Convex and toric geometry to analyze complex dynamics in chemical reaction systems [Elektronische Ressource] / vorgelegt von Anke Sensse

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• Publié par: otto-von-guericke-universitat_magdeburg
• Publié le: 01 janvier 2005
• Nombre de lectures: 22
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Convex and toric geometry to analyze complex
dynamics in chemical reaction systems
DISSERTATION
zur Erlangung des akademischen Grades
doctor rerum naturalium
(Dr. rer. nat.)
genehmigt durch
die Fakult at fur? Naturwissenschaften
der Otto-von-Guericke-Universit at, Magdeburg
vorgelegt von Dipl.-Biophys. Anke Sensse
geb. am 14. April 1976 in Kronberg im Taunus
Gutachter: Prof. Dr. G. Ertl
Prof. Dr. R. Friedrich
PD Dr. M. Hauser
Eingereicht am: 25. Januar 2005
Verteidigt am: 11. Juli 2005Dedicatedtothememoryof KARINGATERMANNTable 1: List of mathematical symbols
C field of complex numbers
R field of real numbers
Z ring of integer numbers
C[x] ring of polynomials with coefficients in the field of complex numbers
I ideal
def;torI deformed toric ideal
V(I) variety of an ideal
def;torV(I ) variety of a deformed toric variety
F family of polynomials depending on the parameters h; jh;j
F family of polynomials with a certain sign pattern in the coefficientsP N
D family of all matrices emerging from the matrix AA
if it is multiplied by a positive definite diagonal matrix
S family of all matrices having the same sign pattern as the matrix AA
N stoichiometric matrix
t th” stoichiometric coefficient of the i h-species in the j -reactionij
0 th? ; ? coefficients of the i -species in the reaction equationsi i
• kinetic matrix
Jac Jacobian
fi coefficient of the characteristic polynomial of the Jacobiani
fJac part of the convex Jacobian matrix
thH i -minor of the Hurwitzi
thv rate of the i -reactioni
thk rate constant of the i -reactioni
tx steady state concentration of the i h-speciesi0
th inverse steady state concentration of the i h-speciesi0
E minimal generating vector of the cone of nonnegative stationary reaction ratesi
Table 2: List of examples
Example 1 Extended Sel’kov oscillator 2 Electrooxidation of formic acid (galvanostatic, one current carrier)
Example 3o of acid (galv allt carriers) 4 Electrooxidation of formic acid (potentiostatic, one current carrier)
Example 5 Methylene blue sulfide oscillator
2+ 6 Ca -oscillations in the cilia of olfactory sensory neurons
Example 7 Sel’kov oscillator (and several extensions)
¡ 2+ 3+ 8 Peroxidase-oxidase reaction system (O , Per and Per subsystems)2Contents
1 Introduction 2
2 Methods and theoretical background 6
2.1 Basics of chemical reaction kinetics . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Algebraic structures in the space of reaction rates . . . . . . . . . . . . . . 8
2.2.1 Set of reaction rates as deformed toric variety . . . . . . . . . . . . 10
2.2.2 Set of stationary reaction rates as a convex cone . . . . . . . . . . . 15
2.2.3 Intersection of the deformed toric variety and the convex cone . . . 18
2.3 Graph theory for reaction networks . . . . . . . . . . . . . . . . . . . . . . 20
3 Stability criteria and the role of feedback-loops 25
3.1 Bifurcation analysis in a high-dimensional parameter space . . . . . . . . . 25
3.1.1 Routh-Hurwitz criterion . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Feedback-loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4 Electrocatalytic oxidation of formic acid 35
4.1 Reaction networks and kinetics . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2 Curves of stationary solutions . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2.1 Geometric considerations . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3.1 Intersection function of the curve of stationary solutions
and the plane of conservation relations . . . . . . . . . . . . . . . . 46
4.3.2 Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.4 Extensions to the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.5 Results, discussions and outlook . . . . . . . . . . . . . . . . . . . . . . . . 52
5 Sources for instability 54
5.1 Destabilizing feedback-loops . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.2 Realistic examples for different types of instability . . . . . . . . . . . . . . 61
6 Activator-inhibitor systems 66
6.1 Feedback-loops for bistability and periodicity . . . . . . . . . . . . . . . . . 66
6.1.1 Minimal bistable oscillators . . . . . . . . . . . . . . . . . . . . . . 67
6.1.2 Examples for bistable oscillators . . . . . . . . . . . . . . . . . . . . 70
ii6.1.3 Regions of bistability and oscillations in parameter space . . . . . . 71
6.2 Feedback-loops for chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.2.1 Extended bistable oscillators . . . . . . . . . . . . . . . . . . . . . . 77
6.2.2 Codimension-two bifurcations . . . . . . . . . . . . . . . . . . . . . 77
6.2.3 Shil’nikov’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.2.4 Homoclinic chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.3 Results, discussion and outlook . . . . . . . . . . . . . . . . . . . . . . . . 84
7 Enzymatic oxidation of NADH 86
7.1 The peroxidase-oxidase reaction system . . . . . . . . . . . . . . . . . . . . 87
7.2 The PO-reaction as an extended activator-inhibitor system . . . . . . . . . 89
7.2.1 Shil’nikov chaos in the PO system . . . . . . . . . . . . . . . . . . 90
7.3 Results, discussion and outlook . . . . . . . . . . . . . . . . . . . . . . . . 98
8 Summary and Outlook 100
8.1 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Bibliography 123
1Chapter 1
Introduction
Chemical reaction kinetics is a challenging topic which attracts more and more scientific
interest, especially, since nonlinear dynamical phenomena have been found in an increas-
ingly large variety of electrochemical and biochemical systems. These systems very often
show a high degree of complexity, not only because of the high number of components
but also because of the great quantity of parameters to take into account. With regard
to this it seems essential to strain an interdisciplinary investigation and to make a whole
arsenal of tools and methods work together to solve each problem with its own particular
difficulties.
The aim of this work, presented as doctorial thesis, is not the addition of some further
models to a long list of successful modelings of different reaction systems. This report is
primarilyconcernedwiththeimprovementandrefinementofrecentlyintroducedmethods
and algorithms to deal with polynomial systems. It will be demonstrated how they can
becombinedwitholdertools fromstoichiometricnetworkanalysis and afew spontaneous
tricks to solve various models for complex reaction systems of intriguing interest. Those
are above all the catalytic reaction systems, which naturally enable nonlinear reaction
behavior on the one hand and on the other hand, they represent the large majority of
systems of scientific and industrial importance. Representatives of these systems can be
found in many reaction systems of our environment. In this work two main examples are
presented, each of them stemming from another field of chemistry. In an electrocatalytic
reaction the catalytic nature is due the electrode in combination with the electric field,
in an biochemical reaction sequence the catalysis is usually exhibited by enzymes.
The procedures and conceptions, which will be presented, stem from algebraic geom-
etry, a theory hardly applied in the analysis of dynamical systems so far.
One pioneering work in theoretical chemistry stems from Bruce Clarke [11], who de-
rivedhisso-calledstoichiometric network analysis byconceptsfromconvexgeometryand
related the network’s topology to the stability of the according kinetic system. The ba-
sic idea of this analysis is the transformation of the set of stationary solutions from the
concentration space into the space of reaction rates, which represents a convex polyhe-
dral cone. Making use of the special convexity properties of this space Clarke derived
the Jacobian matrix and executed the stability analysis in this space. However in this
theory, which provides very efficient methods for a fast recognition of unstable networks
2in practice, one important detail was missing. The properties of the mapping from the
set of stationary concentrations onto the set of stationary reaction rates and of its inverse
have not been elucidated. There was no direct way to transfer the results obtained in the
set of stationary reaction rates back to the set of stationary concentrations.
This gap between the two spaces has been closed two decades later, when supplemen-
tary restrictions for the reaction rates have been found [30]. If the stationary reaction
rates satisfy these restrictions in the space of reaction rates, there is a surjective mapping
back to the set of stationary concentrations.
In order to realize the existence of these restrictions and even for their calculation a
huge theoretical background from toric geometry must be introduced. Using mass action
ratelawthereactionratesrepresentmonomialsinthealgebraicsense,whichofferstheway
for ideal theoretic considerations. The construction of different ideals, the change of an
idealbasisandthechoiceofabasiswithspecialeliminationpropertiesaretheprocedures
to show that the image of the monomial mapping from the concentration space into the
space of reaction rates must lie in the variety of