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The 24 Possible Algebraic Definitions of a 6-Dimensional Binary VectorSpace in the Set of the 64 Triplets of the Standard Genetic Code

Marco V. Josa,*

, Eberto R. Morgadob

, Tzipe Govezenskya

,

Gabriel GarduoSotoa

aTheoretical Biology Group, Instituto de Investigaciones Biomdicas, UniversidadNacional Autnoma de Mxico, Mxico D.F. 04510, Mxico.bFacultad de Matemtica, Fsica y Computacin, Universidad Central Marta

Abreu de Las Villas, Santa Clara, Cuba.

Abstract. Herein, we analyze the 24 different ways of constructing a 6-dimensional binary vectorspace of the 64 triplets of the Standard Genetic Code. We also analyze the transformations that

lead to any model into each of the remaining models. In all the cases the definition of a givenordering begins with a matching of the set { }N C, U, A,G= of the 4 nucleotide RNA bases with the

set { }00,01,10,11

(4)

of the 4 binary duplets. Since the set N has the structure of a Galois Field

of 4 elements, the 6-dimensionalGF 2 hypercube becomes a in a 3-dimensional vector space. We partition the set N into 3 classes: strong-weak, amino-keto andpyrimidine-purine. For each of these classifications there are 2 classes of 3-dimensional cubes over

that preserve the correspondence of the algebraic complementarity. A new type of distance

( ) defined in the cube as a 3-dimensional vector space over the finite field GF is

introduced. The cube has been calledthe Hotel of Triplets, or Genetic Hotel, because of its

resemblance with a three-floor building. As the permutations that convert elements of a family into

elements of the other one are not isometries with respect to the distance D in the 3-dimensionalcube, the corresponding transformation may change the shape and the size of a condominium

of the Hotel. The prism RNY in the cube is geometrically highlighted and compared in the

Genetic Hotel for the different orderings.

NNN GF(4)

GF

D

N

(4)

NNN (4)

NNN

NNN

Keywords: strong-weak, amino-keto, and purine-pyrimidine base classification; algebraic andgeometrical structures of the genetic code; genetic hotels; Klein 4-group

1 Introduction

In earlier works [1-11], different ways in which a structure of a binary 6-dimensionalvector space in the set of the 64 triplets of the Standard Genetic Code have beendefined. In all the cases the definition begins with a matching of the set

{ }N C, U,A,G= of the four nucleotide RNA bases with the set { }00,01,10,11 of the

four binary duplets.Potentially this matching can be performed in 4! =24 differentways. Herein, we analyze the 24 different ways of carrying out that construction,

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together with the transformations that lead to any model into each of the remainingones.The bitwise module two addition, or XOR operation, in the set of duplets, inducesin the set a structure of abelian group, the so called Klein 4 group.

Consequently, the addition in induces an addition in the set of

the 64 triplets, which, together with the obvious definition of the product of thescalars 0 and 1 by all the triplets, conforms a structure of a 6-dimensional vector

spaces over the binary field

N

N N N N NNN =

{ }2 0,1= , isomorphic to ( , also called the 6-

dimensional hypercube. It also has a structure of a Boolean lattice or Booleanalgebra.

)6

2

The so-called Hamming distance, provides to the hypercube ( )6

2a structure of

metric space, and it induces a distance in the 2 vector space . The

Hamming distance induces in an

NNN

( )6

2 valued inner product, being the field

of real numbers, and hence it carries over the properties of orthogonality or

perpendicularity.

1.1 The binary partitions of the set NIt is well known that a set of 4 elements has exactly 3 different partitions as union

of 2 subsets, each of 2 elements. In the case of our set { }N C, U,A,G= the only 3

partitions are: { } { }1: N C,G U,A = , { } { }2: N C,A U,G = and,

{ } { }3: N C, U A,G =

{

. For every selected ordering we will use the

correspondence of a partition of with the partition of the setN

}22( ) 00,01,10,11=

2

2( )

, as determined by algebraic complementarity. We understand

by algebraic complementarity, the one that exists between 2 duplets of zeros andones, when each is the bitwise Boolean negation of the other one. Then the

partition of by algebraic complementarity is { } { }00,11 01,10 . The 3 partitions

are related with chemical properties of the nucleotides. The partition 1

corresponds to the biological classification of nucleotide bases in strong { }S:C,G

(those which form 3 hydrogen bonds between them), and weak { }W:U,

2

A (those

which form 2 hydrogen bonds between them). In the case of , algebraiccomplementarity is consistent with the chemical classification of nucleotides into:

amino nucleotides: { }M:C,A and keto nucleotides: { }K:U,G . Finaly, if 3 is

used, algebraic complementarity is assigned to nucleotide bases of the same

chemical kind: pyrimidines: { }Y:C,U , and purines: { }R :A,G . A selected list of

the different matchings that have been used according to this chemicalcategorization of the bases is provided in Table I:

Table I

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Orderings

00 01 10 11 References

U C A G Swanson [10]

A G U C Jimnez-Montao et al.[2]; Klump [6]

C U G A Karasev and Soronkin [5]U C G A Stambuk [11]

C A U G He et al[1]; Joset al.[3]

C U A G Snchez et al. [7,8]; Jos et al.[4]

G A U C Snchez et al.[7,8]

G U A C Snchez et al.[9]

1.2 The question about isometry

When we consider the 24 ways of selecting the matching of the set of nucleotideswith the set of duplets of zeros and ones two natural questions arise: Do thesedifferent vector spaces lead to the same results? Are they not only pairwiseisomorphic but also isometric?

All the possible hypercubes are isomorphic and isometric with the hypercube

, and both, isomorphism and isometry, are transitive relations. Hence we can

assert that all of the obtained hypercubes are not only pairwise isomorphic, butalso pairwise isometric. The real questions are the following: Do the 24

permutations of a primary order

( )6

2

( )C,U,A,G induce isometric transformations of the

original hypercube, with respect to its Hamming distance? Do they induceisomorphic affine transformation of that hypercube?The question about isometry is related to the type of distance or metric we aredealing with. In the 6-dimensional structure of the Z 2-vector space the Hammingdistance is used. The Hamming distance between two triplets is induced by theHamming distance between their associated sextuples of zeros and ones.

1.3 The Hamming distance

The Hamming distance between 2 sextuples is defined as the number of bits thathave different values. Similarly, the Hamming distance between 2 triplets isinduced by the Hamming distance between their associated sextuples. The

hypercube can be envisaged as a graph, where the vertexes are the

sextuples and the edges are the binary subsets,

( )6

2

{ }1 2 3 6 1 2 3 4 5 6, , , ) , , , , ,(x x x , (y y y y y y )

1 5 6 1 2 3 4 5 6H((x , x , x , x ), (y , y , y , y , y , y ))

1 2 3 4 6(y , y , y , y )

4 5, ,x x x

2 3 4, x , x

5, y , y

, such that the Hamming distance,

, between 2 sextuples that differ in only

one bit is equal to 1. In this case, we say that both sextuples are adjacent. Fromthis, it follows that every sextuple has exactly 6 other sextuples that are adjacent to

it. The Hamming distance between the sextuples and

can be expressed, arithmetically, as the integer

1 2 3 4 5 6(x ,x , x ,x , x ,x )

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1 1 2 2 3 3 4 4 5 5 6 6x y x y x y x y x y x y + + + + +

)6

2

, where the addition and

subtraction are the usual operations in the set of integers. The Hamming

distance also induces in ( anZ

1 2 3 4 5 6(y , y , y , y , y , y ) x

valued inner product, being the field of

real numbers, and together with it, the properties of orthogonality or

perpendicularity. This inner product of the sextuples andis defined as the integer

1, x3 3y

2 3 4 5 6(x , x , x , x , x )2 4 4 5 5x x y x y1 1y 2 6 6x y x y+ + + + + ,

where the operations are the ordinary product and addition in the set of integers.

1 N

C 0

2)2(

0 U 01 A 10 G 11)(C,U,A

, )+,G

( )2

2 2 2=

2 22=

2 Different families of hypercubes that are determined by the initial matchingbetween nucleotides and duplets

2.1 The hypercubes that are obtained by matchings that are consistent with

the correspondence of the partition of with the partition of byalgebraic complementarity

Let us suppose, that our initial matching is , , , . It

means that we select the initial ordering as . Then, the Cayley Table of

the abelian group (N turns out to be:

Table II

+ C U A G

C C U A G

U U C G A

A A G C U

G G A U C

As we said above, Table II corresponds to the so called Klein 4-group, which is afour abelian group in which every element, different of the neutral, has order 2. Itmeans that every element is its own inverse. All the Klein 4-groups are pairwiseisomporphic. Actually, the Klein 4-group is the smallest finite group that is notcyclic, but it is the direct product of 2 binary cyclic groups. Then, the direct product

with the module 2 addition, is its canonical group theoretical

representation. The Klein 4-group has the property that every binary set that

contains the null element C, defines a binary subgroup. It means that the 2 classesof any of the three binary partitions are the member classes of the factor groupover the binary subgroup that contains C. The Klein 4-group can be visualized as a

2-dimensional binary vector space, isomorphic to ( )2

. It also has the

property that any permutation of its 3 non-null elements induces an automorphism,that is, a bijective endomorphism. It is equivalent to the fact that any ordered pair ofnon-null elements is a basis of the vector space, 2 of them being the onlyorthonormal basis with respect to the Hamming inner product.

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2.2 The role of translations as representative of mutations

The concept of translation in an additive group. In any additive group (G, )+ atranslation is a transformation of the form: being a fixed element

of . Obviously, the translation T , associated to the neutral element is the identityfunction of . Every translation is a bijective function and the set , of all

the translations in , is closed under the composition. In fact, the ordered pair

, where denotes the composition of functions, is an abelian subgroup of

the symmetric group (S , of all the bijective transformations of the set G . The

correspondence is an injective homomorphism from

XT :Y Y X, + X

G

(G

0

GI G T(G)

G

X

(T ), )

(G), )

XT (G, )+ in its symmetricgroup , being the image of this homomorphism the group of

translations. Then, every abelian group is isomorphic to its group of translations.

(S(G), ) (T(G), )

The biological concept of mutation. In any chain of RNA nucleotides a mutationis the substitution of any nucleotide bases by another one, from the set . Amutation where a nucleotide is changed to another of the same chemical type, thatis, pyrimidine to pyrimidine, or purine to purine, is called a

N

transition. A mutationwhere a nucleotide is changed to another of different chemical type, that is,pyrimidine to purine, or purine to pyrimidine, is called a transversion. Moreover, wehave 2 different kinds of transversions: first, those that convert a nucleotide to

another of the same class, according to the partition 2 , the amino-ketoclassification, and second, those that convert a nucleotide to another of the same

class, according to the partition 1 , the strongweak classification. Then, the latterare transformations that convert nucleotides into their complementary.

The role of translations. The translations and T of the abelian group

may be taken as representatives of all the different kinds of mutations. Thetranslations , and

C U AT ,T ,T , G

(N, )+

1,

NI T= C

,

UT

G

, associated to pyrimidines, represent transitions. The

first of them, the identity I , pointwise preserves the structure of the 3 partitions

2 and 3 of , while the second, T , preserves the members of the partitionN U

3 , of the pyrimidine-purine classification, and interchange the members of the

other 2 partitions 1 and 2 . The translations T andA

AT

GT , associated to purines,

represent transversions. The first of them, , preserves the structure of the

partition 2 , of the aminoketo classification, and interchange the members of theother 2, whereas the translation, , preserves the members of the partitionGT 1 , ofthe strong-weak classification, and interchange the members of the other 2

partitions 2 and 3 . The foregoing description is illustrated in the followingDiagram I:

Diagram I

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CUT

U

AT

TG

AT

AUT

G

The action of the three non-trivial translations are pointed out in Diagram I. Notethat one of them is a transition and the other 2 are transversions:

Horizontal:Transition TU (preserves the pyrimidine-purine classification)

Vertical:Transversion (preserves the amino-keto classification)AT

Diagonal :Transversion GT (preserves the strong- week classification)

3 Embedding of the abelian group (N, )+ in the 2-dimensional vector

space

2

The abelian group can also be regarded as a bidimensional vector space

over the binary field

(N, )+

{ }2=

0,1

12=P

M

, where the only orthonormal basis, according to theinner product, induced by the Hamming distance, are and . Then, the

only isometric linear transformations, or orthogonal transformation, are those with

matrices and . The matrix is obtained from the identity

matrix by the interchange of the first and the second rows. As it is well known,these kind of matrices, called elementary permutation matrices, when multiplied tothe right by another matrix , perform on the same transformation, in thesecases permutations of the rows 1 and 2. If we identify the elements and

with the canonical unitary vectors e (

(U, A) (A, U)

2

1 0I

0 1

=

0 1

1 0

1

12P

,0)

M

1

A 10=U 0= , 1= and 2e (0,1)= in the Euclidean

coordinated plane the2 = 2 vector space is embedded in the plane,not as a subspace, but as a subset, of the plane. The 4 elements C, , and ,

correspond, respectively, to the vertexes , and (1 of a unit

square situated in the first quadrant of the plane. In the plane , the matrix

represents a reflexion, or axial symmetry, with respect to the line equation

N

),(1,0)

A,

,1)2

U G

12P

y x

(0,0),(0,1

= ,the angle bisector of the first and the third quadrants. As it is well known, theisometric transformation of any metric vector space are the compositions of itsN

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linear isometric transformation with all the translations. Then, in our present case,the group of isometric transformation would have 2x4=8 elements. The 8isometries are described in the following Table III:

Table IIIType of transformation Symbol Action over N Parity as permutation

of the set N

Linear and translation2 CI T= (C,U,A,G) (C,U,A,G) Even

TranslationUT (C, U,A,G) (U,C,G,A) Even

TranslationAT (C,U,A,G) (A,G,C,U) Even

Translation TG (C,U,A,G) (G,A,U,C) EvenLinear

12P (C, U,A,G) (C,A,U,G) Odd

CompositionU 1T P 2 (C,U,A,G) (U,G,C,A) Odd

Composition A 1T P 2 (C,U,A,G) (A,C,G,U) OddComposition

G 1T P 2 (C, U,A,G) (G, U,A,C) ) Odd

Remark:Notice that the linear transformation is the same as the translation .

Hence, the linear transformation is the same as the composition . Then,

there are only 8 different hypercubes that are determined, respectively, by thedifferent orderings: CUAG, UCGA, AGCU, GAUC, CAUG, UGCA, ACGU, GUAC.The first 4 are even permutations of the primary ordering CUAG, while theremainders are odd permutations. It produces a partition of the set into 2 classes:The class of those obtained by translations (even permutations), and the class

obtained by the linear transformation or the composition of with a

translation (odd permutations). It is straightforward to notice that every isometry ofthe vector space , induces exactly an isometry of the 6-dimensional hypercube

. Then, the 8 selected permutations of the set

2I CT

12P C 1T P

12P

2

12P

N

( )6

2 { }N C, U= ,A,G , induce

isometries of the metric space N N N of all the triplets, with respect to itsHamming distance.

4 The hypercube as a 3-dimensional vector space over the Galois

Field of 4 elements.- The Genetic Hotel2

GF(4)

NNN

Under the matching (C, U,A,G) (00,01,10,11) if we identify every nucleotide withthe correspondig integer, in the decimal representation of the binary duplets, thatis, the numbers 0, 1, 2, and 3, the addition table of the group becomes

(Table IV):

(N, )+

Table IV

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+ 0 1 2 3

0 0 1 2 3

1 1 0 3 2

2 2 3 0 13 3 2 1 0

The set can be endowed with a structure of a field, by means of a suitabledefinition of a product. In a forthcoming work [4], it has been made in such a way,that the resulting table for multiplication is (Table V):

N

Table V

C U A GC C C C CU C U A G

A C A G U

G C G U A

And, under the numerical notation, Table V becomes (Table VI):

Table VI

0 1 2 30 0 0 0 0

1 0 1 2 3

2 0 2 3 1

3 0 3 1 2

The obtained field is an algebraic extension, of degree 2, of the binary field{ }2 0,1= of 2 elements. It is up to isomorphisms, the so called Galois Field

of four elements. Since the set has the structure of the Galois Field , the

6-dimensional hypercube becomes a GF in a 3-dimensional vector

space.

GF(4)

4)N

NNN

GF(

2 (4)

4.1 Embedding of the cube in the 3-dimensionalNNN space 3

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The interpretation of the duplets 00, 01, 10 and 11 as the binary representation of

the integers 0, 1, 2 and 3, allows us to insert the set as a subset of the

Euclidean 3-dimensional space,

3N NNN= 3 = , being the field of real

numbers. The eight triplets CCC, GCC, CGC, GGC, CCG, GCG, CGG, GGG,

correspond to the vertexes of a cube or regular hexahedron, with 3 of its edgeslying over the coordinated axis. This cube has edges of length 3 and it is the unionof 27 unitary cubes, which are copies of the unitary cube whose vertexes are:CCC, CCU, CUC, CUU, UCC, UCU, UUC, UUU, and that we denote as . The

set is a 3-dimensional

YYY

YYY 2Z

YYY

vector subspace of , if we see it as a 6-

dimensional vector space. We will call it the elementary subcube of the cube

. We recall that the cube is a subgroup of the additive group

NNN

2Z

NNN (NNN, )+ of the vector space , but it is not aN NNN N subspace, since it is not closedunder multiplication of the scalars A or G of the field by vectors of . The

cube has been called the Hotel of Triplets, or Genetic Hotel, because of its

resemblance with a three-floor building [4].

N YYY

NNN

4.2 Definition of a distance in the cube NNN

In the 3-dimensional vector space NNN a new type of distance has been

defined, which is consistent with the graph-theoretical concept of a distance

between vertexes. For 2 triplets (X and of we define the

distance between them as the nonnegative integer

GF(4)

1 2 3X X )

1 2 3(YY Y ) NNN

1 1X Y +

2 2 3 3X Y X Y +

Z

)

,

where the operations are the ordinary addition and subtraction in the set ofintegers. The latter definition of distance in is similar to the one used for the

definition of the Hamming distance in a hypercube . This distance gives usthe minimal number of edges or unitary segments between the 2 triplets. Obviouslythe maximal distance in is equal to 9.

NNN

2(

n

Z

NNN

Remark: Note that our distances defined in the set are a restriction of a

distance defined in the Euclidean 3-dimensional

NNN

space . Although it is notthe usual Euclidean distance, it is nonetheless a distance, according to the generaldefinition of this concept.The cube can be envisaged as a graph, where the

vertexes are the triplets and the edges are the binary subsets

3

NNN

{ }1 2 3 1 2 3(X X X ),(YY Y ) ,

such that the distance, D is equal to 1. In this case, we say that

both triplets are adjacent. It is easy to prove that, they are adjacent if, and only if,they differ in only one component and these different components are consecutiveunder the selected order (C . Our defined distance between two vertexes,

in a graph theoretical approach, coincides with the minimal length of a pathbetween both vertexes.

1 2 3 3((X X X ), ))

,U,A,G

1 2(Y Y Y

)

4.3 The situation with respect to the distance defined in the cube as

a 3-dimensional vector space over the finite field of 4 elementsD NNN

GF(4)

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In our geometric model all the edges are contained in horizontal lines, parallel tothe vectorial lines or CN , or in vertical lines, that are parallel to the vectorial

line

NCC C

CCN ,We recall the fact that the insertion of the vector N space

into the space , does not mean that it is a vector subspace, since the

operations are different, as the scalars fields are also different.Now, we observethat the odd permutation:

3N NNN=

3

C C, U A, A U,G G , that is, the matching, which induces a linear transformation of the matrix of

the vector space , then a group homomorphism of the additive group

, is not homomorphic for the product

(C, (C,

N

(N,

U,A,G

2)+

) A,U,G)12

P

of the Galois Field GF . In fact, ittransforms the neutral element U of in the element A, which is not the neutral.

Hence, it does not induce a linear transformation of the cube , if it is seen as

an vector space. We see, for example, that it transforms the vector

into the vector UCC, which is not equal to the vector

.On the other hand, the odd permutation CUA , that is

isometric with respect to the Hamming distance, is not an isometry with respect tothe distance . In fact, it converts the unitary vectors UCC, CUC and CCU into thevectors ACC, CAC, and CCA, that have length 2.

(4

CA

)

UG

N

NNN

G

NUCC)ACC)

A ( CC=A ( CC=

D

A

G

As a remarkable result, we have that the 4 even permutations of the ordered set, induce isometries with respect to the distance , but the odd

permutations do not.

(C, U,A,G) D

5 Criteria for the selection of the ordering in the set of the four nucleotides

5.1 The two different classes of 3-dimensional cubes over the Galois Field, that preserve the correspondence of the strong-weak classification

with the algebraic complementarity.

GF(4)

S far we have obtained two different families of 3-dimensional cubes:

1 : The 4 cubes obtained from the ordering CUAG, UCGA, AGCU and GAUC.

2 : The 4 cubes obtained from the ordering CAUG, UGCA, ACGU, and GUAC.

From the above results, we conclude that there are only 8 different ways ofdefining a structure of a 6-dimensional hypercube, or 6-dimensional vector space

over the binary field , in such a way that biological complementarity of

nucleotides is consistent with the algebraic complementarity of the associated

duplets of zeros and ones. They also lead to the only 8 ways up to isomorphisms,of defining a structure of vector space, being the set of DNA nucleotides,

endowed with the structure of the Galois Field of 4 elements. We can also

assert that the 4 permutations of the primary ordering CUAG, induced by additivetranslations, that is, the orderings CUAG, UCGA, AGUC and GAUC, induceisometries of the original 3-dimensional cube , which are also bijective affinetransformations. The other 4 permutations, induced by compositions of thetranslations with the odd permutation CUA , neither induce isometric

2

N N(4)

CA

GF

NNN

G UG

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transformations with respect to the distance , nor affine transformations withrespect to vector space structure. Notwithstanding, the 8 permutations induceisometric transformation of the binary 6-dimensional vector space , with

respect to its Hamming distance, and they are also affine transformations with

respect to the vector space structure. As the permutations that convert

elements of a family into elements of the other one are not isometries with respectto the distance in the 3-dimensional

D

NNNN

2

D N cube, the corresponding transformationmay change the shape and the size of a condominium of the Hotel. Thus, forexample, the set RNY that is a prism in the cube , with edges of length 3, 1,

1, is transformed, under the permutation (C,

NNN

,G)U, A (C, A, U, G) , into a biggerprism of lengths 3, 2, 2 (see Figs. 1 and 2).

G 10,

} {M : C,A K =

M

NNN

1 0

1 1

A

N

21A

) ( )

N

12P

1 1

0 1

C,U,A,

D

G C, U,

21A

Figures 1 and 2

5.2 Orderings where the amino-keto classification is consistent withalgebraic complementarity

Next, we will select the order of nucleotides in such a way that algebraicallycomplementary duplets correspond to nucleotides of the same class: the amino orthe keto class. First, we select the ordering CUGA, that is, the matching : C 00,

and AU 0 1 , that is consistent with the partition

{ }2: N : U,G

N

(amino-keto). If we examine the bijection that

transforms the initial ordering CUAG into the ordering CUGA, we see that it is, in

the vector space , the linear transformation with matrix2

M

D

=21

. This

matrix is obtained from the identity matrix by the addition of the first row to thesecond one, having the only non-null element that is off the main diagonal in theplace (2,1). As it is well known, in these kind of matrices the so-called additionelementary matrices, when they are multiplied to the right by another matrix ,perform on the same transformation. Then, it fixes the vectors C and U, andinterchanges the vectors A and G. In the present case the linear transformation

represented by is not isometric with respect to the Hamming distance, since it

interchanges the vectors of module 1 and module 2, respectively. For the samereason, we can say that it is not an isometry with respect to the distance ,because it interchanges A of module 2 with G of module 3. It is easy to prove thatthis transformation is not linear with respect to the structure of the vector space

of the set .

1, 1

As a conclusion, we see that the permutation ( G,A

=

is neither

isometric with respect to the original Hamming distance, induced by the orderingCUAG, nor isometric with respect to the distance in the associated vector

space. The linear isometry, originally represented by the matrix , which

interchanges U and A, is now represented by the matrix , which

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interchangesU and G, and fixes C and A. Now, applying the new ordering CUGA

all the isometries of the vector space we obtain the following Table VII:2 N

I T= )A

UT )A

AT )A

GT )A

12A )A

UT A

)A

AT A )A

GT A )A

Table VII

Type of transformation Symbol Action over N Parity as permutationof the set N

Linear and translation2 C ( ( )C, U, G, C, U, G, A Even

Translation ( ( )C, U, G, U, C, A, G Even

Translation ( ( )C, U, G, A, G, U, C Even

Translation ( ( )C, U, G, G, A,C, U Even

Linear ( ( )C, U, G, C, G, U, A Odd

Composition12

( ( )C, U, G, U, A, C, G Odd

Composition12 ( ( )C, U, G, A, U, G, C Odd

Composition12 ( ( )C, U, G, G, C, A, U Odd

5.2.1 The 2 different classes of 3-dimensional cubes over the Galois Field, that preserve the correspondence of the aminoketo classification

with algebraic complementarity

GF(4)

From Table VII we see that we have 2 other different families of 3-dimensional

cubes:

3: The 4 cubes obtained from the orderings CUGA, UCAG, AGUC and GACU.

4: The 4 cubes obtained from the orderings CGUA, UACG, AUGC, and GCAU.Here we have again that, as the permutations that convert elements of a family intoelements of the other one are not isometries with respect to the distance D in the3-dimensional cube, the corresponding transformation may change the shapeand the size of a figure. Thus, for example, the set RNY that is a prism in the cube

with edges of length 3, 1, 1, is converted in itself under the permutation

, and under the permutation

N

C, U

NNN

C,U,( ) (A,G ,G,A ) ( )C, U, G, A (C, G, A) U, , is

converted into the union of 2 cubes, one of sides with lengths 3 and the other with

sides of lengths 1 (see Figs. 3 and 4).

Figures 3 and 4

5.3 Orderings where the pyrimidine-purine classification is consistent withalgebraic complementarity

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Now we will select the order of nucleotides in such a way that algebraicallycomplementary duplets correspond to nucleotides of the same class: the

pyrimidine ( or the purine classes)Y ( )R

,

. First, we select the ordering CAGU, that

is, the matching: C 00 , A 01 G 10, U 11, that is consistent with the

partition { } { }3: N Y :C, U R : A,G , =

U,

(pyrimidine-purine). If we examine the

bijection that transforms the initial ordering CUAG into the ordering CAGU, we see

that it is a linear transformation with matrix in the vector space

. The matrix

1 1

1 0

12

N

=12A P 2

N

12

12P

( )A,G

D

is obtained from the identity matrix by permutation of its

columns: when it is multiplied to the left by another matrix M , it performs on the

same transformation. The matrix

M

12 12A P performs a cyclic permutation over the

vectors U, A and G. In the present case the linear transformation represented by

is not isometric with respect to the Hamming distance, since it converts A of

module 1 into G of module 2, and G, of module 2 into U of module 1. For the same

reason, we can say that it is not an isometry with respect to the distance ,because it converts U of module 1 into A of module 2, A of module 2 into G ofmodule 3 and G, of module 3 into U of module 1. As a conclusion, we see that the

permutation is neither an isometry with respect to the

original Hamming distance, induced by the ordering CUAG, nor an isometry withrespect to the distance in the associated

12A P

D

(C,A,G,U )C,

vector space. The linear isometry,originally represented by the matrix , which interchanges U and A, is now

represented by the matrix

12P

1 0

1 1=21A ,which interchangesA and G, and fixes C

and U. Now, according to the new ordering CAGU all the isometries of the

vector space are shown in the following Table VIII:2 NTable VIII

Type of transformation Symbol Action over N Parity as permutationof the set N

Linear and translation2 CI T= ( ) ( )C, A, G, U C, A, G, U Even

TranslationUT ( ) ( )C, A, G, U U, G, A, C Even

TranslationAT ( ) ( )C, A, G, U A, C, U, G Even

Translation GT ( ) ( )C, A, G, U G, U, C, A EvenLinear

12A ( ) ( )C, A, G, U C, G, A, U Odd

CompositionU 1T 2A ( ) ( )C, A, G, U U, A, G, C Odd

CompositionA 1T 2A ( ) ( )C, A, G, U A, U, C, G Odd

CompositionG 1T 2A ( ) ( )C, A, G, U G, C, U, A Odd

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5.3.1 The two different classes of 3-dimensional cubes over the Galois Field, that preserve the correspondence of the pyrimidine-purine

classification with the algebraic complementarity

GF(4)

From Table VIII we see that we have 2 other different families of 3-dimensionalcubes:

5: The 4 cubes obtained from the orderings CAGU, UGAC, ACUG and GUCA.6: The 4 cubes obtained from the orderings CGAU, UAGC, AUCG, and GCUA.Here again, we have that, as the permutations that convert elements of a familyinto elements of the other one are not isometries with respect to the distance inthe 3-dimensional cube, the corresponding transformation may change theshape and the size of a figure. So, for example, the set RNY that is a prism in thecube , with sides of length 3, 1, 1, is converted in a prism of sides of lengths

3, 2, 2, under the permutation

D

N

NNN

( ) ( )C, U, A, G C, A, G, U

)U

, and under both

permutations and( ) (C, U, C,A,G,A,G ( ) ( )UCUAG CGA , is transformed

into the union of 2 prisms, one of sides with lengths 3, 1, 3 and the other withsides of lengths 1, 1, 3 (see Figs. 5 and 6).

Figures 5 and 6

6 Conclusions

In this work we have analyzed the 24 possible matchings of the set of the 4

nucleotides with the 4 binary set of duplets. The set was partitioned into 3classes: strong-weak, amino-keto and pyrimidine-purine. For each category, thereare 2 classes, each with 4 cubes, over , that preserve the correspondence of

the algebraic complementarity. The cube has been christened as The Hotelof Triplets, or Genetic Hotel, because of its resemblance with a three-floor building.

As the permutations that convert elements of a family into elements of the otherone are not isometries with respect to the distance in the 3-dimensional

cube, the corresponding transformation may change the shape and the size ofa condominium of the Hotel. The prism RNY adopts different shapes in the set

depending upon the selected ordering.

N

D

GF(4)

NNN

N

NNN

A priorithere is any biological reason to prefer one ordering over others. Assuming

a primaeval genetic code, such as the RNY code, different evolutionarymechanisms can lead to the same Standard Genetic Code [3]. Different orderingscould be better than others in terms of the properties (e.g. physicochemical,evolutionary, and symmetrical) that are under study and that one finds convenientto be highlighted.

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Acknowledgments

M. V. Jos was financially supported by PAPIIT IN205307, UNAM, Mxico and bythe Macroprojecto: Tecnologas para la Universidad de la Informacin y laComputacin (MTUIC). E. M. was supported by the MTUIC and by Universidad

Central Marta Abreu de las Villas, Santa Clara, Cuba.

References

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

He, M., Petoukhov, S. V., Ricci, P. E. Genetic code, Hamming distance andstochastic matrices.Bull. Math. Biol.66: 14051421 (2004).Jimnez-Montao, M.A., de la Mora-Basez, C.R., Pschel, T. 1996. Thehypercube structure of the genetic code explains conservative and non-conservative amino acid substitutions in vivo and in vitro. Biosystems39:117125 (1996).Jos M. V., Morgado E. R., Govezensky T.An extended RNA code and itsrelationship to the standard genetic code: an algebraic and geometricalapproach. Bull. Math. Biol. 69: 215-243 (2007).Jos M. V., Morgado E. R., Govezensky T. Genetic hotels for the Standardgenetic code: Three dimensional algebraic model. I. In preparation.Karasev V. A., Soronkin S. G. Topological structure of the genetic code.Russ. J, Gen.33: 622-628 (1997).Klump H. H. The physical basis of the genetic code: The choice betweenspeed and precision.Arch. Biochem. Biophys.301: 207-209 (1993).Snchez, R., Morgado, E., Grau, R. A genetic code Boolean structure. I.The meaning of Boolean deductions. Bull. Math. Biol. 67:114 (2005).Snchez, R., Morgado, E., Grau, R. Gene algebra from a genetic codealgebraic structure. J. Math. Biol. 51, 431-457 (2005).Snchez, R., Grau, R., Morgado E. A novel Lie algebra of the genetic codeover the Galois field of four DNA bases. Math. Biosci.202: 156-174 (2006).Swanson R. A unifying concept for the amino acid code. Bull. Math. Biol.46:187-203 (1984).Stambuk N. Universal properties of the genetic code. Croatica Chemica

ACTA73:1123-1139 (2000).

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Figure 1. The prism RNY in the cube under the ordering CUAGNNN

AGU

AGC

GGU

GGC

AAU

AAC

GAU

GAC

AUU

AUC

GUU

GUC

ACU

ACC

GCU

GCC

CUAG

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Figure 2. The prism RNY in the cube under the ordering CAUGNNN

GC

GC

AGU

AGC

AUU

AUC

GGU

GGC

AAU

AAC

GUU

GUC

ACU

ACC

GAU

GAC

U

C

CAUG

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Figure 3.The prism RNY in the cube under the ordering CUGANNN

GAU

GAC

AAU

AAC

GGU

GGC

AGU

AGC

GUU

GUC

AUU

AUC

GCU

GCC

ACU

ACC

CUGA

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Figure 4. The prism RNY in the cube under the ordering CGUANNN

GAU

GAC

GUU

GUC

AAU

AAC

GGU

GGC

AUU

AUC

GCU

GCC

AGU

AGC

ACU

ACC

CGUA

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Figure 5. The prism RNY in the cube under the ordering CAGUNNN

AUU

AUC

GUU

GUC

AGU

AGC

GGU

GGC

AAU

AAC

GAU

GAC

ACC

ACU

GCU

GCC

CAGU

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Figure 6. The prism RNY in the cube under the ordering CGAUNNN

GUU

GUC

AUU

AUC

GAU

GAC

AAU

AAC

GGU

GGC

AGU

AGC

GCC

GCU

ACU

ACC

CGAU

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