On constructing folding heteropolymers.

Simplified models of the protein-folding process have led to valuable insights into the generic properties of the folding of heteropolymers. On the basis of theoretical arguments, Shakhnovich and Gutin [(1993) Proc. Natl. Acad. Sci. USA 90, 7195-7199] have proposed a specific method to generate folding sequences for one of these. Here we present a model of folding in heteropolymers that is comparable in simplicity but different in spirit to the one studied by Shakhnovich and Gutin. In our model, the proposed recipe for constructing folding sequence fails. We find that, as a rule, the construction of folding sequences is impossible to achieve by looking at the native conformation only. Rather, competing conformations have to be taken into account too. An evolutionary algorithm that generates folding sequences by optimizing both stability of the native state and folding time is described. Remarkably, this algorithm produces, among others, sequences that fold reproducibly to metastable states.

Estimation of evolutionary distances under stationary and nonstationary models of nucleotide substitution

Estimation of evolutionary distances has always been a major issue in the study of molecular evolution because evolutionary distances are required for estimating the rate of evolution in a gene, the divergence dates between genes or organisms, and the relationships among genes or organisms. Other closely related issues are the estimation of the pattern of nucleotide substitution, the estimation of the degree of rate variation among sites in a DNA sequence, and statistical testing of the molecular clock hypothesis. Mathematical treatments of these problems are considerably simplified by the assumption of a stationary process in which the nucleotide compositions of the sequences under study have remained approximately constant over time, and there now exist fairly extensive studies of stationary models of nucleotide substitution, although some problems remain to be solved. Nonstationary models are much more complex, but significant progress has been recently made by the development of the paralinear and LogDet distances. This paper reviews recent studies on the above issues and reports results on correcting the estimation bias of evolutionary distances, the estimation of the pattern of nucleotide substitution, and the estimation of rate variation among the sites in a sequence.