Exploring antibody-antigen binding affinity using degree-based topological indices: a graph-theoretical approach

The binding affinity between an antibody and an antigen, measured by the equilibrium dissociation constant $\left(K_d\right)$ and the associated Gibbs free energy of binding $\left(\Delta G\right)$ , is key to immune recognition and the design of therapeutic antibodies. This study explores whether degree-based topological indices, derived from molecular graph models of fifteen antibody-antigen Fab complexes from the Protein Data Bank, contain structural information related to binding affinity. Fifteen indices were calculated using an edge-partitioning technique and analyzed against two thermodynamically based response variables, $p{K}_d$ and experimental $\Delta G$ , employing linear, quadratic, and cubic models, while keeping the untransformed $K_d$ as an additional point of reference. Cubic models reached statistical significance against both $p{K}_d$ and $\Delta G$ independently, while linear and quadratic models did not reach significance for either response variable. Additionally, a size benchmark comparing total atom count and edge count against all three response variables reproduced the performance of the best performing topological index at every model order, confirming that the observed associations reflect molecular size rather than connectivity-specific structural information. The results demonstrate that degree-based topological indices primarily reflect overall molecular size rather than connectivity-specific structural information. This work establishes a solid foundation for developing more focused hybrid descriptor methods for analyzing antibody-antigen affinity.