One of the main steps of kinetic modeling of biochemical pathways is a development of kinetic models of individual enzymes, consisting in derivation of rate equations and verification of parameters using available experimental data.  Exactly during this step of modeling metabolic regulation of biochemical pathway is taken into account at the expense of rate equations modification and bringing into them additional expressions describing enzyme activation or inhibition.

Approach for a development of individual enzymes models has same stages as such one for models of biochemical pathways. We use four levels of different specification for the development of enzyme models. Choice of specification level for enzyme models depend on three causes: (i) complexity of catalytic cycle, (ii) complexity of regulatory mechanism and (iii) amount of experimental data characterizing these processes. It should be noted that problem of specification level choice is a general question for all types of modeling in biology, because during model development researcher should always take into account maximum of object’s properties and from other side verify them with sufficient number of experimental data. Because of this we have used four levels of specification of individual enzyme models:

1. Using Cleland’s classification with minimal number of parameters in rate equation [Cleland W., 1963];
2. Using Cleland’s classification with detail description of enzyme regulatory and catalytic properties [Demin O. et al., 2008];
3. Using generalization of Monod-Wyman-Changeux model proposed by Popova and Sel’kov [Popova S.V. and Sel’kov E.E., 1975];
4. Using separate ODE system for mathematical description of multienzyme complexes functioning [Demin O. et al., 2008];

The first level of specification is simpler the other ones [Cornish-Bowden A., 2001]. Such rate equations (eq. 1) are used, when studying enzyme has simple catalytic cycle and hasn’t metabolic regulation. For example, in case of the most of isomerization reactions we use general Michaelis-Menthen rate equation. In case of multisubtrates enzyme we use this level of specification, when experimental data characterizing enzyme functioning represent only certain kinetic parameters (i.g. K_{m_app} (apparent Michaelis constants), V_m, k_cat or K_eq).

(Eq. 1)

where S_i – enzyme substrate, Р_i – enzyme products, K_{m_app} – apparent Michaelis constants, K_eq – equilibrium constant and G(S_i; P_i; K_{m_app}) is a function describing all possible enzymes states and enzyme-substrate/products complexes.

At this level of spicification we reduce number of parameters in rate equation to the minimum. For example in such case we make an assumption that dissosiation constant are equal to Michaelis constants. Using this assumption we can use for model verification K_{m_app} taken from literature or databases for full model verification. If one or another parameter can’t be verified because of a lack of experimental information we assume that its value should be out of sensitivity range for physiological concentrations of substrates or products. For example, in case of Michaelis constants we assume that values of such parameters should be more than 100 fold higher then maximal observed concentration of this compound.

The second level of specification is used for enzymes with more complicated catalytic or regulatory mechanism and absence of any allosteric properties. Moreover, for a development of such models quantitative experimental data which describe studying enzyme by more detail range are necessary.  As a rule in such case we use data about enzyme kinetic at various initial conditions or dependences of initial enzyme rate from concentration of substrates, products, effectors or other system conditions. Rate equations, which we use at this specification levels, are generalized equations of Cleland’s classification deriving with King-Altman or quasisteady-state methods [Cleland W., 1963; Cornish-Bowden A., 2001]. Such rate equations have more parameters and more complicated formalism against the first level of specification. One more difference that in the second level rates equations we use true values of K_m and K_d, which are independent from experiment conditions and are estimated from the best coincidence between model curves and experimental points. 

(Eq. 2)

where S_i – enzyme substrate, Р_i – enzyme products, K_{m_true} and K_{d_true} – true Michaelis and binding constants respectively, K_eq – equilibrium constant and G(S_i; P_i; K_{m_true}; K_{d_true}) – is a function describing all possible enzymes states and enzyme-substrate/products complexes.

The third level of specification is a mathematic description of allosteric enzymes. Catalytic mechanism of such enzymes could not be fully described in framework of Michaelis-Menten’s or Cleland’s approach, because such allsoteric properties as sigmoid dependences of initial enzyme activity on substrate, products or effectors concentration, maximal points on such dependences and different variants of cooperativity between these compounds require more complicated models [Kurganov B.I., 1968]. The major part of allsoteric properties are results of oligomeric enzyme organization. There are several such modeling approaches for description oligomeric enzymes.  The most well known among them are Hill model [Cornish-Bowden A., 2001], Monod-Wyman-Changeux model (MWC) [Monod J. et al., 1965], Koshland-Nemethy-Filmer model [Koshland D.E., et al., 1966] and Frieden-Kurganov model [Frieden C. et al., 1967; Kurganov B.I., 1968].  The popular approaches to understand kinetics and regulation of individual enzymes are the first and the second one, because Koshland-Nemethy-Filmer model has extremely complicated formalism and large number of parameters, while Frieden-Kurganov model focused on the dissociating enzymes. This enzyme property is peculiar to large number of enzymes, but it has clear experimental observation infrequently. So, the last two models often can’t be verified by experimental data entirely. Hill model is the simplest, although it doesn’t take into account physical grounds conditioning allosteric properties such as concerned transition theory or dissociation enzyme monomers [Cornish-Bowden A., 2001]. Because of this Hill equation in the standard form can describe only one allosteric effect sigmoid dependences of initial enzyme activity on substrate concentration and doesn’t take into account all other effects. So, we think that more powerful approach of mathematical description of allosteric enzymes functioning is generalizations of MWC model [Monod J. Et al., 1965]. In the framework of this approach from the one hand main part of allsoteric properties are taken into account and from other hand MWC model hasn’t such complicated rate equation against Koshland-Nemethy-Filmer model.  Principal assumptions of MWC approach are: (i) each subunit can exist in two conformational states, and each state is characterized by different dissociation constants of the substrates; (ii) all subunits of the oligomeric enzyme are expected to be only in the same conformational state, and for all the monomers transitions between different conformational states occur simultaneously (the concerted transition theory) [Monod J. Et al., 1965].  This implies that the number of different conformational states of the oligomeric enzyme is equal to the number of conformational states of its single subunit (pic. 1).  For example, if each monomer can be in two states, the catalytic cycle of the oligomeric enzyme should include only two conformational states, R (relax – with higher catalytic ability) and T (tense – with less catalytic ability).

Pic. 1. General scheme of Monod-Wyman-Changeux model. R0 – free enzyme in the R-state; Т0 – free enzyme in the T-state; Lo – equilibrium constant between R- and T-states; Ef – allosteric effector; KefR and KefТ – dissociation constants for R- and Т-states of enzyme, respectively. Subgraf S ↔ P represents scheme of association/dissociation of sustartes and rpoducts with an enzyme.

Allosteric regulation of an enzyme in the framework of this approach is taken into account via binding of effectors to regulatory sites of an enzyme and formation of additional complexes (Fig. 1).  The ratio between R- and T-states influences the enzyme rate because of different catalytic properties of the R- and T-forms of the enzyme. An activator has higher affinity to the R-form and shifts the balance towards the more active R-state, leading to an increase of reaction rate; an inhibitor, in turn, binds to the T-form and shifts the balance towards the T-state decreasing the activity of the enzyme. The modulating property of the effector (activation or inhibition) will depend only on the ratio of its constants of dissociation from different forms of the enzyme (Pic. 1).

There were several improvements of MWC model. The most complete of them is a generalization proposed by Popova and Sel’kov [Popova S.V. and Sel’kov E.E., 1975]. This variant of the MWC model allows us to take into account the following characteristics of the enzyme functioning:

• reaction reversibility;
• catalytic cycles of the enzyme monomer including kinetic mechanisms with more than one substrate and product and  competitive inhibition;
• R(relax) and T(tense) states of the enzyme differ from each other not only in substrate affinity and Michaelis constants, but also in catalytic constants.

It should be noted that such enzymes properties is very important for adequate coincidence of model and experimental data for large number of oligomeric enzymes with complicated metabolic regulation. In accordance with generalized MWC model proposed by Popova and Sel’kov [Popova S.V. and Sel’kov E.E., 1975] rate equation for such enzyme can be written in the following manner:

(Eq. 3)

where f is a rate equation for single subunit in R-state,  f’ is the same for T-state, Q is a function that determines the ratio between R and Т forms of the enzyme, E_R is a concentration of a enzyme without substrates and products in R-state, E_T is a concentration of a enzyme without substrates and products in T-state, L_o is a constant of transition between R and Т states of the enzyme, Ef is a concentration of the allosteric effector, K_ef^R and K_ef^Т are constants of dissociation of the effector for R and Т forms of the enzyme, respectively, n is a number of subunits of the enzyme.

The forth level of specification is used for detail description of enzymes, which functioning rate can’t be described with single equation and enzyme mechanism requires taking enzyme transient states as variables [Demin O. et al., 2008]. So, in such case enzyme model becomes a separate ODE system, where variables are not only concentration of substrates and products but concentration of their complexes with enzyme and other its transition states, which it undergoes during catalytic cycle. Such models are used for mathematical description of multienzyme complexes, components of respiratory chains [Demin O.V., et al., 2001], other transmembrane carriers [Metelkin E., et al., 2006] and enzymes with several activities [Demin O. et al., 2008].

Influence of pH on enzyme activity we take into account by standard method proposed by Cornish-Bowden [Cornish-Bowden A., 2001]. This method is based on the assumption that there are two proton-binding sites in an active center of a protein.  It implies that each state of the enzyme can be found in three forms: deprotonated, monoprotonated and diprotonated.  Only monoprotonated form is active. In such case Eq. 1 and 2 get additional function Z_ph (eq. 5).

(Eq. 4)

where K_{h_1} – proton dissociation constant in monoprotonated state and K_{h_2} – in diprotonated state. All over designations are same as in previous equations.

Z_ph has same form for the main part of the enzyme rate equations. However, sometimes it is necessary to modify it, for example when enzyme has different pH optimums for forward and reverse reactions or pH has influence on the K_m. In this case with the help of the additional parameters it is possible to take into account these enzyme peculiarities.

You can find more information about methods of individual enzyme models development in our publications.

1. Peskov K., Demin O. Kinetic model of phosphofructokinase-2 from Escherichia coli. Voprisi Bio Med Pharm Chem (Moscow). (2010) №5, 11-23.
2. Mogilevskaya E., Peskov K., Metelkin E., Plyusnina T., Lebedeva G., Goryanin I., Demin O. Kinetic modeling of E. coli enzymes: integration of in vitro data. IN: Systems Biology and Biotechnology of Esherichia coli. pp. 177 - 207, (2009), (Editor: Sang Yup Lee), Springer Netherlands.
3. Demin O., Goryanin I. Kinetic Modelling in Systems Biology. Taylor & Francis (United States), (2008), pp.360
4. Peskov K., Goryanin I., Demin O. Kinetic Model of Phosphofructokinase-1 from Escherichia coli. J Bioinform Comput Biol (2008) 6(4): 843-67.
5. Mogilevskaya E. A., Lebedeva G. V., Goryanin I. I., Demin O. V. Kinetic model of Escherichia coli isocitrate dehydrogenase functioning and regulation. Biophysics (Moscow), (2007), 52(1), p. 47-56.
6. Goryanin II, Lebedeva GV, Mogilevskaya EA, Metelkin EA, Demin OV. Cellular kinetic modeling of the microbial metabolism. Methods Biochem Anal (2006) v.49 p.437-488
7. Mogilevskaya E.A., Lebedeva G.V., Demin O.V. Kinetic model of E. coli 2-ketoglutarate dehydrogenase functioning. Russian Biomedical Journal Medline.ru (2006), 7(44) p. 442-449.
8. Demin O.V., Plyusnina T.Y., Lebedeva G.V., Zobova E.A., Metelkin E.A., Kolupaev A.G., Goryanin I.I., Tobin F. Kinetic modelling of the E. coli metabolism. IN: Topics in Current Genetics p.31-67, 2005 , Eds. Alberghina L., Westerhoff H.V. , Springer
9. Mogilevskaya E.A., Lebedeva G.V., Demin O.V. Kinetic Model of E. coli Citrate synthase functioning. IN: Proceedings of The 12th International Conference MATHEMATICS. COMPUTER. EDUCATION. V.3 p.934-944, 2005
10. Demin O.V., Dronov S., Goryanin I.I., Lebedeva G.V. Kinetic model of imidazole glycerol phosphate synthetase of Escherichia coli . Biochemistry (Moscow) (2004), 69 (12), 1625-1638.

Moreover, at our website you can find materials describing results of application of this method to modeling of enzyme functioning.