| INVESTIGATION

Predicting Amino Acid Substitution Probabilities UsingSingle Nucleotide Polymorphisms

Francesca Rizzato,* Alex Rodriguez,* Xevi Biarnés,† and Alessandro Laio*,‡,1

*Scuola Internazionale Superiore di Studi Avanzati (SISSA), 34136 Trieste, Italy, †Laboratory of Biochemistry, Institut Químic deSarrià (IQS), Universitat Ramon Llull (URL), 08017 Barcelona, Spain, and ‡The Abdus Salam International Centre for Theoretical

Physics (ICTP), 34151 Trieste, Italy

ORCID ID: 0000-0002-6028-9980 (F.R.)

ABSTRACT Fast genome sequencing offers invaluable opportunities for building updated and improved models of protein sequenceevolution. We here show that Single Nucleotide Polymorphisms (SNPs) can be used to build a model capable of predicting theprobability of substitution between amino acids in variants of the same protein in different species. The model is based on asubstitution matrix inferred from the frequency of codon interchanges observed in a suitably selected subset of human SNPs, andpredicts the substitution probabilities observed in alignments between Homo sapiens and related species at 85–100% of sequenceidentity better than any other approach we are aware of. The model gradually loses its predictive power at lower sequence identity.Our results suggest that SNPs can be employed, together with multiple sequence alignment data, to model protein sequence evolution.The SNP-based substitution matrix developed in this work can be exploited to better align protein sequences of related organisms, torefine the estimate of the evolutionary distance between protein variants from related species in phylogenetic trees and, in perspective,might become a useful tool for population analysis.

KEYWORDS protein sequence evolution; SNP; substitution matrices; protein sequence alignment; substitution rate variability

PREDICTING substitution probabilities between aminoacids is of enormous practical and conceptual importance.

For instance, phylogenetic analyses are rooted on the capa-bility of accuratelypredicting substitutionprobabilities,whichare used to determine evolutionary relationships betweenspecies, and to infer the time scales of speciation events.Substitution probabilities are also used to spot, among a setof sequences, those who have more likely evolved from acommon ancestor (identification of homologs). Moreover,substitution probabilities can be used to classify of the wholeproteome space in protein families of common function (Finnet al. 2015). Substitution probabilities are traditionallylearned from alignments of proteins in different species. Theyare then used to build substitution matrices, which measurethe relative probability that two amino acids are aligned be-cause they descend from a common ancestor rather than by

chance. The simplest procedure to infer such amatrix is basedon counting amino acid substitutions in given ranges of se-quence identity (Henikoff and Henikoff 1992), while othermodels are rooted on a Markov model of evolution (Dayhoffet al. 1978; Gonnet et al. 1992; Jones et al. 1992;Whelan andGoldman 2001; Schneider et al. 2005; Kosiol et al. 2007; Leand Gascuel 2008).

A source of information for inferring substitution proba-bilitiesmoreandmoreaccurately is offeredby the recentboostin DNA sequencing (Ronaghi et al. 1996; Bentley et al. 2008;Wheeler et al. 2008), which provides an increasing amount ofsequenced genomes (1000 Genomes Project Consortiumet al. 2010, 2015). The analysis of genomes has allowednot only the enormous enlargement of the database of pro-tein families, but also the identification of a large numberof Single Nucleotide Polymorphisms (SNPs) (Sherry et al.2001)—isolated nucleotide variations in the DNA se-quence that are commonly present among the individualsof a species. SNPs are nowadays at the center of clinicalresearch (Giacomini et al. 2007), and are a key tool tounderstand population dynamics, such as migration andselection (Kingman 1982; Rosenberg and Nordborg 2002).

Copyright © 2017 by the Genetics Society of Americadoi: https://doi.org/10.1534/genetics.117.300078Manuscript received November 23, 2016; accepted for publication July 18, 2017;published Early Online July 26, 2017.Supplemental material is available online at www.genetics.org/lookup/suppl/doi:10.1534/genetics.117.300078/-/DC1.1Corresponding author: Scuola Internazionale Superiore di Studi Avanzati (SISSA),via Bonomea 265, 34136 Trieste, Italy. E-mail: laio@sissa.it

Genetics, Vol. 207, 643–652 October 2017 643

A question that naturally arises is what is the relationshipbetween the polymorphisms observed within a species andthe substitutions observed between variants of the sameproteins in different species. The interplay between thesephenomena has drawn the attention of several recent stud-ies (Wilson et al. 2011; De Maio et al. 2015). The picturethat emerges from these works, and more in general fromthe literature on polymorphisms, is that SNPs should not beconfused with amino acid substitutions (Sawyer and Hartl1992; Fay et al. 2001; Tamuri et al. 2012). The probability offixation of a polymorphism depends on its positive or nega-tive impact on the fitness of its carrier. However, its destiny,especially for nondetrimental polymorphisms, is stronglyconnected to that of the population hosting it. In otherwords, polymorphisms lie in the “no man’s land” after theoccurrence of a random mutation and before its definitivefixation or rejection. If their nature and their statistics aresimilar to those of random mutations, to those of fixed sub-stitutions observed in proteins of different species, or toneither, is still the object of investigation.

Here, we shed some light on this issue: we show that anappropriately selected subset of SNPs encodes an informationcompatible with the substitutions observed in alignments andcan be used to predict substitution probabilities in pairwisealignments at high sequence identity. We select from thedbSNP database (Sherry et al. 2001) only the human poly-morphisms characterized by no known clinical significance,andwhich are present in at least 1% of the population, havingtherefore a high probability of being nearly neutral. Wedemonstrate that a protein evolution model based on a sub-stitution matrix inferred from this subset of SNPs, and tak-ing into account substitution rate variability (Miyazawa2011a; Rizzato et al. 2016), can be used to predict the tran-sition probabilities between amino acids observed in align-ments. The accuracy of this model in the range of 85–100%of sequence identity is better than that achieved using anyother substitution matrix we are aware of, including LG (Leand Gascuel 2008), WAG (Whelan and Goldman 2001), JTT(Jones et al. 1992) and BLOSUM90 (Henikoff and Henikoff1992) for amino acid models or ECM (Kosiol et al. 2007) forcodon models. This result is particularly significant becausethe SNP-based model presented here, in contrast to all theother models against which it is benchmarked, is not learnedfrom alignments. Therefore, one would expect it to be dis-advantaged with respect to the others in the prediction ofsubstitution frequencies from alignments.

We also demonstrate that the SNP-based substitution ma-trix can be exploited to build phylogenetic trees, especially forHomo sapiens and related species. We therefore foresee it willbe useful in fine-grain resolution of phylogenies in evolution-ary genetics and population analysis.

We will also show that the accuracy of the SNP-basedmodel slowly degrades at medium and low sequence identi-ties, possibly due to the relatively poor statistics of the SNPdatabase, or to a missing ingredient in the model of proteinsequence evolution used in thiswork. This indicates thatmore

research is needed to develop a model capable of describingprotein sequence evolution at an arbitrary evolutionary dis-tance. However, the results presented here provide evidencethat two traditionally separated data domains, SNPs, andsequence alignments, can be employed together to build sucha model.

Materials and Methods

Download and selection of SNP data

From the dbSNP database (Sherry et al. 2001), we selectedthe polymorphisms in the coding part of the human genome.We did not consider those SNPs that are likely to entail apositive or negative structural modification, having a knownclinical significance (benign, likely benign, pathogenic, andlikely pathogenic).

The SNPs were downloaded on the January 14, 2017at www.ncbi.nlm.nih.gov/snp/advanced. Query: (“homosapiens”[Organism]) AND ((“snp”[SNP Class]) OR “multinucleotidepolymorphism”[SNP Class]) AND “missense”[FunctionClass] NOT(((((“benign”[Clinical Significance]) OR “likelybenign”[Clinical Significance]) OR “likely pathogenic”[ClinicalSignificance]) OR “pathogenic”[Clinical Significance]) OR “noinfo”[Validation Status]) for the nonsynonymous polymor-phisms, and the same querywith “synonymous codon” in placeof “missense” for the synonymous polymorphisms. In thismanner we gathered 3,228,872 polymorphisms, 1,144,770 ofwhich preserving the original amino acid (synonymous) and2,084,102 changing it (nonsynonymous, or missense).

From the SNP database to codon substitutions

The identity of the codons involved in a SNPwas derivedwiththe following procedure. We downloaded the SNPs both inflatfile format and in brief format.From the flatfile format we extracted the following data:

The label of that SNP.

Its Global Minor Allele Frequency (GMAF), which refersto the frequency of the second most common allele of theconsidered polymorphism.

The two (or more) alleles, the corresponding aminoacids, and the indication on the frame of the codon inwhich the different nucleotides were found. For example,a frame equal to one means that the polymorphism islocated on the first letter of a codon.

For each of the entries,we selected fromthe brief format theentry with the correct label, and read the nucleotide sequencetwo nucleotides before and after the polymorphism.We recon-structed the codons involved in the polymorphisms from aknowledge of the allele, the frame, and the amino acid. Sincethe sample may be taken in two possible orientations, weselected the orientation consistent with the genetic code. Wedivide the entries into three subsets according to their GMAF: aset of common polymorphisms, characterized by GMAF. 0:2;

644 F. Rizzato et al.

a set of medium-rare polymorphisms, characterized by0:01,GMAF# 0:02; and a set of rare polymorphisms, char-acterized by GMAF# 0:01: By this procedure, we obtained10,585 synonymous and 9772missense SNPswith GMAF. 0:2;30,319synonymousand32,601missense for,0:01,GMAF# 0:02and 1,064,390 synonymous and 1,989,525 missense forGMAF# 0:01:

As usually done for alignments, the substitution processwas treatedas stationary, and the substitutions fromcodon c tocodon c9 were assumed to be as frequent as those from c9 to c.So, the 64 3 64 matrix of occurrences nSNP2c was built bysetting both nSNPc;c9 and nSNPc9;c to one-half of the observed inter-changes between the pair of codons fc; c9g From the matrix ofoccurrences,we computed the frequency of codon interchanges:

fSNP2cc;c9 ¼

nSNP2cc;c9P

dP

e6¼dnSNP2cd;e

(1)

This codon matrix can be summed to a 20 3 20 matrixof interchange frequencies between amino acids by:f SNPAB ¼ P

fc2AgP

fc92BgfSNP2cc;c9 ; where fc 2 Ag is the set of co-

dons coding for amino acid A.

Substitutions involving more than one nucleotide

Nucleotide substitutions are not necessarily isolated, but canhappen simultaneously on neighboring nucleotides. Esti-mates of the fraction of multiple simultaneous substitutionsrange between the 0.3% (Smith et al. 2003) and 3% (Schrideret al. 2011). The number of multiple simultaneous substitu-tions satisfying the query mentioned above is 387. All theseentries have a GMAF ,0.01; therefore, for this subset ofpolymorphisms, we will not quantify the effect of the allelefrequency as will do for SNPs in the Results section.

Computing the transition probabilities for codons andamino acids

From the frequencies of interchanges f SNP2c; we define thenondiagonal entries of the substitution rate matrix QSNP by:

QSNPc;c9 ¼

fSNP2cc;c9

fc(2)

where fc is the frequency of codon c, which we assume to beequal to those tabulated for the human codon usage bias(Nakamura et al. 2000). The diagonal entries of QSNP are,instead, defined as QSNP

c;c ¼ 2P

c9 6¼cQSNPc;c9 : By construction,P

c;c9ðfcQSNPc;c9 Þ ¼ 1: Therefore, the evolutionary time is mea-

sured in units of expected substitutions per site.FromQSNP; or any other genericQmatrix on codons or aminoacids, we compute the transition probability from c to c9 aftera time t by following Rizzato et al. (2016):

PGc;c9ðtÞ ¼� Z N

0er�t�QGaðrÞdr

�c;c9

(3)

where r is the substitution rate, and GaðrÞ is its probabilitydistribution, assumed to be G-shaped (Yang 1993, 1994). We

remark that this procedure is not the standard G-correctionimplemented in the standard algorithms for phylogenetictree reconstruction (Stamatakis 2006; Yang 2007; Priceet al. 2010): here, no information is available on the rate atdifferent sites, so rate variability is treated by averaging overits among-site probability distribution. The G distribution de-pends on the shape parameter, a, which is known to varyfrom protein family to protein family, and, inside the sameprotein family, also in time (Gaucher et al. 2001, 2002; Lopezet al. 2002). In order to take into account the variation of awith evolutionary time, we apply the procedure in Miyazawa(2011a, p. 5). In this procedure, the parameter a is assumedto vary linearlywith time. Therefore, at time t,aðtÞ ¼ t � a0=t0;where t0 is a reference evolutionary time and a0 is the value ofa at that time. The ratioa0=t0 is the only free parameter of thisapproach, and here was set optimizing the value of a at 92.5%of sequence identity, thus obtaining a0=t0 ¼ 1:225:

We also computed the substitution probabilities in themore traditional framework of identical rates on all sites(SNP-no-G model). In this case, the transition probabilityfrom c to c9 at time t is given by:

Pno2Gc;c9 ðtÞ ¼ �

et�Q�c;c9 (4)

From transition probabilities between codons, one estimatesthe transition probabilities between amino acids as follows:

PA;BðtÞ ¼P

fc2AgP

fc92Bg fcPc;c9ðtÞfA

(5)

where fc 2 Ag stands for the set of codons coding for aminoacid A.

The expected sequence identity between an initial se-quence and its evolution after a time t can be deduced fromEquation 3 or Equation 4 by:

seqIDðtÞ ¼XA

fAPA;AðtÞ (6)

This equation implicitly specifies the time at which an evolu-tionarymodel should be comparedwith data characterized bya given sequence identity.

Selection of alignments and computation ofexperimental substitution frequencies

To test how well the evolutionary models described abovepredict the transition probabilities in the alignments, wecomputed the frequencies of amino acid interchanges,f alignAB ðseqIDÞ; from alignments at various sequence identi-ties, grouping them in windows of 3% of sequence iden-tity from 50 to 99%. These frequencies were learned fromUniRef (Suzek et al. 2007), an arrangement of the UniProtdatabase (The UniProt Consortium 2015) that clus-ters sequences above a sequence identity threshold. Wedownloaded from UniRef the clusters at 50% of sequenceidentity with at least one human sequence. The align-ments were downloaded on July 23, 2015 from UniRef

Predicting Amino Acid Substitution Probabilities Using SNPs 645

at www.uniprot.org/help/uniref with query: [query:count:[2TO *] length:[50 TO *] taxonomy:Homo sapiens (Human)[9606] AND identity:0.5 ].

From each cluster, we detected the human sequence andaligned it with the other sequences. Sequences were alignedlocally by the algorithmwater (Smith andWaterman 1981) inthe emboss software package (Rice et al. 2000). Only ungap-ped fractions of the pairwise alignments with # 80 residueswere considered. In order to avoid weighting bigger clustersmore, from each cluster we collected at most one alignmentper window of sequence identity. For each sequence identitywindow the average sequence identity (seqID) was com-puted, and the mismatches in the alignments for every aminoacid pair ðA;BÞ were counted. The substitution process wastreated as an equilibrium one: nalignAB ðseqIDÞ and nalignBA ðseqIDÞwere set equal to one-half of the number of mismatches be-tween the unordered pair of amino acids A and B. The entrieswhere nalignAB ðseqIDÞ, 5 were neglected, being affected by toolarge statistical errors.

From the frequency of interchanges in alignments atseqID$ 98%; we also computed a Q matrix with an analo-gous procedure to that used to compute QSNP from f SNP

(Equation 2).

Download and implementation of benchmark models

We compared our evolutionary model with JTT (Jones et al.1992), LG (Le and Gascuel 2008), ECM unrestricted (Kosiolet al.2007),WAG(Whelan andGoldman2001), andBLOSUM90(Henikoff and Henikoff 1992). Here, we describe our implemen-tation of these models.

The JTT matrix was deduced from the counts (Table 1) inthe original paper (Jones et al. 1992) by the same proceduredescribed there. The correspondingQmatrix was obtained byinverting Equation 4.

The LGQmatrix was reconstructed from the website of oneof its authors (www.atgc-montpellier.fr/download/datasets/models/lg_LG.PAML.txt).

The ECM-unrestricted Q matrix was reconstructed fromthe Supplementary material of the original paper (Kosiolet al. 2007), and was chosen instead of its restricted versionbecause it was proved by the authors to give better results.

The WAG Qmatrix was reconstructed from the website ofone of its authors (www.ebi.ac.uk/goldman/WAG/wag.dat).

The BLOSUM90 matrix was downloaded from the siteof NCBI (www.ncbi.nlm.nih.gov/IEB/ToolBox/C_DOC/lxr/source/data/BLOSUM90). To adapt BLOSUM scores to ourdefinition (see Equation 9), which is deprived of any prefac-tor and computed by a natural logarithm, wemultiplied themby lnð2Þ=2:

In all cases, except BLOSUM, we derived the substitutionprobabilities at the desired times from Equation 4 for theno-G version, or by Equation 3 in the þG version.

Maximum likelihood tests

We also benchmarked the quality of our model by computingthe likelihood of phylogenetic trees with respect to other

substitution models. Maximum likelihood tests were per-formed using the PAML software package (Yang 2007) onreference alignments and phylogenetic trees downloadedfrom the Phylome Database (PhylomeDB v4, data accessedon February 1, 2016) (Huerta-Cepas et al. 2014). PhylomeDBprovides a collection of highly accurate gene phylogenies fora wide variety of gene families and species. PhylomeDB phy-logenies are derived from maximum likelihood tree infer-ence, alignment trimming, and evolutionary model testing.

The reference Primates andModel SpeciesMetaphylomes,seeded on H. sapiens, were downloaded from this database(Phylome IDs: 098 and 0500, respectively). We only ana-lyzedmultiple sequence alignments (and their correspondingphylogenetic trees) characterized by 500–900 pairs of pro-tein sequences, with average sequence identity in the range45–99%. This filter produces two reduced sets of 111 and230 alignments, respectively, for the two phylomes.

The PAML software package was used to compute max-imum likelihoods, keeping tree topologies intact and opti-mizing branch lengths. The standard G-correction, asimplemented in the PAML software, was included, andthe free parameter a was optimized during the maximumlikelihood estimations for all the tested models. We com-puted maximum likelihood values for each of these refer-ence phylogenies using the SNP; the JTT (Jones et al.1992), the WAG (Whelan and Goldman 2001), and theLG (Le and Gascuel 2008) evolutionary models.

The QSNP matrix used here is a reduction on amino acids ifcompared to that in Equation 2:

QA;B ¼P

fc12AgP

fc22Bg fc1Qc1;c2

fA(7)

Data availability

The QSNP matrix obtained with our analysis is available asSupplemental Material, File S2, along with the obtained like-lihoods for the two analyzed phylomes (File S3 and File S4).Furthermore, File S1 contains all the remaining supplemen-tarymaterials. All the data analyzed in this paper were down-loaded directly from public databases (dbSNP, UniRef, andPhylomeDB).

Results

The effect of global minor allele frequency

In this work, we propose to use a selection of SNPs to build anevolutionary model of protein sequences. Before estimating asubstitution ratematrix from theSNPs,weattempt to quantifythe effect of the fitness-related natural selection on the fre-quency of polymorphisms observed in the database. With thisin mind, we divide our collection of SNPs into three subsetscorresponding to different GMAF (see Materials and Meth-ods). SNPs with a large GMAF are common, and are thereforemore likely to get fixed by natural selection. Rare polymor-phisms, characterized by a low GMAF, more likely resemble

646 F. Rizzato et al.

the properties of a randommutation on which fitness-relatednatural selection has not yet completed its action. We com-pare the properties of the subset at medium GMAF (label M,0:01,GMAF# 0:2), and the subset at high GMAF (label H,GMAF. 0:2). We compute the frequency of substitutions fc;c9between codons c and c9 in both datasets, and the corre-sponding error according to the Poisson statistics. For eachpair of different codons, we compute:

eM;Hc;c9 ¼ fMc;c9 2 fHc;c9ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

s2�fMc;c9

�2 þ s2�fHc;c9

�r (8)

where thedenominator is anestimateof the statistical erroronthe difference at the numerator. Figure 1 shows the cumula-tive distribution function (cdf) of ec;c9: In the same figure, weplot the cdf of a Gaussian distribution with average 0 and SD1. The similarity between the two curves shows that the dif-ference (eM;H) between the H andM datasets can be ascribedto statistical errors. We can then conclude that, for the subsetof the SNPs chosen by our procedure, there is no evident biasin the frequency of observing a polymorphism at to mediumand high range of GMAF. We then repeated the same test,comparing the set of SNPs at medium+ high GMAF and a setwith GMAF# 0:01 (label low, L). The cdf of eL;MþH

c;c9 estimatedfor the sets L and M+H is also shown in the figure, and issignificantly different from the cdf of the considered Gauss-ian. This implies that the differences in the frequency of sub-stitutions between the L and the M + H sets cannot be fullyascribed to statistical errors. This, according to our interpre-tation, means that the fitness-related natural selection hasroughly completed its work for GMAF. 0:01; while thishas not yet happened for rarer polymorphisms. Another pos-sible interpretation is that selection has not completed itsaction even for polymorphisms at large GMAF, and that thesetsM andH are indistinguishable by chance. However, in thelight of the results that we are going to present, this last in-terpretation does not seem likely.

The frequency of substitutions from SNPs andfrom alignments

We then compared f SNPAB ; the frequency of substitutions betweenamino acids A and B in our SNP dataset, and, f alignAB ðseqIDÞthe frequency from pairwise alignments at sequence identityseqID$ 98%; namely at very high sequence identity (seeMaterials and Methods). In Figure 2, we plot the entry-by-entry comparison of these frequencies, both with error barsestimated according to Poisson statistics. The red pointscorrespond to the amino acid pairs whose codons differ byonly one nucleotide. For these entries, fSNPAB is estimatedconsidering only substitutions with GMAF. 0:01: For theseentries, the correlation is good, even if the statistical error isnot large enough to explain entirely the deviations from thediagonal: statistical errors cover �30% of the difference.

Thebluepoints correspond toaminoacidpairswhose codonsdifferby twoor threenucleotides.Forexample,histidine iscoded

byCATandCAC, andphenylalaninebyTTTandTTC.Therefore,at least two substitutions are necessary to transform histidine tophenylalanine. Multiple simultaneous substitutions are pre-sent in dbSNP, even if they are almost invariably associatedwith a lowGMAF, and are then excludedwhen selecting onlyentries with GMAF. 0:01: We therefore estimated the fre-quency of polymorphisms associated with two nucleotidesubstitutions in the same codon without applying the filteron GMAF. The fraction of multiple nucleotide polymor-phisms obtained in this manner is 0.46% of the total, anumber of the same order of magnitude of the estimate inSmith et al. (2003), but low with respect to other estimates(Averof et al. 2000; Schrider et al. 2011). These entries areaffected by large statistical errors, and possibly even by sys-tematic errors, since they include entries with a very lowGMAF. However, a mild correlation seems to show up.

This analysis indicates that the frequencies of amino acidsubstitutions estimated from the dbSNP database and fromthe alignments are correlated, butwith deviations that cannotbe fully ascribed to statistical errors. The deviations areparticularly severe for entries associated to multiple nucleo-tide substitutions. In the next section we will show that theinconsistencies observed in Figure 2 can be largely accountedfor by taking mutation rate variability into account.

Prediction of substitution probabilities

We now show that, by taking mutation rate variability intoaccount, the frequencies of interchanges between codonsderived from our selection of SNPs can be used to predictsubstitution probabilities in alignments. As described inMaterials and Methods, we derive from fSNP2c a substitutionrate matrix on codons, QSNP. From this matrix, the transitionprobabilities between codons and amino acids are estimatedby assuming that the substitution rates are G-distributed(Equation 3).

To evaluate the quality of a model of protein sequenceevolution we analyze the scores:

smodelAB ðseqIDÞ ¼ log

"PmodelA;B ½tðseqIDÞ�

fmodelB

#(9)

where PA;BðtÞ is the transition probability from amino acid Ato amino acid B in the evolutionary time t corresponding tothe sequence identity seqID: fB is the frequency of amino acidB, and the log shall here be intended as the natural logarithm.To check if the dynamics predicted by our model is a gooddescriptor of the real one, we compare sSNPþG

AB with the equiv-alent score extracted from ungapped pairwise sequencealignments at the same sequence identity (see Materialsand Methods).

In Figure 3A, we show the entry-by-entry comparison ofsalignAB in alignments at�92.5% of sequence identity and sSNPþG

ABderived from our SNPþ G model (Equation 3) at the samesequence identity. For the SNP model, the parameter a of theG distribution (seeMaterials andMethods) was set to a ¼ 0:25after optimization (see Figure S5 in File S1). In such plots, a

Predicting Amino Acid Substitution Probabilities Using SNPs 647

good model is characterized by points lying along the liney ¼ x; with the smallest deviations. Amino acid pairs whoseinterchange may be determined by a single nucleotide changeare identified by red circles, while those pairs for which this isnot possible are labeled by blues crosses. The points lie nearthe line y ¼ x in both subsets, proving that the SNPþ Gmodelcorrectly predicts the substitution probabilities in the align-ments at 92.5% of sequence identity for both single and mul-tiple nucleotide substitutions.

In Figure 3B, we compare salignAB (x-axis) and sSNP2no2GAB ;

obtained by estimating the transition probabilities with Equa-tion 4, namely, by assuming that protein sites evolve with thesame rate. Contrary to panel (A), here the comparison is notvery good, with deviations comparable to those observed inFigure 2. Even if the points lie in the proximity of the liney ¼ x; the scores for the amino acid interchanges wheredouble or triple mutations are necessary are systemati-cally underestimated. It is evident that taking the variabil-ity of substitution rates makes the SNP-based model moreaccurate.

In the other panels of Figure 3, we compare salignðseqID ¼92:5%Þ with the scores from some popular models for proteinsequence evolution: JTT (Jones et al. 1992) in panel (C), LG(Le and Gascuel 2008) in panel (D), the codon matrix ECM-unrestricted (Kosiol et al. 2007) in panel (E), and BLOSUM90(Henikoff and Henikoff 1992) in panel (F). For each model,time has been chosen in order to attain a sequence identity of92.5% (see Materials and Methods). For JTT, LG, ECM, andalso for theWAGmatrix (Whelan and Goldman 2001) that wewill consider below, we evolved the respective Q matricesboth with and without considering the variation of ratesacross sites, similarly to what done for QSNP: We checkedthat, in all those cases, computing transition probabilities byaveraging over the distribution of the rates (Equation 3)worsens the performances rather than improving them(see Figure S1 in File S1). Therefore, in the comparison in

Figure 3, and in all the following comparisons, we use theno2G version (Equation 4). This may seem counterintuitive,since it is well known that the G-correction tends to improvephylogenetic estimations with all these models. However,the G distribution is only included on average, withoutassociating a specific rate to each site (Miyazawa 2011a;Rizzato et al. 2016).

It is clear from Figure 3 that, at the sequence identity of92.5%, none of the analyzedmodels estimates the probabilityof multiple substitutions more accurately than SNPþ G:while the SNP-no-G underestimates them, the standard mod-els (JTT, LG, ECM-unrestricted and BLOSUM) overestimatethem. BLOSUM90, in particular, dramatically fails to repro-duce experimental data. This is somehow expected. In fact,while all the other models considered here are based on anevolutionary model of substitutions, BLOSUM90 is learnedfrom conserved blocks of multiple sequence alignments(Henikoff and Henikoff 1991), whose maximum sequenceidentity is 90% and without any explicit lower bound. As aconsequence, while BLOSUM90 is perfectly suited to scorealignments at medium and low sequence identity, it is notoptimal for scoring those at high sequence identity. Thesame argumentation can be extended to any other BLOSUMmatrix.

Toassess thequalityof the score inamorequantitativeway,we computed the average distance from the diagonal of thepoints in an entry-by-entry plot (as those in the panels ofFigure 3) divided by the variance of the data:

dmodel ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDsmodelAB 2salignAB

2EA6¼B

smodelsalign

vuut(10)

where s ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihðsAB2hsABiA 6¼BÞ2iA 6¼B

q:

Lower values of d imply better predictions. In Figure 4, weplot the value of d for all the models in the sequence identity

Figure 1 Effect of GMAF on SNP frequencies. Analysisof the statistical consistency between SNP frequenciesat high (label H: GMAF. 0:2), medium (label M:0:01,GMAF#0:2), and low (label L: GMAF#0:01)GMAF. Magenta line: the cumulative distribution func-tion (cdf) of the relative difference between the M andthe H datasets defined in Equation 8. Green line: thecdf of the relative difference between the L and the M + Hdatasets. Black line: the cdf of a Gaussian distribution withzero mean and unit variance.

648 F. Rizzato et al.

range 75–100%. In particular, the thick dashed line describesthe performance of the SNPþ G model obtained with fixedvalue of a ¼ 0:25: In the sequence identity range of 80–100%, this model performs comparably or even better thanthe JTT model, and outperforms the other tested models.However, in the lower sequence identity range, this modelloses much of its predictive power (see also Figure S3 in FileS1). This may be due to the approximation implicit in Equa-tion 3 that rates remain constant in time, which is known notto be true (Fitch and Markowitz 1970; Penny et al. 2001;Lopez et al. 2002). Gaucher et al. (2001), (2002) observedthat this phenomenon leads to a growth of the shape param-eter awith evolutionary time. In order to approximately takethis effect into account, we allowed a to vary linearly in timeaccording to the model in Miyazawa (2011a) (see Materialsand Methods), and obtained the performance described bythe thick solid line in Figure 4. We here take a0 ¼ 0:25 atan evolutionary time corresponding to a sequence identity of92.5%. Therefore, themodel is identical to SNPþ G at fixed aat this sequence identity, while uses larger values of a atlower sequence identity. Even if this approach describes thetime variability of the rates only approximately, it improvesthe performances of the SNPþ G model without adding anyextra parameter.

Likelihood tests on phylogenetic trees

In order to further benchmark the robustness of our SNPþ G

model of protein sequence evolution, we perform likelihoodratio tests, using the PAML software package (Yang 2007), ofour and other popular models, using reference datasets ofphylogenetic trees and multiple sequence alignments on aminoacids retrieved from the Phylome Database (Huerta-Cepas et al.2014). Here, we use a reduced rate matrix on amino acids in-stead of that on codons, as described inMaterials and Methods.Two phylomes are considered: one containing homologous

sequences of closely related species (only primates), andanother covering a wide diversity of species (mammals,birds, insects, plants, fungi, bacteria. . .). On a collection of111 multiple sequence alignments of primates and theircorresponding phylogenetic trees, our model provides thebest likelihood values over the four tested models (SNP,JTT, WAG, and LG) for 48% of the phylogenetic treesbeing assessed (see File S3). Surprisingly, these results areobtained at practically any level of average sequence iden-tity (40–99%) in this first dataset (see Table 1). This indi-cates that our model can also be considered for phylogeneticinference from multiple alignments of sequences in evolu-tionary close species, such as the primate phylome testedhere. These tests are performed for each model by includingthe standard G-correction implemented in the PAML soft-ware and, during the optimization, branch lengths are opti-mized keeping tree topologies fixed. We are confident thatthe quality of these results could be further improved ifusing our substitution rate matrix on codons instead of itsreduction on amino acids, since they lead to different dy-namics (Kosiol and Goldman 2011; Miyazawa 2013). Whenthe same test is performed on a phylome that covers a widevariety of species (labeled asMultiple species in Table 1), thelikelihood improves only for alignments at very high aver-age sequence identity (see File S4). Indeed, our model isderived only from data at extremely high sequence identityand from the same species (Homo sapiens SNPs). However,also in the multiple species phylome, half of the phylogenyreconstructions in the range of high sequence identity canbe improved with our SNP model.

Discussion

Our results indicate that a selection of relatively common andnearly neutral SNPs provide information on the probability of

Figure 2 Substitution frequencies in SNPs andalignments. Comparison of the substitution fre-quencies between amino acids in our selection ofSNPs (f SNPAB ) and in ungapped pairwise alignmentsat seqID$98% with one of the sequences labeledas human (f alignAB ). Each point corresponds to a pairðA;BÞ of amino acids and its x-value is given byf SNPAB ; while its y-value is f alignAB : The error bars showthe statistical errors estimated according to Poissonstatistics. The frequencies of the few entries thatare not present in SNPdb are conventionally set tothe minimum sizable value of (6 � 1026) in order tomake them visible in log scale. Red points: theentries corresponding to amino acid pairs thatcan mutate into each another by a single nucleo-tide substitution. Here f SNP are estimated by con-sidering only substitutions with GMAF.0:01:Blue points: the entries of amino acid pairsencoded by codons differing by two or three nu-cleotides. Here, f SNP are estimated by consideringsubstitutions with any value of GMAF, since noentry with GMAF.0:01 is present in dbSNP.Black line: y ¼ x:

Predicting Amino Acid Substitution Probabilities Using SNPs 649

amino acid substitution quantitatively consistent with align-ments at high sequence identity. In particular, we verified thatour selection of SNPs, when combined with an adequatetreatment of the substitution rate variability, can be usedsuccessfully to learn scoring matrices and score alignmentsat high sequence identity (80–100%), with better perfor-mances with respect to the scoring matrices used nowa-days. The capability of the SNPþ G model of reproducingsubstitution probabilities is remarkable if we consider that,at variance with all the other considered substitution ma-trices, it does not require learning the transition probabil-ities from alignments.

An important result of our analysis is that substitutionfrequencies learned from SNPs can be used to predict aminoacid substitution probabilities only if rate variability is takeninto account (Rizzato et al. 2016). Indeed, the performanceobtained by a SNP-based model in which this effect is notincluded is very bad (see Figure 4). On the contrary, takinginto account rate variability worsens the performance formore standard models such as JTT, LG, WAG, and ECM(Figure S1 in File S1). A possible interpretation of theseresults is that standard models are learned from alignmentsat medium sequence identity. Then, the instantaneous rate

matrices of these models are consistent with an effectiveMarkovian dynamics, in which the effects of the degenera-tion of the genetic code (Kosiol and Goldman 2011) and ofthe rate variability (Rizzato et al. 2016) are averaged out.Instead, the Qmatrix obtained from the SNPs describes evo-lution at extremely short evolutionary times. As a conse-quence, to give accurate results, it should be evolved witha dynamics that accounts for the among-site variability ofthe rates. In order to verify this scenario, we computed asubstitution rate matrix from pairwise alignments with se-quence identity .98%, and analyze its performances whenusing Equation 3 and Equation 4, respectively (Figure S6in File S1). As for the Q matrix learned from SNPs, we find aclear improvement when accounting for the rate variability.This suggests that instantaneous substitution matrices learnedat very short evolutionary times improve their performanceswhen the rate variability is accounted for.

The observed worsening of the performances at lowersequence identities (see Figure 4 and Figure S3 in File S1)is likely to be related to the combination of several effects.First of all, in our model, the transition probabilities are esti-mated by extrapolating at large evolutionary time the infor-mation collected from the SNPs, namely at very short

Figure 3 Comparison of models with data from alignments at 92.5% of sequence identity. Comparison of the scores salignAB from alignments in UniRef(x-axis) and the score smodel

AB of different models of evolution (y-axis). The scores are all computed at a sequence identity of 92.5%. Each pointcorresponds to a pair of amino acids (A;B). A different point style is adopted to distinguish amino acid pairs whose interchange can be determined bya single nucleotide change (red circles) from the pairs where at least two nucleotides must mutate (blues crosses). The scores in which the first andthe second amino acids coincide are not shown for the sake of visibility. Each panel corresponds to a different model: (A) SNPþ G; (B) SNP-no-G (notethat the y axis is not the same as in the other panels); (C): JTT model (Jones et al. 1992); (D): LG model (Le and Gascuel 2008); (E): ECM-unrestrictedmodel (Kosiol et al. 2007); (F): BLOSUM90 (Henikoff and Henikoff 1992). Dashed line: y ¼ x

650 F. Rizzato et al.

evolutionary times. This procedure unavoidably boosts theeffect of errors in the training set of SNPs. A second weak-ness of our model is the treatment of multiple instantaneoussubstitutions, whose importance has already been estab-lished by other works (Miyazawa 2011b). These multiplesubstitutions are here approximately taken into account bymeasuring their frequency in the dbSNP database. However,the multiple substitutions in the database are almost invari-ably associated with a very low GMAF. Therefore, for thesesubstitutions, it was not possible to check the effect of theGMAF on their frequency by the procedure followed forisolated substitutions, and this might introduce systematicerrors. To investigate if a nonoptimal treatment of multipleinstantaneous substitutions induces the worsening of per-formances at low sequence identities, we compared the per-formances of our SNP-derived model with a modifiedversion of the JTT matrix where multiple substitutions arenot allowed (Figure S4 and Figure S2 in File S1). The trendof the performance of the SNP model, and of the modified-JTT model, as a function of sequence identity are very sim-ilar. This indicates that the degradation of the performancesat low sequence identity may indeed be ascribed to a non-optimal treatment of multiple instantaneous substitutions.A last possible source of error in the model is our treatmentof the among-site rate variability, which is taken into ac-count only on average by Equation 3, and by the approachdescribed in Miyazawa (2011a). Indeed, a separate optimi-zation at different sequence identity of the parameter in theG distribution improves the predictions, even if only margin-ally. This indicates that the model in Miyazawa (2011a) onlyapproximates the true evolutionary dynamics, in which, forexample, substitution rates at different sites are known to becorrelated (Fitch and Markowitz 1970).

We also demonstrated that the SNP-based substitutionmatrix providesmarginally better estimations of phylogeneticrelationships in species closely related to H. sapiens, even

when codon information is not available. The different per-formances between the two considered phylomes can be as-cribed to the fact that both the primate phylome and the SNPsare specific for H. sapiens and share features, such as theequilibrium probability of amino acids, which improve thepredictive power.

Even if, at high sequence identity, homology can be de-tected also by generic models, the use of more specific modelsfor highly similar sequences can correct small local mis-alignments and errors in the alignment scores, and in thecalculation of pairwise distances. Scoring the alignmentswith the approach introduced in this work may thereforebecome relevant in the framework of massive human ge-nome sequencing projects aimed at deciphering humangenetic variations among populations (1000 GenomesProject Consortium et al. 2015).

Acknowledgments

We would like to acknowledge fruitful discussions withEdoardo Sarti, Stefano Zamuner, and Michele Allegra. Thiswork was supported by Associazione Italiana per la Ricercasul Cancro 5 per mille (grant 12214 to A.R. and A.L.).

Figure 4 Performance of the different models for se-quence identity in 75–100%. The average differencefrom the prediction of a model and the observations inthe alignment dmodel (Equation 10) as a function ofthe sequence identity. The thick dashed line isobtained by the SNP + G model, with a fixed valueof the a parameter of the G distribution. The thicksolid line is obtained by varying a linearly in timeaccording to the model in Miyazawa (2011a). Thethin lines correspond to the other models we consid-ered: JTT, LG, ECL, WAG, and SNP without G correc-tion (key on the figure).

Table 1 Number and fraction of phylogenies with improvedlikelihood values for the SNP model compared to other threemodels (JTT, WAG, and LG) for two different phylomes: primatephylome and multiple species phylomea

Avrg.Seq. Id. (%) Primates (%) Multiple Species (%)

40–99 53 (48) 15 (15)90–99 12 (43) 6 (50)80–89 12 (46) 7 (29)70–79 15 (60) 2 (11)50–69 12 (43) 0 (0)40–49 2 (50.0) 0 (0.0)a See Materials and Methods, File S3, and File S4 for further details.

Predicting Amino Acid Substitution Probabilities Using SNPs 651

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