Robustness of the Estimator of the Index of Dispersionfor DNA Sequences

Rasmus Nielsen1

Department of Integrative Biology, University of California, Berkeley, California 94720

Received September 4, 1996; revised November 25, 1996

If substitutions in DNA sequences follow a Poissonprocess, the ratio of the variance in the number ofsubstitutions to the mean number of substitutions (theindex of dispersion) should equal 1. In this paper, therobustness of the commonly applied estimator of theindex of dispersion in replacement sites and silentsites to various assumptions regarding DNA evolutionis explored using simulation methods. The estimate ofthe index of dispersion may be strongly biased if theassumptions of the model of substitution are violated.However, the results of this study support the conclu-sions of studies by Gillespie and Ohta that the processof substitution in replacement sites is overdispersed.This result contradicts those of a recent study andshows that the high index of dispersion for replace-ment sites is not an artifact caused by the method ofestimation. r 1997Academic Press

INTRODUCTION

One of the most celebrated predictions of the neutraltheory of molecular evolution is that the number ofsubstitutions between lineages should follow a Poissonclock [see, for example, Kimura (1983)]. Deviationsfrom a Poisson clock can be quantified by the ratio ofthe variance in the number of substitutions to themeannumber of substitutions: the index of dispersion. Undera Poisson process, the variance equals the mean, so theexpected value of the index of dispersion is 1. If valuessignificantly larger than 1 are observed, a (strictly)neutral model of evolution is rejected. The index ofdispersion has commonly been estimated by Rm 5

Var (Ni)/E(Ni), where Ni is the inferred number ofsubstitutions in the ith lineage of a star phylogeny(Kimura, 1983; Gillespie, 1986; Gillespie, 1989). How-ever, this approach is problematic for three reasons:First, Rm is biased because an estimate of the variancedivided by an estimate of the mean is not the same asan estimate of the variance divided by the mean.

Second, the phylogenetic tree relating the species maynot be a perfect star phylogeny. This problem wasaddressed by Gillespie (1989) by the application ofthree taxon phylogenies and by the application ofweighting factors for branch lengths. The application ofthree taxon phylogenies guarantees that the correcttopology is assumed since there is only one possibleunrooted topology for three taxa. The weighting factorfor a particular lineage is calculated as the meannumber of substitutions in the lineage (averaged overall loci) divided by one-third the total number ofsubstitutions in all three lineages. Application of theseweighting factors is supposed to correct for lineageeffects (effects that create differences in the expectednumber of substitutions between lineages such asdeviations from a star phylogeny and generation timeeffects). However, this approach requires thatmany lociare included in the analysis in order to estimate theappropriate weighting factors. Furthermore, appropri-ate weighting requires that the assumed model of DNAevolution is correct.The third problem with the application of the estima-

tor Rm is that the number of substitutions occurring oneach lineage cannot be observed but must be estimated.This estimation is performed by first estimating thenumber of nucleotide differences between all pairs ofsequences, correcting these estimates for multiple hits,and then inferring the number of substitutions on eachbranch (Gillespie, 1989). However, the correction formultiple substitutions results in an increase in thevariance in the inferred number of substitutions abovethat of a Poisson (Bulmer, 1989). This procedure alsorequires that the assumed model of DNA evolution iscorrect. Deviations from the model could affect Rm in avariety of ways, depending on how the assumptions areviolated.Gillespie (1986, 1989), andmore recently Ohta (1995),

estimated the index of dispersion using the methoddiscussed above for a variety of loci for the human–artiodactyl-rodent phylogeny. They subsequently com-pared the index of dispersion in replacement and silentsites. High values of the index of dispersion are ob-served both for replacement substitutions and for silent1 E-mail: rasmus@mws4.biol.berkeley.edu; FAX (510) 643-6264.

MOLECULAR PHYLOGENETICS AND EVOLUTION

Vol. 7, No. 3, June, pp. 346–351, 1997ARTICLE NO. FY970411

3461055-7903/97 $25.00Copyright r 1997 by Academic PressAll rights of reproduction in any form reserved.

substitutions. However, only the values obtained forreplacement substitutions appear to be significantlydifferent from 1 (Gillespie, 1989). Furthermore, thevalues obtained for replacement substitutions are largerthan the values obtained from silent substitutions.Therefore, both authors reject strict neutrality forreplacement substitutions and suggest that varyingdegrees of positive selection and selection againstslightly deleterious mutations are acting on the in-cluded loci. This is an important result because itrepresents the only set of large scale studies for whichneutrality can be positively rejected as the dominatingfactor in protein evolution.Goldman (1994) criticizes Gillespie’s (1986, 1989)

study. He claims that the observed high values of theindex of dispersion are an artifact caused by theassumption of a star phylogeny. While Goldman doesacknowledge that weighting factors are applied in thestudies of Gillespie, he states that ‘‘Gillespie’s analysismay have placed too much reliance on the ability todetermine accurately the weights wi, representing lin-eage effects. . . .’’ Goldmanmakes no attempt to directlyinvestigate the effect of weighting but concludes thatRm ‘‘provides no evidence for failure of Poisson processmodels.’’The aim of this study is to investigate if violations of

the assumed model of substitution or the structure ofthe underlying phylogenetic tree alone can explain theresults of Gillespie and Ohta. Through extensive com-puter simulations, this paper examines if values of theestimate of the index of dispersion for replacement siteswould be higher than the values obtained for silentsites under any possible neutral model of DNA diver-gence. Higher estimates of the index of dispersion forreplacement than for silent sites are central toGillespie’s (1991) assertion that ‘‘silent substitutionsare mostly mutation limited while replacement substi-tutions are not.’’

SIMULATIONS

Three taxon phylogenies are generated by randomlymutating DNA sequences according to a Poisson pro-cess. The models of sequence evolution assumed in thisstudy allow for a transition/transversion (ts/tv) bias,variation of the mutation rate according to a gammadistribution, and differences in the rate of replacementand silent substitutions. The rates of silent and replace-ment substitutions are modeled in two ways. In thefirst model, a site is either completely constrained (noreplacement substitutions allowed) or completely vari-able (replacement substitutions are just as likely assilent substitutions). This model is referred to as the‘‘neutral sites model.’’ In the neutral sites model, theinfinitesimal rate of transition from base i to base j in

position v of the sequence is

qij 5 5avdvkpj if transition

avdvpj if transversion,(1)

where pj is the frequency of base j, av follows a gammadistributionwith shape parametera, k is the transitions/transversion (ts/tv) ratio, and dv is 1 if the substitutionis a silent substitution or if v is a neutral site and is 0 ifv is a site with constraints. Before the simulations, thefirst 1 2 R sites in the sequence are assigned to beconstrained and the remaining R sites are assigned tobe neutral. R is the relative rate of replacement substi-tution and can be interpreted as the ratio of the rate ofreplacement substitution to silent substitution peropportunity for change.In the second model it is assumed that the rate does

not vary among replacement sites but is reduced by afactor ofR in each site. In other words, the infinitesimalrate of transition from base i to j in site v is

qij 5 5avkpj for silent transitions

avpj for silent transversions

avkRpj for replacement transitions

avRpj for replacement transversions.

(2)

This model is referred to as the ‘‘constant selectionmodel.’’ In the constant selection model R has the sameinterpretation as in the neutral sites model. These twomodels have been chosen because they represent thetwo possible extremes regarding the distribution ofreplacement rates under neutrality. All other models ofthe action of purifying selection should, in principle, liesomewhere between these two extremes.Data are simulated in several steps. First, an ances-

tral sequence is created by drawing nucleotides from aspecified distribution (p) and the site-specific rate (av)is determined for each site by randomly drawing from agamma distribution with shape parameter a. Second,the number of substitutions on each branch is deter-mined by drawing from a Poisson distribution and thesubstitutions are subsequently assigned one by oneaccording to the models described above. Third, afterthe three nucleotide sequences are generated, thenumber of substitutions between all pairs of sequencesis estimated using the method of Nei and Gojobori(1986). This method was chosen to mimic the procedureapplied by Gillespie. However, it should be noted thatmore appropriate methods are available under severalof the assumption sets simulated in this paper (e.g.,Goldman and Yang (1994), Li (1993)). The method ofNei and Gojobori cannot be applied to pairs of se-quences with more than 3⁄4 nucleotide differences persite because a log correction is performed. If suchvalues occur in the simulations, the number of nucleo-

347THE INDEX OF DISPERSION

tide differences is arbitrarily set to 3⁄4 the number ofnucleotides minus 1. This will create a strong bias inthe estimation of the index of dispersion toward smallervalues when the level of divergence is very high.Therefore, no simulation results for such high levels ofdivergence are reported. This does not change theconclusions of this study because such high levels ofdivergence are never observed in replacement sites inreal data.The entire procedure (steps 1–3) is repeated 20 times

and the weighting factors and the average index ofdispersion in replacement and silent sites over the 20loci are calculated by Gillespie’s (1989) method usingreplacement weights for replacement substitutions andsilent weights for silent substitutions. No correction forthe increase in the variance due to the correction formultiple hits is performed in the simulations since thiscorrection was not performed by Gillespie (1989). Thesimulation results obtained here should therefore becomparable to the empirical results obtained byGillespie (1989). The simulations are scaled accordingto the expected number of substitutions on the entiretree (u) and results, averaged over 100 simulations, willbe presented for the expectation of Rmr (the estimatedaverage index of dispersion for replacement substitutions),the expectation of Rms (the estimated average index ofdispersion for silent substitutions), the expected ratioof the two expectations, r (Rmr/Rms), and the tailprobability of observing the value for replacement sitesobserved by Gillespie P6.95 [P(Rmr $ 6.95)]. The index rwill be the primary factor of concern since this ratio iscrucial to the conclusions of Gillespie (1989, 1991).Because the weights are calculated by averaging over

20 loci, it is of interest to examine what happens whenthe value of the parameters vary between loci. In somesimulations (Table 2), several parameters are allowedto vary between loci. First, the proportion of loci thatevolve according to a neutral sites model and a constantselection model is varied (mixed model). In the mixedmodel simulations, a locus evolves according to aneutral sites model with probability 0.5 and accordingto a constant selection model with probability 0.5. Thisrepresents an extreme degree of variation in the distri-bution of replacement rates between loci. The overallrate is varied by setting u , exp (u21). Likewise, thetransition/transversion bias is varied by setting k , 11

exp [(k 2 1)21], for the distribution of rates a , 0.1 1

exp [(a 2 0.1)21], and for the ratio of replacement tosilent rates R , exp (R21) and max 5R6 5 1.0. In all cases‘‘exp’’ signifies an exponential random variable and a,u, R, and k are the means of a, u, R, and k, respec-tively. The lower bounds for k and a are set for practicalreasons. The upper bound for R is set to 1.0 since underneutrality the rate of replacement substitution is notexpected to be higher than the rate of silent substitu-tion. The distributions above are chosen rather arbi-

trarily because very little information regarding thedistribution of these parameters among loci is avail-able.

RESULTS AND DISCUSSION

Using the estimation method described above,Gillespie obtained an estimate of the index of disper-sion of 6.95 for replacement sites and 4.64 for silentsites. In this section, I examine whether data simulatedunder the neutral theory could generate values of theindex of dispersion as high as those observed byGillespie. I will demonstrate that the observed valuesof the index of dispersion are not expected under any ofthe neutral models examined in this study.First, simulations under the constant selectionmodel

with a perfect star phylogeny (i.e., all three brancheshave the same length), no transition/transversion bias,equal base frequencies, and no rate variation wereperformed for 300 nucleotides with R 5 0.2 and R 5 1.0(Fig. 1). Notice that in accordance with the results ofGillespie (1989) and Goldman (1994), the index ofdispersion increases with the divergence time. This iscaused primarily by the increase in the variance in theestimated number of substitutions due to the applica-

FIG. 1. Rms and Rmr for different values of u (the total expectednumber of substitutions) when R 5 1 and R 5 0.2 (R 5 the ratio ofreplacement to silent rates). Notice that when R 5 0.2, the totaldivergence in silent sites is much higher for a particular value of u.Consequently, Rms is much higher than Rmr for the same value of uwhen R 5 0.2 than when R 5 1. Each value represents the averageobtained from 100 simulations, each including 20 sets of three300-bp-long sequences. In all cases P6.95 < 0.0.

348 RASMUS NIELSEN

tion of a correction formula. Notice also that when R 5

1.0, the index of dispersion increases approximatelyequally fast for replacement (Rmr) and silent substitu-tions (Rms). However, in the following it will be as-sumed that the rate of replacement substitution islower than the rate of silent substitution. This is areasonable assumption in the present context, becauseit is empirically observed for all of the examined lociand since higher rates in replacement sites than insilent sites are not expected under neutrality. When therate of replacement substitutions is lower than the rateof silent substitutions, the expectation of Rms increasesmuch faster than the expectation of Rmr (Fig. 1).Likewise, no values of Rm $ 6.95 are observed in eitherof these sets of simulations. For simplicity, simulationresults in the following will be reported primarily interms of the expectation of r (Rmr/Rms).Next, a model involving transition/transversion bias

and rate variation along the sequence was examined(Fig. 2, ts/tv). The effect of the ts/tv bias on r appears tobe minor. Again for all simulated values P6.95 < 0.0.However, when there is strong rate variation in the

underlying mutation rate, r stays close to one (Fig. 2,rate var.). There are two reasons for this effect. First,both silent and replacement sites escape the upwardbias due to the correction of multiple substitutionsbecause the expected number of nucleotide differencesremains low. Second, when the rate of substitutionvaries between sites, the variance in the number ofnucleotide differences is lower than that without ratevariation (i.e., rate variation tends to homogenize theobserved number of nucleotide differences betweenlineages). These two effects together imply that ob-served values of the index of dispersion much higherthan one are very unlikely under this model of strongrate variation among sites.The preceding simulations assumed that purifying

selection has the same effect on each site. This may notbe a reasonable assumption given that the constraintsmost likely vary from site to site. The opposite extremeto a constant selection model is that each replacementsite is either completely invariable or completely neu-tral (a neutral sites model). Under neutrality, the truedistribution of rates for any particular locus should liesomewhere in between these two extremes. Notice (Fig.2, neutral sites) that a pattern similar to the one for theconstant selection model is obtained for the neutralsites model, but the reduction in r is even lower in theneutral sites model. In fact, under a neutral sites modelwith the chosen parameters, saturation reduces Rmr.This is exactly the same effect observed when thebiological mutation rate varies. However, under theneutral sitesmodel, the rate varies in replacement sitesbut not in silent sites and r is lower than in the absenceof rate variation.Obviously, the estimator of the index of dispersion is

sensitive to the particular model of sequence evolution.However, since the rate of substitution in a neutralmodel with purifying selection will always be lower inreplacement sites, the expectation of rwill be below 1. rwill remain close to one only in the case where bothRmrand Rms are close to one. This conclusion does notappear to be sensitive to the particularities of themodelof DNAevolution.

Nonstarness

Next, to evaluate the efficiency of the weightingprocedure, it was assumed that the phylogeny is not atrue star phylogeny. Instead, it was assumed that onebranch is three times as long as the two remainingbranches. Results for u 5 30 and u 5 300 are shown inTable 1.In all but one case (a 5 0.1, k 5 1, equal base frequen-

cies, and u 5 30) the expectation of r is lower than one.When r was close to or slightly higher than one, it wasunder values of the parameters at which both Rmr andRms are close to one.The maximum expected index of dispersion for re-

placement sites (not shown) is obtained for a constant

FIG. 2. The ratio of the estimate of the index of dispersion forreplacement sites to the estimate of the index of dispersion for silentsites, r. The five sets of simulations are R 5 1 (no ts/tv bias, no ratevariation, a constant selection model, and R 5 1); R 5 0.2 (no ts/tvbias, no rate variation, a constant selection model, and R 5 0.2); ts/tv(a ts/tv bias with k 5 3.0, no rate variation, a constant selectionmodel, and R 5 0.2); neutral sites (no ts/tv bias, no rate variation, aneutral sites model and R 5 0.2); and rate var. (no ts/tv bias, ratevariation with a 5 0.1, a constant selection model, and R 5 0.2). Thevalues represent the average obtained from 100 simulations, whereeach simulation includes 20 sets of three 300-bp-long sequences. Inall cases P6.95 < 0.0.

349THE INDEX OF DISPERSION

selection model with a 5 `, k 5 1, equal base frequen-cies, and u 5 300). This is exactly the model assumed inthe log correction. For the parameter values mentionedabove, the expected index of dispersion for replacementsites is 2.1, which is far from the value of 6.95 observedby Gillespie. In fact, in all of the simulations not asingle value of the average index of dispersion higherthan or equal to 6.95 was observed. Clearly, nonstar-ness alone does not explain the results obtained byGillespie (1986, 1989). The ad hoc weighting schemeapplied by Gillespie is surprisingly efficient in correct-ing for differences in branch length. The finding byGoldman (1994) that there is no evidence for a failure ofthe Poisson process may be caused largely by the factthat Goldman considered only single loci. The effect ofapplying multiple loci and estimating weighting factorsis twofold. First, as demonstrated in these simulations,the estimates of Rm will be robust to the assumptionsregarding structure of the phylogeny. Second, power isgained since the branch lengths for each locus do notneed to be estimated independently (i.e., the number offree parameters is reduced). To realize this, consider arenewal process with varying rates over time butindependent increments. Under this type of renewalprocess the test should reject the null hypothesis ofconstant rates. Under such a process the distribution ofthe conditional number of substitutions in each branchis given by a Poisson variable with mean e0t l(s) ds,where t is the absolute length of the branch and l(s) isthe rate at time s (see, for example, Ross, 1993, p. 236).In other words, the total number of substitutions willappear Poisson distributed when only one replicate isconsidered and there will be no evidence for overdisper-sion of the substitutional process. However, Gillespie’stest (which includes averaging over loci) will still havepower to reject a constant rate Poisson model as long asl(s) varies between loci as expected under models ofevolution by positive selection. The lack of significanceobserved by Goldman (1994) in several cases may

simply be an effect of the loss of degrees of freedomresulting from the estimation of branch lengths locusby locus.

Locus-Specific Effects

In the preceding simulations it was assumed that allparameters had the same values in all 20 loci. Next, letus assume that these parameters vary among the 20loci. In the following simulations R, k, a, u, and themodel determining the action of purifying selectionmay be random variables distributed as discussedunder Simulations. For simplicity, the base frequenciesare assumed to be constant and equal.The results of these simulations are presented in

Table 2. Evidently, the expected index of dispersion isconsistently lower for replacement substitutions thanfor silent substitutions. Also, notice that increaseddivergence still results in a reduction in r. However, incontrast to the preceding simulations, values of P6.95 .

0.0 are now occasionally observed. Values as high as theempirical observed will occur for replacement substitu-tions in a constant selection model (especially in theabsence of rate variation in the mutation rate). How-ever, in these cases r is considerably below 1. In fact, inthese simulations, not once did values of Rmr $ 6.95and Rms # 4.64 occur at the same time. Unfortunately,it is far from obvious how the true distribution of theaforementioned parameters varies among loci in realdata. It is therefore not possible to rule out locus-specific effects conclusively. However, the simulationsabove strongly suggest that not even locus-specificeffects in combination with unequal branch lengths willcause the observed values of Rms and Rmr.

CONCLUSION

Simulations including rate variation, a transition/transversion bias, unequal base frequencies, differentmodels of selective constraints, and deviations from astar phylogeny were performed. Generally, the esti-mate of the index of dispersion is highly sensitive to theunderlying model of sequence evolution. This impliesthat we can have little confidence in the precise valuesof Rm cited. In fact, the estimator of the index ofdispersion should not be used to examine the evolutionin a single loci. However, for reasonable degrees ofdivergence, values of Rmr larger than or equal to theempirical observed values averaged over 20 loci arefound only rarely when averaging over loci. Further-more, r was slightly larger than 1 in only one case, andin all simulations r decreases with divergence. It doesnot appear that values of Rmr as large as 6.95 can beobserved with any measurable probability at the sametime as Rms # 4.64 if the rate of substitution is muchlower in replacement sites than in silent sites. Inprobabilistic terms, the probability of making the jointobservation of Rms # 4.64 and Rmr $ 6.95 is estimated

TABLE 1

Estimates of the Expectation of r

Neutralsitesmodel

Constantselectionmodel

u 5 30 u 5 300 u 5 30 u 5 300

a 5 `, k 5 1,p 5 51⁄4, 1⁄4, 1⁄4, 1⁄46 0.762 0.061 0.776 0.091a 5 `, k 5 3,p 5 51⁄4, 1⁄4, 1⁄4, 1⁄46 0.708 0.069 0.749 0.097a 5 `, k 5 3,p 5 50.1, 0.2, 0.3, 0.46 0.713 0.037 0.747 0.074a 5 0.1, k 5 1,p 5 51⁄4, 1⁄4, 1⁄4, 1⁄46 0.762 0.582 1.100 0.819a 5 0.1, k 5 3,p 5 51⁄4, 1⁄4, 1⁄4, 1⁄46 0.813 0.591 0.933 0.804a 5 0.1, k 5 3,p 5 50.1, 0.2, 0.3, 0.46 0.863 0.637 0.991 0.809

Note.Each value represents the average obtained from 100 simula-tions, each including 20 sets of three 300-bp-long sequences. k 5 1implies no ts/tv bias and a 5 ` implies no rate variation. In all casesP6.95 < 0.0.

350 RASMUS NIELSEN

to be less than 0.01 independent of the assumed modelof DNA evolution. The results of Gillespie (1989) andOhta (1995) cannot be explained by a simple neutralmodel of evolution regardless of which assumptions areintroduced concerning the model of DNA divergence.Their observations are so extreme that they cannot bedisregarded, despite the obvious inadequacies of theapplied statistical estimator. It remains the case thatthe evolution in replacement sites appears overdis-persed. The only other plausible conclusion is that theevolution in silent sites is strongly underdispersed. Theelevated index of dispersion for replacement sites re-mains one of the single most important observations toexplain in molecular evolution.

ACKNOWLEDGMENTS

I thank J. P. Huelsenbeck and S. Schrodi for comments anddiscussion. This work was supported in part by NIH Grant GM40282to M. Slatkin and by personal grants to R. N. from the DanishResearch Council.

REFERENCES

Bulmer, M. (1989). Estimating the variability of substitution rates.Genetics 123: 615–619.

Gillespie, J. H. (1986). Rates of molecular evolution. Annu. Rev. Ecol.Syst. 17: 637–665.

Gillespie, J. H. (1989). Lineage effects and the index of dispersion ofmolecular evolution.Mol. Biol. Evol. 6: 636–647.

Gillespie, J. H. (1991). ‘‘The Causes of Molecular Evolution,’’ OxfordUniv. Press, Oxford.

Goldman, N. (1994). Variance to mean ratio, R(t), for Poissonprocesses on phylogenetic trees.Mol. Phylogenet. Evol. 3: 230–239.

Goldman, N., and Yang, Z. (1994). A codon based model of nucleotidesubstitution for protein-coding DNA sequences. Mol. Biol. Evol.11(5): 725–736.

Kimura, M. (1983). ‘‘The Neutral Theory of Molecular Evolution,’’Cambridge Univ. Press, Cambridge.

Li, W.-H. (1993). Unbiased estimation of the rates of synonymous andnonsynonymous substitution. J. Mol. Evol. 36: 96–99.

Nei, M., and Gojobori, T. (1986). Simple methods for estimating thenumber of synonymous and nonsynonymous nucleotide substitu-tions.Mol. Biol. Evol. 3: 418–426.

Ohta, T. (1995). Synonymous and nonsynonymous substitutions inmam-malian genes and the nearly neutral theory.J.Mol. Evol. 40: 56–63.

Ross, S. M. (1993). ‘‘Introduction to Probability Models,’’ AcademicPress. San Diego.

TABLE 2

The Estimate of the Expectation of r and P6.95

r/P6.95

Neutral sites model Constant selection model Mixed model

u 5 30 u 5 300 u 5 30 u 5 300 u 5 30 u 5 300

a 5 0.5, k 5 3.0 0.836/0.0 0.535/0.0 0.954/0.0 0.713/0.0 0.814/0.0 0.207/0.0RV: Ra 5 0.5, k 5 3.0 0.803/0.0 0.560/0.0 0.911/0.0 0.830/0.0 0.750/0.0 0.234/0.0RV: R 1 ua 5 0.5, k 5 3.0 0.758/0.0 0.211/0.0 0.770/0.0 0.337/0.0 0.759/0.0 0.274/0.0RV: R 1 u 1 a

a 5 0.5, k 5 3.0 0.823/0.0 0.590/0.0 0.899/0.0 0.776/0.0 0.750/0.0 0.235/0.0RV: R 1 u 1 k

a 5 0.5, k 5 3.0 0.681/0.0 0.191/0.0 0.807/0.0 0.319/0.0 0.757/0.0 0.286/0.0RV: R 1 u 1 a 1 k

a 5 `, k 5 1.0 0.680/0.0 0.077/0.0 0.751/0.0 0.125/0.0 0.737/0.0 0.121/0.0RV: Ra 5 `, k 5 1 0.621/0.0 0.124/0.0 0.625/0.0 0.278/0.16 0.592/0.0 0.218/0.04RV: R 1 ua 5 `, k 5 3.0 0.556/0.0 0.110/0.0 0.586/0.0 0.185/0.04 0.584/0.0 0.180/0.03RV: R 1 u 1 k

Note. Each value represents the average obtained from 100 simulations, each including 20 sets of three 300 bp long sequences. RV impliesthat the parameters are random variables following distributions discussed in the text. k 5 1 implies no ts/tv bias, and a 5 ` implies no ratevariation.

351THE INDEX OF DISPERSION