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Qualitative modeling in computational systems biology
Citation for published version (APA):Musters, M. W. J. M. (2007). Qualitative modeling in computational systems biology. Technische UniversiteitEindhoven. https://doi.org/10.6100/IR629275
DOI:10.6100/IR629275
Document status and date:Published: 01/01/2007
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Qualitative Modeling in ComputationalSystems Biology
Applied to Vascular Aging
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de
Technische Universiteit Eindhoven, op gezag van de
Rector Magnificus, prof.dr.ir. C.J. van Duijn, voor een
commissie aangewezen door het College voor
Promoties in het openbaar te verdedigen
op dinsdag 18 september 2007 om 16.00 uur
door
Mark Wilhelmus Johannes Maria Musters
geboren te Breda
Dit proefschrift is goedgekeurd door de promotor:
prof.dr.ir. P.P.J. van den Bosch
Copromotor:
dr.ir. N.A.W. van Riel
CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN
Musters, Mark W.J.M.
Qualitative modeling in computational systems biology : applied to vascular aging / by
Mark Wilhelmus Johannes Maria Musters. - Eindhoven : Technische Universiteit
Eindhoven, 2007.
Proefschrift. - ISBN 978-90-386-1564-6
NUR 954
Trefw.: nietlineaire differentiaalvergelijkingen / kunstmatige intelligentie /
regelsystemen ; parameterschatting / fysieke veroudering.
Subject headings: nonlinear differential equations / piecewise linear techniques /
parameter estimation / ageing.
This thesis was prepared by using the LATEX typesetting system.
Cover design by Christoph Brach [email protected], http://www.nutsdesign.net
Printed by Gildeprint drukkerijen, Enschede.
Qualitative Modeling in ComputationalSystems Biology
Applied to Vascular Aging
Samenstelling kerncommissie:
prof. dr. ir. P.P.J. van den Bosch promotor TU/e
dr. ir. N.A.W. van Riel copromotor TU/e
prof. dr. P.A.J. Hilbers lid kerncommissie TU/e
dr. ir. H. de Jong lid kerncommissie INRIA Rhone-Alpes
prof. dr. ir. C. Th. Verrips lid kerncommissie UU
Contents
1 Introduction 1
1.1 Challenges in Systems Biology . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Top-Down versus Bottom-Up . . . . . . . . . . . . . . . . . . . . . 4
1.2 Aging of the Vascular System . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Extending Longevity in Mythology . . . . . . . . . . . . . . . . . . 5
1.2.2 Understanding Aging: the Scientific Approach . . . . . . . . . . . . 5
1.2.3 Changes in Biochemical Networks during Vascular Aging . . . . . . 6
1.3 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Project Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.2 Research Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Analysis of Bistable Systems 11
2.1 Basic Knowledge about Feedback Loops, Circuits and Systems . . . . . . . 11
2.2 Nonlinear Dynamics in Biology . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Mathematical Model of ECM Remodeling . . . . . . . . . . . . . . . . . . 14
2.4 Graphical Study of Bistability . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.1 Breaking the Feedback Loop . . . . . . . . . . . . . . . . . . . . . . 16
2.4.2 Solving the Steady-States Symbolically . . . . . . . . . . . . . . . . 16
2.4.3 Deriving Restrictions on Parameter Values . . . . . . . . . . . . . . 17
2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Qualitative Analysis of Nonlinear Biochemical Networks 19
3.1 General Description of the Procedure . . . . . . . . . . . . . . . . . . . . . 19
3.1.1 Approximation of Nonlinear Function with PWA Functions . . . . . 20
3.1.2 Selection of PWA Parameters . . . . . . . . . . . . . . . . . . . . . 25
3.1.3 Detection of Equilibrium Points and Performing Stability Analysis . 27
3.1.4 Construction of Qualitative Transition Graphs . . . . . . . . . . . . 29
3.2 Example: an Artificial Biochemical Network . . . . . . . . . . . . . . . . . 29
3.2.1 PWA Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.2 Determination of the Equilibrium Points . . . . . . . . . . . . . . . 31
3.2.3 Dynamical Behavior at the Equilibrium Points and Stability Analysis 33
v
CONTENTS
3.3 Transition Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 Analysis of the Transforming Growth Factor-β1 pathway 41
4.1 Physiology of the TGF-β1 Signaling Pathway . . . . . . . . . . . . . . . . . 41
4.1.1 Isolation of the R-SMAD Loop . . . . . . . . . . . . . . . . . . . . 42
4.2 Qualitative Analysis of the Transforming Growth Factor-β1 Pathway . . . 44
4.2.1 Model Reduction of the TGF-β1 Pathway . . . . . . . . . . . . . . 44
4.2.2 Quasi-Steady-State Approximation of the TGF-β1 Model . . . . . . 48
4.2.3 From Nonlinear to Piecewise-Affine . . . . . . . . . . . . . . . . . . 51
4.2.4 Equilibria and Stability Analysis . . . . . . . . . . . . . . . . . . . 52
4.2.5 Transition Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5 Signal Transduction of the Unfolded Protein Response 65
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2 Protein Folding of the von Willebrand Factor . . . . . . . . . . . . . . . . 65
5.2.1 Translation and Translocation . . . . . . . . . . . . . . . . . . . . . 66
5.2.2 Protein Folding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.3 The Unfolded Protein Response . . . . . . . . . . . . . . . . . . . . . . . . 67
5.3.1 Signal Transduction in the UPR . . . . . . . . . . . . . . . . . . . . 67
5.4 Mathematical Model of Signal Transduction during the UPR . . . . . . . . 69
5.5 Qualitative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.5.1 From Nonlinear to Piecewise-Affine . . . . . . . . . . . . . . . . . . 72
5.5.2 Equilibrium Points in the UPR Model . . . . . . . . . . . . . . . . 74
5.5.3 Transition Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.5.4 Comparison with Experimental Data . . . . . . . . . . . . . . . . . 80
5.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6 System Identification with Parameter Constraints 87
6.1 The Biochemical Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.2 Qualitative Phase Space Analysis . . . . . . . . . . . . . . . . . . . . . . . 90
6.2.1 Nonlinear to PWA Conversion . . . . . . . . . . . . . . . . . . . . . 90
6.2.2 Transition Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.2.3 Constrained Nonlinear Parameter Estimation . . . . . . . . . . . . 92
6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7 Hybrid System Identification 97
7.1 General Identification Procedure . . . . . . . . . . . . . . . . . . . . . . . . 97
vi
Contents
7.1.1 Model Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.1.2 Identification and Classification of a Hybrid Model . . . . . . . . . 98
7.2 PWA Identification of the Biochemical Oscillator . . . . . . . . . . . . . . 99
7.2.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8 Conclusion and Discussion 103
8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8.2 Future Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
A Nomenclature 107
A.1 List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
A.2 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
A.2.1 Latin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
A.2.2 Greek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Summary 127
Samenvatting 129
Dankwoord 131
About the Author 133
vii
Contents
viii
1Introduction
KNOWLEDGE of health and the need of solutions for addressing diseases have
always fascinated humanity. Understanding the biochemical processes within
cells,“the building blocks of life”, has become indispensable. During the 20th century,
our understanding of cellular biology has increased at an astonishing rate. Over the last
decades, it can mainly be attributed to breakthroughs in the research field of molecular
biology. This research field deals with the use of techniques from various research areas
on solving biological problems. For example, molecular biology provided the necessary
high throughput methods for unraveling the complete human genome in 2001 [96, 173].
This was an important step towards a better comprehension of the “blueprint of life” at
that time, but it has recently become clear that dynamical information provides more
insights about human physiology and pathological phenomena. A human can be viewed
as a system with 1014 cells [72], each containing approximately 25, 000 genes [96, 173]
and intertwined signaling networks operating over distinct spatio-temporal scales [132].
These data emphasize that comprehension of life’s complexity is impossible by intuitive
reasoning alone; computational approaches that integrate the available information into
a single framework have therefore become indispensable [15, 91].
A mathematical model is a description of a system in terms of mathematical equa-
tions [25]. A plethora of mathematical formalisms exists to describe physiological pro-
cesses: discrete, continuous, deterministic, stochastic and combinations of these model-
ing frameworks. Some of these frameworks are suitable for specific situations. For in-
stance, stochastic models are used for processes that are in essence dominated by random
events [156], e.g. binding of a substrate to a receptor; it can result in different outcomes
for the same initial conditions. Since the deterministic approximation of the biochemical
reaction systems becomes more difficult for reactions with a few number of molecules,
stochastic models are therefore more appropriate for describing these biochemical reac-
tions. They provide a thorough description, based on several fundamental properties
from physics. Unfortunately, computational complexity increases exponentially for sys-
1
Chapter 1
tems with more substrates. For a detailed overview of the wide variety of computational
methods, the reader is referred to a review of de Jong [26].
At a cellular level, mathematical models are frequently deterministic descriptions as
they provide a good balance between computational effort and accuracy. Tools from
system identification enable building mathematical models of a dynamic system based on
measured data [170]. The parameters of a given model are subsequently adjusted until the
predicted dynamics coincide as good as possible with the measured signals. To acquire
mathematical models of biochemical networks and to obtain system-level understanding of
the biochemical interactions of these networks have gained much popularity in the research
community over the last few decades [157] and has been called systems biology [85]. Large-
scale and mechanism-based models have gradually earned a central role in this research
field. It has resulted in a variety of models, ranging from signal transduction [31, 68, 147]
to genetic networks [29, 67, 70].
1.1 Challenges in Systems Biology
In their quest for the identification and validation of drug targets and biomarkers, phar-
maceutical companies like AstraZeneca, Bayer AG, Eli Lilly, Merck, Novartis, Organon,
Pfizer and Roche have become interested in the use of the in silico1 concept. However,
there are several challenges that system biologists have to face when constructing com-
puter models of biochemical systems from both the biological and technical perspective:
Experimental Issues
1. The amount of quantitative information is limited in biology. In Fig. 1.1 typi-
cal biological data is shown for which the intensity of each band represent the
qualitative level of a given species in arbitrary units during a certain amount of
time. These data have limited quantitative significance, since they do not represent
physiologically-relevant quantities. Even if these data could be linked to substrate
concentrations, the intensities are extremely sensitive to external influences during
the experiments. Another example is the large pool of qualitative -omics data2
in biology, valuable information which is usually omitted in traditional parameter
estimation. Although by improving measuring techniques much progress has been
made in obtaining data on a cellular level, accurate quantitative information is
only available for a few well-characterized systems [159].
1in silico refers to “performed on computer or via computer simulations”.2-omics data represent data obtained by transcriptomics, proteomics and metabolomics. These are
methods to derive information from messenger-RNA (mRNA), proteins and metabolic analysis profiles,respectively, in a massive parallel way (i.e., currently dozens of metabolites/proteins and thousands of
2
Section 1.1
Figure 1.1 – Typical nuclear extracts (a. and b.) obtained from electrophoretic mobilityshift array (EMSA) of the IκB-NF-κB signaling module [70].
2. The large variations in quantitative measurements due to differences between spe-
cies, individuals and experimental procedures [117, 118, 119]. Experimental data
are also polluted with noise which yields extremely large standard deviations,
thus relatively low reliability. Furthermore, generating experimental data is labor-
intensive and expensive with complicated experimental protocols, which limits the
number and reproducibility of data points considerably.
Technical Challenges
1. Interactions between components are inherently nonlinear due to the chemical law
of mass action [37], which hampers system analysis considerably.
2. Nonlinear system identification requires initial estimates for the unknown param-
eters which has to be relatively close to the “true values”3 to avoid converging to
a wrong solution. During parameter estimation, the parameters of a given model
are varied and the dynamics of the model are simulated. The simulation results
are compared with experimental data to see whether the parameters result in the
most accurate approximation of the experimental for the given model structure.
If not, the procedure is repeated for a different set of parameter values, until the
error between simulated results and experimental results have been minimized to
the lowest error, i.e., the global minimum. However, parameter estimation can
sometimes converge to a suboptimal solution (local minimum) which should be
avoided.
How do Researchers Deal with these Challenges?
Several strategies have been developed to tackle these issues, each with their advantages
and disadvantages:
mRNA levels).3“true value” is a parameter that would produce the best simulated model fit, compared to the
experimental data.
3
Chapter 1
• Mathematical models are frequently implemented with arbitrary parameter values
to obtain a certain realization of the system behavior [82, 97, 135, 137]. Practi-
cal relevance of these models is of marginal importance since a different choice in
parameter values can result in completely different system dynamics [162].
• Mathematical models have been constructed for various biochemical networks [22,
23, 51, 70, 82, 143, 167]. These are, in general, large and complex nonlinear struc-
tures and have been fitted on insufficient data points which produces models with
low predictive power [176].
• A principal part of experimental data is obtained from bacteria and yeast which
are less complex structures than mammalian cells and hence, extrapolating these
data to mammals, is therefore not always allowed. Even if parameters are obtained
from mammalian cells, parameters can vary several orders of magnitude depending
on the experimental procedures [119].
• A specific class of biochemical networks can be studied which shows some mathemat-
ically advantageous properties for which mathematical analysis can be performed.
For example, genetic regulatory networks contain switch-like behavior. Under these
conditions, Boolean abstractions are a suitable modeling class [161, 164]. However,
most biochemical networks do not contain these hard switches. The large disconti-
nuities in discrete formulations introduce system behavior which is not observed in
smoother approximations.
1.1.1 Top-Down versus Bottom-Up
Systems biology makes a distinction between two modeling approaches: top-down and
bottom-up.4 In a top-down modeling approach, the organism is analyzed as a whole and
broken down into smaller, computable entities [66]. This makes modeling and simulation
of multicellular systems feasible, but the sparsity in data forms a difficulty (Fig. 1.2).
Bottom-up modeling is based on reductionism and reconstructs pathways of basic units
like proteins and genes. The post-genomic era has contributed much to the amount of
qualitative information, but the numerous pathways and their complexity in a (single)
cell hamper the translation of results to a physiological multicellular level. The challenge
in human systems biology is therefore to combine information from -omics experiments
(bottom-up) with qualitative information of the physiology (top-down).
We may conclude that computer modeling of biochemical networks is not a trivial task.
Meanwhile it has become more and more important in the post-genomic era, dominated
by qualitative information. This thesis shows that qualitative information can be utilized
to describe and analyze biochemical networks. Biochemical reactions involved in vascular
4Different definitions of bottom-up and top-down appear in literature, see [170] for an overview.
4
Section 1.2
Organism
Tissues
Cells
Proteins
Genes
phenotypes
TOP-DOWN
BOTTOM-UP
-omics profiling
available data
Figure 1.2 – Graphical representation of the top-down and bottom-up approach withrespect to the level of biological organization [93].
aging, strongly related to cardiovascular diseases, will be studied to verify our developed
procedures.
1.2 Aging of the Vascular System
1.2.1 Extending Longevity in Mythology
Mythical stories and history show a profound interest in extending longevity by means of
food or drinking. Ancient Greek mythology is interlarded with tales about divine food,
called ambrosia, that conferred immortality on whoever it consumes. In Asia, the Chinese
emperor Qin Shi Huang sent his alchemist Xu Fu to find the elixir of life, but he discovered
Japan instead of completing his mission. On the American continent, the Incas believed
that the creator god Viracoca rose up from the sacred lake Titicaca in Peru and created
the world. One of its islands, Isla del Sol, contains three separate springs which is believed
to represent a fountain of eternal youth. In the same region, the Spanish conquistador
Juan Ponce de Leon (Fig. 1.3) started his journey in 1512 to find the Fountain of Youth in
the New World. Celtic mythology tells about the quest of the Knight of the Round Table
to search for the Holy Grail, which was believed to give an immortal life to the finder.
There are indications that overproducing certain enzymes involved in nutrient withdrawal
assists in prolonging life in yeast [77] and worms [165] significantly. Besides increasing
life span, quality of life is an additional aspect that should be considered. It is therefore
an intriguing research question whether healthy aging could be pursued for mammals as
well by means of nutrients.
1.2.2 Understanding Aging: the Scientific Approach
Aging is the accumulation of changes responsible for the sequential alterations that ac-
company advancing age and the associated increases in the chance of disease and death
[6]. Although this process concerns the whole senior world population, little is known
5
Chapter 1
Figure 1.3 – Juan Ponce de Leon during his quest [41].
about the effects of aging on the cellular level and how aging cells lead to aging of the or-
ganism. In Western society, the main causes of death are cardiovascular diseases that are
related to aging processes, e.g. atherosclerosis and heart failure. For a more healthy aging,
it is therefore of great importance to understand the highly complex processes involved
in vascular aging. It is a fact that more than 300 theories circulate about aging [108].
General agreement exists over the crucial role of oxidative stress in aging [65], also known
as reactive oxygen species (ROS). Unfortunately, the primary mechanism that underlies
stress-induced aging processes has not been revealed yet. Over the last few decades, sev-
eral aging theories have been proposed in the literature which include telomere shortening
[64, 128, 129], hormones [32], DNA damage/repair [57, 100], caloric restriction [104], mito-
chondria [112] and protein quality control [9, 73, 134, 180]. Identifying and disentangling
the crucial biochemical networks in vascular aging is therefore one of the challenges in the
research field of aging.
1.2.3 Changes in Biochemical Networks during Vascular Aging
Alterations in cellular processes during aging have been considered as a possible indicator
for pathological phenomena [93]. Based on experimental evidence, three aging-related
biochemical networks were selected
Extracellular Matrix
The extracellular matrix (ECM) is a complex network of biomolecules like polysaccharides
and proteins secreted by the cell and serves as a structural element in tissues. The link
between vascular aging and a change in the ECM structure has been reported in litera-
6
Section 1.3
ture [75], but has also experimentally been demonstrated by disturbed levels of mRNA
encoding for proteases, which are enzymes that degrade the ECM mesh. Proteases are
assumed to play a role in cell-controlled composition changes of the ECM (“remodel-
ing”) [97], as will be shown in chapter 2.
Transforming Growth Factor-β1
From experimental data [17], the Transforming Growth Factor-β1 (TGF-β1) pathway is
most likely involved in vascular aging. TGF-β1 is a cytokine that binds to receptors on the
endothelial cells. Many cellular processes have been imputed to this substrate [30], e.g.
the cell cycle [182], vasodilatation [144] and ECM formation [123, 160]. More physiological
details of this network are given in chapter 4.
Unfolded Protein Response
Proteins that are secreted outside the cells need to be folded in the endoplasmic reticulum
(ER) for proper functioning. The unfolded protein response (UPR) is the quality control
mechanism that monitors the folding procedures. However, during aging the UPR seems
to fail in performing its task which leads to an accumulation of misfolded proteins [168]
that can form the basis of various diseases [148]. A general description of the UPR in
mammalian cells is provided in chapter 5.
1.3 Problem Statement
1.3.1 Project Description
This thesis is a result of a project in which Unilever Research Vlaardingen, Aurion, Uni-
versity of Utrecht and Eindhoven University of Technology have cooperated. The aim of
the project was to increase our knowledge of vascular aging by applying genomics, pro-
teomics and metabolomics, which involve data from the genome, proteins and metabolic
products, respectively. The contribution of Eindhoven University of Technology within
this project was to integrate the experimental data in computer models and evaluate the
outcome, providing valuable insights in the development and screening of new compo-
nents for functional foods by means of systems biology. Eventually this would lead to a
selection of natural ingredients that could slow down vascular aging processes, which con-
tributes to healthy aging. The research has financially been supported by the Netherlands
Organization for Scientific Research, grant R 61-594, and by Senter grant TSGE1028.
7
Chapter 1
1.3.2 Research Goals
As mentioned in section 1.1, analysis of biological aging processes with computer models
requires alternative approaches due to the limited amount of quantitative information.
This leads to the main research question of this thesis:
Primary Goal − Develop mathematical procedures to extract information from
typical nonlinear biochemical models that contain little quantitative information.
Main purpose: assistance in (qualitative) system analysis and improved parameter
estimation.
The problem has been reformulated even more strictly: if we assume that initially no
quantitative information is available, how much valuable information can be extracted by
means of (qualitative) analysis of the system dynamics given the topology of the network
and the type of kinetic interactions? And could qualitative analysis be used to analyze
the global dynamics of the system? In this thesis, two symbolic strategies have been
advocated for the analysis of nonlinear biochemical networks:
1. Graphical analysis of monotone systems that exhibit bistability, i.e., dynamics with
two stable steady-states or equilibrium points. This method was initially developed
for the determination of bistable behavior [3], but could be adapted to constrain
specific parameters. This method is the subject of chapter 2.
2. General qualitative piecewise-affine (PWA) analysis. PWA systems are a type
of hybrid systems, i.e., a mathematical modeling paradigm that combines both
discrete and continuous features into one unifying framework. We used this PWA
procedure to approximate a biological process by a number of simpler functions,
which results in a more refined modeling framework for qualitative analysis [141].
This method is applicable to a wide range of biochemical networks and will be
introduced in chapter 3.
These two strategies contribute to a reduction in parameter search space and assist in
system identification.
As the overall project was aimed at a better understanding of vascular aging processes,
the system analysis tools were applied to various biochemical processes involved in vascular
aging. Therefore another goal has been defined:
Secondary Goal − Apply the developed methodologies to typical biochemical
networks that are involved in vascular aging processes.
The graphical analysis was applied to a model of ECM remodeling (chapter 2). The
qualitative PWA analysis will be applied to analyze the TGF-β1 pathway (chapter 4)
8
Section 1.4
and UPR module (chapter 5). Furthermore, this methodology also facilitated parameter
estimation of a putative biochemical oscillator in chapter 6.
1.4 Related Work
The project encompasses several research areas. An overview of work related to the topics
in this thesis will be given below.
Modeling Aging Processes
Studying vascular aging with computer models is a recent development. Top-down ap-
proaches are limited to scaling laws [158, 177, 178, 179] and telomere shortening [128,
129, 174], whereas the bottom-up method is more popular in biochemical networks, like
cytosolic protein folding [135], and the “network theory of aging” [84]. The UPR module
of Saccharomyces Cerevisiae has been modeled by [106], but the mammalian UPR is
more complex [103]. Similar to the UPR in the ER is the heat shock response in the
cytosol of the cell, for which a stochastic model was created [135]. Most parameter values
in this model were chosen arbitrarily and, consequently, the physiological significance is
low. The TGF-β1 pathway is relatively new. Recently a mathematical model of this
pathway was proposed with parameters varying over several ranges [24]. However, the
inhibitory influence of specific substrates [30] was not included in this model. Therefore
mathematical models of the TGF-β1 pathway and UPR response were constructed from
scratch by performing an extensive literature search and translating the obtained infor-
mation in differential equations [117]. To our knowledge, such models have not yet been
developed.
Nonlinear System Identification and Parameter Estimation
System identification of nonlinear systems requires an initial estimate of the parameter
values [102] and these are, as a consequence, often arbitrarily chosen. The parameters
of the model are tuned by Levenberg-Marquardt, Gauss-Newton or multiple shooting
method [95, 116] algorithms by minimizing the cost function J(θ) of the error between
the true parameter set θ and the model fit θ. The solution with the lowest cost function
J(θ), the global minimum, results in the best model fit. Due to poor quality of the data
and erratic initial estimates, the solution can also converge to a local minimum. Another
common way to study the effect of parameters on the output of a model is to apply
sensitivity analysis. The parameters are varied across a range of parameter values to check
the effect of each individual parameter on the system. However, this method demands
a proper estimate of the parameter values as well. Defining upper and lower bounds on
9
Chapter 1
the parameter values could provide a good solution, but becomes computationally too
complex for large variations in the parameter values.
Hybrid Systems in Biology
Although piecewise-linear functions have been used in biology for decades [50], the appli-
cation of hybrid systems in biology is currently reviving this re. Applications are limited
to specific processes with clear switching characteristics [2, 12, 14, 20, 29, 35, 48, 99, 181]
and therefore primarily aimed at genetic regulatory networks. Here we show that hy-
brid systems are not limited to these mathematical functions, but can also be applied to
systems that display “soft switching” such as Michaelis-Menten kinetics. Hybrid system
identification of biochemical networks is in its infancy and has primarily been focussed on
genetic regulatory networks [35] or requires initial estimates of the parameter values [171].
Hybrid system identification can be performed in different ways, see [76] for an overview.
In chapter 6, a hybrid system identification procedure is presented which makes use of a
linear least-squares methodology and therefore no initial estimate of the parameter values
is required.
1.5 Thesis Outline
This thesis develops and significantly extends existing methods to cope with the limita-
tions in biological modeling due to scarcity of biological data. In chapter 2 the class to
model nonlinear biochemical networks is defined and graphical analysis is applied to a
nonlinear model of the ECM. A mathematical model of the formation and degradation
of ECM is derived from the literature [97]. An improved version of the graphical analysis
procedure is proposed and used to derive the so-called Michaelis constants of the ECM
model. Although graphical analysis yields constraints for specific parameters, the class
of systems is limited to monotonic networks that satisfy certain criteria [3]. Chapter 3
introduces a PWA method that is more general applicable. In chapter 4 the TGF-β1
pathway is tested with this procedure and the predicted qualitative behavior agrees with
the experimental data. A model of the UPR, a bit larger in complexity, is developed and
analyzed similarly in chapter 5. In chapter 6, a biochemical oscillator is introduced [16, 54]
and analyzed with the qualitative procedure from chapter 3. It yields a set of constraints
on the parameter values, which have been incorporated in nonlinear system identification.
This method results in better estimates of the parameter values compared to traditional
unconstrained system identification. In chapter 7, a hybrid system identification frame-
work is developed that can provide accurate initial estimates of the parameter values.
This thesis concludes with the main conclusions and future perspectives.
10
2Analysis of Bistable Systems
Based upon experiments, it has been suggested that the Transforming Growth Factor-β1
(TGF-β1) pathway plays a significant role in the pathological phenomena of vascular ag-
ing [75]. One of the processes controlled by TGF-β1 is the “remodeling” of the extracellular
matrix (ECM), an extensive network of biomolecules like glycoproteins and proteoglycans.
The ECM is a layer that surrounds cells and is essential for inter- and intracellular com-
munication, serves as a protective layer and contributes to cell-cell adhesion. Changes
in TGF-β1 expression imply disturbances in the ECM structure as well. Disturbances in
ECM remodeling have directly been related to cardiovascular diseases [75].
2.1 Basic Knowledge about Feedback Loops, Circuits
and Systems
Before a model of the ECM is introduced, some basics on systems theory are required for
understanding the structure of biochemical networks. In general, biochemical networks
contain one or multiple feedback loops, i.e., some proportion of an output signal of a
system is passed (fed back) to the input. This is a common way to control the dynamical
behavior. A single feedback loop consists of a single element and has an enhancing or
inhibiting effect on the input signal. Regulatory elements A, B and C form a feedback
circuit if the level of A exerts an influence on the rate of production of B, whose level
influences the rate of production of C, whose level in turn influences the rate of production
of A [162]. Two classes of feedback circuits exist. Either each element in the circuit exerts
a positive action on its own future evolution (activation), or the sum of all elements in
the circuit exerts a negative action on this evolution (repression); a circuit is positive
or negative if the parity of the number of negative elements in the circuit is even or
odd, respectively [162]. Negative feedback circuits are common in biology. This circuit
class is mainly responsible for stability of an attractor, i.e., a set in the phase space
11
Chapter 2
which is asymptotically approached in the course of dynamics. Positive feedback circuits
have been observed in biological switches [8, 40], the cell cycle [153], and “memory”
effects [101]. Adding several negative and/or positive feedback circuits together results in
negative- or positive-feedback systems, depending on the number of negative and positive
feedback circuits [164]. The graphical study in this chapter is tailored for positive-feedback
systems. These systems have in common that they have multiple steady-states, also called
multistability. Depending on the history of the system and a given set of initial conditions,
the system can converge to a different attractor. A specific subset of multistability is
bistability, in which two stable nodes are present.
2.2 Nonlinear Dynamics in Biology
Biochemical networks are comprised of several biochemical processes, in which special
proteins, called enzymes, enhance the rate of these reactions [46]. For a simple irreversible
conversion of a substrate S into a product P, facilitated by an enzyme E, the chemical
reaction is given by
E + Sk1
GGGGGBFGGGGG
k−1
ESk2
GGGA E + P, (2.1)
ES enzyme-substrate complex,
k1 association rate constant of E with S,
k−1 dissociation rate constant,
k2 conversion rate constant of ES into P.
Making the assumptions that all parameters are larger than zero, ES quickly reaches a
constant value and that the total amount of enzyme ET in the system is fixed, leads to a
single expression for the enzymatic conversion rate f(x):
f(x) =k2ETx
k−1+k2
k1+ x
=Vmaxx
Km + x, (2.2)
x substrate concentration,
Vmax maximal rate of conversion,
Km Michaelis constant.
This equation is also known as the Michaelis-Menten equation [111], its graph is shown
in Fig. 2.1a. Sometimes binding of a substrate molecule to a single enzyme can increase
its affinity to other substrate molecules. This positive cooperativity has been described by
the Hill equation
f(x) =Vmaxx
r
Krm + xr
, (2.3)
12
Section 2.3
00
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
r = 2r = 5
r = 10r = 100
Kmx x
Km
Vmax
f x( ) f x( )
Vmax
Vmax
2
(a) (b)
Figure 2.1 – (a) Michaelis-Menten curve for Vmax = 1 and Km = 0.1 and (b) Hill kineticsfor various cooperativity coefficients (r = 2, 5, 10 or 100), Vmax = 1 and Km = 0.5.
r cooperativity coefficient.
Note that if r = 1, one obtains the standard Michaelis-Menten equation, and for r À 1,
the system resembles a biological switch, see also Fig. 2.1(b). Michaelis-Menten and Hill
kinetics are typical monotonically increasing functions. This property is a necessity for
the graphical analysis procedure, which will be applied to a mathematical model of ECM
remodeling that is composed of ordinary differential equations (ODEs) [122, 124]. This
type of mathematical models has been widely used to simulate concentrations of substrates
by time-dependent variables in order to describe the dynamics of various biochemical
networks [26]. They have the following mathematical form
dxi
dt=
∑j
fj(x), i = 1, . . . , Nx, j = 1, . . . , Nf , (2.4)
x [x1, . . . , xNx ]T ≥ 0, vector with state variables,
fj(x) ≥ 0, rate equation,
Nx total number of states,
Nf total number of rate equations,
t time.
It falls outside the scope of this thesis to provide an extensive overview of modeling dy-
namical systems with ODEs. More information about this topic can be found in standard
engineering textbooks, e.g. [25].
13
Chapter 2
2.3 Mathematical Model of ECM Remodeling
It was hypothesized that there exists a delicate balance between ECM construction and
degradation, which can be simplified to a two- or three-state model, which are defined as
the closed loop and open loop model, respectively [97]. In this chapter, we will focus on
the dimensionless closed loop model (Fig. 2.2) that is composed of so-called bisubstrate
Michaelis-Menten (see chapter 3 for more details) and Hill equations:
dx1
dt=
(c0 − x1)x2
1 + c0 − x1
− k1x1
Km1 + x1
, (2.5)
dx2
dt=
k2xr1
Krm2 + xr
1
− k3x22
Km3 + x2
, (2.6)
x1 concentration of proteolysis fragments,
x2 concentrations protease,
c0 maximal initial ECM concentration,
k1 maximal rate of ECM formation,
k2 maximal rate of protease production,
k3 maximal rate of proteolysis activity,
Km1 Michaelis constant of ECM formation,
Km2 Michaelis constant of protease production,
Km3 Michaelis constant of proteolysis activity.
Numerical exploration of the phase space by varying the parameter values in Eqs. 2.5 and
2.6 has shown that the closed model can generate mono- or bistable behavior, depending
on the parameter values. The closed mode contains indeed a positive feedback circuit
(Fig. 2.2), namely the stimulation of protease production which results in a bistable
system that depends on the choice of parameter values, see Fig. 2.3.
2.4 Graphical Study of Bistability
A method to study system dynamics is by numerically exploring all parameter combina-
tions to check whether bistability occurs. Additionally, a graphical procedure has been
developed to verify whether a given system, composed of monotone functions, exhibits
bistable behavior [3]. A more in-depth view of the mathematical aspects of the graphical
analysis of monotone functions can be found in [4]. The graphical analysis consists of
several steps which are listed below.
14
Section 2.4
Protease
Extracellularmatrix
Proteolysisfragments
c0 - x1 x1
x2
η
ω
Substrate
Degradationproducts
Reaction
Enzymaticstimulation
k1
k2
k3
Figure 2.2 – Closed model of ECM formation and degradation [97]. The scissors indicatethe place where the feedback circuit is cut for analysis purposes. Input of the system isgiven by ω, output of the system is η, the strength of feedback is ν = ω
η . Graphical notationis in agreement with the proposed style of Kitano [86].
0 0.2 0.4 0.6 0.8 1 1.20
0.02
0.04
0.06
0.08
0.1
stable node
stable node
saddle
dx dt1/ = 0
dx dt2/ = 0
0 0.2 0.4 0.6 0.8 1 1.20
0.02
0.04
0.06
0.08
0.1dx dt2/ = 0
dx dt1/ = 0
stable node
(a)
x1
x2 x2
x1
k K1 m2= 0.05, = 0.1
(b)
k K1 m2= 0.06, = 0.1
Figure 2.3 – Distinct dynamics for different sets of parameter values in the ECM model ofLarreta-Garde and Berry [97]: (a) Bistable dynamics if k1 = 0.05, but (b) only one stablenode if k1 = 0.06. The other parameter values are c0 = 0.1, k2 = k3 = 0.1, Km1 = Km2 =0.1, and Km3 = 1.
15
Chapter 2
0 0.2 0.4 0.6 0.8 1 1.20
0.02
0.04
0.06
0.08
0.1
x2
x1
k K1 m2= 0.05, = 0.1081
saddle
stable nodes
0 0.02 0.04 0.06 0.08 0.10
0.02
0.04
0.06
0.08
0.1
0.12
ω
η
ν = 0.925
ν = 1
dx dt2/ = 0
dx dt1/ = 0
(a) (b)
Figure 2.4 – (a) Graphical representation of g(ω). The dashed and dotted line indicateν = 1 and ν = 0.925, respectively. (b) Bistable dynamics for the limit value of Km2.
2.4.1 Breaking the Feedback Loop
First the feedback loop of the model is cut open, see Fig. 2.2. The ECM model is
transformed into an open loop system with input ω and output η. For mathematical
reasons [4], this open loop system has to satisfy two critical properties to allow graphical
analysis:
1. It has a monostable steady-state response to constant inputs (a well-defined steady-
state input/output characteristic),
2. No possible negative feedback circuits are present in the system.1
Rewriting Eqs. 2.5 and 2.6 in terms of input ω yields
dx1
dt=
(c0 − x1)x2
1 + c0 − x1
− k1x1
Km1 + x1
, (2.7)
dx2
dt=
k2ωr
Krm2 + ωr
− k3x22
Km3 + x2
. (2.8)
2.4.2 Solving the Steady-States Symbolically
Next step in the procedure is to derive a mathematical expression x1 = η = g(ω) in steady-
state. In Eqs. 2.7 and 2.8, this is done by solving dx1
dt= dx2
dt= 0 and subsequently deriving
η = g(ω). This leads to a complicated symbolical function g(ω), the corresponding graph
is calculated for the parameter set in the caption of Fig. 2.3. The results are shown in
Fig. 2.4(a).
1Especially this property is a rather strong claim in practice, as most biochemical networks do containnegative feedback circuits. This will be one of the discussion points in section 2.5.
16
Section 2.5
In the previous subsection the closed loop system was converted into an open loop.
Recovering the closed loop system from the open loop description can be performed by
putting ω = η. However, one could also study the effect of a feedback law ω = ν×η, with
ν: the influence of output η on input ω. In Fig. 2.2, ν can be coupled to the contribution of
proteolysis fragments on the formation of protease. Possible bistability within this system
can be derived graphically: if the line η = ων
has three intersections with g(ω), bistability
is guaranteed [3]. Note that the three intersections in this model structure indicate the
position of two stable nodes and one saddle. Fig. 2.4(a) shows that bistability is present
for unitary feedback (ν = 1). For ν < 1, the slope of the line becomes steeper; for
ν < 0.925 only one intersection point can be found, which corresponds to a single stable
node. Consequently ν ≥ 0.925 is a necessary constraint for this specific model to ensure
bistability, for parameter values as stated in the caption of Fig. 2.3.
2.4.3 Deriving Restrictions on Parameter Values
Bistability and related limit cycles are imputed properties of the ECM for the normal
homeostatic situation [97]. Graphical analysis in the previous subsection has shown that
for ν < 0.925 no bistable behavior is observed. The graphical method of [3] is therefore
improved by using this information to put bounds on the parameter value Km2. Rewriting
the nonlinear function with input ω in Eq. 2.8 in terms of η and ν yields
k2ωr
Krm2 + ωr
=k2(νη)r
Krm2 + (νη)r
=k2η
r
(Km2
ν
)r+ ηr
. (2.9)
Hence, after closing the open loop model, the feedback law only influences the value of
Km2. The graph in Fig. 2.4(a) shows that bistability is present for 0.925 < ν < ∞. This
implies that a bistable system can be found for 0 < Km2 ≤ 0.10.925
= 0.1081. Numerical
validation, displayed in Fig. 2.4(b), confirms these results for k1 = 0.05, c0 = 0.1, k2 =
k3 = 0.1, Km1 = 0.1, Km2 = 0.1081, and Km3 = 1.
2.5 Discussion
The graphical method is designed for nonlinear monotone functions that frequently arise
in biochemical networks and can be applied to define limits on the Michaelis constant of
the feedback loop. It is a graphical alternative for standard methods to explore each equi-
librium point numerically [3, 4]. The novelty in this chapter is the extension of an existing
graphical procedure to derive bounds on the parameter values to guarantee bistability.
By means of trial-and-error, the location to open the feedback system turns out to be es-
sential for applying the graphical procedure successfully. In the ECM model, for instance,
17
Chapter 2
only one cut can be analyzed properly, other open loop models were insoluble due to
the limitations in solving mathematical equations symbolically. Despite this restriction,
a five-state model of the mitogen-activated protein kinase (MAPK) cascade was analyzed
in the paper of Angeli and Sontag [3] to illustrate the basics of their graphical method.
However, the negative feedback circuit within the MAPK pathway [82] was replaced by a
positive feedback circuit in order to satisfy the properties listed in subsection 2.4.1. Bio-
chemical networks without negative feedback circuits are more the exception to the rule,
because negative feedback usually induces stability. This underlines the specificity of the
graphical procedure. Another drawback of the graphical method is that one still requires
numerical values for most parameters, like the majority of the classical mathematical
models. Therefore, a more general approach for the analysis of biochemical networks,
that relies only on qualitative information, is desired. This will be the subject of the next
chapter.
18
3Qualitative Analysis of Nonlinear Biochemical
NetworksAnalysis of nonlinear biochemical networks with little quantitative information is not
restricted to systems with positive feedback circuits as shown in chapter 2. Various
methods have been proposed to analyze biochemical networks in general, but the majority
requires quantitative information. In general, qualitative modeling has been common in
the field of artificial intelligence for several decades [94, 126, 141, 150, 163, 164], but
has been limited to second order systems. Expansion of this theory to larger, multi-
dimensional, biological models has been complicated. Therefore applications on biological
examples forced the qualitative research community towards special model classes with
beneficial features [12, 50, 28, 109, 130]. Excluding genetic regulatory networks, most
biochemical networks do not fulfill these strict requirements. Here we elaborate on the
qualitative modeling approach to analyze nonlinear biochemical regulatory networks, by
extending it to a significantly larger class of systems. This novel procedure contributes
to our understanding of nonlinear biochemical networks by means of qualitative system
analysis (chapters 4 and 5) and parameter estimation (chapter 6).
3.1 General Description of the Procedure
The deterministic modeling approach with ordinary differential equations (ODEs), as
introduced in the previous chapter, was used as basic principle of simulating biochemical
networks:dxi
dt=
∑j
fj(x), i = 1 . . . Nx, j = 1 . . . Nf , (3.1)
x [x1, . . . , xNx ]T ≥ 0, vector with state variables,
fj(x) ≥ 0, rate equations,
Nx total number of states,
Nf total number of rate equations,
t time.
19
Chapter 3
The procedure consists of three consecutive steps.
1. Approximation of nonlinear functions with piecewise-affine (PWA) functions. Non-
linear functions are common in biochemical models, which complicates system iden-
tification and analysis. PWA functions are a collection of linear functions separated
by switching planes. PWA functions can be used to approximate the nonlinearities.
2. Detection of equilibrium points and performing stability analysis. Equilibrium
points give an indication of the qualitative behavior of the system.
3. Construction of qualitative transition graphs. A transition graph contains all pos-
sible trajectories of a system and provides valuable knowledge about the dynamics.
Biochemical networks are often large systems that operate at different time scales. Reduc-
ing the complexity of these networks by means of model reduction contributes to a more
compact and comprehensible representation of the original model. As a consequence,
specific parts of the biochemical network can be studied in more detail. Model reduction
will be explained in the next chapter, when this procedure is applied to a model of the
Transforming Growth Factor-β1 (TGF-β1) pathway.
3.1.1 Approximation of Nonlinear Function with PWA Func-
tions
The rate equations fj(x) in Eq. 3.1 are linear and nonlinear functions of the state vector
x. The nonlinear functions are based on Michaelis-Menten kinetics. Such models are
generally too complicated to analyze, so we concentrated on a PWA approximation of the
nonlinear functions [94, 126, 141].
Michaelis-Menten Kinetics
Fig. 3.1 shows an example of how a classical Michaelis-Menten function is approximated
by two PWA segments. In reference [141] an iterative method was presented to select the
correct number of segments, but it primarily applies to complex trigonometric functions.
Nonlinear functions in biochemical networks are quite often monotonic, which are less
complex functions. Therefore two segments, separated by a switching plane α, were
assumed to be sufficient to capture the nonlinear behavior of these functions.1 Note
that using multiple segments would lead to a more refined approximation, but would
simultaneously increase the computational burden. For example, the Michaelis-Menten
equation, f(x), is mathematically defined by
f(x) =Vmaxx
Km + x, (3.2)
1The only exception on this assumption will be bisubstrate Michaelis-Menten inhibition, which willbe approximated by three segments.
20
Section 3.1
Figure 3.1 – Standard Michaelis-Menten equation (solid line) and its PWA approximation(dashed line). The first segment of the PWA function for 0 < x < α is linear; the secondsegment is constant for α < x < ∞. The functions f(x) and ϕ(x) intersect at xIP. Theshaded area between these functions represents the area of the cost functions J1, J2 and J3.
x substrate concentration,
Vmax maximal rate of conversion,
Km Michaelis constant.
The PWA function, ϕ(x), is a two-segment approximation of f(x)
ϕ(x) =
Vmaxxα
if x < α,
Vmax if x ≥ α,(3.3)
α switching plane.
Hill Kinetics
The Hill equation is an extension of Michaelis-Menten kinetics [46]
f(x) =Vmaxx
r
Krm + xr
, (3.4)
r cooperativity coefficient.
Ramp functions have recently been used as a gentle approach to approximate Hill functions
with low cooperativity coefficients (r < 10) [11]. A three-segment PWA function would be
necessary to generate a ramp function. For large values of the cooperativity coefficient r,
21
Chapter 3
0 0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
Vmax
φ( )x
f x( )
f x( )
Km
φ( )x
x
Figure 3.2 – Hill function f(x) for r = 100 (solid line) and its PWA approximation (dashedline).
the Hill function shows the characteristics of a biological switch, see Fig. 3.2. In this thesis,
only systems with large cooperativity coefficients are considered for the sake of simplicity,
so a two-segment approximation is adequate. The following piecewise-constant function
is therefore an appropriate description
ϕ(x) =
0 if x < Km,
Vmax if x ≥ Km.(3.5)
Inhibition
Inhibition of biochemical processes can be modeled as well with a Michaelis-Menten de-
scription
f(x) =VmaxKI
KI + x, (3.6)
KI Michaelis constant of the inhibitor.
The maximum rate of this function is Vmax and decreases monotonically towards zero for
x →∞, see Fig. 3.3. The corresponding PWA approximation is
ϕ(x) =
Vmax
(1− x
α
)if x < α,
0 if x ≥ α.(3.7)
Bisubstrate Michaelis-Menten
Conversion of nonlinear to PWA functions is not only confined to monosubstrate reactions.
Bisubstrate reactions can be approximated with planes. A Michaelis-Menten equation
22
Section 3.1
0 1 α 3 4 50
0.2
0.4
0.6
0.8
Vmax
φ( )x
f x( )
x0 KI α
f x( )
φ( )x
Figure 3.3 – Inhibited Michaelis-Menten function (solid line) and its PWA approximation(dashed line). The circle indicates the point of intersection between f(x) and ϕ(x).
with two substrates, x1 and x2, can be derived and is formulated as (if x1 has a linear
effect and x2 Michaelis-like kinetics)
f(x1, x2) =Vmaxx1x2
Km + x2
. (3.8)
Inspection of the 3D-plot (Fig. 3.4) shows that Eq. 3.8 is a combination of linear and
Michaelis-Menten dynamics. Two planes are therefore assumed to be sufficient for a
PWA description of this function.
ϕ(x1, x2) =
Vmaxx1 if αx1 − x2 < 0,
Vmaxx2
αif αx1 − x2 ≥ 0.
(3.9)
Competitive Inhibition
Some ligands could function as inhibitor of Michaelis-Menten kinetics. A special form
of Michaelis-Menten inhibition is competitive inhibition, in which an inhibitor molecule
binds reversibly to the enzyme at the same site as the substrate. It is mathematically
formulated as
f(x1, x2) =Vmaxx1
Km
(1 + x2
KI
)+ x1
, (3.10)
x1 substrate concentration,
x2 concentration of the inhibiting substrate,
KI Michaelis constant of the inhibition reaction.
23
Chapter 3
fx
x(
,)
12
φ(
,)
xx
12
(a) (b)
Figure 3.4 – (a) Plot of bisubstrate Michaelis-Menten kinetics, and (b) its two-segmentPWA approximation.
A three-segment PWA function is assumed to be a reasonable approximation of Eq. 3.10
ϕ(x1, x2) =
Vmaxx1
α1if x1
α1+ x2
α2< 1 ∧ x2 < α2,
Vmax
(1− x2
α2
)if x1
α1+ x2
α2≥ 1 ∧ x2 < α2,
0 if x2 ≥ α2,
(3.11)
α1 switching plane of the Michaelis-Menten function,
α2 switching plane of the inhibition reaction.
Fig. 3.5 shows the plots of the rate equations. For a more detailed description of the
equations above and other enzymatic kinetic reactions, we refer to standard biochem-
istry literature, e.g. [46]. The nonlinear functions in this section are a small selection of
nonlinearities in biochemical networks which can be approximated with PWA functions.
Extending this PWA approximation to other nonlinear functions is certainly possible
with the theory presented here. However, one should bare in mind that complex non-
linear functions demand a more refined segmentation and, consequently, requires more
computational effort.
24
Section 3.1
Vmax
Km
Vmax
KI
α1
α2x2
x2
x1 x1
fx
x(
,)
12
φ(
,)
xx
12
(a) (b)
Figure 3.5 – (a) Nonlinear function of competitive inhibition, and (b) its three-segmentPWA approximation.
3.1.2 Selection of PWA Parameters
To our knowledge, no research has established a link between the physiological parameters
of a nonlinear model and the parameters of its PWA approximation. Therefore, the most
optimal location of the switching planes with respect to the nonlinear parameters to
obtain the best PWA approximation has to be determined. A minimal difference between
the nonlinear function f(x) and its PWA approximation ϕ(x), must be derived. The
integral of the difference between these functions, the so-called cost function J , has to
be minimized. Cost functions are, as a rule, quadratic functions for penalizing large
errors between two functions [102]. However, an explicit solution for α as function of the
parameters of the nonlinear representation is computationally impossible to derive from a
quadratic cost function. Therefore the modulus of a linear cost function is chosen instead.
The nonlinear Michaelis-Menten equation from Eq. 3.2 and its PWA approximation in
Eq. 3.3 are selected to illustrate the derivation. The total cost function, Jtot, of f(x) and
ϕ(x) is mathematically described by
Jtot =
∫ xmax
0
∣∣∣f(x)− ϕ(x)∣∣∣dx, (3.12)
xmax maximum value of x.
25
Chapter 3
The function f(x) intersects with ϕ(x) at x = xIP and can be subdivided in three separate
cost functions
Jtot = J1 + J2 + J3. (3.13)
The three cost functions are
• For 0 < x < xIP: the nonlinear function f(x) is larger than the linear segment in
PWA function ϕ(x),
J1 =
∫ xIP
0
(Vmaxx
Km + x− Vmaxx
α
)dx, (3.14)
• For xIP < x < α: f(x) is smaller than the linear segment of ϕ(x),
J2 =
∫ α
xIP
(Vmaxx
α− Vmaxx
Km + x
)dx, (3.15)
• For α < x < xmax: f(x) is smaller than the constant segment of ϕ(x),
J3 =
∫ xmax
α
(Vmax − Vmaxx
Km + x
)dx. (3.16)
Fig. 3.1 depicts the graphs of f(x) and ϕ(x). An analytical expression for xIP is required
to solve Eq. 3.13. It is derived by calculating the intersection point between f(x) and the
linear segment of ϕ(x)
VmaxxIP
Km + xIP=
VmaxxIP
α⇒ xIP = α−Km. (3.17)
Combining Eq. 3.13 and 3.17 gives an analytical expression for Jtot
Jtot =1
2Vmax
(α− 2K2
m
α+ . . .
. . . 2Km
(log
Km
α− log
α
α + Km
+ logKm + xmax
α + Km
)), (3.18)
Eq. 3.18 is subsequently minimized as function of the PWA parameter α with Mathematica
arg minα
Jtot ⇒ dJtot
dα=
(α2 − 4αKm + 2K2m) Vmax
2α2= 0. (3.19)
26
Section 3.1
Solving this quadratic expression results in two solutions: α =(2 − √
2)Km and α =(
2 +√
2)Km. With the assumption that α > Km, one solution remains
α =(2 +
√2)
Km, (3.20)
or Km =(1−
√2
2
)α. A plot of f(x) and its PWA approximation ϕ(x) with α =
(2 +
√2)Km is given in Fig. 3.1. These derivations were repeated for the other nonlinear
functions as well. An overview of the nonlinear functions, their PWA approximations and
the link between the auxiliary parameters (α, α1, α2) and their nonlinear counterparts
(Km and KI) are listed in Table 3.1.
3.1.3 Detection of Equilibrium Points and Performing Stability
Analysis
A hybrid approximation Φ(x) of the original model in Eq. 3.1 can be formulated after the
nonlinear functions have been substituted for the PWA approximations. The switching
planes of the PWA approximation Φ(x) divide the phase space in discrete states, or modes,
denoted by q1, . . . , qNq with Nq: total number of modes. Each mode is governed by its own
characteristic set of continuous, linear ODEs and has an invariant associated to it, which
describes the conditions that the continuous state has to satisfy at this mode. Symbolic
expressions of the equilibria can be derived by solving Φ(x) = 0, for each individual mode
q. The equilibrium points have to satisfy the invariants and biochemical constraints, like
x > 0 (positive systems). This procedure leads to a collection of qualitative equilibrium
points that have to satisfy certain existence conditions.
The dynamical behavior at the equilibrium points are given by the eigenvalues λ
of the Jacobian matrix Jm, i.e., a matrix of all first order partial derivatives of the state
vector [81]. The eigenvalues λ give an indication of the dynamical behavior (see Table 3.2)
at a given equilibrium point and can be calculated with standard linear algebra [69, 81, 92]:
det(λINx − Jm) = 0, (3.21)
INx Nx ×Nx identity matrix.
Next step is to verify whether the hybrid system under study is stable. Even if stability is
guaranteed in a certain mode, this does not automatically imply that the complete hybrid
system is stable [19]. Global stability can be calculated by means of finding a Lyapunov
function, see [10, 98] for a thorough description of the methodology. We remark that
topological classification of the equilibria and Lyapunov stability can only be performed
27
Chapter 3
Tab
le3.1
–O
verviewof
nonlinearfunctions
andtheir
correspondingP
WA
approximations.
Nam
ef(x
)ϕ(x
)Lin
kf(x
)an
dϕ(x
)
Mich
aelis-M
enten
Vm
axx
Km
+x
{Vm
axx
αif
x<
α
Vm
ax
ifx≥
αα
=(2
+√
2 )K
m
Inhib
itionVm
axK
I
KI +
x
{V
max
(1−xα )
ifx
<α
0if
x≥α
α=
4.09KI
Hill
Vm
axx
r
Krm
+x
r
{0
ifx
<K
m
Vm
ax
ifx≥
Km
Direct
link
forlarge
r
Bisu
bstrate
M-M
Vm
axx1x2
Km
+x2
{V
max x
1if
αx
1 −x
2<
0Vm
axx2
αif
αx
1 −x
2 ≥0
α=
(2+√
2 )K
m
Com
petitive
Inhib
ition
Vm
axx1
Km (
1+
x2
KI )
+x1
Vm
axx1
α1
ifx1
α1
+x2
α2
<1∧
x2
<α
2
Vm
ax (
1−x2
α2 )
ifx1
α1
+x2
α2 ≥
1∧x
2<
α2
0if
x2 ≥
α2
α1
=(2
+√
2 )K
m
α2
=4.09K
I
28
Section 3.2
Table 3.2 – Eigenvalues of a two-state system and its relation to system behavior.
Eigenvalues Topological classification
Re(λ1,2) < 0 Stable node
Re(λ1,2) > 0 Unstable node
Re(λ1) < 0 < Re(λ2) Saddle point
λ1,2 = a± b i with a = 0 Limit cycle
λ1,2 = a± b i with a < 0 Stable focus
λ1,2 = a± b i with a > 0 Unstable focus
on relatively simple systems due to the complexity of solving symbolic equations. To
understand the system dynamics for larger systems, qualitative transition graphs become
therefore more important.
3.1.4 Construction of Qualitative Transition Graphs
If the motion of the continuous state would lead to violation of the conditions given by the
invariant, a transition must take place to a mode that satisfies this motion. A transition
graph contains all possible trajectories of a system and provides valuable knowledge about
the dynamics. It can be derived by calculating all possible mode transitions. Mode
transitions occur if the inner product of the tangent t(x) of the continuous state with the
normal of the switching plane n is larger than zero [141]
t(x) · n > 0. (3.22)
Hence, the transitions are expressed as symbolic inequalities, assigned to the variable Γ.
Since the dynamics are linear in each mode, Eq. 3.22 can be evaluated at the vertices of
the switching planes [58, 59, 88]. This leads to relatively simple inequalities for all vertices
of the switching planes. Since the dynamics in each mode are linear, this is sufficient to
predict the dynamics of the complete phase space. It yields multiple qualitative transition
graphs that have to satisfy a set of Γs.
3.2 Example: an Artificial Biochemical Network
To illustrate the above procedure, an artificial biochemical network with Michaelis-Menten
kinetics has been selected. The biochemical network consists of two substrates, x1 and
x2, which mutually induce each others production (Fig. 3.6).
29
Chapter 3
x1
Substrate
Degradationproducts
Reaction
Enzymaticstimulation
x2
f1
f2
k2
k4
Figure 3.6 – Graphical interaction scheme of two substrates, x1 and x2, that mutuallyinduce each others production. Notation taken from [86].
3.2.1 PWA Approximation
This system can be described by a nonlinear model of two coupled differential equations:.
dx1
dt= f1(x2)− k2x1, (3.23)
dx2
dt= f2(x1)− k4x2, (3.24)
with Michaelis-Menten functions
f1(x2) =k1x2
Km1 + x2
, (3.25)
f2(x1) =k3x1
Km2 + x1
, (3.26)
x1, x2 substrate concentrations,
k1 maximal rate of x2, induced by x1,
k2 degradation rate constant of x1,
k3 maximal rate of x1, induced by x2,
k4 degradation rate constant of x2,
Km1 Michaelis constant of x1 production,
Km2 Michaelis constant of x2 production.
30
Section 3.2
α2
α1
x1
x2
0 x1
max
x2
max
q1 q2
q3 q4
Figure 3.7 – Phase plane diagram, divided in four different modes. The dotted lines arethe switching planes.
This artificial biochemical network is a second order model. The PWA approximations of
the nonlinear functions f1(x2) and f2(x1) are ϕ1(x2) and ϕ2(x1), respectively:
ϕ1(x2) =
k1x2
α1if x2 < α1,
k1 if x2 ≥ α1,(3.27)
ϕ2(x1) =
k3x1
α2if x1 < α2,
k3 if x1 ≥ α2,(3.28)
α1 switching plane of f1(x),
α2 switching plane of f2(x).
3.2.2 Determination of the Equilibrium Points
Switching planes x1 = α2 and x2 = α1 divide the phase space into four distinct modes
(q1, · · · , q4), as shown in Fig. 3.7. The mathematical description of the hybrid approxi-
mation Φ(x) of the nonlinear system is
Φ(x) =dx
dt=
[k1x2
α1− k2x1;
k3x1
α2− k4x2]
T for mode q1,
[k1x2
α1− k2x1; k3 − k4x2]
T for mode q2,
[k1 − k2x1;k3x1
α2− k4x2]
T for mode q3,
[k1 − k2x1; k3 − k4x2]T for mode q4,
(3.29)
31
Chapter 3
with invariants
q1 : 0 ≤ x1 < α2 ∧ 0 ≤ x2 ≤ α1,
q2 : α2 ≤ x1 ≤ xmax1 ∧ 0 ≤ x2 ≤ α1,
q3 : 0 ≤ x1 < α2 ∧ α1 ≤ x2 ≤ xmax2 ,
q4 : α2 ≤ x1 ≤ xmax1 ∧ α1 ≤ x2 ≤ xmax
2 .
(3.30)
xmax1 maximum concentration of x1,
xmax2 maximum concentration of x2.
For each mode, an analytical expression of the equilibrium points can be derived by
putting the derivatives in Eq. 3.29 to zero. Pseudo equilibrium points are considered as
well. These equilibrium points are located outside the invariant of the corresponding
mode [141]. The positions of the (pseudo) equilibrium points are symbolic expressions of
the parameters k1, k2, k3, k4, α1, and α2. For example, consider an equilibrium point in
mode q1. The equilibrium point in the differential equations of mode q1 in Eq. 3.29 are
put to zero to calculate the position of the equilibrium point
dx1
dt=
k1x2
α1
− k2x1 = 0 ⇔ x1 =k1x2
α1k2
, (3.31)
dx2
dt=
k3x1
α2
− k4x2 = 0 ⇔ x2 =k3x1
α2k4
. (3.32)
Hence, (x1, x2) = (0, 0) is a real equilibrium point in mode q1 and satisfies the invariant
of mode q1, see Eq. 3.30. This procedure is repeated for the equilibria in modes q2, q3 and
q4, the results are listed in Table 3.3.
Table 3.3 – Dynamical behavior of the PWA system for different parameter constraints.
Mode Equilibrium point Classification Existence conditions
q1 (0, 0)Stable node k1k3
α1α2k2k4< 1
Saddle point k1k3
α1α2k2k4> 1
q2
(k1k3
α2k2k4, k3
k4
)Stable node k1k3
α1α2k2k4> 1 ∧ α1 ≥ k3
k4∧ α2 < k1
k2
q3
(k1
k2, k1k3
α1k2k4
)Stable node k1k3
α1α2k2k4> 1 ∧ α1 < k3
k4∧ α2 ≥ k1
k2
q4
(k1
k2, k3
k4
)Stable node k1k3
α1α2k2k4> 1 ∧ α1 < k3
k4∧ α2 < k1
k2
32
Section 3.3
3.2.3 Dynamical Behavior at the Equilibrium Points and Sta-
bility Analysis
First we consider mode q1 again. The Jacobian matrix for mode q1 is defined by
Jq1 =
(−k2
k1
α1
k3
α2−k4
). (3.33)
Eq. 3.33 yields two symbolic expression for the eigenvalues:
λ1,2 =−(k2 + k4)±
√D
2with D = (k2 + k4)
2 − 4
(k2k4 − k1k3
α1α2
). (3.34)
Symbolic calculations show that the equilibrium point at (0, 0) is a stable node fork1k3
α1α2k2k4< 1 and a saddle point for k1k3
α1α2k2k4> 1, see Table 3.2. Similarly, the proce-
dure is repeated for q2, q3 and q4 . The equilibria have to satisfy the invariants from
Eq. 3.30 and biochemical constraints. These impose additional constraints on the pa-
rameter values. An overview of the equilibria and corresponding constraints are listed in
Table 3.3. The next step is to check the stability of the complete hybrid system. Let the
Lyapunov function [10] be defined as
V =1
2
((x1 − xeq
1 )2 + (x2 − xeq2 )2
), (3.35)
with inequality constraint
dV
dt= (x1 − xeq
1 )dx1
dt+ (x2 − xeq
2 )dx2
dt< 0, (3.36)
xeq1 x1 evaluated at equilibrium point,
xeq2 x2 evaluated at equilibrium point,
to guarantee global stability. For mode q1, Eq. 3.36 is evaluated at equilibrium point (0, 0)
for all modes. The result shows that the inequality in Eq. 3.36 is always satisfied for all
(positive) parameter values. Hence, global stability has been proven for this system. The
stability of the equilibria in the other modes was also confirmed according to the same
procedure.
3.3 Transition Analysis
Qualitative transition analysis is required to derive the direction of the trajectories within
the system and to construct the possible transition graphs. In Fig. 3.8, the nullclines are
drawn for the four modes. The nullclines divide the total phase space in different flow
33
Chapter 3
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
(a) (b)
(c) (d)
stable node in q1 stable node in q2
stable node in q3 stable node in q4
Figure 3.8 – Nullclines (solid lines) and vector fields of the trajectories. Filled circle: stablenode; open circle: saddle point. The vector field is obtained by simulating the PWA modelin Eq. 3.29 with parameters (a) k1 = k3 = 1, k2 = k4 = 4, (b) k1 = 7, k2 = 8, k3 = 4,k4 = 10, (c) k1 = 4, k2 = 10, k3 = 9, k4 = 10, (d) k1 = k3 = 3, k2 = k4 = 4. The otherparameter values were in all simulations: α1 = α1 = 0.5, xmax
1 = xmax2 = 1.
34
Section 3.3
domains. The sign of the derivatives of the trajectories is the same in each individual flow
domain. For example, Fig. 3.8(a) is composed of three flow domains, whereas Fig. 3.8(b)
- (d) consist of four flow domains each. The directions of the trajectories in each flow
domain can be calculated on the vertices of the switching planes α1 and α2. For example,
mode q1 in Fig. 3.7 is separated from modes q3 and q2 by x1 = α2 and x2 = α1, respectively.
A trajectory in q1, directed towards q2, crosses x1 = α2 with normal n1→2 = [1, 0]T . The
tangent in q1, Φ1(x), is a mathematical function of the parameters and states
Φ1(x) =
(−k2x1 + k1x2
α1
k3x1
α2− k4x2
). (3.37)
One can deduce from Eqs. 3.22 that trajectories in q1 directed towards q3 are only present
ifk1x2
α1k2
> x1. (3.38)
The same procedure has been applied to switching plane x2 = α1 with n1→3 = [0, 1]T ,
which resulted in the inequality constraint
k3x1
α2k4
> x2. (3.39)
Solving Eqs. 3.38 and 3.39 on their respective switching planes results in complicated and
generally unsolvable algebraic equations [120]. The solutions were therefore calculated
on the vertices of the switching plane [11, 13, 14, 58]. This is sufficient to capture the
complete dynamics on the switching plane as the state equations are linear [59]. The
vertices of switching plane x1 = α2 are located at p1 = (α2, 0) and p3 = (α2, α1), see
Fig. 3.9(a). Substitution of these vertex coordinates in Eq. 3.38 and 3.39 yields the
following constraints on the parameters
Γ0 = k2α2 < 0, (3.40)
Γ1 =k1
k2
> α2, (3.41)
for (x1, x2) = (α2, 0) and (x1, x2) = (α2, α1), respectively. Eq. 3.40 can never be fulfilled
for positive parameter values. No trajectories can therefore traverse from mode q1 to q2
at p1 = (α2, 0), irrespective of the parameter value. Trajectories on vertex p3 can traverse
from q1 towards q2, but have to satisfy Γ1 in Eq. 3.41. As a consequence, trajectories
from q2 directed towards q1 have to meet the complementary constraints. These are
Γ∗0 = k2α2 > 0 and Γ∗1 = k1
k2< α2 for p1 and p3 , respectively. The asterisk indicates the
complementary set, i.e., Γ∗ = NOT(Γ). As Γ∗0 is always satisfied, trajectories at p1 can
only move from q2 to q1 and not vice versa.
35
Chapter 3
Table 3.4 – Inequality constraints, belonging to the mode transitions.
Variable Inequality set
Γ0 k2α2 < 0 ⇒ neverΓ∗0 k2α2 > 0 ⇒ alwaysΓ1 α2 < k1
k2
Γ∗1 α2 > k1
k2
Γ2 α1 < k3
k4
Γ∗2 α1 > k3
k4
Table 3.5 – Mode transitions at the vertices of the artificial biochemical network andcorresponding inequality sets.
Transition p1 p2 p3 p4 p5
q1 → q2 Γ0 × Γ1 × ×q1 → q3 × Γ0 Γ2 × ×q2 → q4 × × Γ2 Γ2 ×q3 → q4 × × Γ1 × Γ1
The complete phase space of the model is analyzed according to the procedure de-
scribed above. This yields six different sets of constraints: Γ0, Γ∗0, Γ1, Γ∗1, Γ2 and Γ∗2. The
results are summarized in Fig. 3.9(b) and Tables 3.4 and 3.5.
Not all sets are simultaneously present in the same transition graph. The sets Γ1 and
Γ2 are complementary to Γ∗1 and Γ∗2, respectively. This gives four unique combinations
of Γs and, hence, four possible transition schemes for the model, displayed in Fig. 3.10.
These transition graphs are qualitatively identical to the vector field in Fig. 3.8.
3.4 Discussion
In this chapter, a procedure has been presented to describe and analyze nonlinear models
as a hybrid system. The focus is on nonlinear processes that are typically observed in
biochemical networks. The methodology is based on three consecutive steps. To demon-
strate the procedure, a model of an artificial biochemical network is chosen. This second
order model is small and can be analyzed with standard phase plane analysis, but in this
case it has an exemplary role. It shows that qualitative analysis can predict the nature
of the transition graphs very well. However, stability analysis and determination of the
dynamical behavior at the equilibria can become complicated for larger systems (more
than three state variables), but are not a requirement for system analysis: qualitative
transition analysis can already provide some essential information about the system dy-
namics. Stability analysis might be performed by exploiting the monotonicity of nonlinear
36
Section 3.4
α2
α1
x1
x2
0
Γ1
Γ1*
Γ2
Γ2*
Γ2
Γ2*
q3
Γ1*
Γ1q4
q2q1
α2
α1
x1
x2
0
q3
q2q1
q4
x
x
n1 2 Γ0*
Γ0*
x x
x
p1
p2 p3 p4
p5
xx x
x
x
(a) (b)
Figure 3.9 – (a) Normal vector of switching plane, directed from q1 towards q2. The verticesof this plane are marked with ×. (b) Complete transition graph of the artificial biochemicalnetwork. The arrows show the direction of the trajectories at all vertices, the constraintsare indicated as well.
functions or using tools like Surface Lyapunov Functions [52, 53], in which exponential
stability can be proven for a large class of systems.
The main advantage of qualitative PWA analysis is the practical relevance in the field
of systems biology. Most methods require models with numerical parameter values for
system analysis [51, 82, 116] or are designed for biochemical networks with advantageous
properties [3, 27]. Contrary to these methods, qualitative PWA analysis is a general pro-
cedure that covers an extensive class of nonlinear biochemical networks, without requiring
quantitative information. The dynamics in each mode is linear, thus studying dynamics
at the vertices of the switching planes is sufficient to make statements about the dynam-
ics of the complete system. Vertex analysis yields simple inequalities. Even for larger
biochemical networks, the complexity does not increase exponentially, which contributes
to the practical relevance of qualitative PWA analysis. To further enhance the practical
usefulness, the symbolic calculations have been automated: determination of the loca-
tions of the equilibrium points and transition graph reconstruction are fully automated in
Mathematica. Quantifier elimination [166] or the algorithms of other qualitative modeling
programs like Robust Verification of Gene Networks (RoVerGeNe) [11] and the Genetic
Network Analyzer (GNA) [27] might be of great value to improve our implementation.
An artificial network is analyzed in this chapter, but it can be used for physiolog-
ically relevant biochemical networks as well. In the next chapter, this method will be
37
Chapter 3
x x
x
x
x
x x
x
x
x
x x
x
x
x
x x
x
x
x
(a) (b)
(c) (d)
q2 q2
q2 q2
q3
q3q3
q3
Figure 3.10 – All possible transitions schemes of the model for inequality sets (a) Γ∗1 ∧Γ∗2,(b) Γ1∧Γ∗2, (c) Γ∗1∧Γ2, and (d) Γ1∧Γ2. We remark that these transition graphs qualitativelycorrespond with the vector fields in Fig. 3.8.
38
Section 3.4
demonstrated for model of the TGF-β1 pathway.
39
Chapter 3
40
4Analysis of the Transforming Growth
Factor-β1 pathwayThis chapter was based on an extended version of paper [120]
The Transforming Growth Factor-β1 (TGF-β1) pathway plays a crucial role in prolifera-
tion, differentiation, apoptosis and motility [7]. It can be targeted by 42 known ligands,
which leads to transciptional control of more than 300 target genes [79]. As already men-
tioned in chapter 1, significant alterations in the TGF-β1 pathway have been observed in
aging endothelial cells. Also the experimental findings of other research groups suggest
the implication of TGF-β1 in vascular wall aging [55, 56, 107] and pathogenesis of chronic
vascular diseases [78, 110]. The qualitative analysis procedure from the previous chapter
will be applied to this biologically relevant example.
4.1 Physiology of the TGF-β1 Signaling Pathway
TGF-β1 is an extracellular cytokine that binds to membrane receptors and initiates the
activation of receptor-regulated SMAD proteins (R-SMADs) [30]. The signal transduction
pathway of TGF-β1 is initiated upon binding of the active form of TGF-β1 to the TGF-β1
receptor [30]. This leads to dimerization of two TGF-β1 receptors (RI and RII), which
are constitutively internalized by the formation of endosomes (membrane-bound compart-
ments inside cells) [33, 113]. The fate of these endosomes depends on whether a complex of
the protein SMAD ubiquitination regulatory factor 2 (SMURF2) and inhibitory-SMADs
(I-SMADs) is bound to the activated receptors, which targets the endosomes for degra-
dation [115]. If the endosomes are not degraded, they contribute to the phosphorylation
of R-SMADs, which are present in the cytosol of the cell (internal fluid of the cell). This
process is enzymatically stimulated by the activated receptors. Phosphorylated R-SMADs
quickly form homodimers, which subsequently bind to a common-mediator SMAD (co-
SMAD) to produce a oligotrimer: the so-called SMAD-complex [105, 154]. This complex
is transported via the nuclear pores into the nucleus in which it promotes the outward
41
Chapter 4
transport of nuclear I-SMADs [30, 74]. The SMAD-complex binds to specific regions on
the DNA to promote gene transcription [30], including the ones involved in ECM forma-
tion [160], the reduction in cell cycle time [182], the regulation of nitric oxide [144], the
production of latent-TGF-β1 [83, 169] and I-SMADs [60]. The latter two can be seen
as auto-induction (positive feedback circuit) and -inhibition (negative feedback circuit)
of the TGF-β1 pathway, respectively [114]. The phosphorylated R-SMAD complexes be-
come dephosphorylated after their regulatory role is accomplished. These substrates are
subsequently transported back as individual SMAD proteins into the cytosol to complete
the R-SMAD loop.
A putative positive feedback in the TGF-β1 pathway is the auto-induction loop with
latent-TGF-β1. This protein is the inactive form of TGF-β1 and present in the ECM. It
is converted into the active form when exposed to radiation, oxidative stress, hormones
or other factors [142]. It is still unknown whether the secreted latent-TGF-β1 is used by
the cell itself or transported to the ECM of neighboring cells. Its affinity to the TGF-β1
receptor might play a role in the effect of latent-TGF-β1 on the cellular response, as was
demonstrated for the endothelial growth factor (EGF) [31].
Negative feedback is regulated by I-SMADs in this biochemical network. Normally,
I-SMADs are located inside the nucleus and are transported into the cytosol upon R-
SMAD activation. In the cytosol, they inhibit the R-SMAD signaling loop by binding
to the activated TGF-β1 receptor. This reduces the phosporylation rate of R-SMADs
by competitive inhibition and degrades the receptor. I-SMADs have a limited survival
time and are eventually degraded. A second form of inhibition is the binding of I-SMADs
to SMURF2. This complex binds to activated receptors in endosomes and target these
vesicles for degradation. The interaction graph of the total TGF-β1 pathway is displayed
in Figure 4.1, following the conventions of [86].
4.1.1 Isolation of the R-SMAD Loop
Fig. 4.1 shows that a plethora of biochemical processes are involved in the TGF-β1 path-
way. Timeseries data of R-SMAD profiles have been the only set of experimental data
we have at our disposal. Therefore the study of the TGF-β1 pathway will be centered
around the R-SMAD loop and additional assumptions are made to prepare the TGF-β1
model for model reduction.
• The positive feedback loop with latent TGF-β1 is discarded, since we assume that
this process is negligible. Also controversy exists about the contribution of auto-
induction in the TGF-β1 pathway. Instead a constant input of active TGF-β1 has
been postulated.
• The assumption is made that the constant presence of active TGF-β1 generates a
42
Section 4.1
TGF- β1
TGF- β1
RI/RII RI RII
TGF- β1
RI/RII
TGF- β1
RI/RII
SMURF2/I-SMAD
SMURF2/I-SMAD
SMURF2/I-SMAD
SMURF2/I-SMAD
SMURF2/I-SMAD
SMURF2
I-SMAD
R-SMAD
R-SMAD
P
co-SMAD
R-SMAD
P
co-SMADSMAD-complex
P
R-SMAD
P
co-SMADSMAD-complex
P
I-SMAD
latentTGF- β1
extracellular space
Substrate
Receptor
Phosphor
Degradationproducts
Geneticregulation
Reaction
Inhibition
Enzymaticstimulation
P
R-SMADco-SMADSMAD-complex
Figure 4.1 – Schematical representation of the TGF-β1 pathway in the vascular endothelialcell.
43
Chapter 4
continuous flow of endosomes.
• The exact role of co-SMAD is unrevealed yet. Most likely it plays a role in genetic
regulation. Exclusion of co-SMAD would produce a simpler model structure and is
not rate-limiting in the whole process, as shown by [24].
These assumptions lead to a reduced model, see Fig. 4.2. This will be the basis for the
quasi-steady-state approximation in the next section.
4.2 Qualitative Analysis of the Transforming Growth
Factor-β1 Pathway
The procedure for qualitative analysis follows the same steps as for the toy example in
chapter 3. In this case model reduction has been applied. Although reducing the number
of state equations is not required for the analysis of this model, it decreases the complexity
of the model significantly by capturing the system dynamics within the time window of
interest. The dynamics of available experimental data, see Fig. 4.3 [125], show that
these operate in a minutes/hours time range. The preliminary experimental data [125] in
Fig. 4.3 on primary mouse hepatocytes [87] might contain a limit cycle. Analysis of the
frequency spectrum confirms this observation (Fig. 4.4).
To focus on a time window of minutes and hours, the adapted procedure for the
qualitative analysis of the TGF-β1 pathway will become:
1. Perform model reduction by means of quasi-steady-state approximation. Providing
additional information on the qualitative parameter values reduces the number of
possibilities in dynamical behavior.
2. Derive the PWA model of the system.
3. Analyze the dynamical behavior in each mode. Determine the position of the
equilibrium points located and what symbolic inequalities do these equilibria have
to fulfill. In addition, stability of the PWA model can be performed.
4. Construct all possible qualitative phase planes on basis of the information above,
yielding the qualitative constraints that they have to satisfy.
4.2.1 Model Reduction of the TGF-β1 Pathway
A nonlinear mathematical model can be formulated on basis of the biochemistry of the
TGF-β1 interaction graph [46], see Fig. 4.2. The nonlinear deterministic model of the
44
Section 4.2
TGF- β1
RI/RII
I-SMAD
R-SMAD
R-SMAD
P
co-SMADSMAD-complex
P
extracellular space
Substrate
Receptor
Phosphor
Degradationproducts
Geneticregulation
Reaction
Inhibition
Enzymaticstimulation
I-SMAD
R-SMAD
P
co-SMADSMAD-complex
P
R-SMADco-SMADSMAD-complex
INPUT
y1 y2
y3
y4
y5y6
y7
P
u
k1
k2
k3
k4
k5
k6k7
k8
k8
k9
Figure 4.2 – Interaction graph of the TGF-β1 pathway with the focus on the R-SMADloop.
45
Chapter 4
0
0.5
1
1.5
0 100 200 300 400 500 600
Phosphorylated R-SMADE
xp
ress
ion
(ar
bit
rary
un
its)
time (min)
Figure 4.3 – Experimental data of phosphorylated R-SMADs. Expression data (diamonds)show that phosphorylated R-SMAD levels oscillate when a constant extracellular TGF-β1
load is applied [125].
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
frequency (Hz)
peak = oscillations
mag
nit
ude
Figure 4.4 – The peak in the graph shows that the experimental data in Fig. 4.3 can exhibitoscillatory behavior.
46
Section 4.2
TGF-β1 pathway consists of seven coupled ordinary differential equations [120]:
dy1
dt= u− k1y1 − k2y1y7
Km1 + y1
, (4.1)
dy2
dt= k3y5 − k4y1y2
Km2
(1 + y7
KI
)+ y2
, (4.2)
dy3
dt=
k4y1y2
Km2
(1 + y7
KI
)+ y2
− k5y3, (4.3)
dy4
dt= k5y3 − k6y4, (4.4)
dy5
dt= k6y4 − k3y5, (4.5)
dy6
dt=
k7yr4
Krm3
+ yr4
− (k8 + k9) y6, (4.6)
dy7
dt= k9y6 − k8y7, (4.7)
y1 activated TGF-β1 receptor concentration,
y2 unphosphorylated R-SMAD concentration,
y3 phosphorylated R-SMAD concentration,
y4 phosphorylated R-SMAD complex concentration in the nucleus,
y5 unphosphorylated R-SMAD complex concentration in the nucleus,
y6 I-SMAD concentration in the nucleus,
y7 I-SMAD concentration in the cytosol,
u input of TGF-β1 supply,
k1 normal receptor decay constant,
k2 maximal decay of I-SMAD induced receptor degradation,
k3 transport rate constant of R-SMADs out of the nucleus,
k4 maximal phosphorylation rate of R-SMADs,
k5 translocation rate constant of phosphorylated R-SMAD complexes into the
nucleus,
k6 dephosphorylation rate constant of SMAD complex,
k7 maximal rate of I-SMAD production,
k8 I-SMAD decay constant,
k9 translocation rate constant of nuclear I-SMAD to cytosolic I-SMAD,
Km1 Michaelis constant of I-SMAD induced receptor degradation,
Km2 Michaelis constant of phosphorylated R-SMAD complex formation,
Km3 Michaelis constant of I-SMAD production,
KI Michaelis constant of I-SMAD binding to the receptor,
r cooperativity coefficient.
47
Chapter 4
Table 4.1 – Timescales for various biochemical processes.
Biochemical process Time (in sec)
Molecular motion 10−15
Translation (per protein) 10−5 − 10−3
Kinase/phosphatase reactions 10−3
Protein conformational changes 10−3
Cell-scale protein diffusion (active) < 100
Protein folding 100
Biomolecular binding 10−1 − 101
Cell-scale protein diffusion (passive) 100 − 101
Cell migration 100 − 102
Receptor internalization 102
Transcriptional control 102
Amino acids → proteins in ER 103
DNA replication 103
Protein secretion by Golgi apparatus 103 − 104
Cellular growth > 104
4.2.2 Quasi-Steady-State Approximation of the TGF-β1 Model
Like the TGF-β1 pathway, typical systems biology models consist of large sets of coupled
differential equations. Frequently these are too large given the amount and quality of
available data from experiments to guarantee a reliable parameter estimation [176]. Re-
ducing the model size is desired, but under the requirement that it captures the essential
processes. Model reduction tools have been widely used, but primarily for linear models [5]
or systems with quantitative information [89, 127]. For nonlinear biochemical networks
with a lack in quantitative information, none of these model reduction techniques is re-
ally suitable with the exception of one: the quasi-steady-state approximation [42, 151]. It
makes use of the differences in time scales of various processes [131, 145, 175]. Table 4.1
gives an overview of the various time scales in which biochemical processes operate [132];
our time window of interest covers the minute and hour range. This a priori knowledge
enables a distinction between slow and fast varying (state) variables. By assuming that
the fast dynamics are instantaneously in equilibrium, only the relatively slow dynam-
ics are preserved and, consequently, the model size will be reduced. A more thorough
explanation of such quasi-steady-state approach will be given below.
48
Section 4.2
From Eqs. 4.1 to 4.7, the following SMAD pools can be defined:
unphosphorylated R-SMAD pool yuRS = y2 + y5, (4.8)
phosphorylated R-SMAD pool ypRS = y3 + y4, (4.9)
total R-SMAD pool ytRS = yuRS + ypRS, (4.10)
total I-SMAD pool ytIS = y6 + y7, (4.11)
The pools ytRS and ytIS are assumed to be constant, based on mass conservation laws.
Model reduction can be applied to Eqs. 4.1 - 4.7, the relatively slow dynamics will be
preserved. First, we consider the state equations of y2 and y5 in Eqs. 4.2 and 4.5, respec-
tively. A distinction between slow and fast varying rates on basis of a priori knowledge
is made by dividing the large rates by ε, given that ε ¿ 1; the rate constants of these
functions are indicated with an apostrophe. In addition, the reasonable assumption is
made that translocation rate constants of SMADs across the nuclear membrane (k3, k5
and k9) are large compared to the other rates
dy2
dt=
k′3y5
ε− k4y1y2
Km2
(1 + y7
KI
)+ y2
, (4.12)
dy5
dt=k6y4 − k′3y5
ε. (4.13)
The dynamics of yuRS are
dyuRS
dt= k6y4 − k4y1y2
Km2
(1 + y7
KI
)+ y2
. (4.14)
Ask′3y5
εÀ k6y4, Eq. 4.13 can be rewritten as y5
dt≈ −k′3y5
ε, preserving the slow dynamics
limε→0
dy5
dt= −k′3y5
ε⇒ y5 = 0
Eq. 4.8−−−−→ yuRS = y2. (4.15)
Combining Eqs. 4.14 and 4.15 results in a reduced expression for the dynamics of yuRS
dyuRS
dt≈ dy2
dt= k6y4 − k4y1y2
Km2
(1 + y7
KI
)+ y2
. (4.16)
Similarly, expressions for the slow dynamics ofdypRS
dtand dytIS
dtcan be derived
dypRS
dt≈ dy4
dt=
k4y1y2
Km2
(1 + y7
KI
)+ y2
− k6y4, (4.17)
49
Chapter 4
I-SMAD
extracellular space
Substrate
Phosphor
Degradationproducts
Reaction
Inhibition
P
R-SMAD
P
co-SMADSMAD-complex
Px4
x7
k8
k9
k4
k6
Figure 4.5 – The components and their interactions of the reduced model of the TGF-β1
pathway.
dytIS
dt≈ dy7
dt=
k8yr4
Krm3
+ yr4
− k9y7. (4.18)
Eqs. 4.16 and 4.17 are combined with Eq. 4.10. By assuming that the TGF-β1 receptor
sustains constant activity (y1 = constant), the complete model is reduced to two states.
The reduced model is converted into a dimensionless system by defining x4 = y4
ytRSand
x7 = y7
ytIS.
dx4
dt= k∗4
1− x4
K∗m2
(1 + x7
K∗I
)+ 1− x4
− k∗6x4, (4.19)
dx7
dt= k∗8
xr4
Krm3
+ xr4
− k∗9x7. (4.20)
The asterisk-superscript of the parameters is left out in the remainder of this chapter for
simplicity reasons. Figure 4.5 shows the interaction graph of the reduced model.
50
Section 4.2
4.2.3 From Nonlinear to Piecewise-Affine
The nonlinear expressions in Eqs. 4.19 and 4.20 are defined as
f1(x4, x7) =k4(1− x4)
1− x4 + Km2
(1 + x7
KI
) , (4.21)
f2(x4) =k8x
r4
Krm3
+ xr4
, (4.22)
respectively, and will be converted into piecewise continuously differentiable functions.
A nonlinear function can be approximated by a PWA function of two or more segments
[141], as shown in chapter 3. The nonlinear function f1(x4, x7) represents competitive
inhibition of R-SMAD phosphorylation. Fig. 3.5(a) gives an impression of the 2D-plot of
this function for some arbitrary selected parameter values. The parameter k4 determines
the maximum of this surface. A PWA approximation of f1(x4, x7) is
ϕ1(x4, x7) =
k4(1− x7
α2) if x4 − α1
α2x7 ≤ 1− α1 ∧ x7 < α2,
k4
α1(1− x4) if x4 − α1
α2x7 > 1− α1 ∧ x7 < α2,
0 if x7 ≥ α2.
(4.23)
Fig. 3.5(a) and (b) display typical graphs of the nonlinear function and its PWA coun-
terpart, respectively. The function f2(x4) is the rate of I-SMAD (x7) expression, induced
by the SMAD complex (x4). Genetic regulation is assumed to function like a switch [2].
Its PWA description ϕ2(x4) is therefore
ϕ2(x4) =
0 if x4 < Km3 ,
k8 if x4 ≥ Km3 .(4.24)
The final model equations Φ(x) can be constructed by replacing the nonlinear functions
f1(x4, x7) and f2(x4) in Eqs. 4.21 and 4.22 with ϕ1(x4, x7) and ϕ2(x4), respectively. The
resulting PWA model dxdt
= Φ(x) consists of maximally six discrete modes (q1, . . . , q6) of
two linear state equations (dx4
dtand dx7
dt), in matrix form given by
Φi(x) =dx
dt= Aix + Bi, (4.25)
Ai, Bi the coefficient matrices of mode qi, listed in Table 4.2.
51
Chapter 4
Table 4.2 – Coefficient matrices of the TGF-β1 model.
Mode A-matrix B-matrix
q1 A1 =
(−k6 − k4
α2
0 −k9
)B1 =
(k4
0
)
q2 A2 =
(−(k6 + k4
α1) 0
0 −k9
)B2 =
(k4
α1
0
)
q3 A3 =
(−k6 00 −k9
)B3 =
(00
)
q4 A4 =
(−k6 − k4
α2
0 −k9
)B4 =
(k4
k8
)
q5 A5 =
(−(k6 + k4
α1) 0
0 −k9
)B5 =
(k4
α1
k8
)
q6 A6 =
(−k6 00 −k9
)B6 =
(0k8
)
The invariants of the six modes are
if q1 : x4 − α1
α2
x7 ≤ 1− α1 ∧ x7 < α2 ∧ x4 < Km3 , (4.26)
if q2 : x4 − α1
α2
x7 > 1− α1 ∧ x7 < α2 ∧ x4 < Km3 , (4.27)
if q3 : x7 ≥ α2 ∧ x4 < Km3 , (4.28)
if q4 : x4 − α1
α2
x7 ≤ 1− α1 ∧ x7 < α2 ∧ x4 ≥ Km3 , (4.29)
if q5 : x4 − α1
α2
x7 > 1− α1 ∧ x7 < α2 ∧ x4 ≥ Km3 , (4.30)
if q6 : x7 ≥ α2 ∧ x4 ≥ Km3 . (4.31)
The switching planes are x4 − α1
α2x7 = 1− α1 and x4 = Km3.
4.2.4 Equilibria and Stability Analysis
The PWA representation in Eq. 4.25 enables qualitative analysis. For Km3 > 1− α1, the
system is divided in exactly six modes by the switching planes, see Fig. 4.6. We remark
that three modes are present for Km3 < 1− α1, but only the more complicated situation
with six modes will be analyzed here.
Proposition 4.2.1. The model contains a single steady-state or has a limit cycle, de-
52
Section 4.2
x4
x7
1-α1 Km3
α2
q1
q2
q3
q4
q5
q6
x
x
x
x x x
x
p1 p2
p3
p4 p5 p6
p7
Figure 4.6 – Modes of the TGF-β1 model. The switching planes divide the model in sixmodes. The vertices of the switching planes are numbered (p1, . . . , p7) and indicated withcrosses.
pending on the choice of parameter values.
Symbolic expressions for all equilibrium points in each mode (q1, ..., q6) have been
derived (see chapter 3 for the procedure). The equilibrium points have to satisfy their
invariants, see Eqs. 4.26 - 4.31. For example, the equilibrium point in mode q1 is derived
by solving Eq. 4.25 for mode q1
Φ1(x) = A1x + B1 = 0, (4.32)
with x = [x4, x7]T , which leads to (x4, x7) = (k4
k6, 0). As the equilibrium is located in q1, it
has to satisfy the invariant x4− α1
α2x7 ≤ 1−α1∧x7 < α2∧x4 < Km3 and this consequently
leads to the existence condition k6 ≥ k4
1−α1. This procedure is repeated for al modes and
confirmed by numerical studies, see Fig. 4.7. A summary of all equilibrium points and
their associated existence conditions is listed in Table 4.3. Note that modes q3 and q6
have pseudo equilibrium points and for the set
k6 <k4(1−Km3)
α1Km3
∧ k9 ≤ k4k8
α2(k4 − k6Km3), (4.33)
no equilibrium point is present. Numerical verification shows that the model exhibits limit
cycle behavior under the parameter constraints in Eq. 4.33, see Fig. 4.7(e). Furthermore,
the existence conditions for each individual equilibrium point are complementary to each
53
Chapter 4
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
stable node in q1
stable node in q5
x4 x4
x4 x4
x7
x7 x7
x7
α2
1-α1 Km3
α2
1-α1 Km3
α2
1-α1 Km3
α2
1-α1 Km3
stable node in q2
stable node in q4
(a) (b)
(c) (d)
Figure 4.7 – Vector field of the PWA model with parameter values that satisfy the existenceconditions for a stable node (dot) in (a) q1: k4 = 0.8, k6 = 5, k8 = 0.5 and k9 = 0.4 ; (b) q2:k4 = 0.8, k6 = 2.5, k8 = 0.5 and k9 = 0.4; (c) q4: k4 = 0.8, k6 = 0.5, k8 = 0.5 and k9 = 1.6;and (d) q5: k4 = 0.8, k6 = 0.5, k8 = 0.5 and k9 = 10. In addition, these vector fields werecomputed with parameter values α1 = 0.75, α2 = 0.5, and Km3 = 0.5. The direction of thesolution trajectories are visualized with arrows.
54
Section 4.2
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
x4
x7
α2
1-α1 Km3
limit cycle
(e)
¿ continuation of Figure 4.7 À (e) Phase plane of the PWA model with parameter valuesthat satisfy the existence conditions for a limit cycle: k4 = 0.8, k6 = 0.6, k8 = 0.5 andk9 = 0.4.
other, i.e., the existence conditions of the equilibrium points in Table 4.3) do not overlap.
This implies that no multiple equilibrium points can occur.
Proposition 4.2.2. If the parameter values and states satisfy the biochemically induced
constraints given in Table 4.3, provided that it has an equilibrium point in one of the
modes, the TGF-β1 model converges to a single stable node that is globally uniformly
asymptotically stable (GUAS).
As shown in Proposition 4.2.1, a single equilibrium point can be present in modes q1,
q2, q4 or q5. The exact location of this equilibrium is dependent on the specific existence
conditions for parameter values (Table 4.3). The stability is guaranteed if a Lyapunov
function V (x) can be found for all modes. This is an essential requirement to guarantee
GUAS. Let the Lyapunov function be defined as V = 12((x4 − xeq
4 )2 + (x7 − xeq7 )2), with
xeq4 and xeq
7 : equilibrium value for x4 and x7, respectively. Hence, dVdt
= (x4 − xeq4 )dx4
dt+
(x7 − xeq7 )dx7
dt< 0 is required to guarantee stability. For q1, this inequality can be solved
given the equilibrium point of q1: (xeq4 , xeq
7 ) = (k4
k6, 0). This condition is always satisfied in
mode q1; stability is therefore guaranteed. The procedure above is applied to the other
modes as well, which shows stability with a common Lyapunov function in all modes.
Similarly, stability of the equilibrium points in modes q2, q4, and q5 can also be proven.
Consequently, the complete system is guaranteed to be GUAS, regardless of the parameter
values.
55
Chapter 4
Table 4.3 – Equilibrium points and corresponding existence conditions.
Mode Equilibrium point Existence condition
q1
(k4
k6, 0
)k6 ≥ k4
1−α1
q2
(k4
k4+α1k6, 0
)k4(1−Km3)
α1Km3≤ k6 < k4
1−α1
q3 (0, 0) invalid
q4
(k4(α2k9−k8)
α2k6k9, k8
k9
)k6 < k4(1−Km3)
α1Km3∧ k4k8
α2(k4−k6Km3)< k9 ≤ k8(k4+α1k6)
α2(k4+k6(α1−1))
q5
(k4
k4+α1k6, k8
k9
)k6 < k4(1−Km3)
α1Km3∧ k9 > k8(k4+α1k6)
α2(k4+k6(α1−1))
q6
(0, k8
k9
)invalid
4.2.5 Transition Analysis
Proposition 4.2.3. The TGF-β1 model contains 14 mode transitions for which 14 dif-
ferent sets of constraints are valid.
The transition q1 → q2, for example, is only feasible if the solution of Φ1(x) in
Eq. 4.25 traverses the switching plane x4 − α1
α2x7 = 1 − α1 from q1 → q2 with normal
n1→2 = [1 − α1
α2]T The dot product of n1→2, and Φ1(x) yields the inequality k4 − k6x4 −
(k4−α1k9
α2)x7 > 0. Subsequent substitution of the vertex coordinates p1 = (1 − α1, 0)
and p3 = (Km3,α2(Km3+α1−1)
α1) in this inequality gives Γ1 : k6 < k4
1−α1and Γ2 : k6 <
k4(1−Km3)+α1k9(Km3+α1−1)α1Km3
, respectively. This procedure is repeated for all transitions, for
which the results have been summarized in Table 4.4 (description of the inequalities) and
Table 4.5 (at which vertices these inequalities hold). This yielded a collection of seven
different sets of constraints on the parameter values (Γ0, · · · , Γ6). However, there are also
seven complementary sets (Γ∗0, · · · , Γ∗6). For example, Γ∗1 is described by k6 > k4
1−α1. One
can conclude that 14 inequalities are therefore required to describe all 14 mode transitions.
Proposition 4.2.4. Transitions q3 → q1 at vertices p4 and p5, transition q4 → q1 at
vertex p5 and transition q6 → q3 at vertices p5 and p7 are always possible.
Transitions q1 → q3, transition q1 → q4, and q3 → q6 have the same set of inequality
constraints on the parameters: Γ0. In other words, these transitions are not possible for
the vertices mentioned here. This automatically implies that the reverse mode transitions
should be valid.
Proposition 4.2.5. Given the existence conditions for the equilibrium points and the
transitions, all possible transition graphs of the TGF-β1 model are shown in Fig. 4.8.
56
Section 4.2
Table 4.4 – Inequality sets of the TGF-β1 model.
Name Inequality set
Γ0 α2k9 < 0 ∨ k6Km3 < 0 ⇒ never
Γ1 k6 < k4
1−α1
Γ2 k6 < k4(1−Km3)+α1k9(Km3+α1−1)α1Km3
Γ3 Km3 < k4
k4+α1k6
Γ4 k6 <α1(k9− k8
α2)+k9(Km3−1)+
k4(1−Km3)α1
Km3
Γ5 k6 < α1(k9 − k8
α2)
Γ6 α2 < k8
k9
Table 4.5 – Transitions in the TGF-β1 model at the vertices.
Transition p1 p2 p3 p4 p5 p6 p7
q1 → q2 Γ1 × Γ2 × × × ×q1 → q3 × × × Γ0 Γ0 × ×q1 → q4 × × Γ3 × Γ0 × ×q2 → q5 × Γ3 Γ3 × × × ×q3 → q6 × × × × Γ0 × Γ0
q4 → q5 × × Γ4 × × Γ5 ×q4 → q6 × × × × Γ6 Γ6 ×
57
Chapter 4
x4
x7
q1
q2
q3
q4
q5
q6
x x
x
x
{Γ5, }Γ6*
x
{Γ5* 6, }Γ
x
{Γ5* 6*, }Γ
x
{Γ2, }Γ4
x
{Γ2, }Γ4*
x
{Γ2* 4*, }Γ
x
9 combinations of inequalities:
Γ0*
Γ1*Γ3*Γ3*
Γ3*
Γ3*
Γ0*
x
{Γ0* 6, }Γ
x
{Γ0* 6*, }Γ
Γ0* Γ0*
{ Γ5Γ Γ Γ Γ Γ Γ0* 1* 2 3* 4 6*, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1* 2 3* 4 * 6, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1* 2 3* 4 * 6*, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1* 2 3* 4* 6*, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1* 2 3* 4* * 6, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1* 2 3* 4* * 6*, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1* 2* 3* 4* 6*, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1* 2* 3* 4* * 6, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1* 2* 3* 4* * 6*, , , , , , }
stable node in mode q1
(a)
Figure 4.8 – (a) Transition graph of all possibilities that satisfy the constraints for anequilibrium point in mode q1. The location of the mode containing the equilibrium point iscolored gray. The sets between curly brackets, for example {Γ1, Γ2} is shorthand notationfor Γ1∧Γ2. If a vertex has multiple mode transitions, the various possibilities are displayed.
58
Section 4.2
stable node in mode q2
x4
x7
q1
q2
q3
q4
q5
q6
x x
x
x
{Γ5, }Γ6*
x
{Γ5* 6, }Γ
x
{Γ5* 6*, }Γ
x
{Γ2, }Γ4
x
{Γ2, }Γ4*
x
{Γ2* 4*, }Γ
x
Γ3* Γ3*
Γ3*
Γ0*
Γ1 Γ3*
Γ0*
x
{Γ0* 6, }Γ
x
{Γ0* 6*, }Γ
Γ0* Γ0*
9 combinations of inequalities:
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3* 4 6*, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3* 4 * 6, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3* 4 * 6*, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3* 4* 6*, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3* 4* * 6, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3* 4* * 6*, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2* 3* 4* 6*, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2* 3* 4* * 6, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2* 3* 4* * 6*, , , , , , }
(b)
¿ continuation of Figure 4.8 À (b) Transition graph of all possibilities that satisfy theconstraints for a stable node in mode q2.
59
Chapter 4
stable node in mode q4
x4
x7
q1
q2
q3
q4
q5
q6
x x
x
x
x
x
{Γ5, }Γ6*
x
{Γ5* 6*, }Γ
{Γ2, }Γ4
x
{Γ2, }Γ4*
x
Γ3Γ3
Γ1 Γ3
Γ0* Γ6*
Γ0*
Γ0*
4 combinations of inequalities:
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4 6*, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4 * 6*, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4* 6*, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4* * 6*, , , , , , }
(c)
¿ continuation of Figure 4.8 À (c) Transition graph of all possibilities that satisfy theconstraints for a stable node in mode q4.
60
Section 4.2
stable node in mode q5
x4
x7
q1
q2
q3
q4
q5
q6
x x
x
x
{Γ5, }Γ6*
x
{Γ5* 6*, }Γ
x
{Γ2, }Γ4
x
{Γ2, }Γ4*
x
{Γ2* 4*, }Γ
x
Γ3
Γ0*
Γ1 Γ3
Γ0*
6 combinations of inequalities:
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4 6*, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4 * 6*, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4* 6*, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4* * 6*, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2* 3 4* 6*, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2* 3 4* * 6*, , , , , , }
Γ3
Γ3
x
Γ0*
Γ6*
(d)
¿ continuation of Figure 4.8 À (d) Transition graph of all possibilities that satisfy theconstraints for a stable node in mode q5.
61
Chapter 4
limit cycle
x4
x7
q1
q2
q3
q4
q5
q6
x x
x
x
{Γ5, }Γ6*
x
{Γ5* 6, }Γ
x
{Γ5* 6*, }Γ
x
Γ0*
Γ1 Γ3
Γ0*
x
{Γ0* 6, }Γ
x
{Γ0* 6*, }Γ
Γ0* Γ0*
9 combinations of inequalities:
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4 6*, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4 * 6, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4 * 6*, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4* 6*, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4* * 6, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2 3 4* * 6*, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2* 3 4* 6*, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2* 3 4* * 6, , , , , , }
{ Γ5Γ Γ Γ Γ Γ Γ0* 1 2* 3 4* * 6*, , , , , , }
{Γ2, }Γ4
x
{Γ2, }Γ4*
x
{Γ2* 4*, }Γ
x
Γ3 Γ3
Γ3
(e)
¿ continuation of Figure 4.8 À (e) Transition graph of all possibilities that satisfy theconstraints for a limit cycle.
62
Section 4.3
Each equilibrium point has to satisfy specific existence conditions (Table 4.3). Simi-
larly, each transition has been constrained as well (Tables 4.4 and 4.5). A combination of
these inequalities provides information whether a transition is feasible for a given equilib-
rium point. To illustrate this, consider an equilibrium point in mode q1. The existence
condition for an equilibrium point in q1 is k6 ≥ k4
1−α1, see Table 4.3. The vertex p1 of
the switching plane between q1 and q2 has to obey the inequality k6 < k4
1−α1, which is
incompatible with the existence conditions for the equilibrium point in mode q1. So the
transition q1 → q2 is not feasible if an equilibrium point is present in mode q1, the tran-
sition q2 → q1 is always valid in this case. This verification process is automated and has
been repeated for all equilibrium point and transitions. There are seven sets of inequalities
and an identical number of complementary sets. Γ0 is never fulfilled, while Γ∗0 is present
in all transition graphs. Excluding Γ0 reduces to six sets of inequalities per transition
graph. The maximum number of combinations with such amount of inequalities is equal
to the number of subsets, namely 26 = 64 combinations are possible in theory. However,
not all subsets are valid in practice as will be shown for an equilibrium point in q1. The
inequality constraints of the equilibrium points in Table 4.3 rule out specific transitions.
For example, the inequality constraints of an equilibrium point in mode q1 exclude Γ1
from the transition graphs. Transitions are verified for all equilibrium points with these
extra restrictions and resulted in multiple transition graphs per equilibrium point. All
possible combinations are displayed in Fig. 4.8. Some vertices contain multiple transitions
which can be directed in opposite direction. To take this into account, the resultant of
both arrows is taken. Some mode transitions are therefore not set perpendicular to the
switching plane the trajectories cross.
4.3 Discussion
A model of the TGF-β1 pathway was developed from scratch and analyzed with the
qualitative analysis procedure of biochemical networks. In the literature, another model of
the TGF-β1 has been reported: Clarke and co-workers [24] created a complex deterministic
model of the SMAD signaling pathway, but they excluded negative feedback by I-SMADs.
The troubling tendency of such large quantitative models is that many parameters values
are needed. These parameter values are often obtained by in vitro evidence or arbitrarily
chosen, which make the results automatically less reliable. Our model shows that a
simple second order model can exhibit both a stable equilibrium or a limit cycle. Unlike
conventional methods, no quantitative information is demanded to do these observations.
Fig. 4.4 can be interpreted as a limit cycle which suggests that the parameters should
obey Eq. 4.33. There are indications that during aging, the TGF-β1 pathway becomes
insensitive to external TGF-β1 influences [107], which would correspond to an equilibrium
63
Chapter 4
with low amounts of phosphorylated SMAD-complexes, e.g. a stable node in mode q1. A
shift from limit cycle to a stable node in q1 can be achieved by adapting the parameters
in Eq. 4.33 until these satisfy the existence condition of mode q1 (see Table 4.3). In this
case, increasing Km2 (deduced from α1), Km3, and/or k6 leads to a stable node, but a
sufficient decrease in k4 provides the same answer. These restrictions assist in parameter
estimation by reducing the parameter search space. Additional experimental data of
I-SMAD (to determine the behavior of x7) would be desirable, but are unfortunately
not available. More details about the TGF-β1 pathway, e.g. the trajectories in the
phase space, would rule out even more parameters by imposing more inequalities on the
parameter values. As shown above, qualitative PWA analysis enables the researcher to do
predictions of a system without having quantitative information. Researchers can profit
from this by designing new experiments to explore specific parts of the dynamics. Besides
assisting parameter estimation and experimental design, a third advantage is the analysis
of pathological phenomena. Diseases can lead to alterations in the dynamics of a system.
If a healthy situation is compared with the diseased state, one can verify which parameters
are changed in the system and examine those in more detail.
The nonlinear functions have been approximated with continuous PWA functions, ex-
cept for Hill kinetics. The piecewise-constant approach to model Hill functions is a rough
approximation and could result in some discrepancies at the vertices. Smoother continu-
ous (but not necessarily differentiable) functions could prevent these phenomena [11]. In
this chapter, a non-continuous PWA function was incorporated to keep the analysis of the
TGF-β1 pathway as simple as possible. Therefore, the choice has been made to approxi-
mate the Hill function with two piecewise-constant segments instead of the three-segment
PWA approach.
64
5Signal Transduction of the Unfolded Protein
Response
5.1 Introduction
Proteins are large biomolecules and built from various types of amino acids, which are
organic molecules that contain both an amino and a carboxyl group. The formation of
proteins takes place by annealing amino acids on specialized particles in the cell, called
ribosomes. The sequence of amino acids determines the primary shape of the protein (con-
formation), but sometimes the protein needs additional folding and other post-processing
steps before it can be used. The fate of proteins determines the location of folding and
post-processing: proteins that remain inside the cell are folded in the cytosol, extracellular
proteins are processed in the endoplasmic reticulum (ER). The ER contains a plethora of
specialized proteins that assist in protein folding, but are highly sensitive to reactive oxy-
gen species (ROS) [168]. This makes protein folding in the ER a possible target for ROS
and could even result in programmed cell death (apoptosis) [138]. It has therefore been
suggested that aging could have a significant impact on protein folding in the ER as well.
This chapter will study protein folding of the von Willebrand factor (vWF). The vWF is
a protein that mediates adhesion of platelets to sites of vascular injury [133]. Secretion
of the vWF is an important physiological function of endothelial cells. The focus will be
on the control part of the protein folding process that initiates processes to reduce stress
in the ER: the unfolded protein response (UPR). First we propose a mathematical model
of the UPR. This model is characterized by the many unknown parameters. Therefore
qualitative analysis can be applied to do some statements about the dynamics.
5.2 Protein Folding of the von Willebrand Factor
One of the cell types that produces the vWF is the endothelial cell. The vWF is also
folded in the ER of its cell. After correct folding, additional post-processing takes place
65
Chapter 5
in the Golgi apparatus. The Golgi apparatus is a system of stacked, membrane-bounded,
flattened sacs involved in modifying, sorting, and packaging macromolecules for secre-
tion [1]. Several pro-vWF molecules are linked here in a tail-to-tail configuration to a
single vWF protein. After the Golgi apparatus, the vWF is stored in Weibel-Palade
bodies which secrete the vWF in response to external signals. The individual steps of
post-translational processing of the vWF will be described in more detail below.
5.2.1 Translation and Translocation
Proteins are formed on a single or multiple ribsomes with an annealing rate of 20 amino
acids per second per ribosome [1]. This process is called translation. The pro-vWF
molecules contain a special signal peptide, which is a short sequence of amino acids that
determines the eventual location of a protein in the cell during translation. This signal
peptide is recognized by a signal-recognition particle (SRP) which is bound to a special
SRP receptor on the ER. The translated protein is subsequently transported (translocated)
in the ER. Immediately after transport, the protein Binding Protein (BiP) binds to the
hydrophobic regions of the surface of an unfolded protein (uPr) to maintain the protein
in a folding-competent state. After addition of adenosine 5’-triphosphate (ATP, principal
carrier of chemical energy in cells), uPr is released from BiP [34, 47] which triggers the
successive steps in protein folding.
5.2.2 Protein Folding
Protein folding starts with adding a oligosaccharide structure, a collection of sugar resi-
dues, to the uPr. This so-called N -glycosylation has four roles [38]:
1. It defines the attachment area for the surface of the protein,
2. The attachment area is shielded from surrounding proteins,
3. It stabilizes the conformation,
4. It provides a biological timer on the folding status of the protein.
Two sugar residues are clipped off the uPr before it enters the CNX/CRT cycle, which
folds the protein in the right configuration. Protein Disulfide Isomerase (PDI) catalyze
several folding steps in this procedure [49, 136].1 After pro-vWF is folded, it is monitored
whether the protein is folded correctly. If not, the enzyme uridine diphosphate (UDP)-
glucose:glycoprotein glucosyl transferase (UGGT) adds new sugar residues to the protein.
This targets the pro-vWF for another folding cycle in the CNX/CRT machinery. After
several cycles the protein is folded correctly. Misfolded or incompletely assembled proteins
are retained in the ER, either bound to BiP [90, 146] or in aggregates [34] that are
1Disulfide bonds are important for stabilization of tertiary structure and for their assembly into mul-timeric structures [152]. PDI is essential in facilitating disulfide bond-dependent folding [139].
66
Section 5.3
subsequently degraded. Therefore, pro-vWF exit from the ER can only occur when folding
and the pro-vWF assembly are successfully completed. The next step is to link two pro-
vWF molecules to construct a dimer, a structure composed of two protein of the same
kind. Again the CNX/CRT cycle is involved. The correctly folded dimer is transported to
the Golgi apparatus in which it forms the final product: the vWF. Fig. 5.1 is a graphical
representation of the folding of the vWF [133].
5.3 The Unfolded Protein Response
Two quality control mechanisms are incorporated in the folding process to ensure high
yields of correctly folded proteins [155]:
1. The protein folding procedure is only continued if sufficient ATP is present to
release BiP from the uPr [43, 44].
2. Folding occurs in several cycles. If the uPr has passed the CNX/CRT cycle several
times, without proper folding, the uPr is targeted for ER associated degradation
(ERAD).
Errors in these two processes are detected and trigger the UPR. The UPR acts on various
processes to reduce ER stress, including promotion of ERAD, increased expression of BiP,
decrease in transcription and translation of proteins, and stimulation of the antioxidant
response.
5.3.1 Signal Transduction in the UPR
The UPR is activated upon deficiency of free BiP, i.e., BiP not bound to uPrs, which trig-
gers the stress sensors. BiP maintains these stress sensors in the inactive state [63, 80],
so that BiP insufficiency initiates the UPR. In the mammalian UPR, three stress sensors
are involved: inositol requiring kinase-1 family (Ire1α and Ire1β), activating transcrip-
tion factor 6 (ATF6), and (PKR)-like endoplasmic reticulum kinase (PERK) [148, 149].
The latter two stress sensors are predominant in mammals under normal physiological
conditions [103]. Although the stress sensors both respond to BiP deficiency, there is a
difference in lag time before they become fully activated [140]. First, activation of the
PERK pathway leads to the phosphorylation of PERK. This activated PERK phosphory-
lates translation initiation factor eIF2α [62] that inhibits the translation of uPr. Another
role of eIF2α is the formation of the protein GADD34 which assists in the dephosphory-
lation of eIF2α [61] and can be seen as negative feedback. Second, ATF6 is translocated
to the Golgi complex and cleaved. The cleaved ATF6 contributes to the transcriptional
activation of BiP [148, 149]. Fig. 5.2 shows graphically how signal transduction in the
mammalian UPR takes place. Note that the precise interaction between BiP and uPr
67
Chapter 5
amino acids
ribos
ome
receptor
BiP
BiPtrans
latio
n
UPR
BiP
UPR
sign
altr
ansd
ucti
onUPR
+ATP
G M
N-glycosylated uPr
Erp
57
CN
X
G M
foldingcycl
e+
PDI
aggregate
G
cor rectly folded
ER
AD
UPR
Proteindegradation
+ATP
Golgipost-processing
correctlyfolded protein
Figure 5.1 – Schematical overview of protein folding and post-processing of the von Wille-brand factor.
68
Section 5.4
is lumped in the production and degradation of uPr. ERAD and folding by means of
BiP/PDI are not separately included in this model.
5.4 Mathematical Model of Signal Transduction dur-
ing the UPR
We can conclude from the previous chapter that the UPR is indispensable for accurate
protein folding. An interesting part of this regulation process is the signal transduction of
the UPR and how it affects the protein folding process. Two related modeling attempts
have been reported. First, a model of the UPR in yeast (Saccharomyces Cerevisiae)
was created [106]. The UPR response in this organism differs considerably from the
mammalian UPR [103]. Namely, S. cerevisiae controls the UPR pathway mainly by one
”stress sensing” branch: the Ire1p-pathway. Second, a stochastic model of the heat shock
process in the cytosol was created by [135], which is closely related to the UPR. Most
parameter values in this model were chosen arbitrarily, so the results were of limited
physiological significance. A kinetic deterministic model was constructed of the signal
transduction pathways in the UPR, based on the physiology of subsection 5.3.1:
dx1
dt= f1(x4)− f2(x1, x7), (5.1)
dx2
dt= f3(x1)− f4(x2), (5.2)
dx3
dt= −f5(x2, x3) + f6(x4, x5), (5.3)
dx4
dt= f5(x2, x3)− f6(x4, x5), (5.4)
dx5
dt= f7(x4)− f8(x5), (5.5)
dx6
dt= f9(x1)− f10(x6), (5.6)
dx7
dt= f10(x6)− f11(x7), (5.7)
x1 uPr,
x2 PERK,
x3 unphosphorylated eIF2α,
x4 phosphorylated eIF2α,
x5 GADD34,
x6 ATF6,
x7 spliced ATF6.
69
Chapter 5
ATF6
unfoldedprotein
PERK
eIF2α eIF2α
GADD34
P
ATF6
Substrate
Splicedsubstrate
Phosphor
Degradationproducts
Transcriptional control
Reaction
Transcriptionalinhibition
Enzymaticstimulation
Delayedtranslation
P
x1
x2
x3 x4
x5
x6
x7
f1 f2
f3
f4
f5
f6
f7f8
f9
f10
f11
Figure 5.2 – Interaction graph of the PERK and ATF6 branches in the mammalian UPR.The molecular interaction between BiP and the uPr is omitted for sake of simplicity; theinfluence of attenuation of translation by PERK and the transcriptional activation of BiPby ATF6 are represented by inhibition of uPr formation and promotion of uPr degradation,respectively.
70
Section 5.4
The corresponding rates are
f1(x4) =k1KI
KI + x4
, (5.8)
f2(x1, x7) =k2x1x7
Km1 + x7
, (5.9)
f3(x1) = k3x1, (5.10)
f4(x2) = k4x2, (5.11)
f5(x2, x3) =k5x2x3
Km2 + x2
, (5.12)
f6(x4, x5) =k6x4x5
Km3 + x5
, (5.13)
f7(x4) = k7x4, (5.14)
f8(x5) = k8x5, (5.15)
f9(x1) = k9x1, (5.16)
f10(x6) = k10x6, (5.17)
f11(x7) = k11x7, (5.18)
k1 maximal rate of uPr production,
k2 maximal degradation rate of uPr,
k3 induction rate constant of uPr on PERK activation,
k4 degradation rate constant of PERK,
k5 maximal phosphorylation rate of eIF2α,
k6 maximal dephosphorylation rate of phosphorylated eIF2α,
k7 induction rate constant of phosphorylated eIF2α on GADD34,
k8 degradation rate constant of GADD34,
k9 induction rate constant of uPr on ATF6 activation,
k10 ATF6 splicing rate constant,
k11 degradation rate constant of ATF6,
KI Michaelis constant of inhibiting uPr production,
Km1 Michaelis constant of uPr degradation,
Km2 Michaelis constant of eIF2α phosphorylation,
Km3 Michaelis constant of eIF2α dephosphorylation.
The original model consists of seven state equations and should be reduced to make it
more comprehensible for analysis. Therefore we assumed the following:
• Activation of GADD34 effect takes place (k7) after ATF6 has been activated [140]; it
only plays a role for prolonged activation of the UPR, under severe stress conditions.
71
Chapter 5
Therefore a constant rate of eIF2α dephosphorylation was assumed. Thus dx5
dt= 0
and f adapted6 (x4) = k]
6x4.
• Total eIF2α pool remains constant, which is mathematically represented by x3 +
x4 = 1 for normalized concentrations.
• uPr activates ATF6 splicing directly, but slowly, with a rate k]2.
• Association and dissociation rates of BiP with PERK is fast [149], so x2 is replaced
by a linear function of x1.
These assumptions reduced the state equations with the quasi-steady-state approximation
and were subsequently normalized to a third-order model.2
dx∗1dt
=k∗1K
∗I
K∗I + x∗4
− k∗2x∗1x∗7
K∗m1 + x∗7
(5.19)
dx∗4dt
=k]
5x∗1(1− x∗4)
K∗m2 + x∗1
− k]6x∗4 (5.20)
dx∗7dt
= k]2x∗1 − k∗11x
∗7 (5.21)
In remainder of this chapter, the ∗ and ] were omitted to make the text more readable.
Fig. 5.3 visualizes the modes of the PWA approximation.
5.5 Qualitative Analysis
The complete qualitative analysis will not be discussed in detail, since this has already
been done in chapters 3 and 4.
5.5.1 From Nonlinear to Piecewise-Affine
Eqs. 5.19 - 5.21 contain three nonlinear equations, i.e.,
f adapted1 (x4) =
k1KI
KI + x4
, (5.22)
f adapted2 (x1, x7) =
k2x1x7
Km1 + x7
, (5.23)
f adapted5 (x1, x4) =
k5x1(1− x4)
Km2 + x1
, (5.24)
2Remark that the intermediate steps of this model reduction has been left out as the same proceduredescribed in chapter 4 is followed.
72
Section 5.5
x
x
x x =7 2α
x =4 1α
x =1 3α
x
x
x
xx
x
x
x
x
x
x
x
x
p1
p3
p15
p4
p5
p6
p7p8
p9
p10
p11 p12p13
p14
p16
p2
Figure 5.3 – Phase space of the PWA approximation of the UPR model. The phasespace is divided by three switching planes: x4 = α1 (dash), α2x1 − x7 = 0 (dot), andx1 + α3x4 = α3 (dash-dot). The vertices of the various modes are marked with crosses andnumbered (p1, . . . , p16).
73
Chapter 5
which were reformulated as the following PWA functions
ϕ1(x4) =
k1(1− x4
α1) if x4 < α1,
0 if x4 ≥ α1,(5.25)
ϕ2(x1, x7) =
k2x1 if α2x1 − x7 < 0,
k2x7
α2, if α2x1 − x7 ≥ 0,
(5.26)
ϕ3(x1, x4) =
k5x1
α3if x1 + α3x4 < α3,
k5(1− x4) if x1 + α3x4 ≥ α3,(5.27)
respectively. The PWA model is in matrix form given by Eq. 4.25. The coefficient matrices
Ai and Bi are given in Table 5.1. The invariants of the eight modes (q1, . . . , q8) are listed
in Table 5.2 and visualized in Fig. 5.4.
5.5.2 Equilibrium Points in the UPR Model
Determining the position of the equilibrium points in the model is the next step in the
procedure.
Proposition 5.5.1. The UPR model has a single steady-state in mode q1, q3, q5 or q7.
The steady-state solutions for all modes have been calculated, see Table 5.3. The equi-
librium points have to satisfy the invariants in Table 5.2, yielding four possible equilibrium
points in modes q1, q3, q5 and q7 with existence conditions
α2 <k9
k11
∧((
α1 ≤ k5
k5 + k6
)∨
(α1 >
k5
k5 + k6
∧ α3 >k1(α1(k5 + k6)− k5)
α1k2k6
)),
(5.28)
α2 >k9
k11
∧((
α1 ≤ k5
k5 + k6
)∨
(α1 >
k5
k5 + k6
∧ α3 >α2k1k11(α1(k5 + k6)− k5)
α1k2k6k9
)),
(5.29)
α2 <k9
k11
∧ α1 >k5
k5 + k6
∧ α3 <k1(α1(k5 + k6)− k5)
α1k2k6
, (5.30)
α2 >k9
k11
∧ α1 >k5
k5 + k6
∧ α3 <α2k1k11(α1(k5 + k6)− k5)
α1k2k6k9
, (5.31)
respectively. The sets of these existence conditions cover all parameter combinations and
74
Section 5.5
Table 5.1 – Coefficient matrices of the eight modes of the UPR model.
Mode A-matrix B-matrix
q1 A1 =
−k2 − k1
α10
k5
α3−k6 0
k9 0 −k11
B1 =
k1
00
q2 A2 =
−k2 0 0
k5
α3−k6 0
k9 0 −k11)
B2 =
000
q3 A3 =
0 − k1
α1− k2
α2k5
α3−k6 0
k9 0 −k11
B3 =
k1
00
q4 A4 =
0 0 − k2
α2k5
α3−k6 0
k9 0 −k11
B4 =
000
q5 A5 =
−k2 − k1
α10
0 −(k5 + k6) 0k9 0 −k11
B5 =
k1
k5
0
q6 A6 =
−k2 0 00 −(k5 + k6) 0k9 0 −k11
B6 =
0k5
0
q7 A7 =
0 − k1
α1− k2
α2
0 −(k5 + k6) 0k9 0 −k11
B7 =
k1
k5
0
q8 A8 =
0 0 − k2
α2
0 −(k5 + k6) 0k9 0 −k11
B8 =
0k5
0
75
Chapter 5
Mode q1 Mode q2
Mode q3 Mode q4
(a) (b)
(c) (d)
Figure 5.4 – Modes of the UPR model: (a) mode q1, (b) mode q2, (c) mode q3, and (d)mode q4.
76
Section 5.5
Mode q5 Mode q6
Mode q7 Mode q8
(e) (f)
(g) (h)
¿ continuation of Fig. 5.4 À (e) mode q5, (f) mode q6, (g) mode q7, and (h) mode q8.
77
Chapter 5
Table 5.2 – Invariants of all modes.
Mode Invariant
q1 x4 < α1 ∧ α2x1 − x7 < 0 ∧ x1 + α3x4 < α3
q2 x4 ≥ α1 ∧ α2x1 − x7 < 0 ∧ x1 + α3x4 < α3
q3 x4 < α1 ∧ α2x1 − x7 ≥ 0 ∧ x1 + α3x4 < α3
q4 x4 ≥ α1 ∧ α2x1 − x7 ≥ 0 ∧ x1 + α3x4 < α3
q5 x4 < α1 ∧ α2x1 − x7 < 0 ∧ x1 + α3x4 ≥ α3
q6 x4 ≥ α1 ∧ α2x1 − x7 < 0 ∧ x1 + α3x4 ≥ α3
q7 x4 < α1 ∧ α2x1 − x7 ≥ 0 ∧ x1 + α3x4 ≥ α3
q8 x4 ≥ α1 ∧ α2x1 − x7 ≥ 0 ∧ x1 + α3x4 ≥ α3
have no overlap; no multiple equilibrium points can coexist. Calculation of Lyapunov
stability and symbolic eigenvalues could not be performed on this system. Therefore no
statements could be made about the nature and stability of these equilibria.
5.5.3 Transition Analysis
Proposition 5.5.2. Transition analysis shows that 12 transition graphs are feasible in
UPR model, which yields 9 qualitatively different transition graphs.
Three switching planes with normal vectors n1→2 = [0, 1, 0]T , n1→3 = [α2, 0,−1]T
and n1→5 = [1, α3, 0]T divide the phase space into eight different modes and 16 ver-
tices (p1, . . . , p16, see Fig. 5.3). The mode transitions at each vertex is calculated with
Eq. 3.22 and the coefficient matrices in Table 5.1. The transitions and their parameter
constraints were deduced similarly. The results are summarized in Tables 5.4 and 5.5; the
*-superscript in Table 5.5 indicates the complementary set of a specific set.
Table 5.4 shows that five inequality sets (Γ1, . . . , Γ5) are required to derive all possible
transition graphs, but some inequality sets belong to a subset of another inequality con-
straint. For example, Γ1 ⊂ Γ2 means that Γ1 is only valid if Γ2 is satisfied. This implies
that {Γ1, Γ2}, {Γ∗1, Γ2}, and {Γ∗1, Γ∗2} are possible combinations. Other subsets are:
Γ5 ⊂ Γ1 ⇒ {Γ5, Γ1}, {Γ∗5, Γ1}, {Γ∗5, Γ∗1}, (5.32)
Γ5 ⊂ Γ2 ⇒ {Γ5, Γ2}, {Γ∗5, Γ2}, {Γ∗5, Γ∗2}, (5.33)
Γ4 ⊂ Γ3 ⇒ {Γ3, Γ4}, {Γ∗3, Γ4}, {Γ∗3, Γ∗4}. (5.34)
These subsets reduce the number of possible combinations from 32 = (25) to 12. The
transition graphs have been plotted in Fig. 5.5 and show that the 12 inequality constraints
78
Section 5.5
Tab
le5.
3–
Equ
ilibr
ium
poin
tsin
the
mod
esof
the
UP
Rm
odel
.
Mode
Equilib
rium
poi
nt
(x1,x
4,x
7)
q 1
(α
1α
3k1k6
k1k5+
α1α
3k2k6,
α1k1k5
k1k5+
α1α
3k2k6,
α1α
3k1k6k9
k11(k
1k5+
α1α
3k2k6)
)
q 2in
valid
q 3
(α
1α
2α
3k1k6k11
α2k1k5k11+
α1α
3k2k6k9,
α1α
2k1k5k11
α2k1k5k11+
α1α
3k2k6k9,
α1α
2α
3k1k6k9
α2k1k5k11+
α1α
3k2k6k9
)
q 4in
valid
q 5
( k1(α
1(k
5+
k6)−
k5)
α1k2(k
5+
k6)
,k5
k5+
k6,
k1k9(α
1(k
5+
k6)−
k5)
α1k2k11(k
5+
k6)
)
q 6in
valid
q 7
( α2k1k11(α
1(k
5+
k6)−
k5)
α1k2k9(k
5+
k6)
,k5
k5+
k6,
α2k1(α
1(k
5+
k6)−
k5)
α1k2(k
5+
k6)
)
q 8in
valid
79
Chapter 5
Table 5.4 – Inequality sets that determine the transitions.
Name Inequality set
Γ0 k1 < 0 ⇒ infeasible
Γ1 α2 > α3
k1
Γ2 α3 < k1
Γ3 α1 < α3−k5
α3−k5−k6∧
((k5 < 1 ∧ α3 < k5) ∨ (k5 > 1 ∧ α3 < 1)
)
Γ4 k5 > 1 ∧ α1 < k5−1k5+k6
Γ5 α2 > 1k1
lead to 9 qualitatively different transition graphs. The arrows in Fig. 5.5 represent the
mode transitions. For example, Fig. 5.5(a) shows that trajectories from q6 can only leave
mode q6 by entering q2. If the parameter constraints of Fig. 5.5(c) are taken, trajectories
can also move in mode q5.
Proposition 5.5.3. Mode transitions q2 → q1, q4 → q2, q4 → q3, q6 → q2, q8 → q4, and
q8 → q6 are always present.
Table 5.5 shows that the inequality sets at the vertices of mode transitions q1 → q2,
q2 → q4, q2 → q6, q3 → q4, q4 → q8, and q6 → q8 are all equal to Γ0 which cannot
be satisfied. Consequently, the mode transitions in the opposite direction are therefore
always valid.
5.5.4 Comparison with Experimental Data
Experimental results of the mammalian UPR are scarce and have recently been obtained
by [36]. Various types of ER stress were applied to suddenly increase and maintain a high
level of uPr (x1) to induce the UPR. The data show that ER stress leads to a quick response
of the PERK branch (x4) if stress-inducing agents like thapsigargin (Tg) or dithiothreitol
(DTT) are used. Experiments with tunicamycin (Tm) and Tg show a delayed response of
the ATF6 pathway (x7) , see Fig. 5.6. This lag in time is in complete agreement with work
of Rutkowski and co-workers [140]. The experimental findings can be matched with the
transition graphs. A large initial amount of uPr (x1) and low activity levels of eIF2α (x4)
and spliced ATF6 (x7) can be taken as point of departure for the model. Mode q7 would
therefore be the most appropriate initial point to simulate the experiment of DuRose
[36]. According to Fig. 5.6, x4 should increase prior to a rise in x7, which corresponds to
80
Section 5.5
Tab
le5.
5–
Tra
nsit
ions
atth
eve
rtic
esan
dco
rres
pond
ing
para
met
erco
nstr
aint
s.
Tra
nsi
tion
p 1p 2
p 3p 4
p 5p 6
p 7p 8
p 9p 1
0p 1
1p 1
2p 1
3p 1
4p 1
5p 1
6
q 1→
q 2×
××
××
Γ0×
×Γ
0×
Γ0
Γ0
××
××
q 1→
q 3Γ∗ 0×
Γ1×
×Γ
0×
×Γ
0×
××
××
××
q 1→
q 5×
×Γ
2×
Γ2×
××
Γ0×
×Γ
0×
××
×q 2→
q 4×
××
××
Γ0×
×Γ
0×
××
×Γ
0×
×q 2→
q 6×
××
××
××
×Γ
0×
×Γ
0×
Γ0
×Γ
0
q 3→
q 4×
××
××
Γ0
Γ0×
Γ0×
××
××
××
q 3→
q 7×
Γ2
Γ2×
××
Γ0×
Γ0×
××
××
××
q 4→
q 8×
××
××
×Γ
0×
Γ0×
××
×Γ
0×
×q 5→
q 6×
××
××
××
×Γ
3Γ
4×
Γ3
Γ4
××
×q 5→
q 7×
×Γ
1Γ
5×
××
×Γ
0Γ
0×
××
××
×q 6→
q 8×
××
××
××
×Γ
0Γ
0×
××
Γ0
Γ0
×q 7→
q 8×
××
××
×Γ
3Γ
4Γ
3Γ
4×
××
××
×
81
Chapter 5
{ *, }Γ0 Γ Γ Γ Γ Γ1 2 3 4 5, , , , *
q1
q2
q3
q4
q5
q6
q7
q8
{ *, }Γ0 Γ Γ Γ Γ Γ1 2 3 4 5, , , ,
(a)
Figure 5.5 – Transition graph for a given set of inequality constraints.
a transition of q7 → q8. Figs. 5.5(c), (f) and (i) do not have this mode transition so the
parameter constraints with Γ∗3∧Γ∗4 are not valid. From a biological perspective, one expects
that the main aim of the UPR is to reduce x1, x4 and x7 to a minimum which corresponds
to mode q3, see Fig.5.5(c). However, the sustained ER stress in the experiments of [36] by
artificially increasing the uPr levels creates a different response. The equilibrium point
in mode q5 has the most similarities with large levels of uPr, ATF6 and phosphorylated
PERK, although mode q6 would intuitively have been more appropriate if an equilibrium
point had been located in that mode. An equilibrium point in q5 can only be obtained if
the parameters satisfy the restrictions in Eq. 5.30, which reduces the number of possible
parameter values. The parameter constraints in Eq. 5.30 also rule out inequality set Γ4 in
Table 5.4, implying that the qualitative transition graphs in Fig. 5.5(a), (b) and (e) are no
valid options. As mentioned before, one of the roles of the UPR is to reduce the amount
of uPr in the ER and, as a consequence, the levels of phosphorylated PERK and spliced
ATF6 to a minimum. In the PWA model, this is represented by mode q3. Experimental
data [36] do not reproduce this physiologically relevant situation due to limitations in the
current measuring techniques. Therefore qualitative analysis can give additional insights
by exploring the qualitative transition graphs and to predict the outcome. Selection
of the correct initial conditions on basis of these qualitative transition graphs assists in
experimental design.
82
Section 5.6
{ *, }Γ0 Γ Γ Γ Γ Γ1 2 3 4 5*, , , , *{ *, }Γ0 Γ Γ Γ Γ Γ1 2 3 4 5, , *, *,
{ *, }Γ0 Γ Γ Γ Γ Γ1 2 3 4 5, , *, *, *
q1
q2
q3
q4
q5
q6
q7
q8
q1
q2
q3
q4
q5
q6
q7
q8
{ *, }Γ0 Γ Γ Γ Γ Γ1 2 3 4 5, , , *,
{ *, }Γ0 Γ Γ Γ Γ Γ1 2 3 4 5, , , *, * { *, }Γ0 Γ Γ Γ Γ Γ1 2 3 4 5*, *, , , *
q1
q2
q3
q4
q5
q6
q7
q8
q1
q2
q3
q4
q5
q6
q7
q8
(b) (c)
(d) (e)
¿ continuation of Fig. 5.5 À The gray arrows emphasize the change in transition directionscompared to Fig. 5.5(a).
83
Chapter 5
q1
q2
q3
q4
q5
q6
q7
q8
{ *, }Γ0 Γ Γ Γ Γ Γ1 2 3 4 5*, , *, *, *
q1
q2
q3
q4
q5
q6
q7
q8
{ *, }Γ0 Γ Γ Γ Γ Γ1 2 3 4 5*, , , *, *
{ *, }Γ0 Γ Γ Γ Γ Γ1 2 3 4 5*, *, , *, *
q1
q2
q3
q4
q5
q6
q7
q8
q1
q2
q3
q4
q5
q6
q7
q8
{ *, }Γ0 Γ Γ Γ Γ Γ1 2 3 4 5*, *, *, *, *
(f) (g)
(h) (I)
¿ continuation of Fig. 5.5 À The gray arrows emphasize the change in transition directionscompared to Fig. 5.5(a).
84
Section 5.6
0
20
40
60
80
100
0 1 2 3 4 5
time (hours)
%cl
eaved
AT
F6
Tg
Tm
DTT
0
20
40
60
80
100
0 1 2 3 4 5
time (hours)
%P
ER
Kphosp
hory
late
d
Tg
Tm
DTT
(a)
(b)
Figure 5.6 – Expression profiles of (a) phosphorylated PERK (indication of phosphorylatedeIF2α, x4) and (b) cleaved ATF6 (x7) [36]. Alternate types of ER stress were applied foreach branch of the UPR. Thapsigargin (Tg): knocks out the energy requirements for proteinfolding; tunicamycin (Tm) inhibits protein glycosylation; and dithiothreitol (DTT) disruptsor prevents protein disulfide bonding.
85
Chapter 5
5.6 Discussion
The UPR is a collection of processes that monitors the quality of protein folding in the
ER and, if necessary, takes appropriate action if this quality tends to diminish. A math-
ematical model of the signaling cascade of the mammalian UPR is created and analyzed
with the qualitative procedure from chapter 3. The analysis of the UPR model yields 9
different types of qualitative transition graphs. Several transition graphs showed qualita-
tively good resemblance with experimental data [36], while others were less likely. This
information could be used to select specific inequality sets of the parameter values, which
reduces the parameter search space during system identification, as will be shown in
chapter 6. Current methods for determining the stability and typological classification
of equilibria are inadequate for multidimensional qualitative systems, but graphical anal-
ysis of the transitions in the phase space provide some indications about the nature of
the system. Still, a solid theoretic framework would be desired. We believe that two
approaches are very promising. First the use of Surface Lyapunov functions (SuLF), in
which the stability of any PWA system can be verified by analyzing a hybrid system with
specialized surfaces [52, 53] might be an option. Second, the monotonic characteristics of
the nonlinear functions in biochemical networks have some advantageous properties, as
was shown in chapter 2, and could help in developing mathematical proofs for stability.
86
6System Identification with Parameter
ConstraintsSo far qualitative information has been used to analyze the system dynamics. However,
qualitative information is not restricted to system analysis alone. In this chapter, qualita-
tive information will be applied to improve parameter estimation by putting constraints on
the parameter values, obtained by qualitative analysis (chapter 3). These restrictions can
be included in a constrained nonlinear optimization procedure to reduce the parameter
search space for parameter estimation. A mathematical model of a biochemical oscillator
was used as a test case [16, 121].
6.1 The Biochemical Oscillator
A simple model of a biochemical oscillator was introduced by Goodwin [54], which de-
scribes the genetically regulated enzymatic conversion of a substrate into a product, see
Fig. 6.1:
dx1
dt=
k1KI
KrI + xr
3
− k2x1, (6.1)
dx2
dt= k3x1 − k4x2, (6.2)
dx3
dt= k5x2 − k6x3, (6.3)
x1 mRNA concentration,
x2 enzyme concentration,
x3 product concentration,
k1 maximum rate of mRNA production,
k2 degradation rate constant of x1,
k3 rate constant of enzyme induction,
87
Chapter 6
DNA mRNA enzyme
substrate
product
Figure 6.1 – Model of enzymatic substrate conversion, as proposed by Goodwin [54]. Thefirst step is mRNA transcription from DNA, which leads to the formation of an enzyme.This enzyme catalyzes the conversion of a substrate into a product that inhibits the mRNAtranscription of the enzyme (bar). Note that the degradation of mRNA, enzyme and productare omitted from this graph. The degradation of the product was assumed to be enzyme-mediated [16].
k4 degradation rate constant of x2,
k5 rate constant of product formation,
k6 degradation rate constant of x3,
KI Michaelis constant of mRNA inhibition,
r cooperativity coefficient.
For r → ∞, the Hill equation shows ideal relay characteristics. Eqs. 6 - 8 describe
a basic oscillator, but periodic cycles are only obtained for r > 8 [39]. It has been
argued that this large value is less likely in this situation from a physiological point of
view [16]. Goodwin’s model [54] was therefore slightly modified by Bliss et al. [16]. The
latter assumed a low cooperativity coefficient (r = 1) and introduced enzyme-catalyzed
degradation of the product, described by Michaelis-Menten kinetics [111]. This resulted
in the following differential equations
dx1
dt=
k1KI
KI + x3
− k2x1, (6.4)
dx2
dt= k3x1 − k4x2, (6.5)
dx3
dt= k5x2 − k6x3
Km + x3
, (6.6)
k6 maximum rate of x3 degradation,
Km Michaelis constant of x3 degradation.
This model was assumed to represent the actual system and will be used as in silico test
case. To provide experimental data for the identification procedure, the nonlinear model
in Eqs. 6.4-6.6 was simulated over a time span of 150 units. Within this period, 50 samples
(N = 50) were collected for x1, x2 and x3 at equidistant discrete time instants. Normally
88
Section 6.2
0 50 100 1500
50
100
150
200
250
300
350
mRNA
enzyme
product
x1
σ = 2.5
x2
x3
t
Figure 6.2 – Simulated experiment of the biochemical oscillator with parameter valuesk1 = 150, KI = 1, k2 = 0.1, k3 = 0.1, k4 = 0.1, k5 = 0.1, k6 = 10, and Km = 1.5. Whitenoise with standard deviation σ = 2.5 was added to the output. The response of mRNA(x1), enzyme (x2) and product concentration (x3) are plotted as function of time t.
distributed white noise e(k) was added to each sample. This resulted in the following
equation for the measurement data (k = 1, . . . , N)
y(k) = x(k) + e(k), (6.7)
y(k): measured output.
The parameter values of the simulated model are k1 = 150, KI = 1, k2 = 0.1, k3 = 0.1,
k4 = 0.1, k5 = 0.1, k6 = 10, and Km = 1.5. Four simulations were performed with an
varying noise level that have standard deviations σ = 0, 1, 2.5 and 5. A typical plot of
the dynamics for σ = 2.5 (Fig. 6.2) shows that the model, for the given set of parameter
values, oscillates. Although the data feigns damped oscillations in Fig. 6.2, the trajectories
converge to a limit cycle over a longer time period.
89
Chapter 6
6.2 Qualitative Phase Space Analysis
6.2.1 Nonlinear to PWA Conversion
The nonlinear model in Eqs. 6.4-6.6 contains two nonlinearities that are functions of x3:
f1(x3) =k1KI
KI + x3
(6.8)
f2(x3) =k6x3
Km + x3
. (6.9)
These functions are approximated with two PWA functions (see chapter 3):
ϕ1(x3) =
k1
(1− x3
α1
)if x3 < α1,
0 if x3 ≥ α1,(6.10)
ϕ2(x3) =
k6x3
α2if x3 < α2,
k6 if x3 ≥ α2.(6.11)
After normalization, these approximations yield the following set of state equations
Φi(x) =dx
dt= Aix + Bi, (6.12)
with x = [x∗1, x∗2, x
∗3]
T of which the coefficient matrices Ai and Bi are listed in Table 6.1.1
The parameters of the normalized PWA approximation are k∗1 = k1
xmax1
, K∗I = KI
xmax3
, k∗2 = k2,
k∗3 =k3xmax
1
xmax2
, k∗4 = k4, k∗5 =k5xmax
2
xmax3
, k∗6 = k6
xmax3
, and K∗m = Km
xmax3
. A graphical representation
of the state space is displayed in Fig. 6.3(a).
6.2.2 Transition Analysis
The next step is to determine the transitions between the three modes. As the modes are
stacked on one another (Fig. 6.3), maximal four mode transitions are possible: q1 → q2,
q2 → q3, q2 → q1 and q3 → q2. Transition analysis (chapter 3) shows that three inequality
constraints are sufficient to describe all transitions at the vertices of the modes. These
constraints are listed in Tables 6.2 and 6.3 and visualized in Fig. 6.3(b). It can be deduced
from Fig. 6.3(b) that the trajectories are always directed towards q1 at vertices p1, . . . , p4.
Graphical analysis of Fig. 6.3(b) shows that a limit cycle is feasible if inequalities Γ1 and
Γ2 are satisfied.
1The additional assumption was made that α1 < α2; simulations showed that changing the order ofthese switching planes did not have any qualitative effect on the end results.
90
Section 6.2
Table 6.1 – Coefficient matrices and invariants of the three-mode biochemical oscillatormodel.
Mode A-matrix B-matrix Invariant
q1 A1 =
−k∗2 0 − k∗1
α1
k∗3 −k∗4 0
0 k∗5 − k∗6α2
B1 =
k∗100
x∗3 < α1 < α2
q2 A2 =
−k∗2 0 0k∗3 −k∗4 0
0 k∗5 − k∗6α2
B2 =
000
α1 < x∗3 < α2
q3 A3 =
−k∗2 0 0k∗3 −k∗4 00 k∗5 0
B3 =
00−k∗6
α1 < α2 < x∗3
x
x
x
x
x
x
x
x
p1
p2
p3 p4
p5 p6
p7
x1
x2
x3
x1max
x2max
x3max
p8
Γ1
Γ1
Γ2
Γ2
Γ0*
Γ0*
Γ0*
Γ0*
x1
x2
x3
x1max
x2max
x3max
q3 α1
α2
q2
q1
(a) (b)
*
*
*
*
*
*
Figure 6.3 – (a) 3D phase space of a model of the biochemical oscillator. The switchingplanes divide the phase space in three modes (q1, q2, and q3) (b) Mode transitions withcorresponding sets of constraints on the parameter values.
91
Chapter 6
Table 6.2 – Mode transitions at the vertices and corresponding symbolic constraints.
Transition p1 p2 p3 p4 p5 p6 p7 p8
q1 → q2 Γ0 Γ0 × × Γ1 Γ1 × ×q2 → q3 × × Γ0 Γ0 × × Γ2 Γ2
Table 6.3 – Inequality sets of the biochemical oscillator model.
Name Inequality set PWA Inequality set nonlinear
Γ0 absent absent
Γ1α1
α2<
k∗5k∗6
1.20KI
Km<
xmax2 k5
k6
Γ2 k∗5 > k∗6 xmax2 k5 > k6
6.2.3 Constrained Nonlinear Parameter Estimation
Transition analysis provides the inequality constraints on the parameter values to guar-
antee a limit cycle. These constraints are converted in parameters of the nonlinear model
(see Table 6.3) with the information from subsection 6.2.1 and chapter 3 to make the
qualitative information compatible for the nonlinear estimation procedure. The values of
xmax2 can roughly be obtained from Fig. 6.2 as a slightly larger value than the maximum
of x2: 200. The nonlinear function is discretized to a data set x(k) in such a way that the
cost function J(θ) can be defined as
J(θ) =N∑
k=1
(x(k, θ)− y(k)
)2
, (6.13)
θ set of estimated parameter values,
N number of measurements,
y(k) experimental data.
92
Section 6.3
Table 6.4 – Results of the parameter identification procedure with the traditional approachand the adapted one that uses qualitative information.
Parameter Real Traditional Qualitative procedure
k1 150 136.25± 40.03 149.96± 11.83
KI 1 0.992± 0.36 1.161± 0.42
k2 0.1 0.101± 0.025 0.110± 0.022
k3 0.1 0.147± 0.081 0.116± 0.035
k4 0.1 0.151± 0.096 0.116± 0.038
k5 0.1 0.094± 0.039 0.107± 0.018
k6 10 9.27± 4.65 10.85± 2.06
Km 1.5 1.27± 0.73 1.62± 0.19
The qualitative constraints in Table 6.3 have been incorporated in the following optimiza-
tion problem
θ = arg minθ>0
J(θ) subject to
k6 − 200k5 < 0,
1.20KI
Km
− 200k5
k6
< 0.
(6.14)
Problem 6.14 is solved in MATLAB with the function fmincon. This parameter iden-
tification procedure iss compared with a traditional unconstrained nonlinear estimation
procedure (lsqnonlin). Both methods require an initial estimate of the parameter values
and therefore several sets of initial estimates θini are selected as input, ranging from 0.5×to 1.5× θ. For each θini, the optimal solution for θ has been calculated. A data set with
σ = 2.5 is used to test both identification approaches. The results are listed in Table 6.4;
estimated k1’s are plotted in Fig. 6.4 for various θini. The parameter values are, in gen-
eral, well estimated for both methods if the initial estimate of the parameters were chosen
close to the actual parameter values (see Fig. 6.4). However, the constrained procedure
performed better if this estimate was less accurate, yielding a smaller standard deviation
on most parameter values (see Table 6.4). We therefore may conclude that incorporating
additional qualitative information in the parameter estimation procedure in this example
outperforms traditional nonlinear identification procedures in terms of accuracy.
93
Chapter 6
0.5 1.50
20
40
60
80
100
120
140
160
180
200
real
traditional
novel
initial value of k1
esti
mat
edk 1
k1true
k1true
k1true
Figure 6.4 – Parameter estimation of k1 for various initial estimates on a data set withσ = 2.5. On the x-axis, the initial estimation is chosen as a fraction of the true value ktrue
1 .Note that the precise estimate cannot be found, which can probably be ascribed to the noisein the data set.
94
Section 6.3
6.3 Discussion
In this chapter, qualitative PWA analysis is applied to a model of a hypothetical biochem-
ical oscillator. Qualitative analysis of the PWA model showed that the approximation
of this results in a specific set of parameter restrictions for which limit cycle behavior is
guaranteed. This extra information fulfills an assisting role in the nonlinear parameter
estimation procedure and contributes to more accurate parameter estimations compared
to conventional nonlinear estimation procedures without constraints, but both procedures
still require a reasonably well initial estimate of the parameter values. In the next chapter,
it is shown that hybrid system identification can provide a fairly good initial estimate,
which can improve parameter estimation even more.
95
Chapter 6
96
7Hybrid System Identification
This chapter was based on paper [121]
In the previous chapter it was shown that the initial estimate of the parameter values
influences identification (see Fig. 6.4). An appropriate initial estimate of the param-
eters is therefore desired, but we do not know which one is appropriate. We develop a
PWA parameter estimation procedure that approximates the nonlinearities with two PWA
functions. Parameter estimation is performed by shifting the switching plane of the PWA
model, thereby classifying the data to a specific mode by minimizing the cost function of
the estimation procedure. The advantage of this method is that no initial estimation of
the parameter values is needed, unlike traditional nonlinear least-squares methods [102]
or the multiple shooting method [18] that require this information. Furthermore, this
novel parameter estimation procedure requires considerably less calculation time than
nonlinear least-squares, multiple shooting methods, and a previously proposed method
with one-step ahead prediction [171]. Again, the mathematical model of a biochemical
oscillator was used as a test case [121].
7.1 General Identification Procedure
7.1.1 Model Class
We consider the following class of discrete time systems:
x(k + 1) =
g1(x(k), θ1) if xi(k) < α,
g2(x(k), θ2) if xi(k) ≥ α,(7.1)
where x ∈ Rn, xi ∈ R denotes the ith component of the state vector x, and the switching
plane α ∈ R. Functions g1(x(k), θ1) and g2(x(k), θ2) are assumed to be smooth nonlinear
97
Chapter 7
functions of the parameters x and θ. The system in Eq. 7.1 is a bimodal hybrid system, in
which the currently active mode is determined by the value of one of the state components.
All states are measured. We assume that both modes are excited in the data set. The
identification problem for Eq. 7.1 consists of determining values of parameters (θ1, θ2 and
α) on basis of measurements x(k), with k = 1, . . . , N . The difficulty of the identification
problem stems from the fact that the switching threshold is not known a priori.
7.1.2 Identification and Classification of a Hybrid Model
For a given α we can define two sets of data:
χ1(α) = {x(k)|xi(k) < α}, (7.2)
χ2(α) = {x(k)|xi(k) ≥ α}. (7.3)
We consider the cost function J of the form:
J(α, θ1, θ2) =∑
x(k)∈χ1(α)
‖x(k + 1)− g1(x(k), θ1)‖2 +
∑
x(k)∈χ2(α)
‖x(k + 1)− g2(x(k), θ2)‖2 . (7.4)
The identification problem can now be formulated as:
{α, θ1, θ2} = arg min{α,θ1,θ2}>0
J(α, θ1, θ2). (7.5)
Note that J(α, θ1, θ2) is not continuous in α. Hence, minimization methods that require
computation of the gradient of J (i.e., steepest descent) cannot be applied. Also note
that if the value of α = αknown is known, the optimization problem in Eq. 7.5 reduces to
{θ1, θ2} = arg min{θ1,θ2}>0
J(αknown, θ1, θ2), (7.6)
which is a smooth nonlinear least squares problem, and can be solved (locally) for θ1 and
θ2 using classical methods. This automatically classifies each data point to mode q1 and
q2.
One way to solve Eq. 7.5 is by “gridding”: we choose a set αmin = α1 < α2 < . . . <
αm = αmax of values (“grid”) for α, and solve Eq. 7.6 for every value in this set. We
chose αmin = mink xi(k) and αmax = maxk xi(k). The advantage of this method is that
the optimal value of the parameter α can always be found with the required precision for
a suitably selected grid. The disadvantage is a larger computational burden. However,
98
Section 7.2
f x1 3( )
f x2 3( )
β11
β
β12
β γ21
-γ11
-γ12
q2
x3
f
β11
β
β12
β
-γ11
-γ12
q1 q2
φ1
3(
)x
α x3
q2q1 q2
φ2
3(
)x
α x3
γ22β2
2β2
β21
(a) (b) (c)
Figure 7.1 – Conversion of nonlinear to PWA for hybrid identification. (a) The two non-linear functions f1(x3) and f2(x3) were approximated by two PWA functions, namely (b)PWA approximation of f1(x3), and (c) PWA approximation of f2(x3). We assumed that theswitching plane α was located at the same position for both functions. The PWA model hasconsequently two modes: q1 and q2. The superscript of the parameters refer to the modenumber.
the data set is relatively small, so gridding can be performed.
7.2 PWA Identification of the Biochemical Oscillator
In previous chapters the PWA approximation of nonlinear functions consisted of a linear
and a constant part. This rough approximation is suitable to explore the complete phase
space, but is too rough for describing the dynamics in a certain region of the phase space.
A more refined approach is required for PWA identification and therefore we approximate
the nonlinear functions f1(x3) and f2(x3) in Eq. 6.8 and Eq. 6.9, respectively, with two
linear segments. In previous chapter, the order of threshold was not important. Therefore
the switching plane x3 = α was assumed to be identical for both functions, also to simplify
the identification and classification methods, which divide the PWA approximation in two
modes. The mode with invariant x3 < α was classified as mode q1; mode q2 was assigned
to the invariant set x3 ≥ α, see Fig. 7.1.
The corresponding continuous-time PWA model is given by the following equations:
dx
dt=
A1x + B1 if x3(t) < α,
A2x + B2 if x3(t) ≥ α,(7.7)
with x: [x1, x2, x3]T ; the coefficient matrices A1, A2, B1 and B2 are listed in Table 7.1.
99
Chapter 7
Table 7.1 – Coefficient matrices and invariant sets of the biochemical oscillator model oftwo modes.
Mode A-matrix B-matrix Invariant
q1 A1 =
−k2 0 −γ1
1
k3 −k4 00 k5 −γ1
2
B1 =
β11
0−β1
2
x3 < α
q2 A2 =
−k2 0 −γ2
1
k3 −k4 00 k5 −γ2
2
B2 =
β21
0−β2
2
x3 ≥ α
7.2.1 Methods
Since the system was composed of two linear modes, a linear least-squares problem was
solved with the MATLAB command lsqlin. The estimated parameters of the PWA
model were collected in vectors θ1 and θ2 as follows:
θ1 = [β11 , γ
11 , β
12 , γ
12 , k2, k3, k4, k5], (7.8)
θ2 = [β21 , γ
21 , β
22 , γ
22 , k2, k3, k4, k5]. (7.9)
The bootstrap method (100 iterations) [18] was applied to determine the variance of these
parameters. The grid has the following dimensions: αmin = 0, αmax = 146.6, and step size
= 0.1.
7.2.2 Results
The experimental data of the simulated nonlinear biochemical oscillator with varying
noise levels were classified and identified with the hybrid identification method. The
results are displayed in Fig. 7.2 for σ = 0 and 5, the estimated parameter values for σ =
0, 1, 2.5 and 5 are listed in Table 7.2. The classification of the data to q1 and q2 satisfies
the expectations: the measurements are assigned to q1 for low x3 values, whereas the
other data correspond to q2. Since a PWA approximation was applied, not all estimated
parameter values could be verified with the original nonlinear parameters used in the
model of biochemical oscillator, i.e., k1, KI, k6, and Km. The estimated parameter values
of k2, k3, k4 and k5 in Table 7.2 were in good agreement with the true parameter values
(Table 6.4).
100
Section 7.3
0
200
400
0
100
200
300
0 50 100 150
0
100
200
x1
x2
x3
0
200
400
0
100
200
300
0 50 100 150
0
100
200
(a)
(b)
x1
x2
x3
t
t
Figure 7.2 – Results of the hybrid identification method for noise levels of (a) σ = 0, and(b) σ = 5.
101
Chapter 7
Table 7.2 – Estimated parameter values for the four data sets with varying noise levels.
Parameter σ = 0 σ = 1 σ = 2.5 σ = 5
β11 55.077 55.236± 1.6 55.944± 4.5 52.035± 4.9
γ11 2.915 2.921± 0.2 3.097± 0.1 2.368± 0.7
β12 5.566 5.574± 0.3 5.615± 0.8 6.236± 0.9
γ12 0.220 0.218± 0.04 0.215± 0.1 0.095± 0.1
β21 7.502 7.607± 0.34 7.645± 0.9 7.734± 1.6
γ21 0.052 0.053± 0.002 0.053± 0.005 0.052± 0.01
β22 9.274 9.313± 0.1 9.289± 0.3 9.235± 0.4
γ22 0.005 0.005± 0.001 0.005± 0.002 0.004± 0.004
k2 0.111 0.111± 0.003 0.111± 0.009 0.114± 0.01
k3 0.103 0.103± 0.001 0.102± 0.002 0.102± 0.003
k4 0.103 0.103± 0.001 0.102± 0.002 0.102± 0.003
k5 0.100 0.1± 0.001 0.099± 0.002 0.098± 0.003
α 12.367 12.122± 1.2 12.510± 2.6 14.339± 4.1
7.3 Discussion
A hybrid system identification procedure was applied: a two-segment PWA approxima-
tion of the nonlinear model is fitted on the data to estimate the parameters in the two
modes. The estimates of the parameters are quite accurate and can be used for nonlinear
parameter estimation procedures. The parameters which are not linked to the nonlinear
functions can be used as initial estimate of the parameters for standard nonlinear param-
eter estimation. For the hybrid identification procedure, we assume that all states are
observable, which is not always the case. As future work, it needs to be verified whether
parameter values of the PWA approximation can be used for estimating the parameter
values of the original nonlinear functions. The challenge is to apply this technique to
larger biological systems with multiple modes.
102
8Conclusion and Discussion
For a thorough understanding of biomedical phenomena it is essential to understand
the underlying cellular dynamics. Mathematical models have shown to integrate data
and information from various sources to solve numerous biomedical research questions
[71, 85]. The expectation is that the role of such models will become more important in
the near future. Performing accurate quantitative measurements in mammalian cells re-
mains the main bottleneck in present research. Reliable data are essential for formulating
mathematical models with high predictive power. In addition, mathematical models of
bioregulatory networks often contain nonlinear functions to describe the various biologi-
cal processes. System analysis and parameter estimation of such models is a cumbersome
task, especially due to the scarcity of quantitative data. Therefore, systems biologists are
exploring alternative ways to improve system analysis and parameter estimation, also to
assist the experimentalists to design most informative experiments [95, 116, 170, 172]. So
far, existing methods are primarily applied to the analysis of a few very specific biochem-
ical networks for which experimental data are available.
8.1 Conclusions
Two research goals have been defined in the introduction.
Primary Goal − Develop mathematical procedures to extract information from
typical nonlinear biochemical models that contain little quantitative information.
Main purpose: assistance in (qualitative) system analysis and improved parameter
estimation.
103
Chapter 8
Graphical Analysis of Bistable Systems
Nonlinear ordinary differential equations (ODEs) are used to describe the dynamical be-
havior of various biochemical networks. In chapter 2, a graphical study of a specific class
of monotone systems is adapted to yield constraints on parameter values associated with
a certain desired (experimentally observed) dynamic behavior. This info could subse-
quently be used to improve parameter estimation. This graphical procedure is tested on
an existing model of extracellular matrix (ECM) remodeling [97]. The closed-loop ECM
model is converted to an open-loop system. Reclosing the loop and adding an adaptable
feedback component puts bounds on a specific parameter value to guarantee bistability,
which is expected behavior allowing constraints on the parameter values. This strategy
is limited to a restricted class of systems that contains only positive feedback circuits,
which limits the applicability of this methodology in practice.
Qualitative PWA Analysis
A qualitative method to analyze a general class of biochemical regulatory networks as
piecewise-affine (PWA) functions has been developed. Nonlinear functions are approxi-
mated with two or three PWA functions, which enables qualitative analysis of the system.
The main contribution of qualitative analysis is its practical relevance to analyze nonlinear
deterministic networks with little quantitative information. Recent work on qualitative
analysis of large biochemical networks is so far limited to nonlinear functions approxi-
mated by piecewise-constant [12, 28, 29, 99] or, very recently, ramp functions [11]. The
work in this thesis extends qualitative analysis with (multidimensional) PWA approxima-
tions; virtually all nonlinear, biochemical, deterministic models can be approximated as a
collection of PWA functions, including metabolic networks and signal transduction path-
ways. The PWA approximation divides the phase space of the nonlinear model in several
modes by means of switching planes. Qualitatively different types of dynamic behavior
appear depending on if and how the state trajectories of the system move through the
modes of the phase space. Analysis of the dynamics at the vertices of the switching planes
leads to a set of, relatively simple, inequalities for the parameters to ensure that certain
mode transitions can occur. The complexity of the analysis is primarily determined by
the number of PWA approximations (i.e., the number of nonlinearities in the model) and
the number of segments used, not by the number of differential equations. The procedure
was demonstrated by using a second order model of TGF-β1. This low order model al-
lowed easy graphical representation. General qualitative methods developed in the area of
artificial intelligence are only applicable to 2D models [126, 141, 164, 163]. Our approach
can be applied to larger systems, as was shown for the UPR model in chapter 5.
Qualitative PWA analysis enables qualitative sensitivity analysis by studying the ef-
104
Section 8.1
fect of changing a specific parameter. It provides valuable information when studying
pathological phenomena to verify what parameters can cause an observed change in sys-
tem dynamics. Traditional sensitivity analysis requires quantitative parameter estimates,
which are usually not available.
Furthermore, qualitative PWA analysis provides (relative) bounds on the parameter
values. This information can be used to reduce the parameter search space and lead to
more accurate parameter estimates than conventional methods, as shown in chapter 6.
Initial estimates can influence the outcome of parameter estimation. Therefore a hybrid
parameter estimation is developed in chapter 7 to provide a rough initial estimate of the
parameter values, which is subsequently applied to a model of the biochemical oscillator.
In short, we can conclude that qualitative PWA analysis is currently the most general
(all types of kinetics and large-scale models) method for qualitative analysis of nonlinear
biochemical networks.
Secondary Goal − Apply the developed methodologies to typical biochemical
networks that are involved in vascular aging processes.
There is a wide variety of biochemical processes that can be linked to aging in gen-
eral [21, 108] or, more specifically, vascular aging [45]. Based on limited experimental
evidence within the project, three relevant biochemical networks have been studied:
1. Remodeling of the ECM (chapter 2), for which a mathematical model has been
derived from the literature [97]. Bistability is expected for this model and only
possible if certain constraints are satisfied.
2. The TGF-β1 signaling pathway (chapter 4). An extensive literature search re-
sulted in a complex kinetic model with positive and negative feedback circuits. By
means of the quasi-steady-state approximation, this can be reduced to a second
order model. System analysis shows that, depending on the choice of parameter
values, a single stable node or a limit cycle is present. Experimental findings in
hepatocytes [125] demonstrate that oscillatory behavior can be observed. During
aging, the TGF-β1 pathway of endothelial cells become less sensitive to external
TGF-β1 [107], which can be explained by an increase in the Michaelis constants
for phosphorylation R-SMAD complex formation, Michaelis constant of I-SMAD
production, dephosphorylation rate constant of SMAD complex or a decrease in
maximal phosphorylation rate of R-SMADs. These are interesting targets for in-
depth analysis.
3. The mammalian UPR in the ER (chapter 5). A kinetic model has been build on
basis of the literature. The PERK and ATF6 branch are isolated as these signaling
pathways indirectly control the regulation of protein formation and degradation.
105
Chapter 8
Equilibrium points were identified by the qualitative analysis procedure. The time
span of the experimental validation [36] has most likely been too short to draw
conclusions about the system behavior, although model simulations coincided quite
well with the experimental data of DuRose et al. [36] for certain conditions of the
parameter values.
8.2 Future Perspectives
• It is shown in genetic regulatory networks that important system features can be
extracted from qualitative information [12, 27, 161], although the focus has primarily
been on systems with piecewise constant functions. The qualitative analysis in
this thesis is not restricted to piecewise-constant, but can deal with PWA systems
as well. Automated procedures to analyze qualitative models, like GNA [27] and
RoVerGeNe [11], are indispensable for larger systems that are common in biology.
So far the procedure has been automated, but should be extended with an automatic
verification and visualization procedure to make data from large, multidimensional
systems easier to grasp.
• The theory about the stability of PWA systems is another point of attention. Cur-
rent approaches for stability analysis of hybrid systems is very limited, and aimed
at hybrid systems with abundant quantitative information. Surface Lyapunov func-
tions (SuLF), in which the stability of any PWA system can be verified by analyzing
a hybrid system with specialized surfaces, could be a promising and solid framework
to analyze the limit cycles and equilibrium points in this thesis [52, 53]. Also exploit-
ing the monotonicity of the nonlinearities in biochemical networks can contribute
to make mathematical statements about stability.
• Sensitivity analysis is frequently applied in systems biology: each parameter is
checked for its impact on the global system dynamics by varying it within a given
parameter range. For nonlinear models, this behavior is highly dependent on the
base value of all parameters (the operating point) and how large the change is.
Qualitative analysis can assist in this process by excluding parameters that have
less significance.
• The development of the qualitative method in this thesis was motivated by biochem-
ical networks, but complicated nonlinear behavior in other engineering fields can
have the same type of functions and lack in quantitative information. Qualitative
PWA analysis can therefore be applied to other research fields as well.
106
Appendix A
Nomenclature
A.1 List of Abbreviations
ADP adenosine 5’-diphosphate
ATF6 activating transcription factor 6
ATP adenosine 5’-triphosphate
BiP binding protein
CNX calnexin
co-SMAD common-mediator SMAD
CRT calreticulin
DNA deoxyribonucleic acid
DTT dithiothreitol
ECM extracellular matrix
eIF2α eukaryotic initiation factor 2α
EMSA electrophoretic mobility shift array
ER endoplasmic reticulum
ERAD ER associated degradation
GADD34 growth-arrest and DNA damage-inducible protein
GNA genetic network analyzer
IRE1 inositol requiring kinase-1
I-SMAD inhibitory SMAD
IκB inhibitor of κB
MAPK mitogen-activated protein kinase
mRNA messenger ribonucleic acid
NF-κB nuclear factor κB
ODE ordinary differential equation
PDE partial differential equation
PDI protein disulfide isomerase
PERK (PKR)-like endoplasmic reticulum kinase
PKR protein kinase regulated by RNA
107
Nomenclature
pRS phosphorylated R-SMAD pool
PWA piecewise-affine
ROS reactive oxygen species
RoVerGeNe robust verification of gene networks
R-SMAD receptor-regulated SMADs
SMURF2 SMAD ubiquitination regulatory factor 2
SRP signal recognition particle
SuLF surface Lyapunov functions
Tg thapsigargin
TGF-β1 Transforming Growth Factor-β1
Tm tunicamycin
tIS total I-SMAD pool
tPA tissue-type plasminogen activator
tRS total R-SMAD pool
UGGT uridine diphosphate-glucose:glycoprotein glucosyl transferase
uPA urokinase-type plasminogen activator
UPR unfolded protein response
uPr unfolded protein
uRS unphosphorylated R-SMAD pool
vWF von Willebrand factor
A.2 Symbols
A.2.1 Latin
A coefficient matrix of the linear part of a state space model
B coefficient matrix of the constant part of a state space model
f(x) rate equation
Jm Jacobian matrix
J(θ) cost function
k rate constant
Km Michaelis constant
KI Michaelis constant of an inhibitor
n normal at a certain switching plane
N total number of samples
Nf total number of rate equations
Nq total number of modes
108
Nomenclature
Nx total number of states
p vertex on switching threshold
q mode of a hybrid system
r cooperativity coefficient
t time
t(x) tangent of a switching threshold
V (x) Lyapunov function
Vmax conversion rate at maximal substrate concentration
x dimensionless state variable
xeq state variable evaluated at equilibrium point
xIP intersection point of f(x) with ϕ(x)
xmax maximal value of x
y regular state variable
y(k) measured output data
A.2.2 Greek
α switching threshold
β constant in PWA approximation
γ slope of PWA approximation
Γ set of constraints
Γ∗ complementary set of Γ
ε small factor, required for the quasi-steady-state approximation
η output of a system
λ eigenvalue
ν feedback strength = ωη
σ standard deviation
θ set of true parameter values
θ set of estimated parameter values
θini initial estimate of the parameter values
ϕ(x) PWA approximation of f(x)
Φ(x) set of PWA state equations
χ data set
ω input of a system
109
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Summary
126
Summary
The human body is composed of a large collection of cells,“the building blocks of life”. In
each cell, complex networks of biochemical processes contribute in maintaining a healthy
organism. Alterations in these biochemical processes can result in diseases. It is therefore
of vital importance to know how these biochemical networks function. Simple reasoning
is not sufficient to comprehend life’s complexity. Mathematical models have to be used
to integrate information from various sources for solving numerous biomedical research
questions, the so-called systems biology approach, in which quantitative data are scarce
and qualitative information is abundant.
Traditional mathematical models require quantitative information. The lack in ac-
curate and sufficient quantitative data has driven systems biologists towards alternative
ways to describe and analyze biochemical networks. Their focus is primarily on the anal-
ysis of a few very specific biochemical networks for which accurate experimental data are
available. However, quantitative information is not a strict requirement. The mutual
interaction and relative contribution of the components determine the global system dy-
namics; qualitative information is sufficient to analyze and predict the potential system
behavior. In addition, mathematical models of biochemical networks contain nonlinear
functions that describe the various physiological processes. System analysis and parame-
ter estimation of nonlinear models is difficult in practice, especially if little quantitative
information is available.
The main contribution of this thesis is to apply qualitative information to model and
analyze nonlinear biochemical networks. Nonlinear functions are approximated with two
or three linear functions, i.e., piecewise-affine (PWA) functions, which enables qualitative
analysis of the system. This work shows that qualitative information is sufficient for the
analysis of complex nonlinear biochemical networks. Moreover, this extra information can
be used to put relative bounds on the parameter values which significantly improves the
parameter estimation compared to standard nonlinear estimation algorithms. Also a PWA
parameter estimation procedure is presented, which results in more accurate parameter
estimates than conventional parameter estimation procedures. Besides qualitative analysis
with PWA functions, graphical analysis of a specific class of systems is improved for
a certain less general class of systems to yield constraints on the parameters. As the
applicability of graphical analysis is limited to a small class of systems, graphical analysis
is less suitable for general use, as opposed to the qualitative analysis of PWA systems.
The technological contribution of this thesis is tested on several biochemical networks
127
Summary
that are involved in vascular aging. Vascular aging is the accumulation of changes respon-
sible for the sequential alterations that accompany advancing age of the vascular system
and the associated increase in the chance of vascular diseases. Three biochemical networks
are selected from experimental data, i.e., remodeling of the extracellular matrix (ECM),
the signal transduction pathway of Transforming Growth Factor-β1 (TGF-β1) and the
unfolded protein response (UPR).
The TGF-β1 model is constructed by means of an extensive literature search and con-
sists of many state equations. Model reduction (the quasi-steady-state approximation)
reduces the model to a version with only two states, such that the procedure can be visual-
ized. The nonlinearities in this reduced model are approximated with PWA functions and
subsequently analyzed. Typical results show that oscillatory behavior can occur in the
TGF-β1 model for specific sets of parameter values. These results meet the expectations
of preliminary experimental results. Finally, a model of the UPR has been formulated and
analyzed similarly. The qualitative analysis yields constraints on the parameter values.
Model simulations with these parameter constraints agree with experimental results.
128
Samenvatting
Het menselijk lichaam bestaat uit een grote hoeveelheid cellen, “de bouwstenen van het
leven”. In iedere cel dragen complexe netwerken van biochemische processen bij aan de
handhaving van een gezond organisme. Veranderingen in deze biochemische processen
kunnen ziektes tot gevolg hebben. Het is daarom van groot belang te achterhalen hoe
deze biochemische netwerken functioneren. Eenvoudig redeneren is niet voldoende om de
complexiteit van het leven te doorgronden. Wiskundige modellen zijn nodig om informatie
van verschillende bronnen te integreren, de zogenaamde systeembiologie aanpak, waarin
kwantitatieve informatie schaars en kwalitatieve informatie meer voor handen is.
Traditionele wiskundige modellen vereisen kwantitatieve informatie. Het gebrek aan
voldoende en nauwkeurige kwantitatieve data hebben systeembiologen gedwongen alter-
natieve methodes te verkennen om biochemische netwerken te beschrijven en te analyseren.
Hun focus is vooral gericht op de analyse van slechts enkele, zeer specifieke biochemische
netwerken waarbij nauwkeurige experimentele data beschikbaar zijn. Kwantitatieve infor-
matie is echter niet strict noodzakelijk om het potentiele systeemgedrag te analyseren. De
onderlinge interactie en de relatieve bijdrage van de individuele componenten bepalen de
globale dynamica van een systeem. Kwalitatieve informatie is soms voldoende. Daarnaast
bevatten wiskundige modellen van biochemische netwerken vaak niet-lineaire functies die
de verschillende fysiologische processen beschrijven. In de praktijk bemoeilijkt dit systee-
manalyse en parameterschatting aanzienlijk, zeker als er weinig kwantitatieve informatie
beschikbaar is.
De belangrijkste bijdrage van dit proefschrift is kwalitatieve informatie toe te passen
om niet-lineaire biochemische netwerken te modelleren en te analyseren. De niet-lineaire
functies worden benaderd met twee of drie lineaire functies, d.w.z. stuksgewijs lineaire
functies, die kwalitatieve analyse van het systeem mogelijk maken. Dit werk toont aan dat
kwalitatieve informatie vaak voldoende is voor systeemanalyse van complexe niet-lineaire
biochemische netwerken. Bovendien kan deze extra informatie worden gebruikt om de
relatieve grenzen van de parameters op te stellen. Dat verbetert de parameterschatting
significant t.o.v. standaard niet-lineaire parameterschatting methodes. Daarnaast is er
ook een stuksgewijs lineaire parameterschatting procedure ontwikkeld die eveneens betere
parameterschattingen oplevert dan traditionele methodes. Behalve kwalitatieve analyse
met stuksgewijs lineaire functies, is er een grafische analyse methode aangepast die ook het
parameterschatten verbetert. De toepasbaarheid van deze grafische methode is beperkt
tot een kleine klasse van systemen wat deze minder geschikt maakt voor algemeen gebruik.
129
Samenvatting
De bijdrage van dit proefschrift is toegepast op verschillende biochemische netwerken
die betrokken zijn bij vaatwandveroudering. Vaatwandveroudering is de verandering in het
vasculaire systeem die samengaat met een toename in leeftijd, met de daarbij behorende
toenemende kans om ziektes van de vaatwand te krijgen. Drie biochemische netwerken zijn
geselecteerd waarvan experimentele data bekend was: modellering van de extracellulaire
matrix (ECM), de signaaltransductie route van Transforming Growth Factor-β1 (TGF-
β1) en de unfolded protein response (UPR). Grafische analyse is uitgevoerd op het ECM
modellering proces, zodat er grenzen aan een parameterwaarde gesteld kunnen worden
om het verwachte systeemgedrag te garanderen. Het TGF-β1 model is opgesteld d.m.v.
een uitgebreid literatuuronderzoek. Model reductie (de quasi-steady-state benadering)
reduceert dit model naar een versie met twee toestanden. De niet-lineariteiten in dit
gereduceerde model zijn benaderd met stuksgewijs lineaire functies en vervolgens wordt
dit model geanalyseerd. De resultaten tonen aan dat oscillaties kunnen ontstaan in het
TGF-β1 model voor specifieke parameterwaarden. Deze resultaten zijn in overeenstem-
ming met voorlopige experimentele resultaten. Tenslotte is het UPR model opgesteld en
geanalyseerd op een soortgelijke manier. De kwalitatieve analyse brengt restricties op
de parameterwaarden voort. Simulaties met deze restricties komen overeen met experi-
mentele resultaten.
130
Dankwoord
Vier jaar onderzoek verrichten zit er op en staat op papier. Dus mag ik eindelijk het
vrolijkste en meest gelezen hoofdstuk van dit proefschrift schrijven: het dankwoord. Ten
eerste wil ik mijn copromotor bedanken. Natal, onze samenwerking begon in 2002 toen
ik vetzuurtransport in de hartspiercel begon te bestuderen in Seattle, daarna afstuderen
bij CS en vervolgens promoveren. Bij jou kon ik altijd terecht als ik even een goed inzicht
kon gebruiken. Daarnaast heb je me vrij gelaten in het onderzoek, waardoor ik deze
vier jaar met plezier heb beleefd. Paul, beste promotor, je kritische en eerlijke houding
hebben ervoor gezorgd dat ik dit proefschrift op tijd heb kunnen voltooien en dat ik ook
tevreden kan terugkijken op het eindresultaat. Gedurende mijn promotie heb ik voor
enkele maanden onderzoek gedaan bij de Helix-groep, INRIA Rhone-Alpes, Frankrijk.
Hidde, hartelijk dank voor je gastvrijheid, grote interesse in dit project en de vele hulp
die ik heb mogen ontvangen gedurende die periode. I would like to thank the people
at the Helix group for introducing me in the French life. I especially want to mention
Samuel and Delphine (hopefully Quentin is doing well) as my helpful roommates and
for their hospitality, Philippe for the nice diners and the badminton evenings in Crolles.
Finally a big thank you for Thomas: the amusing bike trips, the weekend in Saint Etienne,
and the French cursing lessons are things I will never forget. Merci! I hope to welcome
you and Gaelle in the Netherlands in the near future. Axel and Jan, we had some nice
talks in Italy. I will certainly visit you soon in Germany. Verder wil ik Theo Verrips,
Jan Andries Post, Branko Braam en Arie Verkleij bedanken voor hun enthousiasme en
bruikbare ideeen over de unfolded protein response aan het begin van dit project. Peter
Hilbers, bedankt voor het lezen van mijn proefschrift en voor de suggesties. Jorn, we
hebben slechts kort samengewerkt voordat je naar de VS emigreerde, maar bedankt voor
de hulp die je me hebt gegeven in het begin van mijn promotie. Daniel, jouw stage en
afstudeerwerk hebben mij geınspireerd om kwalitatief modelleren beter te bestuderen.
Daarnaast zijn de stafleden en (oud-)promovendi onmisbaar geweest voor de goede sfeer
op de afdeling: Aleksandar, (SpongeBob) Bart, Femke, Heico, Jasper (professor Preisig),
John, Leo, Michal, Michiel, Nelis (het is zo stil in mij...), Patrick (Patty), Patricia, en
skate maharadja Satya. Mircea, thank you for being an excellent office mate. Barbara,
bij vragen over de Engelse taal, keuvelen over Britse seriemoordenaars, vakanties, en
problemen met m’n reisdeclaraties kon ik altijd bij je terecht. Udo, computerproblemen
werden door jou snel verholpen, waardoor ik mijn onderzoek kon blijven doen. Andrej
en paranimf Maarten N., de laatste loodjes wegen het zwaarst... maar ze worden een
131
Dankwoord
stuk lichter als je met drie man aan het overwerken bent op een maandagnacht met harde
trance op de achtergrond. Erg bedankt voor deze en vele andere leuke momenten in de
laatste maanden van mijn promotie.
Naast promoveren kon ik voor een gezonde hoeveelheid sport bij BC Bavel terecht.
Ik wil iedereen bij BC Bavel bedanken voor een erg leuke tijd. Helaas zal ik verhuizen
en daarom afscheid moeten nemen van BC Bavel, maar ik kom zeker nog een paar keer
langs. Joep, collega-promovendus, sportief dat je me keer op keer uitdaagde voor een potje
badminton in Geldrop en de vele flessen La Chouffe die ik daaraan heb overgehouden.
Jeroen, dankzij jou ben ik me meer gaan interesseren voor fitness. Onze gesprekken over
jouw fotografie-droom zullen me bijblijven.
Gedurende mijn studie en promotie heb ik in het leukste studentenhuis van Eindhoven
gewoond. De laatste jaren bruiste het van de gezelligheid en was het heerlijk om thuis te
komen. Het gaat te ver om iedereen te bedanken, maar toch wil ik er een paar uitlichten.
Remco en Tjitske, bedankt voor de leuke spelletjesavonden, halve marathon “wandelinget-
jes”, jullie bezoek in Grenoble en de ontelbare mooie momenten samen; Virginie en Bart,
de koffie smaakt altijd uitstekend, de wintersport was super en jullie waren altijd be-
trokken bij de gezellige feesten in het huis; Andrea, dankje voor jouw oprechte interesse
en de mini-trip naar Oostenrijk; Lex, Lieke, Marijke, en Pascal: ik heb altijd genoten
van jullie gesprekken; Christoph, dankjewel voor het ontwerpen van het meest creatieve
gedeelte van dit proefschrift: de kaft. Halve huisgenoot en paranimf Maarten v.d. V., we
hebben veel meegemaakt in de afgelopen vier jaar en ook vele goede gesprekken gehad,
waarvoor ik je enorm dankbaar ben. Ik heb er veel aan gehad. Daarnaast wil ik nog al
mijn vrienden bedanken voor de leuke tijd samen.
Anouk & Jochem, Petra & Patrick, Esther & Gijs, de enkele keren dat ik jullie zag
vond ik het erg gezellig. Wim en Anneke, door jullie warmte en steun heb ik voor mijn
gevoel een extra thuis erbij gekregen. Jan, Lia en Toon en de kids, jullie betrokkenheid
was enorm. Jullie hebben altijd voor me klaargestaan en gestimuleerd om door te zetten.
Nu maar hopen dat een flinke dosis wijsheid bij die doctors-titel wordt geleverd. Roland,
broertje, de dagelijkse mailtjes met updates over ons hectische leven heeft ons in die vier
jaar meer naar elkaar toegebracht....het is dus toch nog goed gekomen met ons en dat zal
zeker nog lang blijven als het aan mij ligt. Lieve Vivien, tenslotte wil ik jou als laatste
bedanken. Ondanks de afstand voelde je door onze gesprekken precies aan wanneer ik
jouw hulp kon gebruiken tijdens mijn promotie. Bovendien heb je mij laten inzien dat
ontspanning ook nodig is, vooral in de laatste paar zware maanden. Hierdoor kijk ik met
een zeer goed gevoel terug naar die periode en verwacht ik zeker dat er nog een ontzettend
mooie tijd voor ons samen te wachten staat.
Iedereen bedankt voor zijn/haar bijdrage aan dit proefschrift!
132
About the Author
Mark Musters was born on May 20th, 1980 in Breda,
The Netherlands. He studied Biomedical Engineering
at Eindhoven University of Technology (TU/e) from
1998 - 2003. In 2002 he did an internship for three
months at the bioengineering group of Jim Bassingth-
waighte, University of Washington, Seattle, USA, for
which he received the Dr. E. Dekker grant from the
Netherlands Heart Foundation. The subject of this
internship was to develop a computational model of fatty acid transport across the sar-
colemma of the cardiomyocyte. Mark continued this work in a graduation project, which
resulted in a book chapter. After obtaining his Master of Science degree in Biomedical
Engineering, Mark pursued his PhD degree at the Department of Electrical Engineering,
TU/e. The project was a cooperation between the University of Utrecht, Unilever Re-
search, Aurion and TU/e. The role of the TU/e was to provide computational assistance
in understanding vascular aging processes, the so-called systems biology approach. Espe-
cially the lack of quantitative information formed the main computational challenge in this
project. A qualitative approach was therefore chosen to analyze the mathematical models
of the biochemical networks involved in vascular aging. To become more acquainted with
qualitative modeling, Mark was a visiting PhD student from February - May 2006 in the
group of Hidde de Jong, INRIA Rhone-Alpes, Saint Ismier cedex, France (funded by an
NWO grant).
133