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2023年4月19日发(作者：中国儿童文学网)Atheoryofoptimal
diﬀerentialgeneexpression
WolframLiebermeister
lieberme@
MaxPlanckInstituteforMolecularGenetics,Ihnestr.73,14195Berlin./∼agklipp
BCB-BerlinCenterforGenomeBasedBioinformatics
Kinetic
Modeling
Group
IntroductionMetabolicsystems
Thelarge-scalestructureofgeneregulationhasbeenintenselystudiedbymeasuringgeneexpressionon
genomicscale.Clusteranalysesandlinearmodelsofgeneexpressiondatahaverevealedgroupsofcoregulated
genesthatoftensharebiologicalfunctions.Conversely,expressiondatahavebeenusedforannotating
genesandreconstructingmetabolicpathways[2],butinapurelyheuristicalway.Herewederivearelation
betweenexpressionandfunctionfromaprincipleofoptimalregulation[5].Theproposedmodelpredictsa
relationbetweenoptimaldiﬀerentialexpression(afterexternalperturbations),function(quantiﬁedbyresponse
coeﬃcients),andtheregulatorymechanism,asitisfoundinoperons.Asaconsequence,optimalgene
expressionproﬁlesandregulatorynetworksmayportraythetopologyofthemetabolicnetwork.
Ametabolicsystemischaracterizedby
•StoichiometricmatrixNandthekernelmatrixKofstationaryﬂuxes,fulﬁllingNK=0
•ElasticitiesL:linearinﬂuencesofindependentmetaboli滇组词 tesonisolatedreactions
Metabolicresponsecoeﬃcients
describethe(linearised)systemicresponsetoparameterperturbations.
JS
,Cdescribethelinearinﬂuenceofﬂuxchangeofreactionk•ThecontrolcoeﬃcientsC
ikik
onglobalstationaryﬂuxJorconcentrationS
ii
SJ
•Theoremsofmetaboliccontroltheory[3]→constraintsoncontrolcoeﬃcientsC,C
Themodel
y

x
Modelquantities
•x:regulatoryvariables(geneexpression,..)
ThesummationandconnectivitytheoremsCL=0andCK=0yield
JS
−1pTYT
()(C)Mgeneralpropertiesofoptimalregulationpatternsdx=−F
xx
p
AssumethatﬁtnesscurvaturematrixFandparameter-elasticitiesarediagonal:
xx
T
•Ifﬁtnessdependsonlyon:TheoremdxL=0
Fluxes
F
•y(x,):“output”variables(metabolites,ﬂuxes,..)
•F(x,y):ﬁtnessfunction(reproductionrate,..)
•:perturbationparameters(nutrients,temperature,..)
→Regulationproﬁlesfornadjacentreactions(sharingametabolite)
areconﬁnedtoa(n−1)-dimensionalsubspace.
•Ifﬁtnessdependsonlyon:TheoremdxK=0
Concentrations
T
→regulationvaluesoveranystationaryﬂuxmodesumtozero.
TheregulatorsxalwaysadaptthemselvestotheperturbationinordertomaximizeaﬁtnessfunctionF.
Optimalresponsetoperturbations
y
F(x,y)
y(x, +d )
y(x, )
Example:asimplemetabolicnetwork
S1
E1
S2
S3
E4E3
S4
E9
S7S5
E5
−0.10.1−0.4
0.1
0.60.6
0.7−0.7
−0.7
0.4
0.4
change of Jchange of Jchange of J
126
E2
1.0
−0.1
1.0
0.7
0.3
−0.4
Localdescriptionbyderivatives:
yy
•Responsecoeﬃcients:R=∂y/∂x,R=∂y/∂
x
•Fitnessderivatives:F=∂F/∂x,F=∂F/∂y,F=
xyxx
∂F/∂x,F=∂F/∂y
2222
yy
•DeﬁneregulatoryﬁtnessG(x,)=F(x,y(x,))
−0.1
E7E6E8
S6S8
0.1
−0.6
0.3
−0.3
0.2
−0.2
−0.6
−1.0
1.0
dx
G(x, )=F(x,y(x, ))
x
•
Initially,thesystemisinalocallyoptimalstatewhere
y
T
G=F+(R)F=0
xxy
x
yy
T
G=F+(R)FR
xxxxyy
xx
hasnegativeeigenvalues
•Smallperturbation(ofy,x,F,...)
y
•Findresponsedxtoreachanewoptimalstate
Relatingexpressiontocontrolcoeﬃcients
can. var. of CCcan. var. of expression
x
•Condition:G=0beforeandafterperturbation
x
Diﬀerentkindsofperturbations
yyyy
xAchievingaﬁxedchangedy=Rddx=F(R)(RF(R))dy
xxxx
T−1−1T−1
xxxx
ThescaledexpressionproﬁleFdxisalinearcombinationofregulatoryproﬁles.
xx
Singlevaluexperturbed:Asinglecomponentx
ii
becomesconstrainedtoaﬁxedvaluex+dx
ii
Perturbationsdofydx=−G((R)Fdy+(dR)F)
1
−1
dxGdx=
(G)
−1
xx
xx
ii
yy
TT−1

xx
xx
yyy
Comparingsimulatedﬂuxcontrolcoeﬃcientstogeneexpressiondata(Gaschetal.[1])bycanonicalanalysis.
Theﬁrstcomponentsfound(shownabove)aresigniﬁcantlysimilar.
Superposedresponsetotheperturbationofvariablesandresponsecoeﬃcients.
Modelpredictions
Gene书法学习视频 ralpredictions
•Expressionpatternsreﬂectthemetabolicresponsecoeﬃcients
•ReciprocalbehaviourforsmallperturbationsindeletionorRNAiexperiments.
1
Reciprocalresponseinknock-outexperiments
•Knock-outexperiment→expressiondatamatrixM
GAL2
(columns:genesknockedout,rows:thesamegenes,measured,log-values)
0.8
0.6
0.4
•Relationbetweendiﬀerentialexpressionandﬁtnesslossafterdeletions.
1−1−
whereDisdiagonalandG•Modelprediction:M=DG
xxxx
issymmetric.
GAL1
GAL7
GAL100
GAL4
GAL80
GAL3
1−
GestimatedfromIdekeretal.[4]knock-outsingalactosepathway
xx
Predictionsformetabolicsystems
assumingexpression∝enzymaticactivity
0.2
−0.2
−0.4
−0.6
−0.8
−1
Ifperturbinggeneiaﬀectstheexpressionof赞美书香气息的诗句 genej,theop-
positeshouldalsohold.
•Thepredictedcompensationshouldalsoappearinphylogenetic
gene
proﬁles.
•Iftheﬁtnessdependsonlyonﬂux有关中秋节的诗歌或短文 es,andelasticitiesrepresentonlystoichiometry:
correlatedexpressionofneighbourenzymes
•Iftheﬁtnessdependsonlyonconcentrations:
theexpressionproﬁle,summedoveranystationaryﬂux,vanishes.
•Ifasetofmreactionscontrolsn<mindependentﬂuxes:
itsexpressionpatternshouldbeconﬁnedtoan-dimensionalsubspace.
Optimallinearfeedback
Discussion
Theoptimalitypostulateforperturbationsd
canbeimplementedbyalinearfeedback.
•Theapproachisto
limited
-Smallperturbations
-Physiologicalconditions(optimalbehaviourisbased“trainingconditions”duringevolution)
•Theapproachis:
general
Onlyrequirement:Metabolicresponsecoeﬃcientsmustbedeﬁned
•Time-dependentperturbationsofastationarystatecanbetreatedanalogously.
•Quantitativetestsarediﬃcult,because
-Relativelyfewresponsecoeﬃcientscanbemeasured(butsomepropertiesareknown)
-Fitnessfunctionisnotknown
•Resultssuggesttousesparselinearmodels(e.g.,ICA)formicroarraydataanalysis.
R

y

y
RR
y
x
x
x
y
F
y
Tx−1
R=−F(R)F
yxx
x
yy
yy
−1xx
•Theresultingreactiondx=(1−RR)RRdisoptimal
yy
x
•Thefeedbackconnectionsarerelatedtotheresponsecoeﬃcients(≈functionsofaregulators)
•Nonlinearsystems(signallingpathwaysetc.)maylocallyimplementthelinearresponse.
Acknowledgement:ThisworkwasfundedbytheEuropeancommission,grantNo.503269
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References
[1]A.P.Gaschetal.Genomicexpressionprogramsintheresponseofyeastcellstoenvironmentalchanges.MolecularBiologyoftheCell,11:4241–4257,2000.
[2]D.Hanisch,A.Zien,R.Zimmer,andT.Lengauer.Co-clusteringofbiologicalnetwork元日王安石赏析解释 sandgeneexpressiondata.Bi尖组词一年级 oinformatics,18(90001):145S–154,2002.
[3]R.HeinrichandS.Schuster.Theregulationofcellularsystems.Chapman&Hall,1996.
[4]T.Idekeretal.Integratedgenomicandproteomicanalysesofasystematicallyperturbedmetabolicnetwork.Science,292:929–934,2001.
[5]W.Liebermeister,E.Klipp,S.Schuster,andR.Heinrich.Atheoryofoptimaldiﬀerentialgeneexpression.BioSystems,76:261–278,2004.
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