Titulo: The Standard Genetic Code and its relation to mutational pressure: Robustness and equilibrium criteria.

José Luis Hernández Cáceres¹-², Rolando Hong¹, Carlos Martínez Ortiz¹, Miguel Sautié Castellanos¹, Kiria Valdés1 and Ramón Guevara Erra².

¹Bioinformatics Section, Center for Cybernetics Applications to Medicine, CECAM-ISCMH, Havana, Cuba.
²The Abdus Salam International Center for Theoretical Physics (AS-ICTP), Trieste, Italy.

Índice general


Under the assumption of even point mutation pressure on the DNA strand, rates for transitions from one amino acid into another were assessed. Nearly 25% of all mutations were silent. About 48% of the mutations from a given amino acid stream either into the same amino acid or into an amino acid of the same class. These results suggest a great stability of the Standard Genetic Code respect to mutation load. Concepts from chemical equilibrium theory are applicable into this case provided that mutation rate constants are given. It was obtained that unequal synonymic codon usage may lead to changes in the equilibrium concentrations. Data from real biological species showed that several amino acids are close to the respective equilibrium concentration. However in all the cases the concentration of leucine nearly doubled its equilibrium concentration, whereas for the stop command (Term) it was about 10 times lower. The overall distance from equilibrium for a set of species suggests that eukaryotes are closer to equilibrium than prokaryotes, and the HIV virus was closest to equilibrium among 15 species. We obtained that contemporary species are closer to the equilibrium than the Last Universal Common Ancestor (LUCA) was. Similarly, non-preserved regions in proteins are closer to equilibrium than the preserved ones. We suggest that this approach can be useful for exploring some aspects of biological evolution in the framework of Standard Genetic Code properties .

¹This paper was prepared during a working visit by JLHC to the AS-ICTP, where he is a Senior Associate, and was approved as a preprint under the number IC/2004/100 (www.ictp.it/ ?pub_off)



Results from the theory of linear differential equations have boosted many fields of natural sciences, from quantum mechanics to microbiology. Particularly useful is that result stating that for a system of elements if the linear rate of transition between elements may be defined, it is possible to determine the quantities (proportions, concentrations) of each element after a long time of spontaneous evolution . For computing these "equilibrium" concentrations, the state of the system at any given moment is not needed to be known. This result finds application in such areas as the mass action law for chemical equilibrium. In this paper an attempt is made to apply the same ideas to the transition between the codons corresponding to amino acids in a codifying region of DNA. Since the bulk of naturally occurring mutations are point mutations involving only one letter change in the codon, the first version of our model will consider this type of mutations only 1. As it is known, given the Standard Genetic Code (SGC), a point mutation may lead or not lead to a change in the amino acid codified by the unmutated codon. Under certain plausible assumptions it is possible to mathematically model the process. Not only amino acids are codified by the SGC, and there is no especial rule for the transitions, thus we regard the stop command (Term), codified by three codons, as a "virtual amino acid".
Hopefully, this model can shed some light into certain aspects of biological evolution using relatively simple ideas. Thus, for example, the degree of closeness to equilibrium may provide information about the pace at which evolution is proceeding. One may also try to predict how this evolution will continue, or even how the amino acid composition at the eve of evolution looked like.
On the other hand, most of the attempts to characterize the genetic code consider it robustness with respect to mutations 1-4. However, an absolutely robust code will leave no room for evolution and biodiversity 4. Measures for the robustness of the SGC have been proposed by others as a way to show its optimality 1-2, especially in the framework of trying to understand how the genetic code appeared on Earth. The approach introduced here may serve for evaluating both SGC's robustness, as well as its evolution potential.


Assumptions of the model.

We assume the following:
1. If an amino acid is codified by more than one codon, each synonymic codon is used with the same frequency. In a coming section, the case will be considered of the effects of unequal codon usage.
2. All point mutations occur with the same probability. It means that the chance for a change from a basis k into another basis l in the DNA is identical to any other base change.
3. Mutation probability remains the same all along the DNA strand. No preferences for position in the codon are considered.
4. The stop command (Term) is formally regarded as an "amino acid" since there is a probability in the space of codons for any amino acid to become a stop command and vice-versa via a point mutation.
5. The transition rate is proportional to the number of point mutations leading from the ith amino acid toward the jth one.

Arguments supporting the plausibility of some of the assumptions (1-5), may be found in the paper by Maeshiro and Kimura 1.

In our analysis, all possible mutations are considered for each position at the codon. A change in a single nucleotide of the codon leads to a codon change, but not necessarily to a change in the coded amino acid. Thus the mutation correspond to a "change" from phenylalanine (Phe) to phenylalanine; in other words, to a silent mutation. Otherwise, the mutation corresponds to a change form phenylalanine (Phe) to leucine (Leu).
Once all possible mutations for each position at the codon were taken into account, via simple addition of corresponding amino acid changes we obtain the transition matrix .
Furthermore, for a given amino acid its concentration will change according to:

At equilibrium, the "concentrations" will reach constant values, thus , and the system is easily workable out on the following constraint:


Constructing a new matrix from the elements of , and computing the corresponding determinants, it is possible to find out .

Data and programs
Data were downloaded from the codon usage database at www.zakusa.jp. Special programs were developed by one of us (CMO) for estimating the transition matrix M from the data, as well as for computing equilibrium composition and the real amino acid composition of each species.


The initial part of this section is devoted to the properties of the transition matrix M and its relevance for the equilibrium concentrations. If the case were that all elements of the matrix M are identical ( =1, for all i, j), then at equilibrium the concentration of each amino acid will be constant: = (1/21) =0.0476.
Taking into account (1), it is easy to show that the farther the system is from equilibrium the faster will be its evolution towards it.

Some properties of the transition matrix.
Some easy to interpret properties may be drawn from the matrix M (Table I). As apparent, nearly 25% of all point mutations are silent (corresponding to diagonal elements of M). Roughly 48% of all mutations are silent or toward an amino acid from the same class (e. g. from Valine to Alanine, of from Glutamate to Aspartate and vice versa).
The number of zeros in the matrix M (57% of the total) suggests the presence of a large amount of amino acid changes that cannot be achieved via a single mutation. This is reflected in the so-called "Matrices of Mutation Costs for Amino acids".
Matrix M is symmetric, and it endows it with some properties as it can be seen later.

In their studies, Freeland and Hurst 2 selected a huge number of hypothetical codes, with codons corresponding to different amino acids. Their codon assignment was not completely random, since they divided the ''codon space'' (i.e., the 64 possible codons) into the 21 non-overlapping sets of codons observed in the natural code, each set comprising all codons specifying a particular amino acid in the natural code (20 sets for the amino acids and 1 set for the 3 stop codons). They generated each alternative code by randomly assigning each of the 20 amino acids to one of these sets. Whereas all three stop codons remain invariant in position for all alternative codes.
As result of their procedure each of the generated "genetic codes" is topologically similar to the SGC. If one keeps the same topology (even when the coded amino acids are different), this warranties that always 25% of the mutations will be silent. In this sense, no matter how huge the selected number of codes is, it corresponds to a limited subset of all the possible codes.

Table I Matrix M obtained from the proposed model with no codon usage preference. Notice the abundance of large elements on the diagonal, corresponding to silent mutations.

Equilibrium concentrations.

A plausible property expected from M's symmetry is that at equilibrium all the = (1/21) =0.0476. At the same time, small departures from symmetry lead to an uneven distribution of amino acid compositions at equilibrium.

Effect of codon usage.

Matrix was obtained on the assumption 1 being valid. For some amino acids this seems to be the case, for example the codons corresponding to phenylalanine in Homo sapiens (differing less than in 20%). However, the synonymic representation is very uneven for some amino acids as Leucine, where the codons UUA and CUG differ in their usage by more than a factor of five.
Codon usage changes from species to species. Considering uneven codon usage makes the matrix M asymmetric, and this is reflected in the equilibrium concentrations..
We rearranged the matrix for the case of codon usage (codon usage tables were downloaded from www.zakusa.jp) on the condition that the sum of frequencies for a given amino acid equals to the number of corresponding synonyms. We realize that this is not the only possible assumption, but it may be illustrative for understanding the consequences of codon usage manipulations. The corresponding equilibrium concentrations are shown in Table II. As it can be noticed, considering the uneven codon usage can lead to noticeable departures from the even distribution corresponding to the case with identical codon usage. In the case of the Human Immunodeficiency Virus, at equilibrium, the most abundant amino acid (glutamate) exceeds the less abundant (triptophane) by a factor of 1.77. At the same time, these results suggest that the effect of codon usage in a realistic scenario never will lead to departures greater than 35% respect to the theoretical value of  0.047. To elucidate whether or not codon usage differences do play a convenient role for keeping one or another distribution of amino acids remains an open question.

Table II. Equilibrium concentration for different species. Codon usage frequencies were taken into account.

Behavior of real data.
The amino acid compositions of many species are today available at the web (e. g. at the "Zakusa" website). This allows exploring the predictions of our model with real data from many species. Here we approach the question of how far different species are from equilibrium.
Figure 1 shows the result of our analysis for 3 different species. As it can be seen, several amino acids are close to their theoretical equilibrium values. This is the case of phenylalanine, Isoleucine, aspartate, and glutamine in humans. However, some amino acids systematically diverge from equilibrium, regardless the species. This is the case of leucine, whose proportion is always higher than the equilibrium value and the stop command (Term), which always is below. An explanation for the case of the stop codon is that we are looking at functional proteins occurring in real species. If a gene is functional, it will have one or few stop codons. Otherwise it would lead to an unviable protein. Besides, special reparation mechanisms must be developed during evolution for preventing mutations leading to the stop codon. It is not excluded that a similar mechanism is operating for preventing any mutation leading to the substitution of Leucine and other amino acids that retain a high biological importance for the protein function. The need to explore the consequences of a selective mutation repairing mechanism emerges as a task related to our model.
On the other hand, for each species there is different degree of divergence from equilibrium.
We computed the area of the difference between real values and theoretical equilibrium for 15 different species.
This has been represented in fig 2. These results suggest that eukaryotes and prokaryotes are separated by this criterion. A special place corresponds to HIV, for which this distance s low compared to both eukaryotes and prokaryotes.

LUCA and preserved sequences
. The availability of large proteins data bases allows comparing proteins with similar function that are phylogenetically distant. This has allowed, via a top-down approach, to predict the better conserved sequences in evolution as well as the protein composition of the Last Universal Common Ancestor (LUCA) 5. If our model is valid, the preserved sequences 6 as well as LUCA must be farther from equilibrium respect to the non-preserved sequences and the contemporary species respectively. The data behaved in accordance with predictions, as they are summarized in table III.

Non-preserved sequences 0,368
Preserved sequences 0,5325
Modern 0,4251
LUCA 0,5236
Table III. Distances from equilibrium for contemporary and early sequences.

A closer look at the changes in amino acid composition with respect to LUCA revealed that in all but five of the amino acids the changes were in accordance with the equilibrium criteria derived from our model 7. An interesting coincidence is that the five non-corresponding amino acids according to different criteria 8 were not present in the prebiotic environment, and may be regarded as late debutants into biological evolution 7.


Attempts to make congruent the evolution theory with the fact that there are mechanisms for (hopefully unchanged) inheritance have amazed more than one mind 8. The present paper is not an attempt to cope with some of the formidable tasks posed on that field. However, a very candid view to the problem, with a simple model of the SGC under the pressure of evenly distributed mutations can outline some plausible scenarios describing where amino acid composition of the coded proteins can evolve to. Thus, our model shows how differences in codon usage can lead to different amino acid compositions at the end stage of this "evolution". According to this model real species differ in their distance from equilibrium. The fact that the HIV, a species where no or little reparation takes place during the replication process, is closer than other species to equilibrium may find an explanation. Perhaps this may provide some clues to unexplored aspects of the HIV future as a species. Becoming closer to equilibrium the species will loose its mutation potential and the variability will be reduced, perhaps attenuating one of the most difficult to fight aspect of the HIV.
In chemistry, the concept of equilibrium is clear: concentrations remain constant whereas at molecular level transitions are taking place all the time. It is not so simple to conceive what the concept of "equilibrium" introduced here can mean. It may be conceived as a stage where no further changes in amino acid composition will take place, though mutations in individual proteins will be there. However, the overall rate of amino acid changes will be much lower. This does not mean that there will be no changes in species' phenotypic properties. The example of sickle cell anemia illustrates how substantial the changes induced by a single point mutation can be.
The study of real data has allowed to describe how amino acids are being changed in different proteins 10. The "PAM" and "BLOSOM" Matrices may be regarded as very useful tools summarizing how these mutations occur. From our model it is possible to predict how a PAM matrix will look like. The degree of similarity between prediction and reality can tell us about the validity of this simple assumptions as well as the possible weight of other factors not considered in it.
The origin of the genetic code has been a matter of discussion already for almost 40 years 9, 11. Arguments and results found so far can either support or disregard any of the hypotheses available. Our model cannot clarify this question, however, other questions, as that of the putative amino acid composition millions of years ago can be treated in the framework of our model. With knowledge about the real mutation rates it is also possible to suggest the future evolution of the amino acid composition for each species.
Thus, we find that the SGC is endowed not only with properties that allow minimal changes under the effect of mutation load. It also can provide useful information about important aspects of evolution (as the case is of amino acid composition of proteins), as well as the pace at which these changes are taking place under conditions of unchanged mutation load. Other aspects, as a description of the expected evolution of amino acid composition or its previous behavior may be outlined in the framework of this approach. Our predictions, compared with data obtained by other methods about the composition of early biological forms are encouraging at this stage.

Acknwledgements: JLHC is a Senior Associate of the AS-ICTP and would like toacknowledge the Centre for financial support. Authors thank professors Julian Chela-Flores and Roberto Cruz-Rodes (From the AS-ICTP) for encouragement and suggestions, and Ricardo Franklin (Nuclear Physics Faculty at Havana) for useful discussions. This work was prepared during JLHC's stay at the AS- ICTP in September 2004.




1. Maeshiro, T and Kimura M. The role of robustness and changeability on the origin and evolution of genetic codes. Proc. Natl. Acad. Sci. U S A. 28; 95 (9): 5088-5093 (1998)

2. Freeland, S. and Hurst, L. The genetic code is one in a million. J. Mol. Evol. 47, 238:248 (1998).

3. Ardell, D. On error-minimization in a sequential origin of the standard genetic code. J. Mol.Evol. 47, 1:13 (1998).

4. V.R. Chechetkin Block structure and stability of the genetic code Journal of Theoretical Biology 222, 177-188 (2003).

5. Carl Woese The universal ancestor. Proc. Natl. Acad. Sci. U S A. 95, 6854-6859 (1998).

6. Arun K. A., Kuo-Bin L. and Praveen I. Rapid detection of conserved regions in protein sequences using wavelets. In Silico Biology 4, 0013 (2004);

7. D. J. Brooks and J. R. Fresco. Increased Frequency of Cysteine, Tyrosine, and Phenylalanine Residues Since the Last Universal Ancestor. Mol. Cell. Proteomics 1, 125 - 131 (2002).

8. D. J. Brooks, J. R. Fresco, A. M. Lesk, and M. Singh. Evolution of Amino Acid Frequencies in Proteins Over Deep Time: Inferred Order of Introduction of Amino Acids into the Genetic Code. Mol. Biol. Evol., 19, 1645 - 1655 (2002).

9. Crick, F. The origin of the genetic code. J. Mol. Biol. 38, 367-379 (1968).

10. Henikoff S, Henikoff JG.Amino acid substitution matrices from protein blocks. Proc Natl. Acad. Sci. U S A.; 89:10915-10919 (1992).

11. DiGiulio, M. Reflections on the origin of the genetic code: a hypothesis. J.Theor. Biol. 191, 191:196 (1998).


Fig 1. Comparison between real amino acid composition (cyan), and equilibrium composition with (yellow) and without (dark blue) codon usage correction. From top to bottom: HIV, Human and the bacterium Agrobacter.

Figure 2. Distance from equilibrium for a group of species.