More Proofs for `Determination of Stability with respect to Positive Orthant for a Class of Positive Nonlinear Systems'

This is a supplement material for a published article by the authors.In the published paper `Determination of Stability with respect to Positive Orthant for a Class of Positive Nonlinear Systems,' IEEE Trans. on Automatic Control, vol. 53, no. 5, pp. 1329-1334, 2008, by the authors, some proofs are omitted due to the space limitation of the journal. In this note, we present those omitted proofs

Dissipation in noisy chemical networks: The role of deficiency

We study the effect of intrinsic noise on the thermodynamic balance of complex chemical networks subtending cellular metabolism and gene regulation. A topological network property called deficiency, known to determine the possibility of complex behavior such as multistability and oscillations, is shown to also characterize the entropic balance. In particular, only when deficiency is zero does the average stochastic dissipation rate equal that of the corresponding deterministic model, where correlations are disregarded. In fact, dissipation can be reduced by the effect of noise, as occurs in a toy model of metabolism that we employ to illustrate our findings. This phenomenon highlights that there is a close interplay between deficiency and the activation of new dissipative pathways at low molecule numbers.Comment: 10 Pages, 6 figure

Steady state and (bi-) stability evaluation of simple protease signalling networks

Signal transduction networks are complex, as are their mathematical models. Gaining a deeper understanding requires a system analysis. Important aspects are the number, location and stability of steady states. In particular, bistability has been recognised as an important feature to achieve molecular switching. This paper compares different model structures and analysis methods particularly useful for bistability analysis. The biological applications include proteolytic cascades as, for example, encountered in the apoptotic signalling pathway or in the blood clotting system. We compare three model structures containing zero-order, inhibitor and cooperative ultrasensitive reactions, all known to achieve bistability. The combination of phase plane and bifurcation analysis provides an illustrative and comprehensive understanding of how bistability can be achieved and indicates how robust this behaviour is. Experimentally, some so-called “inactive” components were shown to have a residual activity. This has been mostly ignored in mathematical models. Our analysis reveals that bistability is only mildly affected in the case of zero-order or inhibitor ultrasensitivity. However, the case where bistability is achieved by cooperative ultrasensitivity is severely affected by this perturbation

A mathematical model for breath gas analysis of volatile organic compounds with special emphasis on acetone

Recommended standardized procedures for determining exhaled lower respiratory nitric oxide and nasal nitric oxide have been developed by task forces of the European Respiratory Society and the American Thoracic Society. These recommendations have paved the way for the measurement of nitric oxide to become a diagnostic tool for specific clinical applications. It would be desirable to develop similar guidelines for the sampling of other trace gases in exhaled breath, especially volatile organic compounds (VOCs) which reflect ongoing metabolism. The concentrations of water-soluble, blood-borne substances in exhaled breath are influenced by: (i) breathing patterns affecting gas exchange in the conducting airways; (ii) the concentrations in the tracheo-bronchial lining fluid; (iii) the alveolar and systemic concentrations of the compound. The classical Farhi equation takes only the alveolar concentrations into account. Real-time measurements of acetone in end-tidal breath under an ergometer challenge show characteristics which cannot be explained within the Farhi setting. Here we develop a compartment model that reliably captures these profiles and is capable of relating breath to the systemic concentrations of acetone. By comparison with experimental data it is inferred that the major part of variability in breath acetone concentrations (e.g., in response to moderate exercise or altered breathing patterns) can be attributed to airway gas exchange, with minimal changes of the underlying blood and tissue concentrations. Moreover, it is deduced that measured end-tidal breath concentrations of acetone determined during resting conditions and free breathing will be rather poor indicators for endogenous levels. Particularly, the current formulation includes the classical Farhi and the Scheid series inhomogeneity model as special limiting cases.Comment: 38 page

Stability and robust behaviour across classes of biological and chemical models

This thesis describes three applications of the theory of continuous autonomous dynamical systems. The focus of the thesis is on qualitative, as opposed to numerical, analysis. The applications examined are biological and chemical, and as such there are signi�cant uncertainties in any mathematical representation of them. While the qualitative relationships that de�ne a biological or chemical system may be well understood, it is often di�cult to obtain accurate measurements of the parameters that govern each interaction, due to inherent variability and/or experimental constraints. For this reason, a model that avoids dependence on numerical values while still accurately re�ecting the qualitative structure of the system it represents is potentially of use in gaining a greater understanding of how the system can behave. Conversely, if a purely qualitative model allows certain behaviour that is never experimentally observed, this may highlight the importance of certain parameter values for the system's real world behaviour. The �rst application presented is a model of electron transport in mitochondria, the second is a model of an inter-cellular gap junction, and the third represents a set of reactions occurring in a continuous �ow stirred tank reactor. For each application, a reasonable set of qualitative assumptions is found under which there is a unique steady state to which all initial conditions converge, regardless of precise numerical values. Uniqueness of steady states is proved using results on the injectivity of functions, and degree theory. The convergence criteria are constructed using two di�erent areas of dynamical systems theory. The �rst of these is the theory of monotone �ows, while the second is a group of results known as �autonomous convergence theorems�. The theory of monotone �ows is fairly well known, and relies on �nding conditions under which trajectories of a dynamical system preserve a partial ordering, thereby limiting the possibly asymptotic behaviour of the system. The autonomous convergence theorems appear much less well known; they work by �nding a norm under which trajectories approach each other, either in phase space or in a related exterior algebra space. Both theories are discussed in detail, along with some extensions