Pooling Design and Nonadaptive Group Testing: Important Tools for DNA Sequencing (Series on Applied Mathematics) (Series on Applied Mathematics)

249 Pages·2006·9.032 MB·English

by  Dingzhu Du, Frank K Hwang

Series on
Applied Mathematics
Volume 18
POOLING DESIGNS
AND NONADAPTIVE
GROUP TESTING
Important Tools for
DNA Sequencing
Ding-Zhu Du
Frank K Hwang
World Scientific
POOLING DESIGNS
AND NONADAPTIVE
GROUP TESTING
Important Tools for DNA Sequencing
SERIES ON APPLIED MATHEMATICS
Editor-in-Chief: Frank Hwang
Associate Editors-in-Chief: Zhong-ci Shi and U Rothblum
Vol. 1 International Conference on Scientific Computation
eds. T. Chan and Z.-C. Shi
Vol. 2 Network Optimization Problems — Algorithms, Applications and
Complexity
eds. D.-Z. Du and P. M. Pandalos
Vol. 3 Combinatorial Group Testing and Its Applications
by D.-Z. Du and F. K. Hwang
Vol. 4 Computation of Differential Equations and Dynamical Systems
eds. K. Feng and Z.-C. Shi
Vol. 5 Numerical Mathematics
eds. Z.-C. Shi and T. Ushijima
Vol. 6 Machine Proofs in Geometry
by S.-C. Chou, X.-S. Gao and J.-Z. Zhang
Vol. 7 The Splitting Extrapolation Method
by C. B. Liem, T. Lu and T. M. Shih
Vol. 8 Quaternary Codes
by Z.-X. Wan
Vol. 9 Finite Element Methods for Integrodifferential Equations
by C. M. Chen and T. M. Shih
Vol. 10 Statistical Quality Control — A Loss Minimization Approach
by D. Trietsch
Vol. 11 The Mathematical Theory of Nonblocking Switching Networks
by F. K. Hwang
Vol. 12 Combinatorial Group Testing and Its Applications (2nd Edition)
by D.-Z. Du and F. K. Hwang
Vol. 13 Inverse Problems for Electrical Networks
by E. B. Curtis and J. A. Morrow
Vol. 14 Combinatorial and Global Optimization
eds. P. M. Pardalos, A. Migdalas and R. E. Burkard
Vol. 15 The Mathematical Theory of Nonblocking Switching Networks
(2nd Edition)
by F. K. Hwang
Vol. 16 Ordinary Differential Equations with Applications
by S. B. Hsu
Vol. 17 Block Designs: Analysis, Combinatorics and Applications
by D. Raghavarao and L. V. Padgett
Vol. 18 Pooling Designs and Nonadaptive Group Testing — Important Tools
for DNA Sequencing
by D.-Z. Du and F. K. Hwang
POOLING DESIGNS
AND NONADAPTIVE
GROUP TESTING
Important Tools for DNA Sequencing
Ding-Zhu Du
University of Texas at Dallas, USA
and
Xi'an Jiaotong University, China
Frank K Hwang
National Chiao Tung University, Taiwan, ROC
\[p World Scientific
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
POOLING DESIGNS AND NONADAPTIVE GROUP TESTING
Important Tools for DNA Sequencing
Series on Applied Mathematics — Vol. 18
Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd.
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Preface
Group testing has been well-known for its various applications in blood testing,
chemical leakage testing, electric shortage testing, coding, multi-access channel com
munication ..., among others, which are well-documented in our book Combinatorial
Group Testing and Its Applications (1993, 2nd ed. 2000). However, its applica
tion into molecular biology, especially in the design of screening experiment, is quite
recent and still under tremendous development (a group testing algorithm for this
application is often referred to as pooling design). For example, in our 1993 book,
we mentioned this application only on one page. In the 2000 edition we covered this
topic in three chapters. But now we have to write a whole book to report its progress,
and to make painful choices of what to include.
The new application also brings new problems to the theory of group testing. We
mention the three fundamental differences.
1. To achieve a given screening objective, the number of biological experiments
required is huge and each such experiment is time-consuming (compared to electronic
testing). Thus it is of utmost importance to use parallel, or nonadaptive, testing
where all experiments can be performed parallelly, or at least in a few rounds. Tra
ditionally, the theory of group testing has been focused on sequential testing, due
to its requirement of fewer number of tests. But the focus of pooling design is now
shifted to nonadaptive kind. Since a nonadaptive design is just a binary matrix, many
mathematical tools can now be brought to group testing for judicious use.
2. A biological experiment is known for its unreliability, as versus electronic
testing which carries high accuracy. Therefore error-detection and error-correcting,
long ignored in the traditional group testing theory, have to be dealt with squarely.
A very lucky and unexpected turn-out is that not only the newly developed theory
of pooling design can deal with errors, but in a highly structured way in the sense
that treatment of errors is not of the ad-hoc type and all such treatments in different
models have the same features.
3. Attempts have been made before to set up traditional group testing in graph
models, but such attempts are superfluous in the sense that only terminology, but no
v
VI Preface
substance, is graph-theoretic. In the complex model of pooling design, we now have
a genuine extension of group testing to graph testing from a real need in molecular
biology. Not only the theory of group testing is enriched, the extension also provides
many new interesting and challenging problems to graph theory.
A one-semester graduate course on the first nine chapters of this book was taught
at the National Chiaotung University at Hsinchu in Spring 2005. Chapters 6 and 7
were taught as a 3-week short course at the National Center of Theoretical Sciences.
The second author thanks the institutions for financial support and the students for
participation. We wish to thank Dr. Hong Zhao, Dr. Ying Liu and Dr. My Thai for
their help during preparation of this book.
Contents
Preface v
Chapter 1 Introduction 1
1.1 Group Testing 1
1.2 Nonadaptive Group Testing 4
1.3 Applications in Molecular Biology 7
1.4 Pooling Designs for Two Simple Applications 13
1.5 Pooling Designs and Mathematics 15
1.6 An Outline of the Book 17
References 18
Chapter 2 Basic Theory on Separating Matrices 21
2.1 d-Separable and d-Separable Matrices 21
2.2 d-Disjunct Matrices 25
2.3 The Minimum Number of Pools for Given d and n 27
2.4 Combinatorial Bounds for d-Disjunct Matrices with Constant Weight 30
2.5 Asymptotic Lower and Upper Bounds 33
2.6 (d, r)-Disjunct Matrices 40
2.7 Error-Tolerance 45
References 52
Chapter 3 Deterministic Designs 55
3.1 t-Designs and i-Packing 55
3.2 Direct Construction 57
3.3 Explicit Construction of Selectors 59
3.4 Grid Designs 60
3.5 Error-Correcting Code 63
3.6 Transversal Designs 65
vii
viii Contents
3.7 The d = 2 Case 70
References 79
Chapter 4 Deterministic Designs from Partial Orders 83
4.1 Subset Containment Designs 83
4.2 Partial Order of Faces in a Simplicial Complex 84
4.3 Monotone Graph Properties 90
4.4 Partial Order of Linear Spaces over a Finite Field 92
4.5 Atomic Poset 95
References 97
Chapter 5 Random Pooling Designs and Probabilistic Analysis 99
5.1 Introduction to Random Designs 99
5.2 A General Approach to Compute Probabilities of Unresolved Clones — 100
5.3 Random Incidence Designs 106
5.4 Random k-Set Designs 109
5.5 Random r-Size Designs Ill
5.6 Random Distinct fc-Set Designs 113
5.7 Intersection Pooling Designs 115
5.8 Subset Containment Designs in Extended Use 118
5.9 Edge-Representative Decoding with r = 2 and d = 3 122
5.10 Some Trivial 2-Stage Pooling Designs 126
References 128
Chapter 6 Pooling Designs on Complexes 131
6.1 Introduction 131
6.2 A Construction of (H : d; ,z)-Disjunct Matrix 134
6.3 (d, r; ^-Disjunct Matrix 136
6.4 Constructions for (d, r; ^-Disjunct Matrices 139
6.5 Random Designs 144
6.6 Trivial Two-stage Pooling Designs for Complete r-graphs 148
6.7 Sequential Algorithms for H 152
T
References 162
Chapter 7 Contig Sequencing 165
7.1 Introduction 165
7.2 Some Probability Analysis of a /c-subset 166
7.3 Sequential Algorithms 170
7.4 Nonadaptive Algorithms for Matching 172
7.5 The 3-Stage Procedure 184
References 188
Contents ix
Chapter 8 The Inhibitor Model 189
8.1 Introduction 189
8.2 1-Round Algorithm 190
8.3 Sequential and fc-Round Algorithms 193
8.4 Some Other Inhibitor Models 195
References 198
Chapter 9 Hyperplane Designs 201
9.1 Introduction 201
9.2 m-Dimensional Arrays 203
9.3 A K x K Decomposition of K 207
T c n
9.4 Efficiency 215
9.5 Other Transversal Designs 219
9.6 Two Recent Applications 221
References 222
Chapter 10 Non-unique Probe Selection 225
10.1 Introduction 225
10.2 Complexity of Pooling Designs 227
10.3 Complexity of Minimum Pooling Designs 228
10.4 Approximations of Minimum Pooling Designs 231
References 231
Index 233