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Invariantmanifoldsin amodelfor theGABA
receptor
Martin Caberlin∗ Nilima Nigam† SanjiveQazi‡
January8, 2003
Abstract
The reactionschemeof GABA interactingwith the ligand-gatedion-
channeldemonstratesnumericalstiffness. We comparedvariousexplicit
andimplicit numericalmethodsto solve for thereactionscheme,andfound
implicit methods(BackwardEuler, ode23s)performedordersof magnitude
betterthanexplicit methods(Forward Euler, ode23,RK4, ode45)in terms
of stepsizerequiredfor stability, numberof stepsand cputime. Interest-
ingly, we observed the existenceof low dimensionalinvariant manifolds
∗Dept.of MathematicsandStatistics,McGill. email:[email protected]†Dept.of MathematicsandStatistics,McGill. email:[email protected]‡Parker-HughesCancerCenter. email:[email protected]
1
uponsolving thesystemof 8 ordinarydifferentialequationsfor theGABA
reactionscheme.Theemergenceof simpledirectrelationshipsbetweenthe
receptorstatesmayhavesomeimportantrolesin biologicalsignaltransmis-
sion.
1 Intr oduction
Receptor mediatedsignal transmissionin chemical synapsesis a fundamental
componentof neuronalcommunication.GABAA receptorsconductchlorideions
to generateinhibitory post-synapticcurrents.Networksof inhibitory neuronsare
thoughtto playanimportantrole in modulatingneocorticaloscillationsandhence
brain function (Connors& Amital, 1997;Castro-Alamancos& Connors,1996).
Receptor gatedion-channelssuch asthe GABAA receptorcanexist in multiple
conformationalstates:bound,unbound,open,closedanddesensitized. Sincethe
bindingof thetransmitterto thereceptoris thefirst constraintin thepassingof in-
formationfrom one neuronto thenext, we wereinterestedin therelationshipbe-
tweenthebindingeventandthesubsequenttransitionthroughthedifferentstates
of thereceptor. TheGABAA reactionschemewasmodeledvia basicbiochemical
processes.It is essentiallyan adaptationof the fundamentalMichaelis-Menten
mechanismof enzyme-substrateinteractions,andis characterizedby a systemof
2
ordinarydifferentialequations.
Our numericalinvestigationrevealedthatthemodelexhibits stiffness; a char-
acterizationof stiffnesswill be provided. In this particularsituation,this phe-
nomenonmeansthatthedynamicalsystemreducesfrom beingeight-dimensional
(correspondingto eightchemicalconcentrations)to seven,andthento five. This
dimensionreductionhasbiological implications,which we conjecture uponand
would like to investigatefurther.
Stiff dynamicalsystemsareexceedinglydifficult to integrateusingstandard
explicit methods, andrequirevery small time-stepsin orderto maintainnumeri-
cal stability. The useof adaptive algorithmsdoesnot significantlyalleviate this
problem;theintegratorselectsanunreasonablysmallallowablestep-size.Heuris-
tically, the time-stepis selectedto resolve the behavior of the fastesttransient
in the system.However, this transientmay not be the dominantmodeandmay
contribute negligibly to the over-all dynamics ofthe system.The net result is a
significantwasteof computationaleffort. A well-known strategy to numerically
integratestiff systemsis to usean implicit method. In Section2, we first provide
a working definitionof stiffness,andthendemonstratethroughsimpleexamples
thedifference betweenusingexplicit andimplicit methods.
Webeganour investigationby first usingstandard(explicit) algorithmsto nu-
3
merically simulatethe model. During the experimentswe encounteredstability
problemswhich necessitatedthe useof stiff solvers. The successof theselat-
ter algorithmsled us to characterizethe modelasstiff. Oncewe hadstableand
accuratealgorithmsto study the model,we observed the existenceof invariant
manifolds.
The organizationof this paperis asfollows. We begin with a brief descrip-
tion of themodelin question;for furtherdetails we refer the interestedreaderto
(Qazi & Trimmer, 1999). In Section2, we describestiffnessandpresentexam-
pleswhich demonstratethepower of implicit methods.In section3, we present
the resultsof our investigationsof the GABA model. We endthis paperwith a
tentativebiologicalexplanationfor theobservedphenomena,aswell assuggested
experimentsto verify ourfindings.
2 The GABA reactionscheme
GABA, which is γ-aminobutyric acid, is an inhibitory neurotransmitter (Alberts
et al., 1994). The synapseis the site at which neuronalsignalsaretransmitted.
Thepredominantmechanismof signaltransmissionat thesynapseis via chemi-
cal exchangefrom thepresynapticcell to thepostsynapticcell (Qazi& Trimmer,
4
1999).Thenarrow gapbetweentheseis calledthesynapticcleft, into which neu-
rotransmitters,such asGABA, arereleased.Uponrelease,the neurotransmitteris
rapidly removedfrom the synapticcleft. In orderto understandthechemicalsig-
nal, the postsynapticcell mustbe equippedwith GABA binding receptors.One
suchexampleis the GABAA receptor. The GABA reactionschemeconsidered
heremodelsthe variousconcentrationsof GABA, boundandunboundreceptor,
anddesensitizedandopenstatesafterthesimulatedreleaseof aGABA pulse.
TheGABA reactionschemeis basedon theclassicalbimolecularreactionof
Michaelis-Menten,suitablyadaptedto accountfor reversiblereactionsandmulti-
ple receptorstates.TheMichaelis-MentenreactiondescribesasubstrateR, react-
ing with anenzymeN , which form acomplex RN , soonconvertedinto aproduct
P andtheenzyme(Murray, 1990).Schematically, thereactionlookslike
R + Nk−1
k1
RNk2
→ P + N, (1)
wherek1, k−1 andk2 are the rate constantsof the reaction. The Law of Mass
Action saysthat the reactionrate is proportional to the productof the concen-
tration of thereactants(we usethesamelettersto denoteboth thereactantsand
their respective concentrations,with theexception[RN ] = C). This leadsto the
5
Michaelis-Mentenequations,
R = −k1RN + k−1C, E = −k1ES + (k−1 + k2)C,
C = k1ES − (k−1 + k2)C, P = k2C.
The GABA schemeassumesall the reactionsare reversible(ie. all arrows in
the reactions(1) are doublepointed). One of the modificationsof Michaelis-
Mentenin theGABA schemestemsfromthefactthattheendproductP is abound
receptor. In this way, GABA ≡ N is not a catalyst- it is used upin thereaction.
So thesecondhalf of thereactionis not present.A furtheradaptationis that the
GABA schemeinvolvesprocessesin which the receptorR hasmultiple GABA
bindingsites.Theappropriatechangesbecomeapparentafter closeexamination
of theschematicdiagramsin Table1. TheGABA reactionvariablescanbefound
in Table2.
As Qazietal. explain,anadequatemodelof theGABA interactionwith there-
ceptormusthaveat leasttwobindingsites,twoopenstatesandtwodesensitization
states(Qazi& Trimmer, 1999; Joneset al., 1998;Shenet al., 2000;Mozrzymas
et al., 1999). Transitionsfrom the slow desensitizedDs to the fastdesensitized
stateDf have alsobeenincludedin subsequentmodels(Qazi& Trimmer, 1999;
6
Table1: Schematicrepresentationof GABA reactions(Joneset al., 1998;Qazi& Trim-mer, 1999).
BindingReactions StateTransitions
R + Nkoff
2kon
B1 B1
a1
b1
O1
B1 + N2koff
kon
B2 B1
d1
r1
Ds
Ds + Nq
p
Df B2
a2
b2
O2
B2
d2
r2
Df
Ds
p -�q
Df
R2kon -�koff
B1
d1
6
r1
? kon -�2koff
B2
d2
6
r2
?
O1
a1
6
b1
?O2
a2
6
b2
?
Table2: GABA schemevariables
R ReceptorconcentrationN Ligand(GABA) concentrationB1 First boundstateconcentrationB2 SecondboundstateconcentrationDf FastdesensitizedstateconcentrationDs Slow desensitizedstateconcentrationO1 First openstateconcentrationO2 Secondopenstateconcentration
Joneset al., 1998;Mozrzymaset al., 1999). Thesearerepresentedin the seven
statemodelshown in Table1. Qazi et alia assertthat the seven statemodel is
theminimumnecessaryto predictGABA responses(Qazi& Trimmer, 1999). It
is a modelwhich hasbeenwell establishedin the literature(Qazi et al., 1998;
Jonesetal., 1998; Jones& Westbrook,1996;Shenetal., 2000;Mozrzymasetal.,
7
Table3: GABA reactionschemeequations
R = Ψ1(koff , 2kon, R(t)) (2)
N = Ψ1(koff , 2kon, R(t)) + Ψ2(2koff , kon, B1(t)) + pDs(t)N(t) (3)
B1 = −Ψ1(koff , 2kon, R(t)) + Ψ2(2koff , kon, B1(t)) − F1(t) − G1s(t) (4)
B2 = −Ψ2(2koff , kon, B1(t)) − G2f (t) − F2(t) (5)
Df = H(t) + G2f (t) (6)
Ds = −H(t) + G1s(t) (7)
O1 = F1(t) (8)
O2 = F2(t) (9)
Ψj(k1, k2, f(t)) := k1Bj(t) − k2f(t)N(t) (10)
Fj(t) := bjBj(t) − ajOj(t) (11)
Gj`(t) := djBj(t) − rjD`(t) (12)
H(t) := qDs(t)N(t) − pDf (t) (13)
1999).
TheGABA reactionschemeis modelledby thesystemof ODE (2) - (13) in
Table3, andthemodelparametersaregivenin Table4. At this stage,thereis no
a priori reasonto suspectthat thesystemis stiff, thoughit seemslikely thatdue
to the numberof processespresent,somemay occuron fastertime-scalesthan
others.Wediscussthenumericalsimulationsof thismodelin Section4.
8
Table4: GABA reactionschemeparameters
a1 = 1100 sec−1 b1 = 200 sec−1
a2 = 142 sec−1 b2 = 2500sec−1
r1 = 0.2 sec−1 r2 = 25 sec−1
p = 2 sec−1 q = 0.01 sec−1
d1 = 13 sec−1 d2 = 1250 sec−1
kon = 5 · 106 M−1·sec−1 koff = 131 M−1·sec−1
3 Numerical stiffness
We shallbegin this sectionwith a sampledynamicalsystemwhich exhibits stiff-
ness,andseekto convincetheaudiencethatmerelyusinghigher-orderor adaptive
step-sizealgorithmsdoesnot alleviatethe problem,aslong asthesemethodsare
explicit. This motivates thediscussionon numericalstiffness,andwe describe
well-known fixes for the problem.Readerswho are familiar with the issuesof
stiffnessareencouragedto skip to Section4.
To begin with, we areinterestedin computingthesolutiony(x) to thesystem
of ODE
y′(x) = f(x, y(x)), (14)
y(a) = α, x ∈ [a, b].
We notethattheIVP maybea singleequation,or a system.In thelattercasethe
9
solutiony(x) is a vector. In Henrici’s notation(Haireret al., 1993),therecursion
usedto advancethesolutionof (14) from xn to xn+1 is
yn+1 = yn + hnΦ(xn, yn, hn), (15)
whereyn is thenumericalapproximationto theexactsolutiony(xn). Thefunction
Φ is calledtheincrementfunctionof themethod. For example,whenΦ(xn, yn) =
f(xn, y(xn)), thealgorithmobtainedis theusualForwardEuler.
3.1 Stiffness- what is it?
Webegin thissectionwith asimpleexampleto illustratethephenomenonof “stif f-
ness”.
Example1 Consider theIVP
y′ = −10000y − et + 10000, y(0) = 1, t ∈ [0, 5]. (16)
Theexactsolutionof thisproblemis givenby
y(t) = −1
10001et + 1 +
1
10001e−10000t.
10
We seefrom theexpressionabove that thelastexponentialterm 1
10001e−10000t
shouldbecomenegligible in comparisonto 1. Indeed,whent ≈ .072, this term
contributeslessthan10−15 to theoverall computation.
In orderto comparetherelativeefficiency of variousalgorithmsin theexample
above, we requiredthat therelative errorof thecomputedsolutionstayedwithin
10−6 during the interval [0, 5]. For eachalgorithm,we reducedthe step-sizeor
error toleranceuntil thedesiredaccuracy was achieved. We presentthestep-size
used,aswell asthenumberof time-stepstakento integrateover theinterval [0, 5].
Thealgorithmsusedwere:
1. Forward Euler: The simplestexplicit integratorpossible,low-order, and
fixedstep-length.
2. ODE23: MATLAB’ s adaptivestep-size,second-orderalgorithm
3. RK4: A fixedstep-length,explicit fourth-orderalgorithm
4. ODE45: MATLAB’ s adaptivestep-size,fourth-orderalgorithm
In Table 3.1, we compile someresultsusing variousstandardexplicit numeri-
cal algorithms. All explicit solvers, with or without variablestep-size,work in
the sameway. That is, the computationof the approximatesolutionat xn+1 de-
pends onthevaluesof thesolutionalreadycomputed.Eachexperimentwascon-
11
Table5: Resultsof usingexplicit numericalmethodsonexample1.Thetruesolu-tion at t = 5 is y(5) = .9851601681...
algorithm no. of steps maxstep cputime Y(5)ForwardEuler 100000 5×10−5 339.8 .9851609
ode23 19902 3.803×10−4 67.37 .9851602RK4 50000 1×10−4 91.31 .9851616ode45 60273 1.088×10−4 109.41 .9851602
ductedin MATLAB, on a Linux workstationwith a 1.3GHzAthlon processor.
The cputimesreportedarein seconds.The algorithmswould blow up, yielding
NaN , if thestep-sizesweremuchlarger. We seethat thealgorithmsarechoos-
ing a step-sizesmall enoughto accuratelyresolve the fastesttransient- in other
words,the insignificantterm 1
10001e−10000t governsthechoiceof step-size!!This
is neithera problemof accuracy, nor of adaptivity. Thepoorperformanceof all
thesealgorithmsontheexamplepointsto adeeperissue.Merelyusinghigherand
higherorderalgorithms,evenif they areadaptive,doesnotsuffice; thedynamical
system16 exhibits behavior thatconfoundseachof thesemethods.In particular,
all of thesealgorithmshavestabilityproblems;if thestepsizeis notexcruciatingly
small,thecomputedsolutions“blow up”. Wedescribethis systemasstiff .
Heuristically, oneusefuldefinitionof stiff dynamicalsystemsis: A systemin
whichoneor severalpartsof thesolutionvaryrapidly (asanexponential,e−kx for
example,with k large),while otherpartsof thesolutionvary muchmoreslowly
12
(asan oscillator, or linearly) is characterizedasa stiff system.Thesearepreva-
lent in thestudyof dampedoscillators,chemicalreactionsandelectricalcircuits.
Essentially, the derivativeor partsthereofchangeveryquickly.
Theprecisedefinitionof numericalstiffnessis difficult to state;however, the
phenomenonis onethat is widely observedwhile numericallyapproximatingso-
lutionsof differential equations(asseenabove). For our purposes,we adoptthe
following definitionof stiffnessdueto Lambert(Lambert,1991):
Definition 1 If a numericalmethodwith a finite region of absolutestability, ap-
plied to a systemwith anyinitial conditions,is forcedto usein a certain interval
of integration a steplengthwhich is excessivelysmall in relation to thesmooth-
nessof theexactsolutionin that interval, then thesystemis saidto bestiff in that
interval.
This definition allows us to concedethat the stiffnessof a systemmay vary
overtheinterval of integration.Theterm‘excessively small’ requiresclarification.
In regionswherethefasttransientis still alive,weexpectthesteplengthto remain
small.However, in regionswherethefast(or slow) transientshavedied,wewould
expectthesteplengthto increaseto asizereasonableto the problem.
13
A “cure” for stiffnessis theuseof implicit algorithms.Thesearealgorithms
wheretheapproximationYn+1 to the truesolutiony at tn + h is givenby
Yn+1 = Yn + hΦ(Yn+1, t),
whereΦ is chosenaccordingto the systembeingsolved, the accuracy desired,
etc. Note that the updateYn+1 is now the solution of a (typically) non-linear
system.Thus,eachstepof ouralgorithmrequiresanonlinearsolve,anexpensive
proposition.However, sincethedomain ofstabilityof implicit algorithmsis much
larger thanthat of explicit algorithms,moststate-of-the-artstiff solversemploy
adaptive implicit algorithms.For example,thebuilt-in MATLAB solversode23s
andode23tb areimplicit Runge-Kuttamethodswith variablestepsizecontrol.
Thesavingsin termsof computationaltime is well worth theeffort of solvingthe
nonlinearproblem.To drive this messagehome,let us returnto theexamplewe
beganthissectionwith, anduseacoupleof implicit solvers:
1. Backward Euler : the implicit analogof Forward Euler, this methodre-
quiresafixedstep-size.
2. MATLAB’ s ODE23s: this methodusesan adaptive step-size,and is a
second-orderalgorithm.
14
Table6: Someimplicit algorithmsusedto computeexample1. Comparethesewith similar explicit algorithmsin Table3.1.
algorithm no. of steps maxstep cputime Y(5)BackwardEuler 2500 .002 .11 .9851898
ode23s 179 0.1094 0.61 0.9851601
Table3.1 summarizestheperformanceof theseimplicit algorithms.Despitethe
nonlinearsolve, thesealgorithmsoutperformthosein Table3.1 by a coupleof
orders ofmagnitudebothin termsof numberof time-steps,andcputimetaken.
4 The GABA reactionscheme
Thesenumericalexperimentswereconductedafterwehadcomputationallystud-
iedacompletelyunrelatedsystem,theSantillan-Mackey modelof genetranscrip-
tion (Caberlin,2002;Caberlinetal., 2002).Thelatterwasastiff-delaydifferential
systemof 4 reactants,andwe foundinvariantmanifoldspresentin theunderlying
dynamics.To our surprise,our numericalinvestigationsof the GABA reactions
led usto observe similar invariants.In locatinginvariantmanifoldsin theGABA
system,werelieduponobservationsfrom (Caberlin,2002;Caberlinetal., 2002).
Wetypically rantheGABA modelwith theinitial conditionsin Table7. When
describingexperiments,weshallspecifytheinitial GABA concentration,N0.
15
Table7: Initial conditionsfor theGABA scheme
R0 = 1 × 10−6 MN0 ∈ (0.5, 4096) × 10−6 MAll otherinitial variablessetto zero.
4.1 Stiffnessin the GABA reactionscheme
In an attemptto reproduceFigure 4 of (Qazi & Trimmer, 1999), we first ran
ode45 on the GABA system(2) - (13). We set the initial GABA concentra-
tion to N0 = 5 × 10−7 M and ran theexperimentfor t between0 and1 second.
We foundode45 to be unstable.It startsthe integrationwith a steplengthof
2.5× 10−4s. Neithersettingtheinitial stepsizeto 1.0× 10−4s,nor themaximum
stepsizeto1.0×10−3smakesthesolutionstable.Thiscanbeseenin Figure1 with
theaforementionedmaximumstepsize.With amaximumstepsizeof 5 × 10−4s,
ode45 integratesthe systemwell in 2000 steps. With smallerstepsizes,the
plots aregraphicallyindistinguishable atthis resolution. Note the saturationof
plusesfor theexplicit integratorsin Figure2, evenaftertheinitial transientphase.
This is typical of solving stiff systemsusingexplicit methods. With ode23s,
we find goodconvergencewith themaximumsteplengthsetto 0.01s.This step
lengthis used byode23s throughouttheinterval, completingtheintegrationin
100steps,a twenty-foldimprovementoverode45.
16
Figure1: Instabilityof ode45 runningGABA model.
Invariantmanifoldswerediscoveredin theGABA model.Sincethereareeight
dependentvariables, there are 28 possibletwo dimensionalphasespaceplots
and 56possiblethreedimensionalphasespaceplots. With this alone, it would
be quite tediousto identify invariant manifolds. We noticedthat the time plots
for thevariableswhich becamelinearly relatedon the invariantmanifoldsof the
Santillan-Mackey systemcloselyresembledeachotherqualitatively. Using this
asa startingpoint,we plottedthevariableswhosetime plots lookedqualitatively
similar in theGABA scheme.To our surprise,we found twoinvariantmanifolds
- again describedby linear relationshipsbetweenthe variables. In Figure3 we
seethevariablesB1 andO1 pair up. Therelationshipis givenapproximatelyby
O1 ≈ 0.182B1. Thesolutionreachesthis manifoldon theorder of10−2 seconds.
The variablesB2, O2 andDf alsobecomedependent, asseenin Figure4. The
17
Figure2: GABA system,left: ode45, MaxStep= 5 × 10−4s,2000steps.right: ode23s, varyingMaxStep.
relationshiphereis givenapproximatelybyDf ≈ 48.4B2 ≈ 2.73O2. Thesolution
reachesthismanifoldontheorder oftenthsof seconds(around0.3s- 0.5s).In all,
the systemgoesfrom eight dimensionsto seven on the order of1/100seconds,
andthendown to dimensionfive, tenthsof secondslater. An exhaustive search
in the remainingvariablesrevealsthat thesearetheonly invariantmanifoldsthe
GABA reactionsystem.
As for theSantillan-Mackey model,theseinvariantmanifoldobservationspro-
videakey experimentaltestof themodel.
18
Figure3: left: InvariantB1-O1 manifold. right: Rotationin theB1O1-plane.
4.2 Roleof desensitizationin generatinginhibitory curr ents.
Therateatwhich thereceptorentersthedesensitization statewill affect theshape
of inhibitory currents,andthis may be a meansby which endogenousenzymes
regulate receptoractivation (Jones& Westbrook,1997; Bianchi et al., 2001).
Desensitizationtendsto prolong the inhibitory currentand keepsthe transmit-
19
Figure4: top: InvariantB2-O2-Df manifold.bottoml: Rotationin Df O2-plane.bottomr: Rotationin B2O2-plane.
ter in the boundstateof the receptor. This would be expectedto affect output
of neuronalcircuits. GABA inhibitory currentsareknown to beimportantin the
generationof synchronizedoscillationsin thecortex, thalamusandhippocampus
(Connors& Amital, 1997; Staak& Pape,2001). Low frequency rhythmsin the
neocortexrangefrom 5-10 Hz (0.1-0.2s) coincidingwith the appearanceof the
20
lower dimensionalmanifoldsin the GABA reactionscheme.Thelinearcoupling
of the bound,desensitizedand openstatesof the receptorbetween0.1 - 0.5 s
occursataphysiologicallyrelevanttimescaleto affect timing of neuralcircuit ac-
tivity. This suggeststhatsimplerelationshipsmayexist betweenthestatesof the
receptorat low frequenciesof receptoractivation. Thenumberof channelopen-
ingswill bedirectly proportionalto theamount oftransmittertraversingthrough
the boundand desensitizationstates. So, if the transmitteris rapidly removed
someof it will still be boundto the receptorflipping throughbound,openand
desensitizedstates,andthe amount bufferedwill be directly proportionalto the
receptorin the openstatejust prior to the free GABA removal. Therefore,the
systemwill havesomememoryof theactivity at thepoint freeGABA is removed.
Theexistenceof sucha simple linearbuffer mayhave importantimplicationsfor
the transmissionof informationthroughtheGABA receptor. Sucha buffer may
aid in theroutingof signalsfrom one neuronto another. To investigatethis pos-
sibility we arenow building a modelin which pulsesof GABA aredeliveredto
thereceptor. We will monitor theamount ofGABA bufferedandto seeif this is
relatedto theaccuratetransferof frequency informationthroughthereceptor.
It is experimentallypossibleto deliver high concentrationpulsesof GABA
ontoneuronsandmeasuretheinhibitory potentialsthataregenerated.A rangeof
21
frequenciescould be deliveredto the cell with the GABA receptorswhile mea-
suringthefidelity of theresponseusinginformationtheoreticapproaches.These
responsescanbe comparedto the effect of GABA in the presenceof drugsthat
changethekinetic parametersin theGABA model(9, 12, 16). Thequantitative
relationshipbetweentherelative statesof thereceptorandfrequency transfercan
thenbeusedto assesthe roleof desensitization statein information processing.
5 Acknowledgements
Martin Caberlinand Nilima Nigam gratefully acknowledgethe supportof this
work by theNaturalSciencesandEngineeringCouncilof Canada.MC wassup-
portedthroughaPGS-Agrant,andNN wasfundedby anNSERCAppliedMath-
ematicsgrant. Sanjive Qazigratefullyacknowledgesthe institutionalsupportof
theParkerHughes CancerCenter.
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