The Essential Spectral Radius and Asymptotic Properties of Transfer Operators

Abs t r ac t The statistical behavior of deterministic and stochastic dynamical sys-tems may be described using transfer operators, which generalize the no-tion of Frobenius-Perron and Koopman operators. Since numerical tech-niques to analyse dynamical systems based on eigenvalues problems for the corresponding transfer operator have emerged, bounds on its essential spectral radius became of interest. This article shows that they are also of great theoretical interest. We give an analytical representation of the essential spectral radius in L\fj,), which then is exploited to analyse the asymptotical properties of transfer operators by combining results from functional analysis, Markov operators and Markov chain theory. In par ticular, it is shown that an essential spectral radius less than 1, uniform constrictiveness and some "weak form " of the so-called Doeblin condition are equivalent. Finally, we apply the results to study three main prob-lem classes: deterministic systems stochastically perturbed deterministic systems and stochastic systems K e y w o r d s, uniformly constrictive, asymptotically stable, exact, asymptotically pe

Solving the Chemical Master Equation for Monomolecular Reaction Systems Analytically

The stochastic dynamics of a well-stirred mixture of molecular species interacting through different biochemical reactions can be accurately modelled by the chemical master equation (CME). Research in the biology and scientific computing community has concentrated mostly on the development of numerical techniques to approximate the solution of the CME via many realizations of the associated Markov jump process. The domain of exact and/or efficient methods for directly solving the CME is still widely open, which is due to its large dimension that grows exponentially with the number of molecular species involved. In this article, we present an exact solution formula of the CME for arbitrary initial conditions in the case where the underlying system is governed by monomolecular reactions. The solution can be expressed in terms of the convolution of multinomial and product Poisson distributions with time-dependent parameters evolving according to the traditional reaction-rate equations. This very structured representation allows to deduce easily many properties of the solution. The model class includes many interesting examples. For more complex reaction systems, our results can be seen as a first step towards the construction of new numerical integrators, because solutions to the monomolecular case provide promising ansatz functions for Galerkin-type methods

Lumping of Physiologically-Based Pharmacokinetic Models and a Mechanistic Derivation of Classical Compartmental Models

In drug discovery and development, classical compartment models and physiologically based pharmacokinetic (PBPK) models are successfully used to analyze and predict the pharmacokinetics of drugs. So far, however, both approaches are used exclusively or in parallel, with little to no cross-fertilization. An approach that directly links classical compartment and PBPK models is highly desirable. We derived a new mechanistic lumping approach for reducing the complexity of PBPK models and establishing a direct link to classical compartment models. The proposed method has several advantages over existing methods: Perfusion and permeability rate limited models can be lumped; the lumped model allows for predicting the original organ concentrations; and the volume of distribution at steady state is preserved by the lumping method. To inform classical compartmental model development, we introduced the concept of a minimal lumped model that allows for prediction of the venous plasma concentration with as few compartments as possible. The minimal lumped parameter values may serve as initial values for any subsequent parameter estimation process. Applying our lumping method to 25 diverse drugs, we identified characteristic features of lumped models for moderate-to-strong bases, weak bases and acids. We observed that for acids with high protein binding, the lumped model comprised only a single compartment. The proposed lumping approach established for the first time a direct derivation of simple compartment models from PBPK models and enables a mechanistic interpretation of classical compartment models

Physiologically Based Pharmacokinetic Modelling: A Sub-Compartmentalized Model of Tissue Distribution

We present a sub-compartmentalized model of drug distribution in tissue that extends existing approaches based on the well-stirred tissue model. It is specified in terms of differential equations that explicitly account for the drug concentration in erythrocytes, plasma, interstitial and cellular space. Assuming, in addition, steady state drug distribution and by lumping the different sub-compartments, established models to predict tissue-plasma partition coefficients can be derived in an intriguingly simple way. This direct link is exploited to explicitly construct and parameterize the sub-compartmentalized model for moderate to strong bases, acids, neutrals and zwitterions. The derivation highlights the contributions of the different tissue constituents and provides a simple and transparent framework for the construction of novel tissue distribution models

Software Supported Modelling in Pharmacokinetics

A powerful new software concept to physiologically based pharmacokinetic (PBPK) modelling of drug disposition is presented. It links the inherent modular understanding in pharmacology with orthogonal design principles from software engineering. This concept allows for flexible and user-friendly design of pharmacokinetic whole body models, data analysis, hypotheses testing or extrapolation. The typical structure of physiologically-based pharmacokinetic models is introduced. The resulting requirements from a modelling and software engineering point of view and its realizations in the software tool MEDICI-PK [9] are described. Finally, an example in the context of drug-drug interaction studies is given that demonstrates the advantage of defining a whole-body pharmacokinetic model in terms of the underlying physiological processes quite impressively: A system of 162 ODEs is automatically compiled based on the specification of 7 local physiological processes only

Precise Switching of Flagellar Gene Expression in Escherichia Coli by the FlgM–FliA Regulatory Network

A remarkable feature of flagellar synthesis in Escherichia coli is that gene expression is sequential and coupled to the assembly process. The interaction of two key proteins, the flagellar sigma factor FliA and its anti-sigma factor FlgM serves as a major checkpoint in the assembly process that temporally separates middle and late gene expression. While the sequential nature within each gene class has been studied using large-scale transcriptional data, much less is known about the timing controlled by the checkpoint mechanism. In this article, we analyze timing, sensitivity and robustness of the FlgM–FliA core regulatory mechanism based on quantitative molecule data and a detailed stochastic as well as reduced deterministic reaction kinetics model. We find that the pool of free anti-sigma factor FlgM, accumulated during middle gene expression, acts as a molecular timer that determines the delay between successful completion of the hook basal body subunit and the start of expression of flagellar filament proteins. Furthermore, we find that the number of free FliA molecules needs to be tightly controlled for a precise switch from middle to late gene expression. A sensitivity analysis based on the reduced reaction kinetics model reveals that the checkpoint mechanism is very sensitive to changes in levels of competing sigma factors, allowing the bacterium to rapidly adapt to a changing environment. In addition, we find that the reduced model also shows a high sensitivity to the effective synthesis rates of FliA and FlgM. However, this high sensitivity does not generally carry over to the original parameters of transcriptional and translational processes in the detailed model. As a consequence, care has to be taken whenever interpreting results from the robustness analysis of reaction kinetic models comprising lumped or effective parameters