Mathematical Biosciences and Engineering (Sep 2022)
Modeling ligand-macromolecular interactions as eigenvalue-based transition-state dissociation constants may offer insights into biochemical function of the resulting complexes
Abstract
A ligand when bound to a macromolecule (protein, DNA, RNA) will influence the biochemical function of that macromolecule. This observation is empirical and attributable to the association of the ligand with the amino acids/nucleotides that comprise the macromolecule. The binding affinity is a measure of the strength-of-association of a macromolecule for its ligand and is numerically characterized by the association/dissociation constant. However, despite being widely used, a mathematically rigorous explanation by which the association/dissociation constant can influence the biochemistry and molecular biology of the resulting complex is not available. Here, the ligand-macromolecular complex is modeled as a homo- or hetero-dimer with a finite and equal number of atoms/residues per monomer. The pairwise interactions are numeric, empirically motivated and are randomly chosen from a standard uniform distribution. The transition-state dissociation constants are the strictly positive real part of all complex eigenvalues of this interaction matrix, belong to the open interval (0,1), and form a sequence whose terms are finite, monotonic, non-increasing and convergent. The theoretical results are rigorous, presented as theorems, lemmas and corollaries and are complemented by numerical studies. An inferential analysis of the clinical outcomes of amino acid substitutions of selected enzyme homodimers is also presented. These findings are extendible to higher-order complexes such as those likely to occur in vivo. The study also presents a schema by which a ligand can be annotated and partitioned into high- and low-affinity variants. The influence of the transition-state dissociation constants on the biochemistry and molecular biology of non-haem iron (Ⅱ)- and 2-oxoglutarate-dependent dioxygenases (catalysis) and major histocompatibility complex (Ⅰ) mediated export of high-affinity peptides (non-enzymatic association/dissociation) are examined as special cases.
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