Peter H. Sellers

The theory and computation of evolutionary distances: Pattern recognition

A method of finding pattern similarities between two sequences is given. Two portions, one from each sequence, are similar if they are close in the metric space of evolutionary distances. The method allows a complete list to be made of all pairs of intervals, one from each of two given sequences, such that each pair displays a maximum local degree of similarity, and, if the lengths of the given sequences are m and n, then the procedure takes on the order of mn computational steps. This result lends itself to finding similarities by computer between pairs of biological sequences, such as proteins and nucleic acids.

An algorithm for the distance between two finite sequences

In a biological problem, which will be described later, it is necessary to compute the distance or degree of dissimilarity between two finite sequences. A mathematical definition of this distance was brought to my attention by S. M. Ulam, and an algorithm for computing it will be presented here. If m and IZ are the lengths of the two sequences and m < n, then the number of computational steps in the algorithm is m%, where each step consists of selecting the largest of three known numbers. In Section 2 it will be shown how the algorithm can be changed to compute the modifications of this distance which are required in the biological context.

Pattern recognition in genetic sequences by mismatch density

A new development is introduced here in the use of dynamic programming in finding pattern similarities in genetic sequences, as was first done by Needleman and Wunsch (1969). A condition of pattern similarity is defined and an algorithm is given which scans any set of similarities and screens out those which fail to meet the condition. When the set to be scanned contains every pair of segments, one from each of two given sequences of lengthsm andn (i.e. every possible location for a pattern similarity), then it completes the scan in a number of computational steps proportional tom·n, leaving those pairs of segments which satisfy the similarity condition. The algorithm is based on the concept of match density, as suggested by Goad and Kanehisa (1982).

Identifying periodic occurrences of a template with applications to protein structure

Abstract Consider a template P of size m in which each character matches many different characters with various degrees of perfection. Given a text T of size n , we present a simple and practical algorithm that finds the substring of T , which best matches some substring of P n ( P n is the concatenation of an arbitrary number of copies of P ). The algorithm produces the matched pair and their alignment in O( mn ) time.

Analysis of the Possible Mechanisms for a Catalytic Reaction System

Publisher Summary This chapter focuses on the enumeration of all possible mechanisms for a complex chemical reaction system based on the assumption of given elementary reaction steps and species. The procedure presented for such identification is been directly applied to a number of examples in the field of heterogeneous catalysis. Application to other areas is clearly indicated. These would include complex homogeneous reaction systems, many of which are characterized by the presence of intermediates acting as catalysts or free radicals. Enzyme catalysis should also be amenable to this approach. The subject of reaction mechanism also has a bearing on other fundamental problems of physical chemistry. In carrying out the procedure for determining mechanisms that is presented in the chapter, one obtains a set of independent chemical reactions among the terminal species in addition to the set of reaction mechanism. Consideration of a chemical system in terms of unique direct reaction mechanisms required to produce observable rates of change of terminal species has distinct advantages, especially when multiple overall reactions are involved. The required necessary assumptions regarding possible elementary reaction steps are becoming increasingly accessible through modern tools for surface spectroscopy and fundamental theories of chemical kinetics of elementary reaction steps.

The characterization of complex systems of chemical reactions

A method is presented for the enumeration of possible overall chemical reactions and mechanisms, based on an initial choice of elementary reactions connecting a set of chemical species. Such a procedure furnishes an important tool for the study of complex reaction systems. The method is presented in such a way as to be readily implemented by computer solution, which is indispensable for many systems.

Identifying Periodic Occurrences of a Template with Applications to Protein Structures

We consider a string matching problem where the pattern is a template that matches many different strings with various degrees of perfection. The quality of a match is given by a penalty matrix that assigns each pair of characters a score that characterizes how well the characters match. Superfluous characters in the text and superfluous characters in the pattern may also occur and the respective penalties for such gaps in the alignment are also given by the penalty matrix. For a text T of length n, and a template P of length m, we wish to find the best alignment of T with Pn, which is the concatenation of n copies of P, (m will typically be much smaller than n). Such an alignment can simply be obtained by solving a dynamic programming problem of size O(n2m), and ignoring the periodic character of Pn. We show that the structure of Pn can be exploited and the problem reduced to essentially solving a dynamic programming of size O(mn). If the complexity of computing gap penalties is O(1), (which is frequently the case), our algorithm runs in O(mn) time. The problem was motivated by a protein structure problem.

Stanislaw M. Ulam's Contributions to Theoretical Theory

S. M. Ulams contributions to biology are surveyed. The survey covers cellular automata theory, population biology, Fermi-Pasta-Ulam results, pattern recognition, and sequence similarity.

Shortcuts, diversions, and maximal chainsin partially ordered sets

An algorithm is described for finding the maximal weight chain between two points in a locally finite partial order under the restriction that all but x (or fewer) successive pairs in the chain belong to a given subset of the partial order relation. Applications of the method in molecular genetics, critical path scheduling, and other fields are discussed.

Sets of Nonnegative Matrices with Positive Inhomogeneous Products

Let X be a set of k×k matrices in which each element is nonnegative. For a positive integer n, let P(n) be an arbitrary product of n matrices from X, with any ordering and with repetitions permitted. Define X to be a primitive set if there is a positive integer n such that every P(n) is positive [i.e., every element of every P(n) is positive]. For any primitive set X of matrices, define the index g(X) to be the least positive n such that every P(n) is positive. We show that if X is a primitive set, then g(X)⩽2k−2. Moreover, there exists a primitive set Y such that g(Y) = 2k−2.