Modeling, analysis, and control of biological oscillators

Dit proefschrift is gewijd aan de studie van ritmes, zogenaamde "oscillatoren". In het bijzonder houdt het zich bezig met modellering, analyse en regeling van biologische oscillatoren. Het is verdeeld in twee delen, waarbij Deel I is gewijd aan de toepassing van de controletheorie op endocrinologie, en Deel II is toegewezen aan de toepassing van dynamische systemen op de microbiologie. Deel I ontwikkelt drie wiskundige modellen van endocriene regulatie. Het eerste model beschrijft de diurnale patronen van de cortisol. Via een analytische aanpak ontwerpen we een impulsieve controller om de timing en amplitude van secretoire gebeurtenissen te identificeren, terwijl de bloedcortisolspiegels beperkt zijn tot een specifiek circadiaans bereik. Het tweede model beschrijft in het algemeen de controlemechanismen in de hypothalamus-hypofyse-assen, gecontroleerd door de hersenen. Voor dit model, dat een uitbreiding is van de conventionele Goodwin's oscillator met een extra niet-lineaire feedback, stellen we de relatie vast tussen zijn lokaal gedrag op het evenwichtspunt en zijn globale gedrag. Het laatste model beschrijft de pulsatieve secretie van de hypothalamus-hypofyse-assen. Dit model, verkregen uit een impulsieve versie van de oscillator van Goodwin, heeft een bijkomende affiene feedback. Voor dit model presenteren we voorwaarden voor het bestaan, de uniciteit en de positiviteit van een type periodieke oplossing. Deel II bestudeert een biochemisch oscillatormodel (bekend als " Frzilator "), dat de overgangsfase van sociaal gedrag van myxobacteriën beschrijft , een soort bodembacterie. Dit deel bestudeert de Frzilator vanuit twee verschillende perspectieven, namelijk reguliere verstoring en geometrische singuliere verstoring

Design of Intermittent Control for Cortisol Secretion Under Time-Varying Demand and Holding Cost Constraints

We take the release of stress hormone cortisol as a part of an intermittent control feedback system in contrast to the existing continuous models. By modeling cortisol secretion as an impulsive system, we design an impulsive controller as opposed to a continuous controller for adjusting cortisol levels while maintaining the blood cortisol levels within bounds that satisfy circadian demand and cost constraints. Methods: We develop an analytical approach along with an algorithm for identifying both the timing and amplitude of the control. Results: The model and the algorithm are tested by two examples to illustrate that the proposed approach achieves impulsive control and that the obtained blood cortisol levels render the circadian rhythm and the ultradian rhythm consistent with the known physiology of cortisol secretion. Conclusions: The approach successfully achieves the desired circadian impulsive control, which has great potential to be used in personalizing the medications in order to control the cortisol levels optimally. Significance: This type of bioinspired intermittent controllers can be employed for designing noncontinuous controllers in treating Addisonian disease, which is caused by the adrenal deficiency

Impulsive model of endocrine regulation with a local continuous feedback

Whereas development of mathematical models describing the endocrine system as a whole remains a challenging problem, visible progress has been demonstrated in modeling its subsystems, or axes. Models of hormonal axes portray only the most essential interactions between the hormones and can be described by low-order systems of differential equations. This paper analyzes the properties of a novel model of a hypothalamic-pituitary axis, portraying the interactions in a chain of a release hormone (secreted by the hypothalamus), a tropic hormone (produced by the pituitary gland) and an effector hormone (secreted by a target gland). This model, unlike previously published ones, captures two prominent features of neurohormonal regulation systems, namely, the pulsatile (episodic) production of the release hormone and a complex non-cyclic feedback mechanism that maintains the involved hormone concentrations within certain biological limits. At the same time, the discussed model is analytically tractable; in particular, the existence of a so-called 1-cycle featured by a single pulse over one period is proven mathematically

Geometric analysis of oscillations in the Frzilator model

A biochemical oscillator model, describing developmental stage of myxobacteria, is analyzed mathematically. Observations from numerical simulations show that in a certain range of parameters, the corresponding system of ordinary differential equations displays stable and robust oscillations. In this work, we use geometric singular perturbation theory and blow-up method to prove the existence of a strongly attracting limit cycle. This cycle corresponds to a relaxation oscillation of an auxiliary system, whose singular perturbation nature originates from the small Michaelis-Menten constants of the biochemical model. In addition, we give a detailed description of the structure of the limit cycle, and the timescales along it

Parameter-robustness analysis for a biochemical oscillator model describing the social-behavior transition phase of myxobacteria

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 behaviors of myxobacteria. Myxobacteria are a particular group of soil bacteria that have two dogmatically different types of social behavior: 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 behaviors, 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 behavior 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 behavior 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 behavior of the solution