How Atom-Bond Connectivity Indices Guide COVID-19 Drug Discovery
Exploring the intersection of mathematical chemistry and antiviral research
In the relentless battle against COVID-19, while virologists and epidemiologists capture headlines, a quiet revolution has been unfolding in the intersection of mathematics and chemistry. Imagine predicting how a potential drug might behave in the human body simply by analyzing the mathematical representation of its molecular structure. This isn't science fiction—it's the fascinating world of topological indices, numerical descriptors that capture essential information about molecular structures.
Among these, the Atom-Bond Connectivity (ABC) index and its evolving versions have emerged as powerful tools in the global race to understand and combat SARS-CoV-2. This article explores how these mathematical concepts are contributing to antiviral drug research and what they reveal about promising COVID-19 therapeutic candidates.
In chemical graph theory, molecules are represented as mathematical graphs where atoms become vertices and chemical bonds become edges. Topological indices are numerical values derived from these molecular graphs that remain unchanged regardless of how the graph is drawn or positioned. They serve as molecular descriptors that capture essential structural information, allowing researchers to predict chemical behavior without expensive and time-consuming laboratory experiments 3 .
Atoms as vertices, bonds as edges - transforming chemistry into mathematics.
Numerical descriptors that capture essential molecular structure information.
First introduced by Harold Wiener in 1947 to study boiling points of alkane isomers, these indices have since evolved into sophisticated tools for Quantitative Structure-Property Relationship (QSPR) and Quantitative Structure-Activity Relationship (QSAR) studies 1 4 . In the context of COVID-19, they've been particularly valuable for screening potential drug candidates and understanding why certain medications show promise against SARS-CoV-2.
The classic ABC index was introduced by Estrada et al. in 1998 and is defined for a molecular graph as:
where dᵢ and dⱼ represent the degrees (number of connections) of adjacent atoms i and j 1 6 . This mathematical formula essentially quantifies the branching pattern of a molecule, which correlates with its stability and thermodynamic properties.
As research advanced, scientists developed specialized versions of the ABC index:
Enhances the discrimination power between similar molecular structures 2
Based on innovative "ve-degree" (number of different edges incident to vertices in the closed neighborhood) and "ev-degree" (number of vertex neighbors of edges) concepts that capture more nuanced structural information 3
Incorporate self-loops in molecular graphs to distinguish hetero-atoms from carbon atoms in hetero-conjugated molecules 1
| Index Type | Key Features | Applications |
|---|---|---|
| Classic ABC | Based on vertex degrees | Predicting thermodynamic properties |
| Exponential ABC | Enhanced discrimination power | Studying trees and cyclic structures |
| ve-degree ABC | Considers neighborhood edges | Analyzing complex drug structures |
| Self-loop ABC | Accounts for hetero-atoms | Modeling sophisticated pharmaceuticals |
Table 1: Evolution of Atom-Bond Connectivity Indices
Researchers have applied ABC indices and other topological descriptors to several medications repurposed for COVID-19 treatment:
One significant study focused on Hydroxyethyl Starch conjugated with Hydroxychloroquine (HCQ-HEC). By computing ve-degree and ev-degree based topological indices, including the ABC index, researchers gained insights into the physicochemical properties of this potential therapeutic agent 3 .
A comprehensive 2023 study examined eight COVID-19 drugs using eccentricity-based topological descriptors. The researchers performed QSPR/QSAR analysis, identifying regression models that connected topological descriptors with physicochemical properties and biological activity (IC₅₀ values) of these medications 4 .
"The calculation of the topological index of a medication structure enables scientists to have a superior comprehension of the physical science and bio-organic attributes of drugs" 3 .
One illuminating example of ABC indices in action comes from research on Hydroxyethyl Starch conjugated with Hydroxychloroquine (HCQ-HEC) 3 :
HCQ-HEC represented as a molecular graph
Traditional and ve-degrees calculated for each vertex
Multiple topological indices computed
Indices analyzed for property correlations
Using these degrees, they computed multiple topological indices, including the ve-degree atom-bond connectivity index:
where Λ_ve(u) and Λ_ve(v) represent the ve-degrees of adjacent vertices.
The study successfully calculated multiple topological descriptors for the HCQ-HEC structure, creating a mathematical fingerprint of this potential therapeutic agent. While the specific numerical results are highly technical, their importance lies in:
Establishing values for comparing modified structures
Enabling predictions of properties like boiling points and molar refraction 1
Providing a framework for optimizing molecular structure
| Index Category | Specific Indices Computed | Structural Information Captured |
|---|---|---|
| Zagreb Indices | First Zagreb α-index, First Zagreb β-index | Molecular branching & complexity |
| Connectivity Indices | Randic index, Sum-connectivity index | Bond patterns and distribution |
| ABC Indices | Atom-bond connectivity index | Stability and energy relationships |
| Geometric Indices | Geometric-arithmetic index | Structural proportions and symmetry |
Table 2: Topological Indices Calculated for HCQ-HEC Bioconjugate
Researchers in this field employ a diverse array of computational tools and concepts:
Programs that can convert molecular structures into mathematical graphs and compute various topological indices
As referenced in dual-target inhibitor research, this method "uses drug-like molecules as docking probes for feature extraction of target protein pockets" 5
Cutting-edge research uses "3D equivariant conditional generative neural networks" to generate potential drug molecules with desired properties 5
The toolkit is built on several key mathematical concepts:
| Concept | Definition | Role in Molecular Analysis |
|---|---|---|
| Vertex Degree | Number of direct connections to an atom | Basic structural information |
| ve-Degree | Edges in closed neighborhood | Captures local environment effects |
| ev-Degree | Vertex neighbors of an edge | Provides bond-centered perspective |
| Self-loops | Edges connecting vertices to themselves | Models hetero-atoms in pharmaceuticals |
Table 3: Mathematical Concepts in Advanced ABC Indices
The application of atom-bond connectivity indices to COVID-19 therapeutics represents more than just a niche mathematical exercise—it demonstrates a fundamental shift in how we approach drug discovery. By quantifying molecular structure through mathematical descriptors, researchers can navigate the vast chemical space of potential therapeutics more efficiently and rationally.
As these methods continue to evolve, particularly with the integration of artificial intelligence and machine learning 5 , we can expect mathematical chemistry to play an increasingly prominent role in responding to future health crises. The COVID-19 pandemic has accelerated the adoption of these computational approaches, potentially forever changing how we discover and optimize life-saving medications.
From the classic ABC index to its sophisticated descendants, these mathematical tools provide a powerful lens through which we can understand and combat complex biological threats—proving that sometimes, the most potent weapons in medicine are not just chemical compounds, but mathematical ideas.
Mathematical approaches like ABC indices are transforming drug discovery from a trial-and-error process to a rational, predictive science.