**Christian N. S. Pedersen**

**March 2000**

In this thesis we are concerned with constructing algorithms that
address problems with biological relevance. This activity is part of a
broader interdisciplinary area called computational biology, or
bioinformatics, that focus on utilizing the capacities of computers to
gain knowledge from biological data. The majority of problems in
computational biology relate to molecular or evolutionary biology, and
focus on analyzing and comparing the genetic material of organisms. A
deciding factor in shaping the area of computational biology is that
biomolecules that DNA, RNA and proteins that are responsible for
storing and utilizing the genetic material in an organism, can be
described as strings over finite alphabets. The string representation
of biomolecules allows for a wide range of algorithmic techniques
concerned with strings to be applied for analyzing and comparing
biological data. We contribute to the field of computational biology
by constructing and analyzing algorithms that address problems with
relevance to biological sequence analysis and structure prediction.

The genetic material of organisms evolve by discrete mutations, most prominently substitutions, insertions and deletions of nucleotides. Since the genetic material is stored in DNA sequences and reflected in RNA and protein sequences, it makes sense to compare two or more biological sequences in order to look for similarities and differences that can be used to infer the relatedness of the sequences. In the thesis we consider the problem of comparing two sequences of coding DNA when the relationship between DNA and proteins is taken into account. We do this by using a model that penalizes an event on the DNA by the change it induces on the encoded protein. We analyze the model in details and construct an alignment algorithm that improves on the existing best alignment algorithm in the model by reducing its running time with a quadratic factor. This makes the running time of our alignment algorithm equal to the running time of alignment algorithms based on much simpler models.

If a family of related biological sequences are available it is natural to derive a compact characterization of the sequence family. Among other things, such a characterization can be used to search for unknown members of the sequence family. A widely used way to describe the characteristics of a sequence family is to construct a hidden Markov model that generates members of the sequence family with high probability and non-members with low probability. In this thesis we consider the general problem of comparing hidden Markov models. We define novel measures between hidden Markov models and show how to compute them efficiently using dynamic programming. Since hidden Markov models are widely used to characterize biological sequence families, our measures and methods for comparing hidden Markov models immediate apply to comparison of entire biological sequence families.

Besides comparing sequences and sequence families, we also consider problems of finding regularities in a single sequence. Looking for regularities in a single biological sequence can be used to reconstruct part of the evolutionary history of the sequence or to identify the sequence among other sequences. In this thesis we focus on general string problems motivated by biological applications because biological sequences are strings. We construct an algorithm that finds all maximal pairs of equal substrings in a string, where each pair of equal substrings adheres to restrictions on the number of characters between the occurrences of the two substrings in the string. This is a generalization of finding tandem repeats and the running time of the algorithm is comparable to the running time of existing algorithms for finding tandem repeats. The algorithm is based on a general technique that combines a traversal of a suffix tree with efficient merging of search trees. We use the same general technique to construct an algorithm that finds all maximal quasiperiodic substrings in a string. A quasiperiodic substring is a substring that can be described as concatenations and superpositions of a shorter substring. Our algorithm for finding maximal quasiperiodic substrings has a running time that is a logarithmic factor better than the running time of the existing best algorithm for the problem.

Analyzing and comparing the string representations of biomolecules can reveal a lot of useful information about the biomolecules, but knowing the three-dimensional structures of the biomolecules often reveal additional information that is not immediately visible from their string representations. Unfortunately it is difficult and time consuming to determine the three-dimensional structure of a biomolecule experimentally, so computational methods for structure prediction are in demand. Constructing such methods is also difficult and often results in the formulation of intractable computational problems. In this thesis we construct an algorithm that improves on the widely used mfold algorithm for RNA secondary structure prediction by allowing a less restrictive model of structure formation without an increase in the running time. We also analyze the protein folding problem in the two-dimensional hydrophobic-hydrophilic lattice model. Our analysis shows that several complicated folding algorithms do not produce better foldings in the worst case, in terms of free energy, than an existing much simpler folding algorithm.

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