ORIGINAL RESEARCH ARTICLEpublished: 16 April 2014

doi: 10.3389/fevo.2014.00011

The impact of incorporating molecular evolutionary modelinto predictions of phylogenetic signal and noiseZhuo Su1, Zheng Wang1,2, Francesc López-Giráldez1,3 and Jeffrey P. Townsend1,2,4*

1 Department of Ecology and Evolutionary Biology, Yale University, New Haven, CT, USA2 Department of Biostatistics, Yale University, New Haven, CT, USA3 Yale Center for Genomic Analysis, Yale University, New Haven, CT, USA4 Program in Computational Biology and Bioinformatics, Yale University, New Haven, CT, USA

Edited by:

Enrique P. Lessa, Universidad de laRepública, Uruguay

Reviewed by:

Charles Bell, University of NewOrleans, USAJames H. Degnan, University ofNew Mexico, USA

*Correspondence:

Jeffrey P. Townsend, Department ofBiostatistics, Yale School of PublicHealth, 135 College St. #222,New Haven, CT 06510, USAe-mail: jeffrey.townsend@yale.edu

Phylogenetic inference can be improved by the development and use of better modelsfor inference given the data available, or by gathering more appropriate data giventhe potential inferences to be made. Numerous studies have demonstrated the crucialimportance of selecting a best-fit model to conducting accurate phylogenetic inferencegiven a data set, explicitly revealing how model choice affects the results of phylogeneticinferences. However, the importance of specifying a correct model of evolution forpredictions of the best data to be gathered has never been examined. Here, we extendanalyses of phylogenetic signal and noise that predict the potential to resolve nodes ina phylogeny to incorporate all time-reversible Markov models of nucleotide substitution.Extending previous results on the canonical four-taxon tree, our theory yields an analyticalmethod that uses estimates of the rates of evolution and the model of molecular evolutionto predict the distribution of signal, noise, and polytomy. We applied our methods toa study of 29 taxa of the yeast genus Candida and allied members to predict thepower of five markers, COX2, ACT1, RPB1, RPB2, and D1/D2 LSU, to resolve a poorlysupported backbone node corresponding to a clade of haploid Candida species, aswell as 19 other nodes that are reasonably short and at least moderately deep in theconsensus tree. The use of simple, unrealistic models that did not take into accounttransition/transversion rate differences led to some discrepancies in predictions, butoverall our results demonstrate that predictions of signal and noise in phylogenetics arefairly robust to model specification.

Keywords: model selection, signal, noise, phylogenetic informativeness, phylogenetic inference, maximum

likelihood, Bayesian estimation

INTRODUCTIONPhylogenetic inferences can be improved either by improving themodels applied to data, or by improving the quality of the data.Enormous progress has been made in the development of real-istic, powerful evolutionary models for phylogenetic inference,and studies have demonstrated that using correct evolutionarymodels on a same data set can be essential to making correctinferences (e.g., Sullivan and Swofford, 1997; Kelsey et al., 1999;Ripplinger and Sullivan, 2010). Nonetheless, still larger gains inaccuracy of inference could be obtained by optimal selection ofthe data to be gathered to address the phylogenetic question athand (Goldman, 1998; Townsend et al., 2008; Tekle et al., 2010;Fong and Fujita, 2011; Moeller and Townsend, 2011; Crawleyand Hilu, 2012; Horreo, 2012; Silva et al., 2012; Walker et al.,2012; Granados Mendoza et al., 2013). Furthermore, given theadvance of next-generation sequencing methods, such as RADSequencing (Baird et al., 2008) and anchored hybrid enrichment(Lemmon et al., 2012), which allow generation of sequence datafrom hundreds to thousands of putatively orthologous loci, itis becoming increasingly important to either selectively generateor selectively filter the immense data sets generated by next-generation sequencing methods. Thus, inference can be based on

the most useful or least misleading loci for resolving the phy-logeny at hand. However, it remains to be explored the degree towhich model selection impacts the determination of optimal locifor phylogenetic inference.

Less accurate and even inconsistent inferences can result fromincorrect model assumptions (Felsenstein, 1978; Huelsenbeckand Hillis, 1993; Penny et al., 1994; Bruno and Halpern, 1999;but see Rzhetsky and Sitnikova, 1996; Yang, 1997; Posada andCrandall, 2001). Typically, under-parameterization of modelswill lead to underestimates of some high-magnitude compo-nents of the evolutionary process (e.g., Huelsenbeck and Hillis,1993; Gaut and Lewis, 1995; Sullivan and Swofford, 1997, 2001),components that then become a source of homoplasy and mis-leading inference. While the apparent fit of a model to thedata can always be improved by adding additional parameters,over-parameterization without sufficient data can lead to largestochastic errors and misestimation of individual parameters,simultaneous with a higher computational demand (Rannala,2002; Lemmon and Moriarty, 2004). To find a balance betweenthese issues, a variety of statistical approaches have been devel-oped to select models for maximum likelihood (ML) analysisand Bayesian estimation, such as the hierarchical likelihood-ratio

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ECOLOGY AND EVOLUTION

Su et al. Models, phylogenetic signal, and noise

test (hLRT; Frati et al., 1997; Sullivan et al., 1997; Posada andCrandall, 1998), relative Akaike information (Akaike, 1973), rela-tive Bayesian information (Schwarz, 1978), and Decision Theory(Minin et al., 2003; Abdo et al., 2004). Multiple models areoften supported by different selection methods and alternativelysupported models often do lead to incongruent optimal treetopologies.

Nevertheless, ML analysis with alternatively supported mod-els rarely yields significantly different evolutionary inferences,because different outcomes attributable to alternatively supportedmodels are mostly limited to nodes that are poorly resolved bythe data set under any model (Ripplinger and Sullivan, 2008).Because the prospects for improvement of phylogenetic infer-ence by use of optimal data are so promising, obtaining a bet-ter idea of their robustness to model selection is paramount.Recently, several analytical approaches to quantifying the phy-logenetic power of molecular characters have been introduced.Goldman (1998) pioneered an analytical method that uses theFisher information matrix to determine optimal evolutionaryrates and taxon-sampling strategies to maximize the informationto resolve a divergence in a phylogeny. Susko (2011) calculated vialarge sequence-length approximations the probabilities of correctphylogenetic reconstruction for ML estimation for a single poorlyresolved internode. For a proposed discrete data set, Townsendet al. (2012) applied empirical estimates of site substitution ratesand an s-state Poisson model of molecular evolution (s ≥ 2) topredict the probabilities of a data set leading to correct, incor-rect, and polytomous resolution of a quartet-taxon phylogeny.This analytical approach makes it possible to address the unre-solved question of whether choice in model selection affectspredictions of phylogenetic informativeness. Not only does phy-logenetic informativeness depend critically on estimates of therate of evolution of the characters to be applied to a phyloge-netic problem (Graybeal, 1994; Naylor and Brown, 1998; Yang,1998; Rokas and Holland, 2000; Townsend, 2007; Townsend andLeuenberger, 2011; Townsend et al., 2012), but also the impactof characters on inference would likely depend upon the modelof evolution to which they presumably adhere. To investigate thisquestion, we generalize the Townsend et al. (2012) phylogeneticsignal and noise analysis based on the s-state Poisson model, toincorporate any time-reversible homogenous Markov model ofnucleotide substitution. This generalization allows the signal andnoise analysis to incorporate more realistic, higher parameter-ized assumptions of molecular evolution than the s-state Poissonmodel.

After developing this theory, we apply it to an examplestudy, analyzing 29 taxa of the yeast genus Candida and alliedteleomorph genera using phylogenetic signal and noise analy-sis based on 14 commonly used nucleotide substitution models(Figure 1). The yeast genus Candida and allied teleomorph generabelong to the order of Saccharomycetales. Members of the genusCandida are ubiquitous in numerous natural and artificial habi-tats (Kurtzman and Fell, 1998). Several species of Candida, such asC. parapsilosis, C. tropicalis, and C. albicans, are among the mostcommon human pathogenic fungi. Relationships among differ-ent genera or families of the Saccharomycetales have not yet beenfully resolved, especially for relationships along the backbone of

the trees. In a recent study of a six-gene yeast phylogeny witha focus on medically important Candida species by Diezmannet al. (2004), major clades were recognized with statistical sup-port in the order, but several backbone nodes of the clade 2,which included haploid Candida species such as C. guillermondii,C. lusitaniae, Debaryomyces hansenii, and other Candida speciesthat are not invasive pathogens for humans, were not well sup-ported. Similarly, in another five-gene yeast phylogeny study of70 taxa of the genus Candida and allied sexually reproducing gen-era (Tsui et al., 2008), the backbone node corresponding to theclade of a similar set of haploid Candida taxa was not well sup-ported. However, in a recent study using data selected from 706orthologs for only 17 yeast species, the haploid Candida cladewas strongly supported by peptide sequences but not by codingnucleotide sequences from the selected genes (Butler et al., 2009).In our example study, we assay five markers: the second subunit ofthe mitochondrial cytochrome oxidase gene (COX2), the largestsubunit (RPB1) and the second largest subunit (RPB2) of theRNA polymerase II gene, the actin gene (ACT1), and the D1/D2LSU rRNA gene, for their power to resolve a backbone node cor-responding to a clade of haploid Candida species, which has beenpoorly resolved in previous studies, as well as 19 other nodes fea-turing a relatively short, deep internode in the consensus tree. Bycomparing the five genes’ predicted probabilities of resolution forthe 20 assayed nodes based on all alternative models, we elucidatethe degree to which choice in model selection affects predictionsof phylogenetic signal and noise.

THEORYAs in Townsend et al. (2012), we model phylogenetic signal andnoise as accurate and inaccurate parsimony-informative sites forresolution of a four-taxon tree. A site is considered parsimony-informative for resolving the four-taxon tree if it exhibits anAABB pattern of character states at the branch leaves, meaningthat at the given site, two of the branch leaves feature an identi-cal character state (suggesting these two branches belong to thesame clade) while the other two branch leaves share a differentcharacter state. The four-taxon tree has three possible tip-labeledsubtrees, all of which can be supported by an AABB pattern, butonly one of the three subtrees matches the actual four-taxon treetopology (c.f. Figure 1 in Townsend et al., 2012). Any other pat-tern yields no net contribution per site to phylogeny resolutionunder maximum parsimony and very minimal impact per siteunder ML analysis and Bayesian methods; thus any non-AABBpattern is conservatively assumed to make no contribution awayfrom a polytomy by our approach.

Each Markov model of nucleotide substitution can be math-ematically expressed as a four-by-four instantaneous rate matrixQ, in which the element qij is the rate of change from nucleotidebase i to base j during an infinitesimal time period dt, wherej �= i. We use 1, 2, 3, and 4 in matrix indices to denote thefour nucleotide character states in the order of Thymine (T),Cytosine (C), Adenine (A), and Guanine (G). To date almost allof the nucleotide substitution models proposed in the literatureare time-reversible, meaning that the overall rate of change frombase i to base j in a given length of time is assumed to be the sameas that from base j to base i, i.e., πiqij = πjqji(j �= i). For the most

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Su et al. Models, phylogenetic signal, and noise

FIGURE 1 | Relationship among 14 homogenous time-reversible

models of nucleotide substitution. Arrows indicate parameterrestrictions that reduce a more general, higher-parameterized model to amore restricted, lower-parameterized one. For each case of distinctsubstitution types corresponding to a pair of models, the model outside thebrackets allows unequal equilibrium base frequencies and the model insidethe brackets assumes equal equilibrium base frequencies. These modelsare: JC (Jukes and Cantor, 1969), which is equivalent to the s-state Poissonmodel for nucleotides (s = 4); K2P (Kimura 2-Parameter, a.k.a. K80; Kimura,1980); TrNef (Tamura and Nei, 1993 model with equal base f requencies);K3P (Kimura 3-Parameter, a.k.a. K81; Kimura, 1981); TIMef (TransItionalModel with equal base f requencies); TVMef (TransVersional Model withequal base f requencies); SYM (SYMmetric, Zharkikh, 1994); F81(Felsenstein, 1981); HKY (Hasegawa et al., 1985; Felsenstein described anessentially identical model, F84, using a different parameterization—seeKishino and Hasegawa, 1989; Felsenstein and Churchill, 1996; implementedin the software DNADIST in PHYLIP—see Felsenstein, 1993); TrN (Tamuraand Nei, 1993); K3Puf (K3P model with unequal base f requencies, a.k.a.K81uf); TIM (TransItional Model: rAC = rGT �= rAT = rCG �= rAG �= rCT);TVM (TransVersional Model: rAG = rCT �= rAC �= rAT �= rCG �= rGT); andGTR (General Time Reversible, Tavaré, 1986; Rodriguez et al., 1990).

general time-reversible Markov process of nucleotide characters,the General Time Reversible (GTR) model, the rate matrix can bewritten as

Q = μ

⎡⎢⎢⎣

∗ aπC bπA cπG

aπT ∗ dπA eπG

bπT dπC ∗ f πG

cπT eπC f πA ∗

⎤⎥⎥⎦ , (1)

in which the diagonal elements (∗) are qii = −∑j �=i qij such that

elements across each row sum to zero. The product of the meaninstantaneous substitution rate, μ, and a relative rate parameter,a,b,c,d,e, or f, yields a rate parameter. The frequency parameters,πT , πC , πA, and πG, represent the equilibrium frequencies of thefour nucleotide bases, thereby obeying πT + πC + πA + πG = 1.Imposing further constraints on the model parameters reducesEquation 1 from the GTR model to lower-parameterized models(Table 1).

The average substitution rate per nucleotide character, λ, canbe expressed in terms of the rate and frequency parameters as

λ =∑i �=j

πiqij = 2μ(aπTπC + bπTπA + cπTπG + dπCπA

+ eπCπG + f πAπG). (2)

For a nucleotide site with a substitution rate λ and a branch oflength t, λt gives the expected number of substitutions accumu-lated at the site between the two ends of the branch. Becausemodeling of signal and noise probability functions depends onthe site substitution rate λ, the rate matrix needs to be written asa function of λ, Q(λ), by solving Equation 2 for μ, yielding

μ = λ

2(aπTπC + bπTπA + cπTπG + dπCπA + eπCπG + f πAπG). (3)

Then μ from Equation 3 may be substituted into Equation 1. Tocalculate the probabilities of character state changes over a branchof a finite length, we need to calculate the matrix of substitutionprobability P(λ, t), via Equation 4:

P(λ, t) = eQ(λ)t, (4)

in which the element pij(λ, t) represents the probability of changefrom nucleotide base i to base j at a site with a substitution rateλ over a branch of length t. To solve for pij(λ, t), we apply eigendecomposition to the instantaneous rate matrix Q(λ):

Q(λ) = U�U−1, (5)

in which U is a four-by-four matrix whose i-th column is thei-th eigenvector of Q(λ) and � is a four-by-four diagonal matrixwhose i-th diagonal element is the i-th eigenvalue of Q(λ). Thematrix exponential in Equation 4 can thus be evaluated as

P(λ, t) = eQ(λ)t = Ue�tU−1. (6)

For lower-parameterized models (i.e., JC, F81, K2P, HKY, TrNef,TrN, and K3P models), analytical solutions exist for Equation 6

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Su et al. Models, phylogenetic signal, and noise

Table 1 | Parameterization of 14 nucleotide substitution models.

Number of Models of equal Models of unequal

substitution equilibrium equilibrium

types base frequencies base frequencies

πT = πC = πA = πG = 14 πT �= πC �= πA �= πG

1 JC F81

a = f = b = e = c = d ; equal substitution rate

2 K2P HKY

a = f �= b = e = c = d ; transition and transversion

3 TrNef TrN

a �= f �= b = e = c = d ; 2 transitions and transversion

3 K3P K3Puf

a = f �= b = e �= c = d ; transition and 2 transversions

4 TIMef TIM

a �= f �= b = e �= c = d ; 2 transitions and 2 transversions

5 TVMef TVM

a = f �= b �= e �= c �= d ; transition and 4 transversions

6 SYM GTR

a �= f �= b �= e �= c �= d ; 6 unequal substitutions

(also see Yang, 1994). Regardless of model complexity, Equation 6can also be evaluated numerically after substitution of estimatedvalues of the relative rate and base frequency parameters of anygiven model.

We utilize the substitution probabilities calculated viaEquation 6, appropriate for any time-reversible Markov model ofnucleotide substitution, to quantify probabilities for the evolu-tion of character states at a single site with a substitution rate of λ

across an ultrametric four-taxon tree with an internode of lengthto and four subtending branches of equal length T (Figure 2).

We denote the ancestral character states at the two endsof the internode as M and N and the character states at theterminal leaves of the four branches as C1, C2, C3, and C4,respectively, (Figure 2). The pattern of character states that isconsistent with the correct subtree, denoted as τ3, is C1 = C2,C3 = C4, C1 �= C3. The rooting of the tree does not affect anal-ysis outcomes for time-reversible models; consequently, we canwrite out the potential evolutionary histories of character statesby arbitrarily starting from the ancestral character state M onone end of the internode. M can be any of the four charac-ter states, and the probability of each possible state occurringthere is given by its corresponding nucleotide base frequencyπM , where M = 1, 2, 3, or 4. Likewise, on the other end of theinternode, N can be any of the four states, and the probabil-ity of observing each possible state at N from M is pMN(λ, to),where N = 1, 2, 3, or 4. Moving along the four branches tothe terminal leaves, the probability of observing every possible

FIGURE 2 | The canonical phylogenetic quartet in an ultrametric form,

with an internode of length to and four subtending branches of equal

length T. The ancestral character states at the two ends of the internodeare denoted as M and N. The character states at the terminal leaves of thefour branches are denoted as C1, C2, C3, and C4. The pattern of characterstates at a site consistent with the true topology of the four-taxon tree, τ3,is C1 = C2, C3 = C4, C1 �= C3. A site pattern that matches C1 = C3,C2 = C4, C1 �= C2 or C1 = C4, C2 = C3, C1 �= C2 is consistent with one ofthe two possible incorrect subtrees of the four-taxon tree: τ1 or τ2,respectively.

permutation of character states at the leaves conditional on M andN is given by pMC1(λ, T)pMC2(λ, T)pNC3(λ, T)pNC4(λ, T), whereC1, C2, C3, C4 = 1, 2, 3, or 4.

Phylogenetic signal leads to the true topology τ3, whichdemonstrates the synapomorphic pattern of character states,C1 = C2, C3 = C4, C1 �= C3. Phylogenetic signal arises if a char-acter state change has occurred during the deep short internode(i.e., M �= N) and the signal is preserved if zero subsequent char-acter state change has occurred on the subtending lineages toobscure the ancestral character states (i.e., C1 = C2 = M, C3 =C4 = N). In the meantime, noise due to homoplasy can alsolead to the pattern of character states that is consistent withτ3, through character state changes along subtending lineagesthat randomize the character states to the desired pattern at thebranch terminal leaves regardless of the presence of signal over theinternode (c.f. Figure 1 in Townsend et al., 2012). Therefore, theprobability of a site supporting the true topology τ3 with intern-ode length to and length of subtending branches T due to eithersignal or homoplasy, is equal to the sum of probabilities associ-ated with these two scenarios that both lead to the synapomorphicpattern of C1 = C2, C3 = C4, C1 �= C3:

y(λ; to, T)

=4∑

M=1

4∑N=1

4∑C1=C2=1

∑C3=C4 �=C1

πMpMN(λ, to)pMC1(λ, T)pMC2

(λ, T)pNC3(λ, T)pNC4(λ, T). (7)

A homoplasious site pattern of character states that is consis-tent with either of the two incorrect quartet topologies, τ1 and

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Su et al. Models, phylogenetic signal, and noise

τ2, matches C1 = C3, C2 = C4, C1 �= C2, or C1 = C4, C2 = C3,C1 �= C2, respectively, (Figure 2). Both patterns are a result ofhomoplasy along the subtending branches that randomize thecharacter states regardless of the presence of signal over theinternode. The probabilities of a site supporting either of theincorrect topologies (τ1 and τ2) with the defined internode andbranch lengths can be evaluated as the sum of probabilities asso-ciated with all possible scenarios of character state changes thatlead to the respective homoplasious pattern, C1 = C3, C2 = C4,C1 �= C2, or C1 = C4, C2 = C3, C1 �= C2:

x1(λ; to, T)

=4∑

M=1

4∑N=1

4∑C1=C3=1

∑C2=C4 �=C1

πMpMN(λ, to)pMC1

(λ, T)pMC2(λ, T)pNC3(λ, T)pNC4(λ, T), (8)

x2(λ; to, T)

=4∑

M=1

4∑N=1

4∑C1=C4=1

∑C2=C3 �=C1

πMpMN(λ, to)pMC1

(λ, T)pMC2(λ, T)pNC3(λ, T)pNC4(λ, T). (9)

The probability of a site supporting any pattern that is not con-sistent with τ1, τ2, or τ3 (i.e., any non-AABB pattern) is given by

z(λ; to, T) = 1 − y(λ; to, T) − x1(λ; to, T) − x2(λ; to, T). (10)

Following Townsend et al. (2012), we consider n nucleotide sites,each with its own distinct rate of substitution λi, 1 ≤ i ≤ n. Forthis set of sites, Equations 7–10 calculate the probabilities ofeach nucleotide site i supporting the correct topology (y), thetwo incorrect topologies (x1 and x2), and polytomy (z) for anultrametric four-taxon tree with branch lengths to and T basedon any time-reversible Markov model of nucleotide substitu-tion. The greater the number of sites that support the correcttopology there are compared to the numbers of sites that sup-port either of the incorrect topologies, the more likely the wholeset of sites will lead to correct resolution of the four-taxon tree.The probabilities of the whole set of n sites leading to correct,incorrect, and polytomous resolution of the four-taxon tree canthen be evaluated either by building a histogram of potential lev-els of resolution by repeated serial Monte Carlo across sites (c.f.Townsend et al., 2012), or by use of an accurate analytical approxi-mation that yields essentially identical results (Equations 11–13 ofTownsend et al., 2012; c.f. Appendix II of Townsend et al., 2012).Compared to the Monte Carlo approach, the analytical approx-imation demands a significantly less amount of computationpower and has no stochastic element to its result.

MATERIALS AND METHODSThe existing COX2, RPB1, RPB2, ACT1, and D1/D2 LSUsequences of Schizosaccharomyces pombe and Zygoascus helleni-cus (chosen as the outgroups; Diezmann et al., 2004) and 29taxa of the yeast genus Candida and allied teleomorph generawere downloaded from GenBank (Data Sheet 1). The sequence

alignments of the five markers were generated from SATé (Liuet al., 2009) and used for all subsequent analyses without manualadjustment except necessary trimming of the highly gapped endsand indels using MacClade 4.0 (Maddison and Maddison, 2000).The best-fit nucleotide substitution models for the five genesequences were selected with the Akaike information criterion(AIC) measure and four gamma categories via ModelGeneratorv0.85 (Keane et al., 2006). The search for the best-fit model waslimited to the 14 models in Figure 1 without and with �, I,and I +�.

One single data set which consisted of the 31 taxa and 3617characters was prepared with five partitions from alignments ofthe five markers (597 characters from COX2, 574 from RPB1,936 from RPB2, 979 from ACT1, and 531 from D1/D2 LSU; DataSheet 2) and was deposited in TreeBase under the accession num-ber S14200. Bayesian phylogenetic analysis of the five-gene dataset was performed via MrBayes 3.1.2 (Ronquist and Huelsenbeck,2003) using the Metropolis-coupled Markov chain Monte Carlomethod under the optimal model for each gene partition selectedby ModelGenerator, by running four chains with 2,000,000 gen-erations. At every 100th generation trees were sampled, and thefirst 1000 trees sampled prior to the likelihoods converging tostable values were discarded as burn-ins. A 50% majority con-sensus tree was computed with the remaining Bayesian trees, andsignificantly supported nodes were defined to exhibit a Bayesianposterior probability (BPP) ≥ 0.95.

A chronogram for the combined five-gene data set wasobtained based on the consensus tree topology across the con-verged Bayesian posterior computed by MrBayes 3.1.2. S. Pombeand Z. hellenicus were removed from the consensus tree, leav-ing a tree with a basal trichotomy. This species phylogeny wascalibrated by constraining the new basal split to a minimumage of 160 Myr and a maximum age of 180 Myr (cf. Mirandaet al., 2006). Divergence times were estimated by penalized like-lihood with a truncated Newton algorithm in r8s version 1.71(Sanderson, 2006), setting the r8s smoothing parameter to 4.1, avalue obtained by following the r8s program manual instructions(available at http://loco.biosci.arizona.edu/r8s/).

Rates of substitution for each individual site in the five-genedata set were independently estimated based on each of the14 alternative models in Figure 1 by submitting the sequencealignment with the obtained chronogram to PhyDesign (López-Giráldez and Townsend, 2011), to execute ML estimation ofnucleotide substitution rates via HyPhy (Pond et al., 2005).ModelGenerator v0.85 was executed to supply the values of rel-ative rate and frequency parameters for each of the five markersbased on each of the models. The node corresponding to the hap-loid Candida clade and 19 other nodes featuring a relatively short,deep internode in the chronogram were selected to be assayed forsignal and noise analysis, and the internode length to and sub-tending branch length T of each of the 20 assayed nodes wereabstracted from the chronogram.

For each assayed node, four representative taxa were selectedso that each internode corresponded to a four-taxon problemin which the lengths of the four subtending branches of theinternode were as close to one another as possible. In caseswhere the four subtending branches of a given node had unequal

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Su et al. Models, phylogenetic signal, and noise

lengths, T was approximated as the length of the shortest ofthe four subtending branches. This approximation was justifiedbased on the observation that in all of the assayed nodes, thesubtending branches were all similar in length and significantlylonger than the internode. Using the estimated site substitu-tion rates for the five gene sequence alignments based on the14 models and the length measures of the 20 assayed nodes,the probabilities of each of the five markers providing incor-rect, polytomous, and correct resolution of each of the 20 nodesbased on the 14 alternative models were calculated via the ana-lytical method for signal and noise analysis (via Equations 7–10per nucleotide site and Equations 11–13 of Townsend et al.,2012 per data set). These calculations were implemented viaWolfram Mathematica 7 (Wolfram Research, Inc.). The WolframMathematica 7 implementation of the signal and noise analysiscalculations is available for download with instructions from thePhyDesign website (López-Giráldez and Townsend, 2011; http://phydesign.townsend.yale.edu/).

RESULTSThe best-fit models selected by AIC were TVM +� for COX2;GTR + I +� for RPB1, RPB2, and ACT1; and GTR +� for D1/D2LSU. In the computed consensus tree (Figure 3), the backbonenode corresponding to the haploid Candida clade was poorly sup-ported (BPP = 0.72). Of the other 19 assayed nodes, all exceptone (BPP = 0.57) were significantly supported (Data Sheet 3).Subtending branch lengths (Data Sheet 4) were extracted from theobtained chronogram based on the consensus tree (Data Sheet 3).

To prioritize different markers for resolving a given node, itis necessary to compare their probabilities of incorrect resolu-tion, polytomy, and correct resolution simultaneously to makethe optimal marker selection based on the specific experimen-tal design goal at hand. Here we assumed that the GTR modelpredicts the correct ranking of the five markers’ powers for resolv-ing a given node, because the best-fit model selected by AIC wasthe GTR model for four of the five markers, and the TVM modelselected for COX2 is highly similar to the GTR model in terms ofparameterization complexity. Based on the GTR model, the prob-abilities of correct resolution, polytomy, and incorrect resolutionwere similar across the five markers for the backbone node corre-sponding to the haploid Candida clade; among the five markers,RPB2, which had the highest probability of correct resolutionand the lowest probability of incorrect resolution for the node(Figure 4), was the optimal marker.

To efficiently report the results for all of the 20 assayed nodes,comparisons of only the probabilities of correct resolution werepresented, because the probabilities of incorrect resolution weresimilar across the five markers for any given node (e.g., Figure 4).Within every assayed marker, signal and noise analysis basedon the 14 alternative models predicted highly consistent prob-abilities of the marker correctly resolving the haploid Candidaclade, albeit with several modest deviations (Figure 5). For RPB1,RPB2, ACT1, and D1/D2 LSU, the JC and F81 models predictedhigher probabilities of correct nodal resolution than those pre-dicted when analyzed with the other 12 models. For COX2, theprobabilities of correct nodal resolution predicted by the JC, K2P,and TrNef models were considerably higher than those by the

other eleven models (Figure 5). Exactly the same overall trendsand deviations were observed from signal and noise analysis of theother 19 nodes, regardless of whether the node was significantlysupported or not.

For all of the 20 assayed nodes, the GTR model ranked the fivemarkers in the same order of RPB2 (highest probability of cor-rect resolution), ACT1, COX2, RPB1, and D1/D2 LSU (lowest).For every single assayed node (e.g., Figure 5), half or more of the14 models predicted the correct ranking of the five markers andat least 10 models predicted the correct optimal marker, RPB2.Moreover, only the JC, F81, K2P, and TrNef models ever failed topredict the correct optimal marker to resolve one of the assayednodes, while the other ten more complex models never failed topredict the optimal marker for any of the 20 assayed nodes.

By comparing the parameter values of the 14 models estimatedfor the five markers (Data Sheet 5), we observed that within eachmarker, there was an exact match between the models whose pre-dicted probabilities of correct nodal resolution were inconsistentwith those by the other models and the models whose estimatedrate or frequency parameter values deviated significantly fromthose of the other models. In RPB1, RPB2, ACT1, and D1/D2LSU, a shared salient feature of the underlying evolutionaryprocesses is that transitions happened much faster than transver-sions. In particular, the transition between T and C occurred atthe fastest rate among all substitutions. For example, based onGTR, the optimal model for these four markers, rTC/rTG was 9.1for ACT1, 4.9 for RPB1, 4.5 for RPB2, and 3.2 for D1/D2 LSU(Data Sheet 5). Because the JC and F81 models restrict all sub-stitutions to occur at the same rate, these two models failed toaccount for the high transition/transversion bias that the othermodels accounted for with regard to these four markers. In COX2,the sole mitochondrial gene analyzed, the distinct characteristic ofthe underlying evolutionary process was that the equilibrium fre-quencies of T and A were much higher than those of C and G. Forexample, in the best-fit TVM model for COX2, πT = 0.34, πC =0.16, πA = 0.32, and πG = 0.18 (Data Sheet 5). The JC, K2P, andTrNef models, which constrain base frequencies to equality, failedto account for the much higher T and A frequencies in COX2.Meanwhile, the K3P, TIMef, TVM, and SYM models, which alsoassume equal base frequencies, fit the COX2 marker increasinglybetter by allowing a progressively higher value of rTA to compen-sate these models’ inability to account for the higher frequenciesof T and A. In contrast, because the JC, K2P, and TrNef modelsare further restricted to assume equal transversion rates, the threemodels could not compensate for their poor fit to the high T andA frequencies by raising the transversion rate between T and A.

DISCUSSIONWe have extended the Townsend et al. (2012) phylogenetic sig-nal and noise analysis by incorporating all time-reversible Markovmodels of nucleotide substitution into the prediction of thepower of a data set for resolving a quartet-taxon phylogeny. Byimplementing diverse models of molecular evolution, we haveenhanced the precision of signal and noise analysis. If an opti-mal model for a given data set can be established with confidence,the probability distribution of signal, noise, and polytomy fora node of phylogenetic interest that can be achieved from the

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FIGURE 3 | The consensus tree topology obtained from the Bayesian analysis of the combined five-gene data set. Numbers at nodes represent Bayesianposterior probabilities. The internode marked with a bold, italic “X” corresponds to the poorly resolved backbone node of the haploid Candida clade (BPP = 0.72).

data set can now be calculated using the most realistic esti-mates of the optimal model parameters for the data set. It hasbeen shown by an empirical study of a previous method ofphylogenetic informativeness based on a Poisson model that, for

real and simulated data, predictions from the method correlatewith tree distances between true and estimated trees and withbootstrap probabilities (López-Giráldez et al., 2013). Given thisstudy’s finding that better-fit (typically more complex) models

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Su et al. Models, phylogenetic signal, and noise

FIGURE 4 | Stacked probabilities of correct resolution (bottom, light

green), polytomy (middle, blue), and incorrect resolution (top, dark

red) of the backbone node of the haploid Candida clade based on the

GTR model, for five markers.

FIGURE 5 | Probabilities of correct resolution for the backbone node of

the haploid Candida clade for RPB2 (diamond), ACT1 (square), COX2

(triangle), RPB1 (cross), and D1/D2 LSU (star) calculated for each of

the the 14 alternative models.

yield more accurate predictions of which loci are most informa-tive, the generalization to more complex models developed hereinis thus expected to further enhance performance. Additionally,signal and noise analysis based on even the most complex time-reversible model of nucleotide substitution, the GTR model, canbe implemented almost as easily and rapidly as analysis based onthe simplest models. The generalized signal and noise analysisthus yields a highly convenient tool of model-based phylogeneticinformativeness analysis.

Our results suggest that there are not major discrepanciesbetween predictions by signal and noise analysis based on dif-ferent models. In the example study of representative Candidaspecies and allied members using COX2, RPB1, RPB2, ACT1,and D1/D2 LSU, we found that the prediction of the five mark-ers’ probabilities of correct resolution for 20 assayed nodes wasfairly robust to 14 alternative models commonly used in model-based phylogenetic studies. However, the results did suggest thatdeviations in relative predicted probabilities of correct nodal

resolution could result from assuming the simplest models thatpoorly capture the underlying evolutionary process—in particu-lar, transition/transversion bias and base frequency composition.If there are particularly high- or low-magnitude variables in theunderlying evolutionary process, the lowest-parameterized mod-els, namely the JC, F81, K2P, and TrNef models, tend to fit thedata set poorly because these models’ restricted settings are oftenunable to account for heterogeneity of rates in the Q matrixor for heterogeneity of base composition. Consequently, signaland noise analysis applied with these precariously simple mod-els tends to predict a ranking that can deviate moderately fromthe correct ranking of the utility of the assayed markers. Overall,if there are grounds for supporting one model over another fora gene, due to the result of a model selection test on preliminarydata or on a sister clade, it is advisable to eliminate overly sim-ple models that are unlikely to reflect the evolutionary processfor the data to be gathered. Nevertheless, our results suggestedthat predictions of signal and noise analysis are highly robustacross models that incorporate transition/transversion bias andheterogeneity of base composition. Robustness to most modelsexhibited by the generalized signal and noise analysis is reassuringin that it would be challenging to confidently assert a model fordata that has yet to be gathered, although preliminary data, datafrom the closest genome-sequenced species, or data from sisterclades can help to do so.

Considering the importance of model selection to phyloge-netic inference (e.g., Sullivan and Swofford, 1997; Kelsey et al.,1999; Ripplinger and Sullivan, 2010), it is instructive to speculateregarding this perhaps surprisingly modest impact of evolution-ary models on predictions of signal and noise analysis. In phy-logenetic inference problems, it is frequently crucial to select theoptimal—frequently the most complex—molecular evolutionarymodels, because the state identities at ancestral nodes are highlyconstrained by the state identities at the tips of divergent lineages;thus inference of topology at ancestral nodes is sensitive to whichevolutionary model is assumed (Sullivan and Joyce, 2005). In con-trast, phylogenetic informativeness analysis provides predictionsof the utility of data that has not yet been gathered, and operateswithout constraint by the unknown state identities at the tips ofdivergent lineages. Phylogenetic informativeness depends insteadon how likely synapomorphic and homoplasious site patternsmay arise and maintain along the tree branches.

Thus, the substitution rate of a character is the most importantfactor predicting the likelihood of synapomorphy and homoplasy(e.g., Swofford et al., 1996; Goldman, 1998; Yang, 1998; Shpakand Churchill, 2000; Bininda-Emonds et al., 2001; Sanderson andShaffer, 2002; Lin and Danforth, 2004; Danforth et al., 2005;Mueller, 2006; Townsend, 2007; Jian et al., 2008; Regier et al.,2008). The model only has a significant impact on informative-ness in cases where extreme actual values of model parameterslead to inaccurate estimation of average rate when analyzed undera simpler model. The substitution rate of a character given byEquation 2 is a weighted average of the frequency and rateparameters in a molecular evolutionary model. In other words,among the more complex models, differences in signal and noiseestimation are unlikely to arise because the effect of some param-

eters assuming higher values is very likely countered by the effect

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of other parameters assuming lower values. Therefore, althoughspecifying different evolutionary models will lead to differences inthe individual parameter values, the average rate of substitutionis not expected to drastically differ for most character sets withrealistic parameter values. Consequently, predictions based onphylogenetic signal and noise analysis, which are chiefly depen-dent on the rate of evolution, do not tend to experience significantchanges when selecting among the diversity of more complexevolutionary models.

We may gain a further intuitive understanding of the impactof model selection on phylogenetic informativeness analysis byexamining how more complex evolutionary models modify thespace of character states. In Townsend et al. (2012), the proba-bility functions of support for the correct subtree and incorrectsubtrees per site were based on character state spaces consistingof exactly s character states under the Poisson model of molecu-lar evolution, where s ≥ 2 (c.f. Equations 4–7 in Townsend et al.,2012). Signal and noise probability modeling based on differentnumbers of character states leads to different probability func-tions (c.f. Figure 2 in Townsend et al., 2012). Plotting the ratioof the probability of support for the correct subtree to supportfor an incorrect subtree demonstrates that larger character statespaces have a higher ratio of support for the correct subtree toan incorrect subtree (Figure 6). This analysis demonstrates that anontrivial increase of the character state space leads to a higherprediction of the power of a given data set for resolving the nodeof interest.

How, then, does specifying more complex molecular evolu-tion models relate to the character state space? Consider howthe “effective” character state space changes when the K2P modelreplaces the JC model to describe the same data. In the JC model,the four nucleotide states occur at equal frequencies and convertto each other at identical rates. The character state space thusconsists of exactly four states that have equal scope to impactto the evolutionary process. Now, for the K2P model, consideran extreme case where transitions occur at a fast or ordinaryrate, while transversions occur at a negligible rate. With negligibletransversions, a purine cannot convert to a pyrimidine and vice

FIGURE 6 | Ratio of the probability of support for the correct topology

to the probability of support for either incorrect topology over time T,

for a site with a substitution rate λ = 1 and an internode length

t0 = 0.1, from Equations 4 and 5 in Townsend et al. (2012). Theprobability ratio curves are depicted for state spaces of 2, 3, 4, 5, and 20states, respectively, from bottom to top.

versa. Only transitions between purines or between pyrimidinescan occur. Effectively, this character state space under the extremeK2P model consists of two dominating separate subspaces of Aand G and of C and T. Because no intrinsic factors distinguishthese two subspaces, the character state space for the K2P modelis effectively reduced to two by this extreme parameterization.Generally, for more realistic transversion rates, the effect of intro-ducing the K2P model will not be as severe as in the extreme caseconsidered. But in all nondegenerate parameterizations, the effectof using the K2P model instead of the JC model for the same dataset remains a slight decrease of the effective character state space.

This extreme example illustrates the general effect of speci-fying more complex Markov models on the effective characterstate space. Given any fixed data set, extreme parameter values aremore possible in models with more parameters. As a result, thosecharacter states that are involved in substitutions occurring as aconsequence of particularly high model parameters will begin toform subspaces that dominate the character state space for eachcharacter. Other character states undergoing substitutions withlow model parameters, representing shifts away from one dom-inant, high-rate-change character state subspace to another, willstill occur, but their overall frequencies will be lower. As a result,when the molecular evolutionary model grows in complexity, cer-tain character state subspaces, by virtue of their association withhigher model parameter values, will dominate the evolution of thecharacter and thus effectively reduce the character state space inthe short term. In most realistic data sets, there is always a degreeof heterogeneity in model parameter values when described by anoptimal model. Therefore, specifying the simplest models, suchas the JC, K2P, and F81 models, can fail to sufficiently accountfor the heterogeneity in the evolutionary process and thus leadto a nontrivial increase of the effective character state space. Inthese cases, it is necessary to assume a model of sufficient com-plexity to account for the heterogeneity in model parameters, todecrease the effective character state space to its actual size, and tothereby accurately predict the utility of the site for phylogeneticinference. Thus, specifying increasingly more complex evolution-ary models is predicted to lead to decreasingly small impacts onthe predictions of signal and noise analysis.

This prediction was verified by empirical observations fromthe Candida species study. In RPB1, RPB2, ACT1, and D1/D2LSU, both the JC and F81 models were inadequate to accountfor the high transition/transversion bias in the underlying evo-lutionary processes and thus caused a nontrivial and inaccurateincrease of the “effective” character state space. As a result, prob-abilities of correct nodal resolution predicted by the two modelswere consistently higher than those by the other 12 more complexmodels that could account for the transition/transversion bias. Inthe case of COX2, significant heterogeneity in base equilibriumfrequencies meant that specifying the JC, K2P, and TrNef models,which all assume equal frequencies, also led to a nontrivial andinaccurate increase of the “effective” character state space, thusconsistently leading to higher predictions of probabilities of cor-rect nodal resolution. Although the K3P, TIM, TVM, and SYMmodels also assume equal base frequencies, these more param-eterized models were able to progressively compensate for thisinaccuracy in terms of the effective character state space, giving

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Su et al. Models, phylogenetic signal, and noise

way to estimates of an higher transversion rate between T andA, thereby achieving an effective state space closer to the actualsize. Correspondingly, these four models predicted probabilitiesof correct nodal resolution that were relatively more consistentwith probabilities predicted by models of unequal base frequen-cies. In all five markers, probabilities of correct nodal resolutionpredicted by the ten models more complex than the JC, F81, K2P,and TrNef models were highly consistent.

Our results suggest that sequence evolving in accordance withsimpler models at informative rates will tend to exhibit a slightlyhigher probability of correct resolution than sequence evolvingin accordance with more complex models. However, it does notmean that simply applying simpler models to inference analysiswill yield greater actual phylogenetic utility. Simpler models canoften predict a higher probability of correct resolution becausesimpler models tend to average out heterogeneity of rate (bymutation class, or by site). Therefore, if the loci are evolving atrates that are at least moderately appropriate for the problem athand, using a simple model tends to push the inferred rates closerto the average, (coincidentally) ideal rate, and increase predictedinformativeness. However, when a more complex model wouldbe a better fit, this increase is, of course, unrealizable in practice;or to the extent that it is realizable, it will be inaccurate or conveyinappropriately high support.

Lastly, the signal and noise analysis of representative Candidaspecies and allied members offers several insights for resolving thehaploid Candida clade. Due to extremely reduced single-cellularmorphology and highly divergent and adaptive ecology, yeastshave been a problematic group that relies on molecular data forunderstanding their evolutionary histories, which are critical incontending with many yeast pathogens. Because of their medi-cal and industrial importance and generally small genome size,many Candida species have been sequenced for genome data (e.g.,Butler et al., 2009). However, more data would not necessarilyguarantee a fully resolved phylogeny for these yeast species, asdemonstrated in Butler et al. (2009). Furthermore, there are morethan 350 Candida species in the Saccharomycetales (Kirk et al.,2008), and it will remain some time before genome data can begathered for most of them. It should be noted that as regards toour chronogram, lacking a fossil record, the estimation of abso-lute divergence times for yeast species is difficult (Taylor andBerbee, 2006). Most diversification events sampled in this studyare likely earlier than the splits among Saccharomyes species thatare thought to have evolved about 20 million years ago with alevel of nucleotide divergence similar to that found between birdsand human (Dujon, 2006). However, an accurate time scale isirrelevant to the conclusions obtained here, all of which can beexactly replicated on a purely relative time scale (e.g., Moeller andTownsend, 2011). Other sources of heterogeneity include the longevolutionary history potentially accompanied with inconsistentsubstitution rates of different gene sets that could be associ-ated with unique ecological shifts such as association with hosts(Kurtzman and Robnett, 2003; Scannell et al., 2011). The emer-gence and maintenance of haploid lineages via what appears tobe long-term asexual reproduction in some Candida species isassociated with shifting rates of evolution (Ben-Ari et al., 2005;Webster and Hurst, 2012), contributing additional complexity to

the solution of the yeast phylogeny. Results of signal and noiseanalysis on the five assayed markers under alternative modelsdisclosed some issues behind the unresolved phylogeny of thehaploid Candida clade and pointed out an efficient approachtoward future phylogenetic experimental design on this groupof fungi. We demonstrated that signal and noise analysis rankedthe five assayed markers in the order of RPB2 (most informa-tive), ACT1, COX2, RPB1, and D1/D2 LSU (least informative)for resolving the evolution events at the time when the haploidclade split from the diploid pathogens. The current five-gene datadid not provide enough information to resolve the relationshipbetween the haploid clade and its sibling pathogens. The hap-loid Candida clade might thus be best resolved by focusing ondata collection for the more informative markers such as RPB2.In the future, it could be especially worthwhile to apply signaland noise analysis across numerous loci obtained from diversegenome sequences to identify markers even more informativethan RPB2 for the historic epoch in question, and to resolve thehaploid Candida clade.

ACKNOWLEDGMENTSWe sincerely thank Dylan Greaves for helpful discussion of thetopic.

SUPPLEMENTARY MATERIALThe Supplementary Material for this article can be found onlineat: http://www.frontiersin.org/journal/10.3389/fevo.2014.00011/abstract

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Conflict of Interest Statement: The authors declare that the research was con-ducted in the absence of any commercial or financial relationships that could beconstrued as a potential conflict of interest.

Received: 31 January 2014; paper pending published: 01 March 2014; accepted: 27March 2014; published online: 16 April 2014.Citation: Su Z, Wang Z, López-Giráldez F and Townsend JP (2014) The impact ofincorporating molecular evolutionary model into predictions of phylogenetic signal andnoise. Front. Ecol. Evol. 2:11. doi: 10.3389/fevo.2014.00011This article was submitted to Phylogenetics, Phylogenomics, and Systematics, a sectionof the journal Frontiers in Ecology and Evolution.Copyright © 2014 Su, Wang, López-Giráldez and Townsend. This is an open-accessarticle distributed under the terms of the Creative Commons Attribution License(CC BY). The use, distribution or reproduction in other forums is permitted, providedthe original author(s) or licensor are credited and that the original publication in thisjournal is cited, in accordance with accepted academic practice. No use, distribution orreproduction is permitted which does not comply with these terms.

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