Inte rnational Journal of Engineering Technology, Manage ment and Applied Sciences
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[
A Comparison between the Adjacency Matrix Energy and
Dissociation Energy for Amino Acids
Mathavi Manisekar
Associate Professor, Department of Physics,
Fatima College (Autonomous),
Madurai, Tamilnadu, India.
S. Lalitha
Reader and Head (Retd.), Department of Physics,
Fatima College (Autonomous),
Madurai, Tamilnadu, India.
Abstract:
The structure of molecules can be easily depicted by graphs known as chemical graphs. The chemical
graph G of a molecule can be represented by different matrices. The most important matrix representation of
a graph is the Adjacency matrix. The sum of the absolute eigen values of the matrix is known as its energy.
Amino acids are the essential building blocks of the protein chain in human body. Chemical graphs of Amino
acids can be represented by the adjacency matrices. This work is an attempt to show that the dissociation
energies of amino acids bear a constant ratio with the energies of the corresponding adjacency matrices and
hence to suggest the unit for matrix energy.
Key words: Chemical Graph, adjacency matrix, dissociation energy, amino acids.
Introduction:
The atoms of a molecule have a binary relation. Two atoms in a given molecule are either bonded
or not bonded. Molecules can be represented by chemical graphs using this property of existence or non
existence of bonds between atoms. In a chemical graph, the atoms are considered as vertices and the bonds are
considered as edges. The chemical graph simplifies the complex picture of a molecule by depicting only the
connections between the various pairs of atoms in the molecule and enables one to predict the physical and
chemical properties of the molecule [1]. Other structural features like bond lengths and bond angles may be
added as weights.
The chemical graphs can be represented by different matrices like adjacency matrix, distance
matrix, reciprocal distance matrix, detour matrix and so on. Each one of these matrices may lead to a number
of molecular descriptors which may be related to the physical and chemical properties of the molecules.
A.T.Balaban [2] has demonstrated the versatility of graph theory in the analysis of various chemical systems.
Topological indices are structural invariants based on modeling of chemical structures by chemical graphs.
E.Estraade, G.Patlewicz and E.Uriarte[3] have shown the effectiveness of Topological indices in drug design
and discovery. Topological indices have been used for QSAR modeling and selection of mineral collectors
[4]. Topostructural and topochemical indices have been employed in the development of QSAR models of the
aryl hydrocarbon receptor binding potency of a set of dibenzofurons [5].
The adjacency matrix A (G) of a labeled graph G with N vertices is the NxN square symmetric
matrix containing information about the internal connectivity of the vertices in G. It is defined as,
1 if and only if , i and j are bonded.
A ij=
2 if there is a double bond between i and j
0, otherwise
The adjacency matrix of a graph can be subjected to a linear transformation leading to a
diagonal form. Elements of such a diagonal matrix are known as the Eigen values of the matrix. The set of
Eigen values is called the spectrum of the graph and it is an important graph invariant. The limits of a graph
spectrum is decided by the maximum degree of a vertex in the graph. . The sum of the absolute eigen values
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Mathavi Manisekar, S. Lalitha
Inte rnational Journal of Engineering Technology, Manage ment and Applied Sciences
www.ijetmas.com August 2015, Volume 3, Issue 8, ISSN 2349-4476
[
of the matrix is known as its energy. The graph spectral parameters like eigen values and eigenvectors of
matrices associated with chemical graphs can be used to get information about the structure of the molecules
and this area is known as Graph spectral analysis[6]. The graph spectral theory is equivalent to Huckel theory
of molecular orbitals. The Huckel theory is based on the internal connectivity of atoms in a molecule. It
describes a molecular orbital as a linear combination of atomic orbitals. The eigen vectors of the adjacency
matrix are identical with the Huckel molecular orbitals, known as topological orbitals. The Huckel
Hamiltonian is a function of the adjacency matrix [7]. The eigen values of the marked and weighted graphs of
amino acids have been found to have correlation with the atomic masses of the constituent atoms [8]. The
conservation of graph energy during the formation of di-peptides has been proved using chemical graph
theory [9].
Amino acids are important organic compounds composed of amine (-NH2 ) and carboxylic
acid (-COOH) functional groups, along with a side chain specific to each amino acid. In the form of proteins,
amino acids comprise the major component of human muscles, cells and other tissues. There are around
twenty standard proteinogenic amino acids, which combine into peptide chains to form the building blocks of
a vast array of proteins. The physicochemical properties of amino acids can be studied using chemical graph
theory. Since the adjacency matrix represents the connectivity by means of bonds between the atoms of the
molecule, it can be related to the dissociation energy of the molecule which is the sum of the energy needed to
break the various bonds existing in the molecule. The method of arriving at the energy of the chemical graph
is explained through the chemical graph for Glycine.
Determination of chemical graph energy for Glycine
Glycine is one of the twenty standard amino acids mainly found in human body whose composition
and adjacency matrix A(G) are given below. The atoms are labeled as shown below.
Fig.1: Molecular graph of Glycine
A(G) =
The eigen values of this matrix are given below.
- 2.6616 , - 1.7925, - 1, - 0.8384, 0 , 0, 0.8384, 1, 1.7925, 2.6616
The energy of this matrix is 12.59.
The dissociation energy of an amino acid is calculated by adding the dissociation energies of all
the bonds [10] between the atoms of the amino acid. The dissociation energy of Glycine is 5305.57 kJ/mol or
1268.1 kcal/mol. The ratio of dissociation energy in kcal/mol to matrix energy is 100.7. The dissociation
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Mathavi Manisekar, S. Lalitha
Inte rnational Journal of Engineering Technology, Manage ment and Applied Sciences
www.ijetmas.com August 2015, Volume 3, Issue 8, ISSN 2349-4476
[
energy and the matrix energy for all the twenty amino acids have been calculated and tabulated ( Table 1) .
From the table, it can be seen that the ratio of dissociation energy to the matrix energy is a constant for all
amino acids.
Table 1: The dissociation energy and the matrix energy for all the twenty amino acids
S.No.
Amino acid
Adjacency
Dissociation
Ratio
Matrix
energy
b/a
Energy
kcal/mole (b)
(a)
1
Alanine
15.62
1577.6
101
2
Glycine
12.59
1268.1
100.7
3
Valine
21.83
2196.7
100.6
4
Leucine
24.97
2506.0
100.4
5
Isoleucine
24.65
2506.0
100.4
6
Proline
20.91
2213.8
106
7
Serine
17.40
1856.6
107
8
Threonine
20.46
2166.3
106
9
Cysteine
17.40
1751.7
101
10
Tyrosine
33.28
3214.7
97
11
Aspartic acid
21.61
2261.8
105
12
Glutamic acid
24.76
2571.3
104
13
Histidine
27.30
2752.1
101
14
Phenylalanine
30.11
2935.6
98
15
Lysine
27.64
2766.6
100
16
Arginine
31.74
3156.8
100
17
Methionine
22.70
2389.9
105
18
Glutamine
25.55
2552.6
100
19
Asparagine
22.41
2243.0
100
20
Tryptophan
38.91
3670.6
94
The following graph shows the linear relationship between the adjacency matrix energy and the
dissociation energy for Glycine.
Fig.2: The linear relationship between the adjacency matrix energy and the dissociation energy for Glycine.
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Mathavi Manisekar, S. Lalitha
Inte rnational Journal of Engineering Technology, Manage ment and Applied Sciences
www.ijetmas.com August 2015, Volume 3, Issue 8, ISSN 2349-4476
[
Conclusion:
The adjacency matrix of a molecule represents the connectivity of the various atoms of the
molecule. In this paper, the energy of the adjacency matrix is seen to be 1/100 of the dissociation energy of
the molecule. This suggests that the unit for matrix energy is
deci calories/mole.
References:
[1] Nenad Trinajstić, Chemical Graph Theory, Volu me I, CRC Press,Inc.
[2] A. T. Balaban, Chemical Applications of graph theory, Academic Press, 1976.
[3] E.Estraade, G.Patlewicz and E.Uriarte, Ind.J.Chem., Vo l.42A, 2003,pp.1315.
[4] Natarajan, P.Kamalakannan and I.Nirdosh, Ind.J.Chem., Vol.42A, 2003.
[5] S.S.Basak, D.M ills, M.Murnataz and K.Balasubramanian, Ind.J.Chem., Vo l.42A, 03,pp.1385.
[6] Saraswath Visveshwara,.V.Brinda and N.Kannan, Journal of Theoretical and Computational Chemistry, Vo l.1,
N0.1.(2002) 000-000, World Scientific Publishing Co mpany .
[7] Živković. T., Trinajstić, N., and Randić. M., Croat , Chem. Acta,49, 89, 1977.
[8] S.Lalitha, Mathavi Manisekar, Mathematical and Experimental Physics, Narosa Publishing House Pvt.Ltd., 2010,
India.
[9] Mathavi Manisekar and S.Lalitha, International Journal of mathematics and Soft Computing, Volu me 03, Issue 2,
May 2013, Page:11-15.
[10] CRC Handbook of Chemistry and Physics
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Mathavi Manisekar, S. Lalitha