Amino Acid metabolism conflicts with protein diversity.

The 20 protein-coding amino acids are found in proteomes with different relative abundances. The most abundant amino acid, leucine, is nearly an order of magnitude more prevalent than the least abundant amino acid, cysteine. Amino acid metabolic costs differ similarly, constraining their incorporation into proteins. On the other hand, a diverse set of protein sequences is necessary to build functional proteomes. Here, we present a simple model for a cost-diversity trade-off postulating that natural proteomes minimize amino acid metabolic flux while maximizing sequence entropy. The model explains the relative abundances of amino acids across a diverse set of proteomes. We found that the data are remarkably well explained when the cost function accounts for amino acid chemical decay. More than 100 organisms reach comparable solutions to the trade-off by different combinations of proteome cost and sequence diversity. Quantifying the interplay between proteome size and entropy shows that proteomes can get optimally large and diverse.

Introduction

The 20 proteinogenic amino acids are present in nature in different amounts, spanning nearly an order of magnitude (UniProt Consortium 2013). The most abundant amino acid in both Swissprot and TrEMBL databases is leucine, whereas tryptophan and cysteine are the least abundant. According to statistical studies, natural protein sequences are indistinguishable from strings of amino acids chosen at random with the abovementioned abundances (Weiss et al. 2000). Amino acid relative abundances are fairly well conserved across organisms, suggesting that a single underlying principle might determine the amino acid composition of proteomes.

Some 40 years ago Dyer (Dyer 1971; Gupta 2005) suggested that protein sequences could be the result of transcription and translation of random DNA sequences. The amino acid distribution arises from the interplay between the genomic GC content, codon assignment, and redundancy of the genetic code. We will refer to this as the genetic code model and describe it in more detail below. Despite its simplicity the calculated amino acid relative abundances correlate fairly well with the observed ones, although with prominent outliers (Dyer 1971; Gupta 2005).

The “cost minimization principle” suggests that organisms minimize the cost of protein biosynthesis (Seligmann 2003; Heizer et al. 2011). A linear relationship between amino acid abundance and amino acid molecular weight or amino acid metabolic cost is supported by a reasonably high Pearson coefficient of correlation (Seligmann 2003; Heizer et al. 2011). However, the linear relationship is presented as such rather than justified from first principles (Seligmann 2003; Heizer et al. 2011) and cost minimization alone predicts that proteins would be homopolymers of the cheapest amino acid. On the other hand, natural protein folds cannot be encoded with homopolymers, as described by the energy landscape theory of protein folding (Bryngelson and Wolynes 1987). A sufficiently large alphabet is needed to encode the diversity of known proteins (Wolynes 1997, Shakhnovich 1998). Precisely how cost minimization and sequence diversity requirements balance each other is not known.

Here, we explicitly treat the trade-off between two competing forces: The minimization of the metabolic cost of amino acid biosynthesis and the maximization of the number of sequences that can be generated in a proteome from a given amino acid composition. From this basic hypothesis, we deduce a mathematical relationship between amino acid metabolic cost and the logarithm of amino acid abundances. This simple relationship describes the data remarkably better than both the genetic code model and the linear cost-abundance model.

Theory

A Linear Relationship

A naive idea suggests that the probability that an amino acid is incorporated in proteins might reflect the energetic cost of producing the amino acid (with less costly amino acids used more frequently) while maintaining the flexibility to code as many polypeptide chains as possible. Previous work suggested that the relative abundance of amino acids in proteomes is linearly related to the energetic costs of making the amino acids (Seligmann 2003; Heizer et al. 2011). Here, we suggest that it is more appropriate to look for a linear relationship between the logarithms of the relative abundances and the energetic costs. We derive this relationship through a maximization principle.

Given probabilities , representing the relative abundances of the 20 amino acids in a proteome, the number of probable peptide chains of length n in a proteome can be calculated from Shannon’s information theory as e, where is the entropy (Shannon 1948; Shannon and Weaver 1949). The average energetic cost of amino acids in a cell is , where e is the energetic cost of ith amino acid.

The maximization of the number of probable sequences in a proteome and the simultaneous minimization of metabolic cost are equivalent to maximizing the function: The maximum of this function has the property that at a given energetic cost the entropy is highest, that is the flexibility of a proteome to produce different polypeptide chains is greatest. Conversely, at a given entropy the energy consumed by producing proteins is minimized. These properties hold for any choice of units for the energies and the entropy.

Maximizing f predicts a linear relationship with negative slope between the logarithms of the relative abundances and the energetic costs. We maximize the function f by differential calculus given a constraint, namely that the sum of the relative abundances equals unity, . The gradient of the function should be a constant multiple of the gradient of the constraint, the Lagrange multiplier λ. Taking the partial derivative with respect to P of equation (1) and the constraint gives for each i: The value of the intercept can be derived from the constraint: which implies that . This gives the linear relation: between the logarithm of the relative abundance and the energetic cost referred to above, with slope −1 when the energetic cost e is given in the “correct” natural unit e. Taking the exponential of equation (2) gives the relative abundance of the ith amino acid p in terms of the costs in unit e:

The formula is reminiscent of the Gibbs distribution in physics.

The Slope of the Linear Relationship

As the “correct” natural unit e for the energetic costs e, , is not known, we can assume that the energetic costs c used in the examples below are given in terms of some other unit c satisfying ce for some , and are thus linear multiples of these theoretical e: (or e = mc) for . An important fact is that—under the linear relationship derived in the previous section—not only is the relationship linear for this other choice of unit c (i.e., for any other computed energetic cost), with slope −m instead of −1, but also the relative abundances p are invariant under this change of scale: or equivalently,

In particular, if we use energetic costs c measured in unit c, and the observed slope in terms of this unit c is −m, then letting e c we recover what we have called the “correct” natural unit e. We note that is analogous to the thermodynamic temperature in statistical mechanics. When we only have observed data, the slope of the best fitting straight-line approximating the data may depend on the scaling in some other way. That is if we multiply e by to get c, , the slope of the best linear approximation may not multiply by −m. If it does multiply by −m for all m, we say that the best straight-line approximation is scale invariant. In this article, we use the reduced major axis (RMA) regression, which is scale invariant (see Materials and Methods). As such, the predicted relative abundances are independent of the scaling of the costs.

Results

Amino Acid Relative Abundances in Proteomes

We estimate amino acid relative abundances in proteomes in two data sets. Data set DS1 was derived from 108 fully sequenced and annotated genomes from the three domains of life (Tekaia and Yeramian 2006). We translated coding regions into protein sequences and counted the frequency of occurrence of each amino acid, assuming that all proteins are equally abundant (supplementary table S1, Supplementary Material online). Data set DS2 was derived from the PaxDB database for protein abundances (Wang et al. 2012). We considered 17 organisms for which protein sequence and relative abundance data are available for more than 50% of the proteome. We used integrated data sets for the whole organism whenever possible (supplementary table S2, Supplementary Material online).

For both data sets, we tested several models for amino acid relative abundances. The results are shown in table 1, figure 1, and supplementary figure S1, Supplementary Material online, below and described in detail in the next sections.

Table 1.

Pearson’s Correlation Coefficients for Correlation of Amino Acid Relative Abundances with Amino Acid Metabolic Cost and a Model Based on the Genetic Code.

Model	DS1	DS1 (no C)	DS2	DS2 (no C)
Cost(ATP) versus abundance	−0.46	−0.51	−0.58	−0.64
Cost(ATP) versus ln(abundance)	−0.52	−0.64	−0.62	−0.75
Cost(ATP/time) versus abundance	−0.72	−0.68	−0.80	−0.76
Cost(ATP/time) versus ln(abundance)	−0.86	−0.83	−0.91	−0.90
Genetic code model versus ln(abundance)	0.71	0.76	0.62	0.66

Note.—The two columns labeled with (no C) are the results of the same calculations excluding the amino acid cysteine.

F

Correlation of the logarithm of amino acid relative abundances in proteomes with metabolic cost in units of ATP molecules per amino acid molecule (A and D), with metabolic cost in units of ATP molecules per amino acid molecule corrected by amino acid decay (B and E) and with the genetic code model (C and F). (A)–(C) correspond to data set DS1, (D)–(F) correspond to data set DS2. Data points for the amino acid cysteine are shown as empty symbols, the rest of the amino acids are shown as black symbols. The lines are RMA regressions to all data points.

Correlation of Amino Acid Relative Abundances with Metabolic Cost

We test two linear relationships between amino acid relative abundances and the metabolic cost, measured in ATP molecules per molecule of amino acid. The first linear relationship correlates (plain) relative abundances with costs, whereas the second one correlates the logarithms of the relative abundances with costs.

We used the cost estimation from (Akashi and Gojobori 2002), shown in table 2. Amino acid biosynthesis pathways are highly conserved across organisms, as indicated by the high correlation between published estimations of metabolic cost (supplementary text, table S1, Supplementary Material online, and Barton et al. 2010). Differences in cost estimations do exist, such as between aerobic and anaerobic organisms (supplementary text, table S1, Supplementary Material online). However, the main conclusions of this work are independent of the cost estimation used (supplementary text, tables S2 and S3, Supplementary Material online).

Table 2.

Amino Acid Metabolic Cost.

Amino Acid	Cost	Decay(1/time)	Cost(ATP/time)
A	11.7	1	12
C	24.7	30	741
D	12.7	9	114
E	15.3	5	77
F	52	4	208
G	11.7	1	12
H	38.3	14	536
I	32.3	2	65
K	30.3	8	242
L	27.3	2	55
M	34.3	13	446
N	14.7	10	147
P	20.3	3	61
Q	16.3	8	130
R	27.3	4	109
S	11.7	6	70
T	18.7	6	112
V	23.3	2	47
W	74.3	12	892
Y	50	7	350

Note.—Costs in units of ATP molecules per amino acid molecule are from Akashi and Gojobori (2002), costs in units of ATP molecules per amino acid molecule corrected by amino acid decay are from this work. The estimation of amino acid reactivity and decay rates (in relative units) is described in the supplementary text, Supplementary Material online.

Some organisms in DS1 and DS2 lack the biosynthetic pathways for some amino acids, rendering them essential. If an amino acid is essential, it is obtained from the environment and may be then used for protein synthesis or catabolized. Similarly, if an amino acid is not essential, it may or may not be produced by a cell. The amount of energy that can be obtained from catabolizing an essential amino acid is similar to the amount of energy that is needed for its synthesis (Swire 2007). Thus, the incorporation of essential and nonessential amino acids in proteins involves similar energy choices.

The plain amino acid relative abundances show a statistically significant correlation with the amino acid metabolic cost (in ATP units) for both data sets, with Pearson coefficients of correlation r of −0.46 and −0.58 (table 1 and supplementary fig. S1 and , Supplementary Material online). The correlation is also observed for individual organisms in DS1 and DS2 regardless of genomic GC content (fig. 2, black lines in [A] and [B]). These results are in agreement with previous proposals (Seligmann 2003; Heizer et al. 2011).

F

Correlation of amino acid relative abundances in proteomes with metabolic cost in units of ATP molecules per amino acid molecule (black line: plain abundances; blue line: logarithm of the abundances), with metabolic cost in units of ATP molecules per amino acid molecule corrected by amino acid decay (red line: plain abundances; green line: logarithm of the abundances) and with the genetic code model (dashed line). (A) corresponds to data set DS1, (B) corresponds to data set DS2. The data are shown as a function of genomic GC content in the x axis.

However, the theoretical model we put forward suggests that the correlation should improve if we consider the logarithm of the amino acid relative abundances instead of the relative abundances themselves. This is indeed the case, as the r values decrease to −0.52 and −0.62 for DS1 and DS2 (table 1 and fig. 1A and D). The correlation r values decrease for most individual organisms in DS1 and DS2 regardless of genomic GC content (fig. 2, blue lines in [A] and [B]). We conclude that the theoretical model presented here describes the data better than the previously reported empirical relationship between amino acid costs and relative abundances.

Correlation of Amino Acid Relative Abundances with Metabolic Cost Corrected by Amino Acid Decay

Amino acids undergo spontaneous chemical reactions in physiological conditions and degrade over time. Therefore, the metabolic burden of amino acids should consider amino acid decay rates as well as production cost. As the experimental determination of the particular amino acid degradation rate is an extremely difficult task and we could not find a suitable set of amino acid decay rates in the literature, we have deduced a semiquantitative reactivity ranking from previous publications and common knowledge of amino acid chemistry (described in detail in the supplementary text, Supplementary Material online). We have taken into account nucleophilicity, redox reactivity, and other biologically relevant reactions (Creighton 1983) (table 2). The physiological relevance of this proposed ranking is supported by the presence of energy-consuming enzymatic pathways that protect proteins against chemical decay (Moskovitz et al. 1997; Reissner and Aswad 2003; Stadtman 2006; Ströher and Millar 2012). When a cell divides, the offspring cells inherit the same amino acids as the parent cell had. The descendant cells have to be energy efficient on average for the descendant line to survive. Thus, the average may be taken over very long time intervals and the amino acid costs in units of ATP/time should be evolutionary relevant regardless of the proliferation rate of the cells under consideration.

Amino acid production cost and decay rates can be multiplied to yield the amino acid production cost in units of ATP/time (table 2). Plain amino acid production cost can be understood as the energy the cell spends in making a molecule of a given amino acid. On the other hand, this new quantity has units of power and can be understood as the energy the cell spends per unit of time in order to keep a constant concentration of a given amino acid, that is, the energy flux through the metabolism of that amino acid (Lotka 1922).

We reassess the relationship between amino acid relative abundance and metabolic cost, as measured by energy flux in units of ATP/time. We observe a clearly improved correlation between amino acid energy costs in units of ATP/time and both amino acid relative abundances and their logarithms (table 1). In the case of the correlation with amino acid relative abundances, the r values increase to −0.72 and −0.79 for DS1 and DS2 (supplementary fig. S1 and , Supplementary Material online), regardless of genomic GC content (fig. 2, red lines in [A] and [B]). For the correlation with the logarithm of amino acid relative abundances, the r values further rise to −0.86 and −0.91 for DS1 and DS2 (fig. 1B and E). The correlation is better for most individual organisms in both data sets regardless of genomic GC content (fig. 2, green lines in [A] and [B]). Thus, taking into account the simultaneous maximization of proteome entropy and minimization of cost improves the correlation also when amino acid costs are measured in units of ATP/time.

The amino acid cysteine is very reactive, has a low relative abundance (empty symbols in fig. 1 and supplementary fig. S1, Supplementary Material online), a low cost in ATP units and a high cost in ATP/time units (table 2). Consequently, its relative abundance is much better predicted when cost is considered in units of ATP/time (table 1, figs. 1 and 2). We have recalculated the correlations for all models excluding cysteine in order to determine whether the improvement in the r values is due only to this singular, very reactive amino acid (table 1). The main conclusions of this work are valid for the remaining 19 amino acids as well. As before, the r value improves when we consider the logarithm of the relative abundances instead of the relative abundances. Also, the r value increases when we consider amino acid costs in units of ATP/time.

We interpret that the proposed theoretical model, together with the amino acid costs in units of ATP/time, is a very good descriptor of amino acid relative abundances in proteomes. Compared with the initial proposal of a linear relationship between amino acid relative abundances and amino acid costs in units of ATP, the r value improved from −0.46 to −0.86 (DS1) and from −0.58 to −0.91 (DS2).

Correlation of Amino Acid Relative Abundances with the Genetic Code Model

The genetic code model relates amino acid relative abundance with the transcription and translation of random DNA sequences of a given GC content (Dyer 1971; Gupta 2005). To evaluate this model with DS1 and DS2, we retrieved the genomic GC content for each genome from Kryukov et al. (2012) and used it to calculate the expected relative abundances for all 61 amino acid coding triplets. We then translated the triplets into amino acids and obtained the expected amino acid relative abundances in each proteome. This metabolism-agnostic model shows a good correlation between calculated and observed amino acid relative abundances (table 1 and fig. 1C and F). The r values are 0.71 and 0.62 for DS1 and DS2. The correlation is also observed for individual organisms in the database regardless of genomic GC content (fig. 2, dashed lines in [A] and [B]). However, the r values are worse than for the metabolic flux model when amino acid costs are measured in units of ATP/time (table 1). This holds regardless of genomic GC content (fig. 2). The r value is closer to −1 for the metabolic flux model in 105 of the 108 organisms in DS1 (fig. 2A) and for the 17 organisms in DS2 (fig. 2B). This conclusion is also valid if the amino acid cysteine is excluded from the calculations (table 1). We interpret that amino acid relative abundances are better explained when we take into account the simultaneous maximization of proteome entropy and minimization of cost.

The Trade-Off between Amino Acid Metabolic Cost and Protein Sequence Diversity in Natural Proteomes

We postulate a model in which living organisms maximize a target function f that equals the entropy of the amino acid distribution in the proteome h minus the average metabolic cost of an amino acid . This gives rise to a trade-off between both terms. Figure 3 displays this trade-off for all organisms in DS1 (white symbols) and DS2 (black symbols). The figure also shows the expectation for the genetic code model (red symbols); figure 3A shows that most natural proteomes present lower metabolic costs than the genetic code model. Similarly, the entropies of natural proteomes are in the same order as the genetic code model or higher (fig. 3B). Finally, the target function f takes higher values in most natural proteomes than in the genetic code model (fig. 3C).

F

Trade-off between amino acid metabolic cost and proteome sequence diversity. (A) Genomic GC content dependence of the average metabolic cost per amino acid. (B) Genomic GC content dependence of the proteome entropy. (C) Genomic GC content dependence of the target function f. (D) Trade-off between amino acid metabolic cost (x axis) and proteome sequence diversity measured as entropy (y axis). The contour lines indicate the value for the target function, and the triangles correspond to the trade-off model using the values of m for DS1 and DS2 from figure 1B and E. All panels display the 107 organisms in data set DS1 (white symbols), the 17 organisms in data set DS2 (black symbols), and the genetic code model (red symbols). (D) includes genomic GC contents between 0.15 (lower right corner) and 0.75 (lower left corner). The y axis legend to the right of (B) and (D) illustrates the number of probable peptide chains of length 100 given by , where h is the entropy (Shannon 1948; Shannon and Weaver 1949).

Figure 3D plots the entropy h of the amino acid distribution of a proteome against the average amino acid metabolic cost in units of ATP/time. The contour lines indicate constant values of the target function f. The expectation for the trade-off model is also displayed (triangles). Interestingly, each organism reaches the value of f by a different combination of proteome entropy and cost, with the costs varying as much as 20%. The values of both entropy and cost lie within a restricted range. We interpret that the amino acid relative abundances in natural proteomes significantly deviate from the prediction of the genetic code model in a direction that simultaneously minimizes cost and maximizes sequence diversity, that is, toward a better solution to the trade-off between metabolic cost and sequence diversity.

Figure 3C and D also shows that most proteomes in DS1 and DS2 have near-constant values of the target function f. The values of f are close to the expected values for the trade-off model calculated using equations (1) and (3), the costs in table 2, and the values of m for DS1 and DS2 from figure 1B and E (triangles). This observation suggests that all organisms are close to a maximum in f, which is consistent with the maximization principle we have employed. At a maximum of f, the derivative is zero so the nearby values of the target function are nearly constant.

Discussion

Previous models for amino acid relative abundances in proteomes were based on the minimization of protein synthesis metabolic cost (Seligmann 2003; Heizer et al. 2011). However, the encoding and exploration of protein structure and function requires sequence diversity. We propose that the maximization of protein sequence diversity conflicts with the minimization of metabolic flux through amino acids in a proteome, biasing proteome composition. The mathematical formulation of this concept gives rise to a trade-off that unites the two phenomena without introducing further priors and describes proteome composition with remarkable accuracy (table 1, figs. 1 and 2).

Amino acids undergo spontaneous chemical reactions, as such the estimation of cost must take amino acid decay into account (table 2). We show that this leads to a more accurate description of amino acid distributions in proteomes (table 1). Consideration of both sequence diversity and amino acid turnover may also help in studying the relationship of amino acid metabolic cost with protein abundance (Akashi and Gojobori 2002; Swire 2007; Raiford et al. 2008, 2012), with amino acid substitution rates (Barton et al. 2010; Heizer et al. 2011) and with the sequence properties of specific protein classes (Alves and Savageau 2005; Subramanyam et al. 2006; Perlstein et al. 2007; Smith and Chapman 2010).

Amino acid abundances are fairly well conserved across organisms, yet do show some variation (Lightfield et al. 2011) that is not accounted for by the organism-independent metabolic flux model. The unexplained variability in amino acid abundances is largest for cysteine and lowest for threonine, aspartic acid, and leucine in both data set DS1 and data set DS2 (supplementary tables S4 and S5, Supplementary Material online). The performance of the model presented here is slightly worse for extreme values of genomic GC content (fig. 2). This, together with the reasonable success of the genetic code model in explaining amino acid abundances (figs. 1 and 2), suggests that taking into account both amino acid metabolic cost and the genetic code may help future studies of proteome composition. Other possible sources of across-organism variability in amino acid abundances are variations in the metabolic costs and decay rates as a function of growth temperature and oxygen tolerance. Regarding oxygen tolerance, lowering the contribution of redox reactions to amino acid decay does not improve the description of proteomes from anaerobic organisms (data not shown). In the case of cysteine, specific factors such as sulfur availability and disulfide bond formation (Beeby et al. 2005) may play a role as well. However, the low variability of the other sulfur-containing amino acid, methionine (supplementary tables S4 and S5, Supplementary Material online) does not support the importance of sulfur availability.

The model we put forward allows for a direct comparison between proteomes on a common basis (fig. 3). All natural proteomes fall along a line in the entropy-cost plane. This result arises from the observed amino acid relative abundances and the estimated metabolic costs and is independent from the mathematical shape of the relationship between abundances and costs. If the metabolic costs are organism-independent, this would indicate that there are multiple biological solutions to the entropy-cost trade-off. Some proteomes have a lower average per amino acid cost and lower sequence diversity, whereas attaining higher sequence diversity is accompanied by a higher average per amino acid cost (fig. 3).

If the distribution of amino acids is equiprobable, the average metabolic cost per amino acid is 221 in units of ATP/time (table 2). For the average relative amino acid abundances in data sets DS1 and DS2, the average metabolic cost drops to 129 in units of ATP/time. In other words, the metabolic cost of making a protein of length 100 from equiprobable amino acids is the same as the metabolic cost of making a protein of length 170 from the amino acid abundances in data sets DS1 and DS2.

How large is the reduction in proteome sequence diversity associated with this reduction in proteome cost? The number of probable proteins of length 100 is e, where h is the entropy. In the case of equally probable amino acids, nats and the number of probable proteins of length 100 is . For the average relative amino acid abundances in data sets DS1 and DS2, nats and the number of probable proteins of length 100 is . Thus, the number of probable proteins of length 100 is reduced by a factor of 105 in natural proteomes relative to the equiprobable case. In itself, this is a sharp restriction in sequence space. However, it is interesting to compare the 10125 remaining possibilities with the number of sequences explored by terrestrial life since its origin (Dryden et al. 2008). This number lies between 1020 and 1050, implying that natural proteomes are making use of only a small fraction of the available sequence space. To sum up, we suggest that the cost-diversity trade-off allows for the efficient synthesis of large proteomes while not severely restricting protein diversification.

Materials and Methods

According to Sokal and Rohlf (1995, table 15.1) and many other authors, we chose to use here the RMA regression (or least products regression) to fit the data, which is symmetric in both variables, reflects better the best line fitting the data when both variables are subject to errors and is scale invariant as mentioned in Theory. The RMA regression computes the line for m, b minimizing the function

Denoting for the means, it is known that in our case

As usual, the Pearson product–moment correlation coefficient r, , given by the formula (and satisfying that r2 equals the usual R2 coefficient of determination), is used to measure how well the data fit the line: In our case of negative slope, the closer r is to −1 the better it is.

Supplementary Material

Supplementary tables S1 and S2, figure S1, and the supplementary text are available at Molecular Biology and Evolution online (http://www.mbe.oxfordjournals.org/).