Diploma Thesis - IFATifat · Diploma Thesis IFAT-SYS 18 Diploma Thesis Investigation of Interleukin-6 classic- and trans-signaling and therapeutic treatments using set-based model

Diploma Thesis IFAT-SYS 18

Diploma Thesis

Investigation of Interleukin-6 classic- and

trans-signaling and therapeutic treatments using

set-based model analysis methods

Nadine Strobel

Supervisor: Dr.-Ing Stefan Streif

27.08.2012

Systems Theory &

Automatic Control

Institute for Automation Engineering

Laboratory for

Systems Theory and Automatic Control

Prof. Dr.–Ing. Rolf Findeisen

Contents

Abbreviations and Symbols 11

Abstract 13

Zusammenfassung 15

1. Introduction 171.1. Interleukin-6 signaling and clinical relevance . . . . . . . 171.2. Classic- vs. trans-signaling . . . . . . . . . . . . . . . . 18

1.2.1. IL-6 classic-signaling . . . . . . . . . . . . . . . . 191.2.2. IL-6 trans-signaling . . . . . . . . . . . . . . . . 201.2.3. Inhibition through Tocilizumab . . . . . . . . . . 211.2.4. Inhibition through soluble gp130 . . . . . . . . . 211.2.5. From receptor phosphorylation to cell proliferation 221.2.6. The whole network: from IL-6 stimulation to cell

proliferation . . . . . . . . . . . . . . . . . . . . 231.3. Parameter estimation . . . . . . . . . . . . . . . . . . . 25

1.3.1. A set-based approach . . . . . . . . . . . . . . . 251.3.2. A workflow for set-based estimation methods . . 25

1.4. Motivation and aim . . . . . . . . . . . . . . . . . . . . 281.5. Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2. Mathematical modeling 312.1. Experimental data . . . . . . . . . . . . . . . . . . . . . 312.2. Development of the mathematical models . . . . . . . . 34

2.2.1. Model assumptions and simplifications . . . . . . 34

3

4 Contents

2.3. Model implementation and estimation . . . . . . . . . . 372.3.1. Implementation of the models . . . . . . . . . . 372.3.2. Set-based parameter estimation . . . . . . . . . . 382.3.3. Validation and invalidation of mathematical models 392.3.4. Global parameter estimation . . . . . . . . . . . 40

3. Modeling of IL-6-signaling and set-based model analyses 433.1. Additional semi-quantitative/qualitative data . . . . . . 433.2. Modeling workflow and set-based estimation results . . . 46

3.2.1. IL-6 classic-signaling . . . . . . . . . . . . . . . . 473.2.2. Combining classic- and trans-signaling . . . . . . 51

3.3. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 59

4. Individualized therapeutic treatments with sgp130Fc 654.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 654.2. Mathematical procedure . . . . . . . . . . . . . . . . . . 664.3. Ranges of sgp130Fc to inhibit trans-signaling specifically 684.4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 70

5. Conclusions 775.1. Short summary . . . . . . . . . . . . . . . . . . . . . . 775.2. Outlook and final remarks . . . . . . . . . . . . . . . . . 78

Appendix 81A. Model equations . . . . . . . . . . . . . . . . . . . . . . 81

A.1. Model 0 . . . . . . . . . . . . . . . . . . . . . . 81A.2. Model 1Trim . . . . . . . . . . . . . . . . . . . . 82A.3. Model 1Hex . . . . . . . . . . . . . . . . . . . . 82A.4. Model 2Trim . . . . . . . . . . . . . . . . . . . . 84A.5. Model 2TrimToc . . . . . . . . . . . . . . . . . . 86A.6. Model 2TrimRed . . . . . . . . . . . . . . . . . 87A.7. Model 2-1Trim (final) . . . . . . . . . . . . . . . 87

B. Model parameters and definitions . . . . . . . . . . . . . 88C. Constraints for set-based estimation . . . . . . . . . . . 92

C.1. Data point weightening . . . . . . . . . . . . . . 92C.2. Found dependencies between model parameters

due to additional semi-quatitative/qualitative data 93D. Validation of Model 2-1Trim . . . . . . . . . . . . . . . 94

Contents 5

Bibliography 95

Eidesstattliche Erklärung

Ich versichere an Eides Statt durch meine eigenhändige Unterschrift, dassich die vorliegende Arbeit selbstständig und ohne fremde Hilfe angefer-tigt habe. Alle Stellen, die wörtlich oder dem Sinn nach auf Publika-tionen oder Vorträgen anderer Autoren beruhen, sind als solche ken-ntlich gemacht. Ich versichere außerdem, dass ich keine andere als dieangegebene Literatur verwendet habe. Diese Versicherung bezieht sichauch auf alle in der Arbeit enthaltenen Zeichnungen, Skizzen, bildlichenDarstellungen und dergleichen.

Die Arbeit wurde bisher keiner anderen Prüfungsbehörde vorgelegt undauch noch nicht veröffentlicht.

Magdeburg, 27.08.2012

7

Danksagung

Mein besonderer Dank gilt meinem Betreuer Dr.-Ing. Stefan Streif. Erstand mir während meiner Zeit als Diplomandin stets hilfreich zur Seiteund trug mit seinen kritischen Diskussionen maßgeblich zu den Ergeb-nissen dieser Arbeit bei.

Desweiteren danke ich Prof. Dr.-Ing. Rolf Findeisen für seine Er-stgutachtertätigkeit sowie Dr.-Ing. Jürgen Ihlow für seine stellvertretendeFunktion als Gutachter.

Weiterhin danke ich Prof. Dr. rer. nat. Fred Schaper undDr. rer. nat. Anna Dittrich vom Lehrstuhl für Systembiologie in Magde-burg für die vielen und hilfreichen Diskussionen sowie für die Bereitstel-lung experimenteller Daten.

Ich danke Anna, Stephie und Marianne für das Korrekturlesen meinerArbeit.

Schließlich danke ich noch meinem Verlobten Jochen für seine stetigeUnterstützung und sein großes Interesse an meiner Arbeit.

9

Abbreviations and symbols

Methods, proteins and biological components/substances

IL-6 interleukin-6gp80 glycoprotein 80gp130 glycoprotein 130sgp80 soluble glycoprotein 80sgp130 soluble glycoprotein 130Ba/F3 immortalized murine bone marrow-derived pro-B-cell lineToc TocilizumabODE ordinary differential equationJAK janus kinaseSTAT signal transducers and activators of transcriptionMAPK mitogen-activated protein kinasePBS phosphate buffered salineDMEM Dulbeccos’s modified eagle mediumFBS cell staining bufferSP scatchard plotSPR surface plasmon resonanceELISA enzyme-linked immunosorbent assayLC liver cirrhosis

Symbols

11

12 Contents

u system inputw weightening factorχ cost functionp parameter setp̂ parameter set that minimizes χ

ny amount of system outputs ynt amount of data points tyi experimental data pointyi simulated data pointσi,j standard deviationx0 individual start conditions for optimization

Others

FDA Food and Drug Administration (USA)p. phosphorylatednorm. normalizedtot. totallb lower boundub upper bound

Abstract

BackgroundInterleukin-6 (IL-6) is a key regulator of inflammatory processes. TheIL-6 receptor complex contains two subunits, glycoprotein 80 (gp80)and glycoprotein 130 (gp130). Both subunits exist as membrane-boundand soluble proteins. Classic-signaling is induced by binding of IL-6 tothe membrane bound receptors. Soluble gp80 (sgp80) induces trans-signaling which plays a crucial role in chronic inflammation and can-cer. Interestingly, soluble gp130 (sgp130) acts as an inhibitor of trans-signaling. Based on this observation a new anti-inflammatory designerprotein (sgp130Fc) was developed to inhibit trans-signaling. However,recent experimental data indicate that sgp130Fc can also inhibit classic-signaling.

QuestionsRelated to the identification of therapeutic treatments we use mathe-matical modeling and ask:

a. Which components and reactions are essential for classic- andtrans-signaling in IL-6 and sgp130Fc treated cells?

b. Can sgp130Fc concentrations be predicted for given pathophysi-ological concentration of IL-6 and sgp80 which affect specificallytrans- but not classic-signaling?

c. How can one make predictions with respect to treatments even inthe presence of uncertain experimental data and parameters?

13

14 Contents

ResultsUsing the set-based analyses methods iteratively different models couldbe ruled out as inconsistent with the uncertain experimentally determineddata. Finally, a consistent model combining classic- and trans-signalingwas identified and guaranteed outer approximations of consistent pa-rameter values were derived. The model predicts concentration rangesof sgp130Fc at pathophysiological levels of IL-6 and sgp80 at whichsgp130Fc can be used to guarantee specific blockage of trans-signaling.

ConclusionIL-6 is an important factor for inflammatory disease propagation andsgp130Fc is a promising anti-inflammatory drug. However, experimentaldata and the analysis of mathematical models underpin the hypothesisthat it not just inhibits trans- but also classic-signaling. The modelsallow predictions of potential individualized therapeutic treatment sce-narios considering specific concentration of IL6 and sgp80 as found inthe individual patient.

Kurzzusammenfassung

HintergründeInterleukin-6 (IL-6) ist ein wichtiger Regulator von Entzündungsprozessen.Der IL-6-Rezeptor besteht aus zwei Untereinheiten, Glykoprotein 80(gp80) und Glykoprotein 130 (gp130), welche jeweils membranständigoder löslich vorliegen können. Der sogenannte klassiche Signalweg wirddurch die Bindung von IL-6 an den membranständigen Rezeptor in-duziert. Lösliches gp80 (sgp80) induziert den trans-Signalweg, welchemeine bedeutende Rolle bei chronischen Entzündungsprozessen und Krebszugewiesen wird.Lösliches gp130 (sgp130) agiert als Inhibitor des trans-Signalweges.Basierend auf dieser Beobachtung wurde ein rekombinantes Protein en-wickelt (sgp130Fc), welches spezifisch den trans-Signalweg inhibiert.Eine neue Studie weist allerdings auf, dass sgp130Fc ebenso den klas-sichen Signalweg inhibieren kann.

FragestellungenAufgrund der obigen Erkenntnisse nutzen wir systembiologische Ansätzeund beantworten folgenden Fragen:

a. Welche Komponenten des IL-6-Signalweges sind essentiell für dieAktivierung des klassichen- und des trans-Signalweges?

b. Können sgp130Fc Konzentrationen vorhergesgat werden, die unterpathophysiologischen IL-6 und sgp80 Konzentrationen spezifischden trans- aber nicht den klassischen-Signalweg inhibieren?

c. Wie können die Unsicherheiten der Messdaten sowie der Modellpa-rameter bei der Modellierung und den Modellvorhersagen berück-sichtig werden?

15

16 Contents

Ziele und ErgebnisseZiel dieser Arbeit ist ein verbessertes Verständnis der IL-6-induzierten Signalwegtransduktion. Hierfür wurden mathematische Mod-elle aufgestellt, mit Hilfe dessen grundlegende und essentielle Kompo-nenten bzw. Reaktionen bei der IL-6 Signaltransduktion identifiziertwerden konnten. Verschiedene Modellhypothesen und Parameterkombi-nationen sollten getestet werden. Über systembiologische Ansätze unddurch mathematische Darstellung biologischer Prozesse konnten neueVorhersagen und Hypothesen getroffen werden. Über einen neuen, men-genbasierten Ansatz wurden außerdem die Unsicherheiten der Parametersowie der Zustände und Messdaten im Modell berücksichtigt.Zusammenfassend wurde ein Modell entwickelt, welches Konzentra-tionsbereiche von sgp130Fc unter pathophysiologischen IL-6 und sgp80Konzentartionen vorhersagt, in denen sgp130Fc spezifisch den trans-aber nicht den klassichen-Signalweg inhibiert.

Chapter 1

Introduction

This chapter gives an overview of IL-6 signaling and its role in diseasepropagation. Furthermore, possible therapeutic inhibitors are introducedand the aims of systems biology approaches are presented.

1.1. Interleukin-6 signaling and clinicalrelevance

Interleukin-6 (IL-6) is a multipotent cytokine that is involved in many cel-lular functions like proliferation, angiogenesis, apoptosis, differentiationand inflammation. In hepatocytes, for instance, IL-6 induces the expres-sion of acute phase proteins such as C-reactive protein [18], amyloidsand fibrinogen and reduces substances like albumine and cytochromeP450 [43]. Furthermore, IL-6 induces maturation of megakaryocytesto platelets, activates haematopoietic stem cells and promotes tumorgrowth and proliferation (Fig. 1.1) [46]. Dysregulation of IL-6 produc-tion leads to numerous pathological states. It has been shown that highand persistent serum levels of IL-6 foster diseases like rheumatoid arthritis[8], Crohn’s disease [30] and multiple sclerosis [5]. Furthermore, studieshave shown a correlation between IL-6 production and tumor prolifera-tion. Especially in the case of prostate cancer IL-6 plays a critical role[14].

17

18 1 Introduction

Figure 1.1.: Multipotent properties of IL-6 [46].

1.2. Classic- vs. trans-signaling

The mechanisms by which IL-6 initiates signal transduction are well stud-ied [17]. However, some facts remain still unclear.IL-6 has been shown to interact with the non-signaling receptor subunitglycoprotein 80 (gp80) on target cells causing the recruitment of thesignal-transducing receptor subunit glycoprotein 130 (gp130). Then, ahexameric signaling complex containing two molecules of each IL-6, gp80and gp130 is formed [3]. The assembly of the receptor complex causesthe phosphorylation of tyrosine residues in the cytoplasmatic chain ofgp130 through janus kinases (JAKs). Activated JAKs which are con-stitutively bound to gp130 induce the activation of downstream signal-ing cascades such as JAK-STAT (STAT: signal transducer and activatorof transcription), MAPK (mitogen activated protein kinase) and PI3K(phosphatidylinositol 3-kinase), reviewed in [17].Pathway activation via the membrane-bound receptor gp80 is known asclassic-signaling. But not all cells express gp80 on their surfaces. Viasoluble gp80 (sgp80) IL-6 is able to activate the so called trans-signalingpathway. For a detailed description of IL-6 classic- and trans-signalingand the relationship between receptor activation and cell growth refer to

1.2 Classic- vs. trans-signaling 19

Subsections 1.2.1 - 1.2.6.It has been shown that trans-signaling plays an important role in differ-entiation and development of neuronal cells [26], embryonic stem cells[36], T-cells [1] and haematopoietic progenitor cells [33]. Several studiescould also show that the proinflammatory activities of IL-6 as it is for ex-ample the case in rheumatoid arthritits depend mainly on trans-signaling[2, 15, 24]. As rheumatoid arthritis is a widespread disease medical stud-ies focus on substances that are able to inhibit IL-6 pathway activation,especially trans-signaling and to reduce the negative effects of an imbal-anced IL-6 production.In the following the two pathways and the corresponding inhibitors ofIL-6-signaling are introduced.

1.2.1. IL-6 classic-signaling

During classic-signaling IL-6 binds to membrane-bound gp80 and inducesthe binding of the receptor subunit gp130. Then, the IL-6:gp80:gp130complex hexamerizes and gp130 becomes tyrosine phosphorylated. How-ever, the real stoichiometry of the IL-6 receptors is still not clear. A studyby Schroers et al. suggested, for instance, that first a dimer of gp130binds to one IL-6:gp80 complex and then at higher ligand concentra-tions a second IL-6:gp80-complex is recruted [42]. Nevertheless also atetrameric constitution (IL-6:gp80:(gp130)2) was proposed [16].Fig. 1.2 demonstrates IL-6 classic-signaling and receptor assembly as de-scribed in [3]. According to Boulanger et al. first, the dimer complexIL-6:gp80 associates with one molecule gp130 forming a trimer. Then,a second trimer IL-6:gp80:gp130 is bound leading to complex hexamer-ization.

20 1 Introduction

Figure 1.2.: IL-6 classic-signaling.

1.2.2. IL-6 trans-signaling

During trans-signaling IL-6 binds to the soluble receptor subunit sgp80inducing the association with gp130 and hexamerization of the cytokine-receptor-complex (Fig. 1.3) [37]. There exist also a designer protein thatexerts enhanced activity. Hyper-IL-6 is a recombinant protein in whichIL-6 is fused to sgp80 by a flexible peptide linker. The protein is ableto bind to gp130 and induces signaling in cells which do not expressmembrane-bound gp80 thus mimicing IL-6 trans-signaling. The fusionprotein is 100-1000 times more effective than the single proteins IL-6and sgp80 [9].

Figure 1.3.: IL-6 trans-signaling.


1.2.3. Inhibition through Tocilizumab

Tocilizumab (Toc, name of drug: RoActemra) is a humanized mon-oclonal antibody that inhibits binding of IL-6 to its related receptorsgp80 and sgp80. Thus, Toc inhibits both classic- and trans-signaling(Fig. 1.4) [52]. Toc has been approved by the Food and Drug Admin-istration (FDA) for the treatment of rheumatoid arthritis in adults [32].The antibody inhibits IL-6-signaling globally and is not able to suppressnegative effects of an imbalanced signaling specifically [7]. Therefore,the application of Toc is controversial.

Figure 1.4.: Schematic representation of inhibition through Toc.(A) Classic- and (B) trans-signaling.

1.2.4. Inhibition through soluble gp130

Soluble gp130 (sgp130) was found to be a natural inhibitor of IL-6 trans-signaling [21]. Contrary to Toc sgp130 does not bind to sgp80 alone butto the IL-6:sgp80 complex. Therefore, sgp130 was initially thought notto interfere with classic-signaling. Based on these observations a new de-signer protein called sgp130Fc was developed that inhibits trans-signalingmore effective than natural sgp130 does [38]. After IL-6 and sgp80 haveassociated sgp130Fc binds to the complex causing that the membran-bound receptor gp130 cannot bind to the IL-6:sgp80 complex anymore.Further trans-signaling is inhibited (Fig. 1.5). At present sgp130Fc is inclinical development and will enter phase I in early 2013 [38].Garbers et al. investigated the inhibitory effects of sgp130Fc experimen-tally and postulated the hypothesis that sgp130Fc has also an inhibitory

22 1 Introduction

effect on IL-6 classic-signaling depending on the intracellular [IL−6][sgp80] -ratio

(see Chapter 4) [12].

Figure 1.5.: Inhibition through sgp130Fc in IL-6 trans-signaling.

1.2.5. From receptor phosphorylation to cellproliferation

IL-6 cell stimulation leads to the activation of two important pathways:the JAK-STAT signaling pathway and the MAPK-signaling cascade.During JAK-STAT signaling first, tyrosine residues become phosphory-lated through JAKs. Then, cytoplasmatic STAT transcription factors arerecruited, become activated, and translocate as dimers in the nucleus.Subsequently they induce the expression and regulation of different tar-get genes. Targets are, for instance, genes for cell proliferation, infectionsand inflammations [17]. Also genes that encode for negative regulatorsof the JAK-STAT pathway like PTP (protein tyrosine phosphatases) andSOCS (suppressors of cytokin signaling) are expressed [41].The MAPK-pathway is a linear signaling cascade however, not well un-derstood so far. The pathway involves Gab1 adapter proteins (GRB2-associated-binding protein 1) and SHP2 (protein tyrosine phosphatase2). In the following kinases like Ras (small GTPase), Raf (Ras associ-ated factor 1), MEK (mitogen activated protein kinase kinase) and ERK(extracellular signal regulated kinase) become activated. Activation ofERK leads subsequently to the phosphorylation and thus translocation ofdifferent transcription factors in the nucleus. Activation of the MAPK-signaling cascades governs the growth, proliferation, differentiation andsurvival of many cell types [4].Fig. 1.6 shows schematically the connection between phosphorylation


and activation of cytokine-receptor complexes and cell proliferation dueto IL-6 stimulation.

Figure 1.6.: Activation of downstream processes after cell stimulationwith IL-6.

Both pathways, classic- and trans-signaling initiate important down-stream processes that again activate, for instance, target genes for cellproliferation.

1.2.6. The whole network: from IL-6 stimulation tocell proliferation

Fig. 1.7 presents schematically the network of the IL-6 induced receptoractivation. The figure respresents IL-6 classic- and trans-signaling withrespect to different cell stimuli and inhibitors. The mechanisms weredescribed in the subsections before. Soluble substances are represented

24 1 Introduction

Fig

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1.3 Parameter estimation 25

by circles and membrane-bound species by rectangles. Double arrowsdescribe kinetic reactions while one-way arrows indicate the activation offurther pathways. Receptor phosphorylation and activation, respectivelyand cell proliferation are connected via the activation of the JAK-STATand/or the MAPK signaling pathways (highlighted in yellow) and furthergene expression.

1.3. Parameter estimation

1.3.1. A set-based approach

To obtain a better understanding and deeper inside into complex biolog-ical processes mathematical modeling has become a very important tool[19]. By abstracting, simplifying and simulating signaling cascades ormetabolic networks one can gain important and useful knowledge aboute.g. the dynamical behavior of a cell or an organism. Mathematicalmodeling is not a trivial task as a lot of kinetic parameters are unknownand have to be estimated through sophisticated methods. Besides manyother methods a set-based estimation approach can be taken. Set-basedmethods allow to take into account all uncertainties of the data andparameters. By searching for whole sets of parameters and checking theconsistency of the model with the (uncertain but bounded) experimentaldata statements about the guaranteed invalidity of the model can bemade [39]. Please also refer to Chapter 2, Section 2.3.2.Recently, the MatLab-based toolbox ADMIT (Analysis, Design andModel Invalidation Toolbox) has been developed which implements theset-based estimation method [45].

1.3.2. A workflow for set-based estimation methods

Parameter estimation and especially identification strongly depend on thedata and prior knowledge available. Therefore, in the following (Fig. 1.8)a workflow for set-based parameter estimation is proposed describing theestimation of the mathematical models step by step. Starting with asteady-state model one has to consider all provided information on themodel and biological background (prior knowledge). This means, forinstance, lower and upper bounds on the model parameters, variables

26 1 Introduction

and experimental data. Outer bounds for measurements are derivedfrom the experimentally determined standard deviations. Through outer-approximation the invalidity of the chosen model can be investigated. Ifthe outer-approximation of the feasible solution set is empty the model issaid to be guaranteed invalid. In the case of invalidity one has to considereither another model structure meaning that the chosen kinetics may beto simple or the extension of the model by adding new equations and/orexperimental data. If the solution set is found not to be empty the modelis said to be not invalid. If a model is not invalid a bisectioning algorithmcan be applied identifying consistent parameter regions and excludingthose that are not consistent with the provided data. Furthermore, theconvex hull is computed.By adding semi-quantitative/qualitative statements like

Ki ≥ Kj (1.1)

and/or

Kj ≤ Kk, (1.2)

we can derive new constraints (linear/non-linear) that again can leadto the invalidation of whole parameter regions (see Section 3.1.1). Inthe next steps the addition of new data and the extension of the modelis necessary to improve the results and to prove model hypotheses andinvalidity again.

1.3 Parameter estimation 27

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28 1 Introduction

1.4. Motivation and aim

IL-6 is a very important mediator of inflammatory diseases and tumorproliferation. Although many studies about IL-6 have been made byusing molecular biological methods the dynamics of pathway activationand signal transduction are still not fully understood.Also many systems biology approaches of IL-6 signaling were made [44,34, 48]. Nevertheless, there exist no models which investigate processesat the receptor level of the two different pathways (classic and trans)with respect to the inhibition through sgp130Fc.The aim of this work is to derive mathematical models that are able toreproduce experimental data and to dissolve processes at the receptorlevel more detailed. Furthermore, the mathematical models shall giveanswers to the following questions:

a. Which components and reactions are essential for classic- andtrans-signaling in IL-6 and sgp130Fc treated cells?

b. Can sgp130Fc concentrations be predicted for given pathophysi-ological concentration of IL-6 and sgp80 which affect specificallytrans- but not classic-signaling?

c. How can one make predictions with respect to treatments even inthe presence of uncertain experimental data and parameters?

Especially answering b) can lead to promising results in the clinical ap-plication of sgp130Fc. Trans-signaling correlates with harmful diseaseslike prostate cancer, rheumatoid arthritis and multiple sclerosis. As rem-edy, the desginer protein sgp130Fc was developed and is currently underclinical investigations. As in contrast to Toc sgp130Fc is able to in-hibit IL-6-trans-signaling specifically the protein is very promising. If onecould make targeted statements about the range of effects of sgp130Fcthe impacts of the diseases above could be weakened and even cured.

1.5. Outline

The purpose of this thesis is a better understanding of IL-6 mediatedpathway activation on a systems level. In the first part (Chapter 2) we

1.5 Outline 29

describe the experimental data available for model fitting and invalida-tion. Furthermore, necessary model assumptions as well as the methodsto analyze the mathematical models are described. In the followingpart (Chapter 3) we present a modeling workflow for the derivative ofthe final model. Thereby, we rule out wrong model hypotheses usinga set-based estimation approach. In Chapter 4 model predictions andanalyses concerning the inhibitory effect of soluble gp130Fc on classic-and trans-signaling will be made. Chapter 5 gives a summary of thethesis and an outlook for further investigations.

Chapter 2

Mathematical modeling

In this chapter the experimental data available for mathematical model-ing and model validation is described and shown. Furthermore, necessaryassumptions are explained and the methodologies to analyze the derivedmodels are described.

2.1. Experimental data

Experimental data for simulation and parameter estimation were takenfrom the work by Garbers et al. [12]. Ba/F3 cells (an immortalizedmurine bone marrow-derived pro-B-cell line) stably expressing gp130 andgp80 were exposed to different concentrations of IL-6, sgp80, Toc andsgp130Fc (Fig. 2.2 and Table 2.1). The biological readout was measuredafter two days through the CellTiter-Blue Viability Assay. In this assaythe number of viable cells is determined and hence cell proliferation [6].Figure 2.1 demonstrates the setup of the dose-dependent experimentsfrom Garbers et al.. First, Ba/F3 cells were washed three times with ster-ile PBS (phosphate buffered saline) and resuspended in DMEM (Dulbec-cos’s modified eagle medium) containing 10 % FBS (cell staining buffer).The final concentration amounted 5 ·103 cells per well in a 96-well plate.Then, the cells were stimulated with varying concentrations of eitherIL-6, sgp80, Toc, sgp130Fc or combinations at the intitial point t0 = 0(measured in days). After 2 days cell proliferation was determined. Theprocedure was performed for eight different input concentrations in each

31

32 2 Mathematical modeling

Figure 2.1.: Experimental set-up for the determination of cell proliferationin Ba/F3-gp180-gp130 cells.

experimental setup. In Table 2.1 the provided experimental data by Gar-bers et al. and the corresponding figures in [12] are listed. Furthermore,Fig. 2.2 presents the experimental data graphically.Garbers et al. hypothesized that sgp130Fc is also able to inhibit classic-signaling depending on the concentration levels of IL-6 and sgp80.Therefore, Garbers et al. used different [IL−6]

[sgp80] ratios as experimental

inputs (Table 2.1, Fig. 4A, 5A, 5B and 5C in [12]) and determined dif-ferent cell proliferation curves which is presented in Fig. 2.2 D. In D theinhibitory profile of sgp130Fc is presented. Ba/F3 cells were culturedwith 10 [ ng

mL ] IL-6 plus 200 [ ngmL ] sgp80 (black), 3.3 [ ng

mL ] IL-6 plus 1000[ ng

mL ] sgp80 (green), 10 [ ngmL ] IL-6 plus 10 [ ng

mL ] sgp80 (blue) and 10 [ ngmL ]

IL-6 plus 50 [ ngmL ] sgp80 (red). One can see that especially if sgp80 ex-

ceeds the IL-6 level significantly cell proliferation is inhibited strongest.In the case of [IL−6]

[sgp80] = 1 cell growth is inhibited least.

2.1 Experimental data 33

Table 2.1.: Overview of experimental data taken by [12]. Only data fromthe experiments with the Ba/F3-gp130-gp80 cell line were taken.

Inputs Figure in [12]

Toc, IL-6 1BToc, IL-6, sgp80 1CIL-6, sgp80 1Dsgp130Fc

[IL6][sgp80] = 0.05 4A[IL6]

[sgp80] = 0.0033 5A[IL6]

[sgp80] = 1 5B[IL6]

[sgp80] = 0.2 5C

IL-6 unpublished

Furthermore, in Fig. 2.2 A and B the inhibitory profiles of Toc in IL-6 signaling are shown. The cells were cultured with 10 [ ng

mL ] IL-6 andincreasing concentrations of Toc. Higher Toc concentrations lead tofewer cell proliferation in classic- (A) as well as in trans-signaling (B).In plot C a sgp80-dependent proliferation assay is presented. The cellswere cultured with 10 [ ng

mL ] IL-6 and increasing concentrations of sgp80.If increasing sgp80 concentrations are titrated cell growth increases, too.Plot E shows an IL-6-dependent proliferation assay. Here, an ascendingcell proliferation can be observed with increasing IL-6 concentrations.Note that the experimental data as well as inhibitor, receptor and totalconcentrations were normalized to their corresponding maximum.


0

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norm

.)

0.005 0.01 0.025 0.1 0.125 0.25 0.5 10.5

0.6

0.7

0.8

0.9

1.0

ce

ll p

rolif

era

tio

n

(

norm

.)

sgp80_tot (norm.)

0

0.2

0.4

0.6

0.8

1.0

1.2

sgp130Fc_tot (norm.)

0

0.2

0.4

0.6

0.8

1.0

0 0.00050.005 0.025 0.05 0.25 0.5 1

IL-6_tot (norm.)

ce

ll p

rolif

era

tio

n

(

norm

.)

A B

C D

ce

ll p

rolif

era

tio

n

(

norm

.)

EFigure 2.2.: Stimulus-dependent measurements as described in [12]. Ab-solute values were normalized to their corresponding maximum.

2.2. Development of the mathematical

models

2.2.1. Model assumptions and simplifications

To obtain a minimal model that is able to describe IL-6 pathway activa-tion on the receptor level different assumptions had to be made. Theseare listed in the following.

1. Assumptions on reaction kinetics.a) The whole system is considered to be ideally and well dis-tributed.

2.2 Development of the mathematical models 35

b) This allows the neglection of local gradients and the use ofmass-action-kinetics:

A + Bk1−⇀↽−kh

1

AB (2.1)

A and B are reactants forming the complex AB. AB in turn candissociate to A and B. The related kinetic constants are denotedby kh

1 and k1.c) The equations are written as ordinary differential equations(ODEs):

d[AB]

dt= k1 · [A] · [B] − kh

1 · [AB] (2.2)

where [A], [B] and [AB] describe concentrations of species A, Band AB.d) All reactions are considered to be reversible.e) Stochastic effects are neglected.

2. As the biological readout was measured two days after stimulationthe cells are considered to be in a steady-state.a) There exist no changes in concentrations over time and thereforeODEs with the left hand side equal to zero are implemented forall state variables:

0 =d[AB]

dt(2.3)

We now refer to Eq. (2.3) as steady-state-equation.b) The expression of gp80, sgp80 and gp130 is considered to beconstant over two days. Thus, receptor internalization is neglected.c) The determined cell proliferation is considered to be equivalentto the phosphorylation and activation of the cytokine-receptor-complexes (see Section 1.2.5).

Simplification of recepetor hexamerizationAs decribed in Chapter 1 the real stoichiometry of the IL-6 signalingreceptor assembly is still not clear [42, 16]. To keep the model structureas simple as possible and to reduce the amount of model parameterswe simplify and aggregate these steps. Fig. 2.3 shows exemplarily thesimplification of receptor hexamerization in classic-signaling. The high-lighted turquoise box and the dotted line imply the aggregation of thehexamer formation.


Figure 2.3.: Simplification of receptor hexamerization in IL-6 classic-signaling.

Signal transfer from receptor phosphorylation to cell growthAs demonstrated in Fig. 2.4 we have to deal with different time-scalesin the modeling approach presented.Shortly after IL-6 stimulation (< 5 minutes) cytokine-receptor-complexesare formed and gp130 residues become tyrosine phosphorylated. Afterone minute one can detect phosphorylated STAT3 and/or STAT1 in thecells (JAK-STAT pathway). However, cell proliferation was measuredafter two days. We have no knowledge and data about the transfer-behavior from receptor phosphorylation to, for instance, STAT activa-tion, further gene-regulation and cell proliferation (grey boxes and ques-tion marks in Fig. 2.4). Thus we simplify these steps assuming that thesignal from the receptor level to cell proliferation is transferred linearly.

2.3 Model implementation and estimation 37

Figure 2.4.: Different time scales from cell stimulation with IL-6 to cellproliferation.

2.3. Model implementation and estimation

2.3.1. Implementation of the models

The models were implemented as static equations in MatLab 7.11.0(Mathworks, U.S.) and ADMIT, respectively and reaction rates werechosen to be polynomial.


2.3.2. Set-based parameter estimation

The set-based model estimation approach was recently developed [39].The method allows to take uncertain experimental data and variablesinto account and searches for whole parameter regions which can repro-duce the experimental data. ADMIT provides an environment for theset-based estimation method and finds outer-approximates of the modelvariables and parameters [45]. Eq. (2.4) shows examplarily the imple-mentation of the model equations in ADMIT. As mentioned we considersteady-state-equations thus, the equations are time-independent. Werather implemented stimulus-dependent equations:

0 = k1 · [A(sgp130Fc)] · [B(sgp130Fc)]

−kh1 · [AB(sgp130Fc)] (2.4)

In Eq. (2.4) the variables A, B and AB are functions of the stimulusconcentration, e.g. sgp130Fc and the kinetic parameters are describedby k1 and kh

1 , respectively.If the solution set of the model parameters is found to be empty thederived model is guaranteed invalid and has to be rejected. This is incontrast to local estimations in which a solver finds no sets of param-eters but a possible parameter combination. However, they are mostlynot uniquely determinable. If the solution set is found not to be emptythe model can be used for further extensions. However, a guaranteedstatement about model validity cannot be made at this point. In the set-based approach outer approximates on the model parameters are found.These approximates are over estimated and can still include invalid solu-tions which are regarded at first to be valid (false positive). A guaranteedstatement about model validity can be done after, for instance, a Monte-Carlo sampling or an inner-approximation. An inner-approximation onlyincludes valid solutions of the parameter sets and excludes invalid ones.In the following we refer to models for which the solution set is emptyas invalid and for the others as not invalid.


2.3.3. Validation and invalidation of mathematicalmodels

The aim of systems biology approaches is not only the derivative ofmathematical models that are consistent with given experimental databut mainly the derivative of models that can give predictions concern-ing specific biological behaviors. Fig. 2.5 demonstrates the procedure insystems biology approaches. Based on existing experimental data onetranslates the complexity of the ’reality’ (e.g. cell signaling cascades)into an abstract one by representing biological processes through math-ematical equations (e.g. ODEs). Then, the model equations are fittedto the experimental data available and the model is either validated orinvalidated. In the case of invalidity no parametrization can be foundbeing able to reproduce the experimental data. Therefore, the model hasto be revised as, for instance, the choosen kinetics may be not suitable todescribe the processes. In the case of a validity a possible parametriza-tion can be found being able to reproduce the experimental data withina certain range of tolerance. One possibility to define the range of toler-ance can be, for instance, to determine the one-fold standard deviationof the experimental data. Through further in silico experiments new hy-potheses can be generated that have then to be tested in the laboratory.There are several possibilities to validate a mathematical model. In alocal estimation approach, for instance, a vector with valid parametervalues is found in the neighborhood of a selected parameter start vector[20]. In a global optimization the whole parameter space is scanned forpossible valid parameter values [29]. The global optimization method ispreferred if the number of model equations and thus parameters is highand the optimization problem hard to solve.To make predictions on specific therapeutic treatments with sgp130Fc itis necessary to validate the final mathematical model (Model 2-1Trim,Fig. (3.2)) within the set-based estimation approach. Therefore, a re-peated random sampling of the model parameters in a choosen intervalis performed in ADMIT. If a possible candidate is found the initial pa-rameter bounds can be replaced by the randomly generated parametervalues:

Kiniti = [lb, ub] → Kvalid

i = [value]


in which lb denotes the lower bound and ub the upper bound of thei-th parameter K. To proceed all model parameters have to be set to ’ofInterest’ and the state variables have to be set to ’of interest’ for the firstinhibitor concentration. The values for the remaining variables are foundthrough constraint propagation. By testing if the randomly generatedparameter constellation returns a feasible solution we can rule out wrongsamples and find those that are able to reproduce the data. Havingreplaced the bounds and thus validated the model with the experimentaldata we can make predictions concerning therapeutic treatments withsgp130Fc. Please refer also to Appendix D for the determined and validparameter combination of Model 2-1Trim (final).

Figure 2.5.: Development of mathematical models.

2.3.4. Global parameter estimation

The set-based method requires strict bounds on data and parameters.Sometimes model invalidity results from model outliers. These are single


data points which fall not within the experimentally standard deviation.To identify such outliers we also estimate the derived models through aconventional global estimation approach. Afterwards we can weightenseveral data points in the set-based estimation method with an additionalerror such that a not invalid solution set can be found. By means ofFig. 2.6 data point weightening is explained. Fig. 2.6 shows the norma-

1e−7 1e−6 1e−5 1e−4 1e−3 1e−2 1e−1 10

0.2

0.4

0.6

0.8

1

p_

IL6

_sg

p8

0_

gp

13

0

(n

orm

.)


Figure 2.6.: Identification of data points which do not fall within theexperimentally determined one-fold standard deviation in Model 2-1Trim.

lized quantity of phosphorylated receptor complexes in trans-signalingdepending on the total amount of sgp130Fc. One can see that thedata points at 10−3, 10−1 and 1 sgp130_tot (norm.) lie outside theircorresponding standard deviations (errorbars). Thus, we choose a greaterstandard deviation compared with the experimentally determined one asouter bounds for these points. Please refer to Appendix A to find theweightened standard deviations.The global parameter estimation method is based on the Maximum-Likelihood. The cost function can be described as followed:

χ2(p) =

ny∑i=1

nt∑j=1

(yi(ti) − yi(ty, p, xo)

σ2i,j

)2

(2.5)

In Eq. (2.5), ny describes the amount of outputs, nt the amount ofdata-points, yi the experimental data-points, yi the simulated data and


σi,j the standard deviations. Simulated points depend on measurement-points ti, the parameter set p and start values x0.In Eq. (2.6) p̂ describes the parameter set that minimizes the cost func-tion χ2(p):

p̂ = argpminχ2(p) (2.6)

χ2(p) is a constant containing all deviations between experimental andsimulated data.We choose the solver simannealingSB which is based on the simulatedannealing algorithm [23]. Simulated annealing is a genetic algorithm forfinding the global minimum of a cost funtion. The algorithm starts withan initial parameter vector p0 and determines a so called temperature.This temperature describes the deviations of simulated and experimentaldata. Then, in the next step a new parameter vector is generated ran-domly and the resulting temperature is compared with the one before.Thus, the algorithm systematically lowers the temperature storing thebest point found so far. SimannealingSB is part of the Systems BiologyToolbox 2 in MatLab [40].

Chapter 3

Modeling of IL-6-signaling andset-based model analyses

In this chapter a modeling workflow is presented. The workflow describesthe derivative of the final model which depicts IL-6-classic- and trans-signaling and the inhibitory effect of sgp130Fc in Ba/F3-gp80-gp130cells. Furthermore, the models are analyzed in the set-based estimationframework.

3.1. Additional semi-quantitative/qualitativedata

The set-based estimation method allows guaranteed statements aboutmodel invalidity while taking uncertain but bounded data into account.However, successful estimation results also depend on prior knowledgeabout, for instance, binding affinities of biological species like IL-6 andgp80.To improve the estimation results we took data from the literature inwhich different binding affinties were experimentally determined (Table3.1). We used the information from Table 3.1 to conclude additionalrelationships between the model parameters (see Appendix A.3). For

43

44 3 Modeling of IL-6-signaling and set-based model analyses

Table 3.1.: Binding affinities from literature. Abbreviations: SP:scatchard plot; ELISA: enzyme-linked immunosorbent assay; SPR: sur-face plasmon resonance.

Binding affinity of Value [nM] Method Ref.

IL-6 to gp80 0.5 SP [54]IL-6:gp80 to gp130 0.015 SP [54]Toc to gp80 2.54 ELISA [28]IL-6 to sgp80 1.7 · 10−8 SPR [51]

2.2 · 10−10 SPR [11]5.5 · 10−8 SPR [22]

sgp130Fc to IL-6:sgp80 5.97 · 10−7 SPR [47]

instance, binding of IL-6 to the membrane-bound receptor gp80 is de-scribed through Eq. (3.13) and the association of gp80 with Toc throughEq. (3.19). K1,cl, KToc,cl and Kh

Toc,cl denote the corresponding kineticconstants (refer to Appendix A for detailed parameter descriptions anddefinitions). The determined values from Zohlnhöfer et al. as well asMihara et al. show that the binding affinity of IL-6 to gp80 is higherthan the one of Toc to gp80 [54, 28]. Thus, the following constraint canbe formulated:

K1,cl ≤Kh

Toc,cl

KToc,cl(3.1)

Furthermore, we conclude from Table 3.1 that the binding affinity of IL-6to sgp80 is also higher than the affinity of Toc to gp80:

Kh1,tr

K1,tr≤

KhToc,cl

KToc,cl(3.2)

In Eq. (3.2) Kh1,tr and K1,tr are parameters for the binding and dissocia-

tion of IL-6 to soluble gp80. Fig. 3.1 represents the bisectioning resultsof the model parameters after adding the constraints above (Eq. (3.1),Eq. (3.2)). The y- and x-axes represent the outer-bounds of the cor-responding model parameter. While blue squares represent not invalid

3.1 Additional semi-quantitative/qualitative data 45

parameter regions, white regions correspond to invalid ones. One cannotice that without prior knowledge on the parameters no sets of pa-rameters for Kh

Toc,cl and KToc,cl or K1,tr and K1,cl could be determinedbeing guaranteed invalid presented in (A) and (B). By adding the de-rived constraints guaranteed invalid sets of parameters could be ruled outwhich is shown in (C) and (D). Based on plots (C) and (D) additionaldependencies as constraints on the parameters could be derived. Hence,invalid sets of parameters can be excluded step by step. Please refer toAppendix A for the derived constraints on the model parameters and toSection 3.3 for a further discussion.

Figure 3.1.: Bisectioning results for parameters in Model 2TrimToc. Bi-sectioning over (A) Kh

Toc,cl and KToc,cl, no constraints were added (B)

K1,tr and K1,cl, no constraints were added (C) KhToc,cl and KToc,cl as

well as (C) K1,tr and K1,cl after constraints (3.1) and (3.2) were added.


3.2. Modeling workflow and set-basedestimation results

To obtain models that are able to reproduce experimental data a work-flow for modeling was created describing all model extensions and inputs(Fig. 3.2).

Figure 3.2.: Overview of implemented models and model extensions.

We derived several models which allowed us to test different hypothe-ses. All model parameters and initial conditions were assumed to beuncertain. A model was discarded if the outer-approximation of the fea-sible solution set was found to be empty and thus no parameter set wasfound being consistent with the experimental data. In Fig. 3.2 modelswith the numbers ’0’ and ’1’ describe classic-signaling while ’2’ stands for

3.2 Modeling workflow and set-based estimation results 47

the combination of classic- and trans-signaling. Red checkmarks denotemodels which are not invalid. The little star on top of Model 2-1Trim(final) indicates that the model was validated within the estimated outerbounds. Starting with Model 0 and Model 1Trim (classic-signaling) dif-ferent models were derived combining classic- and trans-signaling withdifferent model inputs. For model fitting we took experimental data fromthe work by Garbers et al. [12]. Please refer also to Fig. 2.2 for dataexplanation and to the next section (3.2.1) for further model explana-tions. All equations, parameters and parameter definitions for Model 0until Model 2-1Trim (final) are described in Appendix A.

3.2.1. IL-6 classic-signaling

Initial constraintsWe fitted the following models to the provided data in [12] and addedthe experimentally determined one-fold standard deviation as bounds onthe measurements (errorbars in Fig. 2.2, Eq. (3.3)).

[lb, ub]Mi= Mi ± δi (3.3)

Note that in a normal distribution 68% of the observations fall withinthe one-fold standard deviation. Thus, we use the global estimationapproach to identify those data points which fall not within the one-foldstandard deviation (ouliers) and weighten them with an additional error(Chapter 2, Section 2.3.3). Please refer to Appendix C.1 to find theweightened standard deviations.As we have no knowledge about parameter intervals we set them to

[lb, ub]Ki= [0.01, 100] (3.4)

Furthermore, the initial bounds on the model variables are defined as

[lb, ub]Vi= [0, 1], (3.5)

where lb denotes the lower bound, ub the upper bound and i the i-thmeasurement M, standard deviation δ, parameter K and variable V.


Model 0 (initial)The initial mathematical model describes IL-6 classic-signaling thus,the binding of IL-6 (IL6 in model equations) to receptor subunit gp80.Recpetor assembly with gp130 and gp80 is not explicitly represented.Therefore, we aggregated binding of gp130 to the IL-6:gp80 complex(IL6_gp80) and assume that gp130 species are represented by gp80.Eq. (3.6) describes the association of IL-6 to the receptor mathemati-cally.

0 =d[IL6]

dt= K1 · [Rcomplex] − [IL6] · [gp80] (3.6)

Rcomplex denotes the complex consisting of IL-6 and the correspond-ing receptor species. Once, cytokine-receptor-complexes are formedtyrosine residues become phosphorylated, mathematically representedby p_Rcomplex. The corresponding steady-state-equation is shown inEq. (3.7).

0 =d[p_Rcomplex]

dt= Kp · [Rcomplex] − [p_Rcomplex] (3.7)

The model was fitted to the IL-6-dependent proliferation assay in whichBa/F3 cells expressing gp130 and gp80 were stimulated with varying IL-6concentrations ranging from 0 - 200 [ ng

mL] (Table 2.1 and Fig. 2.2 E).

For Model 0 we found that within the choosen parameter bounds theparameter set is empty. No outer-approximates of the solution set couldbe determined. Model 0 is therefore guaranteed invalid. In all, it seemsas if the initial model is too simple to describe IL-6 classic-signaling.Therefore, we extended Model 0 and included the stepwise receptor as-sociation in Model 1Trim (Fig. 3.2).

Model 1TrimIn Model 1Trim binding of the IL-6:gp80 complex to the receptorsubunit gp130 was implemented. That means that the activated re-ceptor is represented by the phosphorylated IL-6:gp80:gp130 complex(p_IL6_gp80_gp130). Eq. (3.8) describes the association and dissocia-tion of gp130.


0 =d[gp130]

dt= K2 · [IL6_gp80_gp130]

−[IL6_gp80] · [gp130] (3.8)

Gp130 binds to the complex consisting of IL-6 and gp80 and forms thetrimer IL-6:gp80:gp130 (IL6_gp80_gp130). Then, gp130 becomes ty-rosine phosphorylated which is represented by p_IL6_gp80_gp130 inEq. (3.9).

0 =d[p_IL6_gp80_gp130]

dt= Kp · [IL6_gp80_gp130]

−[p_IL6_gp80_gp130] (3.9)

In this model we neglected receptor hexamerization and used data fromFig. 2.2 E for model fitting.For Model 1Trim we obtained a non-empty solution set meaning thatthe model is not invalid. Through outer-bounding and bisectioning wecould estimate outer-approximates of the model variables and param-eters and excluded regions being inconsistent with the measurements.Fig. 3.3 shows the results of the outer-approximation of the state vari-ables in Model 1Trim. The plots represent the outer-bounds for IL-6(A), gp130 (B) and p_IL6_gp80_gp130 (C) in dependency of the totalamount of IL-6. Plot (D) shows the estimated bounds for the modelparameters K1, K2 and Kp. Plots (A) and (B) show each unboundedspecies. While for increasing total concentrations of IL-6 the amount offree IL-6 molecules rises the quantitiy of free gp130 subunits decreases.High IL-6 levels lead to an increased cell proliferation thus, the amountof free gp130 molecules decreases as more IL-6:gp80:gp130 complexesare formed. In plot (C) one can recognize that the outer-bounds (exper-imental determined standard deviation) for p_IL6_gp80_gp130 couldnot be improved further. In (D), one can see that only the lower boundof Kp could be improved. Kp is a kinetic parameter and and describesreceptor phosphorylation and dephosphorylation, respectively. We as-sume that the model output p_IL6_gp80_gp130 is equivalent to themeasured cell proliferation (Chapter 2, Section 2.2.1) and fit the vari-able to the experimental data in Fig. 2.2 E. Thus, Kp can be improved


best.To summarize briefly we found a not invalid solution set for Model1Trim. As we do an outer-approximation we cannot make any guar-anteed statement about the validity of the model at this point. Theouter-approximation of a model includes still invalid solutions. Modelvalidation will only be done for the final model.As we are also interested in understanding receptor hexamerization weimplemented in the following a model that describes hexamerization dueto IL-6 stimulation.

10−6

10−4

10−2

100

0

0.2

0.4

0.6

0.8

1.0

IL-6_tot (norm.) IL-6_tot (norm.)

IL-6

(n

orm

.)

gp

13

0 (

no

rm.)

IL-6_tot (norm.)

p_

IL6

_g

p8

0_

gp

13

0

(

norm

.)

K1 K2 Kp

lb

ub 100100100

0.01 0.01 6.7810−6 10−4 10−2 100

0

0.2

0.4

0.6

0.8

1.0

10−6

10−4

10−2

100

0

0.2

0.4

0.6

0.8

1.0

A B

C DFigure 3.3.: Outer-bounding for the state variables in Model 1Trim. (A)gp130, (B) IL6, (C) p_IL6_gp80_gp130 in dependency of the totalamount of IL-6. (D) outer-bounds of the model parameters K1, K2 andKp. Red, bold lines indicate not invalid and black, thin lines invalid sets.Abbreviations: lb, lower bound; ub, upper bound.

Model 1HexIn an extension of Model 1Trim, Model 1Hex, receptor hexamerizationwas investigated by modeling the dimerization of the IL-6:gp80:gp130complex. For simplification we aggregate the dimerization definingthat one trimerized complex IL-6:gp80:gp130 forms the complex IL-6:gp80:gp130hex by binding a second trimer. In Eq. (3.10) the steady-


state-equation for the formation and degradation of the IL-6:gp80:gp130complex (IL6_gp80_gp130) ist described.

0 =d[IL6_gp80_gp130]

dt= [IL6_gp80] · [gp130]

−K2 · [IL6_gp80_gp130]

−Khex · [IL6_gp80_gp130]

+Khhex · [IL6_gp80_gp130hex] (3.10)

Once, hexamerized IL-6:gp80:gp130 complexes (IL6_gp80_gp130hex)are formed the gp130 subunits become tyrosine phosphorylated.Eq. (3.11) describes the steady-state-equation for phosphorylated andhexamerized IL-6:gp80:gp130 (p_IL6_gp80_gp130hex).

0 =d[p_IL6_gp80_gp130hex]

dt= Kp · [IL6_gp80_gp130hex]

−[p_IL6_gp80_gp130hex] (3.11)

Model 1Hex was again fitted to the IL-6 dependent proliferation assayas in Model 0 and Model 1Trim.Model 1Hex was also found to be not invalid (Fig. 3.2). However, incontrast to Model 1Trim this model has two more kinetic parameters tobe estimated (Khex and Kh

hex). To keep the model structure as simpleas possible and to keep the amount of kinetic parameters low we usedModel 1Trim for further extensions.

3.2.2. Combining classic- and trans-signaling

To derive a model which combines IL-6 classic- and trans-signaling weextended Model 1Trim by trans-signaling.

Model 2TrimWe extendend Model 1Trim by the binding of IL-6 to the soluble IL-6receptor sgp80. Eq. (3.12) describes the steady-state-equation for theassociation and dissociation of sgp80.


0 =d[sgp80]

dt= K2,tr · [IL6_sgp80] − [IL6] · [sgp80] (3.12)

To combine classic-and trans-signaling mathematically the steady-state-equations for IL-6 (Eq. (3.6)) as well as for gp130 (Eq. (3.8)) have tobe extended by trans-signaling.

0 =d[IL6]

dt= K1,cl · [IL6_gp80] − [IL6] · [gp80]

+Kh1,tr · [IL6_sgp80] − K1,tr · [IL6] · [sgp80] (3.13)

Eq. (3.13) describes the binding of IL-6 to its receptors gp80 andsgp80, respectively. Hence, two different IL-6-recepetor species areformed: IL6:gp80 and IL6:sgp80. Both associate in the following withthe receptor subunit gp130 forming the trimers IL-6:gp80:gp130 and IL-6:sgp80:gp130. Eq. (3.14) describes the steady-state equation for gp130.

0 =d[gp130]

dt= K2,cl · [IL6_gp80_gp130]

−[IL6_gp80] · [gp130]

+Kh3,tr · [IL6_sgp80_gp130]

−K4,tr · [IL6_sgp80] · [gp130] (3.14)

The corresponding phosphorylated and activated receptor complexes arerepresented by p_IL6_gp80_gp130 (classic-signaling, Eq. (3.15)) andp_IL6_sgp80_gp130 (trans-signaling, Eq. (3.16)).

0 =d[p_IL6_gp80_gp130]

dt= Kp,cl · [IL6_gp80_gp130]

−[p_IL6_gp80_gp130] (3.15)

0 =d[p_IL6_sgp80_gp130]

dt= Kp,tr · [IL6_sgp80_gp130]

−[p_IL6_sgp80_gp130] (3.16)


In Eq. (3.15) and (3.16) Kp,cl and Kp,tr represent the phosphorylationconstants in classic- and trans-signaling, respectively.For fitting of trans-signaling we took data from Fig. 2.2 C into account.Here, Ba/F3 cells were stimulated with different sgp80 concentrationsvarying from 1-200 [ ng

mL ].In the set-based estimation approach a not invalid solution set was foundfor Model 2Trim (Fig. 3.2). As we try to keep the model structure assimple as possible and to keep the amount of model parameters low wepresent in the following Model 2TrimRed. Here we derived a reducedand more simplified version of Model 2Trim.

Model 2TrimRedIn this model the gp130 population is represented by the receptor sub-units gp80 and sgp80, respectively (equivalent to Model 0 (initial)).In Eq. (3.17) and (3.18) the formation and degradation of phosphorylatedcytokin-receptor-complexes in classic- and trans-signaling are describedas steady-state-equations.

0 =d[p_Rcomplex,classic]

dt= Kp,cl · [Rcomplex,classic]

−[p_Rcomplex,classic] (3.17)

0 =d[p_Rcomplex,trans]

dt= Kp,tr · [Rcomplex,trans]

−[p_Rcomplex,trans] (3.18)

Rcomplex,classic and Rcomplex,trans denote the signal tranducing com-plexes in classic- and trans-signaling, respectively which consist of IL-6and the corresponding receptor species. The model was fitted to thedata from Fig. 2.2 C and E.For Model 2-1TrimRed the outer-approximation of the solution set wasfound to be empty (Fig. 3.2). It seems that simplification of gp130receptor association with IL-6-receptor-complexes is not an adequate as-sumption to represent IL-6 pathway activation. Thus, we use Model2Trim for further model extension.In the next step we are interested in the inhibitory effect of Toc in IL-6-signaling. Therefore we implemented Model 2TrimToc which is described


in the following.

Model 2TrimTocTo obtain Model 2TrimToc we extended Model 2Trim by the inhibitoryeffect of Toc in both pathways, classic- and trans-signaling.Eq. (3.19) and (3.20) describe the steady-state-equations for gp80 andsgp80, respectively.

0 =d[gp80]

dt= K1,cl · [IL6_gp80] − [IL6] · [gp80]

−KToc,cl · [gp80] · [Toc] + KhToc,cl · [gp80_Toc] (3.19)

0 =d[sgp80]

dt= K2,tr · [IL6_sgp80] − [IL6] · [sgp80]

−KToc,tr · [sgp80] · [Toc] + KhToc,tr · [sgp80_Toc] (3.20)

The receptor subunits gp80 and sgp80 bind to IL-6 forming cytokin-receptor complexes at which gp130 can associate. The monoclonal anti-body Toc is able to bind to both receptors forming complexes of gp80:Toc(gp80_Toc) and sgp80:Toc (sgp80_Toc). Subsequently gp130 is notable to bind to gp80:Toc and sgp80:Toc complexes, respectively thusfurther receptor internalization and signaling is inhibited.The model was fitted to the data from Fig. 2.2 A and B. Here, Ba/F3cells were stimulated with different concentrations of Toc varying from10−3 -104[ μg

mL ].Model 2TrimToc returned a not invalid set of parameters (Fig. 3.2).In Fig. 3.4 the bisectioning results of different parameters are shown.While for (A), (B) and (C) additional constraints could be derived (re-fer to Appendix A.3) to exclude guaranteed invaldi sets of parametersnothing could be determined for the parameters K6,tr and K5,tr. It isnoticable that especially the upper bounds of Kh

Toc,cl and KToc,cl couldbe identified and improved. This is due to the derived constraints andprior knowledge based on Table 3.1.We are mainly interested in therapeutic tretaments with the inhibitorsgp130Fc. Based on the previous findings we again used Model 2Trimfor further model extension by the effects of sgp130Fc in Ba/F3 cells.


Important model equations and the results from the set-based analysismethod are described in the following.

Figure 3.4.: Bisectioning results of several parameters in Model 2Trim-Toc. (A) Kp,tr and Kp,cl, (B) K4,tr and K3,tr, (C) Kh

Toc,cl and KToc,cl

and (D) K6,tr and K5,tr Blue squares denote feasible parameter regionswhile white regions describe infeasible ones.

Model 2-1Trim (final)Model 2-1Trim is the final model and describes the inhibitory effect ofsgp130Fc in trans-signaling. As already explained in Chapter 1, Section1.2.4 sgp130Fc competes with the membran-bound gp130 as it also bindsto the complex of IL-6 and sgp80 forming the trimer IL-6:sgp80:sgp130Fc(IL6_sgp80_sgp130Fc). Eq. (3.21) describes association and dissocia-tion of sgp130Fc with the dimer complex IL-6:sgp80 mathematically.

0 =d[sgp130Fc]

dt= Ksgp130Fc · [IL6_sgp80_sgp130Fc]

−[IL6_sgp80] · [sgp130Fc] (3.21)

For model fitting in trans-signaling we took data from Fig. 2.2 D into ac-count. Here, Ba/F3 cells were stimulated with different concentrations


of sgp130Fc varying from 10−3 -104[ μgmL ].

Garbers et al. investigated the effect of sgp130Fc in Ba/F3 cells depend-ing on the concentrations of IL-6 and sgp80 [12]. We therefore have

different constraints for the ratio of [IL−6][sgp80] as additional model inputs

(Table 2.1 and Fig. 2.2D):

• [IL6][sgp80] = 0.05, (Fig. 2.2 (D) black curve)

• [IL6][sgp80] = 0.0033, (Fig. 2.2 (D) green curve)

• [IL6][sgp80] = 1, (Fig. 2.2 (D) blue curve)

• [IL6][sgp80] = 0.2 (Fig. 2.2 (D) brown curve).

By iterative cycles of parameter estimation we first took experimentaldata from Figure 4A in [12] into account identifying infeasible outer-approximates on the model parameters and variables. Then, the datasets from Figures 5A, 5B and 5C were added one after another and theestimation and bisectioning procedures were applied again.For Model 2-1Trim (final) a not invalid solution set was found. Fig. 3.5shows selected bisectioning plots for the model parameters. One can seethat for the parameters K2,cl and K1,cl as well as for K1,tr and Kh

1,tr setsof parameters could be determined that are guaranteed invalid. There-fore, linear dependencies can be formulated as additional constraintsto eliminate these regions (see Appendix A and Section 3.3 for a dis-cussion). For the parameters Kp,tr and Kp,cl the lower bounds couldbe improved from the initial bound of 0.01 to 9.29 and 24.16, respec-tively. Therefore, linear dependencies can be formulated as additionalconstraints to eliminate these regions (see Appendix C.2 and Section3.3 for a discussion). For the parameters Kp,tr and Kp,cl the lowerbounds could be optimized from the initial bound of 0.01 to 9.29 and24.16, respectively. The reason is the same as for Model 1Trim (Section3.2.1). As we fit our model outputs p_IL6_gp80_gp130 in classic andp_IL6_sgp80_gp130 in trans-signaling to the experimental determinedcell proliferation (Fig. 2.2 E and D) the phosphorylation constants canbe improved better than the other parameters. We have, for instance,no measuremnts on model states like IL6_sgp80_gp130 or gp130. Thusthe corresponding kinetic parameters cannot be improved significantly.


Figure 3.5.: Bisectioning results of parameters in Model 2-1Trim (final).(A) K2,cl and K1,cl (B) Kp,tr and Kp,cl, (C) K1,tr and Kh

1,tr and (D)Ksgp130, K2,cl and K1,cl While blue squares describe not invalid param-eter regions, white regions describe guaranteed invalid ones.

In the following plot (Fig. 3.6) the outer-approximates (red lines) of themodel variables IL-6 (A), gp130 (B), p_IL6_gp80_gp130 (C), sgp80(D), IL6_sgp80_gp130 (E) and p_IL6_sgp80_gp130 (F) are shown.Whereas the variables in (A), (B) and (C) are plotted in dependencyof the total amount of IL-6, (D), (E) and (F) are shown in depen-dency of the total amount of sgp130Fc. For the output variablesp_IL6_gp80_gp130 and p_IL6_sgp80_gp130 we used the experimen-tal determined standard-deviation as outer and initial bounds in the set-based estimation approach. One can see that these bounds could notbe improved further. For the other variables we set the initial boundsto [0,1] (Eq. (3.7)). As we used the IL-6 dependent proliferation assayfor model fitting in classic-signaling (Fig. 2.2 E) the amount and outer-bounds of free IL-6 molecules (IL-6 (norm.)) could be improved verywell in (A). In (B) the outer-bounds of model variable gp130 (norm.) isshown. The plot demonstrates that with increasing IL-6 concentrationsthe proportion of unbounded (non-complexed) gp130 receptor subunits


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FFigure 3.6.: Outer-bounding for state variables in Model 2-1Trim. (A)IL-6, (B) gp130 and (C) p_IL6_gp80_gp130 in dependency of thetotal amount of IL-6, (D) sgp80, (E) IL6_sgp80_gp130 and (F)p_IL6_sgp80_gp130 in dependency of the total amount of sgp130Fc.Red, bold lines indicate not invalid sets the respective variable whereasblack, thin lines indicate invalid ones.

decreases. This is in agreement with the proliferation assay as high IL-6levels lead to an increased cell proliferation. As the cell proliferation iscorrelated to the amount of phosphorylated cytokin-receptor complexesthe level of unbounded gp130 has to decrease. In plot (D) and (E) theouter-bounds of sgp80 and IL6_sgp80_gp130 in dependency of the to-tal amount of sgp130Fc is shown, respectively. One can see that for theformer the initial bounds could not be improved. However, for the state

3.3 Discussion 59

variable IL6_sgp80_gp130 the plot demonstrates that for high concen-trations of sgp130Fc (10−2-100) the upper bounds could be improved.This shows that with increasing inhibitor concentrations the formationof the trimer is inhibited. The outer-bounds on p_IL6_gp80_gp130 andp_IL6_sgp80_gp130 could not be improved further.

Validation of Model 2-1Trim (final)As described and explained in Chapter 2, Section 2.3.3 we have to val-idate the final model for further model predictions. The following plot(Fig. 3.7) shows the estimation results for Model 2-1Trim (final). Whilesimulated data is represented by a solid line, measurements are shownby a dotted line. Errorbars indicate experimentally determined standarddeviations. Data for model fitting were taken by [12] (Fig. 2.2 D, blackcurve). One can see that the simulated data lie within the experimentallydetermined errorbars. Thus, the sampled parmeter vector (see AppendixD) is able to reproduce the data.

Figure 3.7.: Validation of Model 2-1Trim (final).

3.3. Discussion

We presented different mathematical models for classic- and trans-signaling in IL-6 stimulated Ba/F3 cells expressing gp130 and gp80 ontheir cell surfaces. To explain how the final model was derived a modelingworkflow was developed and different hypotheses were tested. For model


estimation and identification we used a set-based estimation method al-lowing to take uncertain but bounded variables and measurements intoaccount. Wrong model hypotheses as well as guaranteed invalid sets ofparameters could be ruled out.

Model uncertainties and additional semi-quantative/qualitativedataThrough the development of the final model step by step we could madeguaranteed statements about the invalidity of different models rulingout wrong model hypotheses. During set-based parameter estimationall uncertainties are captured and taken into account. The estimationresults at hand depend on prior model knowledge. For instance, priorknowledge about binding affinities of serveral model species which canbe easily added as constraints to the optimization task.During estimation outer-bounds of the model parameters and variablesare found and regions that are not consistent with the data are sys-tematically discarded. Thus, no valid solution is lost. However, aswe do an outer-approximation we cannot make any guaranteed state-ment about the model validity. Estimated parameter sets through outer-approximation are over-estimated. Thus, the solution set can still in-clude invalid solutions which are falsely regarded as valid (false positive).Therefore, the next steps would consist to apply an inner-approximation.The inner-approximation returns sets of parameters which do not includefalse positive solutions however, guaranteed inner parameter bounds.

To use more of prior knowledge we took data from literature in whichthe binding affinities of several model species were determined. We thenderived constraints which were added to the estimation problems stepby step. Through a bisectioning algorithm over several pairs of param-eters we then could exclude parameter regions being inconsistent withthe experimental data and thus guaranteed invalid.As demonstrated in Fig. 3.2 B, Fig. 3.5 and Fig. 3.6 additional con-straints in a linear and/or non-linear way could be derived. For instance,we found a linear dependeny for the parameters K1 and K2 (denotationModel 1Trim, Fig. 3.3) and for K1,cl and K2,cl (denotation Model 2-1Trim (final), Fig. 3.6 A), respectively. However, we have to ask whetherthe additional constraints based on our literature search make sense withrespect to the biological meaning of the model parameters. K1 and K2

3.3 Discussion 61

and K1,cl and K2,cl, respectively are the equilibrium constants of theassociation/dissociation of IL-6 to gp80 and the IL-6:gp80 complex togp130. In this case the linear dependency follows from the sequenceof receptor formation and activation after IL-6 stimulation. First, IL-6binds to gp80 and then gp130 is recruted. This means that the order inwhich the receptors are assembled is defined and depend on each other.Based on the bisectioning result in Fig. 3.5 (B) and (C) and Fig. 3.6 (C)we could also derive a linear dependency between K4,tr and K3,tr, Kh

Toc,cl

and KToc,cl and Kh1,tr and K1,tr. The kinetic constants K4,tr and K3,tr

are parameters for the binding and dissociation for IL-6:sgp80 to gp130in trans-signaling. Thus, the linearity between these two parameters isnot suprising as we know that forth and back reactions are not decou-pled and depend on each other. The reason for the linear dependencybetween the parameters Kh

Toc,cl and KToc,cl as well as Kh1,tr and K1,tr is

the same as described above. The kinetic parameters are constants forthe association and dissociation of Toc to the membrane-bound receptorgp80 and IL-6 to the soluble receptor sgp80, respectively. Thus, they arecoupled at each other.One has also to keep in mind that the binding affinities from Table 3.1were mainly determined in vitro. The question is whether those valuesare comparable with in vivo data. Surely not but as the determinationof binding affinities in vivo is not a trivial task they are first steps inorder to obtain prior knowledge on model parameters in the set-basedestimation approach.

Binding to the receptor subunit gp130 is essential for IL-6 pathwayactivationWe started with the initial model (Model 0) modeling the associationof IL-6 to the receptor subunit gp80 in which the transducing recep-tor subunit gp130 is represented by gp80 species. Model 0 was thenextended by binding of the IL-6:gp80 complex to the receptor subunitgp130 (receptor trimerization, Model 1Trim). In the set-based estima-tion approach Model 0 was found to be guaranteed invalid meaning thatno parameter sets could be found reproducing the measurements. Model1Trim turned out not to be invalid. Receptor hexamerization was alsotested (Model 1Hex) and was also found not to be invalid. However,as we try to keep the model minimal we used Model 1Trim for further


extension. Compared to Model 1Trim, Model 1Hex has two more kineticparameters to be estimated and identified.The results from the set-based estimation for Model 0 and Model 1Trimimply that binding of the IL-6:gp80 complex to the receptor subunitgp130 is an essential step and component for pathway activation in IL-6stimulated cells.Model 2Trim and Model 2-1Trim present a combination of both path-ways, classic- and trans-signaling. Especially Model 2-1Trim is of im-portance as the model describes the inhibitory effects of sgp130Fc intrans-signaling. Both models turned out not to be invalid. To reducethe amount of model parameters we implemented Model 2-1TrimRedwhich neglects the association with the receptor subunit gp130 in classic-and trans-signaling. However, the model turned out to be invalid as noparametrization could be found that was consistent with the measure-ments. This finding underpins again the hypothesis that binding to gp130is a crucial step in IL-6 pathway activation and receptor internalization.Furthermore, our finding is also in agreement with, for instance, a studyby Quaiser et al. [34]. The authors propose a workflow for identifiabil-ity testing in the early JAK-STAT signaling for model simplifications.For consisteny of the model with experimental data association of theIL-6:gp80 complex with gp130 was necessary and could not be simplified.

Toc and its anti-inflammatory effectsBesides the final model, Model 2-1Trim in which the inhibitory effect ofsgp130Fc in trans-signaling is implemented we also derived a model thatinvestigates the inhibitory effects of Toc in IL-6-signaling.Toc was approved as treatment for rheumatoid arthritis. However, itwas demonstrated that the inhibitor not only suppresses inflammationprocesses but also positive functions of IL-6 [7]. Further investigationson a systems level would give new approaches towards a better under-standing of Toc, its function in suppressing inflammations and relatednegative side effects.

Model refinements and outlookFor further model refinements and improvements it is important to iden-tify the Black Boxes and to dissolve the different time-scales in oursystem by measuring, for instance, STAT1 and STAT3 activation in de-pendency of cytokin-receptor formation.

3.3 Discussion 63

Since binding of IL-6 to its related receptors gp80 and sgp80, respec-tively is critical for pathway activation it would be helpful to investigatealternative reaction kinetics. With the help of power-laws, for instance,one could model receptor cooperativity by estimating non-integer kineticorders [49]. However, in a power-law model the number of unknown pa-rameters can be higher than in a conventional kinetic model. Thereforethe optimization problem becomes bigger and more time-consuming tosolve.As Model 1Trim was able to reproduce the experimental data we usedthis model for further extensions. However, the investigation of Model1Hex in which receptor hexamerization is implemented would also bean approach to improve the IL-6 mediated pathway activation. Despitehexamerization is not fully understood one knows that after stimulationwith IL-6 complexes are formed consisting out of more than only onesubunit of each gp80 and gp130.As described in Chapter 3 we only used experimental data determinedwith the cell line Ba/F3-gp130-gp80. To improve the understandingof IL-6 pathway activation in Ba/F3 cells one should take the othercell line Ba/F3-gp130 into account. Ba/F3-gp130 cells do not ex-press gp80 receptor subunits but gp130. New experimental data byDr. Anna Dittrich (unpublished, OvGU Magdeburg, Chair of SystemsBiology, Prof. Dr. Fred Schaper) imply, for instance, that the regulationof the receptor expression in these two cell lines may undergo differentmechanisms. Thus, assumption 2b) from Chapter 2 has to be refined asthe concentrations of sgp80, gp80 and gp130 cannot be considered tobe constant over two days.

ConclusionTo conclude, we derived the final Model 2-1Trim which describes theinhibtion through sgp130Fc in IL-6 trans-signaling. The model wasfound to be consistent with the provided experimental data by Garberset al. and is therefore a good basis for further model investigation.Sgp130Fc is a promising inhibitor of IL-6-signaling und suppresses in-flammation processes which can lead to harmful diseases like rheumatoidarthritis, multiple sclerosis or cancer. It was initially found not to interferwith IL-6 classic-signaling however, Garbers et al. hypothesize the con-trary and stated that also classic-signaling is affected depending on the[IL−6][sgp80] -ratio. In patients who suffer from the diseases above it was found


that especially the concentration level of sgp80 increases dramatically.To make predictions concerning therapeutic effects of sgp130Fc undervarying sgp80 and IL-6 levels is therefore of great interest.

Chapter 4

Individualized therapeutictreatments with sgp130Fc

Sgp130Fc is a promsing anti-inflammatory drug as it was initially shownto specifically inhibting harmful IL-6 trans-signaling. This is in contrastto other substances that have already been authorised to the market.Tocilizumab, for example, inhibits both pathways, classic- and trans-signaling. Thus, it suppresses the IL-6 pathway globally and positivefunctions like its role in metabolism and immune system, too [7]. How-ever, it was shown that sgp130Fc also affects IL-6-classic-signaling de-pending on the pathophysiological levels of IL-6 and sgp80.This chapter aims to make predictions concerning an individual therapywith sgp130Fc and its treatment in patients. Thus, leading to a bet-ter understanding of the inhibitory effect of the designer protein underindividual conditions.

4.1. Motivation

Garbers et al. postulated that inhibition of classic-signaling by sgp130Fcis possible if sgp80 exceeds the level of IL-6 [12]. Then, sgp130Fc isable to shift the balance of free IL-6 molecules to the side of inac-tive IL-6:sgp80:sgp130Fc complexes (Fig. 1.5). Trapping of IL-6:sgp80complexes by sgp130Fc results in a complete elimination of free IL-6molecules and thereby to an indirect inhibition of the classical pathway.

65

66 4 Individualized therapeutic treatments with sgp130Fc

However, at which pathophysiological [IL−6][sgp80] ratios sgp130Fc is able to

affect classic-signaling is not determined so far.It is well examined that different types of tissues underlie different gp130and sgp80 expression levels and strongly vary from individual to individ-ual. As the therapeutic results of a treatment with sgp130Fc dependson these conditions an individualized medicine is necessary.In the follwing we analyse different ranges of sgp130Fc, IL-6 and sgp80.Based on our final model, Model 2-1Trim (Chapter 3, Section 3.2.2),we are using the introduced set-based estimation approach to define aconcentration range of sgp130Fc and a [IL−6]

[sgp80] ratio at which sgp130Fcblocks trans- but not classic-signaling and therefore specifically sup-presses inflammatory processes.

4.2. Mathematical procedure

To make predictions concerning different concentration levels ofsgp130Fc, sgp80 and IL-6 at which sgp130Fc specifically inhibits trans-signaling we have to determine new constraints.First, we introduce the model parameters Kclassic and Ktrans. Bothare specific parameters for IL-6-signaling which describe the quantityof phosphorylated receptor-complexes in classic- and trans-signaling, re-spectively. They depend on p_IL6_gp80_gp130 (phosphorylated recep-tor complex in classic-signaling) and p_IL6_sgp80_gp130 (phosphory-lated receptor complex in trans-signaling) as follows:

p_IL6_gp80_gp130 ≥ Kclassic (4.1)

p_IL6_sgp80_gp130 ≤ Ktrans (4.2)

Note that data, variables and inhibitor concentrations were normalizedto their corresponding maximum. Thus, the initial bounds for Kclassic

and Ktrans are set to:

Kclassic = [0, 1] (4.3)

Ktrans = [0, 1] (4.4)

4.2 Mathematical procedure 67

Fig. 4.1 illustrates the meaning of Eq. (4.1) and (4.2) graphically. Whilethe lower bound for the quantity of activated receptor-complexes in theclassical pathway is restricted by the corresponding parameter Kclassic

(red bold line), the upper bound for the quantity of activated receptor-complexes in trans-signaling is determined by Ktrans (blue bold line).Therefore, we determine an intersection O in which both pathways,classic- and trans-signaling overlap (Fig. 4.1, pink area and Eq. (4.5)).

O = |p_IL6_gp80_gp130 ∪ p_IL6_sgp80_gp130| (4.5)

Figure 4.1.: Graphical description for model predictions concerning ther-apeutic treatments with sgp130Fc. O: intersection of classic- and trans-signaling.

Within the intersection O and the pink area trans-and classic-signalingare both affected by sgp130Fc. The red area in Fig. 4.1 describes con-centration ranges at which

Ktrans ≥ Kclassic (4.6)


If this scenario occurs than classic-signaling is most affected by sgp130Fc.The blue area in Fig. 4.1 describes concentration ranges at which

Ktrans < Kclassic (4.7)

To define a concentration range of sgp130Fc at which only trans- butnot classic-signaling is affected we have to estimate the parameters suchthat Eq. (4.7) is satisfied. Then, only the trans-signaling pathway isaffected by sgp130Fc but not the classical one.

4.3. Ranges of sgp130Fc to inhibittrans-signaling specifically

We introduced two additional constraints (Eq. (4.1) and (4.2)) to figureout at which concentration ranges of sgp130Fc the inhibitor specificallyblocks IL-6 trans-signaling. The results are represented in the following.We determined outer-bounds for sgp130Fc, IL-6 and sgp80 in the set-based estimation approach at which the level of phosphorylated receptorcomplexes in trans-signaling is smaller than the level of phosphorylatedreceptor complexes in classic-signaling. Thus, in this scenario classic-signaling is not suppressed by sgp130Fc (Table 4.1, Fig. 4.2).

Table 4.1.: Prediction results concerning specific levels of IL-6 and sgp80and the input sgp130Fc. Abbreviations: norm.: normalized.

Concentration level (norm.) Concentration ratio

IL-6 = [0.25, 0.33]

sgp80 = [0.78, 0.99] [IL−6][sgp80] = [0.25, 0.42]

p_IL6_gp80_gp130 = [0.12, 0.37]p_IL6_sgp80_gp130 = [10−3, 0.21]sgp130Fc = [10−4, 1]Ktrans = [0, 0.37]Kclassic = [0, 0.37]

First, we found that for a specific inhibition of trans-signaling the level

4.3 Ranges of sgp130Fc to inhibit trans-signaling specifically 69

of sgp130Fc has to be within the range of [10−4, 1] (norm.). Theabbreviation ’norm.’ describes thereby the normalization of the inter-val to the maximal sgp130Fc concentration which was titrated to theBa/F3 cells in [12]. Thus the predicted sgp130Fc interval correspond

to a concentration range of [1, 104] [ng][mL] . Furthermore, the level of

sgp80 needs to be greater than that of IL-6. This is in agreement withmany pathophysiological situations at which the concentration of sgp80in the serum is also higher than that of IL-6 [11, 35]. At a concentra-

tion range of sgp130Fc ranging from [1, 104] [ng][mL] the ratio of [IL−6]

[sgp80]

varies from 25 - 42 %. This interval includes pathophysiological con-centrations with elevated serum levels of IL-6 and sgp80 compared tophysiological ranges. Fig. 4.2 shows the lower and upper bounds for IL-6(A), sgp80 (B), p_IL6_gp80_gp130 (C) and p_IL6_sgp80_gp130 (D)in dependency of sgp130Fc (equivalent to sgp130Fctot as input vector).While for IL-6 the outer-bounds could be determined to [0.25, 0.33] forsgp80 we found a range of [0.78, 0.99]. The outer-approximates forp_IL6_gp80_gp130 and p_IL6_sgp80_gp130 were estimated to [0.12,0.37] and [0.003, 0.21], respectively. For the parameters Kclassic andKtrans we found outer-bounds of [0, 0.37]. Please refer to the nextsection for a discussion.


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Figure 4.2.: Model predictions concerning therapeutic treatments withsgp130Fc. Estimated outer bounds for (A) IL-6, (B) sgp80, (C)p_IL6_gp80_gp130, (D) p_IL6_sgp80_gp130 in dependency of thetotal amount of sgp130Fc. (E) Optimized bounds for the parametersKclassic and Ktrans. The concentration range for sgp130Fc was deter-mined to sgp130Fc = sgp130Fctot = [10−4, 100].

4.4. Discussion

The therapeutic application of sgp130Fc to reduce dysregulatedinflammationsIn the past several studies concluded that inflammatory diseases like mul-tiple sclerosis, rheumatoid arthritis and Crohn’s disease accompany with

4.4 Discussion 71

altered levels of sgp80 and IL-6 compared to those in healthy individuals[50, 30]. It is well known that the anti-inflammatory activities of IL-6 aremediated by classic- whereas the pro-inflammatory responses are medi-ated by trans-signaling [2, 15, 24]. This is improtant since therapeuticinhibitors of IL-6-signaling like Toc and sgp130Fc have been developedto regulate abnormal signaling in the classical and trans pathway, respec-tively. The clinical application of Toc and the investigation of sgp130Fchave lead to promising results so far. However, Toc was shown to inhibitthe IL-6 pathway globally. Thus, positive functions of IL-6 signaling arealso affected and the application of Toc remains controversial.Sgp130Fc was recently found not only to inhibt trans- but also classic-signaling depending on the concentration levels of IL-6 and sgp80 [12].As clinicians are especially interested in the prevention of inflammationsresulting from trans-signaling in an individualized manner our aim wasto make predictions concerning concentration levels of IL-6, sgp80 andsgp130Fc at which the latter one specifically inhibts trans-signaling. Wefirst derived a mathematical model which combines IL-6 classic- andtrans-signaling and which describes the inhibitory effect of sgp130Fc inthe trans-signaling pathway. Through parameter sampling we could val-idate the model and perform systems biology anlyses concerning theimpact of concentration ranges of sgp130Fc.We first could limit the outer-bounds of sgp130Fc to [10−4, 1] (norm.).

This corresponds to a concentration range of [1, 104] [ng][mL] in [12]. Then,

the systems biology analyses revealed an interval of [0.25, 0.33] as outer-bounds of IL-6 and an interval of [0.78, 0.99] as outer-bounds of sgp80.Within the corresponding ratio bounds of [0.25, 0.42] classic-signalingand its anti-inflammatory processes are unaffected by sgp130Fc.The results demonstrate that the level of sgp80 is about three-fold higherthan that of IL-6 and transcends that considerably. This is again in agree-ment with serum levels in several diesease patterns [50, 30, 53, 13]. We

could not determine any [IL−6][sgp80] ratio at which only the trans-signaling

pathway is affected by sgp130Fc if the sgp130Fc conncentration rangewas smaller then 10−4 (norm.) and 1 [ng]

[mL] , respectively. One reasoncould be that at lower concentrations the amount of sgp130Fc moleculesis too less to trap complexes out of IL-6 and sgp80 thus inhibition of theclassic- and trans-signaling pathways is impossible.The results are of importance as we can make predictions concerning


therapeutic treatments with sgp130Fc in the individualize patient. Atargeted sgp130Fc application corresponding to the specific [IL−6]

[sgp80] ra-tio could reduce harmful effects of diseases like multiple sclerosis andrheumatoide arthritis due to trans-signaling. Furthermore, the resultscan also lead to targeted applications of sgp130Fc in cancer therapy. Inprostate cancer the disease correlates with increased IL-6 production andactivity of trans-signaling, respectively [14]. It could be, for instance, ex-amined how far a specific medication of the designer protein could lead toa decreased tumor proliferation. Depending on the patient’s tumor pro-gression first, the serum levels of sgp80 and IL-6 have to be determined.Then, an individualized treatment with sgp130Fc can be made and hasto be adjusted to the respective stadium of tumor growth. However,the results are still in silico and based on data generated in cell cultureexperiments. In the next step the model predictions have to be testedin the laboratory. There is a great need to design experiments testingscenarios with varying [IL−6]

[sgp80] ratios at certain sgp130Fc concentrationsranges. Later on the results from the wet lab can be added to the finalmodel to refine the previous predictions. Based on the experimental re-sults and if necessary the model has to be refined and/or extended again.

Comparison of the model predictions with sgp80 as well as IL-6levels during several diseasesThe results from several studies determining sgp80 and IL-6 levels duringvarious diseases revealed that the expression of both changes comparedto healthy individuals. The level of sgp80 mostly increases during dis-eases however, not significantly [50, 30, 53, 13]. In the study by Frielinget al. even a decrease was determined [10].In the case of IL-6 Giannitrapani et al. determined a considerable 4.6-fold elevation of the IL-6 concentration in patients suffering from hepa-tocellular carcinoma compared to the controls. Moreover, Nancey et al.measured in average a 1.6-fold increase of the IL-6 level in patients whoare affected by active Crohn’s disease compared to the healthy state.To compare the concentration ratios of [IL−6]

[sgp80] in both studies with theprediction results in our study the determined concentrations in the pa-tients have to be corrected. The data from literature is given in [ng]

[mL] . Toobtain data given in molar ratios we have to relate the concentrations forIL-6 and sgp80 to their corresponding molecular weights. Whereas for

4.4 Discussion 73

IL-6 a molecular weight of 21-28 kDa was found, for sgp80 a weight of50-55 kDa was determined [31, 25]. We took the average and calculatedthe factor by which the patient’s data has to be corrected to 2.14 (Table4.2).

Table 4.2.: Data correction to compare the determined values from lit-erature and the prediction results.

Molecular weight [kDa] Ref. Correction factor

IL-6 21-28 [31]sgp80 50-55 [25] 2.14

Our results show that the application of sgp130Fc strongly depends onthe serum concentrations of IL-6 and sgp80. The pathophysiologicalmolar ratios determined in most of the studies above are significantlysmaller than the resulting ratio based on our model predictions. OnlyGiannitrapani et al. determined a [IL−6]

[sgp80] ratio greater than the one pre-

dicted [13] (Table 4.3).

The determined pathophysiological [IL−6][sgp80] ratios from Table 4.3 are not

in accordance with our prediction results. One reason could be thatfor simplicity we neglected local gradients und assumed the system tobe ideally and well distributed. However, in vivo this does not occur.Especially the focus of inflammation in the patient’s serum show signifi-cantly elevated concentrations of IL-6 and sgp80. So far our final model,Model 2-1Trim, is not able to reproduce such scenarios. Furthermore, weimplemented differential equations assumed to be in steady-state. How-ever, receptor assembly in classic- and trans-signaling occur within thefirst minutes after IL-6 stimulation. As an outlook the development ofa dynamical model that is able to reproduce receptor internalization de-pending on the stimuli is necessary. It is obvious that pathophysiological[IL−6][sgp80] ratios greatly scatter between different diseases and patients. Thisunderpins that only an individualized treatment with the trans-signalinginhibitor sgp130Fc leads to the desired results and has to be adpated tothe individualized disease propagation.


Table 4.3.: Pathophysiological concentrations for sgp130Fc, IL-6 andsgp80 taken from literature. Abbreviation: LC: liver cirrhosis.

[IL−6][sgp80] sgp130Fc [ng]

[mL] Disease Ref.

Literatur 0.002 Crohn’s disease [30]0.70 liver carcinoma [13]0.0014 [500, 800] hepatitis C (LC) [27]0.019 sepsis [10]

Prediction [0.25, 0.42] [1, 104]

Possible treatment approachesIn the study by Migita et al. the concentration of natural occuring sgp130was determined to [500, 800] [ng]

[mL] at a [IL−6][sgp80] ratio of 0.0014 [27] (Table

4.3). As already known sgp130 is also able to inhibit IL-6 trans-signalinghowever not effective as sgp130Fc. During therapeutic treatment withsgp130Fc one should also consider the serum concentration of the nat-ural occuring inhibitor. An excessive presence of soluble gp130 species(natural occuring + sgp130Fc) may cause converse effects in the pa-tient. For example, we determined the range of sgp130Fc to [10−4, 1]

and [1, 104] [ng][mL] , respectively. We hypothesize that this is the con-

centration range at which only trans- signaling is affected by sgp130Fc(Fig. 4.3). However, what happens at higher sgp130Fc concentrations?In Fig. 4.3 the determined ranges of sgp130Fc are illustrated graphically.Below a concentration of 1 [ng]

[mL] neither trans- nor classic-signaling are

affected by sgp130Fc (white, ∅). Whereas within the range of 1-104

[ng][mL] only trans-signaling is suppressed by sgp130Fc (blue), beyond a

concentration of 104 [ng][mL] both signaling pathways are affected (pink).

If a certain dose of sgp130Fc is applied to the patient it could happenthat in addition with natural occuring sgp130 the concentration rangegreater than 104 [ng]

[mL] is reached at which also the classic pathway isaffected by sgp130Fc. During an individualized therapeutic treatmentone has to keep this in mind. It would be also possible to apply a com-bined treatment. For instance, depending on the [IL−6]

[sgp80] ratio clinicanscould administer sgp130Fc in combination with doses of sgp80. Thus

4.4 Discussion 75

shifting the [IL−6][sgp80] ratio at which sgp130Fc is able to inhibit harmful

trans-signaling.

Figure 4.3.: Schematic presentation of predicted concentration ranges atwhich IL-6 classic- and trans-signaling are affected by sgp130Fc. Abbre-viation: ∅: no trans- and classic-signaling affected.

ConclusionsTo conclude, the results of the systems biology analyses in this chap-ter are first steps towards individualized therapeutic treatments withsgp130Fc in patients suffering from serious diseases like multiple scle-rosis, rheumatoid arthritis and several types of cancer. A uniform ad-ministration of the designer protein is not possible as the therapy resultsstrongly depend on the indivdual concentration levels of IL-6 and sgp80in the serum. Our literature search demonstates that one can not deter-mine generalized IL−6

sgp80 ratios in diseases as they scatter from one personto another.Thus, there is a great demand on further investigations concerning atherapeutic usage of sgp130Fc not only to prevent dangerous inflamma-tory diseases but also to at least reduce the resulting side effects.

Chapter 5

Conclusions

5.1. Short summary

In the first part of the thesis we established a mathematical model whichdescribes IL-6-classic- and trans-signaling and the inhibitory effect ofsgp130Fc in Ba/F3-gp80-gp130 cells on a receptor level. For mathemat-ical modeling, parameter estimation and model validation we took datafrom Garbers et al. into account who determined proliferation curvesdepending on different stimuli like IL-6, sgp80, Toc and sgp130Fc [12].Through a set-based estimation approach we could find two models thatare not consistent with the experimental data and thus guaranteed in-valid. Model 1Trim and Model 2TrimRed were found to be invalid.Thus, we concluded that receptor trimerization and association of IL-6:gp80 and IL-6:sgp80 complexes with gp130, respectively is a crucialstep in IL-6-signaling. We finally derived Model 2-1Trim and could find aparametrization for that the model was consistent with the experimentaldata. The model was used to determine individualized therapeutic con-centration ranges of sgp130Fc at which the inhibitor specifically blocksIL-6 trans-signaling. The predictions were not consistent with data fromliterature. This is not suprinsingly as the pathophysiological IL-6 andsgp80 concentrations were taken from patients who suffer from diseaseslike sepsis or liver chirrosis. In such diseases IL-6 and sgp80 accumulatein the patient. However, due to the model assumptions in Chapter 2the final model is not able to explain local gradients. Thus, we cannot

77

78 5 Conclusions

simulate such scenarios so far.

5.2. Outlook and final remarks

We took data presented in [12] for model fitting and validation. However,the estimation results concerning the parameter bounds were not satisfy-ing. We found for each not invalid model that only the lower bounds ofthe corresponding phosphorylation constants (Kp,cl and Kp,tr) in classic-and trans-signaling could be improved. Garbers et al. determiend thecell proliferation of Ba/F3 cells depending on different stimuli. As wehave no measurements about the activation of further pathways like theJAK-STAT pathway we assumed that the determined cell proliferationis equivalent to phosphorylated cytokin-receptor complexes. Thus, mostof the information flow into the phosphorylation reactions in both path-ways due to the data available and model assumptions. Therefore, Kp,cl

and Kp,tr could be improved best. Although we found data in literatureto derive additional constraints on several model parameters the outer-bounds could not improved well.It is obvious that the derived steady-state model is not most suitable todescribe IL-6-signaling on a receptor level as we know that receptor inter-nalization and assembly are highly dynamic processes. Furthermore, weassume a linear transfer behavior from cell stimulation to proliferation.However, it is likely that this is not the case. At the moment Model2-1Trim is not able to describe the different time-scales and dynamicsdue to receptor assembly and internalization. In the next steps andto refine Model 2-1Trim it is necessary to dissolve the mentioned timescales and to identify the black boxes in the network model (Fig. 2.4).Thus, the derivative of a dynamical model is necessary. The availabilityof experimental data on the transfer behaviors from cell stimulation toreceptor formation and/or from receptor formation to JAK-STAT acti-vation would be very helpful. We then would obtain a better insightinto IL-6-classic-and trans-signaling and considerable differences in thedynamics.The establishment of the final model and the corresponding predic-tions provide an initial understanding of inhibition of trans- and classic-signaling through sgp130Fc in IL-6-stimulated Ba/F3 cells. The designer

5.2 Outlook and final remarks 79

protein is of great interest as it specifically prevents harmful inflamma-tions due to trans-signaling. However, there is a great need for furtherinvestigations on a systems as well as experimental level.

Appendix

A. Model equations

A.1. Model 0

0 =d[IL6]

dt= K1 · [R_complex] − [IL6] · [gp80]

0 =d[p_R_complex]

dt= Kp · [R_complex] − [p_R_complex]

with the following conserved moieties:

IL6_total = [IL6] + [R_complex] + [p_R_complex]

gp80_total = [gp80] + [R_complex]

+ [p_R_complex]

81

82 5 Conclusions

A.2. Model 1Trim

To derive Model 1Trim, the initial model was extended by trimerizationwith the receptor subunit gp130 and by the following equations:

0 =d[gp130]

dt= K2 · [IL6_gp80_gp130]

−[IL6_gp80] · [gp130]

0 =d[p_IL6_gp80_gp130]

dt= Kp · [IL6_gp80_gp130]

−[p_IL6_gp80_gp130]


IL6_total = [IL6] + [IL6_gp80] + [IL6_gp80_gp130]

+ [p_IL6_gp80_gp130]

gp80_total = [gp80] + [IL6_gp80]

+ [IL6_gp80_gp130] + [p_IL6_gp80_gp130]

gp130_total = [gp130] + [IL6_gp80_gp130]

+ p_IL6_gp80_gp130

A.3. Model 1Hex

0 =d[IL6_gp80_gp130]

dt= [IL6_gp80] · [gp130]

−K2 · [IL6_gp80_gp130]

−Khex · [IL6_gp80_gp130]

+Kphex · [IL6_gp80_gp130hex]

0 =d[p_IL6_gp80_gp130_hex]

dt= Kp · [IL6_gp80_gp130_hex]

−[p_IL6_gp80_gp130_hex]

A Model equations 83



+ [IL6_gp80_gp130]_hex

+ [p_IL6_gp80_gp130_hex]


+ [IL6_gp80_gp130] + [IL6_gp80_gp130_hex]

+ [p_IL6_gp80_gp130_hex]


+ [IL6_gp80_gp130_hex]

+ p_IL6_gp80_gp130_hex

84 5 Conclusions

A.4. Model 2Trim

0 =d[IL6]

dt= K1,cl · [IL6_gp80]

−[IL6] · [gp80]

+Kh1,tr · [IL6_sgp80]

−K1,tr · [IL6] · [sgp80]

0 =d[sgp80]

dt= K2,tr · [IL6_sgp80]

−[IL6] · [sgp80]

0 =d[gp130]

dt= [IL6_gp80_gp130]

−K2,cl · [IL6_gp80] · [gp130]

+K3,tr · [IL6_sgp80_gp130]

−K4,tr · [IL6_gp80] · [gp130]

0 =[IL6_sgp80_gp130]

dt= K5,tr · [IL6_sgp80] · [gp130]

−K6,tr · [IL6_sgp80_gp130]

−Kp,tr · [IL6_sgp80_gp130]

+[p_IL6_sgp80_gp130]

0 =d[p_IL6_gp80_gp130]

dt= Kp,cl · [IL6_gp80_gp130]

−[p_IL6_gp80_gp130]

0 =d[p_IL6_sgp80_gp130]

dt= Kp,tr · [IL6_sgp80_gp130]

−[p_IL6_sgp80_gp130]




+ [IL6_sgp80] + [IL6_sgp80_gp130]

+ [p_IL6_gp80_gp130] + [p_IL6_sgp80_gp130]


+ [IL6_gp80_gp130] + [p_IL6_gp80_gp130]

sgp80_total = [sgp80] + [IL6_sgp80]

+ [IL6_sgp80_gp130] + [p_IL6_sgp80_gp130]


+ [p_IL6_gp80_gp130] + [IL6_sgp80_gp130]

+ [p_IL6_sgp80_gp130]

Toc_total = [Toc] + [gp80_Toc]

86 5 Conclusions

A.5. Model 2TrimToc

d[gp80]

dt:= 0 = K1,cl · [IL6_gp80]

−[IL6] · [gp80]

−KToc,cl · [gp80] · [Toc]

+KhToc,cl · [gp80_Toc]

d[sgp80]

dt:= 0 = K2,tr · [IL6_sgp80]

−K3,tr · [IL6] · [sgp80]

−[sgp80] · [Toc]

+KToc,tr · [sgp80_Toc]

d[sgp80_Toc]

dt:= 0 = [sgp80] · [Toc]

−KToc,tr · [sgp80_Toc]


gp80_total = [gp80] + [gp80_Toc] + [IL6_gp80]

+ [IL6_gp80_gp130] + [p_IL6_gp80_gp130]

sgp80_total = [sgp80] + [sgp80_Toc] + [IL6_sgp80]

+ [IL6_sgp80_gp130] + [p_IL6_sgp80_gp130]

Toc_total = [Toc] + [gp80_Toc] + [sgp80_Toc]


A.6. Model 2TrimRed

0 =d[IL6]

dt= K1,cl · [R_complex, cl]

−[IL6] · [gp80]

+Kh1,tr · [R_complex, tr]

−[IL6] · [sgp80]

0 =d[p_R_complex, cl]

dt= Kp,cl · [R_complex, cl]

−[p_R_complex, cl]

0 =d[p_R_complex, tr]

dt= Kp,tr · [R_complex, tr]

−[p_R_complex, tr]


IL6_total = [IL6] + [R_complex, cl] + [p_R_complex, cl]

+ [R_complex, tr] + [p_R_complex, tr]

gp80_total = [gp80] + [R_complex, cl]

+ [p_R_complex, cl]

sgp80_total = [sgp80] + [R_complex, tr]

+ [p_R_complex, tr]

A.7. Model 2-1Trim (final)

d[sgp130]

dt:= 0 = Ksgp130 · [IL6_sgp80_sgp130]

−[IL6_sgp80] · [sgp130]


sgp130_total = [sgp130] + [IL6_sgp80_sgp130]

88 5 Conclusions

B. Model parameters and definitions

B Model parameters and definitions 89

Tab

le5.1

.:K

inet

icpa

ram

eter

s,de

fini

tion

san

dfind

ings

for

Mod

el0,

Mod

el1T

rim

and

Mod

el1H

exSym

bol

Defi

niti

onD

escr

ipti

onV

alue

Ref

.C

oncl

usio

ns

K1

kh 1

k1

Bin

ding

affini

tyof

IL-6

togp

80.

0.5

nM[5

4]K

To

ck

Toc

k1

On-

rate

ofbi

ndin

g

Toc

togp

80.

Kh T

oc

KT

oc

=2.5

4nM

[28]

Kh T

oc

≥K

To

c

Kh T

oc

kh T

oc

k1

Off

-rat

eof

bind

ing

Toc

togp

80.

Kh T

oc

KT

oc

≥K

1

K2

kh 2

k2

Bin

ding

affini

tyK

1≥

K2

ofIL

-6:g

p80

togp

130.

0.01

5nM

[54]

Kh T

oc

KT

oc

≥K

2

Kp

kp

kh p

Rec

ipro

caleq

uilib

rium

cons

tant

ofph

osph

oryl

ated

IL-6

:gp8

0:gp

130

com

plex

.K

hex

khex

k2

On-

rate

ofre

cept

orhe

xam

eriz

atio

n

Kp hex

kh hex

k2

Off

-rat

eof

rece

ptor

hexa

mer

izat

ion

90 5 Conclusions

Tab

le5.2

.:K

inet

icpa

ram

eter

s,de

fini

tion

san

dfind

ings

for

Mod

el2T

rim

,2-

1Tri

mT

ocan

d2-

1Tri

mSym

bol

Defi

niti

onD

escr

ipti

onV

alue

Ref

.C

oncl

usio

ns

K1,c

lk

h 1,c

l

k1,c

ls.

a.s.

a.s.

a.s.

a.

KT

oc,c

lk

Toc,c

l

k1,c

ls.

a.s.

a.s.

a.s.

a.

Kh T

oc,c

l

kh T

oc,c

l

k1,c

ls.

a.s.

a.s.

a.s.

a.

K2,c

lk

h 2,c

l

k2,c

ls.

a.s.

a.s.

a.s.

a.

Kp,c

lk

p,c

l

kh p,c

l

s.a.

s.a.

s.a.

s.a.

KT

oc,t

rk

h Toc,t

r

KT

oc,t

rB

indi

ngaffi

nity

ofT

octo

sgp8

0.

K1,t

rk

1,t

r

k1,c

lO

n-ra

teof

bind

ing

IL-6

tosg

p80.

Kh 1,t

r

kh 1,t

r

k1,c

lO

ff-r

ate

ofbi

ndin

gK

h 1,t

r

K1,t

r=

1.7

·10

−8

nM[5

1]

IL-6

tosg

p80.

Kh 1,t

r

K1,t

r=

2.2

·10

−1

0nM

[11]

Kh 1,t

r

K1,t

r=

5.5

·10

−8

nM[2

2]K

h 1,t

r≤

K1,t

r

K2,t

rk

h 1,t

r

k1,t

rB

indi

ngaffi

nity

ofIL

-6to

sgp8

0.K

h 1,t

r

K1,t

r=

K2

,tr

K3,t

rk

h 2,t

r

k2,c

lO

n-ra

teof

bind

ing

IL-6

:sgp

80to

gp13

0

B Model parameters and definitions 91

Tab

le5.2

.:K

inet

icpa

ram

eter

s,de

fini

tion

san

dfind

ings

for

Mod

el2T

rim

,2-

1Tri

mT

ocan

d2-

1Tri

mK

4,t

rk

2,t

r

k2,c

lO

ff-r

ate

ofbi

ndin

g

IL-6

:sgp

80to

gp13

0

K5,t

rk

2,t

r

kh p,t

r

K6,t

rk

h 2,t

r

kh p,t

r

K3,t

r

K4,t

r=

K6,t

r

K5,t

r

Kp,t

rk

p,t

r

kh p,t

r

Rec

ipro

caleq

uilib

irum

cons

tant

ofph

osph

oryl

ated

IL-6

:sgp

80:g

p130

com

plex

.

Ksg

p130

kh sg

p130

ksg

p130

Bin

ding

affini

tyof

Ksg

p130F

c≤

K1,c

l

sgp1

30Fc

toIL

-6:s

gp80

.5.9

7·1

0−

7nM

[47]

Ksg

p130F

c≤

K2,c

l

92 5 Conclusions

C. Constraints for set-based estimation

C.1. Data point weightening

Table 5.3.: Data points and weightening factors. Abbreviations:STDexp.: experimentally determined standard deviation, STDerr: stan-dar deviation with additional error.

Model Plot Data point STDexp STDerr

Model 1Trim Fig. 2.2 E 0.005 0.02 0.101 0.03 0.10

Model 2-1Trim Fig. 2.2 D 10−3 0.01 0.04

(final) [IL−6][sgp80] =0.2 10−1 0.04 0.17

1 0.02 0.18Fig. 2.2 D 10−1 0.08 0.22[IL−6][sgp80] =0.05 1 0.11 0.21

C Constraints for set-based estimation 93

C.2. Found dependencies between model parametersdue to additional semi-quatitative/qualitativedata

Table 5.4.: Additional constraints on model parameters for set-based op-timization based on bisectioning results

Constraints

Model 1Trim K2 ≤ K1 + 24.9975Model 2TrimToc Kp,tr ≥ 38.5

Kp,cl ≥ 46.71Kp,tr ≥ −0.42 · Kp,cl + 72.16K4,tr ≥ K3,tr − 24.9975

KhToc,cl ≤ −19.83 · KToc,cl + 219.83

KhToc,cl ≥ 9.16 · KToc,cl − 27.46

Kh1,tr ≥ 4.46 · K1,cl − 74.88

Model 2-1 Trim (final) K4,tr ≥ K3,tr − 12.50K2,cl ≤ K1,cl + 24.9975

Kh1,tr ≥ 4.46 · K1,cl − 62.44

Ksgp130 ≤ K1,cl + 24.9975Ksgp130 ≤ K2,cl + 24.9975

94 5 Conclusions

D. Validation of Model 2-1Trim

Table 5.5.: Result of parameter sampling

Symbol Value

K1,cl 13.89K2,cl 30.28Kp,cl 19.88K1,tr 60.38Kh

1,tr 27.22K2,tr 19.88K3,tr 1.53K4,tr 74.68K5,tr 26.10K6,tr 54.08Kp,tr 24.58Ksgp130 27.30

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