Alignment of molecular sequences seen as random path analysis

We propose a generating functional method--random path analysis (RPA)--that generalizes the classical dynamic programming (DP) method widely used in sequence alignments. For a given cost function, DP is a deterministic method that finds an optimal alignment by minimizing the total cost function for all possible alignments. By allowing uncertainty, RPA is a statistical method that weights fluctuating alignments by probabilities. Therefore, DP maybe thought of as the deterministic limit of RPA when the fluctuations approach zero. DP is the method of choice if one is only interested in optimal alignment. But we argue that, when information beyond the optimal alignment is desired, RPA gives a natural extension of DP for biological applications. As an algebraic approach, RPA is computationally intensive for long sequences, but it can provide better parametric control for developing analytical or perturbational results and it is more informative and biologically relevant. The idea of RPA opens up new opportunities for simulational approaches and more importantly it suggests a novel hardware implementation that has the potential of improving the way a sequence alignment is done. Here we focus on deriving a mathematically rigorous solution to RPA both in its combinatorial form and in its graphical representation; this puts DP in logical perspective under a more general conceptual framework.