Derangement model of ligand-receptor binding

TL;DR

This paper introduces a derangement-based model to analyze ligand-receptor binding, providing formulas and simulations to understand how ligands find optimal receptors amidst competition, highlighting the importance of search-limited dynamics.

Contribution

It extends derangement theory to ligand-receptor systems, deriving formulas, partition functions, and equilibrium states, and classifies system behaviors based on temperature-dependent binding regimes.

Findings

Identification of two distinct binding regimes based on temperature.

Derivation of formulas for counting partial derangements in ligand-receptor systems.

Simulation results showing the dominance of search-limited behavior in biologically relevant conditions.

Abstract

We introduce a derangement model of ligand-receptor binding that allows us to quantitatively frame the question "How can ligands seek out and bind to their optimal receptor sites in a sea of other competing ligands and suboptimal receptor sites?" To answer the question, we first derive a formula to count the number of partial generalized derangements in a list, thus extending the derangement result of Gillis and Even. We then compute the general partition function for the ligand-receptor system and derive the equilibrium expressions for the average number of bound ligands and the average number of optimally bound ligands. A visual model of squares assembling onto a grid allows us to easily identify fully optimal bound states. Equilibrium simulations of the system reveal its extremes to be one of two types, qualitatively distinguished by whether optimal ligand-receptor binding is the dominant form of binding at all temperatures and quantitatively distinguished by the relative values of two critical temperatures. One of those system types (termed "search-limited," as it was in previous work) does not exhibit kinetic traps and we thus infer that biomolecular systems where optimal ligand-receptor binding is functionally important are likely to be search-limited.