KaKs_Calculator: Calculating Ka and Ks Through Model Selection and Model Averaging

Abstract

KaKs_Calculator is a software package that calculates nonsynonymous (Ka) and synonymous (Ks) substitution rates through model selection and model averaging. Since existing methods for this estimation adopt their specific mutation (substitution) models that consider different evolutionary features, leading to diverse estimates, KaKs_Calculator implements a set of candidate models in a maximum likelihood framework and adopts the Akaike information criterion to measure fitness between models and data, aiming to include as many features as needed for accurately capturing evolutionary information in protein-coding sequences. In addition, several existing methods for calculating Ka and Ks are also incorporated into this software. KaKs_Calculator, including source codes, compiled executables, and documentation, is freely available for academic use at http://evolution.genomics.org.cn/software.htm.

Introduction

Calculating nonsynonymous (Ka) and synonymous (Ks) substitution rates is of great significance in reconstructing phylogeny and understanding evolutionary dynamics of protein-coding sequences across closely related and yet diverged species ^{1., 2., 3.}. It is known that Ka and Ks, or often their ratio (Ka/Ks), indicate neutral mutation when Ka equals to Ks, negative (purifying) selection when Ka is less than Ks, and positive (diversifying) selection when Ka exceeds Ks. Therefore, statistics of the two variables in genes from different evolutionary lineages provides a powerful tool for quantifying molecular evolution.

Over the past two decades, several methods have been developed for this purpose, which can generally be classified into two classes: approximate method and maximum likelihood method. The approximate method involves three basic steps: (1) counting the numbers of synonymous and nonsynonymous sites, (2) calculating the numbers of synonymous and nonsynonymous substitutions, and (3) correcting for multiple substitutions. On the other hand, the maximum likelihood method integrates evolutionary features (reflected in nucleotide models) into codon-based models and uses the probability theory to finish all the three steps in one go (4). However, these methods adopt different substitution or mutation models based on different assumptions that take account of various sequence features, giving rise to varied estimates of evolutionary distance (5). In other words, Ka and Ks estimation is sensitive to underlying assumptions or mutation models (3). In addition, since the amount and the degree of sequence substitutions vary among datasets from diverse taxa, a single model or method may not be adequate for accurate Ka and Ks calculations. Therefore, a model selection step, that is, to choose a best-fit model when estimating Ka and Ks, becomes critical for capturing appropriate evolutionary information ^{6., 7.}.

Toward this end, we have applied model selection and model averaging techniques for Ka and Ks estimations. We use a maximum likelihood method based on a set of candidate substitution models and adopt the Akaike information criterion (AIC) to measure fitness between models and data. After choosing the best-fit model for calculating Ka and Ks, we average the parameters across the candidate models to include as many features as needed since the true model is seldom one of the candidate models in practice (8). Finally, these considerations are incorporated into a software package, namely KaKs_Calculator.

Algorithm

Candidate models

Substitution models play a significant role in phylogenetic and evolutionary analyses of protein-coding sequences by integrating diverse processes of sequence evolution through various assumptions and providing approximations to datasets. We focused on a set of time-reversible substitution models ^{9., 10., 11., 12., 13., 14., 15., 16.} as shown in Table 1 ^{17., 18.}, ranging from the Jukes-Cantor (JC) model, which assumes that all substitutions have equal rates and equal nucleotide frequencies, to the general time-reversible (GTR) model that considers six different substitution rates and unequal nucleotide frequencies. Subsequently, we incorporated the parameters in each nucleotide model into a codon-based model ^{19., 20.}. As a result, a general formula of the substitution rate q_ij from any sense codon i to j (i ≠ j) is given for all candidate models (19):

$$q_{ij}=\{\begin{array}{ll}0\hfill & \mathrm{if}i\mathrm{and}j\mathrm{differ}\mathrm{by}\mathrm{more}\mathrm{than}\mathrm{one}\mathrm{difference}\hfill \\ k_{xy}{\pi }_j\hfill & \mathrm{if}i\mathrm{and}j\mathrm{differ}\mathrm{by}\mathrm{a}\mathrm{synonymous}\mathrm{substitution}\mathrm{of}x\mathrm{for}y\hfill \\ \omega k_{xy}{\pi }_j\hfill & \mathrm{if}i\mathrm{and}j\mathrm{differ}\mathrm{by}\mathrm{a}\mathrm{nonsynonymous}\mathrm{substitution}\mathrm{of}x\mathrm{for}y\hfill \end{array}$$

where π _j is the frequency of codon j, ω is the Ka/Ks ratio, and k_xy is the ratio of r_xy to r_CA, x, y ∊{A, C, G, T} (Table 1). For example, in the JC model, k_xy and π_j are equal to 1 owing to equal substitution rates and equal nucleotide frequencies assumed. In the Hasegawa-Kishino-Yano (HKY) model, k_TC and k_AG become equivalent to the transition/transversion rate ratio and π_j can be estimated from sequences, similar to the method reported by Goldman and Yang (19). Other models can be accommodated by making obvious modifications. Therefore, we could acquire maximum likelihood scores in various values generated from individual candidate model by implementing the codon-based models in a maximum likelihood framework ^{19., 20.}.

Table 1.

Candidate Models for Model Selection and Model Averaging in KaKs_Calculator

Model	Description (Reference)	Nucleotide frequency	Substitution rate*
JC F81	Jukes-Cantor model (9) Felsenstein’s model (10)	Equal Unequal	rTC = rAG = rTA = rCG = rTG = rCA
K2P HKY	Kimura’s two-parameter model (11) Hasegawa-Kishino-Yano model (12)	Equal Unequal	rTC = rAG ≠ rTA = rCG = rTG = rCA
TNEF TN	TN model with equal nucleotide frequencies Tamura-Nei model (13)	Equal Unequal	rTC ≠ rAG ≠ rTA = rCG = rTG = rCA
K3P K3PUF	Kimura’s three-parameter model (14) K3P model with unequal nucleotide frequencies	Equal Unequal	rTC = rAG ≠ rTA = rCG ≠ rTG = rCA
TIMEF TIM	Transition model with equal nucleotide frequencies Transition model	Equal Unequal	rTC ≠ rAG ≠ rTA = rCG ≠ rTG = rCA
TVMEF TVM	Transversion model with equal nucleotide frequencies Transversion model	Equal Unequal	rTC = rAG ≠ rTA ≠ rCG ≠ rTG ≠ rCA
SYM GTR	Symmetrical model (15) General time-reversible model (16)	Equal Unequal	rTC ≠ rAG ≠ rTA ≠ rCG ≠ rTG ≠ rCA

^*r_ij indicates the rate of substitution of i for j, where i, j ∊ {A, C, G, T}.

Model selection

AIC (21) has been widely used in model selection aside from other methods such as the likelihood ratio test (LRT) and the Bayesian information criterion (BIC) (8). AIC characterizes the Kullback-Leibler distance between a true model and an examined model, and this distance can be regarded as quantifying the information lost by approximating the true model. KaKs_Calculator uses a modification of AIC (AIC_C), which takes account of sampling size (n), maximum likelihood score (lnL_i), and the number of parameters (k_i) in model i as follows:

$${\mathrm{AIC}}_{\mathrm{C}i}={\mathrm{AIC}}_i+\frac{2k_i(k_i+1)}{n-k_i-1}=-2\ln L_i+2k_i+\frac{2k_i(k_i+1)}{n-k_i-1}$$

AIC_C is proposed to correct for small sampling size, and it approaches to AIC when sampling size comes to infinity. Consequently, we could use this equation to compute AIC_C for each candidate model and then identify a model that possesses the smallest AIC_C, which is a sign for appropriateness between models and data.

Model averaging

Model selection is merely an approximate fit to a dataset, whereas a true evolutionary model is seldom one of the candidate models (8). Therefore, an alternative way is model averaging, which assigns each candidate model a weight value and engages more than one model to estimate average parameters across models. Accordingly, we first need to compute the Akaike weight (w_i, where i = 1, 2,…, m) for each model in a set of candidate models:

$$w_i=\frac{\exp \left[-\frac{1}{2}\left({\mathrm{AIC}}_{\mathrm{C}i}-\min {\mathrm{AIC}}_{\mathrm{C}}\right)\right]}{{\displaystyle {\sum }_{j=1}^m}\exp \left[-\frac{1}{2}\left({\mathrm{AIC}}_{\mathrm{C}j}-\min {\mathrm{AIC}}_{\mathrm{C}}\right)\right]}$$

where min AIC_C is the smallest AIC_C value among candidate models. We can then estimate model-averaged parameters. Taking k_TC as an example, a model-averaged estimate can be calculated by:

$$k_{\mathrm{TC}}=\frac{{\displaystyle {\sum }_{i=1}^m}\left[w_i\times I(k_{\mathrm{TC},i})\times k_{\mathrm{TC},i}\right]}{{\displaystyle {\sum }_{i=1}^m}\left[w_i\times I(k_{\mathrm{TC},i})\right]}$$

where k_TC,i is k_TC in model i and

$$I(k_{\mathrm{TC},i})=\{\begin{array}{ll}1\hfill & \mathrm{if}{\mathrm{r}}_{\mathrm{TC}}\ne {\mathrm{r}}_{\mathrm{CA}}\mathrm{in}\mathrm{model}i\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}$$

Application

KaKs_Calculator is written in standard C++ language. It is readily compiled and run on Unix/Linux or workstation (tested on AIX/IRIX/Solaris). In addition, we use Visual C++ 6.0 for graphic user interface and provide its Windows version that can run on any IBM compatible computer under Windows operating system (tested on Windows 2000/XP). Compiled executables on AIX/IRIX/Solaris and setup application on Windows, as well as source codes, example data, instructions for installation and documentation for KaKs_Calculator is available at http://evolution.genomics.org.cn/software.htm.

Different from other existing tools ^{22., 23.}, KaKs_Calculator employs model-selected and model-averaged methods based on a set of candidate models to estimate Ka and Ks. It integrates as many features as needed from sequence data and in most cases gives rise to more reliable evolutionary information (see the comparative results on simulated sequences at http://evolution.genomics.org.cn/doc/SimulatedResults.xls) (24). KaKs_Calculator also provides comprehensive information estimated from compared sequences, including the numbers of synonymous and nonsynonymous sites and substitutions, GC contents, maximum likelihood scores, and AIC_C. Moreover, KaKs_Calculator incorporates several other methods ^{19., 25., 26., 27., 28., 29., 30., 31.} and allows users to choose one or more methods at one running time (Table 2).

Table 2.

Methods Incorporated in KaKs_Calculator

Approximate method
Method	Mutation model#1			Reference
	Step 1	Step 2	Step 3
NG	JC	JC	JC	26
LWL	JC	K2P	K2P	28
MLWL	K2P	K2P	K2P	30
LPB	—*	—*	K2P	25., 29.
MLPB	—*	—*	K2P	30
YN	HKY	HKY	HKY	27
MYN	TN	TN	TN	31
Maximum likelihood method
Method	Mutation model#2			Reference
GY	HKY			19., 20.
MS	a model that has the smallest AICC among 14 candidate models			Model-selected method proposed in this study
MA	a model that averages parameters across 14 candidate models			Model-averaged method proposed in this study

^#1The approximate method involves three basic steps: Step 1: counting the numbers of synonymous and nonsynonymous sites; Step 2: calculating the numbers of synonymous and nonsynonymous substitutions; Step 3: correcting for multiple substitutions.

^#2The maximum likelihood method uses the probability theory to finish the three steps in one go (4).

^*No specific definition of synonymous and nonsynonymous sites or substitutions.

Although there exist 203 time-reversible models of nucleotide substitution (8), model selection in practice is often limited to a subset of them (32), and thus model averaging can reduce biases arising from model selection. Therefore, model-averaged methods should be preferred for general calculations of Ka and Ks. Some planned improvements include application of model selection and model averaging to detect positive selection at single amino acid sites, which requires high-speed computing for maximum likelihood estimation, especially when an adopted model becomes complex.

In conclusion, KaKs_Calculator incorporates as many features as needed for accurately extracting evolutionary information through model selection and model averaging, therefore it may be useful for in-depth studies on phylogeny and molecular evolution.

Authors’ contributions

ZZ designed and programmed this software, and drafted the manuscript. JL carried out computer simulations to generate sequences. XQZ performed test for earlier versions of the software. JW and GKSW contributed in conceiving this software and participated in software design. JY supervised the study and revised the manuscript. All authors read and approved the final manuscript.

Competing interests

The authors have declared that no competing interests exist.