Drummond et al. (2005) have recently introduced a theory for why highly expressed genes evolve slowly. Translation is error prone, and inactivated or misfolded proteins resulting from mistranslation impose substantial costs on the cell (Bucciantini et al. 2002), costs that increase with expression level. One way in which the cost associated with a highly expressed gene is reduced is translational accuracy (Akashi 1994, 2001), whereby the gene is encoded with optimal codons whose translation is less error prone than the translation of other codons. Translational accuracy can explain why the rate of synonymous evolution d_S is correlated with expression level, codon adaptation index, or protein abundance. However, it cannot explain why the rate of nonsynonymous evolution d_N shows an even stronger correlation with these quantities (Drummond et al. 2005). Selection for translational accuracy can reduce the translational error rate by a factor of 4–9 (Precup and Parker 1987), but even optimally coded genes that are highly expressed may produce a large amount of erroneous polypeptides. Therefore, Drummond et al. (2005) suggest that highly expressed genes should be under additional selection to be tolerant to translation errors, that is, the polypeptides produced from these genes should fold properly even if they were erroneously translated. Recent work on protein biochemistry has shown that proteins can differ widely in their tolerance to missense substitutions and that properly chosen point mutations can dramatically increase the tolerance of a protein to additional substitutions (Bloom et al. 2005). Drummond et al.'s hypothesis, referred to as selection for translational robustness, predicts a constraint on the nonsynonymous rate of evolution, whereas selection for translational accuracy predicts primarily a constraint on the synonymous rate of evolution. We must assume that both selective constraints will operate on genes that are frequently translated.

Genes that are highly tolerant to translational missense errors must also, by definition, be highly tolerant to missense substitutions. However, the translational robustness hypothesis predicts that these genes will nevertheless be strongly conserved under evolution. An example of a gene that exhibits this paradoxical behavior is Rubisco, an extremely abundant protein that fixes carbon dioxide during photosynthesis. Rubisco is strongly conserved across phyla, but appears to tolerate most missense substitutions in the laboratory without loss of fold (Spreitzer 1993; Kellogg and Juliano 1997).

Second, we studied the effect of varying α on K(A) (Figure 2). The variable α influences mainly the asymptotic limit of K(A) for small A [with K(A) increasing with α], but does not affect how quickly K(A) decays for large A.

Third, we studied the behavior of the expected fitness and the expected number of neutral sites for varying levels of A. The expected fitness is ~1 for both very low and very high A, but drops below 1 for the intermediate values of A for which K(A) starts to decay (Figure 3a). The expected number of neutral sites is αL/(α + 1) for very small A and increases to 1 for large A (Figure 3b). We can understand the different evolutionary regimes of low A and high A as follows: For low A, since very little protein is synthesized, the cost associated with misfolded proteins following erroneous translation is negligible. Therefore, the expected fitness in this regime is 1. Since the cost of misfolding is negligible, the number of neutral sites is not under selection in this regime and it settles to the value at which the mutations increasing n and those decreasing n exactly balance each other. For high A, on the other hand, the cost of translation-induced misfolding is tremendous. Therefore, at high A, the population converges to the single optimal sequence with n = L (or n = l if some sites can never be neutral). Every mutation that reduces n by even 1 is highly deleterious and therefore will virtually never go to fixation, even in a small population. For l = L, the optimal sequence (which has n = l) pays no cost whatsoever under mistranslation, and the expected fitness is again 1. For intermediate A, the cost of translation-induced misfolding is significant but not debilitating. As a result, n is elevated in comparison to its low-A limit, but the expected fitness falls below 1.

Finally, the calculation in the appendix predicts that the evolutionary rate should be independent of N if we plot it as a function of the expected number of neutral sites E[n] rather than a function of A. Figure 4 shows that this prediction is indeed accurate. We find that there are two distinct regimes for the evolutionary rate. For small E[n], the evolutionary rate increases with E[n]. This increase is caused by the increased availability of neutral mutations with growing E[n]. However, even though we can calculate what the evolutionary rate would be for arbitrarily small E[n], in equilibrium E[n] will never be below its limiting value for small A, αL/(α + 1). For large E[n], the behavior of K is reversed, and now it decreases with increasing E[n]. The decay comes about because in this regime, even though there are many mutations that do not disrupt the fold of a properly translated protein, these mutations increase the amount of mistranslated, misfolded proteins and thus are selected against. The quantity E[n] can get arbitrarily close to L (or l for m > 0), and therefore K can decay to almost zero if A is sufficiently large.

The model abstracts a previous thermodynamic model of protein mutational tolerance introduced by Bloom et al. (2005) in which mutations may leave unperturbed or destabilize the protein's native structure (common) or stabilize it (uncommon). Increases in stability tend to increase the number of sites at which substitutions can be tolerated, so-called neutral sites, while decreases in stability usually cause misfolding or decrease the number of neutral sites. In this work, we have modeled neutral sites directly. In doing so, we allow only stepwise changes in neutral sites, sacrificing treatment of large stability changes that might radically alter the number of neutral sites and the potential stability dependence of mutational effects to gain analytical tractability.

Our results show a clear example of the paradox cited in the Introduction (Figure 4): Given selection against the costs of protein misfolding, genes simultaneously become more mutationally tolerant (larger number of neutral sites, n) but evolve slower as if fewer mutations were tolerated. The paradox's resolution requires disentangling two kinds of mutational tolerance, one of which captures the likelihood of loss of protein function due to a mutation while the other quantifies the fitness cost of that mutation, the cost that ultimately determines evolutionary rate. When protein misfolding imposes minimal fitness costs, as is the case with low-expression proteins, the proportion of mutations that preserve protein function governs the rate of evolution (Figure 4, left). However, at high expression levels, fitness costs of mutations that preserve protein function grow large and can become the dominant determinant of evolutionary rate (Figure 4, right).

This observation has an important corollary. When selection for translational robustness is weak, functional loss is likely the main determinant of fitness costs. Thus, our results suggest that evolutionary conservation of sites in low-expression proteins may be more likely to indicate functional importance than similar conservation at sites in high-expression proteins.

Our simple model produces an exponential decline in evolutionary rate with increasing expression level, whereas in yeast, a power law better describes this relationship (Drummond et al. 2005). Several possibilities may explain the discrepancy. First, our model assumes a symmetric binomial distribution of the number of neutral sites, but the distribution for real proteins may be skewed or heavy tailed. Second, the cost of additional misfolded proteins may not be independent of the number of already misfolded proteins. For example, misfolded proteins form toxic aggregates (Bucciantini et al. 2002), and aggregation is not a linear function of protein concentration. Finally, differences in protein structure and function between high- and low-expression proteins may play a role. Drummond et al. (2005) have previously examined differences between functionally and structurally similar paralogs and found a similar power-law relationship. However, more subtle but important differences may separate paralogs and influence their evolution.

Our results here demonstrate that profound differences in protein evolutionary rates can arise even in the absence of functional and structural differences and when variables such as protein length, the translation error rate, and the underlying distribution of the number of neutral sites are held constant. In real genomes, of course, all these features vary and some, perhaps all, are under selection. The value of the model is its utility in explaining why highly expressed proteins evolve slowly across taxa (Drummond et al. 2005).

Interestingly, our model reveals two evolutionary-rate regimes (Figures 1 and and2),2), one in which rates remain relatively constant with increasing protein production and another in which rates decline precipitously. In yeast, virtually all genes appear to fall in the latter regime, as the evolutionary rate declines almost immediately from low to high expression (Drummond et al. 2005), raising the possibility of genome-level selection in this direction. If yeast protein synthesis levels reflect organismal needs and cannot be freely modulated, as seems likely, and protein synthesis costs dominate cellular energy consumption, as evidence suggests (Princiotta et al. 2003), the remaining genome-level target for selection is on the fitness cost per misfolded protein, c. Decreasing c pushes genes away from the decline to where cost differences become negligible, whereas increasing c pushes genes toward the decline, amplifying the cost difference between high- and low-expression proteins. One way to decrease c is to drive translation errors down to negligible levels. Another is to maintain a quality-control apparatus (e.g., chaperones and proteases) with so much excess bandwidth that cost differences associated with variability in protein misfolding become negligible. Evidence suggests that both strategies for decreasing c impose significant fitness penalties. Decreasing translational error rates can be easily accomplished, often with a single ribosomal mutation (Alksne et al. 1993), but ribosomal accuracy and growth rate are often negatively correlated, presumably through the intrinsic speed/accuracy trade-off inherent in ribosomal proofreading (Kurland 1992). Maintenance of a chaperone fleet large enough to dilute out misfolding cost differences would divert enormous cellular resources for little benefit, and the massive induction of chaperones after heat shock suggests that cellular chaperone levels do not have much remaining bandwidth under normal conditions. Overall, it seems plausible that the steep decline observed in yeast's evolutionary-rate–expression relationship reflects a balance favoring a relatively high cost per misfolded protein c. Costly translational accuracy and quality control machinery may be reduced so long as the increased errors and reduced folding assistance can be compensated for; translational robustness provides that compensation—essentially for free—but is ultimately limited by mutation pressure away from robust sequences and by the fundamental intolerance of proteins to at least some errors.

Our model distinguishes between the number of polypeptides produced per gene, x, and the abundance of functional proteins, A, yet our approximate solution essentially equates these quantities with only minor accuracy loss. The approximation works for two reasons. First, misfolded polypeptides impose a negligible cost for low-abundance proteins, while for high-abundance proteins, misfolded polypeptides are rare because of selection for translational robustness. We expect these nontrivial results to hold for many organisms. Second, in our model, the number of translation events x, the primary determinant of the number of mistranslated proteins, is estimated accurately by A, a situation unlikely to hold for most organisms. Protein abundance reflects a balance of ongoing translation and turnover (Greenbaum et al. 2003), such that a high abundance can result either from moderate translational frequency and long protein half-life or from rapid translation and short half-life. Because half-lives can vary over orders of magnitude (Hargrove and Schmidt 1989), abundance and translation frequency may only weakly correlate in real organisms. Among protein abundance, mRNA expression level, and translation frequency, we hypothesize that the latter, even though difficult to measure, will best predict evolutionary rate.

Bürger et al. (2005) recently studied a question closely related to this article, asking why phenotypic mutation rates (corresponding to the translational error rate in this article) are much higher than genotypic mutation rates. Within the framework of their model, Bürger et al. (2005) found very little pressure for reduction of phenotypic mutation rates below a certain threshold. Even though we keep the translational error rate constant in our model, we can consider a change in the number of neutral sites n as a change in the phenotypic mutation rate and thus compare our results to those of Bürger et al. (2005). In contrast to their conclusions, we find that the fraction of neutral sites, n/L, quickly rises to the maximum possible for highly expressed genes, thus reducing the phenotypic mutation rate to zero except when some sites cannot be made neutral. We believe that the differences in results are caused by differences in the way in which we and Bürger et al. (2005) treated costs of erroneously translated proteins in our models. Bürger et al. (2005) consider the total cost of protein synthesis, but do not consider additional penalties imposed by misfolded proteins, not only for their recognition and cleanup by the quality-control system but also for their innate toxicity (Bucciantini et al. 2002). Clearly, if we neglect these unique costs, then the only pressure to reduce the phenotypic mutation rate is to reduce the cost of synthesis for misfolded proteins, and this pressure will be weak if this cost is only a small proportion of the total cost of protein synthesis. In our model, on the other hand, we have focused exclusively on costs of misfolded proteins apart from their synthesis costs, implicitly assuming that the total cost of protein synthesis is approximately equal to the cost of synthesis of functional proteins and that the benefit of the functional proteins will pay for their synthesis. We believe that there is indeed a strong selective pressure to reduce the phenotypic mutation rate for highly expressed genes, but that it is cheaper for cells to evolve translationally robust genes than to evolve highly accurate transcription and translation machinery.