A discrete scattering series representation for lattice embedded models of chain cyclization

Abstract

In this paper we develop a lattice based model of chain cyclization in the presence of a set of occupied sites V in the lattice. We show that within the approximation of a Markovian chain propagator the effect of V on the partition function for the system can be written as a time-ordered exponential series in which V behaves like a scattering potential and chainlength is the timelike parameter. The discrete and finite nature of this model allows us to obtain rigorous upper and lower bounds to the series limit. We adapt these formulas to calculation of the partition functions and cyclization probabilities of terminally and globally cyclizing chains. Two classes of cyclization are considered: in the first model the target set H may be visited repeatedly (the Markovian model); in the second case vertices in H may be visited at most once(the non-Markovian or taboo model). This formulation depends on two fundamental combinatorial structures, namely the inclusion–exclusion principle and the set of subsets of a set. We have tried to interpret these abstract structures with physical analogies throughout the paper.