9 1988 by The Humana Press Inc. All fights of any nature whatsoever reserved. 0163-4992/88/1200-0271504.60 Some Mathematical Aspects of Mapping DNA Cosmids JAMES L. CORNE3-FE 1'2 AND CHARLES DELISI *'1'3'4 'Laboratory of Mathematical Biology, National Cancer Institute, National Institutes of Health, Bethesda, ~riD 20892; 2Department of /Vlathematics, Iowa State University, Ames, IA 50011; 3Office of Health and Environmental Research, US Department of Energy, Washington, DC ABSTRACT A number of experimental and mathematical problems must be solved before high resolution physical maps of mammalian chromo- somes can be reliably determined. Such a map might consist of an or- dered set of nonsequenced, overlapping DNA fragments 20,000- 40,000 bases long, produced by digestion of a chromosome, using two restriction enzymes. Map construction requires assigning a sig- nature to each fragment that differentiates it unambiguously from ev- ery other fragment, and then devising a computationally efficient al- gorithm that will provide a unique ordering of the fragments. In the first part of this paper we present a polynomial time algorithm that yields a unique map, and is largely independent of the method for assigning signatures. In the next section we analyze the distribution of lengths of re- striction digest fragments and discuss the implications for the algor- ithm, including the expected number of map gaps. Finally, we dis- cuss a specific method for assigning signatures proposed by Hans Lehrach, based on which of a panel of probes binds to a given frag- ment. In particular we examine the effects of fragment length hetero- geneity on the theoretical .optimum length and number of probes, and the extent to which false signatures might be obtained by non- specific binding. We conclude that the Lehrach strategy is effective *Author to whom all correspondence and reprint requests should be addressed. 4Current address: Dept. of Biomath. Science, Mount Sinai Medical School, New York, I'W 10028 Cell Biophysics 271 Vol. 12, 1988