Based on our experience in kinetic modeling of coupled multiple metabolic pathways we propose a generic rate equation for the dynamical modeling of metabolic kinetics. Its symmetric form makes the kinetic parameters (or functions) easy to relate to values in database and to use in computation. In addition, such form is workable to arbitrary number of substrates and products with different stoichiometry. We explicitly show how to obtain such rate equation exactly for various binding mechanisms. Hence the proposed rate equation is formally rigorous. Various features of such a generic rate equation are discussed. For irreversible reactions, the product inhibition which directly arise from enzymatic reaction is eliminated in a natural way. We also discuss how to include the effects of modifiers and cooperativity.
Based on our experience in kinetic modelling of coupled multiple metabolic pathways, we propose a generic rate equation for the dynamical modelling of metabolic kinetics. It is symmetric for forward and backward reactions. It's Michaelis-Menten-King-Altman form makes the kinetic parameters (or functions) easy to relate to experimental values in database and to use in computation. In addition, such uniform form is ready to arbitrary number of substrates and products with different stiochiometry. We explicitly show how to obtain such rate equation rigorously for three well-known binding mechanisms. Hence the proposed rate equation is formally exact under the quasi-steady state condition. Various features of this generic rate equation are discussed. In particular, for irreversible reactions, the product inhibition which directly arise from enzymatic reaction is eliminated in a natural way. We also discuss how to include the effects of modifiers and cooperativity.
The advances in modern biology require large-scale mathematical modelling [1,2,3,4]. One such important task is the kinetic modelling of coupled metabolic pathways [5,6,7,8,9,10,11,12], to understand how an organism lives. While it is true that biological processes are based on well-studied chemical reactions, a modeler immediately encounters several formidable difficulties. First, kinetic equations describing enzymatic reactions are often complicated due to numerous parameters [13,14], even without considering stochastic effects [15]. Moreover, each enzymatic equation appears different from others depending on the enzyme mechanism involved. Therefore, modelling a simple biological process with relatively detailed chemical and biological information can be a daunting task [16,17]. Thus, those rigorous chemical reactions in a coupled metabolic network do not lend themselves easily to metabolic simulations with many reactions. Nevertheless, many databases such as Brenda [18,19], KEGG [20], and MetaCyc [21,22] have been documenting our progress in knowledge of enzyme behavior. If we assume that this difficulty could be overcome with great care and effort in using the appropriate rate equations with the aid of enzyme databases, there exists a second difficulty. Most chemical reaction parameters are not measured under the living conditions, how does one know they are applicable to a real organism? If not, how could one make the appropriate and necessary adjustments in a complicated though exact rate equation? More seriously, even if those in vitro parameters are relevant, it is not possible to measure all of the rate constants in the rigorous rate equation, whose number can easily be tens of thousands for multiple coupled pathways [23,24] in ever changing physiological conditions, the third difficulty. The question naturally arises: Is there a way to address those difficulties?
In this paper, based on our kinetic modelling experience we introduce a generic rate equation capable of reproducing exactly the full rigorous rate equations irregardless of the enzyme mechanism for the full rate equation. The equation takes into account thermodynamic constraints. It generalizes easily to any reactions containing an arbitrary number of substrates and products. The effects of modifiers (activators and inhibitors) as well as cooperativity can be included. Thus, it appears to be capable of addressing the above difficulties.
The outline of the rest paper is as follows. In the next section, Sec. 2, we introduce the generic enzymatic rate equation. In Sec. 3, we show how the full rigorous mechanistic rate equation can be written exactly into the form we proposed using three important cases as examples. In Sec. 4, we discuss the case when cooperativity is involved. In Sec. 5, we discuss a useful ansatz for the functions f 1 and f 2 and also thermodynamic considerations in the generic equation, Eq. (2.2). We also discuss relationship to another proposed rate equation that is different from ours and how to include the effects of modifiers. We conclude in Sec. 6.
Generic Enzymatic Rate Equation under Living Conditions 3
In this section we introduce the generic rate equation.
To further illustrate difficulties and to demonstrate usefulness of an answer to the questions posed in the introduction, we consider the well known Michaelis-Menten equation (see Eq. (3.11)). Once we are dealing with bisubstrate enzymes beyond the simple Michaelis-Menten equation [25] for a single substrate and product, the rate equations becomes unwieldy due to the rapid growth in the number of parameters in the equation. This is despite the fact that the Michaelis-Menten equation has been shown to hold even for a fluctuating single enzyme [26]. Is it possible to fully measure the 18 terms in the steady state [27,28] rate equation for a random Bi-Uni mechanism (c.f. Eq. (3.13))? The answer has been no so far experimentally. It has been recognized that enzymes have not been characterized to such gr
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