QuantitativeUnderstandinginBiologyModuleIV:ODEsLectureI:IntroductiontoODEs

OrdinaryDifferentialEquationsWebeganourexplorationofdynamicsystemswithalookatlineardifferenceequations.Thisisaniceintroductiontodynamicsystemsforafewreasons.First,linearsystemsare“well-behaved”inthattheycanbetreatedanalytically.Second,theuseofdiscretetimestepsindifferenceequationsallowedustoside-stepafairamountofcalculus.

Unfortunately,therealworldisnotsoconvenient,sowe’llneedtogearuptodealwithbothnon-linearsystemsandwithtruedifferentialequations.We’lltrytokeepthenecessarycalculustoaminimum,butsomewillbeneeded.Fortunately,manyoftheoperativeconceptsaresimilar,soalthoughthemathmaybealittlemoreintimidating,youarenowinagoodpositiontounderstandit.

We’llbeginbyconsideringoneofthesimplest,andmostessential,dynamicsystems,andcompareandcontrastittoitsdiscretecounterpart.Let’sreturntoanexamplewelookedatpreviously,thatofgrowthproportionaltocurrentamountorsize.Thediscreteformofthismodelis…

𝑥!!! = 𝜆 ∙ 𝑥!

…andthesolutionis…

𝑥! = 𝑐 ∙ 𝜆!

Toexpresstheideaofarateofgrowthproportionaltocurrentsizeoramountusingdifferentialequations,wewouldwrite…

𝑑𝑥𝑑𝑡

= 𝜆 ∙ 𝑥

Solvingthis(usingcalculus)givesus…

𝑑𝑥𝑥= 𝜆 ∙ 𝑑𝑡

𝑑𝑥𝑥= 𝜆 ∙ 𝑑𝑡

ln 𝑥 = 𝜆 ∙ 𝑡 + 𝐶

…whereCisanarbitraryconstantoftheintegration.

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Thiscanbesimplifiedintotheconventionalform…

𝑥 = 𝑒!∙!!!

𝑥 = 𝑒!∙!𝑒!

𝑥 = 𝑐 ∙ 𝑒!∙!

Inthelaststepwehaveredefinedthearbitraryconstantforourconvenience.

Toreview,wehave

Discrete Continuous𝑥! = 𝑐 ∙ 𝜆! 𝑥 = 𝑐 ∙ 𝑒!∙!

Notethatinbothcasestheevolutionofourstatevariableissomevalueraisedtoapowerthathastodowithtime.Inthediscretecase,theexponentisthenumberoftimestepsinourmodelsimulation;thisisadimensionlesscountingnumber.Inthecontinuouscase,theexponentisa“scaledtime”.Itisthetimeofthesimulation,butin‘unitsof1/λ’.Youwillseethismotifveryofteninthisformandinoneother…

𝑥 = 𝑐 ∙ 𝑒! !

Inthislatterform,τisthe‘timeconstant’ofthesystem.Ithasunitsoftime,justliket,sotheratioisdimensionless.Theratiot/τissometimesreferredtoasthe“reducedtime”.

Notethatthevaluesoftheλsinthediscreetandcontinuouscaseshavedifferentinterpretations.Inthediscretemodel,avalueofλ=1yieldsasteady-statesolution;thesystemneither‘explodes’nor‘fizzles’.Whenλ>1,thesystemwillgrow,andwhen0<λ<1,thesystemwilldecay.

Inthecontinuoustimemodel,theequilibriumconditionisreachedwhenthederivativeiszero,orwhenλ=0.Negativevaluesofλgivedecayingsystemsandpositivevaluesofλgivegrowingsystems.

ExponentialDecayTheexponentialdecayisprobablythemostfundamentalmotifyou’llseewhenyoustudydifferentialequationsanddynamicsystemsingeneral.Manydynamicsystemsinnaturethatreachanequilibriumapproachitbyanexponentialdecay.Toemphasizethenatureofthedecayweusuallywrite…

𝑥 = 𝑥! ∙ 𝑒!! !

…ratherthantakingτtobenegative.

Asabove,weoftenwriteexponentialswithatimeconstantτ.τmakesourexponentdimensionless,anditisinterpretedasspecifyinghowlongittakestoreachequilibriuminoursystem.Analternativemeasureofthisthatyoumaybefamiliarwithisthehalf-life,t1/2.Ahalf-lifeandatimeconstantrepresentthesamecharacteristicofadynamicsystem,andonecaneasilyinterconvertthem.

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Asanexampleofhowhalf-lifeandtimeconstantarerelated,we’llrevisitproteindynamics.Let’sconsiderasystemwhereweonlyareconcernedwithproteindegradationandnotproteinproduction.Wesaythattheproteinconcentrationdecaysexponentially,whichmeansthattherateofdegradationisproportionaltotheamountofproteinpresent.Thisisareasonablemodelingassumptioninmanycontexts.Wecanwrite…

𝑑𝑝𝑑𝑡

= −1𝜏𝑝

…andthesolutionis…

𝑝 = 𝑝!𝑒!!!

Arecentpapercalculatedproteindecayratesforahundredproteinsinmammaliancells(Edenetal.Nature,2011).Theyfoundintheirsystemthattheaverageproteinhalf-lifewas9hours.Let’sfindtheτforthishalf-life.Thisoccurswhentheproteinconcentrationis½ofitsinitialvalue,orwhenp=½p0

12𝑝! = 𝑝!𝑒

!!! !!

12= 𝑒!

!! !!

ln12

= −𝑡! !

𝜏

𝜏 = −𝑡! !

ln 12

Inourcaset1/2=9hrs,sowehave…

𝜏 = −9 ℎ𝑟𝑠

ln 12

= 12.98 ℎ𝑟𝑠 ≅ 13 ℎ𝑟𝑠

Nowthatyouseethatτisjustalineartransformationofthehalf-life.Itisworthgettingfamiliarwithhowquicklyexponentialdecayoccurswithrespecttoτ.Ifweplotreducedxversusreducedtime…

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t/τ x/x0

0 1

0.693112

1.098613

2 0.1353

3 0.0498

…wecanobservethat:

• Thedecayfallstohalfitsinitialvalueinabout0.7timeconstants.• Thedecayfallstoone-thirdofitsinitialvalueinabout1.1timeconstants.• Intwotimeconstants,thevaluehasdecayedtoabout13%ofitsinitialvalue.• Inthreetimeconstants,thevaluehasdecayedtoabout5%ofitsinitialvalue.

Notethatanexponentialdecaynever“finishes”asitasymptoticallyapproacheszero.However,asitisusefultothinkaboutdynamicevents,wecanmakeanarbitrarydefinitionabouthowmuchofadecayhastohavebeenachievedbeforeaneventis‘over’.Ifwedefinetheendofaneventasthetimewhenitreaches95%recoverytotheequilibriumvalue,weseethateventstakeaboutthreetimeconstants.Ifyouwanttouseadifferentcutoff,thatisOK.Whatisimportantisthatthelengthofaneventisontheorderofafewtimeconstants.Forexample,ifyoudefineyourcutoffas99.44%recovery,you’llfindthateventslast5.2timeconstants.Thislineofreasoningleadsustotheveryimportantrealizationthatτtellsus“howlongstufftakestohappeninoursystem”.

Notethatexponentialdecaysdon’talwaysdecaytowardszero,anddon’talwaysdecayfromhighervaluestolowervalues.Forexample,considerthesystem

𝑑𝑥𝑑𝑡

= −𝜆 ∙ 𝑥 − 𝑥!

Thegeneralsolutiontothissystemis…

𝑥 = 𝑥! + 𝑐 ∙ 𝑒!!∙!

Afewparticularsolutionstothissystemfordifferentvaluesofx0,c,andtareshowninthefigurebelow.Frominspectionoftheplotonly,youshouldbeabletoestimateτforthered(bottom)andblue(middle)curves.Thegreen(upper)curvemaybealittletrickierbecauseitisdifficulttoestimatebyeyewhatvaluethecurveisasymptoticallyapproaching;howmightyoureliablydetermineτforthiscase?[Ans:fromtoptobottom,τ=2,0.5,1].

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FormulatingSystemsofODEsTheartofmodelingdynamicsystemsusuallybeginswithadescriptionofasystemandatransformationofthatdescriptionintoasetofdifferentialequations.Ifthesystemconsidersspatialvariabilityinadditiontotemporalvariability,youwillusuallyendupwithasetofpartialdifferentialequations.Forthissectionoftheclasswewillfocusonsystemsthatdonotconsiderspatialvariability,andthereforewillresultinordinarydifferentialequations.

Let’sconsideranexampleofhowasystemofODEscanbeformulatedfromasystemdescription.We’llreturntoourbasicenzyme-substratesystem:

𝐸 + 𝑆!!,!!! 𝐸𝑆

!! 𝐸 + 𝑃

Wecanwritethedifferentialequationofeachspeciesbysimpleapplicationofthelawofmassaction.

𝑑[𝐸]𝑑𝑡

= −𝑘! 𝐸 𝑆 + 𝑘!! 𝐸𝑆 + 𝑘![𝐸𝑆]

𝑑[𝑆]𝑑𝑡

= −𝑘! 𝐸 𝑆 + 𝑘!! 𝐸𝑆

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𝑑 𝐸𝑆𝑑𝑡

= 𝑘! 𝐸 𝑆 − 𝑘!! 𝐸𝑆 − 𝑘![𝐸𝑆]

𝑑 𝑃𝑑𝑡

= 𝑘![𝐸𝑆]

Weshouldnotethattheabovedifferentialequationsassumethesystemisatconstantvolume.Asthisisnotachemistryclass,wewilljustacceptthis.

Tosimulatetheabovesystem,weneedthemodelparameters.Inthiscase,therearethree:k1,k-1,andk2.Wealsoneedinitialconditionsforthestatevariables.Inthiscase,fourwouldbeneeded:[E]0,[S]0,[ES]0,and[P]0.

RunningaSimulation:BasicTheoryThemostcommonwayofthinkingaboutrunningasimulationofasystemofODEsistheEulermethod.We“discretize”thesystembyturningthedifferentialequationsintothedifferenceequationanalogs.Thediscreteanalogofthelastequationinoursystemwouldbe…

Δ 𝑃Δ𝑡

= 𝑘! ⋅ [𝐸𝑆]

𝑃 !!! − 𝑃 ! = 𝑘! ⋅ 𝐸𝑆 ! ⋅ Δ𝑡

𝑃 !!! = 𝑃 ! + 𝑘! ⋅ 𝐸𝑆 ! ⋅ Δ𝑡

Asimilartechniquecanbeappliedtotheotherequationsinthesystem.Thesedifferenceequationscanbesolvedquitestraightforwardlybysimpleiteration.Ifweareluckyandtheresultantsystemofequationsislinearandcanbecastinmatrixform,wecanbringthetechniqueswelearnedintheprevioussectiontobear.Unfortunately,thisisnotthecaseinthissystem.Thesecondequationdiscretizesasfollows:

Δ 𝑆Δ𝑡

= −𝑘! 𝐸 𝑆 + 𝑘!![𝐸𝑆]

𝑆 !!! = 𝑆 ! − 𝑘! 𝐸 ! S ! ⋅ Δ𝑡 + 𝑘!! 𝐸𝑆 ! ⋅ Δ𝑡

Sadly,theproductofthetwostatevariablesprecludesalinear,matrixformulationofthesystem.

Oncewehavediscretizedourdifferentialequations,wecanchooseasufficientlysmallΔtanduseacomputertoiterateovertime.

Thismethod,elegantandsimpleasitis,hasapitfall.Thedevilisinthephrase‘sufficientlysmall’.Oftensufficientlysmallmeansreally,reallysmall.Sometimesitisimpracticallysmall.Thebiggestimpactofthisisoftencomputertime;youmayneedlotsofstepstoiterateyoursystem.However,toosmallcangetyouintotroublewiththefloatingpointaccuracyofyourcomputertoo.Additionally,thesimpleEulerintegrationmethodcanintroducesystematicerrors.

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Toseetheproblem,let’sconsidertheexampleofasimpleexponential

𝑑𝑥𝑑𝑡

= 1 − 𝑥

Inthiscaseweknowtheanalyticalsolutiontobe(forthecasewherex0=0)

𝑥 = 1 − 𝑒!!

Thisisanexponentialapproachtox=1.Thediscreteversionofthedifferentialequationis

𝑥!!! = 𝑥! 1 − Δ𝑡 + Δ𝑡

Itiseasytoshowthatforeverystep,nomatterhowsmallΔtis,thisequationwillalwaysover-predictthechangeinx.

ItcanbeshownthatEulerintegrationhaserrorsthatareO(Δt2).Thus,togettoa“reasonable”answer,onemayneedtochooseaverysmallΔt.

RunningaSimulation:PracticalTheoryTherearemanyschemesthattrytofixtheproblem.Theso-calledpredictor-correctormethodsuseanEuler-likemethodtopredictthenewvalueofx.Thederivativeisthenevaluatedatthenewvalue,andisusedtocorrectthepredictioninsomemanner.

ProbablythemostimportantpracticalmethodforintegratingsystemsofODEsisthefourthorderRunge-Kuttamethod;thisisoftenabbreviatedasRK4.ItcanbeshownthattheRK4methodhasanerrorofO(Δt5).WhilethereareseveralRKalgorithmsofdifferentorders,RK4isthemostcommonlyused.IfsomeonesaystheyareusingtheRunge-Kuttaalgorithmwithoutspecifiyingtheorder,youcanbesuretheyareusingRK4(p<0.0001).TheequationsforRK4aregivenhereforcompletenessonly…

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Anotherimportantenhancementistheideaofvariablestepsize.Theideaisthatifthingsarenotchangingveryquicklyinyoursimulation,youcantakelargerstepsizes.Thiscanbeahugetime-saver.Implementationofthelogictoachievevariablestepsizesisbeyondthescopeofthiscourse.

RunningaSimulation:PracticalMethodsThegoodnewsisthatMatlabimplementsRK4withvariablestepsizesoyoudon’thavetoworryaboutthedetails.ThekeytogettingMatlabtodotheheavyliftingisthatyouneedtowriteaMatlabfunctionthatgivesthederivativesofeachstatevariable.Thisfunctionisstoredinam-file.Forexample,thefollowingm-filewillgivethederivativetoanexponentialapproachtoaspecifiedvaluewithaspecifiedtimeconstant(File:test1_func.m):

function [ dydt ] = test1_func(t, y, y0, tau) %TEST1_FUNC Simple exponential decay for testing RK4 ODE solver % Provides derivatives for an exponential decay to y0 with time constant tau. dydt = [ -(y - y0)/tau ];

Then,inMatlab,withthisonthepath,wewrite…

start = 0; y0 = 5; tau = 1; tspan = [0 10]; [t y] = ode45(@test1_func, tspan, start, [], y0, tau); plot(t,y)

Itisalsoinstructivetoinspecttheoutputarraystandy.You’llseethatthetimestepisnotuniform.

Warning:plot(y)willgiveyouerroneousresults!

Amorecomplexexample–ourMichaelis-Mentenmodel(file:mm.m):

function [ dydt ] = mm( t, y, k1, k_1, k2 ) %MM derivatives for Michaelis Menten kinetics % MM kinetics: E + S <----k1,k_1----> ES --k2--> E + P % state vars are: y(1) = S, y(2) = E, y(3) = ES, y(4) = P % Fetch variables S = y(1); E = y(2); ES = y(3); P = y(4); dydt = [ -k1 * S * E + k_1 * ES; -k1 * S * E + k_1 * ES + k2 * ES; k1 * S * E - k_1 * ES - k2 * ES; k2 * ES ];

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AndtheninMatlab:

start = [1, 0.1, 0, 0]; k1 = 0.1; k_1 = 0.1; k2 = 0.3; tspan = [0 1000]; [t, y] = ode45(@mm, tspan, start, [], k1, k_1, k2); plot(t,y); hold on; legend('S', 'E', 'ES', 'P'); Observehowquicklythecomplexformsandreachesa“quasi-equilibrium”concentration.Thenobservethattheproductformswithamuchslowertimeconstant.Imaginehowmanysimulationstepsyouwouldneedifthewholesimulationwererunwithatimestepnecessarytogetthefirstbitofthesimulationcorrect.