A Bounded Rate Model for Mass Action Multiple Binding Processes: Stability Analysis

Abstract

In several biological processes a multiple binding reaction occurs, between a univalent ligand L and a n-valent macromolecule M, described by the scheme:

$$\begin{array}{*{20}{c}} {M + L\underset{{{{k}_{{{{d}_{1}}}}}}}{\overset{{{{k}_{{{{a}_{1}}}}}}}{\longleftrightarrow}}ML} \\ {ML + L\underset{{{{k}_{{{{d}_{2}}}}}}}{\overset{{{{k}_{{{{a}_{2}}}}}}}{\longleftrightarrow}}M{{L}_{2}}} \\ { \cdots \cdots \cdots \cdots } \\ {M{{L}_{{n - 1}}} + L\underset{{{{k}_{{{{d}_{n}}}}}}}{\overset{{{{k}_{{{{a}_{n}}}}}}}{\longleftrightarrow}}M{{L}_{n}}} \\ \end{array}$$

((1.1))

where ${\text{k}}_{{\text{a}}_{\text{i}} }$ and ${\text{k}}_{{\text{d}}_{\text{i}} }$ are respectively the association and dissociation rate constants. This kind of multiple reaction may either occur in solution (like the one between oxygen and the large respiratory proteins, e.g. hemoglobin in the vertebrates or hemocyanin in the invertebrates) or on cell surface (like the one between hormones and their cell receptors or the one between antigen and antigenic receptors on lymphocyte surface).