Kinetic Modeling: Definitions

From a modeling perspective, biochemical networks are a set of chemical species that can be converted into each other through chemical reactions. The focus of biochemical network models is usually on the levels of the chemical species and this usually requires explicit mathematical expressions for the velocity at which the reactions proceed.

The most popular representation for these models uses ordinary differential equations (ODEs) to describe the change in the concentrations of the chemical species.

Each chemical species in the network is represented by an ODE that describes the rate of change of that species along time. The ODE is composed by an algebraic sum of terms that represent the rates of the reactions that affect the chemical species. For a chemical species X :

where s_i is a stoichiometry coefficient that is the number of molecules of X consumed or produced in one cycle of reaction i, with a positive sign if it is produced or negative if consumed, and v_i is the velocity of reaction i . Obviously, for reactions that do not produce or consume X the corresponding s_i is zero.

The velocity of each reaction is described by a rate law that depends on the concentrations of the reaction substrates, products, and modifiers.

These ODE models can be used to simulate the dynamics of the concentrations of the chemical species along time given their initial values.

 

Kinetic simulation of biochemical systems involves the following basic steps:

1.  Identification of the kinetic problem: This step involves the determination and output variables as well as the intermediates. Key species and physical properties to be simulated are determined and tabulated.

2. Model formulation: Although that does not have to be the case, most studies employ the reaction rate equations (RRE) to model biochemical systems. In the RRE, one simply defines the changes in the concentrations (or equivalently the number of molecules) as a function of time and location. We note that constraints can be embedded in RRE using a Lagrange undetermined multipliers type formalism.

3. Choosing a method: At this step a decision needs to be made as to whether to adopt a deterministic or stochastic formulation. Although it is still relatively uncommon, a hybrid approach which moves between deterministic and stochastic regimes is another possibility. In the deterministic approach, a reasonable time step is chosen adaptively, based on the estimated local error according to a differential equation model. In contrast, discrete stochastic simulations model each individual reaction event. Using the probabilities of the reactions (called the reaction propensities), one determines the occurrence statistics of the involved reactions. This is generally done in one of two ways: In next reaction type methods, the sequence of the reactions are chosen according to the reaction propensities and the reaction times are computed to of the elapsed time step and compute how many times the involved reactions occur according to their probabilities.

4. Simulation: Integrating the RRE deterministically involves choosing an appropriate algorithm from a wide selection of sophisticated numerical methods for systems of ODEs and DAEs. Highly efficient and reliable software is readily available, although one must usually determine whether or not the problem is stiff and then choose the software accordingly. Roughly speaking, an ODE or DAE system is stiff if it involves a wide range of time scales and the fastest of those scales correspond to stable processes. In chemically reacting systems, the wide range of time scales can come from some reaction rates being several orders of magnitude or more greater than others. A solver which is designed for non-stiff systems will run very slowly (because it needs to choose time steps on the scale of the fastest process in the system to maintain stability for its explicit formulas), if it is applied to a stiff problem. The cure for stiffness is to approximate the differential equation using implicit methods. The software that is available for the accurate solution of stiff ODE and DAE systems always makes use of implicit methods. Stochastic simulation algorithms are not as advanced as the deterministic integrators and they do not scale well with the model size and complexity. Even though recent algorithmic improvements have led to impressive gains, stochastic simulations are still too slow for most realistic systems. However, as discussed in the previous section, fluctuations can be large in biological systems and, more importantly, stochastic variations may have implications for biological functions and responses. Therefore, when feasible, stochastic simulation methods should be preferred. This is particularly true for processes that involve small number of regulatory molecules with large variations in their expression levels.

5. Trajectory analysis: Time courses obtained during the simulations are catalogued and combined to derive the statistical distributions of the desired output/prediction quantities, such as the concentrations of experimentally monitored species and the resulting material/mass flow in the system or the probability distribution of the reactions. Similarly, from simulation results for different model parameter sets, one can study the sensitivity to parameter variations.

Note: Texts and figures in this page have been chosen from the following references: Kinetic Modeling of Biological Systems (Methods Mol Biol. 2009, 541, 311–335)

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