ESSAY
- 1. Cybernetic Modeling Arash Habibi-Soureh Graduate Mentor: Frank DeVilbiss Faculty Supervisor: Doraiswami Ramkrishna
- 2. In the 1960’s, JacquesMonod wasstudyingthe growthof unicellularbacteriawhengiven certainsubstrates(food).Whathe found wasthatdependingonthe type of substrate being administeredtothe bacteria,particulargrowthpatternscouldbe predicted.Itwasnotedthatthe bacteriagrewfastestwhenfedglucose,andslowerwhenadministereddifferentsubstratessuchas pyruvate or xylose.Mostinterestinglywaswhatoccurredwhenthis bacteriumwasadministered pyruvate firstandthenglucose;itwasdiscoveredthatthe bacteriawouldsuppressalreadyexisting pyruvate catabolic(digestion) enzymes toproduce catabolicenzymesforglucose!Suchresults suggestedthatbacteriawhenintroducedtoaparticularenvironmentof substrates wouldfollow a digestionpatternthatwouldyieldthe greatestpossible biomassgrowth.Thisbacterial economycanbe seeninour humaneconomyintermsof how besttoutilize apaycheck.Like substrateswe receive moneyanddependingonwhatoccursin ourenvironmentwe have certainfactorsthatdetermineshow we bestutilize ourmoney.UsingMonod’sobservations,we asengineerscanuse mathematicsto model a cell’sbiomassgrowthwhengivencertainsubstrates. From growthdata fromcellularbiomassgrowthonsingle substrates,the implementationof mathematicscanmake modelsthatwill allow ustopredictcellularbiomassgrowthontwoormore differentsubstrates;we call thiscyberneticmodeling. Asapposedtokineticmodeling efforts,which accounts onlyforthe catabolicrates of each substrate,cyberneticmodelingaccountsfor the cellular economydescribesbyMonod. The premise of thismodel is the use of cyberneticvariablesthatwill act as our control variables,thenidentifyingenzymaticsynthesisandbiomassgrowthkinetics whichboth use concentrationandcyberneticvariables.Basedonpreviousexperimental datafromcellulargrowth on multiple substrates,the cyberneticmodel accountsforthe followingassumptions: 1. Givenmultiplesubstrates,onwhichgrowthratesare distinctlydifferent,the microbespreferto grow onthe fastestsubstrate withthe growthcurve showingmutiauxial behavior 2. Growth behaviorrangesfromsimultaneousutilizationof multiplesubstrates(whengrowth ratesare nearlythe same) tosequential utilizationwithintermediatelagperiods 3. The growth rate on a mixture of substratesisnevergreaterthanthe maximumof the growth rateson individualsubstrates 4. While growingonsingle substrate,if afastersubstrate isadded,the microbesinhibitthe activity of alreadyavailableenzymesforthe slower substrate
- 3. The followingparametervalues are alsoaccountedfor: Mu max isthe maximum specificgrowth fromaparticularsubstrate,Kare Michaelisconstantsand are basedon the concentrationof substrate,Yisthe total drybiomassweight,alphais the enzymaticsynthesisrate constant,andbetais the enzymaticdecay rate constant;alphaand beta are assumedtobe the same forall keyenzymes. The growthrate parametercan be representedas . It can be notedhere thatall parametersinthe cyberneticmodel are basedon data fromcellulargrowthona single substrate. The basic ideaof the model isthat a particularsubstrate will be utilizedbythe biomasswhen catalyzedbya keyenzyme (denotedbyS,B, andE, respectively) ascanbe seenfromthe given reactionequation: .The rate equationforthisreaction sequence canbe givenas: , where c isthe biomassconcentration,sisthe concentrationof the substrate,ande is the specificlevel of enzyme. By incorporatingthe effectof dilutionof the specificenzyme leveldue tobiomassgrowthand the constant proteindecayinthe cells,the rate equationfore can be givenby: . In thisrate equation,the cyberneticvariable u representsthe control actionsof the cellular regulationmechanismsof catobolicrepressionand induction;likewise,the mechanismsof catabolicinhibitionandactivationthatcontrolsthe activities of existingenzymesare representedbythe cyberneticvariable v. The optimal allocationof
- 4. resourcestoan alternative isproportionaltothe amountof returnsobtainedfromthatalternative, therebythe fractional allocationof limitedresourcesis: .Furthermore,the actual growthrate that the cell isexpectedtoexperience includingcatabolicinhibitionscanbe givenby: . The rate by whichsubstratesare actuallyutilizedbythe cell maybe represented as: . The rate of total biomassgrowthmaybe representedas: . The rates of u and v as well asthose of e, s,and c constitute the cyberneticmodel forcellulargrowth on multiple substartes. Whenthe ratesof e,s, and c are solvedinMATLABusingordinary differentialequation(ODE) solvers,the rate bywhichourreactiontakesplace (r) can thenbe calculatedandmodeled. As can be seenbythe graph on the leftof cellulargrowthonglucose andxylose,K.Oxytoca will utilize glucosefirst asitisthe substrate the cell utilizesthe quickest.Asthe cellulareconomy experiences“diminishingreturns”fromthe glucose,enzymesforthe utilizationof xylosewillbegin as describedbythe diauxie phenomenon,thisisshowninthe middleof the graphbythe hump knownas the diauxie lagphase.Afterthe diauxielagphase will the presence of utilizationbymeans of xylose will be apparent. The same canbe describedforthe graphon the right representing cellulargrowthof K.Oxytocaon three substrates,butinsteadof one lagphase there will be two.
- 5. We can furtherenhance ourmodelingcapabilitiesbyutilizingthe hybridcyberneticmodel.The hybridcyberneticmodel uses biologicalinformationof anorganismandmathematicstopredict metabolicbehavior.If we assume steady-stateconditions,flux-basedapproachessuchas elementaryflux modescanbe implementedtopredictall cellularmetabolicfluxesof interest,such as biomassgrowth,products,wastes,andsoforth. Furthermore,measurementsbasedonaccessible fluxescanbe usedto calculate forfluxesthatare not. From the steady-state intracellular metabolite assumption,we canrepresentmetabolismalgebraicallyas: ,where Sis the full stoichiometricmatrix,wisthe flux vectorof n componentsforall reactionsconsideredin metabolism, andbrepresentsaflux exchange vector. Each row vectorof S isassociatedwiththe balance onan intracellularmetabolite; eachcolumn vectorrepresentsstoichiometriccoefficientsof speciesparticipatinginaspecificreactionwitha positive signforproductsanda negative signforreactantsanda 0 for nonparticipants.Since we can use measurements of external components toestimate externalfluxes,bymassconservation, internal componentsof anorganismmustbe enforcedbyapseudo-steadystate approximationso that all internal fluxeswhichare coupledstoichiometricallytothe externalfluxescanbe estimated. Mathematicallythispseudo-steadystate approximationisgivenas: ,where Smis the stoichiometricmatrix representingonlythe intracellularspecies;inthe Smatrix shownabove,Sm will be all rowsnotencompassedbythe redbox.It shouldbe notedthatsuch a pseudo-steadystate approximationassumesthatdilutionof metabolitesdue tobiomassgrowthcanbe neglected comparedto othertermsinthe massbalance.
- 6. The vector of extracellularvariablesisgivenby: ,wheresisa vector of n componentsof substrates,pisthe vectorof n fermentationproducts,andcis biomass,all expressed inconcentrationperunitvolume of culture;intracellularvariablesmaybe representedwitham matrix of n componentsinunitsof amountpercell.By substitutingoursteady-state equation,a general dynamicmodel canrepresentthe reactionrate vectorr: . Usingthe pseudo- steadystate approximationoninternalmetabolites,we have: .To thisend,if we introduce amatrix Px representingthe projectionof the flux vector rontothe vectorrx of exchange fluxes,we candescribe ourdynamicmodel withthe equation: ,where rrepresents the regulatedfluxes. In orderto model more detailedmetabolicnetworkstopredictmostfermentationbyproducts throughmetabolism, the cyberneticmodel frameworkhasbeenadaptedfrompreviousmodels.By requiringidentificationof metabolicsubunitseachcomprisingelementarypathways,the subunits are to compete forthe total resourcesavailable forenzyme synthesis.Thisconceptof elementary flux modesreferstovariousroutinesinthe networkthroughwhichexternal substrateinconverted intovariousfermentationproductsandbiomassgrowth. Furthermore,byrankingthe various elementaryflux modestobe includedinthe metabolismbythe relative weightsof the global cyberneticvariables,abiochemicalpathwayfromthe substratestothe downstreamproductscan be shown.Inthis model,the global cyberneticvariables,basedonaglobal objective,will provide the relative weightsof the differentelementarymodes. In the hybridmodel,we use the elementarymode decompositiontoexpressthe reactionrate vectorby: , where Zis the matrix containingnrowseach representingan elementarymode vectorrelatingreactionsinthe mode tothe uptake rate of substrate throughthat mode.
- 7. By the steady-state assumption,the regulatedreactionrate vectorisexpressedthroughthe regulateduptake vectorrm.The cyberneticvariablesuandv are formulatedtomaximizethe overall glucose uptake rate througheachmode by meansof matchingand proportional laws: The differential equationof state vectorx can be givenas: , where V is the diagonal matrix of v,the uptake rate vectorrm featuresnenzymeswhose levelsare includedin the vectorem. The differentialequationforsuchenzymesare givenas: , where alphaisthe constantfor enzyme synthesis,betaisenzyme degradationrate,andrgis the sum of the growth ratesthroughall modes: . In orderto completelydefine ourhybridcyberneticmodel,nextwe mustaccount for the kineticexpressionsforthe substrate uptake ratesandthe enzyme synthesis rates.
- 8. The reactionrate of each elementaryflux mode will be afunctionof glucose concentrationand itsenzyme level.The numberof reactionratesisthe same as the numberof total elementaryflux modes,nz. , where Kisthe saturationconstant,e is the enzyme levelforthe i-thelementaryflux mode,andkmax isthe maximumuptake rate of the i- th elementaryflux mode.Forthe reducednetwork, itshouldbe mentionedthatrm9will be since itis solelydependentonthe single reactionrate of formate decomposition.The enzymesynthesisreactionforthe i-thelementaryflux mode,re isalsoonlya functionof glucose: ,where KandKe are given. All initial conditionsof extracellularmetabolitesare measured,all growth-associatedenzyme levelsare setat .9, and all maintenance-associatedenzymelevelsare setat .8. As can be seen,the hybridcyberneticmodel performedwellwithrespectto experimental observations.Thiscanbe justifiedbasedonthe qualityof fitwiththe model andexperimental data.
- 9. Sometimestreatingalarge-sizednetworkwiththe hybridcyberneticmodel resultsin overparameterization.The use of lumpedhybridcyberneticmodelingisanadaptationof cybernetic modelingthatcansolve suchissueswithonlylimiteddataavailable.Lumpedhybridcyberneticmodeling classifiesdifferentelementarymodesintodifferentfamiliesbasedonparticularcriteria, suchaswhat substrate(s) will be consumedthrougheachfamily.The elementarymodesare thenlumpedwithineach family,andthe lumpedelementarymodesare thenincorporatedintothe hybridcyberneticmodel framework.Thereby,the uptakeflux issplitamonglumpedelementarymodesinsteadof wouldbe individualelementarymodes. Itisassumedthatfluxesthroughindividualelementarymodes isa functionof structural return-on-investment(sROI) whichinturnisproportional tothe yieldof vital productssuch as biomassandATP. The dynamicmass balance of extracellularmetabolitesisgivenas: ,where x is the vectorof n concentrationsof extracellularcomponentsincludingbiomass(c),Sx isthe (nx x nr) stoichiometricmatrix,andris the vectorof nr intracellularandexchange fluxes.Byapplicationof quasi- steady-state approximations,the flux vectorcanbe representedbyaconvex combinationof elementary modesgivenby: , where Zf isa (nr x nf) lumpedelementarymode matrix andrf isthe vector of nf uptake fluxesthroughlumpedelementarymodes;eachcolumnof Zf isassumedtobe normalized withrespecttoa reference substratesothe rf impliesuptake fluxesthroughlumpedelementarymodes. The uptake flux throughthe Jthfamilyof elementarymodescanbe describedas: , where the subscriptJindicates the index of elementarymode family,vf isacyberneticvariable controllingenzymeactivity,ef isrelativeenzyme level,andrf isthe unregulatedflux termgivenby kinetics.Relativeenzymelevel,ef,isgivenby: .Efj isdeterminedbythe differential equation: ,where the fourright-handtermsare constitutive synthesisrate,inducible synthesisrate,degradationrate, and dilutionrate bygrowth, respectively,uf isthe cyberneticvariableregulatingthe inductionof enzymesynthesis,andrfe isthe kineticinducible enzymesynthesisrate.The maximumlevel atsteady-state isgivenby:
- 10. , where Ybis biomassyield,rf isthe maximumspecificuptake rate,and rfe is the maximuminducible enzymesynthesisrate of the Jth lumpedelementarymode.Bymeansof matchingand proportional laws,the cyberneticvariablesuandv are computedby: , where pj isthe return-on-investment(ROI),oftentakenas carbon uptake flux orgrowthrate. Computationof lumpedelementarymodescanbe illuminatedthroughthe givenflowchart:
- 11. The computationof elementarymodesismuchthe same wayas in the hybridcyberneticmodelinthe beginning.However,the questionof vital productsiswhatwill separate the formermethodfromthe lumpedelementarymode method.Withthismethod,the subdivisionof elemenarymodesintobiomass producingmodesandATPproducingmodesare classifiedandthenlumpedtoproduce ourlumped modes. Furthermore, tuningparameters are introduced intothe formulationthatuse dynamic experimental datato distill outthe significantelementarymodes inafamilythatwere obscuredby lumping. The tuningtaskisorganizedasfollows: 1. In the absence of yieldorflux data,all coefficientsare settozero.Underthissetting,new formulationscomute lumpedelementarymodesfromstoichiometryalone. 2. The coeffecientstobe tunedare selecteddependingonavalable experiemental data 3. In tuningthe coefficientsthe weightsof elementarymodesintheirlumpingare notnegative By thistuningmanner,the errorbetweenmodelestimationandmeasureddatashall be minimized; thiswill onlystrenghthenourmodel predictions. The lumpedhybridcyberneticmodel inthisresearchwasdone withthe anaerobicgrowthof E. coli strains.For a systemof a single lumpedelementarymode,model equationsare givenby: , where xgisthe concentrationof glucose,xi isthe i- the extracellularmetabolite,andYi andis yield.Furthermore: , where biomass(c) istreatedasan extracellularmetabolite.Otheroarametersare fixedasseen before: .Since the effectof enzyme synthesisis limitedwithinacertainrange,we can make suchalpha andbeta generalizations;the Michaelis constantK is alsoa fixedvalue asitssensistivityisreflectedineperiementsof low substarte concentrations.AsYi istheorizedtobe givenfromlumpedelementraymodes,the remaining
- 12. parameterkmax isto be optimizedfromdynamicfermentationdata.Itshouldbe notedthat all lumpedhybridcyberneticmodelingwascompletedviaAUMICsoftware,developedatPurdue University. As can be seen,the hybridcyberneticmodel performedwellwithrespecttoexperimental observations.Thiscanbe justifiedbasedonthe qualityof fitwiththe model andexperimental data as was seenwithAUMIC’soptimizationtools. Cyberneticmodelingisnotonlyaninterestingwaytopredictthe behaviorof the growing biological world,butalsohasvital real-worldapplicationsaswell.The applicationsof thisresearch can extendtobiofuels,pharmaceuticals,foodindustry,andthe listgoeson.Notonlyare the uses numerous,butthe technologyisgreenandpossiblyrenewable whichisaverydesirable traitfor future industry.Inthe future the workshall continue toimprove modelingperformance sothatsuch desirable expectationsdescribedabove canbe yielded.
- 13. Works Cited Kim,JinIl,JefferyD.Varner,andDoraiswami Ramkrishna. A Hybrid Modelof AnaerobicE. Coli GJT001: Combination of Elementary Flux Modesand CyberneticVariables.Publication.WestLafayette:Purdue University,2008. Print. Kompala,DhinakarS.,Doraiswami Ramkrishna,NormanB.Jansen,andGeorge T.Tsao. Investigation of Bacterial Growthon Mixed Substrates:ExperimentalEvaluation of CyberneticModels.Publication.West Lafayette:Purdue University,1985.Print. Song,Hyun-Seob,andDoraiswami Ramkrishna. CyberneticModelsBased on Lumped Elementary Modes Accurately Predict Strain-SpecificMetabolicFunction.Publication.WestLafayette:PurdueUniversity, 2010. Print.