Physical understanding of complex multiscale biochemical models viaalgorithmic simplification: Glycolysis in Saccharomyces cerevisiae
Panayotis D. Kourdis a, Ralf Steuer b,c, Dimitris A. Goussis a,∗a Department of Mechanics, School of Applied Mathematics and Physical Sciences, National Technical University of Athens, 157 73 Athens, Greeceb Institute for Theoretical Biology, Humboldt University of Berlin, Invalidenstraße 43, 10115 Berlin, GermanycManchester Interdisciplinary Biocentre, The University of Manchester, 131 Princess Street, Manchester M1 7DN, UK
a r t i c l e i n f o
Article history:Received 15 September 2009Received in revised form3 June 2010Accepted 4 June 2010Available online 10 June 2010Communicated by A. Mikhailov
Large-scale models of cellular reaction networks are usually highly complex and characterized by a widespectrum of time scales, making a direct interpretation and understanding of the relevant mechanismsalmost impossible. We address this issue by demonstrating the benefits provided by model reductiontechniques.We employ the Computational Singular Perturbation (CSP) algorithm to analyze the glycolyticpathway of intact yeast cells in the oscillatory regime. As a primary object of research for manydecades, glycolytic oscillations represent a paradigmatic candidate for studying biochemical functionand mechanisms. Using a previously published full-scale model of glycolysis, we show that, due to fastdissipative time scales, the solution is asymptotically attracted on a low dimensional manifold. Withoutany further input from the investigator, CSP clarifies several long-standing questions in the analysis ofglycolytic oscillations, such as the origin of the oscillations in the upper part of glycolysis, the importanceof energy and redox status, as well as the fact that neither the oscillations nor cell–cell synchronizationcan be understood in terms of glycolysis as a simple linear chain of sequentially coupled reactions.
The increasing complexity of mathematical models in biologyand genetics necessitates the development of particular algorith-mic tools to facilitate the understanding of the underlying physicalprocesses and their interactions [1–5]. In this respect, a numberof methodologies have been developed for constructing simplifiedor reduced models that are of low dimension but retain all signifi-cant features of the full model. Such algorithms have recently beensuccessfully employed for the analysis of a variety of problems inbiochemistry; see for example Refs. [6–17].Reduction of large and complex nonlinear mathematical
models is mainly based on the presence of very fast time scales,which quickly become exhausted, allowing slower time scalesto characterize the evolution of the physical process. During theslow evolution, these fast time scales do not directly affect theprogress of the system, but simply constrain its evolution ina low dimensional surface — characterized thus as dissipative.This situation is usually known as stiffness [18] and the low
dimensional surface, where the evolution of the system ischaracterized by the slow time scales, is referred as a slowmanifold [19–21]. The motion on this manifold can be describedin terms of a simplified system, i.e. a system which is free of thecomponents in the originalmodel that produce the fast time scales.These fast components participate in a number of equilibria, whichare enforced by the action of the fast time scales. The simplestexpressions for these equilibria are provided by the quasi-steadystate or partial equilibrium assumptions. However, in most casesmore complicated expressions are required [17,22].A large number of algorithms for model reduction and
simplification rely on the magnitude of the existing fast/slowtime scale gaps and they produce either both the manifold andthe model governing the slow evolution [6,22–28] or simply themanifold [29,30]. These algorithms provide either leading orderaccuracy [6,24,25,27,28] or higher order accuracy in an iterativefashion [22,23,26,29,30], the measure of the accuracy providedbeing the magnitude of the gap between the fast and the slowtime scales. Excellent reviews of such algorithms can be found inRefs. [31,32].One of the major advantages that the simplification algorithms
provide is the identification of the various components in acomplex mathematical model that are the most influential in theformation of the manifold and/or in the slow evolution alongthe manifold [9,33]. These are the components that must attract
P.D. Kourdis et al. / Physica D 239 (2010) 1798–1817 1799
particular attention when interested in the long term simulationof complex biochemical problems.Among these components are those participating in the various
equilibria developing due to the action of fast time scales, thosethat drive the slow evolution, etc. The identification of thesecomponents is very important for the construction of models thatare (i) reliable, (ii) of reduced dimensions (due to the system’sevolution on the low dimensional manifold) and (iii) non-stiff(since on the manifold the fast time scales are not present).In addition, this identification will facilitate the detection ofvarious means for controlling the physical process under study,by adjusting the manifold or the way in which the system evolveson it.This kind of knowledge can be acquired only when the
fast and slow dynamics of the problem can be identified andseparately examined. Of course, given the size and complexity ofthe mathematical models of current interest, the separation offast/slow dynamics can only be achieved algorithmically.Here, the Computational Singular Perturbation (CSP) algo-
rithm [22,23] will be employed for the analysis of the gly-colytic pathway of intact yeast cells in a Continuous-flow StirredTank Reactor (CSTR) [34,35]. Glycolysis, the breakdown of glu-cose with concomitant formation of ATP , is present in almost allorganisms and represents a paradigmatic and one of the mostthoroughly studied pathways in biochemistry. Under particularconditions, the concentrations of metabolic intermediates of gly-colysis exhibit sustained oscillations, thus providing a uniquemodel system to use for analyzing and understanding the inter-nal regulation of the pathway. Glycolytic oscillations have beenobserved for several decades using yeast extracts as well as intactcells, and a variety of kinetic models have been proposed for de-scribing the kinetic mechanisms that are required for the emer-gence of sustained oscillations [34–38]. Our objective is to perform,on the basis of the full-scale model of the glycolytic pathway de-veloped by Hynne et al. [34], the analysis of the fast and slow dy-namics, the construction of the low dimensional manifold, as wellas the construction of a simplified minimal model governing theevolution on the manifold. It is demonstrated that the insights ob-tained by algorithmic simplification of the full-scale model are inexcellent agreement with biochemical assumptions employed inthe construction of small-scale and medium-scale models.The construction of simplified models of glycolysis on the basis
of time scale analysis has already been addressed for the non-oscillatory regime, using either complex [6] or simple models [16].Considering a mechanism consisting of 24 variables and 21reactions, an 18-step simplified model was developed in Ref. [6],which exhibited good accuracy. Using linearly logarithmic kinetics,it was demonstrated in Ref. [16] that a number of fast variables canbe computed in termsof the slowones, allowing for the eliminationof the respective variables from the simplifiedmodel. In both cases,these algorithms provide limited accuracy, require some kind ofinput from the investigator and do not straightforwardly allow foran intuitive interpretation of the results — shortcomings that theCSP method seeks to overcome.CSP was initially developed to treat large and complex
chemical kinetic mechanisms in the context of combustionproblems [22,23,33,39–43], but later was employed for theanalysis of other physical problems, including processes in thecell [9]. CSP is fully algorithmic, requiring no input from theinvestigator, apart from the detailed model and the accuracy thatthe simplified model is required to provide. The CSP algorithm canidentify the variables affected by the fast time scales, the processesparticipating most in establishing the various equilibria under theaction of the fast time scales, and the processes controlling the slowevolution. In addition, CSP can identify the interactions among thevariables during the slow motion on the manifold.
Table 1The reactions of the full-scale model of glycolysis, adapted from Hynne et al. [34].For the metabolite abbreviations see Fig. 1. The term buffer denotes exchange ofliquid in the tank reactor. The reaction system gives rise to two mass conservationrelationships.
For themodel of glycolysis and the oscillatory regime examinedhere, it will be shown that the long term evolution of the22-dimensional model takes place along a limit cycle that extendsin a three-dimensional subspace of the phase space. This featureis the result of (i) the existence of two conservation laws, (ii) thedevelopment of ten dissipative fast time scales, which force thetrajectory to move on a ten-dimensional manifold and (iii) theeffective decoupling on the manifold of three dimensions fromthe remaining seven, the later being practically decoupled fromall other dimensions of the problem as well. A summary ofpreliminary results was presented in Ref. [44].The structure of the paper is as follows. First the dynamics of
the problemwill be explored with an emphasis placed on the typeof the developing time scales and their significance to a possibleconstruction of a simplified model. After a brief presentation ofthe CSPmethod, the size of possible reduced ordermodels that canbe constructed and the accuracy that these models provide will bediscussed. Next, the dynamics and the couplings on the manifoldwill be analyzed. Finally, themajor features of the simplifiedmodelwill be discussed.
2. The full-scale model of glycolysis
We focus on the full-scale model of the glycolytic pathwayin S. cerevisiae developed by Hynne et al. [34], one of themost comprehensive models of the glycolytic pathway currentlyavailable, a schematic overview of which is depicted in Fig. 1. Themodel involves 24 reactions among 22metabolites, as summarizedin Table 1.The model describes yeast glycolysis in a suspension of intact
yeast cells as a two-phase system with a common homogeneousextracellular and homogeneous intracellular phase, respectively. Itis assumed that CSTR conditions prevail in which glucose, cyanideand a suspension of starved yeast cells flow into the reactor at aconstant rate. The volume is kept fixed by removing the surplusliquid. The governing equations are of the form
dydt= Q−1
(S1R1 + · · · + SKRK
)= g(y) (1)
1800 P.D. Kourdis et al. / Physica D 239 (2010) 1798–1817
Fig. 1. The full-scale model of glycolysis developed by Hynne et al. [34]. Themodel involves 24 reactions, including transport reactions and dilution of theextracellular medium. The subscript x denotes extracellular metabolites. Themetabolite abbreviations are: Glc , glucose; G6P , glucose-6-phosphate; F6P ,fructose-6-phosphate; FBP , fructose-1,6-bisphosphate; DHAP , dihydroxyacetonephosphate; GAP , glyceraldehyde-3-phosphate; BPG, 1,3-bisphosphoglycerate; PEP ,phosphoenol pyruvate; Pyr , pyruvate; ACA, acetaldehyde; EtOH , ethanol;Glyc , glyc-erol; ATP , adenosine-triphosphate; ADP , adenosine-diphosphate; AMP , adenosine-monophosphate; NADH , nicotinamide adenine dinucleotide (reduced); NAD+ ,nicotinamide adenine dinucleotide (oxidized).
where the elements of the N-dimensional column vector y are theconcentrations of the metabolites (in mM), t is time (in min), theN-dimensional column state vector Sk and the scalar Rk denotethe stoichiometric vector and rate, respectively, of the kth reactionwithN = 22 andK = 24 for themodel considered. For the detailedexpressions for the reaction rates and the definition of the variousconstants we refer the reader to the original publication [34]. TheN × N matrix Q is diagonal, its entries equal either to unity for theintracellular metabolites or to the ratio of the extracellular volumeto the total volumeof intracellular cytosol, yvol, for the extracellularones.After an initial transient, the model exhibits either a stationary
or an oscillatory (limit cycle) state, depending on the values ofthe kinetic parameters and the initial conditions. In the following,we adopt the parameters and initial conditions from [34], unlessotherwise stated. In particular, we set the mixed flow glucoseand cyanide concentrations [Glcx]0 = 24.0 mM and [CN+x ]0 =5.60 mM and the ratio yvol = 59. Using these parameters thesystem eventually evolves into the oscillatory state regime. Thenumerical solution of the original and simplified models was
Fig. 2. The evolution of theNADH concentration (mM)with time during the period0 < t < 100 min. On the right: magnification of the part when fully oscillatorymotion is established.
Fig. 3. The trajectory on the [NADH]–[ACA] and the [Glc]–[ATP] planes, during theperiod 450 < t < 500 min.
Fig. 4. The trajectory on the [Glcx]–[PEP] and the [Glc]–[GAP] planes, during theperiod 450 < t < 500 min.
obtained with the LSODE package, which is based on the variableintegration step implicit BDF algorithms [18].The oscillatory behavior of the glycolysis model is displayed
in Fig. 2 for the evolution of the concentration of nicotinamideadenine dinucleotide (NADH) in the period 0 < t < 100 min, thebehaviors of the other metabolites being similar. The oscillatorymotion develops as various transient components die out and ischaracterized by a period Tch = 0.64 min and a frequency ωch =2π/T = 10 min−1, approximately.Inspecting the solution in the phase space confirms that the
solution is attracted towards a limit cycle. As is depicted in Fig. 3 forthe interval 450 < t < 500 min in which all fast initial transientsare exhausted, fully oscillatory motion is established along a limitcycle at sufficiently long times. The structure of the cycle suggeststhat it might be governed by a low dimensional system.Although some projections of the limit cycle, such as those
shown in Fig. 3, suggest that the cycle might be describable as atwo-dimensional system, other projections, such as those shownin Fig. 4, suggest a higher dimensionality of the system.A common cause for the trajectory to evolve in a low
dimensional domain of the phase space is the development ofvarious equilibria, the simplest and most familiar ones beingthe Quasi-Steady State Approximation and Partial EquilibriumApproximation (in the following QSSA and PEA, respectively).
P.D. Kourdis et al. / Physica D 239 (2010) 1798–1817 1801
Fig. 5. The evolution of the rate of change of [BPG] and [AMP] concentrations(mM/min) with time (min) along with that of the related reaction rates.
The possibility that such equilibria develop in the problemconsidered here is further implied by the fact that, past theinitial transient, the rate of change of the concentration of variousmetabolites is much smaller than that indicated by the magnitudeof the reaction rates contributing to these rates of change. Themagnitude of the cancellations occurring is demonstrated inFig. 5 for the concentration of the metabolites BPG and AMP , thegoverning equations of which are
d[BPG]dt
=(R8f − R8b
)−(R9f − R9b
)(2)
d[AMP]dt
= −(R24f − R24b
)(3)
where Rkf and Rkb denote the forward and backward rate of the kthreaction. Comparing the magnitude of the reaction rates and thatof the rate at which the concentration of BPG changes, Fig. 5 showsthat significant cancellations are taking place among the forwardand backward rates of Reactions 8 and 9. These cancellations resultin an evolution of [BPG] according to Eq. (2), much slower than thatsuggested by the magnitude of each of the reaction rates involved.Similarly, Fig. 5 shows that significant cancellations among theforward and backward rates of Reaction 24 result in an evolutionof [AMP], according to Eq. (3), which is much slower than what themagnitudes of the two rates suggest.Such cancellations are usually produced by the action of fast
dissipative time scales, much faster than the characteristic onesof the system’s behavior. Due to these fast time scales, a numberof equilibria develop among various processes in the model. Thenumber of these equilibria equals the number of fast time scales.Fast dissipative time scales relate to the eigenvalues largest inmagnitude of the system’s Jacobian, J = grad(g), the real partof which is negative and much larger in magnitude than theimaginary part.In the problem considered here, at each point in time the ten
eigenvalues with the largest magnitude are real and negative, thenext two forming a complex pair. The evolution of the real andimaginary parts of this eigenvalue pair is displayed in Fig. 6. It isshown that the imaginary part dominates over the real part andis approximately equal to the frequency of the oscillatory motion,λi = O(ωch = 10 min−1). Since this time scale can be consideredas the characteristic of the system, these findings suggest that themaximum number of equilibria that can be established by fastdissipative time scales is ten.The kth time scale is introduced here as
τk =
(√λ2kr + λ
2ki
)−1(4)
where the kth eigenvalue is defined as λk = λkr + iλki, withthe subscripts ‘‘r ’’ and ‘‘i’’ denoting real and imaginary parts,respectively. Fig. 7 displays the evolution of the ten dissipativefastest time scales, τ1 to τ10, along with the characteristic of the
Fig. 6. The evolution in time (min) of the real (solid) and imaginary (broken) partsof the complex eigenvalue pair, λ11,12 = λr ± iλi (min−1).
Fig. 7. The evolution in time (min) of the twelve fastest time scales.
system’s evolution time scale:τchar = τ11 ≡ τ12. (5)It is shown that the largest time scale gap develops between τ1 andτ2, while the gap between τ10 and τchar is relatively small.The significance of these ten fast dissipative time scales in
forcing the trajectory to evolve in a subdomain of the phase spaceand in producing a simplifiedmodel governing the slow oscillatorymotion will be discussed next, after a brief presentation of the CSPmethod.
3. The CSP method
According to the CSPmethod [22,23,42–47], Eq. (1) is cast in theformdydt= a1f 1 + a2f 2 + · · · + aN−1f N−1 + aN f N (6)
where ak and f k denote the CSP basisN-dimensional column vectorand amplitude, respectively, of the kth mode. The amplitudes aredefined as
f k = bk • g(y) (7)where the dual N-dimensional row vectors bk satisfy the orthog-onality condition bk • an = δkn. According to Eq. (6), the elementsof the vectors ai have the units of the corresponding elements inthe state vector y and the amplitudes f i have units of inverse time.Assuming that the M fastest time scales are of dissipative natureand are much faster than the rest, Eq. (6) can be cast as
dydt= ar fr + asfs (8)
where the terms ar fr (fast modes) and asfs (slow modes) relateto the M fast and K = N − M slow, respectively, time scales.
1802 P.D. Kourdis et al. / Physica D 239 (2010) 1798–1817
The various quantities in Eq. (8) are defined as
ar =(a1 . . . aM
)br =
b1...
bM
(9)
as =(aM+1 . . . aN
)bs =
bM+1...
bN
(10)
fr = brg(y) =
f1
...
f M
(11)
fs = bsg(y) =
fM+1
...
f N
. (12)
When the fast dissipative time scales become exhausted, thecorresponding amplitudes become negligibly small:
fr ≈ 0 (13)
so Eq. (8) simplifies to
dydt≈ asfs (14)
denoting that the system evolves on an N − M-dimensionalmanifold emerging as the fast processes equilibrate under theaction of the exhausted fast dissipative time scales. On themanifold,which is defined by the M components of Eq. (13), the solutionevolves according to the slow time scales, as stated by Eq. (14).For the glycolysis model considered here, N = 22 and, as wasdiscussed previously,Mmax = 10.Part of the error committed by neglecting the fast components
in the simplified model, Eq. (14), can be corrected by applyingat the start of each time step the Homogeneous Correction (HChereafter):
ynew = y− arτrr fr (15)
and using ynew , instead of y, as the initial value of the state vectorfor the new integration step [22,23,39]. This correction moves thestate vector closer to the manifold, along the fast directions ar . InEq. (15) theM ×M matrix τrr is the inverse of the matrix
λrr =
(dbr
dt− br J
)ar .
The slow modes accounted for in Eq. (14) include the so-calledconservation modes, which relate to infinitely slow time scales dueto conservation laws, thus making no contribution to the system’sevolution [42]. For the model of the glycolytic pathway consideredhere, two such conservation modes exist:
as can be verified by the stoichiometry of the reactions in Table 1.These modes produce the row vectors, say, b21 and b22, whichproduce f 21 ≡ 0 and f 22 ≡ 0. Such vectors are constructedas follows. The elements of b21 are all zero, except those thatcorrespond to the concentrations [NAD+] and [NADH], which areset equal to 1. Similarly, the elements of b22 are all zero, exceptthose that correspond to the concentrations [ATP], [ADP] and[AMP], which are also set equal to 1. In this way, the vanishingamplitudes f 21 and f 22 express the fact that the time derivative ofthe sums [NAD+]+[NADH] and [ATP]+[ADP]+[AMP] equal zero.
In essence, these two conservation laws state that the solutionof the detailed model, Eq. (1), evolves inside an N − 2 = 20-dimensional space.The CSP basis vectors, ai and bi, are computed using two
distinct iterative procedures [22,23]. The br -refinement increasesiteratively the accuracy bywhich themanifold, as expressed by Eq.(13), is approximated by increasing the accuracy of the set of basisvectors br . Each br -refinement yields
fk+1,r = O(εM fk,r
)(18)
where fk,r = bk,rg(y), k denotes the number of refinementsperformed and
εM = τM/τM+1 < 1 (19)
provides a measure of the fast/slow time scales gap [22,23,45].In Eq. (18), the symbol O(·) indicates that each element ofthe vector fr becomes O(εM) smaller with each br -refinement.The ar -refinement guarantees that only slow time scales areencountered in the simplifiedmodel, Eq. (14). One such refinementis sufficient [45,46].Given that the simplified model is valid as long as the fast
components are negligible, the M fast time scales are declaredexhausted when
|τM+1ar fr | < εrel|y| + εabs (20)
where εrel and εabs denote the relative and absolute error allowedalong a slow time step, εrel being a scalar and εabs being a vector.According to Eq. (20), the error produced by the simplified model,Eq. (14), over the time period of interest, say T , is ofO(fr), assumingthat ar = O(1) and T = O(1).According to Eq. (18), it follows that the accuracy provided by
the simplified model Eq. (14) depends on both:
• the size of the gap among the fast and the slow time scales, asindicated by the magnitude of εM , and• the number of br -refinements employed.Next, it will be demonstrated that these two quantities, along
with the accuracy that the simplified model is required to provide,directly determine the degree to which the original model can besimplified.Since we are interested in the long term behavior of the system,
in the following the initial conditions considered are such thatthe initial value of the state vector y(0) lies on the limit cycle,with all possible initial transients due to the fast time scales beingexhausted.
4. On the dimension of the fast subspace
As stated in the previous section, the manifold is defined by theM vanishing components of Eq. (13), where M is the number ofexhausted fast dissipative time scales.Figs. 8 and 9 show the evolution of the fast amplitudes, f kwhere
k = 1, . . . ,M , for the cases M = 6 and M = 10, respectively,the M = 10 case relating to the maximum dimension that thefast subspace might span, since the 11th time scale is not of adissipative nature but of an oscillatory one and characteristic of theperiod of the limit cycle, as was shown in Section 2.The amplitudes shown in Figs. 8 and 9 were computed by
employing either one or two br -refinements, in both cases withoutor with the HC.In the M = 6 case, Fig. 7 shows that the fast/slow time
scale ratio εM=6 = τ6/τ7 is of O(10−1), indicating that eachbr -refinement will decrease the six fast amplitudes by O(εM=6),providing thus an increased accuracy in the approximation of thecorresponding manifold by this amount.These estimates are verified by the results displayed in Fig. 8,
where it is shown that when setting M = 6, the use of one
P.D. Kourdis et al. / Physica D 239 (2010) 1798–1817 1803
Fig. 8. M = 6. The evolution in time (min) of the six fast CSP amplitudes: first row: one br -refinement, without HC; second row: two br -refinements, without HC; thirdrow: one br -refinement, with HC; fourth row: two br -refinements, with HC.
br -refinement with no use of the HC produces fast amplitudeswhich are O(10−2). This finding suggests that an error of similarmagnitude will be committed by using the simplified modelEq. (14). If two br -refinements are employed with no use of theHC, the six fast amplitudes become O(10−3), their magnitudedropping by O(εM=6), which sets the accuracy improvement whenan additional br -refinement is employed.
The implementation of the HC is shown to be very effective,reducing the magnitude of the six fast amplitudes to O(10−5)when using one br -refinement and to O(10−6) when using twobr -refinements, i.e. producing in both cases an accuracy improve-ment ofO(10−3). This significant improvement in accuracymust beattributed to the high degree of quasi-linear character exhibited bythe six fast modes.
1804 P.D. Kourdis et al. / Physica D 239 (2010) 1798–1817
Fig. 9. M = 10. The evolution in time (min) of the ten fast CSP amplitudes: first row: one br -refinement, without HC; second row: two br -refinements, without HC; thirdrow: one br -refinement, with HC; fourth row: two br -refinements, with HC.
In the M = 10 case, εM=10 = τ10/τ11 is of O(5 × 10−1). As aresult, the effect of each br -refinementwill be half as effective as intheM = 6 case. Indeed, as the results displayed in Fig. 9 show, themagnitudes of the ten fast amplitudes drop from O(5.0 × 10−1),computed after one br -refinement, to O(1.0 × 10−1), after anadditional br -refinement is implemented. The application of theHC reduces themagnitude of the amplitudes by about three ordersto O(10−5); the level of quasi-linearity manifested in the M = 6case is shown to persist in theM = 10 case as well.
Considering the evolution around the limit cycle and thedesiredlevel of accuracy being specified, these findings suggest:
• the maximum simplification levels allowed, and• the tools required to achieve them
when attempting to construct a simplified model.For example, if one is interested in a smaller than O(10−2)
error when traveling one cycle length, the M = 6 simplifiedmodel can be employed when using either one br -refinement and
P.D. Kourdis et al. / Physica D 239 (2010) 1798–1817 1805
Fig. 10. M = 6. Top: the evolution in time (min) of [ATP] and [GAP] computed fromthe full and the simplified models, the latter when employing one br -refinementand without or with the homogeneous correction. Bottom: the relative error of[ATP] and [GAP]when comparing the solutions of the full and simplified models.
the HC or two br -refinements and no HC. For a similar accuracy,the M = 10 simplified model can also be employed, with theimplementation of the HC inevitable now, no matter whether oneor two br -refinements are implemented. On the other hand, if themaximum possible reduction is desired, i.e. M = 10, the findingsdisplayed in Fig. 9 indicate that two br -refinements along with theHC provide an O(10−5) error, higher accuracy requiring additionalbr -refinements.
5. Accuracy of the simplified models
The influences of (i) the number of br -refinements employedand (ii) the use of the homogeneous correction determine directlythe accuracy of the corresponding simplified models, since bothtend to reduce the values of the fast amplitudes. In this sectionsolutions obtained on the basis of the original model will becompared to those obtained on the basis of the simplified modeland the relative error will be reported, the later defined for avariable x as
REx =|xoriginal − xsimplified||xoriginal|
.
All simplified models were of fixed dimensions (N − M = 16or 12, i.e. M = 6 or 10) and they were constructed with onear -refinement. For simplicity, the HC, when used, was applied onlyat the beginning of each time step and not in the subintervalswhere theRHSwas evaluated. The initial conditions are on the limitcycle, so the fast time scales are exhausted and no fast transientstake place.Fig. 10 demonstrates the influence of the HC in the accuracy
of the solution from the simplified model for the M = 6 caseconstructed with one br -refinement. Considering [ATP] and [GAP],it is shown that, when no HC is applied, the relative errorgenerated is between O(10−2) and O(10−1), in accordance withthe magnitude of the fast amplitudes omitted; see Fig. 8. Theapplication of the HC at the start of each time step is shown tosignificantly reduce the error in both the amplitude and the phaseof [ATP] and [GAP].Fig. 11 shows the effect of an additional br -refinement, still
for the M = 6 case. As expected from the magnitude of the fastamplitudes displayed in Fig. 8, it is shown that, when no HC is
Fig. 11. M = 6. Top: the evolution in time (min) of [ATP] and [GAP] computed fromthe full and the simplified models, the latter when employing two br -refinementsand without or with the homogeneous correction. Bottom: the relative error of[ATP] and [GAP]when comparing the solutions of the full and simplified models.
Fig. 12. M = 6, 10. Top: the evolution in time (min) of [ATP] and [GAP]computed from the full and the simplified models, the latter when employing twobr -refinements and thehomogeneous correction. Bottom: the relative error of [ATP]and [GAP]when comparing the solutions of the full and simplified models.
used, an O(10−1) reduction in the relative error is obtained whentwo br -refinements are employed instead of one, i.e. an O(εM=6)accuracy improvement is achieved. As in the previous case whereone br -refinement was employed, Fig. 11 demonstrates that theapplication of the HC reduces the error by a similar amount, i.e. byabout an order.The influence of the number of dissipative time scales
considered exhausted, M , on the accuracy obtained with theresulting simplified model is displayed in Fig. 12, the reportedresults being obtained with two br -refinements and the HC. It isshown that theM = 6 simplifiedmodel provides an O(10−1)moreaccurate solution than the M = 10 model. Given that εM=6 =O(10−1) and εM=10 = O(5×10−1), half of this superior accuracy inperformance of theM = 6model is explained by the fast/slow timescale gap τ6/τ7 being wider than the τ10/τ11 one, which relates totheM = 10 model. The additional accuracy in performance of the
1806 P.D. Kourdis et al. / Physica D 239 (2010) 1798–1817
Table 2The CSP pointer for the ten fastest modes (t = 50 min).
M = 6model is due to the higher effect of theHCwhen it is appliedto the first six amplitudes as compared with when it applies to thefirst ten, as is demonstrated in Figs. 8 and 9.Clearly, for the M = 10 model to provide similar accuracy to
theM = 6 model, more br -refinements are required.At this point, we emphasize that a simplifiedmodel constructed
using directly the conventional QSSAprovides amuch less accuratesolution compared with that provided by any simplified modelproduced by the CSP algorithm. A demonstration of this claim ispresented in Appendix A.
6. CSP diagnostics on theM = 10 manifold
The CSP method, by decomposing the original system of Eq. (1)into a fast and a slow component, as shown in Eqs. (13) and (14),allows for the acquisition of significant biophysical understandingof themechanisms involved that are responsible for the generationof the low dimensional manifold and drive the system on thismanifold.Here and in the subsequent section several fundamental in-
sights are presented that can be acquired by employing var-ious CSP tools. In the following, we examine the M = 10case, which assumes all ten dissipative time scales faster thanτchar to be exhausted. This assumption amounts to the con-sideration of a 10-dimensional manifold, given that the solu-tion of the 22-dimensional problem effectively evolves inside a20-dimensional phase space due to the two conservation lawsEqs. (16) and (17). As in the previous section, all numerical resultsthat are reported here were obtained with initial conditions lyingon the low dimensionalmanifold, so that all fast time scales are ex-hausted. The CSP data are obtained after two br -refinements andone ar -refinement.
6.1. The CSP pointer
For the M = 10 case, the ten fastest time scales are assumedexhausted and the manifold is described by ten equations of theformf k = bk • g(y) = qk1f R
1f+ · · · + qk24f R
24f
+ qk1bR1b+ · · · + qk24bR
24b≈ 0 (21)
where the second equality follows from Eq. (1), qkif = −qkib =
bkQ−1Si and k = 1, . . . ,M . Each of theseM equations is associatedwith a (usually small) number of variables, the dynamics of which
are affected by the corresponding kth fast time scale and theyparticipate in the expression for the rates Rif or Rib, contributingsignificantly to the cancellations occurring [40,43]. These variablescan be identified with the help of the CSP pointer [33,42,43]:
Dk = diag[akbk
]=[a1kb
k1, . . . , a
Nk bkN
](22)
where a1kbk1+· · ·+a
Nk bkN = 1. A value of a
ikbki close to unity identifies
the ith variable associated with the kth CSP mode and the relatedtime scale, as discussed previously.The non-negligible values of the CSP pointer for the ten fastest
modes are displayed in Table 2, at t = 50 min, a point in timewhere all these modes are exhausted, as displayed in Fig. 9, sincethe simulations reported here started from the limit cycle. It isshown that the first mode points to [BPG], the second mode pointsto [GAP], the thirdmode points to both [AMP] and [ADP], the fourthmodepoints to [PEP], the fifthmodepoints to [F6P], the sixthmodepoints to [NADH] and to a lesser degree to [NAD+], the seventhmode points to [DHAP], the eighth mode points to [ACA], the ninthmode points to [Glc] and the tenth mode points to [EtOH]. The factthat these variables are most related to the fast dynamics as thesystem evolves around the limit cycle is in excellent agreementwith previous findings for glycolytic oscillations. For example,an (a priori) QSSA was employed for [BPG] in the (medium-scale) model of Wolf et al. [37], based on available biochemicalinformation. Likewise, themetabolites [GAP] and [AMP] are knownto participate in the rapidly equilibrating reactions 7 and 24,respectively.Experience shows that when the CSP pointer identifies a single
variable (as in all except the third and sixth modes), the relatedconstraint f i ≈ 0 to leading order resembles a QSSA, while whenmore than one variable is identified the constraint to leading orderresembles a PEA [47]. These issues will be further clarified whenthe reactions most responsible for shaping the manifold will beidentified.Regarding the third and sixth modes, the fact that they both
point to two variables is due to the two conservation modes,Eqs. (16) and (17). The variables pointed at by the third mode,[AMP] and [ADP], both participate in the conservation mode,Eq. (16), along with the variable [ATP]. Similarly, the variablespointed at by the sixth mode, [NADH] and [NAD+], are the onlyparticipants in the conservation mode, Eq. (17). Clearly, when oneof the variables in a conservation law undergoes a fast transient,at least one additional variable in the same conservation law mustundergo a similar fast transient.
P.D. Kourdis et al. / Physica D 239 (2010) 1798–1817 1807
6.2. The participation index
The influence of each of the 24 reactions, listed in Table 1, on theshape of the manifold, can be determined by evaluating their con-tribution to each of the ten equations that describe the manifold,Eqs. (21). Usually, in each of Eqs. (21) only a relatively small num-ber of reactions participate in the cancellations occurring, with thecontribution of the remaining reactions being negligible. The im-portant reactions can be identified with the CSP Participation In-dex [33,42,43]:
Pki =qki R
i
24∑j=1|qkjf Rjf | +
24∑j=1|qkjbRjb|
(23)
where k denotes the mode (k = 1, . . . ,M), i denotes the forwardor backward direction of the nth reaction (i.e., i = 1, . . . , 48 forn = 1, . . . , 24),while by definition |Pk1 |+· · ·+|P
k48| = 1. Since only
exhausted modes are considered, for which f k ≈ 0, the followingrelation also holds: Pk1+· · ·+P
k48 ≈ 0. As a result, a relatively large
value of Pki indicates a large contribution of the ith reaction to theconstraint developed along the kth CSP basis vector ak, imposed bythe kth fast time scale being exhausted.Fig. 13 displays the evolution of the most significant Participa-
tion Indices Pki for the ten exhausted modes (k = 1, . . . , 10).Consider the first mode, for which Fig. 13 shows that f 1 ≈ 0 is
due to the equilibration of the eighth and ninth reactions, i.e.,
f 1 = q18f R8f+ q19f R
9f+ q18bR
8b+ q19bR
9b≈ 0 (24)
where q18f = −q18b and q
19f = −q
19b, all being approximately equal
to unity (e.g., at t = 100min q18f = 1.00006 and q19f = −1.00030).
The terms in Eq. (24) relating to the remaining reactions wereomitted, since their contribution was negligible. Given Eq. (2)which governs the evolution of [BPG], this result implies that thedecay of the first amplitude f 1 relates directly to the establishmentof the steady state approximation for [BPG], in accordancewith theCSP pointer results shown in Table 2, which selects [BPG] as thevariable pointed at for the first mode.Similarly, for the second mode Fig. 13 shows that f 2 ≈ 0 is due
to the equilibration of the sixth, seventh and eighth reactions, i.e.,
f 2 = q26f R6f+ q27f R
7f+ q28f R
8f
+ q26bR6b+ q27bR
7b+ q28bR
8b≈ 0 (25)
where q26f = −q26b, q
27f = −q
27b and q
28f = −q
28b, all being
approximately equal to unity (e.g., at t = 100 min q16f = 1.0122,q17f = 1.0000 and q
18f = −1.0061). Given that the CSP pointer
selects [GAP] as the variable pointed at and that the governingequation for [GAP] is
d[GAP]dt
=(R6f − R6b
)+(R7f − R7b
)−(R8f − R8b
)(26)
it is clear that the decay of the second amplitude f 2 relates directlyto the establishment of the steady state approximation for [GAP].The departure from unity of the coefficients qik in Eqs. (24) and
(25) and the presence in these relations of additional reaction rates(neglected in Eqs. (24) and (25) and not shown in Fig. 13 due totheir minor contribution) are both higher order corrections in theestablished equilibria [46]. These equilibria are expressed fully byf 1 ≈ 0 and f 2 ≈ 0, the leading order of which is the QSSA of [BPG]and [GAP], respectively. A discussion of these issues is presented inAppendix B.Consider now the third mode, for which the CSP pointer
selects [AMP] and [ADP] as the variables pointed at. Using the
Participation Index and considering terms that produce |P3i | >0.02, the amplitude of the mode simplifies to
f 3 = q39f R9f+ q324f R
24f+ q39bR
9b+ q324bR
24b≈ 0 (27)
where q39f = −q19b and q
324f = −q
124b, theminor contributions from
reactions nos 3 and 10 being neglected here, in accordancewith theresults displayed in Fig. 13. The corresponding Participation Indicesare
These findings suggest that the relation f 3 ≈ 0 is to leadingorder the result of the equilibration of the forward and backwarddirections of reaction no. 24, R24f ≈ R24b, which involves boththe variables pointed at [AMP] and [ADP]. However, additionalreactions introduce higher order contributions to the decay of f 3,mainly reaction no. 9 and to a lesser degree reactions nos 3 and 10,all of which involve [ADP]. The origin of these additional terms,which introduce significant higher order contributions to thepartial equilibration of reaction no. 24, is discussed in Appendix B.The influence of these additional higher order terms on the
shape of the manifold is displayed in Fig. 14, where the solution onthe [AMP]–[ADP] plane is shown, computed with both the originalrates R9f and R9b and the ones perturbed by 5% in magnitude,i.e. 1.05 ∗ R9f or 1.05 ∗ R9b. As expected, the resulting responseis of the order of the 5% perturbation on a term contributing 6%–8%in the cancellations occurring among the additive terms in theexpression for f 3, Eq. (27).Similar conclusions can be reached for the physical meaning
of the decay of the remaining amplitudes, f 4 to f 10. For example,Fig. 13 shows clearly that the amplitude decay of the fourth modeis associated with the establishment of the QSSA for the variable[PEP], which is the corresponding variable pointed at (see Table 2),and its evolution is governed by the equation
d[PEP]dt=(R9f − R9b
)− R10, (28)
the reactionno. 10being irreversible. In addition, Fig. 13 shows thatthe decay of the amplitudes f 5 and f 6 is related to the QSSA for thevariables F6P and NADH , which are the corresponding variablespointed at and are governed by the equations
d[F6P]dt
=(R4f − R4b
)− R5 (29)
d[NADH]dt
=(R8f − R8b
)− R12 − R15. (30)
The contributions of the forward and backward rates of reactionsnos 6, 9 and 24, shown in Fig. 13 in the cancellations occurringamong the additive terms of f 5 and f 6 are due to higher ordereffects.Investigation of all Participation Indices of all ten exhausted
amplitudes f i (i = 1, . . . , 10) reveals that the reactions nos 2, 3, 4,5, 6, 7, 8, 9, 10, 11, 12, 13, 18 and24 exhibit significant contributionsin the equations describing themanifold, Eq. (21), producing valuesof |Pki | > 0.01.
6.3. The G matrix
The version of CSP presented in [47] introduced a numberof tools that allow for further insight into the biochemicalmechanisms of glycolytic oscillations. According to this version,the state vector y is decomposed as
yr =
y1
...
yM
ys =
yM+1
...
yN
(31)
1808 P.D. Kourdis et al. / Physica D 239 (2010) 1798–1817
Fig. 13. M = 10. The evolution in time (min) of the most significant Participation Indices Pki for the fastest ten exhausted modes (i = 1, . . . , 10).
where yr and ys are M- and N − M-dimensional column vectors,respectively. The M elements in yr are those which are mostrelated to the M fastest time scales, as identified by the CSP
pointer, ys containing the remaining N − M elements of thestate vector y. The vector field g is decomposed accordingly,as
P.D. Kourdis et al. / Physica D 239 (2010) 1798–1817 1809
gr =
g1
...
gM
gs =
gM+1
...
gN
. (32)
In addition, theM × N dimensional Grs matrix is introduced:
Grs =
∂y1
∂yM+1. . .
∂y1
∂yN...
...
∂yM
∂yM+1. . .
∂yM
∂yN
(33)
which can be locally computed in closed form and in an iterativemanner [47]. The matrix Grs is meaningful only when the statevector evolves on the manifold and provides the response of thevariables in yr to perturbations of the variables in ys. With thesequantities, Eq. (13) that defines the manifold can be cast as
gr = Grs gs (34)
stating that the rate of change of theM components in yr is a linearfunction of the rate of change of the N −M components of ys. Theterms Gij g
j that contribute most to the value of the ith componentof gr will identify the major dependencies of the rate of change ofthe fast variables in yr on the rate of change of the remaining onesin ys.Considering the M = 10 case and results according to the
CSP pointer discussed in Section 6.1, the best selection for the tenelements in yr are the variables that are most related to the tenfastest time scales, i.e.,
With this definition of yr and ys, it is shown that themajor termscontributing most in the RHS of Eq. (34), as the solution evolvesaround the limit cycle, are those related to the rates of changeof only five variables in ys, namely [ATP], [G6P], [ADP], [FBP] and[NAD+]. The contribution of the rate of change of the remainingcomponents in ys is negligible. This is demonstrated in Fig. 15where the evolution of the rate of change of each component in yris displayed along with that of the non-negligible additive terms inthe corresponding components in the RHS of Eq. (34). In particular,it is shown that the following six very accurate relations hold:
gGAP = GGAPFBP gFBP
gDHAP = GDHAPFBP gFBP
gAMP = GAMPADP gADP+ GAMPATP g
ATP
gBPG = GBPGFBP gFBP+ GBPGATP g
ATP+ GBPGADP g
ADP
gF6P = GF6PG6P gG6P+ GF6PATP g
ATP+ GF6PADP g
ADP
gGlc = GGlcG6P gG6P+ GGlcATP g
ATP+ GGlcADP g
ADP
as well as the following two approximate ones:
gACA ≈ GACAFBP gFBP
gEtOH ≈ GEtOHFBP gFBP .
The rates of change of the remaining elements in yr , namely gPEPand gNADH , involve the contributions of the rates of change ofall five variables [ATP], [G6P], [ADP], [FBP] and [NAD+], with thecontribution of the rate of [FBP] being the most significant.
Fig. 14. The solution on the [AMP]–[ADP] plane during the period 450 < t <500 min, using the original and the perturbed rates R9f and R9b .
The fact, shown in Fig. 15, that the rate of change of the tenvariables in yr can be expressed as a linear combination of therate of change of only the five components in ys1 suggests thatthey are not affected by the rate of change of the remainingseven components in ys2. We note that the decomposition alsoreflects differences in the topology of the pathway. While thevector ys1 primarily contains proper intermediates of glycolysis([G6P], [FBP]), as well as indicators of the redox and energystatus ([ATP], [ADP], [NAD+]), the vector ys1 mainly consists ofmetabolites in the extracellular medium. Notable exceptions areglycerate ([Glyc]) and pyruvate ([Pyr]), which will be discussed inmore detail below.To further validate the assertion that the rate of change of the
variables in yr are not affected by the remaining seven componentsin ys2, we compare the time evolution of the original model withthat of a perturbed model, as shown in Fig. 16. Specifically, theperturbed model consists of the original model in which themagnitude of the rate of change of the variables in ys2 is increasedby 20% for all times after t = 25 min. In other words, in theperturbedmodel the governing equations for yr and ys1 are similarto the ones in the original model, while the governing equation forys2 is initially, up to t = 25 min, similar to that of the originalmodel, say,
dys2
dt= gs2(yr , ys1, ys2) = gs2(y) (35)
being replaced for all subsequent times, t ≥ 25 min, by theequation
dys2
dt= 1.20 gs2(yr , ys1, ys2) = 1.20 gs2(y). (36)
The results displayed in Fig. 16 show that the perturbationimposed from t = 25 is immediately felt by the components inys2, such as [Glyc] and [Pyr]. As regards the components in yr , suchas [BPG] and [GAP], Fig. 16 verifies that the imposed perturbationhas no effect on them.Moreover, Fig. 16 indicates that the imposedperturbation has no effect on the components of ys1, such as[ATP] and [NAD+]. Again, these findings are in good agreementwith previous work on the mechanisms responsible for glycolyticoscillations. In particular, itwas previously pointed out that neitherthe oscillations nor the cell–cell synchronization via extracellularacetaldehyde ([ACAx]) can be understood as a simple cause-and-effect chain running through the intermediates of the glycolytic
1810 P.D. Kourdis et al. / Physica D 239 (2010) 1798–1817
Fig. 15. M = 10. The evolution in time (min) of the M components in the LHS of Eq. (34) and the most important additive terms of the corresponding components in theRHS, i.e. g i (i = 1, . . . ,M), and of the largest Gikg
k (k = 1, . . . ,N −M).
pathway itself [37]. For example, in an in silico experiment byWolfet al. [37], the concentrations of several glycolytic intermediateswere clamped to fixed (stationary) values — nonetheless the
oscillations persisted. To understand this puzzling phenomenon,it was suggested that the oscillations mainly originate aroundthe phosphofructokinase reaction (no. 5 in Fig. 1) and are
P.D. Kourdis et al. / Physica D 239 (2010) 1798–1817 1811
24 26 28 30
time
4.36
4.40
[Gly
c]
24 26 28 30time
1
16.7485
16.7495
[Pyr
]
24 26 28 30time
2.4×10-4
2.8×10-4
3.2×10-4
[BP
G]
24 26 28 30time
0.115
0.120
0.125
[GA
P]
24 26 28 30time
2.0
2.2
[AT
P]
24 26 28 30time
0.64
0.66
[NA
D+ ]
Fig. 16. The effects of a 20% perturbation in the magnitude of the rate of change ofthe five components in ys2 imposed from t = 25 on the solution.
propagated by oscillations in energy and redox status. Our resultscorroborate this assertion, as the metabolites in the vector ys1exactly correspond to the situation described above, whereaspyruvate has only a negligible influence on the oscillations.
6.4. Discussion and summary
In this section is was demonstrated that the development of theten-dimensional manifold is the result of ten linearly independentequilibria among some of the reaction rates in the full-scale modelof the glycolytic pathway. To leading order, these equilibria takethe form of either QSSA or PEA, their final forms being morecomplex expressions with the addition of higher order terms.Furthermore, it was shown that along the limit cycle the rates ofchange of certain variables were important participants in theseequilibrations, while those of others were of no importance.In addition, it was demonstrated that the rate of change of
the ten variables in yr can be regulated by varying the rate ofchange of the five variables in ys1. Taking into account the twoconstraints, Eqs. (16) and (17), and the fact that the two variablesin ys1, [ADP] and [NAD+], are also affected by the fast time scales,this dependence of yr can be further reduced to the rate of changeof only three variables in ys1, namely [ATP], [G6P] and [FBP],which are slow variables since they are not pointed at by theCSP pointer of the ten fast modes; see Table 2. The limit cycle inthe ([ATP], [G6P], [FBP])-space is shown in Fig. 17. These findingsare in excellent agreement with previous results on the origin ofglycolytic oscillations.It was further demonstrated that perturbations in the rate of
change of the seven variables in ys2 had no significant effect on theevolution of the other variables in the model, namely those in ys1and, through them, those in yr .An explanation of the effective decoupling of the variables in ys2
from the remaining variables in the model is given by the results
4.2
4.4
2.0 2.2
4.5
4.8
5.1
5.4
FBP
ATP
G6P
FBP
Fig. 17. The limit cycle in the ([ATP], [G6P], [FBP])-space.
Fig. 18. The evolution of the three variables in ys1 (left) and all ones in ys2 (right)on the manifold.
Fig. 19. The evolution of the three variables in ys1 on the manifold, when allvariables in ys2 are either allowed to evolve (‘orig’) or held constant (‘mod’).
displayed in Fig. 18. It is shown that on the cycle all seven variablesin ys2 exhibit a variation which is either very small (like that of[Pyr] and [Glyc]) or negligible (like that of [Glcx], [Glycx], etc.) —in contrast to the three controlling variables in ys1, [ATP], [G6P]and [FBP], which exhibit a significant variation. This difference inamplitudes is also observed experimentally [37,34], which led tothe hypothesis that the oscillations mainly propagate via energyand redox cofactors, rather than via the backbone of glycolysis.A further demonstration of the negligible influence of the
variables in ys2 on all other variables when the solution evolveson the limit cycle is presented in Fig. 19. In that figure, the solutionof the original problem is compared to that of a modified problem,in which the variables in yr and ys1 are allowed to evolve as in theoriginal problem, but the variables in ys2 are not allowed to evolve— in analogy to the computational experiments of Wolf et al. [37].The conclusion is that the motion on the manifold around
the limit cycle can be regulated by controlling the evolution of
1812 P.D. Kourdis et al. / Physica D 239 (2010) 1798–1817
just three variables, [ATP], [G6P] and [FBP]. Since these variablesare not affected by the fast time scales, their evolution will bedetermined by linear relations among the reaction rates in themodel operating in the slow subdomain of the phase space,according to Eq. (14).Finally, it was shown that, of the 24 reactions in the glycolysis
model, only 15 of them exhibited a significant participation inthe ten equilibria that gave rise to the ten-dimensional manifold,the remaining nine reactions having negligible influence. Thesignificance of the 24 reactions in the evolution of the glycolysisprocess around the limit cycle will be examined next.
7. The simplified model
The slow motion around the limit cycle is governed by thesimplified model equation (14), which is free of the componentsof the problem that are responsible for the development of the Mfastest dissipative time scales. Considering the M = 10 case, thesimplified model is of the form
dydt≈ a11f 11 + · · · + a20f 20. (37)
Given the original form of the governing equations, Eq. (1), thismodel can be recast as
dydt≈ Q−1
(a11f 11 + · · · + a20f 20
)(38)
where Q−1ak = ak. In this form the vectors ak (k = 11, . . . , 20)can be viewed as the stoichiometry of the simplified problem andthe amplitudes f k as the corresponding rates.An important issue that arises from Eq. (38) is the identification
of the original reactions that aremost responsible for the evolutionof the trajectory along the limit cycle, according to the slow timescales.From the definition of the CSP amplitudes f k, Eqs. (7) and (21),
the contribution of each direction of the 24 reactions, listed onTable 1, in the RHS of Eq. (37) or (38) can be evaluated with theCSP Importance Index [33,42,43]:
Ini =
20∑s=11ans q
siRi
20∑s=11
(24∑j=1|ans q
sjf Rjf | +
24∑s=1|ans q
sjbRjb|
) (39)
where n (n = 1, . . . ,N) refers to the 22 variables in y, i (i =1f , . . . , 24f , 1b, . . . , 24b) refers to the forward and backwarddirections of the 24 reactions in the detailed model. By definition|I1n | + · · · + |I
48n | = 1, so the magnitude and sign of the value
of Ini denote the degree and the manner in which the ith reactioncontributes to the evolution of the nth variable as the trajectorymoves around the limit cycle.The significant Importance Indices for the three controlling
variables in ys1, [ATP], [G6P] and [FBP] are displayed in Fig. 20.It is shown that reactions nos 3, 4, 5, 6, 8, 11, 12, 22, 24 exhibita significant influence on the motion of the trajectory along thelimit cycle, the effect of the remaining reactions being negligible.In Section 6.2 itwas shown that these reactions, with the exceptionof reaction no. 22, contribute significantly to the shape of the ten-dimensional manifold.Such information is of relevance if one seeks to devise ways
to control the structure of the limit cycle in the phase space. Forexample, suppose it is desired to alter the shape and period of thelimit cycle, leaving the manifold intact. The results presented sofar suggest that such a goal can be achieved by manipulating therate of reaction no. 22, [ATP] + [G6P] → [ADP] + storage. The
Fig. 20. The evolution in time (min) of the largest Importance Indices Iki for the slowvariables FBP , G6P and ATP , for the M = 10 simplified model, when the solutionevolves around the limit cycle.
outcome of such an experiment is demonstrated in Fig. 21, wheretrajectories on the [ATP]–[G6P] plane for three different values ofthe rate constant of the 22nd reaction, k22, are displayed.It is shown that decreasing the value of k22 results in larger
limit cycles and periods, a 20% decrease resulting in a period Tch =0.76min, i.e. an 18% increase relative to the originalmodel value. Incontrast, increasing the value of k22 tends to produce smaller limitcycles and periods, leading to the disappearance of the oscillatorymotion, as is the case with a 20% increase in the value of k22.In order to investigate the origin of this behavior, the fastest
twelve time scales were computed for the two perturbed valuesof k22 and are displayed in Fig. 22. The ten fastest of these relate tothe ten exhausted CSP modes that describe the ten-dimensionalmanifold and the two slower ones that characterize the slowmotion on the manifold.Taking into account the time scales for the original value of
k22, shown in Fig. 7, the results displayed in Fig. 22 suggest thefollowing. The small variations in the value of k22 considered didnot alter the magnitude of the ten exhausted fastest time scales or
P.D. Kourdis et al. / Physica D 239 (2010) 1798–1817 1813
1.5 2.0 2.5 3.0
[ATP]
4
5
6
7
[G6P
]
k 22
0.8 k 22
1.2 k 22
Fig. 21. The trajectories on the [ATP]–[G6P] plane for three different values of therate constant of the 22nd reaction, k22 .
48 49 50time
τi
0.8 k22
48 49 50time
τi
1.2 k22
1e-06
1e-04
1e-02
1e+00
1e-06
1e-04
1e-02
1e+00
Fig. 22. The twelve fastest time scales, for two perturbations in the value of therate constant of the 22nd reaction: left: 0.8k22; right: 1.2k22 . The 11th and 12thtime scales are denoted in red and blue, respectively. (For interpretation of thereferences to colour in this figure legend, the reader is referred to the web versionof this article.)
their nature; their dissipative character is also not altered. Rather,it is mostly the characteristic time scale, τchar = τ11 = τ12, whichis affected. In particular, decreasing values of k22 resulted in slowerτchar , whereas increasing values of k22 had the opposite effect.An additional feature of a 20% perturbation in the value of k22,
which is crucial for the behavior displayed in Fig. 21, is relatedto the nature of the time scales τ11 and τ12. In particular, a20% decrease leaves the corresponding eigenvalues, λ11 and λ12,qualitatively similar to those in the unperturbed case, as shownin Fig. 6, the two eigenvalues forming a complex conjugate pair,the real part of which oscillates around zero. In contrast, as shownin Fig. 23, a 20% increase in the value of k22 produces a real partwhich is negative everywhere around the cycle, thus assigning adissipative character to the corresponding time scales and τ11 andτ12.Apparently, the oscillation of the real part of the complex
eigenvalue pair around zero, as happens for the original value ofk22 and for a 20% decrease in its value, is essential for sustainingthe oscillatory behavior of the model. When this oscillation doesnot develop and the real part attains negative values, as occurs for a20% increase in the value of k22, the dissipative character of τ11 andτ12 allows a decrease in the dimensions of themanifold (increase inthe value ofM), further than the ten dimensions. In fact, as Fig. 21shows, in the 1.2k22 case the manifold eventually becomes zero-dimensional, since all time scales slower than τ11 and τ12 are alsodissipative.At this point it is worth indicating that this drastic effect of
reaction no. 22 in altering the dynamics of the glycolysis model,as indicated in Fig. 21, is based not simply on its magnitude, but onits projection in the slow subdomain. In fact, as Fig. 24 shows, itsrate is much smaller than the rate of many other reactions in theglycolysis model.
48 49 50time
λr
0.8 k 22
1.0 k 22
1.2 k 22
-4
0
4
Fig. 23. The evolution in time (min) of the real part of the complex eigenvalue pair,λ11,12 = λr ± iλi (min−1).
48 49 50
time
0
100
200
300
Ri
i = 2fi = 4fi = 6fi = 9fi = 13fi = 22i = 24f
Fig. 24. The evolution in time (min) of the rate of reaction no. 22 along with therates of selected reactions.
8. Conclusions
We have presented an algorithmic simplification of a complexbiochemical reaction network to obtain insights into the functionand relevant interactions in a paradigmatic cellular system. Thestarting point was a full-scale model of yeast glycolysis [34], stillone of the most comprehensive and best validated models ofa biochemical pathway and a paradigmatic model for systemsbiology. The model incorporates fast (of dissipative nature) andslow time scales, which were analyzed separately.It was shown that the solution of the 22-dimensional model
evolves around a limit cycle, which is located on a ten-dimensionalmanifold. On that manifold, seven of its ten dimensions decouplefrom the remaining three. The time evolution along the limitcycle can thus be described in terms of a three-dimensionalsystem involving only the metabolites ATP , G6P , and FBP . Thesefindings are in excellent agreement with previous work onglycolytic oscillations, which showed that the oscillations aremainly confined to the upper part of the glycolytic pathway —suggesting that the oscillations originate in the vicinity of thephosphofructokinase (PFK) reaction.We demonstrated that our algorithmic approach recovers sev-
eral well-known assertions about glycolytic oscillations withoutrequiring further specific biochemical knowledge or further in-put from the investigator. It was shown that glycolytic oscillationscannot be understood as a simple cause-and-effect chain throughthe intermediates of glycolysis. In particular, pyruvate (Pyr), thekey intermediate of glycolysis, was shown to have negligible in-fluence on the limit cycle. This corroborates previous results of
1814 P.D. Kourdis et al. / Physica D 239 (2010) 1798–1817
Wolf et al. [37], who clamped the concentrations of metabolicintermediates to fixed values, without impeding the oscillations.Rather, the oscillations are propagated by the action of the cofac-tors that correspond to the redox and energy status of the pathway,ATP/ADP and NAD+/NADH .A similar reasoning applies to cell–cell synchronization of
glycolytic oscillations. The reproducible observation of limit cycleoscillations implies synchronization between individual cells,which is believed to be mediated by external acetaldehyde(ACAx). However, most current models fail to reproduce the fastsynchronization between cells. Several studies showed that thetransient time until synchronization is achieved is significantlylonger in silico than observed in vivo [34,36,37], with acetaldehydeas the only coupling agent. Indeed, though not explicitly modeled,these results are confirmed by our analysis. External acetaldehydewas shown to have a limited influence with respect to thetime evolution along the limit cycle; thus its putative role asthe sole mediator of cell–cell synchronization warrants furtherexperimental investigations.It was shown that the ten-dimensional manifold, on which
the limit cycle lies, is described by ten algebraic relations amongvarious reaction rates, the significant contributions originatingfrom 14 specific reactions. It was shown in Section 6.2 that,to leading order, these relations resemble the well-knownQSSA and PEA for selected variables and reactions, respectively.However, as was shown in Appendix A, these approximationsalone are inadequate for producing an accurate reduced model.Therefore, the simplified model had to incorporate higher ordercorrections, in order to produce accurate results. As was shown inAppendix B, such corrections involve reactions not participating inthe particular QSSA or PEA and increase drastically the accuracy inapproximating the manifold.In essence, these higher order corrections can be viewed as
the asymptotic expressions for the amount by which the fastvariables (or fast reactions) assumed in QSSA (or in PEA) failfrom achieving full steady state (or full equilibrium) status. Inthe simplified model, the fast reactions which participate in therelations denoting QSSA or PEA are substituted by their asymptoticexpressions, which involve slow reactions. It was shown inSection 7 that as a result, the rate of change of the three controllingmetabolites ATP , G6P , and FBP in the simplified model depends onthe rate of reactions no. 8 and no. 11, neither of which involvesthese three metabolites.In summary, our analysis demonstrates the usefulness of al-
gorithmic simplification using the CSP method to obtain insightsabout the functioning and interactions in cellular pathways. In par-ticular, and in accordancewith the goals of systems biology, the re-cent advances in molecular biology for generating transcriptomic,proteomic and metabolomic data in large quantities are expectedto result in the construction of increasingly detailed models of cel-lular systems. We expect that algorithmic simplification of largereaction networks will play an increasing role in the analysis – andeventually understanding – of the interactions taking place in liv-ing cells.For physical problems exhibiting oscillatory behavior, such as
the one analyzed here, the identification of the components ofthe reaction network that sustain the dynamics of the oscillatorymotion and determine the size of its period is of great significance.Since such a study requires the analysis of intermediate timescales (in particular, the fastest of the slow ones), the fast/slowdecomposition reported here is the first step towards its successfulcompletion.
Acknowledgements
PDK and DAG gratefully acknowledge the support of a 5EBE2007 internal grant by NTUA. RS is supported by the programFORSYS-Partner of the BMBF as well as by the transnationalresearch initiative ‘Systems Biology of Microorganisms’ (SysMO).
Appendix A. A simplifiedmodel constructed on the basis of theQSSA and PEA
To compare our results, a conventionally simplified modelcan be constructed using the QSSA with the algorithm reportedin [48,49]. With the aim of constructing a ten-step model, thisalgorithm requires the identification of ten fast variables and anequal number of fast reactions. The stoichiometry of the simplifiedmodel is then constructed such that it is free of the fast variables,the corresponding rates being free of the rates of the fast reactions.Taking into account the CSP pointer results presented in
Section 6.1, we consider the ten fast variables [BPG], [GAP], [AMP],[PEP], [F6P], [NADH], [DHAP], [ACA], [Glc] and [EtOH] to bein a steady state. According to the reactions in Table 1, theseassumptions are expressed by the relationships
where Rk denotes the net reaction rate of the kth reaction, i.e. Rk =Rk,f − Rk,b.The ten-step simplified mechanism is constructed using the
ten linearly independent equations (A.1) and the two conservationlaws Eqs. (16) and (17). The ten components of Eq. (A.1) areemployed for the evaluation of the ten variables in the steadystate. The two conservation laws, Eqs. (16) and (17), are used forthe computation of the variables [NAD+] and [ADP], respectively,shown in Table 2 to be related to the fast time scales. The simplifiedmodelwill apply to the remaining ten variables [ATP], [G6P], [FBP],[Glyc], [Pyr], [Glcx], [EtOHx], [Glycx], [ACAx], [CN−x ], which werealready shown to relate to the slow time scales of the problem.Following the algorithm reported in [48,49], let us further
assume that the simplified model is free of the ten fastestlinearly independent reactions that are most responsible for thegeneration of the ten components of the algebraic equation (A.1).These reactions are identified as those consuming most (i.e., theyexhibit the largest magnitude around the limit cycle) the relatedten variables in the steady state. Following this procedure, thesimplified model is constructed so that reactions nos 2, 4, 6, 7, 9,10, 12, 13, 18 and 24 are absent in the simplified model.After making these assumptions, the following simplified
model is constructed:d[ATP]dt
= −R3 − R5 + 2R8 − R22 − R23
d[G6P]dt
= R3 − R5 − R22
d[FBP]dt= R5 − 0.5(R8 + R15)
d[Glyc]dt
= R15 − R16
d[Pyr]dt= R8 − R11
d[Glcx]dt
=(R1 − R3
)/yvol (A.2)
P.D. Kourdis et al. / Physica D 239 (2010) 1798–1817 1815
Fig. A.1. A comparison of the solution from the original model and theconventionally reduced one, referring to the reduced model: top: fast variablescomputed from the algebraic equation (A.1); bottom: slow variables from thedifferential Eq. (A.2).
d[EtOHx]dt
=(R8 − R14 − R15
)/yvol
d[Glycx]dt
=(R16 − R17
)/yvol
d[ACAx]dt
=(−R8 + R11 + R15 − R19 − R20
)/yvol
d[CN−x ]dt
=(−R20 + R21
)/yvol
which, as expected, applies to the ten slow variables and is free ofthe ten fast reactions.The level of accuracy provided by this simplified model is
demonstrated in Fig. A.1, for variables computed either from thealgebraic (top) or the differential (bottom) equations, Eqs. (A.1) and(A.2), respectively.It is shown that, while the solution from the original model
evolves in a steady oscillatorymotion, the solution of the simplifiedmodel undergoes a transient before reaching a different oscillatorymotion. A comparison of the two solutions clearly demonstratesthat considerable error is introduced by the simplifiedmodel, bothin the magnitude of the various concentrations and in the periodof oscillation.
Appendix B. On the higher order correction to the QSSA andPEA
It was shown in Section 6.2 that – to leading order – theamplitudes f i of the exhausted modes resemble either Quasi-Steady State Approximations (QSSA) or Partial EquilibriumApproximations (PEA). Furthermore, it was shown that these ap-proximations were amenable to higher order corrections, whichare provided by the inclusion in the corresponding expressions ofreaction rates not participating in the QSSA or PEA.Here the origin of the additional terms, which provided higher
order corrections to the QSSA and PEA, will be examined. Only thefirst-order correction will be considered, the discussion of higherorder corrections being an extension of the material that follows.The first and third fastest modes of the M = 10 case will beexamined, the former based on the QSSA of [BPG] and the latteron the PEA of reaction no. 24. Similar comments can be stated forthe remaining exhausted modes.
Consider the QSSA of [BPG]:
S0 = R8 − R9 ≈ 0 (B.1)
where if the reaction is reversible Rk = Rkf − Rkb and thesuperscripts f and b denote the forward and backward directions,respectively. If an expansion of the vector field g(y) in the form ofEq. (6) was employed, the scalar S0 in Eq. (B.1) could be consideredas the amplitude corresponding to a row basis vector b0 whichhas all of its elements equal to zero, except the one correspondingto [BPG] which is set equal to 1, i.e. S0 = b0 • g(y). Thefirst br -refinement, the implementation of which increases theaccuracy in approximating the manifold according to Eq. (18), isequivalent to differentiating Eq. (B.1)with respect to time [32]. Thisdifferentiation yields
The various parameters in the expressions above are definedin Ref. [34]. The scalar S1 can be considered as the amplitudeof a mode, corresponding to a row basis vector b1 after onebr -refinement, i.e. S1 = b1 • g(y), the starting point of thisrefinement being b0. Assuming that the mode examined here isexhausted, it is expected therefore that setting S1 = 0 will providea more accurate approximation of the manifold than when settingS0 = 0.Values of the scalars S0 and S1 as well as those of the various
additive terms on the RHS of Eqs. (B.1) and (B.2) at t =100 min are displayed in Table B.1. The computed values of S0 =2.2231 × 10−4 and S1 = 5.0207 × 10−7 indicate that a moreaccurate representation of the manifold is indeed provided by theexpression S1 = 0 in comparison to S0 = 0. This improvementin accuracy is provided by small contributions from a number ofreactions that do not participate in the QSSA of [BPG], notablyreactions nos 6, 10 and 14. It is exactly with the weight reflectedin the values shown in Table B.1 that these reactions appear in thecomponents of the Participation Index of the first mode, the majorones being shown in Fig. 13.
1816 P.D. Kourdis et al. / Physica D 239 (2010) 1798–1817
Table B.1The additive terms in S0 and S1 at (t = 100 min).
As indicated by the results displayed in Table B.2, the computedvalues P0 = 0.5002 and P1 = 0.1879 × 10−2 prove that therelation P1 = 0 provides a higher order accuracy in approximatingthe manifold than the relation P0 = 0. As the values of theadditive terms in the expression for S1 displayed in Table B.2 show,this accuracy improvement is the result of the involvement in theexpression of S1 of a number of reactions, in additional to theforward and backward directions of reaction no. 24. The additionalreactions having a significant presence are reactions nos 9, 10,3 etc. This feature agrees very well with the Participation Indexfor the third fastest mode, shown in Fig. 13, where it is shownthat contributions from reactions nos 24f, 9b and 3 balance thecontribution from reactions nos 24b, 9f and 10.
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