He M., Petoukhov S. Mathematics of Bioinformatics: Theory, Methods and Applications

GENETIC MATRICES AND THE DEGENERACY OF THE GENETIC CODE 193

FIGURE 8.13 Genomatrix

P

231

CAGU

.

CCC CA C ACC AAC CCA C AA ACA AAA

CGC CUC AGC AUC CGA CUA A GA AUA

GCC GAC UCC UAC GCA GA A UCA UAA

GGC GUC UGC UUC GGA GUA UG A U UA

CCG CA G ACG AAG CCU C AU ACU AAU

CGG CUG AGG AUG CGU CUU A GU AUU

GCG GAG UCG UAG GCU GA U UCU UAU

GGG GUG UGG UUG GGU GUU UG U U UU

FIGURE 8.14 Genomatrix

P

312

CAGU

.

CCC ACC CCA ACA C AC AAC C AA AAA

GCC UCC GCA UCA GAC UA C GAA UAA

CCG ACG CCU ACU C AG AA G CAU AAU

GCG UCG GCU UCU GAG UA G GAU UAU

CGC AG C CGA AGA CUC A UC CUA AUA

GGC UG C GGA UGA GUC UU C GUA UUA

CGG AG G CGU AGU CUG A UG CUU AUU

GGG U GG GGU UGU GUG UU G GUU UUU

FIGURE 8.15 Genomatrix

P

132

CAGU

.

CCC CA C CCA CAA ACC A AC ACA AAA

CGC CUC CGA CUA A GC AU C A GA AUA

CCG CA G CCU CAU ACG A AG ACU AAU

CGG CUG CGU CUU AGG AU G A GU AUU

GCC GAC GCA GA A UCC U AC UCA UAA

GGC GUC GGA GUA UGC UU C U GA UUA

GCG GAG GCU GA U UCG U AG UCU UAU

GGG GUG GGU GUU UG G U UG UG U U UU

FIGURE 8.16 Genomatrix

P

213

CAGU

.

CGC CGA AGC AGA CUC CUA A UC AUA

CGG CGU AGG AGU CUG CUU A UG AUU

GGC GGA UGC UGA GUC GUA U UC UUA

GGG GGU UGG UGU GUG GUU U UG UUU

CCC CCA ACC ACA C AC CAA A AC AAA

CCG CCU ACG ACU C AG CA U A AG AAU

GCC GCA UCC UCA GAC GA A UAC UAA

GCG GCU UCG UCU GAG G AU U AG UAU

FIGURE 8.17 Genomatrix

P

321

CAGU

.

CCC ACC CA C AAC CCA ACA C AA AAA

GCC UCC GAC UAC GCA UCA GAA UAA

CGC AG C CUC AUC CGA AGA CUA AUA

GGC U GC GUC UUC GGA UGA GUA UUA

CCG ACG CA G AAG CCU ACU C AU AAU

GCG UCG GAG UAG GCU UCU GAU UAU

CGG AG G CUG AUG CGU AGU CUU AUU

GGG U GG GUG UUG GGU UGU GUU UUU

194 MATRIX GENETICS, HADAMARD MATRICES, AND ALGEBRAIC BIOLOGY

algorithm was discovered (Petoukhov, 2005b ) which inverts the color of some

cells in these matrices and leads to the generation of a new set of mosaic

genomatrices (Figures 8.20 and 8.21 ) related to the Walsh functions and to

Hadamard matrices. If the black (white) color of each cell of these new matri-

ces is interpreted such that this cell contains the element + 1 ( − 1), these numeri-

cal matrices demonstrate the following:

• Each row (or each column in some cases) of these new genomatrices

coincides with one of the Walsh functions.

• A set of all rows (or columns in some cases) of each genomatrix is a

complete set of eight different Walsh functions, and each genomatrix is

one of the Hadamard 8 × 8 matrices.

Later we describe this genetic algorithm, called the U - algorithm because it

is constructed on the basis of the special molecular status of uracil U (or

thymine T). But next we review Hadamard matrices.

8.3 THE GENETIC CODE AND HADAMARD MATRICES

By deﬁ nition, a Hadamard matrix of dimension n is the n × n matrix H ( n ) with

elements + 1 and − 1. It satisﬁ es the condition

HnHn nI

n

() ()

=

T

(8.1)

where H ( n )

T

is the transposed matrix and H ( n ) is the n × n identity matrix.

Some Hadamard matrices of dimension 2

k

are formed, for example, by the

recursive formula H (2

k

) = H (2)

(

k

)

= H (2) 䊟 H (2

k

− 1

) for 2 ≤ k ∈ N , where 䊟

denotes the Kronecker (or tensor) product and ( k ) the Kronecker exponentia-

tion, k and N are integers, and H (2) is as shown in Figure 8.18 . In this chapter

we will indicate by black (white) all cells in the Hadamard matrices that

contain the element + 1 (the element − 1, correspondingly).

Rows of a Hadamard matrix are mutually orthogonal; that is, every two

different rows in a Hadamard matrix represent two perpendicular vectors, a

FIGURE 8.18 Family of Hadamard matrices H (2

k

) based on the Kronecker product.

Matrix cells with elements + 1 are shown in black.

1 1 1 1

1 1 -1 1 - 1 1

H

(2) = -1 1 ; H(4) = -1 -1 1 1

1 -1 -1 1

H(2

K-1

) H(2

K-1

)

H(2

K

) = -H (2

K-1

) H(2

K-1

)

THE GENETIC CODE AND HADAMARD MATRICES 195

scalar product of which is equal to 0. The element − 1 can be disposed in any

of four positions in a Hadamard matrix H (2).

A Kronecker product of two Hadamard matrices is a Hadamard matrix as

well. A permutation of any columns or rows of a Hadamard matrix leads to a

new Hadamard matrix. Hadamard matrices and their Kronecker powers are

used widely in spectral methods of analysis and processing of discrete signals

and in quantum computers. A transform of a vector a¯ by means of a Hadamard

matrix H gives the vector u¯ = H a¯ , generally called the Hadamard spectrum of

the vector a¯ . A greater analogy exists between Hadamard transforms and

Fourier transforms (Ahmed and Rao, 1975 ). In particular, the fast Hadamard

transform exists in parallel with the fast Fourier transform. An entire class of

multichannel spectrometers with Hadamard transforms is known (Tolmachev,

1976 ), where the principle of tape masks (or chain masks) is used, reminiscent

of the principles of a chain construction of genetic texts in DNA. Hadamard

matrices are used widely in the theory of coding. For example, they are con-

nected with Reed – Muller error - correcting codes and with Hadamard codes

(Peterson and Weldon, 1972 ), with the theory of compression of signals and

images, with a realization of Boolean functions by means of spectral methods,

with the theory of planning of multiple - factor experiments, and in many other

branches of mathematics.

Rows of Hadamard matrices called Walsh functions (see Chapter 1) are

used for a spectral presentation and for a transfer of discrete signals (Ahmed

and Rao, 1975 ; Geramita, 1979 ; Yarlagadda and Hershey, 1997 ). Walsh func-

tions can be represented in terms of products of Rademacher functions

r

n

( t ) = sign(sin 2

n

π t ), n = 1, 2, 3, … , which accept the two values + 1 and − 1 only

(here “ sign ” is the function that gives the sign of the argument). When united

in square matrices, sets of numerated Walsh functions form systems that

depend on features of the union. Figure 8.19 shows two examples of systems

FIGURE 8.19 Examples of two systems of Walsh functions which are used frequently

in the theory of digital signal processing. Left: the Walsh – Hadamard system. Right: the

Walsh – Paley system. Each black (white) cell contains the number + 1 ( − 1). Quantities

of changes of the signs + and − are shown for each row and each column. (From

Trahtman and Trahtman, 1975 .)

0 0

7 1

3 3

4 2

1 7

6 6

2 4

5 5

0 7 3 4 1 6 2 5 0 1 3 2 7 6 4 5

196 MATRIX GENETICS, HADAMARD MATRICES, AND ALGEBRAIC BIOLOGY

of such functions, which are used widely in the theory of digital signal

processing.

Hadamard matrices are connected with Walsh – Hadamard transforms,

which are the most famous among nonsinusoidal orthogonal transforms and

which can be calculated by means of mathematical operations of addition and

subtraction only [for more details, see the literature (Ahmed and Rao, 1975 ;

Trahtman and Trahtman, 1975 ; Yarlagadda and Hershey, 1997 )]. Hereafter we

use the simpliﬁ ed designations of matrix elements in illustrations of Hadamard

matrices: the symbol “ + ” or a black matrix cell represents the element + 1; the

symbol − or a white matrix cell represents the element − 1. The theory of dis-

crete signals pays special attention to quantities of changes of the signs +

and − along each row and each column in Hadamard matrices. These quanti-

ties are connected through the important notion of sequency as a generaliza-

tion of the notion of frequency (Ahmed and Rao, 1975 , p. 85).

Biological organisms are sets of biochemical molecules. A wide use of

Hadamard matrices in analytical chemistry can be found, for example, in the

work of Pan (2007) , which draws special attention to applications of Hadamard

matrices to enhance signal - to - noise ratio. This is explained in a simple example

of weighing. The basic idea is connected with weighing of objects in groups,

but not separately, for a more accurate determination of their individual

weights. In the example of four objects, we can weigh them in two different

ways. First, we can weigh each of them individually by means of a single pan

spring balance well calibrated to give us correct values Ψ

1

, Ψ

2

, Ψ

3

, and Ψ

4

for

objects 1, 2, 3, and 4 with a small random error e . Second, we can weigh all

four objects in groups by means of a two - pan balance to arrive at their general

weights η

1

, η

2

, η

3

, and η

4

in the next four weighings, with appropriate random

errors e

1

, e

2

, e

3

, and e

4

:

η

112341

=++++ΨΨΨΨe

η

212342

=−+−+ΨΨΨΨe

η

312343

=+−−+ΨΨΨΨe

η

412344

=−−++ΨΨΨΨe

Here a measurement with a negative value means that the object is placed

on the opposite pan of the balance. From these equations one can easily cal-

culate the values Ψ

1

, Ψ

2

, Ψ

3

, and Ψ

4

, and this ﬁ nal result will be much more

precise than in the previous case of weighing each object individually [for

details, see the work by Pan (2007) ]. The disposition of the signs + and − i n

this system of four equations is identical to their disposition in the relevant

Hadamard 4 × 4 matrix. In this way, applications of Hadamard transforms

enhance the signal - to - noise ratio.

Some of the normalized Hadamard matrices are unitary operators (e.g.,

2

− 0.5

[1 1; 1 − 1]). They serve as one of the important instruments in creating

THE GENETIC CODE AND HADAMARD MATRICES 197

quantum computers, which utilize Hadamard gates (as the evolution of the

closed quantum system is unitary) (Nielsen and Chuang, 2001 ). Next we dem-

onstrate connections of Hadamard matrices with the Kronecker families of

genetic matrices described above.

Algebraic biology already includes examples of applications of Walsh func-

tions (alongside other systems of basic functions) to the spectral analysis of

various aspects of genetic algorithms and sequences (Forrest and Mitchell,

1991 ; Geadah and Corinthios 1977 ; Goldberg, 1989 ; Lee and Kaveh, 1986 ;

Shiozaki, 1980 ; Vose and Wright, 1998 ; Waterman, 1999 ). The book by

Zalmanzon ( 1989 , p. 416) contains a review of investigations made by various

authors about Walsh orthogonal functions in physiological systems of supra-

cellular levels as well. We investigate whether structures of the genetic code

have such direct relations with Hadamard matrices, which can justify system-

atic applications of Walsh – Hadamard functions to spectral and other analyses

of many inherited biological structures at various levels. In this section we put

forth evidence regarding connections of Hadamard matrices with the genetic

code in its Kronecker matrix forms of presentation.

The genetic alphabet with its four letters A (adenine), C (cytosine), G

(guanine), and U/T (uracil in RNA and thymine in DNA) is characterized by

a phenomenological disturbance of symmetry related with the special status

of the letter U/T:

• The three nitrogenous bases A, C, and G have one amide (amino group),

NH

2

, but the fourth basis, U/T, does not have this amide (Figure 2.1 ).

• The letter U is replaced by the letter T in genetic sequences only at transi-

tions from RNA to DNA, and vice versa, for unknown reasons (in con-

trast to the three letters A, C, and G, which are not replaced).

• This special status of U/T leads to a special U - algorithm, which trans-

forms a wide set of genetic 8 × 8 matrices of 64 triplets into appropriate

Hadamard 8 × 8 matrices.

Here we should mention the importance of amino group NH

2

. The amino

group of amino acids bears a base function that provides recognition of an

amino acid by an enzyme (Chapeville and Haenni, 1974 ). The importance of

nitrogen compounds in molecular genetics is reﬂ ected in such names as “ amino

acids ” (organic acids containing amino groups), “ nitrogenous bases ” ; the “ N -

end ” of a nucleotide circuit, with which protein synthesis always begins; and

so on. All proteins are polyamides. A lack of proteins in food leads to a number

of heavy infringements in the nitrogenous exchange. Beginning with works by

Gierer and Mundry (1958) and Schuster and Schramm (1958) , it has been

known that action of nitrous acid, NHO

2

, on RNA leads to the amino - mutation

of RNA. More precisely, this action deletes the amino group NH

2

at the nitrog-

enous bases A and C and leads ﬁ nally to a replacement of the nitrogenous

bases A and C by the bases G and U, respectively: A → G and C → U. These

198 MATRIX GENETICS, HADAMARD MATRICES, AND ALGEBRAIC BIOLOGY

amino - mutations are utilized traditionally to demonstrate molecular mecha-

nisms as the origin of genetic mutations. Nitrogenous acid exists only in diluted

water solutions, which are similar to solutions in biological organisms.

The phenomenological division of the four - letter genetic alphabet in the

ratio 3 : 1 reminds one of the division of four components of the simplest

Hadamard matrix H (2) = [ + 1 + 1 ; − 1 + 1] in the same ratio with three compo-

nents + 1 and a single component − 1 (Figure 8.18 ). Taking it into account that

the genetic alphabet can be represented in the form of a 2 × 2 matrix

P = [C A; U G] by analogy with the Hadamard matrix H (2) (Figure 8.18 ).

Let us return now to the U - algorithm, which transforms the genomatrices

of 64 triplets (Figures 8.2 and 8.4 to 8.17 ) into relevant Hadamard 8 × 8 matri-

ces. A deﬁ nition of the U - algorithm is the following:

• Each triplet in the black - and - white genomatrices (Figures 8.2 and 8.4 to

8.17 ) should change its own color into the opposite color each time the

letter U stands in an odd position (in the ﬁ rst or third position) inside

the triplet.

• Each triplet, which is disposed in a black (white) cell of such a changed

genomatrix is interpreted as the number + 1 ( − 1 ) .

For example, according to the U - algorithm, the cells with the triplets UCA

and GAU change their sign once, while the cell with the triplet UAU changes

its sign twice, which means that the sign of this cell is unchanged. As a result

of such U - algorithmic sign changes, a new mosaic in each of the same genoma-

trices appears. Such a mosaic is identical to the corresponding mosaic of a

Hadamard matrix. Actually, if each black triplet (white triplet) in these geno-

matrices is replaced by the number + 1 ( − 1), numeric matrices are formed

(Figures 8.20 and 8.21 ). One can easily check that these new numeric matrices

satisfy deﬁ nition (8.1) of Hadamard 8 × 8 matrices: H (8) H (8)

T

= 8 I

n

. One can

suppose that this U - algorithm (of inverting the signs every time the letter U

or T appears in an odd position of triplets) is connected with the biological

mechanism of mutual replacement of the letters U and T at transition from

RNA to DNA, and vice versa.

One should note a special feature of the genetic Hadamard matrices in

Figure 8.20 : A quantity of changes of the signs + and − is equal to 14 for each

of the halves of these matrices (we say upper, lower, left, and right halves).

Such a “ symmetrical ” feature is typical of many other genetic Hadamard

matrices (see some additional examples in Figure 8.21 ). One can call Hadamard

matrices with such a feature balanced Hadamard matrices . One can check that

each of the 4 × 4 quadrants of these Hadamard 8 × 8 matrices is a balanced

Hadamard matrix as well. This feature distinguishes the genetic Hadamard

matrices described from the Hadamard matrices in Figure 8.19 , which are

widely used in technical applications. For some reason nature has chosen the

genetic code, which is connected with balanced Hadamard matrices. One can

THE GENETIC CODE AND HADAMARD MATRICES 199

ﬁ nd additional details on this theme in the literature (He and Petoukhov, 2009 ;

Kappraff and Petoukhov, 2009 ; Petoukhov, 2008a – d ; Petoukhov and He, 2009 ).

All such Hadamard matrices represent various basic systems of orthogonal

functions, which are coordinated with the structural peculiarities of molecular

systems of the genetic code. They can be utilized in genetic systems for spectral

methods of genetic information processing with the use of noise - immunity

coding, of compression of signals, and of other useful possibilities that

Hadamard matrices and Walsh functions possess.

FIGURE 8.20 Some examples of balanced Hadamard matrices, which are produced

from the six genomatrices indicated by means of the U - algorithm. The black cells cor-

respond to the elements + 1 and the white cells correspond to the elements − 1. Numbers

of changes of the signs + and − (or changes of colors) are shown for each row and each

column.

P

CAUG

123

: for P

CAUG

231

:

3

7

4

3

1

4

6

0

2

6

5

2

0

5

7

1

5 2 6 1 5 2 6 1

5 2 5 2 4 3 4 3

P

CAUG

312

: for P

CAUG

132

:

1

7

6

1

2

4

5

2

7

6

0

4

5

3

3 3 4 4 4 4 3 3

3 4 4 3 3 4 4 3

for

for P

CAUG

213

:

for P

CAUG

321

:

1

3

6

4

2

7

5

0

3

2

4

5

0

6

7

1

5 2 5 2 6 1 6 1

5 5 2 2 4 4 3 3

200 MATRIX GENETICS, HADAMARD MATRICES, AND ALGEBRAIC BIOLOGY

More can be added about cyclic relations inside the set of described geno-

matrices. For example, the set of genomatrices

P

123

CAUG

,

P

231

CAUG

,

P

312

CAUG

,

P

321

CAUG

,

P

213

CAUG

,

P

132

CAUG

(see Figures 8.2 , 8.4 , and 8.5 ) was produced on the basis of two

subsets of cyclic shifts of positions in triplets: 1 – 2 – 3 → 2 – 3 – 1 → 3 – 1 – 2 → 1 – 2 –

3 and 3 – 2 – 1 → 2 – 1 – 3 → 1 – 3 – 2 → 3 – 2 – 1. These cyclic relations are presented

graphically in Figure 8.22 .

FIGURE 8.21 Additional examples of the 12 balanced Hadamard matrices which are

produced from the 12 genomatrices of triplets by means of the U - algorithm. Black cells

correspond to elements + 1, and white cells correspond to elements − 1.

for P

GCAU

123

: for P

GCAU

231

: for P

GCAU

312

:

for P

GCAU

132

: for P

GCAU

213

: for P

GCAU

321

:

for P

CAGU

123

: for P

CAGU

231

: for P

CAGU

312

:

for P

CAGU

312

:

for P

CAGU

213

:

for P

CAGU

321

:

GENETIC MATRICES AND MATRIX ALGEBRAS 201

It is obvious that an analogous scheme can be demonstrated for appropriate

subsets of Hadamard genomatrices as well. Such cyclic relations among dif-

ferent genomatrices can be used in mathematical models of inherited cyclic

processes in living organisms.

8.4 GENETIC MATRICES AND MATRIX ALGEBRAS OF

HYPERCOMPLEX NUMBERS

Complex and hypercomplex numbers, which are utilized in physics and math-

ematics, possess matrix forms for their representation. The notion of number

is the main notion of mathematics and mathematical natural sciences. In view

of this, an investigation of a possible connection of the genetic code to multi-

dimensional numbers in their matrix presentations should be undertaken.

Algebras of complex numbers

zx x=+

01

**

1i

and hypercomplex numbers

xx x

kk011

** *

1i i+++

are well known. It is also known that complex and hyper-

complex numbers have presentation as well as matrix forms of linear and

FIGURE 8.22 Two cyclic sequences of the mosaic genomatrices

PPPP

123 231 312 123

CAUG CAUG CAUG CAUG

→→→

and

PPPP

321 213 132 321

CAUG CAUG CAUG CAUG

→→→

, which

arise as a result of relevant cyclic permutations of positions in triplets: 1 – 2 – 3 → 2 – 3 –

1 → 3 – 1 – 2 and 3 – 2 – 1 → 2 – 1 – 3 → 1 – 3 – 2. The number over each matrix shows a rele-

vant type of permutation of positions in all triplets. (From Petoukhov, 2008e .)

1-2-3

2-1-3

1-3-2

3-1-2 2-3-1

3-2-1

202 MATRIX GENETICS, HADAMARD MATRICES, AND ALGEBRAIC BIOLOGY

vector forms. For example complex numbers z = x * 1 + y * i (where 1 is the real

unit and i is the imaginary unit: i

2

= − 1) possess the matrix form of presenta-

tion shown in Figure 8.23 . By the way, complex numbers are utilized in com-

puters in this matrix form.

The quaternions by Hamilton

Qx x x x=+++

0112233

*** *

1i i i

(where

iii

1

2

3

2

1===−

,

ii ii i

12 21 3

**

=− =

,

ii ii i

13 31 2

**

=− =−

,

ii ii i

23 32 1

**

=− =

), which are uti-

lized widely in both physics and mathematics, have a matrix form of presenta-

tion as well. Figure 8.24 shows this form and its decomposition in the basic

elements 1 , i

1

, i

2

, and i

3

in their matrix forms of presentation. In addition, the

multiplication table of the basic elements 1 , i

1

, i

2

, and i

3

is demonstrated.

FIGURE 8.23 Top: complex numbers in their matrix form of presentation and their

decomposition on the basic elements 1 and i , which are shown in their matrix forms of

presentation as well. Matrix cells with positive coordinates are indicated in black and

cells with negative coordinates in white. Bottom: multiplication table of the basic ele-

ments 1 and i .

z = x

0

*1 + x

1

*i = x

0

x

1

= x

0

*1 0

0 1

+ x

1

*0 1

-1 0

-x

1

x

0

1 i

11 i

i i -1

FIGURE 8.24 Top: quaternions by Hamilton in the matrix form of presentation; cells

with positive coordinates are shown in black and cells with negative coordinates are

shown in white. Middle: the decomposition of quaternions in their matrix form in the

basic elements 1 , i

1

, i

2

, and i

3

, which are shown in the matrix form of presentation.

Bottom: multiplication table of these basic elements.

Q = x

0

*1 + x

1

*i

1

+ x

2

*i

2

+ x

3

*i

3

=

x

0

x

1

x

2

x

3

=

-x

1

x

0

-x

3

x

2

-x

2

x

3

x

0

-x

1

-x

3

-x

2

x

1

x

0

=x

0

*

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

+ x

1

*

0 1 0 0

-1 0 0 0

0 0 0 -1

0 0 1 0

+ x

2

*

0 0 1 0

0 0 0 1

-1 0 0 0

0 -1 0 0

+x

3

*

0 0 0 1

0 0 -1 0

0 1 0 0

-1 0 0 0

1 i

1

i

2

i

3

1 1 i

1

i

2

i

3

i

1

i

1

-1 i

3

-i

2

i

2

i

2

-i

3

-1 i

1

i

3

i

3

i

2

-i

1

-1

He M., Petoukhov S. Mathematics of Bioinformatics: Theory, Methods and Applications

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