We conducted three sets of pairwise...

We conducted three sets of pairwise comparisons: human versus chimpanzee, human versus mouse, and mouse versus rat. Figure 4 shows the distributions (smoothed histograms) of posterior means and the MLEs of t and ω in those comparisons. In the human–chimpanzee comparison, the Bayesian ω estimates are slightly shifted to the right compared with the MLEs for low ω values and shifted to the left for high ω values. The mean, median, and 25% and 75% percentiles of the Bayesian estimates are 0.369, 0.320, and (0.180, 0.500) whereas those of the MLEs are 0.307, 0.193, and (0.062, 0.411) (table 3). The human and chimpanzee genes are very similar and the patterns are similar to those observed in computer simulation for low t values. Moreover, there are 377 and 2,507 gene alignments in which

= 0 and

= 0, respectively, as well as 2 and 423 alignments where

= ∞ and

= ∞, respectively. The Bayesian method does not produce any such extreme estimates. The number of genes in which the ω estimate is >1 is 1,121 for ML and 299 for the Bayesian method (table 4). The discrepancy is the result of two effects, a short evolutionary distance and a short sequence length, both indicating a lack of information and greater influence from the prior. Genes with

> 1 tend to be small (median sequence length 313 codons, compared with 454 codons for all genes). For example, one gene among those 1,121 with

> 1 has

= 1.22 (95% confidence interval—CI 0.37–4.01) and posterior mean

= 0.93 (95% credibility interval—CI 0.36–2.43). This gene has a length of 262 codons and has a small evolutionary distance with

= 0.043 (95% CI 0.024–0.077) and

= 0.047 (95% CI 0.027–0.082), so that the prior has an impact. Another gene has

= 1.27 (95% CI 0.75–2.16) and

= 1.13 (95% CI 0.60–2.13). This gene is 257 codons in length and the ML and Bayesian distance estimates are 0.17 (95% CI 0.13–0.24) and 0.18 (95% CI 0.13–0.24), respectively. The second gene has a similar length to the first but because the sequence distance is greater, the prior is much less important. In a third gene, of length 1,019 codons, the MLEs are

= 0.041 (95% CI 0.030–0.056) and

= 1.27 (95% CI 0.77–2.07), compared with the Bayesian estimates

= 0.042 (95% CI 0.031–0.057) and

= 1.13 (95% CI 0.59–2.14). In this case, the effect of the prior is unimportant, because the gene is long.

Fig. 4.

Distributions (smoothed histograms) of Bayesian and ML estimates of t and ω from mammalian and bacterial pairwise gene comparisons. Numbers of genes analyzed in each comparison are shown in the right part of the figure.

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Table 3.

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Descriptive Statistics of Bayesian (top, underlined) and ML (bottom) Estimates of t and formula from Pairwise Comparisons of Protein-Coding Genes from Mammalian Species and Bacterial Strains.

		ω							t
		Mean	SD	Quartiles			N₀	N_∞	Mean	SD	Quartiles			N₀	N_∞
	No. of Genes	Mean	SD	25%	50%	75%	N₀	N_∞	Mean	SD	25%	50%	75%	N₀	N_∞
Human–chimpanzee	14,215	0.369	0.246	0.180	0.320	0.500	0	0	0.025	0.072	0.013	0.019	0.028	0	0
Human–chimpanzee	14,215	0.307	0.418	0.062	0.193	0.411	2507	423	0.022	0.042	0.010	0.016	0.025	377	2
Human–mouse	14,624	0.130	0.125	0.044	0.093	0.176	0	0	0.812	0.574	0.503	0.691	0.958	0	0
Human–mouse	14,624	0.126	0.157	0.040	0.089	0.170	221	0	0.849	1.252	0.499	0.686	0.952	0	30
Mouse–rat	13,359	0.168	0.168	0.055	0.118	0.228	0	0	0.242	0.179	0.163	0.215	0.281	0	0
Mouse–rat	13,359	0.159	0.180	0.046	0.108	0.215	509	0	0.238	0.232	0.161	0.212	0.278	0	3
Escherichia coli K-12–E.coli O157	2,619	0.179	0.170	0.055	0.116	0.252	0	0	0.080	0.354	0.026	0.043	0.068	0	0
Escherichia coli K-12–E.coli O157	2,619	0.099	0.174	0.001	0.034	0.110	912	31	0.073	0.527	0.020	0.038	0.064	121	6
E. coli K-12–Salmonella typhimurium LT2	2,619	0.037	0.042	0.016	0.025	0.042	0	0	2.261	1.546	1.153	1.836	3.129	0	0
E. coli K-12–Salmonella typhimurium LT2	2,619	0.025	0.042	0.006	0.018	0.032	164	0	5.052	8.481	1.087	1.748	4.066	0	217

		ω							t
		Mean	SD	Quartiles			N₀	N_∞	Mean	SD	Quartiles			N₀	N_∞
	No. of Genes	Mean	SD	25%	50%	75%	N₀	N_∞	Mean	SD	25%	50%	75%	N₀	N_∞
Human–chimpanzee	14,215	0.369	0.246	0.180	0.320	0.500	0	0	0.025	0.072	0.013	0.019	0.028	0	0
Human–chimpanzee	14,215	0.307	0.418	0.062	0.193	0.411	2507	423	0.022	0.042	0.010	0.016	0.025	377	2
Human–mouse	14,624	0.130	0.125	0.044	0.093	0.176	0	0	0.812	0.574	0.503	0.691	0.958	0	0
Human–mouse	14,624	0.126	0.157	0.040	0.089	0.170	221	0	0.849	1.252	0.499	0.686	0.952	0	30
Mouse–rat	13,359	0.168	0.168	0.055	0.118	0.228	0	0	0.242	0.179	0.163	0.215	0.281	0	0
Mouse–rat	13,359	0.159	0.180	0.046	0.108	0.215	509	0	0.238	0.232	0.161	0.212	0.278	0	3
Escherichia coli K-12–E.coli O157	2,619	0.179	0.170	0.055	0.116	0.252	0	0	0.080	0.354	0.026	0.043	0.068	0	0
Escherichia coli K-12–E.coli O157	2,619	0.099	0.174	0.001	0.034	0.110	912	31	0.073	0.527	0.020	0.038	0.064	121	6
E. coli K-12–Salmonella typhimurium LT2	2,619	0.037	0.042	0.016	0.025	0.042	0	0	2.261	1.546	1.153	1.836	3.129	0	0
E. coli K-12–Salmonella typhimurium LT2	2,619	0.025	0.042	0.006	0.018	0.032	164	0	5.052	8.481	1.087	1.748	4.066	0	217

Note.—The F61 model is used for codon frequencies. Results for ML have been calculated after removing the infinite estimates. N₀ is the number of genes with the MLE formula or formula = 0, whereas N_∞ is the number of genes with the MLE formula or formula = ∞.

Table 3.

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Descriptive Statistics of Bayesian (top, underlined) and ML (bottom) Estimates of t and formula from Pairwise Comparisons of Protein-Coding Genes from Mammalian Species and Bacterial Strains.

		ω							t
		Mean	SD	Quartiles			N₀	N_∞	Mean	SD	Quartiles			N₀	N_∞
	No. of Genes	Mean	SD	25%	50%	75%	N₀	N_∞	Mean	SD	25%	50%	75%	N₀	N_∞
Human–chimpanzee	14,215	0.369	0.246	0.180	0.320	0.500	0	0	0.025	0.072	0.013	0.019	0.028	0	0
Human–chimpanzee	14,215	0.307	0.418	0.062	0.193	0.411	2507	423	0.022	0.042	0.010	0.016	0.025	377	2
Human–mouse	14,624	0.130	0.125	0.044	0.093	0.176	0	0	0.812	0.574	0.503	0.691	0.958	0	0
Human–mouse	14,624	0.126	0.157	0.040	0.089	0.170	221	0	0.849	1.252	0.499	0.686	0.952	0	30
Mouse–rat	13,359	0.168	0.168	0.055	0.118	0.228	0	0	0.242	0.179	0.163	0.215	0.281	0	0
Mouse–rat	13,359	0.159	0.180	0.046	0.108	0.215	509	0	0.238	0.232	0.161	0.212	0.278	0	3
Escherichia coli K-12–E.coli O157	2,619	0.179	0.170	0.055	0.116	0.252	0	0	0.080	0.354	0.026	0.043	0.068	0	0
Escherichia coli K-12–E.coli O157	2,619	0.099	0.174	0.001	0.034	0.110	912	31	0.073	0.527	0.020	0.038	0.064	121	6
E. coli K-12–Salmonella typhimurium LT2	2,619	0.037	0.042	0.016	0.025	0.042	0	0	2.261	1.546	1.153	1.836	3.129	0	0
E. coli K-12–Salmonella typhimurium LT2	2,619	0.025	0.042	0.006	0.018	0.032	164	0	5.052	8.481	1.087	1.748	4.066	0	217

		ω							t
		Mean	SD	Quartiles			N₀	N_∞	Mean	SD	Quartiles			N₀	N_∞
	No. of Genes	Mean	SD	25%	50%	75%	N₀	N_∞	Mean	SD	25%	50%	75%	N₀	N_∞
Human–chimpanzee	14,215	0.369	0.246	0.180	0.320	0.500	0	0	0.025	0.072	0.013	0.019	0.028	0	0
Human–chimpanzee	14,215	0.307	0.418	0.062	0.193	0.411	2507	423	0.022	0.042	0.010	0.016	0.025	377	2
Human–mouse	14,624	0.130	0.125	0.044	0.093	0.176	0	0	0.812	0.574	0.503	0.691	0.958	0	0
Human–mouse	14,624	0.126	0.157	0.040	0.089	0.170	221	0	0.849	1.252	0.499	0.686	0.952	0	30
Mouse–rat	13,359	0.168	0.168	0.055	0.118	0.228	0	0	0.242	0.179	0.163	0.215	0.281	0	0
Mouse–rat	13,359	0.159	0.180	0.046	0.108	0.215	509	0	0.238	0.232	0.161	0.212	0.278	0	3
Escherichia coli K-12–E.coli O157	2,619	0.179	0.170	0.055	0.116	0.252	0	0	0.080	0.354	0.026	0.043	0.068	0	0
Escherichia coli K-12–E.coli O157	2,619	0.099	0.174	0.001	0.034	0.110	912	31	0.073	0.527	0.020	0.038	0.064	121	6
E. coli K-12–Salmonella typhimurium LT2	2,619	0.037	0.042	0.016	0.025	0.042	0	0	2.261	1.546	1.153	1.836	3.129	0	0
E. coli K-12–Salmonella typhimurium LT2	2,619	0.025	0.042	0.006	0.018	0.032	164	0	5.052	8.481	1.087	1.748	4.066	0	217

Table 4.

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The Numbers of Genes with formula Estimate Greater or Less than 1 Using the Bayesian and ML Methods.

Data			Bayesian
Data			< 1	> 1	N_L
Human–chimpanzee		< 1	13,094	0	78
		> 1	822	299
		N_B		3
Human–mouse		< 1	14,617	0	2
		> 1	1	6
		N_B		2
Mouse–rat	ML	< 1	13,313	0	5
		> 1	10	36
		N_B		2
Escherichia coli K-12–E. Coli O157		< 1	2,574	0	0
		> 1	43	2
		N_B		0
E. coli K-12–Salmonella typhimurium LT2		< 1	2,617	0	0
		> 1	2	0
		N_B		0

Data			Bayesian
Data			< 1	> 1	N_L
Human–chimpanzee		< 1	13,094	0	78
		> 1	822	299
		N_B		3
Human–mouse		< 1	14,617	0	2
		> 1	1	6
		N_B		2
Mouse–rat	ML	< 1	13,313	0	5
		> 1	10	36
		N_B		2
Escherichia coli K-12–E. Coli O157		< 1	2,574	0	0
		> 1	43	2
		N_B		0
E. coli K-12–Salmonella typhimurium LT2		< 1	2,617	0	0
		> 1	2	0
		N_B		0

Note.—N_L is the number of genes with statistically significant formula based on the LRT at the 5% level (one-sided with critical value 2.71) in the likelihood method, whereas N_B is the number of genes with P(ω > 1 | x) > 0.95 in the Bayesian analysis.

Table 4.

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The Numbers of Genes with formula Estimate Greater or Less than 1 Using the Bayesian and ML Methods.

Data			Bayesian
Data			< 1	> 1	N_L
Human–chimpanzee		< 1	13,094	0	78
		> 1	822	299
		N_B		3
Human–mouse		< 1	14,617	0	2
		> 1	1	6
		N_B		2
Mouse–rat	ML	< 1	13,313	0	5
		> 1	10	36
		N_B		2
Escherichia coli K-12–E. Coli O157		< 1	2,574	0	0
		> 1	43	2
		N_B		0
E. coli K-12–Salmonella typhimurium LT2		< 1	2,617	0	0
		> 1	2	0
		N_B		0

Data			Bayesian
Data			< 1	> 1	N_L
Human–chimpanzee		< 1	13,094	0	78
		> 1	822	299
		N_B		3
Human–mouse		< 1	14,617	0	2
		> 1	1	6
		N_B		2
Mouse–rat	ML	< 1	13,313	0	5
		> 1	10	36
		N_B		2
Escherichia coli K-12–E. Coli O157		< 1	2,574	0	0
		> 1	43	2
		N_B		0
E. coli K-12–Salmonella typhimurium LT2		< 1	2,617	0	0
		> 1	2	0
		N_B		0

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