We extend a structural result by A. Dress and M. Steel, "Convex tree realizations of partitions," Applied Math Letters, 5 (1993), pp. 3–6., to show that the three-state Perfect Phylogeny problem reduces in polynomial time to the classic 2-SAT problem. We also give a more expanded exposition of the proof of the structural result from Dress and Steel. We hope this note will encourage additional researchers to try to solve the central open question: finding simple efficient solutions to the k-state Perfect Phylogeny problem for k > 3.

We present a comprehensive survey of combinatorial algorithms and theorems about lattice protein folding models obtained in the almost 15 years since the publication in 1995 of the first protein folding approximation algorithm with mathematically guaranteed error bounds. The results presented here are mainly about the HP-protein folding model introduced by Ken Dill in 1985. The main topics of this survey include: approximation algorithms for linear-chain and side-chain lattice models, as well as off-lattice models, NP-completeness theorems about a variety of protein folding models, contact map structure of self-avoiding walks and HP-folds, combinatorics and algorithmics for side-chain models, bi-sphere packing and the Kepler conjecture, and the protein sidechain self-assembly conjecture. As an appealing bridge between the hybrid of continuous mathematics and discrete mathematics, a cornerstone of the mathematical difficulty of the protein folding problem, we show how work on 2D self-avoiding walks contact-map decomposition can build upon the exact RNA contacts counting formula by Mike Waterman and collaborators leading to renewed hope for analytical closed-form approximations for statistical mechanics of protein folding in lattice models. We also include in this paper a few new results, research directions within reach of rigorous results, and a set of open problems that merit future exploration.

With increasing amounts of interaction data collected by high-throughput techniques, understanding the structure and dynamics of biological networks becomes one of the central tasks in post-genomic molecular biology. Recent studies have shown that many biological networks contain a small set of "network motifs," which are suggested to be the basic cellular information-processing units in these networks. Nevertheless, most biological networks have stochastic nature, due to the intrinsic uncertainties of biological interactions and/or experimental noises accompanying the high- throughput data. The building blocks in these networks thus also have stochastic properties. In this paper, we study the problem of identifying stochastic network motifs that are derived from families of mutually similar but not necessarily identical patterns of interactions. Motivated by existing methods for detecting sequence motifs in biopolymer sequences, we establish Bayesian models for stochastic biological networks and develop a group of Gibbs sampling strategies for finding stochastic network motifs. The methods are applied to several available transcriptional regulatory networks and protein-protein interaction networks, and several stochastic network motifs are successfully identified.