Stochastic kinetic modelling

Mathematical models are widely applied in biosciences and different modeling routes can be taken to describe biological systems. The type of model to use depends completely on the objective of the study. Models can be dynamic or static, deterministic or stochastic. Kinetic models are commonly used to study transient states of the cell such as the cell cycle [101] or signal transduction pathways [102], whereas stoichiometric models are generally used when kinetics parameters are unknown and steady state systems is assumed [48, 103]. 

In this paper the outlines of anomalous stochastic kinetic models are discussed. These models are derived by relaxing certain traditional assumptions. Furthermore, a very specific example is given to illustrate that spatial inhomogenity can lead to time inhomogenity by some lumping procedure. 

The occurence of deviance from the usual assumptions (i.e. ordered structure of time, space and energy) leads to anomalous stochastic kinetic models. 

In the past, the equivalence between the size distribution generated by the Smoluchowski equation and simple statistical methods [9, 12, 40-42] was a source of some confusion. The Spouge proof and the numerical results obtained for the kinetics models with more complex aggregation physics, e.g., with a presence of substitution effects [43,44], revealed the non-equivalence of kinetics and statistical models of polymerization processes. More elaborated statistical models, however, with the complete analysis made repeatedly at small time intervals have been shown to produce polymer size distributions equivalent to those generated kinetically [45]. Recently, Faliagas [46] has demonstrated that the kinetics and statistical models which are both the mean-field models can be considered as special cases of a general stochastic Markov process. 

After the kinetic model for the network is defined, a simulation method needs to be chosen, given the systemic phenomenon of interest. The phenomenon might be spatial. Then it has to be decided whether in addition stochasticity plays a role or not. In the former case the kinetic model should be described with a reaction-diffusion master equation [81], whereas in the latter case partial differential equations should suffice. If the phenomenon does not involve a spatial organization, the dynamics can be simulated either using ordinary differential equations [47] or master equations [82-84]. In the latter case but not in the former, stochasticity is considered of importance. A first-order estimate of the magnitude of stochastic fluctuations can be obtained using the linear noise approximation, given only the ordinary differential equation description of the kinetic model [83-85, 87]. 

Along with the isomerism of linear copolymers due to various distributions of different monomeric units in their chains, other kinds of isomerisms are known. They can appear even in homopolymer molecules, provided several fashions exist for a monomer to enter in the polymer chain in the course of the synthesis. So, asymmetric monomeric units can be coupled in macromolecules according to "head-to-tail" or "head-to-head"—"tail-to-tail" type of arrangement. Apart from such a constitutional isomerism, stereoisomerism can be also inherent to some of the polymers. Isomers can sometimes substantially vary in performance properties that should be taken into account when choosing the kinetic model. The principal types of such an account are analogous to those considered in the foregoing. The only distinction consists in more extended definition of possible states of a stochastic process of conventional movement along a polymer chain. 

For example, under kinetic modeling of "living" anionic copolymerization in the framework of the terminal model, a macromolecule is associated with the realization of a certain stochastic process. Its states (a,r) are monomeric units, each being characterized along with chemical type a and also by some label r. This random quantity equals the moment when this monomeric unit entered in a polymer chain as a result of the addition of o-type monomer to the terminal active center. It has been... 

The text reviews the methodology of kinetic analysis for simple as well as complex reactions. Attention is focused on the differential and integral methods of kinetic modelling. The statistical testing of the model and the parameter estimates required by the stochastic character of experimental data is described in detail and illustrated by several practical examples. Sequential experimental design procedures for discrimination between rival models and for obtaining parameter estimates with the greatest attainable precision are developed and applied to real cases. 

The CME is the equation for the probability function p, or equivalently if the system s volume is constant, for the probability function p(ni, n9, , jv, t)where is the number of molecules of species i. With given concentrations (ci, c2, , c v) at a time t, deterministic kinetic models give precisely what the concentrations will be at time t + St. According to the stochastic CME, however, the concentrations at t + St can take many different values, each with certain probability. 

The stochastic model is a kinetic model whose rate and time constants can be related to the affinity energy of adsorption. The Frenkel equation connects the adsorption energy, E , and the average stationary phase residence time of one adsorption event Tg [106] ... 

Cavazzini et at. showed that the above Monte Carlo model of nonlinear chromatography is equivalent to the Thomas kinetic model of second order Langmuir kinetics [70]. The solution of the Thomas model for a Dirac impulse injection is given by Eq. 14.65. When the chromatographic process is modeled at the molecular level with the stochastic model, the Thomas model becomes [70] ... 

Thermal and acid-catalyzed deprotection kinetics of PBOCST and PTBMA was monitored by UV and IR spectroscopy, respectively [515], and compared very favorably with models based on a stochastic kinetics simulator (CKS)... 

---