Modeling the glycolysis: an inverse problem approach

Jacques Demongeot Andrei Doncescu University J. Fourier University Paul Sabatier Grenoble Toulouse TIMC-IMAG UMR CNRS 5525 LAAS UPR CNRS 8001 Faculty of Medicine Avenue Colonel Roche 38700 La Tronche France 31000 Toulouse France J.Demongeot@imag.fr adoncesc@laas.fr

Abstract We show in this paper that the metabolic chain can be supposed a potential-Hamiltonian system in which the dynamical flow can be shared between gradient dissipative and periodic conservative parts. If the chain is branched and if we know the fluxes at the extremities of each branch we can deduce information about the internal kinetics (e.g. place of allosteric and Michaelian step with respect to those of branching paths, cooperatively) from minimal additional measurements inside the black box constituted by the system. We will treat as example the glycolysis with the pentose pathway whose fluxes measurements are done at the pyruvate and pentose levels. Keywords: metabolic networks, enzymatic kinetics, inverse problem, potential-Hamiltonian decomposition, generalized control strength coefficients

1. Introduction Today the biomathematics represents the key field to explain the functionality of the life science. To analyze a biological system it is necessary to find out new mathematical models allowing to explain the evolution of the system in a dynamic context or to dread doing of a simple manner the complex situations where the human experience overtakes the mathematical reasoning. Several dynamic biological models are composed of a large number of coupled nonlinear differential equations with the particular property that some characteristic quantities are involved in many of these equations. Remarkable quantities are often of great significance from both the physical and mathematical points of view and so, it can seem judicious to strengthen their role in the analysis of the model by use of suitable transformations, in aim of reducing the mathematical and numerical complexity of the problem under consideration. All reactions that allows to go from a molecule to the other is a metabolic pathway, the most known is the glycolyses. 2. The glycolysis with the pentose-phosphate pathway The metabolic networks dynamics are in their enzymatic part ruled by the classical kinetics (essentially Michaelis-Menten, Hill and allosteric ones and combined in functioning-dependent structures [31]). If we limit the

modelling to these kinetics, we can highly simplify their mathematical treatment. We will first consider the example of the glycolysis and after generalize to networks involved in morphogenesis and energetics.

Figure 1. The glycolysis with the pentose pathway. E1 denotes the 4 enzymes of the high glycolysis (hexokinase HK, phosphohexoseisomerase PHI, phosphofructokinase PFK and aldolase AL), E2 denotes the glyceraldehyde-3P-dehydrogenase, E3 denotes the 3 enzymes of the low glycolysis (phosphoglycerate-kinase, phosphoglycerate-mutase and enolase and pyruvate-kinase), E4 denotes the pyruvate-kinase and E5 the 3 enzymes of the oxydative part of the pentose pathway (glucose-6P-dehydrogenase, 6P-gluconolactonase, phospho-gluconatedehydrogenase) Let us define by x1, x2, x3 and x4 the concentrations of respectively the successive main metabolites of the glycolysis: glucose, glyceraldehyde-3-P, 1,3-biphospho-glycerate and phospho-enol-pyruvate. If we consider that all steps summarizing the glycolysis in Figure 1 are Michaelian and reversible, except the first (the enzymatic complex E1 includes the allosteric irreversible kinetics of the phospho-fructo-kinase PFK having the cooperativity n; see [27,31] for this complex kinetics) and the last (both

2009 International Conference on Advanced Information Networking and Applications Workshops

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DOI 10.1109/WAINA.2009.135

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kinase and dehydrogenases of enzymatic complexes E4 and E5 are irreversible), we will consider the differential system (S) ruling the glycolysis with the pentose pathway until ribulose-5-P: dx1/dt=J–V1x1

n/(1+x1n)

dx2/dt=V1x1n/(1+x1

n)-V2x2/(1+x2)+LαV2x3/(1+x3) dx3/dt=αV2x2/(1+x2)-V3x3/(1+x3)-LαV2x3/(1+x3) +L'V3x4/(1+x4) dx4/dt=V3x3/(1+x3)-V4x4/(1+x4)-L'V3x4/(1+x4) Let us consider now the change of variables: yi=xi

1/2 and dyi/dxi=xi

-1/2/2, the yi's are ruled by the system (S'): dy1/dt=[J–V1y1

2n/(1+y12n)]/2y1

dy2/dt=[V1y12n/(1+y1

2n)-V2y22/(1+y2

2)+LαV2y32/(1+y3

2)]/2y2

dy3/dt=[αV2y22/(1+y2

2)-V3y32/(1+y3

2)-LαV2y32/(1+y3

2) +L'V3y4

2/(1+y42)]/2y3

dy4/dt=[V3y32/(1+y3

2)-V4y42/(1+y4

2)-L'V3y42/(1+y4

2)]/2y4 If we consider now the potential P and the Hamiltonian energies Hi defined by [17-20]: P= -JLog(y1)/2+V1Log(1+y1

2n)/4n+V2Log(1+y22)/4+

(V3+ LαV2)Log(1+y32)/4+(V4-+L'V3)Log(1+y4

2)/4; H1=V1Log(1+y1

2n)/4n;H2=αV2Log(1+y22)/4+LαV2Log(1+y3

2)/4H3= V3Log(1+y3

2)/4+L'V3Log(1+y42)/4,

we have: dy1/dt=-∂P/∂y1+∂H1/∂y2; dy2/dt=-∂P/∂y2-∂H1/∂y1+∂H2/∂y3

dy3/dt=-∂P/∂y3+∂H3/∂y4-∂H2/∂y2; dy4/dt=-∂P/∂y4-∂H3/∂y3.

The Hamiltonian energies Hi are conservative, because:

dHi/dt=∂Hi/∂yi dyi/dt + ∂Hi/∂yi+1 dyi+1/dt = 0,

when the potential part of the flow vanishes. Hence when Then we have: V1x*1

n/(1+x*1n) = J

V2x*2/(1+x*2)-LαV2x*3/(1+x*3) = J (1-α)V2x*2/(1+x*2) = J1 V3x*3/(1+x*3) = J2(V4+L'V3)/(V4-L'V3) αV2x*2/(1+x*2)=J2(V3+LαV2)(V4+L'V3)/V3(V4-L'V3)- J2L'V3/(V4-L'V3); V3x*3/(1+x*3)=V4x*4/(1+x*4)+ L'V3x*4/(1+x*4); (V4-L'V3)x*4/(1+x*4) = J2 Hence we calculate α by using the following formula: J1α/(1-α)=J2(V3+LαV2)(V4+L'V3)/V3(V4-L'V3)- J2L'V3/(V4-L'V3)

we are sharing the velocity field dy/dt, where y={yi}i=1,4, between two parts, one gradient dissipating the potential P and the other Hamiltonian conserving the energies Hi's allows, when the attractor is a limit cycle (which is the case if we add fructose-2,6-diphosphate (F26P2) or ADP (see Figure 3 top) as activator of the complex E1 [12,13]), to share the parameters of the system (S') into 3 sets: the parameters in P modulating the amplitude, the parameters in the Hi's modulating the frequency and the parameters both in P and in the Hi's (cf. Figure 2). When the attractor is a fixed point, the first enzymatic complex E1 has a stable stationary state defined by V1x*1

n/(1+x*1n)=J.

If this value x*1 is reached the first because J and V1 are larger than the other maximal velocity parameters Vi's (i>1), then the system (S') becomes (S''): dy2/dt = -∂P*/∂y2+∂H2*/∂y3

dy3/dt = -∂P*/∂y3-∂H2*/∂y2+∂H3*/∂y4 dy4/dt = -∂P*/∂y4-∂H3*/∂y3, where: P*= -JLog(y2)/2+V2Log(1+y2

2)/4+(V3+LαV2) Log(1+y3

2)/4+(V4-+L'V3)Log(1+y42)/4

H*2= -αV2Log(1+y22)/4+LαV2Log(1+y3

2)/4 H*3= -V3Log(1+y3

2)/4 +L'V3Log(1+y42)/4

Figure 2. potential (left) and Hamiltonian (right) systems In this case, the parameters appearing only in P*, like J or V4 are modulating the localisation of the fixed point (cf. [ ] for a general approach of the potential-Hamiltonian decomposition in 2d-differential systems). 2. The inverse problem Let us suppose that we measure the output fluxes J1 and J2. Then from the system (S) we can calculate the sharing parameter α which regulates the pentose pathway and the low glycolysis dispatching: V1x*1

n/(1+x*1n) = J

V2x*2/(1+x*2)-LαV2x*3/(1+x*3) = V1x*1n/(1+x*1

n) (1-α)V2x*2/(1+x*2) = J1 αV2x*2/(1+x*2) = V3x*3/(1+x*3) + LαV2x*3/(1+x*3) -L'V3x*4/(1+x*4) When all the n fluxes of a metabolic system have reached their stable stationary value, then we can define the notion of control strength Cik exerted by the metabolite xi on the flux Φk of the kth step by: Cik=∂LogΔΦk/∂LogΔxi [21,28,34] and we have: ∀ k=1,n, Σi=1,n Cik = 1.

P H

dx/dt = -?P/?x dy/dt = -?P/?y

dx/dt = ?H/?y dy/dt = -?H/?x

P H

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Figure 3. Effectors (activators and inhibitors) of the PFK inside the enzymatic complex E1 (top); reduction of parametric instability region by taking into account the fraction of metabolites fixed to the PFK (when V1 is small), where σ1 is the entry flux of fructose-6-phosphate (F6P) and ρ is the ratio KR,F6P/KR,ADP between the association constants of F6P and ADP to the active allosteric form R of the PFK (middle); experimental evidence of oscillations in the large parametric instability region (V1 large) [12] It can be used to prove that the most regulating molecules in glycolysis are the energetic molecules ATP and ADP (which conversely are mainly produced by glycolysis: for example, 84% of the ATP production in yeast is provided by glycolysis [29]). When oscillations occur, we can use similar variables in order to control the period τk and the amplitude Ampk of the kth step flux [6,7,32]: Tik=∂Logτk/∂Logxi and Aik==∂LogAmpk/∂Logxi If ξ is the eigenvalue of the Jacobian matrix of the differential system for whose stationary state has bifurcated in a limit cycle (Hopf bifurcation), then τk=2π/Imξ, and we have in the 2-dimensional case: Imξ=(ΔP)2-4(C(P)+C(A)+∂2P/∂x1∂x2(∂2H/∂x2

2

-∂2H/∂x12) +∂2H/∂x1∂x2(∂2P/∂x1

2-∂2P/∂x22)1/2/2,

where ΔP=∂2P/∂x1

2+∂2P/∂x22 is the Laplacian of P,

and where

C(P)=∂2P/∂x12∂2P/∂x2

2-(∂2P/∂x1∂x2)2 is the mean Gaussian curvature of the surface P, both taken at the stationary state of the 2-dimensional differential system. 4. A pure potential case In certain cases the Hamiltonian part vanishes, as in the case of n-switches [9-11] used for modelling plant growth [14,15] and only P rules the network dynamics: dx1/dt=J1+Vx1

n/(1+x1n+ x2

n)-k1x1 dx2/dt=J2+Vx2

n/(1+x1n+ x2

n)-k2x2, and P is defined by: P=-J1Log(y1)/2-J2Log(y2)/2-Vlog[(1+y1

2n+y22n)1/4n] +ky1

2/4+ ky22/4, with

yi=xi1/2, and

dy1/dt=[J1+Vy12n/(1+y1

2n+ y22n)-k1y1

2]/2y1, dy2/dt=[J2+Vy2

2n/(1+y12n+ y2

2n)-k2y22]/ 2y2.

In this case, minima of P are fixed points of the dynamics. Figure 4. Attractor landscape of a 2-swiches with 2 fixed points located at the minima of P (left); Waddington's morphogenetic chreode (right) [33] The successive fluxes are equal to Φk= -∂P/∂yk, and ΔΦk= -Σm=1,n Δym∂2P/∂yk∂ym, hence, because P is C3: Cik= ∂LogΔΦk/∂LogΔxi = 2yi(∂LogΔΦk/∂LogΔyi) +2Δyi(∂LogΔΦk/∂Logyi) = 4yiΔyi(∂LogΔΦk/∂yi) =-4yiΔyi(Σm=1,nΔym∂3P/∂yk∂ym∂yi)/(Σm=1,nΔym∂2P/∂yk∂ym) =-4yiΔyiΔ(∂2P/∂yk∂yi)/Δ(∂P/∂yk)=-4yiΔLog(∂2P/∂yk∂yi), and we have in our example (Figure 4): ∂2P/∂yi

2=-V[(2n-1)yi2n-2(1+yi+1

2n)+yi4n-2]/2(1+y1

2n+y22n)2 +Ji/2yi

2+ki and ∂2P/∂y1∂y2 = ∂2P/∂y2∂y1 = Vny1

2n-1y22n-1/(1+y1

2n+y22n)2.

For exemple, C21=-4y1Δy2[(2n-1)(1+y1

2n)-(2n2+1)y22n-1]/ (1+y1

2n+y22n).

5. A pure Hamiltonian case In some cases, the metabolic system is purely Hamiltonian, as in the case of Volterra system [17,18,32]: dx1/dt=x1(a-bx2); dx2/dt=x2(cx1-d), and H is defined by: H=-cey1+dy1-bey2+ay2, with yi=Logxi, dy1/dt=a-bey2; dy2/dt=c ey1-d. In this case, the orbits of the dynamics are the contour lines of H (Figure 5).

P

x2

x1

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Figure 5. Orbits "center" landscape of Volterra system 6. A control approach The system’s stability is an important property of the model, as it is unlikely that a cell remains in an unstable steady state. Moreover, a stable model of a steady state is an essential prerequisite for control analysis, as control analysis quantifies the system’s infinite reaction on infinite deviations. Evidence of the dynamic reaction system’s stability can be gained by an investigation of the eigenvalues of the jacobian matrix. Furthermore, these eigenvalues contain information on how fast a deviation from the given steady state will disappear. The time constants here are defined as the reciprocal values of the real parts of the eigenvalues. They represent the time a perturbation needs to decline to e-1 of its starting value. In addition, potential non-zero imaginary parts of the eigenvalues point out the system’s ability to oscillate. Statistical optimization process utilize linear estimation techniques (least square estimation) to produce models that describe the research space. All real parts of the eigenvalues are found to be negative, indicating a stable dynamic system behaviour. The small simulated time and the presence of the analytical functions may impede a return to the steady state. The range of time constants in the dynamic system presented in this paper is from t1=0.29ms to t2=120s. This large range of time constants reflects the high stiffness of the system. The highest time constant yet is still four orders of magnitude smaller than the time constant for the dilution by growth, which is 10h. This finding indicates that growth has a negligible influence on the systems capability to return to steady state after a perturbation. Among the eigenvalues of the dynamic system, there are two pairs of conjugate complex eigenvalues. Oscillations are due to two natural frequencies, which can be calculated from the imaginary part Im of the complex

eigenvalues λ by ( )πλ

2Im

. For the presented system the

natural frequencies are f1=0.022s-1, synonymous with a period of 46s, and f2=0.014s-1 (period of 71s). Control characteristics From the dynamic model, flux control coefficients (FCC) of the reactions in the system on the glucose uptake by the phosphotransferase system were determined. The results are given in Fig. 7.

Figure 7. Flux Control Coefficients on PTS

The highest control on glucose uptake is carried by the PTS, but only with a FCC of 0.4. Almost as high are the values of the flux control coefficients of R5PI, PFK, G6PDH, PDH, and the absolute value of the FCC of Ru5P. Thus, both of the glucose degrading pathways depicted in this model are remarkably involved in the control of glucose uptake. In the Embden Meyerhof Parnas pathway, PFK is the enzyme with the highest control. In the ppp, G6PDH exerts a significant control on glucose uptake, but also R5PI and Ru5P, whereas the absolute values of the flux control coefficients of R5PI and Ru5P are virtually the same. Moreover, the fact that the FCC of Ru5P is negative suggests that the important role of R5PI and Ru5P is to ensure a balanced production of Xyl5P and Rib5P in order to achieve a high flux through TKa, TKb and TA. This suggestion would also explain the negative FCC of RPPK. Furthermore, the removal of pyr by PDH has a high control on glucose uptake. The main role of this reaction in the control is to remove carbon from the reaction system. It also lowers the pep/pyr ratio, which plays an important role in the PTS kinetics, but for the whole system this effect is negligible as shown below by the response coefficients of pep and pyr. Although with a small absolute value, the negative FCC of GAPDH is a surprise, as this reaction is crucial for the degradation of glucose. It can be explained in a equivalent way as the high flux control by R5PI and Ru5P. A lowered expression level of GAPDH seems to enhance the fluxes in the nonoxidative part of the ppp. The elasticity of PDH with respect to pyruvate is 3.6. The elasticities of R5PI and Ru5P with respect to Ribu5P are 1. The elasticities of the other 3 reactions with the highest absolute values of flux control coefficients are given in Fig.8.

Figure. 8. Elasticises coefficients

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The PTS rate is determinated by both substrates and products, but the high g6p elasticity stands out. The f6p concentration exerts the main influence on the PFK rate, followed by a noticeable inhibition by pep. In this model, the role of atp in the PFK rate expression is almost negligible. The elasticities of G6PDH indicate that there is a noticeable influence of the anabolic reduction charge,

which is defined as nadphnadp

nadph

CCC+

, on the G6PDH rate.

The roles of pep and pyr in glucose uptake are ambiguous, although both metabolites are directly involved in the phosphotransferase system. While pep is a substrate of PTS, it has a nonnegligible inhibitory effect on PFK. Pyruvate, which is a product of PTS, is also substrate of PDH. Both PFK and PDH exert a significant control on glucose uptake. 7. A general approach We can generalize the inverse approach to genetic networks [1-5] in which we study the influence of boundary conditions [8] analogous to the entry flux conditions considered in this paper. If the influence of a change in these boundary conditions is weak we call them robust or structurally stable [16,22]. The difference with metabolic networks is essentially due to the existence of multiple attractors especially limit cycles [23,25], but these last ones are very useful to create specific spatial patterns (e.g. necessary for the speciation with an easy individual phenomenologic recognition), if we add to the genetic control a metabolic part with various effectors and a diffusion of these effectors, called also morphogens in the case of embryologic processes [26]. 8. Conclusion We have shown in this paper that it was possible to partially do a reverse identification of some kinetic parameters in metabolic processes knowing only the input and output fluxes of the system. We have used in particular the potential-Hamiltonian decomposability of the enzymatic chains dynamics which represents a new approach for simplifying their kinetic equations. References [1] J. Aracena, S. Ben Lamine, M.A. Mermet, O. Cohen and J. Demongeot. Mathematical modelling in genetic networks : relationships between the genetic expression and both chromosomic breakage and positive circuits. IEEE Trans. Syst. Man Cybernetics, 33:825-834, 2003. [2] J. Aracena, J. Demongeot and E. Goles. Mathematical modelling in genetic networks. IEEE Trans. Neural Networks, 15:77-83, 2004. [3] J. Aracena, J. Demongeot and E. Goles. Fixed points and maximal independent sets on AND-OR networks. Discr. Appl. Math., 138:277-288, 2004. [4] J. Aracena, J. Demongeot and E. Goles. On limit cycles of monotone functions with symmetric connection graphs. Theoret. Comp. Sci., 322:237-244, 2004. [5] J. Aracena and J. Demongeot. Mathematical Methods for Inferring Regulatory Networks Interactions: Application to Genetic Regulation. Acta Biotheoretica, 52:391-400, 2004.

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