes (POULTON 1995 ; MARCHINGTON et al. 1997 ; MEIRELLES and SMITH 1998 ).

View larger version (22K): In this window In a new window Download PPT slide   Figure 2. Diagrammatic views of some possible scenarios for development of gonads (G) and gametes from the zygote (Z) during ontogeny. Values of a denote the number of cell replications that take place before separation of a common daughter cell that leads to development of separate gonads, b is the number of cell divisions that transpire before the gonads begin development, and c cell replications take place subsequent to the sharing of a common daughter cell for the separate gametes within gonads.

Correlations of organelle genes within and among cells may be due to either identity by state (IBS) or identity by descent (IBD; COCKERHAM 1969 ). IBS refers to genes that are identical, but there is no measure of recent common ancestry that leads to that identity. IBD denotes genes that are identical because of known processes of inheritance through which the genes are copies of one another. Whereas both modes of identity are important for assessing the status of gene diversity, it is only the IBD that confers meaningful information relative to the original variation at some (sometimes arbitrary) starting point (COCKERHAM 1973 ). The original genetic variation in the zygote is V_z = q₀(1 - q₀), where q₀ is the frequency of an organelle gene at time zero (zygote; cf., COCKERHAM 1967 , COCKERHAM 1969 , COCKERHAM 1973 ; CHESSER 1991A , CHESSER 1991B ). All identity of genes in the initial generation (zygote) is due to IBS, because there is no known ancestral history of genes. Without any systematic processes that may result in some genes leaving more descendants than others (thus, increasing IBD), then the expected genetic variance within and among cells of the developing embryo will be unchanged from the original variance, V_z. Therefore, it is important to define the random and systematic processes that lead to accumulations of organelle genes that are IBD if gene dynamics and spatial patterns are to be understood. The patterns of genetic variance derived from this process should accurately reflect empirical results if the assumptions of the model are not violated (COCKERHAM 1973 , pp. 681–682; see also SUGG et al. 1996 ; DOBSON et al. 1997 ). Throughout this article, therefore, I use the term "gene correlation" in derived expressions. However, these correlations could also be defined as the "probability of identity by descent." First, I address those processes that affect changes from the amount of organelle genetic variation that an individual inherits to that which it transmits to progeny by way of gametes. Second, I consider mating and dispersal tactics that influence the variation among individuals and the dynamics of organelle genes within populations.

I define ₁, ₂, and ₃ as the expected correlation of frequencies for organelle genes within gametes, among gametes within gonads (e.g., ovary or testis), and among gametes from different gonads, respectively (cf., COCKERHAM 1969 , COCKERHAM 1973 ). Each of these expected correlations (or probability of IBD) is assumed to be zero at the time the zygote is initially formed. Because I am addressing genetic changes that may take place within an individual, all identity of genes in the initial stage is considered to be IBS, and the initial correlations of genes are zero even if they actually may not be (COCKERHAM 1973 , pp. 681–682). Figure 3 depicts a simple scenario for the accumulation of organelle gene correlations within and between a pair of gonads during ontogeny. Before divergence of cells leading to the separate gonads (a cell divisions) all cells covary in the amount of change in gene diversity. After divergence, each separate cell line leading to formation of the gonads covaries for the loss of gene diversity up to the time of gamete formation (b cell divisions). Loss of gene diversity within cells continues independently for the c cell divisions required for gamete formation. Thus, organelle genes for cells in separate cell lines leading to gametes for the different gonads diverge (presumably at random) for b + c cell divisions. Correlations within and among organelle genes may accumulate within an embryo because of different rates of replication for organelles, isolation (and random differentiation) of cell lines during ontogeny and reductions or increases in organelle numbers during cytokinesis. These organelle gene correlations are measured relative to the original genetic variation in the zygote, and not to other individuals in the population. Assuming no recombination of organelle genes, the gene correlations between pairs of organelles within and between cells are either zero or one, indicating that organelles are identical by descent (1), or not related during ontogeny (0; COCKERHAM 1969 ; SUGG et al. 1996 ). The resultant correlations are relative to the original variation in the zygote (V_z) such that the amount of variation remaining at any time t is (1 - ^(t)₁)V_z within gametes, (1 - ^(t)₂)V_z among gametes within the same gonad, and (1 - ^(t)₃)V_z among gametes from different gonads (cf., COCKERHAM 1969 , COCKERHAM 1973 ). The probability of heteroplasmic gametes, therefore, is V_z(1 - ₁). Of course, when variation within the population disappears, no individual variation exists (V_z = 0), and the process of vegetative segregation is irrelevant except for the occurrence of novel mutations. Thus, the gene correlations denote the expected proportion of the original genetic variation in the zygote that has been lost within gametes, within gonads, and among gonads, due to random or systematic processes during ontogeny.

View larger version (57K): In this window In a new window Download PPT slide   Figure 3. Graphic representation of the accumulation of gene correlations within gametes (1), among gametes within the same gonad (2), and among gametes of different gonads (3) as a function of the number of cell divisions since divergence of sister cells. The symbols a, b, and c represent the number of cell divisions until the cell lines diverge to separate gonads, separate gametocytes, and separate gametes, respectively. Organelle genes will covary for variation lost during the phases in ontogeny that they share in common. Divergence of cell lines will result in divergence among the lines for the genetic variation that remained at the time of divergence. Rates of divergence are a function of the dominant eigenvalue () and the number of cell divisions occurring during each phase (see text).

Following Figure 2 and Figure 3I assume there are a cell divisions before the bifurcation of cell lines that give rise to the primordial gonad tissues, b cell divisions occur before the gonad tissues are formed, and c cell divisions occur in the formation, maturation, and eventual contribution to progeny of the gametes within each gonad. Estimation of the rate of decay of organelle diversity is a function of the number of organelles within cells. For mitochondria, it appears that a reduction in number occurs before gamete formation, and that the numbers increase rapidly after zygote formation (KOEHLER et al. 1991 ). For simplicity, I assume that a zygote instantaneously increases the number of organelles (m_z) to result in an average of r replicates per organelle in the cell, and that the number of organelles per cell thereafter is m_zr. The replication of organelles increases the average correlation of organelle genes within the zygote because a measurable fraction of organelles will now be identical by common descent (sister organelles). The change in gene correlation within a zygote (_z) after organelle replication is shown in Figure 4. Therefore, after replication of organelles the average correlation of genes within the cell is

(1)

where r_i denotes the number of replicates of the ith organelle, ²_r is the variance in the number of replicates across organelles, and rm_z (rm_z - 1)/2 is the number of pairs of different organelles within cells (see Figure 4; cf., CHESSER et al. 1993 ).

View larger version (27K): In this window In a new window Download PPT slide   Figure 4. This graphic depicts the replication of organelles of a particular type within a cell (such as the zygote). The original cell contains 5 primary organelles. After replication, the cell contains 10 organelles, but variation in the number of replicates is evident. The initial pair-wise correlations of organelle genes are all zero. After replication, however, some organelles are identical by common descent and have correlations of unity.

Organelles also replicate before cell division and the organelles will be distributed between the two daughter cells during cytokinesis. Hereafter, the z subscript will be dropped because cell divisions have progressed beyond the zygote. Therefore, it is assumed that each organelle replicates, yielding a total of mr organelles to be equally divided between the two daughter cells during cytokinesis (e.g., BIRKY 1983 , BIRKY 1987 ). The possible pairings of homologous and analogous organelles after replication are precisely the same as shown in the example in Figure 4. Therefore, the distribution of organelle pairs IBD (sister organelles) incorporated into a daughter cell assumes a binomial distribution. For organelle genes it is possible that an organelle is joined by a sister organelle in the same daughter cell (Pr = ), or that each enters a separate daughter cell (Pr = ). Because mitosis results in two sister cells, there are a total of (mr/2)((mr/2) - 1)/2 possible pairings of organelles within a daughter cell, each with a probability of ¹/₂. Of these pairings, however, the expected number of pairs that are IBD (correlation is unity) is (see middle term of Equation 1) . Therefore, the expected average correlation of organelle genes within daughter cells is

(2a)

where t indicates the progression of cell divisions. The rate of change of gene correlations is determined by the dominant eigenvalue (), which is defined by 1 - , whereas the effective size of organelles during cell divisions (n_e) is (1 - )^-1. The effective number of organelles is defined as the hypothetical number of organelles undergoing identical and exact replication and segregation processes (¹/₂ to each daughter cell) that would produce a particular rate of change in organelle genetic variance. Using Equation 2a, these two variables become

(2b)

After t cell divisions the expected correlation of organelle genes within cells is determined as

(2c)

Because the number of organelles within cells is often very large (PIKO and MATSUMOTO 1976 ), the following equations will also suffice except in cases of severe reductions of organelles or nonrandom clonal expansion:

(2d)

If the number of mitochondria within cells remains constant throughout future cell divisions, then r = 2, and n_e = . Obviously, when the variance in replicates over organelles is zero, the effective number of organelles is approximately equal to twice the number of organelles before replication. If the variance equals the mean replication number (²_r = 2) , then the n_e is approximately the number of organelles before replication.

BIRKY 1983 , BIRKY 1987 , BIRKY 1994 suggested that often sister organelles, such as mitochondria, may remain in close proximity in the cytoplasm and therefore migrate to the same daughter cell during cytokinesis. Nonrandom sorting of sister organelles in this manner will lead to greater proportions of organelle genes IBD present within cells during vegetative segregation than if migration were random. Likewise, repulsion of sister organelles within the cytoplasm may yield lower frequency of organelle genes IBD. If y is the probability that two sister organelles will migrate to the same daughter cell after cytokinesis (y > ¹/₂ yields positive association, y < ¹/₂ is negative association or avoidance; 0 y 1) then

(3a)

and

(3b)

Obviously, complete avoidance of sister organelles (y = 0) will result in infinite effective numbers of organelles and no loss of organelle gene diversity. A y value of unity will halve the effective number and thereby double the rate of loss of genetic diversity within cells over that of random transfer of organelles between cells.

Reduction in organelle numbers will also affect the correlations between organelle genes within cells. It would appear that Equation 2aEquation 2bEquation 2cEquation 2d and Equation 3aEquation 3b could be used for organelle reductions as well. Organelles that are eliminated would have r_i = 0 while those maintained would yield r_i = 1, assuming no replication of remaining organelles. This scenario, however, will always result in the term, ²_r + r(r - 1) = 0 , and thus, = 1 and infinite n_e. These results imply that organelle reductions would result in no loss of genetic variance. Expectations of gene correlations from the above scenario, however, are not consistent with the binomial probabilities assumed in Equation 2aEquation 2bEquation 2cEquation 2d and Equation 3aEquation 3b because organelles with zero replicates are not candidates for selection within cells. Assuming no recombination of organelle genes, the number of pairs of organelles IBD within a cell before reduction in organelle number is expected to equal . If the number of organelles is reduced from m to m - R, assigning P = as the number of pairs of organelles and i as the number of airs IBD, the average correlation (₁) of genes between organelles within a cell becomes

(4)

It is clear from this equation that reductions in organelle numbers at the zygote stage would not affect the expected gene correlations for the zygote cell. Because ₁ at the zygote stage is zero, the expected gene correlation after reduction would remain at zero. Although this expression is undefined when R = m - 1, it is obvious that reduction to a single organelle will result in a total loss of genetic variance. Figure 5 shows that the expected genetic variation for organelles within cells is virtually unchanged by reduction in numbers of organelles unless the number remaining (m - R) is less than about five. Therefore, for bottlenecks in organelle numbers to serve as the primary means of promoting homoplasmy (₁ = 1.0) of gametes (and thus, zygotes if there is uniparental inheritance; AVISE 1994 , p. 62), the reduction must result in very few organelles in gametes (m - R < 5; R > 995 in Figure 5). This is consistent with estimates for postulated bottlenecks during oogenesis for human mtDNA. HAUSWIRTH and LAIPIS 1985 reported that the reduction yielded between 1 and 6 mitochondria, and that the segregating unit could be a single mitochondrion. Of course, nonrandom selection of organelles could take place in germ cell lines, or the value of ₁ could be very nearly unity before organelle reduction. Subsequent cell division beyond the stage of organelle reduction will result in a as defined in Equation 2aEquation 2bEquation 2cEquation 2d or Equation 3aEquation 3b, but with a reduced value of m, leading to further increases in ₁. However, reductions are presumed to occur primarily in germ cell generations and, thus, near the final stage of cell division (HAUSWIRTH and LAIPIS 1985 ; AVISE 1994 ).

View larger version (18K): In this window In a new window Download PPT slide   Figure 5. Accumulation of organelle gene correlations within gametes as a function of the correlation before organelle reduction (R) and the number of organelles remaining after reduction has transpired (m - R). In all examples the initial number of organelles in a cell was 1000. Note that substantial change in gene correlations does not occur unless the remaining number of organelles is very low (5).

Homoplasmy of gametes may be promoted by differential replication of organelles rather than reduction in numbers. It is possible that some organelles become quiescent during cytokinesis, while others undergo rapid proliferation. Thus, one or more organelles may replicate manyfold (r_i 1), whereas others would undergo no replication (r_i = 1). The most rapid rate of loss of gene diversity within cells would be realized if all daughter organelles were the product of a single organelle. In this case, the variance in replicate number would equal (m - 1)(r - 1)², where m is the original number of organelles within the parental cell, and r is the average number of organelle replicates. Assuming random vegetative segregation of organelles into the separate daughter cells, the eigenvalue would become (from Equation 2b) = 1 - and n_e = . Differential replication of this sort can result in dramatic reductions in organelle n_e and a rapid approach to homoplasmy over successive cell divisions. For example, consider a maternal cell that contains 100 mitochondria. One mitochondrion yields 101 replicates while the remaining 99 yield only one each (they do not duplicate themselves). The average number of replicates per organelle (r) is 2, and the variance in replicate number is 99. Although there are 200 mitochondria segregating between the two cells, the effective number is only 3.9. After sixteen generations of cell divisions, approximately 99% of the original organelle gene diversity will be lost.

Reductions of organelle numbers, clonal expansion, or random segregation processes within cells will not affect the expectation of correlations between organelle genes between gametes within gonads (₂) or between gametes of different gonads (₃). This can be confirmed using m as the reduced number of organelles within gametes. The possible number of organelle pairs between cells is P = m², and i is the number of pairs of organelles that are identical by common descent, giving the following expected gene correlations:

(5)

Thus, although organelle reductions and random segregation events may alter gene diversity within cells, remaining variation among separate lineages of cells will be conserved. Even if reductions of organelle numbers or losses in diversity during vegetative segregation result in homoplasmy of gametes, there may be measurable variation between progeny produced by gametes from different gonads.

If there are total gonads (Figure 2), with the jth gonad producing g_j of a total of viable progeny produced by a female, then the expectations for the correlation of organelle genes within and among gametes are (assuming negligible effects of initial increases or terminal reductions in organelle numbers)

(6a)

Estimation of the correlation among gametes among gonads (₃) must be restricted to include only pairwise comparisons of gametes from different gonads. Therefore, a triangular matrix such as those shown in Figure 3 must be employed. Each element of the matrix (zeros and ones), however, will be represented by a rectangular submatrix rather than by the expected correlation between a pair of organelles. Each submatrix will contain the expected organelle gene correlation between pairs of gametes produced by two different gonads. Thus, for the jth and ith gonads, the submatrix would contain g_jg_i expected correlation values (zeros and ones). The organelle genes will covary only for the portion of their ontogeny before bifurcation of cell lines to form different gonads (Figure 2 and Figure 3). This expression includes the variance in progeny numbers produced by gonads (²_g)

(6b)

The proportion of progeny resulting from gametes produced by the same gonad (_g) depends on the mean () and variance (²_g) of the number of progeny produced by individual gonads

(7)

Using these expressions, the weighted average correlation of organelle types between different gametes produced by a parent is

(8)

Variations in the number of gametes contributed by gonads may affect the average correlation of genes within gametes. Thus, assigning _1(i) as the expected correlation of organelle genes within gametes of the ith gonad, the weighted average is ₁ = ()^-1g_i1(i) .

The proportion of genetic variation found among organelles within gametes (f_oT; the subscript T represents the total genetic variance remaining within the pool of all gametes), organelles among gametes within gonads (f_oG), and among gonads (f_GT) can be computed as

(9)

Using Equation 6aEquation 6b, these f-statistics for differentiation of organelle genes during ontogeny are

(10a)

with each denoting the weighted averages shown in Equation 6a and Equation 6b. Equation 10a conforms to the expectation that (1 - f_oT) = (1 - f_oG)(1 - f_GT) (cf., WRIGHT 1951 ).

Equation 10a can be simplified for organisms that have paired gonads (e.g., mammals) and in which the average numbers of cell divisions for intervals b and c are independent quantities (cov(b,c) = 0). In such cases,

(10b)

This equation also applies when comparing a pair of tissues separately. In that case, the average values of b and c would apply to a specific pair of tissues. An empirical analysis of organelle gene diversity between or among tissues will not yield absolute values of a, b, or c; however, if decay rates per cell division (1 - ) are equal for lineages leading to gametes in separate tissues, the relative number of cell divisions for b and c can be estimated using various combinations of the fixation indices, as

(11)

The average number of cell divisions that had taken place before separation of cells giving rise to the gonads (a) cannot be estimated without knowledge of the original variance in the zygote.

Thus far, emphasis has been placed on comparisons of genetic variation within and among gametes. The rationale for such an emphasis is that the expectations of gene correlations within and among gametes are necessary for determining the transitions to subsequent generations. These equations are relevant, however, for comparing cells of any tissues where ₃ is the fraction of genetic variation that has already been lost before the tissues diverged from a common daughter cell. When considering the variation of organelle types for particular cells, such as gametes, relative to the original variation in the zygote, then ₃ 0, f_oZ = ₁, f_GZ = ₂, and f_oG = . In heteroplasmic organisms such relationships could be useful for estimating the rates of loss for organelle genotypes during particular stages of ontogeny and for phylogenetic relationships among cells of various tissues. Various tissues, such as kidneys, lungs, eyes, and leaves, may exhibit different interval lengths for a, b, and c (obviously without contribution to progeny), and thus have different patterns of differentiation from those seen in gametes (MARCHINGTON et al. 1997 ). Because = 1 - , three different estimates of n_e are possible if interval lengths and intraorganism f-statistics are known. For organisms with independent interval lengths during ontogeny, these are

(12)

These three effective numbers are expected to be equal to one another if the processes of segregation and replication do not change throughout ontogeny. Comparisons of rates of divergence of various tissues can be useful for determining which phases of ontogeny may undergo shifts in segregation and/or replication events, thereby affecting the magnitude of heteroplasmy. Because particular diseases are often associated with mitochondrial heteroplasmy (POULTON 1995 ), this information may assist in predicting the processes that influence the expression of variant organelle genes and which organs may be at greater risk.

Cytoplasmic organelle DNA, particularly mitochondrial DNA, are believed to experience high mutation rates due to lack of known repair mechanisms for errors that arise during replication (WILSON et al. 1985 ; AVISE 1994 ). Mutations to organelle genes may occur during cell proliferation from the zygote to the derived gonads and gametes (BENTZEN et al. 1988 ; BENDALL et al. 1996 ). Correlations of organelle genes within gametes and between gametes within and among gonads will usually be reduced by random mutations. When mutation rate (µ) per cell replication is included, the recursive equation for the correlation of a random pair of organelle genes within a cell is

(13)

where (1 - µ)² is the probability that neither of the random pair of genes has mutated. Assuming that ⁽⁰⁾₁ = 0 , after a + b + c cell divisions (zygote to gamete ontogeny), the correlation of organelle genes within a gamete is (ignoring terms with µ²)

(14)

The correlation of organelle genes among gametes within the same gonad is similar to Equation 14 above, except through a + b, rather than a + b + c cell divisions. After separation from a common daughter cell, the organelle genes for cells leading to the gametes may in fact diverge because of mutations that erode the correlations accrued before that time. The probability that neither of the pair of identical organelles has mutated after c cell divisions is (1 - µ)^2c. Therefore, the value of ₂ becomes

(15)

Likewise, the correlation of organelle genes among gametes from different gonads will typically have greater divergence with ongoing mutation because b + c cell divisions separate the cells from their common daughter cell. Therefore,

(16)

These equations regarding mutation rate per cell division may seem laborious and superfluous, especially since µ could be defined as the probability of mutation during ontogeny (zygote to gamete release), yielding much simpler forms like ₁ = ₁(1 - µ)², ₂ = ₂(1 - µ)², ₃ = ₃(1 - µ)². If mutation rates and effective sizes of cellular organelles remain constant during all phases of ontogeny, then all lines depicting organelle gene correlations in Figure 3 will be shifted downward proportionally. Thus, the divergence of organelle genes between tissues will still be relative to the times since divergence of cell lines (Equation 10aEquation 10b and Equation 11). The utility of either model for mutation is dependent on the mechanisms under study. If evaluation of cells is to be done after the processes of vegetative segregation are complete, then the latter are sufficient. However, studies of cell differentiation and organelle gene dynamics during intermediate stages of ontogeny will probably require consideration of mutation events over finer time scales such as per cell division (Equation 14Equation 15Equation 16). WILSON et al. 1985 suggest that mutations that arise during replication of organelles are primarily responsible for their rapid evolution. Mutations of organelle genes, however, may accumulate over time in tissues that do not undergo mitosis (JAZIN et al. 1996 ). If such mutations accumulate linearly over time, and if it is assumed that the mutations produce novel rearrangements, correlations within cells, between cells within tissues, and between cells of different tissues would be ~₁(1 - µ)^2t, ₂(1 - µ)^2t, and ₃(1 - µ)^2t, respectively; t references a particular time frame over which mutation rate is estimable, and ₁, ₂, and ₃ are gene correlations that had accrued before the time of mitotic quiescence of the tissue.

The previous equations including mutation do not include possible increases or reductions in cellular organelles. If vegetative segregation of organelles is considered to undergo a three-step process: (1) rapid organelle proliferation in the zygote or shortly thereafter, (2) proliferation of cells over a + b + c cellular divisions with relatively constant processes of organelle distribution between sister cells, and (3) rapid reduction in organelle number at the time of gamete formation, then, using equations above, the expected correlation of organelle genes within cells at the termination of this process is determined by combining Equation 4 and Equation 14, yielding

(17)

Expectations for ₂ and ₃ remain unchanged from Equation 15 and Equation 16.

The equations presented above are intended to represent a probabilistic set of scenarios for organelle differentiation during ontogeny. Estimates for cellular differentiation, effective numbers of organelles, and relative divergence times for cell lines during ontogeny may serve as null models for assessing processes affecting somatic and gametic organelle gene variation. It is likely that many alternative arrangements can be envisioned for partitioning organelle diversity and conveyance of genetic variation to progeny (see BIRKY 1983 , BIRKY 1987 , BIRKY 1994 for reviews). Of course, if the losses of organelle variation are not constant over cell divisions, then the correlations could be represented simply as ₁ = 1 - _a+b+c, ₂ = 1 - _a+b, and ₃ = 1 - _a with each 1 - depicting the proportion of organelle diversity lost during the complete phase of ontogeny depicted by the subscripts. Exact functions for losses of organelle variation are probably dependent on the taxon considered and are not necessary to develop the general mathematical models presented in the next section. For expressions derived below, I am not concerned with the specific details of losses during ontogeny and present equations in terms of ₁, ₂, ₃, and . The manner in which gonads develop and gametes are formed and distributed to progeny may differ considerably between the sexes. Therefore, in equations developed below I use superscripts P and M to denote paternal and maternal gametes, respectively.

	SPATIAL AND TEMPORAL DYNAMICS

TOP ABSTRACT ORGANELLE DIVERSITY IN ONTOGENY SPATIAL AND TEMPORAL DYNAMICS DISCUSSION LITERATURE CITED

To determine the dynamics of organelle genes through space and time, I use several parameters and methods developed in models for nuclear genes and homoplasmy and uniparental inheritance (CHESSER et al. 1993 ; SUGG and CHESSER 1994 ; CHESSER and BAKER 1996 ) with some refinements. A synopsis of some parameters is provided below:

is the probability that random pairs of progeny born within breeding groups are born from the same mother, where k, ²_k are the mean and variance, respectively, of number of progeny per female;

is the probability that random females within breeding groups mate with the same male, where l, ²_l are the mean and variance, respectively, of numbers of female mates per male;

is the probability that progeny born by a single mother are the product of the same male, where p, ²_p are the mean and variance, respectively, of number of male mates per female;

is the probability that randomly selected progeny within a group are the product of the same male parent (CHESSER and BAKER 1996 ; assumes zero covariance between family size and number of mates per female, family size and number of mates per male, and number of mates per female and number of mates per male); and d_m, d_f are the rates of dispersal of males and females, respectively, among breeding groups within the population, D = d_m + d_f - d_md_f.

In previous models using variance in group size (CHESSER and BAKER 1996 ) I assumed the probability that progeny were either male or female to be one-half ( _f = _m), and the probability that a randomly selected pair of individuals are from the same breeding group was

(18)

with n and ²_n denoting the mean and variance of numbers of females per breeding group. Thus, the number of males and females born within breeding groups was equal. Not all males may breed, however, and _m is calculated over all available males and not simply the number of breeding males (CHESSER and BAKER 1996 ). If the ratios of males and females born within breeding groups vary over the population, then the probability that randomly selected individuals, regardless of sex, were natives of the same breeding group is

(19)

where n_m and n_f are the average numbers of males and females per group and _nm,_nf is the covariance of the number of male and female progeny born within groups. The covariance of male and female numbers over the breeding groups is ( (n_{f(i) - nf})) . For males and females considered separately,

(20)

The probability R that mating occurs between a sire and dam born in different native breeding groups is

(21)

Inbreeding, including potential mating between siblings, half-siblings, or progeny related by group coancestry, would be expected to take place with a probability of 1 - R. Equation 19 is obviously not necessary for characters that are inherited uniparentally; however, it could be incorporated into equations of CHESSER and BAKER 1996 for nuclear gene transitions, substituting R for (1 - )D and D - R for D. Those equations for either uniparentally transmitted genes or diparental genes multiplied by d_m or d_f would use the corresponding equation in (20), above.

To determine the transitions of organelle types over generations I consider that the proportion of organelles contributed to a zygote by a sire is denoted by x whereas that contributed by the dam is 1 - x. Because both males and females may convey organelles to the zygotes, the correlation of types within individuals must be included, unlike uniparental models. The form of this correlation is very similar to inbreeding (F) as in diploid, sexually reproducing models (CHESSER et al. 1993 ) because the gene correlations represent the probabilities of identity by common descent or coancestry (COCKERHAM 1969 , COCKERHAM 1973 ). However, for organelle inheritance a proportion x of the types will be transmitted by the father while 1 - x are contributed by the mother and IBD of organelles within progeny may be conveyed by either or both parents. The average amount of heteroplasmy in a zygote produced by a mating is 1 - F. Thus, "inbreeding" will accumulate within zygotes by the expression

(22)

with superscripts P and M referencing paternal and maternal types, respectively, and _t and _t denoting the average coancestry of organelle genes within and among breeding groups, respectively (CHESSER 1991A , CHESSER 1991B ). The measures of coancestry, or gene correlations (COCKERHAM 1969 , COCKERHAM 1973 ), are relative to the genetic variation in the initial generation [V₀ = ()^A_i=1p_(0)i(1 -p_(0)i)] ; A is the total number of different alleles]. Therefore, the proportion of variation remaining within and among breeding groups in the tth generation is V₀(1 - _t) and V₀(1 - _t), respectively. Equation 22 incorporates the potential loss of genetic variance during ontogeny of the zygote and the development of gametes; for example, at conception the correlation of organelle types in a male is F_t, yet the correlation of organelle types in his gametes, upon maturation, will be F_t (1 -^(P)₁) + ^(P)₁ due to losses in variation during cell divisions in ontogeny and gametogenesis. The zygotes of progeny born in the population will possess a fraction of the initial organelle diversity (V₀) represented as V₀ (1 - F_t+1).

Using _mm, _mf, and _ff to reference the coancestry between male parents, male and female parents, and female parents within breeding groups, respectively, the coancestry for organelle types among progeny born within groups is

(23)

The origination of the two parents may be from the same or different breeding groups, depending on dispersal rates for each sex. The coancestry of male parents is complicated by the fact that _m of the progeny born share the same sire via polygyny and multiple paternity (CHESSER 1991A , CHESSER 1991B ; SUGG and CHESSER 1994 ; CHESSER and BAKER 1996 ). The correlation of organelle types within a male zygote is F_t, and the subsequent correlation among his gametes is F_t(1 - ^(P)) + ^(P) . Therefore, the coancestry among male parents within groups is

(24)

Likewise, _f is the proportion of progeny born within groups that share the same mother (CHESSER et al. 1993 ), and the coancestry of organelle types among mothers within groups is

(25)

Coancestry between mothers and fathers within groups is

(26)

Substituting Equation 24, Equation 25, and Equation 26 into 23, the coancestry of organelles for progeny born within breeding groups is determined as

(27)

Expanding R, the coancestry is

(28)

the correlation of organelle types among individuals from different breeding groups () in the total population is

(29)

which, upon expansion of R, becomes

(30)

From the expressions above a transition matrix, T, for F, , and can be derived. For brevity, I present the expressions without expansion of R,

(31)

Constants are accumulated into a column vector, C, for the accumulation of F, , and as

(32)

such that {F_t+1, _t+1, _t+1} = T {F_t, _t, _t} + C. Equations for the transfer of organelle genes to progeny have not taken mutation into account. Mutation would serve to reduce coancestry between individuals. Assigning the mutation rate over a generation time as (a + b + c)µ, a matrix M is defined as

(33)

and the transition of gene correlations becomes {F_t+1, _t+1, _t+1} = (T · M){F_t, _t, _t} + (1 - )²C. The accumulation of values for F, , and over generations is readily determined by iteration of the matrix multiplication and scalar addition using initial values of zero for each of the state variables after parameter values have been assigned.

For each generation the proportion of remaining genetic variations for organelles found among breeding groups (F_LS), within individuals relative to the variation within their breeding group (F_IL), and within individuals relative to the total variation within the subpopulation (F_IS) (WRIGHT 1969 ) are determined by

(34)

(CHESSER 1991A ; COCKERHAM 1973 ).

Iteration of the matrix multiplication and vector addition also enables the determination of effective population sizes for organelle genes for each generation. As has previously been shown (CHESSER et al. 1993 ) effective population sizes can be derived for each hierarchical level at which gene correlations occur. For uniparentally inherited genes the rate of change of gene diversity per generation is 1/N_e, whereas rates of loss for diploid nuclear genes transmitted by sexual reproduction would be 1/2N_e. The expressions for inbreeding, coancestral, and intergroup effective size for organelle genes that may be inherited proportionally from either or both parents are

(35)

In time, the effective sizes become identical at a value referred to as the asymptotic effective size, Nˆ_e (CHESSER et al. 1993 ). Therefore, the expected rate of change of variation for organelle genes per generation is 1/(1 + 4x(1 - x))Nˆ_e.

Clearly, with the proliferation of techniques available to assess gene diversity for genes with variable modes of transmission and disparous contributions by parents, the concept of the effective population size becomes unclear. WRIGHT's (1931) original intent was to describe the number of individuals of an ideal population that would confer the same rate of change of genetic variation as that of a particular nonideal population in question. For most applications of effective population size, it is indeed the rate of change of gene diversity that is sought and the N_e serves only as a vehicle to infer rates of genetic change. This inference is obscured when some rates are determined by 1/(2N_e), such as nuclear genes, and others by 1/N_e or 1/(1 + 4x(1 - x)N_e). For example, organelle genes transmitted by a single parent may have the same effective size as nuclear genes; despite the equality of effective sizes, however, the rate of loss of organelle diversity will be twice that of nuclear genes (BIRKY et al. 1989 ; CHESSER and BAKER 1996 ). Also, if N_e for organelles is the number of females (N_f; AVISE 1994 ) and N_e for diploid nuclear genes is the number of breeders (N_f + N_m), then the rate of change of organelle variation is four times that for nuclear genes (BIRKY et al. 1989 ). As such, the conceptual intent of N_e is obscured by the necessity to transform the values in accordance with the mode of inheritance and proportional contributions by parents to determine rates of change. To avoid confusion, I refer to the effective rates of change of genetic variation (R_e) that are determined for any genetic character as

(36)

referencing the rate of loss of genetic variation within individuals due to inbreeding (inbreeding rate), rate of loss of genetic variation within breeding groups (coancestral rate), and rate of loss of overall genetic variation (intergroup rate). The effective number of gametes (G_e) is simply the reciprocal of R_e, giving

(37)

The G_e is defined as the number of gametes of an ideal population that would confer the same rate of change in genetic variation as in a nonideal population in question. However, there is no need to transform the G_e value for genetic characters that have different modes of inheritance. All values for rates of change, effective numbers of gametes, and F-statistics are readily attained by iterative procedures described above. Complexity of the transition matrix prohibits exact analytical solutions.

Equation 31, Equation 34, and Equation 36 can be used to determine the rate of loss of gene diversity in the population for any generation as

(38)

Similarly, the effective rate of inbreeding is

(39)

If it is assumed that ^(P)₁ ^(M)₁ = ₁ the above equation can be reduced to

(40)

and, under the same assumption, the rate of change of genetic variation for organelles within breeding groups, the coancestral rate, is

(41)

The complexit