General recurrence theory of ligand binding on a three-dimensional lattice

摘要

Several general conclusions are obtained for the finite linear three-dimensional lattice models which are built by replicating two-dimensional layers in the third dimension. It is shown that the linear lattice, although less symmetrical, has a simpler structure than the circular lattice where the periodical boundary condition is imposed. If the ligands bound in one layer interact with ligands on its d neighbor layers, the size of the transfer matrix M for the linear lattice is equal to the number of unique binding configurations in d consecutive layers, which is usually smaller than the size of the original transfer matrix M' that determines the recurrence relation of the circular lattice. In certain situations a significant reduction of the matrix size can be achieved. Matrix M' contains all the eigenvalues of matrix M in addition to other eigenvalues if the binding configurations are degenerate. The global partition functions as well as the contracted partition functions at either end of the linear lattice obey the same unique and minimum recurrence relation determined by the secular equation of M. The two ends of the linear lattice, which break the symmetry of the circular lattice, actually make the linear lattice simpler than the circular lattice. The reduced size of the transfer matrix (or the order of the recurrence relation) for the linear lattice not only makes the model more accessible, but also allows the model to describe linear systems more accurately by making the model closer to the system under study. The general theory is applied to several lattices with simple geometries that are of interest in biology and statistical mechanics.