Estimation of evolutionary parametersusing short, random and partial sequencesfrom mixed samples of anonymousindividualsSteven H. Wu1,2* and Allen G. Rodrigo2,3
Abstract
Background: Over the last decade, next generation sequencing (NGS) has become widely available, and is now thesequencing technology of choice for most researchers. Nonetheless, NGS presents a challenge for the evolutionarybiologists who wish to estimate evolutionary genetic parameters from a mixed sample of unlabelled or untaggedindividuals, especially when the reconstruction of full length haplotypes can be unreliable. We propose two novelapproaches, least squares estimation (LS) and Approximate Bayesian Computation Markov chain Monte Carloestimation (ABC-MCMC), to infer evolutionary genetic parameters from a collection of short-read sequencesobtained from a mixed sample of anonymous DNA using the frequencies of nucleotides at each site only withoutreconstructing the full-length alignment nor the phylogeny.
Results: We used simulations to evaluate the performance of these algorithms, and our results demonstrate that LSperforms poorly because bootstrap 95 % Confidence Intervals (CIs) tend to under- or over-estimate the true valuesof the parameters. In contrast, ABC-MCMC 95 % Highest Posterior Density (HPD) intervals recovered from ABC-MCMC enclosed the true parameter values with a rate approximately equivalent to that obtained using BEAST, aprogram that implements a Bayesian MCMC estimation of evolutionary parameters using full-length sequences.Because there is a loss of information with the use of sitewise nucleotide frequencies alone, the ABC-MCMC 95 %HPDs are larger than those obtained by BEAST.
Conclusion: We propose two novel algorithms to estimate evolutionary genetic parameters based on theproportion of each nucleotide. The LS method cannot be recommended as a standalone method for evolutionaryparameter estimation. On the other hand, parameters recovered by ABC-MCMC are comparable to those obtainedusing BEAST, but with larger 95 % HPDs. One major advantage of ABC-MCMC is that computational time scaleslinearly with the number of short-read sequences, and is independent of the number of full-length sequences inthe original data. This allows us to perform the analysis on NGS datasets with large numbers of short readfragments. The source code for ABC-MCMC is available at https://github.com/stevenhwu/SF-ABC.
Keywords: Markov chain Monte Carlo, Next generation sequencing, Short read sequences, Approximate Bayesiancomputation, Evolutionary genetics
* Correspondence: [email protected] Institute, Arizona State University, Tempe, AZ 85287, USA2Department of Biology, Duke University, Box 90338, Durham, NC 27708, USAFull list of author information is available at the end of the article
BackgroundOver the last decade, next generation sequencing(NGS) has become widely available, and is now thesequencing technology of choice for most researchers.NGS produces sequences that are relatively short,varying between 50 bp to 400 bp depending on thespecific platform [1–3]. Researchers use NGS in sev-eral different ways. In this manuscript, we considerthe use of NGS in evolutionary studies, where shortread fragments are obtained from longer, amplifiedtarget sequences in mixed samples of unlabeled oruntagged (= “anonymous”) individuals. These types ofsamples are often collected from viral or bacterialpopulations. The traditional sampling protocol forevolutionary studies that rely on sequences frommany individuals has been to use Sanger sequencingtechnology to obtain the sequence(s) of one (or more)DNA fragment(s) from each individual in the sampleseparately. This is typically followed by a multiplesequence alignment and reconstruction of the phyl-ogeny or genealogy, perhaps with the simultaneousinference of relevant evolutionary parameters [4, 5].With NGS, short read fragments are typically
shorter than the fragment of interest. When NGS isapplied to a mixed collection of DNA from severalindividuals, the challenge for the evolutionary biolo-gist is the absence of an alignment of full-lengthsequences, each corresponding to an individual in thesample [6]. And without an alignment of full-lengthsequences, how does one estimate the evolutionaryparameters of interest?One approach is to attempt to reconstruct the full
length haplotypes, and use an alignment of these re-constructed haplotypes in standard evolutionary ana-lyses [7–9]. There are several programs that attemptto reconstruct the full-length haplotypes from shortread fragments obtained from mixed, unlabeled,collections of individuals e.g., ShoRAH, ViSpA, Pre-dictHaplo and Qure [10–13]. Analyses have shownthat with many data sets, reconstruction of haplotypescan be unreliable, producing either too many haplo-types and/or sequences that have relatively low iden-tity to the original sequences [14–16]. Consequently,a researcher who chooses to use these reconstructedfull-length haplotypes with any program that requiresfull-length alignments, will be implicitly integratingthe errors of haplotype reconstruction into their esti-mation of evolutionary parameters.We propose two alternative approaches to infer
population genetic parameters from a collection ofshort-read sequences obtained from a mixed sampleof anonymous DNA using the frequencies of nucleo-tides at each site only. To our knowledge, there isno existing method capable of estimating these
parameters without reconstructing the full lengthsequence alignment nor the phylogeny. A similarapproach had been proposed by Johnson and Slatkinpreviously [17], but their method focuses on samplesof very large numbers of individuals where each readin an alignment of short-reads is assumed to comefrom a separate genome. In contrast, the methodswe propose assumes either (1) that a relatively shortfragment of the genome is the target of NGS, and/or(2) there are relatively few individual organisms inthe sample. In essence, these assumptions ensurethat each site for each individual is covered by mul-tiple reads, so that the frequency of nucleotides ateach site can be estimated with reasonable accuracy.In samples of viruses and bacteria, for instance, theamplified region is small, say, a few kilobases long,and the number of genomes in a single PCR reactionis often unknown ranging from the tens to thethousands. With microbial populations, typically vi-ruses, there is also the opportunity to collect serialsamples from the same fast evolving species over aperiod of time. For this reason, the methods we havedeveloped estimate the population genetic parame-ters θ ∝ Nμ, the effective population size scaled bymutation rate, and μ, the mutation rate per site perunit time [18, 19].We describe two algorithms to estimate these
parameters, Least Squares (LS) estimation [20] andApproximate Bayesian Computation Markov chainMonte Carlo (ABC-MCMC) estimation [21–23]. Oneadvantage of the ABC-MCMC approach over moretraditional MCMC approaches is that ABC-MCMCdoes not require formulation or computation of alikelihood; instead, the method relies on the use ofsummary statistics derived from simulated data toaccept or reject a proposal in the Markov chain. Thedetails of these two algorithms are described in thenext section.We used simulations to evaluate the performance of
these algorithms, and we compared these to theresults obtained with BEAST [8], a program that iscommonly used to simultaneously infer phylogeniesand evolutionary parameters using Bayesian MCMCinference with full-length sequences. Our simulationsdemonstrate that LS point estimates are unbiased, butproduce bootstrap intervals that typically over- orunderestimate true parameter values. In contrast,ABC-MCMC is able to estimate evolutionary parame-ters without reconstructing full-length haplotypes,producing 95 % Highest Posterior Density (95 %HPD) intervals that have equivalent coverage to thoseobtained by MCMC with full-length alignments; how-ever, there can be up to a 10-fold difference betweenthe lowest and highest bound of each 95 % HPD.
Wu and Rodrigo BMC Bioinformatics (2015) 16:357 Page 2 of 12
MethodsAs noted above, both algorithms apply to samples ofshort-read sequences obtained from a collection of longertarget sequences from mixed and unlabeled individuals ina population. The first method is based on least squares(LS) estimation. As we show below, the second methodapplies the LS results as a pre-processing step prior to be-ginning the ABC-MCMC. Both methods estimate evolu-tionary parameters using only the proportion of eachnucleotide at each site, and do not require reconstructionof full length haplotypes nor the phylogeny/genealogy. Ifserial samples are available, sequences from a later time-point share common ancestors with those from the earliertimepoint. We assume that samples collected from eachtimepoint are sequenced separately with NGS technology.As we noted earlier, we assume that each short read
sequenced by NGS will be shorter than the full length hap-lotypes, and we will obtain many more short read frag-ments than the original number of haplotypes. We alsoassume that a reference or consensus sequence for the tar-geted region is available, and we are able to align each shortread fragment to a unique location on the reference se-quence. After the short reads are aligned to the reference,we can count the frequency of each nucleotide at each site.In practice, the frequency of each nucleotide will be influ-enced by the sequencing error from NGS [1]. For the sim-plicity of the algorithms, we assume that the short-readshave been error-corrected prior to analysis.
Least squares (LS) estimationFor serial samples, we can estimate the intra-timepoint(within a single sample) and inter-timepoint (betweensamples from two different timepoints) average pairwisesequence diversity. Both inter-timepoint diversity (Dinter)and intra-timepoint diversity (Dintra) are calculatedusing the proportion of each nucleotide at each site.The intra-timepoint diversity (Dintra,s,t) for site s at time tis calculated as:
Dintra;s;t ¼ 1−X
j∈A;C;G;T
Fs;j;t2 ð1Þ
where Fs,j,t is the proportion of nucleotide j at site s attime t.Similarly, the inter-timepoint diversity (Dinter,s,t1,t2)
between time t1 and t2 at site s is calculated as
Dinter;s;t1;t2 ¼ 1−X
j∈A;C;G;T
Fs;j;t1Fs;j;t2 ð2Þ
Once the average pairwise diversity for each site iscalculated, the mean intra-timepoint diversity for anyspecified timepoint is given by:
Dintra;t ¼ 1n
Xns¼1
Dintra;s;t ð3Þ
and the mean inter-timepoint diversity between timet1 and t2 is:
Dinter;t1;t2 ¼ 1n
Xns¼1
Dinter;s;t1;t2 ð4Þ
where n is the number of sites. If the sequences areobtained from T timepoints, there will be T × (T − 1)/2 esti-mates of average diversity, of which T will be intra-timepoint diversities.The LS method uses both inter- and intra-timepoint
diversity to estimate effective population size and mu-tation rate, based on the method described by [20].Population genetics tells us that in any given samplefrom a constant-sized population of a set of se-quences of a neutrally evolving locus, average pairwisesequence diversity (measured as the average propor-tional distance between any two sequences in thesample) is an estimate of θ, which is proportional tothe product of mutation rate and effective populationsize, with the proportionality constant determined bywhether the population is haploid (proportionalityconstant = 2, θ = 2Nμ) or diploid (proportionality con-stant = 4, θ = 4Nμ).To estimate the parameters of interest, let μ be the mu-
tation rate per unit of time, θ be the effective populationsize scaled by μ, and Δt the time between two samplingevents. Note that both μ and Δt are scaled to the sameunit of time; typically this will be chronological time but,rarely, time in generations may be available. Under a con-stant population size, and a constant mutation rate, thereare only two parameters to be estimated, θ and μ. Asnoted above, θ is estimated by Dintra and θ + μΔt is esti-mated by Dinter. Once estimates of θ and μ are obtained,we can estimate kN = θ/μ, where k is the unspecified pro-portionality constant.We can construct a least squares regression by let-
ting Y be a vector of all Dinter and Dintra, and X bean indicator variable that identifies whether/how θand μ contribute to the expectation of Dinter or Dintra.For the constant population, constant mutation ratemodel, the indicator value for θ is always 1 and the indica-tor of μ is just Δt. We are then able to fit Y =XΒ usingleast-squares, where the Β is the LS estimator of param-eter θ and μ. For example, if there are three timepointsTA, TB, and TC, and the intervals between each adjacentpair is 200 units of time apart, we can construct a set oflinear equations to express the relationship between theseparameters, as follows:
Wu and Rodrigo BMC Bioinformatics (2015) 16:357 Page 3 of 12
Using the least squares method to solve for B= (X’X)−1X’Y, we obtain our estimation of θ and μ.The model can be extended to multiple population sizes
and/or multiple mutation rates. If there are three time-points TA, TB, and TC, then we can estimate the effectivepopulation size for each timepoint using the intra distancewithin each timepoint. Let θA, θB, and θC be the scaledpopulation sizes for timepoint TA, TB and TC. ΔAB is thetime difference between timepoint TA and TB, and ΔBC isthe time difference between timepoint TB and TC.
Dintra−A
Dintra−B
Dintra−C
¼¼¼
θAθBθC
Dinter−AB
Dinter−BC
Dinter−AC
¼¼¼
θA þ μΔAB
θB þ μΔBC
θA þ μΔAB þ μΔBC
ð7Þ
Therefore we update the matrix
X ¼
1 0 00 1 00 0 1
000
1 0 00 1 01 0 0
ΔAB
ΔBC
ΔAB þ ΔBC
2666664
3777775
B0 ¼ θA θB θC μ½ �
ð8Þ
Alternatively, for a constant population size andmultiple mutation rates, the model can be specific as:
Dintra−A
Dintra−B
Dintra−C
¼¼¼
θθθ
Dinter−AB
Dinter−BC
Dinter−AC
¼¼¼
θ þ μABΔAB
θ þ μBCΔBC
θ þ μABΔAB þ μBCΔBC
ð9Þ
Therefore we update the matrices X and B as
X ¼
111
0 00 00 0
111
ΔAB 00 ΔBC
ΔAB ΔBC
2666664
3777775
B0 ¼ θ μAB μBC½ �
ð10Þ
To obtain confidence intervals for our estimates, weused a bootstrap procedure in which we generated1000 pseudoreplicate datasets by resampling sites withreplacement along the alignment of short-read se-quences. We did this separately for each timepoint inour simulated datasets. For each pseudoreplicate werecalculated the site frequencies, Fs,j,t,, and we wereable to estimate N and μ; 95 % Confidence Intervalswere obtained by taking values corresponding toupper and lower 2.5 % percentiles of ordered boot-strap estimates.
ABC-MCMC (Approximate Bayesian Computation -Markov chain Monte Carlo)Markov chain Monte Carlo Bayesian inference is acomputationally intensive technique for recoveringthe posterior probability of parameters of interest,while taking account of prior knowledge (includingthe degree of uncertainty) about these parameters. IfP(D|ϕ) (often referred to as the likelihood of ϕ), isthe probability of obtaining the data given a set ofparameters, ϕ, and P(ϕ) is the prior information wehave about the joint distribution of these parameters,then the posterior distribution, P(ϕ|D), is proportionalto the product of likelihood and prior, P(ϕ|D) ∝P(D|ϕ)P(ϕ). Often it is very difficult to obtain theposterior distribution function analytically. The ele-gance of MCMC resides in its ability to derive therelative posterior distribution by using a proposal dis-tribution to randomly generate a Markov chain of po-tential states (= values that the parameters of interestcan take), and sampling from this chain by acceptingor rejecting states in proportion to the posteriordensity. MCMC works because although it is not easyto calculate the distribution of P(ϕ|D), it is easy toobtain P(ϕ*|D), for a specific value of ϕ*. Conse-quently by repeatedly sampling ϕ* in the correct pro-portions, the distribution of P(ϕ|D) is approximated.Once a sample of sufficient size is obtained, it be-comes possible to derive estimates for the parametersof interest.One common implementation of MCMC uses the
Metropolis-Hastings algorithm [24, 25], which can bedescribed by the following steps.
Wu and Rodrigo BMC Bioinformatics (2015) 16:357 Page 4 of 12
Step 1: Begin with initial parameter values ϕi.Step 2: Propose a new parameter value ϕ* using the
proposal distribution q(ϕ*|ϕi).Step 3: Calculate the acceptance ratio, α, using the
following formula:
α ¼ min 1;P ϕ�jDð Þq ϕijϕ�� �P ϕijD� �
q ϕ�jϕi� �
( )
Generate μ from U(0, 1) and accept ϕi + 1 = ϕ* ifµ < α.Otherwise set ϕi + 1 = ϕi.
Step 4 Set i = i + 1 and repeat Step 1.
The algorithm is repeated until the Markov chain sam-ples from the target distribution, typically the (joint)posterior distribution of the parameter(s).Approximate Bayesian Computation (ABC) is a
simulation-based algorithm for Bayesian inference[21–23]. ABC does not require the calculation of thelikelihood P(D|ϕ); instead, it uses the agreement be-tween summary statistics obtained from D, and thoseobtained from simulations of data under differentvalues of ϕ to obtain the relative posterior probabilitydistribution P(ϕ|D). If the summary statistic, S, is(nearly) sufficient then P(D|ϕ, S) ≅ P(D|S). ABC isused precisely because it circumvents the need to cal-culate a challenging or intractable likelihood. Here,we describe our ABC-MCMC procedure as follows:
1. Calculate a set of p sufficient statistics, So = (S1o,⋯, Sp
o),on the observed dataset, Do.
2. If i = 1, draw the initial parameter value ϕi fromprior distribution P(ϕ). If i > 1, then propose the newparameter value ϕ* from q(ϕ* | ϕi), where q(. | ϕi) isthe proposal distribution with mean equal to ϕi.
3. Simulate the dataset (D*) using the modelparameter values ϕ* and calculate the sufficientstatistics S* = (S1
* ,⋯, Sp* ) based on D*. If the
distance between the sufficient statistics on thesimulated dataset and the observed statistics,d(S*, So), is greater than a threshold ε, then rejectthe proposed parameter values and set ϕi + 1 = ϕi. Ifthe difference is less than ε, then set ϕi + 1 = ϕ* withprobability α, calculated by the modifiedMetropolis-Hasting ratio:
α ¼ min 1;q ϕijϕ�� �
P ϕ�ð Þq ϕ�jϕi� �
P ϕi� �
( )
The distance, d(S*, So), is calculated as:
4. Set i = i + 1 and go back to step 2 for a large numberof iterations.
Although ABC requires that sufficient statistics (ornearly sufficient statistics) are used, it is non-trivialobtaining appropriate sufficient statistics. For this rea-son, Fearnhead et al. [26] proposed a semi-automaticmethod to generate “nearly” sufficient statistics for ABCby using a linear combination of commonly-used sum-mary statistics. The linear combination is obtained byregressing the summary statistics against known valuesof ϕ from simulated training datasets. The regressionequation serves as the new summary statistic. The out-line of their algorithm is as follows:
1. Define a training region in parameter space that isrepresentative of the parameter values one expectsto obtain in relatively high densities in the posteriordistribution.
2. Draw parameter value ϕT from this training regionand simulate a dataset, DT, based on ϕT. Calculate psummary statistics, ST = (S1
T,⋯, SpT), on DT. There
are no hard-and-fast rules about which summarystatistics to use, but for our data, there are obviouscandidates, including average pairwise intra- andinter-sample diversity, number of variable sites etc.(see below for a list of summary statistics used).
3. Repeat Step 2 for k iterations, where k≫ n.Therefore, there are k values of ϕT drawn from thetraining region, and for each ϕT there are a set of nsummary statistics.
4. For each parameter, ϕT∈ ϕT, regress the values ofϕT against the set of summary statistics, ST, obtainedfrom all simulations. As noted, the LS regressionequations that are obtained are a linear combinationof summary statistics in S, and serves as a single-valued sufficient statistic for each ϕ∈ ϕ, and usedas S* in the ABC-MCMC algorithm above.
Implementation of the full algorithmFor each dataset, we calculate the proportion of each nu-cleotide at each site of the alignment of short-read se-quences to a reference sequence. Our LS estimationprocedure is applied to obtain point estimates of the pa-rameters of interest. These point estimates are used asguidelines to help us define the training region forFearnhead et al.’s algorithm. We set the training regionas a uniform distribution with mean equal to the LS esti-mate with upper and lower bounds set to 5 fold aboveand below the mean, i.e. from (0.2× to 5×).Linear combinations of the following summary statis-
tics are used as sufficient statistics in the regression. Allsummary statistics are calculated using the proportion ofeach nucleotide at each site.
d S�; ;Soð Þ ¼Xpj¼1
S�j−So
jð Þ2So
j
Wu and Rodrigo BMC Bioinformatics (2015) 16:357 Page 5 of 12
1. Mean sequence distances, measured within eachtimepoint and between each pair of timepoints usingEquations 3 and 4.
2. Variances for the sequence distances between eachsite within each timepoint and between each pair oftimepoints.
Varintra;t ¼Xn
s¼1Dintra;s;t−�Dintra;t
� �2n
ð11Þ
Varinter;t1;t2 ¼Xn
s¼1Dinter;s;t1;t2−�Dinter;t1;t2
� �2n
ð12Þ
3. A chi-squared distance is calculated with the followingsteps:a. Setup 20 frequency categories at 0.05 intervals
ranging from 0 to 1.b. For each time point, assign each site to a
frequency category that encompasses theproportion of the most frequent nucleotide forthat site, and repeat this for all sites.
c. Finally, calculate the proportion of sites in eachcategory to obtain a relative frequency spectrumFSt . We also create a reference frequencyspectrum FSR, in which all categories have thesame proportion, i.e. 0.05 each. Calculate the chi-square distance between the reference frequencyspectrum and the frequency spectrum for eachtime point using the following formula,
X20category cð Þ¼1
FSt;c− FSR;c� �2
FSR;cð13Þ
4. Divide the pattern of the proportion of eachnucleotide between any two timepoints into fourdifferent categories, and record the number of sitesin each category:a. Category 1: Sites are identical across both
timepoints.b. Category 2: Sites in which one nucleotide is fixed
in one timepoint, and the same nucleotide is inthe majority in the other timepoint.
c. Category 3: There are mutations in bothtimepoints but the same nucleotide is mostfrequent in both timepoints (in other words, thenucleotide with the highest frequency in onetimepoint is also the nucleotide with the highestfrequency in the second timepoint).
d. Category 4: All others sites.
By applying the Fearnhead et al. algorithm, we obtainlinear equations as sufficient statistics for N and μ, theseare used in the ABC-MCMC above.
Two priors are specified for ABC-MCMC: the prior dis-tribution for N, the effective population size, is p Nð Þ ¼ 1
N ,and, the prior for mutation rate, μ, is a uniform distributionbetween [0,1]. The full algorithm is summarized in Fig. 1.We have found that allowing N and μ to vary inde-
pendently along the MCMC chain results in inefficientmixing, with higher autorcorrelation between samples ofa given interval. For this reason, we used block updating,where both N and μ are updated at each generation ofthe chain [27, 28].
Simulation analysisTwo sets of simulation analyses were performed. The firstset tested the performance of the LS method with a rangeof evolutionary parameters. One hundred datasets weresimulated by Bayesian Serial SimCoal [29]. The effectivepopulation size was fixed at 3000 and mutation rate fixedat 10−5 mutations per site per generation. By exploring dif-ferent combinations of other parameters, we were able toestimate the performance of this algorithm. The numberof sequences per timepoint were set to 5, 10, 20 and 40.
Fig. 1 Flow chart of the full ABC-MCMC algorithm
Wu and Rodrigo BMC Bioinformatics (2015) 16:357 Page 6 of 12
We used 3, 5 and 10 timepoints in our simulations, with100, 200, 400 and 600 generations between each timepoint. Tables 1, 2 and 3 show the combination of parame-ters used in the simulations. For each combination, 100datasets were simulated.The second set of simulations compared the per-
formance between our ABC-MCMC implementationand the Bayesian MCMC approach implemented inthe software Bayesian Evolutionary Analysis SamplingTrees (BEAST) [8].Based on the results of the first simulation analysis,
one hundred datasets were simulated using BayesianSerial SimCoal [29]. We choose the following parame-ters for this simulation, there were three timepointswith inter-timepoint intervals set at 400 generations,the number of sequences per timepoint was fixed at40, the mutation rate was fixed at 10−5 mutations persite per generation, and the effective population sizewas fixed at 3000.For our LS and ABC-MCMC methods, only the
relative frequency of nucleotides per site was availableas data. For BEAST analyses, the simulated full-lengthsequence alignment was used. BEAST ran for 10 mil-lion iterations, and ABC-MCMC ran for 1 million it-erations with 100,000 thousand iterations in the pre-processing stage. For both BEAST and ABC-MCMC,three independent chains were run for each dataset tocheck for convergence. Samples were recorded every1000 iterations in order to reduce the autocorrelationand the first 10 % of the samples were discarded asburn-in. The trace plots were checked manually forconvergence and the 95 % Highest Posterior Densityregion (HPD) for each parameter was calculated usingTracer [30].
ResultsSimulation result 1: least square estimationA series of simulations with different parameter set-tings were tested. Table 1 reports the means of LS
estimates of population size and mutation rate ob-tained over 100 simulations. The results demonstratethat LS estimation requires sufficient numbers ofsequences to obtain reasonable estimates of the pro-portion of each nucleotide for each site. When thenumber of full-length sequences is low (n < 10),change in one nucleotide at a site has a major effecton the nucleotide frequencies at that site; therefore, itis difficult for the LS algorithm is to estimate the trueparameters. In contrast, as the number of sequences inthe sample increases, estimation improves markedly.In Tables 2 and 3, the number of generations between
two consecutive timepoints has only a minimal effect on
Table 1 LS results with different number of sequences
Wu and Rodrigo BMC Bioinformatics (2015) 16:357 Page 7 of 12
the estimation efficiency, hence all simulations with inter-timepoint intervals longer than 200 generations have simi-lar performances. The number of timepoints does nothave a major effect either.The relative efficiency of LS against the other two
methods (ABC-MCMC and full-length BayesianMCMC with BEAST) can be compared with the sec-ond set of simulations, where data were generatedwith 3 timepoints, with 40 sequences per timepointand with intra-timepoint intervals of 400 generations.In these simulations, comparison of bootstrap 95 %Confidence Intervals of population size (Fig. 2) andmutation rate (Fig. 3) estimates, however, revealed anunflattering picture of LS estimation performance.Only a few LS 95 % Confidence Intervals enclose thetrue parameter values. Although the LS estimates areunbiased (i.e., their average over all simulations equalsthe true value), any given estimate performs poorly.
Simulation results 2: BEAST and ABC-MCMC estimationBEAST was used to analyze 100 datasets with aconstant-sized coalescent model with the parametersdescribed above, to obtain estimates of both popula-tion size and mutation rate. Each analysis ran for 10million iterations initially, and samples were storedevery 1000 iterations. After manually inspecting thetrace plots for convergence, some dataset were re-analysed with 100 million iterations. The 95 % HPDfor mutation rate contains the true value 93 timesand effective population size 89 times.
ABC-MCMC analyses were performed on the same100 datasets with 1 million iterations and sampleswere stored every 1000 iterations. In addition, another100,000 iterations of preprocessing simulations wereused to estimate the sufficient statistics. An exampleof the regression equations used as sufficient statisticsis in.Some datasets failed to converge for ABC-MCMC
and were reanalyzed with 10 million iterations. Twoout of 100 datasets failed to converge even with 10million iterations, and these are excluded from theanalyses. Figure 4 is an example of the trace plot forboth mutation rate and effective population size. Theplot indicates that the MCMC chain mixes well. Fig-ure 5 gives an example of the posterior density of theeffective population size recovered, plotted against theprior distribution. Given the difference between theprior and posterior density, it is apparent that thereis sufficient signal in the data to shift the posteriordistribution of effective population size away from theprior distribution.The 95 % HPD for mutation rate included the true value
87 times out of 98 analyses, and the 95 % HPD for popula-tion size included the true value 91 times out of 98 ana-lyses. The results of both BEAST and ABC-MCMC aresummarized in Table 4. Based on the number of 95 %HPDs that include the true values, both BEAST and ABC-MCMC perform similarly. However, the actual 95 % HPDintervals from ABC-MCMC are wider than the 95 % HPDfrom BEAST (see Figs. 2 and 3): on average the 95 %
Fig. 2 Plot of 95 % Confidence Intervals and 95 % Highest Posterior Densities of population size recovered using the LS bootstrap, ABC-MCMCand BEAST. The green lines are the 95 % CIs of LS bootstraps, the red lines are the 95 % HPDs of ABC-MCMC, and the blue lines are the 95 %HPDs obtained using BEAST. The true value of population size is shown as a solid black line. Note that the vertical axis is measured on a log scale
Wu and Rodrigo BMC Bioinformatics (2015) 16:357 Page 8 of 12
HPDs for effectively population size are 8-fold wider andthose for mutation rate are 5-fold wider.
Discussion and conclusionsIn this paper, we propose two new algorithms to esti-mate evolutionary genetic parameters by using only the
frequency of nucleotides at each site as input. The least-squares method provides a fast way to estimate effectivepopulation size and mutation rate, but our results indi-cate that LS estimates (and their bootstrap confidenceintervals) tend to under- or over-estimate the true pa-rameters. Consequently, we cannot recommend our LS
Fig. 3 Plot of 95 % Confidence Intervals and 95 % Highest Posterior Densities of mutation rate recovered using the LS bootstrap, ABC-MCMC andBEAST. The green lines are the 95 % CIs of LS bootstraps, the red lines are the 95 % HPDs of ABC-MCMC, and the blue lines are the 95 % HPDsobtained using BEAST. The true value of mutation rate is shown as a solid black line
Fig. 4 Trace plot from ABC-MCMC for both effective population size and mutation rate after removing the first 10 % of the generations asburn-in. This demonstrates that the MCMC chain mixes well
Wu and Rodrigo BMC Bioinformatics (2015) 16:357 Page 9 of 12
algorithm as a standalone method for obtaining esti-mates of evolutionary parameters. Our LS method,nonetheless, provides a useful baseline for the trainingregion which we need to use to derive nearly sufficientstatistics for our ABC-MCMC procedure.Parameter values recovered by ABC-MCMC are com-
parable to those obtained using BEAST, albeit withmuch wider 95 % HPDs. The performance of ABC-MCMC, relative to that of BEAST is unsurprising, be-cause BEAST has access to the full-length alignment ofall sequences in the sample. When the full-length align-ment is summarized by the proportion of nucleotides ateach site, there is inevitably a loss of information. Add-itionally, our methods do not reconstruct phylogenies ofthe sequences, further reducing estimation efficiency.Despite these significant constraints, it is interesting thatthe sitewise nucleotide frequencies are able to provide
enough information to obtain meaningful estimates ofthe parameters of interest.The methods we have developed assume that the ref-
erence sequence is available and short reads can bealigned to the reference accurately. Obviously the per-formance of this approach is dependent on the quality ofthe reference sequence and how well these short readsare aligned to it. As noted above, in our simulations wehave obtained site nucleotide frequencies from the full-length sequences; we expect that with real NGS data,the accuracy with which we estimate the sitewise fre-quencies of each nucleotide will depend on sequencingerror. Preprocessing the raw reads for quality, read-length, and identity to the reference sequence is likely toremove a significant amount of this error. In this case, itis unlikely that any remaining errors will distort the sitefrequencies enough to have a noticeable effect on theestimates. Also, all insertions and deletions (indels) have
Fig. 5 The prior and posterior distributions for the effective population size from ABC-MCMC
Table 4 Summarized the number of times that 95 % HPD includes the true value for both ABC-MCMC and BEAST
Algorithm ABC-MCMC BEAST
No. of Converged Dataset 98 100
No. of 95 % HPD for μ includes the true value (1e-5) 87 93
No. of 95 % HPD for N includes the true value (3000) 91 89
Mean Effective Population Size (Lower quartile, upper quartile) 2457.4 (1503.0, 3637.9) 2851.688 (2623.965, 3134.936)
Wu and Rodrigo BMC Bioinformatics (2015) 16:357 Page 10 of 12
been ignored in all simulations and summary statistics.However, our implementation of the ABC-MCMCmodel allows indels at each site as a fifth “nucleotide” orstate. We have not yet applied this to our analysis.One major advantage of using only the site frequencies
is that it can be applied to an arbitrary number of se-quences in the original sample. In our simulations, wedid not simulate short-read sequences; instead, we usedthe full-length sequences to derive the nucleotide fre-quencies at each site. Nonetheless, the amount of timerequired to estimate the proportion of each nucleotidefor each site will scale linearly to the number of short-read sequences only, regardless of the number of full-length sequences from which these were derived, and itis likely to be a very fast calculation. In contrast,methods that rely on building phylogenies from full-length alignments must contend with the superexponen-tial growth in the number of possible trees as the num-ber of sequences increases. For a realistic NGS dataset,the number of reconstructed full-length haplotypes canbe large. Consequently, our methods can be used onNGS datasets with large numbers of short read frag-ments, obtained from a large number of full-lengthsequences.In this paper, we have only developed methods to esti-
mate effective population size and mutation rate. Popula-tion geneticists are also interested in other evolutionaryparameters, including migration rates and recombinationrates. We believe that ABC-MCMC, like other BayesianMCMC methods, provides a flexible framework to con-struct more complex evolutionary models. But the use ofsitewise nucleotide frequencies alone means that we lackthe finer-grained information afforded by a genealogy orphylogeny of full-length sequences; consequently, we arenot certain how much complexity can be added to themodels before the sitewise nucleotide frequencies we usein our methods lose all signal. This is certainly an area thatwe intend to explore. The source code for ABC-MCMC isavailable at https://github.com/stevenhwu/SF-ABC.
Competing interestsThe authors declare that they have no competing interests.
Authors’ contributionsSHW and AGR conceived and designed the model, and wrote themanuscript. Both authors read and approved the final manuscript.
AcknowledgmentsSHW and AGR were supported by a Duke University research grant. Wethank Yuantong Ding for sharing her results on the use of other short-readhaplotype assembly programs, and other members of the Rodrigo lab forhelpful comments and ideas as we developed these methods.
Author details1Biodesign Institute, Arizona State University, Tempe, AZ 85287, USA.2Department of Biology, Duke University, Box 90338, Durham, NC 27708,USA. 3The National Evolutionary Synthesis Center, Durham, NC 27705, USA.
Received: 28 April 2015 Accepted: 30 October 2015
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