Computer simulation is a routine approach for evaluating computational procedures of different algorithms. In molecular phylogeny, one major approach for simulating DNA sequence evolution is to generate an ancestral sequence for the root of a tree and "evolve" it along the tree building process according to substitution models, branch lengths, and substitution parameters 1132333435. This approach can be implemented in the evolver program in the PAML (Phylogenetic Analysis by Maximum Likelihood 36) package, which usually uses nucleotide or amino acid sequence data to simulate evolving protein-coding sequences. To assess the advantages of γ-MYN in comparison with YN and MYN, we generated three groups of simulated sequences with the PAML package: (1) equal codon frequencies, (2) human frequencies (based on human protein-coding genes from the ENSEMBL database) 37 and (3) rice frequencies (based on rice protein-coding genes) 38. We also generated 2,000 sequence pairs with 1,200 bp in length for examining the effect of different parameters.

In general, a better method should have relatively minimal deviations from real values with near infinite amount of data and within a reasonable range of all relevant parameters. In reality, we have to define both data in a limited way and parameter ranges within reasonable boundaries. In this exercise, we use ω = 0.3, 1, and 3 to represent negative, neutral, and positive mutations, respectively 3, and fix parameter t to 0.6 for initial assessment. Since genuine values for κ often range from 1.5 to 5, we always fix κ = 3.75 as typical. Considering that γ-MYN differentiates κ_Y from κ_R, we always fix one of them to 3.75 and allow the other varying from 1 to 10. We then analyze ω among data generated with YN, MYN, and γ-MYN against κ_R (fixing κ_Y = 3.75), using the three codon frequencies under different selective pressures (Figure 1A–I). We observed that γ-MYN produces less deviated ω from the standard data under negative selection as we perform analyses for different species. Although γ-MYN performs in a very similar way as MYN does, it is obviously better than YN under either positive or neutral selections. Since biased codon frequencies often have opposite effects as compared to the bias of transition/transversion rate ratio, ignoring codon frequency bias can lead to an overestimation of ω 13. Using empirical data from human and rice, which represent distinct codon usages, we also did not detect any effect among different codon frequencies (Figure 1A–I). Since most of the evolutionary studies tend to calculate evolutionary rates between closely-related species, future research should focus more on the effect of different parameters and the improvement of calculations under negative selection.

MYN method assumes that different sites in a sequence evolve in the same way and at the same rate. It is obvious that such an assumption does not happen in the real world for most proteins and their genes. For instance, mutation rates are not the same in nuclear and organellar genomes among different species 39. In addition, sequence variations among portions or domains of proteins mutate differently from a fixed mutation rate due to their specific structural and functional constraints for different genes under different selective pressures. Therefore, we introduced a parameter α in MYN method so that each substitution rate across sites is assumed to follow γ-distribution.

Since α is an unknown random variable and its variations may lead to changes of probability density of γ-distribution as well as deviations of γ-MYN method, we chose different parameters to force it to deviate from real values under different selective pressures (Figure 2). For a qualitative survey, the order of estimated values of ω, in the cases of κ_R = 1, 2, and 3, is: YN <γ-MYN < MYN; the order of estimated values of ω for the rest cases, κ_R = 4, 5, 6, 7, 8, 9, and 10, is: γ-MYN<MYN <YN. Furthermore, we observed that estimated ω do not change much as α varies when expected ω = 1 or 3, and γ-MYN again performs better when ω = 0.3 than it does when ω = 1 or 3. Because most calculated ω values indicate negative selection, and variation of α has stronger influence under negative selection, we analyzed the variation of ω in a range of 0.1 to 0.9 to evaluate the effect of α on ω. We obtained different optimal α values when ω varies from 0.1 to 0.9, and plotted different γ distribution densities (Figure 3). Each curve appears reaching its maximum and goes down with an increasing substitution rate. The peaks of the curves shift to the left and become lower in density when optimal α values decrease from 4.8 to 1.5; the decrease is attributable to the increase of ω (selective pressure) from 0.1 to 0.9. Furthermore, selective pressure shows significant effects on α, with an increase in the probability at the lowest and highest substitution rates across all sites. Because γ-MYN produces less biases than both YN and MYN do when ω varies from 0.1 to 0.9 under α = 4 (data not shown), we chose α = 4 as a typical value for our analyses.

The parameter t represents divergence time between two sequences. To test the effect of t on our method, we use human codon frequency (2,000 pairs of sequences with 400 codons for each case), and vary t from 0.1 to 1. Since γ-MYN does not change much in comparison with MYN under positive selection and neutral selection, we only consider the three obvious conditions of negative selection when κ_R = 10 and κ_Y = 1 are fixed: ω = 0.2, 0.3 and 0.4 (Figure 4). Although YN, MYN, and γ-MYN all have a nearly identical overall trend when t varies from 0.1 to 1, and they all tend to overestimate ω for negative selection, γ-MYN deviates less from the expected values. Despite the fact that γ-MYN also overestimates ω and the overestimation becomes less obvious as t increases, while the overestimation of both YN and MYN becomes severer.

We used the same data (2,000 pairs of human codon sequences with 400 codons for each case) and methods to test the effects of κ_R and κ_Y. We plotted the average estimates of ω from YN, MYN, and γ-MYN methods against κ_Y = κ_R for the parameter combinations: the expected ω values vary as 0.2 and 0.3 when α = 4 (Figure 5). While the curves produce from YN and MYN methods superimpose each other when κ_R (=κ_Y) varies from 1 to 10, γ-MYN deviates clearly less from the expected ω. We found that γ-MYN still performs better than the other two methods, whereas MYN is degraded to YN when κ_Y is equal to κ_R. The result suggests that the assumption of variable substitution rates among different sites is necessary to Ka and Ks calculations.

Usually the effect of S% (the fraction of synonymous sites in a sequence) is considered as a factor of method evaluation. Changes in ω in relation to S% are often evaluated based on the effect of S% on the deviation of Ka and Ks. Therefore, an overestimated S% may give rise to underestimation of Ks and overestimation of Ka, resulting in overestimation of ω. Likewise, underestimation of S% may also lead to overestimation of Ks and underestimation of ω. It has been reported that S% has enormous influence on Ka and Ks under negative selection but has neglectable effect under positive selection 40. We used human sequences to examine the effect of S% on γ-MYN (Table 2), fixing κ_Y to 3.75. As κ_R increases, the value of S% generated from our method exhibits minor fluctuations under different negative selections, when compared to that from YN. But the difference between γ-MYN and MYN is minute under this condition. In more details, the order of estimated values of S%, in the cases of κ_R = 1, 2, and 3, is: YN < MYN <γ-MYN; in the rest cases, the order of estimated values of S%, when κ_R = 5, 6, 7, 8, 9 and 10, is: MYN <γ-MYN < YN. We did not observe any obvious trend under the condition of κ_R = 4. Therefore, γ-MYN is deemed insensitive to S% changes.

The length of homologous genes subjected to an analysis usually varies in actual calculation. In order to evaluate the effect of variable sequence lengths, we use two groups of simulated rice sequences under the conditions of (1) ω = 0.2, κ_R = 10, κ_Y = 1, t = 0.6, and α = 4; and (2) ω = 0.3, κ_R = 10, κ_Y = 1, t = 0.6, and α = 4. We then calculate the average estimated ω when the number of codons varied from 100 to 1,000 (Table 3). It appears that all three methods overestimate ω regardless the number of codons in the datasets. In particular, despite the fact that all three methods give rise greater biases for shorter sequences (<300 codons), γ-MYN performs better than the other two methods. We also found that the performance of γ-MYN is getting better faster than the other two methods as the number of codon increases.

We used three ortholog datasets for the test, 14,323 from human-dog, 16,066 from human-mouse, and 12,351 from human-chimp. For a more comprehensive display, we examined the cumulative percentage of κ_R-κ_Y (Figure 6), showing different transitional substitutions with unequal frequencies. For example, the cumulative percentages for κ_R - κ_Y > 0.4 for human-dog, human-mouse and human-chimp orthologs are 52.27%, 52.66%, and 24.47% and those for κ_R - κ_Y < -0.4 are 25.36%, 24.31%, and 21.87%, respectively. In the rest cases, for |κ_R - κ_Y| ≤ 0.4, they are 22.37%, 23.02%, and 53.66% for the three ortholog groups. We found that the value for human-chimp is more than twice as much as that of human-dog (or human-mouse), and the reasons are attributable to a close evolutionary relationship between human and chimpanzee 41.

To evaluate the performance of γ-MYN, we compared a set of values for several key parameters (S%, Ka, Ks, and ω) generated with γ-MYN and three other selected methods in a straightforward way, considering three cases of κ_R - κ_Y > 0.4, κ_R - κ_Y < -0.4, and |κ_R - κ_Y| ≤ 0.4 (Table 4). We chose the value of 0.4 as a threshold so that the three cases can stand for three groups of κ_R under the condition of κ_Y = 3.75: (1) κ_R = 5, 6, 7, 8, 9 and 10; (2) κ_R = 1, 2, and 3; (3) κ_R = 4. Other than YN and MYN, we also used a maximum likelihood method proposed by Goldman and Yang (denoted as GY) 33.

The results showed a few interesting trends. First, GY performs in a similar way as YN does as compared to MYN and γ-MYN; it is consistent with our previous simulation results, as they share a common consideration of transition/transversion rate bias and nucleotide frequencies bias 1242. Second, the trends of ω estimates with the three methods, YN, MYN and γ-MYN, are consistent with our simulation results. In the cases of |κ_R - κ_Y| ≤ 0.4 and κ_R - κ_Y > 0.4, when κ_R = 4, 5, 6, 7, 8, 9, and 10, the order of estimated values of ω is: γ-MYN < MYN < YN. When confined to κ_R - κ_Y < -0.4, when κ_R = 1, 2, and 3, YN underestimates ω and MYN overestimates ω as compared to γ-MYN. Taking ω estimates as an example, they are 0.1695, 0.1800, and 0.1713 for human-dog orthologs, 0.1465, 0.1566, and 0.1468 for human-mouse orthologs, and 0.5790, 0.6312 and 0.6309 for human-chimp orthologs, calculated with YN, MYN, and γ-MYN, respectively. These findings are in agreement with our simulation studies. Third, the orders of S% estimates with the three methods (YN, MYN and γ-MYN) are also consistent with our simulation results. For example, when κ_R - κ_Y > 0.4, κ_R = 5, 6, 7, 8, 9 and 10, YN overestimates S% and MYN underestimates S% as compared to γ-MYN. In the case of κ_R - κ_Y < -0.4, when κ_R = 1, 2, and 3, the order of estimated values of S% is: YN < MYN <γ-MYN. Fourth, we took one gene as an example to show the outperformance of our new method over others. Among the human-chimp orthologs, ω values of an immunoglobulin interleukin-1-related receptor (NP_068577) are listed as 1.02406, 0.622611, 0.59999, and 0.843755, when γ-MYN, MYN, YN, and GY are used for the calculation, respectively. Obviously, only γ-MYN is able to detect positive selection for this gene, and others failed. This gene has been studied previously based on a population genetics analysis of extended-haplotype-homozygosity in Northeast Asians, and a possible positive selection scheme was proposed for it 43. This result is in accordance with the result of large-scale scanning on positively selected genes between human and chimpanzee genomes 4445.

A C++ program implementing γ-MYN method is included in the updated KaKs_Calculator 42, available upon request. And we tested the running time with YN, MYN, γ-MYN, and GY, using the three testing datasets (14,725 human-dog, 16,368 human-mouse, and 15,646 human-chimp gene pairs). Table 5 shows the time consumption for each method to compute Ka/Ks ratios from the three datasets and their average running time. On average, γ-MYN takes 600 folds less time than GY does, and YN, MYN, and γ-MYN perform similarly in time consumption. We believe that γ-MYN may become a useful tool for large-scale studies, when ML-based methods (such as GY) are deemed time-consuming.

A major limitation of Ka/Ks methods, mentioned in literatures, is their poor ability for detecting positive selection (adaptive selection) 646474849505152. To detect positive selection at sites requires that ω value averaged over all branches is >1 and to detect positive selection along lineages requires ω value averaged over all sites is >1 6. Therefore, Ka/Ks methods are only useful to weight average selection pressure over sites and branches. They may not be able to detect positive selection for some highly conserved proteins that are mostly invariable but become fragile when a single site alters. Other detrimental cases include transmembrane domains where high variability may not change its physiochemical property. To overcome the weakness, there have been methods developed, such as Likelihood Ratio Test (LRT) 53545556 implemented in PAML 57 and Hyphy software 58, to identify positive selection 45596061, which tend to be qualitative. An obvious pitfall of these methods is that they do not weigh the relative degree of two genes under negative selection. Ka/Ks methods can perform better in this regard 38626364 as they tend to be more quantative. In addition, Ka/Ks methods can be readily extended from our current work to detect the sequence alternations that lead to protein structure changes and positive selection, in combination with other techniques, such as ancestral sequence reconstruction 656667 and primary 6869 or tertiary windowing 7071. Therefore, we believe that the two different lines of methods (LRT-like methods and Ka/Ks methods) should also be useful under appropriate conditions.

With the introduction of the parameter α, our method γ-MYN shows significant improvements when compared with the other two related methods in both simulation and tests on real data. As γ-MYN assumes that evolutionary rate at each site follows γ-distribution, we found that the parameter α has observable effects under different evolution rates. For instance, when ω >= 1, γ-MYN remains stable. We also observed that selective pressure can overwhelm the variable substitution rates across sites and it becomes the most influential factor when increasing dramatically. Therefore, when we consider strong positive selection and neutral evolution, the effect of variable substitution rates across sites can be somewhat neglected. In addition, more parameters often lead to increase of complexity of an algorithm, resulting in the decrease of efficiency. However, we hold the view that proper introduction of parameters is worthwhile.

It has been noticed that the majority of the evolutionary selections are actually negative in nature, and the statement is confirmed by our analyses on real data. When ω varies from 0.1 to 1, we selected optimal α to minimize biases and found that γ-MYN is very sensitive when α changes under negative selection. Furthermore, the optimal α becomes smaller when ω becomes larger under negative selection, and the effects of various substitution rates across sites become evident under negative selection, emphasizing the importance of our method in the calculation under negative selection.

As to γ-Tamura-Nei model, it usually leads to higher variations, especially in phylogeny analyses. However, we found some multi-level observations in both simulation and real data testing (see in additional file 1). The difference of variations of ω between YN and γ-MYN (or MYN) seems to be correlated to the value of κ_R - κ_Y, when results in Tables S2-S6 were examined together. For instance, when κ_R-κ_Y < 0, ω values vary more when γ-MYN is used than YN is. As κ_R-κ_Y increases, the ω variation values from γ-MYN decrease, leading to lower numbers than those of YN. The variations of ω calculated with γ-MYN are slight higher than those yielded from MYN in most cases but not all. These results reflect the distinction between the usage of γ-Tamura-Nei model (or Tamura-Nei model) in KaKs computation and that of phylogeny reconstruction.

We begin our discussion with how α parameter in γ-MYN improves MYN method. It is known that ignoring rate variation among sites leads to underestimation of both the sequence distance and the transition/transversion rate ratio κ (both κ_R and κ_Y) 4. κ is used not only in estimating S and N but also in generating a transition probability matrix for estimating S_d and N_d. If we derive an approximate formula for ω = Ka/Ks ≈ (N_d/N)/(S_d/S) (the symbol of "≈" is used to emphasize the absence of correction for multiple substitutions), ω is composed of two parts: N_d/N and S_d/S. For purifying selection, synonymous substitutions occur more frequently than nonsynonymous ones so we should only focus on Ks (S_d/S). Since κ decrease is related to the reduction of substitution rate between two codons, underestimation of κ leads to underestimation of S_d. In addition, nucleotide transitions between two codons are more likely to be synonymous especially at the third codon positions, underestimation of κ leads to underestimation of S. However, the influence of κ on S_d is significantly stronger than that on S. As a consequence, an underestimated κ, when used in MYN, may give rise to underestimation of S_d/S, resulting in overestimation of ω as compared with γ-MYN. Our theoretical deductions are consistent with both simulation (Figure 2) and real data (Table 4).

In addition, we found the optimal α values fall between 1 and 5 (Figure 3). In these cases, the distribution of gamma values is bell-shaped, meaning that most sites have intermediate rates around 1 whereas a few sites have either very low or very high rates 4. When selection pressure increases, the number of sites of intermediate rates decreases (Figure 3). In particular, when α approaches infinity, the distribution diminishes into the model of a single rate for all sites, which is used in MYN method. If α ≤ 1, the distribution has a highly skewed L-shape, suggesting that most sites have either very low rates of substitution or are nearly "invariable" with possible substitution hotspots. Furthermore, estimates of α from real data in many species over multiple sequences show increases from 0.26 to 3.0, and this relatively wide window allows us to explore the spectrum of different substitution rates over different sites 4

Divergence time (t) is another parameter important for the estimation of Ka and Ks. When divergence time reaches the extremes, the compared sequences among genes often vary considerably and their corresponding protein structures may changed over greater evolutionary time scale. Therefore, under such conditions it may become meaningless to calculate the substitution rates of such genes. However, most methods for calculating Ka and Ks use homologous genes for estimating substitution rates among closely-related species or within close lineages, and observable selections are mostly negative. Since our method γ-MYN has better performance than other methods when ω < 1, it provides a useful alternative for more comprehensive Ka and Ks calculations.

Our past work has testified that methods for estimating Ka and Ks should be used cautiously and one should not draw simple conclusions on gene evolution from Ka and Ks analyses based on a single method. Therefore, we recommend a method based on model selection and model averaging 1242, and γ-MYN has just brought a new choice into such endeavours. Our method does not challenge other methods such as GY (Goldman-Yang) method, a typical maximum likelihood (ML) that has been always considered to be the method of choice 1333. It has been suggested in the literature that GY and YN both give rise to similar estimates on Ka and Ks primarily due to the fact that they both take account of major dynamic features of DNA sequence evolution, including transition/transversion rate and nucleotide/codon frequency biases 1242. As γ-MYN performs better than other methods under certain conditions, despite the fact that its advantages seemed less obvious under other conditions, we believe that γ-MYN may become a useful tool for large-scale sequence analysis when ML-based methods are deemed time-consuming.

In Markov-chain models of codon substitution, the codon triplet is considered as the unit of evolution, and a Markov chain is used to describe substitutions from one codon to another codon 4. In detail, the state space of the chain is the sense codons with regard to the canonical genetic code. Stop codons are not allowed inside a functional protein and are not considered. Although there are several mutation (substitution) models that take different sequence variation features into account, in this report we limit our discussions to the Tamura-Nei Models 24 (see Table S1 in additional file 2 for details).

γ-MYN also needs a transition probability matrix similar to YN and MYN. We assigned the substitution rate q_ij from any codon i to j (i ≠ j) to generate a transition probability matrix as follows:

The diagonal elements of the transition probability matrix, Q = {q_ij}, are determined based on the mathematical requirement that the row sums equal to zero. The matrix is normalized with the result that the sum over non-diagonal terms is 1.

To generate the transition probability matrix, we need to estimate κ_R and κ_Y. Similar to YN and MYN, we calculated four nucleotide frequencies (g_T, g_C, g_A, g_G), proportions of transitional differences between purines (T_R), and between pyrimidines (T_Y), and the proportion of transversional differences (V) from compared sequences:

where g_R = g_A + g_G and g_Y = g_T + g_C. Note that α is the square of the inverse of the variation coefficient in the gamma function.

We then used equation 3 to estimate κ_R and κ_Y.

The detailed procedures for deducing κ_R and κ_Y were summarized in additional file 2. We also made other modifications accordingly, such as using κ_R and κ_Y to estimate S and N, generating relevant transition probability matrix (Equation 1), considering different transitional evolution pathways to count S_d and N_d, and correcting for multiple substitutions when estimating Ka and Ks (Equation 4; 24).

When compared to MYN and YN, our γ-MYN method considers that the rate of nucleotide substitution λ approximately follows γ-distribution. In fact, if the rate of nucleotide substitution λ is the same for all sites considered, the model becomes the model that is used in MYN.

γ-MYN uses an iterative approach to estimate Ka and Ks. Before iteration, γ-MYN computes nucleotide frequencies (regarding to the three codon positions), κ_R, κ_Y, S, and N from the sequences to be analyzed. Based on the F3 × 4 model 36, codon frequencies are calculated by multiplying each nucleotide frequencies. κ_R and κ_Y are estimated from four-fold degenerate sites at the third codon position and non-degenerate sites. S and N are calculated by using κ_R, κ_Y, and codon frequencies. γ-MYN chooses initial values for t and ω as starting point for iteration. It generates a transition probability matrix that represents substitution probabilities from one codon to another by using ω, t, κ, and codon frequencies. This transition probability matrix is subsequently used to deduce S_d and N_d and for new estimates of ω and t. γ-MYN repeats the calculation for another transition probability matrix, until the algorithm converges.

We used simulated sequences generated from hypothetical common ancestral sequences for our comparative analysis by randomly choosing codons (61 excluding stop codons) from the ancestral sequences according to codon frequencies that were derived from three empirical datasets: (1) equal codon frequencies 13, (2) human codon frequencies deduced from 39,420 human protein-coding genes from ENSEMBL database (Release 35; 37) and (3) rice codon frequencies deduced from 19,079 rice protein-coding genes 38 (see in additional file 1).

In addition to codon frequencies, we also have to fix or choose ranges of other parameters for the simulation, including sequence length, divergence time (t), two ratios of transitional rate between purines (κ_R) and between pyrimidines (κ_Y) to transversional rate, and selective pressure ω. Although ω varies from gene to gene, ω = 1 and 3 can be regarded as "typical values" for neutral mutation and positive selection, respectively 31367, which are observable from real datasets. Since most calculated ω values indicate negative selection and variation of parameter α has stronger influence under negative selection, we analyzed the variation of ω in a range of 0.1 to 0.9 for the evaluation of effects of α on ω. To accurately examine the effect of one parameter and to avoid stochastic errors arising from other factors, we generated 2,000 pairs of sequences. Three orthologous gene sets were downloaded from NCBI's HomoloGene database (Build 61), which contained 14,725 human-dog, 16,368 human-mouse, and 15,646 human-chimp gene pairs 72. We considered "NA" occurrence (in any of Ka, Ks or ω) as unreliable data and filtered the orthologous pairs (extremes in sequence homology) that have such labels, and 14,323 human-dog, 16,066 human-mouse, and 12,351 human-chimp gene pairs were remained. The datasets were used for comparing γ-MYN with other methods.

Kateryna Makova (assisted by Mr. Chungoo Park), Center for Comparative Genomics and Bioinformatics, Department of Biology, 305 Wartik Lab, Penn State University, University Park, PA, 16802

In this paper, the authors suggest a new algorithm, called γ-MYN, to accurately estimate nonsynonymous and synonymous substitution rates (Ka and Ks) of protein-coding DNA.

The γ-MYN considers the variation of substitution rates among different sites in a sequence, which is overlooked by existing methods, and shows that their unequal substitution rates affect Ka and Ks.

David A. Liberles, Department of Molecular Biology, University of Wyoming, Laramie WY 82071, USA

"γ-MYN: A new algorithm for estimating Ka and Ks with consideration of variable substitution rates" by Wang, Wan, Zhang, and Yu describes a more model rich version of the original Yang and Nielsen 2000 model 13. Previous work added parameters to differentiate between purine transitions and pyrimidine transitions 14. The current work adds a gamma distribution on top of the previously described work.

The stepwise addition of parameters to the Yang and Nielsen approach reflects an attempt to add increasing layers of biological realism. Differentiating between purine and pyrimidine transitions is driven by potential underlying forces like codon bias to the extent that it is correlated across codons in a gene and the chemistry (specificity) of DNA damaging agents, DNA polymerase, and DNA repair enzymes (see 73 for example). The biological link to the gamma distribution is somewhat less clear in the way that it has been applied. Nucleotide sequences as well as amino acid sequences typically show support for a gamma distribution characterizing rates across sites. At the nucleotide level, this is typically related to two components: differences in substitution rate in the different codon positions due to the nature of the genetic code as well as amino acid level constraint on the protein. The former category is already modeled with ω, potentially creating some degree of redundancy between the ω and α parameters. Modeling α at the amino acid level (translated codons) would not suffer from the redundancy and likely accounts for the improvement in performance by γ-MYN.

Zhaolei Zhang, Banting & Best Dept. of Medical Research (BBDMR), Department of Molecular Genetics, University of Toronto, 160 College St., Room 608, Donnelly CCBR Building, Toronto, ON M5S 3E1, Canada

"γ (gamma)-MYN: A new algorithm for estimating Ka and Ks with consideration of variable substitution rates"

Authors: Da-Peng Wang, Hao-Lei Wan, Song Zhang, and Jun Yu.

Shamil Sunyaev, Harvard Medical School, Boston, MA, United States

This manuscript presents a new method to compute Ka and Ks. The authors incorporated Tamura-Nei model into the Yang-Nielsen approach. This extension of the method would be of interest to experts in molecular evolution. I have a few comments and suggestions.

1) The manuscript would benefit from a much clearer justification of the method and discussion of its applicability. Tamura-Nei model was developed for the control region of the mitochondrial genome. Is the model with the uniform selective constraint (reflected by parameter omega) and raw mutation rates following gamma distribution realistic for nuclear protein coding genes? The dominant source of mutation rate variation in mammalian genes (used by the authors for testing the method) is likely to be the context-dependency, predominantly due to CpG contexts. Is the gamma distribution model capable of capturing this variation? Different rates for transitions between purines and pyrimidines imply strong strand bias. Is this a realistic assumption in nuclear genes and does it justify incorporation of additional parameters? Also, Tamura-Nei distance is known to have higher variance. How is this reflected in the performance of the method? I would suggest discussing these issues in the introduction and discussion sections.