The genetic code is very close to a global optimum in a model of its origin taking into account both the partition energy of amino acids and their biosynthetic relationships

We used the Moran's I index of global spatial autocorrelation with the aim of studying the distribution of the physicochemical or biological properties of amino acids within the genetic code table. First, using this index we are able to identify the amino acid property - among the 530 analyzed - that best correlates with the organization of the genetic code in the set of amino acid permutation codes. Considering, then, a model suggested by the coevolution theory of the genetic code origin - which in addition to the biosynthetic relationships between amino acids took into account also their physicochemical properties - we investigated the level of optimization achieved by these properties either on the entire genetic code table, or only on its columns or only on its rows. Specifically, we estimated the optimization achieved in the restricted set of amino acid permutation codes subject to the constraints derived from the biosynthetic classes of amino acids, in which we identify the most optimized amino acid property among all those present in the database. Unlike what has been claimed in the literature, it would appear that it was not the polarity of amino acids that structured the genetic code, but that it could have been their partition energy instead. In actual fact, it would seem to reach an optimization level of about 96% on the whole table of the genetic code and 98% on its columns. Given that this result has been obtained for amino acid permutation codes subject to biosynthetic constraints, that is to say, for a model of the genetic code consistent with the coevolution theory, we should consider the following conclusions reasonable. (i) The coevolution theory might be corroborated by these observations because the model used referred to the biosynthetic relationships between amino acids, which are suggested by this theory as having been fundamental in structuring the genetic code. (ii) The very high optimization on the columns of the genetic code would not only be compatible but would further corroborate the coevolution theory because this suggests that, as the genetic code was structured along its rows by the biosynthetic relationships of amino acids, on its columns strong selective pressure might have been put in place to minimize, for example, the deleterious effects of translation errors. (iii) The finding that partition energy could be the most optimized property of amino acids in the genetic code would in turn be consistent with one of the main predictions of the coevolution theory. In other words, since the partition energy is reflective of the protein structure and therefore of the enzymatic catalysis, the latter might really have been the main selective pressure that would have promoted the origin of the genetic code. Indeed, we observe that the beta-strands show an optimization percentage of 94.45%, so it is possible to hypothesize that they might have become the object of selection during the origin of the genetic code, conditioning the choice of biosynthetic relationships between amino acids. (iv) The finding that the polarity of amino acids is less optimized than their partition energy in the genetic code table might be interpreted against the physicochemical theories of the origin of the genetic code because these would suggest, for example, that a very high optimization of the polarity of amino acids in the code could be an expression of interactions between amino acids and codons or anticodons, which would have promoted their origin. This might now become less sustainable, given the very high optimization that is instead observed in favor of partition energy but not polarity. Finally, (v) the very high optimization of the partition energy of amino acids would seem to make a neutral origin of the ability of the genetic code to buffer, for example, the deleterious effects of translation errors very unlikely. Indeed, an optimization of about 100% would seem that it might not have been achieved by a simple neutral process, but this ability should probably have been generated instead by the intervention of natural selection. In actual fact, we show that the neutral hypothesis of the origin of error minimization has been falsified for the model analyzed here. Therefore, we will discuss our observations within the theories proposed to explain the origin of the organization of the genetic code, reaching the conclusion that the coevolution theory is the most strongly corroborated theory.

New nonexistence results on perfect permutation codes under the hamming metric

<p style='text-indent:20px;'>Permutation codes under the Hamming metric are interesting topics due to their applications in power line communications and block ciphers. In this paper, we study perfect permutation codes in <inline-formula><tex-math id="M1">\begin{document}$ S_n $\end{document}</tex-math></inline-formula>, the set of all permutations on <inline-formula><tex-math id="M2">\begin{document}$ n $\end{document}</tex-math></inline-formula> elements, under the Hamming metric. We prove the nonexistence of perfect <inline-formula><tex-math id="M3">\begin{document}$ t $\end{document}</tex-math></inline-formula>-error-correcting codes in <inline-formula><tex-math id="M4">\begin{document}$ S_n $\end{document}</tex-math></inline-formula> under the Hamming metric, for more values of <inline-formula><tex-math id="M5">\begin{document}$ n $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ t $\end{document}</tex-math></inline-formula>. Specifically, we propose some sufficient conditions of the nonexistence of perfect permutation codes. Further, we prove that there does not exist a perfect <inline-formula><tex-math id="M7">\begin{document}$ t $\end{document}</tex-math></inline-formula>-error-correcting code in <inline-formula><tex-math id="M8">\begin{document}$ S_n $\end{document}</tex-math></inline-formula> under the Hamming metric for some <inline-formula><tex-math id="M9">\begin{document}$ n $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ t = 1,2,3,4 $\end{document}</tex-math></inline-formula>, or <inline-formula><tex-math id="M11">\begin{document}$ 2t+1\leq n\leq \max\{4t^2e^{-2+1/t}-2,2t+1\} $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M12">\begin{document}$ t\geq 2 $\end{document}</tex-math></inline-formula>, or <inline-formula><tex-math id="M13">\begin{document}$ \min\{\frac{e}{2}\sqrt{n+2},\lfloor\frac{n-1}{2}\rfloor\}\leq t\leq \lfloor\frac{n-1}{2}\rfloor $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M14">\begin{document}$ n\geq 7 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M15">\begin{document}$ e $\end{document}</tex-math></inline-formula> is the Napier's constant.</p>