Qualitative Modeling in Computational Systems Biology

Qualitative modeling in computational systems biology

Citation for published version (APA): Musters, M. W. J. M. (2007). Qualitative modeling in computational systems biology. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR629275

DOI: 10.6100/IR629275

Document status and date: Published: 01/01/2007

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Download date: 26. Sep. 2021 Qualitative Modeling in Computational Systems 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 Computational Systems 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 Rhˆone-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 1 Introduction

NOWLEDGE of health and the need of solutions for addressing diseases have K 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 equations [25]. A plethora of mathematical formalisms exists to describe physiological processes: discrete, continuous, deterministic, stochastic and combinations of these modeling frameworks. Some of these frameworks are suitable for specific situations. For instance, 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 reactions. 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 computer 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 typical 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 mobility shift array (EMSA) of the IκB-NF-κB signaling module [70].

2. The large variations in quantitative measurements due to diﬀerences between species, 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 parameters 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 structures 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 mathematically 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

phenotypes Organism BOTTOM-UP Tissues Cells Proteins TOP-DOWN Genes -omicsprofiling

availabledata

Figure 1.2 – Graphical representation of the top-down and bottom-up approach with respect 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 León(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 accompany 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 Le´onduring his quest [41].

about the eﬀects of aging on the cellular level and how aging cells lead to aging of the organism. 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, several 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], mitochondria [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 (“remodeling”) [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, proteomics 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 components 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 contributes 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 approaches 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 information 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 θˆ. 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 application 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 hybrid 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 limitations 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 framework is developed that can provide accurate initial estimates of the parameter values. This thesis concludes with the main conclusions and future perspectives.

10 2 Analysis 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 aging [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 communication, 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 k1 k2 E + S FGGGGGGGGGGB ES GGGA E + P, (2.1) k−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):

k E x V x f(x) = 2 T = max , (2.2) k−1+k2 + x Km + x k1 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 r Vmaxx f(x) = r r , (2.3) Km + x

12 Section 2.3

1 Vmax Vmax1

0.8 0.8 f() x f() x 0.6 0.6 Vmax

2 0.4 0.4 r =2 r =5 r =10 0.2 0.2 r =100 0 0 0 Km Km x x (a) (b)

Figure 2.1 – (a) Michaelis-Menten curve for Vmax = 1 and Km = 0.1 and (b) Hill kinetics for 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

dx X i = f (x), i = 1,...,N , j = 1,...,N , (2.4) dt j x f j

T x [x1, . . . , xNx ] ≥ 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 dynamical 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:

dx (c − x )x k x 1 = 0 1 2 − 1 1 , (2.5) dt 1 + c0 − x1 Km1 + x1 r 2 dx2 k2x1 k3x2 = r r − , (2.6) dt Km2 + x1 Km3 + x2

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 combinations 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

k ω 3

Protease

c0 -x1 x1

Extracellular Proteolysis matrix fragments

k1 η

Substrate Reaction

Degradation Enzymatic products stimulation

Figure 2.2 – Closed model of ECM formation and degradation [97]. The scissors indicate the place where the feedback circuit is cut for analysis purposes. Input of the system is ω given by ω, output of the system is η, the strength of feedback is ν = η . Graphical notation is in agreement with the proposed style of Kitano [86].

k1=0.05, K m2 =0.1 k1=0.06, K m2 =0.1 0.1 0.1 dx2/ dt =0 dx2/ dt =0 0.08 stablenode 0.08 x1 x1 dx1/ dt =0 0.06 0.06 dx1/ dt =0

0.04 0.04 saddle 0.02 0.02 stablenode stablenode 0 0 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2

x2 x2 (a) (b)

Figure 2.3 – Distinct dynamics for diﬀerent sets of parameter values in the ECM model of Larreta-Garde and Berry [97]: (a) Bistable dynamics if k1 = 0.05, but (b) only one stable node 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

k1=0.05, K m2 =0.1081 0.12 0.1 dx/ dt =0 ν =0.925 2 0.1 0.08 η 0.08 x1 saddle dx/ dt =0 0.06 1 0.06 0.04 0.04 0.02 ν =1 0.02 stablenodes 0 0 0 0.02 0.04 0.06 0.08 0.1 0 0.2 0.4 0.6 0.8 1 1.2 ω x2 (a) (b)

Figure 2.4 – (a) Graphical representation of g(ω). The dashed and dotted line indicate ν = 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

dx (c − x )x k x 1 = 0 1 2 − 1 1 , (2.7) dt 1 + c0 − x1 Km1 + x1 r 2 dx2 k2ω k3x2 = r r − . (2.8) dt Km2 + ω Km3 + x2

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-

dx1 dx2 state. In Eqs. 2.7 and 2.8, this is done by solving dt = 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 contain negative 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 ω can be derived graphically: if the line η = ν 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

r r r k2ω k2(νη) k2η = = ¡ ¢r . (2.9) Kr + ωr Kr + (νη)r Km2 r m2 m2 ν + η

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 0.1 implies that a bistable system can be found for 0 < Km2 ≤ 0.925 = 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 equilibrium 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 essential 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 3 Qualitative Analysis of Nonlinear Biochemical Networks

Analysis 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, multidimensional, 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: dx X i = f (x), i = 1 ...N , j = 1 ...N , (3.1) dt j x f j

T x [x1, . . . , xNx ] ≥ 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 identification 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 possible 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

V x f(x) = max , (3.2) Km + x 1The only exception on this assumption will be bisubstrate Michaelis-Menten inhibition, which will be 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 second segment is constant for α < x < ∞. The functions f(x) and ϕ(x) intersect at xIP. The shaded 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)   Vmaxx if x < α, ϕ(x) = α (3.3)  Vmax if x ≥ α,

α switching plane.

Hill Kinetics

The Hill equation is an extension of Michaelis-Menten kinetics [46]

r Vmaxx f(x) = r r , (3.4) Km + x 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

Vmax

0.8 f() x 0.6 φ()x 0.4 φ()x

0.2 f() x

0 0.2 0.4K 0.6 0.8 1 m x

Figure 3.2 – Hill function f(x) for r = 100 (solid line) and its PWA approximation (dashed line). 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   0 if x < Km, ϕ(x) = (3.5)  Vmax if x ≥ Km.

Inhibition

Inhibition of biochemical processes can be modeled as well with a Michaelis-Menten description V K f(x) = max I , (3.6) KI + x

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  ¡ ¢  x Vmax 1 − if x < α, ϕ(x) = α (3.7) 0 if x ≥ α.

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

Vmax φ()x 0.8 f() x f() x φ()x 0.6

0.4

0.2

0 0 1 α 3 4 5 0 KI α 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)

Vmaxx1x2 f(x1, x2) = . (3.8) Km + x2

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.   Vmaxx1 if αx1 − x2 < 0, ϕ(x1, x2) = (3.9)  Vmaxx2 α if αx1 − x2 ≥ 0.

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 Vmaxx1 f(x1, x2) = ³ ´ , (3.10) x2 Km 1 + + x1 KI

x1 substrate concentration, x2 concentration of the inhibiting substrate,

KI Michaelis constant of the inhibition reaction.

23 Chapter 3

1 2

x x

(,)

f x x

(a) (b)

Figure 3.4 – (a) Plot of bisubstrate Michaelis-Menten kinetics, and (b) its two-segment PWA approximation.

A three-segment PWA function is assumed to be a reasonable approximation of Eq. 3.10   Vmaxx1 x1 x2  if + < 1 ∧ x2 < α2,  α1 ³ ´ α1 α2 x2 x1 x2 ϕ(x1, x2) = Vmax 1 − if + ≥ 1 ∧ x2 < α2, (3.11)  α2 α1 α2  0 if x2 ≥ α2,

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 biochemistry 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 nonlinear functions demand a more reﬁned segmentation and, consequently, requires more computational eﬀort.

24 Section 3.1

V max Vmax

1 2

(,)

x x

(,) f x x x1 φ x1 Km α1 KI x2 α2 x2 (a) (b)

Figure 3.5 – (a) Nonlinear function of competitive inhibition, and (b) its three-segment PWA 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

Z xmax ¯ ¯ ¯ ¯ Jtot = ¯f(x) − ϕ(x)¯dx, (3.12) 0 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),

Z xIP µ ¶ Vmaxx Vmaxx J1 = − dx, (3.14) 0 Km + x α

• For xIP < x < α: f(x) is smaller than the linear segment of ϕ(x),

Z α µ ¶ Vmaxx Vmaxx J2 = − dx, (3.15) xIP α Km + x

• For α < x < xmax: f(x) is smaller than the constant segment of ϕ(x),

Z xmax µ ¶ Vmaxx J3 = Vmax − dx. (3.16) α Km + x

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)

IP IP Vmaxx Vmaxx IP IP = ⇒ x = α − Km. (3.17) Km + x α

Combining Eq. 3.13 and 3.17 gives an analytical expression for Jtot Ã 1 2K2 J = V α − m + ... tot 2 max α ! ³ max ´ Km α Km + x ... 2Km log − log + log , (3.18) α α + Km α + Km

Eq. 3.18 is subsequently minimized as function of the PWA parameter α with Mathematica

2 2 dJtot (α − 4αKm + 2Km) Vmax arg min Jtot ⇒ = = 0. (3.19) α dα 2α2

26 Section 3.1

³ √ ´ Solving this quadratic expression results in two solutions: α = 2 − 2 Km and α = ³ √ ´ 2 + 2 Km. With the assumption that α > Km, one solution remains

³ √ ´ α = 2 + 2 Km, (3.20)

³ √ ´ ³ 2 or Km = 1 − 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 Inhibition Competitive M-M Bisubstrate Hill Inhibition Menten Michaelis- Name al 3.1 Table f K V K V K V V K K max max max max ( m m m r x I m ³ + + + V ) + 1+ x K x x x x x max 1 x r r I x 2 vriwo olna ucin n hi orsodn W approximations. PWA corresponding their and functions nonlinear of Overview – K x 2 2 x I 1 ´ + x 1        ( ( ( ( ϕ ( if 0 V if V 0 V V if 0 V V V V x max max max max max max max max Link ) α α α 1 x x x x 2 1 ³ ¡ 1 1 1 if if if − − K < x x if if α < x x α x ≥ α x αx αx ≥ 2 2 ¢ ´ K 1 1 α m m − − if if if x x x α < x x α α x x 2 2 ≥ 1 1 1 1 2 ≥ < + + ≥ α 0 0 α α α x x 2 2 2 2 2 ≥ < 1 1 ∧ ∧ x x 2 2 α < α < 2 2 α large for link Direct α α α α = 4 = = 2 1 4 = = ¡ ¡ f + 2 + 2 . 09 ¡ ( + 2 x . 09 K and ) √ √ K I √ 2 2 I ¢ ¢ 2 K K ¢ ϕ K m m ( x m ) r

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

k2 x1 x2 k4

Substrate Reaction

Degradation Enzymatic products stimulation

Figure 3.6 – Graphical interaction scheme of two substrates, x1 and x2, that mutually induce 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:.

dx 1 = f (x ) − k x , (3.23) dt 1 2 2 1 dx 2 = f (x ) − k x , (3.24) dt 2 1 4 2 with Michaelis-Menten functions

k1x2 f1(x2) = , (3.25) Km1 + x2 k3x1 f2(x1) = , (3.26) Km2 + x1

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

max x2

q3 q4 x2

α1

q1 q2

0 α2 max x1 x1

Figure 3.7 – Phase plane diagram, divided in four diﬀerent modes. The dotted lines are the 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:   k1x2 if x2 < α1, ϕ (x ) = α1 (3.27) 1 2  k1 if x2 ≥ α1,   k3x1 if x1 < α2, ϕ (x ) = α2 (3.28) 2 1  k3 if x1 ≥ α2,

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 approximation Φ(x) of the nonlinear system is   k1x2 k3x1 T [ − k2x1; − k4x2] for mode q1,  α1 α2   k1x2 T dx [ − k2x1; k3 − k4x2] for mode q2, Φ(x) = = α1 (3.29) dt  k3x1 T [k1 − k2x1; − k4x2] for mode q3,  α2  T [k1 − k2x1; k3 − k4x2] for mode q4,

31 Chapter 3 with invariants

q1 : 0 ≤ x1 < α2 ∧ 0 ≤ x2 ≤ α1, max q2 : α2 ≤ x1 ≤ x ∧ 0 ≤ x2 ≤ α1, 1 (3.30) max q3 : 0 ≤ x1 < α2 ∧ α1 ≤ x2 ≤ x2 , max max q4 : α2 ≤ x1 ≤ x1 ∧ α1 ≤ x2 ≤ x2 .

max x1 maximum concentration of x1, max x2 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 k1x2 k1x2 = − k2x1 = 0 ⇔ x1 = , (3.31) dt α1 α1k2 dx2 k3x1 k3x1 = − k4x2 = 0 ⇔ x2 = . (3.32) dt α2 α2k4

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

k1k3 Stable node α α k k < 1 q1 (0, 0) 1 2 2 4 Saddle point k1k3 > 1 α1α2k2k4 ³ ´ k1k3 k3 k1k3 k3 k1 q2 , Stable node > 1 ∧ α1 ≥ ∧ α2 < α2k2k4 k4 α1α2k2k4 k4 k2 ³ ´ k1 k1k3 k1k3 k3 k1 q3 , Stable node > 1 ∧ α1 < ∧ α2 ≥ k2 α1k2k4 α1α2k2k4 k4 k2 ³ ´ k1 k3 k1k3 k3 k1 q4 , Stable node > 1 ∧ α1 < ∧ α2 < k2 k4 α1α2k2k4 k4 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 Ã ! k1 −k2 α1 Jq = . (3.33) 1 k3 −k4 α2

Eq. 3.33 yields two symbolic expression for the eigenvalues: √ µ ¶ −(k2 + k4) ± D 2 k1k3 λ1,2 = with D = (k2 + k4) − 4 k2k4 − . (3.34) 2 α1α2

Symbolic calculations show that the equilibrium point at (0, 0) is a stable node for k1k3 < 1 and a saddle point for k1k3 > 1, see Table 3.2. Similarly, the proce- α1α2k2k4 α1α2k2k4 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 parameter 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

1 ¡ ¢ V = (x − xeq)2 + (x − xeq)2 , (3.35) 2 1 1 2 2 with inequality constraint

dV dx dx = (x − xeq) 1 + (x − xeq) 2 < 0, (3.36) dt 1 1 dt 2 2 dt

eq x1 x1 evaluated at equilibrium point, eq x2 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

stablenodein q1 stablenodein q2 1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

(a) (b)

stablenodein q3 stablenodein q4 1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Figure 3.8 – Nullclines (solid lines) and vector ﬁelds of the trajectories. Filled circle: stable node; open circle: saddle point. The vector ﬁeld is obtained by simulating the PWA model in 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 other max max parameter values were in all simulations: α1 = α1 = 0.5, x1 = x2 = 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. T A trajectory in q1, directed towards q2, crosses x1 = α2 with normal n1→2 = [1, 0] . The tangent in q1,Φ1(x), is a mathematical function of the parameters and states Ã ! k1x2 −k2x1 + α1 Φ1(x) = . (3.37) k3x1 − k4x2 α2

One can deduce from Eqs. 3.22 that trajectories in q1 directed towards q3 are only present if k1x2 > x1. (3.38) α1k2 T The same procedure has been applied to switching plane x2 = α1 with n1→3 = [0, 1] , which resulted in the inequality constraint

k3x1 > x2. (3.39) α2k4

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)

k1 Γ1 = > α2, (3.41) k2 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

∗ ∗ k1 Γ = k2α2 > 0 and Γ = < α2 for p1 and p3 , respectively. The asterisk indicates the 0 1 k2 ∗ ∗ complementary set, i.e., Γ = NOT(Γ). As Γ0 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 ⇒ never ∗ Γ0 k2α2 > 0 ⇒ always k1 Γ1 α2 < k2 ∗ k1 Γ α2 > 1 k2 k3 Γ2 α1 < k4 ∗ k3 Γ α1 > 2 k4

Table 3.5 – Mode transitions at the vertices of the artiﬁcial biochemical network and corresponding 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,Γ0,Γ1,Γ1,Γ2 and Γ2. 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 Γ1 and Γ2, 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 demonstrate 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 dynamics. Stability analysis might be performed by exploiting the monotonicity of nonlinear

36 Section 3.4

* p Γ1 x 5 x q q4 q q 3 3 Γ1 4 x2 x 2 Γ2 Γ2 * p2 p3 p4 Γ1 α1xx x α1x x x Γ1 n12 * * * Γ0 Γ2 Γ2

q1 p1 q q q x 2 1 x 2 0 α 0 * 2 Γ0 α2 x1 x1 (a) (b)

Figure 3.9 – (a) Normal vector of switching plane, directed from q1 towards q2. The vertices of this plane are marked with ×. (b) Complete transition graph of the artiﬁcial biochemical network. The arrows show the direction of the trajectories at all vertices, the constraints are 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 procedure 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 dynamics 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 physiologically relevant biochemical networks as well. In the next chapter, this method will be

37 Chapter 3

x x q3 q3

x x x x x x

q2 q x x 2

(a) (b)

x x q3 q3

x x x x x x

q2 q x x 2

∗ ∗ Figure 3.10 – All possible transitions schemes of the model for inequality sets (a) Γ1 ∧ Γ2, ∗ ∗ (b) Γ1 ∧Γ2, (c) Γ1 ∧Γ2, and (d) Γ1 ∧Γ2. We remark that these transition graphs qualitatively correspond with the vector ﬁelds in Fig. 3.8.

38 Section 3.4

demonstrated for model of the TGF-β1 pathway.

39 Chapter 3

40 4 Analysis of the Transforming Growth Factor-β1 pathway

This chapter was based on an extended version of paper [120]

The Transforming Growth Factor-β1 (TGF-β1) pathway plays a crucial role in proliferation, 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 mentioned 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 degradation [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 formation [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 become 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 pathway. 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

extracellularspace TGF- β1

latent TGF- β1 TGF- β1

RI/RII RI RII SMURF2/ I-SMAD

TGF- β TGF- β1 1 R-SMAD

RI/RII RI/RII SMURF2/ SMURF2/ I-SMAD I-SMAD

R-SMAD P

I-SMAD SMAD- co-SMAD SMURF2 P complexR-SMAD P co-SMAD

SMAD- co-SMADcomplex I-SMAD R-SMAD SMAD- co-SMAD P complexR-SMAD P

Substrate P Phosphor Reaction Enzymatic stimulation Degradation Genetic Receptor Inhibition products regulation

Figure 4.1 – Schematical representation of the TGF-β1 pathway in the vascular endothelial cell.

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

INPUT u

TGF- β 1 extracellularspace

RI/RII

y1 k1 y2 R-SMAD k3 k k2 4

y7 SMAD- I-SMAD co-SMADcomplex y3 k8 P R-SMAD P k9 k5

k 8 SMAD- co-SMADcomplexR-SMAD y5 I-SMAD y6 SMAD- k7 co-SMAD k6 P complexR-SMAD P y4

Substrate P Phosphor Reaction Enzymatic stimulation Genetic Receptor Degradation Inhibition products regulation

Figure 4.2 – Interaction graph of the TGF-β1 pathway with the focus on the R-SMAD loop.

45 Chapter 4

PhosphorylatedR-SMAD 1.5

0.5

Expression(arbitraryunits) 0 0 100 200 300 400 500 600 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].

0.25

0.2

0.15 peak=oscillations

magnitude 0.1

0.05

0 0 0.05 0.1 0.15 0.2 0.25 frequency(Hz)

Figure 4.4 – The peak in the graph shows that the experimental data in Fig. 4.3 can exhibit oscillatory behavior.

46 Section 4.2

TGF-β1 pathway consists of seven coupled ordinary diﬀerential equations [120]:

dy1 k2y1y7 = u − k1y1 − , (4.1) dt Km1 + y1 dy2 k4y1y2 = k3y5 − ³ ´ , (4.2) dt y7 Km 1 + + y2 2 KI

dy3 k4y1y2 = ³ ´ − k5y3, (4.3) dt y7 Km 1 + + y2 2 KI dy 4 = k y − k y , (4.4) dt 5 3 6 4 dy 5 = k y − k y , (4.5) dt 6 4 3 5 r dy6 k7y4 = r r − (k8 + k9) y6, (4.6) dt Km3 + y4 dy 7 = k y − k y , (4.7) dt 9 6 8 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 dynamics 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, respectively. A distinction between slow and fast varying rates on basis of a priori knowledge is made by dividing the large rates by ², given that ² ¿ 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

dy k0 y k y y 2 = 3 5 − ³ 4 1 2´ , (4.12) dt ² y7 Km 1 + + y2 2 KI dy k0 y 5 =k y − 3 5 . (4.13) dt 6 4 ²

The dynamics of yuRS are

dyuRS k4y1y2 = k6y4 − ³ ´ . (4.14) dt y7 Km 1 + + y2 2 KI

0 0 k3y5 y5 k3y5 As ² À k6y4, Eq. 4.13 can be rewritten as dt ≈ − ² , preserving the slow dynamics

0 dy5 k3y5 Eq. 4.8 lim = − ⇒ y5 = 0 −−−−→ yuRS = y2. (4.15) ²→0 dt ²

Combining Eqs. 4.14 and 4.15 results in a reduced expression for the dynamics of yuRS

dyuRS dy2 k4y1y2 ≈ = k6y4 − ³ ´ . (4.16) dt dt y7 Km 1 + + y2 2 KI

dypRS dytIS Similarly, expressions for the slow dynamics of dt and dt can be derived

dypRS dy4 k4y1y2 ≈ = ³ ´ − k6y4, (4.17) dt dt y7 Km 1 + + y2 2 KI

49 Chapter 4

extracellularspace

I-SMAD

x7 k4

k8 SMAD- k6 co-SMAD P complexR-SMAD P x4

P Phosphor Reaction Substrate Degradation Inhibition products

Figure 4.5 – The components and their interactions of the reduced model of the TGF-β1 pathway.

r dytIS dy7 k8y4 ≈ = r r − k9y7. (4.18) dt dt Km3 + y4

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. y4 The reduced model is converted into a dimensionless system by deﬁning x4 = and ytRS y7 x7 = . ytIS

dx4 ∗ 1 − x4 ∗ = k4 ³ ´ − k6x4, (4.19) dt ∗ x7 Km 1 + ∗ + 1 − x4 2 KI r dx7 ∗ x4 ∗ = k8 r r − k9x7. (4.20) dt Km3 + x4

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

k4(1 − x4) f1(x4, x7) = ³ ´, (4.21) x7 1 − x4 + Km 1 + 2 KI r k8x4 f2(x4) = r r , (4.22) Km3 + x4 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   x7 α1 k4(1 − ) if x4 − x7 ≤ 1 − α1 ∧ x7 < α2,  α2 α2 k4 α1 ϕ1(x4, x7) = (1 − x4) if x4 − x7 > 1 − α1 ∧ x7 < α2, (4.23)  α1 α2  0 if x7 ≥ α2.

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   0 if x4 < Km , ϕ (x ) = 3 (4.24) 2 4  k8 if x4 ≥ Km3 .

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 dx resulting PWA model dt = Φ(x) consists of maximally six discrete modes (q1, . . . , q6) of dx4 dx7 two linear state equations ( dt and dt ), in matrix form given by

dx Φ (x) = = A x + B , (4.25) i dt i i

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 µ ¶ µ ¶ k4 −k6 − α k4 q1 A1 = 2 B1 = 0 −k9 0 µ ¶ µ ¶ k4 k4 −(k6 + α ) 0 α q2 A2 = 1 B2 = 1 0 −k9 0 µ ¶ µ ¶ −k6 0 0 q3 A3 = B3 = 0 −k9 0 µ ¶ µ ¶ k4 −k6 − α k4 q4 A4 = 2 B4 = 0 −k9 k8 µ ¶ µ ¶ k4 k4 −(k6 + α ) 0 α q5 A5 = 1 B5 = 1 0 −k9 k8 µ ¶ µ ¶ −k6 0 0 q6 A6 = B6 = 0 −k9 k8

The invariants of the six modes are

α1 if q1 : x4 − x7 ≤ 1 − α1 ∧ x7 < α2 ∧ x4 < Km3 , (4.26) α2 α1 if q2 : x4 − x7 > 1 − α1 ∧ x7 < α2 ∧ x4 < Km3 , (4.27) α2

if q3 : x7 ≥ α2 ∧ x4 < Km3 , (4.28) α1 if q4 : x4 − x7 ≤ 1 − α1 ∧ x7 < α2 ∧ x4 ≥ Km3 , (4.29) α2 α1 if q5 : x4 − x7 > 1 − α1 ∧ x7 < α2 ∧ x4 ≥ Km3 , (4.30) α2

if q6 : x7 ≥ α2 ∧ x4 ≥ Km3 . (4.31)

α1 The switching planes are x4 − x7 = 1 − α1 and x4 = Km3. α2

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

x p7

q3 q6 x7

p4 p5 p6 α2 x x x q4 q1 p3 x q5 p1 q p2 x 2x 1-α1 Km3 x4

Figure 4.6 – Modes of the TGF-β1 model. The switching planes divide the model in six modes. The vertices of the switching planes are numbered (p1, . . . , p7) and indicated with crosses.

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)

T k4 with x = [x4, x7] , which leads to (x4, x7) = ( , 0). As the equilibrium is located in q1, it k6 α1 has to satisfy the invariant x4 − x7 ≤ 1−α1 ∧x7 < α2 ∧x4 < Km and this consequently α2 3 k4 leads to the existence condition k6 ≥ . This procedure is repeated for al modes and 1−α1 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

k4(1 − Km3) k4k8 k6 < ∧ k9 ≤ , (4.33) α1Km3 α2(k4 − k6Km3) 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

stablenodein q1 stablenodein q2 1 1

0.8 0.8 x7 x7 0.6 0.6 α2 α2 0.4 0.4

0.2 0.2

0 0 0 0.21-α1 0.4Km30.6 0.8 1 0 0.21-α1 0.4Km30.6 0.8 1 x4 x4 (a) (b)

stablenodein q4 stablenodein q5 1 1

0.8 0.8 x7 x7 0.6 0.6 α2 α2 0.4 0.4

0.2 0.2

0 0 0 0.21-α1 0.4Km30.6 0.8 1 0 0.21-α1 0.4Km30.6 0.8 1 x4 x4 (c) (d)

Figure 4.7 – Vector ﬁeld of the PWA model with parameter values that satisfy the existence conditions 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 were computed with parameter values α1 = 0.75, α2 = 0.5, and Km3 = 0.5. The direction of the solution trajectories are visualized with arrows.

54 Section 4.2

limitcycle

0.8 x7

0.6 α2 0.4

0.2

0 0 1-0.2α1 0.4Km30.6 0.8 1 x4 (e)

¿ continuation of Figure 4.7 À (e) Phase plane of the PWA model with parameter values that satisfy the existence conditions for a limit cycle: k4 = 0.8, k6 = 0.6, k8 = 0.5 and k9 = 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 1 eq 2 eq 2 GUAS. Let the Lyapunov function be defined as V = 2 ((x4 − x4 ) + (x7 − x7 ) ), with eq eq dV eq dx4 x4 and x7 : equilibrium value for x4 and x7, respectively. Hence, dt = (x4 − x4 ) dt + eq dx7 (x7 − x7 ) dt < 0 is required to guarantee stability. For q1, this inequality can be solved eq eq k4 given the equilibrium point of q1:(x , x ) = ( , 0). This condition is always satisfied in 4 7 k6 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 ³ ´ k4 k4 q1 , 0 k6 ≥ k6 1−α1 ³ ´ k4 k4(1−Km3) k4 q2 , 0 ≤ k6 < k4+α1k6 α1Km3 1−α1

q3 (0, 0) invalid ³ ´ k4(α2k9−k8) k8 k4(1−Km3) k4k8 k8(k4+α1k6) q4 , k6 < ∧ < k9 ≤ α2k6k9 k9 α1Km3 α2(k4−k6Km3) α2(k4+k6(α1−1)) ³ ´ k4 k8 k4(1−Km3) k8(k4+α1k6) q5 , k6 < ∧ k9 > k4+α1k6 k9 α1Km3 α2(k4+k6(α1−1)) ³ ´ k8 q6 0, invalid k9

4.2.5 Transition Analysis

Proposition 4.2.3. The TGF-β1 model contains 14 mode transitions for which 14 different sets of constraints are valid.

The transition q1 → q2, for example, is only feasible if the solution of Φ1(x) in

α1 Eq. 4.25 traverses the switching plane x4 − x7 = 1 − α1 from q1 → q2 with normal α2 α1 T n1→2 = [1 − ] The dot product of n1→2, and Φ1(x) yields the inequality k4 − k6x4 − α2 k4−α1k9 ( )x7 > 0. Subsequent substitution of the vertex coordinates p1 = (1 − α1, 0) α2 α2(Km3+α1−1) k4 and p3 = (Km3, ) in this inequality gives Γ1 : k6 < and Γ2 : k6 < α1 1−α1 k4(1−Km3)+α1k9(Km3+α1−1) , respectively. This procedure is repeated for all transitions, for α1Km3 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

∗ ∗ ∗ k4 seven complementary sets (Γ , ··· , Γ ). For example, Γ is described by k6 > . One 0 6 1 1−α1 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

k4 Γ1 k6 < 1−α1

k4(1−Km3)+α1k9(Km3+α1−1) Γ2 k6 < α1Km3

k4 Γ3 Km3 < k4+α1k6

k8 k4(1−Km3) α1(k9− )+k9(Km3−1)+ α2 α1 Γ4 k6 < Km3

k8 Γ5 k6 < α1(k9 − ) α2

k8 Γ6 α2 < 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

stablenodeinmode q1

{Γ0*,}Γ6 {Γ0*,}Γ6*

Γ0* Γ * x 0 x

x {Γ ,}Γ * {Γ *,}Γ {Γ *,}Γ * Γ0* 5 6 5 6 5 6

q3 q6 x7 x x x

x q4 {Γ2,}Γ4 {Γ2,Γ4*} {Γ2*,Γ4*} Γ0* q1 q5 Γ3* Γ * x 3 x x Γ1* q Γ3* x 2x Γ * 3 x4

9combinationsofinequalities:

{ΓΓΓΓΓΓΓ0*,,,,,,}1* 2 3* 45 6* {ΓΓΓΓΓΓΓ0*,,,,,,}1* 2 3* 4**5 6* {ΓΓΓΓΓΓΓ0*,,,,,,}1* 2 3* 4 5* 6 {ΓΓΓΓΓΓΓ0*,,,,,,}1* 2* 3* 4* 5 6* * * * * * {ΓΓΓΓΓΓΓ0 ,,,,,,}1 2 3 4 5 6 {ΓΓΓΓΓΓΓ0*,,,,,,}1* 2* 3* 4**5 6 * * * * * {ΓΓΓΓΓΓΓ0 ,,,,,,}1 2 3 4 5 6 {ΓΓΓΓΓΓΓ0*,,,,,,}1* 2* 3* 4**5 6* {ΓΓΓΓΓΓΓ0*,,,,,,}1* 2 3* 4**5 6

(a)

Figure 4.8 – (a) Transition graph of all possibilities that satisfy the constraints for an equilibrium point in mode q1. The location of the mode containing the equilibrium point is colored gray. The sets between curly brackets, for example {Γ1, Γ2} is shorthand notation for Γ1 ∧ Γ2. If a vertex has multiple mode transitions, the various possibilities are displayed.

58 Section 4.2

stablenodeinmode q2

{Γ0*,}Γ6 {Γ0*,}Γ6*

Γ0* Γ * x 0 x

x * {Γ *,}Γ {Γ *,}Γ * Γ0* {Γ5,}Γ6 5 6 5 6

q3 q6 x7 x x x

x q4 {Γ2,}Γ4 {Γ2,}Γ4* {Γ2*,}Γ4* Γ0* q1 q5 Γ * Γ3* 3 x x x q Γ3* x 2x Γ1 Γ3* x4

9combinationsofinequalities:

{ΓΓΓΓΓΓΓ0*,,,,,,}1 2 3* 45 6* {ΓΓΓΓΓΓΓ0*,,,,,,}1 2 3* 4**5 6* {ΓΓΓΓΓΓΓ0*,,,,,,}1 2 3* 4 5* 6 {ΓΓΓΓΓΓΓ0*,,,,,,}1 2* 3* 4* 5 6* * * * * {ΓΓΓΓΓΓΓ0 ,,,,,,}1 2 3 4 5 6 {ΓΓΓΓΓΓΓ0*,,,,,,}1 2* 3* 4**5 6 * * * * {ΓΓΓΓΓΓΓ0 ,,,,,,}1 2 3 4 5 6 {ΓΓΓΓΓΓΓ0*,,,,,,}1 2* 3* 4**5 6* {ΓΓΓΓΓΓΓ0*,,,,,,}1 2 3* 4**5 6

(b)

¿ continuation of Figure 4.8 À (b) Transition graph of all possibilities that satisfy the constraints for a stable node in mode q2.

59 Chapter 4

stablenodeinmode q4

x {Γ ,}Γ * {Γ *,}Γ * Γ0* 5 6 5 6

q3 q6 x7 x x

Γ0* x x q4 {Γ2,}Γ4 {Γ2,}Γ4* Γ0* Γ6* q1 q5 Γ3 Γ3 x x q x 2x Γ1 Γ3 x4

4combinationsofinequalities:

{ΓΓΓΓΓΓΓ0*,,,,,,}1 2 3 45 6* {ΓΓΓΓΓΓΓ0*,,,,,,}1 2 3 4* 5 6* {ΓΓΓΓΓΓΓ0*,,,,,,}1 2 3 4 5* 6* {ΓΓΓΓΓΓΓ0*,,,,,,}1 2 3 4**5 6*

(c)

¿ continuation of Figure 4.8 À (c) Transition graph of all possibilities that satisfy the constraints for a stable node in mode q4.

60 Section 4.2

stablenodeinmode q5 x {Γ ,}Γ * {Γ5*,}Γ6* Γ0* 5 6

q3 q6 x x7 x Γ * x 0 x q4 * {Γ2,}Γ4 {Γ2,}Γ4* {Γ2*,}Γ4* Γ0* Γ6 q1 q5 Γ3 Γ3 x x x q Γ3 x 2x Γ1 Γ3 x4

6combinationsofinequalities:

{ΓΓΓΓΓΓΓ0*,,,,,,}1 2 3 45 6* {ΓΓΓΓΓΓΓ0*,,,,,,}1 2 3 4**5 6* {ΓΓΓΓΓΓΓ0*,,,,,,}1 2 3 4 5* 6* {ΓΓΓΓΓΓΓ0*,,,,,,}1 2* 3 4* 5 6* * * * {ΓΓΓΓΓΓΓ0 ,,,,,,}1 2 3 4 5 6 {ΓΓΓΓΓΓΓ0*,,,,,,}1 2* 3 4**5 6*

(d)

¿ continuation of Figure 4.8 À (d) Transition graph of all possibilities that satisfy the constraints for a stable node in mode q5.

61 Chapter 4

limitcycle

{Γ0*,}Γ6 {Γ0*,}Γ6*

Γ0* Γ * x 0 x

x * {Γ *,}Γ {Γ *,}Γ * Γ0* {Γ5,}Γ6 5 6 5 6

q3 q6 x7 x x x

x q4 {Γ2,}Γ4 {Γ2,}Γ4* {Γ2*,}Γ4* Γ0* q1 q5 Γ3 Γ3 x x x q Γ3 x 2x Γ1 Γ3 x4

9combinationsofinequalities:

{ΓΓΓΓΓΓΓ0*,,,,,,}1 2 3 45 6* {ΓΓΓΓΓΓΓ0*,,,,,,}1 2 3 4**5 6* {ΓΓΓΓΓΓΓ0*,,,,,,}1 2 3 4 5* 6 {ΓΓΓΓΓΓΓ0*,,,,,,}1 2* 3 4* 5 6* * * * {ΓΓΓΓΓΓΓ0 ,,,,,,}1 2 3 4 5 6 {ΓΓΓΓΓΓΓ0*,,,,,,}1 2* 3 4**5 6 * * * {ΓΓΓΓΓΓΓ0 ,,,,,,}1 2 3 4 5 6 {ΓΓΓΓΓΓΓ0*,,,,,,}1 2* 3 4**5 6* {ΓΓΓΓΓΓΓ0*,,,,,,}1 2 3 4**5 6

(e)

¿ continuation of Figure 4.8 À (e) Transition graph of all possibilities that satisfy the constraints 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 equilibrium point. To illustrate this, consider an equilibrium point in mode q1. The existence

k4 condition for an equilibrium point in q1 is k6 ≥ , see Table 4.3. The vertex p1 of 1−α1 k4 the switching plane between q1 and q2 has to obey the inequality k6 < , which is 1−α1 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 transition 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 Γ0 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 continuous (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 approximate the Hill function with two piecewise-constant segments instead of the three-segment PWA approach.

64 5 Signal 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 (conformation), 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 oxygen 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 secretion [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 residues, 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 triggers 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 transcription 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 dephosphorylation 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

UPR

r e ce UPR ptor

signal BiP transduction ribosome anslation tr BiP BiP + ATP aminoacids

UPR

GM N-glycosylateduPr

Erp5

CNX I D 7 P aggregate + e

y f

o c

l d g i

n GM c o

r r e c tl y f olded correctly G foldedprotein

+ ATP RAD E UPR

Protein degradation Golgi post-processing

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 during 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:

dx 1 = f (x ) − f (x , x ), (5.1) dt 1 4 2 1 7 dx 2 = f (x ) − f (x ), (5.2) dt 3 1 4 2 dx 3 = −f (x , x ) + f (x , x ), (5.3) dt 5 2 3 6 4 5 dx 4 = f (x , x ) − f (x , x ), (5.4) dt 5 2 3 6 4 5 dx 5 = f (x ) − f (x ), (5.5) dt 7 4 8 5 dx 6 = f (x ) − f (x ), (5.6) dt 9 1 10 6 dx 7 = f (x ) − f (x ), (5.7) dt 10 6 11 7 x1 uPr, x2 PERK, x3 unphosphorylated eIF2α, x4 phosphorylated eIF2α, x5 GADD34, x6 ATF6, x7 spliced ATF6.

69 Chapter 5

f f2 1 unfolded protein f3 x1 f9 f4 x2 PE RK ATF6 x6

f5 f10

eIF2α x3 eIF2α x4 P ATF6 x7 f 6 f11 f8 f7 GADD34

Substrate P Phosphor Reaction Enzymatic stimulation Spliced Degradation Delayed Transcriptional substrate products translation inhibition

Transcriptionalcontrol

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; the inﬂuence of attenuation of translation by PERK and the transcriptional activation of BiP by ATF6 are represented by inhibition of uPr formation and promotion of uPr degradation, respectively.

70 Section 5.4

The corresponding rates are

k1KI f1(x4) = , (5.8) KI + x4 k2x1x7 f2(x1, x7) = , (5.9) Km1 + x7

f3(x1) = k3x1, (5.10)

f4(x2) = k4x2, (5.11)

k5x2x3 f5(x2, x3) = , (5.12) Km2 + x2 k6x4x5 f6(x4, x5) = , (5.13) Km3 + x5

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

dx5 Therefore a constant rate of eIF2α dephosphorylation was assumed. Thus dt = 0 adapted ] and f6 (x4) = k6x4.

• 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 k2.

• 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

∗ ∗ ∗ ∗ ∗ ∗ dx1 k1KI k2x1x7 = ∗ ∗ − ∗ ∗ (5.19) dt KI + x4 Km1 + x7 ∗ ] ∗ ∗ dx4 k5x1(1 − x4) ] ∗ = ∗ ∗ − k6x4 (5.20) dt Km2 + x1 dx∗ 7 = k] x∗ − k∗ x∗ (5.21) dt 2 1 11 7

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.,

adapted k1KI f1 (x4) = , (5.22) KI + x4 adapted k2x1x7 f2 (x1, x7) = , (5.23) Km1 + x7 adapted k5x1(1 − x4) f5 (x1, x4) = , (5.24) Km2 + x1

2Remark that the intermediate steps of this model reduction has been left out as the same procedure described in chapter 4 is followed.

72 Section 5.5

p x 16

p11 p x x 12 p 13x p p5 15 x x x7 = α 2

p14 x p10 x p 9 p p6 x 4 x x x p7 x p3 x x4 = α 1 p8 x p2 p1 x x1 = α 3

Figure 5.3 – Phase space of the PWA approximation of the UPR model. The phase space is divided by three switching planes: x4 = α1 (dash), α2x1 − x7 = 0 (dot), and x1 + α3x4 = α3 (dash-dot). The vertices of the various modes are marked with crosses and numbered (p1, . . . , p16).

73 Chapter 5 which were reformulated as the following PWA functions   x4 k1(1 − ) if x4 < α1, ϕ (x ) = α1 (5.25) 1 4  0 if x4 ≥ α1,   k2x1 if α2x1 − x7 < 0, ϕ2(x1, x7) = (5.26)  k2x7 , if α2x1 − x7 ≥ 0,  α2  k5x1 if x1 + α3x4 < α3, ϕ (x , x ) = α3 (5.27) 3 1 4  k5(1 − x4) if x1 + α3x4 ≥ α3, 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 equilibrium 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 Ãµ ¶ k9 k5 α2 < ∧ α1 ≤ ∨ k11 k5 + k6 µ ¶! (5.28) k5 k1(α1(k5 + k6) − k5) α1 > ∧ α3 > , k5 + k6 α1k2k6

Ãµ ¶ k9 k5 α2 > ∧ α1 ≤ ∨ k11 k5 + k6 µ ¶! (5.29) k5 α2k1k11(α1(k5 + k6) − k5) α1 > ∧ α3 > , k5 + k6 α1k2k6k9

k9 k5 k1(α1(k5 + k6) − k5) α2 < ∧ α1 > ∧ α3 < , (5.30) k11 k5 + k6 α1k2k6

k9 k5 α2k1k11(α1(k5 + k6) − k5) α2 > ∧ α1 > ∧ α3 < , (5.31) k11 k5 + k6 α1k2k6k9 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     k1 −k2 − 0 k α1 1 k5 q1 A1 =  −k6 0  B1 =  0  α3 k9 0 −k11 0     −k2 0 0 0 k5 q A =  −k6 0  B = 0 2 2 α3 2 k9 0 −k11) 0     0 − k1 − k2 k α1 α2 1 k5 q3 A3 =  −k6 0  B3 =  0  α3 k9 0 −k11 0     0 0 − k2 0 α2 k5 q4 A4 =  −k6 0  B4 = 0 α3 k9 0 −k11 0     k1 −k2 − 0 k α1 1 q5 A5 =  0 −(k5 + k6) 0  B5 = k5 k9 0 −k11 0     −k2 0 0 0 q6 A6 =  0 −(k5 + k6) 0  B6 = k5 k9 0 −k11 0     0 − k1 − k2 k α1 α2 1 q7 A7 =  0 −(k5 + k6) 0  B7 = k5 k9 0 −k11 0     0 0 − k2 0 α2 q8 A8 =  0 −(k5 + k6) 0  B8 = k5 k9 0 −k11 0

75 Chapter 5

Mode q1 Mode q2

(a) (b)

Mode q3 Mode q4

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

(e) (f)

Mode q7 Mode q8

(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.

T T Three switching planes with normal vectors n1→2 = [0, 1, 0] , n1→3 = [α2, 0, −1] T and n1→5 = [1, α3, 0] divide the phase space into eight different modes and 16 vertices (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 constraint. For example, Γ1 ⊂ Γ2 means that Γ1 is only valid if Γ2 is satisﬁed. This implies ∗ ∗ ∗ that {Γ1, Γ2}, {Γ1, Γ2}, and {Γ1, Γ2} are possible combinations. Other subsets are:

∗ ∗ ∗ Γ5 ⊂ Γ1 ⇒ {Γ5, Γ1}, {Γ5, Γ1}, {Γ5, Γ1}, (5.32) ∗ ∗ ∗ Γ5 ⊂ Γ2 ⇒ {Γ5, Γ2}, {Γ5, Γ2}, {Γ5, Γ2}, (5.33) ∗ ∗ ∗ Γ4 ⊂ Γ3 ⇒ {Γ3, Γ4}, {Γ3, Γ4}, {Γ3, Γ4}. (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 ´ 9 k 6 k 2 9 k k 3 6 k α 1 1 k α 3 + α 2 11 α k 1 5 ´ k α ´ ) 1 ) 6 k 5 k 2 k 2 α k 9 − ) , 3 ´ ) k 6 6 ) 6 9 α k 5 k k 1 k + k 1 6 ) α + 5 k 6 k 5 − k + 3 2 k ) k ( 5 k 6 ( α 2 + 11 k 3 1 1 k 5 k k 1 α α α 1 k 5 + k ( 1 ( ( 5 k α 1 α 1 k 11 k 11 ( k + 2 k ) k 1 2 2 7 α α 11 , α k ( k , 1 1 6 9 5 , x 6 k α α k k 2 4 k 1 1 5 k k + k k 5 3 5 2 , x , k α k 1 1 α 6 1 k , k x , α 5 1 ) + 9 k + 5 α 5 k 5 k k 6 k − k 1 , ) ) 2 k 6 ) 11 6 k 5 k k k , 3 k 6 + ) 6 α + k 5 6 − k 1 5 1 ) k k 2 k α k 6 ( 6 ( k + 3 9 k k 1 + 3 5 k 1 α α α + k 2 2 11 k ( 1 5 ( k 3 k α 2 k α 1 – Equilibrium points in the modes of the UPR model. 5 11 1 α ( k k k α 1 1 + 1 α 1 1 5 α α α k k k ( 2 2 1 1 k α k α ³ invalid ³ invalid ³ invalid ³ invalid Table 5.3 1 2 3 4 5 7 8 6 Mode Equilibrium point ( q q q q q q q q

79 Chapter 5

Table 5.4 – Inequality sets that determine the transitions.

Name Inequality set

Γ0 k1 < 0 ⇒ infeasible

α3 Γ1 α2 > k1

Γ2 α3 < k1 ³ ´ α3−k5 Γ3 α1 < ∧ (k5 < 1 ∧ α3 < k5) ∨ (k5 > 1 ∧ α3 < 1) α3−k5−k6

k5−1 Γ4 k5 > 1 ∧ α1 < k5+k6 1 Γ5 α2 > k1

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 0 16 p × Γ 0 15 p Γ ×× × × × 0 0 0 0 14 p Γ Γ Γ Γ × × × 4 13 p × ×× × × × × × × Γ 0 0 0 3 12 p Γ Γ Γ Γ 0 11 p × × × × × × Γ × ×× × × × × × × × 0 4 0 4 10 p × × × × × × × × × ×× × × × × ×Γ × × × × × × × × ×× × × × × Γ Γ Γ × × × × 0 0 0 0 0 0 0 0 3 0 3 0 9 p Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ 4 8 p × × × Γ 0 0 0 3 7 p × × × × × × Γ Γ Γ Γ 0 0 0 0 6 p Γ × × × Γ Γ Γ 2 5 p Γ × × × × 5 4 p × × × × × × Γ 1 2 2 1 3 p Γ Γ Γ Γ 2 2 p × Γ – Transitions at the vertices and corresponding parameter constraints. ∗ 0 1 p ×Γ ×× ×× × ×× × × × × ×× × × ×× × × × × × × ×× × × × ×× × × × ×× × × × × × × × × × × × × × × × × × × × × 5 7 8 3 4 8 2 6 6 7 8 4 Table 5.5 q q q q q q q q q q q q → → → → → → → → → → → → 1 3 4 1 2 6 1 2 5 5 7 3 q q q q q q Transition q q q q q q

81 Chapter 5

{Γ0*,}ΓΓΓΓΓ1,,,, 2 3 4 5

{Γ0*,}ΓΓΓΓΓ1,,,, 2 3 4 5*

q2 q6

q1 q5

q4 q8

q3 q7

(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 Γ3∧Γ4 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*,}ΓΓΓΓΓ1,, 2 3*, 4*, 5 {Γ0*,}ΓΓΓΓΓ1*,,,, 2 3 4 5* {Γ0*,}ΓΓΓΓΓ1,, 2 3*, 4*, 5*

q2 q6 q2 q6

q1 q5 q1 q5

q4 q8 q4 q8

q3 q7 q3 q7

(b) (c)

{Γ0*,}ΓΓΓΓΓ1,,, 2 3 4*, 5 {Γ *,}ΓΓΓΓΓ*, *,,, * {Γ0*,}ΓΓΓΓΓ1,,, 2 3 4*, 5* 0 1 2 3 4 5

q2 q6 q2 q6

q1 q5 q1 q5

q4 q8 q4 q8

q3 q7 q3 q7

(d) (e)

¿ continuation of Fig. 5.5 À The gray arrows emphasize the change in transition directions compared to Fig. 5.5(a).

83 Chapter 5

{Γ0*,}ΓΓΓΓΓ1*,, 2 3*, 4*, 5* {Γ0*,}ΓΓΓΓΓ1*,,, 2 3 4*, 5*

q2 q6 q2 q6

q1 q5 q1 q5

q4 q8 q4 q8

q3 q7 q3 q7

(f) (g)

* * * * * * {Γ0*,}ΓΓΓΓΓ1*, 2*,, 3 4*, 5* {Γ0 ,}ΓΓΓΓΓ1, 2, 3, 4, 5

q2 q6 q2 q6

q1 q5 q1 q5

q4 q8 q4 q8

q3 q7 q3 q7

(h) (I)

¿ continuation of Fig. 5.5 À The gray arrows emphasize the change in transition directions compared to Fig. 5.5(a).

84 Section 5.6

100

40 Tg Tm 20

% PERK phosphorylated DTT 0 0 1 2 3 4 5 time (hours) (a)

100

40 Tg

% cleaved ATF6 Tm 20 DTT 0 0 1 2 3 4 5 time (hours) (b)

Figure 5.6 – Expression proﬁles of (a) phosphorylated PERK (indication of phosphorylated eIF2α, x4) and (b) cleaved ATF6 (x7) [36]. Alternate types of ER stress were applied for each branch of the UPR. Thapsigargin (Tg): knocks out the energy requirements for protein folding; tunicamycin (Tm) inhibits protein glycosylation; and dithiothreitol (DTT) disrupts or 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 mathematical 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 qualitatively 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 analysis 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 6 System Identiﬁcation with Parameter Constraints

So far qualitative information has been used to analyze the system dynamics. However, qualitative information is not restricted to system analysis alone. In this chapter, qualitative 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 describes the genetically regulated enzymatic conversion of a substrate into a product, see Fig. 6.1:

dx1 k1KI = r r − k2x1, (6.1) dt KI + x3 dx 2 = k x − k x , (6.2) dt 3 1 4 2 dx 3 = k x − k x , (6.3) dt 5 2 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

substrate

DNA mRNA enzyme

product

Figure 6.1 – Model of enzymatic substrate conversion, as proposed by Goodwin [54]. The ﬁrst 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 mRNA transcription of the enzyme (bar). Note that the degradation of mRNA, enzyme and product are 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 k1KI = − k2x1, (6.4) dt KI + x3 dx 2 = k x − k x , (6.5) dt 3 1 4 2 dx3 k6x3 = k5x2 − , (6.6) dt Km + x3 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

σ =2.5 350 mRNA enzyme 300 product x 1 250 x2 200 x3

150

100

0 0 50 100 150 t

Figure 6.2 – Simulated experiment of the biochemical oscillator with parameter values k1 = 150, KI = 1, k2 = 0.1, k3 = 0.1, k4 = 0.1, k5 = 0.1, k6 = 10, and Km = 1.5. White noise 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:

k1KI f1(x3) = (6.8) KI + x3

k6x3 f2(x3) = . (6.9) Km + x3 These functions are approximated with two PWA functions (see chapter 3):  ³ ´  x3 k1 1 − if x3 < α1, ϕ (x ) = α1 (6.10) 1 3  0 if x3 ≥ α1,   k6x3 if x3 < α2, ϕ (x ) = α2 (6.11) 2 3  k6 if x3 ≥ α2. After normalization, these approximations yield the following set of state equations

dx Φ (x) = = A x + B , (6.12) i dt i i

∗ ∗ ∗ T 1 with x = [x1, x2, x3] of which the coeﬃcient matrices Ai and Bi are listed in Table 6.1. ∗ k1 ∗ KI ∗ The parameters of the normalized PWA approximation are k1 = max , KI = max , k2 = k2, x1 x3 max max ∗ k3x1 ∗ ∗ k5x2 ∗ k6 ∗ Km k3 = max , k4 = k4, k5 = max , k6 = max , and Km = max . A graphical representation x2 x3 x3 x3 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.

1 The additional assumption was made that α1 < α2; simulations showed that changing the order of these 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 oscillator model. Mode A-matrix B-matrix Invariant   k∗   −k∗ 0 − 1 k∗ 2 α1 1  ∗ ∗    ∗ q1 A1 =  k3 −k4 0  B1 = 0 x3 < α1 < α2 k∗ 0 k∗ − 6 0 5 α2

 ∗    −k2 0 0 0  ∗ ∗    ∗ q2 A2 = k3 −k4 0 B2 = 0 α1 < x3 < α2 k∗ 0 k∗ − 6 0 5 α2     ∗ −k2 0 0 0  ∗ ∗    ∗ q3 A3 = k3 −k4 0 B3 = 0 α1 < α2 < x3 ∗ ∗ 0 k5 0 −k6

Γ2 Γ2 x p7 p8 α Γ x max 2 max 1 x3 x3 Γ1 * * x x x3 3 q p5 x p 3 α1 p3 6 x p4 * x Γ0 max q xmax * x 2 2 x Γ0 2 p1 * x x* x2 Γ* p2 2 q1 0 * Γ0

max max * x * x1 x1 1 x1 (a) (b)

Figure 6.3 – (a) 3D phase space of a model of the biochemical oscillator. The switching planes divide the phase space in three modes (q1, q2, and q3) (b) Mode transitions with corresponding 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 ∗ max α1 k5 1.20KI x2 k5 Γ1 < ∗ < α2 k6 Km k6 ∗ ∗ max Γ2 k5 > k6 x2 k5 > k6

6.2.3 Constrained Nonlinear Parameter Estimation

Transition analysis provides the inequality constraints on the parameter values to guarantee 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 max x2 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(θˆ) can be deﬁned as

N X ³ ´2 J(θˆ) = x(k, θˆ) − y(k) , (6.13) k=1

θˆ 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 approach and 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 optimization problem

θˆ = arg min J(θ) subject to θ>0

k6 − 200k5 < 0, (6.14) 1.20K 200k I − 5 < 0. Km k6

Problem 6.14 is solved in MATLAB with the function fmincon. This parameter identification 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× ˆ 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 general, 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

200

180

160

k 140

120

estimated 100

40 real 20 traditional novel 0 true true true 0.5 k1 k1 1.5 k1

initialvalueof k1

Figure 6.4 – Parameter estimation of k1 for various initial estimates on a data set with true σ = 2.5. On the x-axis, the initial estimation is chosen as a fraction of the true value k1 . Note that the precise estimate cannot be found, which can probably be ascribed to the noise in the data set.

94 Section 6.3

6.3 Discussion

In this chapter, qualitative PWA analysis is applied to a model of a hypothetical biochemical 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 7 Hybrid 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 parameters 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:   g1(x(k), θ1) if xi(k) < α, x(k + 1) = (7.1)  g2(x(k), θ2) if xi(k) ≥ α,

n th where x ∈ R , xi ∈ R denotes the i 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:

X 2 J(α, θ1, θ2) = kx(k + 1) − g1(x(k), θ1)k + x(k)∈χ (α) 1 X 2 kx(k + 1) − g2(x(k), θ2)k . (7.4)

x(k)∈χ2(α)

The identiﬁcation problem can now be formulated as:

ˆ ˆ {α,ˆ θ1, θ2} = arg min J(α, θ1, θ2). (7.5) {α,θ1,θ2}>0

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

ˆ ˆ known {θ1, θ2} = arg min J(α , θ1, θ2), (7.6) {θ1,θ2}>0 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

q1 q2 q1 q2 1 f1() x 3 β1 1 2 -γ1 β2 2 γ2

)

f f2() x 3 3

x 2 x

() 2 -γ ( 1

1 1 2 3 β γ2 φ 1 φ 1 β2

x3 α x3 α x3 (a) (b) (c)

Figure 7.1 – Conversion of nonlinear to PWA for hybrid identiﬁcation. (a) The two nonlinear 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 the switching plane α was located at the same position for both functions. The PWA model has consequently two modes: q1 and q2. The superscript of the parameters refer to the mode number.

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 A1x + B1 if x3(t) < α, = (7.7) dt  A2x + B2 if x3(t) ≥ α,

T with x:[x1, x2, x3] ; 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 of two modes. Mode A-matrix B-matrix Invariant     1 1 −k2 0 −γ1 β1 q1 A1 =  k3 −k4 0  B1 =  0  x3 < α 1 1 0 k5 −γ2 −β2     2 2 −k2 0 −γ1 β1 q2 A2 =  k3 −k4 0  B2 =  0  x3 ≥ α 2 2 0 k5 −γ2 −β2

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:

1 1 1 1 θ1 = [β1 , γ1 , β2 , γ2 , k2, k3, k4, k5], (7.8)

2 2 2 2 θ2 = [β1 , γ1 , β2 , γ2 , 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

400 x1 200

300 200 x2 100 0

200

x3 100 0 0 50 100 150 t (a)

400 x 1 200

300 200 x2 100 0

200

100 x3 0 0 50 100 150 (b) 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

1 β1 55.077 55.236 ± 1.6 55.944 ± 4.5 52.035 ± 4.9 1 γ1 2.915 2.921 ± 0.2 3.097 ± 0.1 2.368 ± 0.7 1 β2 5.566 5.574 ± 0.3 5.615 ± 0.8 6.236 ± 0.9 1 γ2 0.220 0.218 ± 0.04 0.215 ± 0.1 0.095 ± 0.1 2 β1 7.502 7.607 ± 0.34 7.645 ± 0.9 7.734 ± 1.6 2 γ1 0.052 0.053 ± 0.002 0.053 ± 0.005 0.052 ± 0.01 2 β2 9.274 9.313 ± 0.1 9.289 ± 0.3 9.235 ± 0.4 2 γ2 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 approximation 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 parameter 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 8 Conclusion 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 remains 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 biological 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 biochemical networks for which experimental data are available.

8.1 Conclusions

Two research goals have been deﬁned in the introduction.

103 Chapter 8

Graphical Analysis of Bistable Systems

Nonlinear ordinary differential equations (ODEs) are used to describe the dynamical behavior 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 subsequently 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 approximated 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 approximated by piecewise-constant [12, 28, 29, 99] or, very recently, ramp functions [11]. The work in this thesis extends qualitative analysis with (multidimensional) PWA approximations; virtually all nonlinear, biochemical, deterministic models can be approximated as a collection of PWA functions, including metabolic networks and signal transduction pathways. 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 allowed 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 system 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 general [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 resulted 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 functions (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 exploiting 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 biochemical 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 ² small factor, required for the quasi-steady-state approximation η output of a system λ eigenvalue ω ν feedback strength = η σ standard deviation θ set of true parameter values θˆ 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

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125 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 accurate and sufficient quantitative data has driven systems biologists towards alternative ways to describe and analyze biochemical networks. Their focus is primarily on the analysis 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 dynamics; 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 parameter 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 responsible 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 consists 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 visualized. 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 informatie is echter niet strict noodzakelijk om het potentiëlesysteemgedrag 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 systeemanalyse 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 experimentele 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 Rhône-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 Etiënne, and the French cursing lessons are things I will never forget. Merci! I hope to welcome you and Gaëllein 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 ideeënover 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 ge¨ınspireerd 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 betrokken 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 sarcolemma 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 Rhˆone-Alpes, Saint Ismier cedex, France (funded by an NWO grant).

133