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Conspiracy in bacterial genomes

The rank ordered distribution of the codon usage frequencies for 123 bacteriae is best fitted by a three parameters function that is the sum of a constant, an exponential and a linear term in the rank n. The parameters depend (two parabolically) from the total GC content. The rank ordered distribution of the amino acids is fitted by a straight line. The Shannon entropy computed over all the codons is well fitted by a parabola in the GC content, while the partial entropies computed over subsets of the codons show peculiar different behavior, exhibiting therefore a first conspiracy effect. Moreover the sum of the codon usage frequencies over particular sets, e.g. with C and A (respectively G and U) as i-th nucleotide, shows a clear linear dependence from the GC content, exhibiting another conspiracy effect.


Introduction
The genetic code is degenerate, refering to the fact that almost all the amino acids are encoded by 'synonymous' codons, and this degeneracy is primarily found in the third position of the codon. Some codons are used much more frequently than others to encode a particular amino acid, the pattern of codon usage varying between species. In the last few years, the number of available data for coding sequences has considerably increased [1], allowing for analysis to look for regularities, correlations and general features over the whole exonic region. Recently, from an analysis of the rank distribution for codons, in RNA coding sequences 1 performed in many genes for several biological species, the existence of a universal, i.e. biological species independent, distribution law for codons for the eukaryotic code has been remarked [2]. Indeed, it was pointed out that the rank of codon usage probabilities follows a universal law, the frequency function of the rank ordered codons being very nicely fitted by a sum of an exponential, a linear part and a constant. Such a universal behaviour suggested the presence of general biases, one of which was identified with the total exonic GC content (denoted throughout the paper by y GC , 0 ≤ y GC ≤ 1), which is well known to play a strong role in the evolutionary process. In fact, the values of the parameters appearing in the fitting expression, plotted versus the total exonic GC content of the biological species, were reasonably well fitted by a parabolic function of the GC content. It is worthwhile to recall that the determination of the kind of law the codon rank distribution follows is extremely interesting in the investigations of the nature of the evolutionary process, which has acted upon the codon distribution, i.e. the eventual presence of a bias.
Possible origins of codon bias have been recognized either as the result of natural selection or as the result of mutational pressure acting on the whole genome, also known as neutral evolution theory [3,4]. Experimental evidence has showed now that both effects should occur (see e.g. [5] for the influence of the biased mutation pressure in the perspective of the neutral theory of molecular evolution, [6] for exhibiting the role of external selective forces in the case of thermophilic bacteria, [7] for an analysis of genome-wide codon bias of bacterial species, [8] for a discussion on the relative role of mutational bias and translational selection for codon usage in many genomes, including bacteria ones, [9] for a model explaining trends in codon and amino-acid usage vs. GC content). Quantitative model of directional mutational pressure has been proposed to measure to some extent the relative role of selection constraints and neutrality [10,11]. Many efforts have also been made to study codon bias among genes of a given genome, in particular the influence of the gene expression level [12,13,14], of the tRNA abundance [15,16], of translational accuracy [17], see also [9] and references therein.
It seemed to us interesting to continue our previous analysis on a 'trend across species' basis by focusing on the bacteria, one main interest of the present study arising from the wide variation of the total GC content ranging from 25% to 75 %, whereas the GC variation inside a bacterial genome is much smaller. Our results is that the rank ordered distribution of the codon usage frequencies for bacteria is best fitted by a three parameters function, which is the sum of a constant, an exponential and a linear term in the rank n. Two of the three parameters depend parabolically on the total exonic GC content. As the sum, over suitable sets below defined, of the codon usage frequencies is well fitted by a straight line in the GC content, we say that conspiracy effect has to be present. Moreover, we conjecture the existence of an averaged discrete symmetry in the codon usage frequencies, reasonably 1 The DNA is constituted by four bases, the adenine (A), the cytosine (C), the guanine (G) and the thymine (T), this last one being replaced by the uracile (U) in the messenger RNA. A codon is defined as an ordered sequence of three bases. Coding sequences in RNA are characterized by their constituting codons.
confirmed by the data. We also calculate the rank ordered distribution of the 20 amino acids, which is satisfactorily fitted by a straight line in y GC .
We compute the Shannon entropy (as defined in [18]) and find that its behaviour in function of the exonic GC content is a parabola, whose apex is around the value 0.50 of the GC content, which is expected for the behaviour of the Shannon entropy for two variables. Moreover, the Shannon entropies for the codons, whose orders in rank are, respectively, in the ranges 1-15, 16-25 and 26-61, have peculiar features, which we comment below.
The study of this paper was based on a sample of 109 bacteria and 14 archaea, with a codon statistics larger than 300 000. The data were taken from Codon Usage Tabulated from GenBank [1] (see also http://www.kazusa.or.jp/codon/), release 138 for eubacteria and mainly release 144 for archaea.

Codon usage probabilities distribution
Let us define the usage probability for the codon XZN (X, Z, N ∈ {A, C, G, U}) where n XZN is the number of times the codon XZN has been used in all considered processes, for a given biological species, and N tot is the total number of codons used in the same processes. It follows that our analysis and predictions hold for biological species with sufficiently large statistics of codons. For each biological species, codons are ordered following the decreasing order of the values of their usage probabilities, i.e. codon number 1 corresponds to the highest value, codon number 2 is the next highest, and so on. We denote by f (n) the probability P (XZN ) of finding XZN in the n-th position. Of course the same codon occupies in general two different positions in the rank distribution function for two different species. By plotting f (n) versus the rank we confirm that the data are best fitted by the kind of function we found in [2], i.e. the sum of an exponential function, a linear function and a constant: In the following analysis we shall not consider the three stop codons, whose function is very peculiar. Therefore the four parameters α, η, β and γ are constrained by the normalisation condition where ξ < 1 is a number that is computed for any biological species from the codon usage frequency for the 61 encoding codons (note that the result is almost unchanged if the data are normalized to one summing over the 64 coding codons). 2 In the following, the parameters for the different fits have been computed using a best-fit procedure, the curve fit being based on the Levenberg-Marquardt algorithm [19]. The χ 2 coefficient is defined by where y i are the actual values, y ′ i are the calculated ones, y and y ′ are the means of the actual values and of the calculated ones respectively. Recall that R 2 represents the ratio of the explained variance on the total variance. Fits can be considered as satisfactory for values of R 2 0.8.
In figs. 1-4, we report the plot of f (n) as a function of n for a few bacteria: Borrelia burgdorferi (GC = 28.8 %), Bacillus subtilis (GC = 44.4 %), Escherichia coli (GC = 51.8 %), and Ralstonia solanacearum (GC = 67.5 %). One remarks that the codon rank distributions show mainly a not far from uniform or slowly decreasing behaviour for most of the codons, and a peak of a small number of highly 'overrepresented' codons (codon bias contribution). 3 Note that the fits of the rank ordered distribution f (n) by a Yule law or Zipf law are unsatisfactory. Indeed, as emphasized in [20], when the majority of points resides in the tail of the distribution, it is necessary to fit the whole range of data. From the previous discussion, we expect the parameters to depend on the total GC content of the genes region (here the total exonic GC content). We have investigated this dependence and we report in figs. 5-8 the plots of the parameters α, β, γ and η versus the total exonic GC content. In terms of the total exonic GC content y GC of the biological where the goodness of the fit being given by and where the goodness of the fit being given by The η parameter shows a 'λ-like' behaviour in terms of y GC , centered on the value y GC = 0.50, while the values of the β parameter are mainly in the range 3 10 −4 ≤ β ≤ 5 10 −4 . Note that the errors on β become large when y GC is far from the mean value 0.50. Of course we are not able to predict which codon occupies the n-th rank. Finally, let us remark that the total exonic GC content y GC has to satisfy the consistency condition where the sum is over the set J of integers to which the 56 codons containing G and/or C nucleotides belong and d i is the multiplicity of these nucleotides inside the i-th codon.

Amino-acid rank distribution
It is natural to wonder if some kind of universality is also present in the rank distribution of amino acids. From the available data for codon usage, we can immediately compute (using the bacterial code) the frequency of appearance of any amino acid F (n) (1 ≤ n ≤ 20) in the whole set of coding sequences. The calculated values as a function of the rank are satisfactorily fitted by a straight line, In terms of the total exonic GC content, the parameters B and F for eubacteria show a parabolic dependence: the goodness of the fit being given by and the goodness of the fit being given by One remarks that the most frequent amino acid is always above the line. This can be easily understood in the light of eq. (2). Indeed, the most frequent amino acids get, in general, a contribution of the exponential term of (2) with a low value of n. Of course, the frequency of an amino acid is correlated to frequencies of its encoding codons given by (2). If the ranks of the encoding codons were completely random, we do not expect that their sum should take equally spaced values, as is the case in a regression line. Therefore, we can infer, for the biological species whose amino-acid frequencies are very well fitted by a line, the existence of some functional constraints (conspiracy effect) on the codon usage. An analysis of the influence of the total GC content on the amino-acids composition of the protein of 59 bacteria has been reported in [21], where references to previous works on the subject can be found, see also [10,11]. The figures, reported in [21], qualitatively agree with the linear behavior given by eq. (13) and show sensible quantitative differences with the expected frequencies, computed on the basis of the neutral model, where the frequency of an amino acid is the sum of the synonymous codon frequencies, the frequency of a codon being computed as the product of the probability of the three nucleotides (mean field model). This analysis already hints in the direction of the presence of functional constraints.
However, the behaviour predicted by (2) fits the experimental data very well, while the shape of the distribution of amino acids seems more sensible to the biological species. In fact, one can remark on many plots of the amino-acid distributions the existence of one or two plateaux, which obviously indicate an equal probabilities of use for some amino acids. Presently, we have neither any arguments to explain the uniform distribution of amino acids from the ranked distribution of the corresponding codons nor we know of any explanation of this pattern of distribution.

The Shannon entropy
We compute the Shannon entropy, related to the codon rank distribution and plot it versus the total exonic GC content y GC for the sample of 109 eubacteria and 14 archaea, see fig. 11. The contribution of the stop codons is negligeable and represents less than 0.5% of the total entropy. The Shannon entropy is very well fitted by a parabola, where the goodness of the fit being given by Note that the parabola has its apex for y GC ≈ 0.50, which is expected for the behaviour of the Shannon entropy for two variables (here GC and AU). We have computed the partial Shannon entropies for the codons whose orders in rank are, respectively, in the ranges 1-15, 16-25 and 26-61 and we report them in fig. 11 versus the GC content. In any of the three sets, we have put the codons whose contributions to the entropy, with respect the GC content, have similar behaviour. Indeed, the codons of the first set are primarily influenced by the exponential term in the rank distribution (2), leading for the partial entropy S 1−15 to a parabolic behaviour with positive curvature and minimum at 50% GC content. For the codons of the intermediate set, the exponential term and the last two terms are of the same order of magnitude, hence the partial entropy S 16−25 is almost uniform with respect to the GC content. For the codons of the last set, the exponential term is completely negligeable, and since f (n) is very small, −f (n) log 2 f (n) ≃ f (n)/ ln 2. The trend of the partial entropy S 26−61 is thus essentially given by the behaviour of the γ parameter, hence the parabolic shape with negative curvature and maximum at 50% GC content. The fact that the Shannon entropy is a parabola shows obviously that the codon distribution is not uniform: the uniform distribution corresponds to the maximal entropy S = log 2 N , independent of the GC content, where N is the number of considered codons (here N = 61). The concavity of the entropy is also well understood in this context in terms of deviations to the uniform distribution when y GC is far from the mean value 50%. On the contrary, the partial entropy S 1−15 , which is associated to the 'overrepresented' codons, exhibits a convex shape. This feature can also be understood by realizing that in the case of y GC far from 50%, the overrepresented codons distribution would be closer to the expected value for a non biased distribution than in the case of y GC ≈ 50%.

The mystery of the straight lines
In recent papers [22,23], it has ben remarked, for 129 eubacteria and for 19 archaea, that the codonspecific nucleotide frequency P i X , i.e. the sum of the codon usage for any given nucleotide X in any fixed i-th position, that is, for example (2) (3) Figure 11: Shannon entropy and partial entropies vs. GC content: (1) total entropy, (2) partial entropy rank 1-15, (3) partial entropy rank 16-25, (4) partial entropy rank 26-61 is fitted, versus the GC content, by a straight line, with coefficients depending on the position and on the nature of the nucleotide and slightly different for eubacteria and archaebacteria (see Table 1 for eubacteria and Table 2 for archaea for the values of the slope and the axis intercept in the case of our sample of bacteria). 4 There are 12 codon-specific nucleotide frequencies, which are constrained by three normalisation conditions The fact that the sum of the codon position-specific nucleotide frequencies is a linear function of GC is a mysterious feature, which becomes more mysterious in the light of the results of Sec. 2, where we have remarked that the parameters appearing in the rank ordered frequency (2) are parabolic functions of GC. A conspiracy has to be present between the f (n), see eq. (2), for any bacteria such that the sum of f (n) over the 16-dimensional sets Q, depending on the considered bacteria and corresponding the codon position-specific frequencies produces a linear function of GC, that is Things become still more mysterious taking into accounts what we have remarked, guided by the mathematical structure of the crystal basis model of the genetic code [24]. Let us recall that in this model the four nucleotides are assigned to the 4-dim fundamental irreducible representation (1/2, 1/2) of U q→0 (so(4)) ≡ U q→0 (sl(2) ⊕ sl(2) = U q→0 (sl H (2) ⊕ sl V (2)), (the lower labels H and V denote the two commuting sl (2) 4 Note that the difference of behaviour between eubacteria and archaea could stem from the fact that eubacteria (at least in our sample) are mainly mesophilic while archaea are essentially thermophilic.
Eq. (23) can be rewritten, for instance for i = 1, as (27) and, looking at the codon matrix, we remark that the l.h.s., respectively the r.h.s., is, up to ∓1/2, the sum of the two 4 × 4 matrices obtained by mirror inversion with respect to the center of the matrix, that is the exchange C → A, G → U. 5 So we make the following conjecture. The sum of codon usage frequencies over a set of r codons (4 ≤ r ≤ 16), placed on the same side of the diagonals of the matrix, plus the mirror set is described by a straight line in y GC with a very small slope coefficient a (|a| < 0.1) and a constant term b equal to r/32. In other words, the sum of codon usage frequencies over a set of r codons, placed on the same side of the diagonals of the matrix with a plus the numerical factor −r/16, is equal, up a factor r/32, to the opposite of the sum of the codon usage frequencies over the set of r mirror codons. The mirror symmetry with respect to the secondary diagonal (resp. principal diagonal) is equivalent, in terms of U q→0 (sl H (2) ⊕ sl V (2)) to the change of the sign of the third component of J 3,H and J 3,V (resp. of the sign of the third component of J 3,V ). Denoting P i (XY Z) (resp. P i (XY Z)) the codon usage frequency of the i-th codon XY Z (resp. of the mirror codon XY Z), we make the ansatz that the following equation holds k∈I P k (XY Z) + P k (XY Z) = a y GC + r 32 (28) where I is r-dimensional set of codons placed on the same side of the principal diagonal or secondary diagonal of the codon matrix . The mirror codon XY Z with respect to the secondary diagonal (resp. to the principal diagonal) has opposite values of J 3,H and J 3,V of the XY Z codon (respectively, opposite values of J 3,V ). One can see from Table 3 that indeed the values of the axis intercept b are very close to the conjectured values r/32 (4/32 = 0.125, 8/32 = 0.250, 12/32 = 0.375). So our conjecture seems supported by the experimental data and suggests the existence of an averaged symmetry C → A, G → U or C → G, U → A for the codon usage frequencies. 6 Of course when 2r is equal to the number of the codons in the considered set, the above conjecture is trivially the normalisation condition. An analysis over our sample of 109 eubacteria, randomly drawing 4, 8, 12 codons in the 16dim set of codons with a specified nucleotide in a fixed position, shows that the straight line behaviour is surprisingly present already for 4 codons (see Table 3), where we present few examples which confirm the conjecture. In this table, N denotes the number of random drawings of n different codons belonging to a given set I (e.g. in the first line of Table 3 to behave linearly in terms of the GC content y GC , with slope a and axis intercept b. We observe that the distribution of the coefficients a and b is peaked around mean values a and b with standard deviations σ a and σ b , which are reported in Table 3. Two particular subsets have also been considered, I p = {UNN ′ , CUN, CCC, CCU, CCA, CAC, CAU, CAA, AUN, ACU, AAU} and I s = {CNN ′ , GCN, GGC, GGG, GGU, GUC, GUG, GUU, UCN, UUC, UGC}, corresponding respectively to codons above the principal and secondary diagonals of the codon matrix (26). We have summed the experimental codon usage of these codons and of the n mirror codons with respect to the secondary diagonal. As it can be read from Tables 1 and 3, no real meaningful difference appears between n < 16 and n = 16 for the first six sets I of Table 3, which correspond to fix the C or G codons (hence the A or U codons) in first, second and third position respectively. Note that the number of possible different configurations is 1946 for n = 4 or 12 and 12870 for n = 8. Surprisingly, this behavior is more evident for codons with fixed nucleotides in third position, that is for codons encoding different amino acids, strongly suggesting strong constraints between amino acids.
Finally, let us remark that such a behaviour does not show up any longer if one chooses for I sets which violates the diagonal symmetry (for example the first four lines or the first four columns of the codon matrix (26), or the whole set of codons) and considers random drawings in the same conditions as above.
So, we derive the conclusion that the sum of the codon usage frequencies over a suitable set of codons is a linear function of GC, showing a conspiracy effect between the f (n) similar to the one of eq. (24), that is where now Q is any 2r-dimensional set of codons satisfying the above discussed characteristics.
The results above were based on sums of codons of the type P (XY Z) + P (XY Z), the bar meaning the mirror codon obtained by the mirror symmetry C ↔ A, G ↔ U. It is interesting to see what happens if one considers instead sums of codons and their reverse complementary codons, i.e. P (XY Z) + P ( ZY X) where the hat means the complementary rule C ↔ G, A ↔ U (in other words, summing codons on one strand and complementary codons on the other strand). A preliminary analysis shows that the behaviour (28) also holds, but the slope and axis intercept ranges are wider and the mean slope is not close to zero. Such an analysis could be instructive because it might shed some light on the DNA strand asymmetry [28,29,30,31]. However, in the present work, we analyse the coding sequences extracted from the CUTG database, for which one does not known whether the coding sequence comes from a gene on the leading strand or on the lagging strand, and the intergenic (non coding) sequences were obviously not considered. Hence we are unable to comment on this point for the moment.

Discussion and conclusion
The distribution of the experimental codon probabilities for the total exonic region for a large sample of bacteria is well fitted by the law (2). The spectrum of the distribution is universal, but the codon, which occupies a fixed level, depends on the biological species. The universal form of the function f (n) strongly suggests the presence of a bias of very general origin. Indeed, in [32], a mutation model has been proposed where the intensity of the mutation matrix, essentially in one point mutation, depends from the variation of the labels identifying the states of the irrep. of U q→0 (sl(2) ⊕ sl(2)) describing, in the crystal basis [24], the codons. The numerically computed stationary solution of the master equation for the distribution of the 64 codons is nicely fitted by a function of the form eq. (2). The model depends from the choice of the form of the fitness function and from the values of the arbitrary parameters appearing in the mutation matrix, but unrealistic choice of these values destroys the goodness of the fit. The remark of [23] on the mystery of the straight line in the codon position-specific frequencies as function of the total exonic GC content, raises, on the light of the previously discussed form of the rank ordered distribution, an even more intriguing question: which is the mechanism which ensures that the sum of the frequencies f (n), given by eq. (2) over a set of codons, whose rank distribution generally depends on the biological species, is a linear function in y GC ? From this property, we say that there is a conspiracy between the different codons. These effects are more evident when we sum the frequencies of the codons with fixed third nucleotide, which implies constraints on the amino-acids distribution. We have also discussed in detail the structure of the Shannon entropy. It is commonly stated in the literature that the non-coding part of DNA exhibits more correlation than the coding part, which is in contradiction with what one would naively expect as the coding part is more subjected to functional constraints. The results of our work points out the existence of strong correlations in the exonic part, which very likely witness the existence of functional bias, worthwhile of further analysis, and which should be better interpreted and understood.
It has been known for some time that the plots of the first/second codon position GC content versus the third codon position GC content (or versus the total GC content) show straight line correlations [5,11,9,33]. The straight line behaviour of the codon position-specific frequencies remarked in [23] (see also Tables 1 and 2) of course implies these correlations, but is stronger. In [9], a four-parameter model is developed to explain the 64 codons and the 20 amino-acids usage frequencies in function of the GC content. Considering the sums of the eight codons with the same GC content in position (which correspond to the columns of our codon matrix (26)), a satisfactory agreement between experimental values and theoretical curves is found. The authors conclude that the GC content primarily drives the codon usage rather than the inverse. Our results, concerning the GC dependence of the codon usage, although settled on different grounds, go in the same direction.
As a last remark, let us make some comments on genes. Although the GC content varies much less inside a genome than across genomes for bacteria, it is instructive to see whether the straight line effect also occurs at the gene level. In the case of E. coli K12 for example, for which the mean GC content is 51.8%, most of the genes correspond to values of GC close to the mean value, but the tails of the P i X distributions (X = C, G, U, A and i = 1, 2, 3) show a trend towards straight lines. Surely, a more detailed study deserves attention. Analogous trend for Homo sapiens has been observed in [33].
Finally we have conjectured the existence of a quasi-symmetry with respect to the principal and secondary diagonal of the codon matrix, written in the form suggested by the crystal basis model of the genetic code [24]. Clearly these discrete symmetries can be formulated without any reference to the crystal basis, but they appear naturally in this mathematical modelisation. So, as conclusive remark, let us point out that crystal basis model seems to provide the kinematical variables useful to describe some properties of the genetic code. As it is well known, the use of appropriate variables in mathematics, in physics, and, very likely, also in biology is an essential step for facing effectively a problem.