No abstract

To understand the sophisticated control mechanisms of the human's endocrine system is a challenging task that is a crucial step towards precise medical treatment of many disfunctions and diseases. Although mathematical models describing the endocrine system as a whole are still elusive, recently some substantial progress has been made in analyzing theoretically its subsystems (or axes) that regulate the production of specific hormones. Secretion of many vital hormones, responsible for growth, reproduction and metabolism, is orchestrated by feedback mechanisms that are similar in structure to the model of simple genetic oscillator, described by B.C. Goodwin. Unlike the celebrated Goodwin model, the endocrinal regulation mechanisms are in fact known to have non-cyclic structures and involve multiple feedbacks; a Goodwin-type model thus represents only a part of such a complicated mechanism. In this paper, we examine a non-cyclic feedback system of hormonal regulation, obtained from the classical Goodwin's oscillator by introducing an additional negative feedback. We establish global properties of this model and show, in particular, that the local instability of its unique equilibrium implies that almost all system's solution oscillate; furthermore, under additional restrictions these solutions converge to periodic or homoclinic orbits.

The use of systems-based pharmacological modeling approaches to characterize mode-of-action and concentration-effect relationships for drugs on specific hemodynamic variables has been demonstrated. Here, we (i) expand a previously developed hemodynamic system model through integration of cardiac output (CO) with contractility (CTR) using pressure-volume loop theory, and (ii) evaluate the contribution of CO data for identification of system-specific parameters, using atenolol as proof-of-concept drug. Previously collected experimental data was used to develop the systems model, and included measurements for heart rate (HR), CO, mean arterial pressure (MAP), and CTR after administration of atenolol (0.3-30 mg/kg) from three in vivo telemetry studies in conscious Beagle dogs. The developed cardiovascular (CVS)-contractility systems model adequately described the effect of atenolol on HR, CO, dP/dtmax, and MAP dynamics and allowed identification of both system-and drug-specific parameters with good precision. Model parameters were structurally identifiable, and the true mode of action can be identified properly. Omission of CO data did not lead to a significant change in parameter estimates compared to a model that included CO data. The newly developed CVS-contractility systems model characterizes shortterm drug effects on CTR, CO, and other hemodynamic variables in an integrated and quantitative manner. When the baseline value of total peripheral resistance is predefined, CO data was not required to identify drug-and system-specific parameters. Confirmation of the consistency of system-specific parameters via inclusion of data for additional drugs and species is warranted. Ultimately, the developed model has the potential to be of relevance to support translational CVS safety studies.

We develop a tool based on bifurcation analysis for parameter-robustness analysis for a class of oscillators and, in particular, examine a biochemical oscillator that describes the transition phase between social behaviours of myxobacteria. Myxobacteria are a particular group of soil bacteria that have two dogmatically different types of social behaviour: when food is abundant they live fairly isolated forming swarms, but when food is scarce, they aggregate into a multicellular organism. In the transition between the two types of behaviours, spatial wave patterns are produced, which is generally believed to be regulated by a certain biochemical clock that controls the direction of myxobacteria's motion. We provide a detailed analysis of such a clock and show that, for the proposed model, there exists some interval in parameter space where the behaviour is robust, i.e. the system behaves similarly for all parameter values. In more mathematical terms, we show the existence and convergence of trajectories to a limit cycle, and provide estimates of the parameter under which such a behaviour occurs. In addition, we show that the reported convergence result is robust, in the sense that any small change in the parameters leads to the same qualitative behaviour of the solution.