Evaluation of linear chain partition functions by consideration of sequence conditional probabilities

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Abstract
A simple, systematic treatment for the class of linear chain models whereby each unit can exist in one of several states, giving rise to a partition function that is a sum over all lengths and orders of sequences of states, is given for increasingly long range interactions between the sequences. The technique consists of constructing relations for sequence conditional probabilities which give rise to a system of linear equations for the a priori sequence probabilities and the unit partition function; matrix inversion eliminates the a priori probabilities leaving a relation for the unit partition function. When the unit partition function is calculated, one can immediately evaluate the a priori sequence probabilities from which all other average quantities can be calculated (thus making it unnecessary to calculate average quantities by taking derivatives of the partition function). A short derivation is given of the generating function method of treating independent sequences and it is shown that the partition function for finite chains can be expressed solely in terms of generating functions (circumventing the need to construct eigenvectors). The technique is then used to treat the case where there is an arbitrary interaction between adjacent sequences with application to titration and chain folding models. The extension to higher order interactions is shown to follow in a similar fashion.