Feedback loops with time scales [Elektronische Ressource] / Stefan Brandt

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• Mathematics

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• Publié par: technische_universitat_munchen
• Publié le: 01 janvier 2010
• Nombre de lectures: 67
• Langue: Deutsch

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¨ ¨TECHNISCHE UNIVERSITAT MUNCHEN
Zentrum Mathematik
Feedback Loops With Time Scales
Stefan Brandt¨ ¨TECHNISCHE UNIVERSITAT MUNCHEN
Zentrum Mathematik
Feedback Loops With Time Scales
Stefan Brandt
Vollst¨andigerAbdruckdervonderFakult¨atfur¨ MathematikderTechnischenUniversit¨at
M¨ unchen zur Erlangung des akademischen Grades eines
Doktors der Naturwissenschaften (Dr.rer.nat.)
genehmigten Dissertation.
Vorsitzender: Univ.-Prof.Dr.Peter Rentrop
Prufer¨ der Dissertation: 1. Univ.-Prof.Dr.Johannes Muller¨
2. Univ.-Prof.Dr.Jurgen¨ Scheurle
3. Ao. Univ.-Prof.Dr.Peter Szmolyan
¨Technische Universit¨at Wien, Osterreich
(Schriftliche Beurteilung)
DieDissertationwurdeam01.September2009beiderTechnischenUniversit¨atMunc¨ hen
eingereicht und durch die Fakult¨at fur¨ Mathematik am 10. Februar 2010 angenommen.Zusammenfassung
R¨ uckkopplungsschleifen (”Feedback Loops”) bilden ein zentrales Motiv in biologischen
regulatorischen Netzwerken. Sie spielen außerdem eine wichtige Rolle im Auffinden von
Hysterese-Effekten und/oder oszillativem Verhalten. In dieser Arbeit entwickeln wir ein
allgemeines Model fur¨ gekoppelte Feedback Loops, wobei wir einen schnellen positiven
miteinemlangsamennegativenFeedbackLoopverbinden.DasEinfuhren¨ derverschiede-
nen Zeitskalen erlaubt eine tiefere mathematische Analyse eines drei-dimensionalen Pro-
totyps. Wir beweisen die Existenz einer Linie von Hopf-Bifurkationen, einer Linie von
homoklinen Orbits und Canard Orbits. Desweiteren untersuchen wir vier verschiedene
generische Systeme in einem assoziierten planaren Fall numerisch. Als eine m¨oglich An-
wendung gekoppelter Feedback Loops analysieren wir ein Model fur¨ die Blutgerinnung.
Abstract
Feedback loops are a central motive in biological regulatory networks, playing an impor-
tant role finding hysteresis effects and oscillatory behavior. In this thesis we develop a
general system of coupled feedback loops, combining a fast positive feedback loop with a
slow negative one. The introduction of different time scales allows a deeper investigation
of a three-dimensional prototype. We prove the existence of a line of Hopf bifurcations, a
line of homoclinic orbits, and canard cycles. Furthermorewe give a numerical description
of four generic cases in an associated planar case. As an application of coupled feedback
loops we investigate a model of the blood coagulation system.
vAcknowledgement
The last years have been an amazing, challenging and enjoyable experience. A lot of
people contributed to that, and even if it is impossible to mention all of them here, I
want them to know, that I am very thankful for all their help, all their encouragement
and for numerous interesting discussions.
First and foremost, I sincerely thank my advisor Prof. Dr. Johannes Muller¨ for
introducing me into the field of biomathematics, for the constant support and for the
continued helpful advice.
The research for this thesis was carried out at the Helmholtz Center Munich -
German Research Center for Environmental Health. I am very grateful to Prof. Dr.
Rupert Lasser for giving me the possibility to work at the Institute of Biomathematics
and Biometry and for the excellent working conditions there.
A special thank goes to Prof. Dr. Peter Szmolyan for the opportunity to visit
him at the TU Wien and for the fruitful discussions during my stay there.
Out of all the members at the IBB I wish to express my gratitude to Christina,
Wolfgang, Burkhard, Moritz, Stefan, Kristine and Alexandra. A special thank goes to
Georg for providing a constant support of coffee and for having an endless patience
listening to my problems.
Last but not least I thank my parents, who always supported me. Their patience,
understanding and encouragement were a great help, when it was most required.
viiContents
1 Introduction 1
2 Fundamental Setting And Requisites 3
2.1 BiologicalMotivation. ............................ 3
2.2 Tools From Geometric Singular Perturbation Theory . . . . . . . . . . . 8
3 Coupled Feedback Loops 12
3.1 GeneralSystem ................................ 12
3.1.1 FastAndSlowSystem ........................ 12
3.1.2 TheLayerProblem .......................... 14
3.2 SimplifiedSystem............................... 18
3.2.1 BifurcationsInTheLayerProblem ................. 20
3.2.2 A Picture Of The Flow For ε=0 .................. 24
4 Two-dimensional Approach 30
4.1 ANewTimeScale .............................. 30
4.2 TheCanardPoint. 33
4.2.1 TheFlowNearTheCanardPoint 36
4.2.2 The Chart K ............................. 382
4.2.3 The K 411
4.2.4 OutlinesOfProofs .......................... 46
4.3 TheHomoclinicPoint ............................ 49
4.4 GlobalAspectsOfTheFlow......................... 52
5 Three-dimensional Approach 59
5.1 LocalFlowNearTheCanardPoint ..................... 59
5.2 PeriodicOrbitsI ............................... 61
5.3 HomoclinicOrbit . .............................. 64
5.4 PeriodicOrbitsII............................... 68
6 Global Behavior Of Prototypical Examples 75
6.1 Takens - Bogdanov Bifurcation . . . . . . . . . . . . . . . . . . . . . . . 75
6.2 Unstable TB Orbits, A >0 . ........................ 78
6.3 U TB Orbits, A <0 . 80
6.4 Stable TB Orbits, A <0 . .......................... 83
6.5 Stable TB Orbits, A >0 . 85
ix7 Application: The Extrinsic Coagulation System 89
7.1 Motivation. .................................. 89
7.2 KeyPlayers . ................................. 92
7.3 ThresholdBehavior:AMinimalModel ................... 95
7.4 MinimalModelForTheWholeStory . 104
7.5 Realistic coagulation models . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.6 Discussion ................................... 115