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15
Four Strikes against Physical Mapping of DNA
 JOURNAL OF COMPUTATIONAL BIOLOGY
, 1993
"... Physical Mapping is a central problem in molecular biology ... and the human genome project. The problem is to reconstruct the relative position of fragments of DNA along the genome from information on their pairwise overlaps. We show that four simplified models of the problem lead to NPcomplete ..."
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Cited by 57 (8 self)
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Physical Mapping is a central problem in molecular biology ... and the human genome project. The problem is to reconstruct the relative position of fragments of DNA along the genome from information on their pairwise overlaps. We show that four simplified models of the problem lead to NPcomplete decision problems: Colored unit interval graph completion, the maximum interval (or unit interval) subgraph, the pathwidth of a bipartite graph, and the kconsecutive ones problem for k >= 2. These models have been chosen to reflect various features typical in biological data, including false negative and positive errors, small width of the map and chimericism.
On the Complexity of DNA Physical Mapping
, 1994
"... The Physical Mapping Problem is to reconstruct the relative position of fragments (clones) of DNA along the genome from information on their pairwise overlaps. We show that two simplified versions of the problem belong to the class of NPcomplete problems, which are conjectured to be computationa ..."
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Cited by 40 (7 self)
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The Physical Mapping Problem is to reconstruct the relative position of fragments (clones) of DNA along the genome from information on their pairwise overlaps. We show that two simplified versions of the problem belong to the class of NPcomplete problems, which are conjectured to be computationally intractable. In one version all clones have equal length, and in another, clone lengths may be arbitrary. The proof uses tools from graph theory and complexity.
Genomics via Optical Mapping II: Ordered Restriction Maps
 Journal of Computational Biology
, 1996
"... In this paper, we describe our algorithmic approach to constructing ordered restriction maps based on the data created from the images of population of individual DNA molecules (clones) digested by restriction enzymes. The goal is to devise mapmaking algorithms capable of producing highresolution, ..."
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Cited by 33 (19 self)
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In this paper, we describe our algorithmic approach to constructing ordered restriction maps based on the data created from the images of population of individual DNA molecules (clones) digested by restriction enzymes. The goal is to devise mapmaking algorithms capable of producing highresolution, highaccuracy maps rapidly and in a scalable manner. The resulting software is a key component of our optical mapping automation tools and has been used routinely to map cosmid, lambda and BAC clones. The experimental results appear highly promising. 1 Genomics and Optical Mapping Optical mapping [CAH+95, CJI+96, HRL+95, JRH+96, MBC+95, SCH+95, SLH+93, WHS95] is a single molecule methodology for the rapid production of ordered restriction maps from individual DNA molecules. Ordered restriction maps were constructed originally from yeast chromosomes by using fluorescence microscopy to visualize restriction endonuclease cutting events on individual fluorochromestained DNA molecules [SCH+95,...
DNA physical mapping and alternating Eulerian cycles in colored graphs
 Algorithmica
, 1995
"... Abstract. Smallscale DNA physical mapping (such as the Double Digest Problem or DDP) is an important and difficult problem in computational molecular biology. When enzyme sites are modeled by a random process, the number of solutions to DDP is known to increase exponentially as the length of DNA in ..."
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Cited by 20 (1 self)
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Abstract. Smallscale DNA physical mapping (such as the Double Digest Problem or DDP) is an important and difficult problem in computational molecular biology. When enzyme sites are modeled by a random process, the number of solutions to DDP is known to increase exponentially as the length of DNA increases. However, the overwhelming majority of solutions are very similar and can be transformed into each other by simple transformations. Recently, Schmitt and Waterman [SW] introduced equivalence classes on the set of DDP solutions and raised an open problem to completely characterize equivalent physical maps. We study the combinatorics of multiple solutions and the cassette transformations of Schmitt and Waterman. We demonstrate that the solutions to DDP are closely associated with alternating Eulerian cycles in colored graphs and study order transformations of alternating cycles. We prove that every two alternating Eulerian cycles in a bicolored graph can be transformed into each other by means of order transformations. Using this result we obtain a complete characterization of equivalent physical maps in the SchmittWaterman problem. It also allows us to prove Ukkonen's conjecture on word transformations preserving qgram composition. Key Words. Graph theory, DNA mapping, DNA sequencing.
A Partial Digest Approach to Restriction Site Mapping
 Bulletin of Mathematical Biology
, 1993
"... We present a new, practical algorithm to resolve the experimental data in restriction site analysis, which is a common technique for mapping DNA. Specifically, we assert that multiple digestions with a single restriction enzyme can provide sufficient information to identify the positions of the rest ..."
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Cited by 19 (2 self)
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We present a new, practical algorithm to resolve the experimental data in restriction site analysis, which is a common technique for mapping DNA. Specifically, we assert that multiple digestions with a single restriction enzyme can provide sufficient information to identify the positions of the restriction sites with high probability. The motivation for the new approach comes from combinatorial results on the number of mutually homeometric sets in one dimension, where two sets of n points are homeometric if the multiset of n(n \Gamma 1)=2 distances they determine are the same. Since experimental data contains error, we propose algorithms for reconstructing sets from noisy interpoint distances, including the possibility of missing fragments. We analyze the performance of these algorithms under a reasonable probability distribution, establishing a relative error limit of r = \Theta(1=n 2 ) beyond which our technique becomes infeasible. Through simulations, we establish that our techni...
A Lower Bound on the Number of Solutions to the Probed Partial Digest Problem
 Advances in Applied Mathematics
, 1993
"... The Probed Partial Digestion mapping method partially digests a DNA strand with a restriction enzyme. A probe, which attaches to the DNA between two restriction enzyme cutting sites, is hybridized to the partially digested DNA, and the sizes of fragments to which the probe hybridizes are measured. T ..."
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Cited by 14 (2 self)
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The Probed Partial Digestion mapping method partially digests a DNA strand with a restriction enzyme. A probe, which attaches to the DNA between two restriction enzyme cutting sites, is hybridized to the partially digested DNA, and the sizes of fragments to which the probe hybridizes are measured. The objective is to reconstruct the linear order of the restriction enzyme cutting sites from the multiset of measured lengths. In many cases, more than one underlying linear ordering is consistent with a multiset of measured lengths. This article shows that a multiset of N measured lengths can have as many as \Omega\Gamma N t ) solutions for any t ! i \Gamma1 (2) where i(t) is the Riemann Zeta Function and i \Gamma1 (2) ß 1:73. 1 Introduction The Probed Partial Digestion (or PPD) mapping scheme is used to generate physical maps of large DNA strands using restriction enzymes and probes. A DNA strand can be viewed as a finite sequence over the alphabet of four letters fA, C, G, Tg. A re...
A Case Study In Experimental Design Applied To Genetic Algorithms With Applications To DNA Sequence Assembly
 American Journal of Mathematical and Management Sciences
, 1997
"... Experimental design and response surface methodology is applied to tuning the parameters of an optimization program employing genetic algorithms. Attention is directed to the combinatorially challenging DNA sequence assembly problem. Fine tuning of a 10K size test problem leads to a considerably imp ..."
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Cited by 9 (0 self)
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Experimental design and response surface methodology is applied to tuning the parameters of an optimization program employing genetic algorithms. Attention is directed to the combinatorially challenging DNA sequence assembly problem. Fine tuning of a 10K size test problem leads to a considerably improved solution to a 35K problem of sequence assembly that is of significant biological interest. Key Words and Phrases: genetic algorithms; design of experiments; response surface methods; DNA sequence assembly. R.J. PARSONS & M.E. JOHNSON 1. Introduction Design and analysis of experiments and response surface methods have prospered in this century through successful applications in agriculture initially and in industry in more recent decades. Great strides in the development of these methods have taken place owing to improvements in computing capability. Although the semiconductor industry upon which our computing platforms depend has particularly benefited from design of experiments and ...
The Restriction Mapping Problem Revisited
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES (JCSS
, 2002
"... In computational molecular biology, the aim of restriction mapping is to locate the restriction sites of a given enzyme on a DNA molecule. Double digest and partial digest are two wellstudied techniques for restriction mapping. While double digest is NPcomplete, there is no known polynomial alg ..."
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Cited by 6 (0 self)
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In computational molecular biology, the aim of restriction mapping is to locate the restriction sites of a given enzyme on a DNA molecule. Double digest and partial digest are two wellstudied techniques for restriction mapping. While double digest is NPcomplete, there is no known polynomial algorithm for partial digest. Another disadvantage of the above techniques is that there can be multiple solutions for reconstruction. In this
Genomic Mapping by End Characterized Random Clones: A Mathematical Analysis
 Genomics
, 1995
"... Physical maps can be constructed by "fingerprinting" a large number of random clones and inferring overlap between clones when the fingerprints are su#ciently similar. Lander and Waterman (Genomics (1988) 2: 231239) gave a mathematical analysis of such mapping strategies. The analysis is useful for ..."
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Cited by 5 (0 self)
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Physical maps can be constructed by "fingerprinting" a large number of random clones and inferring overlap between clones when the fingerprints are su#ciently similar. Lander and Waterman (Genomics (1988) 2: 231239) gave a mathematical analysis of such mapping strategies. The analysis is useful for comparing various fingerprinting methods. Recently it has been proposed to fingerprint or characterize ends of clones rather than the entire clone. Such fingerprints, which include sequenced clone ends, require a deeper mathematical analysis than that of LanderWaterman. This paper studies clone islands, which can include uncharacterized regions, and also the islands that are formed entirely from the ends of clones. 1 1
Solving the Double Digestion Problem as a MixedInteger Linear Program
, 2001
"... The double digestion problem for DNA restriction mapping is known to be NPcomplete. Several approaches to the problem have been used including exhaustive search, simulated annealing, branchandbound. In this paper, we consider a mixedinteger linear programming formulation of the problem and show ..."
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Cited by 1 (1 self)
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The double digestion problem for DNA restriction mapping is known to be NPcomplete. Several approaches to the problem have been used including exhaustive search, simulated annealing, branchandbound. In this paper, we consider a mixedinteger linear programming formulation of the problem and show that with this formulation the problem can be solved efficiently to a fairly large size using stateoftheart integer programming techniques. In particular, we present computational results obtained by using the CPLEX mixedinteger linear programming software on a set of randomly generated, largescale double digestion problems.