Evolution of Protein Molecules

Information Theory and Reliable Communication

We introduce NeighborNet, a network construction and data representation method that combines aspects of the neighbor joining (NJ) and SplitsTree. Like NJ, NeighborNet uses agglomeration: taxa are combined into progressively larger and larger overlapping clusters. Like SPLITSTREE, NeighborNet constructs networks rather than trees, and so can be used to represent multiple phylogenetic hypotheses simultaneously, or to detect complex evolutionary processes like recombination, lateral transfer and hybridization. NeighborNet tends to produce networks that are substantially more resolved than those made with SPLITSTREE. The method is efficient (O(n3) time) and is well suited for the preliminary analyses of complex phylogenetic data. We report results of three case studies: one based on mitochondrial gene order data from early branching eukaryotes, another based on nuclear sequence data from New Zealand alpine buttercups (Ranunculi), and a third on poorly corrected synthetic data.

NeighborNet: An Agglomerative Method for the Construction of Planar Phylogenetic Networks

Sphere Packings, Lattices and Groups

An on-line algorithm is presented for constructing the suffix tree for a given string in time linear in the length of the string. The new algorithm has the desirable property of processing the string symbol by symbol from left to right. It always has the suffix tree for the scanned part of the string ready. The method is developed as a linear-time version of a very simple algorithm for (quadratic size) suffixtries. Regardless of its quadratic worst case this latter algorithm can be a good practical method when the string is not too long. Another variation of this method is shown to give, in a natural way, the well-known algorithms for constructing suffix automata (DAWGs).

https://web.stanford.edu/~mjkay/Ukkonen.pdf

On-line construction of suffix trees

Constructing evolutionary trees for species sets is a fundamental problem in computational biology. One of the standard models assumes the ability to compute distances between every pair of species, and seeks to find an edge-weighted treeT in which the distanced ij T in the tree between the leaves ofT corresponding to the speciesi andj exactly equals the observed distance,d ij . When such a tree exists, this is expressed in the biological literature by saying that the distance function or matrix isadditive, and trees can be constructed from additive distance matrices in0(n 2) time. Real distance data is hardly ever additive, and we therefore need ways of modeling the problem of finding the best-fit tree as an optimization problem.

A robust model for finding optimal evolutionary trees

We consider the problem of determining the three-dimensional folding of a protein given its one-dimensional amino acid sequence. We use the HP model for protein folding proposed by Dill (1985), which models protein as a chain of amino acid residues that are either hydrophobic or polar, and hydrophobic interactions are the dominant initial driving force for the protein folding. Hart and Istrail (1996a) gave approximation algorithms for folding proteins on the cubic lattice under the HP model. In this paper, we examine the choice of a lattice by considering its algorithmic and geometric implications and argue that the triangular lattice is a more reasonable choice. We present a set of folding rules for a triangular lattice and analyze the approximation ratio they achieve. In addition, we introduce a generalization of the HP model to account for residues having different levels of hydrophobicity. After describing the biological foundation for this generalization, we show that in the new model we are able to achieve similar constant factor approximation guarantees on the triangular lattice as were achieved in the standard HP model. While the structures derived from our folding rules are probably still far from biological reality, we hope that having a set of folding rules with different properties will yield more interesting folds when combined.

Local rules for protein folding on a triangular lattice and generalized hydrophobicity in the HP model

Most known two-dimensional matching algorithms have a rectangular text (image) and rectangular pattern (template) as their input. These matching algorithms perform a row by row analysis followed by some column processing. These techniques are only efficient if all the rows are of equal length, hence the necessity for rectangular patterns.

Efficient matching of nonrectangular shapes

We consider the problem of fitting an n x n distance matrix D by a tree metric T. This problem is NP-hard for most reasonable distance functions between D and T. Recently, an approximation algorithm was presented (Agarwala et al., 1996) which achieves a factor of 3 approximation to the L infinity best fitting tree. We call this method the Single Pivot (SP) heuristic. Within the biology community, the so-called Neighbor-Joining (NJ) heuristic (Saitou and Nei, 1987) has wide acceptance. In this paper, we introduced a new Double Pivot (DP) heuristic, which is an extension of the SP heuristic, and show that DP outperforms NJ on biological and random data.

Numerical taxonomy on data: experimental results.

Multiple sequence alignment (MSA) is important in functional, structural and evolutionary studies of sequence data. Much research has focussed on posing MSA as an optimization problem, and several optimization criteria have been explored. In this paper, we discuss biological and mathematical problems that arise in cost function design for the multiple sequence alignment problem. In particular, we focus on tree alignment, which is often viewed as the most ``biological'''' of the rigorous approaches to MSA. We point out several important pitfalls in current optimization approaches to MSA and identify characteristics for good cost function design. We address some extra design issues specific to approximation algorithms. We hope these ideas will lead to future research on a biologically realistic and mathematically rigorous approach to MSA.

/pdf/on-the-design-of-optimization-criteria-for-multiple-sequence-bj22zuici0.pdf

Martin Farach

Papers

A robust model for finding optimal evolutionary trees

Local rules for protein folding on a triangular lattice and generalized hydrophobicity in the HP model

Efficient matching of nonrectangular shapes

Numerical taxonomy on data: experimental results.

On the Design of Optimization Criteria for Multiple Sequence Alignment