

  • Jean Lehmann ,


  • Michel Cibils,

    Affiliation: Section de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

  • Albert Libchaber



  • Jean Lehmann, 
  • Michel Cibils, 
  • Albert Libchaber
  • Published: June 3, 2009
  • DOI: 10.1371/journal.pone.0005773




A major issue about the origin of the genetic system is to understand how coding rules were generated before the appearance of a family of coded enzymes, the aminoacyl-tRNA synthetases. Each of these ~20 different enzymes has a binding pocket specific for one of the 20 encoded amino acids, and also displays an affinity for a particular tRNA, the adaptor for translation [Fig. 1(a)]. These adaptors are characterized by their anticodons, a triplet of base located on a loop. The synthetases establish the code by attaching specific amino acids onto the 3′ ends of their corresponding tRNAs, a two-step process called aminoacylation [1]. The first step (activation) involves an ATP, and leads to the formation of a highly reactive intermediate, aa–AMP (aa = amino acid). The second step consists of the transfer of the amino acid from AMP onto the 3′ end of the tRNA. Those tRNAs can subsequently participate in the translation of RNA templates, during which codons about to be translated are tested by the anticodons of incoming tRNAs. When anticodon-codon complementarity occurs, an amino acid is added onto the nascent protein through the formation of a new peptide bond [2].

Figure 1. Model of the elementary translation system.



How could a translation system operate in the absence of the synthetases? Recent works have shown that particular RNA stem-loops of ~25 bases can self-catalyze the covalent binding of amino acids onto their own 3′ ends [3], [4]. These RNAs however require aa–AMP as a substrate because they cannot manage the activation step in their present form. In addition, they show little specificity for the amino acids, raising the question of how a code could be generated by them. Some answers will likely be provided by the activation step if possible to implement on these small RNAs. This issue is not examined in the present paper.





It is assumed that the association rate constants k+ of all anticodon-codon couples are alike. This situation is expected since it already occurs in the context of anticodon-anticodon interactions [6], which is similar. In a first approximation, only complementary matchings are considered. We therefore have k+ = constante, k−(I) for codon I and k−(II) for codon II. As for the kinetics of peptide bond formation (kcat), an earlier work showed that this variable may strongly depend on the side-chains of the amino acids [5]. Accordingly, two constants (kcat(1) and kcat(2)) are defined.





With the above hypotheses, let us define the probability of codon I being translated by amino acid 1:(1)One can similarly write , and verify that the normalization condition is satisfied. It is convenient for the analysis to relate the different kinetic constants and concentrations with ratios. We denoteby defining the kinetic constants in order to have a≥1 and c≤1. Then, expression (1) becomes(2)Similarly, the probability of codon II being translated by amino acid 2 is(3)



A few general statements can be made about relation (4). With a, b, c, d>0 to be physically meaningful, the coding regime Σ cannot be observed if the amino acids are characterized by identical kcat (a = 1) or if they are at the same relative concentrations (d = 1). In both cases, the only possibility is and a = d = 1. If both codons I and II display identical k (c = 1), we also get , and .

Amino acid requirements for the coding regime

An examination of expressions (2) and (3) when b varies within the interval ]0, ∞[shows that the coding regime (4) can be satisfied only if c<1 and(5)



Figure 2. Relative abundance of primitive amino acids.



Figure 3. Effect of the local environment on a reaction rate.



Figure 4. Amino acids side-chains and kinetics of peptide bond formation.



Codons for the coding regime

Let us now examine how well any two codons (related by c = k−(I)/k−(II)) can satisfy relation (4). To a low value of c corresponds a high value of Σ (see below). The ΔG0 values of all complementary anticodon-codon interactions of the genetic code span approximately from −2 to −6 kcal mol−1 at T = 310 K [5]. These estimates refer to crude anticodon-codon interactions: they do neither include additional ΔG0 contributions occurring within modern ribosomes [9], nor the effect of tRNA anticodon loop refinement taking place in the present-day genetic system [15][16]. With Keq = k+/k = exp(−ΔG0/RT), and with k+ = constant (see above), the lowest value of c lies somewhere between 10−2 and 10−3.

A particular solution of Σ is obtained when ac = 1 which is physically relevant: for a given a, Σ increases relatively rapidly as c decreases between 1 and 1/a [see Fig. 5(a–b)]. The turning point c = 1/a sets up the maximal value of the “easy gains” for Σ, i.e. gains which do not require an unrealistic ΔΔG0 value between the two codons. When c<1/a, Σ increases less and less, until it reaches a maximal value. Furthermore, a simplification in the algebraic treatment of relation (4) occurring when ac = 1 (implying b = kcat(1)/k−(I) = kcat(2)/k−(II)) allows us to determine that has a maximum with respect to d when . Since , this maximum is also characterized by b = 1. This result shows that the above estimates of a and d are well compatible with a polymerization process in a coding regime. Taking these values (a = 102; d = 10−1) together with c = 10−2, one gets .

Figure 5. Regimes of the polymerization process.

(a) Effect of the value of the ratio kcat(1)/k−(I) on the polymerization process: p(1|I) and p(2|II) as a function of b for three significant values of d when a = 102 and c = 10−2. When d = 1, either homopolymerization of aa2 (b<1), or random polymerization (b≫1) are observed. When d = 0.1, a coding regime Σ = 0.835 is observed at the transition between two types of homopolymerization, which occurs at b = 1. When d = 0.01, either random polymerization (b≪1) or homopolymerization of aa1 (b>1) are observed. (b) Level of coding Σ as a function of c when a = 102 and d = 10−1. Three significant values of Σ are indicated. The inset shows a plot similar to (a) for three particular values of c (reported from the main graph).



Connecting amino acids with codons

The b parameter makes a connection between two kinetic constants of different origins: kcat(1) is determined by an amino acid (bound to an RNA) while k−(I) is determined by RNA. Among the two constants, only k−(I) could be tuned by the translation machinery, through the selection of the tRNA anticodon loop. Anticodons of two bases (if structurally possible) are expected to have a high k−(I) because of the low anticodon-codon ΔG0 and the importance of thermal fluctuations, while the opposite trend is anticipated for larger anticodons. An examination of Fig. 5(a) from low to high values of b reveals that a coding regime occurs at b~1, in the transition between two types of homopolymerization. Below b = 1, the outcome of polymerization is controlled by kcat; above this value, it is controlled by the relative concentrations. The transition region is sensitive to small differences in kinetic constants precisely because kcat(1)k−(I). The existence of a correlation in the genetic code reflecting a dependence between these two kinetic constants (Fig. 3 in ref. [5]) supports our analysis, and allows us to connects b~1 with 3-nt anticodons, this size being structurally associated with 7-nt loops [17].

An elementary form of the genetic code

The above analysis shows that Σ~0.8 is achieved with the two categories {Ala, Gly} and {Val, Asp} when c = 10−2. Since any two codons of the existing genetic code can be characterized by c values much closer to 1 (implying a random polymerization), it can be concluded that an initial coding system with already 64 codons and 20 amino acid could not work. For several reasons, it has been proposed that the original set of anticodons and codons was limited to 5′GNC3′, where N is U, C, G or A [18][20]. Remarkably, these four codons encode the above four amino acids, the two categories {Ala, Gly} and {Val, Asp} being associated resp. with {GCC, GGC} and {GUC, GAC}. Both codons in each category display identical anticodon-codon ΔG0 estimates, and the ΔΔG0 between the two categories is ~1.9 kcal mol−1 [5], implying a c value of ~0.045.

Fig. 5(b) shows the level of coding Σ as a function of c with a = 102 and d = 10−1. In this configuration, the maximal value that Σ can theoretically reach (c→0) is da/(da+1)≈0.91. The plot shows that Σ is sensitive to small variations of c within the region 1–0.01 (i.e. ac≥1), where it rapidly increases as c decreases. As for the “GNC system” (c = 0.045), Σ reaches a value as high as 0.77. Below 0.01, c rapidly falls to 0 as Σ tends to its maximal value.


Perturbation of the coding regime by mismatches

This Section discusses briefly the issue of mismatches. Let us consider the possibility that codon I is “read” by tRNA II and codon II is “read” by tRNA I. In Fig. 1(b), this implies that a red anticodon may also bind a blue codon and vice-versa. These interactions are called “mismatches”.

Let us assume that the association rate constant (k+) of these mismatches is identical to the one of complementary anticodon-codon interactions. The corresponding dissociation rate constant (k−(mm)) will usually be much higher than k−(II) (the highest of the two dissociation rate constants of complementary interactions). Experimental data about the G·U “wobble” base-pair however suggest that these two constants could also be similar [22].

In addition to a, b, c, and d previously defined, let us define an additional parameter: e = k−(II)/k−(mm). Let us now rewrite while including the additional terms due to mismatches:(1')

Let us consider a system in which [tRNA(I)] = [tRNA(II)] (i.e. the number of copies of the two tRNAs are identical). Following the hypothesis about the (non-selective) self-aminoacylation process described above, one has, therefore,andRewriting expression (1') with the parameters a, b, c, d, e, one gets(2')Similarly, one gets for :(3')

It can be verified that when e = 0 (i.e. the dissociation rate constant of the mismatches has a very large value), one gets the initial relations (2) and (3).

Since relations (2') and (3') cannot be further simplified, it is not straightforward to analyze Σ* with the new expressions and . However, Σ* can be numerically examined, and it turns out that the results are rather similar to those for Σ as long as e≤1.

Fig. 6 is similar to Fig. 5(a), but it includes de perturbation introduced by e. It can be seen that even when k−(mm) = k−(II) (i.e. e = 1), Σ* is not dramatically different from Σ, and when k−(mm) = 10 k−(II) (i.e. e = 0.1), the effect of mismatches on Σ* becomes totally negligible. Another way to examine the effect of the perturbation is to keep b set to 1 (with ac = 1), and establish the difference (Table 1).

Figure 6. Effect of mismatches on the polymerization process.

Probabilities p(1|I)* and p(2|II)* as a function of b for five values of e≤1 (a = 102; b = 1; c = 10−2; d = 0.1). This diagram shows that the intersection between p(1|I)* and p(2|II)* (which defines the coding regime Σ*) is only slightly affected by mismatches, as long as the dissociation constant of these mismatches (k−(mm)) remains equal (e = 1) or higher (e<1) than the highest dissociation constant of the complementary matches (k−(II)).


Table 1. Perturbation of the coding regime by mismatches.






One should consider the issue of the initiation of protein synthesis in the system described here. This step is critical since a small amino acid may only weakly stabilize the initial tRNA on the ribosome cofactor. Large hydrophobic amino acids such as Leucine or Isoleucine are possible candidates since they are found in prebiotic synthesis experiments (Fig. 2). Also, the ester bond connecting these amino acids to the 3′ end of the tRNA is less prone to hydrolysis as compare with other amino acids [29], which might be critical for initiation. Another possibility is that a dipeptide already present at the 3′ end of the first tRNA [3] may help initiate translation.

In conclusion, our results show that the properties of amino acids and RNA can naturally impose a partially coded polymerization along RNA templates. We also found that the associated coding mechanism is remarkably robust against mismatches. When supplied with “meaningful” RNA sequences, translation systems of this kind should be capable of generating pools of proteins a small fraction of which will be functional. The feed-back action of these proteins on the translation itself may further increase its efficiency, allowing more codons to be added to its repertoire. In this evolutionary perspective, it can be speculated that a critical effect of emerging synthetases will be to establish only the [amino acid – tRNA] configurations that are fit for translation, a “learning” action that RNA alone cannot logically achieve.

Author Contributions



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