Under words “kinetic modeling” we will imply the system of mechanistic ordinary differential equations, which defines state of concerned system of chemical reaction in the every time point, i.e. assign the metabolites concentration changing as time function. These equations are written down by following way

dX/dt = V_{production X} – V_{consumption X}

and named equations of chemical kinetic. Here X - any metabolites concentration, V_{production X} and V_{consumption X} – total rates of its production and consumption.

Kinetic model of metabolic or signal conductive pathway development involves several stages:

1) Development and research of stoichiometric model

On this stage is established the full list of all cellular intermediates, enzymes, molecules, cofactors etc, including in the system. Moreover, all possible interactions between system compounds are elucidated (enzymatic and non-enzymatic). The result is kinetic scheme (directed graph tree) of metabolic system development, defining possible interactions between all system compounds.

2) Derivation of reaction rate equations, i.e., finding of algebraic expressions, describing the rates of individual reaction dependence on system intermediates concentrations.

For every reaction detailed and reduced description is developed. Detailed description represents development of full catalytic cycle of corresponding enzyme, including all its possible states: free enzyme, enzyme complex with substrates and products, modified enzyme states, enzyme complex with effectors (inhibitors and activators) etc. During the scheme of catalytic enzyme cycles development use information available from literature and electronic databases about catalytic reaction mechanisms and features of this enzyme activity regulation. At deficiency of experimentally confirmed information the mechanism of biochemical reaction can be assumed reasoning from belonging of enzyme to the one or another class (for example, to the class of amidotransferases, dehydrogenases, etc).

After estimation of enzyme catalytic cycle, the system of mechanistic ordinary differential equations is written down; this describes the alteration of all possible enzyme states in various time points. Redused description of biochemical reaction is a result of simplification of this differential equations system and represents evident dependence of enzymatic reaction rate on total enzyme concentration, concentrations of substrates, products and reaction effectors, as well as on corresponding kinetic reaction parameters (Michaels constants K_m, inhibition constants K_m, constants of dissociation K_d, elementary constants of individual catalytic stages rate). During performing of system reduction and derive corresponding equation of enzymatic reaction rate we use several approximations:

• approximation of quasi-stationary state for the differential equation system, describing catalytic cycle. The equitation of reaction rate derives as stationary rate of products production in the corresponding catalytic cycle.
• approximation of “quick equilibration” i.e. taking into account the hierarchy of times of individual catalytic stages going, that allows to reduce the system of differential equations in compliance with Tikhonov theorem.

3) Finding the numerical values of kinetic parameters, entering into reaction rates equations.

The numerical values of kinetic parameters (K_m, K_i, K_d and reaction rate constants) may be received by following way:

• be found in the literature
• be found in the electronic databases, containing the information about kinetic of enzymatic processes. At present time only such databases as EMP, BRENDA, WIT can provide such information about kinetic parameters of individual enzymatic reactions.
• be estimated reasoning from condition of the best coincidence of experimentally measured dependence of initial reaction rate on concentration of substrates, products and effectors with corresponding theoretical dependences of reaction rate equation.
• be estimated reasoning from condition of the best coincidence of substrate and products concentration dependences in time course, received in kinetic experiments, with equations system solutions describing these parameters.

All sources and methods of kinetic parameters values estimation mentioned above contain or use only experimental information received in vitro. By our estimations, the two first sources contain no more than 30% (in dependence on organism, tissue, and biochemical system selected) of all necessary information about kinetic constant values. Experimentally measured and published in the literature dependences described in the two last methods of kinetic constants values estimation allow us to find the numerical values approximately 10-40% another parameters, figuring in the model of selected biochemical system. In that way, for estimation of remained 30-60% kinetic parameters is necessary to carry out additional in vitro experiments. The results of such experiments should be families of kinetic curves describing the dependences substrate and products on time course for every enzymatic reaction, kinetics of which is insufficiently described in the literature.

4) Development the system of differential equation describing the behavior of selected biochemical system on time course

This system of differential equations as looks in the following way:

dX/dt = N*V(X; e; K),

where x = [x₁,… x_m] vector of concentrations of m biological compounds, included into the model (metabolites, cofactors, etc); V = [v₁,… v_n] vector of n rate equations; e = [e₁,… e_n] – vector of enzymes concentrations, taking part in this metabolic system; K = [K₁,… K_p] – vector of kinetic parameters of system; N – stoihiometric matrix of system with dimension m, n, corresponding to kinetic scheme of the system, received on the first stages of kinetic model development.

5) Numerous solution and research of regulatory and dynamic properties of kinetic model of selected biochemical system

The system of mechanistic ordinary differential equations, describing any biochemical system is numerically integrated by means of particularized program packet DBSolve7. On this stage the research of dynamic and regulatory properties of system are carried out. In particular, the numbers of stationary states in the system is defined and its stability is investigated. The dependence of stationary concentrations of metabolites in the system on parameters of inputs and outputs of systems is investigated as well as on parameters of cofactors reproduction and on inhibition (activation) various reaction. Moreover by means of methods of metabolism control theory the distribution of stationary fluxes control in the system between various enzymes is researched and sensitivity of a system to the change of other kinetic parameters is analyzed.

6) Verification of the model (or its separate fragments) with experimental data available in vivo

On this stage the model checkout happens and specification of several their parameters in an effort to reproduce main features of behavior known from the experiment of modeling biological object, observing in vivo. Such information for the model verification may serve the data of concentration dependence on time course for key metabolites as a result of adding in the system defined compounds as well as data about metabolic changes in consequence of artificially made alterations in the cell genome.

7) After the model was developed, investigated and verificated, it may be applied to the modeling of various kind of biological scripts or for prediction consequences of various interferences in the metabolism.

Various examples of such kind researches you can find on the pages of our web site.It is very important that by means of our method not only a number of interesting conclusions about functioning of various organisms metabolism have been received (you can make sure of it in the list of our publications or on the pages of web site) but also applied tasks connected with bioengineering, biotechnology, pharmacology and pharmaceutics have been solved.