Ribosome pausing and nonsense error model

Below, we outline the assumptions behind our ribosome PAusing and NonSense Error (PANSE) model and the formal likelihood function they lead to.

PANSE model assumptions.

For a given gene g, m_g represents its equilibrium mRNA transcript concentration within a cell and represents the average ribosome initiation rate for each of its mRNAs. Gene g consists of n_g codons and the elongation and nonsense error rates at codon position i are c_g,i and b_g,i, respectively. We assume the elongation rate c_g,i varies between sites such that InverseGamma where the shape parameter and scale parameter can vary between the 61 sense codons. The across-site heterogeneity of a given sense codon’s elongation rates reflects a variety of factors that alter ribosome elongation [22,47]. Although the nonsense error rate b_g,i likely varies with codon and position, for simplicity we focus solely on the variation between codons and, thus, treat the codon-specific nonsense error rate b as a constant, having the same value across all codons of type c. Based on these assumptions, it is possible to model the distribution of footprints as coming from a negative binomial distribution, i.e., with , , where . The composite parameter is a rescaling of a codon’s specific rate parameter by the ratio of the partition function for the RFP distribution Z from which the footprints are sampled and the total number of observed footprints in a dataset . Thus, Y/Z is a measure of the sampling efficiency of the experiment.

Given Y_g,i represents the observed number of ribosome footprints (RFP) at codon i of gene g, it follows that the likelihood of observing Y_g,i is,

(1)

where is a rescaling of the mRNA_g specific ribosome initiation rate by the density of mRNA_g transcripts within the cell, m_g. Assuming independence between elongation steps, it follows that the probability of a ribosome successfully elongating at codons 1 through is,

(2)

where represents the PDF of the InverseGamma distribution for the ribosome elongation rate for codon i.

Because the nonsense error rate (NSE rate) b between sense codons is likely to be several orders of magnitude less than a given c_g,i and we will be working with our likelihood function on the log scale, we can greatly speed up our evaluation of the log of Eqn. (1) by approximating using a 2nd order Taylor Series approximation around a mean NSE rate for all codons . Doing so gives,

Analysis of ribosome profiling data

Raw ribosome profiling for S. cerevisiae reads were downloaded from the Sequence Read Archive (SRA Run Accession: SRR1049521) [27]. These data were generated using a flash-freeze protocol to halt ribosome elongation, avoiding many of the technical artifacts caused by cycloheximide [27,48]. The raw reads were processed using riboviz2 [49] and a transcriptome FASTA file containing annotated protein-coding sequences from the Saccharomyces Genome Database (Release R64-2–1), as well as 250 nucleotide upstream and downstream of the start and stop codons, respectively. Aligned reads of length 28–30 were extracted and assigned to codons in the A-site, assuming a 15-nucleotide offset relative to the 5’-end of the read [44] (S1 Fig). We note that using different A-site offsets based on the riboWaltz R package had little impact on parameter estimation (S2 Fig).

A flat file was created that includes the number of ribosome footprints (i.e., counts) assigned to each codon within a gene. This file was used as the primary input into our implementation of PANSE. We employed many filtering criteria to remove genes that violate the assumptions of the model (see S1 Text and S3 Fig). This led to a final dataset consisting of RFP counts for 3,112 S. cerevisiae protein-coding genes, representing approximately 50% of genes found in the genome.

Fitting the pausing and nonsense error model

The PANSE model was fit via a Markov chain Monte Carlo (MCMC) algorithm to the codon-level RFP counts for the genes included in the final data set. The MCMC was run for 50,0000 iterations, keeping every 5th iteration to reduce autocorrelation. Convergence was assessed by comparing the results of two separate MCMCs that were started at random points in parameter space. Posterior means and 95% highest density intervals (HDIs) were calculated for each parameter of PANSE based on the MCMC traces. Consistent with our previous work [7], gene-specific initiation rates are assumed to follow a lognormal distribution with a mean initiation rate of 1. This is accomplished by fixing the mean of the lognormal distribution to be , where is the standard deviation of the lognormal distribution. The shape and scale parameters, and , for each codon-specific elongation rate c, and the codon-specific NSE rates b, were assumed to have a flat prior with ranges (0, 100), (0, 100), and (1E-100, 1E-1) on the natural scale. We note these are particularly broad distributions, but we wanted to ensure that adequate parameter space could be explored. We fit PANSE (1) assuming no nonsense errors occur, (2) assuming uniform NSE rates across codons, and (3) allowing NSE rates to vary across codons. These three model fits were compared using the Deviance Information Criterion (DIC) [50] to determine statistical support for variation in NSE rates across codons.

The ribosome profiling data we used were prepared with a flash-freeze protocol to halt elongation, but there is still an observable increase in ribosome density in the first 200 codons. Although this increased density was less than observed in ribosome profiling measurements using cycloheximide to halt elongation, it is unclear if this reflects true biology or a technical artifact [27]. As it is plausible that ribosome counts for the first 200 codons are impacted by unknown technical biases, we fit PANSE masking the ribosome counts for this region in the likelihood calculation for each gene. However, we did include these codons in our calculation of the expected probability of elongation up to codon i .

We compared parameter estimates of PANSE to (1) independent empirical data such as mRNA abundances and tRNA-based proxies for elongation rates [27], and (2) parameter estimates from the Ribosomal Overhead Cost version of the Stochastic Evolutionary Model of Protein Production Rates (ROC-SEMPPR), which estimates protein production rates and codon-specific selection coefficients from codon usage bias patterns [7]. See previous work regarding parameter estimation with ROC-SEMPPR for S. cerevisiae [7,51].

Analyzing variation in NSE rates b across codons

To better understand the variation in NSE rates b across codons, we performed a linear regression of the posterior mean NSE rates b with properties of the codon. This included the number of stop codons 1 nucleotide mutation away from the codon (0, 1, or 2), the missense error probability of the codon [52], and whether or not the codon is prone to frameshifts [42]. We weighted each codon in the regression by the standard deviations estimated from the MCMC traces.

Quantifying variation in translation completion probabilities across genes

We used our estimates of elongation and NSE rates, c and b, for each of the 61 sense codons to calculate the translation completion probability . This allowed us to apply this formula to all genes in the S. cerevisiae genome, regardless of whether or not it was included in the model fit. We assessed how variation in translation completion probabilities varied across genes as a function of length and gene expression measured as mRNA abundances (in units of RPKM) taken from previous work [27]. Additionally, we compared how well our inferences of translation completion probabilities made directly from empirical ribosome profiling data compared to theoretical estimates based on simulations from a Totally Asymmetric Exclusion Process (TASEP) model of translation [34].

Quantifying elongation probability variation within and across genes

As before, c_i and b_i represent the expected elongation and NSE rates, respectively, of the codon at position i. Given that there are only two possible outcomes in our model (elongation or a nonsense error), it follows that the probability a ribosome elongates the growing peptide chain at codon i is,

The probability a ribosome reaches codon i by successfully elongating at the previous codons is,

To evaluate how elongation probabilities vary with codon position i, we calculated the average probability a ribosome elongates at a given position as the geometric mean of the elongation probability across genes. We then regressed the log(Position i) with the average elongation probability to quantify how the elongation probability changes as a function of position. To test if the observed slope was greater than expected under the null model of no selection against nonsense errors, we generated 1000 different permuted sets of genes for each of 3 different possible nulls: (1) the synonymous codons of an amino acid were permuted within a gene (randomized by amino acid), (2) amino acids and codons were permuted within a gene’s coding region (randomized by CDS), and (3) codons were permuted across genes (randomized across CDS). The slope estimated from the real set of genes was then compared to each of the 3 null distributions, with the p-value = (k + 1)/(1000 + 1), where k is the number of occurrences in which a slope from a permuted set of genes was greater than the slope from the real set of genes. A similar analysis was performed on the real data after binning genes based on mRNA abundances into the lower quartile (low expression), interquartile (moderate expression), and upper quartile (high expression) on the log scale.

Identifying codons enriched in the 5’-end

To identify codons enriched in the 5’-end and 3’-end of the CDSs (i.e., near the start and stop codon, respectively), we used both an absolute and a relative definition of the termini. The absolute definition considered the termini to be the first and last 100 codons of a coding sequence. In this case, we restricted our analysis to coding sequences with a minimum of 250 codons. The relative definition considered the termini to be the first and last 25% of codons along a coding sequence. In this case, no minimum length cutoff was used.

To identify codons enriched in the 5’-end, an empirical expectation was determined by calculating the frequency of each codon, relative to its synonyms, in the “middle” (i.e., neither of the termini) of the CDS across all genes. For each codon, we used a one-sided binomial test to determine if the observed frequency in the 5’-ends was greater than expected by chance based on its observed frequency in the middle of the CDS. A similar enrichment test was used for the 3’-ends.

Calculating the cost of nonsense errors

To better understand the potential importance of nonsense errors in translation, we calculated the expected cost of producing a complete, functional protein (in terms of the number of hydrolyzed NTPs) from a given transcript that includes the impact of nonsense errors. For simplicity, we excluded the cost of amino acid synthesis from our calculations.

Conceptually, we break the cost of protein synthesis into two overlapping sets of categories: direct vs. indirect costs and fixed vs. variable costs. Direct costs include the NTP used in assembling the small and large ribosome subunits on the mRNA and each elongation step by the ribosome, whereas indirect costs are based on the synthesis cost of the translation infrastructure. In terms of fixed and variable costs, fixed costs are the direct and indirect costs of translating a protein in the absence of a nonsense error, i.e., costs incurred every round of successful translation. Variable costs are the expected direct and indirect costs of translation that are wasted when a nonsense error occurs – these costs are variable because they are only incurred if a nonsense error occurs. As a result, the total cost of producing a particular protein is the sum of fixed and variable costs, each of which is comprised of direct and indirect costs.

From our model, it follows that

(3)

Where a₁ = 4 NTP and a₂ = 4 NTP are the direct costs of translation initiation and elongation, respectively, in terms of hydrolyzed phosphate bonds [18].

Combining the contribution of fixed and variable direct costs of protein synthesis yields,

(4)

Similarly, combining the contribution of the fixed and variable indirect costs of protein synthesis yields,

(5)

The term represents the indirect cost of ribosome pausing up to codon i. Because RPF data lacks information on the absolute rate of ribosome elongation, we rescaled our estimates of pausing times such that the average elongation rate across the 61 sense codons was ∼9.3 codon/sec or, equivalently, sec/codon. Assuming the average mRNA has a CDS of 400 codons, the same average elongation rate as above, and that at any given time 80% of the ribosome population is engaged in translating mRNAs [27], it follows that the expected initiation rate c₀ is sec (see S1 Text for more details).

The parameter C converts the indirect cost of ribosome pausing (in units of seconds) into their equivalent costs in terms of NTP and has the units of NTP/sec. Although we are unaware of any empirical estimates of C, we used two different approaches to estimate C that vary by less than one order of magnitude. One estimate of C is based on selection coefficients estimated from a ROC-SEMPPR analysis of the S. cerevisiae genome and yields C_ROC = 0.71 NTPs/sec). The second estimate of C is based on the assembly cost (in NTPs) and average lifespan (in seconds) of the ribosome and yields = 5.5 NTPs/sec. See S1 Text for detailed descriptions of these calculations. Given these two independent estimates of C are the only ones we have, we treat C_ROC and as lower and upper bounds on C, respectively. Thus, the average cost of ribosome pausing is approximately or 0.59 NTP/codon for C_ROC and , respectively, suggesting indirect costs are a fraction of the direct cost of 4 NTP/codon.

In addition to using our cost estimates directly, we also used them to test the hypothesis that the order of synonymous codons along a gene shows evidence of adaptation to reduce the cost of nonsense errors. To do so, we generated a null distribution of the expected cost of translation for each gene when there is no adaptation to reduce the cost of nonsense errors by permuting the order of synonymous codons. These permuted sequences had the exact same amino acid sequence and codon usage as the sequence observed in the genome, but differed in the order of synonyms. For each permutation, we calculated its expected protein production cost using Eqn. (3). We then compared the mean protein production cost of our population of permuted sequences to that of the protein production cost of the observed sequence. We then scored each observed sequence as having a protein production cost either greater than (0) or less than (1) the expected cost under the null hypothesis. To test whether the observed sequences are biased towards lower costs than expected under the null hypothesis, we performed a binomial test (H₀: p = 0.5). This allowed us to test if the number of sequences with an expected cost less than the mean expected cost across the permuted sequences was greater than expected by chance. Because the gross cost of nonsense errors scales with the gene’s protein production rate, we performed a logistic regression to test if the probability that a gene has a lower cost than expected increases with gene expression.

Analysis of additional S. cerevisiae ribosome profiling data

To complement our analysis of the Weinberg et al. data, which is generally considered to be high quality, we also analyzed data from two more recent ribosome profiling experiments: Wu et al. 2019 (SRA: SRR7241903) [29] and Ferguson et al. 2023 (SRA:SRR23242245, SRR23242246) [53] for comparison. We also analyzed a wild-type replicate (SRA:SRR5766382) and an elp1 deletion strain replicate (SRA: SRR5766388) from Chou et al. 2017 [28] to confirm that changes to tRNA function (in this case, deletion of a tRNA modification enzyme) primarily impact elongation rates c and not NSE rates b. These datasets were processed using the riboviz2 pipeline, with minor differences in the read lengths selected and in defining the A-site (see S1 Text). Finally, to estimate stop codon “NSE rates,” we introduced a slight modification to the Weinberg et al. data, allowing up to 15 codons in empirically determined 3’-UTRs (obtained from [54]).

NSE rates vary across codons

Theoretical and computational studies often ignore nonsense errors or assume a uniform (background) NSE rate b across all codons [21,30,33,34]. However, empirical studies indicate nonsense errors occur at an appreciable frequency [9,32], with targeted studies suggesting background NSE rates b could vary across codons [35,36]. This naturally leads to two questions: (1) do we see evidence of nonsense errors in ribosome profiling data, and (2) do we see evidence that NSE rates b (and not just NSE probabilities) vary by codon? To answer question (1), we compared the most complex model (NSE rates b vary across codons) to the simplest model (no nonsense errors) using a standard model comparison approach based on the Deviance Information Criterion (DIC) [50,58]. We find that the variable NSE rate b model better fits the ribosome profiling data compared to the no nonsense error model by 2381 DIC units. This indicates that the effects of nonsense errors are detectable in ribosome profiling data and that changes to ribosome density along transcripts are not solely due to differences in codon waiting times.

To answer question (2), we compared the model allowing NSE rates b to vary across codons to a model that assumed all codons had a uniform NSE rate. The uniform NSE rate model still allows for differences in NSE probabilities Pr(NSE) through differences in codon elongation rates. We find that the model allowing for variation in NSE rates b across codons is 187 DIC units better than the model assuming uniform NSE rates, indicating strong support for differences in the NSE rates across codons. This suggests the existence of factors of codons aside from elongation rate that alter the probability of nonsense errors Pr(NSE). NSE rates b varied over multiple orders of magnitude, ranging from 8.87 × 10⁻⁶ to 1.98 × 10⁻³ (Fig 2A). When accounting for differences in elongation rates c, the mean NSE probabilities Pr(NSE) across codons were on the order of 10⁻⁴, consistent with previous estimates in E. coli and S. cerevisiae [19–23].

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Fig 2. The background NSE rates b vary across codons.

NSE rates b are reported on a log₁₀ scale. (A) Posterior means and 95% highest density intervals (HDIs) of NSE rates b estimated for each codon. Colors indicate the number of nucleotide mutations away a codon is from a stop codon. Solid and dashed black lines indicate the NSE rate b posterior mean and 95% HDI for a model fit sharing information across codons (i.e., assuming no variation in NSE rates). To be consistent with our previous work, we separated the amino acid serine into two blocks denoted S and Z. Bolded x-axis labels indicate codons that are frameshift competent [42]. (B) Comparison of NSE rates b with empirical missense error probabilities estimated from mass spectrometry data [52]. The Spearman rank correlation is reported. (C) Comparison NSE rates b between 11 frameshift competent codons (as defined in [42]) and the 50 other sense codons. A Welch’s two-sample t-test is reported. (D) Regression coefficient estimates from a weighted multiple regression of codon properties and the NSE rates b. Variables include the number of stop codon neighbors at each position, whether or not the codon was prone to frameshifts (frameshift competent), and the missense error probability of the codon. The intercept reflects the mean NSE rate b of the reference class of codon. Error bars represent 95% confidence intervals. An * indicates statistical significance (p < 0.05).

https://doi.org/10.1371/journal.pgen.1012162.g002

Importantly, our model is agnostic to the specific mechanisms associated with nonsense errors. Notable mechanisms that can lead to nonsense errors are release factor binding of sense codons, peptidyl-tRNA drop-off, and frameshift errors [59]. Mismatches between the codon and anticodon resulting from near-cognate or even non-cognate tRNA binding (i.e., missense errors) can increase the chances of peptidyl-tRNA drop-off and frameshift errors [40,43,60]. Consistent with this, we find that our NSE rate b estimates from ribosome profiling are positively correlated with codon-specific missense error probabilities estimated from a large number of mass spectrometry measurements in S. cerevisiae (Spearman rank correlation , , Fig 2B) [52]. Previous work in S. cerevisiae identified 11 sense codons that were particularly prone to causing frameshifts when located in the P-site of a ribosome, particularly when followed by a slow codon [42]. Based on our model estimates, these 11 “frameshift competent” codons had generally higher NSE rates b than the other 50 sense codons (on the log₁₀-scale, mean of -3.58 vs. -4.44, Welch’s two-sample t-test, p = 0.0015, Fig 2C).

To better understand how these factors associated with higher NSE rates b contribute to the observed variation, we regressed our estimates against the number of stop codon neighbors (the number of stop codons that are a single nucleotide mutation away from a codon), the average missense error probability, and whether or not the codon is frameshift competent (Fig 2D, S3 Table). We note that we treated the number of stop codon neighbors as a categorical variable, as it is always either 0, 1, or 2. As this is regressing multiple categorical variables against the log₁₀(NSE rate b), each of these coefficients represents a change relative to a reference class defined by the intercept.

One might expect that having more stop codon neighbors increases the NSE rate b of a codon, as there would seem to be a greater chance of being mistakenly recognized by a release factor. Indeed, we find that codons with a single stop codon neighbor at the 3rd position of the codon have an NSE rate b almost 1 order of magnitude greater than codons with no stop codon neighbors at the third position (, p = 0.0012). We did not observe such a relationship for codons with 2-stop codon neighbors at the third position (, p = 0.884). In contrast, codons with a stop codon neighbor at either the 1st or 2nd nucleotide position showed no significant difference in NSE rates b compared to codons with no stop codon neighbors at these positions. This likely reflects that the ribosome more closely monitors codon-anticodon base pairing at the first 2 positions [61]. Consistent with the independent tests (Fig 2B,2C) and in line with the suspected role of missense errors in contributing to peptidyl-tRNA drop-off and frameshifts, we find a positive effect of a codon’s missense error probability (, ) and frameshift competency (, ) on NSE rates b.

Nonsense errors are an unlikely explanation for the “5’-ramp”

Ribosome profiling measurements often exhibit increased ribosome densities at the 5’-end of the CDS relative to the remainder of the mRNA [27]. We simulated ribosome profiling data based on the PANSE model using the parameters estimated from the real data. We observe a moderate correlation between the real and simulated ribosome counts (Spearman rank correlation , . On a position-by-position basis (ignoring the first 200 codons), the average log-fold difference between the real and simulated data is approximately 0.015 (One-sample t-test , S5 Fig), suggesting our PANSE model slightly underestimates the number of counts by about 1.5%, on average.

Based on the metagene profile of the ribosome densities for the real and simulated data, we observe good agreement in the post-200^th codon region (Fig 3A, unshaded region). This is expected because this was the region used for calculating the likelihood of the data during model fitting. In contrast, the model poorly predicts ribosome densities near the 5’-end (a.k.a the 5’-ramp region), which were excluded during the model fitting. The gradual decrease in ribosome densities along the first 200 codons is far more drastic in the real data than expected based on the model parameters estimated using the remainder of the CDSs (Fig 3A, shaded region). This suggests other factors are at play in the 5’-ramp that impact ribosome densities, including that this is a technical artifact [27].

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Fig 3. The effects of increased ribosome density at the 5’-end of coding sequences.

(A) Comparing metagene profiles for real ribosome profiling data (purple) and simulated ribosome profiling data based on the PANSE model (yellow). The first 200 codons are shaded to emphasize these codons were excluded in the likelihood calculation during model fitting. (B) Comparison of NSE rate b estimates for each of the 61 sense codons based on either the first 200 codons or the remainder of the gene.

https://doi.org/10.1371/journal.pgen.1012162.g003

The discrepancy between real and simulated data raises the question of the necessary NSE rates b to generate the dramatic drop in ribosome density at the 5’-end. We fit PANSE to only the first 200 codons to determine if the parameters estimated were biologically plausible. Although initiation rates and elongation rates c are in good agreement with estimates from the post-200 codon fits (S6 Fig), the NSE rates b are significantly greater, with a mean NSE rate of approximately 0.003 across codons (Fig 3B). These higher NSE rates b translate to a higher average NSE probability Pr(NSE) of approximately 0.004 across the 61 sense codons. An NSE probability Pr(NSE) of 0.004 per codon means only 45% of initiated ribosomes are expected to make it to the 200^th codon, which is unrealistically low. Taken together, our results suggest the 5’-ramp in ribosome profiling data is, at best, only partially due to nonsense errors.

The probability that translation is completed varies greatly between transcripts

Based on our best model fit, codons differ in their elongation rates c and NSE rates b, meaning they also differ in their NSE probability (i.e., )). As such, variation in codon usage across genes is expected to lead to variation in the probability of a ribosome completing translation . Across all genes, the median probability of a ribosome completing translation is approximately 0.92 (interquartile range 0.87 – 0.95). A key factor determining the probability of a ribosome completing translation is the number of codons within a transcript’s CDS. Even if the probability of a nonsense error at any given codon is rare, longer CDSs afford more opportunities for a nonsense error. The length of a CDS and its probability of experiencing a nonsense error are highly correlated, as expected (Fig 4A, Spearman rank correlation , ). Unsurprisingly, the S. cerevisiae proteins YHR099W, YKR054C, and YLR106C are the only transcripts considered with a greater than 50% chance of a ribosome experiencing a nonsense error. This likely has little to do with codon usage, as their respective transcripts have > 3700 codons, such that there are many opportunities for any given ribosome to drop off during translation.

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Fig 4. Variation in the probability of a single ribosome completing translation .

Colors represent from ROC-SEMPPR (i.e., ) for each gene. (A) Relationship between CDS length (in number of codons) and the probability of a nonsense error occurring, . (B) Comparison of the probability of no nonsense error occurring with simulation-based estimates of drop-off resilience [34]. The size of the point represents the length (in number of codons) of the gene’s coding region.

https://doi.org/10.1371/journal.pgen.1012162.g004

A previous study estimated the probability that a ribosome completes translation using a Totally Asymmetric Exclusion Process (TASEP) simulation parameterized by polysome profiling data (translation initiation rates), tRNA counts (codon-specific elongation rates), and a single NSE rate estimate from E. coli [34]. We compared our estimates of translation completion probability based on a model fit to empirical data to theoretical expectations based on this TASEP simulation (referred to as “drop-off resilience”), finding them to be highly correlated (Fig 4B, Spearman rank correlation , ). We still observe a high correlation between the empirical and theoretical estimates when conditioning on CDS length (partial Spearman rank correlation, , ), indicating length is not the only cause of the similarity between the two estimates of translation completion probabilities. Estimates of from PANSE are generally greater than expected based on these TASEP simulations, which assumed a uniform NSE rate b across all sense codons. The NSE rate b used for the TASEP simulations was taken from a study that implicitly assumed the drop in ribosome density along transcripts was solely due to nonsense errors [21]. As we and others [22] have shown, the 5’-ramp region inflates estimates of NSE rates. Regardless, the high correlation between and theoretical expectations suggests our PANSE model generally captures the across-transcript variability in the probability a ribosome completes translation based solely on empirical data.

Evidence supports adaptation to reduce nonsense errors

As protein synthesis is energetically costly, natural selection is expected to result in genome-level patterns consistent with adaptations to reduce the costs of nonsense errors. We specifically examine two key adaptations: (1) the reduction in frequency of codons with higher NSE probabilities Pr(NSE) along a CDS (5’ to 3’ direction), and (2) an anti-correlation between the frequencies of codons with higher NSE probabilities and gene expression.

Evidence that adaptation increases with position.

As the energetic investment in producing a protein increases as amino acids are added to the peptide chain, natural selection against nonsense errors is expected to be weakest near the start of translation, resulting in position-specific patterns of codon usage in which nonsense error-prone codons are biased toward the 5’-end of a gene’s CDS [17,18,30,33,62]. As such, codons with higher NSE probabilities Pr(NSE) are expected to be enriched in the 5’-ends relative to the remainder of the CDS. To control for the different lengths of CDSs, we assigned codons to the 5’-end based on their relative positions (i.e., their actual position divided by the number of codons in a given CDS) using a relative position cutoff of 0.25. The “middle” of a gene’s CDS was defined as codons falling between 0.25 and 0.75. By comparing synonymous codon frequencies in the 5’-ends to the middle region, we identified codons enriched at the 5’-end (one-sided binomial test, Benjamini-Hochberg adjusted p < 0.05). As expected, codons enriched in the 5’-ends have higher Pr(NSE) than codons showing no difference between the 5’-end and the middle regions (Wilcoxon rank sum test, , Fig 5A). We observe no such pattern if comparing the Pr(NSE) of codons enriched in the 3’-ends to those not enriched relative to the middle of CDSs (Wilcoxon rank sum test p = 0.67, Fig 5B). We obtain a similar result if we define the termini as the first and last 100 codons of each gene’s CDS (S7 Fig). We emphasize that the first 200 codons of each CDS were not considered in the actual parameter estimation (i.e., were not considered in the likelihood calculation).

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Fig 5. Selection against nonsense errors is weakest at the 5’-ends.

(A) Comparison of codon-specific Pr(NSE) (on the log scale) of codons enriched in the 5’-end (the first 25% of codons for each sequence) vs. those not significantly enriched relative to the middle regions of the CDSs. Wilcoxon rank sum test p-value reported. (B) Same as in (A), but using the last 25% of codons. (C) Geometric mean of the probability of successful elongation by the ribosome up to the 500^th codon across all CDSs (purple). Solid lines and equations represent the linear regressions relating the log(Position) of a codon to the change in probability of elongation for the real sequences and the various nulls. For the null regression lines, the mean slope and the mean intercept across the 1000 permuted sequences were used.

https://doi.org/10.1371/journal.pgen.1012162.g005

Natural selection against nonsense errors is expected to strengthen along a CDS, thereby increasing the probability of successful elongation. Except for the first 10 codons, the probability of elongation increases along the CDS before appearing to plateau (Fig 5B). This increase in the probability of elongation as a function of codon position is consistent with adaptations to reduce the energetic cost of nonsense errors. Regarding the apparent decrease in the probability of elongation for the first 10 codons, we note that these results are qualitatively consistent with previous findings that observed a decrease in codon bias immediately following the start codon, followed by a gradual increase [62]. The change in the probability of elongation observed in the true sequences is much greater than observed for the null expectations generated by permuting codons (Figs 5B, S8). This is consistent with natural selection against nonsense errors being generally weaker at the 5’-end.

Evidence that adaptation increases with gene expression.

As highly expressed genes generally undergo more rounds of translation (i.e., take up a larger portion of the cell’s energy budget), a lower probability of completing translation in a highly expressed gene will lead to more wasted NTP. Thus, selection against nonsense errors should increase with gene expression, such that highly expressed genes generally have higher probabilities of completing translation . In our comparisons of CDS length and the probability a nonsense error occurs, we observed a clear gradient in this relationship: highly-expressed genes tend to have lower probabilities of experiencing a nonsense error compared to CDSs of similar length but lower expression (Fig 4A). Using mRNA abundances [27] as an independent measure of gene expression, we find a positive correlation between the probability of completing translation and gene expression (Spearman rank correlation , , Fig 6A). This analysis does not control for CDS length, indicating highly expressed genes are more likely to be successfully translated regardless of length. By using a partial Spearman correlation to control for length, we find that the correlation between gene expression and the probability of completing translation increased (partial Spearman rank correlation , ). As expected, our results are consistent with selection against nonsense errors being generally stronger in high-expression genes compared to moderate- or low-expression genes of similar length.

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Fig 6. Highly-expressed genes are better adapted to avoid nonsense errors.

(A) Comparing gene expression (mRNA abundance RPKM) with the probability of a ribosome completing translation . Histograms on the x and y-axes represent the distributions of the corresponding variables. Colors indicate the length of the CDS.(B) Comparing relative NSE probabilities Pr(NSE) among synonymous codons to selection coefficients from ROC-SEMPPR. A higher value of indicates a codon that is disfavored by selection relative to a reference codon (the alphabetically last codon for each amino acid). Error bars represent the 95% HDIs. (C) Same as in Fig 5B but separating genes into bins based on mRNA abundances (RPKMs). Solid lines represent the linear regression relating codon log(position) to the probability of elongation. Regressions (all coefficients with ) and Spearman rank correlations are reported.

https://doi.org/10.1371/journal.pgen.1012162.g006

ROC-SEMPPR assumes selection for on codon usage is uniform along a CDS, but codons with higher NSE probabilities are expected to be avoided in high-expression genes due to the increased energetic costs of experiencing frequent translation errors. Under selection against nonsense errors, we expect ROC-SEMPPR’s selection coefficients to be correlated with relative NSE probabilities (i.e., between synonymous codons). We find that and differences in NSE probabilities Pr(NSE) are well-correlated (Spearman rank correlation , Fig 6B), indicating that synonymous codons with higher NSE probabilities Pr(NSE) are avoided in high-expression genes.

Consistent with the avoidance of synonymous codons with higher NSE probabilities Pr(NSE) in high-expression genes, we observed that the probability of successful elongation by position is greater in high-expression genes (Fig 6C), suggesting these genes are better adapted to reduce the cost of nonsense errors. Independent of gene expression, we observed that the probability of successful elongations increases along the transcripts.

The energetic costs of nonsense errors are likely substantial

Although nonsense errors are rarer on average than missense errors [9,59], they may be more costly due to the high probability that the truncated protein is non-functional. We calculated the expected cost of translation (in terms of NTP) (see Materials and Methods) that accounts for the direct ATP cost of translation initiation and peptide elongation, the indirect overhead cost due to ribosome pausing, and the direct and indirect costs of nonsense errors. The direct and indirect costs associated with translation initiation and elongation are the fixed costs, as these will be incurred every time a functional protein is produced. In contrast, direct and indirect costs associated with nonsense errors are variable costs as these will only be incurred if a nonsense error occurs (see Eqn. 3). For the cost of ribosome pausing during translation C, we focus on a rough estimate based on ribosome production costs and half-lives as this estimate is more directly tied to ribosome assembly (see S1 Text). Unsurprisingly, the majority of NTP used during translation is associated with fixed direct costs of translation initiation and ribosome elongation (Fig 7A); however, this is not expected to impact the evolution of codon usage because fixed direct costs do not vary across codons.

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Fig 7. Expected cost of translation in terms of NTP across genes.

(A) Relative costs of translation in terms of direct costs (translation initiation and peptide elongation), indirect costs (overhead costs of ribosome pausing), and variable costs (cost of nonsense errors). Genes are ordered based on total energetic flux: the product of the total costs and protein production rates (as estimated by ROC-SEMPPR, i.e., ). (B) The log fold-difference between the variable costs (i.e., the direct and indirect costs of nonsense errors) and the fixed indirect cost of elongation (i.e., ribosome pausing). The mean log fold difference is -0.56 (One-sample t-test . (C) Relationship between gene expression (mRNA RPKM) and the expected cost of translation per codon. The Spearman rank correlation and the associated p-value are reported. (D) Relationship between gene expression and the probability of whether or not a gene has a lower observed energetic cost than expected based on the permuted nulls. Histograms represent the distribution of gene expression for genes with and without evidence of adaptation against nonsense errors. The solid line and the equation represent the result of a logistic regression. The logistic regression slope is statistically significant (p = 0.00024), but the intercept is not (p = 0.19). (E) Same as in (D), but with length as the predictor variable. Both the logistic regression slope () and intercept () are statistically significant.

https://doi.org/10.1371/journal.pgen.1012162.g007

Studies on the evolution of codon usage have primarily focused on the cost of pausing, either explicitly or implicitly. Based on NTP/s, the fixed indirect cost (i.e., the cost of ribosome pausing) is generally greater than variable costs (both variable direct and variable indirect costs) (Fig 7A,7B). However, variable costs (i.e., the direct and indirect costs associated with a nonsense error) are usually within an order of magnitude of the fixed indirect costs (i.e., the total cost of ribosome pausing for producing a single protein) for the majority of genes under consideration (≈86%, Fig 7B). We note this is likely a conservative estimate of variable costs. Using NTP/s, which is based on comparing ROC-SEMPPR’s selection coefficients to ribosome elongation rates c, we find the variable costs to be generally greater than the fixed indirect costs (S9 Fig). Consistent with adaptation to reduce the cost of translation, indirect costs (both fixed and variable) generally decrease as a function of total energetic flux (S10 Fig). We suspect our estimates of C_ROC and reflect reasonable bounds on the true energetic cost of ribosome pausing. Combined, our results suggest the costs of nonsense errors are comparable to the cost of ribosomal pausing.

As natural selection to reduce energetic costs is expected to be stronger in high-expression genes, we expect the cost of a gene to decrease as gene expression increases. Consistent with this expectation, we observe the expected cost per codon (i.e., , where n is the number of codons) is negatively correlated with gene expression (Fig 7C). To test for evidence of adaptation to reduce the cost of nonsense errors, we generated a null distribution for the expected translation cost for each gene based on 1000 permuted sequences of synonymous codons. These permutations could be viewed as nulls reflecting the absence of natural selection against NSEs, as permutations do not alter the codon-order invariant fixed translation costs. As codon usage for S. cerevisiae is at selection-mutation-drift equilibrium [6], we are effectively testing against a null that assumes adaptive codon usage via natural selection that varies with gene expression but not with position within a gene (e.g., natural selection against ribosome pausing). We find the true expected cost of a sequence is less than the mean expected cost of the permuted sequence for 59% of genes, which is greater than the 50% expected by random chance (two-sided binomial test, ). The average difference between the true cost and the mean cost of the permuted sequences is approximately 4 NTPs, roughly the same cost as initiating another ribosome or the direct cost of translation elongation.

We have already seen evidence that highly expressed genes are better adapted to reduce the cost of nonsense errors. Furthermore, as nonsense errors are more likely to occur in longer genes, gene length may be another predictor of adaptation to reduce the cost of nonsense errors. By classifying genes as either those that were better adapted relative to the permuted sequences (i.e., 1 if and 0 otherwise), we find highly expressed and longer genes are more likely to be adapted to reduce the cost of nonsense errors (Fig 7D,7E).

Parameter estimates across S. cerevisiae ribosome profiling datasets are consistent

Ribosome profiling, as with any high-throughput experiment, can be subject to technical and biological variation [55]. There are many protocols for performing ribosome profiling, each with possible biases [27,48,63]. To ensure that our model parameter estimates were generally consistent across measurements, we fit PANSE to data from two independent ribosome profiling measurements: Wu et al. [29] and Ferguson et al. [53]. Although these measurements were performed in S. cerevisiae, they used different protocols.

Comparing the metagene plots for the transcripts considered in each of the datasets, Weinberg et al. and Wu et al. exhibit the 5’-ramp; however, this effect is nearly absent from the Ferguson et al. data, with a slight depletion in ribosomes in the region between the 75^th and 100^th codon (S11A Fig). We emphasize that none of these reads were included in the actual likelihood estimation. The Ferguson et al. data had the fewest transcripts under consideration (1,918 as compared to 2,785 for Wu et al., and 3,112 for Weinberg et al.). The Ferguson et al. data were also much sparser (S11B Fig), even if only considering the same sequences across all datasets (S11C Fig). Despite this, we see overall good agreement between the NSE rate b estimates of these two datasets with the Weinberg et al. estimates, despite the reduced number of reads (Spearman rank correlation and for Wu et al. and Ferguson et al., respectively, S11D, S11E Fig).

Notably, the Ferguson et al. dataset has much noisier estimates (S11E Fig), but this is unsurprising given that it is a much smaller dataset. When assuming NSE rates b across sense codons when analyzing Ferguson et al., the NSE rate b is lower, but comparable, to the same analysis using the Weinberg et al. data: 9.29 × 10⁻⁵ (95% HDI: 8.27 × 10⁻⁵ − 1.03 × 10⁻⁴) vs. 1.41 × 10⁻⁴ (95% HDI: 1.34 × 10⁻⁴ − 1.47 × 10⁻⁴). Although estimates of ribosome waiting times w = 1/c (inverse of the elongation rates) are correlated with a proxy based on the tRNA gene copy number, this effect is weakest for the Ferguson et al. data (S11F Fig). Additionally, the distribution of ribosome waiting times w appears narrower than in the Weinberg et al. and the Wu et al. data (S11F Fig). Overall, our results suggest we are detecting a consistent signal of nonsense errors across these independent datasets, but the reduced number of reads in Ferguson et al. data significantly weakens this signal.