QSPR-analysis of amino acids with fuzzy Topological indices

Introduction

Let G = (V, E) be a graph with |V | = n and |E| = m. The open neighborhood of a vertex v ∈ V is N (v) = {u/ uv ∈ E}. The degree of a vertex v is d v = |N (v)|. A molecular graph is a labeled graph in which the atoms and bonds of the molecule are represented by the vertices and edges, respectively. Chemical graph theory is a branch of mathematical chemistry, which is found to be useful in study of QSAR (Quantitative Structure-Activity Relationships) and QSPR (Quantitative Structure-Property Relationship) analysis. Numerous studies have been made relating to the above mentioned fields by using what are called molecular descripttors.

There are mainly three types of descriptors: 1. Degree based molecular descriptors 2. Distance based molecular descriptors 3. Eigenvalue based molecular descriptors A pair of molecular descriptors known as the first Zagreb index M 1 (G) and the second Zagreb index M 2 (G), are the first degree-based molecular descriptors and they first appeared in the topological formula for the total π-electron energy of conjugated molecules that has been derived by Gutman and Trinajstic´ [7] , the first and second Zagreb indices are defined as:

Nowadays, due to various applications of fuzzy graph theory, a huge number of researchers are working on topological indices [2, 3, 5, 6, 9-11, 15, 16] . Kalathian et al. [12] have studied the first index for fuzzy graphs and is defined as follows:

To maintain the similarity of the first Zagreb index for crisp graphs, Islam and Pal [10] redefined the first Zagreb index in the following way.

The second fuzzy Zagreb index [3] is defined as:

The forgotten Zagreb index for fuzzy graph was put forward in [15] . It is defined as:

Amino Acids are the organic compounds that combine to form peptide and pro-teins, hence they are referred to as the building blocks for peptide and proteins. Amino acids are organic compounds containing the basic amino groups (-NH 2 ) and carboxyl groups (-COOH).Altogether, there are twenty amino acids, which are involved in the Construction of proteins. Hellberg [13] , developed three principal properties for 20 amino acids and variation in amino acid sequence within sets of peptides is described by three principal properties hydrophilicity (z 1 ), bulk (z 2 ) and electronic properties (z 3 ). These are derived from principal components analysis (PCA) of a matrix of 29 physicochemical variables for 20 amino acids.

Apart from this two descriptor are derived from the side chain surface area and atomic charge. One descriptor is isotropic surface area (ISA), proposed and developed by Grigoras [6] as the portion of the solute molecule which is accessible to nonspecific interactions with the solvent, water. The other one is Electronic charge index (ECI), ECI is the sum of absolute value of CNDO/2 charges of the side chain atoms. ISA and ECI were derived and QSAR methodologies were used to correlate structural variation with variation in activity for the sets of peptides. One physical property of amino acid is molecular weight (MW), molecular weight is a measure of the sum of the atomic weight values of the atoms in a molecule. Molecular weight is used in chemistry to determine stoichiometry in chemical reactions and equations. Molecular weight is commonly abbreviated by M.W. or MW. Molecular weight is either unitless or expressed in terms of atomic mass units (amu) or Daltons (Da).

The following molecular descriptors are considered for the study.

Data set

Intercorrelation analysis between molecular descrip-tors

The intercorrelation between 03 molecular descriptors (FZ 1 , FZ 2 (G) and FF 1 (G)) is car-ried out in table 6.4 and the values for pair of indices is judged by correlation coefficient r. For r > 0.97 it is highly intercorrelated, for r > 0.9 it isj appreciably intercorrelated, for r < 0.9 it isj weakly intercorrelated and for r < 0.5 it is not intercorrellated [14] . For physical property z 1 , the regression models are:

Intercorrelation of molecular descriptors

References

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