- Research article
- Open Access
Bayesian semiparametric regression models to characterize molecular evolution
- Saheli Datta^{1}Email author,
- Abel Rodriguez^{2} and
- Raquel Prado^{2}
https://doi.org/10.1186/1471-2105-13-278
© Datta et al.; licensee BioMed Central Ltd. 2012
- Received: 15 December 2011
- Accepted: 11 October 2012
- Published: 30 October 2012
Abstract
Background
Statistical models and methods that associate changes in the physicochemical properties of amino acids with natural selection at the molecular level typically do not take into account the correlations between such properties. We propose a Bayesian hierarchical regression model with a generalization of the Dirichlet process prior on the distribution of the regression coefficients that describes the relationship between the changes in amino acid distances and natural selection in protein-coding DNA sequence alignments.
Results
The Bayesian semiparametric approach is illustrated with simulated data and the abalone lysin sperm data. Our method identifies groups of properties which, for this particular dataset, have a similar effect on evolution. The model also provides nonparametric site-specific estimates for the strength of conservation of these properties.
Conclusions
The model described here is distinguished by its ability to handle a large number of amino acid properties simultaneously, while taking into account that such data can be correlated. The multi-level clustering ability of the model allows for appealing interpretations of the results in terms of properties that are roughly equivalent from the standpoint of molecular evolution.
Keywords
- Nonsynonymous Substitution
- Dirichlet Process
- Ancestral Sequence
- Nonsynonymous Change
- Truncation Level
Background
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[2–4]. Methods which use changes in physicochemical amino acid properties have thus been proposed in the study of evolution. For example,[5–7] 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.
Methods
Dirichlet process mixture models
The Dirichlet process (DP) was formally introduced by[12] as a prior probability model for random distributions G. A DP(ρ, G_{0}) prior for G is characterized by two parameters, a positive scalar parameter ρ, and a parametric base distribution (or centering distribution) G_{0}. ρ can be interpreted as the precision parameter, with larger values of ρ resulting in realizations of G that are closer to the base distribution G_{0}.
where${\delta}_{{\varphi}_{l}}(\xb7)$ denotes a point mass at ϕ_{ l }. The locations ϕ_{ l } are i.i.d. draws from G_{0}, while the corresponding weights w_{ l } are generated using the following “stick-breaking” mechanism. Let w_{1}=v_{1}and define${w}_{l}={v}_{l}\prod _{r=1}^{l-1}(1-{v}_{r})$ for l=2,3,…, where {v_{ l }:l=1,2,…} are i.i.d. draws from a Beta(1, ρ) distribution. Defining the weights in this way ensures$\sum _{l=1}^{\infty}{w}_{l}=1$. Furthermore, the sequences {v_{ l }:l=1,2,…} and {ϕ_{ l }:l=1,2,…} are independent.
One advantage of DP mixture models over other approaches to clustering and classification is that they allow us to automatically estimate the number of components in the mixture. Indeed, from the Pólya urn representation of the process it should be clear that, although the number of potential mixture components is infinite, the model implicitly places a prior on the number of components that, for moderate values of ρ, favors the data being generated by an effective number of components K^{∗}=max_{i≤n}{ξ_{ i }}<n.
The model
Our data consist of observed and expected amino acid distances derived from a DNA sequence alignment, a specific phylogeny, a stochastic model of sequence evolution, and a predetermined set of physicochemical amino acid properties. In the analyses presented here, we disregard uncertainty in the alignment/phylogeny/ancestral sequence level since our main focus is the development and implementation of models that allow us to make inferences on the latent effects that several amino acid properties may have on molecular evolution for a given phylogeny and an underlying model of sequence evolution. Extensions of these analyses that take into account these uncertainties are briefly described in Conclusions. For further discussion on this issue, see also[10].
In order to calculate the observed distances, we first infer the ancestral sequences under a specific substitution model and a given phylogeny. In our applications, we use PAML version 3.15[20] and the codon substitution model of[21], which accounts for the possibility of multiple substitutions at a given site. Nonsynonymous substitutions are then counted by comparing DNA sequences between two neighboring nodes in the phylogeny. The observed mean distance, denoted as y_{i,j} for site i and property j, is obtained as the mean absolute difference in the property scores due to all nonsynonymous substitutions at site i. Only those sites with at least one nonsynonymous change from the ancestral level are retained for further analysis.
We consider a hierarchical regression model that relates x_{i,j} to y_{i,j}and allows us to compare the expected and observed distances at the codon level for several properties simultaneously with the following rationale. If a given site i is neutral with respect to property j, then y_{i,j}≈x_{i,j}. If property j is conserved at site i, then y_{i,j}<<x_{i,j} and finally, if property j is radically changing at site i, then y_{i,j}>>x_{i,j}.
where${n}_{i}^{O}$ is the observed number of nonsynonymous changes at a particular site i and β_{i,j} and${\sigma}_{i,j}^{2}$ are the regression coefficient and variance parameter associated with site i and property j. The mixture model accounts for the fact that some of the${y}_{\mathrm{ij}}^{\ast}$s can be equal to zero as some nonsynonymous changes do not change the value of the property being measured (e.g., Aspargine, Aspartic acid, Glutamine, Glutamic acid all have the same hydropathy score).
To complete the model, we need to describe a model for the matrix of regression coefficients β_{i,j}. There are a number of possible models for this type of data which utilize Bayesian nonparametric methods; some recent examples include the infinite relational model (IRM)[22, 23], the matrix stick breaking process (MSBP)[24], and the nested infinite relational model (NIRM)[14, 15].
is a random distribution such that${\Pi}_{k}={v}_{k}\prod _{s<k}(1-{v}_{s})$, v_{ k }∼Beta(1,ρ), and${\mathit{\theta}}_{k}^{\ast}\sim {H}_{k}$. Indeed, the discrete nature of F ensures that ties among the θ_{ j }happen with non-zero probability.
with${w}_{l,k}={u}_{l,k}\prod _{r<l}\{1-{u}_{r,k}\}$, u_{l,k}∼Beta(1,γ_{ k }) for every k, and φ_{l,k} are independently drawn from the baseline measure G_{0,l,k}.
where${p}_{1}\left({\vartheta}_{l,k}^{2}\right)\sim $Inv-Ga(a_{ κ },b_{ κ }),$p\left({\varphi}_{l,k}\right|{\vartheta}_{l,k}^{2})\sim \mathsf{\text{N}}({\alpha}_{k},{\vartheta}_{l,k}^{2}/{V}_{0})$ and${p}_{2}\left({\vartheta}_{l,k}^{2}\right)\sim $Inv-Ga$({a}_{\sigma}^{\ast},{b}_{\sigma}^{\ast})$. Here ϕ_{l,k} and${\vartheta}_{l,k}^{2}$ respectively denote the unique values β_{i,j}and${\sigma}_{i,j}^{2}$ can take, whereas λ is the prior probability that ϕ_{l,k}has the value zero (i.e., the properties associated with this cluster are strongly conserved at this cluster of sites).
Note that our model implies that both sites and properties are exchangeable a priori. If no additional prior information is available, this type of assumption seems reasonable. However, a posteriori, it is possible to have sites behave differently in different clusters.
To complete the model we place hyperpriors on all parameters of the resulting model. Conjugate priors are chosen for ease of computation. α_{ k } denotes the mean for the ϕ_{l,k}s that are different from zero belonging to a specific cluster of properties k and is assumed to have a N(m_{ α },C_{ α }) prior for all k. The DP concentration parameters ρ and γ_{ k } are assumed to follow Ga(a_{ ρ },b_{ ρ }) with mean a_{ ρ }/b_{ ρ }, and Ga(a_{ γ },b_{ γ }) with mean a_{ γ }/b_{ γ } for all k, respectively. λ, which is the prior probability for the point mass at 0 in G_{0lk}, follows a Beta(a_{ λ },b_{ λ }). The specific choice of hyperparameters is discussed later as part of each data analysis. In general, we use Ga(1,1) priors for the DP concentration parameters and a N(1,C_{ α }) prior for α_{ k }to correspond to our assumption of neutrality a priori for the properties.
Related work
We compare results from our proposed method with results from a few currently available methods that aim to characterize molecular evolution while also taking into account changes in amino acid properties, namely, the regression model in[10], TreeSAAP[25], and EvoRadical[9].
In[10], the first level of the model is the regression equation on${y}_{i,j}^{\ast}$ as in equation (1), but it implicitly assumes independence among properties and independence among sites unlike our current model. The model in[10] is suitable for use when a few mostly independent amino acid properties are being analyzed whereas the new semiparametric model is better suited to the analysis of a large number of possibly correlated properties.
TreeSAAP uses the methods of[6] to classify nonsysnonymous substitutions into one of M categories, with higher numbered categories corresponding to sites showing radical changes and lower numbered categories used for sites showing conserved changes for a given property. For the analysis considered here, we used 8 categories where categories 6, 7, and 8 corresponded to sites showing radical changes, and categories 1 and 2 to sites showing conserved changes. Nonsynonymous changes are inferred from the ancestral reconstruction using the nucleotide substitution models in baseml implemented in PAML. We used a Bonferroni correction to correct for multiple comparisons.
EvoRadical implements the models of[9], which use partitions of amino acids to parameterize the rates of property-conserving and property-altering codon substitutions in a maximum likelihood framework. The model considers three types of substitutions: synonymous, property-conserving nonsynonymous and property-altering nonsynonymous which is a slight improvement from[8]. For analyses with multiple properties, one has to create different partitions for the different properties and run EvoRadical for each property.
Posterior simulation
Various algorithms exist for posterior inference of DP mixtures - some of the most popular ones use (i) the Pólya urn characterization to marginalize out the unknown distribution(s)[26, 27], (ii) a truncation approximation to the stick-breaking representation of the process which paves the way for the use of methods employed in finite mixture models[28, 29], (iii) reversible jump MCMC or split-merge methods[30, 31]. Some other recent approaches have also used variational methods[32] and slice samplers[33].
We use an extension of the finite mixture approximation discussed in[28] for its ease of implementation. Truncating F at a sufficiently large K, we write${F}^{\left(K\right)}=\sum _{k=1}^{K}{\Pi}_{k}{\delta}_{{\theta}_{k}^{\ast}}$, with the weights Π_{ k }and locations${\theta}_{k}^{\ast}$ generated as described earlier in this Section. Next we introduce configuration variables {ζ_{ j }} such that, for k=1,…,K, ζ_{ j }=k if and only if${\mathit{\theta}}_{j}={\theta}_{k}^{\ast}$. Similarly for G_{ k }, we truncate at a sufficient level L, and introduce another set of configuration variables {ξ_{i,k}} where ξ_{i,k}=l, with l=1,…,L, if and only if${\theta}_{i,k}^{\ast}={\phi}_{l,k}$. Additional details about the algorithm are provided in the Appendix.
To determine the truncation levels K and L, we follow[29]. In particular, note that conditional on ρ (the DP concentration parameter), the tail probability$\sum _{k=K}^{\infty}{\Pi}_{k}$ has expectation {ρ/(1 + ρ)}^{K−1}. Using prior guesses for ρ and acceptable tolerance levels for the tail probability to be small, one can then solve for the truncation level K. In our analyses, we used K and L in the range of 25 to 35. These values are in line with those used in other applications (for example, see[34]).
Results and discussion
Empirical exploration via simulation studies
We present two simulation studies to check the performance of the model under different scenarios. Additional simulation scenarios that may be of interest are available as an Additional file1.
Simulation study 1
The setup for the first simulation is as follows. We generate values for the distinct regression coefficients (ϕ_{l,k}) from a N(1,0.25). The number of distinct regression coefficients depends on the particular clustering structure for the corresponding simulation. Once we obtain the regression coefficients, we generate observations y_{i,j} from N(ϕ_{l,k}x_{i,j},σ^{2}=0.001). The x_{i,j}s are obtained from the lysin data set described below with analyses for 32 properties, which implies J=32 and I=94.
We fitted the model in The Model subsection to the${y}_{i,j}^{\ast}$s and${x}_{i,j}^{\ast}$s, with the following modifications: (i) the NIRM is imposed on β_{i,j}, so φ_{l,k}=ϕ_{l,k} and (ii) ϕ_{l,k}∼G_{0}where G_{0}∼N(α,τ^{2}). We used K=25 and L=25 for the simulations. The MCMC algorithm was run with the following hyperpriors: ρ∼Ga(1,1), γ_{ k }∼Ga(1,1) for all k, α∼N(1,0.25). σ^{2}∼Inv-Ga(100, 10) and τ^{2}∼Inv-Ga(2,4) were chosen such that the prior means corresponded to the true values for these hyperparameters. Results are based on 15000 iterations, with the first 5000 discarded as burn-in. Convergence was assessed by running two chains where each chain was initialized by randomly assigning the β_{i,j}s to different partitions. Posterior summaries based on the two chains were consistent with each other.
This scenario corresponds to the type of situation we expect on most real datasets: properties will cluster into groups and, within each group of properties, clusters of sites with similar responses can be clearly identified. Our results suggest that, as expected, the model is capable of identifying these multiple clusters with high accuracy and therefore accurately estimate the value of the regression coefficients. Other scenarios, including extreme cases where all properties belong to a common cluster while sites belong to one of several clusters, and cases where each property has a different effect on amino acid rates are available as Additional file1.
To investigate the effect of the truncation levels and the priors on our model, we performed sensitivity analysis by varying the truncation levels as well as the different hyperparameters. Increasing the truncation level to 35 did not affect the results and the estimated posterior means of the β s showed close agreement with the true values. The analyses was also fairly robust to the choice of the priors, since varying the hyperparameters had almost no effect on the results. Decreasing the prior variance of τ^{2} makes the results marginally better, i.e., posterior means of the β_{i,j}s,${\widehat{\beta}}_{i,j}$s, are slightly closer to the true values.
Simulation study 2 - data simulated from a biological model
In our second simulation study the model is evaluated in the context of biological sequences generated from an evolutionary model. In particular, a Markov model was used to generate 20 sequences of 90 codons each. For the first one-third of the sites (sites 1-30) we used transition probabilities obtained from the codon-substitution model of[21] with equal equilibrium probabilities for all 61 codons. For the second one-third of the sites (sites 31-60), we modified the transition probability matrix from the previous step by increasing the probabilities of transitions between codons that have small distances for volume and decreasing the probabilities of transitions between codons that have large distances for volume - this was done to encourage only those changes that conserve volume in this part of the sequences. Finally, for the last one-third of the sites (sites 61-90), we modified the original transition probability to encourage radical changes in hydropathy. Thus, we increased some transition probabilities between codons that have very different hydropathy scores and decreased a few of those that have similar hydropathy scores. Note that, since the equilibrium probabilities are either uniform or roughly uniform across all sites, the correlation structure across properties is retained in the expected distances, which simplifies the interpretation of the results.
Once we obtained the sequences, we generated ancestral sequences using PAML, version 3.15,[20] and calculated observed and expected distances y_{i,j} and x_{i,j} for five properties, namely, hydropathy (h), volume (M_{ v }), polarity (p), isoelectric point (p H_{ i }) and partial specific volume (V^{0}). Of these, h and p are correlated and so are M_{ v }and V^{0}.
Our model was fitted with K=25 and L=25 as truncation levels. The prior distributions were the same as the ones used for our previous simulation. Results are based on 15000 iterations, of which the first 5000 were burn-in. There did not seem to be any obvious problems with convergence, which was assessed by visual inspection of trace plots of some of the parameters.
Comparing results between models in [[10]] and the new semiparametric model, for the data simulated under a biological model
Parametric regression[10] | Semiparametric regression | |
---|---|---|
30 sites with largest posterior mean${\widehat{\beta}}_{i,j}$ for h | 4, 5, 6, 10, 14, 18, 19, 21, 22, 23, 24, 33, 48, 52, 54, 59, 62, 64, 65, 67, 71, 74, 75, 77, 80, 81, 82, 84, 85, 89 | 4, 5, 14, 18, 19, 21, 24, 33, 37, 39, 48, 52, 54, 59, 62, 64, 65, 67, 71, 72, 74, 75, 77, 80, 81, 82, 84, 85, 86, 89 |
30 sites with lowest posterior mean${\widehat{\beta}}_{i,j}$ for M_{ v } | 5, 6, 7, 9, 16, 19, 24, 25, 26, 27, 28, 31, 32, 36, 44, 49, 51, 58, 59, 60, 61, 64, 65, 67, 79, 80, 82, 83, 85, 88 | 5, 6, 7, 9, 16, 18, 19, 24, 25, 26, 27, 28, 31, 32, 34, 36, 38, 49, 58, 59, 60, 61, 64, 65, 67, 79, 80, 83, 84, 88 |
Sites identified as significant by TreeSAAP for the different properties for the simulation study based on a biological model
Property | Radically changing (1.645) | Radically changing (3.695) | Conserved (1.645) | Conserved (3.695) |
---|---|---|---|---|
h | 5, 59, 65, 67, 71, 74, 81, 82, 89 | 74 | 36, 83 | None |
p | 21, 24, 37, 64, 65, 67, 71, 74, 75, 81, 82, 89 | None | 7, 18, 36, 49, 55 | None |
M _{ v } | 10, 33, 66 | None | 5, 18, 36, 49 | None |
V ^{0} | 10, 13, 33, 66 | None | 18, 36 | None |
p H _{ i } | 39, 55, 72 | None | 11, 64, 72 | None |
Finally, we analyzed the sequences generated previously with EvoRadical using two different partitions[8] - one for p and the other for M_{ v }. We chose to run Evoradical with p instead of h, since a partition of the amino acids for polarity was already available in[8]. Additionally, given that h and p are correlated, we expect to see somewhat similar results for these two properties.
Sites that have high posterior probabilities (>0.95) of belonging to each site class for the different partitions for EvoRadical for the simulated data
Property | ω≤1,γ ≤1 | ω≤1,γ >1 | ω>1,VVγ ≤1 | ω>1,γ >1 |
---|---|---|---|---|
p | None | None | None | 1, 2, 5, 7, 10, 11, 12, 13, 14, 18, 19, 20, 26, 27, 30, 32, 33, 34, 36, 37, 42, 43, 47, 53, 57, 59, 61, 62, 63, 64, 66, 67, 68, 69, 72, 73, 74, 75, 77, 82, 83, 86, 87, 88, 90 |
M _{ v } | None | None | None | 2, 7, 9, 18, 19, 20, 22, 27, 31, 32, 36, 38, 53, 55, 61, 62, 64, 67, 72, 74, 86 |
Illustration with Lysin data
Our proposed model was applied to the sperm lysin data set which consisted of cDNA from 25 abalone species with 135 codons in each sequence[35]. Sites with alignment gaps were removed from all sequences, which resulted in 122 codons for the analysis presented here. The phylogeny of[35] and the codon substitution model M8 in PAML, version 3.15,[20] was used to generate the ancestral sequences. The model M8 uses a discretized beta distribution to model ω values between zero and one with probability p_{0} and allows for an additional positive selection category with ω>1 and probability p_{1}.
List of 32 amino acid properties used in the analysis
AAindex accession number (if available) | Property | Symbol | AAindex accession number (if available) | Property | Symbol |
---|---|---|---|---|---|
KYTJ820101 | Hydropathy | h | ∗ | Helical contact area | C _{ a } |
GRAR740103 | Molecular volume | M _{ v } | ZIMJ680104 | Isoelectric point | p H _{ i } |
MANP780101 | Surrounding hydrophobicity | H _{ p } | OOBM770103 | Long-range non-bonded energy | E _{ l } |
ZIMJ680103 | Polarity(Zimmerman) | p _{ zim } | ∗ | Mean r.m.s. fluctuation displacement | F |
CHOP780201 | Alpha-helical tendencies | P _{ α } | FASG760101 | Molecular weight | M _{ w } |
GRAR740102 | Polarity(Grantham) | p | ∗ | Normalized consensus hydrophobicity | H _{ nc } |
PONP800108 | Average number of surrounding residues | N _{ s } | COHE430101 | Partial specific volume | V ^{0} |
∗ | Power to be at the C-terminal | α _{ c } | WOEC730101 | Polar requirement | P _{ r } |
GRAR740101 | Composition | c | ∗ | Power to be at the middle of alpha-helix | α _{ m } |
∗ | Compressibility | K ^{0} | ∗ | Power to be at the N-terminal | α _{ n } |
FAUJ880113 | Equilibrium constant (ionization of COOH) | p K ^{ ′ } | MCMT640101 | Refractive index | μ |
CHOP780202 | Beta-structure tendencies | P _{ β } | OOBM770102 | Short and medium range non-bonded energy | E _{ sm } |
ZIMJ680102 | Bulkiness | B _{ l } | PONP800107 | Solvent accessible reduction ratio | R _{ a } |
∗ | Buriedness | B _{ r } | ∗ | Thermodynamic transfer hydrophobicity | H _{ t } |
∗ | Chromatographic index | R _{ F } | OOBM770101 | Total non-bonded energy | E _{ t } |
CHAM830101 | Coil tendencies | P _{ c } | CHOP780101 | Turn tendencies | P |
Strongly conserved sites ( ${\widehat{\beta}}_{\mathit{ij}}\mathbf{<}\mathbf{0}\mathbf{.}\mathbf{4}$ ) for lysin data for different clusters
Cluster | Site number |
---|---|
1 | 96 |
2 and 3 | 22, 28, 35, 51, 111, 117, 128 |
11, 17, 18, 19, 24, 25, 27, 29, 33, 35, 42, 43, 47, 49, 51, | |
4 | 53, 58, 64, 66, 68, 69, 71, 73, 79, 81, 88, 94, 96, 98, 100, |
101, 104, 105, 110, 111, 114, 115, 117, 121, 122, 129, 131 |
The results are fairly robust to the choice of different hyperparameter values. Note that the scale factor for${\vartheta}_{l,k}^{2}$ ultimately affects the variation in the β_{i,j}values, and it is advisable to choose it so that the prior variance for the unique β_{i,j}s is not too large.
Conclusions
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.
Appendix: details about the Gibbs sampler
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.
where${\vartheta}_{l,k}^{2}$ is${\vartheta}_{l,k}^{2}$ if ϕ_{l,k}=0 or is${\vartheta}_{l,k}^{2}/{n}_{i}^{O}$ if ϕ_{l,k} is different from zero. Π_{ k } is sampled in two parts: first, by generating v_{ k }from a$\mathsf{\text{Beta}}(1+{m}_{k},\rho +\sum _{s=k+1}^{K}{m}_{s})$ 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${\Pi}_{k}={v}_{k}\prod _{s=1}^{k-1}(1-{v}_{s})$.
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$\mathsf{\text{Beta}}(1+{n}_{l,k},{\gamma}_{k}+\sum _{r=l+1}^{L}{n}_{\mathrm{lr}})$ 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${w}_{l,k}={u}_{l,k}\prod _{r=1}^{l-1}(1-{u}_{r,k})$.
where${m}_{\xi ,k}^{\ast}$ is the number of unique row indicators ξ_{i,k}, for a specific cluster of columns k.
with the individual expressions obtained as follows.
where${m}_{\varphi}=\left(\frac{{\alpha}_{k}{V}_{0}+\sum _{{\Omega}_{l,k}^{i,j}}{n}_{i}^{O}{y}_{i,j}^{\ast}{x}_{i,j}^{\ast}}{{V}_{0}+\sum _{{\Omega}_{l,k}^{i,j}}{n}_{i}^{O}{x}_{i,j}^{\ast 2}}\right)$ and${C}_{\varphi}=\frac{{\vartheta}_{l,k}^{2}}{{V}_{0}+\sum _{{\Omega}_{l,k}^{i,j}}{n}_{i}^{O}{x}_{i,j}^{\ast 2}}$.
.
Software availability
The R code implementing the models in the paper is freely available at http://www.ams.ucsc.edu/~raquel/software/.
Declarations
Acknowledgements
RP and SD were supported by the NIH/NIGMS grant R01GM072003-02. AR was supported by the NIH/NIGMS grant R01GM090201-01.
Authors’ Affiliations
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