Uncertainty and robustness analysis of biochemical reaction networks via convex optimisation and robust control theory [Elektronische Ressource] / vorgelegt von Steffen Waldherr
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Uncertainty and robustness analysis of biochemical reaction networks via convex optimisation and robust control theory [Elektronische Ressource] / vorgelegt von Steffen Waldherr

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150 Pages
English

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Published 01 January 2009
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Uncertainty and robustness analysis
of biochemical reaction networks
via convex optimisation and robust
control theory
Von der Fakult at Konstruktions-, Produktions-, und Fahrzeugtechnik und
dem Stuttgart Research Centre for Simulation Technology der Universit at
Stuttgart zur Erlangung der Wurde eines Doktors der
Ingenieurwissenschaften (Dr.–Ing.) genehmigte Abhandlung
Vorgelegt von
Ste en Waldherr
aus Friedrichshafen
Hauptberichter: Prof. Dr.-Ing. Frank Allgower
Mitberichter: Prof. Pablo A. Iglesias, PhD
Prof. Dr.-Ing. Elling W. Jacobsen
Tag der mundlic hen Prufung: 30. September 2009
Institut fur Systemtheorie und Regelungstechnik
Universit at Stuttgart
2009Acknowledgments
The results presented in this thesis are based on my work as a research assistant at the
Institute for Systems Theory and Automatic Control (IST) of the University of Stuttgart
from 2005 to 2009. During that time, I was also a PhD student in the Graduate School
Simulation Technology of the University of Stuttgart, and a visiting researcher at the
Automatic Control laboratory of the Swedish Royal Institute of Technology.
I want to thank Prof. Frank Allg ower for motivating me to work in the eld of systems
biology, and for his supervision during my doctoral studies. His great enthusiasm has
certainly been one of the driving forces in this work. I am also grateful to Prof. Elling
Jacobsen for inviting me to spend some time doing research in his group, on which some
of the results in this thesis are based, and want to thank him and Prof. Pablo Iglesias for
being on my thesis committee.
Theinteractionswithmanycolleaguessharingrelatedinterestswascrucialtomywork.
First, I want to mention Thomas Ei ing, Madalena Chaves, Prof. Rolf Findeisen, and
Prof. Peter Scheurich for guiding me into the eld of systems biology, each with their
own unique approach. I also want to thank other members of the IST systems biology
group, namely Christian Breindl, Marcello Farina, Jan Hasenauer, Prof. Jung-Su Kim,
Solvey Maldonado, and Monica Schliemann for very helpful discussions about the work
presentedhereandrelatedtopicsthroughoutthemajorpartofmydoctoralstudies. Also,
my research bene ted greatly from interactions with other groups. In particular, I want
to mention Malgorzata Doszczak from the experimental side, and Stefan Streif and Prof.
Fabian Theis for their collaboration in applications of my results. In addition to those
alreadymentioned, IwanttothankProf.ChristianEbenbauer, UlrichMunz, Prof.Nicole
Radde, Marcus Reble, Markus Rehberg, Daniella Schittler, Simone Schuler, and Gerd
Schmidt for helpful comments on this thesis.
I also want to thank all my former and current colleagues at the IST for creating a
very enjoyable and stimulating working atmosphere. It was always a pleasure doing both
academic and non-academic activities in this group. Last but not least, I want to thank
Annie and my parents for their love and support in all these years.
Stuttgart, October 2009 Ste en Waldherr
IIIIf a man will begin with certainties, he shall end in doubts; but if he will be
content to begin with doubts, he shall end in certainties.
Francis Bacon
Pro cience and Advancement of LearningContents
Index of notation VII
Deutsche Kurzfassung X
1 Introduction 1
1.1 Research motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2h topic overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Contribution of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Dynamical models for biochemical reaction networks 9
2.1 Basic modelling techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Local sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Classical models in systems biology . . . . . . . . . . . . . . . . . . . . . . 12
3 Uncertainty and robustness analysis of steady states 20
3.1 Introduction and problem statement . . . . . . . . . . . . . . . . . . . . . 20
3.2 Steady state infeasibility certi cates via semide nite programming . . . . . 24
3.3 Uncertainty analysis for steady states . . . . . . . . . . . . . . . . . . . . . 28
3.4 Robustness analysis for states . . . . . . . . . . . . . . . . . . . . . 31
3.5 Summary and discussion of the steady state analysis . . . . . . . . . . . . 38
4 Robustness analysis of qualitative dynamical behaviour 39
4.1 Introduction and problem statement . . . . . . . . . . . . . . . . . . . . . 39
4.2 Robustness analysis based on Jacobian uncertainty . . . . . . . . . . . . . 41
4.3 Ros analysis via Positivstellensatz infeasibility certi cates . . . . . 47
4.4 Summary and discussion of dynamical analysis . . . . . . . . . . . . . . . . 56
5 Locating bifurcation points in high-dimensional parameter spaces 58
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2 Loop breaking and steady state stability properties . . . . . . . . . . . . . 59
5.3 Bifurcation search via feedback loop breaking . . . . . . . . . . . . . . . . 65
5.4 Application to biochemical signal transduction . . . . . . . . . . . . . . . . 69
5.5 Summary and discussion of the bifurcation search method . . . . . . . . . 75
6 Kinetic perturbations for robustness analysis and sensitivity modi cation 76
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.2 Theory of kinetic perturbations . . . . . . . . . . . . . . . . . . . . . . . . 77
6.3 Robustness analysis with kinetic perturbations . . . . . . . . . . . . . . . . 83
VContents
6.4 Local sensitivity modi cations via kinetic perturbations . . . . . . . . . . . 89
6.5 Summary and discussion of the perturbation approach . . . . . . . 92
7 Construction and analysis of a TNF signal transduction model 93
7.1 Introduction to TNF signal transduction . . . . . . . . . . . . . . . . . . . 93
7.2 Development of a model for the anti-apoptotic TNF network . . . . . . . . 94
7.3 Analysis of oscillatory behaviour . . . . . . . . . . . . . . . . . . . . . . . . 99
7.4 Sensitivity modi cation by kinetic perturbations . . . . . . . . . . . . . . . 106
7.5 Discussion of the TNF network model analysis . . . . . . . . . . . . . . . . 108
8 Conclusions 110
8.1 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
A Proofs 114
A.1 Proof of Lemma 4.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
A.2 Proof of Proposition 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
A.3 Proof of Theorem 5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
B TNF network model summary 118
B.1 Molecular species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
B.2 List of reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
B.3 Nominal parameter values . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Bibliography 125
VIIndex of notation
Acronyms
Acronym Description
GMA generalised mass action
MAPK mitogen activated protein kinase
ODE ordinary di erential equation
SDP semi-de nite program
TNF tumor necrosis factor
Notation
Symbol Description
inA inertia of square matrix A
nn ndiagx∈R diagonal matrix with entries of x∈R on the diagonal
R non-negative real numbers+
R[x] ring of polynomials in the vector variable x overR
kS space of real symmetric kk matrices
M0 matrix or vector M is elementwise non-negative
P ()0 matrix P is positive (semi-)de nite
AB the setA is a subset of the setB (not necessarily proper)
A\B relative complement of the set B in the set A
nnI identity matrix inRn
Model and uncertainty description
Symbol Description
∈R exponent for i–th species in j–th reaction for GMA networksij
∂FnnA∈R system’s Jacobian A= =SV
∂x
n+q∈R state–parameter pair
n q nF :R R →R ODE right hand side F =Sv
k ∈R reaction rate constant for j–th reactionj +
M ∈R Michaelis-Menten saturation parameter for j–th reactionj +
qp∈R vector of reaction rate parameters
qp˜∈R perturbed parameter vector
ϕ∈R adjustable parameter
qPR set of parameter vectors
nmS∈R stoichiometric matrix
VIIIndex of notation
mv(x,p)∈R vector of reaction rates
mv˜(x,p)∈R perturbed reaction rate vector
∂vmnV(x,p)∈R reaction rate Jacobian V =
∂x
nx∈R vector of state variables
nX R set of state vectors
Model analysis
Symbol Description
(A ,B ,C ) state space representation of linearised open loop systemo o o
∈N number of critical frequencies (elements ofR)
mn, ∈R unscaled, scaled kinetic perturbation
nf(x,u,p)∈R vector eld for open loop system
ig( ,jω ())∈R transfer function value at i–th branch of critical frequenciesc
G( ,s )∈C transfer of input–output system
h(x)∈R output function for open loop system
(2k 2)kK∈R matrix for the representation of a ne state and parameter constraints
n+qMR q–dimensional manifold of steady state–parameter pairs
∈R robustness radius for qualitative dynamical behaviour
n+q n:R →R functional representation of manifoldM
1Q(),R()∈C transfer function decomposition G=QR
RR realness locus of a transfer function
%∈R robustness radius for steady state values
s∈C complex frequency variable
n∈R change in local sensitivity with respect to ϕ
nq, ∈R unscaled, scaled local sensitivity of steady state
T ∈R[] multiplier polynomial for Handelman representation
u, y∈R input, output for open loop system
kU ∈S matrix for the sum of squares representation of ODE right hand side
k TW ∈S matrix derived from the dyadic product
ω∈R frequency variable
nx (p)∈R steady state in dependence of parameterss
k∈R vector of monomials
n+q ( p)R set of steady state–parameter pairs for parameter p
Y ∈R[] polynomial for equality constraints
Z∈R[] polynomial for inequality constraints
VIIIAbstract
In the area of systems biology, dynamical models of biochemical reaction networks are
used to derive model-based predictions about the related biological processes. This thesis
provides new methods to study how parametric uncertainty a ects such predictions. The
focus of this study is on predictions about the steady states and the type of dynamical
behaviour, such as bistability or oscillations.
Concerning steady states, the problem of uncertainty analysis is investigated. For
a given extent of parametric uncertainty, the objective is to compute bounds on the
variations in the steady states. In view of an underlying feasibility problem, a method
based on semide nite programming is developed to solve this problem. The approach is
also applied to compute a measure for the robustness of the location of steady states in
the presence of parametric uncertainty.
Regarding the e ect of parametric uncertainty on the type of dynamical behaviour,
the robustness problem is considered. A robustness measure is de ned by the extent
of parametric uncertainty for which no local bifurcations occur. An approach to solve
the robustness problem with frequency domain methods is investigated. The proposed
feedback loop breaking method allows to characterise parametric uncertainties for which
the type of dynamical behaviour is robust. On the one hand, a lower bound on the
correspondingrobustnessmeasureiscomputedbyprovidingPositivstellensatzinfeasibility
certi cates for the underlying equations. On the other hand, the feedback loop breaking
concept is adopted for the design of a bifurcation search algorithm in a high-dimensional
parameter space. The results of the search algorithm thereby provide an upper bound on
the robustness measure.
In addition, the novel concept of kinetic perturbations is introduced. This is a class of
speci cparametricuncertaintieswhichareparticularlyusefulfortheanalysisofbiochem-
ical reaction networks. It is shown that a robustness analysis is performed e ciently for
kinetic perturbations by use of the structured singular value. As a side result, the direct
relation between kinetic perturbations and changes to the sensitivity of steady states in
a biochemical reaction network is demonstrated.
To complement the methodological results, a novel model for a speci c biochemical
signal transduction system within the TNF induced signalling network is constructed.
Themodelisanalysedwithmethodsdevelopedinthisthesis. Inadditiontoanillustrative
application of the new methods, the ndings of this analysis also provide new biological
insight into TNF signal transduction.
IX