An optimal formulation of the matrix method in statistical mechanics of one-dimensional interacting units: Efficient iterative algorithmic procedures

Statistical mechanical calculations for one-dimensional interacting units have undergone great development in many fields of macromolecular science. The partition function is the most rigorous description of an ensemble of molecules in equilibrium. Specific methodological formulations and algorithms were developed to handle numerically the inclusion of long-range effects (as compared to usual nearest neighbor Ising models) and known sequence-dependent heterogeneities of biological macromolecules. The most successful approach in formulating these problems in a very general and tractable framework was the so-called matrix method. Despite several improvements, it was claimed that in practical applications this approach had fundamental limitations inherent to any “matrix” formulation. We show here that a new conceptual formulation allows us to overcome these limitations completely. We propose a general iterative procedure that combines the theoretical advantages of the matrix method with the possibility of highly optimized and efficient numerical algorithms.