## Prime Number

As regards the genetic code, one should juxtapose the abstract syntax of the arithmetical language with some well-defined sets of units in the genetic code. In other words, these units should be added and their sums within each set should be written temporarily as we are accustomed in decimals. Theoretically, some original number notations should replace these decimals as soon as we uncover the genetic code's primordial number system, if there is one. The art of number notation since the Babylonian sexagesimal system till our computer codes should help us in solving this problem. However, anyone who follows this approach would find unexpectedly that the choice among a variety of number systems is no longer necessary. Surprisingly, the genetic code really privileges a number system and, even more unusual, the system is the decimal one. This means that the decimal system is probably of the same age as the genetic code.

Arithmetic provided the genetic code with artifactural differences through a particular feature of one of its criterions for divisibility. As soon as one discovered the decimal system, all criterions of divisibility arose instantly and became attributes of the system. The decimals divisible by prime number (PN) 037 in Fig. 2 show the particular feature that attracts the attention of a reckoner looking for some numeration system in the genetic code. This feature exists in the decimal system and vanishes in other systems. For instance, the particular notations of decimals 111,

Fig. 2 The place-value decimal system represented through digital symmetry of the numbers divisible by prime number (PN) 037. This arithmetical syntactic feature is an innate attribute of the genetic code. The PN 037 notation with a leading zero emphasizes zero's equal participation in the digital symmetry. Numbers written by identical digits are devised by PN 037 x 3 = 111 and appear regularly. The sum of identical digits within each notation gives the quotients of these numbers divided by PN 037. Analogous summation for the numbers written by unique digits is equal to the central quotient in the column. This feature allows one to perform integer division by PN 037 through addition. Numbers written by unique digits are linked by cyclic permutations in the columns. The directions of permutations in the neighboring columns are mirror symmetrical.

Consider the decimal criterion for divisibility by PN 037 on decimal number 13901085 as an example. Use a triplet frame to apportion digits among digital triplets 139, 010, and 850. Note the triplet frame is indifferent about its three possible positions: insert zero(s) at the flank(s) to complete the digital triplets. The three-digit checksum of the digital triplets 139 + 010 + 850 is equal to 999. The particular notation 999 symbolizes divisibility by PN 037. Make a carry from thousand's unit, if any, into unit's unit. This operation preserves both the three-digit notation of the checksum and the criterion symbolism. For example, decimal number 21902631 has the checksum 021 + 902+631 = 1554. The carry into unit's unit changes the checksum into 001 + 554 = 555, which is the three-digit symbol of divisibility by PN 037. Therefore, only a three-digit adding machine is needed to reveal the divisibility by PN 037 of any number, irrespectively of how large this number would be. Add or take away one PN 037 additionally to those checksums that do not take the particular notation after the triplet summation. A subsequent appearance of the particular notation confirms the divisibility by PN 037, whereas its absence is evidence of indivisibility

222, and 333 look like ordinary numbers 157, 336, and 515 in the octal system. The feature not only excites our sensation of beauty but may simplify some computational procedures too. Moreover, decimalization of the genetic code may be a special case of the general computational power of genomes and their molecular machinery. In fact, the only reason for a number system to appear is for arithmetic calculations. We shall return to this issue in the final Section 17.

### 5 The Genetic Code Itself

Based on the above considerations, we present here a sort of a brief survey of the genetic code. The record in Fig. 3 is the universal genetic code. The genetic code is the only known alphabet that arose without participation of the human mind. What is more, the code's origin became a synonym of the origin of life. In fact,

Fig. 3 The universal genetic code. The genetic code contains 64 triplets of four DNA nitrogenous bases Thymine, Cytosine, Adenine, and Guanine onto which 20 amino acids and 2 syntactic signs, Start and Stop, are mapped. There are two kinds of bases, i.e. pyrimidines (T and C) and purines (A and G). Each strand of DNA molecule and its axially oriented triplets of bases have the 5' and 3' ends. The unique direction of triplet reading is 5' ^ 3'. The triplets that code for identical amino acids form a synonymic series denoted by initial brace. The names of the 20 amino acids and their trigram abbreviations are specified for each synonymic series. The amino acid molecules are shown in Fig. 4. The quantity of triplets in the series is referred to as degeneracy. The degeneracy is described by the Roman figures. There are the standard degeneracies IV, III, II, and I in the genetic code. Three amino acids serine (Ser), leucine (Leu), and arginine (Arg) have each two separate synonymic series with degeneracy II and IV, but not single series with degeneracy VI (Rumer, 1966). Beginning and termination of a protein synthesis is called Start and Stop. The sign Start is combined with amino acid methionine (Met). There is no amino acid for the sign Stop in the genetic code.

The universal genetic code has two triplets TGT and TGC that code for amino acid cysteine (Cys) and one triplet TGA that codes for Stop. There is another version of the genetic code conditionally called Euplotes octocarinatus code version (Marshal et al., 1967; Meyer et al., 1991; Grimm et al., 1998). Its triplets TGT, TGC, and TGA code for cysteine and are denoted by the closing brace. The E. octocarinatus code version is more symmetrical due to the same degeneracy within both 5' AT and 5' TG solid series. There are 24 synonymic series in the E. octocarinatus genetic code version. Both code versions are considered in this chapter there is no way to write or read any gene when no code is available. Thanks to its immutability, the universal genetic code is the most accurate information messenger from the time of genesis to the present. It is generally accepted that the code only translates any gene from a triplet sequence of DNA into an amino acid sequence of a protein. This routine work, however, is not its unique capability.

A triplet as a universal codon is a primary attribute of the genetic code. A single hydrogen atom of the glycine side chain and the tryptophan double ring as well as an "emptiness" of syntactic sign Stop are coded universally by the triplet of bases. In fact, the constant length of the codon is a prerequisite to equip the code with regular degeneracy. In its turn, the degeneracy proves to be one of the crucial systematization parameters of the new order in the genetic code.

The degeneracy possesses numerical values. These are integer ordinals IV, III, II, and I. Two rules govern degeneracy. The first says that any degeneracy resides within a solid set of four triplets beginning with the same two bases. For example, glycine in the bottom right-hand corner is coded by the solid set of four synonymic triplets GGT, GGC, GGA, and GGG. It is said that degeneracy of glycine synonymic series equals IV. There are eight solid series with degeneracy IV in the genetic code. The second rule concerns the synonymic series of degeneracy II named the broken series. A pair of such series halves its four solid triplets so that these triplets form two pairs ending with either pyrimidines or purines. For example, the solid set in the top left-hand corner is halved between phenylalanine and leucine. Phenylalanine possesses the TTT and TTC pair ending with pyrimidines T and C. Another TTA and TTG pair ending with purines A and G is passed to leucine. There is also a kind of broken series with degeneracy III and I. Methionine and tryptophan each are coded by a single triplet ending with base G.

Note here that triplet ATG, which codes for methionine, is simultaneously a syntactic sign Start. It begins any protein gene. The semantically antisymmetrical syntactic sign Stop terminates genes. Its synonymic pair of triplets TAA and TAG carefully follows the second rule though this pair codes for "emptiness." It is not a coincidence that these two syntactic signs are special symbols among routine amino acid coding. Both these signs appear below as semantic cues inserted into the genetic code similar to the cartouches with Pharaohs' names inserted in the hieroglyphics of the Rosetta Stone (see Section 13.2).