Theory of multivalent binding in one and two-dimensional lattices

Area of Science:

• Biophysics
• Statistical Mechanics
• Molecular Biology

Background:

• Ligand binding to lattices is fundamental in molecular biology, influencing processes like gene regulation.
• Existing models, such as the McGhee-von Hippel model, often simplify lattice geometry and binding interactions.
• Understanding binding properties requires accurate theoretical frameworks that account for lattice dimensionality and cooperativity.

Purpose of the Study:

• To develop an exact analytical solution for ligand binding to linear lattices using contracted partition function theory.
• To investigate ligand binding to two-dimensional toroidal lattices, modeling systems like DNA and protein helices.
• To compare binding behavior in one-dimensional versus two-dimensional systems and assess the implications for experimental data interpretation.

Main Methods:

• Application of contracted partition function theory to derive a recursion relation for the system's partition function.
• Development of a generating function for exact analytical solutions of ligand binding.
• Derivation of site-specific properties through simple transformations of analytical expressions.

Main Results:

• An exact analytical solution for ligand binding to linear lattices is obtained, encompassing general cooperativity and binding site coverage.
• The McGhee-von Hippel model is shown to be a special case of the derived solution in the limit of infinite lattice sites.
• Non-cooperative binding to a 2D torus can mathematically mimic cooperative binding to a 1D lattice when the number of sites covered (m) equals the number of sites per section (s).

Conclusions:

• The developed theory provides a versatile and exact method for analyzing ligand-ligand binding interactions on various lattice structures.
• The geometry of the lattice and site interactions significantly influence binding properties, necessitating careful interpretation of experimental data.
• Caution is advised when interpreting Scatchard plots using 1D models for systems with inherently 2D geometry, especially with limited site coverage.