The Kolmogorov-equations for the stochastic model of the Michaelis-Menten reaction is explicitly solved by a method in the special case when the number of enzyme molecules is exactly one, which can also be extended for cases with a few enzyme molecules. The important differences between the stochastic and deterministic approaches and their biological significance is analyzed. Beside the symbolic solution of the time course of the irreversible reaction the equilibrium is also described for the reversible case. Comparisons are made with the usual steady state approximation.

Pharmacokinetic models simplify biological complexity by dividing the body into interconnected compartments. The time course of the chemical's amount in each compartment is then expressed as the solution to a system of ordinary differential equations. Difficulties arise when the model contains more variables and parameters than comfortable for mathematical and computational treatment. To overcome such difficulties the new and powerful tools of lumping are applied, which are aimed at reducing a differential system by aggregating several variables into one. The lumped model is a differential equation system, whose variables are interpretable in terms of variables of the original system. The reduced model is usually required to satisfy some constraints: it may be necessary to keep state variables of interest for prediction unlumped. For this purpose constrained lumping methods have are also available. After presenting the theory, we study, here, through practical examples, the potential of such methods in toxico/pharmacokinetics. We first simplify a 2-compartment pharmacokinetic model by symbolic lumping. We then explore the reduction of a 6-compartment physiologically based pharmacokinetic model for 1,3-butadiene with numerical constrained lumping.

The concepts and methods of systems biology are extended to neuropharmacology in order to test and design drugs for the treatment of neurological and psychiatric disorders. Computational modelling by integrating compartmental neural modelling techniques and detailed kinetic descriptions of pharmacological modulation of transmitter-receptor interaction is offered as a method to test the electrophysiological and behavioural effects of putative drugs. Even more, an inverse method is suggested as a method for controlling a neural system to realise a prescribed temporal pattern. In particular, as an application of the proposed new methodology, a computational platform is offered to analyse the generation and pharmacological modulation of theta rhythm related to anxiety.

We analyze the concept of Self-Organized Criticality in the case of dynamical systems, in particular for nonlinear reactive systems including arbitrary numbers of species and chemical reactions. We find mathematical conditions that allow for the existence of this behaviour. It is shown that these quite restrictive conditions can nevertheless be satisfied for physical reasons, in particular in chemical kinetics or population dynamics. It can be concluded that SOC, although apparently highly non-generic, should be observed in many natural dynamical systems. These results are applied for homogeneous and inhomogeneous catalytic chemical systems.

Polynomial differential equations showing chaotic behavior are investigated using polynomial invariants of the equations. This tool is more effective than the direct method for proving statements like the one: the Lorenz equation cannot be transformed into an equation which would be a mass action type kinetic model of a chemical reaction.

This paper deals with ratio-dependent predator-prey systems with delay. We will investigate under what conditions delay cannot cause instability in higher dimension. We give an example when delay causes instability.

Exact linear lumping has earlier been defined for a finite dimensional space, that is, for the system of ordinary differential equations y'=f o y as a linear transformation M for which there exists a function f^ such that y^:=My itself obeys a differential equation y^'=f^ o y^. Here we extend the idea for the case when the values of y are taken in a Banach space. The investigations are restricted to the case when f is linear. Many theorems hold for the generalization of exact lumping, such as necessary and sufficient conditions for lumpability, and relations between the qualitative properties of the original and the transformed equations. The motivation behind the generalization of exact lumping is to apply the theory to reaction-diffusion systems, to an infinite number of chemical species, to continuous components, or to stochastic models as well.

It is shown that none of the proper or improper orthogonal transformations transforms the Lorenz-equation into a kinetic equation, i.e. into an equation representing reasonable chemistry. It is also shown that none of the proper orthogonal transformations transforms a model by Rössler into a kinetic model either. The importance of the presence of negative cross-effects is hereby emphasized.

The Lotka-Volterra model is shown to be the simplest unique model among those having the same linearized form. We also show that the two-dimensional Explodator is not unique in its own class: there are four models possessing the same linearized form. Finally, we propose a method for the construction of formal chemical models having prescribed properties (i. e. having prescribed a tzpe of one or more equilibrium points.)

The mesoscopic nonequilibrium thermodynamics of a reaction-diffusion system is described by the master equation. The information potential is defined as the logarithm of the stationary distribution. The Fokker-Planck approximation and the Wentzel--Kramers--Brillouin method give very different results. The information potential is shown to obey a Hamilton-Jacobi equation, and from this fact general properties of this potential are derived. The Hamilton-Jacobi equation is shown to have a unique regular solution. Using the path integral formulation of the Hamilton-Jacobi approximation of the master equation it is possible to calculate rate constants for the transition from one well to another one of the information potential and give estimates of mean exit times. In progress variables, the Hamilton-Jacobi equation has always a simple solution which is a state function if and only if there exists a thermodynamic equilibrium for the system. An inequality between energy and information dissipation is studied, and the notion of relative entropy is investigated. A specific two-variable system and systems with a single chemical species are investigated in detail, where all the defined relevant quantities can be calculated explicitly.

A two-valued piecewise-linear, constant slope, 1D transformation is defined as a simplified model for hysteresis. By using Góra's S-matrix and classical results on Rényi's beta-transformation x|-->beta x (mod 1) the density function of the induced absolutely continuous ergodic measure is determined. Interesting simulation results reveal the chaos established.

Polynomial equations so important in different fields of applications, especially in chemistry, are investigated. The literature is reviewed with the aim of possible applications of pure mathematical results in the field of chemical kinetics. Finally an example is treated in detail showing how to transform a mass action type kinetic equation into a homogeneous quadratic mass action type kinetic equation. The problem of transforming back is also discussed.

The induced kinetic differential equations of reactions with mass action type kinetics are characterized within the class of polynomial differential equations: it is shown that a polynomial differential equation is kinetic if and only if it contains no negative cross effect. A constructive proof is given to prove sufficiency of the condition.

Starting from a special model of chirality we show how the mathematical theory of formal reaction kinetics and  the  program  package based  on  it  can  be  utilized  when  analyzing  the  variations  of models. The speciality of the package is that it highly utilizes the advantages of Mathematica,  

it automatically writes down and numerically solves the kinetic differential equation,  

it can also simulate the usual stochastic model,   

it  calculates  the  graph-theoretic  and  linear  algebraic  quantities  which  allow  to deduce statements on the qualitative behavior of the concentration of the different species.