Mechanistic, multi-scale model to improve treatment of anemia in chronic kidney diseas

34 Pages·2017·6.95 MB·English

RESEARCHARTICLE
Whole-body iron transport and metabolism:
Mechanistic, multi-scale model to improve
treatment of anemia in chronic kidney disease
JoydeepSarkar1¤a,AlkaA.Potdar2¤b,GeraldM.Saidel3*
1 PricewaterhouseCoopersLLP,NewYork,NY,UnitedStatesofAmerica,2 DepartmentofBiomedical
Engineering,CaseWesternReserveUniversity,Cleveland,OH,UnitedStatesofAmerica,3 Departmentof
BiomedicalEngineering,CaseWesternReserveUniversity,Cleveland,OH,UnitedStatesofAmerica
a1111111111 ¤a Currentaddress:HolmuskUSA,Durham,NC,UnitedStatesofAmerica
a1111111111 ¤b Currentaddress:F.WidjajaFoundationInflammatoryBowel&ImmunobiologyResearchInstitute,
DepartmentofMedicine,Cedars-SinaiMedicalCenter,LosAngeles,CA,UnitedStatesofAmerica
a1111111111
*[email protected]
a1111111111
a1111111111
Abstract
Ironplaysvitalrolesinthehumanbodyincludingenzymaticprocesses,oxygen-transport
OPENACCESS viahemoglobinandimmuneresponse.Ironmetabolismischaracterizedby~95%recycling
andminorreplenishmentthroughdiet.Anemiaofchronickidneydisease(CKD)ischarac-
Citation:SarkarJ,PotdarAA,SaidelGM(2018)
Whole-bodyirontransportandmetabolism: terizedbyalackofsynthesisoferythropoietinleadingtoreducedredbloodcell(RBC)for-
Mechanistic,multi-scalemodeltoimprove mationandaberrantironrecycling.TreatmentofCKDanemiaaimstonormalizeRBCcount
treatmentofanemiainchronickidneydisease.
andserumhemoglobin.Clinically,thevariousfluxesofirontransportandaccumulationare
PLoSComputBiol14(4):e1006060.https://doi.
org/10.1371/journal.pcbi.1006060 notmeasuredsothatchangesduringdisease(e.g.,CKD)andtreatmentareunknown.
Unwantedironaccumulationinpatientsisknowntoleadtoadverseeffects.Currentwhole-
Editor:DanielA.Beard,UniversityofMichigan,
UNITEDSTATES bodymodelslackthemechanisticdetailsofirontransportrelatedtoRBCmaturation,trans-
ferrin(TfandTfR)dynamicsandassumepassiveironeffluxfrommacrophages.Hence,
Received:March31,2017
theyarenotpredictiveofwhole-bodyirondynamicsandcannotbeusedtodesignindividual-
Accepted:February27,2018
izedpatienttreatment.Forprediction,wedevelopedamechanistic,multi-scalecomputa-
Published:April16,2018 tionalmodelofwhole-bodyironmetabolismincorporatingfourcompartmentscontaining
Copyright:©2018Sarkaretal.Thisisanopen majorpoolsofironandRBCgenerationprocess.Themodelaccountsformultipleformsof
accessarticledistributedunderthetermsofthe ironinvivo,mechanismsinvolvedinironuptakeandreleaseandtheirregulation.Further-
CreativeCommonsAttributionLicense,which
more,themodelisinterfacedwithdrugpharmacokineticstoallowsimulationoftreatment
permitsunrestricteduse,distribution,and
reproductioninanymedium,providedtheoriginal dynamics.Wecalibratedourmodelwithexperimentalandclinicaldatafrompeer-reviewed
authorandsourcearecredited. literaturetoreliablysimulateCKDanemiaandtheeffectsofcurrenttreatmentinvolving
DataAvailabilityStatement:Allrelevantdataare combinationofepoietin-alphaandirondextran.Thisinsilicowhole-bodymodelofiron
withinthepaperanditsSupportingInformation metabolismpredictsthatayearoftreatmentcanpotentiallyleadto90%downregulationof
files.Codeforthemathematicalmodelisincluded
ferroportin(FPN)levels,15-foldincreaseinironstoreswithonlya20%increaseinironflux
inthesupplementarymaterial.
fromthereticulo-endothelialsystem(RES).Modelsimulationsquantifiedunmeasurediron
Funding:Theauthorsreceivednospecificfunding
fluxes,previouslyunknowneffectsoftreatmentonFPN-levelandironstoresintheRES.
forthiswork.
Thismechanisticwhole-bodymodelcanbethebasisforfuturestudiesthatincorporateiron
Competinginterests:ThefirstauthorJoydeep
metabolismtogetherwithrelatedclinicalexperiments.Suchanapproachcouldpavethe
SarkarworksasaconsultantwithPricewaterhouse
wayfordevelopmentofeffectivepersonalizedtreatmentofCKDanemia.
Coopers(PwC),buttheworkpresentedinthe
manuscriptisnotrelatedtohisworkatPwC,was
PLOSComputationalBiology|https://doi.org/10.1371/journal.pcbi.1006060 April16,2018 1/34
Computationalmodelofanemiaofchronickidneydisease
doneonpersonaltime,anddoesnotinvolveany
financialorotherformsofconflictofinterest.
Authorsummary
Ironismostlyrecycledwithverylittlelossorreplenishment.Thismodelsimulatesthe
complexregulatorynetworkthatmaintainsironwithinhealthylimits.Irondiseasesare
typicallycharacterizedbybreakdownofsuchregulatorypathways(e.g.synthesisoferyth-
ropoietin,inhibitionofironreleasebyhepcidin,etc.)thatleadtopathologicallackofiron
ordepositionofiron.Mostironmetabolismresearchfocusesonthespecificrolesofthe
regulatoryproteins(e.g.HFE,ceruloplasmin),butherewechosetofocusontheinterplay
betweenferroportinandhepcidin.Thecurrentliteraturelacksanintegratedwhole-body
viewofironmetabolismwithkeyfluxesthatareessentialforinvestigatingtherolesofreg-
ulatoryproteinwithinfeedbacknetworksandmolecularpathways.Theseaspectscanbe
investigatedbysimulationswithourtop-down,mechanisticcomputationalmodel.Our
modelsimulationssuggestpossibleimprovementsintreatmentofanemiaofchronickid-
neydisease.Furthermore,thismodelcanprovideaplatformforfuturedevelopmentsof
powerfulpredictivetoolsthatcanbeusedtoacceleratedrugdevelopmentofiron-disorder
diseases.
Introduction
Ironisessentialforawidevarietyofbiologicalfunctions.Itsmostcriticalphysiologicalfunc-
tionisassociatedwiththeoxygencarryingcapacityofhemoglobin.Ironalsoplaysimportant
rolesinmitochondrialredoxreactions(cytochromes)andinhealthyimmunefunction[1,2].
Whilethereisalargefocusonthemultitudeofdiseasesimplicatedduetoirondeficiency,
excessiveironisalsoverytoxic[1].Hence,ironlevelsinthebodymustbeverytightly
regulated.
Inanadultmaleindevelopedcountries,thebloodironlevelis4-5gofwhich50–60%is
associatedwithhemoglobin.Mostironisstoredintheliver,spleenandotherorgansinferric
(Fe3+)formboundtothestorageprotein,ferritin(FN)[2,3].Circulatinginplasmaisasmall
amount(3–4mg)ofFe3+boundtotheprotein,apo-(Tf).Theliversensesserumironthrough
thetransferrinreceptor2pathwaysandregulatesthesynthesisandsecretionofhepcidininto
blood[4,5].Inbodyfluids,ferrous(Fe2+)ionsarehighlyunstableatphysiologicalpH.Since
Fe2+ionsarehighlyreactive,theymustbetightlycontrolledtopreventdamage.Therefore,
Fe2+isconvertedtoFe3+,whichbindstoironchelators(Tf,FN)exceptatverylowpH,e.g.,
insideendosomes[3].
About25mgofironisrecycledperdayviathehemoglobinsynthesisanddegradationcycle.
Only4–5%ofthisironislostandneedstobereplenishedthroughabsorptionfromdiet[6,7].
Aschematicrepresentationoftherecyclingprocessinironmetabolismhasbeendepictedin
literature[8].Macrophagesofthereticulo-endothelialsystem(RES)degradesenescentred
bloodcells(RBCs)toreleaseironthatistransportedbyTf.Thematuringerythroblastsinthe
bonemarrowthenutilizethisironforincorporationintohemoglobinbeforeenteringblood.
Overthelastdecade,manynewdetailsaboutirontransporthavebeendiscovered.Cerulo-
plasmin(Cp),aferroxidaseassociatedwithmacrophageironrelease[9–12],isessentialin
intestinalirontransport.Hepcidin(Hepc),adefensin[13–16],regulatesironreleasefrom
macrophagesthroughdirectdegradationofferroportin(FPN)[17].Thereisacomplexnet-
workofdifferentenzymesandhormonesthatprovidesanintricatecontrolofironmetabo-
lism.Teasingoutthespecificimportanceofdifferentmoleculesunderdifferentconditionsis
PLOSComputationalBiology|https://doi.org/10.1371/journal.pcbi.1006060 April16,2018 2/34
Computationalmodelofanemiaofchronickidneydisease
verychallenging.Mathematicalmodelingofkeyexperimentaldatacanhelpprovideanswers
tosuchcomplexproblems.
Mathematicalmodelsofirontransporthavebeendevelopedandrefinedsincetheearly
1970’s.Mostofthesemodelsarebasedonferrokineticstudies[18,19]andfocusonflux
changesandrelativeabundanceofironindifferentorgans.Also,mathematicalmodelsofspe-
cificprocesssuchasthemolecularcontrolofHepcsynthesisoritseffectonserumironhave
beendeveloped[20,21].Typically,modelsofwhole-bodyironmetabolismconsiderironina
singlemolecularform,whichispassivelytransportedaccordingtoitsconcentrationgradient.
However,ironreleasefrommacrophagesrequiresfacilitatedtransport[22]ratherthansimple
passivediffusion[23,24].Despitethekeyroleofironinerythropoiesis[25,26],modelshave
notconsidereditinformationofironhemoglobin.Quantitativeunderstandingofironmetab-
olismandrecyclingundernormalanddifferentpathologicalconditionsrequiresmathematical
modelsthatintegratemechanisticdetailsofironuptake,releaseandtransportbetweenthe
majorpools,thedynamicsoftheseprocesses,aswellasthefeedbackregulationmechanisms
controllingthem.Withamulti-scalemodel,processesthatoccuratthemolecularandcellular
levelscanberelatedtoobservedbehaviorsatthetissue-,organ-andwhole-bodylevels.
Recentstudies[27–29]proposemathematicalmodelstobetterpredictthedosingstrategy
ofrecombinanterythropoietin(rEpo)usedtotreatanemiainpatientswithchronickidneydis-
ease(CKD)[30–32].ThecurrentclinicalguidelinesfortreatmentofanemiainCKD[33,34]
alsoincludeadministrationofirondextrantoCKDpatientswhererEpoaloneisnotenough
toimprovethehemoglobinlevels.Evenwithlowtransferrinsaturationandlowserumironin
patientswithCKDanemia,ironuptakethroughthegutdoesnotincreaseandoralironsupple-
mentsaretypicallyineffective[7,33,35].Theoptimalhemoglobinlevelneedstobepersonal-
ized[33].However,currentfocusoncontrolofoverallhemoglobinlevelsdoesnotaccountfor
theimpactoftherapyonironmetabolism,especiallyontherecyclingprocessanddetrimental
effectsofironoverloadindifferenttissues[36,37].
Inthisstudy,wehavedevelopedamulti-scalemodelofironmetabolism,whichintegrates
intracellular,molecularmechanismswithcellularandtissuetransportofiron.Avarietyofper-
turbationscenarioswerecarefullychosentoestimatemodelparametersfromdifferentpartsof
themodel.Consequently,wecansimulatetheimportantclinicaloutputs(e.g.serumiron,total
ironbindingcapacity,serumhemoglobin,RBCcellconcentration)relatedtotherapywhile
simultaneouslyprovidingoutputofchangesintransportfluxesandintracellularspeciesrelated
toironmetabolism.OfspecialclinicalsignificanceisourmodelsimulationofanemiainCKD
patientswithinsufficienterythropoietinandtreatmentswithrEpoandirondextraninfusion.
Moregenerally,thegoalofthismechanisticmathematicalmodelistoinvestigatequantitatively
theresponsesoftheironmetabolismsystemunderdifferentdiseaseconditionsandtreatment
strategiesasaguidefornewerandimprovedtreatments.
Models
Themodelofironmetabolismdevelopedherehasfourscales:whole-body,tissue,cellularand
molecular.Atop-downmodelingmethodologyhasbeenusedtodevelopthismodelproviding
justenoughdetailtosimulatetheinter-tissueironfluxesandchangesduringdiseaseandtreat-
ment.Asystemdiagram(Fig1)showsfourmajortissuecompartments:blood(B),reticular-
endothelialsystem(RES),bonemarrow(BM)andliver(L).Wehaveconsiderederythropoie-
tin(Epo)synthesizedbythekidneysandHepcsynthesizedfromliverasmajorhormonesthat
helpmaintainironhomeostasis.TheBcompartmentconsistsofredbloodcell(RBC)and
plasma(P)phases.Themodelcoversthemajortransportprocessesforironbetweenthese4
compartmentsaswellasthesensingelementsintheliverandkidney.
PLOSComputationalBiology|https://doi.org/10.1371/journal.pcbi.1006060 April16,2018 3/34
Computationalmodelofanemiaofchronickidneydisease
Fig1.Majorcomponentsandprocessesofwhole-bodyironmetabolismincorporatedinthemodel.
https://doi.org/10.1371/journal.pcbi.1006060.g001
Adetailedschematicofthereactionsandtransportprocessesincorporatedinthemodelis
presentedinFig2.IntheRES,wedistinguishintracellular(I),membrane(M),andinterstitial
fluid(ISF)phases.TheBMcompartmentincludescolony-formingprecursorcells(CFUe)that
matureandleadtoerythroblasts(EB).TheLcompartmentinterfaceswiththeBcompartment.
Becausetherateofironlossandreplenishmentbyintestinalabsorptionissmall,thismetabolic
modelfocusesonkeyaspectsofironrecyclingincontrasttopreviousmodelsofironmetabo-
lism[18,20,26,28,38,39].Ourmodelincorporatesdetailedmolecularmechanismsofiron
transport[18]thatdifferentiateactiveandpassivediffusivetransport,aswellasthemajorspe-
ciesoftheironrelevantfortransportandrecycling.Thesemechanisticdetailshelpelucidate
theroleofdifferentproteins,enzymesandhormonesinironhomeostasisanddisease
conditions.
Blood(B)compartment
Inthebloodcompartment,weconsiderRBCdynamicsaswellasiron-relatedmolecular
speciesinRBCandplasma.Inplasma,thesespeciesincludeiron-hemoglobin(HbFe),free
ferriciron(Fe3+),apo-transferrin(Tf),mono-ferrictransferrin(Fe3+Tf),diferrictransferrin
((Fe3+) Tf),ceruloplasmin(Cp),erythropoietin(Epo),andhepcidin(Hepc).
2
Inaconstant-volumeplasmaV ,amassbalanceofmolecularspeciesjthatdiffusesbetween
P
compartmentsandchangesbyreactionrateRj leadstotheplasmaconcentrationCj dynam-
P P
ics:
dCj
V P ¼Jj (cid:0) Jj þV Rj ð1Þ
P dt ISF!P P!EB P P
PLOSComputationalBiology|https://doi.org/10.1371/journal.pcbi.1006060 April16,2018 4/34
Computationalmodelofanemiaofchronickidneydisease
Fig2.Keytransportandreactionprocessesofironmetabolisminvolvingfourmodelcompartments:Blood(B),reticuloendothelialsystem
(RES),bonemarrow(BM)andliver(L).Fourdifferenttypesofarrowsareusedtoconnectcompartmentsandspecieswhichareexplainedinthe
legenddrawninsidethefigure.
https://doi.org/10.1371/journal.pcbi.1006060.g002
PLOSComputationalBiology|https://doi.org/10.1371/journal.pcbi.1006060 April16,2018 5/34
Computationalmodelofanemiaofchronickidneydisease
Here,theISF-plasmadiffusionfluxis
Jj ¼hj ðCj (cid:0) CjÞ j¼Tf;ðFe3þÞTf;ðFe3þÞ Tf;Cp;Fe3þ ð2Þ
ISF!P ISF!P ISF P 2
andtheplasma-erythroblast(EB)diffusionfluxis
Jj ¼hj ðCj (cid:0) Cj Þ j¼Tf;ðFe3þÞTf;ðFe3þÞ Tf ð3Þ
P!EB P!EB P EB 2
Wherehj andhj aremasstransportcoefficients;Cj andCj areconcentrationsinISF
ISF(cid:0) P P(cid:0) EB ISF EB
andEB.ExpressionsforRj (j¼Fe3þ;Tf;ðFe3þÞTf;ðFe3þÞ Tf areprovidedintheTable1.The
P 2
degradationofFe3+referstonon-specificbindingofserumFe3+toplasma.
TheconcentrationsofEpoandHepcincreaseinplasmabyendogenoussynthesisand
decreasebynaturaldegradation:
dCEpo CEpo dCHepc CHepc
P ¼TEpo (cid:0) P ; P ¼THepc(cid:0) P ð4Þ
dt K!P tEpo dt L!P tHepc
whereTEpo andTHepcaretheinputratesfromsynthesisofEpobykidneysandHepcbyliver,
K!P L!P
whichhavebeendiscussedindetaillater(sectiononErythropoietinandHepcidinInputs).
ThesemolecularspecieshaveaveragedecaytimesτEpoandτHepc,respectively.
TheRBCnumberperplasmavolume(N )increasesfromEBdifferentiationand
RBC
decreasesbymacrophagephagocytosisrepresentedbyacellnumberdensitybalance:
dN k
RBC ¼ RBC EBN (cid:0) d N ð5Þ
dt V EB RES RBC RBC
P
whereN istheEBnumberinthebonemarrowcompartment,k isthedifferentiation
EB RBC EB
ratecoefficient.ThedeathratecoefficientbyRESphagocytosisd =1/τ isthe
RES RBC RBC
inverseoftheaverageRBClife-spaninplasma.
AssociatedwiththeRBCisHbFe3+,whoseconcentration(relativetoplasmavolume)
changesbytheentryofdifferentiatingerythroblastscarryingHbFe3+anddecreaseswiththe
removalofRBCbyphagocytosis:
(cid:18) (cid:19)
dCHbFe k N
P ¼ RBC EB EB CHbFe(cid:0) d CHbFe ð6Þ
dt V NSS EB RES RBC P
P EB
whereCHbFeistheHbFe3+concentrationinbonemarrow.Here,thenumberofEBcellsinthe
EB
bonemarrowisscaledbytheirsteady-statevalues,whichareinitialvalues,NSS ¼N ð0Þ.
EB EB
BoneMarrow(BM)compartment
Precursorcell(CFUe)dynamics. TheCFUenumberN (μ,t)changeswithrespectto
CFU
maturity,i.e.,age(μ)andtime(t)[38].Thesecellsproliferate,butarenegligibleinbonemar-
rowreactionortransportprocessesrelatedtoiron.Fromanage-distributedcellnumber
Table1. Reactionsinblood(B)compartment.
ReactionTerm Expression
RTf (cid:0) k CFe3þCTf
P Fe3þ;Tf P P
RFe3þTf k CFe3þCTf (cid:0) k CFe3þCFe3þTf
P Fe3þ;Tf P P Fe3þ;Fe3þTf P P
RPðFe3þÞ2Tf kFe3þ;Fe3þTfCPFe3þCPFe3þTf
RFe3þ (cid:0) dFe3þCFe3þ
P P P
https://doi.org/10.1371/journal.pcbi.1006060.t001
PLOSComputationalBiology|https://doi.org/10.1371/journal.pcbi.1006060 April16,2018 6/34
Computationalmodelofanemiaofchronickidneydisease
balance,weobtain
@N @N
CFU þv CFU ¼b N ; 0(cid:20)m(cid:20)m ð7Þ
@t CFU @m CFU CFU F
wheretherateofagingisν =dμ/dtandtheratecoefficientofproliferationβ isassumed
CFU CFU
constant.ThisisasimplificationofamodelpresentedbyMahaffyetal.[25].Atsteadystate,
CFUenumberis
NSS ðmÞ¼N0 expðb mÞ ð8Þ
CFU CFU CFU
TherateofCFUeagingisrepresentedas:
!
CEpo
v ¼v P ð9Þ
CFU 0 CEpoþKm
P Epo
ThisrelationisbasedonsaturationofhighaffinityEporeceptorsofCFUe.
TheCFUenumberformingatμ=0fromdifferentiatingBurstFormingUnitErythroids
(BFUe)isdependentonthetotalBFUenumberN availablefordifferentiationand
BFUe
CEpoðtÞaccordingtoanempiricalfunction[36]:
P
8 (cid:18) (cid:19)
>>< N CPEpoðtÞ for CEpoðtÞ<CEpoð0Þ
NCFUð0;tÞ¼>>: BFUe CPEpoð0Þ for CPEpoðtÞ(cid:21)CPEpoð0Þ ð10Þ
N P P
BFUe
whereCEpoð0Þisthesteady-statevalue.
P
NoirontransporthappensintheCFUeage-distributedcompartment.Furthermore,all
detailsofthedevelopmentofthereceptorsontheCFUeetc.areignoredforthismodel.At
maturityCFUe’sbecomeerythroblasts(EB)withtheaveragenumberoftransferrinreceptors
ontheirsurfaceandtheaverageintracellulariron-freehemoglobin.Theinternalprocessesare
developedforrecyclingoftransferrinreceptorsandincorporationofironintohemoglobin.
Erythroblast(EB)regionrelations. Atfullmaturityμ ,differentiationtoEBoccursata
F
rate:
(cid:12)
@N (cid:12)
FEB CFU ¼vCFU @mCFU(cid:12)(cid:12) ð11Þ
m¼mF
TheEBnumberincreasesbyCFUedifferentiationanddecreasesbyEBdifferentiationinto
RBC:
dN
EB ¼F (cid:0) k N ð12Þ
dt EB CFU RBC EB EB
Wherek isadifferentiationratecoefficient.Theprocessofmaturationoferythroblasts
RBC EB
intomatureRBChasbeensimplifiedintoalumpedmodelunliketheage-distributedmodel
formaturationofCFU.Withthissimplification,ironuptakebymaturingerythroblastsisrep-
resentedbyasinglesetofreactionsinasingleEBcompartment.Atsteadystate,weobtain
V d
NSS ¼ P RES RBCNSS ð13Þ
EB k RBC
RBC EB
PLOSComputationalBiology|https://doi.org/10.1371/journal.pcbi.1006060 April16,2018 7/34
Computationalmodelofanemiaofchronickidneydisease
Combiningequationsabove,wefind
(cid:12)
@N (cid:12)
FESSB CFU ¼nCFU @mCFU(cid:12)(cid:12) ¼nCFUbCFUNCSSFUðmFÞ¼kRBC EBNESBS ð14Þ
m¼mF
DefiningthecharacteristictimeforCFUe’stobecomeRBC’sasτ =1/k ,wecan
EB RBC EB
expressthesteady-stateCFUnumberasfollows:
NSS ðm Þ¼NSS=ðt n b Þ ð15Þ
CFU F EB RBC CFU CFU
Thesteady-stateCFUnumberisdefinedasfollows:
V d
NSS ¼ P RES RBCNSS ¼V d t NSS ð16Þ
EB k RBC P RES RBC EB RBC
RBC EB
TocomputeNSS ðm ÞandNSS,weobtainestimatesfromtheliteratureforV ,d ,τ ,
CFU F EB p RES RBC EB
β andNSS .FollowingMahaffy[25],wesetν =1.
CFU RBC CFU0
Ironandtransferrindynamics. IntheEBregion,ironistakenfrom(Fe3+)Tfand
(Fe3+) Tfviatransferrinreceptor(TfR).Thereactionsofchemicalspeciesarerepresentedby
2
thefollowingkinetics(Fig2):
TfRþðFe3þÞTf $ðTfRÞFe3þTf
TfRþðFe3þÞ Tf $ðTfRÞðFe3þÞ Tf
2 2
ðTfRÞðFe3þÞTf !TfRþFe3þþTf ð17Þ
ðTfRÞðFe3þÞ Tf !TfRþ2Fe3þþTf
2
Fe3þþHb!HbFe3þ
Here,themechanismofironreleasefromTfandrecyclingofTfandTfRhasbeensimplifiedas
asinglereaction.
InrepresentingtheconcentrationchangesofspeciesjinEBwithtime,weassumeEBvol-
umeisconstant.ConcentrationsinEBincreaseswithCFUeentryanddecreasesbecauseofEB
maturationintoRBCandreactionsintheEBcompartmentasfollows:
(cid:18) (cid:19)
dCj F Bj k N
EB ¼ EB CFU CFU (cid:0) RBC EB EB Cj þRj ; j¼TfR;Hb ð18Þ
dt V V NSS EB EB
EB EB EB
whereBTfR ,BHb aretheconstantamountsofTfRandHbperCFUeenteringtheEBcompart-
CFU CFU
ment[7,40].
SinceHbFe3+isnotinCFUeitsconcentrationchangesaccordingto
(cid:18) (cid:19)
dCHbFe3þ (cid:0) k N
EB ¼ RBC EB EB CHbFe3þ þRHbFe3þ ð19Þ
dt V NSS EB EB
EB EB
ForthespeciesthatdonotenterorleavetheEBcompartment,theconcentrationchangesonly
byreaction:
dCj
EB ¼Rj ; j¼ðTfRÞðFe3þÞTf;ðTfRÞðFe3þÞ Tf;Fe3þ ð20Þ
dt EB 2
Forotherspeciesconcentrations,changescanoccurbyreactionandtransportintooroutof
PLOSComputationalBiology|https://doi.org/10.1371/journal.pcbi.1006060 April16,2018 8/34
Computationalmodelofanemiaofchronickidneydisease
plasma:
dCj J
EB ¼Rj þ EB P; j¼ðFe3þÞTf;ðFe3þÞ Tf;Tf ð21Þ
dt EB V 2
EB
ExpressionsforRj areprovidedintheTable2.
EB
REScompartment
TheREScompartmentisdividedintointracellular(I),membrane(M)andinterstitial(ISF)
regions,whichareassumedtohaveconstantvolumes(Fig2).Ironfromthehemoglobinin
senescentRBC’sthatarephagocytosedbythemacrophagesoftheISFisdeliveredasFe3+into
theintracellularregion.Then,Fe3+isconvertedtoFe2+(labileironpool)byendosomalreduc-
ingagentsandthenitbindstointracellularFPN.Thiscomplexiscarriedbyanenergy-driven
processtothecellmembrane,whereFe2+isconvertedtoFe3+bysequestrationwithferritin
(FN).Atthemembrane,oxygen(O )andceruloplasmin(CpCu2+)oxidizeFe2+FPNtoFe3+
2
FPN.ThedissociationofFe3+FPNallowsFPNtorecycleandFe3+todiffuseintotheISFor
bindtoTftoformFe3+Tfinthemembraneregion.
Formostspecies,thetransportfluxesbetweenMandISFaregovernedbypassivediffusion:
Jj ¼hj ðCj (cid:0) Cj Þ; j¼Tf;ðFe3þÞTf;ðFe3þÞ Tf ð22Þ
M!ISF M(cid:0) ISF M ISF 2
Forafewspecies,thetransportfluxesbetweenRESregionsareenergy-drivenprocesses:
JFPN ¼h CFPN; JFe2þFPN ¼h CFe2þFPN; JFe3þ ¼h CFe3þ ð23Þ
I M I M M M I M I I ISF M ISF M M
IntheIregion,thespeciesjconcentrationschangeaccordingto
dCj Jj (cid:0) Jj
I ¼ I M M I þRj j¼Fe2þ;FPN ð24Þ
dt V I
I
TheFe3+fromthesenescentRBCactsasasourceofFe3+intheintracellular(I)region.Hence
theequationforconcentrationofFe3+intheIregionvariesas:
(cid:18) (cid:19)
dCFe3þ JFe3þ (cid:0) JFe3þ V
I ¼ I M M IþRFe3þ þd CHbFe P ð25Þ
dt V I RES RBC P V
I I
Table2. Reactionsintheerythroblast(EB)compartment.
ReactionTerm Expression
RTEBfR (cid:0) kon;TfR;Fe3þTfCETBfRCEFeb3þTf (cid:0) kon;TfR;ðFe3þÞ2TfCETBfRCEðFBe3þÞ2Tf þkTfR;recycleðCETBfRFe3þTf þCETBfRðFe3þÞ2TfÞ
þk CTfRFe3þTf þk CTfRðFe3þÞ2Tf
off;TfRFe3þTf EB off;TfRðFe3þÞ2Tf EB
RTfRFe3þTf k CTfRCFe3þTf (cid:0) k CTfRFe3þTf (cid:0) k CTfRFe3þTf
EB on;TfR;Fe3þTf EB EB off;TfRFe3þTf EB TfR;recycle EB
RTfRðFe3þÞ2Tf k CTfRCðFe3þÞ2Tf (cid:0) k CTfRðFe3þÞ2Tf (cid:0) k CTfRðFe3þÞ2Tf
EB on;TfR;ðFe3þÞ2Tf EB EB off;TfRðFe3þÞ2Tf EB TfR;recycle EB
RFe3þTf (cid:0) k CTfRCFe3þTf þk CTfRFe3þTf
EB on;TfR;Fe3þTf EB EB off;TfRFe3þTf EB
RðFe3þÞ2Tf (cid:0) k CTfRCðFe3þÞ2Tf þk CTfRðFe3þÞ2Tf
EB on;TfR;ðFe3þÞ2Tf EB EB off;TfRðFe3þÞ2Tf EB
RFe3þ k ðCTfRFe3þTf þCTfRðFe3þÞ2TfÞ(cid:0) k CFe3þCHb
EB TfR;recycle EB EB Fe3þ;Hb EB EB
RHb (cid:0) k CFe3þCHb
EB Fe3þ;Hb EB EB
RHbFe3þ k CFe3þCHb
EB Fe3þ;Hb EB EB
https://doi.org/10.1371/journal.pcbi.1006060.t002
PLOSComputationalBiology|https://doi.org/10.1371/journal.pcbi.1006060 April16,2018 9/34
Computationalmodelofanemiaofchronickidneydisease
TheFPNconcentrationdependsonFPNendogenoussynthesisandloss:
dCFPN JFPN
I ¼ I MþRFPN þSFPN ð26Þ
dt V I I
I
where
CFPNj (cid:0) CFPN
SFPN ¼ I SS I ð27Þ
I t
FPN
Here,τ isthehalf-lifeofFPNandCFPNj isthesteady-stateconcentrationofFPN.The
FPN I SS
modelignoresanytranscriptionalregulationofFPNashasbeenobservedespeciallydueto
hypoxia,irondeficiencyetc.[41,42].IntheMregion,thespeciesjconcentrationschangeas:
dCj Jj þJj (cid:0) Jj
M ¼ M I M ISF I MþRj ð28Þ
dt V M
M
wherej¼Fe2þFPN;Fe3þFPN;Fe3þ;FPN;CpCu2þ;CpCu1þ;Tf;Fe3þTf;Fe3þTf.IntheISFregion,
2
thespeciesjconcentrationchangesas:
dCj N Jj (cid:0) Jj
ISF ¼ scale factor ISF M ISF!PþRj ð29Þ
dt V ISF
ISF
whereJj ¼hj ðCj (cid:0) CjÞ; j¼Tf; ðFe3þÞTf; ðFe3þÞ Tf;CpCu2þ;CpCu1þ ð30Þ
ISF!P ISF(cid:0) P ISF P 2
ThereactionratesforeachchemicalspeciesjinthethreeRESregions(Rj ,Rj,Rj )are
ISF I M
basedonthekineticsasindicatedinFig2anddescribedbelow.WeincorporatedtheHepc
blockingofirontransportfromRESthroughdegradationofbothintracellularandmembrane
FPN[15–17].Intheintracellular(I)region:
Fe3þ $Fe2þ
Fe2þþFPN !Fe2þFPN
ð31Þ
FPNþHepc!(cid:10)
FPN !(cid:10)
where(cid:10)representsdegradationproducts.Inthemembrane(M)region:
4ðFe2þÞFPNþO þ4Hþ⇄4ðFe3þÞFPNþ2H O
2 2
Fe2þFPNþCpðCu2þÞ!Fe3þFPNþCpðCu1þÞ
Fe3þFPN !Fe3þþFPN
Fe3þþTf !ðFe3þÞTf ð32Þ
Fe3þþðFe3þÞTf !ðFe3þÞ Tf
2
4CpðCu1þÞþO þ4Hþ!4CpðCu2þÞþ2H O
2 2
FPNþHepc!(cid:10)
PLOSComputationalBiology|https://doi.org/10.1371/journal.pcbi.1006060 April16,2018 10/34