Analysis and control of Boolean networks: a semi-tensor product approach

474 Pages·3.031 MB·English

by  Cheng, Daizhan;Qi, Hongsheng;Li, Zhiqiang

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Daizhan Cheng (cid:2) Hongsheng Qi (cid:2) Zhiqiang Li
Analysis and
Control of Boolean
Networks
A Semi-tensor Product Approach
Dr.DaizhanCheng ZhiqiangLi
AcademyofMathematicsandSystems AcademyofMathematicsandSystems
Science(AMSS),InstituteofSystems Science(AMSS),InstituteofSystems
Science Science
ChineseAcademyofSciences ChineseAcademyofSciences
100190Beijing 100190Beijing
China,People’sRepublic China,People’sRepublic
[e-mail]
HongshengQi
AcademyofMathematicsandSystems
Science(AMSS),InstituteofSystems
Science
ChineseAcademyofSciences
100190Beijing
China,People’sRepublic
ISSN0178-5354
ISBN978-0-85729-096-0 e-ISBN978-0-85729-097-7
DOI10.1007/978-0-85729-097-7
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Preface
Motivated by the Human Genome Project, a new view of biology, called systems
biology, is emerging [5]. Systems biology does not investigate individual genes,
proteins or cells in isolation. Rather, it studies the behavior and relationships of
all of the cells, proteins, DNA and RNA in a biological system called a cellular
network.Themostactivenetworksmaybethoseassociatedwithgeneticregulation,
whichregulatethegrowth,replication,anddeathofcellsinresponsetochangesin
theenvironment.
How do these genetic regulatory networks function? In the early 1960s Jacob
and Monod showed that any cell contains a number of “regulatory” genes that act
asswitchesandwhichcanturneachanotheronandoff.Thisshowsthatagenetic
networkisof“on–off”type[7].
Boolean networks, first introduced by Kauffman, have become powerful tools
fordescribing,analyzing,andsimulatingcellularnetworks[2,3].Hence,theyhave
received much attention, not only from the biology community, but also from re-
searcherswithbackgroundsinphysics,systemsscience,etc.
The purpose of this book is to present a new approach to the investigation of
Boolean(control)networks.Inthisnewapproach,alogicalrelationisexpressedas
analgebraicequation,andalogicaldynamicalsystem,suchasaBooleannetwork,
is converted into a standard discrete-time linear system. Similarly, a Boolean con-
trol network is converted into a discrete-time bilinear system. In this way, various
tools for solving conventional algebraic equations and dealing with difference or
differentialequationscanbeusedtosolvelogic-basedproblems.Underthisframe-
work,thetopologicalstructuresofBooleannetworksarerevealedviathestructures
of their network transition matrices. The state space, subspaces, etc., are then de-
fined as sets of logical functions. This framework makes the state-space approach
to dynamical (control) systems applicable to Boolean (control) networks. Using
this new technique, we investigate the properties and control design of Boolean
networks. Many basic problems in control theory are studied, such as controlla-
bility, observability, realization, stabilization, disturbance decoupling and optimal
control.
Thefundamentaltoolinthisapproachisanewmatrixproduct,calledthesemi-
tensor product (STP). The STP of matrices is a generalization of the conventional
v
vi Preface
matrixproducttothecasewherethedimension-matchingconditionisnotsatisfied.
Thatis,weextendthematrixproductAB tothecasewherethecolumnnumberof
AandtherownumberofBaredifferent.Thisgeneralizationpreservesallthemajor
propertiesoftheconventionalmatrixproduct.
Using the STP, a logical function can be converted into a multilinear mapping,
called the matrix expression of logical relations. Under this construction, the dy-
namicsofaBooleannetworkcanbeexpressedasaconventionaldiscrete-timelinear
system.Inthelightofthislinearexpression,certainmajorfeaturesofthetopology
of a Boolean network, such as fixed points, cycles, transient time, and basins of
attractors,canbeeasilyrevealedviaasetofformulas.
WhenthecontrolofaBooleannetworkisconsidered,thebilinearsystemrepre-
sentationofaBooleancontrolnetworkmakesitpossibletoapplymosttechniques
developedinmoderncontroltheorytotheanalysisandsynthesisofaBooleancon-
trolnetwork.
Themaincontentsofthisbookareasfollows.
Chapter 1 consists of a brief introduction to propositional logic. This is very
elementaryandinvolvesonlythepropositionallogicrequiredinthisbook.Areader
whoisfamiliarwithmathematicallogiccanskipit.
InChap.2weintroducesomebasicconceptsandpropertiesoftheSTP,whichis
theprincipaltoolusedinthisbook.TheSTPisageneralizationoftheconventional
matrix product in cases where the dimension-matching requirement for the factor
matricesfails.Thisgeneralizationpreservesthemajorpropertiesoftheconventional
matrixproduct.
In Chap. 3 we consider the matrix expression of logical relations. Identifying
T (true)and F (false)withvectors [1,0]T and [0,1]T,respectively,alogicalvari-
able becomes a 2-dimensional vector variable. Using the STP, a logical function
canbeexpressedasamultilinearmappingwithrespecttoitslogicalargumentsso
that each logical function is uniquely determined by a matrix, called its structure
matrix.
Chapter4isdevotedtosolvinglogicalequations.Usingthematrixexpressionof
logicasystemoflogicalequationscanbeconvertedintoalinearalgebraicequation.
Ignoringthecomplexityofcomputation,thesolutionofsystemsoflogicalequations
becomes theoreticallyequivalentto the solution of algebraic equations, which can
beachievedwithstraightforwardcomputation.
Chapter5considersthelinearexpressionofBooleannetworks.Usingthetech-
nique developed in previous chapters, the dynamics of a Boolean network is con-
verted into a conventional discrete-time linear system. In the light of this linear
expression,thetopologicalstructuresofBooleannetworksareinvestigatedviatheir
transition matrices. Formulas are obtained to calculate the fixed points, cycles of
differentlengths,transientperiod,andthebasinofeachattractor.
The input-state structures of Boolean control networks are studied in Chap. 6.
Thecompoundedstructureofcyclesininput-statespaceisobtained.Thisapproach
isappliedtotheanalysisofBooleannetworkswithcascadingstructure.The“rolling
gear” structure of cycles is revealed, which explains the phenomenon that tiny at-
tractorscandeterminethevastorderofthenetwork[4].
Preface vii
Chapter7presentsatechniquetobuildthedynamicmodelofaBooleannetwork
viaobserveddata.Insteadof buildingthe logicaldynamicsof a Booleannetwork,
we first identify its algebraic form, so the conversion of the algebraic form of a
Booleannetworkbacktoitslogicalformisfirstinvestigated.Afterageneralmodel
constructiontechniqueisintroduced,severalspecialcasesarestudied,includingthe
knownnetworkgraphcase,theleastin-degreemodel,theuniformmodel,etc.The
problemofdealingwithdatacontainingerrorsisalsodiscussed.
In Chap. 8 a systematic state-space description is developed. The state space
(and its subspaces) of a Boolean (control) network are defined in a dual way, i.e.,
theyaredefinedassetsoflogicalfunctions.Itisshownthatthisdescriptionisvery
convenientinrevealingthepropertiesofBooleannetworksandinthecontroldesign
ofBooleancontrolnetworks.
Chapter9isdevotedtoBooleancontrolnetworks.Usinglinearexpressions,itis
shownthatBooleancontrolnetworkscanbeconvertedintolinearcontrolsystems.
Some basic control problems such as controllability and observability of Boolean
controlnetworksaretheninvestigatedviatheirequivalentforms forlinearcontrol
systems.
Chapter10considerstherealizationproblemofBooleancontrolnetworks.First,
coordinatetransformationsareconsidered,andthenthe Kalmandecompositionof
Booleaninput–outputnetworksisproposed.UsingtheKalmandecomposition,the
minimumrealizationofaBooleaninput–outputmappingisobtained.
ThestabilityandstabilizationproblemisdiscussedinChap.11.Theapplicable
setfrommetric-basedconvergenceanalysis[6]isenlargedbytheuseofcoordinate
transformations. Based on the analysis of the network transition matrix, necessary
and sufficient conditions are then obtained for stability and stabilization by either
open-loopcontrolorclosed-loopcontrol.Severalexamplesareincluded.
Chapter 12 considers the disturbance decoupling problem. First, the output-
friendlysubspaceisintroduced.Formulasandalgorithmsareprovidedtoconstruct
a minimum regular subspace, which is called the “friend” of output y. The de-
signtechniqueforconstructingthefeedbackandsolvingthedisturbancedecoupling
problemispresented.Toconstructaconstantstabilizingcontrol,thecanalizingmap-
ping,whichisageneralizationofthecanalizingfunction,isproposedanditsmain
propertiesarerevealed.
In Chap. 13 we consider the coordinate-independent geometric structure of
Boolean (control) networks. Based on this structure, the feedback decomposition
of Boolean control networks is studied. The input-state decomposition, including
cascadingandparalleldecompositions,andinput–outputdecompositionofBoolean
control networks are investigated, and necessary and sufficient conditions are pre-
sented.
Chapter14dealswiththemultivaluedlogicwhichcouldprovideamoreprecise
description for real networks such as gene regulation networks, etc. The structure
of k-valued logical networks is first investigated. Controllability and observabil-
ity of k-valued logical networks are then considered. In fact, almost all the argu-
ments and results about Boolean networks can be extended to the k-valued logic
setting.
viii Preface
Chapter 15 considers the optimal control of Boolean control networks. To deal
withBoolean (or k-valued)games with s-memory, higher-order Boolean(control)
networksareintroduced,andtheiralgebraicformsarealsopresented.Theone-to-
onecorrespondencebetweenthecyclesoftheoriginalnetworkandthecyclesofits
algebraicformisestablished.Theoptimalcontrolproblemistheninvestigatedand
theoptimalcontrolisdesigned.
Chapter16introducesausefultool,calledtheinput-stateincidencematrix,which
isanalgebraicdescriptionoftheinput-statetransfergraph.Controllabilityandob-
servability of Boolean control networks are revisited and some further results are
presented. The topological structures of Boolean control networks with free con-
trols are also investigated. Finally, the results are extended to mix-valued logical
dynamicalsystems.
Chapter 17 investigates the identification of Boolean control networks. First, a
newobservabilityconditionisobtainedwhichprovidesawaytoconstructtheinitial
stateofatrajectoryfromitsinput–outputdata.Anecessaryandsufficientcondition
foridentifiabilityisthenpresented.Anumericalalgorithmisproposedforpractical
application.
Chapter 18 considers an application to game theory. We consider a game with
finitely many players and where each player has finitely many possible actions.
When the game is infinitely repeated, a strategy using finite memory becomes a
logical dynamical system. Hence, the results obtained for Boolean or logical net-
works are applicable to finding Nash or sub-Nash solutions for the infinitely re-
peatedgames.
The primary objects of this book are deterministic Boolean networks, but in
Chap.19weprovideabriefintroductiontorandomBooleannetworks.Basiccon-
ceptsarepresentedandthenthesteady-statedistributionofarandomBooleannet-
workisinvestigated.Finally,thestabilizationofarandomBooleannetworkisstud-
ied. Recently, random Boolean networks have been the subject of much research,
andsoadetaileddiscussionisbeyondthescopeofthiswork.
AppendixAexplainsrelevantnumericalcalculations.Asoftwaretoolboxforthe
algorithmsisavailableathttp://lsc.amss.ac.cn/~dcheng/.
AppendixBcontainsproofsofsomekeypropertiesofthesemi-tensorproduct,
whicharetranslatedfrom[1],withthepermissionofSciencePress.
This book is self-contained. The prerequisites for its use are linear algebra and
somebasicknowledgeofthecontroltheoryoflinearsystems.Themanuscriptwas
originally prepared when the first author was visiting Kyoto University. The first
authorwouldliketoexpresshisheartythankstoProfessorYutakaTakahashiforhis
proof-reading and useful suggestions for parts of the manuscript. The manuscript
has been used as lecture notes in a series of seminars organized jointly by the
Academy of Mathematics and Systems Science, Tsinghua University, and Peking
University.Manycolleaguesandstudentsattendingtheseseminarshavecontributed
to this book via useful discussions, suggestions, and corrections. Particularly, Dr.
Yin Zhao helped in the preparation of Chaps. 15–17. Dr. Yifen Mu, Dr. Zhenning
Zhang,Dr.YinZhao,Dr.XiangruXu,andDr.JiangboZhanghelpedwiththefinal
galleyproofofthemanuscript.TheauthorsarealsoindebtedtoMr.OliverJackson
forhiswarmheartedsupport.
References ix
TheresearchpresentedinthisbookwaspartlysupportedbytheChineseNational
NaturalScienceFoundationundergrantnumberG60736022.
Beijing DaizhanCheng
HongshengQi
ZhiqiangLi
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