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Open Access Highly Accessed Methodology article

Codon Deviation Coefficient: a novel measure for estimating codon usage bias and its statistical significance

Zhang Zhang12, Jun Li3, Peng Cui14, Feng Ding14, Ang Li12, Jeffrey P Townsend56 and Jun Yu7*

Author Affiliations

1 Computational Bioscience Research Center (CBRC), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Kingdom of Saudi Arabia

2 Current address: CAS Key Laboratory of Genome Sciences and Information, Beijing Institute of Genomics, Chinese Academy of Sciences, Beijing 100029, China

3 School of Biological Sciences, The University of Hong Kong, Hong Kong, China

4 Current address: Department of Pharmacology and Toxicology and the Cancer Center, Medical College of Wisconsin, Milwaukee, Wisconsin 53226, USA

5 Department of Ecology and Evolutionary Biology, Yale University, New Haven, Connecticut 06520, USA

6 Program in Computational Biology and Bioinformatics, Yale University, New Haven, Connecticut 06520, USA

7 CAS Key Laboratory of Genome Sciences and Information, Beijing Institute of Genomics, Chinese Academy of Sciences, Beijing 100029, China

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BMC Bioinformatics 2012, 13:43  doi:10.1186/1471-2105-13-43

The electronic version of this article is the complete one and can be found online at: http://www.biomedcentral.com/1471-2105/13/43


Received:29 November 2011
Accepted:22 March 2012
Published:22 March 2012

© 2012 Zhang et al; licensee BioMed Central Ltd.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Background

Genetic mutation, selective pressure for translational efficiency and accuracy, level of gene expression, and protein function through natural selection are all believed to lead to codon usage bias (CUB). Therefore, informative measurement of CUB is of fundamental importance to making inferences regarding gene function and genome evolution. However, extant measures of CUB have not fully accounted for the quantitative effect of background nucleotide composition and have not statistically evaluated the significance of CUB in sequence analysis.

Results

Here we propose a novel measure--Codon Deviation Coefficient (CDC)--that provides an informative measurement of CUB and its statistical significance without requiring any prior knowledge. Unlike previous measures, CDC estimates CUB by accounting for background nucleotide compositions tailored to codon positions and adopts the bootstrapping to assess the statistical significance of CUB for any given sequence. We evaluate CDC by examining its effectiveness on simulated sequences and empirical data and show that CDC outperforms extant measures by achieving a more informative estimation of CUB and its statistical significance.

Conclusions

As validated by both simulated and empirical data, CDC provides a highly informative quantification of CUB and its statistical significance, useful for determining comparative magnitudes and patterns of biased codon usage for genes or genomes with diverse sequence compositions.

Keywords:
Codon deviation coefficient; CDC; Codon usage bias; CUB; Statistical significance; Background nucleotide composition; GC content; Purine content; Bootstrapping

Background

Codon usage bias or CUB, a phenomenon in which synonymous codons (that encode the same amino acid) are used at different frequencies, is generally believed to be a combined outcome of mutation pressure, natural selection, and genetic drift [1-5]. Within any given species, genes often exhibit variable degrees of CUBs. Moreover, CUB for an individual gene is related closely with gene expression for translational efficiency and/or accuracy [6-10]. Therefore, the ability to accurately quantify CUBs for protein-coding sequences is of fundamental importance in revealing the underlying mechanisms behind codon usage and understanding gene evolution and function in general.

Over the past few years, a number of measures have been proposed for the quantification of CUB [11-23], leading to investigations on the pattern of CUBs within and across species [24-30]. Since CUB is primarily shaped by selection and mutation [5], different measures are differentially informative with regard to differentiating causes. For instance, there are purely descriptive measures of CUB as caused by the joint effects of mutation and selection, such as, the Effective Number of Codons (Nc or ENC) [13] and the Relative Synonymous Codon Usage [22]. Alternatively, other measures of CUB specifically accord with selection on codon usage associated with translation, such as, the Codon Adaption Index (CAI) [12] and the Frequency of Optimal codons [15]. In addition, a number of studies have attempted to estimate selection on codon usage based on population genetics [31-35].

These existing measures generally fall into two categories, as they compare the observed codon usage distribution of target coding sequence against the distribution based on a reference set of highly-expressed genes (e.g., CAI) or the distribution based on a null hypothesis of uniform usage of different synonymous codons (e.g., Nc). The former measures are highly dependent on their corresponding reference sets (from which preferred codons are derived) and accordingly are limited by the comprehensiveness and accuracy of reference sets. Since reference sets are species-specific, these measures are inappropriate for comparison of CUBs across species [36]. Additionally, they are unreliable in cases where there is inadequate knowledge about the highly-expressed genes for a given species [37], such as for newly sequenced species that have a limited number of annotated genes.

Due to these shortcomings, measures that do not require prior knowledge of reference gene sets have been implemented. These measures assume a null distribution of uniform usage of synonymous codons and estimate the departure of the observed codon usage from the expected. Among them, Nc is one of the most widely used measures [13]. Its variant, Nc' [19], incorporates GC content of coding sequence as background nucleotide composition (BNC) into CUB estimation. Accounting for BNC refines codon usage analysis, providing a comparable metric for analyses within and among species exhibiting various non-uniform BNCs. In the context of protein-coding sequences, for instance, bacteria have diverse BNCs as their GC contents vary widely - from ~20% to ~80%. Even within a single species, genes often differ considerably in background GC content, as in the case of Escherichia coli str. K-12 substr. MG1655, whose genes have GC contents ranging from 26.9% (rfaS; length = 311aa) to 66.8% (yagF; length = 655aa). Therefore, it is crucial to measure the departure of codon usage from the corresponding background composition (instead of the presumed uniform codon usage). Due to its appropriate consideration of BNC, Nc' outperforms other relevant measures [19].

However, all extant measures (including Nc') still have limitations. First, they give a general estimate of CUB, but have not been supplied with straightforward procedures for assessing the statistical significance of the bias in codon usages for any given gene. Genes that vary in length and differ in CUB may exhibit different levels of statistical significance for their codon biases. Assessing statistical significance can strengthen functional relationships ascertained considerably by discounting sampling error in correlated gene sets. Second, no previous measure is fully effective at incorporating BNC into CUB estimation. Although Nc' factors GC content as BNC, it does not account for known variation in BNCs at three different codon positions [38]. In bacteria, for instance, Bartonella quintana str. Toulouse and Clostridium thermocellum ATCC 27405 have very similar GC contents in coding sequences (40.5% and 40.4%, respectively), but their position-specific GC contents are quite different: 53.3% and 47.3% at the first codon position, 38.6% and 34.0% at the second codon position, and 29.5% and 39.9% at the third codon position, respectively. Likewise, genes within a given species can also have heterogeneous BNCs at the three codon positions; in E. coli, for example, there are two genes, emrE and hlyE, that are similar in their overall GC contents (41.5% and 41.1%) but different in positional GC contents: 42.7% and 48.2% at the first position, 46.4% and 32.0% at the second position, and 35.5% and 43.2% at the third position, respectively. Such differences in positional BNCs reflect the outcomes of diverse evolutionary mechanisms (e.g., dinucleotide bias [39], horizontal gene transfer [40], strand compositional asymmetry in bacteria [41], isochore structure in vertebrates [42], etc.), thus conflating the roles of mutation and selection acting at different codon positions. Therefore, incorporation of differential positional BNCs into CUB estimation promises to increase its effectiveness and reliability.

Moreover, GC content is not the sole parameter of BNC. As illustrated in Zhang and Yu [43], joint use of GC and purine contents effectively models nucleotide, codon, and amino acid compositions. In contrast to a broader variation of GC content, purine content varies within a much narrower range fluctuating around 50%, presumably because purines play a determinative role in physicochemical properties of amino acids [44,45]. Similar with GC content, purine content differs not only from one species to another, but also from one gene to another, and even between genes with similar GC contents. For instance, emrE and hlyE in E. coli, which are similar in their overall GC contents, have entirely different purine contents not only at the overall level (45.8% and 55.6%, respectively), but also at three codon positions (54.5% and 68.3% at the first position, 34.5% and 48.2% at the second position, and 48.2% and 50.2% at the third position, respectively). Thus, in addition to GC content, purine content is also a significant feature of BNC.

Here we present a novel measure, Codon Deviation Coefficient (CDC), using it to characterize CUB and to ascertain its statistical significance. CDC takes account of both GC and purine contents, comprehensively addressing heterogeneous BNCs, not only in sequences but also at three codon positions. It adopts the cosine distance metric to quantify CUB and employs the bootstrapping to assess its statistical significance, requiring no prior knowledge of reference gene sets. We describe CDC in detail and provide comparative results in the form of an in-depth evaluation of simulated sequences and empirical data.

Methods

Expected codon usage

CDC considers both GC and purine contents as BNC and derives expected codon usage from observed positional GC and purine contents. We denote the content of the four nucleotides (adenine, thymine, guanine, and cytosine), GC content, and purine content as A, T, G, C, S and R, respectively. As in Zhang and Yu [43], position-dependent nucleotide contents can be formulated in the following way:

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(1)

where Si and Ri are their corresponding observed contents at codon position i and Ai, Ti, Gi, Ci are expected nucleotide contents at codon position i (i = 1, 2, 3). For any sense codon xyz, where x, y, z ∈ {A, T, G, C}, the expected usage πxyz is defined as the product of its constituent expected nucleotide contents x1y2z3, normalized by the sum over all sense codons, viz.

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(2)

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Codon usage bias

Any coding sequence can be represented as a vector of n dimensions, whose entries correspond to n sense codon usages in the sequence. The dimension n equals 61 for the canonical code; although codons ATG and TGG could be set aside due to the absence of synonymous codons, calculation based on a vector of 61 dimensions instead of 59 dimensions makes little substantial difference. To calculate CUB for any given sequence, we employ the cosine distance metric [46] based on the cosine of the angle between the two vectors of n dimensions. Therefore, when both expected (π) and observed (<a onClick="popup('http://www.biomedcentral.com/1471-2105/13/43/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/13/43/mathml/M4">View MathML</a>) codon usage vectors are available for any given sequence, CDC renders a distance coefficient ranging from 0 (no bias) to 1 (maximum bias), to represent CUB, expressed by the deviation of <a onClick="popup('http://www.biomedcentral.com/1471-2105/13/43/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/13/43/mathml/M4">View MathML</a> from π (Eq. 3).

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(3)

Statistical significance of codon usage bias

We implement a bootstrap resampling of N = 10000 replicates for any given sequence to evaluate the statistical significance of non-uniform codon usage. Each replicate is randomly generated according to the sequence BNC (Si and Ri, i = 1, 2, 3) and the sequence length. Consequently, we obtain a bootstrap distribution of N estimates of CUB. A two-sided bootstrap P-value is calculated as twice the smaller of the two one-sided P-values [47]. P ranges from 0 to 1. By convention, a statistically significant CUB is identified by P < 0.05. CDC features its first application of the bootstrap resampling in estimating the statistical significance of CUB. Bootstrapping may also be applicable to other related measures.

Implementation and availability

CDC is written in standard C++ programming language and implemented into Composition Analysis Toolkit (CAT), which is distributed as open-source software and licensed under the GNU General Public License. Its software package, including compiled executables on Linux/Mac/Windows, example data, documentation, and source codes, is freely available at http://cbb.big.ac.cn/software webcite and http://cbrc.kaust.edu.sa/CAT webcite.

Results and discussion

Comparative analysis on simulated data

To evaluate the performance of CDC and compare it against the most powerful extant measure, Nc', as well as Nc, we took an approach based on that of Novembre [19] to simulate coding sequences specifying different positional BNCs and varying sequence lengths. Five sets of position-associated compositions were used to generate simulated sequences (Table 1). It should be noted that CDC ranges from 0 (no bias) to 1 (maximum bias), whereas Nc' and Nc range from 20 (maximum bias) to 61 (no bias). To facilitate comparisons of CDC with Nc' and Nc, we use the formula (61- Nc')/41 and (61- Nc)/41 to rescale their ranges, denoted as scaled Nc' and scaled Nc, respectively, from 0 (no bias) to 1 (maximum bias).

Table 1. Background nucleotide compositions at three codon positions specified in simulations

A good measure should not deviate much from its expectation as the amount of data approaches infinity or any sufficiently large number. Thus, we first simulated sequences with a total of 100,000 codons using five positional composition sets (PCSs) (Table 1). Considering the fact that both GC and purine contents govern BNC, we fixed one of them to be uniform at three codon positions and allowed the other to have various positional compositions. We examined heterogeneous positional compositions for GC (Figure 1A to 1C) and purine (Figure 1D to 1F) contents, respectively. Consistent with expectations, when the PCS was uniform, CDC and scaled Nc' performed similarly, both taking a value close to 0 (Figure 1). When the heterogeneity of positional composition increased for GC content (Figure 1A to 1C), CDC continued to perform well for all cases examined, whereas scaled Nc' and scaled Nc generated biased estimates, especially in cases where there was high heterogeneity in positional BNCs. Similarly, when purine content had heterogeneous positional compositions (Figure 1D to 1F), CDC again exhibited much lower biases than scaled Nc' and scaled Nc. Since Nc ignores BNC, Nc' performed better than Nc when the PCS was non-uniform (Figure 1A, C, D and 1F) and they exhibited comparable estimates only in cases where the PCS was uniform (Figure 1B and 1E). These results agree well with those of Novembre [19]. In addition, when we set heterogeneous positional BNCs for both GC and purine contents, CDC consistently outperformed Nc' and Nc for nearly all the parameter combinations tested (Table 2).

thumbnailFigure 1. Codon usage bias across a variety of positional background nucleotide compositions. Heterogeneous positional background compositions were considered for GC content (panels A to C) and purine content (panels D to E), respectively. The expected values of codon usage bias are zero for all examined cases.

Table 2. Codon usage bias across a variety of positional background compositions for GC and purine contents

To evaluate CDC in a comprehensive manner, we also examined all possible quantitative relationships among positional GC contents (Table 3), although there are identified patterns about quantitative relationships among positional nucleotide compositions (e.g., GC content at the 1st codon position tends to be always larger than that at the 2nd codon position [48]). On the whole, CDC achieved greater power than scaled Nc' and scaled Nc across all examined cases. Scaled Nc' performed better than scaled Nc, consisting again with the analysis reported by Novembre [19]. Similar results were also obtained when we considered all possible quantitative relationships among positional purine contents (Table 4).

Table 3. Codon usage bias across all possible quantitative relationships among positional GC contents

Table 4. Codon usage bias across all possible quantitative relationships among positional purine contents

To examine the effect of variable sequence length on the integrity of CDC, we considered a wide range of sequence lengths from 100 to 3,000 codons. We set both GC and purine contents to be heterogeneous at three codon position using the four non-uniform PCSs (Table 1). To avoid stochastic errors, we repeated simulations 10,000 times for each parameter combination and thus each estimate was determined from 10,000 replicates. Overall, CDC performed better than Nc' and Nc across all sequence lengths examined (Figure 2). When the heterogeneity of BNC increased from low to high, CDC tended to have less biases, whereas Nc' and Nc produced increasingly biased estimates, especially for the case where there was high heterogeneity in positional BNCs (Figure 2D). For short sequences (<300 codons), CDC yielded much lower biases and smaller standard deviations (SD) than Nc' and Nc, although all three measures produced estimates that are somewhat biased. To obtain more reliable estimates of CUB, our results suggest that input sequences should have at least 100 codons in length. When sequence length was decreased below 100 codons, CDC still performed better than Nc' and Nc, although the biases of Nc' and Nc were in opposite directions as compared with those of CDC (Figure 2B to 2D; not apparent in Figure 2A). For long sequences, CDC generated less biased estimates and SDs, whereas Nc' and Nc continued to yield more biased estimates and SDs.

thumbnailFigure 2. Codon usage bias across a range of sequence lengths. Sequences were simulated with the four non-uniform positional composition sets: Low (panel A), Med-1 (panel B), Med-2 (panel C) and High (panel D). Each estimate was determined based on 10000 replicate simulated sequences. The expected values of codon usage bias are zero for all examined cases.

To test the influence of different CUBs on the power of CDC, we evaluated a range of CUBs from low to high. Unlike the previous simulations (which are based on nucleotide compositions), we generated simulated sequences by randomly setting different synonymous codon frequencies and considering variable CUBs with a range from 0.1 to 0.9. We repeated simulations 1,000 times for each case and accordingly each estimate was averaged over 1,000 replicates. On the whole, CDC exhibited greater power in detecting diverse CUBs; compared with Nc' and Nc, the estimated CUBs of CDC were very closer to the expected ones (Table 5). When the expected CUBs varied from low to high, CDC performed consistently to give rise to close estimates. Contrastingly, Nc' and Nc yielded biased CUB estimates across all tested cases and these biases became more pronounced when the expected CUB was extremely low. When the expected CUBs increased from low to high, Nc' and Nc exhibited increasing power in CUB estimation. While they approached the power of CDC when the expected CUB was high, CDC remained more powerful than Nc' and Nc. Taken together, our simulation results demonstrated that CDC is superior to Nc' and Nc.

Table 5. Differences between estimated and expected codon usage biases

Application to empirical data

It is generally acknowledged that CUB correlates closely with gene expression level in both unicellular [6-10] and multicellular [11,49-51] organisms. Different species may have different heterogeneities in positional BNCs. To empirically test CDC and compare it to three popular measures, Nc', Nc and CAI, we collected multiple expression data sets from five different species in this study: (1) Escherichia coli from Bernstein et al. [52] (in LB and M9 media), (2) Saccharomyces cerevisiae from Holstege et al. [53], (3) Drosophila melanogaster from Zhang et al. [54], (4) Caenorhabditis elegans from Roy et al. [55], and (5) Arabidopsis thaliana from Wuest et al. [56] (Additional file 1). We estimated CUB by CDC, scaled Nc', scaled Nc and CAI, and correlated their estimates with gene expression levels in these five species (Table 6).

Additional file 1. Empirical expression data analysis. Correlations between codon usage bias and gene expression level in different expression data sets.

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Table 6. Correlation coefficients of codon usage bias with gene expression level

On the whole, CDC outperformed scaled Nc' and scaled Nc in correlating closely with gene expression level. Although CDC and scaled Nc' produced comparable correlation coefficients in yeast (detailed below), CDC exhibited larger correlation coefficients than scaled Nc' and scaled Nc for all the rest cases (Table 6). When comparing CDC to CAI, we found comparable correlation coefficients in E. coli (LB medium) and yeast, but in general CDC performed better than CAI (Table 6 and Additional file 1). However, it should be noticed that the values of CAI are calculated from expression data (since it requires a reference set of highly-expressed genes), whereas those of CDC are not. When we restricted the above analysis to the top 10% genes referring to their expression levels, CDC continued to perform better than scaled Nc', scaled Nc, and CAI (Additional file 1). In addition, considering the correlation coefficients among these five species, we found that the smallest values always belonged to A. thaliana (regardless of metric used), indicating relatively weaker selection on A. thaliana codon usage by comparison with those of the other four species (Table 6). Such phenomenon was discovered previously in a comparative analysis between A. thaliana and Oryza sativa [57]. Overall, CDC correlated positively with gene expression level, much better than scaled Nc', scaled Nc, and CAI.

As noted, the correlation coefficients produced by CDC and scaled Nc' were similar in yeast but different in others (Table 6). Since CDC takes positional GC and purine contents as BNC and Nc' considers only GC content as BNC and ignores positional heterogeneity, this result can be probably explained by relatively lower heterogeneity of positional BNCs in yeast. To further investigate this possibility, we examined the heterogeneities of positional GC and purine contents in these five species (Figure 3). Consistent with our expectation, heterogeneities of positional GC contents were indeed lower in yeast by comparison with other species (Figure 3A to 3C), especially at the second and third codon positions. In contrast, higher heterogeneities of positional GC contents were apparent in E. coli (Figure 3A and 3B for the first and second codon positions, respectively) and D. melanogaster (Figure 3B and 3C for the second and third codon positions, respectively). These results agree well with the observation that the difference of correlation coefficient between CDC and scaled Nc' in yeast was smaller than that in E. coli or D. melanogaster (Table 6). As a consequence, CDC correlated more closely with scaled Nc' in yeast than in E. coli or D. melanogaster (Figure S13 in Additional file 1). In contrast to GC content, heterogeneities of positional purine contents were relatively smaller and similar among the five species tested, presumably attributable to the fact that GC content ranges more broadly (20%--80%) than purine content (40%--60%) [48,58,59].

thumbnailFigure 3. Heterogeneity of positional background nucleotide compositions in E. coli (2,766 genes in M9 medium), S. cerevisiae (5,142 genes), D. melanogaster (1,651 genes),C. elegans (12,184 genes), and A. thaliana (1,332 genes). Heterogeneities of positional GC contents are represented by absolute differences between overall GC content and its positional contents: GC-GC1 for the first position (panel A), GC-GC2 for the second position (panel B), and GC-GC3 for the third position (panel C), respectively. Likewise, heterogeneities of positional purine content are absolute differences between overall purine (AG) content and its positional contents: AG-AG1 for the first position (panel D), AG-AG2 for the second position (panel E), and AG-AG3 for the third position (panel F), respectively.

We proceeded to calculate CDC values (as well as GC and purine contents) for all E. coli genes (Additional file 2). CDC values ranged from 0.046 to 0.550 and the mean and median values were 0.239 and 0.187, respectively (Figure 4). The majority of genes (69%) exhibited CDC values between 0.15 and 0.25. The gene with the highest CDC value is trpL, a key component in the attenuation system that controls the expression of the trpLEDCBA operon in response to tryptophan availability [60]. However, bootstrap resampling illustrates that the CUB value of trpL gene is not statistically significant (P = 0.77), most likely due to its short length (14 aa), consistent with our simulation results that short sequences tend to have biased CUB estimates. The gene with the highest CDC value and statistical significance in CUB is rpmI (CDC = 0.481), which encodes ribosomal protein L35. In contrast, scaled Nc' and scaled Nc identified rplL (encoding the ribosomal protein L7/L12) and eno (catalyzing the interconversion of 2-phosphoglycerate and phosphoenolpyruvate) genes, respectively, as having the strongest CUBs (Additional file 2).

Additional file 2. Estimates of codon usage bias for all E. coli genes. Codon usage biases of all E. coli genes estimated by CDC, Nc' and Nc.

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thumbnailFigure 4. Comparison of CDC distributions between ribosomal protein (54 RP genes vary from 0.244 to 0.481) genes and all genes (4,144 genes range from 0.046 to 0.550) in E. coli.

Ribosomal protein (RP) genes are, in general, both essential and highly expressed, and it is believed that their CUB values are greater than those of other genes [61]. In the case of E. coli, CDC values for 54 RP genes vary from 0.244 to 0.481, larger than the mean and median values of all E. coli genes (Figure 4). Nearly all RP genes have statistically significant CUBs, with three exceptions (Additional file 3): (1) rpmE: CDC = 0.267, P = 0.1136; encoding RP L31, which may be loosely associated with ribosome [62], (2) rpmF: CDC = 0.329, P = 0.1096; encoding RP L32, which locates near the peptidyltransferase center [63], and (3) rpmJ: CDC = 0.422, P = 0.0564; encoding RP L36, which is non-essential for protein synthesis [64]. These results suggest that an accurate measure such as CDC has the potential to illuminate the evolutionary process that has operated on each gene.

Additional file 3. Estimates of codon usage bias for ribosomal proteins in E. coli. Codon usage biases of ribosomal proteins in E. coli estimated by CDC, Nc' and Nc.

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Conclusions

In summary, we have described a novel measure of CUB, the Codon Deviation Coefficient. As validated by simulated sequences and empirical data, CDC outperforms other measures by providing informative estimates of CUB and its statistical significance. CDC features no necessity for any prior knowledge regarding gene expression or function, properly accounts for BNC, and utilizes a bootstrap assessment to evaluate the statistical significance of CUB. Therefore, CDC promises a significant advance in raw analysis of codon usage, providing the means to better reveal aspects of the historical evolutionary pressures on gene function without the assumptions of underlying reference data sets.

Abbreviations

CUB: Codon Usage Bias; CDC: Codon Deviation Coefficient; BNC: Background Nucleotide Composition; PCS: Positional Composition Set; A: Adenine content; T: Thymine content; G: Guanine content; C: Cytosine content; S: GC content; R: Purine content; Ai, Ti, Gi, Ci, Si, Ri, A, T, G, C, S, R at codon position i, respectively, where i = 1, 2, 3.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

ZZ designed the algorithm, developed the program, and drafted the manuscript. JL participated in the design of the algorithm and analyzed the simulated data. PC collected expression data and analyzed the real data. FD carried out data visualization. AL participated in the software development and testing. JPT helped to draft the manuscript and revised the manuscript. JY supervised the study and revised the manuscript. All authors read and approved the final manuscript.

Acknowledgements

We thank anonymous reviewers for their critical comments and constructive suggestions on this manuscript. We also thank Joe Yu for helpful comments on this work and George Marselis for providing assistance on software hosting. This work was supported by King Abdullah University of Science and Technology (KAUST), Kingdom of Saudi Arabia, and the National Science and Technology Key Project (2008ZX1004-013), 863 Program (2009AA01A130), and 973 Program (2011CB944100) from the Ministry of Science and Technology, the People's Republic of China.

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