VARIANCES AND FREQUENCY DISTRIBUTIONS OF GENETIC DISTANCE IN EVOLUTIONARY PHYLADS

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1 Heredity (1978), 40 (), 5-37 VARIANCES AND FREQUENCY DISTRIBUTIONS OF GENETIC DISTANCE IN EVOLUTIONARY PHYLADS JOHN C. AVISE Department of Zoology, University of Georgia, Athens, Georgia 3060, U.S.A. Received 8.vi.77 SUMMARY Previously introduced deterministic models about mean genetic distances in evolutionary phylads are extended to predictions about variances and frequency distributions of genetic distance. Variance ratios in living members of speciesrich versus species-poor phylads are contrasted under two competing assumptions: (I) genetic differentiation is a function of time, unrelated to the number of cladogenetic events and () genetic differentiation is proportional to the number of speciation events in the group. When genetic distance is a function of time, the ratio of variances of genetic distance in speciose versus depauperate phylads of equal evolutionary age is usually less than one, and generally decreases as the difference in rates of speciation in the two phylads increases. On the contrary, when genetic distance is a function of the number of speciations in the history of a phylad, the ratio of variances in speciose versus depauperate phylads of equal evolutionary age is greater than one, and increases as the difference in rates of speciation in the two phylads increases. The predictions about ratios of skewness and kurtosis of genetic distances are not different in the " time-divergence" and " clad-divergence" models. Nonetheless, genetic distances among living representatives of evolutionary phylads are consistently skewed toward lower genetic distances. This finding suggests that most observed distances between closely related species may be relatively uninformative about the amount of genetic differentiation involved in speciation. I. INTRODUCTION THE majority of techniques for assaying genetic distance between species can for obvious reasons utilise only extant representatives of evolutionary phylads. Researchers employing chromosomes, proteins, or DNA as probes of genetic relationships must rely on inferential procedures for reconstructions of past evolutionary events. Morphologists face similar difficulties, although the problem may be alleviated in groups of organisms with a good fossil record. Several procedures have been developed for forming dendrograms or phylogenetic trees, beginning with matrices of similarity or distance among living species (Edwards and Cavalli-Sforza, 1965; Cavalli-Sforza, 1966; Farris, 197; Sneath and Sokal, 1973). The approach taken in this report is somewhat different. Beginning with evolutionary dendrograms specified by several parameters, such as time of origin and rate of splitting (cladogenesis), distance matrices are constructed. This approach focuses attention on properties of the matrix rather than on properties of the phylogenetic tree alone; hence it may lead to theoretical predictions which can be directly tested against empirical matrices generated by electrophoretic or other techniques. Utilising this method in an earlier report, Avise and Ayala (1975) formulated distinct and testable predictions about expected ratios of mean values of elements in 40/ D 5

2 6 JOHN C. AVISE distance matrices. In particular, when genetic distance between species is a function of time, mean genetic distances among living members of speciose (species-rich) and depauperate (species-poor) phylads of equal evolutionary age are very similar. On the contrary, when genetic distance is a function of the number of speciations in the history of a phylad, the ratio of mean genetic distances separating species in speciose versus depauperate phylads is greater than one, and increases rapidly as the frequency of speciations in one group relative to the other increases. In this paper, these same deterministic models are extended to predictions of the variance and frequency distributions of genetic distance. A current controversy in evolutionary genetics concerns whether most evolutionary change is compressed into speciation episodes (referred to as rectangular evolution, or punctuated equilibria), or whether change is gradual and cumulative in ancestor-descendant sequences of populations through time (phyletic gradualism) (Eldredge and Gould, 197; Harper, 1975; Stanley, 1975, 1976). In order to generate predictions which follow from these hypotheses, models are examined which contrast two extreme alternatives: (1) genetic distance is a function of time, unrelated to the rate of speciation in a phylad, and () genetic distance is a function of rate of speciation. Can distinct and testable predictions be generated about the second, third, and fourth moments of the distributions of genetic distance among living members of evolutionary assemblages?. THE MODELS The rationale and development of the basic models are discussed in detail by Avise and Ayala (1975). Two sets of models are considered, distinguished by alternative assumptions about the major determinants of genetic differentiation. In models 1A and 1B, the amount of genetic distance separating any pair of species is assumed to be a function of the amount of time elapsed since those species last shared a common ancestor. In models A and B, genetic distance is assumed to be a function of the rate of speciation in a phylad (model A distance is proportional to the number of cladogenetic events [dads] separating species; model B distance is proportional to the number of species arising per clad). We are interested in comparing evolutionary phylads of equal age. Barring differences in extinction rates, such phylads may come to possess different numbers of living species by one or both of two factors: (a) models IA and A speciation episodes occur at different rates, or (b) models lb and B different numbers of species arise per speciation episode. With the above assumptions, an infinite number of dendrograms could theoretically be visualised. In order to permit analytic solutions for means and variances of distances in matrices specified by these dendrograms, additional simplifying assumptions were adopted which apply to both models I and : (1) cladogenetic events occur at regular time intervals; () no species become extinct; and (3) there are no convergent or parallel genetic changes in different lineages. As discussed by Avise and Ayala (1975), the models remain realistic because the results appear to be relatively insensitive to relaxations of these underlying assumptions. The following parameters are defined for a phylad X,:

3 MOMENTS OF GENETIC DISTANCE 7 number of time units since the origin of the phylad (counted on some arbitrary absolute scale, such as years); m number of time units between dads; t/m (k is assumed, for simplicity, to be an integer); number of species generated at each clad (li> I, constant for a given phylad); genetic distance between two extant species, a and b; number of extant species in the phylad; number of all pairwise comparisons between extant species in a phylad. Thus (dropping subscripts where there is no ambiguity): (1) = ( 1), () Also, as shown previously by Avise and Ayala (1975), the total distance for all possible pairwise comparisons between extant species in a phylad is, according to models IA or 1 B: 1 i=k Ed(1A, 1B) = ) ui i. (4) \ Ji=1 According to model A, the total distance for all pairwise comparisons is 'i 1 (i 1)P. (5) \ ji=i According to model B, the total distance for all pairwise comparisons between extant species is: C (3) 1 d(b) = 1k i (_T_) i. (6) \ Ji=i 3. RESULTS (i) &fodel 1. Genetic dffèrentiation is proportional to time The variance of genetic distance in a matrix is simply the sum of squares divided by the number of elements. Since a matrix is completely specified by a given dendrogram, the parametric variance can be evaluated rather than estimated, and the appropriate division is c,. The sum of squares is composed of (a) the sum qf distances squared, and (b) the correction term (the square of the sum of distances, divided by ci). From (4), the sum of distances squared is: /z i\ i=k mlk(_1 il'. (7) \ Ji=i

4 8 JOHN C. AVISE The correction term is / / i'" \ (mlk(_t ) \ j=i 8 1k(1k_1) Hence, the variance of genetic distance among living members of an evolutionary phylad, when genetic distance is proportional to time, is: o(1a, 1B) = mlk(-_) i=k 11 (mlk(!tj) :: 1_i) i 1k(1k_1) (9) The variances calculated for a variety of dendrograms specified by various values of k, t, and I are presented in table I. They range from a lowof049(l=6,k=,t=4)toahighof849(l,k3,t= 1). The comparisons of particular interest involve phylads of equal evolutionary age, but characterised by different rates of speciation. For model IA, the number of species arising at each clad in speciose and depauperate phylads is equal (IR = ip), but the time units between dads are fewer in the speciose phylad (mr < mp). For these phylads, comparisons of variances of genetic distance are made by examining values within a column, such as I =, and with equal t, hut different k's (and hence also different values of rn). In table there are a total of 60 such comparisons. The highest ratio of variances of genetic distance in speciose versus depauperate phylads (or/ap) of equal evolutionary age is, according to model IA, 1 06, and the lowest ratio is 00. In general, c decreases as rate of speciation increases among phylads of equal evolutionary age. For model lb. the number of time units between dads is identical in species-rich and species-poor phylads (rnr = nip), but the number of species arising per clad differs (IR > Ip). Comparisons of variances are now made by examining values within a row of table 1. Again, variance of genetic distance decreases as rate of speciation increases among phylads of equal evolutionary age. Hence and decreases as the difference in rate of species formation increases. Formulae using parameters of the models could also be developed for measures of skewness (yr) and kurtosis (y), but they would be large and cumbersome. In practice it is far easier to calculate values directly from the distance matrices using conventional formulas. Values of y and V for various phylads are given in table 1. All distributions are skewed toward lower genetic distances (negative values of yi), and all distributions except those with very few species comparisons are leptokurtic (positive value of Yz)

5 MOMENTS OF GENETIC DISTANCE 9 TABLE I Variances (top entry), measures of skewness (Vi, second entry), and measures of kurtosis (y, third entry) of genetic distance in pairwise comparisons between living species of various phylads, when genetic change is proportional to time (models IA and 1B) r k=t/m = age of phylad; m = time units between cladogenetic events; I = number of species generated per cladogenetic event. The number of extant species per phylad is 1k. K

6 30 JOHN C. AVISE (ii) Iviodel A. Genetic differentiation is proportional to the number of cladogenetic events in the phjlad From (5), the sum of distances squared is: The correction term becomes: 1i ik (j1)1_ (10) Ji=1 (lk(_) i=k =i (t_1)l1) (11) and hence the variance of genetic distance among living members of an evolutionary phylad, when genetic distance is proportional to the number of dads in the phylad, is: o(a) = / /11\i=k j 1k( (i 1)111 )i=1 \. Ji=i 1k(1k_1) lk(lk_1) (1) Variances calculated for a variety of dendrograms specified by values of k and l are presented in table. Again, appropriate comparisons of variances are made by examining values within a column specified by a given 1.. Variance increases as rate of speciation increases. Hence, when genetic change is a function of number of speciations in a phylad, a,rft,p>1, and the ratio generally increases as the difference in rates of speciation in the phylads increases. Measures of skewness and kurtosis according to model A are presented in table. All distributions are again skewed toward lower genetic distance, and most distributions are leptokurtic. In fact, for given schedules of I and k, the y and 7 values are identical (slight differences probably due to rounding) to those for comparable phylads calculated according to models la and lb. (iii) Ivlodel B. Genetic dftrentiation is proportional to the number of species generated per clad Proceeding as before, from (6) the sum of distances squared is: 111 (11) ::

7 MOMENTS OF GENETIC DISTANCE 31 TABLE Variances (top entry), measures of skewness (yr, second entry), and measures of kurtosis (ye, third entry) of genteic distance in pairwise comparisons between living species in various phylads, when genetic change is proportional to the number of cladogenetic events separating species (model A) k l l l l I l l I Symbols as in table 1. The correction term is: (1k_i (11) i=k t 1 1t) 1k(1k_1) (14) and hence the variance of genetic distance among living members of an evolutionary phylad, when genetic distance is proportional to the number of species generated per clad, is: ) f / k 1 71_1\i=k j - \ V v i' ' )i \. \. lk(lk_l) (15) lk(lk_1) Variances calculated for several dendrograms specified by values of k and I are presented in table 3. The appropriate comparisons of variances are made by examining values within a row specified by k. Variance increases dramatically as the number of species generated per clad increases. Hence, when genetic change is a function of number of species generated

8 3 JOHN C. AVISE per clad, o/a> 1, and the ratio rapidly increases as the difference in rates of speciation in the phylads increases. TABLE 3 Variances (top entry), measures of skewness (yl, second entry), and measures of kurtosis (y, third entry) of genetic distance in pairwise comparisons between living species in various phytads, when genetic change is proportional to the number of species generated per clad (model B) k ' l l l 'O l l Symbols as in table 1. Measures of skewness and kurtosis for various phylads according to model B are presented in table 3. All distributions are skewed toward lower genetic distance. However, in this case most distributions are platykurtic. (iv) Relative predictions of "time-divergence " and " clad-divergence " models The previous paper in this series demonstrated that distinct predictions obtain for expected mean distances in evolutionary phylads of equal age, but with different rates of speciation, depending upon whether genetic distance is proportional to time or to rate of speciation (Avise and Ayala, 1975). The present analysis shows that distinct predictions are also generated about the expected variances of genetic distance among living members of these phylads. When genetic distance is a function of time, the ratio of variances of genetic distance in speciose versus depauperate phylads of equal evolutionary age (tr/o) is usually less than one, and generally decreases as the difference in rates of speciation in the two phylads increases. On the contrary, when genetic distance is a function of the number of speciations in the history of a phylad, the ratio of variances in speciose

9 MOMENTS OF GENETIC DISTANCE 33 versus depauperate phylads of equal evolutionary age (cr/cp) is greater than one, and increases as the difference in rates of speciation in the two phylads increases. When genetic distance is proportional to time, J and oç are related as shown in fig. 1. Within the range of t, values shown, the relationship is nearly linear, but the slope of the regression decreases as 1 increases. When genetic distance is proportional to the number of dads separating species, d and o are related as in fig.. Since the relationships of i and d are complex and depend strongly on values of i, k, and t, the predictions about variance ratios in different phylads according to either model 1 or model II FIG. 1. Relationships between mean distance (d) and variance of distance (ci) in phylads with k = and I = to I = 6, when genetic divergence is proportional to time (model I). are not as simple as were predictions about distance ratios. However, predictions of variance ratios in model I versus model are consistent, and in fact, for most pairs of phylads, are even more distinct than predictions about mean distances. Examples of predicted variance ratios in speciose and depauperate phylads of equal evolutionary age, calculated according to model la versus model A, are shown in fig. 3. For model lb versus model B, the predictions are in the same direction but even stronger. In contrast to means and variances of genetic distance, the expected forms of the distribution of distances are similar in both the time-divergence and clad-divergence models. In all cases, distributions are strongly skewed towards lower genetic distances. For models 1A and A, the expected ratios of skewness in species-rich versus species-poor phylads are identical (fig. 3). In most cases, distributions of genetic distance are leptokurtic. d

10 JOHN C. AVISE 4 0d3 I II 1 d FIG.. Relationships between mean distance (d) and variance of distance (a) in phylads with various k and I schedules, when genetic divergence is proportional to the number of cladogenetic events separating species (model A). 4 I C -J uj C 0 0 3,,/, / r S SṢ. S. S S 3 4 S (MODEL A) FIG. 3. Predicted ratios of variances of genetic distance in species-rich (R) versus speciespoor (F) phylads of equal evolutionary age; ordinate-variance ratios when genetic distance is proportional to time, and IR = Ip (model IA); abscissa-variance ratios when genetic distance is proportional to the number of cladogenetic events, and Ip (model A). Plotted points are those which can be calculated from phylads with the k, t, and I values in tables and 3. The dark area along the 1 1 line encloses the ratios of skewness (YSR/v1p) calculated according to the same models (see text).

11 MOMENTS OF GENETIC DISTANCE 35 However, when genetic change is proportional to the number of species generated per clad (model B), the distribution of genetic distances among living species is usually platykurtic. Predictions about the means, variances, and frequency distributions of genetic distance according to the timedivergence and clad-divergence models are summarised in table Dxscussiorc The models predict that if phyletic evolution (anagenesis) and cladogenesis are correlated, the ratio of variances of genetic distance in species-. rich versus species-poor phylads of equal evolutionary age should be greater than one, and should generally increase as the difference in rates of speciation between the two phylads increases. On the contrary, if genetic change in evolution is primarily a function of time, and not correlated with rates of cladogenesis, the ratio of variances of genetic distance among living species should be less than one in species-rich versus species-poor phylads of comparable evolutionary age. The models have been developed in such a way as to permit ready comparisons with empirical distance data derived from electrophoretic or other techniques. One survey strategy employed in attempts to determine the effect of speciation on the magnitude of genetic differentiation involves assaying large numbers of closely related species pairs. A survey of the recent electrophoretic literature found a total of 615 pairwise population comparisons among such closely related species, and only 6 per cent of these comparisons give genetic distances less than 0.10 (Avise et al., 1975). Such results give the intial impression that speciation is normally accompanied by substantial genetic divergence. However, these estimates of genetic distance include differentiation subsequent to as well as during the speciation process, with no criteria to distinguish between contributing factors. Results of the present models demonstrate that among living members of many evolutionary phylads, the frequency distributions of genetic distance are strongly skewed toward lower genetic distances. Even if all speciations entail little genic change, the vast majority of comparisons among, say, congeneric members of a phylad, may be large, while only the relatively few pairs of species with exceptionally small genetic distances provide a true description of genetic differentiation during speciation. A second strategy to determine genetic change during speciation involves assaying populations which at present appear to be in various stages of the speciation process. The most complete study of this type was conducted in the Drosophila willistoni complex, and indicated that about 0 per cent of the genes exhibited major allele frequency changes prior to the completion of reproductive isolation (Ayala et al., 1974). Such studies provide an excellent summary of genetic differentiation in specific groups, but the results might lack generality. First, in only a relatively few organisms can populations presently in various stages of speciation be identified. Second and more important, those cases which can be identified may represent a very biased sample of speciation events those in which reproductive isolation develops gradually over a long period of time. Assume for sake of argument that in most groups of organisms reproductive isolation develops very rapidly, perhaps through sympatric mechanisms, or perhaps in small isolate popu-

12 "3 C) TABLE 4 Summary of the results predicted according to the various models Differences r between Genetic distance proportional to () Number of speciations between species (model A), or species-rich (1) Time number of species generated per clad (model B) C and species- poor phylads d vs d Vi Vs C) Model JA Model A me < mp; dr All negative Usually All negative Usually 1 <1 (skewed toward positive >> I >> 1 (skewed toward positive lower distances) (leptokurtic) lower distances) (leptokurtic) Model lb Model B mr _ mp; All negative Usually 4 All negative Usually <1 (skewed toward positive >> I >> 1 (skewed toward 115> 1 4 negative lower distances) (leptokurtic) lower distances) (platykurtic)

13 MOMENTS OF GENETIC DISTANCE 37 lations. Such types of speciations would probably not be recognised and included in the second survey strategy. The approach taken to the speciation question in thi report is new, and can be applied without bias with respect to how speciation occurred. Phyletic evolution and cladogenesis could be positively correlated because populations evolving at a faster rate speciate more often, or it could be because the process of speciation per Se involved substantial genetic alteration. In either event, the models developed here generate qualitatively distinct predictions about ratios of mean distance and variance of distance in living members of evolutionary phylads exhibiting different rates of speciation, depending on whether genetic change is a function of time or of rate of speciation. The models are simple, general, probably robust, but are not precise. When compared against appropriate data sets, they should permit critical tests of whether time since divergence, or rate of speciation, is the main contributor to genetic differentiation. 5. REFERENCES AVISE, j. c., AND AYALA, F. j Genetic change and rates of cladogenesis. Genetics, 81, AVI5E, J. C., SMITH, J. J., AND AYALA, F. j Adaptive differentiation with little genic change between two native California minnows. Evolution, 9, AYALA, F. J., TRACEY, H. L., HEDGECOCK, D., AND RICHMOND, R. c Genetic differentiation during the speciation process in Drosophila. Evolution, 8, CAVALLI-sFoRzA, L. L Population structure and human evolution. Proc. Roy. Soc. B, 164, EDWARDS, A. W. F., AND CAVALLI-SFORZA, L. L A method for cluster analysis. Biometrics, 1, ELDREDGE, N., AND GOULD, S. j Punctuated equilibria: an alternative to phyletic gradualism. In Models in Paleontology, T. J. M. Schopf, ed., Freeman, Cooper, and Co., pp FARRI5, j. s Estimating phylogenetic trees from distance matrices. Amer. Nat. 106, HARPER, C. W., JR Origin of species in geologic time: alternatives to the Eldredge- Gould model. Science, 190, SNEATH, i'. H. A., AND SOKAL, R. R Alumerical Taxonomy. W. H. Freeman, San Francisco. STANLEY, 5. M A theory of evolution above the species level. Proc..Wat. Acad. Sci., 7, STANLEY, S. M Stability of species in geologic time. Science, 19,