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Genetic Code Boolean Algebras
AUTHORS' NAMES:
ROBERSY SANCHEZ
Research Institute of Tropical Roots, Tuber Crops and Banana (INIVIT).
Biotechnology group.
Santo Domingo.
Villa Clara.
CUBA.
RICARDO GRAU
Center of Studies on Informatics
Central University of Las Villas
Villa Clara
CUBA
EBERTO MORGADO
Faculty of Mathematics Physics and Computation,
Central University of Las Villas
Villa Clara
CUBA
Abstract: - Two dual Boolean algebras are proposed to understand the underlying logic of the genetic code
structure. In such Boolean structures, deductions have physico-chemical meaning. Particularly, the codons with
Adenine as a second base that code to hydrophilic amino acids are not deductible from codons with uracil in the
same position, which code to hydrophobic amino acids. As a result, in the Hasse diagrams of the corresponding
Boolean lattices the symmetric images of codons with adenine as a second base are always codons with uracil as
a second base. Moreover, according to other findings the Hamming distance between two codons and the
Boolean deductions can help us to describe the gene evolution process. In the experimental confrontation it has
also been found that most of the mutations in HIV reverse transcriptase gene with drug resistance correspond to
deductions and have a small Hamming distance related to the wild type HXB2. Some similar results have been
found in the human phenylalanine hydroxylase mutant genes.
Key-Words: - Genetic code; molecular evolution; Boolean algebra; Boolean lattice; Hasse diagrams; amino acid
properties.
1 Introduction
The study of the genetic code order is so far an
attractive puzzling problem. The genetic code is the
biochemical system to establish the rules by which
the nucleotide sequence of a gene is transcribed into
the mRNA codon sequence and by which the mRNA
is translated into the amino acid sequence of the
corresponding protein.
The codon set is an extension of the four-letter
alphabet found in the DNA molecule. These “letters”
are the DNA bases: Adenine, Guanine, Cytosine, and
Thymine, usually denoted by A, G, C, T (in the RNA,
T is changed for U, Uracil). They are paired
according to the following rule: G≡C, A=T (where
each ‘−’ symbolizes a hydrogen bond). It is said that
base G is the complementary base of C, and A is that
of T in the DNA molecule, and vice versa. Moreover,
it is well-known that there is an association between
the second-position base and the hydrophobicity of
the corresponding amino acids. The amino acids that
have U at the second position of their codons are
hydrophobic: {I, L, M, F, V} (the amino acids are
written using the one-letter symbols), whereas those
that have A at the second position are hydrophilic
(polar amino acids): {D, E, H, N, K, Q, Y} [1]. The
non-random organization of the genetic code has
been pointed out and the Manifold hypothesis was
proposed to explain the enigmatic order observed [1]
[2] [3] [4]. However, the origin of this order remains
an enigma.
There have been many attempts to introduce a formal
characterization of the genetic code [5] [6] [7] [8] [9]
[10] [11] [12]. The most relevant formal
characterizations of the genetic code that are closed
to our model involve the binary representation of 4
nucleotide bases. In their original paper JiménezMontaño et. al. [7] [9] suggested a binary
interpretation of the genetic code. These authors
denoted the chemical type: {R, Y} as Ci, where R: {A,
G} are purine and Y: {C, U} are pyrimidines and the
hydrogen bond category as Hi: {W, S}, where W: {A,
U} are weak and S: {C, G} are strong. Then they
assumed that the codon-anticodon interaction energy
obeyed
the
following
hierarchical
order:
C2>H2>C1>H1>C3>H3. Finally they defined the
following correspondences A=00, G=01, U=10,
C=11. As a result, the Boolean hypercube could
represent this binary code of six variables.
Several models with a binary representation of the
bases have been proposed, each one in a different
way. Stambuk’ model leads us to: T = 00, C = 01, G
=10 and A = 11 [13]; Karasev and Stefanov [11] in
their topological model proposed C=00, U=01, G=10,
A=11.
In our paper, two genetic code Boolean Algebras are
presented to reflect the relationship between codon
assignment and the physico-chemical properties of
amino acids. These algebras are defined by using the
same physico-chemical properties used by JiménezMontaño et al. [7]: the hydrogen bond and the
chemical types. In our model a binary representation
G=00; A=01; U=10 is defined; C=11 for one primal
Boolean algebra and C=00; U=01; A=10; G=11 for a
dual Boolean algebra.
The aim of this paper is to describe a new Boolean
structure of the standard genetic code in order to
understand the underlying logic of the genetic code
structure and to reveal the correspondence between
the algebraic considerations and the experimental
data. Different stand points of the theoretical bases of
these results were accepted for publishing in the
journals: ‘Bulletin of Mathematical Biology’ and
‘MATCH Communications in Mathematical and in
Computer Chemistry’. Here we resume them and
present new experimental evidences.
2 Theoretical Model
Since the codons are base triplets and the number of
codons in the genetic code is the number of threeletter variations with repetition from the four-letter
base alphabet, it is natural to build the genetic code
partial order from the partial order of the four DNA
bases. In every Boolean algebra for any two elements
α, β∈ X we have α≤β, if and only if ¬α∨β=1. If
¬
α∨β=1 it is said that β is deduced from α.
Furthermore, if α≤β or α≥β the elements α and β are
said to be comparable. Otherwise, they are said not to
be comparable. The last partial order is defined by
using the hydrogen bond number and the chemical
types of purines {A, G} and pyrimidines {U, C}
bases. Next, the Boolean lattice is built on the base
triplet set (64-codons set) that will be the direct third
power of the Boolean lattice of four DNA bases.
The Boolean lattice of the four bases is built
assuming that the complementary bases in the lattice
are the complementary bases in the DNA molecule
(G≡C and A=T or A=U during the translation of
mRNA). That is, it is required for the bases with the
same number of hydrogen bonds in the DNA
molecule and with different chemical types to be
complementary elements in the lattice. However, this
lattice must have two non-comparable elements, a
maximum, a minimum and two non-comparable
elements. At this point, it is assumed that the desired
Boolean lattice must be the direct third power of the
four-base Boolean lattice. The maximum element in
the Boolean lattice of the genetic code must be the
direct third power of the maximum element in the
Boolean lattice of the four bases and then we come to
the correspondence:
U→UUU, C→CCC, G→GGG, A→AAA.
The Boolean lattice with the desired biological
signification must be selected by taking into account
the physico-chemical properties of these codons and
their respective amino acids. The selection criterion
used here has its support in the following
observations:
1) Both codons GGG and CCC have the same
maximum hydrogen bond numbers. This property is
reflected in the Boolean lattice, so that the GGG
complementary element has to be CCC. Furthermore,
both codons code for small amino acid side chains
with small polarity difference [14] Glycine and
Proline respectively. Then this similar property
determines that these elements are comparable.
A
2) Both codons UUU and AAA have the same
minimum hydrogen bond numbers and then, the
complementary element in the Boolean lattice of
UUU has to be AAA. But, these codons respectively
code for amino acid side chains with extreme
opposite polarities, Leucine (a hydrophobic residue)
and Lysine (having a strong polar group).
Consequently, this opposite property determines the
fact that these elements are not supposed to be
comparable.
B
A 01
C 11
G 00
U 10
U 01
G 11
C 00
A 10
Figure 1. The Hasse diagrams of the four bases
Boolean lattices. A: Primal Boolean lattice and B:
Dual Boolean lattice
These last observations allow us to choose two dual
Boolean lattices of the four bases that will be
conventionally called Primal and Dual Boolean
lattices. From the first observation it might think that
the maximum element in the primal Boolean lattice is
C and the minimum element is G; or that in the dual
Boolean lattice, the maximum element is G and the
minimum element is C. The second observation
means that the elements U and A are not comparable
and therefore, they should not be the maximum or
minimum elements in a Boolean lattice with
biological meaning. So, we have two Boolean lattices
(B(X), ∨, ∧) (primal lattice) and (B’(X), ∧, ∨) (dual
lattice), where X={U, C, G, A}. These Boolean
lattices have their equivalent Boolean algebras. The
Hasse diagrams of the two duals Boolean lattices
obtained are shown in Fig.1. It is obvious that the
primal and dual terms in these Boolean lattices (or
Boolean algebras) will be interchanged and as we
will see later, they do not affect the biological
meaning. To simplify the notation we will refer
simultaneously to both algebras as B(X).
the four DNA bases. Explicitly, the direct product
C=B(X)xB(X)xB(X) is taken as the Boolean algebras
of codons. These algebras are isomorphic to ((Z2)6, ∨,
∧), induced by the isomorphism ϕ: B(X)→(Z2)2 , so,
for instance (in the primal algebra):
In Fig 1 the isomorphisms of the Boolean lattices
B(X) with the Boolean lattices ((Z2)2, ∨, ∧) and ((Z2)2,
∧, ∨), where Z2={0,1} are also represented. These
isomorphisms are derived from the fact that all
Boolean lattices with the same number of elements
are isomorphic. Then, it is possible to represent the
primal lattice by means of the correspondence:
G↔00; A↔01; U↔10; C↔11. Likewise, for the
dual lattice we have: C↔00; U↔01; A↔10; G↔11.
As mentioned above, this is the distance between
nodes in the Hasse diagram. The Hamming distance
between two genes (DH) will be the sum of the
Hamming distance between the respective codons.
That is, for two genes α and β with N codons, we
have:
The Boolean algebras of codons are obtained from
the direct third power of the Boolean algebras B(X) of
GUC∨CAG=CCC ↔001011∨110100=111111
GUC∧CAG=GGG↔001011 ∧110100=000000
¬
(GUC)=CAG↔¬(001011)= 110100
Thus, we start from the source alphabet of the genetic
code, consisting of the four nucleotides of the DNA
and the mRNA and arrive at the second extension of
that alphabet with 26=64 letter-codons of the genetic
code.
The Hamming distance (dH) between two codons,
represented as binary sextuplets corresponds to the
number of different digits between them. That is,
dH(GUC, CAG) = dH (001011, 110100) = 6
dH(CUG, UGA) = dH(001011, 100001) = 3
N
DH (α , β ) = ∑ d H i (α i , β i )
i =1
3 Experimental Confrontations
At this point, we want to show some model
connections with the experimental data to help in the
understanding of the mutation process of molecular
evolution. In both lattices, codons are read in the
5´→3´ direction and anticodons in the 3´→5´
direction following the standard convention.
Consequently the anticodon of the codon 5’GUC3’
represented by 001011 in the primal lattice, is the
triplet 3’CAG5’ similarly represented by 001011 in the
dual lattice or represented by 110100 in the primal
lattice
3.1 The Hasse Diagram of the Genetic Code
In Fig 2 the Hasse diagram of both the primal and
dual genetic code Boolean lattices are simultaneously
shown. For further clarity we have not directed the
edges of the graph. The symmetric properties of this
diagram are determined by the Boolean function:
NOT: XYZ→ ¬(XYZ), so that if the complementary
bases of X1, X2, X3 are the bases X1’, X2’, X3’ (Xi , X’i
∈{A, C, G, U}, i =1,2,3), then the image of codon
X1X2X33’ is codon 5’X’1X’2X’3 3’. From this last
observation, the symmetric image of a codon that
codes to hydrophilic amino acids having codons with
A in the second position is always a codon that codes
to hydrophobic amino acids (codons with U in the
second position). For instance, the symmetric image
of the anti-chain: {GUG, UGG, GGU, GGA, AGG,
GAG} in the Hasse diagram of Fig 2, is the antichain: {CAC, ACC, CCA, CCU, UCC, CUC} taking
the ordered elements one by one. As a result, GUG
has the CAC image and so on.
5’
In general, the codons that code to amino acids with
extreme hydrophobic differences are in different
chains with maximum length. Particularly, codons
with U as a second base will appear in chains of
maximum length where codons with A as a second
base will not.
These results suggest that the algebraic properties of
codons in the Boolean lattices are associated with the
hydrophobic properties of amino acids. Kauzman in
1959 stated that the hydrophobic effects had the main
Figure 2. Hasse diagram of the Boole lattice of the genetic code. Each grayscale denotes a different group of
codons according to the second base
role in the protein process folding because an aqueos
environment surrounds proteins and protein-water
interactions are considered as the leading power in
the folding of the polypeptide chain [15]. Hence, the
algebraic properties of the genetic code Boolean
lattices could help us to understand the hydrophobic
changes in the gene mutation process.
The Hamming distance between two codons in the
Hasse diagram reflects the difference between the
physico-chemical properties of the corresponding
amino acids. In Table 1 the average of the Hamming
distance between the codon sets XAZ, XUZ, XCZ and
XGZ is shown (X, Z ∈{A, C, G, U}). The maximum
distance corresponds to the transversions in the
second base of codons. It is well known that such
transversions are the most dangerous since they
frequently alter the hydrophobic properties and the
biological functions of proteins. Particularly, between
codons of hydrophilic and hydrophobic amino acids
there are larger values of the Hamming distance.
Table 1. The average of the Hamming distance
between the codon sets XAZ, XUZ, XCZ and XGZ.
XGZ XUZ XAZ XCZ
3
3
4
XGZ 2
2
4
3
XUZ 3
4
2
3
XAZ 3
3
3
2
XCZ 4
100
Activity
(%)
A
80
60
40
20
0
0
Fig 3, A and B show the changes in the enzymatic
activities of the HIV protease and the reversetranscriptase mutants versus the Hamming distance
The experimental values were taken from Rose et al.
[16] and Kim et al. [17].
100
Generally, one can observe that as the Hamming
distance between the wild type increases, the mutant
enzyme activity decreases. Such results are a
consequence of the genetic code order. The Hamming
distance between DNA bases is determined by their
physico-chemical properties. The arrangement of
codons in the genetic code is such that the Hamming
distance between codons are connected with the
physico-chemical properties of amino acids
20
2
4
Activity
(%)
6
8
10
Hamming distance
B
80
60
40
0
0
5
10
15
Hamming distance
Figure 3. Changes in the enzymatic activities of
mutants versus the Hamming distance. A: Ability of
HIV mutant proteases to process the Gag polyprotein
[18]. B: DNA polymerase activity of the HIV reversetranscriptase. [19]. Activity changes in the mutants
are normalized with respect to the wild type enzyme
3.2 Boolean Deductions
Both Boolean algebras reflect the well known
experimental results: single-base substitutions are
strongly conservative in regards to amino acid
changes in polarity [3] [18] [19]. Particularly, from
polar amino acids with codons that have A in the
second position it is impossible, by means of
deductions, to obtain any hydrophobic amino acid
with codons having U in the second position (see
above).
It have been point out that the genetic code reduces
the effects of point mutations and minimizes
subsequent transcription and translation errors to
make the reproduction of genetic information
possible [1] [3] [18] [19].
Next, it should be expected that the most frequently
observed mutations minimize the above effect in
proteins. Hence, the fitting of our model with the
experimental fact implies that the most frequent
mutations in protein should be deductible from the
respective wild type
In Table 2 those mutations in HIV-1 reverse
transcriptase that confer drug resistance were taken
into account. One can see that only a small number of
mutations are not comparable.
Likewise, in the human phenylalanine hydroxylase
(PAH) gene most frequent mutations correspond to
deductions (Table 3). The biological function of the
PAH gene is very sensible to small physico- chemical
changes. It is frequently observed that the enzymatic
activity is altered by codon changes that keep the
hydrophilic and hydrophobic properties causing
clinical disorders in patients, for instance, A342T,
D282N and D415N. But the biological function of
PAH enzyme activity is kept. So, we can say that in
the B(X) algebra the deductions have physicochemical and biological meaning.
Table 2. The deductible mutations found in the HIV reverse transcriptase gene conferring drug resistance. Most
of the reported mutations in HIV reverse transcriptase gene are comparables. Mutations that are not comparable
are presented in bold face. In the table there are only single point mutations, but there have been sequential
mutations reported in different combinations.
Amino
acid
changes
Codon
Mutation
Antiviral*
Amino
acid
changes
Codon
Mutation
Antiviral*
A 62 V
A 98 G
D 67 A
D 67 E
D 67 G
D 67 G
D 67 N
E 138 A
E 138 K
E 44 A
E 44 D
E 89 G
E 89 K
F 116 Y
F 77 L
G 141 E
G 190 A
G 190 E
G 190 Q
G 190 S
G 190 T
G 190 V
G 190 V
G 190 V
H 208 Y
I 135 M
I 135 T
K 101 Q
K 103 R
K 103 T
K 70 E
GCC -> GTC
GCA -> GGA
GAC -> GCC
GAC -> GAG
GAC -> GAG
GAC -> GGC
GAC -> AAC
GAG -> GCG
GAG -> AAG
GAA -> GCA
GAA -> GAC
GAA -> GGA
GAA -> GGA
TTT -> TAT
TTC -> CTC
GGG -> GAG
GGA -> GCA
GGA -> GAA
GGA -> CAA
GGA -> TCA
GGA -> ACA
GGA -> GTA
GGA -> GTA
GGA -> GTA
CAT -> TAT
ATA -> ATG
ATA -> ACA
AAA -> CAA
AAA -> AGA
AAA -> ACA
AAA -> GAA
Multi-drug resistant
L-697,661
AZT (zidovudine)
Multi-drug resistant
(+)dOTFC
Multi-drug resistant
AZT (zidovudine)
TSAO
Multi-drug resistant
3TC (lamivudine)
3TC (lamivudine)
PFA (foscarnet)
PFA (foscarnet)
Multi-drug resistant
Multi-drug resistant
UC-16
BI-RG-587,
Multi-drug resistant
Multi-drug resistant
DMP-266 (efavirenz)
DMP-266 HBY 097
DMP-266 BI-RG-587
DMP-266 (efavirenz)
BI-RG-587
AZT, lamivudine,
Delavirdine
Delavirdine
LY-300046 HCl
LY-300046 HCl,
S-1153, UC-42
3TC, PMEA
H 208 Y
I 135 M
I 135 T
K 101 E
K 101 Q
K 103 N
K 103 R
K 103 T
K 70 E
K 70 R
K 70 S
L 100 I
L 210 W
L 214 F
L 74 V
P 119 S
P 157 S
Q 145 M
Q 151 M
Q 161 L
R 211 K
S 156 A
T 139 I
V 106 A
V 108 I
V 118 I
V 179 D
V 179 F
V 75 T
W 88 G
W 88 S
CAT -> TAT
ATA -> ATG
ATA -> ACA
AAA -> GAA
AAA -> CAA
AAA -> AAC
AAA -> AGA
AAA -> ACA
AAA -> GAA
AAA -> AGA
AAA -> AGA
TTA -> ATA
TTG -> TGG
CTT -> TTT
TTA -> GTA
CCC -> TCC
CCA -> TCA
CAG -> ATG
CAG -> ATG
CAA -> CTA
AGG -> AAG
TCA -> GCA
ACA -> ATA
GTA -> GCA
GTA -> ATA
GTT -> ATT
GTT -> GAT
GTT -> TTT
GTA -> ACA
TGG -> GGG
TGG -> TCG
AZT, lamivudine, PFA
Delavirdine, BI-RG-587
Delavirdine, nevirapine
Multi-drug resistant
LY-300046 HCl
Multi-drug resistant
LY-300046 HCl, I-EBU
S-1153, UC-42
3TC, PMEA
AZT (zidovudine)
ddI, d4T
Multi-drug resistant
AZT, lamivudine, PFA
AZT, ph-AZT
1592U89 (abacavir)
F-ddA (lodenosine)
3TC (lamivudine)
Multi-drug resistant
Multi-drug resistant
PFA (foscarnet)
AZT , lamivudine
PFA (foscarnet)
ADAMII, Calanolide A
Multi-drug resistant
DMP-266, trovirdine
3TC (lamivudine)
Multi-drug resistant
TMC125
d4C, d4T, emtricitabine
PFA (foscarnet)
PFA (foscarnet)
*All of the mutation information contained in this printed table was taken from the Los Alamos web site:
http://resdb.lanl.gov/Resist_DB. One mutation could be resistant against multiple drugs.
Table 3. Human phenylalanine hydroxylase (PAH) variants caused by mutational events in the PAH gene. The
majority of these mutations result in deficient enzyme activity and cause hyperphenylalaninemia. The mutations
that are not comparable are presented in bold face. Only has been printed the enzymatic activities reported in the
database PAHdb (see below).
*
Amino
acid
changes
Y204C
A104D
A165P
A246V
A259T
A259V
A300S
A300V
A309D
A309V
A313T
A313V
A322G
A322T
A342P
A342T
A345S
A345T
A373T
A395G
A395P
A403V
A447D
A47E
A47V
C203C
C217G
C217R
C265Y
C334S
C357G
Codon
Mutation
TAT->TGT
GCC->GAC
GCC->CCC
GCT->GTT
GCC->ACC
GCC->GTC
GCC->TCC
GCC->GTC
GCC->GAC
GCC->GTC
CGA->ACA
GCA->GTA
GCC->GGC
GCC->ACC
GCA->CCA
GCA->ACA
GCT->TCT
GCT->ACT
GCC->ACC
GCC->GGC
GCC->CCC
GCT->GTT
GCC->GAC
GCA->GAA
GCA->GTA
TGC->TGT
TGT->GGT
TGT->CGT
TGC->TAC
TGC->TCC
TGC->GGC
PAH
Enzyme
%
80
26
<3
0,3 - 8
75
26
16
<100
9 - 16
<1
Amino
acid
changes
D143G
D145V
D151G
D222V
D282N
D315Y
D338Y
D394A
D394H
D415N
D59V
D59Y
D84Y
E178G
E178V
E182G
E183Q
E205A
E221G
E280K
E330D
E390G
E422K
E56D
E76A
E76G
F161S
F233L
F254I
F263L
F299C
**
Codon
Mutation
GAT->GGT
GAC->GTC
GAT->GGT
GAT->GTT
GAC->AAC
GAT->TAT
GAC->TAC
GAT->GCT
GAT->CAT
GAC->AAC
GAT->GTT
GAT->TAT
GAT->TAT
GAA->GGA
GAA->GTA
GAA->GGA
GAA->CAA
GAG->GCG
GAA->GGA
GAA->AAA
GAG->GAC
GAG->GGG
GAG->AAG
GAG->GAT
GAG->GCG
GAG->GGG
TTT->TCT
TTC->TTA
TTC->ATC
TTC->TTG
TTT->TGT
PAH
Enzyme
%
33
0-4
72 - 114
18
1 - 12
70 - 80
7
<3
*
The amino acid is represented using the one letter symbol.**PAH Enzyme %: PAH enzyme activity in cell lysates (as %
wild type). All of the mutation information contained in this printed table was taken from the PAHdb World Wide Web site:
http://www.pahdb.mcgill.ca/
4 Conclusion
The Boolean structures of the molecular compounds
of the genetic coding system suggest the non-random
organization of the genetic code. These Boolean
structures are very simplified minimal mathematical
models that help us to better comprehend the logic
underlying the genetic code. Explicitly, these
structures reflect a strong connection between genetic
code orders and the physico-chemical properties of
amino acids.
Next, the connections between the algebraic codon
relationships and the physico-chemical properties of
amino acids are the logical consequences of the
above considerations. Such connections are present,
for instance, in the Hasse diagrams of the Boolean
lattices. The symmetric image of a codon with U as a
second base codifying to hydrophobic amino acids is
always a codon with A as a second base codifying to
hydrophilic amino acids. As result the codon
assignment in the standard genetic code is stated in
such a way that no hydrophobic amino acid with
codons having U in the second position can be
obtained through deductions from hydrophilic amino
acids with codons having A in the second position.
Moreover, the Hamming distance between wild type
and mutant genes reflects the differences in their
biological activity levels.
In the experimental data confrontation it has been
found that most frequent mutations in HIV-1 reverse
transcriptase gene resistant to drugs are comparable
to the corresponding codons in the wild type. Similar
situation takes place in the mutants of the
phenylalanine hydroxylase gene where the single
point mutations in codons are frequently comparable.
These new results confirm that the Boolean algebra
could allow us to model the gene mutation process.
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