HAL Id: hal-00531138 https://hal.archives-ouvertes.fr/hal-00531138 Submitted on 2 Nov 2010 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Quantitative Evaluation of Respiration Induced Metabolic Oscillations in Erythrocytes Bjørn Hald, Mads F. Madsen, Sune Danø, Bjørn Quistorff, Preben G. Sørensen To cite this version: Bjørn Hald, Mads F. Madsen, Sune Danø, Bjørn Quistorff, Preben G. Sørensen. Quantitative Evalua- tion of Respiration Induced Metabolic Oscillations in Erythrocytes. Biophysical Chemistry, Elsevier, 2009, 141 (1), pp.41. 10.1016/j.bpc.2008.12.008. hal-00531138
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HAL Id: hal-00531138https://hal.archives-ouvertes.fr/hal-00531138
Submitted on 2 Nov 2010
HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.
Quantitative Evaluation of Respiration InducedMetabolic Oscillations in Erythrocytes
Received date: 22 September 2008Revised date: 15 December 2008Accepted date: 20 December 2008
Please cite this article as: Bjørn Hald, Mads F. Madsen, Sune Danø, Bjørn Quistorff,Preben G. Sørensen, Quantitative Evaluation of Respiration Induced Metabolic Oscilla-tions in Erythrocytes, Biophysical Chemistry (2009), doi: 10.1016/j.bpc.2008.12.008
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Quantitative Evaluation of Respiration Indu edMetaboli Os illations in Erythro ytesBjørn Hald1Dept. of Biomedi al S ien es, Mads F. MadsenaDept. of Biomedi al S ien es,Sune DanøaDept. of Biomedi al S ien es, Bjørn Quistor�Dept. of Biomedi al S ien es,Preben G. SørensenDept. of Chemistry,University of Copenhagen, Copenhagen, DenmarkaPresent address: Topsoe Fuel Cell, Lyngby, Denmark.
1Corresponding author. Address: Dept. of Biomedi al S ien es, University ofCopenhagen, Blegdamsvej 3, 2200 Copenhagen, Denmark, Tel.: (+45) 3532 7403,Fax: (+45) 3532 7070
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Abstra tThe hanges in the partial pressures of oxygen and arbon dioxide (PO2andPCO2
) during blood ir ulation alter erythro yte metabolism, hereby aus-ing �ux hanges between oxygenated and deoxygenated blood. In the studywe have modeled this e�e t by extending the omprehensive kineti modelby Mulquiney and Ku hel (1999) with a kineti model of hemoglobin oxy-/deoxygenation transition based on an oxygen disso iation model developedby Dash and Bassingtwaighte (2004). The system has been studied duringtransitions from the arterial to the venous phases by simply for ing PO2andPCO2
to follow the physiologi al values of venous and arterial blood. Theinvestigations show that the system passively follows a limit y le driven bythe for ed os illations of PO2and is thus inadequately des ribed solely bysteady state onsideration. The metaboli system exhibits a broad distribu-tion of time s ales. Relaxations of modes with hemoglobin and Mg 2+ bindingrea tions are very fast, while modes involving gly olyti , membrane trans-port and 2,3-BPG shunt rea tions are mu h slower. In omplete slow moderelaxations during the 60 s period of the for ed transitions ause signi� antovershoots of important �uxes and metabolite on entrations � notably ATP,2,3-BPG, and Mg 2+. The overshoot phenomenon arises in onsequen e ofa periodi al for ing and is likely to be widespread in nature � warranting aspe ial onsideration for relevant systems.
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RBC Metabolism during Respiration 2Key words: Erythro ytes; Blood Cir ulation; Mathemati al Modelling;Os illations;
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RBC Metabolism during Respiration 3Introdu tionThe erythro yte is a simple ell in the sense that it obtains its energy and re-dox equivalents entirely from anaerobi metabolism of glu ose through gly o-lysis and the pentose phosphate pathway (PPP - see list of abbreviations).Furthermore, it has no geneti material whi h simpli�es modelling onsid-erably. Many kineti models have been developed that have des ribed theinter ellular dynami s fo using on the oxygenated and deoxygenated steadystate on entrations. The numeri ally simplest way to obtain information onsteady state dynami s is through modal analysis (1) (see 'Theory') in whi hthe relaxation times for ea h mode of a steady state are the re ipro als ofthe real part of the eigenvalues of the Ja obian. There is no general methodfor al ulation of expli it expressions for the dependen e of relaxation timeson kineti parameters in many- omponent systems. The general trend anbe illustrated by onsidering a simple example. In a linear irreversible hain→ S1
k1→ S2k2→ S3 · · ·Sn−1
kn−1
→ Sn
kn→the eigenvalues of the Ja obian are −k1,−k2, · · · ,−kn and the orespond-ing relaxation times are τi = 1/ki. In this ase the largest relaxation time isquite natually the re ipro al of the smallest rate onstant. If the rea tions ina hain are reversible the width of the relaxation time distribution in reases.Furthermore, feedba k over distant parts of the network may onsiderablyin rease the relaxation times. In the supplementary material (SI, se tion1.1) the Yates-Pardee model is used to exemplify hanges in time s ale dis-
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RBC Metabolism during Respiration 4tribution to the point where the system be ome unstable and the largestrelaxation time be omes ini�nite. Erythro yte metabolism ontains severallinear hains with reversible rea tions and feedba k, and it is not surprisingthat models of this system have extremely slow modes with relaxation timesof several hours as shown by (2).Compared with the period of ir ulation at rest of around 60 s it is learthat a omplete relaxation to steady state may not o ur if the slow modesare ex ited. Fast modes orresponding to small relaxation times may followenvironmental variations a urately while slow modes never relax ompletelyto the urrent steady state. Consequently, to des ribe physiologi al behaviorthe temporal behavior of the bloodstream must be taken into a ount.Experimental eviden e suggests that steady state gly olyti �ux is higher indeoxygenated blood ompared to oxygenated blood in vitro (3�7), whereasthe opposite to a lesser extent is true for the PPP (3). The �ux hanges havebeen hypothesized to satisfy the demand for NADPH under oxygenated on-ditions and replenish the ATP stores under deoxygenated onditions (3, 8).The �ndings imply that dynami hanges take pla e a ross pulmonary andperipheral tissues where O2 and CO2 rapidly ex hange. In this study we haveextended the steady state model of human erythro yte metabolism developedby Mulquiney and Ku hel (9, 10). Spe i� ally, our extended model in ludesmathemati al des riptions of: 1) The transition kineti s of Hb, building onprevious work by Dash and Bassingthwaighte (11) and 2) The os illatory on entrations of O2 and CO2 in erythro ytes, for ed by the repeated pas-sage of the erythro ytes through the arterial and venous ompartments ofthe ir ulatory system (see 'Methods').
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RBC Metabolism during Respiration 5We estimate the hanges in erythro yte metabolism upon for ed os illatorygas ex hanges. Besides gly olyti �ux hanges, the for ing also auses sig-ni� ant overshoots of a number of important metabolites � notably ATP,2,3-BPG, and Mg 2+ � whi h may have physiologi al onsequen es for ery-thro ytes.The study exempli�es the use of modelling to dis lose an emergent prop-erty of a bio hemi al network, here the overshoot-phenomena observed fromsimulations of the present model.TheoryThe kineti model of erythro yte metabolism in this study en ompasses 63metabolites and 60 rea tions. A s hemati drawing of the extended modelis shown in Figure 1. The red spe ies are able to bind Hb. Spe ies ableto ross the membrane are shown in blue. Extra ellular Gl , La , Pyr andPi are treated as onstants, whereas PO2and PCO2
are for ed. The modelen ompasses des riptions of gly olysis, PPP, 2,3-BPG shunt, Hb and Mg 2+binding rea tions, and the Hb oxy/deoxy transitions as well as the for ing ofPO2and PCO2
. The omplete set of rea tion rate expressions an be found inthe 'Supplementary Information' (SI). Key parameters and initial onditionsof the model are given in tables S3 and S4 in the SI.The model of erythro yte metabolism by Mulquiney and Ku hel (9) was ho-sen be ause it, to our opinion, is the most omprehensive model with respe tto 2,3-BPG metabolism and Hb and Mg 2+ metabolite binding rea tions. Ahuge level of detail is needed in order to assess the possible e�e ts of respira-
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RBC Metabolism during Respiration 6tion on the ore metabolism. The most important allosteri regulations arethe 2,3-BPG and ATP inhibition and AMP a tivation of PFK. Table S5 inthe SI presents the most important allosteri e�e tors of HK, PFK, ALD,and PK.Modal analysisModal analysis of steady states is a general method to get infomation on thetime s ales of a metaboli network. The time s ales are the hara teristi times for metaboli pro esses to rea h ompletion. The relevant pro ess inthis study is the relaxation of an arbitrary state to a stationary state. Ifthe hara teristi time for this pro ess is always smaller than the ir ulationtime of the blood the state of the system will almost always be lose to oneof the stationary states. For initial states lose to the stationary state the hara teristi times are alled the relaxation times. They an be al ulatedfrom the eigenvalues of the Ja obian matrix evaluated at the steady state.The Ja obian matrix is de�ned asJ = N
∂v
∂S(1)where N is the stoi hiometri matrix, and v is a ve tor ontaining all rates aselements, and S is the ve tor of metabolite on entrations. The eigenvaluesof the Ja obian are determined by diagonalizing the Ja obian at the steadystate (in this study Mathemati a (Wolfram In .) was used). The relaxationtimes are de�ned by
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RBC Metabolism during Respiration 7τi =
1|Re(λi)|
, i = 1, 2, 3, ..., n (2)where λi are the orresponding eigenvalues of the Ja obian (1).The modes are the eigenve tors ei of the Ja obian. They an be on-sidered as a basis of n orthogonal ve tors, pointing out the dire tions inthe spa e of metabolite on entrations along whi h the system relaxes ba kto the steady state with the relaxation times τi. The rea tions involvedin this relaxation an be determined by transforming the eigenve tors ofthe Ja obian from on entration spa e to rea tion spa e whi h is a spa ewith rea tion extents as oordinates. This is done most easily by al ulatingderivatives of rea tion rates with resp t to the metabolite on entrations (i.e. al ulating the unnormalized elasti ity-matrix of Metaboli Control Analysis(MCA)), and muliplying this matrix with the eigenve tors of the Ja obian.The transformed eigenve tors are the olumns of∂v
∂S·W (3)where W ontains the eigenve tors of the Ja obian as olumns. Transfor-mations of this type are used in Table 2.As the eigenve tors onstitute a basis, any ve tor, u, in the metabolite spa e an be written as a linear ombination of the eigenve tors u =
∑ciei. The oe� ients, ci, may be al ulated by proje ting u onto the eigenve tors. A oe� ient value ci measures the extent to whi h the relaxation of u an beas ribed the individual relaxation mode. As the Ja obian may be an unsym-metri al matrix � a usual property of Ja obians of open hemi al systems �
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RBC Metabolism during Respiration 8proje tions on the right eigenve tors of the Ja obian have to be performedusing left eigenve tors, denoted ei, i = 1, 2, . . . , n (12):e
iu = ciThis pro edure is utilized in Table 3.MethodsThe extensions to the ore model by Mulquiney and Ku hel (9) in lude theHb transition rea tion, Hbd ⇀↽ Hbo and a model that an for e on entra-tions of sele ted spe ies during blood ir ulation.Hemoglobin TransitionThe rate equation of the Hb (oxy-deoxy) transition is based on the followingthree major assumptions: First, the rea tion kineti s is fast ompared withthe frequen y of ir ulation. Se ond, it is assumed that Hb only undergoestransition between the T and R states. Fully deoxygenated Hb is thus equiv-alent with the T state and vi e versa for the R state � no intermediaries area ounted for. Third, only Hb with no heterotropi ligands atta hed is ableto undergo a transition in the model des ription.With these assumptions the rate equations an be approximated by:
vforward = k · [Hbd] · KHb
vbackward = k · [Hbo]
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RBC Metabolism during Respiration 9where the equilibrium onstant KHb is a fun tion of PO2, 2,3-BPG, pH,PCO2
, and temperature (see (11) and in the SI). The s aling onstant k isset to 105 to enfor e fast transition ( hanging the k-value from 102 to 109does not hange the behavior of the system). Derivation of the rate equationis found in se tion 2 in the SI.Model Des ription of Blood Cir ulationThe modelling of the blood ir ulation is based on the assumption that thebloodstream onsists of just four ompartments, namely: the arteries (be-tween the left ventri le and the organs), the organ apillaries, the veins (be-tween the organs and the right ventri le), and the lung apillaries (betweenthe right and left ventri le of the heart). The hemi al omposition of bloodexperien ed by an erythro yte espe ially PO2and PCO2
os illates in time asit passes through these four ompartments at a onstant velo ity. This for estransitions between oxy- and deoxy-Hb to o ur in the organ apillaries andin the apillaries of the lungs.The model of os illatory for ing employs a 'step-like' fun tion (see se tion3 in the SI) in whi h the phase and amplitude of a for ed spe ies an bearbitrarily hosen. The for ing of PO2and PCO2
an be seen in Figure 2.pH onsiderationsArterio-venous variations in pH are also known to o ur in the blood anddi�eren es are dramati ally enhan ed upon physi al exer ise (13). However,at rest the intra ellular ompartment of erythro ytes is remarkably stableagainst arterio-venous pH variations in the blood, i.e. intra ellular pH only
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RBC Metabolism during Respiration 10 hanges ≈0.05 units in the ourse of ir ulation due to the high bu�er a-pa ity of Hb and the Haldane e�e t (14�16). Consequently, the pH in�uen eon metabolism at rest is very limited. As in lusion would ause onsider-able additional omputational load and questionable rea tion kineti s, pHvariations are negle ted in the model.Integration methodThe set of oupled ODEs was integrated forward in time using CVODE (17)for sti� problems (Ba kward Di�erentiation Formulas � version from sept.1994) with an absolute ǫa = 10−18 and a relative ǫr = 10−10. CVODE ispart of a free suite of numeri al integrators olle tively known as Sundialsand an be downloaded at: www.llnl.gov/CASC/sundials/Typi al simulations were ompleted within a minute on a standard dual orePC.ResultsSimulating the erythro yte system ir ulating periodi ally between the arte-rial and venous ompartments shows that the intra ellular metabolite on- entrations os illate as well, i.e. the system passively follows a limit y ledriven by the for ed os illations of O2 and CO2 tensions. Figure 3 illustratesthe os illations of la tate �ux and the free 2,3-BPG on entration. Generally,the metabolite on entrations follow the steady state values orrespondingto hanges in O2 and CO2. Yet, this is not the ase for many rea tion �uxesas well as on entrations of Hb, Mg 2+ (free or metabolite bound), 2,3-BPG,
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RBC Metabolism during Respiration 11ATP and ADP as shown below.By inserting parameter values hara teristi of the arterial and venous om-partments, respe tively, the system was also allowed to attain either arterialor venous steady state. Comparison of the arterial and venous steady statesrevealed that the sets of on entrations and �uxes are highly similar, as sum-marized in Table S6 and S7 in the SI. (For omparison, data for the arterialsteady state predi ted by the model by Mulquiney and Ku hel and previ-ously published values for the orresponding metabolites are also given inTable S8.)The main result of the study is that the arterio-venous (AV) ratios of a num-ber of important rea tion �uxes and metabolite on entrations in the for edsystem of ir ulating erythro ytes di�er signi� antly from the orrespondingratios for steady state simulations. Nonetheless, most intermediary metabo-lites of the gly olyti and PPP pathways are fairly onstant. Table 1 showsratios between important metabolite on entrations and rea tion �uxes, re-spe tively, al ulated between a) the late arterial and late venous phase onthe limit y le and b) the arterial and venous steady states (Hb spe ies areleft out be ause of their trivial oxy-deoxy ratios). For metabolites it is read-ily seen that limit y le to steady state ratios of 2,3-BPG, ATP, FBP, Mg 2+and Mg 2+ bound spe ies are of signi� an e (ratios between 15 and 40%). Forrea tion �uxes the ratios through most of the gly olyti pathway, 2,3-BPGshunt and membrane transports are even greater (di�eren es between 50 and700%. Di�eren es are also seen for AK, Mg 2+ and Hb binding rea tions buthere the steady state �uxes are zero). Transport of La , Pyr, and Pi anexplain the smaller ratios with respe t to metabolites.
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RBC Metabolism during Respiration 12These AV ratios between the limit y le and steady states are alled over-shoots in the following. Overshoot examples are illustrated in Figure 4. Thedynami s leading to the observed behavior is analyzed through the use ofrelaxation modes.Relaxation ModesCal ulation of the relaxation modes of the steady states revealed that the54 relaxation modes (there are 9 onserved hemi al moieties in the model)of the arterial steady state are highly similar to the modes of the venoussteady state (data not shown). Table 2 presents a short overview of themodes where the relaxation times and most involved rea tions have beengrouped together (more detail an be found in the SI). It an be seen that Hband Mg 2+ binding rea tions are ex lusively present in the very fast modes,while gly olyti , 2,3-BPG shunt, and transport rea tions are omponents ofthe slow modes. These �ndings agree with previous models of erythro ytemetabolism (e.g. (2, 9)).The For ed SystemThe period of blood ir ulation at rest is approximately 60 s (13). Hen e,the time window provided by the ir ulatory for ing is too small to allow omplete relaxation of the system in the arterial or venous steady state. Itwas tested that no instabilities were introdu ed on the limit y le by thefor ed os illations, i.e. no eigenvalue be omes positive (data not shown),and it must be on luded that the limit y le is a passive response to thePO2os illations that primarily drives the system. PCO2
os illations in the
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RBC Metabolism during Respiration 13physiologi al range of 40-46 Torr alone ould not produ e signi� ant �uxos illations in the system. Consequently, shifting the Hb transition equilib-rium through PO2perturbs the system through the Hb-metabolite bindingrea tions, hereby ausing a hange of the attra tor lo ation in metabolitespa e.To estimate whi h modes would be ex ited and to what extent, if in�nite re-laxation after a perturbation was allowed, on entration di�eren es betweena) the early arterial phase of the limit y le and arterial steady state, andb) the early venous phase of the limit y le and venous steady state wereproje ted onto arterial steady state eigenve tors. The results are displayedin Table 3. For both proje tions mode no.54 (the very slow, 2,3-BPG pro-du ing mode) is the preponderant mode that annot a hieve relaxation inthe time frame given by the respiratory for ing.System Dynami sBe ause of the relative loseness to the arterial and venous steady states,the overshoots displayed by the for ed system ompared to the steady states an be explained to a large part in terms of the eigenve tors and asso iatedeigenvalues of the Ja obian. That is, the exa t geometry of the "ex ited"eigenve tors in the phase plane and the magnitude of the orrespondingeigenvalues oupled with the amplitude and "e�e tiveness" (�ux ontrol) ofthe for ed paramters.As an illustrative example we onsider a two-dimensional system for whi htwo relaxation modes � one fast and one slow � relax towards steady state. Asillustrated in Figure 5, periodi hanges of an external parameter translo-
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RBC Metabolism during Respiration 14 ates the steady state , where the magnitude of translo ation depends onthe amplitude of os illations and �ux ontrol of the parameter. Neverthe-less, the properties of the relaxation modes may remain largely unaltered(at any given timepoint only one steady state is present. The two steadystates shown represent the steady state at the two di�erent values of theexternal parameter). The �fast� eigenve tor determines the initial dire tionof relaxation followed by slow relaxation along the dire tion of the �slow�eigenve tor. From Figure 5 it is lear that the angle between the eigen-ve tors, the separation of time s ales, the frequen y of periodi transitions,and the distan e between the steady state lo ations determines the extentof overshoots for some spe ies on the limit y le. For some spe ies, ratiosare larger on the limit y le ompared with ratios between the steady statelo ations.The data in tables 2 and 3 show that the slow 2,3-BPG produ ing rea tionmodes ombined with fast Hb and Mg 2+ binding rea tion modes providethe basis for the observed dynami s in the high dimensional ase of eryth-ro yte metabolism. The slow 2,3-BPG produ ing rea tion mode are ex itedbut annot relax in the timeframes given by the arterial and venous om-partments. Consequently, 2,3-BPG on entration is quasistationary and thenonlinear feedba k e�e ts of 2,3-BPG inhibition of PFK and HK that de linein the venous phase ( oupled with a slight in rease of MgATP at the sametime) explains the observed in rease of gly olyti �ux in the venous phase.
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RBC Metabolism during Respiration 15Dis ussionErythro yte metabolism provides an ex ellent example of a biologi al systemoperating on a multitude of time s ales. The ensuing relaxation modes anbe used as a tool to analyze omplex models in order to reveal fun tionalinsights of the system near and at steady state. The lo ation of a stablesteady state in metabolite spa e depends on external onditions, e.g. thepartial pressure of O2. The main on lusions of this study are the following:1) The periodi for ing of PO2is a primary driving for e behind gly olyti �ux hanges in the ir ulation. 2) The exa t geometry and time s ale disper-sal of fast and slow relaxation modes oupled with PO2
transition frequen yand amplitude bring about an overshoot phenomenon observed for both anumber of �uxes and metabolite on entrations. 3) The overshoots are aug-mented by the nonlinear e�e ts of 2,3-BPG. The overshoot phenomenon, asillustrated in Figure 5, is a general dynami al property of a driven systemwith a broad distribution of time s ales. The requirements are 1) a periodi hange of steady state lo ation due to one or more os illatory parametersand 2) that the dire tions of relaxation toward stationarity (approximatedby eigenve tors) are not perpendi ular. It is thus anti ipated that the phe-nomenon is widespread in nature and must be taken into a ount both whenstudying externally for ed systems and when applying a for ing onto su hsystems.The predi ted overshoots are expe ted to be hard to dete t experimen-tally in vivo due to the fast Hb and Mg 2+ binding rea tions. However, the
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RBC Metabolism during Respiration 16overshoots displayed by Mg 2+, ATP, and 2,3-BPG may have an importantimpa t on erythro yte se ondary metabolism. Mg 2+ and ATP are impor-tant ofa tors and regulators of numerous rea tions in the ell and Mg 2+is believed to regulate membrane stability (7, 18, 19). 2,3-BPG is of en-tral importan e to hemoglobin fun tion and gly olyti �ux hanges. Themodel predi ts that total 2,3-BPG is at a quasi-steady state. In the arte-rial phase most 2,3-BPG is free or bound to Mg 2+. Free 2,3-BPG inhibitsHK, PFK, and Ald. In the venous phase most is bound to Hb ausing arise in Mg 2+ and a de line in enzyme inhibition. Also, the predi tions ofthe arterio-venous on entration ratios ould provide a swit h me hanism totrigger or augment the se ondary oxygenation dependent fun tions withinthe erythro yte (transport a tivities (20), stru tural hanges (21, 22), mem-brane �uidity et .).As is seen, it is paramount to take the dynami s of a system into a ountin order to obtain a faithful representation of the system at hand. Deter-mination of intra ellular �uxes or on entrations of metabolites is (as far aswe know) always arried out relatively long time (on the order of ∼30 minto several hours) after withdrawal of sample blood (see (9, 23) and refen esherein). This is usually done to ensure steady state onditions. Our resultsindi ate that even longer time is required. In modern time NMR-studies(e.g. (8, 16, 24�26)) erythro ytes following removal of bu�y oat are in u-bated up to several hours before measurement. The �ndings of the presentstudy indi ate that standard experimental pro edures and steady state ap-proximations may be unable to apture a number of important features ofthe erythro yte system under the physiologi al onditions of arterio-venous
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RBC Metabolism during Respiration 17oxygenation hanges in the ir ulation. dB3 involvementIn the literature, the oxygen-dependent modulation of erythro yte metabo-lism is assumed to be a result primarily of the interplay between Hb and dB3 (7, 8, 27�31). This study shows that gly olyti os illations are ableto o ur without dB3 involvement, albeit at this state of investigation ourmodel annot explain all the experimentally reported di�eren es betweenoxy- and deoxygenated erythro ytes at steady state.The ytoplasmati domain of Band3 ( dB3) is a plausible e�e tor on eryth-ro yte metabolism. Re ent eviden e (27, 28, 32) suggests that dB3 is ableto bind PFK, ALD, GAPDH, and Hb at the membrane and thereby ini-tiate the formation of a gly olyti enzyme omplex, whi h is expe ted toinhibit atalyti a tivities ompletely (29). In a separate series of in sili oexperiments, we also in luded dB3 binding to Ald, PFK, GAPDH, Hbd,and Hbo (data not shown) using a set of asso iation onstants used in anerythro yte model by Kinoshita et al. (33). The al ulations showed evenhigher shifts of arterio-venous ratios both on the limit y le and between thetwo steady states and we observed in reased overshoots. However, with the dB3 in lusion the free on entrations of ALD, PFK, and GAPDH be ameso low that it must be on luded that dire t use of the published asso iation onstants of dB3-bindings (33�36) must be wrong (see table S2 in the SI).(True asso iation onstants depends heavily on ioni strength and lo al pH).
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RBC Metabolism during Respiration 18Model LimitationsThe most important limitation of our model and all other models on erythro- yte metabolism is the la k of detailed implementations of MgATP, NADHand NADPH/GSH onsuming pro esses. All existing erythro yte models(9, 23, 33, 37, 38) handle su h pro esses by linear �rst order rea tions witha generalized rate onstant. Hen e, the supply is modelled in detail, butmodels of the demand for these important ofa tors are inadequate. Forinstan e, metaboli ontrol analysis reveal that the ATPase and the "spon-taneous" oxidation rea tion of GSH exert almost ex lusive ontrol of the �uxthrough gly olyti and PPP, respe tively (39). In other organisms, for in-stan e in Es heri hia oli, gly olyti �ux is ontrolled by the demand of ATPas well (40). Metaboli models have to re�e t the supply and demand of therelevant spe ies (just as e onomi al models have to a ount for in omes andexpenditures) in order to hint useful answers to physiologi al questions.The kineti model developed by Mulquiney and Ku hel is a huge model omprised of many parameters and bio hemi al details. Thus, the modelmay ontain onsiderable errors and may be shown to �t experimental datausing a di�erent set of parameters. It is di� ult to assess to what extentthe dynami properties would have di�ered with other (hypotheti al) sets of onsistent parameter values. However, it seems reasonable to assume thatthe set of parameter values that �ts the experimental data best also re�e tstrue dynami al properties of the metaboli network in vivo.In the ontext of ir ulating erythro ytes, the limitations imposed by thesimplisti hemoglobin rate expression and the for ing model are not impor-
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RBC Metabolism during Respiration 19tant, as the purpose of the model is not to be fully a ountable of quantitativedetails (virtually impossible), but to display important �ux and metabolite hanges in the ourse of bloodstream traversal.Limitations of our model in lude the la k of proper implementations of vol-ume and pH hanges. As noted in the 'Methods se tion', the minor pH hanges at resting onditions only have a small degree of in�uen e on meta-bolism. Volume hanges are known to o ur in erythro ytes be ause of the hloride shift. The retention of negative harge auses the erythro ytes toswell around 3% in the venous phase (13). The presented model assumesa onstant intra ellular volume. However, a 3% hange of volume will nota�e t the al ulations of this study signi� antly.New experimental resear h suggest that ATP is released upon hypoxi on-ditions (41�44). The orresponding hanges in ATP level seems to be low ompared with the intra ellular on entration. As the time s ale of adenineturnover seems to be around two weeks (9) in the pres en e of glu ose wehave hosen to set the total nu leotide ontent onstant in this study.Validation of the model and the results of this study would require measure-ments of erythro ytes moving between hemi ally well-de�ned arterial andvenous ompartments of a hemi al rea tor. A veri� ation of the long re-laxation times of 2,3-BPG produ tion ould be arried out by simultaneousmeasurements of glu ose uptake rate and the total on entration of intra el-lular 2,3-BPG from freshly drawn erythro ytes in venous and arterial bloodby NMR.
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RBC Metabolism during Respiration 20Con lusionIn this study we have shown that the respiratory for ing of erythro yte me-tabolism results in a) gly olyti �ux os illations and b) overshoots of entralmetabolites. This behavior an be explained by the �xed period and ampli-tude of oxygen ex hange during ir ulation oupled with the slow modes ofgly olysis, metabolite transport and 2,3-BPG produ tion. Systems with dis-persed time s ale distributions oupled to periodi hanges in a steady statelo ation may often produ e a dynami behavior equivalent to the observedbehavior of erythro yte metabolism for ed by respiration.Referen es1. Heinri h, R., and S. S huster, 1996. The Regulation of Cellular Systems.Chapman & Hall (ITP).2. Liao, J., and E. Lightfoot Jr., 1987. Extending the Quasi-Steady StateCon ept to Analysis of Metaboli Networks. J. theor. Biol. 126:253�273.3. Messana, I., M. Orlando, L. Cassiano, L. Penna hietti, C. Zuppi,M. Castagnola, and B. Giardina, 1996. Human erythro yte metabo-lism is modulated by the O2-linked transition of hemoglobin. FEBS390:25�28.4. Hamasaki, N., T. Asakura, and S. Minakami, 1970. E�e t of oxygen ten-sion on gly olysis in human erythro ytes. J Bio hem (Tokyo) 68(2):157�61.
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RBC Metabolism during Respiration 215. Rapoport, I., H. Berger, and S. Rapoport, 1976. Response of the gly o-lysis of human erythro ytes to the transition from the oxygenated to thedeoxygenated state at onstant intra ellular pH. Bio him Biophys A ta428:193�204.6. Murphy, J., 1960. Erythro yte metabolism II. Glu ose metabolism andpathways. J Lab Clin Med 55:286�302.7. Barvitenko, N., N. Adragna, and W. R.E., 2005. Erythro yte SignalTransdu tion Pathways, their Oxygenation Dependen e and Fun tionalSigni� an e. Cell Physiol Bio hem 15:1�18.8. Galtieri, A., E. Tellone, L. Romano, F. Misiti, E. Bello o, S. Fi arra,A. Russo, D. Di Rosa, M. Castagnola, B. Giardina, and I. Messana, 2002.Band 3 protein fun tion in human erythro ytes: e�e t of oxygenation -deoxygenation. Bio himi a et Biophysi a A ta 1564:214�218.9. Mulquiney, P., and P. Ku hel, 1999. Model of 2,3-bisphosphogly eratemetabolism in the human erythro yte based on detailed enzyme kineti equations: equations and parameter re�nement. Bio hem. J. 342:581�596.10. Mulquiney, P., and P. Ku hel, 2003. Modelling Metabolism with Math-emati a Analysis of Human Erythro yte. CRC Press In .11. Dash, R., and J. Bassingthwaighte, 2004. Blood HbO2 and HbCO2disso iation urves at varied O2, CO2, pH, 2,3-DPG and temperaturelevels. Ann Biomed Eng 32(12):1676�93.
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RBC Metabolism during Respiration 2212. Hynne, F., P. Graae Sørensen, and T. Møller, 1993. Complete optimiza-tion of models of the Belousov-Zhabotinsky rea tion at a Hopf bifur a-tion. J. Chem. Phys 98(1):219�230.13. Ganong, W., 2005. Review of Medi al Physiology, 22.th edition.M Graw-Hill Companies.14. Enoki, Y., S. Tomita, N. Maeda, M. Kawase, and T. Okuda, 1972. Asimple method for determination of red ell intra ellular pH. NipponSeirigaku Zasshi 34(11):761�2.15. Freedman, J., and J. Ho�man, 1979. Ioni and osmoti equilibria ofhuman red blood ells treated with nystatin. J Gen Physiol 74(2):157�85.16. Labotka, R. J., 1984. Measurement of intra ellular pH and deoxyhe-moglobin on entration in deoxygenated erythro ytes by phosphorus-31nu lear magneti resonan e. Bio hemistry 23(23):5549�55.17. Cohen, S., and A. Hindmarsh, 1994. CVODE User Guide. Lawren eLivermore National Laboratory report UCRL-MA-118618 .18. Beaven, G., J. Parmar, G. Nash, B. Bennett, and W. Gratzer, 1990.E�e t of magnesium ions on red ell membrane properties. J MembrBiol 118(3):251�7.19. Page, S., M. Salem, and M. Laughlin, 1998. Intra ellular Mg2+ regu-lates ADP phosphorylation and adenine nu leotide synthesis in humanerythro ytes. Am J Physiol (Endo rinol Metab) 274:920�927.
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RBC Metabolism during Respiration 2320. Gibson, J., A. Cossins, and J. Ellory, 2000. Oxygen-Sensitive MembraneTransporters In Vertebrate Red Cells. The Journal of Experimental Bi-ology 203:1395�1407.21. Tuvia, S., S. Levin, and R. Korenstein, 1992. Oxygenation-deoxygenation y le of erythro ytes modulates submi ron ell membrane�u tuations. Biophys J 63:599�602.22. Barbul, A., Y. Zipser, A. Na hles, and R. Korenstein, 1999. Deoxygena-tion and elevation of intra ellular magnesium indu e tyrosine phospho-rylation of band 3 in human erythro ytes. FEBS Letters 455:87�91.23. Joshi, A., and B. O. Palsson, 1990. Metaboli Dynami s in the Hu-man Red Cell. Part I - A Comprehensive Kineti Model. J theor. Biol.141:515�528.24. Berthon, H. A., W. A. Bubb, and P. W. Ku hel, 1993. 13C n.m.r. iso-topomer and omputer-simulation studies of the non-oxidative pentosephosphate pathway of human erythro ytes. Bio hem J 296 ( Pt 2):379�387.25. Petersen, A., S. Risom-Kristensen, J. Ja obsen, and M. Horder, 1990.31P-NMR measurements of ATP, ADP, 2,3-diphosphogly erate andMg2+ in human erythro ytes. Bio himi a et Biophysi a A ta 1035:169�174.26. Messana, I., F. Misiti, S. el Sherbini, B. Giardina, and M. Castagnola,1999. Quantitative determination of the main glu ose metaboli �uxes
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RBC Metabolism during Respiration 24in human erythro ytes by 13C- and 1H-MR spe tros opy. J Bio hemBiophys Methods 39:63�84.27. Weber, R., W. Voelter, A. Fago, H. E hner, C. E., and P. Low, 2004.Modulation of red ell gly olysis: intera tions between vertebrate hemo-globin and ytoplasmi domains of band 3 red ell membrane proteins.Am J Physiol Regul Integr Comp Physiol 287:454�464.28. Campanella, M., H. Chu, and P. Low, 2005. Assembly and regulationof a gly olyti enzyme omplex on the human erythro yte membrane.PNAS 102:2402�2407.29. Low, P., P. Rathinavelu, and M. Harrison, 1993. Regulation of Gly olysisvia Reversible Enzyme Binding to the Membrane Protein, Band 3. J BiolChem 268:14627�14631.30. Low, P., 1986. Stru ture and fun tion of the ytoplasmi domain ofband 3: enter of erythro yte membrane-peripheral protein intera tions.Bio himi a et Biophysi a A ta 864:145�167.31. Low, P., D. Allen, T. Zion he k, P. Chari, B. Willardson, R. Geahlen,and M. Harrison, 1987. Tyrosine phosphorylation of band 3 inhibitsperipheral protein binding. J Biol Chem 262(10):4592�6.32. von Rü kmann, B., and D. S hubert, 2002. The omplex of band3 protein of the human erythro yte membrane and gly eraldehyde-3-phosphate dehydrogenase: stoi hiometry and ompetition by aldolase.Bio him Biophys A ta 1559(1):43�55.
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RBC Metabolism during Respiration 2533. Kinoshita, A., K. Tsukada, T. Soga, T. Hishiki, Y. Ueno, Y. Nakayama,M. Tomita, and M. Suematsu, 2007. Roles of hemoglobin Allostery inhypoxia-indu ed metaboli alterations in erythro ytes: simulation andits veri� ation by metabolome analysis. J Biol Chem 282(14):10731�41.34. Kelley, G., and D. Winzor, 1984. Quantitative hara terization of theintera tions of aldolase and gly eraldehyde-3-phosphate dehydrogenasewith erythro yte membranes. Bio him Biophys A ta. 778(1):67�73.35. Walder, J., R. Chatterjee, T. Ste k, P. Low, G. Musso, E. Kaiser,P. Rogers, and A. Arnone, 1984. The intera tion of hemoglobin withthe ytoplasmi domain of band 3 of the human erythro yte membrane.J Biol Chem. 259(16):10238�46.36. Jenkins, J., D. Madden, and T. Ste k, 1984. Asso iation of phospho-fru tokinase and aldolase with the membrane of the inta t erythro yte.J Biol Chem. 259(15):9374�8.37. Rapoport, T., and R. Heinri h, 1975. Mathemati al Analysis of Multien-zyme Systems. I. Modelling of the Gly olysis of Human Erythro ytes.BioSystems 7:120�129.38. Ni, T., and M. Savageau, 1996. Model Assesment and Re�ning UsingStrategies from Bio hemi al Systems Theory: Appli ation to Metabo-lism in Human Red Blood Cells. J. Theor. Biol 179:329�368.39. Mulquiney, P., and P. Ku hel, 1999. Model of 2,3-bisphosphogly eratemetabolism in the human erythro yte based on detailed enzyme ki-
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RBC Metabolism during Respiration 26neti equations: omputer simulation and Metaboli Control Analysis.Bio hem. J. 342:597�604.40. Koebmann, B., H. Westerho�, J. Snoep, D. Nilsson, and P. Jensen, 2002.The gly olyti �ux in Es heri hia oli is ontrolled by the demand forATP. J Ba teriol. 184(14):3909�16.41. Ellsworth, M., 2003. Red blood ell-derived ATP as a regulator of Skele-tal Mus le Perfusion. Med S i Sports Exer 36(1):35�41.42. Farias III, M., M. Gorman, M. Savage, and E. Feigl, 2004. Plasma ATPduring exer ise: possible role in regulation of oronary blood �ow. AmJ Physiol Heart Cir Physiol 288:1586�1590.43. Sprague, R., A. Stephenson, M. Ellsworth, C. Keller, and A. Lonigro,2001. Impaired release of ATP from RBCs of humans with primarypulmonary hypertension. Exp Biol Med 226(5):434�439.44. Gonzalez-Alonso, J., D. Olsen, and B. Saltin, 2002. Erythro yte and theregulation of humman skeletal mus le blood �ow and oxygen delivery.Role of ir ulating ATP. Cir ulation Resear h 91:1046�1055.45. Boron, W., and E. Boulpaep, 2003. Medi al Physiology. Elsevier (Saun-ders).
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RBC Metabolism during Respiration 27FootnotesA knowledgementsThis work was supported by the European Union through the Network ofEx ellen e BioSim, Contra t No. LSHB-CT-2004-005137 and by the Euro-pean S ien e Foundation Fun Dyn Programme.AbbreviationsMetabolites: Gl : α-D-glu ose; G6P: α-D-glu ose-6-phosphate; FBP: β-D-fru tose 1,6-bisphosphate; 1,3-BPG: 1,3-biphosphogly erate; 2,3-BPG: 2,3-biphosphogly erate; PEP: phosphoenolpyruvate; Pyr: (S)-pyruvate; La : L-la tate; GSH: Glutathione; NADH: Ni otinamide adenine dinu leotide (re-du ed form); NADPH: Ni otinamide adenine dinu leotide phosphate (re-du ed form); Pi: inorgani phosphate; Hbd: deoxy-hemoglobin; Hbo: oxy-hemoglobin; Enzymes; HK: hexokinase (EC 2.7.1.1); PGI: glu ose-6-phos-phate isomerase (EC 5.3.1.9); PFK: 6-phosphofru to kinase (EC 2.7.1.11);ALD: fru tose-bisphosphate aldolase (EC 4.1.2.13); TPI: triose phosphateisomerase (EC 5.3.1.1); GAPDH: gly eraldehyde 3-phosphate dehydroge-nase (phosphorylating) (EC 1.2.1.12); BPGSP: bisphosphogly erate syn-thase/phosphatase (EC 5.4.2.4 and EC 3.1.3.13); BPGSP7: the 2,3-BPGprodu ing elementary rea tion in the shunt; PGK: phosphogly erate kinase(EC 2.7.2.3); PGM: phosphogly erate mutase (EC 5.4.2.1); ENO: phos-phopyruvate hydratase (EC 4.2.1.11); PK: pyruvate kinase (EC 2.7.1.40);LDH: L-la tate dehydrogenase (EC 1.1.1.27); G6PDH: glu ose 6-phosphate
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RBC Metabolism during Respiration 28dehydrogenase (EC 1.1.1.49); Ru5PE: ribulose-5-phosphate epimerase (EC5.1.3.1); TK: transketolase (EC 2.2.1.1); AK: adenylate kinase (EC 2.7.4.3);Lumped enzymati rea tions; ATPase: non-gly olyti energy onsumption;oxNADH: redu ing pro esses requiring NADH; PyrTR: pyruvate transport;La TR: la tate transport; PhosTR: inorgani phosphate transport; Mis el-laneous; dB3: ytoplasmati domain of Band3; MCA: Metaboli ControlAnalysis; PO2: partial pressure of O2; PPP: pentose phosphate pathway; SI:Supplementary information
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RBC Metabolism during Respiration 29Figure LegendsFigure 1.Core metabolism of human erythro ytes. The �gure shows the rea tionsof erythro yte metabolism onsidered in the model. The red spe ies aremetabolites apable of binding to Hb. The blue spe ies are spe ies alsopresent extra ellularly and are transported a ross the ell membrane. CO2is drawn purple as the on entration of CO2 is for ed in the model. (Abbre-viations an be found in the SI).Figure 2.For ed os illations of PO2(solid line) and PCO2
(dashed line). Two periodsof ir ulation is shown, i.e. 120 se onds. The length of the venous, arterialand apillary phases are about 42.5, 13.5 and 2×2.0 se onds, respe tively(45).Figure 3.Limit y le os illations of la tate �ux (red urve) and free 2,3-BPG on en-tration (blue urve) as fun tions of time. The bar at the top of the �gureindi ates the arterial (red) and venous (blue) phases. The period of [2,3-BPG℄ os illations follows the period of for ed oxygen stri tly.Figure 4.Overshoot phenomenon in ATP (a) and 2,3-BPG (b) on entration and PFK( ) and LDH (d) �uxes. The �gure ompares the on entration or �ux on
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RBC Metabolism during Respiration 30the for ed limit y les ( urves) to the steady states ( rosses) at a for ingamplitude of 40 to 95 Torr. The rosses mark the steady state on entrations40, 60, 80, and 95 Torr, respe tively.Table 1.Overshoots of sele ted metabolite on entrations (left) and rea tion rates(right). Arterio-venous ratios are al ulated both on the limit y le (L.C.)and between steady states (St.st.). Values representing the arterial andvenous phases on the limit y le are taken to be the last time point before ashift in PO2( apillary traversal) o urs. BPGSP7: The 2,3-BPG produ ingrea tion in the 2,3-BPG shunt. oxNADH: unspe i�ed redu ing pro essesrequiring NADH. NA: Not Appli able - steady state �ux zero.Table 2.Overview of relaxation modes of the steady states. The relaxation ba k tosteady state upon a perturbation an be systemati ally presented througheigenmode analysis of the steady state. In the 'Largest omponent'- olumns,rea tions whi h have the largest omponents of the rea tion spa e eigenve -tors orresponding to the indi ated relaxations times are shown.Table 3.Full relaxation to steady state after apillary traversal (in the for ed system)requires mostly relaxation of mode no.54. The table shows the normalized oe� ients (coefi/
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RBC Metabolism during Respiration 31state and (b) early venous phase and venous steady state. The total sumfrom the arterial proje tion is 3.5 times larger than the venous proje tion.Figure 5.Example of overshoot produ ing dynami s in a 2-dimensional phase plane.The lo ation of the steady state (marked with a grey dot) depends on exter-nal parameters that display periodi transitions. Thus, two lo ations of thesteady state are shown. The two non-perpendi ular dire tions of relaxation� one fast and one slow � towards the steady state may be largely preservedupon the transition. Initially, the system is driven in the fast dire tion fol-lowed by relaxation along the slow dire tion. As the transition frequen y ishigher than the slow relaxation mode a limit y le shown as a stippled urveemerges. As long as the eigenve tors are not perpendi ular, on entrationdi�eren es are greater (or smaller � as shown here in the verti al dire tion)on the limit y le ompared to the di�eren e between steady state lo ations.Table 4.Distribution of hemoglobin spe ies in late arterial and venous phase. Totalintra ellular on entration of Hb is set to be 7.0mM in the model.
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Table 1: Overshoots of selected metabolite concentrations (left) and reaction rates(right). Arterio-venous ratios are calculated both on the limit cycle (L.C.) and be-tween steady states (St.st.). Values representing the arterial and venous phases on thelimit cycle are taken to be the last time point before a perturbation (capillary traversal)occurs. BPGSP7: The 2,3-BPG producing reaction in the 2,3-BPG shunt. oxNADH:unspecified reducing processes requiring NADH. NA: Not Applicable - steady stateflux zero.
Largestnos. time components1-24 < 2 ms Mostly Hb and Mg2+ binding reactions25-39 < 1 s Mostly PPP reactions40-44 < 10 s Glycolytic, PPP, Hb and Mg2+ binding reactions45-50 < 4 m Glycolysis, 2,3-BPG shunt and Lac, Pyr and Pi transport51-54 8 m - 5 h 2,3-BPG shunt, glycolysis and Lac, Pyr and Pi transport
Table 2: Overview of relaxation modes of the steady states. The relaxation back tosteady state upon a perturbation can be systematically presented through eigenmodeanalysis of the steady state. In the ’Largest component’-columns, reactions which havethe largest components of the reaction space eigenvectors corresponding to the indi-cated relaxations times are shown.
Table 3: Full relaxation to steady state after capillary traversal (in the forced system)requires mostly relaxation of mode no.54. The table shows the normalized coefficients(coe fi/∑coe fi) from projections of concentration differences between: (a) the earlyarterial phase of the limit cycle and the arterial steady state and (b) early venous phaseand venous steady state. The total sum from the arterial projection is 3.5 times largerthan the venous projection.
