By Louis H. Kauffman (auth.), Jill P. Mesirov, Klaus Schulten, De Witt Sumners (eds.)

This IMA quantity in arithmetic and its functions MATHEMATICAL ways TO BIOMOLECULAR constitution AND DYNAMICS is likely one of the volumes according to the complaints of the 1994 IMA Sum mer software on "Molecular Biology" and includes Weeks three and four of the four-week application. Weeks 1 and a pair of seemed as quantity eighty one: Genetic Mapping and DNA Sequencing. We thank Jill P. Mesirov, Klaus Schulten, and De Witt Sumners for organizing Weeks three and four of the workshop and for enhancing the complaints. We additionally take this chance to thank the nationwide Institutes of healthiness (NIH) (National heart for Human Genome Research), the nationwide technology origin (NSF) (Biological Instrumen tation and Resources), and the dept of power (DOE), whose fi nancial help made the summer time software attainable. A vner Friedman Robert Gulliver v PREFACE The innovative growth in molecular biology in the final 30 years opens find out how to complete knowing of the molecular constructions and mech anisms of residing organisms. Interdisciplinary examine in arithmetic and molecular biology is pushed through ever turning out to be experimental, theoretical and computational strength. The mathematical sciences accompany and help a lot of the growth completed by means of scan and computation in addition to offer perception into geometric and topological homes of biomolecular constitution and procedures. This quantity contains a consultant pattern of the papers offered over the last weeks of the month-long Institute for arithmetic and Its purposes summer season 1994 application in Molecular Biology.

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Gel velocity, random knot frequency, and topological ground state energy of a knot. There is clear laboratory evidence that when closed DNA loops are knotted, different types of knots behave differ• Partially supported by NSF DMS-9407132. t Department of Mathematics, University of Iowa, Iowa City, IA 52242. 39 40 JONATHAN SIMON ently in gel electrophoresis [10][12][13][18][49]. The duplex loops are long enough and nicked, so they are conformationally flexible; the differences in gel velocity are detecting different configurations (or time-averaged conformations, as opposed to individual rigid conformations), what the topologists would call different knot types.

365-373. [38] G. Burde and H. Zieschang, Knots, de Gruyter, Berlin, 1985. B. G. Whittington, Extension of a theorem on super-multiplicative functions, J. Phys. A 12 (1979), L245-247. [40] N. Madras and G. Slade, The Self-Avoiding Walk, Birkhiiuser, Boston, 1993. M. A. Welsh, Further results on the rate of convergence to the connective constant for a self-avoiding walk, Q. J. Math. Oxford 13 (1962), 108. J. W. Sumners, E. G. Whittington, Entanglement complexity of self-avoiding walks, J. Phys. A 25 (1992),65576566.

JANSE VAN RENSBURG ET AL. 1. The writhes of these polygons are related by the equation W(A) = W(A') + W(B). The details of the proof of this Lemma are given in [30]. The idea is to make use of the theorem of Lacher and Sumners, so that one can work with linking numbers, and to use an isotopy argument. 1. For every function f(n) = o(v'n), the probability that the absolute value of the writhe of an n-edge polygon is less than f( n) goes to zero as n goes to infinity. ProoF The proof relies on a combination of Kesten's pattern theorem [33], and a coin tossing argument [35].