Phenotypic mutation rates and the abundance of abnormal proteins in yeast

Abstract

Phenotypic mutations are errors that occur during protein synthesis. These errors lead to amino acid substitutions that give rise to abnormal proteins. Experiments suggest that such errors are quite common. We present a model to study the effect of phenotypic mutation rates on the amount of abnormal proteins in a cell. In our model, genes are regulated to synthesize a certain number of functional proteins. During this process, depending on the phenotypic mutation rate, abnormal proteins are generated. We use data on protein length and abundance in Saccharomyces cerevisiae to parametrize our model. We calculate that for small phenotypic mutation rates most abnormal proteins originate from highly expressed genes that are on average nearly twice as large as the average yeast protein. For phenotypic mutation rates much above 5 x 10(-4), the error-free synthesis of large proteins is nearly impossible and lowly expressed, very large proteins contribute more and more to the amount of abnormal proteins in a cell. This fact leads to a steep increase of the amount of abnormal proteins for phenotypic mutation rates above 5 x 10(-4). Simulations show that this property leads to an upper limit for the phenotypic mutation rate of approximately 2 x 10(-3) even if the costs for abnormal proteins are extremely low. We also consider the adaptation of individual proteins. Individual genes/proteins can decrease their phenotypic mutation rate by using preferred codons or by increasing their robustness against amino acid substitutions. We discuss the similarities and differences between the two mechanisms and show that they can only slow down but not prevent the rapid increase of the amount of abnormal proteins. Our work allows us to estimate the phenotypic mutation rate based on data on the fraction of abnormal proteins. For S. cerevisiae, we predict that the value for the phenotypic mutation rate is between 2 x 10(-4) and 6 x 10(-4).

Figure 1. Evolving Phenotypic and Genotypic Mutation Rates

The evolution of phenotypic (solid lines) and genotypic (dotted lines) mutation rates is shown. For each of seven different values for the cost of erroneous proteins, c, we conducted ten simulation runs and calculated an average evolutionary trajectory (see text for more details). As expected, only the phenotypic mutation rate is affected by changes in c. Near u = 2 × 10⁻³ we observe an upper limit for the phenotypic mutation rate. Even large changes in c affect the phenotypic mutation rate only marginally (compare grey and brown lines). This (effective) upper bound for u is the result of a rapid, nonlinear increase in abnormal proteins as a function of u (see Figure 2). The (grey) box indicates the possible range of phenotypic mutation rates (1 × 10⁻⁵ – 5 × 10⁻³, according to Parker [5]) with 5 × 10⁻⁴ (dashed line) as a commonly used estimate for the global error rate.

Figure 2. The Amount of Erroneous Proteins as a Function of the Phenotypic Mutation Rate, u

The solid curve shows the expected numberx̄ of amino acids required to synthesize abnormal proteins according to Equation 7 with values for n_i and y_i from yeast (see Methods and Materials). The dash-dotted curve shows the total number of abnormal proteins. For better comparison, we scaled the number of proteins so that the dash-dotted and solid curves meet at u = 10⁻⁵. The dashed line shows the linear approximation tox̄ (see Equation 8). The dotted line indicates the amount (in amino acids) of functional proteins in a yeast cell, which equals 2.029 × 10¹⁰. Near u = 5 × 10⁻⁴ (the estimate for the global phenotypic mutation rate), the linear approximation begins to deviate noticeably from the exact value. A doubling of u at this point would result in more erroneous than error-free proteins. Another doubling would result in more than seven times as many erroneous than error-free proteins. This nonlinear increase is also observed if one considers the number of abnormal proteins (dash-dotted curve).

Figure 3. Weighted Averages of Protein Length (10) and Expression Level (11)

To determine which protein and expression levels are most relevant for the amount of abnormal proteins in a cell, we calculated weighted averages of protein length (solid line) and expression level (dashed line). As weights, we used the amount of expected abnormal proteins, . For small phenotypic mutation rates, highly expressed proteins are most relevant for the amount of abnormal proteins in a cell. This begins to change at u = 5 × 10⁻⁴, when lowly expressed, large proteins begin to dominatex̄. Inaccurate protein synthesis makes it practically impossible to synthesize these larger proteins error-free.

Figure 4. Relative Contribution of Each Gene to the Amount of Erroneous Proteins in Yeast if u = 5 × 10⁻⁴

The solid line shows the cumulative contribution of each gene tox̄. A steep increase, as seen here, indicates that few genes are responsible for most of the abnormal proteins in a cell. We also plot the cumulative distribution for the three components of : protein length, n_i (dashed line), number of functional proteins, y_i (dotted line), and expected amount of erroneous proteins to synthesize one error-free protein, (dash-dotted line). The curvature of the solid line is similar to the curvature of the dotted line. Hence, most of the abnormal proteins stem from few genes because these genes are also expressed at a very high level.

Figure 5. Selection for Translational Robustness as a Function of Amino Acid Substitution Rate

The top panel shows the average amount of abnormal proteins,x̄, after 500,000 cycles of mutation and selection (see text for more detail). The mid-panel shows the average log-likelihood (LL) of the m-neutralities after selection. For u ≥ 1.92 × 10⁻⁵, the average LLs are significantly lower than expected by chance. The 5 (0.1) quantile is given by −25.024 (−25.040). The total LL decreases noticeably for u _aa > 5 × 10⁻⁴. The lower panel shows the average LL of three groups of 100 proteins. We consider the 100 largest proteins, the 100 most highly expressed proteins, and the 100 proteins with the largest . For 100 proteins, the 5% (0.1%) quantile for the average LL is −25.117 (−25.215). The lower panel shows that the extent of selection for translational robustness increases nonlinearly for large proteins whereas it increases approximately linearly in the other two groups of proteins.

Figure 6. m-neutralities, v_i, After Selection for Three Different Amino Acid Substitution Rates u_aa

The data points are sorted according to the length of the proteins. The increased effectiveness of selection for higher m-neutralities in large proteins for high amino acid substitution rates is clearly visible.

Figure 7. Fraction of Preferred Codons After Selection for Three Different Ribosomal Amino Acid Substitution Rates u_r

The genes are sorted according to the length of the protein. For u_r = 1.37 × 10⁻⁴, selection introduces a codon bias only for few genes (compare with m-neutralities in Figure 6). An increase in the phenotypic mutation rate leads to more intensive selection for preferred codons in large genes than in small genes.

Figure 8. Evolution of Genotypic and Phenotypic Mutation Rates with Different Simulation Parameters (see the section Effect of Initial Values and Parameters on the Simulation Results)

For all five parameter combinations, we use c = 10⁻¹¹. Most simulations reach equilibrium within 1.5 × 10⁷ generations. For π_u = π_μ = 10⁻⁵, it was necessary to extend the simulations to 5 × 10⁷ generations. For simulations that end after 1.5 × 10⁷ generations, we plot straight lines thereafter. These lines serve as visual cues and are at the level of the last value.