Conformational statistics of macromolecules using generalized convolution

A new technique for generating statistical properties of chain-molecule conformations is presented. Conditional probability density functions (PDFs) describing the frequency of occurrence of the relative position and orientation of frames of reference affixed to selected backbone atoms serve as the inputs. Ensemble statistical properties of whole chains are generated by performing multiple generalized convolutions of these conditional PDFs. The formulation is shown to include classical theories such as the hindered and freely rotating chains, the Gaussian random walk, and the rotational isomeric state model. The convolution model is modified to include the long-range effects of excluded volume. An analytical example is used to illustrate the procedure. A general algorithm to calculate the ensemble properties of an arbitrary chain macromolecule is presented. In this algorithm, each of the N degrees of freedom (e.g. torsion angles) is assumed to have K discrete states. Using the convolution procedure, a chain is divided into P statistical units. The computational requirement is reduced from an $\text{O}$(K^N) calculation (corresponding to direct enumeration) to one which is $\text{O}\text{(P(C+K}{}^{\mathrm{N/P}}\text{))}$ where C is the computational complexity of the convolution procedure. In the case of a homopolymer, computations are reduced further to $\text{O}\text{(C}\text{log}\text{(P)+K}{}^{\mathrm{N/P}}\text{).}$