Codon equilibrium I: Testing for homogeneous equilibrium

Summary We present theoretical considerations that suggest that synonymous-codon usage might be expected to be close to an equilibrium distribution given a very homogeneous process of silent substitution. By homogeneous we mean that substitution depends only on the two bases involved, so that 12 base-substitution rates completely describe the silent substitution process. We have developed a method of statistically testing for such homogeneous equilibrium and applied it to reported data on the codon usages of different classes of organisms. Weakly expressed bacterial sequences and both mammalian and nonmammalian eukaryotic sequences deviate significantly from a random pattern of codon usage, in the direction of homogeneous equilibrium. On the other hand, highly expressed bacterial sequences do not exhibit homogeneous equilibrium, which may be correlated with recent experimental results showing that they are optimized to accept the most abundant tRNAs. To examine the effect of amino acid replacements on the homogeneous model of silent substitution, we divided the amino acids with degenerate codes into two classes, those with high mutabilities and those with low, and performed the same analysis on bacterial and eukaryotic data sets. The codon sets of the highly mutable class of amino acids are not further from homogeneous equilibrium than are the codon sets of the class with low mutabilities. We also found for the eukaryotic data that these independent classes of codon sets show very similar equilibrium patterns. The various results suggest a high level of uniformity in the process of silent fixation in the different synonymous-codon sets, especially in eukaryotes.     

Codon equilibrium II: Its use in estimating silent-substitution rates

We study the equilibrium in the use of synonymous codons by eukaryotic organisms and find five equations involving substitution rates that we believe embody the important implications of equilibrium for the process of silent substitution. We then combine these five equations with additional criteria to determine sets of substitution rates applicable to eukaryotic organisms. One method employs the equilibrium equations and a principle of maximum entropy to find the most uniform set of rates consistent with equilibrium. In a second method we combine the equilibrium equations with data on the man-mouse divergence to determine that set of rates that is most neutral yet consistent with both types of data (i.e., equilibrium and divergence data). Simulations show this second method to be quite reliable in spite of significant saturation in the substitution process. We find that when divergence data are included in the calculation of rates, even though these rates are chosen to be as neutral as possible, the strength of selection inferred from the nonuniformity of the rates is approximately doubled. Both sets of rates are applied to estimate the human-mouse divergence time based on several independent subsets of the divergence data consisting of the quartet, C- or T-ending duet, and A- or G-ending duet codon sets. Both rate sets produce patterns of divergence times that are shortest for the quartet data, intermediate for the CT-ending duets, and longest for the AG-ending duets. This indicates that rates of transitions in the duet-codon sets are significantly higher than those in the quartet-codon sets; this effect is especially marked for A----G, the rate of which in duets must be about double that in quartets.     

Codon equilibrium II: Its use in estimating silent-substitution rates

Summary We study the equilibrium in the use of synonymous codons by eukaryotic organisms and find five equations involving substitution rates that we believe embody the important implications of equilibrium for the process of silent substitution. We then combine these five equations with additional criteria to determine sets of substitution rates applicable to eukaryotic organisms. One method employs the equilibrium equations and a principle of maximum entropy to find the most uniform set of rates consistent with equilibrium. In a second method we combine the equilibrium equations with data on the man-mouse divergence to determine that set of rates that is most neutral yet consistent with both types of data (i.e., equilibrium and divergence data). Simulations show this second method to be quite reliable in spite of significant saturation in the substitution process. We find that when divergence data are included in the calculation of rates, even though these rates are chosen to be as neutral as possible, the strength of selection inferred from the nonuniformity of the rates is approximately doubled. Both sets of rates are applied to estimate the human-mouse divergence time based on several independent subsets of the divergence data consisting of the quartet, C- or T-ending duet, and A- or G-ending duet codon sets. Both rate sets produce patterns of divergence times that are shortest for the quartet data, intermediate for the CT-ending duets, and longest for the AG-ending duets. This indicates that rates of transitions in the duet-codon sets are significantly higher than those in the quartet-codon sets; this effect is especially marked for AG, the rate of which in duets must be about double that in quartets.     

Testing for Hardy-Weinberg equilibrium in samples with related individuals

When the classical chi(2) goodness-of-fit test for Hardy-Weinberg (HW) equilibrium is used on samples with related individuals, the type I error can be greatly inflated. In particular the test is inappropriate in population isolates where the individuals are related through multiple lines of descent. In this article, we propose a new test for HW (the QL-HW test) suitable for any sample with related individuals, including large inbred pedigrees, provided that their genealogy is known. Performed conditional on the pedigree structure, the QL-HW test detects departures from HW that are not due to the genealogy. Because the computation of the QL-HW test becomes intractable for very polymorphic loci in large inbred pedigrees, a simpler alternative, the GCC-HW test, is also proposed. The statistical properties of the QL-HW and GCC-HW tests are studied through simulations considering a sample of independent nuclear families, a sample of extended outbred genealogies, and samples from the Hutterite population, a North American highly inbred isolate. Finally, the method is used to test a set of 143 biallelic markers spanning 82 genes in this latter population.     

Relative Codon Adaptation: A Generic Codon Bias Index for Prediction of Gene Expression

The development of codon bias indices (CBIs) remains an active field of research due to their myriad applications in computational biology. Recently, the relative codon usage bias (RCBS) was introduced as a novel CBI able to estimate codon bias without using a reference set. The results of this new index when applied to Escherichia coli and Saccharomyces cerevisiae led the authors of the original publications to conclude that natural selection favours higher expression and enhanced codon usage optimization in short genes. Here, we show that this conclusion was flawed and based on the systematic oversight of an intrinsic bias for short sequences in the RCBS index and of biases in the small data sets used for validation in E. coli. Furthermore, we reveal that how the RCBS can be corrected to produce useful results and how its underlying principle, which we here term relative codon adaptation (RCA), can be made into a powerful reference-set-based index that directly takes into account the genomic base composition. Finally, we show that RCA outperforms the codon adaptation index (CAI) as a predictor of gene expression when operating on the CAI reference set and that this improvement is significantly larger when analysing genomes with high mutational bias.     

Costly dispersal can destabilize the homogeneous equilibrium of a metapopulation

I investigate the stability of the homogeneous equilibrium of a discrete-time metapopulation assuming costly dispersal with arbitrary (but fixed) spatial pattern of connectivity between the local populations. First, I link the stability of the metapopulation to the stability of a single isolated population by proving that the homogeneous metapopulation equilibrium, provided that it exists, is stable if and only if a single population, which is subject to extra mortality matching the average dispersal-induced mortality of the metapopulation, has a stable fixed point. Second, I demonstrate that extra mortality may destabilize the fixed point of a single population. Taken together, the two results imply that costly dispersal can destabilize the homogeneous equilibrium of a metapopulation. I illustrate this by simulations and discuss why earlier work, arriving at the opposite conclusion, was flawed.     

Variance within homogeneous phytoplankton populations, I: Theoretical framework for interpreting histograms

A framework is presented for interpreting frequency distributions of volume or fluorescence as measured by a flow cytometer on homogeneous phytoplankton populations. The framework, based on both laboratory experience and theoretical concepts, is illustrated with the use of a simulation model. Asynchronous, synchronous, and phased populations were simulated, with constant and variable growth patterns over the cell cycle. Though simulations produced a wide variety of histogram shapes, including multimodal distributions, the primary difference between asynchronous and synchronous/phased distributions lies in their temporal variation. Histograms that are constant in time indicate asynchronous populations; when populations are not asynchronous, their histogram shapes vary with a periodicity on the same time scale as the cell cycle. A probability density function for the case of asynchronous populations with a constant growth rate is derived. When fitted to simulated histograms this two-parameter density function yields estimates of the two parameters: mean and variance of cell volume (or mass) at age 0.     

Testing for Selection on Synonymous Sites in Plant Mitochondrial DNA: The Role of Codon Bias and RNA Editing

Since plant mitochondrial genomes exhibit some of the slowest known synonymous substitution rates, it is generally believed that they experience exceptionally low mutation rates. However, the use of synonymous substitution rates to infer mutation rates depends on the implicit assumption that synonymous sites are evolving neutrally (or nearly so). To assess the validity of this assumption in plant mitochondrial genomes, we examined coding sequence for footprints of selection acting at synonymous sites. We found that synonymous sites exhibit an AT rich and pyrimidine skewed nucleotide composition compared to both non-synonymous sites and non-coding regions. We also found some evidence for selection associated with both biased codon usage and conservation of regulatory sequences involved in mRNA processing, although some of these findings are subject to alternative non-adaptive interpretations. Regardless, the inferred strength of selection appears too weak to account for the variation in substitution rates between the mitochondrial genomes of plants and other multicellular eukaryotes. Therefore, these results are consistent with the interpretation that plant mitochondrial genomes experience a substantially lower mutation rate rather than increased functional constraints acting on synonymous sites. Nevertheless, there are important nucleotide composition patterns (particularly the differences between synonymous sites and non-coding DNA) that remain largely unexplained.     

Testing the Hardy-Weinberg equilibrium in the HLA system

A new method of testing the Hardy-Weinberg equilibrium in the human leukocyte antigen (HLA) system is proposed and applied to real data. The derivation is based on the maximum likelihood method and closely related to standard regression theory. The test statistic has a closed representation of residual sum of squares by a projection mapping of data onto the estimated regression plane. Under the Hardy-Weinberg law the noniterative estimates for the gene frequencies are suggested by the use of the projection mapping. The test statistic and gene frequency estimates are shown to be asymptotically equivalent to the maximum likelihood method and to be more efficient than the other suggested test statistic when there are more than two identified alleles.     

The Evolution of Codon Preferences in Drosophila: A Maximum-Likelihood Approach to Parameter Estimation and Hypothesis Testing

Synonymous codon usage in related species may differ as a result of variation in mutation biases, differences in the overall strength and efficiency of selection, and shifts in codon preference—the selective hierarchy of codons within and between amino acids. We have developed a maximum-likelihood method to employ explicit population genetic models to analyze the evolution of parameters determining codon usage. The method is applied to twofold degenerate amino acids in 50 orthologous genes from D. melanogaster and D. virilis. We find that D. virilis has significantly reduced selection on codon usage for all amino acids, but the data are incompatible with a simple model in which there is a single difference in the long-term N _e, or overall strength of selection, between the two species, indicating shifts in codon preference. The strength of selection acting on codon usage in D. melanogaster is estimated to be |N _e s|≈ 0.4 for most CT-ending twofold degenerate amino acids, but 1.7 times greater for cysteine and 1.4 times greater for AG-ending codons. In D. virilis, the strength of selection acting on codon usage for most amino acids is only half that acting in D. melanogaster but is considerably greater than half for cysteine, perhaps indicating the dual selection pressures of translational efficiency and accuracy. Selection coefficients in orthologues are highly correlated (ρ= 0.46), but a number of genes deviate significantly from this relationship. Received: 20 December 1998 / Accepted: 17 February 1999     

Codon optimizer: a freeware tool for codon optimization

Selection plays a major role in the determination of codon usage in all organisms studied so far. In highly expressed genes, a narrow set of codons is used and these codons correspond to the more abundant tRNA species. This minimizes the risk of tRNA depletion during translation. In fact, the codons in a gene may be true bottlenecks, especially in cases where foreign genes are expressed in a host in which the usage of codons in highly expressed genes does not resemble the usage of codons in the species from which the foreign gene originates. In such cases, it has been shown that substitution of rare codons in the introduced gene may increase the yield dramatically. In addition, replacement of rare codons might decrease the chance of misincorporation and protect the protein from premature turnover. Here, a piece of software is announced that calculates a codon-optimized sequence of any gene based on knowledge of highly expressed genes of a host. In addition, it calculates the codon adaptation index of the gene and identifies internal type II restriction sites of the optimized sequence. The program runs under Windows and is available as freeware for use in academia.     

Codon:Anticodon and anticodon:Anticodon interaction. Evaluation of equilibrium and kinetic parameters of complexes involving a G:U wobble

In order to learn about the effect of the G:U wobble interaction we characterized the codon:anticodon binding between triplets: UUC, UUU and yeast tRNA^Phe (anticodon GmAA) as well as the anticodon:anticodon binding between Escherichia coli tRNA^Glu₂, E. coli tRNA^Lys (anticodons: mam⁵s²UUC, and mam⁵s²UUU, respectively) and tRNA^Phe from yeast and E. coli (anticodon GAA) using equilibrium fluorescence titrations and temperature jump measurements with fluorescence and absorption detection. The difference in stability constants between complexes involving a G:U pair rather than a usual G:C basepair is in the range of one order of magnitude and is mainly due to the shorter lifetime of the complex involving G:U in the wobble position. This difference is more pronounced when the codon triplet is structured, i.e., is built in the anticodon loop of a tRNA. The reaction enthalpies of the anticodon:anticodon complexes involving G:U mismatching were found to be about 4 kcal/mol smaller, and the melting temperatures more than 20°C lower, than those of the corresponding complexes with the G:C basepair. The results are discussed in terms of different strategies that might be used in the cell in order to minimize the effect of different lifetimes of codon-tRNA complexes. Differences in these lifetimes may be used for the modulation of the translation efficiency.     

Codon usage patterns in cytochrome oxidase I across multiple insect orders

Synonymous codon usage bias is determined by a combination of mutational biases, selection at the level of translation, and genetic drift. In a study of mtDNA in insects, we analyzed patterns of codon usage across a phylogeny of 88 insect species spanning 12 orders. We employed a likelihood-based method for estimating levels of codon bias and determining major codon preference that removes the possible effects of genome nucleotide composition bias. Three questions are addressed: (1) How variable are codon bias levels across the phylogeny? (2) How variable are major codon preferences? and (3) Are there phylogenetic constraints on codon bias or preference? There is high variation in the level of codon bias values among the 88 taxa, but few readily apparent phylogenetic patterns. Bias level shifts within the lepidopteran genus Papilio are most likely a result of population size effects. Shifts in major codon preference occur across the tree in all of the amino acids in which there was bias of some level. The vast majority of changes involves double-preference models, however, and shifts between single preferred codons within orders occur only 11 times. These shifts among codons in double-preference models are phylogenetically conservative.     

General Rules for Optimal Codon Choice

Different synonymous codons are favored by natural selection for translation efficiency and accuracy in different organisms. The rules governing the identities of favored codons in different organisms remain obscure. In fact, it is not known whether such rules exist or whether favored codons are chosen randomly in evolution in a process akin to a series of frozen accidents. Here, we study this question by identifying for the first time the favored codons in 675 bacteria, 52 archea, and 10 fungi. We use a number of tests to show that the identified codons are indeed likely to be favored and find that across all studied organisms the identity of favored codons tracks the GC content of the genomes. Once the effect of the genomic GC content on selectively favored codon choice is taken into account, additional universal amino acid specific rules governing the identity of favored codons become apparent. Our results provide for the first time a clear set of rules governing the evolution of selectively favored codon usage. Based on these results, we describe a putative scenario for how evolutionary shifts in the identity of selectively favored codons can occur without even temporary weakening of natural selection for codon bias.     

Testing for a trend in tumor prevalence rates: I. Nonlethal tumors

In the analysis of animal carcinogenicity studies, the standard survival-adjusted test for a dose-related trend in the prevalence of nonlethal tumors is the Hoel-Walburg test, which stratifies on age at death by grouping survival times into intervals. An alternative analysis assesses trend on the basis of the likelihood score test under a logistic model for the prevalence function, which adjusts for survival by including age at death as a continuous regression variable. Extensive simulations demonstrate that the test based on modeling the prevalence log-odds as a linear function of age is more powerful than the Hoel-Walburg test, regardless of the intervals used by the latter to stratify the data. Without incorporating a continuity correction, the size of each test often exceeds the nominal level, especially when the mortality patterns differ across dose groups. Corrected versions of the tests operate at conservative levels, where the degree of conservatism varies with the distribution of the data. When the mortality patterns for the dose groups are similar, both tests have essentially the same power to detect a trend in tumor prevalence rates. However, when mortality varies with dose, the logistic regression test with a linear age term is more powerful than the Hoel-Walburg test, and this gain in power increases as the dose-specific mortality patterns become more disparate.     

Testing Hardy-Weinberg equilibrium on allelic data from VNTR loci.

Several methods for testing independence of pairs of alleles in a population that are obtained from a VNTR locus are presented. We assume an exchangeable quasi-continuous distribution of the fragment lengths used to measure the allelic pairs. Bivariate-estimated quantiles computed from the quantiles of the entire data set are then utilized for testing independence. These methods have the advantage of being minimally susceptible to the criticism of (a) the inability of a technology to measure a few small-sized or rather large-sized fragments and (b) inadequate estimation of the homozygotic proportion.