Probabilistic Boolean Networks: The Modeling and Control of Gene Regulatory Networks

275 Pages·2009·16.322 MB·English

by  Ilya Shmulevich, Edward R. Dougherty

Probabilistic Boolean Networks
The Modeling and Control
of Gene Regulatory Networks
I l y a j h i r i u l t v i c h
Edward ft. Stoughe rty
Probabilistic Boolean Networks
Probabilistic Boolean Networks
The Modeling and Control
of Gene Regulatory Networks
Ilya Shmulevich
Institute for Systems Biology
Seattle, Washington
Edward R. Dougherty
Texas A&M University
College Station, Texas
Translational Genomics Research Institute
Phoenix, Arizona
® Society for Industrial and Applied Mathematics • Philadelphia
10 987654321
Library of Congress Cataloging-in-Publication Data
Shmulevich, Ilya, 1969-
Probabilistic boolean networks : the modeling and control of gene regulatory networks /
Ilya Shmulevich, Edward R. Dougherty.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-898716-92-4
1. Genetic regulation—Computer simulation. I. Dougherty, Edward R. II. Society for
Industrial and Applied Mathematics. III. Title.
[DNLM: 1. Gene Regulatory Networks. 2. Models, Genetic. 3. Models, Statistical.
QU 470 S558p 2010]
QH450.S56 2010
572.8'65—dc22
2009034643
This book is dedicated to the memory
of Norbert Wiener,
THE FATHER OF MODERN TRANSLATIONAL SCIENCE.
Contents
Preface xi
1 Boolean Networks 1
1.1 Cell Types and Cellular Functional States............................................... 5
1.2 Relevant Nodes.......................................................................................... 6
1.3 Network Properties and Dynamics............................................................ 8
1.4 Boolean Models of Biological Networks.................................................. 15
1.4.1 The segment polarity network of the fruit fly ............................. 16
1.4.2 Control of the cell cycle .............................................................. 16
1.4.3 T-cell receptor signaling............................................................... 20
1.5 Discretization............................................................................................. 21
1.5.1 Coefficient of determination......................................................... 23
2 Structure and Dynamics of Probabilistic Boolean Networks 27
2.1 Markov Chains as Models of Biological Regulation....................................27
2.2 Definition of Probabilistic Boolean Networks.......................................... 31
2.3 Dynamics: State Transition Probabilities................................................... 34
2.4 The Existence of Steady-State Distributions............................................. 38
2.5 Steady-State Analysis of PBNs.....................................................................40
2.5.1 Steady-state analysis via simulation................................................44
2.5.2 Steady-state probabilities of attractors and basins............................48
2.6 Relationships of PBNs to Bayesian Networks.......................................... 58
2.6.1 Bayesian networks............................................................................58
2.6.2 Independent PBNs and DBNs...................................................... 63
2.7 Mappings between PBNs...............................................................................68
3 Inference of Model Structure 81
3.1 Consistent and Best-Fit Extensions............................................................ 81
3.1.1 Sensitivity regularization............................................................... 85
3.2 Coefficient of Determination as an Inferential Tool.....................................89
3.3 Design of Networks under Data Consistency Requirements.........................95
3.3.1 Contextual data consistency...........................................................100
3.3.2 Optimization of consistency-based design......................................108
3.4 Information Theoretic Approaches..............................................................115
vii
viii Contents
3.4.1 Minimum description length-based network inference from time
series data.........................................................................................116
3.5 Inference of PBNs from Time Series Data..................................................119
3.5.1 Splitting the temporal data sequence into pure subsequences . . . 120
3.5.2 Estimation of switching, selection, and perturbation probabilities .121
3.6 Validation of network inference procedures...............................................122
4 Structural Intervention 125
4.1 Impact of Function Perturbation on State Transitions................................126
4.1.1 Identifying function perturbations..................................................129
4.2 Intervention via Constructive Function Perturbation...................................131
4.2.1 Concept of structural intervention................................................131
4.2.2 Method for constructive function-based intervention....................133
4.2.3 Intervention in a WNT5 A network..................................................134
4.3 Impact of Structural Perturbation on the Steady-State Distribution . . . . 136
4.3.1 Rank-one perturbations....................................................................138
4.3.2 Perturbation in the same row...........................................................142
4.3.3 Extension to multiple rows by iterative computation.......................143
4.3.4 Application to PBNs.......................................................................143
4.4 Structural Intervention via Markov Chain Perturbation Theory.................145
4.4.1 AWNT5ABN................................................................................146
4.5 Long-Run Sensitivity...................................................................................148
4.5.1 Long-run sensitivity with respect to probabilistic parameters ... 148
4.5.2 Long-run sensitivity with respect to regulatory functions..............150
4.5.3 One-predictor function perturbations...............................................151
4.5.4 One-bit function perturbations........................................................153
4.5.5 Function perturbations considering one-gene regulation.................153
4.5.6 Properties of long-run sensitivity.....................................................154
4.5.7 Sensitivity and robustness of control...............................................155
4.5.8 Sensitivity in a mammalian cell-cycle network .............................156
5 External Control 161
5.1 Intervention via One-Time Gene Perturbation............................................161
5.2 Finite-Horizon Control................................................................................165
5.2.1 Control problem .............................................................................165
5.2.2 Solution by dynamic programming..................................................167
5.2.3 Illustrative example..........................................................................168
5.2.4 Finite-horizon control in a melanoma network................................170
5.3 Infinite-Horizon Control.............................................................................171
5.3.1 Optimal control solution: Discounted and bounded cost per stage 173
5.3.2 Optimal control solution: Average cost per stage..........................178
5.3.3 Infinite-horizon control for a WNT5A network.............................181
5.4 Approximation............................................................................................183
5.4.1 A linear model................................................................................183
5.4.2 Intervention in a family of BNs .....................................................184
Contents IX
5.4.3 Imperfect information ....................................................................185
5.4.4 Reduction of a context-sensitive PBN............................................187
5.4.5 Reinforcement learning....................................................................191
5.5 Constrained Intervention.............................................................................194
5.5.1 Constrained intervention in a mammalian cell-cycle network ... 199
5.5.2 Cyclic intervention..........................................................................204
5.6 Robust Control...........................................................................................205
5.6.1 Perturbation bounds ......................................................................206
5.6.2 Mini max robust control....................................................................208
5.6.3 Bayesian robust control....................................................................211
5.6.4 Uncertainty in the switching probabilities......................................212
5.7 Adaptive Infinite-Horizon Control..............................................................215
5.8 Mean-First-Passage-Time Stationary Control.............................................216
5.8.1 Model-free intervention ................................................................218
5.9 Steady-State-Based Control Policies...........................................................221
5.9.1 Steady-state-distribution greedy control policy .............................221
5.9.2 Conservative steady-state-distribution control policy....................222
5.9.3 Performance comparison................................................................224
6 Asynchronous Networks 227
6.1 Asynchronous PBNs...................................................................................227
6.1.1 Deterministic asynchronous PBNs..................................................227
6.1.2 Semi-Markov asynchronous PBNs..................................................229
6.2 Intervention in Asynchronous Networks.....................................................232
6.2.1 Intervention in DA-PBNs.................................................................232
6.2.2 Intervention in SMA-PBNs..............................................................234
6.2.3 Solution for three intertransition interval distributions....................238
6.2.4 Intervention in a mutated mammalian cell-cycle SMA-PBN . . . 239
Bibliography 243
Index 261
Preface
It was around the period of World War II that Arturo Rosenblueth and Norbert Wiener
were taking the first steps in the direction of systems medicine. They formed an interesting
pair: Rosenblueth, a physiologist at the Harvard Medical School, and Wiener, the father
of modern engineering in the United States. For this book, their conception of science is
salient. They wrote, “The intention and the result of a scientific inquiry is to obtain an un­
derstanding and a control of some part of the universe.” [ 1 ] For them, as a research team, the
part of the universe was physiology. An appreciation of their words is important. Under­
standing is not some vague, subjective explanation, but rather the precision of mathematical
systems needed for the representation of relationships between measurable quantities and
future predictions based on those relationships. Control is the ability to change physical
behavior in a manner concomitant with the mathematical system representing the relevant
phenomena.
Rosenblueth and Wiener take an active view of science: it is to change the world.
In contemporary terminology, rather than science, one might say that they were describing
translational science. “Translational science transforms a scientific mathematical model,
whose purpose is to provide a predictive conceptualization of some portion of the physi­
cal world, into a model characterizing human intervention (action) in the physical world.
Whereas the pure scientist typically tries to minimize human interference, translational sci­
ence extends science to include conceptualization of human-originated action in the physi­
cal world and thereby raises epistemological issues relating to the knowledge of this inten­
tional intervention into the natural order. Scientific knowledge is translated into practical
knowledge by expanding a scientific system to include inputs that can be adjusted to affect
the behavior of the system and outputs that can be used to monitor the effect of the external
inputs and feed back information on how to adjust the inputs.” [2] It is this translational
scientific view that Wiener brought into line with modern science during his illustrious ca­
reer. In perhaps the greatest transformation of engineering epistemology since antiquity,
Wiener fundamentally altered the way human beings perceive scientifically based action in
the world. Teaming with Rosenblueth, he brought that transformation into medicine.
Thinking of Wiener, this book should be read in two ways. First, considering the
specific definitions, theorems, and equations, it discusses a particular dynamical model for
gene regulatory networks—probabilistic Boolean networks (PBNs). It covers basic model
properties, inference of model parameters from data, and intervention in the model to in­
crease the likelihood of the network being in desirable states. Taking a wider perspective,
one can view the PBN model as a vehicle in which to elucidate the therapeutic goals of
XI
XII Preface
translational genomics. The PBN model is rather general and includes both determinis­
tic functional aspects and probabilistic characteristics inherent to the modeling of complex
systems. Therefore, it is well suited to serve as a mathematical framework to study basic
issues dealing with systems-based genomics, specifically, the relevant aspects of stochas­
tic, nonlinear dynamical systems. These include long-run dynamical properties and how
these correspond to therapeutic goals, the effect of complexity on model inference and the
resulting consequences of model uncertainty, altering network dynamics via structural in­
tervention, such as perturbing gene logic, optimal control of regulatory networks over time,
limitations imposed on the ability to achieve optimal control owing to model complexity,
and the effects of asynchronicity.
We do not know what models will ultimately be adopted for specific applications,
but we do know that basic translational issues considered in this book will have to be con­
fronted no matter what model is used. In translational science, we are faced with the prob­
lem of controlling complex systems in an environment of uncertainty—just the framework
in which Wiener pioneered. We need to understand how the inevitable issues manifest
themselves in the modeling and control of gene regulatory networks so that therapeutic
control strategies can be derived and, even prior to that, so that appropriate experiments
can be carried out to gain sufficient modeling information.
We have tried to unify the different strands of research that have been pursued over
the last eight years and continue to be pursued. Issues such as inference, network re­
duction, constrained control, greedy control, and asynchronicity are just beginning to be
studied. Moreover, how these are resolved in practice will depend on close interaction
between biologists, physicians, mathematicians, and engineers. Only in that way will the
phenomena, medical outcome, and model be brought into a coherent whole to fulfill the
promise of translational science. We hope that this book provides a step in that direction.
[1] Rosenblueth, A., and N. Wiener, ‘The role of models in science.” Philosophy of
Science, 12,316-321, 1945.
[ 2] Dougherty, E. R., “Translational science: epistemology and the investigative pro­
cess,” Current Genomics, 10 (2), 102-109, 2009.
Acknowledgments
The authors extend their appreciation to the numerous colleagues and students who
contributed to developments discussed in this book. Although there are too many to men­
tion explicitly, we would like to at least mention the names of several whose contributions
play key roles in the present book: Michael M. Bittner, Marcel Brun, Aniruddha Datta,
Babak Faryabi, Ivan Ivanov, Seungchan Kim, Peter Krawitz, Harri Lahdesmaki, Steve
Marshall, Ranadip Pal, Xiaoning Qian, Golnaz Vahedi, Yufei Xiao, and Wei Zhang. We
would also like to acknowledge those organizations that have supported the research be­
hind this book: National Science Foundation, National Human Genome Research Institute,
National Institute of General Medical Sciences, National Cancer Institute, Translational
Genomics Research Institute, Institute for Systems Biology, W. M. Keck Foundation, Uni­
versity of Texas M. D. Anderson Cancer Center, and Texas A&M University. We thank
Babak Faryabi and Xiaoning Qian for proofreading the manuscript and making numerous
valuable suggestions. We are also grateful to our editor at SIAM, Elizabeth Greenspan,
for her constant encouragement, patience, and support. IS would like to extend his sin­
cere thanks to The Helen Riaboff Whiteley Center at the University of Washington Friday
Harbor Laboratories, which provided a superbly peaceful and productive environment to
work on this book. IS also thanks his family, Andrei, Anna, and Janette, for their contin-