Oberwolfach Seminar: Some Mathematical Challenges from Life Sciences

November 16th - 22nd, 2003
Deadline for applications:
October 15, 2003
Peter Schuster, Wien
Peter Stadler, Leipzig
Günter Wagner, Yale University
Current molecular genetics and genome research initiated a new era in biology. An incredibly large amount of data demands new methods in storage, annotation, validation, and retrieval. Bioinformatics is a novel field, which is concerned with the development of algorithms and techniques for these purposes. Yet, little use of this enormous amount of information has been made so far to conceive a comprehensive theoretical biology. The input from mathematics is indispensable. In this seminar we focus on the development of a comprehensive theory of evolution and highlight the relation between genetics and organismic biology by means of concepts from discrete mathematics. In particular, we make use of the ideas of mappings and landscapes in order to formalize and investigate complex relations in structural and evolutionary biology.

Biological evolution, in essence, is based on variation and evaluation of variants. Its dynamics is governed by a fundamental dichotomy: The genotype or genome is the object that undergoes variation while evaluation operates on the phenotype. The genome is a sequence of nucleotides in a DNA molecule. The phenotype on the other hand is a complex organism together with all its properties and its embedding in an environment. The mapping from the space of genotypes onto a phenotype space is central to the mechanism of evolution. Information on molecular details is required for proper modeling of such mappings, and these details have become accessible by now. Indeed, direct analysis of genotype-phenotype maps is possible at present for evolution of molecules in the test-tube as well as for virus evolution. Since a great variety of bacterial genomes is already sequenced we can expect fast progress in this area too. For more complex mappings like those in multi-cellular organisms or even societies only selected partial aspects can be modeled successfully.

In the seminar we shall present an introduction into the notion of landscapes and more general mappings and apply it to the problem of sequence-structure relations of biopolymers. Questions that can be analyzed straightforwardly are, for example, inverse folding, the search for sequences, which fold into predefined structures, or the computation of probabilities to obtain certain structures through folding of randomly chosen sequences. The notion of genotype-phenotype mapping is applied to modeling evolution in phenotype space, and the notion of accessibility and nearness of phenotypes is discussed. Then, we present tools for the mathematical analysis of landscapes and maps like normal mode analysis by means of Laplacians defined on discrete spaces. Topologies and less restricted concepts of pretopologies are applied to statistical neighborhood relations of preimages of phenotypes in genotype space. Eventually, the concept of genotype-phenotype maps is extended to real genomes with many genes including any order of gene interactions. It is shown that "multilinear" models can be applied to study the effects of epistasis on quantitative genetic variation, on the response to selection, and on genetic canalization. These models provide access to a comprehensive measurement theory of fitness.
Recommended reading - Fairly simple introductions
Peter Schuster, Evolution in silico and in vitro: The RNA model. Biological Chemistry 382 (2001), 1301-1314
Peter Schuster, Mathematical challenges from molecular evolution. In: Bj÷rn Engquist and Wilfried Schmid, eds. Mathematics Unlimited - 2001 and Beyond. Pages 1019-1038. Springer-Verlag, Berlin 2001
Peter Schuster and Peter F. Stadler, Networks in molecular evolution. A common theme at all levels. Complexity 8/1 (2003), 34-42
Peter Schuster, Molecular Insights into Evolution of Phenotypes. In: James P. Crutchfield and Peter Schuster, eds. Evolutionary Dynamics: Exploring the Interplay of Selection, Accident, Neutrality, and Function. Pages 163-215. Oxford University Press, New York 2003
Original papers and reviews
Christian M. Reidys, Peter F. Stadler, and Peter Schuster, Generic properties of combinatory maps. Neutral networks of RNA secondary structures. Bull.Math.Biol. 59 (1997), 339-397
Walter Fontana and Peter Schuster, Continuity in evolution. On the nature of transitions. Science 280 (1998), 1451-1455
Walter Fontana and Peter Schuster, Shaping space. The possible and the attainable in RNA genotype-phenotype mapping. J.Theor.Biol. 194 (1998), 491-515
Bärbel M. R. Stadler, Peter F. Stadler, Günter P. Wagner, and Walter Fontana, The topology of the possible: Formal spaces underlying patterns of evolutionary change. J.Theor.Biol. 213 (2001), 241-274
Thomas F. Hansen and G’nter P. Wagner, Modeling genetic architecture: A multilinear theory of gene interaction. Theor.Popul.Biol. 59 (2001), 61-86
Christian M. Reidys and Peter F. Stadler, Combinatorial Landscapes. SIAM Review 44 (2002), 3-54

Mathematisches Forschungsinstitut Oberwolfach   updated: August 15th, 2003