BioSystems 70 (2003) 187–209

Arithmetic inside the universal genetic code
Vladimir I. shCherbak
Department of Applied Mathematics, al-Faraby Kazakh National University, 71 al-Faraby Avenue, Almaty 480078, Kazakhstan, CIS
Received 28 October 2001; received in revised form 23 April 2002; accepted 10 December 2002

Abstract
The first information system emerged on the earth as primordial version of the genetic code and genetic texts. The naturalappearance of arithmetic power in such a linguistic milieu is theoretically possible and practical for producing information
systems of extremely high efficiency. In this case, the arithmetic symbols should be incorporated into an alphabet, i.e. the genetic
code. A number is the fundamental arithmetic symbol produced by the system of numeration. If the system of numeration weredetected inside the genetic code, it would be natural to expect that its purpose is arithmetic calculation e.g., for the sake ofcontrol, safety, and precise alteration of the genetic texts. The nucleons of amino acids and the bases of nucleic acids seem mostsuitable for embodiments of digits. These assumptions were used for the analyzing the genetic code.
The compressed, life-size, and split representation of the Escherichia coli and Euplotes octocarinatus code versions wereconsidered simultaneously. An exact equilibration of the nucleon sums of the amino acid standard blocks and/or side chains
was found repeatedly within specified sets of the genetic code. Moreover, the digital notations of the balanced sums acquired, in
decimal representation, the unique form 111, 222, . . . , 999. This form is a consequence of the criterion of divisibility by 037. The
criterion could simplify some computing mechanism of a cell if any and facilitate its computational procedure. The cooperative
symmetry of the genetic code demonstrates that possibly a zero was invented and used by this mechanism. Such organization of
the genetic code could be explained by activities of some hypothetical molecular organelles working as natural biocomputers ofdigital genetic texts.
It is well known that if mutation replaces an amino acid, the change of hydrophobicity is generally weak, while that of size isstrong. The antisymmetrical correlation between the amino acid size and the degeneracy number is known as well. It is shownthat these and some other familiar properties may be a physicochemical effect of arithmetic inside the genetic code.
The “frozen accident” model, giving unlimited freedom to the mapping function, could optimally support the appearance ofboth arithmetic symbols and physicochemical protection inside the genetic code.
© 2003 Elsevier Ireland Ltd. All rights reserved.
Keywords: Genetic code; Genetic language; Arithmetic power; Natural biocomputer; Mathematical model

1. Introduction
Arithmetic is the only tool for producing information systems of extremely high efficiency. Could life,
being an information phenomenon, use arithmetic forthe sake of control, safety, and precise alterations of
its genetic texts? To answer this question, we can
E-mail address: genecodelab@hotmail.com (V.I. shCherbak).

resort to a similarity among genetic, ordinary, andalgorithmic languages (Eigen and Winkler, 1985).
Ordinary languages fuse the letters and digits into
a common alphabet of signs. A meaning is mapped
onto these signs by virtue of linguistic condition of
abstract character (Yockey, 2000). For digits, thiscondition contains the numeration system. The only
reason for the numeration system to appear is forarithmetic calculations. Hence, the presence of the0303-2647/$ – see front matter © 2003 Elsevier Ireland Ltd. All rights reserved.
doi:10.1016/S0303-2647(03)00066-2

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numeration system inside the alphabet of some language can be an indication that users of that language
know and employ arithmetic.
The genetic code is a sui generis alphabet of genetic texts. The digital nature of these texts was notedfor various reasons (Gamow, 1954; Yockey, 1992,
2000), however, computing has not been noted in
genetic text. As regards to amino acids, their digitalnature is not clear. Hydrophobicity—an important
property of the amino acids—can hardly be digitized(Jiménez-Montaño, 1993; Siemion, 1994). That is
why we have to discern a more fine and invariable embodiment of a digit in amino acids. Thus, the nucleonsof amino acids, i.e. an equivalent of their mass, and thebases of genetic supermolecules (Gamow, 1954) seem
more suitable for “balls” of a biological “abacus”.
An arithmetic power in the linguistic milieu of genetics is possible. The artificial biocomputers working
by using the genetic supermolecules were discussed
theoretically and tested experimentally (Adleman,
1994; Chen and Wood, 2000; Faulhammer et al.,2000). Continuing such ideas, one may assume thatnatural computing can exist as well. For instance,such a computing could be essence of exact geneprocessing and scrambling (Landweber and Gilbert,
1993; Landweber et al., 2000). If that is the case,some cell organelles should work as biocomputers.
Thereby, we have to discover the number systems
with which they work. We can imagine that at the
moment of the origin of the arithmetic power, the biocomputers modified the genetic code itself. Indeed,
any given alphabet and the genetic code in particularshould somehow correspond to the selected number
system. Moreover, the ordinary computers suggestthat some useful feature of the number system cantotally predetermine genetic “soft hardware”.
The number system is not a physicochemical
property and, hence, only our mind can discern thisarithmetic abstract. It was noted that the genetic code
has much in common with the ASCII code (Yockey,
2000), which is computer alphabet of algorithmic
languages. Binary-numerated and sequenced ASCII
symbols reproduce, in turn, the natural series of the
Arabic numerals, starting with a zero, as well asalphabetically, i.e. semantically, arranged Roman letters. In that way, the linguistic essence as well asinternal binary and decimal arithmetic of the ASCII
code become apparent.

Could the similarly arranged genetic code haveanalogous regulation? Specifying some relevant code
sets, Gamow (1954), Rumer (1966), and Hasegawa
and Miyata (1980) did not suppose that some elements of these sets emphasize the decimal notation of
the positional number system. Moreover, remarkablephysical relations support this arithmetic abstract.
As shown below, “decimalization” is superposed
perfectly onto the numerous exact balances of thenucleons as well as onto the semantic symmetry of
the triplets, a zero also acts as a special number.
When we think of the genetic code, we usuallyvisualize certain physicochemical structures and theirinteractions (Knight et al., 1999) or ask for somephysical relations among them (Sukhodolets, 1989;
Rosen, 1999). Although these are burning issues, it
seems that the genetic code is connected more closely
to abstract notions of arithmetic than with notions ofphysics or chemistry. Let us give detailed reasons forthat.

2. A few relevant images of the genetic code
The universal genetic code is shown in Fig. 1. Fromthe mathematical viewpoint, it is the mapping function f:T → A,S of the 64 base triplets (T) and 20
l-α-amino acids (A) with two syntactic signs Start
and Stop (S). The compressed representation of thegenetic code is formed by the series of synonymic
triplets whose 3 bases are compressed on the 1:D
scale; where D is the degeneracy numbers IV, III, II, I(Fig. 1a).
The life-size representation includes 64 pairs of
the triplet-amino acid and triplet-syntactic sign type(Fig. 1b). The regular degeneracy and universal triplet
are the general ordering of the genetic code, whichhave not had a plausible explanation until now.
The split representation (Fig. 1c) completes antisymmetrically its compressed form. The split representation is made up of four base types with 48 identicalmolecules of each type. The simple rule maps 192
separate bases onto 20 amino acids and 2 syntactic
signs. The separate bases can be transformed inversely
into triplets by the conditions specified in Figs. 3 and
6–14.
Let us consider two versions of the genetic code.
These are the commonly accepted Escherichia coli

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Fig. 1. The universal genetic code. There are the E. coli code version with the TGY-Cys, TGA-StopI series and the E. octocarinatus code
version with the TGH-Cys series. (a) The compressed representation of the genetic code includes the synonymic series whose 3 bases
are compressed mathematically to the 1:D scale, where D is the degeneracy number. (b) The life-size genetic code representation includes
64 triplets, onto which 20 amino acids and 2 syntactic signs, Start and Stop, are mapped. (c) The split genetic code representation places
four types of the nitrogen bases outside their triplets. The separate bases are combined into four sets. The amino acids and syntactic signs
are mapped onto these bases by the rule that is demonstrated for the 5 TG fragment of the E. coli version of the genetic code. The
direction from the 5 to 3 position coincides with the top-to-bottom direction (this manner of writing allows some new code symmetries
to be shown graphically).

version with the TGY-Cys, TGA-StopI series and the
Euplotes octocarinatus version with the TGH-Cys series (Marshal et al., 1967; Meyer et al., 1991; Tanand Heckmann, 1998; Grimm et al., 1998; Tan et al.,1999). Only the canonical 20 amino acids are studied. Their ionized forms, post-translational modifications, as well as the cotranslational incorporation of
the SeCys amino acid are beyond the scope of thiswork.
Rumer (1966) partitioned the six time degenerated
series into the degeneracy II and IV series and gathered the “complete” series of degeneracy IV into one

set and the “broken” series into the other (Fig. 2a).
Note that each of these sets gets the same number ofthe life-size triplets, which are mapped in a one-to-one
manner by the unique Rumer’s TCAG → GACTtransformation or by combined TCAG → CTGA andTCAG → AGTC transformation (Fig. 3; Shcherbak,
1989a; Karasev and Sorokin, 1997). These transformations form an algebraically closed Abelian groupof order 4, denoted below by V. Although each ofthese transformations can act independently or be absent, they constitute an ordered assembly. Dankwerts
and Neubert (1975), Bertman and Jungck (1979),

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Fig. 2. The Rumer’s (1966) division of the compressed genetic code (a) and three bisections (b), (c), and (d) suggested by Konopelchenko
and Rumer (1975). Wide bordering shows the common series of the bisection (c) and (d) for the sake of regularities in Fig. 8. The
E. octocarinatus code version is used for the representation of the compressed genetic code here and in the figures below because thedegeneracy is distributed symmetrically among its series.

Jiménez-Montaño et al. (1996), Zhang (1997), andJiménez-Montaño (1999) considered the group structure of the genetic code in this aspect.
Konopelchenko and Rumer (1975) noted other universal bisections of the compressed code. The universality means the simultaneous bisection of the 64
life-size triplets, 8 “complete”, and 16 “broken” series, as well as the correspondence of the bisection
axes to the transformations of group V. Due to natural
distribution of the degeneracy, there are only two such5 bisections (Fig. 2b and c) and the unique Center
bisection (Fig. 2d).
Gamow (1954) suggested one significant divisionof the 64 life-size triplets. He divided the triplets depending on the composition of identical and uniquebases, regardless of their types and positions in the
triplets (Fig. 4).

3. The number system is attribute of computing
If the number system were in the sets specified
above, it could be disclosed by means of some featureof natural numbers produced by the system itself. The
consequences of Pascal’s general criterion of divisibility are but few of such features. Ordinary computers
use the binary notation and even parity, i.e. divisibility by 2, as an accuracy control. Similarly, a cell could
use the perfect positional system with enough small
radix and some criterion of divisibility by the number

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191

Fig. 3. The Rumer’s (1966) division. The series of degeneracy IV and III, II, I are distributed among the two sets in the 1:2 ratios whereas
the life-size triplets are bisected. The triplets are mapped either by the Rumer’s unique transformation or by the combination of two othertransformations of group V. The graphic representation of the genetic code images its E. octocarinatus version; the revealed regularities
are also true for the E. coli version.

Fig. 4. The Gamow’s division of the 64 life-size triplets. The triplets are divided into three sets depending on the base composition of
the triplets. Twenty possible combinations of four bases, three at a time, are placed on triangular substrates. Each of the combinations
generates some fixed triplets. (a) Triplets with two identical bases and one unique base form the largest set. (b) As regards to the othertwo sets, the first one comprises the identical base triplets, while the second one—antisymmetrically—comprises the unique base triplets.
Pairs of circular arrows show Spin → Antispin transformation.

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Fig. 5. Consequences of Pascal’s general criterion of divisibility for three-digit numbers of the decimal and quaternary system that are
divided by the PQ 037 and PQ 0134 = 7, respectively. Digital sequences of these numbers are ordered identically in both systems. Digitalsymmetries are exhibited with a period of (q − 1), where q is the radix of the number system. Numbers made up of the same digits havea period equal to their order n = 3. The sum of the digits within their notation is equal to the quotient of these numbers divided by PQ.
Such sum for numbers made up of different digits is equal to the central quotient in the column. This feature allows one to organize
the operation of integer division by PQ by the operation of addition. The numbers made up of different digits are linked by a cyclic
permutation in the columns. The direction of permutation in the neighboring columns is mirror symmetrical; the letter M and the respective
axis designate this symmetry. The digital regularities are invariant under the inversion of the direction of number notation. The notation of
the numbers 037 and 0134 with a leading zero emphasizes the equal opportunities for zero to participate in the digital symmetries. The
present criterion of divisibility bears a close analogy to the formal parameters of the genetic encoding.

equal to 2 (Mac Dynaill, 2002) and/or larger than 2
to increase the accuracy of the control. This criterion
should give particular shape to the digital sequence sothat the molecular genetic machinery can recognize iteasily. For example, the decimal criterion of divisibility by Prime Quantum (PQ) 037 fulfills this condition(Fig. 5a).
The criterion says: only the decimal numbers,
whose three-digit sum of digital triplets has a homogeneous digital sequence (or the sum acquires such a sequence after adding/subtracting PQ 037), are divided
by PQ 037 (the triplet frame has any of the three possible positions with zeros added at the flanks to form
complete triplets). Let us consider this criterion for thenumber 60150013320229856, for example. The sumof the digital triplets for one of the frames is equal to
the number 601 + 500 + 133 + 202 + 298 + 560 =
2294. In order to get the three-digit sum the operationis repeated 002 + 294 = 296 and, finally, PQ 037 isadded to this sum 296 + 037 = 333. The operationsin other two frames are the same:
060 + 150 + 013 + 320 + 229 + 856 = 1628,
001 + 628 = 629, 629 + PQ 037 = 666;
006 + 015 + 001 + 332 + 022 + 985 + 600 = 1961,
001 + 961 = 962, 962 + PQ 037 = 999.
Only one such summation in any of the frame is sufficient for finding out the divisibility by PQ 037. Notethat the digital sequences 296, 629, 962 are connectedby a cyclic permutation of the digits, which is another“palpable” criterion feature. The criterion is valid for
the PQ = 1n 1n−1 . . . 11 q /n, if the condition (q −1)/n = Int is applied and the n-digit reading frame isused. For example, the systems with radixes q = 4 (PQ
7, Fig. 5b), q = 7 (PQ 19), q = 10 (PQ 37, Fig. 5a),q = 13 (PQ 61), etc. meet all these requirements forn = 3.
The symmetry of the three-digit sequences in Fig. 5
is named below as the Prime Quantum Divisibility(PQD) feature. It gets the particular shape of digital
triplet if the decimal (quaternary) system is used andloses it in the other systems. The PQD feature not
only excites the sensation of beauty but also could
be useful for some molecular organelles working as
biocomputers. This feature may simplify molecularmachinery and facilitates the computational procedure
itself. Such ability is discussed in Section 13.

4. The nucleons and nitrogen bases as digits
The number system might use a quantity of somecode elements to symbolize a number. Such remarkable elements could come to light indirectly in oneof the familiar code regularities and, most probably, in some correlation with degeneracy. Using

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Table 1
The antisymmetrical correlation between the amino acid side chain nucleon number R and the series degeneracy number D (Hasegawa
and Miyata, 1980)
Amino acid Gly Ala Ser

Proa Val Thr Cys

Leu Ile Asn Asp Gln Lys Glu Met His Phe Arg

Tyr Trp

Side chain 1 15 31
42–1 43 45 47
57
57 58 59 72 72 73 75 81 91 100 107 130
nucleonnumber, R
Degeneracy IV IV IV, II IV IV IV IIIb , IIc IV, II III II II II II II I
II II II, IV II I
number, D
The wedge-shaped figure and the line of variable thickness denote a change of the nucleon number and the degeneracy number, respectively.
This correlation is a part of the genetic code cooperative symmetry in Fig. 12.
a Formal borrowing of one nucleon from the Pro side chain in favor of its standard block (see text for details).
b The E. octocarinatus code version.
c The E. coli code version.

degeneracy—a mutual product of both code sets—
Rumer ordered the set of triplets; for symmetry, thedegeneracy should order the amino acid-syntactic
sign set as well. Indeed, Hasegawa and Miyata (1980)
noted that the degeneracy number D and the aminoacid nucleon number N are antisymmetrically correlated (Table 1). Let us use the nucleon as the embodiment of a digit for the amino acids. The most commonand stable isotopes are taken for the calculating thenucleon numbers of the 20 canonical amino acids.
Each of the amino acid molecules is divided universally into two parts. These are the unique side
chain R and the standard block B –CH(NH2 )COOH.
For instance, the nucleon number of the least Gly andthe largest Trp side chain equals 1 and 130. The nucleon number of the syntactic signs StopI and StopII,
which do not have corresponding amino acids, is set
to be zero; the syntactic sign Start associates withthe amino acid Met whose nucleon number is 75, etc.
The one exception to the general structure of aminoacids is Pro. It is called an imino acid because thenitrogen of its standard block has one less hydrogenbonded to it. In the special case of Pro, the division
produces a block, which has 73 nucleons. The formal
borrowing of one nucleon from the Pro side chain in
favor of its block brought the block nucleon number
to the standard 73 + 1 = 074, whereas the side chain
nucleon number becomes 42 − 1 = 41. Seemingly
strange from the physicochemical point of view, theborrowing looks like an ordinary computational trickof hypothetical biocomputers.
Another number symbol may be composed by therules of the positional system of numeration. The nitrogen bases of the genetic supermolecules seem suit-able to the role of the quaternary digits. Gamow (1954)
noted this possibility in his pioneering article on thegenetic code. He wrote that a long number written ina four-digital system could characterize the hereditaryproperties of any given organism. Time has shownthat his statement was taken as a metaphoric one.

5. The arithmetic regularity of the compressed
code representation
Modified by biocomputers, the genetic code should
add a certain formal ordering to its well-known physicochemical regularities. Let us consider the four levelsof this ordering. The logic level includes the universal
conditions by which the code is divided into sets. The
cardinal numbers of these sets are studied at the physical level to find some quantitative relations, whereas
the digital sequences of these numbers are studied at
the linguistic level to find the PQD features. And finally, a cooperative behavior of the code elements isstudied at the level of the ordered sets.
Rumer discovered the ordered manifestation within
the triplet set (Fig. 3). The supplementary manifestation within the amino acid-syntactic sign set is shownin Fig. 6 (Shcherbak, 1993b, 1994). The logic levelcontains the condition of Rumer’s division. The physical and linguistic levels are superimposed exactly inspite of their independence. The physical level is manifested by the squares of the first three Pythagorean
numbers (Fig. 6a); these numbers appear if the nucleon
sums are represented in the parts of PQ 037. Anothermanifestation of the physical level is a balance of thenucleon sums of the standard blocks and side chains

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Fig. 6. The arithmetic regularity of the compressed representation of the genetic code (see Figs. 2a and 3). It appears simultaneously
at the logic, physical, and linguistic level. The logic level contains the condition of the Rumer’s division. The physical level shows thequantitative properties of the nucleon sums. These are (a) the squares of the first three Pythagorean numbers and (b) the block-and-chainbalance. The linguistic level manifests the PQD feature. The raster spot indicates the formal borrowing of one nucleon from imino acid Pro
side chain in favor of its standard block. Numbers, divided by PQ 037 and larger 999, are written ad exemplum 2331≡333 + 2 × 999 tounderline their PQD features. Note that the product of PQ 037 and PQ 0134 is equal to 259, which is the final digital permutation in (a).
The canonical DNA pair of the adenine and thymine residues has also 259 + “0” nucleons and the other pair of the guanine and cytosine
residues has 259 + “1” nucleons; the alone uracile residue has 111 nucleons. The graphic representation of the genetic code images its E.
octocarinatus version; the revealed regularities are also true for the E. coli version.

in Fig. 6b. The linguistic level represents the PQD 037
features of the nucleon sums. Such a multilevel manifestation is referred to as an arithmetic regularity.
The nucleon sum of the standard blocks of the wholecode is larger than that of the side chains. Chains arelarger than blocks only in 7 out of 24 series. Onlya limited set with the number of series less than 24
and containing mostly large amino acids is capableof balancing their own blocks. Owing to requirementsof Eq. (13) (Table 2) the balance together with largeamino acids is located within the degeneracy III, II,
I set and not within the IV set of the genetic code.

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195

Fig. 7. The arithmetic regularity of the compressed representation of the genetic code (see Fig. 2b). (a) The amino acids and syntactic
signs whose triplet series have the pyrimidine base in the 5 triplet position manifest a new block-and-chain balance. (b) The arithmetic
regularity in Fig. 6 and the new balance algebraically predetermine the quantitative relations. The graphic representation of the genetic
code images its E. octocarinatus version; the revealed regularities are also true for the E. coli version.

As a result, the concentration of small amino acids
increases in the degeneracy IV set. Outwardly, thislooks like antisymmetrical correlation between the degeneracy and molecular mass noted by Hasegawa and
Miyata (1980). The regularities in Fig. 6 are a possible
interpretation of this correlation.
A similar response in analogous circumstances indicates a general regularity of the investigated object.
The amino acids of the first 5 bisection (Fig. 3b)
whose series have 5 pyrimidine bases are shownin Fig. 7a. The same elements, arranged differently,
demonstrate a new block-and-chain balance, similar
to the previous one in Fig. 6b. After the balance in
Fig. 6b the PQD 037 feature becomes trivial in an algebraic sense for the block-and-chain type of balancing.
Surprisingly, the PQD 037 feature changes henceforth
the type of manifestation to bypass this limitation.
One manifestation of the arithmetic regularity
makes use of the second 5 and unique Center bisection simultaneously (Fig. 2c and d). These bisections comprise a hierarchy of three balances of thechain-and-chain type (Fig. 8). The ordering procedurein Figs. 7 and 8 were patterned by Verkhovod (1994)
after it was discovered in Fig. 6 (Shcherbak, 1993a,b).
Surprisingly, the borrowing of one nucleon from theimino acid Pro side chain works, while the standard
block itself being the recipient of such a borrowed nucleon, is located outside the considered manifestation.
Mitochondria deviate from the universal code. It
should be noted that none of the numerous deviations(Jukes and Osawa, 1990) disturbs the balance of common series in Fig. 8c.

6. The arithmetic regularity of the life-size code
representation
The triplets having two identical and one uniquebase are shown in Fig. 9 (see also Fig. 4a; Shcherbak,
1996). The arithmetic regularity is manifested within
two bisected subsets whose triplets have either two
identical pyrimidine or two identical purine bases

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Fig. 8. The arithmetic regularity of the compressed representation of the genetic code (see Fig. 2c and d). The side chain nucleon sum of
the series with 5 and Center bases T, G is equal to the chain nucleon number 444 + 999 of the whole compressed code. The same is also
true for the series with bases A, C (see right side of (a) and (b)). These chain-and-chain balances are caused by the counterbalance of 5
T, G and Center A, C arms as well as the 5 A, C and Center T, G arms (see left side of (a) and (b)). The equilibrated hierarchy as awhole is based on the common series balance (see (c)); wide bordering shows these series in Fig. 2c and d. The common series have T
and/or G, and A and/or C bases in both 5 and Center position. The graphic representation of the genetic code images its E. octocarinatus
version; the revealed regularities are also true for the E. coli version.

(Fig. 9a). A new chain-and-chain balance is formedby 999 nucleons. The PQD feature is always significant for this type of balancing; the nucleon borrowingof the Pro amino acid works as well.
It is believed that the 3 base is of less importance forthe biological encoding. It seems that the 5 , Center,and 3 bases are of equal importance for the arithmetic
regularity.
In Fig. 9a the central axis bisects the code in such a
way that the type of unique base is disregarded. Now,
the central axis in Fig. 9b bisects the code by using
the type of unique base, disregarding identical ones.
The sum of nucleons for the triplets with the uniquepyrimidine base is equal to the decimal number 888
and the PQD 037 feature is manifested again.
The subset with purines as identical bases inFig. 9a has an additional manifestation of the arithmetic regularity. This subset consists of three linesof the triplets, which are subdivided symmetrically.
Two of the subdivisions are a pair of halved lines (avertical dash-line indicates a halving) and the third
one is a complete line (shown by wide bordering).
These subdivisions demonstrate a triple balance ofthe chain-and-chain type. Each of its balanced arms
consists of 333 nucleons.
In addition, the complete line has an analogue written by synonymic triplets in Fig. 9b. This balanced
pair of lines owes its existence to the regular degeneracy and, in particular, the nonordinary degeneracy III of the amino acid Ile. This is the reason
why the sum of nucleons in Fig. 9b is given by twosummands 333 + 777. That sum can be transformed
into balance of the (block + chain)-and-chain type,i.e. 777 + 777.
The arithmetic regularity of the life-size code becomes complete with similar arrangement of the rest of

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197

Fig. 9. The arithmetic regularity of the genetic code life-size representation (see Fig. 4a). (a) The triplets with the identical pyrimidine
and identical purine bases manifest the chain-and-chain balance and PQD feature. The triplets with identical purine bases manifest the
triple chain-and-chain balance and the PQD features. (b) The triplets with the unique pyrimidine and unique purine base manifest the PQD
feature. (a) and (b): The synonymic triplets of two balanced lines are denoted by the wide bordering. The graphic representation of thegenetic code images both the E. coli and the E. octocarinatus version.

the triplets. These are the first and second antisymmetrically arranged code sets in Fig. 5b. The antisymmetry underlies their (block + chain)-and-(block + chain)balance in Fig. 10 as well.

7. The arithmetic regularity of the split code
representation
The split representation of the genetic code in
Fig. 1c consists of 64 × 3 = 192 separated bases.
The amino acids and syntactic signs are mapped
onto these bases and listed in Fig. 11. The only

division into the separate Thymine-list and the common Adenine-Guanine-Cytosine-list demonstrates the
PQD 037 feature of the side chain nucleon sums
666 + 2 × 999 and 555 + 7 × 999, respectively.
As a rule, the linguistic level of the arithmetic regularity is superimposed on some balance, for example,of the block-and-chain type. Recall that chains all overthe code may balance only some part of their blocks.
The common A-G-C-list contains 138 standard blocks,
whose nucleon sum is equal to 222 + 10 × 999. The
nucleon sum of the amino acid side chains of bothlists, i.e. the split code on the whole, is exactly thesame number (Shcherbak, 1999).

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Fig. 10. The arithmetic regularity of the genetic code life-size representation (see Fig. 4b). There is a unique way of antisymmetricalapplication of the three group V transformations to the pair of the code sets. Namely, one of the transformations is chosen to be common
for both sets; if in the first set it is paired with one of the remaining transformations, then in the second set it is paired with the otherone. The Spin → Antispin transformation does not affect the triplets in the first set, but—antisymmetrically—it fixes the triplet positionsin the second set. The sets could be coaxially placed in one of the alternative positions. The antisymmetrically positioned smaller andlarger nucleon sums eliminate this alternative. The balance becomes apparent if the common Rumer’s transformation is paired with the
TCAG → CTGA transformation in the first set. The graphic representation of the genetic code images its E. coli version; the revealed
regularities are true for this version.

New block-and-chain balance is similar to the previous balances in Figs. 6b and 7a, but is contrarwiseorganized. There, the fixed set of the chains balances
all their own blocks, whereas here, the fixed set of theblocks balances all the code chains. The PQD feature
becomes again a significant manifestation due to another block-and-chain type of balancing.

of these sets. The systematization principle of the E.
octocarinatus compressed code version is shown in
Fig. 12a (Shcherbak, 1993a). According to the principle, an isolated triplet, an amino acid or a syntactic
sign, is not an object of systematization. Two composite objects play this role. These objects are:(i) the sets of the same degeneracy series;(ii) the same degeneracy series within one of the sets.

8. The systematization principle of thecooperative symmetry of the genetic code
The arithmetic regularity operates with unorderedsets. The following is an attempt to order the elements

Each of these objects is a component part of anotherone (see crossing arrows in Fig. 12a). The objects
are aligned with a monotonic and parallel–oppositionchange of two parameters. These parameters are:

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199

Fig. 11. The arithmetic regularity of the split representation of the genetic code (see Fig. 1c). The Thymine-list and the common
Adenine-Guanine-Cytosine-list demonstrate the PQD features. These features indicate the block-and-chain balance between the amino acidside chains of the whole split genetic code and the fixed set of the standard blocks of the A-G-C-list. The revealed regularities are true
for the E. coli version of the genetic code.

(i) the degeneracy number of the series triplets;(ii) the nucleon number of the series amino acids (syntactic signs).
Essentially, the principle combines Rumer’s division by the series degeneracy and Hasegawa and
Miyata’s series alignment by the masses into an organic whole (see Fig. 3 and Table 1). A reliable
systematization principle must produce a strong andvarious ordering using a weak and uniform condition.
Seemingly, the genetic code exceeds the fragmentary
ASCII arrangement in this respect.
The word calligramme means spatially orderedsymbols of the code elements. The calligramme in
Fig. 12b shows the genetic code cooperative symmetry (Shcherbak, 1988, 1989a,b,c). The common
parameters indicate that the arithmetic regularity

and the cooperative symmetry are tightly boundphenomena.
It seems that a zero—fascinating invention ofarithmetic—is in use by biocomputers. The zerosof the Stop signs were trivial summands in the nucleon sums of the arithmetic regularity. Now, the zeroforms the calligramme as an ordinal number. On thatground, the zero occupies its formally predetermined
position at the beginning of the natural number series
and places its own triplet series at the flank of thedegeneracy II set. Being in this position, these series
entirely maintain cooperative symmetry. It is surprising that the zero, i.e. abstract nothing, operates thesubstantial objects of the code. However, a presumable implication of biocomputers to the modificationof the genetic code offers a satisfactory explanationof that fact.

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Fig. 12. The systematization principle of the genetic code. (a) The centrosymmetrical flow chart indicates equilibrium of the principle
formulation. (b) The application of the principle to the compressed series in Fig. 1a results in the cooperative symmetry of the genetic code.
The calligramme of the cooperative symmetry comprises entirely the regularity of the Rumer’s division in Fig. 3 and the manifestation of
the arithmetic regularity in Fig. 6. The lines of variable thickness and wedge-shaped figures symbolize an alignment with a monotonic
and parallel-opposition change of the degeneracy and nucleon numbers.

9. The symmetry of the 5 sequences
The degeneracy IV sequence is arranged by the mirror, translation, and inversion symmetry (Fig. 13a).
Specifically, 5 bases are invariant under a combined
mirror and inversion operation; a pair of RRYY quartets
forms a minimum pattern of the translation symmetry.
Three symmetry operations arrange the degeneracy
III, II, I sequence, and in doing so the operations are
the same as in the degeneracy IV set (Fig. 13b). A
pair of flanking TATAT boxes is mirror symmetrical.
A pair of the central AGC triplets forms a minimum
pattern of the translation symmetry. The inversion andthe absence of it, i.e. an identical transformation, interconnect the 5 and 3 bases of the degeneracy II set(Fig. 13c). This regularity is projected to the 3 and5 bases of the tRNA antitriplets (Fig. 13d) and for-malizes the wobble pairing rule (Crick, 1966). Since
the tRNA antitriplets of the degeneracy II series have
guanine as the purine 5 base (Jukes, 1983), the orderwith regard to it is unambiguous. If this symmetry is
maintained, there is no dilemma as to the relative position of amino acids Gln and Lys.

10. The symmetry of the Center sequences
Both sequences of the degeneracy IV (RY-representation) set and III, II, I set are mirror symmetrical(Fig. 14a). The sequence of the degeneracy IV set andits projection into the coaxially positioned sequenceof the degeneracy III, II, I set are interconnectedby the inversion symmetry. The coaxial symmetryis manifested due to the unique geometric shape of

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201

Fig. 13. The calligramme of the cooperative symmetry of the genetic code. Both 5 base sequences are ordered by the common set ofsymmetry operations (a), (b), and (c). These are the mirror (designated by letter M), translation (designated by italicized letters RY or
TCAG) and inversion of the Base → Complementary base or Purine base ↔ Pyrimidine base type (designated by color gradient whose
white and black colors symbolize the purine and pyrimidine bases, respectively). (d) The Purine base ↔ Pyrimidine base inversion andidentical transformation TCAG → TCAG formalize the wobble pairing of the 5 and 3 bases of the degeneracy II set. The graphic
representation of the genetic code images its E. octocarinatus version; the revealed regularities are true for this version.

the calligramme. The shape and its coaxial symmetry
decay in the scale of the life-size representation.
The classic mirror symmetry of the degeneracy
IV set is replaced by its semantic variant for apalindromic feature (denoted by ␲) of two separate
triplets in Fig. 14b. (The term palindrome is usedin the linguistic sense as a line of characters, which
are centrosymmetrical and whose semantics is invariant with respect to the directions of reading.)
The symmetry-preserving palindromic feature is anindicator of similar semantic structure within the degeneracy III, II, I set. The point is that both Center

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Fig. 14. The calligramme of the cooperative symmetry of the genetic code. Both sequences of Center bases are ordered by the common
set of symmetry operations. These are the mirror (designated by M), the inversion of the Purine base ↔ Pyrimidine base type, and thesemantic symmetry of the palindromic feature (designated by ␲). The graphic representation of the genetic code images its E. octocarinatus
version; the revealed regularities are true for this version. The graphical symbols are the same as in Fig. 13.

sequences have, similar to the 5 sequences above, acommon set of symmetry operations, and the palindromic one is among them.
As predicted, the palindrome of the degeneracy III,
II, I set is manifested in the triplet reading frame starting with the single base G at the flank (Fig. 14c).
The palindrome has remarkable semantics expressedby the syntactic signs StopII and Start. These signs
are in a semantically antisymmetrical position withrespect to the central homogeneous triplet AAA. The

palindrome uses all syntactic signs of the E. octocarinatus code version: two termination signs StopII and
one twice-repeated initiation sign Start.
The unique palindrome semantics is not the only
reason why the given position of the reading framegets the better of its two other possible positions. This
position is fixed as preferable also by a symmetricintersection between the real triplets and the abstract
symbols of these triplets in the palindrome (shown in
Fig. 14c by wide bordering). One of two directions

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of reading is associated with the intersections and ispreferable in this respect. The semantically arranged
palindrome text, which is made by the code symbols
within the code itself, is striking.
The arithmetic regularity appears surprisingly
again, now through the palindrome semantics. The
summation of nucleons reveals the block-and-chainbalance and manifests the PQD 037 feature in Fig. 14b,d and e (the G residue in Fig. 14b has 150 nucleons).
The calligramme is adapted exactly to the E. octocarinatus code version. The series TGA-StopI, absent in this version, is excluded from the syntactic
signs of the palindrome, while its TGA-Cys substitutesupports the manifestation of the PQD 037 feature in
Fig. 14e.
The ASCII code comprises the arranged Roman
letters and Arabic numeral series for the sake of thecontrol of information. Similarly, the frequently usedconsensus sequences had the best chance to be incorporated into the calligramme. One can try to discern
the symbols of the canonical base pairing in Fig. 13a,the TATA-box initiating transcription in Fig. 13b,
a spacer zone with a poly(A) background between
the Stop and the Start signs of neighboring genes inFig. 14c, etc. One can assume from the symmetry
consideration that the split E. coli code version in
Fig. 11 keeps an unknown calligramme of 222 + 10 ×999 nucleons and 192 bases. The Pythagorean numbers (Fig. 6a), sequence AGC (Fig. 13b), the preferable
directions (Fig. 14c), etc. could be its parameters.

203

the protection against an abrupt change of size. A
calligramme fragment underlies the histograms ofhydrophobicity and molecular size in Fig. 15. The
transversions of the most mutable 3 base are shownfor the largest degeneracy II code set by arrows. Dueto the symmetry of the fragment, the transversions arecentrosymmetrical in their arrangement too. Recall
that the triplet series are aligned by the monotonouschange of the nucleon numbers of their amino acids.
The molecular size of an amino acid is generally
proportionate to its nucleon number. Therefore, themonotony excludes a centrosymmetrical shape ofthe size histogram. And vice versa, the hydrophobicity of an amino acid does not depend explicitly
on its size and nucleon number. This allows the hydrophobicity to give any centrosymmetrical shape tothe histogram, for example, such as that, which was
fixed within the actual code. As a result, when aminoacids are replaced through mutation the change ofhydrophobicity is generally weak, while that of size isstrong.
Freeland and Hurst (1998) showed that only one inevery million random alternative codes generated ismore efficient against mutation than the natural code.
However, those codes hardly ever can get some comparable exact order at random. The natural genetic
code is a unique one rather than one in a million inthis respect. Thus, the natural code is highly efficientat minimizing the effects of mutation, though thatability is to some extent limited by arithmetic inside.
It seems that the universal genetic code is arranged
optimally.

11. The physicochemical effect of the cooperative
symmetry
12. The mathematical model of the genetic code
It is well known that the amino acids, triplet series, and degeneracy of the series are optimally distributed and, as a rule, possible consequences of themutations such as an abrupt change of hydrophobicity,
are, to some extent, weakened (Goldberg and Wittes,
1966; Volkenstein, 1966; Sjöström and Wold, 1985;
Figureau, 1987; Knight et al., 1999). However, Shulz
and Schirmer (1979) ask a question: why is it that another property of amino acids—a geometrical size ofside chain—has no comparable protection though thisproperty and is no less important than hydrophobicity?
One may conclude that the possession of cooperative symmetry does not allow the code to enhance

Ordering completeness and uniqueness needs a simple tool for evaluation. Diophantine equations andsome conditions allow arithmetic regularity and cooperative symmetry to be described formally (Table 2).
Such a system must show to what extent the code
obeys formal ordering. In other words, it has to answer the question: could another arrangement of thecanonical amino acids or other amino acid molecules
having different nucleon numbers produce the sameformal ordering?
The number of unknown variables is larger thanthat of the equations. This is typical of Diophantine

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Fig. 15. Physicochemical effect of the cooperative symmetry of the genetic code is that if mutation occurs, the genetic code is satisfactorily
protected against an abrupt change of hydrophobicity rather than the geometric size. The sequences of the 5 and Center bases of thedegeneracy III, II, I sets (see Figs. 13 and 14) are combined with the histograms of (a) hydrophobicity (Lacey and Mullins, 1983) and (b)molecular volume (Jungck, 1978). Arrows designate the transversion mutations of the 3 position of the triplets as well as the corresponding
substitutions of the amino acid hydrophobicity and geometric size within the degeneracy II set. This effect is true for both E. coli and E.
octocarinatus code version. The graphic representation of the genetic code images its E. octocarinatus version.

equation systems, which have a solution owing tosome side conditions (Table 3).
The roots of an algebraic system may be unique orcontained within certain intervals depending to what
extent the system is determined. The degree of determination becomes a reliable criterion of the evaluationof the code obeying the formal order.
The system was subjected to computational investigation and found completely defined. Its uniqueroots coincided with the nucleon numbers of thecanonical amino acids and syntactic signs as well aswith the actual degeneracy numbers. The f:T → A,Sfunction maps uniquely these roots onto the life-size
triplets of the analytical version of the genetic code
(Table 4).

The roots are more than mere mathematics. Theseare 18 different nonzero numbers. They determine thenecessary minimum of different amino acids so as yet
the arithmetic regularity and cooperative symmetry
could be realized.
The variable B = 74 limits the standard block to
the configuration of the α-, but not β-amino acid,
which is a formal solution of one of the problemsof the code origin. It is obvious that a certain nucleon number is not always a symbol of a single
molecule out of many natural amino acids. However, the root of the variable Gly equal to 1 does notcorrespond to any side chain, but that of the aminoacid glycine. The same is true, at least, for the variables Ala, SerII, and SerIV. Thus, some roots areV.I. shCherbak / BioSystems 70 (2003) 187–209

205

Table 2
The Diophantine equation system comprises Eqs. (1)–(29) and logical conditions; there are auxiliary Eqs. (a) and (b) in this list
The arithmetic regularity of the compressed code representation in Fig. 6
Series compressed in scale 1 : DIV ; Gly + Ala + SerIV + Pro + Val + Thr + LeuIV + ArgIV = 333;
Series compressed in scales 1 : DIII , 1 : DII , and 1 : DI ; (Ile) + (Cys + StopII + SerII + LeuII + Asn + Asp + Gln +Lys + His + Phe + ArgII + Tyr) + (StopI + Met + Trp) = 111 + 999.

(b)

The arithmetic regularity of the compressed code representation in Figs. 6–8; the series with Center base either C or A
Compressed scale 1 : DIV ; 5 T ; SerIV = 31;
Compressed scale 1 : DII ; 5 T ; StopII + Tyr = 107;
Compressed scale 1 : DIV ; 5 G; Ala = 15;
Compressed scale 1 : DII ; 5 G; Asp + Glu = 132;
Compressed scale 1 : DIV ; 5 C; Pro = 41;
Compressed scale 1 : DII ; 5 C; Gln + His = 153;
Compressed scale 1 : DIV ; 5 A; Thr = 45;
Compressed scale 1 : DII ; 5 A; Asn + Lys = 130;

(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)

The arithmetic regularity of the compressed code representation in Figs. 6–8; the series with Center base either T or G
Compressed scale 1 : DIV ; 5 C; LeuIV + ArgIV = 157;
Compressed scale 1 : DIV ; 5 G; Gly + Val = 44;
Compressed scales 1 : DIII ; 1 : DII ; 1 : DI ; 5 T ; LeuII + Phe + Cys + StopI + Trp = 325;
Compressed scales 1 : DIII ; 1 : DII ; 1 : DI ; 5 A; Ile + Met + SerII + ArgII = 263.

(9)
(10)
(11)
(12)

The scale symmetry of the compressed and life-size code representation(Gly + Ala + SerIV + Pro + Val + Thr + LeuIV + ArgIV) × DIV + (Ile) × DIII + (Cys + StopII + SerII + LeuII + Asn + Asp+ Gln + Lys + Glu + His + Phe + ArgII + Tyr) × DII + (StopI + Met + Trp) × DI = 3404, i.e. 92PQ.

(13)

(a)

The arithmetic regularity of the life-size code representation in Figs. 9 and 10
Identical bases (i) = {Y }; unique bases (u) = {R}; 2LeuII + Ile + Tyr + Val + Cys + 2Pro + Thr + His + Ala + ArgIV = 691;
i = {Y }; u = {Y }; Phe + 2LeuIV + 2SerIV + Pro = 308;
i = {A}; 5 or 3 u = {Y }; 2Asn + StopII + Gln = 188;i = {R}; Center u = {Y }; i = {N}; 5 u = {R}; Ile + Thr + Val + Ala = 160;
i = {G}; 5 or 3 u = {Y }; 2Gly + Trp + ArgIV = 232;i = {A}; 5 or 3 u = {R}; Lys + Glu = 145;i = {G}; 5 or 3 u = {R}; Gly + ArgII = 101;i = {R}; Center u = {R}; i = {N}; 5 u = {R}; ArgII + Glu = 173.
Triplets with either three identical or three unique bases; Phe + Lys + SerIV + Ile + His + 2Ala + LeuIV + Cys + StopI+ Asp + Met + Gln + SerII = 703.

(22)

The arithmetic regularity of the split code representation in Fig. 11
Triplets with base(s) T ; Ala + ArgIV + Asn + Asp + 3Cys + Gly + His + 4Ile + 4LeuII + 5LeuIV + Met + 5Phe + Pro+ SerII + 5SerIV + StopI + 2StopII + Thr + Trp + 3Tyr + 5Val = 666 + 2 × 999.

(23)

The arithmetic regularity of the palindromic features in Fig. 14d and e
StopII + StopII + Lys + Start + Start = 222;
SerII + 2Lys + Cys = 222.

(24)
(25)

The standard block nucleon sums of the block-and-chain balances in Figs. 6b, 7a, 11, and 14dSeries compressed in scales 1 : DIII , 1 : DII , 1 : DI ; 15B + bStopII + bStopI = 111 + 999;
Compressed series with 5 Y ; 10B + BPro + bStopII + bStopI = 814;
A-G-C-list of separate bases; 127B + 11BPro + 4bStopII + 2bStopI = 222 + 10 × 999;
Compressed series of palindrom; bStopII + bStopII + B + bStart + bStart = 222.

(26)
(27)
(28)
(29)

(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)

The system includes 33 positive integer variables. The majority of them correspond to the nucleon numbers of the amino acid side chains and standard
blocks. These are the 23 chains designated as italicized three-letter abbreviations; two chains of the syntactic signs Start and StopI = StopII (the samefunctionality equates both Stop variables); two blocks of the amino acid B and imino acid BPro ; and two blocks of the syntactic signs bStart and bStopI = bStopII .
There are also four variables symbolizing the degeneracy numbers DIV , DIII , DII , DI in this list. The logical conditions of the code division are a part ofthe model conditions. They are written in smaller fonts at the defined equations.
Algebra allows one to specify a minimum set of irreplaceable fragments necessary for composing from them the genetic code with arithmetic inside. The
axis of Fig. 2a divides the compressed code into two fragments that are described by auxiliary Eqs. (a), (b), and (26). The simultaneous division by thisand the other three axes of Fig. 2 produces a set of fragments described by Eqs. (1)–(12), (26), and (27).
The compressed and the life-size representations of the code are combined by the scale symmetry (Fig. 1). An algebraic symbol of that is Eq. (13). The
life-size triplets with unique (u) and identical (i) bases demonstrate the arithmetic regularity in Fig. 9. The combined axes of Fig. 9 divide the code into
eight fragments described by Eqs. (14)–(21). The regularities in Figs. 10 and 11 are described by Eqs. (22), (23), and (28).
The arithmetic regularity of the palindrome (Fig. 14) is described by Eqs. (24), (25), and (29).
The unambiguous assignment of the nucleon numbers of four various blocks through four Eqs. (26)–(29) is obvious. But it is not obvious a priori that
there are enough balances of the block-and-chain types in the genetic code so that that assignment takes place.

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Table 3
Side conditions of the Diophantine equation system
Integer numbers of degeneracy; DIV > DIII > DII > DI > 0.
Series compressed in scale 1 : DIV ; 0 ≤ Gly < Ala < SerIV < Pro < Val < Thr < LeuIV < ArgIV.
E. octocarinatus series compressed in scale 1 : DIII ; 0 ≤ Cys < Ile.
E. octocarinatus series compressed in scale 1 : DII ; 0 ≤ StopII < SerII < LeuII < Asn < Asp < Gln < Glu;
Asp < Lys < Glu < His < Phe < ArgII < Tyr.
E. octocarinatus series compressed in scale 1 : DI ; 0 ≤ Start < Trp.
∃x ∈ [2, 14] ∪ [16, 23] ∪ [32, 38].
StopI = StopII

(30)
(31)
(32)
(33a)
(33b)
(34)
(35)
(36)

The sequences of inequalities (30)–(34) are the algebraic symbol of the cooperative symmetry. The classic and semantic symmetries are
mapped onto these sequences in Figs. 13 and 14. These and other system conditions determine the mapping function f:T → A,S of theanalytical version of the genetic code. For example, the palindrome semantic antisymmetry and symmetric intersections in Fig. 14c map
both syntactic function StopII and Start onto particular triplets and roots xi .
Lehman and Jukes (1988) assumed that originally some triplets could temporarily lose encoded amino acids and turn into termination
signs. The cooperative symmetry should extrude such triplets to the flanks of the four sets in Fig. 12b. For this reason, the inequalities(30)–(34) allow the flank variables Gly, Cys, Start, and Met, to have a relic zero, and vice versa, the variables StopI = StopII, to have
non-zero nucleon numbers.
The canonical molecules together with other natural amino acids are incapable of forming an uninterrupted natural number series by nucleon
numbers. Gaps, which are predicted reliably at least for small amino acids (Greenstein and Winitz, 1991), are assigned by condition (35).

associated with certain molecules numerically assigned to them.
The possession of the syntax signs StopI and StopII
in the form of the standard triplet series is fraught

with the danger of early terminations. Lewin (1987)
assumed that for some reason the genetic code could
not push two dangerous triplet series outside of its
bounds in order to replace them with some non-triplet

Table 4
The analytical version of the genetic code shows the system solution in the form of the standard table5 position3 position

Center position
T

C

A

G

T

TTT
TTC
TTA
TTG

091
091
057
057

TCT
TCC
TCA
TCG

031 (Ser)031 (Ser)031 (Ser)031 (Ser)

TAT
TAC
TAA
TAG

107
107
000 (Stop)
000 (Stop)

TGT
TGC
TGA
TGG

047
047
000a (Stop); 047b
130

T
C
A
G

C

CTT
CTC
CTA
CTG

057
057
057
057

CCT
CCC
CCA
CCG

042–1
042–1
042–1
042–1

CAT
CAC
CAA
CAG

081
081
072
072

CGT
CGC
CGA
CGG

100
100
100
100

T
C
A
G

A

ATT
ATC
ATA
ATG

057
057
057
075 (Start)

ACT
ACC
ACA
ACG

045
045
045
045

AAT
AAC
AAA
AAG

058
058
072
072

AGT
AGC
AGA
AGG

031 (Ser)031 (Ser)
100
100

T
C
A
G

G

GTT
GTC
GTA
GTG

043
043
043
043

GCT
GCC
GCA
GCG

015 (Ala)
015 (Ala)
015 (Ala)
015 (Ala)

GAT
GAC
GAA
GAG

059
059
073
073

GGT
GGC
GGA
GGG

001 (Gly)
001 (Gly)
001 (Gly)
001 (Gly)TCAGThe parenthetical amino acid or syntactic sign imply that the numerical value of the corresponding root predetermines the chemical
structure of amino acid or its absence in case of the termination. The PQD feature of the nucleon sum at the levels Cβ , Cδ , etc., and thegeometrically equivalent levels of the ring amino acids in Figs. 6 and 7a allows one to increase in number the parenthetical amino acids.
a The numerical symbol of the E. coli code version.
b The numerical symbol of the E. octocarinatus code version.

V.I. shCherbak / BioSystems 70 (2003) 187–209

consensus sequences there. It seems that the formallyindispensable zero roots of the system could be sucha reason.
The nonzero root of the variable Start and zero rootsof the variables StopI and StopII formally predetermine the essential differences of the initiation and termination, namely, the participation or absence of anamino acid in these acts. This may be an allusion to
why the current translation has enough standard intermediates for the initiation instead of the specific factors for the termination.
Both regular degeneracy and universal triplet are
indispensable conditions of the model. One can speculate that these phenomena owe their origin to thiscircumstance.

13. Discussion
Let us assume that at early stages of the origin oflife it developed the arithmetic power of evaluative
genetic language. The material objects of that power
could be some molecular organelles working like biocomputers. These organelles should modify previous
versions of the primordial genetic code in accord witha selected numeration system and some computing algorithms, for example, such as serial additions (MacDynaill, 2002). Then, the actual genetic code is a final
variant of that modification.
Which of the existing models best describes theabove events? Arithmetic provides specific requirements for selection. First, the arithmetic symbols needa freedom of the mapping function to appear into thecode (Yockey, 2000). Second, any given collective regularity should be a jump phenomenon rather than a result of some gradual changes. Third, a suitable model
should give an opportunity to both arithmetic regularity and physicochemical protection to arise.
Crick et al. (1976) introduced a primordial tRNAs
to fix the product of the “frozen accident” (Crick,1968). These intermediaries give desired freedom
to the mapping function. They release hydrophobicity from hypothetical task to be direct and selective
connector of triplets and amino acids. Porschke’s
experimental data (1985) did not show adequate
recognition, accuracy, and attractive power to considerhydrophobicity as a unique and reliable code maker.
The “released” hydrophobicity could take the protec-

207

tive shape in Fig. 15a with less difficulty. In addition,
the tRNA intermediaries promote discontinuous jumpof the primordial genetic code.
The code might have influenced and been influenced
by the grammatical categories of the evaluative genetic language and arithmetic might be among thesecategories. The genetic machinery relates to discrete
movement of different molecules in space, and consequently, an inertia effect of mass. One can supposethat this effect selected the massive nucleons as theamino acid’s digital signs (Kashkarov et al., 2002).
Gamow (1954) speculated that hereditary properties
could be characterized by a sequence of quaternary
digits. That sequence represented by three-digit numbers could have at certain distances some checksums
divisible e.g. by quaternary PQ 0134 . Theoretically,such arithmetic background may underlie any given
genetic text without limitation of its biological content.
Analyzing its own arithmetic, the background might
check, restore, and alter superimposed biological content by means of calculations omitting the translation.
One can speculate that such total analysis takes placein conjugate chromosomes during enigmatic meiosis
prophase I, whose duration is up to 3 weeks for humanmale beings.
The divisibility by PQ as a validation criterion,
if any, simplifies molecular machinery and facilitates the computational procedure of hypotheticalorganelles working as biocomputers. Perhaps a biocomputer is hardly more complicated than, e.g.
mRNA-polymerase, and probably some known organelles have the capability to perform arithmetic.
A cell should get a computing result as a sequenceof digits on some register. One may expect that themolecular register of biocomputers should be compact
and highly accurate. Indeed, the test of divisibility by
PQ of any given number independently of its value
requires only the three-digit register. If the result iscorrect, the PQD feature gives its symmetrical shape
to the molecular digits on the register. A routine shapeanalysis could convert these digits to an “analogue”
form, and initiate certain biochemical reactions depending on the computing result. The symmetricshape or its absence contributes to a unique identifiability of the correct or incorrect register status.
Note that a simple algorithm allows one to organize
arithmetic division by PQ by summation of digits ofa dividend number (see Fig. 5).

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Although the revealed ordering does not go beyond
the boundary of the genetic code, one can consider
it as a universal pattern applicable to actual genetic
texts. On the other hand, one can assume that only
divisibility by the number 37 might be a final effectwhose PQD 037 feature is only a by-product, whichis indifferent for the genetics, and discernible by thehuman mind whose system is also the decimal one. If
that is the case, some things of the strict physical naturesuch as the balances and structures of the cooperative
symmetry remain valid as yet.

Acknowledgements
Part of this study was made during the author’s visit
to Max-Planck-Institute für biophysikalische Chemie,
Göttingen, Germany. The author is greatly indebted toProfessor Manfred Eigen for his support at that earlystage of the present work and expresses his special
thanks to Ruthild Winkler-Oswatitsch.
The work was financed by the Ministry of Science and Education of Republic of Kazakhstan and
Max-Planck-Gesellschaft, Germany.
References
Adleman, L.M., 1994. Molecular computation of solutions to
combinatorial problems. Science 266, 1021–1023.
Bertman, M.O., Jungck, J.R., 1979. Group graph of the genetic
code. J. Heredity 70, 379–384.
Crick, F.H.C., 1966. Codon-anticodon pairing: the wobble
hypothesis. J. Mol. Biol. 19, 548–555.
Crick, F.H.C., 1968. The origin of the genetic code. J. Mol. Biol.
38, 367–379.
Crick, F.H.C., Brenner, S., Klug, A., Pieczenik, G., 1976. A
speculation on the origin of protein synthesis. Origins Life 7,
389–397.
Chen, J., Wood, D.H., 2000. Computation with biomolecules. Proc.
Natl. Acad. Sci. U.S.A. 97, 1328–1330.
Dankwerts, H.J., Neubert, D., 1975. Symmetries of the genetic
code doublets. J. Mol. Evol. 5, 327–332.
Eigen, M., Winkler, R., 1985. Das Spiel. Piper Verlag, München,
Zürich, pp. 281–316.
Faulhammer, D., Cukras, A.R., Lipton, R.J., Landweber, L.F.,
2000. Molecular computation: RNA solutions to chessproblems. Proc. Natl. Acad. Sci. U.S.A. 97, 1385–1389.
Figureau, A., 1987. Information theory and the genetic code.
Origins Life 17, 439–449.
Freeland, S.J., Hurst, L.D., 1998. The genetic code is one in a
million. J. Mol. Evol. 47, 238–248.

Gamow, G., 1954. Possible relation between deoxyribonucleic acidand protein structures. Nature 13, 318.
Goldberg, A.L., Wittes, R.E., 1966. Genetic code: aspects oforganization. Science 153, 420–424.
Greenstein, J.P., Winitz, M., 1991. Chemistry of the Amino Acids.
John Wiley and Sons, Inc., New York, London.
Grimm, M., Brünen-Nieweler, C., Junker, V., Heckmann, K., Beier,H., 1998. The hypotrichous ciliate Euplotes octocarinatus has
only one type of tRNACys with GCA anticodon encoded on asingle macromolecular DNA molecule. Nucleic Acids Res. 26,
4557–4565.
Hasegawa, M., Miyata, T., 1980. On the antisymmetry of theamino acid code table. Origins Life 10, 265–270.
Jiménez-Montaño, M.A., 1993. On the syntactic structure andredundancy distribution of the genetic code. BioSystems 32,
11–23.
Jiménez-Montaño, M.A., 1999. Protein evolution drives theevolution of the genetic code and vice versa. BioSystems 54,
47–64.
Jiménez-Montaño, M.A., de la Mora-Basañez, C.R., Poeschel, T.,1996. The hypercube structure of the genetic code explainsconservative and non-conservative amino acid substitutions invivo and in vitro. BioSystems 39, 379–384.
Jukes, T.H., 1983. Evolution of the amino acid code: inferencesfrom mitochondrial codes. J. Mol. Evol. 19, 219–225.
Jukes, T.H., Osawa, S., 1990. The genetic code in mitochondria
and chloroplasts. Experimentia 46, 1117–1126.
Jungck, J.R., 1978. The genetic code as periodic table. J. Mol.
Evol. 11, 211–224.
Karasev, V.A., Sorokin, S.G., 1997. Topological structure of thegenetic code. Russian J. Genetics 33, 622–628.
Kashkarov, V.V., Krassovitskiy, A.M., Mamleev, V.S., shCherbak,
V.I., 2002. Random sequences of proteins are exactly balanced
like the canonical base pairs of DNA. In: Proceedings of the10th ISSOL Meeting and 13th International Conference on theOrigin of Life, Oaxaca City, Mexico, June 30–July 4.
Knight, R.D., Freeland, S.J., Landweber, L.F., 1999. Selection,
history and chemistry: the three faces of the genetic code.
Trends Biochem. Sci. 24, 241–247.
Konopelchenko, B.G., Rumer, Yu.B., 1975. The codon
classification inside genetic code. Docl. Acad. Nauk. SSSR 223,
471–474.
Lacey Jr., J.C., Mullins Jr., D.W., 1983. Experimental studies
related to the origin of the genetic code and the process ofprotein synthesis—a review. Origins Life 13, 3–42.
Landweber, L.F., Gilbert, W., 1993. RNA editing as a source ofgenetic variation. Nature 363, 179–182.
Landweber, L.F., Kuo, T.-C., Curtis, E., 2000. Evolution andassembly of an extremely scrambled gene. Proc. Natl. Acad.
Sci. U.S.A. 97, 3298–3303.
Lehman, N., Jukes, T.H., 1988. The genetic code development bystop codon takeover. J. Theor. Biol. 135, 203–214.
Lewin, B., 1987. Genes. John Wiley and Sons, Inc., New York,
London.
Mac Dynaill, D.A., 2002. A parity code interpretation of nucleotide
alphabet composition. Chem. Commun. 18, 2062–2063.
Marshal, R.E., Cascey, T.C., Nirenberg, M., 1967. Fine structureof RNA codewords recognized by bacterial, amphibian andmammalian transfer RNA. Science 155, 820–825.

V.I. shCherbak / BioSystems 70 (2003) 187–209
Meyer, F., Schmidt, H.I., Plümper, E., Hasilik, A., Mersmann,G., Meyer, H.E., Engström, A., Heckmann, K., 1991. UGA istranslated as cysteine in pheromone 3 of Euplotes octocarinatus.
Proc. Natl. Acad. Sci. U.S.A. 88, 3758–3761.
Porschke, D., 1985. Differential effect of amino acid residues on
the stability of double helices formed from polyribonucleotides
and its possible relation to the evolution of the genetic code.
J. Mol. Evol. 21, 192–198.
Rosen, G., 1999. Molecular content relations in the genetic code.
Phys. Lett. A 253, 354–357.
Rumer, Yu.B., 1966. About systematization of the codons of thegenetic code. Dokl. Acad. Nauk. SSSR 167, 1393–1394.
Shcherbak, V.I., 1988. The co-operative symmetry of the genetic
code. J. Theor. Biol. 132, 121–124.
Shcherbak, V.I., 1989a. Rumer’s rule and transformation in thecontext of the co-operative symmetry of the genetic code. J.
Theor. Biol. 139, 271–276.
Shcherbak, V.I., 1989b. Ways of wobble pairing are formalizedwith the co-operative symmetry of the genetic code. J. Theor.
Biol. 139, 277–281.
Shcherbak, V.I., 1989c. The “START” and “STOP” of the genetic
code: why exactly ATG and TAG, TAA? J. Theor. Biol. 139,
283–286.
Shcherbak, V.I., 1993a. The symmetrical architecture of thegenetic code systematization principle. J. Theor. Biol. 162, 395–
398.
Shcherbak, V.I., 1993b. Twenty canonical amino acids of thegenetic code: the arithmetical regularities. Part I. J. Theor. Biol.
162, 399–401.
Shcherbak, V.I., 1994. Sixty-four triplets and 20 canonical aminoacids of the genetic code: the arithmetical regularities. Part II.
J. Theor. Biol. 166, 475–477.
Shcherbak, V.I., 1996. A new manifestation of the arithmeticalregularity suggests the universal genetic code distinguishes
the decimal system. In: Proceedings of the 11th International

209

Conference Origin of Life, Book of Abstracts, Orleans July
5–12, France.
Shcherbak, V.I., 1999. A new manifestation of the decimal systemin the genetic code. In: Proceedings of the 12th International
Conference Origin of Life, Book of Abstracts, San-Diego, July
11–16, USA.
Shulz, G.E., Schirmer, R.H., 1979. Principles of Protein Structure.
Springer-Verlag, New York, Heidelberg, Berlin.
Siemion, I.Z., 1994. Compositional frequencies of amino acids inthe proteins and the genetic code. BioSystems 32, 163–170.
Sjöström, M., Wold, S., 1985. A multivariate study of therelationship between the genetic code and the physical–chemical
properties of amino acids. J. Mol. Evol. 22, 272–277.
Sukhodolets, V.V., 1989. The genetic code as a clue of molecular
evolution. J. Theor. Biol. 141, 379–389.
Tan, M., Heckmann, K., 1998. The two ␥-tubulin-encoding genesof the ciliate Euplotes crassus differ in their sequences, codon
usage, transcription initiation sites and poly(A) addition sites.
Gene 210, 53–60.
Tan, M., Heckmann, K., Brünen-Nieweler, C., 1999. The
micronuclear gene encoding a putative aminoacyl-tRNA
synthetase cofactor of the ciliate Euplotes octocarinatus
is interrupted by two sequences that are removed during
macromolecular development. Gene 233, 131–140.
Verkhovod, A.B., 1994. Alphanumerical division of the universal
genetic code: new divisions reveal new balances. J. Theor. Biol.
170, 327–330.
Volkenstein, M.V., 1966. The genetic code of protein structures.
Biochim. Biophys. Acta 119, 421–424.
Yockey, H.P., 1992. Information Theory and Molecular Biology.
Cambridge University Press, Cambridge, New York.
Yockey, H.P., 2000. Origin of life on earth and Shannon’s theoryof communication. Comput. Chem. 24, 105–123.
Zhang, C.T., 1997. A symmetrical theory of DNA sequences andits applications. J. Theor. Biol. 187, 297–306.