The structural and functional role of a codon in a gene determines its ability to freely change. For example, nonsynonymous (amino acid altering) substitutions may not be tolerated at certain codon sites due to strong negative selection, while at other sites some nonsynonymous substitutions may be allowed if they do not affect key physicochemical properties associated with protein function 1 . Thus, at such preferentially changing sites, more frequent substitutions occur between physicochemically similar amino acids (or codons which lead to those amino acids) than dissimilar ones 2 3 4 . Methods which use changes in physicochemical amino acid properties have thus been proposed in the study of evolution. For example, 5 6 7 use distances to calculate deviations from neutrality for a particular amino acid property. Alternative approaches model the evolution of protein coding sequences as continuous-time Markov chains with rate matrices that distinguish between property-altering and property-conserving mutations as in 8 and 9 . More recently, 10 proposed a Bayesian hierarchical regression model that compares the observed amino acid distances to the expected distances under neutrality for a given set of amino acid properties and incorporates mixture priors for variable selection. The hierarchical mixture priors enable the model in 10 to identify neutral, conserved and radically changing sites, while automatically adjusting for multiple comparisons and borrowing information across properties and sites.

A common feature of all the methods listed above is the implicit assumption that properties are independent from each other in terms of their effect on evolution. A review of the amino acid index database (available for example at http://www.genome.jp/dbget/aaindex.html), which lists more than 500 amino acid properties, shows that a large number of them are highly correlated. Although the correlations we observe in the data can be different from those computed from the raw amino acid scores due to the influence of factors such as codon bias, by ignoring these correlations we are also ignoring the fact that correlated properties may affect a particular site in similar ways. Hence, approaches that do not take into account the correlations in the rates of mutations on different codons do not make use of key information about the relative importance of different physicochemical properties on molecular evolution.

A natural way to account for correlations in the data is by considering a factor structure, see for example 11 . However, selecting the number and order of the factors can be a difficult task in this type of factor models. In addition, the particular structure of the model in 11 makes it difficult to incorporate the effect of the factors on regions that are very strongly conserved. This paper extends the Bayesian hierarchical regression model in 10 by placing a nonparametric prior on the distribution of the regression coefficients describing the effect of properties on molecular evolution. The prior is an extension of the well known Dirichlet process prior 12 13 to model separately exchangeable arrays 14 15 . As in 10 , the main goal of the model described in this paper is to identify sites that are either strongly conserved or radically changing. In order to account for correlations across properties, our model clusters properties with similar effects on evolution, and within each such group, clusters sites with similar regression coefficients and nonparametrically estimates their distribution. In addition to accounting for correlations across properties, this structure allows us to dramatically reduce the number of parameters in the model and generate interpretable insights about molecular evolution at the codon level.

Although the clusters of properties can in principle be considered nuisance parameters that are of no direct interest, in practice posterior inference on the clustering structure can provide interesting insights about the molecular evolution process of a given gene. Indeed, as will become clear in the following sections, our approach incorporates the effect of amino acid usage bias. Hence, any significant differences between the cluster structure estimated from the observed protein-coding sequence alignment and the correlation structure derived from the raw distances between the properties in such cluster can be interpreted a signal of extreme amino acid usage bias in that particular region of the genome.

The rest of the paper is organized as follows. A brief review of DP mixture models along with the details of our model is provided in the Methods section. This section also includes a review of some of the currently available methods for characterizing molecular evolution that take into account changes amino acid properties. The model is then evaluated via simulation studies and illustrated through a real data example. The simulated and real data analyses, as well as comparisons between the proposed semiparametric regression approach and other methods, are presented in Results and discussion. Finally, the Conclusions section provides our concluding remarks.

In this paper, we present a Bayesian hierarchical regression model with a nested infinite relational model on the regression coefficients. The model is capable of identifying sites which show radical or conserved amino acid changes. The (almost sure) discreteness of the DP realizations induces clustering at the level of properties which is analogous to the factor model in 11 , with the advantage being that the nonparametric method automatically determines the appropriate number of clusters. The multi-level clustering ability of the NIRM also induces clustering at the level of sites and allows us to capture skewness and heterogeneity in the distribution of the random effects distribution associated with each cluster of properties.

The main advantage of the models we have described is their ability to simultaneously handle multiple properties with potentially correlated effects on molecular evolution. Our simulations suggest that our models are flexible but robust, being capable of dealing with a range of situations including those where properties are perfectly correlated, as well as those where all properties are uncorrelated. Our semiparametric regression models also work well, particularly in comparison with the regression model in 10 , TreeSAAP and EvoRadical, when applied to DNA sequence data generated from an evolutionary model. In addition, the analysis of the lysin data suggests that the model leads to reasonable results.

The NIRM that is the basis of our model defines a separately exchangeable prior on matrices. This means that the prior is invariant to the order in which properties and sites are included. This is due to the fact that the rows as well as the columns of the parameter of interest are independent draws from a DP. From the point of view of modeling multiple properties, this is a highly desirable property. However, assuming that DNA sites are exchangeable can be questionable. Although this is a potential limitation of our model, we should note that the assumption of independence across sites (which is a stronger assumption than exchangeability) underlies all the methods discussed in the Background section. If information about the 3-dimensional structure of the encoded protein or other sequence specific information that can guide the construction of the dependence model is available, our model could be easily extended to account for this feature. In the absence of such information, exchangeability across DNA sites seems to be a reasonable prior assumption. Indeed, in contrast to the most common independence assumption, our exchangeability assumption allows us to explain correlations at the level of sites.

In our applications, we have used codon substitution models for reconstructing ancestral sequences as we wished to compare our methods to other methods for detecting selective sites that also use codon substitution models, such as those implemented in PAML and EvoRadical. However, it is possible to perform the proposed Bayesian semiparametric analyses using amino acid substitution models instead of codon substitution models. Note that the substitution model is only used in the calculation of the observed distances. First, we infer the ancestral sequences under a specific substitution model and a given phylogeny. We then compute the observed distances for a given property and a given site as the mean absolute difference in property scores due to all nonsynonymous substitutions at that site, where the nonsynonymous substitutions are counted by comparing the DNA sequences between two neighboring nodes in the phylogeny. The reconstructed ancestral sequences, and therefore the observed distances in our model, may differ under different substitution models, but the method can be implemented under any substitution model, including amino acid substitution models. The gain in execution time from using amino acid substitution models instead of codon-based ones could potentially be significant if the uncertainty in the alignment/phylogeny/ancestral level is taken into account.

Finally, it is important to note that the “observed” distances are not really directly observed, but are instead constructed from ancestral sequences and, therefore, subject to error. A simple way to account for this additional level of uncertainty is to modify the computation of expected distances by incorporating the ideas of 37 . This approach was previously employed in 10 , with little impact on the final results.

The truncations and the introduction of the configuration variables imply that (2) and (3) can be written as

ζ j | { Π k } ∼ ∑ k = 1 K Π k δ θ k ∗ ξ i , k | { w l , k } ∼ ∑ l = 1 L w l , k δ φ l , k

with φ _l,k ∼G _0lk and Π _k and w _l,k being the appropriate stick breaking weights. Writing the model as in (5) helps in obtaining the forms of the full conditionals as below.

The column indicators ζ _j for j=1,…,J are sampled from a multinomial distribution with probabilities

P ( ζ j = k | ⋯ ) = q j k ∝ ∑ l = 1 L ∏ { i : ξ i , k = l } Π k N ( y i , j ∗ | ϕ l , k x i , j ∗ , ϑ l , k 2 ) ,

where ϑ l , k 2 is ϑ l , k 2 if ϕ _l,k =0 or is ϑ l , k 2 / n i O if ϕ _l,k is different from zero. Π _k is sampled in two parts: first, by generating v _k from a Beta ( 1 + m k , ρ + ∑ s = k + 1 K m s ) for k=1,…,K−1 and v _K =1, where m _k is the number of columns assigned to cluster k and then, by constructing Π k = v k ∏ s = 1 k − 1 ( 1 − v s ) .

For i=1,…,Iand k=1,…,K, the indicators ξ _i,k are also sampled from a multinomial with probabilities of the form

P ( ξ i , k = l | ⋯ ) = p i , k l ∝ ∏ { j : ζ j = k } w l , k N ( y i , j ∗ | ϕ l , k x i , j ∗ , ϑ l , k 2 ) .

The updated weights w _l,k are sampled in a manner similar to the Π _k , i.e., u _l,k are generated from a Beta ( 1 + n l , k , γ k + ∑ r = l + 1 L n lr ) for l=1,…,L−1 and u _Lk =1, where n _l,k is the number of β _i,j s assigned to atom l of cluster k and then, by constructing w l , k = u l , k ∏ r = 1 l − 1 ( 1 − u r , k ) .

Following 18 , the DP concentration parameters ρand γ _k are sampled in two steps by introducing auxiliary variables η ₁and η ₂. First, sample η ₁from

p ( η 1 | ρ , ⋯ ) = Beta ( ρ + 1 , J )

and then ρfrom

p ( ρ | η 1 , ⋯ ) = a ρ + n ζ ∗ − 1 a ρ + n ζ ∗ − 1 + J ( b ρ − log ( η 1 ) ) × Ga ( a ρ + n ζ ∗ , b ρ − log ( η 1 ) ) + J ( b ρ − log ( η 1 ) ) a ρ + n ζ ∗ − 1 + J ( b ρ − log ( η 1 ) ) × Ga ( a ρ + n ζ ∗ − 1 , b ρ − log ( η 1 ) ) ,

where n ζ ∗ is the number of unique column indicators ζ _j . Similarly, for each k=1,…,K,

p ( η 2 | γ k , ⋯ ) = Beta ( γ k + 1 , I ) ,

p ( γ k | η 2 , ⋯ ) = a γ + m ξ , k ∗ − 1 a ρ + m ξ , k ∗ − 1 + I ( b γ − log ( η 2 ) ) × Ga ( a γ + m ξ , k ∗ , b γ − log ( η 2 ) ) + I ( b γ − log ( η 2 ) ) a γ + m ξ , k ∗ − 1 + I ( b γ − log ( η 2 ) ) × Ga ( a γ + m ξ , k ∗ − 1 , b γ − log ( η 2 ) ) ,

where m ξ , k ∗ is the number of unique row indicators ξ _i,k , for a specific cluster of columns k.

To sample the unique φ l , k = ( ϕ l , k , ϑ l , k 2 ) s given in (4), we introduce a set of indicator variables ψ _l,k which take the value 1 when ϕ _l,k is different from zero. For l=1,…,Land k=1,…,K, ψ _l,k , ϑ l , k 2 and ϕ _l,k are jointly sampled in the following way - ψ _l,k is sampled by integrating out ϕ _l,k and ϑ l , k 2 from its full conditional, ϑ l , k 2 is sampled conditional on ψ _l,k and ϕ _l,k is sampled conditional on both the corresponding ψ _l,k and ϑ l , k 2 , i.e.,

p ( ψ l , k , ϑ l , k 2 , ϕ l , k | ⋯ ) = p ( ψ l , k | ⋯ ) p ( ϑ l , k 2 | ψ l , k , ⋯ ) × p ( ϕ l , k | ψ l , k , ϑ l , k 2 , ⋯ )

with the individual expressions obtained as follows.

First, let Ω l , k i , j = { ( i , j ) : ξ i ζ j = l , ζ j = k } . Then,

p ( ψ l , k | ⋯ ) ∝ λ ∫ ∏ Ω l , k i , j N ( y i , j ∗ | 0 , ϑ l , k 2 ) IG ( ϑ l , k 2 | a κ , b κ ) d ( ϑ l , k 2 ) + ( 1 − λ ) ∫ ∫ ∏ Ω l , k i , j N ( y i , j ∗ | ϕ l , k x i , j ∗ , ϑ l , k 2 / n i O ) × N ( ϕ l , k | α k , ϑ l , k 2 / V 0 ) IG ( ϑ l , k 2 | a σ ∗ , b σ ∗ ) d ( ϕ l , k ) d ( ϑ l , k 2 ) .

p ( ϑ l , k 2 | ψ l , k , ⋯ ) = IG I ∗ J ∗ 2 + a κ , 1 b κ + σ 1 , scale − 1 if ψ l , k = 0 IG I ∗ J ∗ 2 + a σ ∗ , 1 b σ ∗ + σ 2 , scale − 1 if ψ l , k = 1 ,

where I ∗ J ∗ = ∑ i , j 1 { ξ i ζ j = l , ζ j = k } and the update terms are given by σ 1 , scale = ∑ Ω l , k i , j y i , j ∗ 2 2 and σ 2 , scale = α k 2 V 0 2 + ∑ Ω l , k i , j n i O y i , j ∗ 2 2 − ( α k V 0 + ∑ Ω l , k i , j n i O y i , j ∗ x i , j ∗ ) 2 2 ( V 0 + ∑ Ω l , k i , j n i O x i , j ∗ 2 ) .

p ( ϕ l , k | ψ l , k , ϑ l , k 2 , ⋯ ) = 0 if ψ l , k = 0 N ( m ϕ , C ϕ ) if ψ l , k = 1 ,

where m ϕ = α k V 0 + ∑ Ω l , k i , j n i O y i , j ∗ x i , j ∗ V 0 + ∑ Ω l , k i , j n i O x i , j ∗ 2 and C ϕ = ϑ l , k 2 V 0 + ∑ Ω l , k i , j n i O x i , j ∗ 2 .

The full conditional of λ is given by

p ( λ | ⋯ ) ∼ Beta ( a λ + ∑ l , k 1 { ψ l , k = 0 } , b λ + ∑ l , k 1 { ψ l , k = 1 } ) .

Finally, for k=1,…,K, the full conditional of α _k is given by

p ( α k | ⋯ ) ∼ N ( m α ∗ , C α ∗ )

where

C α ∗ = 1 1 C α + ∑ { l : ψ l , k = 1 } V 0 ϑ l , k 2

and

m α ∗ = C α ∗ m α C α + ∑ { l : ψ l , k = 1 } V 0 ϕ l , k ϑ l , k 2

.

The authors declare that they have no competing interests.

SD, AR and RP formulated the model. SD performed the analyses and drafted the manuscript. AR and RP revised the manuscript draft. All authors read and approve the final version of the manuscript.