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Local matching of random restriction maps

Part of: Combinatorial probability Limit theorems

Published online by Cambridge University Press:  14 July 2016

Mengxiang Tang  and
Michael S. Waterman
Mengxiang Tang*
Affiliation:
University of Southern California
Michael S. Waterman*
Affiliation:
University of Southern California
*
∗ Postal address: Mathematics Department, University of Southern California, Los Angeles, CA 90089-1113, USA.
∗ Postal address: Mathematics Department, University of Southern California, Los Angeles, CA 90089-1113, USA.
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Abstract

Optical mapping is a new technique to generate restriction maps of DNA easily and quickly. DNA restriction maps can be aligned by comparing corresponding restriction fragment lengths. To relate, organize, and analyse these maps it is necessary to rapidly compare maps. The issue of the statistical significance of approximately matching maps then becomes central, as in BLAST with sequence scoring. In this paper, we study the approximation to the distribution of counts of matched regions of specified length when comparing two DNA restriction maps. Distributional results are given to enable us to compute p-values and hence to determine whether or not the two restriction maps are related. The key tool used is the Chen-Stein method of Poisson approximation. Certain open problems are described.

Keywords

Restriction mapsmatchingChen-Stein method

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 60F20: Zero-one laws
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 2 , June 2001 , pp. 335 - 356
Copyright
Copyright © Applied Probability Trust 2001 

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References

Anantharaman, T. S., Mishra, B., and Schwartz, D. C. (1997). Genomics via optical mapping II: ordered restriction maps. J. Comput. Biol. 4, 91–118.CrossRefGoogle ScholarPubMed
Arratia, R., and Waterman, M. S. (1985). An Erdõs—Rényi law with shifts. Adv. Math. 55, 13–23.CrossRefGoogle Scholar
Arratia, R., and Waterman, M. S. (1989). The Erdõs—Rényi strong law for pattern matching with a given proportion of mismatches. Ann. Prob. 17, 1152–1169.CrossRefGoogle Scholar
Arratia, R., Goldstein, L., and Gordon, L. (1989). Two moments suffice for Poisson approximations: the Chen–Stein method. Ann. Prob. 17, 9–25.Google Scholar
Arratia, R., Gordon, L., and Waterman, M. S. (1986). An extreme value theory for sequence matching. Ann. Statist. 14, 971–993.Google Scholar
Arratia, R., Gordon, L., and Waterman, M. S. (1990). The Erdõs—Rényi law in distribution, for coin tossing and sequence matching. Ann. Statist. 18, 539–570.Google Scholar
Chen, L. H. Y. (1975). Poisson approximation for dependent trials. Ann. Prob. 3, 534–545.Google Scholar
Huang, X., and Waterman, M. S. (1992). Dynamic programming algorithms for restriction map comparison. Comput. Appl. Biosci. 8, 511–520.Google ScholarPubMed
Karlin, S., and Altschul, S. F. (1990). Methods for assessing the statistical significance of molecular sequence features by using general scoring schemes. Proc. Nat. Acad. Sci. USA 87, 2264–2268.Google Scholar
Karlin, S., Dembo, A., and Kawabata, T. (1990). Statistical composition of high-scoring segments from molecular sequences. Ann. Prob. 18, 571–581.Google Scholar
Karlin, S., Ghandour, G., Ost, F., Tavaré, S., and Korn, L. J. (1983). New approaches for computer analysis of nucleic acid sequences. Proc. Nat. Acad. Sci. USA 80, 5660–5664.Google Scholar
Kohara, Y., Akiyama, K., and Isono, K. (1987). The physical map of the whole E. coli chromosome: application of a new strategy for rapid analysis and sorting of large genomic libraries. Cell 50, 495–508.Google Scholar
Lander, E. S., and Waterman, M. S. (1988). Genomic mapping by fingerprinting random clones: a mathematical analysis. Genomics 2, 231–239.Google Scholar
Lin, J. et at. (1999). Whole-genome shotgun optical mapping of Deinococcus radiodurans. Science 285, 1558–1562.Google Scholar
Neuhauser, C. (1994). A Poisson approximation for sequence comparisons with insertions and deletions. Ann. Statist. 22, 1603–1629.Google Scholar
Rudd, K. E., Miller, W., Ostell, J., and Benson, A. (1990). Alignment of Google Scholar
Rudd, K. E. et at. (1990). Alignment of Escherichia Coli k12 DNA sequences to a genomic restriction map. Nucleic Acids Res. 18, 313–321.CrossRefGoogle ScholarPubMed
Schwartz, D. C., Li, X., Hernandez, L. I., Ramnarain, S. P., Huff, E. J., and Wang, Y. (1993).Google Scholar
Schwartz, D. C. et at. (1993). Ordered restriction maps of Saccharomyces cerevisiae chromosomes constructed by optical mapping. Science 262, 110–114.Google Scholar
Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proc. 6th Berkeley Symp. Math. Statist. Prob. Vol. 11, Probability Theory, eds Le Cam, L. M. et at. University of California Press, Berkeley, pp. 583–602.Google Scholar
Tang, M. (2000a). Topics in computational genome analysis: (I) matching restriction maps and (II) evolution of gene families by duplication. Doctoral Thesis, University of Southern California.Google Scholar
Tang, M. (2000b). Global matching of random restriction maps. Methodol. Comput. Appl. Prob. 2, 183–201.Google Scholar
Waterman, M. S. (1995). Introduction to Computational Biology: Maps, Sequences and Genomes. Chapman and Hall, New York.CrossRefGoogle Scholar
Waterman, M. S., and Raymond, R. (1987). The match game: new stratigraphic correlation algorithm. Math. Geol. 19, 109–127.Google Scholar
Waterman, M. S., and Vingron, M. (1994). Sequence comparison significance and poisson approximation. Statist. Sci. 9, 367–381.Google Scholar
Waterman, M. S., Smith, T. F., and Katcher, H. L. (1984). Algorithm for restriction map comparisons. Nucleic Acids Res. 12, 237–242.Google Scholar