Discrete Mathematics Summary

Monday - Friday, June 27 - July 1, 2005

The twelve members of the Discrete Mathematics Working group are attending an undergraduate class taught by Elizabeth S. Allman, University of Southern Maine, and John A. Rhodes, Bates College. The title of the course is "The Mathematics of Phylogenetic Trees."

Description: Until recently, the inference of the evolutionary history of currently living species was based primarily on painstaking studies of their morphological similarities, together with comparison to the fossil record. Now a vast new source of evolutionary data is available through genetic sequencing. While similarities in DNA sequences among species suggest close ancestral relationships and differences suggest greater evolutionary divergence, how to infer an entire evolutionary tree from biological sequences is a rich mathematical question. This course begins with an overview of the sorts of biological questions of interest, and a look at the nature of biological sequence data. We then develop several of the modern approaches to sequence-based phylogenetics, focusing on the modeling of the process of molecular evolution along a tree. Shortcomings of the various methods and models, both theoretical and practical, will be used to motivate new ones.

Necessary mathematical and biological background will be kept minimal: basic probability and linear algebra are helpful but can be picked up along the way. The course will also include elements of combinatorics, algorithmics, Markov models and statistics, as well as hands-on computer work with real and simulated data.

Reference: Elizabeth S. Allman, John A. Rhodes. Mathematical Models in Biology, an Introduction. Cambridge University Press, 2004

Following the daily one hour lecture, the SSTP working group meets for an additional hour.

On Day 1, Brian Hopkins reviewed some basic mathematical and biological vocabulary to help put the first lecture in context.

On Day 2, the group worked on some of the homework problems which had been assigned that day. Five members of the group participated in the undergraduate homework session for the course later that afternoon. While there we focused on two problems. The first concerned the phylogeny of four taxa which were related according to a tree. The total distance between pairs of taxa along the tree was given in a table. Our goal in each of the parts of the problem was to construct a tree that was consistent with the information given us.

The second problem dealt with trying to ascertain the relationship between the genes of the HIV virus among six subjects. We were given aligned parts of each subject's DNA sequence (80 nucleotides). Our goal was to come up with an approach to try to determine the phylogenetic tree which best matched the given information.

We found these two problems intriguing. We may use these two as bases for our projects during the next two weeks.

On Day 4, John Rhodes spoke with us for an hour after the lecture and answered our questions.

On Day 5, we worked on the homework problems that had been assigned for that day and started the ones due on Tuesday. We discussed the possibility of working on projects similar to the problems that we had seen on Tuesday.

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.