Application of abductive ILP to learning metabolic network inhibition from temporal data

22 Pages·2006·0.63 MB·English

MachLearn(2006)64:209–230
DOI10.1007/s10994-006-8988-x
Application of abductive ILP to learning metabolic
network inhibition from temporal data
AlirezaTamaddoni-Nezhad·RaphaelChaleil·AntonisKakas·StephenMuggleton
Received:11April2005/Revised:6March2006/Accepted:31March2006/Publishedonline:30June
2006
SpringerScience+BusinessMedia,LLC2006
Abstract In this paper we use a logic-based representation and a combination of Abduc-
tion and Induction to model inhibition in metabolic networks. In general, the integration
ofabductionandinductionisrequiredwhenthefollowingtwoconditionshold.Firstly,the
givenbackgroundknowledgeisincomplete.Secondly,theproblemmustrequirethelearning
ofgeneralrulesinthecircumstanceinwhichthehypothesislanguageisdisjointfromthe
observationlanguage.Boththeseconditionsholdintheapplicationconsideredinthispaper.
Inhibitionisveryimportantfromthetherapeuticpointofviewsincemanysubstancesde-
signedtobeusedasdrugscanhaveaninhibitoryeffectonotherenzymes.Anysystemableto
predicttheinhibitoryeffectofsubstancesonthemetabolicnetworkwouldthereforebevery
usefulinassessingthepotentialharmfulside-effectsofdrugs.Inmodellingthephenomenon
ofinhibitioninmetabolicnetworks,backgroundknowledgeisusedwhichdescribesthenet-
worktopologyandfunctionalclassesofinhibitorsandenzymes.Thisbackgroundknowledge,
whichrepresentsthepresentstateofunderstanding,isincomplete.Inordertoovercomethis
incompletenesshypothesesareconsideredwhichconsistofamixtureofspecificinhibitions
of enzymes (ground facts) together with general (non-ground) rules which predict classes
ofenzymeslikelytobeinhibitedbythetoxin.Theforegroundexamplesarederivedfrom
invivoexperimentsinvolvingNMRanalysisoftime-varyingmetaboliteconcentrationsin
Editor:RuiCamacho
A.Tamaddoni-Nezhad((cid:3))·S.Muggleton
Dept.ofComputing,ImperialCollegeLondon
e-mail:[email protected]
S.Muggleton
e-mail:[email protected]
R.Chaleil
Dept.ofBiologicalSciences,ImperialCollegeLondon
e-mail:[email protected]
A.Kakas
Dept.ofComputerScience,UniversityofCyprus
e-mail:[email protected]
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210 MachLearn(2006)64:209–230
rat urine following injections of toxins. The model’s performance is evaluated on training
and test sets randomly generated from a real metabolic network. It is shown that even in
thecasewherethehypothesesarerestrictedtobeground,thepredictiveaccuracyincreases
withthenumberoftrainingexamplesandinallcasesexceedsthedefault(majorityclass).
Experimentalresultsalsosuggestthatwhensufficienttrainingdataisprovided,non-ground
hypotheses show a better predictive accuracy than ground hypotheses. The model is also
evaluatedintermsofthebiologicalinsightthatitprovides.
Keywords AbductiveILP·Bioinformatics·Modellingmetabolicnetworks
1. Introduction
Thecombinationofabductionandinductionhasrecentlybeenexploredfromanumberof
angles(FlachandKakas,2000).Moreover,theoreticalissuesrelatedtothecompletenessof
thisformofreasoninghavealsobeendiscussedbyvariousauthors(Yamamoto,1997;Ito
andYamamoto,1998;Inoue,2001).Someimplementedsystemshavebeendevelopedfor
combiningabductionandinduction(MuggletonandBryant,2000)andothershaverecently
beenproposed(Moyle,2002;Ray,2003).Therehavealsorecentlybeendemonstrationsofthe
applicationofabduction/inductionsystemsintheareaofSystemsBiology(Zupanetal.,2003;
Kingetal.,2004)thoughinthesecasesthegeneratedhypotheseswereground.Ingeneral,
the integration of abduction and induction is required when the following two conditions
hold.Firstly,thegivenbackgroundknowledgeisincomplete.Secondly,theproblemmust
requirethelearningofgeneralrulesinthecircumstanceinwhichthehypothesislanguageis
disjointfromtheobservationlanguage.
The research reported in this paper is being conducted as part of the MetaLog project
(MetaLog). This project aims at building of causal rules concerning the actions of tox-
ins. These rules are built from Nuclear Magnetic Resonance (NMR) data in combination
withbackgroundknowledgederivedfromtheKyotoEncyclopediaofGenesandGenomes
(KEGG)(Ogata,1999).TheNMRspectraprovideinformationconcerningthefluxofmetabo-
liteconcentrationsbefore,duringandafteradministrationofatoxin.
In this paper we use a logic-based representation and a combination of Abduction and
Induction to model inhibition in metabolic networks. Inhibition is very important from
the therapeutic point of view since many substances designed to be used as drugs can
have an inhibitory effect on other enzymes. Any system able to predict the inhibitory ef-
fect of substances on the metabolic network would therefore be very useful in assessing
the potential harmful side-effects of drugs. In our specific problem domain of modelling
the phenomenon of inhibition, we use the frameworks of Abductive Logic Programming
(ALP)(Kakasetal.,1993;KakasandDenecker.,2002)andInductiveLogicProgramming
(ILP)(Muggleton,1991;MuggletonandDeRaedt,1994).Thepurposeofabductionisto
generatehypotheses(intheformofgroundfacts)aboutinhibitedenzymesfromtheNMR
observationsofmetaboliteconcentration.Thepurposeoftheinductionprocessistolearn
generalrulesfromtheseabducedhypotheses.Thesegeneralrulesdescribetheinhibitionof
enzymesintermsofavarietyofproperties,includingchemicalpropertiesoftheinhibitor
andthefunctionalclassofenzymes.Partoftheinformationaboutinhibitionrequiredbythe
inductionprocesscanbeobtainedfromdatabasessuchasBRENDA.1 However,formany
inhibitorstheavailabledatamaynotbeenoughtogenerateanygeneralrule.Theresultsof
abduction,fromthepreviousstage,thenactasinvaluabledatafortheinductionprocess.
1http://www.brenda.uni-koeln.de/.
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MachLearn(2006)64:209–230 211
Inapreviouspaper(Tamaddoni-Nezhadetal.,2004)describingourinitialinvestigation
inthistopicwemodelledtheeffectsofasingletoxin,hydrazine.Thepreviousmodelignored
thetemporalvarianceofmetaboliteconcentrations.Bycontrast,inthispaperwedescribean
extendedstudyinwhichtemporalvariationiscapturedandtheresultingmodelforhydrazine
iscontrastedwiththatofasecondlivertoxin,ANIT.TheNMRdataforHydrazineandANIT
werethefirstdatasetswhichhavebeenmadeavailabletotheprojectbyourcollaborators
whostudiedthesetoxinsaspartoftheCOMETproject(Lindonetal.,2003).
Inourstudy,examplesextractedfromtheNMRdataconsistofmetaboliteconcentrations
(upordownregulationpatternsextractedfromNMRspectraofurinefromratsdosedwith
the toxin) at 8 hours, 24 hours, 48 hours, 72 hours and 96 hours after the injection of the
toxin. Background knowledge consists of known metabolic networks (from KEGG) and
enzymesknowntobeinhibitedbythetoxin.Thisbackgroundknowledge,whichrepresents
thepresentstateofunderstanding,isincomplete.Inordertoovercomethisincompleteness
hypotheses are considered which consist of a mixture of specific inhibitions of enzymes
(ground facts) together with general rules (non-ground) which predict classes of enzymes
likely to be inhibited by the toxin. Hypotheses about inhibition are built using Progol5.0
(MuggletonandBryant,2000).Thepredictiveaccuracyisassessedandcomparedforboth
groundandnon-groundhypotheses.Itisshownthateveninthecasewherethehypotheses
are restricted to be ground, the predictive accuracy increases with the number of training
examples and in all cases exceeds the default (majority class). Experimental results also
suggestthatwhensufficienttrainingdataisprovided,non-groundhypothesesshowabetter
predictiveaccuracythangroundhypotheses.
Therestofthepaperisorganisedasfollows.Section2introducestheunderlyingbiological
problem of our study. Section 3 gives the background to the task of logical modelling of
scientifictheoriesusingabductionandinduction.Theexperimentsoflearninggroundand
non-ground hypotheses for inhibition are then described in Section 4. Section 5 discusses
relatedworkandSection6concludesthepaper.
2. Inhibitioninmetabolicnetworks
Metabolismismadebytheprocessesusedbyorganismstotransformelementsfromtheir
environment into the essential building blocks for life. Moreover, metabolism provides a
sourceofenergyforcellsanddegradestoxiccompoundsinpreparationforexcretion.The
graphoftheseinterlinkedchemicalreactionsisknownasthemetabolicnetwork(Jeongetal.,
2000;Ravasz,2002;AlmandArkin,2003).Thereactionsthattakeplaceinthisnetworkneed
tobefinelycoordinated.Biochemicalreactionsarecatalysedbyhighlyspecialisedproteins
knownasenzymes.Withoutenzymestherateofmostreactionsinlivingorganismswouldbe
tooslowtosustainlife.Enzymesactaskeyscontrollingtheactivationofthedifferentparts
ofthenetworkandarethereforecriticaltoitsoperation(Albertsetal.,1994).
Thecollectionofknownmetabolicnetworks,basedondataaccumulatedthroughyears
ofresearch,isnowaccessiblethroughpublicly-availablemetabolicdatabases,allowingtheir
studyfromanetworkperspective(Papin,2003;Alvesetal.,2002).Evenwiththehelpofthis
newSystemsBiologyapproachtometabolism,wearestillfarfromunderstandingmanyof
itsproperties.Oneofthelessunderstoodphenomena,especiallyfromanetworkperspective,
isinhibition.Somechemicalcompounds,knownasinhibitors,canaffectenzymes,impeding
their function. This in turn affects the normal flux in the metabolic network, the result of
whichisreflectedintheaccumulationordepletionofcertainmetabolites.
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212 MachLearn(2006)64:209–230
Fig. 1 A metabolic sub-network involving metabolites affected by hydrazine. Information on up/down
changesinmetaboliteconcentrationsfromNMRspectraiscombinedwithKEGGmetabolicdiagrams.The
enzymesassociatedwithasinglereaction(solidline)oralinearpathway(dottedline)areshownasasingle
enzymeoralistofenzymes
Inhibition is important from a therapeutic point of view since many substances de-
signed to be used as drugs can have an inhibitory effect on other enzymes. For example,
ParacetamolisaninhibitorofCOX-3cyclo-oxygenase(Swierkoszetal.,2002),preventing
the formation of arachidonic acid into prostaglandins, which are involved in the pain and
fever process (Boutaud et al., 2002). Paracetamol is transformed by the cytochrome P450
dependentenzymesproducingN-acetyl-p-benzo-quinoneimine(NAPQI)whichisnormally
conjugatedwithglutathioneforrenalelimination.Incaseofoverdosetheglutathioneproduc-
tionpathwaysarenotabletoproduceenoughglutathioneandNAPQI,whichisveryreactive,
accumulatesintheliverleadingtoitsfailure(ZimmermanandMaddrey,1995).Anysystem
abletopredicttheinhibitoryeffectofsubstancesonthemetabolicnetworkwouldtherefore
beveryusefulinassessingthepotentialharmfulside-effectsofdrugs.
Inthisworkweuseexperimentaldataontheaccumulationanddepletionofmetabolites
to model the inhibitory effect of various toxins, including hydrazine (NH -NH ), in the
2 2
metabolicnetworkofrats.Figure1showsthemetabolicpathwayssub-networkofinterest
also indicating with “up” and “down” arrows, the observed effects of hydrazine on the
concentration of some of the metabolites involved. This sub-network was manually built
fromtheinformationcontainedintheKEGGmetabolicdatabase(Ogata,1999).Westarted
fromthesetofchemicalcompoundsforwhichthereisinformationfromtheNuclearMagnetic
Resonance(NMR)experimentsonup-anddown-regulationfollowingtoxintreatment.From
this we tried to construct the minimal network representing the biochemical links among
thembytakingtheminimumpathwaybetweeneachpairofcompoundsandcollapsingall
thosepathwaystogetherthroughthesharedchemicalcompounds.Whenthereismorethan
one pathway of similar length (alternative pathways) all of them are included. Pathways
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MachLearn(2006)64:209–230 213
involving “promiscuous” compounds (compounds involved in many chemical reactions2)
are excluded. KEGG contains a static3 representation of the metabolic network (reactions
connectingmetabolites);wheretheexistenceofareactionisonlyconditionedbytheexistence
of at least one gene coding for an enzyme catalysing the reaction. NMR data provides
information on the concentrations of metabolites and their changes with time. The NMR
datausedinthisstudyrepresentvariationsofconcentrationofthemetabolites(relativeto
theirconcentrationbeforeinjectionofhydrazine)whicharemeasuredafter8hours,24hours,
48 hours, 72 hours and 96 hours. The effect of a toxin on the concentrations of chemical
compounds is coded in a binary way. Only up/down changes (increasing/decreasing) in
compoundconcentrationsareincorporatedinthemodel.
In this sub-network the relation between two compounds (edges in the network) can
compriseasinglechemicalreaction(solidlines)oralinearpathway(dottedlines)ofchemical
reactionsinthecaseswherethepathwaybetweenthosecompoundsiscomposedbymore
thanonereactionbutnotinvolvingothercompoundsinthenetwork(branchingpoints).The
directionalityofthechemicalreactionsisnotconsideredinthisrepresentationandinfactitis
leftdeliberatelyopen.Althoughmetabolicreactionsflowinacertaindirectionundernormal
conditions,thismaynotbethecasein“unusual”conditionsliketheonewearemodelling
here (inhibition). Inhibition of a given reaction causes the substrates to accumulate. This
may cause an upstream enzyme to start working backwards in order to maintain its own
substrate/productequilibrium.
The“onetomany”relations(chemicalreactionswithmorethanonesubstrateorproduct)
areindicatedwithafilledcircle.Theenzymesassociatedwiththerelations(singlechemical
reactionsorlinearpathways)areshownasasingleenzymeoralistofenzymes.
3. Logicalmodellingofscientifictheories
Modellingascientificdomainisacontinuousprocessofobservingthephenomena,under-
standing these according to a currently chosen model and using this understanding, of an
otherwisedispersecollectionofobservations,toimprovethecurrentgeneralmodelofthe
domain. In this process of development of a scientific model one starts with a relatively
simplemodelwhichgetsfurtherimprovedandexpandedastheprocessisiteratedover.Any
modelofthephenomenaatanystageofitsdevelopmentcanbeincompleteinitsdescription.
Thetaskthenistouseinformationgiventousbyexperimentalobservationstoimproveand
possiblycompletethisdescription.Thedevelopmentofourtheoriesisthendrivenbytheob-
servationsandtheneedforthesetheoriestoconformtotheobservations.Ourapproachwill
fallmuchinthesamespiritoftheoriesofscientificdiscovery(Popper,1959;Hempel,1965)
inthesensethatthedevelopmentofascientifictheoryisconsideredtobeanincremental
processofrefinementguidedstronglybytheempiricalobservations.
Consideringalogicalapproachtothisproblemofincrementaldevelopmentofascientific
model,philosophersofsciencehaverecognizedtheneedtointroducenewsyntheticforms
of reasoning, alongside with the analytical reasoning form of deduction. As early back as
Aristotle we see two forms of synthetic logical reasoning: Abduction and Induction. In
the nineteenth century, Charles Sanders Peirce (Harstshorne, 1958; Peirce, 1957) sets out
clearlythesethreeformsofsyllogisticreasoning—deduction,abductionandinduction—and
2Selectionofpromiscuouscompoundswasdonemanuallybyconsideringthoseinvolvedinmorethan50
reactions.
3Inreallifetheenzymesaredynamicallyproducedondemandwhenthereactionneedstooccur.
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214 MachLearn(2006)64:209–230
studiestheirrespectiveroleinthedevelopmentofscientifictheories.Morerecently,several
authors(seeforexamplethebooks(Holland,1989;JosephsonandJosephson,1994;Magnani,
2001;FlachandKakas,2000))havestudiedabductionandinductionfromtheperspective
ofArtificialIntelligenceandCognitiveScience.Inparticular,theworkinFlachandKakas
(2000)isdevotedtotheproblemofcomparingthesetwoformsofreasoningandinvestigating
theirpossibleunificationorintegrationforthepurposesofArtificialIntelligence.
Givenatheory,T,thatdescribesourcurrent(incomplete)modelofthescientificdomain
underinvestigation,andasetof(experimental)observations,O,abductionandinductionare
employedintheprocessofassimilatinginthecurrenttheorythenewinformationcontained
intheobservations.Theybothsynthesizenewknowledge, H,thusextendingthemodel,T,
toT ∪H,accordingtothesameformalspecificationof:
–T ∪H |= O,and
–T ∪H isconsistent.
where|=denotestheentailmentrelationoftheformallogicusedintherepresentationofour
theoryandconsistencyrefersalsotothecorrespondingnotioninthislogic.Theparticular
choiceofthisunderlyingformalframeworkoflogicisingeneralamatterthatdependson
theproblemorphenomenathatwearetryingtomodel.Inmanycasesthisistakentobefirst
orderpredicatecalculus,asforexampleintheapproachoftheorycompletioninMuggleton
and Bryant (2000). But other logics can be used, e.g. the non-monotonic logics of Logic
ProgrammingwithNegationasFailureorDefaultLogicwhenthemodellingofourproblem
requiresthislevelofexpressibility.InmanyapproachesofMachineLearninginArtificial
Intelligencewherewewanttouseautomatedformsofourlogic,thechoiceoflogiccanalsobe
drivenbypracticalconsiderationsofavailabilityofeffectivecomputationalmodelsforthese.
Onewaytodistinguishthetwoformsofreasoningistoconsidertheextendtowhichwe
(a-priori)allowthenewknowledge, H,tocomplementthecurrenttheory T.Abductionis
typicallyappliedonamodel,T,inwhichwecanseparatetwodisjointsetsofpredicates:the
observablepredicatesandtheabduciblepredicates.Thebasicassumptionthenisthatour
modelT hasreachedasufficientlevelofcomprehensionofthedomainsuchthatallthein-
completenessofthemodelcanbeisolated(undersomeworkinghypotheses)initsabducible
predicates.TheobservablepredicatesareassumedtobecompletelydefinedinT;anyincom-
pletenessintheirrepresentationcomesfromtheincompletenessintheabduciblepredicates.
Inpractice,observablepredicatesdescribetheempiricalobservationsofthedomainthat
wetryingtomodel.Theobservationsarerepresentedbyformulaethatreferonlytoobserv-
ablepredicates(andpossiblysomebackgroundauxiliarypredicates—seebelow)typically
bygroundatomicfactsontheobservablepredicates.Theabduciblepredicatesdescribeun-
derlying(theoretical)relationsinourmodelthatarenotobservabledirectlybutcan,through
themodelT,bringaboutobservableinformation.Wealsohavebackgroundpredicatesthat
areauxiliaryrelationsthathelpuslinkobservableandabducibleinformation(e.g.theyde-
scribeexperimentalconditionsorknownsub-processesoftheproblemdomainthatweare
modelling).
Havingisolatedtheincompletenessofourmodelintheabduciblepredicates,thesewill
form the basis of abductive explanations for understanding, according to the model, the
specific observations that we have of our scientific domain. Abduction generates in these
explanations(typically)extentionalknowledgethatrefersonlytotheabduciblepredicates
andthatisspecifictotheparticularstateorscenariooftheworldpertainingtotheobservations
explainedandtothegivenmodelT fromwhichtheyweregenerated.Addinganexplanation
tothetheorythenallowsustopredictfurtherobservableinformationbutthisnewpredictive
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MachLearn(2006)64:209–230 215
powerisrestrictedtocomeonlythroughthealreadyknownrulesinourtheorythatlinkthe
observablepredicatestootherpredicates.
Ontheotherhand,inductiveinferencegeneratesintentionalknowledgeintheformofnew
generalrulesthatnowcanprovide,eitherdirectlyorindirectlythroughthecurrenttheoryT
thattheyextend,newinterrelationshipsbetweenthepredicatesofourtheorythatcaninclude
theobservablepredicatesandeveninsomecasesnewpredicates.Theinductivehypothesis
thusintroducesnew,hithertounknown,linksbetweentherelationsthatwearestudyingthus
allowing new predictions on the observable predicates that would not have been possible
beforefromtheoriginaltheoryunderanyabductiveextension.
A cycle of integration of abduction and induction (Flach and Kakas, 2000) emerges
that is suitable for our task of incremental scientific modelling. Abduction is first used
to transform (and in some sense normalise) the observations to an extensional hypothesis
ontheabduciblepredicates.Theninductiontakesthisasinputandtriestogeneralizethis
extentionalinformationtogeneralrulesfortheabduciblepredicatesnowtreatingtheseas
observable predicates for its own purposes. The cycle can then be repeated by adding the
learned information on the abducibles back in the model as new partial information on
the incomplete abducible predicates. This will affect the abductive explanations of new
observationstobeusedagaininasubsequentphaseofinduction.Hencethroughthiscycle
ofintegrationtheabductiveexplanationsoftheobservationsareaddedtothetheorynotin
the(simple)formthattheyhavebeengeneratedbutinageneralizedformgivenbyaprocess
ofinductiononthese.
Asimpleexample,adaptedfromRay(2003),thatillustratesthiscycleofintegrationof
abductionandinductionisasfollows.Supposethatourmodelcontainsthefollowingrule
andbackgroundfacts:
sad(X)iftired(X),poor(X).
tired(oli),tired(ale),tired(kr),
academic(oli),academic(ale),academic(kr),
student(oli),lecturer(ale),lecturer(kr).
whereouronlyobservablepredicateissad/1.
Given the observations O ={sad(ale),sad(kr),not sad(oli)} can we improve our
model? The incompleteness of our model resides in the predicate poor. This is the only
abduciblepredicateinourmodel.Usingabductionwecanexplaintheobservations O via
theexplanation:
E={poor(ale),poor(kr),notpoor(oli)}.
Subsequently,treatingthisexplanationastrainingdataforinductivegeneralizationwecan
generalizethistogettherule:
poor(X)iflecturer(X)
thus(partially)definingtheabduciblepredicate poor whenweextendourtheorywiththis
rule.
The combination of abduction and induction has recently been studied and deployed
in several ways within the context of Inductive Logic programming (ILP). In particular, a
new form of ILP, called Theory Completion introduced in Moyle and Muggleton (1997),
MuggletonandBryant(2000);andMoyle(2000),aims,aswehavedescribedabove,tocom-
pletethecurrenttheorywherethenewlygeneratedpartsofthetheoryneednotbeintheform
ofclausesthatreferdirectlytothepredicatesofthegiventrainingexamples—observations.
TherealizationofTheoryCompletionthroughInverseEntailment(MuggletonandBryant,
2000)canbeseenasaparticularcaseofintegrationofabductiveinferenceforconstructing
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216 MachLearn(2006)64:209–230
down(m1). inhibited(e1,m2,m1).
up(m2). observables ABDUCTION abducibles not inhibited(e3,...). examples INDUCTION class(E,ec1),
. of inhibited enzyms . of inhibition rules cofact(E,...),
. . .
.
BACKGROUND KNOWLEDGE
reaction(m1,e1,m2). inhibits(h,e8).
reaction(X,E,Y), reaction(m3,e2,m4). inhibits(h,e9).
NMR inhibited(E,Y,X), . .
... . .
KEGG BRENDA
Fig.2 AnAbductive/Inductiveframeworkformodellinginhibition.Thepurposeoftheabductionprocessis
togeneratehypothesesaboutinhibitedenzymesfromtheNMRobservationsofmetaboliteconcentration.The
purposeoftheinductionprocessistolearnfromtheabducedhypotheses,whicharegroundfactsofinhibition,
generalrulesabouttheinhibitionofenzymesintermsofchemicalpropertiesoftheinhibitor,functionalclass
ofenzymesetc
a“minimal”clause(calledthebottomclause)andinductiveinferencetogeneralizethisclause
givingthenewclausetobeaddedtothetheory.ThisisimplementedinProgol5.0andapplied
toseveralproblemsincludingthatofthediscoveryofthefunctionofgenesinanetworkof
metabolicpathways(Kingetal.,2004).InMoyle(2000)TheoryCompletionisrealizedinan
ILPsystem,calledALECTO,whichagainintegratesaphaseof,extraction-caseabduction,
totransformeachcaseofatrainingexampletoanabductivehypothesiswithaphaseofin-
ductionthatgeneralizestheseabductivehypotheses.Ithasbeenused(Moyle,2002)tolearn
robotnavigationcontrolprogramsbycompletingthespecificdomainknowledgerequired,
withinageneraltheoryofplanningthattherobotusesforitsnavigation.
Ingeneral,theintegrationofabductionandinductionenhancesthemodeldevelopment
in several ways. It provides a better opportunity to test the correctness of the generated
hypothesesasthiscanincreasethescopeoftesting.Thepredictiveaccuracy(ordegreeof
corroboration)ofanabductiveexplanationcanbeusedasacriteriontoselectamongstdif-
ferentpossibleexplanationsoftheobservations.Butconsideringtheabductiveexplanations
astheyareintheirextensionalformprovidesarelativelyrestrictedscopefortestingofthe
hypotheses.However,whenweintegratethiswithinductionwecangetpredictionsoutside
therealmofthegivenobservationsandtheoryandthusincreasethescopeoftestingofour
hypotheses.Inatightintegrationofabductionandinductionthechoiceofanexplanationin
thefirstabductivephaseofthecycleislinkedtothesecondphaseofhowwelltheexpla-
nationgeneralizesthroughinduction.Theabductiveexplanationsandintentionalhypothesis
generated from them are thus evaluated together where the generalizability (measured by
thepredictiveaccuracybutalsobyothercriteriasuchasthedegreeofcompression)ofthe
explanation feeds back in the choice of abductive explanation. Such frameworks of tight
integrationofabductionandinductionwithILPinclude:Progol5.0(MuggletonandBryant,
2000),ALECTO(Moyle,2000),ACL(KakasandRiguzzi,2000),SOLDR(Yamamotoand
Fronho¨fer,2001),CF-Induction(Inoue,2001)andHAIL(Ray,2003).WewilluseProgol
5.0tocarryouttheexperimentsinourstudyinthispaper.
For our specific problem domain of modelling the phenomenon of inhibition the cycle
ofintegrationofabductionandinductionisshowninFig.2.Thepurposeoftheabduction
processistogeneratehypothesesaboutinhibitedenzymesfromtheNMRobservationsof
metabolite concentration. For this we need to start with a theory which models how the
concentrationofmetabolites(e.g.up-downregulations)isrelatedtoinhibitionofenzymes
(seeSection3.2forsuchamodel).Thepurposeoftheinductionprocessistolearnfromthe
abducedhypotheses,whicharegroundfactsofinhibition,generalrulesabouttheinhibition
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MachLearn(2006)64:209–230 217
ofenzymesintermsofchemicalpropertiesoftheinhibitor,functionalclassofenzymesetc.
Partoftheinformationaboutinhibitionrequiredbytheinductionprocesscanbeobtained
fromdatabasessuchasBRENDA.However,formanyinhibitorstheavailabledatamaynot
beenoughtogenerateanygeneralrule.Theresultsofabduction,fromthepreviousstage,
thenactasinvaluabledatafortheinductionprocess.
3.1. Modellinginabductivelogicprogramming
A framework that allows declarative representations of incomplete theories is that of Ab-
ductiveLogicProgramming(ALP)(Kakasetal.,1993;KakasandDenecker.,2002).Inthis
frameworkamodeloratheory,T,isdescribedintermsofatriple(P,A,IC)consistingofa
logicprogram, P,asetofabduciblepredicates, A,andasetofclassicallogicformulaeIC,
calledtheintegrityconstraintsofthetheory.TheprogramPcontainsdefinitionalknowledge
representingthegenerallawsaboutourproblemdomainthroughacompletedefinitionofa
setofobservablepredicatesintermsofeachother,backgroundpredicates(whichareagain
assumed to be completely specified in P) and a set of abducible predicates that are open.
Abduciblepredicatesappearonlyintheconditionsoftheprogramruleswithnodefinition
in P.Theintegrityconstraints,IC,representassertionalknowledgethatwemayhaveabout
ourdomain,augmentingthemodelin P,butwithoutdefininganypredicates.
GivensuchanALPtheorytheinferenceofabduction(i.e.ofabductiveexplanation)is
thenspecializedaccordinglyinthefollowingway:
Definition1. Given an abductive logic theory (P,A,IC), an abductive explanation for an
observationO,isaset,(cid:2),ofgroundabducibleatomsonthepredicates Asuchthat:
– P∪(cid:2)|= O
LP
– P∪(cid:2)|= IC.
LP
where|= denotesthelogicalentailmentrelationinLogicProgramming4.
LP
Theabductiveexplanation(cid:2)representsahypothesiswhichwhentakentogetherwiththe
modeldescribedinthetheoryT explainshowanonemptyexperimentalobservableOcould
hold.AnabductiveexplanationpartiallycompletesthemodelasdescribedinthetheoryT.
Whentheobservation O isagiventrainingexampleinalearningproblemwecanseethat
suchabductiveexplanationscoincidewiththeconceptofextraction-caseabduction,asused
inKakasandRiguzzi(2000),MoyleandMuggleton(1997),andMoyle(2002),totransform
eachcaseofatrainingexampletoanabductivehypothesis.
Theimportantroleoftheintegrityconstraints,IC,istoimposevalidityrequirementsonthe
abduciblehypotheses(cid:2).Theyaremodularlystatedinthetheory,separatelyfromthebasic
modelcapturedin P,andtheyareusedtoaugmentthiswithanypartialinformationthatwe
mayhaveontheabduciblepredicatesorotherparticularrequirementsthatwemaywantthe
abductivelygeneratedexplanationsofourobservationstohave.Inmostpracticalcasesthe
integrityconstraintsareoftheformofclausalrules:B ∧...∧B → A ∨...∨ A where
1 n 1 k
A ,...,A and B ,...,B arepositiveliterals.Intheseconstraints,kcanbepossiblyzero
1 k 1 n
(wewillthenwritetheconclusionasfalse)inwhichcasetheconstraintisadenialprohibiting
anysetofabduciblesthatwouldimplytheconjunction B ,...,B .
1 n
4Forexample,whentheprogramPcontainsnonegationasfailurethenthisentailmentisgivenbytheminimal
Herbrandmodeloftheprogramandthetruthofformulaeinthismodel.
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218 MachLearn(2006)64:209–230
AnabductiveexplanationpartiallycompletesthemodelasdescribedinthetheoryT.As
mentionedabove,thisisalimitedconservativeextensionofthemodel,duetotheextensional
natureoftheseexplanationsandtheirrestrictedscopeofpredictiontothespecificsituation(s)
oftheobservationstheyexplain.Thesubsequentgeneralization,throughinduction,ofground
abductive explanations to intensional rules for the abducibles provides a greater level of
completingtheinitialtheory(model).Theruleslearnedontheabduciblesprovide(partial)
definitionsoftheabduciblepredicates,tobeaddedintheprogramP,thuspartiallycompleting
theinitiallyincompletemodel.Wecantheniteratetheprocessofscientificanalysisofthe
phenomena,i.e.gathernewobservations,explainthem,generalizetheexplanationsandtest
theinductivehypothesesthusgenerated,untilwearesatisfiedfromthelevelofcompleteness
thatourmodelhasreached.
3.2. ModellinginhibitioninALP
Wewilldevelopamodelforanalyzing(understandingandsubsequentlypredicting)theeffect
oftoxinsubstancesontheconcentrationofmetabolites.Ourrepresentationwilluseasthe
setofobservablepredicatesthesinglepredicate:
concentration(Metabolite,Level,Time)
expressing the fact that at some time, Time, a metabolite, Metabolite, has a certain level
ofconcentration,Levelwhichinthesimplestcasecantakethetwovalues,downorup.In
general,theconcentrationpredicationwouldcontainafourthargument,namelythenameof
thetoxinthatweareexaminingbutwewillassumehereforsimplicitythatwearestudying
onlyonetoxinatatimeandhencewecanfactorthisout.Backgroundpredicatessuchas:
reactionnode(Metabolites1,Enzymes,Metabolites2)
describethetopologyofthenetworkofthemetabolicpathwaysasdepictedinFig.1.For
example,thestatement
reactionnode((cid:6)l−2−aminoadipate(cid:6),(cid:6)2.6.1.39(cid:6),(cid:6)2−oxo−glutarate(cid:6))
expresses the fact that there is a direct path (reaction) between the metabolites l−2−
aminoadipateand2−oxo−glutaratecatalyzedbytheenzyme2.6.1.39.Moregenerally,
wecanhaveasetofmetabolitesoneachsideofthereactionandasetofdifferentenzymes
thatcancatalyzethereaction.
Notealsothatthesereactionsareingeneralreversible,i.e.theycanoccurineitherdirection
andindeedthepresenceofatoxincouldresultinsomereactionschangingtheirdirection
inanattempttocompensate(re-balance)theeffectsofthetoxin.Themodelalsoinvolves
backgroundbiochemicaldataonenzymesandmetabolitesthatwouldbeusedintheprocess
ofinductivegeneralizationoftheabducedhypotheses.
Theincompletenessofourmodelresidesinthelackofknowledgeofwhichmetabolic
reactions are adversely affected in the presence of the toxin. This is captured through the
declarationoftheabduciblepredicate:
inhibited(Enzyme,Metabolites1,Metabolites2,Time)
capturingthehypothesisthatatthetimeTimethereactionfromMetabolites1toMetabolites2
is inhibited by the toxin through an adverse effect on the enzyme, Enzyme, that normally
catalyzesthisreaction.Forexample,
inhibited((cid:6)2.6.1.39(cid:6),(cid:6)l−2−aminoadipate(cid:6),(cid:6)2−oxo−glutarate(cid:6),8)
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