DISSERTATION submitted to the Combined Faculty for the Natural Sciences and for Mathematics of the Ruperto-Carola University of Heidelberg, Germany for the degree of Doctor of Natural Sciences Put forward by Master of Science Joanna Kawka Born in Chorzów, Poland Oral examination: Mathematical modeling of SGK1 dynamics in medulloblastoma tumor cells 1. Advisor: Prof. Dr. Anna Marciniak-Czochra 2. Advisor: Abstract This work is devoted to mathematical modeling of deregulation of the Wnt/β-catenin signaling pathway in medulloblastoma resulting in abnormal dynamics of target genes. Medulloblastoma is a brain tumor, mostly diagnosed in children. It is associated with several molecular genetic alterations. Specific aberrations of chromosome 6q, leading either to the chromosome copy-number loss (monosomy 6) or gain (trisomy 6), occur in two different subtypes of the tumor. The model is a nine-dimensional system of ordinary differential equations and describes nonlinear dynamics of the key ingredients of the signaling process. The model is based on the law of mass action and accounts for a two-compartment architecture of a cell consisting of the nucleus and cytoplasm. The model helps to understand molecular differences between the two medulloblastoma mutation subtypes that are associated with different patient prognosis. Our studies are based on a collaboration with the group of Prof. Dr. med. Stefan Pfister at the Division of Pediatric Neuro-oncology Research Group of the German Cancer Research Center (DKFZ). The model is used to evaluate data from the gene expression microarray data from the clinics in Heidelberg, Boston and Amsterdam. Numerical simulations lead to new biological hypotheses related to a significant role of the regulatory loop SGK1-GSK3β-MYC, a part of the Wnt/β-catenin signaling pathway. Simulations indicate the advantage of using the pharmacological inhibitor of SGK1 in patients with copy-number gain of chromosome 6q. Finally, the simulation results suggest a beneficial use of an adjuvant therapy in a trisomy 6 treatment. Mathematical analysis of the ordinary differential equations system confirms the wellposedness of the model and provides basic properties of the solutions. Supported by numerical analysis, we conclude about global stability of a unique positive equilibrium corresponding to the homeostasis of the system. We also tackle the parameter estimation problem using statistical assessment of the results and Gauss-Newton method. Sensitivity analysis provides insight into the role of model parameters. In particular, it confirms the sensitivity of the system to the parameter of SGK1 degradation. The model provides a powerful tool to study mechanistically the underlying process and to support the experiments. Zusammenfassung Die vorliegende Arbeit befasst sich mit der Deregulation des Wnt/β-catenin Signalweges in Medulloblastomen, und der daraus resultierenden Fehlregulation seiner Zielgene. Das Medulloblastom ist ein meist kindlicher Hirntumor, der mit charakteristischen genetischen Veränderungen einhergeht. Spezifische Aberrationen des Chromosoms 6q führen entweder zu einem Verlust (copy number loss, Monosomie 6) oder zu einem Zugewinn (copy number gain, Trisomie 6) von Genkopien und erlauben die Unterscheidung von zwei verschiedenen Medulloblastomsubtypen. Das in dieser Arbeit entwickelte Modell besteht aus einem neundimensionalen System gewöhnlicher Differentialgleichungen, die die nichtlineare Dynamik der Schlüsselkomponenten des Wnt/β-catenin Signalweges beschreiben. Das Modell beruht auf dem Massenwirkungsgesetz und berücksichtigt die Kompartimentierung eukaryontischer Zellen in Zellkern (Nukleus) und Zytoplasma. Es dient dem Verständnis der molekularen Unterschiede zwischen den beiden erwähnten prognostisch unterschiedlichen Medulloblastomsubtypen. Diese Arbeit basiert auf einer Kooperation mit der Abteilung Pädiatrische NeuroOnkologie (Prof. Dr. med. Stefan Pfister) des Deutschen Krebsforschungszentrums (DKFZ). Das entwickelte Modell wird zur Evaluation von Microarray Genexpressionsdaten verschiedener Kliniken in Heidelberg, Boston und Amsterdam verwendet. Numerische Simulationen führen zu neuen, biologisch relevanten Hypothesen im Hinblick auf die herausragende Rolle der Regulationsschleife SGK1-GSK3β-MYC, die Teil des Wnt/β-catenin Signalwegs ist. Die Simulationen legen den Einsatz pharmakologischer SGK1-Inhibitoren in Patienten mit copy number gain des Chromosoms 6 sowie positive Auswirkungen einer adjuvanten Therapie der Trisomie 6 nahe. Die mathematische Untersuchung des Differentialgleichungssystems bestätigt die Wohlgestelltheit des Modells und liefert grundlegende Eigenschaften seiner Lösungen. Gestützt durch numerische Methoden wird die globale Stabilität des eindeutigen positiven Gleichgewichtszustandes gezeigt, der der Homöostase des Systems entspricht. Mit Hilfe der Gauss-Newton-Methode wird das zugehörige Parameterschätzungsproblem betrachtet und die Ergebnisse werden statistisch untersucht. Eine Sensitivitätsanalyse bietet Einblick in die Auswirkungen verschiedener Modellparameter. Insbesondere bestätigt sie die Sensitivität des Systems bezüglich der Degradierungsrate von SGK1. Das entwickelte Modell stellt ein leistungsfähiges Werkzeug zum mechanistischen Studium der dem Medulloblastom zugrundeliegenden Prozesse dar und liefert Erklärungsansätze für experimentelle Befunde. Acknowledgements Homo doctus in se semper divitias habet "A learned person always has riches within himself" Phaedrus, a Roman fabulist To start is easier than to finish. I would like to give my great gratitude to everyone who helped me to accomplish my PhD thesis. Especially, I would like to thank Prof. Dr. Anna Marciniak-Czochra who gave me invaluable guidance during whole scientific research and showed me mathematics in the everyday world. Nevertheless, this project would not be possible without full support of Prof. Dr. med. Stefan Pfister who I am grateful for medical assistance and time, so costly in his profession. I cannot avoid mentioning his student, Dominik Sturm, who had patience to answer my biological questions. I am also thankful to Dr. Thomas Carraro, who contributed to my PhD thesis while working on the parameter estimation problem, for his helpful tuition and clever advise. Finally, I would like to thank my family, boyfriend and friends for encouragement and support coming from the heart. That, they never stopped believing in me and cheered me up with words and deeds. Contents 1 Introduction 1 2 Biological background 5 5 2.1 Medulloblastoma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Monosomy 6 and trisomy 6 . . . . . . . . . . . . . . . . . . . . 7 2.2 Signaling pathway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.1 Regulation of the Wnt/β-catenin signaling pathway . . . . . . . . 11 2.2.2 Regulation of gene targets of the Wnt/β-catenin signaling pathway 15 3 Systems biology approach 3.1 Processes described by the model . . . . . . . . . . . . . 3.1.1 Transcription and translation . . . . . . . . . . . 3.1.2 Spontaneous degradation . . . . . . . . . . . . . 3.1.3 Degradation through the interaction with proteins 3.1.4 Transport between the nucleus and cytoplasm . . 3.2 Formulation of the mathematical model . . . . . . . . . 3.2.1 From biology to equations . . . . . . . . . . . . 3.2.2 System of ordinary differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Model analysis 4.1 Existence and uniqueness of global solutions in ordinary differential system 4.2 Asymptotic behavior of model solutions . . . . . . . . . . . . . . . . . . 4.2.1 Steady States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Stability of ordinary differential system . . . . . . . . . . . . . . 4.2.3 Linearization of the medulloblastoma model . . . . . . . . . . . i 19 19 19 20 20 22 23 23 26 29 30 31 31 33 33 CONTENTS ii 5 6 7 Numerical simulations 5.1 Data, parameters and numerical tools . . . . . . . . . . . . . . . . . . . . 5.1.1 Patient data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Model calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Graphical user interface . . . . . . . . . . . . . . . . . . . . . . 5.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Monosomy 6, trisomy 6 and control case . . . . . . . . . . . . . 5.2.2 Comparison between the patient data and simulations based on the microarray data . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Discrepancy in dynamics of genes MYC and SGK1 in trisomy 6 and monosomy 6 . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Correlation between the SGK1 mRNA production and cMyc protein level in the nucleus . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 Role of inhibition in the SGK1 protein . . . . . . . . . . . . . . . 5.2.6 Effect of the GSK3β protein stabilization . . . . . . . . . . . . . Parameter estimation and optimal experimental design 6.1 Parameter estimation problem . . . . . . . . . . . . . . . . . . . 6.1.1 General formulation . . . . . . . . . . . . . . . . . . . . 6.1.2 Solving the nonlinear problem . . . . . . . . . . . . . . . 6.2 Confidence region in the form of ellipses . . . . . . . . . . . . . . 6.2.1 Definition of covariance matrix . . . . . . . . . . . . . . 6.2.2 Geometrical interpretation of confidence region . . . . . . 6.3 Parameter estimation in medulloblastoma model . . . . . . . . . . 6.3.1 Parameter sensitivity analysis . . . . . . . . . . . . . . . 6.3.2 Parameter nonlinearities, coupling and measurement error 6.3.3 Variation in parameter values . . . . . . . . . . . . . . . . 6.4 Optimal experimental design . . . . . . . . . . . . . . . . . . . . 6.4.1 Sequential design . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 41 42 43 46 47 47 53 55 67 67 69 73 73 74 76 79 79 79 82 82 89 101 108 109 121 CHAPTER 1 Introduction Increased contribution of mathematical modeling is observed in many sections of modern life. It is involved in natural science, social science and engineering. Well defined models can describe complicated systems, such as signaling cell pathways, and can help in understanding the processes, as well as answering biological questions without time and money consuming experiments. One can monitor the process development and perceive its characteristic attributes. In the context of medulloblastoma, we model intracellular signaling processes belonging to the Wnt/β-catenin pathway, which are decisive for dynamics of the disease. This work is based on systems biology approach to understand medulloblastoma dynamics, its clinical prognosis and to propose possible treatments. Medulloblastoma is a solid brain tumor linked to mutations related to chromosome 6q. The resulting aberration of the number of gene copies leads to abnormal dynamics of the Wnt/β-catenin pathway, which causes increased or decreased synthesis of several proteins influencing in turn cell proliferation, apoptosis and consequently the tumor growth. The aim is to develop a mathematical model which provides an insight into the dynamics of medulloblastoma. Our approach is based on the methods of mathematical modeling, analysis, simulation and parameter estimation. Cancer development and growth of solid tumors is a multiscale process, where the macroscopically observed growth of tumor is a result of the abnormal cell turnover and movement, which is governed by intracellular signaling. In this work, we focus on a mathematical description of the intracellular processes. The Wnt/β-catenin signaling pathway attracts a lot of interests in experimental biology [13], [31], [75]. Since it plays an important role in developmental processes [76] and 1 2 CHAPTER 1. INTRODUCTION cancerogenesis [36], [45], the Wnt/β-catenin signaling pathway and related proteins are a broad area of investigations, also using the tools of mathematical modeling. The pioneers in modeling of the core elements of the pathway were Reinhart Heinrich, Roland Krüger and Ethan Lee in the context of colon cancer (for review, see [40], [42]). In this thesis, we do not focus on the system delineated by Heinrich’s work. We develop a model that involves the ingredients of the pathway which are crucial for medulloblastoma tumor dynamics. Medulloblastoma may be caused by several mutations. We consider monosomy and trisomy of chromosome 6q. The aberration of chromosome 6 in long arm refers to monosomy, when there is a loss of chromosome 6q. In opposite, the gain of chromosome 6q is ascribed to the trisomy. The striking fact is that the two types of medulloblastoma reveal extremely different prognosis. The deregulation of the Wnt/β-catenin signaling pathway has a huge impact on the downstream target genes. The cascade of subsequent abnormal interactions between these genes results in tumorigenesis. In the case of monosomy 6 and trisomy 6, we observe the deregulation of mRNA levels. A question arises, how to investigate such complex problem in the context of modeling. Based on the literature and collaboration with the group of Prof. Dr. med. Stefan Pfister at the Division of Pediatric Neuro-oncology Research Group of the German Cancer Research Center (DKFZ), we develop an ordinary differential system, describing dynamics of the key ingredients of the Wnt/β-catenin signaling system. To describe intracellular dynamics, we consider two-compartment approach taking into account transport between the cell nucleus and cytoplasm. To model the system, we reduce it to its main components. This enables obtaining a clear and comprehensive model, which still describes the complex system. We focus on the interplay between MYC and serum and glucocorticoidinducible kinase 1 (SGK1), which are the products of the Wnt/β-catenin signaling pathway, and glycogen synthase kinase (GSK3β). Numerical simulations of the model yield a better understanding of the process. In particular, the model indicates the importance of the SGK1 gene in the development of medulloblastoma, where target genes of the Wnt/β-catenin signaling pathway show strong aberration. Finally, we employ the gene expression microarray data obtained from the clinics due to cooperation with Stefan Pfister and Dominik Sturm for two types of medulloblastoma, monosomy and trisomy of chromosome 6q. We study dynamics of the system and investigate to what extent the prognosis is related to the deregulation of the MYC and SGK1 mRNAs. After a series of numerical simulations, we formulate biological hypotheses on the significant role of loop SGK1GSK3β-MYC (part of Wnt/β-catenin signaling pathway). We also propose a new therapy based on a pharmacological inhibitor as an adjuvant therapy to the one that is nowadays 3 in use. The outline of this manuscript is the following. We introduce biological background in Chapter 2. We explain the differences between two types of medulloblastoma. Then, we bring closer molecular details of the Wnt/β-catenin signaling pathway and target genes of the pathway. In Chapter 3 we elucidate dynamics of the system. We describe all underlying processes such as transcription, translation, phosphorylation, protein degradation and transport between the nucleus and cytoplasm. Consistently, we formulate mathematical equations based on the biological phenomena and the law of mass action. Chapter 4 is devoted to the mathematical properties of the model. We discuss wellpossedness in the sense of classical theory of ordinary differential equations. We prove global existence and uniqueness of the solutions and present numerical simulations of the system for several sets of initial conditions. Supported by numerical calculations, we show that the solution is globally, asymptotically stable with the equilibrium point classified as focus. Numerical solutions of the model are presented in Chapter 5. Various aspects of particular protein kinetics are demonstrated. We present the solutions and compare them to reveal the biological meaning. The crucial role of SGK1 in the system is explored. The suggested adjuvant therapy shows the influence of the pharmacological inhibitor of the SGK1 protein to the homeostasis in the investigated system. To study the parameter estimation problem, we deliver the commonly used methods in Chapter 6. We perform the sensitivity analysis and run Gauss-Newton’s algorithm to identify the parameters as well as we provide the statistical assessment of the solution. From the sensitivity analysis, we show the importance of the SGK1 gene. At the end of the chapter the identified parameters are refined by the method of sequential optimal experimental design and hence we obtain a set of admissible estimates for the model. The final Chapter 7 encompasses a summary and outline of the future research directions. This completes the content of the manuscript starting from the biology, through the modeling and finishing on the optimal experimental design. 4 CHAPTER 1. INTRODUCTION CHAPTER 2 Biological background A tumor, defined as abnormal mass of tissue, may be benign or malignant. In this chapter we present the nature of a specific brain tumor type Medulloblastoma, statistics and the biological issues that lie beyond the topic. We focus on the Wnt/β-catenin signaling pathway in medulloblastoma and in the end we derive a scheme of the modeled system, where we consider the fundamental biological processes. 2.1 Medulloblastoma Medulloblastoma is a malignant brain tumor that mainly affects children. 85% of all medulloblastoma cases are below the age of eighteen [39]. The mortality during the first two years after diagnosis oscillates between 10% and 15% [38]. The rapidly-growing tumor is localized in the brain area which controls speech, balance, and posture. Depending on the age of the patient the symptoms can be headaches, vomiting, nausea, gait abnormality, eye squint, sensory neuropathy [2], [32], [61]. In the early stage of tumor development the symptoms are often not distinctive and for this reason the disease is not apparent. Fast recog- Figure 2.1: Computer tomography brain nition gives better chance for the children re- scan showing a medulloblastoma tumor (circovery, however symptoms are too general and cled in red). Image thanks to S. Pfister. often the disease is diagnosed in late stage. The diagnosis is confirmed after magnetic resonance imaging (MRI) and furthermore detailed 5 6 CHAPTER 2. BIOLOGICAL BACKGROUND tests are carried out (e.g., analysis of the cerebrospinal fluid). If the medulloblastoma is recognized, the treatment involves resection of the tumor and then, depending on the patient’s stage and molecular aberration, radiotherapy, chemotherapy or both therapies are applied [58], [62]. Biological evidence 2.1 Medulloblastoma is attributed to several mutations in the tumor genome, such as 17q gain, i(17q), MYC/MYCN amplification, 6q gain, 6q loss and Wnt pathway activation ([55], [70], Figure 2.2). Importantly, prognosis depends on the type of mutation causing the disease as well as on the stage of diagnosis (e.g., size and extent of the tumor, etc.). In general, tumors can be either non-invasive or metastasize through the cerebrospinal fluid [54]. Nowadays, the treatment technics are improved, but still the side effects are a big concern. Although many children recover, they can have endocrinological, cognitive and neurological problems, such as speech difficulties. Figure 2.2: Overall survival probabilities for different chromosomal mutations in medulloblastoma. Adapted from Pfister, 2009. 2.1. MEDULLOBLASTOMA 7 2.1.1 Monosomy 6 and trisomy 6 In a healthy system, there are two (almost) identical copies of each chromosome. Each copy has a long and short arm (Figure 2.3). The short arm is described as p-arm, where p is taken from the french word petit - small. The long arm is called q-arm, where q is just Figure 2.3: Diagram of chromosome the next letter from the Latin alphabet. Trisomy is a structure pointing long and short mutation in which there are three copies of chromo- arms. Copyright ⃝motifolio.com c some 6 and in monosomy there is only one copy of chromosome 6. Our research is devoted to understanding the role of signaling pathways in two types of medulloblastoma, carrying either a monosomy or a trisomy of the long arm of chromosome 6q. Monosomy is loss of one chromosome copy (6q loss) and trisomy is gain of one chromosome copy (6q gain). Figure 2.4: A genomic hybridization profiles of tumors with monosomy 6 and trisomy 6. Adapted from Pfister, 2009. The DNA hybridization profiles from the two types of tumor show the discrepancy between chromosomal aberrations (Figure 2.4). Each of these mutations is associated with a strikingly different patient prognosis (Figure 2.5). 8 CHAPTER 2. BIOLOGICAL BACKGROUND Biological evidence 2.2 Medulloblastoma patients harboring a trisomy of chromosome 6q in the tumor genome are found to have a poor prognosis, while tumors characterized by a monosomy of chromosome 6q has a good prognosis following the medical treatment [55]. Monosomy 6 is always found in combination with the β-catenin mutation, leading to constitutive Wnt signaling activation. It was discovered that mutation of β-catenin in medulloblastoma tumor cells was associated with a good prognosis in a pediatric patient [19]. Figure 2.5: Kaplan Meier curves showing the estimated overall survival probabilities for medulloblastoma patients according to the copy-number status of chromosome arm 6q in the tumor. Adapted from Pfister, 2009. The 6q loss and 6q gain appear to induce the deregulation of the expression of MYC and SGK1 in the mutated cells (for biological notation, see Table 2.1). MYC is a transcription factor that regulates the expression of a number of genes and is involved in the biological processes such as cell growth and proliferation [14]. SGK1 is responsible for the intracellular transport and cell survival [7], [68]. Both genes seem to play an important role in the cell’s homeostasis and their deregulation can affect cellular processes finally resulting in carcinogenesis. In the case of monosomy of chromosome 6q, we observe a downregulation of the mRNA level of SGK1, whereas its upregulation is detected in the case of trisomy 6 (Figure 2.1. MEDULLOBLASTOMA 9 Biological notation MYC gene (italic) MYC mRNA (italic) cMyc protein (straight) Table 2.1: Biological notation. Example provided for the MYC gene. 2.6). Interestingly, the SGK1 gene is located on chromosome 6q, which suggests that a disruption in the chromosome balance alters the mRNA level of SGK1. However, the mechanism has not yet been explained. The concomitant upregulation of the mRNA level of MYC is found in both types of medulloblastoma. Since pediatric monosomy 6 of medulloblastoma is always related to the β-catenin mutation, the increase of the β-catenin translocation to the nucleus may explain upregulation of its target gene MYC [39]. The reason of MYC mRNA upregulation in trisomy 6 is not understood. qRT PCR 2009: SGK1 expression levels in primary tumors real-time quantitative PCR (n=41) real-time quantitative PCR (n=41) 16 6 log2-ratio [tumor / control] fold change [tumor / control] 14 12 10 8 6 4 2 4 2 0 -2 -4 0 -6 loss balanced loss gain 0.25 2.68 5.65 gain chr. 6q copy-number status chr. 6q copy-number status mean fold change: balanced mean log2-ratio: -2.79 0.20 2.22 Figure 2.6: Fold change and log2 -ratio of SGK1 levels for 41 medulloblastoma samples. Comparison between different medulloblastoma subgroups: monosomy 6 (6q loss), trisomy 6 (6q gain) and "balanced", which stands for the medulloblastoma with no aberration in chromosome 6. Data thanks to S. Pfister and D. Sturm. CHAPTER 2. BIOLOGICAL BACKGROUND 10 By definition fold change we understand the ratio of mRNA extracted from the tumor cells and control cells. Control cells are normal cells which are not affected by tumor. 2.2 Signaling pathway Signaling pathway is a cascade of biochemical reactions taking place in a cell. It takes place as a response of the cells to signals they receive from environment or from other cells. It processes information from cell membrane to the genome. A membrane is equipped with receptors, which may be occupied by ligands. Those ligands are the molecules responsible for the signaling activation. If ligand binds to the receptor the cascade of protein reactions leads to an activation/deactivation of a particular set of genes (Figure 2.7). Figure 2.7: Scheme of the signaling pathway from a cell surface ligand-receptor to the nucleus. c Copyright ⃝motifolio.com Any changes on the molecular level that occur upon the activation of signal transmi- 2.2. SIGNALING PATHWAY 11 tion is reflected in a cell fate, thus tissue fate (Figure 2.8). Further, different pathways often interact with each other creating a signaling network [24]. The proper work of the network is substantial to maintain the cellular homeostasis. Mutations taking place during cancerogenesis lead to perturbations in the corresponding signaling pathways and hence to abnormal gene expression. Here, we delineate the specific pathway which is called Wnt/β-catenin signaling pathway in its natural and mutated state. Figure 2.8: Scheme of the cell fate dependent on the multiple extracellular signals. Copyright c ⃝motifolio.com 2.2.1 Regulation of the Wnt/β-catenin signaling pathway The Wnt/β-catenin signaling pathway is one of the most important pathways in human cells. The Wnt gene was discovered already 30 years ago [35]. The knowledge about its function and role in the cell increased during the recent years [30]. The so called canonical Wnt pathway describes a cascade of reactions regulating embryonal development and adult tissue maintenance [76]. Substantially, destabilization of the Wnt pathway leads to tumorigenesis and was found in many cancers [36], [45]. 12 CHAPTER 2. BIOLOGICAL BACKGROUND Normal situation in the absence of Wnt ligands Revealed in the former, cells can respond to the extracellular environment via ligands that bind to the receptors in the cell membrane. In the Wnt signaling pathway, if there are no Wnt ligands which could bind to low-density lipoprotein receptor-related proteins (LRP), no activation of cell surface receptor proteins (Frizzled) occurs. The multiprotein destruction complex captures then and phosphorylates β-catenin as the complex cannot be destroyed by inactive Frizzled [25], [71]. Such phosphorylated β-catenin is marked for degradation and in consequence there is no β-catenin translocation to the nucleus, so the transcription of genes, such as SGK1 and MYC, cannot take place (Figure 2.9). Figure 2.9: Schematic diagram of the Wnt signaling pathway in the absence of Wnt ligands. If there is deficiency of Wnt ligands, β-catenin is degraded via a destruction complex. Consistently, no transcription of its target genes is possible. 2.2. SIGNALING PATHWAY 13 Normal situation in the presence of Wnt ligands If on the cell surface Wnt ligands bind to LRP receptors, Frizzled is activated. The consequence of the latter is inhibition of the multiprotein destruction complex, leading to β-catenin accumulation in the cytoplasm. This protein abundance entails the translocation to the nucleus, where β-catenin binds to transcription factors (TCF/LEF) and initiates the transcription of particular genes [45], MYC and SGK1, among others (Figure 2.10). Figure 2.10: Schematic diagram of activated Wnt pathway. In the presence of Wnt ligands the destruction complex is destroyed by Frizzled and hence β-catenin is not phosphorylated, thus not degraded. The subsequent effect of β-catenin accumulation in the cytoplasm leads to its nuclear translocation. Then, the protein binds to TCF/LEF transcription family and initiates the transcription of several genes (e.g., SGK1 and MYC). 14 CHAPTER 2. BIOLOGICAL BACKGROUND Mutation in the Wnt pathway The pathway is, however, not only modulated by the absence or presence of Wnt signaling. There are many mutations which give rise to an aberration (a mutation of any protein from the destruction complex), where the absence of Wnt ligands does not ensure the degradation of β-catenin. In medulloblastoma the mutation influences directly β-catenin, causing its resistance to degradation. The destruction complex cannot bind to the mutated β-catenin and therefore this protein is not degraded. Consecutive accumulation in the cytoplasm triggers β-catenin translocation to the nucleus, which results in abundant transcription of the downstream target genes of the pathway [22], [78], (Figure 2.11). This, together with β-catenin mutation, is always found in pediatric medulloblastoma carrying a monosomy 6. The corresponding medulloblastoma subtype is called WNT subgroup MBs. Figure 2.11: Schematic diagram of the Wnt pathway with mutated β-catenin. The absence of Wnt ligands should lead to β-catenin degradation due to activity of the destruction complex. However, a mutation on residues S33 and S37 (specific monomers of protein) of β-catenin blocks the destructive activity of the complex [57], [64]. β-catenin cannot be phosphorylated and for this reason is not marked for degradation. The increased stability of the protein implies the nuclear translocation and further transcription of SGK1 and MYC. 2.2. SIGNALING PATHWAY 15 2.2.2 Regulation of gene targets of the Wnt/β-catenin signaling pathway In cells that are not deregulated, presence of β-catenin in the nucleus leads to the transcription of MYC [43] and SGK1 [16] at normal level. Then the translation of their mRNA is observed in the cytoplasm. After translation each protein is transported to an appropriate cellular compartment depending on the role in the cell. The SGK1 protein remains in the cytoplasm if no additional signal is activated [21] and cMyc protein moves to the nucleus, where it functions as a transcription factor. SGK1 in the cytoplasm can bind to kinase GSK3β [18] and phosphorylate it, marking for degradation [3]. GSK3β is a protein that can shuttle between the cytoplasm and nucleus, and when it is in the cytoplasm, it can be degraded via an interaction with SGK1 [3], [77]. However, when GSK3β is shifted to the nucleus, it may bind to cMyc and phosphorylate this protein targeting it for degradation [26]. SGK1 and GSK3β return to the previous states after their phosphorylating activity, ready for new binding. Since the transport between the cytoplasm and the nucleus is an essential process in the cell, it is necessary to take into consideration the transport rate as well as difference of the volumes of these two compartments. Additionally, some substrates may exist in different compartments of the cell and in the framework of ordinary differential equation modeling they have to be considered as separate variables. In Figure 2.12, we depict the modeled system. Remark 2.1 We use subscripts t and p to distinguish between the transcripts and the proteins, respectively. We do the same for the nuclear and cytoplasmic localization using subscripts n and c, respectively. When a protein binds to another protein, the process of phosphorylation takes place and then no other activity of these proteins is performed. For this reason, we label such state as occupied, denoting it by subscript occ. 16 CHAPTER 2. BIOLOGICAL BACKGROUND Figure 2.12: Schematic diagram of the modeled biological dynamics. Abbreviations: SGK1t - concentration of the SGK1 mRNA (transcript), SGK1p - cytoplasmic concentration of the SGK1 protein, M Y Ct - concentration of the MYC mRNA (transcript), cM ycpc - cytoplasmic concentration of the cMyc protein, cM ycpn - nuclear concentration of the cMyc protein, GSK3βc - cytoplasmic concentration of the GSK3β protein, GSK3βn - nuclear concentration of the GSK3β protein, GSK3βocc - phosphorylating GSK3β in a complex with cMyc, SGK1occ - phosphorylating SGK1 in a complex with GSK3β. 2.2. SIGNALING PATHWAY 17 Summary of Chapter 2 In this chapter we outlined the biological background that helps in understanding the basic processes in medulloblastoma. We presented medulloblastoma as a brain tumor, mostly found in children and diagnosed with different prognosis. To describe the phenomena found in medulloblastoma on the cellular level, we introduced a molecular overview of the signaling pathway, particularly the Wnt/β-catenin signaling pathway. The chosen key players compose an interesting system, which we want to model to investigate monosomy 6 and trisomy 6 subgroups of the medulloblastoma. 18 CHAPTER 2. BIOLOGICAL BACKGROUND CHAPTER 3 Systems biology approach to understand the dynamics of medulloblastoma In this chapter we develop a mathematical model of medulloblastoma. Our model is focused on a part of the Wnt signaling pathway, whose perturbations are crucial for the disease development and dynamics. We discuss biological phenomena involved in the modeled system. A systematic description of the underlying processes delivers a base to formulate autonomous ordinary differential equations (ODEs) of the interactions between particular molecules. We propose a model having the form of nine nonlinear ODEs. Our model reflects chemical kinetics of the reactants’ concentrations in the nucleus-cytoplasm environment highlighting the discrepancy in prognosis of two types of medulloblastoma associated with a monosomy and trisomy of chromosome 6q. 3.1 Processes described by the model We propose a model of the time dynamics of four major biochemical species involved in the system. We consider β-catenin, SGK1, MYC and GSK3β, accounting for the different stages in their synthesis (mRNA, protein), different intracellular location (nucleus, cytoplasm) and different biological processes, see Figure 2.12. 3.1.1 Transcription and translation Production of MYC and SGK1 is mainly regulated by β-catenin and members of TCF/LEF protein family. TCF/LEF, however, is present in abundance in comparison to other components of the system, therefore for sake of simplicity in our model we neglect its impact. 19 CHAPTER 3. SYSTEMS BIOLOGY APPROACH 20 The main role for the intensity of the transcription of MYC and SGK1 is credited to the amount of the concentration of β-catenin. We assume that both transcription rates are equal to t1 and t3 , respectively, whereas translation is assigned by translation rates s2 and s4 for the SGK1 mRNA and for the MYC mRNA, respectively t 1 β-catenin − → SGK1 mRNA, t3 β-catenin − → MYC mRNA, s2 SGK1 mRNA − → SGK1 protein, s4 MYC mRNA − → cMyc protein. 3.1.2 Spontaneous degradation The spontaneous degradation of the SGK1 mRNA and MYC mRNA is described by coefficients d1 and d3 , respectively. Their proteins are degraded spontaneously at rates d2 for SGK1 and d4 for cMyc. The degradation of GSK3β is activated by various proteins (e.g., Dishevelled, PKB and SGK1 [77]). Neglecting details of the dynamics of Dishevelled and PKB, we model a spontaneous degradation of GSK3β protein as a linear process at a degradation rate do . Thus, we have d 1 SGK1 mRNA − → degraded SGK1 mRNA, d3 MYC mRNA − → degraded MYC mRNA, d2 SGK1 protein − → degraded SGK1 protein, d4 cMyc protein − → degraded cMyc protein, do GSK3β protein − → degraded GSK3β protein. 3.1.3 Degradation through the interaction with proteins We consider two degradation processes mediated by proteins in the investigated system. In particular, one of the cytoplasmic isoforms of SGK1 binds to GSK3β with binding rate d6 and promotes its degradation via phosphorylation. Subsequent dissociation of the [SGK1·GSK3β] complex occurs at a rate p9 , and results in releasing SGK1 protein molecules that are active again. We do not go into details of the process of phosphorylated GSK3β degradation, so we assume that this process is immediate, taking place simultaneously with SGK1 releasing from the complex [3], d 6 SGK1 + GSK3β − → [SGK1·GSK3β], p9 [SGK1·GSK3β] − → SGK1. 3.1. PROCESSES DESCRIBED BY THE MODEL 21 A scheme of these processes is presented in Figure 3.1. Figure 3.1: Steps in the process of GSK3β phosphorylation by SGK1. Figure 3.2: Steps in the process of cMyc phosphorylation by GSK3β. Additionaly, in the nucleus GSK3β binds to cMyc, phosphorylates and leads to cMyc degradation, see Figure 3.2. It is important to note that after dissociation of the complex, GSK3β is ready for new binding and cMyc undergoes a degradation [23], CHAPTER 3. SYSTEMS BIOLOGY APPROACH 22 d 5 GSK3β + cMyc − → [GSK3β·cMyc], p8 [GSK3β·cMyc] − → GSK3β. 3.1.4 Transport between the nucleus and cytoplasm When describing signaling pathway the architecture of the intracellular space can be treated as a two compartment structure consisting of the cell nucleus and the cell cytoplasm. We model transport between two compartments using scaling factor kv which reflects the difference in volume between the nucleus and cytoplasm [41], [44]. Two compartment model of transport processes in cell A volume of the substrate in the nucleus and cytoplasm is assigned by sn and sc , respectively. By the law of mass action the total amount of substrate sn + sc is constant. Additionally, the speed of transport is proportional to the concentration of the transported substrate. Therefore, equations delineating the transport of the substrate concentrations in the nucleus and cytoplasm can be formulated in the form dsc = Kn S n − Kc S c dt / : Vc , (3.1) dsn = Kc Sc − Kn Sn / : Vn , (3.2) dt where Sn and Sc are the nuclear and cytoplasmic substrate concentrations. Kn and Kc are coefficients of the cytoplasmic and nuclear transport, respectively. Dividing equation (3.1) by volume of the cytoplasm Vc and equation (3.2) by volume of the nucleus Vn results in dSc = k n Sn − k c Sc , dt (3.3) dSn = kv kc Sc − kv kn Sn , (3.4) dt where kc = Kc /Vc and kn = Kn /Vc are scaled transport coefficients, and kv = Vc /Vn is a scaling factor. kv factor is used in the following equations to scale the transport rate between the nucleus and cytoplasm. It reflects differences between the size of the nucleus and cyto- 3.2. FORMULATION OF THE MATHEMATICAL MODEL 23 plasm. Since cancer cells are characterized by abnormally large nuclei the coefficient kv changes in cells during the neoplastic transformation. Furthermore, transport coefficients are different for different proteins as well as for different movement directions. GSK3β import to the nucleus is assigned by the rate c6 and its export by the rate c7 . cMyc import rate to the nucleus is ascribed by c4 , k c v 4 cytoplasmic cMyc −− → nuclear cMyc, kv c 6 cytoplasmic GSK3β −− → nuclear GSK3β, c7 nuclear GSK3β − → cytoplasmic GSK3β. 3.2 Formulation of the mathematical model Mathematical approach to describe biochemical phenomena gives a great opportunity to analyze biological system without laborious and cost consuming experiments. Based on the law of mass action, introduced by Guldberg and Waage in 1864, we formulate a system of ODEs describing the biochemical interactions. The law states that the rate of any given chemical reaction at which substance reacts is proportional to the product of molar concentrations of the reactants, which are called active mass [46]. Additionally, velocity of chemical reactions is proportional to the product of reacting substances and to a constant which characterizes each reaction. Taking into account the biochemical kinetics described in Section 3.1, we obtain a system, which describe evolution in time of the investigated substances. 3.2.1 From biology to equations The ODEs represent biological processes, which appear on the intracellular level. To keep the biological meaning of the particular species in this section, we define each model variable using its biological nomenclature. Each equation represents the time change of a variable, which is the concentration of one biochemical species. All variables are functions of time. The first equation (gene activation equation) models the dynamics of SGK1 transcript, dSGK1t (t) = t1 βcat − d1 SGK1t (t). dt (3.5) The production of SGK1 is described by the first term of the right hand side (r.h.s.) in 24 CHAPTER 3. SYSTEMS BIOLOGY APPROACH (3.5). The spontaneous degradation of the SGK1 transcript is modeled by the second term on the r.h.s. We assume β-catenin to be constant, because in a system without mutation we observe its constant influx on the normal level [76]. Modeling of β-catenin mutation is described in Chapter 5. The dynamics of the SGK1 protein in the cytoplasm is described by four terms, dSGK1p (t) = s2 SGK1t (t)−d2 SGK1p (t)−d6 SGK1p (t)·GSK3βc (t)+p9 SGK1occ (t). dt (3.6) The first term of the r.h.s. of (3.6) stands for the translation (protein synthesis). We assume only a spontaneous degradation of SGK1 that is expressed by the next term. Binding of SGK1 to GSK3β is modeled using a bilinear term d6 SGK1p (t) · GSK3βc (t) and it promotes GSK3β degradation via phosphorylation. Protein SGK1 from this complex is occupied and cannot bind to another protein. The last term regards dissociation of the [SGK1·GSK3β] complex and results in releasing the SGK1 molecules after phosphorylating GSK3β. Released SGK1 is freed from the complex and it becomes active again. We model the MYC mRNA in the same way as the SGK1 mRNA. Therefore, we have transcription of MYC and spontaneous degradation terms, dM Y Ct (t) = t3 βcat − d3 M Y Ct (t). dt (3.7) The dynamics of the cMyc protein in the cytoplasm is described by dcM ycpc (t) = s4 M Y Ct (t) − d4 cM ycpc (t) − c4 cM ycpc (t). dt (3.8) The first term corresponds to the cMyc synthesis and the second one to spontaneous degradation. The expression c4 cM ycpc (t) stands for the process of the transport between the nucleus and cytoplasm. Dynamics of the cMyc protein in the nucleus consists of two terms. We model here transport from the cytoplasm using the scaling coefficient kv , and then the degradation of cMyc, which occurs when GSK3β binds to this molecule, dcM ycpn (t) = kv c4 cM ycpc (t) − d5 cM ycpn (t) · GSK3βn (t). dt (3.9) 3.2. FORMULATION OF THE MATHEMATICAL MODEL 25 Now, we write the equation for the cytoplasmic GSK3β as dGSK3βc (t) = PGSK + c7 GSK3βn (t) − c6 GSK3βc (t) − d6 SGK1p (t) · GSK3βc (t) dt − do GSK3βc (t). (3.10) The first term of the r.h.s. of (3.10), PGSK , corresponds to the influx of GSK3β protein due to translation. The second and third term describe import from the nucleus, c7 GSK3βn (t), and transport to the nucleus, c6 GSK3βc (t), respectively. The last two terms stand for the degradation of GSK3β. Here, SGK1 binds to GSK3β and promotes its degradation in the phosphorylation process, whereas the spontaneous degradation proceeds at the rate do . Next, we model the dynamics of the nuclear GSK3β as dGSK3βn (t) = kv c6 GSK3βc (t) − kv c7 GSK3βn (t) − d5 GSK3βn (t) · cM ycpn (t) dt + p8 GSK3βocc (t). (3.11) The first two terms determine the import to the nucleus and the export to the cytoplasm. The third term stands for the loss of GSK3β that binds to cMyc and is involved in the process of phosphorylation. The last term is responsible for GSK3β, which is freed after phosphorylation. The process of phosphorylation in the complex [GSK3βn · cM ycpn ] is modeled analogously to the phosphorylation in the complex [SGK1p · GSK3βc ]. Considering the process of phosphorylation as time consuming, we introduce an additional variable to model it. To distinguish the states in which the proteins are ready for new binding and from those which are still in the complex, we use an additional variable for the latter, i.e., occupied GSK3βocc , which is described by an ODE with two terms in the r.h.s. dGSK3βocc (t) = d5 GSK3βn (t) · cM ycpn (t) − p8 GSK3βocc (t). dt (3.12) The first term corresponds to the phosphorylation process, where GSK3β phosphorylates cMyc, and thus initiates its degradation. The second term indicates the loss of bound GSK3β that is already dissociated from cMyc with its phoshorylating activity restored. We also consider occupied SGK1 as CHAPTER 3. SYSTEMS BIOLOGY APPROACH 26 dSGK1occ (t) = d6 SGK1p (t) · GSK3βc (t) − p9 SGK1occ (t). dt (3.13) The first term stands for the complex arising, where SGK1 phosphorylates GSK3β and the second term is the loss of SGK1 that becomes active again. 3.2.2 System of ordinary differential equations Rewriting equations (3.5) - (3.13) in the terms of new variables u = [u1 , ..., u9 ]T describing each biological species and taking k1 = t1 βcat, k3 = t3 βcat, P = PGSK lead to the following system of equations. Ordinary differential equations model of the medulloblastoma signaling du1 dt du2 dt du3 dt du4 dt du5 dt du6 dt du7 dt du8 dt du9 dt = k1 − d1 u1 , (3.14) = s2 u1 − d2 u2 − d6 u2 u6 + p9 u9 , (3.15) = k3 − d3 u3 , (3.16) = s4 u3 − d4 u4 − c4 u4 , (3.17) = kv c4 u4 − d5 u5 u7 , (3.18) = P + c7 u7 − c6 u6 − d6 u2 u6 − do u6 , (3.19) = kv c6 u6 − kv c7 u7 − d5 u5 u7 + p8 u8 , (3.20) = d 5 u 5 u 7 − p8 u 8 , (3.21) = d 6 u 2 u 6 − p9 u 9 , (3.22) with initial condition ui (0) = uio , where uio ≥ 0 i = 1, ..., 9. We assume positivity of parameters: ci , di , ki , pi , si , kv , P > 0 ∀ i. Variable notations and their biological meanings are listed in Table 3.1. 3.2. FORMULATION OF THE MATHEMATICAL MODEL Key players SGK1t SGK1p M Y Ct cM ycpc cM ycpn GSK3βc GSK3βn GSK3βocc SGK1occ Related model variables u1 u2 u3 u4 u5 u6 u7 u8 u9 27 Biological meaning SGK1 transcript in the nucleus SGK1 protein in the cytoplasm MYC transcript in the nucleus cMyc protein in the cytoplasm cMyc protein in the nucleus GSK3β protein in the cytoplasm GSK3β protein in the nucleus phosphorylating GSK3β protein in the nucleus phosphorylating SGK1 protein in the cytoplasm Table 3.1: Description of model 3.14 - 3.22 variables. Biological nomenclature and their equivalent mathematical notation. Summary of Chapter 3 In this chapter we formulated the ODE system describing the dynamics of medulloblastoma signaling found in the literature and discussed with biologists. To indicate the ratio of the compartments volume of the nucleus and cytoplasm, we introduced scaling coefficient kv , an important factor while formulating the two-compartment model. 28 CHAPTER 3. SYSTEMS BIOLOGY APPROACH CHAPTER 4 Mathematical analysis of the medulloblastoma model In this chapter we present an analysis of the ODEs model of the dynamics of medulloblastoma signaling. We show that the model is well-posed. Model well-posedness 1. Global existence: there exists a solution of u(t) for t ∈ [0, ∞). 2. Uniqueness: the solution of the considered problem is unique. 3. Nonnegativity: the solution is nonnegative for nonnegative initial data. The proposed model (3.14) - (3.22) is a system of autonomous ODEs du = F (u), dt (4.1) with initial condition u(0) = uo . u := [u1 , ..., un ]T , F := [f1 , ..., fn ]T , F : G −→ Rn is Lipschitz continuous and G is an open subset of Rn , n = 9. fi stands for the r.h.s. of the i-th equation of (3.14) - (3.22). The solutions of this Cauchy problem describe dynamics of each reactant concentration. 29 CHAPTER 4. MODEL ANALYSIS 30 4.1 Existence and uniqueness of global solution Theorem 4.1 System (3.14) - (3.22) with nonnegative initial data has a global unique solution u(t), t ≥ 0. Proof 4.1 • Local existence and uniqueness Since the right hand side of the system F(u) is a C 1 function, it is also a locally Lipschitz-continuous function. Then, we apply Picard-Lindelöf Theorem [29] and obtain local existence and uniqueness of the solution. The classical theory of ODEs provides also smoothness of the solution and its continuous dependence on the initial data and model parameters [29]. • Nonnegativity To show nonnegativity of the solution, we check that dui |u =0 ≥ 0 dt i ∀ i, i = 1, ..., 9 (4.2) holds. Then, the set R9+ ∪{0} is positive invariant with respect to the generated flow of the ODE system. Indeed, since the model parameters are positive (see Section 5.1.2), it holds: du1 |u =0 dt 1 du2 |u =0 dt 2 du3 |u =0 dt 3 du4 |u =0 dt 4 du5 |u =0 dt 5 du6 |u =0 dt 6 du7 |u =0 dt 7 du8 |u =0 dt 8 du9 |u =0 dt 9 = k1 ≥ 0, (4.3) = s2 u1 + p9 u9 ≥ 0, (4.4) = k3 ≥ 0, (4.5) = s4 u3 ≥ 0, (4.6) = kv c4 u4 ≥ 0, (4.7) = P + c7 u7 ≥ 0, (4.8) = kv c6 u6 + p8 u8 ≥ 0, (4.9) = d5 u5 u7 ≥ 0, (4.10) = d6 u2 u6 ≥ 0. (4.11) 4.2. ASYMPTOTIC BEHAVIOR OF MODEL SOLUTIONS 31 Using continuity of the solution and nonnegativity of the initial data, i.e., uo ≥ 0, we conclude that solutions cannot leave R9+ ∪ {0}. • Global existence Multiplying equations by properly chosen positive constants and adding them leads to ( ) d s2 s4 u5 u7 u8 k1 s 2 k3 s 4 u1 +u2 + u3 +u4 + +u6 + + +u9 = P + + −d4 u4 −d2 u2 dt d1 d3 kv kv kv d1 d3 d5 − d0 u6 − u5 u7 − d6 u2 u6 . (4.12) kv Due to the nonnegativity of the solution, we obtain ( ) d s2 s4 u5 u7 u8 u 1 + u2 + u 3 + u4 + + u6 + + + u9 ≤ K, (4.13) dt d1 d3 kv kv kv where K is a positive constant. Integrating both sides of (4.13), we obtain the estimate s2 s4 u5 u7 u8 u1 + u2 + u3 + u4 + + u6 + + + u9 ≤ Kt + C, C ∈ R. (4.14) d1 d3 kv kv kv Consequently, we obtain that linear combination of the model variables is bounded for all finite times. Since the solutions are nonnegative, we conclude that they are bounded. Hence, the solutions of the model can be extended globally in time for t ∈ [0, ∞). Observation 4.1 Numerical simulations suggest that the solutions are also uniformly bounded (for simulations, see next chapter). 4.2 Asymptotic behavior of model solutions In this section we determine the stability of the steady states of the ODE system applying linearization [10]. We apply the Hartman-Grobman Theorem [10], which justifies the use of linearization in the study of stability of nonlinear dynamical system. 4.2.1 Steady States Definition 4.1 A solution ū with F (ū) = 0 is called a steady state solution or an equilibrium of (4.1). CHAPTER 4. MODEL ANALYSIS 32 To study the stability of equilibria of the model, we calculate steady state solution ū = [ū1 , ..., ū9 ] of (4.1). This corresponds to the homeostasis in the biological system. Thus, we set F (ū) = 0. (4.15) Theorem 4.2 The model (3.14) - (3.22) has a unique positive steady state. Proof 4.2 By solving the corresponding algebraic equations of (4.15), we obtain a unique positive equilibrium given by Steady state of the model ū1 = ū2 = ū3 = ū4 = ū5 = ū6 = ū7 = ū8 = ū9 = k1 , d1 k1 s2 , d1 d2 k3 , d3 s 4 k3 , d4 + c4 d3 s4 k3 kv c4 c7 k1 s2 d6 ( + do ), d4 + c4 d3 d3 d5 c6 P d1 d2 P d 1 d2 , k1 s2 d6 + do d1 d2 c6 P d 1 d2 , c7 k1 s2 d6 + do d1 d2 kv c4 s4 k3 , p8 d4 + c4 d3 P d1 d2 d6 k1 s2 . p9 d1 d2 k1 s2 d6 + do d1 d2 (4.16) (4.17) (4.18) (4.19) (4.20) (4.21) (4.22) (4.23) (4.24) 4.2. ASYMPTOTIC BEHAVIOR OF MODEL SOLUTIONS 33 4.2.2 Stability of ordinary differential system Definition 4.2 An equilibrium ū of (4.1) is said to be stable if for each ϵ > 0, there exists a δ > 0 such that |u(t) − ū| < ϵ f or all t ≥ 0, whenever |u(0) − ū| < δ. (4.25) Definition 4.3 An equilibrium ū of (4.1) is said to be asymptotically stable if it is stable and there exists a > 0 such that lim |u(t) − ū| = 0, t→∞ whenever |u(0) − ū| < a. (4.26) Intuitively, Definition 4.3 means that an equilibrium ū of (4.1) is asymptotically stable if the trajectories starting near ū approach ū for t → ∞. If ū is not stable, it is said to be unstable. To determine stability of ū, we use the method of linearization. 4.2.3 Linearization of the medulloblastoma model Definition 4.4 The linearization of system (3.14) - (3.22) at the equilibrium ū is given by du = J(u − ū), where J = DF (ū) is the Jacobian matrix of function F at ū. dt The Jacobian J of system (3.14) - (3.22) is of the form: J= −d1 s2 0 0 0 0 0 0 0 0 −d2 − d6 ū6 0 0 0 −d6 ū6 0 0 d6 ū6 0 0 −d3 s4 0 0 0 0 0 0 0 0 −d4 − c4 kv c4 0 0 0 0 0 0 0 0 −d5 ū7 0 −d5 ū7 d5 ū7 0 0 −d6 ū2 0 0 0 −c6 − d6 ū2 − d0 kv c6 0 d6 ū2 0 0 0 0 −d5 ū5 c7 −kv c7 − d5 ū5 d5 ū5 0 0 0 0 0 0 0 p8 −p8 0 0 p9 0 0 0 0 0 0 −p9 (4.27) CHAPTER 4. MODEL ANALYSIS 34 To study the behavior of the linearized system in the neighborhood of ū, we need to assure that the behavior of the linearized system at sufficiently small neighborhood of ū is topologically equivalent to the dynamics of nonlinear system. We apply classical theory given by Theorem 4.3 (Grobman-Hartman [10]) If ū is a hyperbolic equilibrium of (4.1), then there exists a homeomorphism H such that the orbits of (4.1) in the region U of ū are mapped by H to orbits of the linearized system, i.e., du = DF (ū)(u − ū) dt (4.28) for u ∈ U . = F (u) if all Definition 4.5 An equilibrium ū is called a hyperbolic equilibrium of du dt eigenvalues λi,i=1,...,9 of the Jacobian matrix J = DF (ū) have nonzero real part. Theorem 4.4 If ū is a hyperbolic equilibrium of du = F (u) and if all eigenvalues of the dt linear transformation DF (ū) have negative real parts, then ū is asymptotically stable [10]. Since the system of nine equations is not treatable analytically, we use numerical calculations of the eigenvalues of J for different set of parameters. We study (3.14) - (3.22) for a special choice of parameters obtained based on the available data (see Chapter 5 for details). We obtain the following steady states for a healthy and trisomy 6 sample: Steady states Healthy sample [µM] Trisomy 6 sample [µM] ū1 = 0.0001143, ū2 = 0.057, ū3 = 0.000266, ū4 = 0.049, ū5 = 0.1, ū6 = 0.0012, ū7 = 0.0039, ū8 = 0.000489, ū9 = 0.0000332, ū1 = 0.00071, ū2 = 0.35, ū3 = 0.009, ū4 = 1.55, ū5 = 4.92, ū6 = 0.0002, ū7 = 0.00063, ū8 = 0.0039, ū9 = 0.00003, 4.2. ASYMPTOTIC BEHAVIOR OF MODEL SOLUTIONS 35 where in healthy sample - SGK1 mRNA: 1, MYC mRNA: 1, kv : 2 and in the trisomy 6 sample - SGK1 mRNA: 6.19, MYC mRNA: 31.86, kv : 0.5 (for details, see Subsection 5.1.1). The eigenvalues of J (4.27) for the healthy and trisomy 6 samples are: Eigenvalues Healthy sample [s−1 ] Trisomy 6 sample [s−1 ] λ1 = -0.4504, λ2 = -0.6004, λ3 = -0.0204, λ4 = -0.0015 + 0.0005i, λ5 = -0.0015 - 0.0005i, λ6 = -0.0004, λ7 = -0.0007, λ8 = -0.0022, λ9 = -0.0006, λ1 = -2.8606, λ2 = -0.6001, λ3 = -0.1092, λ4 = -5e-05 + 0.0001i, λ5 = -5e-05 - 0.0001i, λ6 = -0.0004, λ7 = -0.0007, λ8 = -0.0022, λ9 = -0.0006. We remark that all eigenvalues λi,i=1,...,9 have negative real parts. Next, we perform simulations for a large range of parameters and we obtain that calculated eigenvalues have negative real parts too. From Theorem 4.3 and 4.4, we conclude that the nontrivial equilibrium of (3.14) - (3.22) is asymptotically stable. Additionally, we classify ū as a focus as there exist λi with Im(λi ) ̸= 0, [72]. For a good representation of solution dynamics we plot solutions u5 and u8 , where we can easily see equilibrium to be a focus for a trisomy 6 sample. Figure 4.1 presents numerical simulations of the solutions u5 and u8 for a healthy sample and Figure 4.2 for a trisomy 6 sample. Observation 4.2 Numerical simulations suggest that the system is also globally asymptotically stable as we start the solution from various initial conditions and we finish at steady states (see Figure 4.3, Figure 4.4 and Figure 4.5). CHAPTER 4. MODEL ANALYSIS 36 −3 1 x 10 0.8 u8 0.6 0.4 0.2 0 0 0.2 0.4 u 0.6 0.8 5 Figure 4.1: Phase portrait with marked equilibrium (black dot) for the state variables u5 and u8 of several different sets of initial conditions (healthy sample - SGK1 mRNA: 1, MYC mRNA: 1, kv : 2, see Chapter 5). Due to eigenvalue analysis, the critical point is classified as focus. −3 6 x 10 5 u8 4 3 2 1 0 0 5 10 u5 15 20 Figure 4.2: Phase portrait with marked equilibrium (black dot) for the state variables u5 and u8 of several different sets of initial conditions (trisomy 6 sample - SGK1 mRNA: 6.19, MYC mRNA: 31.86, kv : 0.5, see Chapter 5). Due to eigenvalue analysis, the critical point is classified as focus. 4.2. ASYMPTOTIC BEHAVIOR OF MODEL SOLUTIONS −4 8 x 10 37 SGK1 mRNA perturbation of initial conditions SGK1 mRNA [ µM ] 7 SGK1 mRNA 6 5 4 3 2 1 0 0 2000 4000 6000 8000 10000 time [ s ] SGK1 in the cytoplasm perturbation of initial conditions 0.16 SGK1 protein [ µM ] 0.14 0.12 SGK1 protein 0.1 0.08 0.06 0.04 0.02 0 0 0.5 1 1.5 time [ s ] −4 8 x 10 2 4 x 10 MYC mRNA perturbation of initial conditions MYC mRNA [ µM ] 7 6 MYC mRNA 5 4 3 2 1 0 0 5000 10000 15000 time [ s ] Figure 4.3: Perturbation of initial conditions for SGK1 mRNA, cytoplasmic SGK1 and MYC mRNA for the healthy sample. Simulations show convergence of solutions to the equilibrium. CHAPTER 4. MODEL ANALYSIS 38 cMyc in the cytoplasm perturbation of initial conditions 0.12 cMyc protein [ µM ] 0.1 cMyc protein 0.08 0.06 0.04 0.02 0 0 5000 10000 15000 time [ s ] cMyc in the nucleus perturbation of initial conditions 0.35 0.3 cMyc protein [ µM ] cMyc protein 0.25 0.2 0.15 0.1 0.05 0 0 0.5 1 1.5 time [ s ] 2 4 x 10 GSK3β in the cytoplasm perturbation of initial conditions 0.01 GSK3β protein [ µM ] 0.009 0.008 GSK3β protein 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 0 5000 10000 15000 time [ s ] Figure 4.4: Perturbation of initial conditions for cytoplasmic cMyc, nuclear cMyc and cytoplasmic GSK3β for the healthy sample. Simulations show convergence of solutions to the equilibrium. 4.2. ASYMPTOTIC BEHAVIOR OF MODEL SOLUTIONS 39 GSK3β in the nucleus perturbation of initial conditions 0.02 GSK3β protein [ µM ] 0.018 0.016 GSK3β protein 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0 0 5000 10000 15000 time [ s ] −4 10 x 10 phosphorylating GSK3β perturbation of initial conditions GSK3β protein GSK3β protein [ µM ] 8 6 4 2 0 −2 0 0.5 1 1.5 2 time [ s ] −5 5 x 10 2.5 4 x 10 phosphorylating SGK1 perturbation of initial conditions 4.5 SGK1 protein SGK1 protein [ µM ] 4 3.5 3 2.5 2 1.5 1 0.5 0 0 2000 4000 6000 8000 10000 time [ s ] Figure 4.5: Perturbation of initial conditions for nuclear GSK3β, phosphorylating GSK3β and phosphorylating SGK1 for the healthy sample. Simulations show convergence of solutions to the equilibrium. 40 CHAPTER 4. MODEL ANALYSIS Summary of Chapter 4 In this chapter we proved existence, uniqueness and nonnegativity of the solutions of the system (3.14) - (3.22). We provided also estimates assuring the global existence. To analyze the stability of the system (3.14) - (3.22), we linearized it and performed stability analysis supported by numerical calculations. We found the model to be locally asymptotically stable and the equilibrium to be a focus. Finally, we conclude that the system has a global unique positive equilibrium which is asymptotically stable. CHAPTER 5 Numerical simulations Our model (3.14) - (3.22) is applied to simulate the behavior of two types of medulloblastoma, characterized by monosomy 6 and trisomy 6, respectively. To validate the model, we use patient data on the SGK1 and MYC mRNAs obtained in the clinics in Heidelberg (Division of Molecular Genetics, German Cancer Research Center (DKFZ) Heidelberg, Germany [37]), Boston (Children’s Hospital Boston, Boston, MA 02115, USA [12]) and Amsterdam (Department of Human Genetics, Academic Medical Center, Amsterdam, the Netherlands [60]) thanks to Stefan Pfister and Dominik Sturm. In this chapter we describe the patient data from the clinics and how we introduce them into our model. We present parameters chosen for the model and the numerical tool used. We outline the dynamics of each population for monosomy 6, trisomy 6 and normal cells. We investigate the system dynamics and we make comparison between the patient data and the results obtained from the simulations. Finally, we highlight crucial features of the particular medulloblastoma subgroups, which lead us to formulate biological hypotheses. Numerical simulations of the mathematical model reveal the importance of SGK1 expression for the prognosis in medulloblastoma. 5.1 Data, parameters and numerical tools In this section we deal with the patient data. We delineate origin of the parameter values and a graphical user interface that we apply to investigate the dynamics of the model. 41 CHAPTER 5. NUMERICAL SIMULATIONS 42 5.1.1 Patient data Each patient sample is profiled by the set of features: gender, age, metastasis progression (mstg), death status and others. We do not apply these data to our model but compare them to the results obtained from the simulations. The gene expression of SGK1 and MYC (that is the SGK1 and MYC mRNAs) is measured using DNA microarray analysis and it is provided by experimentalists. These data are fed to the model. To model monosomy 6 and trisomy 6 associated medulloblastoma, we use different parameterizations of the SGK1 transcript synthesis and MYC transcript synthesis. The parameterization corresponds to the experimentally observed fold change (see Section 2.1.1) of the level of the SGK1 and MYC mRNAs in medulloblastoma tumor cells and in the normal cells. The MYC mRNA fold change in both cases is higher than 1. The SGK1 mRNA fold change is smaller than 1 in monosomy 6 and larger than 1 in trisomy 6. We obtain that fold change of SGK1, which is synthesized on chromosome 6q, in 6q loss and 6q gain is not proportional to the copy-number of chromosome (Table 5.1). Intuitively, in monosomy (one copy of chromosome) the fold change should be equal to about 0.5 and in trisomy (three copies of chromosome) to 1.5. Biological evidence 5.1 Recent experiments show that the nonlinear relation between the number of gene copy and resulting gene expression of SGK1 is linked to the pattern of DNA methylation [56]. To simulate dynamics of both medulloblastoma types, we consequently adjust the SGK1 transcription rate such that the levels of mRNA agree with the patient data. We do the same procedure for the MYC mRNA. The mRNA experimental data which we follow in simulations, are collected from individual patients resected tumors. Exemplary fold changes of monosomy 6 and trisomy 6 samples are given in Table 5.1. Biological evidence 5.2 Monosomy 6 samples are characterized by the SGK1 mRNA downregulation compared to a healthy cells. In trisomy 6 sample we can observe overexpression compared to the healthy cells. MYC mRNA is overexpressed in both medulloblastoma subgroups and in the monosomy 6 case it is even higher. Such dependencies are found in a majority of the collected patient samples. 5.1. DATA, PARAMETERS AND NUMERICAL TOOLS monosomy fold change tumor/control trisomy fold change tumor/control SGK1 mRNA 0.74 MYC mRNA 16.27 SGK1 mRNA 7.11 MYC mRNA 6.45 43 Table 5.1: Examples of microarray data values from the clinics for two types of medulloblastoma. To adjust in the simulations the SGK1 and MYC transcription rates such that levels of mRNAs agree with the patient data, first we determine the basic set of parameters (Table 5.3) for the model of the control case (see Section 2.1.1). Then, to indicate the increase or decrease of the particular SGK1 or MYC mRNA in the tumor cells, we extend the model to account for the mutation introducing coefficients L1 and L3 (parameterization) in two equations, dSGK1t (t) = L1 t1 βcat − d1 SGK1t (t), dt (5.1) dM Y Ct (t) = L3 t3 βcat − d3 M Y Ct (t), dt (5.2) where L1 is responsible for the SGK1 mRNA increase/decrease and L3 is responsible for the MYC mRNA increase. L1 and L3 play the role of control variables, i.e., independent variables that can be manipulated or controlled in an experimental design to understand how they affect the dependent variables of the model. They correspond to the fold change of selected transcripts. Originally, L1 and L3 are equal 1. Furthermore, an important feature observed in medulloblastoma is a relatively small volume of the cytoplasm and quite large volume of the nucleus, which is typical for other malignant cells [50], [65], [51]. In our model, the ratio of two compartments is considered, hence we adjust the scaling factor kv (see Section 3.1.4), accordingly. For normal cells, we assume kv = 2, i.e., the cytoplasm is twice larger than the nucleus by volume. For malignant cells, we assume kv = 0.5, i.e., the volume of the nucleus is twice larger than the cytoplasm. The scaling factor may be different for different cell types. 5.1.2 Model calibration Our model consists of nine nonlinear ODEs and involves nineteen parameters. The parameters describe rates of the basic processes, e.g., translation, transport, association, etc. Orders of the magnitude of parameters are chosen following the Reference [44]. Thus, our CHAPTER 5. NUMERICAL SIMULATIONS 44 parameter magnitudes (places after decimal point) correspond to the analogous processes from [44]. The translation rate is of the order of 10−1 , spontaneous protein degradation is of the order of 10−4 , etc. In the strategy of chosing parameters, we consider the protein size and the length of the transcript (see Table 5.2). More detailed, the size of the cMyc protein is 439 amino Gene Transcript Length Protein Size MYC SGK1 GSK3β 6.001 bases 148.867 bases 273.095 bases 439 amino acids 421-526 amino acids depending on isoform 420 amino acids Table 5.2: Gene description. acids and size of GSK3β is 420 amino acids. Smaller molecules should be transported faster to the nucleus, hence the transport coefficient of GSK3β to the nucleus is equal to c6 = 0.003 s−1 and of cMyc it is c4 = 0.002 s−1 . The transport coefficient of GSK3β to the cytoplasm is equal to c7 = 0.0009 s−1 , consequently it is smaller than the transport coefficient of GSK3β to the nucleus. We assume that GSK3β has crucial phosphorylating activity in the complex with the nuclear cMyc, so there is a bigger demand of GSK3β in the nucleus. Therefore, we assume that GSK3β shuttles faster to the nucleus (c6 > c7 ). Other factors can also influence the transport, but for sake of simplicity in our model we stick to the size rule. Further, the length of the MYC mRNA transcript is 6001 bases and the length of the SGK1 mRNA transcript is 148867 bases. Taking into account the difference in both transcript lengths, we assume that the MYC transcript is produced faster than the SGK1 transcript. As a result, we assume higher transcription rate for MYC. The size of the SGK1 protein isoforms is in the range 421-526 amino acids. The difference in the protein size of SGK1 and cMyc is not large, but the difference in transcripts length is great, hence the translation of MYC should be also faster than the translation of SGK1 (translation of the nucleotide sequence takes time). The bigger difference between the transcript and the protein size, the longer translation process and the smaller the total amount of the new synthesized protein in the same period of time. Thus, we take s4 = 0.4 s−1 as translation rate of MYC and s2 = 0.2 s−1 as translation rate of SGK1. Next, in our model we have a spontaneous degradation for both transcripts and proteins. We want to preserve the assumption that level of the MYC gene, as well transcript as protein, is higher than level of the SGK1 gene (transcript, protein). For this reason, we apply higher degradation rate of the SGK1 gene than MYC for both the transcript and protein. From the molecular point of view transcripts are less stable than proteins, what is reflected in differences of degradation rates. The transcript degradation rate is larger than the pro- 5.1. DATA, PARAMETERS AND NUMERICAL TOOLS 45 tein degradation rate for the same gene. Considering degradation through the interaction with other proteins, we stick to the rule that the probability of meeting of two proteins is higher in the nucleus than in the cytoplasm. The reason is that the nucleus is smaller than cytoplasm in the normal cells and binding process may occur much faster in this compartment. In our system the phosphorylation of the SGK1 protein is indicated as a fast process. Thus, the dissociation of the complex [SGK1·GSK3β] is assumed to be faster than dissociation of the complex [GSK3β·cMyc]. Following the above methodology, we choose the parameters of the model. Symbol Value Description PGSK βcat t1 t3 s2 s4 do d1 d2 d3 d4 d5 d6 kv c4 c6 c7 p8 p9 0.00002 µMs−1 0.4 µM 2 × 10−7 s−1 4 × 10−7 s−1 0.2 s−1 0.4 s−1 0.00005 s−1 0.0007 s−1 0.0004 s−1 0.0006 s−1 0.0002 s−1 0.5 µM−1 s−1 0.3 µM−1 s−1 2 0.002 s−1 0.003 s−1 0.0009 s−1 0.4 s−1 0.6 s−1 constant influx of the GSK3β protein to the cytoplasm transcription factor for the MYC and SGK1 mRNA synthesis transcription rate of SGK1 transcription rate of MYC translation rate of SGK1 translation rate of cMyc spontaneous degradation rate of the GSK3β protein in the cytoplasm spontaneous degradation rate of the SGK1 mRNA spontaneous degradation rate of the SGK1 protein spontaneous degradation rate of the MYC mRNA spontaneous degradation rate of the cMyc protein in the cytoplasm degradation rate of the cMyc protein in the nucleus by GSK3β degradation rate of the GSK3β protein in the cytoplasm by SGK1 scaling coefficient, cytoplasm to the nucleus ratio transport of cMyc to the nucleus coefficient transport of GSK3β to the nucleus coefficient transport of GSK3β to the cytoplasm coefficient dissociation coefficients of the [GSK3β·cMyc] complex dissociation coefficients of the [SGK1·GSK3β] complex Table 5.3: A summary of model parameters for healthy sample (cf. [44]). Statement 5.1 Numerical simulations show robustness of the model behavior with respect to the parameters. The results are qualitatively conserved for a large range of parameter values. 46 CHAPTER 5. NUMERICAL SIMULATIONS 5.1.3 Graphical user interface R We perform numerical simulations with MATLAB⃝ ODE solver ode23s [67], using crude error tolerances to resolve stiff systems [52]. Our system is stiff since the stiff coefficient for basic set of parameters maxi=1,2...m |λi | s= (5.3) mini=1,2...m |λi | is equal to s = 1500 for healthy sample parameterization and s = 7000 for chosen trisomy 6 sample parameterization (see eigenvalues λi in Section 4.2.3). For sake of simplicity, we created graphical user interface (GUI) to make simulations and apply different tests (see Figure 5.1 and 5.2). The advantage of such application is that there is no need to change R the source code of MATLAB⃝ m-file every time when the parameters or other factors of the simulation are changed. The GUI consists of three types of elements: values that user can adjust, values that user gets after performing simulations (not available for changes of the user) and the graph of the solutions. Parameters of the mathematical model, initial values, time settings and number of visualized solutions can be adjusted by the user. A pop-up menu gives additional opportunity to either display the time dependence or the phase portrait. The important data which we get due to simulations visible on GUI are: steady state, eigenvalue and fold change. We present GUI as a convenient tool, where different scenarios of parameters can be easily tested. Figure 5.1: Graphical user interface - numerical solution of trisomy 6 sample - time dependence graph. 5.2. SIMULATION RESULTS 47 Figure 5.2: Graphical user interface - numerical solution of trisomy 6 sample - phase portrait. 5.2 Simulation results In this section we present numerical simulations based on 49 patient samples (20 trisomy samples and 29 monosomy samples). The simulations are done for the initial conditions corresponding to the stationary state values, see (4.16) - (4.24), of the "healthy" system (signaling in "normal" - non-malignant cells). We compare the dynamics between monosomy 6, trisomy 6 and normal cells. We also investigate the correlation between patient data and simulations. Proceeding the simulation studies, we formulate new hypotheses concerning the difference in dynamics of the two types of medulloblastoma based on the regulatory loop SGK1-GSK3β-MYC. 5.2.1 Monosomy 6, trisomy 6 and control case We perform numerical simulations for 6q loss, 6q gain and the control case for chosen patient samples (see Table 5.4) to better understand the dynamics of the medulloblastoma subgroups and the control. We depict these three cases on Figures 5.3, 5.4 and 5.5 for all model variables. CHAPTER 5. NUMERICAL SIMULATIONS 48 fold change SGK1 mRNA fold change MYC mRNA scaling factor monosomy L1 = 0.74 L3 = 16.27 kv = 0.5 trisomy L1 = 7.11 L3 = 6.45 kv = 0.5 control L1 = 1 L3 = 1 kv = 2 Table 5.4: Examples of microarray data values from the clinics for two types of medulloblastoma and control. • Figure 5.3 [A] Dynamics of the SGK1 mRNA: In the trisomy 6 cells we obtain the exponential increase of the SGK1 mRNA and simultaneous exponential decrease in monosomy 6. [B] Dynamics of the SGK1 protein in the cytoplasm: SGK1 protein in the cytoplasm follows its mRNA values, in both monosomy 6 and trisomy 6. [C] Dynamics of the MYC mRNA: We observe the exponential MYC mRNA increase in both types of medulloblastoma. • Figure 5.4 [A] Dynamics of the cMyc protein in the cytoplasm: The behavior of the cytoplasmic cMyc is based on the MYC mRNA level; most often the cytoplasmic cMyc is lower in the trisomy 6 case than in the monosomy 6 case. [B] Dynamics of the cMyc protein in the nucleus: There is a peak of cMyc and then decrease in both types of medulloblastoma. Most often the nuclear cMyc in trisomy 6 is higher than in monosomy 6, opposite to the cytoplasmic cMyc. 5.2. SIMULATION RESULTS 49 [C] Dynamics of the GSK3β protein in the cytoplasm: GSK3β in the cytoplasm is increased in 6q loss and decreased in 6q gain. • Figure 5.5 [A] Dynamics of the GSK3β protein in the nucleus: In both types of medulloblastoma the level of GSK3β in the nucleus is higher than for the cytoplasmic GSK3β. [B] Dynamics of the phosphorylating GSK3β protein in the nucleus: GSK3β increase is observed in monosomy 6 and trisomy 6. The increase is larger in the case of monosomy 6, however in both types of cancer the nuclear cMyc is high. Thus, we have increased phosphorylation activity of GSK3β. [C] Dynamics of the phosphorylating SGK1 protein in the cytoplasm: In 6q gain, there is a peak of the phoshorylating SGK1 and then a decrease. Inversely, in 6q loss we have at first a decrease and then we observe that the SGK1 level increases. CHAPTER 5. NUMERICAL SIMULATIONS 50 −3 1 x 10 [A], Dynamics of SGK1 mRNA 0.9 SGK1 mRNA [ µM ] 0.8 0.7 normal cells 6q loss: SGK1(0.74); MYC(16.27) 6q gain: SGK1(7.11); MYC(6.45) 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 8 time [ s ] 10 4 x 10 [B], Dynamics of SGK1 in the cytoplasm 0.45 0.4 SGK1 protein [ µM ] 0.35 normal cells 6q loss: SGK1(0.74); MYC(16.27) 6q gain: SGK1(7.11); MYC(6.45) 0.3 0.25 0.2 0.15 0.1 0.05 0 0 2 4 6 8 time [ s ] −3 4.5 x 10 10 4 x 10 [C], Dynamics of MYC mRNA 4 normal cells 6q loss: SGK1(0.74); MYC(16.27) 6q gain: SGK1(7.11); MYC(6.45) MYC mRNA [ µM ] 3.5 3 2.5 2 1.5 1 0.5 0 0 2 4 6 time [ s ] 8 10 4 x 10 Figure 5.3: Numerical simulations based on exemplary microarray data values from the clinics. Population dynamics of SGK1 mRNA, cytoplasmic SGK1 and MYC mRNA. Results for 6q gain, 6q loss and normal cells. 5.2. SIMULATION RESULTS 51 [A], Dynamics of cMyc in the cytoplasm 0.8 cMyc protein [ µM ] 0.7 normal cells 6q loss: SGK1(0.74); MYC(16.27) 6q gain: SGK1(7.11); MYC(6.45) 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 8 time [ s ] 10 4 x 10 [B], Dynamics of cMyc in the nucleus 1.8 normal cells 6q loss: SGK1(0.74); MYC(16.27) 6q gain: SGK1(7.11); MYC(6.45) 1.6 cMyc protein [ µM ] 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 2 4 6 8 time [ s ] −3 1.6 x 10 10 4 x 10 [C], Dynamics of GSK3B in the cytoplasm GSK3B protein [ µM ] 1.4 1.2 1 normal cells 6q loss: SGK1(0.74); MYC(16.27) 6q gain: SGK1(7.11); MYC(6.45) 0.8 0.6 0.4 0.2 0 0 2 4 6 time [ s ] 8 10 4 x 10 Figure 5.4: Numerical simulations based on exemplary microarray data values from the clinics. Population dynamics of the cytoplasmic and nuclear cMyc protein and GSK3β in the cytoplasm. Results for 6q gain, 6q loss and normal cells. CHAPTER 5. NUMERICAL SIMULATIONS 52 −3 x 10 [A], Dynamics of GSK3B in the nucleus 5 GSK3B protein [ µM ] 4.5 4 3.5 3 normal cells 6q loss: SGK1(0.74); MYC(16.27) 6q gain: SGK1(7.11); MYC(6.45) 2.5 2 1.5 1 0.5 0 0 2 4 6 8 time [ s ] 2 10 4 x 10 [B], Dynamics of phosphorylating GSK3B in the nucleus −3 x 10 GSK3B protein [ µM ] 1.8 normal cells 6q loss: SGK1(0.74); MYC(16.27) 6q gain: SGK1(7.11); MYC(6.45) 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 2 4 6 8 time [ s ] 3.65 10 4 x 10 [C], Dynamics of phosphorylating SGK1 in the cytoplasm −5 x 10 3.6 SGK1 protein [ µM ] 3.55 3.5 normal cells 6q loss: SGK1(0.74); MYC(16.27) 6q gain: SGK1(7.11); MYC(6.45) 3.45 3.4 3.35 3.3 3.25 3.2 3.15 0 2 4 6 time [ s ] 8 10 4 x 10 Figure 5.5: Numerical simulations based on exemplary microarray data values from the clinics. Population dynamics of protein GSK3β in the nucleus, phosphorylating GSK3β in the nucleus and phosphorylating SGK1 in the cytoplasm. Results for 6q gain, 6q loss and normal cells. 5.2. SIMULATION RESULTS 53 Now, we focus on the long time behavior of particular protein concentration. In both types of cancer the cMyc protein is always higher than control. The cytoplasmic cMyc demonstrates even higher fold change in monosomy 6 than in the case of trisomy 6. On the opposite, the nuclear cMyc level indicates much lower fold change in the monosomy 6 case than in the trisomy 6 case. As a result, we observe significant differences between the two cellular compartments with respect to the type of medulloblastoma tumor cells. Further, the nuclear GSK3β is lower in trisomy 6 than in monosomy 6, even much lower than in the normal cells. Taking into account the cytoplasmic SGK1 level, we have a lower level (even below control) in monosomy 6 compared to trisomy 6. In general, based on the results from the simulation, we can notice the following dependencies that are crucial for the model understanding and may explain discrepancy in medulloblastoma prognosis. Observation 5.1 The results of simulations show following qualitative relations: cytoplasmic GSK3β↓ cytoplasmic SGK1 ↑ −−−−−−−−−−−−→ nuclear GSK3β ↓ → nuclear cMyc ↑ and cytoplasmic GSK3β↑ cytoplasmic SGK1 ↓ −−−−−−−−−−−−→ nuclear GSK3β ↑ → nuclear cMyc ↓. 5.2.2 Comparison between the patient data and simulations based on the microarray data In previous section we investigated concentrations of each species in the case of monosomy 6, trisomy 6 and normal cells. We want to compare the results from the simulations with the patient data. We link the death status with the cMyc protein and find a strong correlation between the death status and high nuclear cMyc level. Trisomy 6 In our analysis we compare samples of high nuclear cMyc (from simulations) with patients assigned to the group of positive death status. We found a positive correlation. It seems that high level of the cMyc protein in the nucleus favors patients death. The comparison indicates also that the critical high nuclear cMyc differs for different clinics. The explanation may lie in short time diagnosis or environmental features promoting longer survival time of the patient (Figure 5.6). 54 CHAPTER 5. NUMERICAL SIMULATIONS Figure 5.6: Trisomy 6: presented data are composed of the patient part and results from the simulations. We notice that for high nuclear cMyc from the simulations (bright grey) there are samples where death status (dark grey) is assigned. Fold change of the cMyc protein equal to zero is due to the rounding a point decimal number and in fact is below 0.5 value. Monosomy 6 In the case of 6q loss we have lower correlation between level of the nuclear cMyc and death status. Nevertheless, we still observe a tendency (Figure 5.7). Generally, the prognosis based on the patient data is much better in the case where level of the nuclear cMyc is substantially lower. General observations Our analysis yields the conclusion that the level of cMyc in the nucleus lends itself as a predictive factor. System dynamics reveal interesting features of monosomy 6 and trisomy 6. We claim that the observations indicate divergence in prognosis for the two types of medulloblastoma. Additionally, we remark that even if the nuclear cMyc in monosomy 6 is higher than in the normal cells (most cases), it is not too high and, hence, the treatment still provides positive effects. In the case of trisomy 6 the nuclear cMyc is too high and treatment cannot overcome the disease. 5.2. SIMULATION RESULTS 55 Figure 5.7: Monosomy 6: presented data are composed of the patient part and results from the simulations. We notice that for high nuclear cMyc from the simulations (bright grey) there are samples where the death status (dark grey) is assigned. Fold change of the cMyc protein equal to zero is due to the rounding a point decimal number and in fact is below 0.5 value. Statement 5.2 There exists a positive correlation between high nuclear cMyc obtained from the simulations and patients death status taken from the patient data (see Figure 5.6 and Figure 5.7). 5.2.3 Discrepancy in dynamics of genes MYC and SGK1 in trisomy 6 and monosomy 6 In this section we investigate differences in the nuclear cMyc. To do so, we simulate particular mRNA of MYC and SGK1. As dynamics of the MYC and SGK1 mRNAs strongly depend on the type of medulloblastoma, we see how they influence the patient prognosis CHAPTER 5. NUMERICAL SIMULATIONS 56 (i.e., the amount of the nuclear cMyc). We simulate different patient samples and compare them for different scenarios. C OMPARISON OF THE SAMPLES WITH SIMILAR VALUES OF THE MYC mRNA IN BOTH TYPES OF MEDULLOBLASTOMA Since MYC regulates many genes, we perform simulations of the monosomy 6 and trisomy 6 model to investigate how SGK1 impacts the MYC production. Numerical simulations show that the change in the MYC transcription rate may result in markedly different levels of the cMyc protein depending on the magnitude of perturbations in the SGK1 dynamics, which may be the reason for the different prognosis (Table 5.5 and Table 5.6). Based on the fold change of the SGK1 and MYC mRNA, we present the dynamics of the cytoplasmic and the nuclear cMyc in Figure 5.8 and Figure 5.9 to hightlight the difference between the two cellular compartments. The cytoplasmic level follows the dynamics of the MYC mRNA and the nuclear cMyc follows the system dynamics. monosomy SGK1 mRNA MYC mRNA fold change 0.54 28.59 tumor/control trisomy SGK1 mRNA MYC mRNA fold change 4.16 29.78 tumor/control cMyc in the nucleus 4 cMyc in the nucleus 31 Table 5.5: Microarray data values from the clinics for two types of medulloblastoma of similar MYC mRNA and different SGK1 mRNA production. monosomy SGK1 mRNA MYC mRNA fold change 0.32 12.33 tumor/control trisomy SGK1 mRNA MYC mRNA fold change 7.92 12.02 tumor/control cMyc in the nucleus 1 cMyc in the nucleus 24 Table 5.6: Microarray data values from the clinics for two types of medulloblastoma of similar MYC mRNA and different SGK1 mRNA production. 5.2. SIMULATION RESULTS 57 [B], Dynamics of cMyc in the nucleus [A], Dynamics of cMyc in the cytoplasm 4 1.5 1 normal cells 6q loss: SGK1(0.54); MYC(28.59) 6q gain: SGK1(4.16); MYC(29.78) 0.5 cMyc protein [ µM ] cMyc protein [ µM ] 3.5 3 2.5 normal cells 6q loss: SGK1(0.54); MYC(28.59) 6q gain: SGK1(4.16); MYC(29.78) 2 1.5 1 0.5 0 0 2 4 6 8 time [ s ] 0 0 10 2 4 6 8 time [ s ] 4 x 10 10 4 x 10 Figure 5.8: Numerical simulations based on exemplary microarray data values from the clinics. Dynamics of [A] cMyc protein in the cytoplasm and [B] cMyc protein in the nucleus in the trisomy 6 case (pink-solid dotted line), monosomy 6 case (blue-dashed line) and healthy tissue (green-solid line), each corresponding to different production rate of SGK1 and similar production rate of MYC for two types of medulloblastoma. [A], Dynamics of cMyc in the cytoplasm [B], Dynamics of cMyc in the nucleus 0.7 3 2.5 0.5 normal cells 6q loss: SGK1(0.32); MYC(12.33) 6q gain: SGK1(7.92); MYC(12.02) 0.4 0.3 0.2 2 normal cells 6q loss: SGK1(0.32); MYC(12.33) 6q gain: SGK1(7.92); MYC(12.02) 1.5 1 0.5 0.1 0 0 cMyc protein [ µM ] cMyc protein [ µM ] 0.6 2 4 6 time [ s ] 8 10 4 x 10 0 0 2 4 6 8 time [ s ] Figure 5.9: Numerical simulations based on exemplary microarray data values from the clinics. Dynamics of [A] cMyc protein in the cytoplasm and [B] cMyc protein in the nucleus in the trisomy 6 case (pink-solid dotted line), monosomy 6 case (blue-dashed line) and healthy tissue (green-solid line), each corresponding to different production rate of SGK1 and similar production rate of MYC for two types of medulloblastoma. 10 4 x 10 58 CHAPTER 5. NUMERICAL SIMULATIONS C OMPARISON OF THE SAMPLES WITH SIMILAR VALUES OF THE MYC mRNA WITHIN EACH MEDULLOBLASTOMA TYPE Due to simulations, samples with similar mRNA of MYC and varying levels of the SGK1 expression within one type of medulloblastoma show different nuclear cMyc levels. Again, we notice that the SGK1 mRNA is responsible for the difference (Table 5.7 and Table 5.8). Additionally, we observe strong influence of higher SGK1 mRNA on the levels of the nuclear cMyc in the case of trisomy 6. We present the dynamics of the cytoplasmic and the nuclear cMyc in Figure 5.10 and Figure 5.11. monosomy SGK1 mRNA MYC mRNA fold change 0.19 17.18 tumor/control monosomy SGK1 mRNA MYC mRNA fold change 0.47 17.20 tumor/control cMyc in the nucleus 1 cMyc in the nucleus 2 Table 5.7: Microarray data values from the clinics for monosomy 6 of similar MYC mRNA and different SGK1 mRNA production. trisomy SGK1 mRNA MYC mRNA fold change 7.92 12.02 tumor/control trisomy SGK1 mRNA MYC mRNA fold change 1.12 13.59 tumor/control cMyc in the nucleus 24 cMyc in the nucleus 4 Table 5.8: Microarray data values from the clinics for trisomy 6 of similar MYC mRNA and different SGK1 mRNA production. 5.2. SIMULATION RESULTS 59 [A], Dynamics of cMyc in the cytoplasm [B], Dynamics of cMyc in the nucleus 0.9 0.4 0.8 0.35 normal cells 6q loss: SGK1(0.19); MYC(17.18) 6q loss: SGK1(0.47); MYC(17.20) 0.6 0.5 0.4 0.3 cMyc protein [ µM ] cMyc protein [ µM ] 0.7 normal cells 6q loss: SGK1(0.19); MYC(17.18) 6q loss: SGK1(0.47); MYC(17.20) 0.3 0.25 0.2 0.15 0.2 0.1 0.1 0 0 2 4 6 8 time [ s ] 10 0.05 0 2 4 6 8 time [ s ] 4 x 10 10 4 x 10 Figure 5.10: Numerical simulations based on exemplary microarray data values from the clinics. Dynamics of [A] cMyc protein in the cytoplasm and [B] cMyc protein in the nucleus in the first monosomy 6 case (pink-solid dotted line), second monosomy 6 case (blue-dashed line) and healthy tissue (green-solid line), each corresponding to different production rate of SGK1 and similar production rate of MYC. [A], Dynamics of cMyc in the cytoplasm [B], Dynamics of cMyc in the nucleus 0.7 3 2.5 0.5 normal cells 6q gain: SGK1(7.92); MYC(12.02) 6q gain: SGK1(1.12); MYC(13.59) 0.4 0.3 0.2 2 normal cells 6q gain: SGK1(7.92); MYC(12.02) 6q gain: SGK1(1.12); MYC(13.59) 1.5 1 0.5 0.1 0 0 cMyc protein [ µM ] cMyc protein [ µM ] 0.6 2 4 6 time [ s ] 8 10 4 x 10 0 0 2 4 6 8 time [ s ] Figure 5.11: Numerical simulations based on exemplary microarray data values from the clinics. Dynamics of [A] cMyc protein in the cytoplasm and [B] cMyc protein in the nucleus in the first trisomy 6 case (pink-solid dotted line), second trisomy 6 case (blue-dashed line) and healthy tissue (green-solid line), each corresponding to different production rate of SGK1 and similar production rate of MYC. 10 4 x 10 60 CHAPTER 5. NUMERICAL SIMULATIONS C OMPARISON OF THE SAMPLES WITH THE MYC mRNA MUCH HIGHER MONOSOMY 6 THAN IN TRISOMY 6 IN We observe that for the patient samples with much higher MYC mRNA production in monosomy 6 than in trisomy 6 we still have nuclear cMyc lower in monosomy 6 than in trisomy 6 (Table 5.9 and Table 5.10). This indicates much higher SGK1 mRNA production in trisomy 6 than in monosomy 6. Intuitively, we would claim that the nuclear cMyc concentration depends mostly on the MYC mRNA production. However, in this case we confirm a strong impact of SGK1 on the nuclear cMyc. The dynamics of cytoplasmic and the nuclear cMyc are presented in Figure 5.12 and Figure 5.13. monosomy SGK1 mRNA MYC mRNA fold change 0.54 28.59 tumor/control trisomy SGK1 mRNA MYC mRNA fold change 7.92 12.02 tumor/control cMyc in the nucleus 4 cMyc in the nucleus 24 Table 5.9: Microarray data values from the clinics for two types of medulloblastoma of MYC mRNA production higher in monosomy 6 than in trisomy 6. monosomy SGK1 mRNA MYC mRNA fold change 0.65 11.76 tumor/control trisomy SGK1 mRNA MYC mRNA fold change 5.7 5.22 tumor/control cMyc in the nucleus 2 cMyc in the nucleus 7 Table 5.10: Microarray data values from the clinics for two types of medulloblastoma of MYC mRNA production higher in monosomy 6 than in trisomy 6. 5.2. SIMULATION RESULTS 61 [A], Dynamics of cMyc in the cytoplasm [B], Dynamics of cMyc in the nucleus 1.4 3 2.5 normal cells 6q loss: SGK1(0.54); MYC(28.59) 6q gain: SGK1(7.92); MYC(12.02) 1 0.8 0.6 0.4 cMyc protein [ µM ] cMyc protein [ µM ] 1.2 normal cells 6q loss: SGK1(0.54); MYC(28.59) 6q gain: SGK1(7.92); MYC(12.02) 1.5 1 0.5 0.2 0 0 2 2 4 6 8 time [ s ] 0 0 10 2 4 6 8 time [ s ] 4 x 10 10 4 x 10 Figure 5.12: Numerical simulations based on exemplary microarray data values from the clinics. Dynamics of [A] cMyc protein in the cytoplasm and [B] cMyc protein in the nucleus in the trisomy 6 case (pink-solid dotted line), monosomy 6 case (blue-dashed line) and healthy tissue (green-solid line), each corresponding to production rate of MYC higher in monosomy 6 than in trisomy 6. [B], Dynamics of cMyc in the nucleus 0.8 0.6 0.7 0.5 normal cells 6q loss: SGK1(0.65); MYC(11.76) 6q gain: SGK1(5.7); MYC(5.22) 0.4 0.3 0.2 0.6 normal cells 6q loss: SGK1(0.65); MYC(11.76) 6q gain: SGK1(5.7); MYC(5.22) 0.5 0.4 0.3 0.2 0.1 0 0 cMyc protein [ µM ] cMyc protein [ µM ] [A], Dynamics of cMyc in the cytoplasm 0.7 0.1 2 4 6 time [ s ] 8 10 4 x 10 0 0 2 4 6 8 time [ s ] Figure 5.13: Numerical simulations based on exemplary microarray data values from the clinics. Dynamics of [A] cMyc protein in the cytoplasm and [B] cMyc protein in the nucleus in the trisomy 6 case (pink-solid dotted line), monosomy 6 case (blue-dashed line) and healthy tissue (green-solid line), each corresponding to production rate of MYC higher in monosomy 6 than in trisomy 6. 10 4 x 10 62 CHAPTER 5. NUMERICAL SIMULATIONS C OMPARISON OF THE SAMPLES WITH SIMILAR SGK 1 mRNA WITHIN MONOSOMY 6 AND TRISOMY 6 We perceive the important role of the SGK1 mRNA in the patient samples of medulloblastoma. However, we cannot disregard the MYC mRNA which itself also has influence on the nuclear cMyc protein. Consistently, we perform simulations for similar SGK1 mRNA production and different MYC mRNA production for samples in monosomy 6 (Table 5.11). Then, we repeat this scenario for the trisomy 6 samples (Table 5.12). Our intuition is confirmed by the tests and suggests that the MYC mRNA production has also an impact on the nuclear cMyc. monosomy SGK1 mRNA MYC mRNA fold change 0.27 38.11 tumor/control monosomy SGK1 mRNA MYC mRNA fold change 0.27 31.06 tumor/control cMyc in the nucleus 3 cMyc in the nucleus 2 Table 5.11: Microarray data values from the clinics for monosomy 6 of similar SGK1 mRNA and different MYC mRNA production. trisomy SGK1 mRNA MYC mRNA fold change 7.92 12.02 tumor/control trisomy SGK1 mRNA MYC mRNA fold change 7.86 3.05 tumor/control cMyc in the nucleus 24 cMyc in the nucleus 6 Table 5.12: Microarray data values from the clinics for trisomy 6 of similar SGK1 mRNA and different MYC mRNA production. The dynamics of the cytoplasmic and nuclear cMyc are presented in Figure 5.14 and Figure 5.15. In the situation of monosomy 6, we notice that even if the MYC mRNA production is high the resultant amount of the cMyc protein in the nucleus is much lower than in the trisomy 6. Therefore, we may explain why prognosis in 6q loss is better than in 6q gain. This shows that high MYC mRNA in monosomy 6, even much higher than in trisomy 6, is not fatal for patients. Numerical simulations confirm the medical observations that good prognosis is correlated with monosomy 6 and poor prognosis is correlated with trisomy 6. 5.2. SIMULATION RESULTS 63 [A], Dynamics of cMyc in the cytoplasm [B], Dynamics of cMyc in the nucleus 2 1.4 1.8 1.2 1.4 1.2 normal cells 6q loss: SGK1(0.27); MYC(38.11) 6q loss: SGK1(0.27); MYC(31.06) 1 0.8 0.6 cMyc protein [ µM ] cMyc protein [ µM ] 1.6 1 normal cells 6q loss: SGK1(0.27); MYC(38.11) 6q loss: SGK1(0.27); MYC(31.06) 0.8 0.6 0.4 0.4 0.2 0.2 0 0 2 4 6 8 time [ s ] 0 0 10 2 4 6 8 time [ s ] 4 x 10 10 4 x 10 Figure 5.14: Numerical simulations based on exemplary microarray data values from the clinics. Dynamics of [A] cMyc protein in the cytoplasm and [B] cMyc protein in the nucleus in the first monosomy 6 case (pink-solid dotted line), second monosomy 6 case (blue-dashed line) and healthy tissue (green-solid line), each corresponding to different production rate of MYC and similar production rate of SGK1. [A], Dynamics of cMyc in the cytoplasm [B], Dynamics of cMyc in the nucleus 0.7 3 2.5 0.5 normal cells 6q gain: SGK1(7.92); MYC(12.02) 6q gain: SGK1(7.86); MYC(3.05) 0.4 0.3 0.2 2 normal cells 6q gain: SGK1(7.92); MYC(12.02) 6q gain: SGK1(7.86); MYC(3.05) 1.5 1 0.5 0.1 0 0 cMyc protein [ µM ] cMyc protein [ µM ] 0.6 2 4 6 time [ s ] 8 10 4 x 10 0 0 2 4 6 8 time [ s ] Figure 5.15: Numerical simulations based on exemplary microarray data values from the clinics. Dynamics of [A] cMyc protein in the cytoplasm and [B] cMyc protein in the nucleus in the first trisomy 6 case (pink-solid dotted line), second trisomy 6 case (blue-dashed line) and healthy tissue (green-solid line), each corresponding to different production rate of MYC and similar production rate of SGK1. 10 4 x 10 CHAPTER 5. NUMERICAL SIMULATIONS 64 C OMPARISON OF THE SAMPLES WITH GSK 3β TRISOMY 6 DYNAMICS IN MONOSOMY 6 AND The graphs presented before show how the nuclear cMyc changes with respect to the SGK1 and MYC mRNA production. However, there is no direct interaction between these two proteins. −3 8 x 10 [B], Dynamics of cMyc in the nucleus [A], Dynamics of GSK3β in the nucleus 4 3.5 normal cells 6q loss: SGK1(0.54); MYC(28.59) 6q gain: SGK1(4.16); MYC(29.78) 6 5 4 3 cMyc protein [ µM ] GSK3β protein [ µM ] 7 3 2.5 1.5 2 1 1 0.5 0 0 2 4 6 time [ s ] 8 10 4 x 10 normal cells 6q loss: SGK1(0.54); MYC(28.59) 6q gain: SGK1(4.16); MYC(29.78) 2 0 0 2 4 6 8 time [ s ] Figure 5.16: Numerical simulations based on exemplary microarray data values from the clinics. [A] GSK3β protein in the nucleus and [B] cMyc protein in the nucleus in the trisomy 6 case (pinksolid dotted line), monosomy 6 case (blue-dashed line) and healthy tissue (green-solid line). High GSK3β protein leads to low nuclear cMyc protein and low GSK3β protein leads to high nuclear cMyc protein. GSK3β protein is a link between SGK1 and cMyc as it interacts with both of the proteins. We perceive that in the case of monosomy 6 the GSK3β protein is higher than in the control case. It is caused by weaker interaction with SGK1, which is downregulated in monosomy 6. On the other side, in the case of trisomy 6 the GSK3β protein decreases under the control level due to the SGK1 upregulation (Figure 5.16). The following changes in GSK3β influence further the nuclear cMyc. The regulatory loop SGK1-GSK3β-MYC demonstrates totally diverse behavior applying to the two types of medulloblastoma. This underlines the crucial role of these proteins in trisomy 6 and monosomy 6. 10 4 x 10 5.2. SIMULATION RESULTS S UMMARY 65 OF THE OBSERVATIONS FROM THE SIMULATIONS - Similar MYC mRNA production in trisomy 6 and monosomy 6 results in different nuclear cMyc according to the SGK1 mRNA production (Figure 5.8, Figure 5.9). - Similar MYC mRNA production in each medulloblastoma subgroup underlies the influence of varying SGK1 mRNA production on the nuclear cMyc (Figure 5.10, Figure 5.11). - Higher MYC mRNA production in monosomy 6 than in trisomy 6 may still result in much lower nuclear cMyc (Figure 5.12, Figure 5.13). - High MYC mRNA in monosomy 6 may still bring positive prognosis (Figure 5.14). - Low MYC mRNA in trisomy 6 may lead to high nuclear cMyc due to influence of SGK1 (Figure 5.15). - Degradation of the nuclear cMyc by GSK3β is higher in the case of monosomy 6 than in the case of trisomy 6 (Figure 5.16). - Dynamics of the regulatory loop GSK3β-SGK1-MYC may explain discrepancy in the nuclear cMyc. Statement 5.3 We identify the crucial role of SGK1 as a driving factor which strongly influences the cMyc level in the nucleus. MYC is assigned to be a proto-oncogene and is found upregulated also in other cancers [15], [53], [63]. Increased MYC leads to upregulated expression of many genes, some of which are involved in the cell proliferation. These changes may result in cancer. High increase in the cMyc protein follows the increase of the SGK1 protein. In monosomy 6 the production of SGK1 is lower than 1, thus the cMyc protein is not increased due to the SGK1 activity. Indeed, we observe that prognosis is much better in this case. Taking into account both types of medulloblastoma, we claim that SGK1, which is correlated with MYC, may play a main role in the survival differential. In Figure 5.17, we present relationship between the cMyc protein in the nucleus, the MYC and SGK1 mRNAs obtained through the numerical simulations of the dynamics of the model with different parameter sets corresponding to the different virtual patients. CHAPTER 5. NUMERICAL SIMULATIONS 66 Change in the nuclear cMyc based on MYC and SGK1 mRNA 3.5 nuclear cMyc 3 2.5 2 1.5 1 0.5 0 0 1 0.8 0.6 0.5 0.4 0.2 0 −3 1 x 10 SGK1 mRNA −3 x 10 MYC mRNA Figure 5.17: Relationship between the cMyc protein in the nucleus, the MYC and SGK1 mRNAs obtained through the numerical simulations of the dynamics of the model with different parameter sets corresponding to the different virtual patients. We note the almost linear relationship between the SGK1 mRNA and the nuclear cMyc when the MYC mRNA is close to 0. 6 5 nuclear cMyc 4 3 2 1 0 0 0.2 0.4 0.6 SGK1 mRNA 0.8 1 −3 x 10 Figure 5.18: Correlation between the level of the cMyc protein in the nucleus and SGK1 mRNA obtained through the numerical simulations for varying values of parameters t1 and t3 . We obtain positive correlation between the SGK1 mRNA expression and levels of the nuclear cMyc protein. 5.2. SIMULATION RESULTS 67 5.2.4 Correlation between the SGK1 mRNA production and cMyc protein level in the nucleus To estimate the correlation between the SGK1 mRNA production and the nuclear cMyc level in the medulloblastoma samples, we perform the following numerical experiment. We fix the point in time for which we test the correlation. We vary the values of the rates, t1 and t3 , by adding to them the pseudorandom values drawn from the standard normal distribution. We plot 500 points which relate to the number of vector entries of the randomly varied parameters. We obtain positive correlation. For higher values of the SGK1 mRNA production we have higher values of the nuclear cMyc. The inference is that the SGK1 production increases cMyc protein level in the nucleus (see Figure 5.18). 5.2.5 Role of inhibition in the SGK1 protein In the case of critically high cMyc in the nucleus, we use the model to study the effect of the reduction of the protein amount. There are several options to obtain such effect. From the practical point of view, the best medical option is to apply a pharmacological inhibitor. The effect of inhibition (decreased activity of the protein) is in our system modeled by an increased degradation of the protein. Numerical simulations predict that increased degradation of the SGK1 protein (Figure 5.19[A]) leads to a significant decrease of cMyc levels in the nucleus. Moreover, using the model we can compare the effects of different pharmacological strategies. Therefore, we also simulate system (3.14) - (3.22) with the cMyc level reduction (Figure 5.19[B]) by increased degradation. Then, we show that indeed inhibition of the SGK1 protein is a more efficient way to decrease the level of cMyc than increased degradation of the cMyc protein. Importantly, the impact of SGK1 on stabilization of the cMyc protein is maintained through the degradation of GSK3β. SGK1 phosphorylates GSK3β resulting in its degradation, hence inhibition of SGK1 leads to increase in the GSK3β concentration. The consequence of the latter is higher degradation of cMyc by GSK3β. The degradation of cMyc itself is not sufficient enough as the upregulated SGK1 still stimulates the cMyc increase in the nucleus. Numerical simulations of the model reveal that the SGK1 inhibition is more efficient in decreasing the cMyc concentration. We obtain that ten times increased SGK1 degradation yields a very good outcome, where level of the nuclear cMyc is almost equal to the control level (Table 5.13). The application of the SGK1 inhibition can be treated as an adjuvant treatment to the traditional one. However, we should not undertake the same actions in monosomy 6, where we already have a low level of SGK1. Further decreasing of SGK1 could lead to unexpected mutations in the cell because of such low SGK1 level (Figure 5.20 and Table 5.14). CHAPTER 5. NUMERICAL SIMULATIONS 68 [A], Dynamics of cMyc in the nucleus [B], Dynamics of cMyc in the nucleus 1.8 1.8 cMyc protein [ µM ] 1.4 1.2 1 0.8 0.6 1.4 1.2 1 0.8 0.6 0.4 0.4 0.2 0.2 0 0 2 4 6 8 time [ s ] 10 4 x 10 normal cells 6q gain: SGK1(7.11); MYC(6.45) 6q gain − 2 times increased cMyc degradation 6q gain − 5 times increased cMyc degradation 6q gain − 10 times increased cMyc degradation 1.6 cMyc protein [ µM ] normal cells 6q gain: SGK1(7.11); MYC(6.45) 6q gain − 2 times increased SGK1 degradation 6q gain − 5 times increased SGK1 degradation 6q gain − 10 times increased SGK1 degradation 1.6 0 0 2 4 6 8 time [ s ] Figure 5.19: Numerical simulations based on exemplary microarray data values from the clinics. Effect of increased degradation [A] of SGK1 (removal of the SGK1 protein) and [B] of cMyc (removal of the cMyc protein); trisomy 6 sample (SGK1 mRNA fold change is equal to 7.11, MYC mRNA fold change is equal to 6.45). Fold change of the nuclear cMyc after SGK1 inhibition SGK1(7.11); MYC(6.45) 11 2 times inhibition 6 5 times inhibition 2 10 times inhibition 1 Fold change of the nuclear cMyc after cMyc inhibition SGK1(7.11); MYC(6.45) 11 2 times inhibition 10 5 times inhibition 8 10 times inhibition 6 Table 5.13: Fold change of the nuclear cMyc after SGK1 inhibition and cMyc inhibition. Trisomy 6 case. Fold change of the nuclear cMyc (no inhibition) SGK1(0.74); MYC(16.27) 3 1 SGK1(0.19); MYC(20.5) SGK1(0.44); MYC(10.02) 1 Fold change of the nuclear cMyc after 2 times SGK1 inhibition SGK1(0.74); MYC(16.27) 1.5 0.5 SGK1(0.19); MYC(20.5) SGK1(0.44); MYC(10.02) 0.56 Table 5.14: Fold change of the nuclear cMyc before and after SGK1 inhibition. Monosomy 6 case. 10 4 x 10 5.2. SIMULATION RESULTS 69 [A], Dynamics of cMyc in the nucleus [B], Dynamics of cMyc in the nucleus after 2 times increased SGK1 degradation 0.45 0.3 0.35 normal cells 6q loss: SGK1(0.74); MYC(16.27) 6q loss: SGK1(0.19); MYC(20.5) 6q loss: SGK1(0.44); MYC(10.06) 0.25 0.3 normal cells 6q loss: SGK1(0.74); MYC(16.27) 6q loss: SGK1(0.19); MYC(20.5) 6q loss: SGK1(0.44); MYC(10.06) 0.25 0.2 0.15 cMyc protein [ µM ] cMyc protein [ µM ] 0.4 0.2 0.15 0.1 0.1 0.05 0 2 4 6 time [ s ] 8 10 0.05 0 4 x 10 2 4 6 8 time [ s ] Figure 5.20: Numerical simulations based on exemplary microarray data values from the clinics. Comparison of the nuclear cMyc [A] before and [B] after SGK1 inhibition in the monosomy 6 different cases. SGK1 degradation increased two times. Observation 5.2 SGK1 inhibition is more efficient than cMyc inhibition to obtain low nuclear cMyc. Treatment in trisomy could be extended to inhibition of SGK1. 5.2.6 Effect of the GSK3β protein stabilization In parallel, we study the dynamics of GSK3β in the case of the cMyc and SGK1 inhibition. Simulations indicate that inhibition of cMyc does not influence the dynamics of GSK3β (Figure 5.21) and that GSK3β stays downregulated compared to the normal cells. It explains why the level of cMyc in the nucleus remains high in spite of inhibition. Simulations of the SGK1 inhibition reveal a positive influence on the levels of GSK3β (Figure 5.22 and Table 5.15), which become elevated. This in turn generates a decrease in cMyc level in the nucleus. The coupling loop SGK1-GSK3β-MYC seems to play an important role in both types of medulloblastoma, showing different properties between monosomy 6 and trisomy 6 with respect to aberrations in the SGK1 and MYC mRNAs. 10 4 x 10 CHAPTER 5. NUMERICAL SIMULATIONS 70 −3 [A], Dynamics of GSK3β in the nucleus [B], Dynamics of cMyc in the nucleus 1.8 4 1.6 3.5 1.4 3 normal cells 6q gain: SGK1(7.11); MYC(6.45) 6q gain − 2 times increased cMyc degradation 6q gain − 5 times increased cMyc degradation 6q gain − 10 times increased cMyc degradation 2.5 2 1.5 cMyc protein [ µM ] GSK3β protein [ µM ] x 10 1.2 1 0.8 0.6 1 0.4 0.5 0.2 0 0 2 4 6 8 time [ s ] 0 0 10 normal cells 6q gain: SGK1(7.11); MYC(6.45) 6q gain − 2 times increased cMyc degradation 6q gain − 5 times increased cMyc degradation 6q gain − 10 times increased cMyc degradation 2 4 6 8 time [ s ] 4 x 10 10 4 x 10 Figure 5.21: Numerical simulations based on exemplary microarray data values from the clinics. Influence of cMyc inhibition [A] on the GSK3β level, [B] on the nuclear cMyc. 1.8 normal cells 6q gain: SGK1(7.11); MYC(6.45) 6q gain − 2 times increased SGK1 degradation 6q gain − 5 times increased SGK1 degradation 6q gain − 10 times increased SGK1 degradation 8 7 GSK3β protein [ µM ] [B], Dynamics of cMyc in the nucleus [A], Dynamics of GSK3β in the nucleus 6 5 4 3 1.4 1.2 1 0.8 0.6 2 0.4 1 0.2 0 0 2 4 6 time [ s ] 8 10 4 x 10 normal cells 6q gain: SGK1(7.11); MYC(6.45) 6q gain − 2 times increased SGK1 degradation 6q gain − 5 times increased SGK1 degradation 6q gain − 10 times increased SGK1 degradation 1.6 cMyc protein [ µM ] −3 x 10 0 0 2 4 6 8 time [ s ] Figure 5.22: Numerical simulations based on exemplary microarray data values from the clinics. Influence of SGK1 inhibition [A] on the GSK3β level, [B] on the nuclear cMyc. Observation 5.3 Inhibition of SGK1 contributes to the balance of a wider range of proteins, e.g., GSK3β and cMyc. 10 4 x 10 5.2. SIMULATION RESULTS Fold change of the nuclear GSK3β after SGK1 inhibition SGK1(7.11); MYC(6.45) 0.14 2 times inhibition 0.28 5 times inhibition 0.7 10 times inhibition 1.4 71 Fold change of the nuclear cMyc after SGK1 inhibition SGK1(7.11); MYC(6.45) 11 2 times inhibition 6 5 times inhibition 2 10 times inhibition 1 Table 5.15: Fold change of GSK3β and cMyc in the case of SGK1 inhibition. Summary of Chapter 5 In this chapter we presented numerical simulations of the ODE model discussed in previous chapters. We performed several tests to investigate the proteins behavior and interactions that occur on the intracellular level. Our study strongly emphasize the crucial role of SGK1 in the process of biological homeostasis in the cell. We suggest that SGK1 is the key gene indicating the discrepancy of prognosis in medulloblastoma and has big influence on the amount of the nuclear cMyc. Inhibition of the SGK1 protein is much more efficient procedure than inhibition of cMyc to decrease the nuclear cMyc. Additionally, reducing the SGK1 protein restores also the balance of the GSK3β protein in the system. All facts assert that high SGK1 leads to negative prognosis and its inhibition brings positive results. We hope that inhibition of the SGK1 protein can be used as an adjuvant therapy for the patients with trisomy 6 diagnosis. 72 CHAPTER 5. NUMERICAL SIMULATIONS CHAPTER 6 Parameter estimation and optimal experimental design This chapter is devoted to the identification of model parameters and application of optimal experimental design. Complexity of the biological model can strongly influence the robustness of the optimal parameter estimation. The goal is to have a predictive model to investigate the biological environment according to different biological signals. To perform good system predictions the parameters should be identified with the smallest possible uncertainties taking into consideration the model dynamics, which reflect the biological behavior. Our goal is to estimate parameters by fitting the model to the experimental data to obtain the minimal costs of experimentation and enhance the quality of estimation. Nowadays, processing techniques have improved a lot and more methods in data collection are found. However, large amount of data do not assure the best experimental set-up to infer estimates that are the closest to the real parameter values. Very often, in non-optimal designs there is less useful information and much more experimental runs must be performed. In consequence of latter, estimation of parameters is a laborious process and sometimes it can happen that parameters are obtained with low precision. With optimal experimental design the scheme is optimized in a way that taken measurements give the best estimate with the lowest cost. 6.1 Parameter estimation problem In this section we approach the parameter estimation problem and the Gauss-Newton method, which provides a numerical solution of the stated problem. We introduce the solution operator, the observation operator as well as the cost functional, which quantifies 73 74 CHAPTER 6. PARAMETER ESTIMATION AND EXPERIMENTAL DESIGN the difference between experimental and simulated data. To define the Gauss-Newton method, we derive the gradient and the approximation of the Hessian of the least squares functional and finally we present a calculation procedure in subsequent steps of the algorithm. 6.1.1 General formulation We consider model (3.14) - (3.22) of biochemical interactions in the form of u̇ = F (u, q) (6.1) in medulloblastoma tumor cells (see Figure 2.12), where q is a finite number of unknown model parameters q := (q1 ,...,qm ) ∈ Q ⊂ Rm with m = 19. The concentration of each species is represented by the solution u, F := [f1 , ..., fn ]T , F : G → Rn , where G := U × Q and F ∈ C 1 (U × Q). Model solutions depend on time and model parameters U : [0, ∞) × Q → Rn with n = 9 . Theorem 4.1 yields U ∈ C 1 ([0, ∞) × Q). Table 6.1 shows the respective parameters used. The experimental measurements are gathered into a vector C̄, which is in the observation space D ⊂ Rl . It is crucial to satisfy l ≥ m, [66], to be able to determine the model parameters. Further, we introduce the solution operator S(q) : Q → U , which maps model parameters to the solution space. The observation operator C(u) : U → D maps the solution to the observation space. Here we use the point measurement C(u) := u(tk ), where tk is the time point in which measurement is taken, k ∈ N. From practical point of view, we are aware that each of the measured values have a random error which we call perturbation p ∈ P ⊂ D. Consequently, the measurement values are C̄ = C̄true + p, i.e., the sum of real value and measurement error. To evaluate the deviation between experimental measurements C̄ and state values C(u) dependent on parameters q, we apply the cost functional J(u, q) : U × Q → R. Finally, we can formulate the perturbed parameter estimation problem in the following form. Problem 6.1 To find the optimal solution for model parameters the minimization of the cost functional under constraints is carried out. That reads { min J(u, q), (u,q)∈(U ×Q) (6.2) s.t. u̇ = F (u, q). From Problem (6.2) it follows that we compare experimental data with data obtained from the model. Introducing the weighted least squares functional and using the reduced formulation [6], i.e., Ĉ(q) := C(S(q)), where u := S(q), we can rewrite the statement (6.2) into ˆ 2 . min 21 J(q) (6.3) D q∈Q 6.1. PARAMETER ESTIMATION PROBLEM Notation for estimates Notation of parameters used in model q1 q2 q3 q4 q5 q6 q7 q8 q9 q10 q11 q12 q13 q14 q15 q16 q17 q18 q19 t1 βcat d1 s2 d2 d6 p9 t3 d3 s4 c4 kv d5 PGSK c7 c6 do p8 d4 75 Biological meaning transcription rate of SGK1 transcription factor for the MYC and SGK1 mRNA synthesis spontaneous degradation rate of the SGK1 mRNA translation rate of SGK1 spontaneous degradation rate of the SGK1 protein degradation rate of the GSK3β protein in the cytoplasm by SGK1 dissociation coefficients of the [SGK1·GSK3] complex transcription rate of MYC spontaneous degradation rate of the MYC mRNA translation rate of cMyc transport of cMyc to the nucleus coefficient scaling coefficient, cytoplasm to the nucleus ratio degradation rate of the cMyc protein in the nucleus by GSK3β constant influx of GSK3β to the cytoplasm transport of GSK3β to the cytoplasm coefficient transport of GSK3β to the nucleus coefficient spontaneous degradation rate of GSK3β in the cytoplasm dissociation coefficients of the [GSK3· cMyc] complex spontaneous degradation rate of cMyc in the cytoplasm Table 6.1: Description of model variables. From left to right: notation used to define parameters to estimate, equivalent variables in ODE model (3.14) - (3.22) and biological meaning. 76 CHAPTER 6. PARAMETER ESTIMATION AND EXPERIMENTAL DESIGN ˆ := Ĉ(q) − C̄ is a reduced cost functional, where Ĉ(q) − C̄ is called a residual vector J(q) 2 and · = (C −1 ·, ·) is a weighted norm with weighting matrix C −1 (see Section 6.2.2), D D D which corresponds to the covariance matrix of the measurements [17], [73]. Additionally, we assume that measurement errors are independent and normally distributed, which justifies the use of the least squares functional [9]. 6.1.2 Solving the nonlinear problem Problem 6.2 Our goal is to solve the nonlinear least squares (LS) problem in reduced formulation ˆ 2 . min 12 J(q) (6.4) D q∈Q We solve the optimization Problem 6.4 applying the Gauss-Newton method iteratively ˆ [34]. To find a local minimum of reduced formulation of J(q), we choose an initial guess of q 0 . Our problem can be stated in the following way. Algorithm 6.1 Calculate q iteratively starting with q 0 and make update by setting q i+1 = q i +αi ∆q i , where αi is the stepsize and ∆q i is the search direction defined by the gradient and approximation of the Hessian in each iteration i, where i = 0, 1, 2... Convergence is assured for a small residual problem and obtained updating q by ∆q, which is expressed by ( )−1 ˆ ˆ ∆q := − ∇2 J(q) ∇J(q), (6.5) ˆ is the gradient of the cost functional J(q) ˆ in the form where ∇J(q) ( ) ˆ := Ĝ(q)T C −1 Ĉ(q) − C̄ . ∇J(q) D Ĝ(q)ij := (6.6) ∂ Ĉi (q) ∂qj is the Jacobian (see Section 6.3.1). ˆ ˆ The term ∇ J(q) is the approximation of the Hessian of the cost functional J(q), where the second order derivatives are neglected [8] 2 ˆ := Ĝ(q)T C −1 Ĝ(q). ∇2 J(q) D (6.7) We write the update of the Gauss-Newton problem by inserting (6.6) and (6.7) in (6.5). Definition 6.1 The Gauss-Newton update { q i+1 = q i + αi ∆q i , ( )−1 ( ) −1 −1 ∆q i = − Ĝ(q)T CD Ĝ(q) Ĝ(q)T CD Ĉ(q) − C̄ . (6.8) 6.1. PARAMETER ESTIMATION PROBLEM 77 In each iteration the condition that the new calculated value of the LS functional is lower than the one calculated in previous iteration is imposed [27] LS(q i+1 ) < LS(q i ). (6.9) If condition (6.9) is not fulfilled, the backtracking line search strategy is used, i.e., in the i-th iteration αi is updated as follows. We choose an initial stepsize α0i = 1 and a ϱ ∈ (0, 1). For iterations j = 0, 1, ... we update αji = α0i ϱj (ϱj is ϱ to the power of j) until the condition (6.9) is fulfilled. If fulfilled, q i+1 is calculated for the new αji . In our application we used ϱ = 0.5. We proceed updates defined in (6.8) till the main breaking condition is satisfied, which is when ∆q is lower than assumed accuracy ϵ1 . The convergence is obtained and next steps would bring only quite low variation in parameter value, see [48]. Remark 6.1 The Gauss-Newton method is a local optimization algorithm. If there are several local minima, several q 0 should be chosen to assure the global solution. The scheme of the Gauss-Newton method algorithm is presented in Table 6.2. We use the Gauss-Newton method in the next sections to find the model parameter estimates systematically, highlighting different aspects of parameter estimation problem. 78 CHAPTER 6. PARAMETER ESTIMATION AND EXPERIMENTAL DESIGN The Gauss-Newton nonlinear least squares algorithm 1 Initialize: 2 initial guess of q 0 , initial residual - calculate LS value at q 0 , ϱ = 0.5 3 Global loop: 4 calculate Ĝ(q) for q i 5 calculate gradient based on Ĝ(q) 6 calculate approximation of the Hessian based on Ĝ(q) 7 calculate ∆q 8 set α0i = 1 9 update q i+1 10 evaluate residuals - calculate LS value at q i+1 11 Local loop: 12 break loop if LS(q i+1 ) < LS(q i ) 13 update αji 14 update q i+1 15 evaluate residuals - calculate LS value at updated q i+1 16 increment local iteration counter j 17 error exit if maximal number of iterations exceeded 18 End local loop 19 increment global iteration counter i 20 error exit if maximal number of iterations exceeded 21 normal exit if (|∆q| < ϵ1 ) 22 End global loop Table 6.2: Pseudocode of the Gauss-Newton algorithm. 6.2. CONFIDENCE REGION IN THE FORM OF ELLIPSES 79 6.2 Confidence region in the form of ellipses The goal of this section is to feature the probabilistic character of the parameter estimation problem and subsequently to show its graphical interpretation. We introduce the formula of covariance matrix that allows us to draw the confidence region. The confidence region is a region of parameter uncertainties around the estimated parameters qi , qj (i ̸= j), i.e., the confidence region is evaluated for two different parameters. This graphical representation is in the form of an ellipse, also denominated here confidence ellipse. The confidence region is a great tool to visualize the reliability of an estimated solution. 6.2.1 Definition of covariance matrix Taking into account the probabilistic representation of perturbations p, we have the Gaussian nature of measurement error eM := p ∼ N (0, σ 2 ), that stands for the normal distribution with expected value of the measurement error equal zero and variance is given by the weighting matrix CD with constant uncorellated variance var(eM ) = σ 2 . We assume that there are no outliers [20]. Definition 6.2 Measurement error follows an accuracy of which measurement is extracted by the experimentalist. To see, how the estimates depend on the measurement error we perform the covariance analysis. In the following [74], the covariance matrix is assumed to be ( )−1 −1 Cov := ĜT CD Ĝ , (6.10) where Ĝ, introduced in Subsection 6.1.2, represents the Jacobian with calculated sensitivities. 6.2.2 Geometrical interpretation of confidence region Based on Cov we can interpret the parameter estimation problem geometrically, taking into account the probabilistic character of the parameter estimation problem [1], [4], [74]. Geometrical representation which approximates the statistical distribution of the estimated parameters is in the form of an ellipse [20], [59]. The ellipse is centered at searched value qi , qj and the uncertainties of estimated parameters are denoted by axes. We represent the ellipses in a coordinate system where √ lengths of principal axes correspond to the square root of eigenvalues λ of the covari√ √ ance matrix Cov for particular parameters. λmax and λmin stand for the longer and shorter ellipse axis, respectively. The ellipses are derived from the concept of a 95% confidence interval for a normally distributed random value [17]. We present the statistical 80 CHAPTER 6. PARAMETER ESTIMATION AND EXPERIMENTAL DESIGN results on parameter error estimation in the form of ellipses in Figure 6.1, where we set i = 1, j = 2. Figure 6.1: Graphical representation of the parameter estimation problem. [A] Measurements sampled in particular time point projected into [B] parameter space Q. The uncertainties of estimated parameters depend on the measurement error and sampling time. Confidence ellipses visualize how the variance of measurement errors influences the error in parameter estimation. The variance of the measured data is expressed in (6.10) by CD , whereas Ĝ is the Jacobian (see Section 6.3.1). Measurements are taken for different time probes tk and therefore the Jacobian depends also on the time points of measurements. If we find good measurement points, the uncertainties of estimated parameters are small. If the uncertainties are below an expected threshold, which depends on the application, then we are satisfied with the estimates. If measurements of two species are)independent and have the same error, the data ( 1 0 covariance matrix is CD = σ 2 . The second option is that, measurements are 0 1 ( ) a 0 with a,b > 0 and a ̸= b. If meaindependent but errors differ, thus CD = σ 2 0 b surements are correlated and sampled with different error the covariance matrix is not ( ) a c diagonal and reads CD = σ 2 with a,b,c,d > 0 and a ̸= b ̸= c ̸= d. The exd b perimentalist may assume or predict the measurement error and may know if data are correlated or not. In our work we assume that measurements are independent and have the same error. Interestingly, depending on the measurement points the ellipses are more or less slant- 6.2. CONFIDENCE REGION IN THE FORM OF ELLIPSES 81 ing in the coordinate system. The shape of ellipses, so alignment of the principal axes with respect to the parameter axes, describe the correlation between parameters. More slant- Figure 6.2: 95% confidence ellipses. [A] The parameters are not correlated, the estimation of parameter q1 is better than estimation of parameter q2 , [B] parameters are not correlated, the error in estimation is the same for both parameters, [C] parameters are correlated and parameter q1 is worse in estimation than parameter q2 . ing ellipses refer to the parameters that are more correlated due to chosen measurement points, what shows that the error in one parameter influences the error in the correlated pa( )−1 −1 rameter. Following that Cov = ĜT CD Ĝ describes coupling between measurement error and probes depending on sampling time, we plot types of the confidence ellipses corresponding to different parameter correlations (Figure 6.2). 82 CHAPTER 6. PARAMETER ESTIMATION AND EXPERIMENTAL DESIGN The graphical method of ellipses is the linearized approximation of the problem, thus to check its reliability, we also perform Monte Carlo simulations using the Gauss-Newton method described in Section 6.1 and compare both approaches in one plot. Definition 6.3 The Monte Carlo method is a numerical method based on random sampling of the data [47]. 6.3 Parameter estimation in medulloblastoma model In this section we investigate different aspects of the parameter estimation problem. The medulloblastoma model consists of nonlinear ODEs and depends on unknown paramaters, which can be estimated if we take advantage of experimental data. We show the behavior and difficulty in estimating the parameters of the medulloblastoma model (3.14) - (3.22). We perform our analysis for two species, i.e., the concentration of the nuclear cMyc protein and the concentration of occupied GSK3β in the nucleus, respectively. We take measurements to identify the parameters of the model. We do not have real experimental data, therefore we use randomly perturbed simulations as surrogate data. In Subsection 6.3.1 we approach the parameter sensitivity analysis problem. Then, in Subsection 6.3.2 we deal with parameter nonlinearities, coupling between parameters and the influence of measurement error on estimation. In Subsection 6.3.3 we raise the problem of impact on estimation of variation in parameter values, which account for the parameters found in the literature, often presented by range in values. At the end of each subsection we make a table with reliable estimates. 6.3.1 Parameter sensitivity analysis Parameter sensitivity analysis [11], [33], is one of the steps to examine the parameter estimation problem. The measurements collected using some specific experimental protocol may not allow for the reliable parameter estimation that can be highly uncertain. We investigate if parameters are weak or strong sensitive to the measurements accounting for a chosen sampling protocol. If we have a good sampling protocol our estimation bring positive results and the behavior of the model for given set of parameters can explain the measurements. For future goals the model with well established parameters can predict dynamics under different inputs. Definition 6.4 Sensitivity analysis in a parameter estimation problem is the research of the sampling protocol, i.e., the time points for which model shows the highest sensitivity. This corresponds to the time point in which the model contributes the highest information to the estimation process. 6.3. PARAMETER ESTIMATION IN MEDULLOBLASTOMA MODEL 83 Problem 6.3 Find model species Ĉi for which the sensitivity is the highest. Find time points for which particular parameters have the highest sensitivity. Definition 6.5 We define a Jacobian sensitivity matrix Ĝ(q)ij of the state Ĉi (q) with respect to parameter q by a change in qj , where i = 1,...,n and j = 1,...,m. Ĝ(q)ij := ∂ Ĉi (q) ∂qj (6.11) The definition can be found in [74]. Differentiation of Ĝ(q)ij gives the system of sensitivities ( ) n ∑ d ∂ Ĉi (q) ∂fi (Ĉ, q, t) ∂ Ĉi ∂fi (Ĉ, q, t) = + . (6.12) dt ∂qj ∂qj ∂qj ∂ Ĉi i=1 Remark 6.2 The condition number of an m × m matrix A is cond(A) := ∥A∥∥A∥−1 and is a measure of how close a matrix is to being singular. If cond(A) is large depending on the operating system, e.g., greater than 1e15, then numerically matrix A is singular. To improve the condition number, we introduce a scaling q = q̄ · qref (also applied in model equations (3.14) - (3.22)) and reformulate (6.12) into d dt ( ∂ Ĉi (q̄ · qref ) ∂(q̄ · qref )j ) = n ∑ ∂fi (Ĉ, q̄ · qref , t) i=1 ∂ Ĉi · ∂fi (Ĉ, q̄ · qref , t) ∂ Ĉi + · qrefj , (6.13) ∂(q̄ · qref )j ∂(q̄ · qref )j where q̄ = 1 is an initial guess, qref = const. is a reference parameter and q is a rescaled parameter. Remark 6.3 For future application when we write about model species, we stick to the notation Ci for the simplicity and to emphasize that measurements are mapped to the observation space, C(ui ) = u(tk ). 84 CHAPTER 6. PARAMETER ESTIMATION AND EXPERIMENTAL DESIGN Numerical simulations To solve the sensitivity problem, we calculate the sensitivities by numerical evaluation of (6.1) and (6.13). To check if our sensitivities are well calculated we use a finite difference method [69] and we compare both numerical solutions in one plot. In our model, where n = 9 and m = 19, we obtain 180 = (19 · 9)sensitivity_eqs + 9model_eqs equations to be simR ulated in MATLAB⃝ . We check for which species and for which parameters the system reveals the highest sensitivity. To find parameters with the highest absolute amplitude of sensitivity, we perform simulations for species C5 (the concentration of the nuclear cMyc protein) found to have the highest sensitivity. In our model q5 = d6 (degradation rate of the SGK1 protein) is the most sensitive parameter with respect to species C5 . The parameters q7 = p9 (dissociation coefficients of [SGK1·GSK3β]) and q17 = do (spontaneous degradation rate of the GSK3β protein in the cytoplasm) show sensitivity close to zero. This suggests that we are not able to estimate all parameters. The absolute amplitude of sensitivity is denoted by AS. The amplitudes of sensitivity of C5 to the parameters are given in Table 6.3. Parameter AS q1 q2 q3 q4 q5 q6 q7 q8 q9 q10 0.32 2.67 3.45 1.67 8.75 1.67 ∼0 2.64 2.38 2.64 Parameter AS q11 q12 q13 q14 q15 q16 q17 q18 q19 0.31 2.41 1.26 1.69 1.02 1.67 ∼0 0.48 0.23 Table 6.3: The absolute amplitude of sensitivity AS for model parameters regarding species C5 . Parameter q5 shows the highest sensitivity. Now, we investigate how the chosen measurement time points influence the accuracy of estimates. We start our analysis with only three parameters. We fix other parameters, by deleting the corresponding row and column from the species Jacobian matrix to calculate the proper Cov to plot ellipses. We choose mc = 3 < m parameters, i.e., parameter q5 with the highest sensitivity, arbitrary parameter q13 = d5 (degradation rate of the cMyc protein in the nucleus by GSK3β) and arbitrary parameter q14 = PGSK (constant influx of the GSK3β protein to the cytoplasm). For mc = 3 we choose l = 3 time points of the 6.3. PARAMETER ESTIMATION IN MEDULLOBLASTOMA MODEL 85 measurements to fulfill l ≥ mc and perform numerical comparison based on the method of ellipses. We assume a measurement error eM = 10% represented by the covariance matrix CD . From practical point of view too little error is not realistic and too large does not bring reliable results. We choose the following time points measured in seconds: t1 = 12000, t2 = 19000, t3 = 20000, t4 = 21000, t5 = 23000, where each of time point represents different amplitudes of sensitivity. Definition 6.6 We call the design parameter the parameter, which describes the sampling protocol. Here, it is the time tk,k=1,...,l , where tl corresponds to the last time point of the experiment and we define the design parameter as ξ ∈ Π, where Π is a design space and ξ = {t1 , ..., tl }. We prepare the experiment design using the design parameter ξ = {tv , t2 , t5 }, where t2 and t5 are fixed. We vary tv,v={1,3,4} to investigate different sensitivity scenarios for t1 , t3 and t4 , where the inequality AS(t1 ) < AS(t3 ) < AS(t4 ) holds (see Figure 6.3 to Figure 6.5), i.e., amplitude of sensitivity in point t1 is lower than amplitude of sensitivity in point t3 , and the amplitude of sensitivity in point t3 is lower than amplitude of sensitivity in point t4 . We describe the design sets: L: tv = t1 chosen for low amplitude AS of sensitivity, M: tv = t3 chosen for middle amplitude AS of sensitivity, H: tv = t4 chosen for high amplitude AS of sensitivity, for our model L: ξ = {12000, 19000, 23000}, M: ξ = {20000, 19000, 23000}, H: ξ = {21000, 19000, 23000}. The sensitivities with different measurement points are plotted in Figure 6.3 to Figure 6.5 and corresponding ellipses for parameter estimation problem are in Figure 6.6 to Figure 6.8. For chosen three parameters we extract data only from species C5 . Remark 6.4 We call set Qad be a set of admissible, in sense possible, parameters q that we can estimate, where qmin < q < qmax , qmin = 0, qmax = 2q. 86 CHAPTER 6. PARAMETER ESTIMATION AND EXPERIMENTAL DESIGN [A], Sensitivity analysis of parameter q5 in species C5 sensitivity for q5, concentration of C5 [ µM ] 4 concentration 2 0 −2 finite difference method sensitivity species density of C5 −4 t1 12000 t 19000 2 −6 t 20000 3 t 21000 −8 4 sensitivity t 23000 5 −10 0 1 2 3 4 5 6 7 time [ s ] 4 x 10 Figure 6.3: Sensitivity with time points for species C5 in analysis of [A] parameter q5 . [B], Sensitivity analysis, of parameter q13 in species C5 2.5 sensitivity for q13 concentration of C5 [ µM ] concentration 2 finite difference method sensitivity species density of C 1.5 5 1 t1 12000 0.5 t2 19000 t 20000 3 0 t4 21000 −0.5 t5 23000 −1 −1.5 0 sensitivity 1 2 3 4 time [ s ] 5 6 7 4 x 10 Figure 6.4: Sensitivity with time points for species C5 in analysis of [B] parameter q13 . On the first glance, we perceive that estimation for three chosen parameters is possible, as the error is smaller than 100%, and hence, we do not obtain negative values for the estimates. For different design sets we obtain better or worse identifications. Therefore, 6.3. PARAMETER ESTIMATION IN MEDULLOBLASTOMA MODEL 87 [C], Sensitivity analysis of parameter q14 in species C5 2.5 sensitivity for q14, concentration of C5 [ µM ] 2 concentration 1.5 1 finite difference method sensitivity species density of C5 0.5 0 t 12000 −0.5 t2 19000 1 t3 20000 −1 t 21000 4 −1.5 −2 0 t5 23000 sensitivity 1 2 3 4 5 6 time [ s ] 7 4 x 10 Figure 6.5: Sensitivity with time points for species C5 in analysis of [C] parameter q14 . [A], Confidence regions 1 + 8.000000E−01 L M q13 H 1 1 − 8.000000E−01 1 − 8.000000E−01 1 q5 1 + 8.000000E−01 Figure 6.6: Comparison of 95% confidence regions: case H, M, L for C5 for design parameters H: ξ = {19000, 23000, 21000}, M: ξ = {19000, 23000, 20000}, L: ξ = {19000, 23000, 12000}. [A] parameters q5 and q13 . we see how choice of the sampling time points is important for the estimation problem. For species C5 the set H with sampling point of high AS, so tv = t3 = 20000, depicts the 88 CHAPTER 6. PARAMETER ESTIMATION AND EXPERIMENTAL DESIGN [B], Confidence regions q14 1 + 8.000000E−01 1 H M L 1 − 8.000000E−01 1 − 8.000000E−01 1 q5 1 + 8.000000E−01 Figure 6.7: Comparison of 95% confidence regions: case H, M, L for C5 for design parameters H: ξ = {19000, 23000, 21000}, M: ξ = {19000, 23000, 20000}, L: ξ = {19000, 23000, 12000}. [B] parameters q5 and q14 . [C], Confidence regions 1 + 1.000000E+00 L M q14 H 1 1 − 1.000000E+00 1 − 1.000000E+00 1 q13 1 + 1.000000E+00 Figure 6.8: Comparison of 95% confidence regions: case H, M, L for C5 for design parameters H: ξ = {19000, 23000, 21000}, M: ξ = {19000, 23000, 20000}, L: ξ = {19000, 23000, 12000}. [C] parameters q13 and q14 . 6.3. PARAMETER ESTIMATION IN MEDULLOBLASTOMA MODEL 89 smallest ellipses, where parameters are estimated with the smallest uncertainty. Following this argumentation, the biggest ellipses are for set L with sampling time point of low AS, i.e., tv = t1 = 12000 is taken for low sensitivity. The estimation is the worst in this case. We notice clear difference in the ellipses size when adapting tv for different sensitivity scenarios. Further investigation of the relationship between particular parameters is appealing as it also corresponds to the absolute amplitude of the sensitivity AS. Parameter q5 in comparison to other parameters is identified with higher accuracy, where the amplitude AS is the highest (Figure 6.6, Figure 6.7). Comparing parameters q13 and q14 we see that q13 is estimated with lower precision (Figure 6.8). This reflects that absolute amplitude AS is lower for q13 than for q14 . Next, it can be noticed that for the measurement error of 10% the error in estimation of parameter q5 , parameter q13 and parameter q14 is less than 100% (see Remark 6.4) for a very little number of measurements l = 3. Estimation of the three chosen parameters for the model is acceptable and parameter estimation problem is not ill-posed, [28]. In next sections we observe that for more measurement points and for two species C5 and C8 (the concentration of occupied GSK3β in the nucleus) the error in parameter estimation is much lower. Choosing measurements points based only on the sensitivity analysis disregarding their correlation does not necessarily bring positive results. The problem with optimal sampling time points is more intricate, as coupling between the parameters must be taken into account. The second issue is that state variables are nonlinear with respect to the parameters. To learn more about this problem we refer to [74]. Short Summary 6.1 In this subsection we tackled the problem of sensitivity analysis. We investigated the influence of different time probes on the estimation process taking into account sensitivity amplitude. Corollary of our study are the following estimates that will be used in the sequential design (see Section 6.4.1) in the context of optimal experimental design: Estimates q5 , q13 , q14 Investigated in Subsection 6.3.1 6.3.2 Parameter nonlinearities, coupling and measurement error In order to maximize the amount of information extracted from the experiments, we have to deal with the following problems. Problem 6.4 Nonlinearities of the system can produce nonconvexity [5] of the least squares functional. In consequence, we may obtain multimodality. 90 CHAPTER 6. PARAMETER ESTIMATION AND EXPERIMENTAL DESIGN Multimodality means that we do not have one global minimum and multiple local minima can exist. We have to also deal with coupling between model parameters. Problem 6.5 If parameter, e.g., q1 is afflicted with an error and is coupled with parameter, e.g., q2 , then coupling of these parameters produces an error in parameter q2 . Problem 6.6 The strength of coupling and nonlinearities rises with increase of the measurement error. Remark 6.5 In our model, the measurement error is set to the samples of species C5 and C8 . Remark 6.6 If for certain parameters the nonlinearities are too big, then we cannot apply the method of ellipses (Section 6.2) to estimate the range of parameters. Numerical simulations To show the role of nonlinearities in parameter estimation, we perform numerical simulations for parameters q2 = t1 (transcription rate of SGK1), q5 = d2 (spontaneous degradation rate of the SGK1 protein), q8 = t3 (transcription rate of MYC), q18 = p8 (dissociation coefficients of the [GSK3β·cMyc] complex) and l = 41 (number of measurements). The method of ellipses and Monte Carlo simulation sampling with 50 points and eM = 10% of measurement error for arbitrary chosen 41 measurement points for species C5 and C8 is applied (Figure 6.9, Figure 6.10 and Figure 6.11). In the Monte Carlo simulation we first choose the measurement time points, so in consequence we have the corresponding measurement for each time point. The next step is to perturb the measurements by adding an error to see how this error influences the parameter estimation error. By Gauss-Newton algorithm we visualize estimates for 50 samples. For estimation in parameters q2 and q8 (see Figure 6.9) we observe non symmetric distribution of the points which means that our model indicates strong nonlinearity with respect to these parameters. In other cases all points are well distributed. For different measurement points the model may or may not indicate vulnerability to nonlinearities, respectively. The error of measurement is eM = 10% and the error in parameter estimation varies according to the particular estimates. Comparing the chosen parameters we can see that accuracy in parameter estimation for q5 and q18 is the biggest, and in fact, much bigger than for any other parameters. To improve accuracy of estimates the number of measurement points can be increased or "not estimable" parameters can be fixed. Further investigation leads us to fixing the parameters q2 and q8 , because they have the biggest influence on the estimate error increase comparing to other tested parameters. If we fix 6.3. PARAMETER ESTIMATION IN MEDULLOBLASTOMA MODEL Confidence region for parameters q2 & q8 91 Confidence region for parameters q2 & q5 with samples from Monte Carlo simulation with samples from Monte Carlo simulation 1 + 3.000000E−01 q 8 q5 1 + 3.000000E−01 1 1 1 − 2.000000E−01 1 − 2.000000E−01 1 q2 1 + 3.000000E−01 1 − 2.000000E−01 1 − 2.000000E−01 1 q2 1 + 3.000000E−01 Figure 6.9: 95% confidence regions and Monte Carlo points, parameters q2 , q5 and q8 , species C5 and C8 . Detected nonlinearity for parameters q2 and q8 . Confidence region for parameters q2 & q18 Confidence region for parameters q5 & q18 with samples from Monte Carlo simulation with samples from Monte Carlo simulation 1 + 3.000000E−01 q 18 q18 1 + 2.500000E−02 1 1 1 − 2.000000E−01 1 − 2.000000E−01 1 q2 1 + 3.000000E−01 1 − 2.000000E−02 1 − 2.000000E−02 1 q5 1 + 2.500000E−02 Figure 6.10: 95% confidence regions and Monte Carlo points, parameters q2 , q5 and q18 , species C5 and C8 . them, the error in other estimates is smaller. We have to be careful during selection of the parameters, as not all parameters are estimable. 92 CHAPTER 6. PARAMETER ESTIMATION AND EXPERIMENTAL DESIGN Confidence region for parameters q5 & q8 with samples from Monte Carlo simulation Confidence region for parameters q8 & q18 with samples from Monte Carlo simulation 1 + 3.000000E−01 q8 q18 1 + 2.500000E−01 1 1 1 − 2.000000E−01 1 − 2.000000E−01 1 q5 1 + 2.500000E−01 1 − 2.000000E−01 1 − 2.000000E−01 1 q8 1 + 3.000000E−01 Figure 6.11: 95% confidence regions and Monte Carlo points, parameters q5 , q8 and q18 , species C5 and C8 . Up to now, we performed simulations for a confidence region linearized about q(1,1), for each parameter. To further check the behavior of the ellipses with respect to nonlinearities, we introduce randomly small perturbations and linearize the confidence region for each parameter about q in the range {0.9-1.1}. We draw both ellipses (for q(1,1) and q(0.9-1.1,0.9-1.1)) in one plot with centers in point (1,1), and investigate the model nonlinearity regarding chosen parameters and measurements. If the difference is big the method of ellipses cannot be applied to estimate the range of searched parameters. We perform simulations for parameters q5 , q9 = d3 (spontaneous degradation rate of the MYC mRNA), q12 = kv (scaling coefficient, cytoplasm to the nucleus ratio), q13 and q14 for measurement error of 10%. The results (Figure 6.12 to Figure 6.16) show only little difference between shape in ellipses, therefore we state that we may estimate these parameters using the method of ellipses. 6.3. PARAMETER ESTIMATION IN MEDULLOBLASTOMA MODEL Confidence regions for parameters q & q 5 9 9 5 q q 1 1 − 4.000000E−01 1 − 4.000000E−01 Confidence regions for parameters q & q 12 1 + 4.000000E−01 12 1 + 4.000000E−01 93 1 q 1 + 4.000000E−01 1 1 − 4.000000E−01 1 − 4.000000E−01 5 1 q5 1 + 4.000000E−01 Figure 6.12: 95% confidence regions for different linearizations, parameters q5 , q9 and q12 , species C5 and C8 ; pink-dashed: linearization about q(1,1), blue-solid: linearization about q(0.91.1,0.9-1.1). Confidence regions for parameters q5 & q13 1 + 4.000000E−01 q14 q13 1 + 4.000000E−01 1 1 − 4.000000E−01 1 − 4.000000E−01 Confidence regions for parameters q5 & q14 1 q5 1 + 4.000000E−01 1 1 − 4.000000E−01 1 − 4.000000E−01 1 q5 1 + 4.000000E−01 Figure 6.13: 95% confidence regions for different linearizations, parameters q5 , q13 and q14 , species C5 and C8 ; pink-dashed: linearization about q(1,1), blue-solid: linearization about q(0.91.1,0.9-1.1). 94 CHAPTER 6. PARAMETER ESTIMATION AND EXPERIMENTAL DESIGN Confidence regions for parameters q & q 9 12 1 1 − 4.000000E−01 1 − 4.000000E−01 Confidence regions for parameters q & q 9 13 1 + 4.000000E−01 q13 q12 1 + 4.000000E−01 1 q 1 + 4.000000E−01 1 1 − 4.000000E−01 1 − 4.000000E−01 9 1 q 1 + 4.000000E−01 9 Figure 6.14: 95% confidence regions for different linearizations, parameters q9 , q12 and q13 , species C5 and C8 ; pink-dashed: linearization about q(1,1), blue-solid: linearization about q(0.91.1,0.9-1.1). Confidence regions for parameters q12 & q13 1 + 4.000000E−01 q14 q13 1 + 4.000000E−01 1 1 − 4.000000E−01 1 − 4.000000E−01 Confidence regions for parameters q12 & q14 1 q12 1 + 4.000000E−01 1 1 − 4.000000E−01 1 − 4.000000E−01 1 q12 1 + 4.000000E−01 Figure 6.15: 95% confidence regions for different linearizations, parameters q12 , q13 and q14 , species C5 and C8 ; pink-dashed: linearization about q(1,1), blue-solid: linearization about q(0.91.1,0.9-1.1). 6.3. PARAMETER ESTIMATION IN MEDULLOBLASTOMA MODEL Confidence regions for parameters q & q 9 Confidence regions for parameters q 13 &q 1 + 4.000000E−01 1 1 − 4.000000E−01 1 − 4.000000E−01 14 q14 q14 1 + 4.000000E−01 95 1 q 9 1 + 4.000000E−01 1 1 − 4.000000E−01 1 − 4.000000E−01 1 q 1 + 4.000000E−01 13 Figure 6.16: 95% confidence regions for different linearizations, parameters q9 , q13 and q14 , species C5 and C8 ; pink-dashed: linearization about q(1,1), blue-solid: linearization about q(0.91.1,0.9-1.1). Next we check, how the variation in measurement error eM reflects the parameter estimation. We present a numerical solution of both Monte Carlo 50 sampling points and ellipses for parameters q5 , q9 , q12 , q13 , q14 . We set l = 41 for eM = 1% of measurement error (species C5 and C8 ) and also for eM = 10% of measurement error (species C5 and C8 ). It is demonstrated that for chosen errors eM and l = 41 the estimation error is below 100%. We find that even for eM = 10% of measurement error we still have a reasonable solution. A measurement error of eM = 10% results in parameter estimation error in all cases less than 40% and for eM = 1% estimation error is less than 5% (Figure 6.17 to Figure 6.26). Remark 6.7 The Monte Carlo simulation for 50 sampling points is compared with the method of ellipses reflecting the 95% confidence regions. It is noticed that the method of ellipses represents a good linearized approximation of the model for different aspects of analysis. 14 CHAPTER 6. PARAMETER ESTIMATION AND EXPERIMENTAL DESIGN 96 Confidence region with 1% in measure error 1 + 4.000000E−01 q9 q9 1 + 5.000000E−02 Confidence region with 10% in measure error 1 1 1 − 4.000000E−02 1 − 4.000000E−02 1 q5 1 + 5.000000E−02 1 − 4.000000E−01 1 − 4.000000E−01 1 1 + 4.000000E−01 q5 Figure 6.17: 95% confidence region and Monte Carlo numerical results for estimation of parameters q5 and q9 for C5 and C8 with [A] 1% and [B] 10% of measurement error. Confidence region with 1% in measure error 1 + 4.000000E−01 q12 q12 1 + 5.000000E−02 Confidence region with 10% in measure error 1 1 1 − 4.000000E−02 1 − 4.000000E−02 1 q5 1 + 5.000000E−02 1 − 4.000000E−01 1 − 4.000000E−01 1 q5 1 + 4.000000E−01 Figure 6.18: 95% confidence region and Monte Carlo numerical results for estimation of parameters q5 and q12 for C5 and C8 with [A] 1% and [B] 10% of measurement error. 6.3. PARAMETER ESTIMATION IN MEDULLOBLASTOMA MODEL Confidence region with 1% in measure error Confidence region with 10% in measure error 1 + 4.000000E−01 q13 q13 1 + 5.000000E−02 97 1 1 1 − 4.000000E−02 1 − 4.000000E−02 1 q5 1 + 5.000000E−02 1 − 4.000000E−01 1 − 4.000000E−01 1 q5 1 + 4.000000E−01 Figure 6.19: 95% confidence region and Monte Carlo numerical results for estimation of parameters q5 and q13 for C5 and C8 with [A] 1% and [B] 10% of measurement error. Confidence region with 1% in measure error 1 + 4.000000E−01 q14 q 14 1 + 5.000000E−02 Confidence region with 10% in measure error 1 1 1 − 4.000000E−02 1 − 4.000000E−02 1 q5 1 + 5.000000E−02 1 − 4.000000E−01 1 − 4.000000E−01 1 q5 1 + 4.000000E−01 Figure 6.20: 95% confidence region and Monte Carlo numerical results for estimation of parameters q5 and q14 for C5 and C8 with [A] 1% and [B] 10% of measurement error. 98 CHAPTER 6. PARAMETER ESTIMATION AND EXPERIMENTAL DESIGN Confidence region with 1% in measure error 1 + 4.000000E−01 q12 q12 1 + 5.000000E−02 Confidence region with 10% in measure error 1 1 − 5.000000E−02 1 − 5.000000E−02 1 q9 1 + 5.000000E−02 1 1 − 4.000000E−01 1 − 4.000000E−01 1 q9 1 + 4.000000E−01 Figure 6.21: 95% confidence region and Monte Carlo numerical results for estimation of parameters q9 and q12 for C5 and C8 with [A] 1% and [B] 10% of measurement error. Confidence region with 1% in measure error 1 + 4.000000E−01 q13 q13 1 + 5.000000E−02 1 1 − 5.000000E−02 1 − 5.000000E−02 Confidence region with 10% in measure error 1 q9 1 + 5.000000E−02 1 1 − 4.000000E−01 1 − 4.000000E−01 1 q9 1 + 4.000000E−01 Figure 6.22: 95% confidence region and Monte Carlo numerical results for estimation of parameters q9 and q13 for C5 and C8 with [A] 1% and [B] 10% of measurement error. 6.3. PARAMETER ESTIMATION IN MEDULLOBLASTOMA MODEL Confidence region with 1% in measure error Confidence region with 10% in measure error 1 + 4.000000E−01 q14 q 14 1 + 5.000000E−02 99 1 1 − 5.000000E−02 1 − 5.000000E−02 1 q9 1 + 5.000000E−02 1 1 − 4.000000E−01 1 − 4.000000E−01 1 q9 1 + 4.000000E−01 Figure 6.23: 95% confidence region and Monte Carlo numerical results for estimation of parameters q9 and q14 for C5 and C8 with [A] 1% and [B] 10% of measurement error. Confidence region with 1% in measure error 1 + 4.000000E−01 q13 q 13 1 + 5.000000E−02 1 1 − 5.000000E−02 1 − 5.000000E−02 Confidence region with 10% in measure error 1 q12 1 + 5.000000E−02 1 1 − 4.000000E−01 1 − 4.000000E−01 1 q12 1 + 4.000000E−01 Figure 6.24: 95% confidence region and Monte Carlo numerical results for estimation of parameters q12 and q13 for C5 and C8 with [A] 1% and [B] 10% of measurement error. 100 CHAPTER 6. PARAMETER ESTIMATION AND EXPERIMENTAL DESIGN Confidence region with 1% in measure error 1 + 4.000000E−01 q14 q14 1 + 5.000000E−02 Confidence region with 10% in measure error 1 1 − 5.000000E−02 1 − 5.000000E−02 1 q12 1 + 5.000000E−02 1 1 − 4.000000E−01 1 − 4.000000E−01 1 q12 1 + 4.000000E−01 Figure 6.25: 95% confidence region and Monte Carlo numerical results for estimation of parameters q12 and q14 for C5 and C8 with [A] 1% and [B] 10% of measurement error. Confidence region with 1% in measure error 1 + 4.000000E−01 q14 q14 1 + 5.000000E−02 1 1 − 5.000000E−02 1 − 5.000000E−02 Confidence region with 10% in measure error 1 q13 1 + 5.000000E−02 1 1 − 4.000000E−01 1 − 4.000000E−01 1 q13 1 + 4.000000E−01 Figure 6.26: 95% confidence region and Monte Carlo numerical results for estimation of parameters q13 and q14 for C5 and C8 with [A] 1% and [B] 10% of measurement error. 6.3. PARAMETER ESTIMATION IN MEDULLOBLASTOMA MODEL 101 Short Summary 6.2 In this subsection we performed numerical simulations in order to deal with interesting aspects of parameter estimation, i.e., nonlinearities, coupling and measurement error. The successive estimates established after these simulations are added to our estimates table: Estimates q5 , q13 , q14 q9 , q12 , q18 Investigated in Subsection 6.3.1 Subsection 6.3.2 6.3.3 Variation in parameter values In the following we consider the parameters q are unknown and that they are to be estimated. The parameters r are known and described with some range in values. Based on the preliminary simulations, we assume that ellipses approximate the behavior of Monte Carlo simulations and therefore we visualize only ellipses. Definition 6.7 We define the known parameters r described with some range as parameters for which laboratory experiments reveal slightly different values of the same parameter. Problem 6.7 The aim of this subsection is to examine how the variation in parameter values combined with the error of measurement eM influences the error in total parameter identification. In biological modeling the knowledge of exact parameter values, e.g., representing the rate constant, is fundamental. Sometimes, we can find in the scientific literature necessary data. Unfortunately, often data is not available or parameters are given in ranges. Evidently, the estimation process is better if we have more information available. Therefore, to use all possible information we can use the parameters with a range in values to make the estimation. To analyze Problem 6.7, we fix some parameters r adding them the ranges of uncertainty and we test the influence of this variation on the accuracy of estimates. To do this, we plot ellipses based on the covariances derived from the Jacobian in the direction of estimated and fixed parameters. Remark 6.8 In this subsection, we redefine the already known covariance Cov (6.10) by Covq . Definition 6.8 The covariance matrix Covq is of the form )−1 ( −1 Covq := ĜTq CD Ĝq , where the covariance is derived in the direction of estimated parameters. (6.14) 102 CHAPTER 6. PARAMETER ESTIMATION AND EXPERIMENTAL DESIGN Definition 6.9 The covariance matrix Covr is of the form )−1 ( T −1 ) ( T −1 )T ( T −1 )−1 ( −1 Ĝq CD Ĝq , Ĝq Ĝq CD Ĝr Cr Ĝq CD Ĝr Covr := ĜTq CD (6.15) where the covariance is derived in the direction of fixed parameters with ranges and matrix Cr describes the range of the fixed parameters. Definition 6.10 To plot the confidence ellipses, we apply the covariance of estimated and fixed parameters in the form Cov := Covq + Covr . (6.16) Numerical simulations We perform simulations to obtain confidence ellipses given by (6.16) for the following parameters q5 , q9 , q12 , q13 and q14 . We introduce variation (uncertainty) from 1% up to 10% with 1% step in the following two fixed parameters with ranges. The first parameter is r1 = βcat (transcription factor for the MYC and SGK1 mRNA synthesis), and the second parameter is r3 = d1 (spontaneous degradation rate of the SGK1 mRNA). The investigation containing variation in fixed parameters is tested (Figure 6.27 to Figure 6.36). Confidence regions, error in fixed parameter r1 1 + 4.000000E−01 q9 q9 1 + 4.000000E−01 1 1 − 4.000000E−01 1 − 4.000000E−01 Confidence regions, error in fixed parameter r3 1 q5 1 + 4.000000E−01 1 1 − 4.000000E−01 1 − 4.000000E−01 1 q5 1 + 4.000000E−01 Figure 6.27: 95% confidence regions for variation in fixed parameters from 1% up to 10% for two parameters r1 and r3 , estimation of parameters q5 , q9 , species C5 and C8 with 10% of measurement error. 6.3. PARAMETER ESTIMATION IN MEDULLOBLASTOMA MODEL Confidence regions, error in fixed parameter r1 Confidence regions, error in fixed parameter r3 1 + 4.000000E−01 q12 q12 1 + 4.000000E−01 103 1 1 − 4.000000E−01 1 − 4.000000E−01 1 q5 1 + 4.000000E−01 1 1 − 4.000000E−01 1 − 4.000000E−01 1 q5 1 + 4.000000E−01 Figure 6.28: 95% confidence regions for variation in fixed parameters from 1% up to 10% for two parameters r1 and r3 , estimation of parameters q5 , q12 , species C5 and C8 with 10% of measurement error. Confidence regions, error in fixed parameter r1 1 + 4.000000E−01 q13 q13 1 + 4.000000E−01 1 1 − 4.000000E−01 1 − 4.000000E−01 Confidence regions, error in fixed parameter r3 1 q5 1 + 4.000000E−01 1 1 − 4.000000E−01 1 − 4.000000E−01 1 q5 1 + 4.000000E−01 Figure 6.29: 95% confidence regions for variation in fixed parameters from 1% up to 10% for two parameters r1 and r3 , estimation of parameters q5 , q13 , species C5 and C8 with 10% of measurement error. 104 CHAPTER 6. PARAMETER ESTIMATION AND EXPERIMENTAL DESIGN Confidence regions, error in fixed parameter r1 3 1 + 8.000000E−01 q14 q14 1 + 4.000000E−01 Confidence regions, error in fixed parameter r 1 1 − 4.000000E−01 1 − 4.000000E−01 1 q5 1 + 4.000000E−01 1 1 − 8.000000E−01 1 − 8.000000E−01 1 q5 1 + 8.000000E−01 Figure 6.30: 95% confidence regions for variation in fixed parameters from 1% up to 10% for two parameters r1 and r3 , estimation of parameters q5 , q14 , species C5 and C8 with 10% of measurement error. Confidence regions, error in fixed parameter r1 1 + 4.000000E−01 q12 q 12 1 + 4.000000E−01 1 1 − 4.000000E−01 1 − 4.000000E−01 Confidence regions, error in fixed parameter r3 1 q9 1 + 4.000000E−01 1 1 − 4.000000E−01 1 − 4.000000E−01 1 q9 1 + 4.000000E−01 Figure 6.31: 95% confidence regions for variation in fixed parameters from 1% up to 10% for two parameters r1 and r3 , estimation of parameters q9 , q12 , species C5 and C8 with 10% of measurement error. 6.3. PARAMETER ESTIMATION IN MEDULLOBLASTOMA MODEL Confidence regions, error in fixed parameter r1 q13 q Confidence regions, error in fixed parameter r3 1 + 4.000000E−01 13 1 + 4.000000E−01 105 1 1 − 4.000000E−01 1 − 4.000000E−01 1 q9 1 + 4.000000E−01 1 1 − 4.000000E−01 1 − 4.000000E−01 1 q9 1 + 4.000000E−01 Figure 6.32: 95% confidence regions for variation in fixed parameters from 1% up to 10% for two parameters r1 and r3 , estimation of parameters q9 , q13 , species C5 and C8 with 10% of measurement error. Confidence regions, error in fixed parameter r1 1 + 8.000000E−01 q14 q 14 1 + 4.000000E−01 1 1 − 4.000000E−01 1 − 4.000000E−01 Confidence regions, error in fixed parameter r3 1 q9 1 + 4.000000E−01 1 1 − 8.000000E−01 1 − 8.000000E−01 1 q9 1 + 8.000000E−01 Figure 6.33: 95% confidence regions for variation in fixed parameters from 1% up to 10% for two parameters r1 and r3 , estimation of parameters q9 , q14 , species C5 and C8 with 10% of measurement error. 106 CHAPTER 6. PARAMETER ESTIMATION AND EXPERIMENTAL DESIGN Confidence regions, error in fixed parameter r1 1 + 4.000000E−01 q13 q13 1 + 4.000000E−01 Confidence regions, error in fixed parameter r3 1 1 − 4.000000E−01 1 − 4.000000E−01 1 q12 1 + 4.000000E−01 1 1 − 4.000000E−01 1 − 4.000000E−01 1 q12 1 + 4.000000E−01 Figure 6.34: 95% confidence regions for variation in fixed parameters from 1% up to 10% for two parameters r1 and r3 , estimation of parameters q12 , q13 , species C5 and C8 with 10% of measurement error. Confidence regions, error in fixed parameter r1 1 + 8.000000E−01 q14 q14 1 + 4.000000E−01 1 1 − 4.000000E−01 1 − 4.000000E−01 Confidence regions, error in fixed parameter r3 1 q12 1 + 4.000000E−01 1 1 − 8.000000E−01 1 − 8.000000E−01 1 1 + 8.000000E−01 q12 Figure 6.35: 95% confidence regions for variation in fixed parameters from 1% up to 10% for two parameters r1 and r3 , estimation of parameters q12 , q14 , species C5 and C8 with 10% of measurement error. 6.3. PARAMETER ESTIMATION IN MEDULLOBLASTOMA MODEL Confidence regions, error in fixed parameter r1 q14 q 1 1 − 4.000000E−01 1 − 4.000000E−01 Confidence regions, error in fixed parameter r3 1 + 8.000000E−01 14 1 + 4.000000E−01 107 1 q13 1 + 4.000000E−01 1 1 − 8.000000E−01 1 − 8.000000E−01 1 q13 1 + 8.000000E−01 Figure 6.36: 95% confidence regions for variation in fixed parameters from 1% up to 10% for two parameters r1 and r3 , estimation of parameters q13 , q14 , species C5 and C8 with 10% of measurement error. In each Figure we depict ellipses starting from 1% of variation in fixed parameters up to 10% of variation with a step of 1%. The smallest ellipse corresponds to the lowest uncertainty in fixed parameters and the biggest ellipse to the largest uncertainty. The results illustrate that the ellipses of the estimated parameters vary more or less with respect to two fixed parameters. Ellipses of parameters {q5 , q9 }, {q5 , q12 }, {q5 , q13 }, {q9 , q12 }, {q9 , q13 }, {q12 , q13 } do not differ much for cases r1 and r3 . Additionally, the variation in fixed parameters does not entail any large increase in estimation uncertainty. Therefore, we say that for particular estimations the variation in fixed parameters does not influence the solution. For other ellipses, i.e., {q5 , q14 }, {q9 , q14 }, {q12 , q14 } and {q13 , q14 } we observe big increase in parameter estimation uncertainty for 10% of variation in fixed parameter r3 . For parameter r3 the accuracy in parameter estimation is much lower than for parameter r1 . Particularly, q14 is vulnerable to the variation in fixed parameter r3 . It follows that quality of estimation in parameter q14 is strongly affected by uncertainty in r3 . Altogether, we find that estimation is sensitive to the uncertainty in r3 . In consequence, it might be necessary to fix such parameter completely, if possible. However, the system is not sensitive to the uncertainty in r1 . Thus, when no information (or even any ranges) about such parameter is available, it can be considered as parameter to estimate. Remark 6.9 The size and shape of ellipses does not change much upon variation of r1 . Hence, we consider parameter r1 to be estimable, since the sensitivity is high enough (Table 6.3). 108 CHAPTER 6. PARAMETER ESTIMATION AND EXPERIMENTAL DESIGN Therefore, we set q1 = r1 and we estimate this parameter too. We can observe that some of the fixed parameters with ranges cannot be taken into account in estimation, if we want to achieve precise estimates. Short Summary 6.3 In this subsection we introduced the covariance Cov composed of the covariance matrices derived in the direction of fixed and estimated parameters. We illustrated changes in the size and shape of the ellipses based on Cov with respect to applied measurement error. Our considerations are finalized by finding one more parameter that can be estimated for our model: Estimates q5 , q13 , q14 q9 , q12 , q18 q1 Investigated in Subsection 6.3.1 Subsection 6.3.2 Subsection 6.3.3 The whole investigation ends up with establishing seven parameters, for which estimation error is below 100% (see Remark 6.4). 6.4 Optimal experimental design The goal of this section is to minimize the variance of estimates according to the design parameter (experimental set-up) ξ ∈ Π. ( )−1 −1 Strictly speaking, we want to minimize the matrix Cov = ĜT CD Ĝ for sampling time, which in our model takes over the role of design parameter. Minimizing the variance corresponds to maximizing the certainty of the information. Definition 6.11 Optimal experimental design (OED) aims to the most significant identification of unknown parameters in the model. Goal 6.1 Goal of OED is the reduction of the confidence region of the model parameters. We improve our estimation by choosing optimal sampling protocol (design parameter ξ) in order to reduce Cov. There are the following optimality criteria to find the optimal sampling protocol (see [17]), which are based on functional Ψ of the covariance Cov: ∏ D-optimal design criterion Ψ(Cov) = λi , i E-optimal design criterion Ψ(Cov) = max(λi ), ∑ T-optimal design criterion Ψ(Cov) = λi , i C-optimal design criterion Ψ(Cov) = max(λi ) , min(λi ) where λi,i=1,...,n are eigenvalues of matrix Cov. 6.4. OPTIMAL EXPERIMENTAL DESIGN 109 6.4.1 Sequential design Remark 6.10 Sequential experimental design leads to parameter refinement in an iterative process. Remark 6.11 In sequential experimental design minimization of covariance matrix Cov is expected for each iteration. The equation for Cov contains Jacobian, which depends on the sensitivities of the state with respect to given parameter values. However, we want to estimate the parameter values which in fact are used in Jacobian. Therefore, we have to specify a prior guess q 0 to be used in Jacobian. We improve sequentially our estimates based on the Jacobian (initial guess) and the specific optimality criteria used. The algorithm presented in Table 6.4 is as follows. We have initial parameter estimate 0 q for ξ 0 = (t1 , t2 , ..., tj ) and measurements C̄ = (C̄1 , ..., C̄j ). We minimize the functional of Cov over the new design and update ξ i+1 by new time point tj+1 .We perform the experiment to extract new sample measurement and calculate parameter estimates. We repeat the procedure untill we obtain satisfying parameter convergence given by ϵ2 in sequential update of the parameter design. Sequential OED algorithm 1 2 3 4 5 6 7 8 9 10 11 Initialize: initial design ξ o = (t1 , t2 , ..., tj ), measurements C̄ = (C̄1 , ..., C̄j ), initial guess of parameter q 0 Loop: tj+1 = arg minξi (Ψ(Cov(ξ i ))) ξ i+1 = (ξ i , tj+1 ) perform experiment to get measurement C̄j+1 calculate parameter estimate q i+1 based on Gauss-Newton algorithm evaluate confidence region based on the ellipses increment global iteration counters j and i normal exit if (|q i+1 − q i | < ϵ2 ) End loop Table 6.4: Pseudocode of sequential OED algorithm. It happens that the initial guess q 0 of the parameters is far away from the solution q, what may result in a large confidence region. In such case, we would need a new strategy and we would have to prepare a new setup of initial guess of parameters and then follow the procedure of estimation process. For highly nonlinear models we can work with many 110 CHAPTER 6. PARAMETER ESTIMATION AND EXPERIMENTAL DESIGN sets of initial parameter guesses in order to find the proper one and even then fail. For further details see [59]. Problem 6.8 The goal is to derive parameter estimates with the highest possible accuracy based on the optimality criteria in the sequential design. We consider all the criteria described in the beginning of Section 6.4. Tests performed in previous sections demonstrate that parameter estimation is not possible for all model parameters, hence we have a subset Qad of admissible parameters. We apply the optimality criteria to that subset Qad = {q1 , q5 , q9 , q12 , q13 , q14 , q18 } and find the minimum of the functional Ψ for a given criteria. For example, for the T-optimality design we search for the minimal value of the trace of covariance matrix Cov in each iteration, etc. Heuristic design To compare numerical solutions built on different design criteria in order to find optimal parameter, we start the first iteration with the following heuristic design preliminary chosen from sensitivity analysis: C5 : ξ 0 = {t1 = 18000, t2 = 19500, t3 = 20500, t4 = 21000, t5 = 22000, t6 = 23000, t7 = 24000}; C8 : ξ 0 = {t1 = 20000, t2 = 21000, t3 = 22000, t4 = 23500, t5 = 24200, t6 = 24500, t7 = 25000}. The measurement error is assumed to be eM = 1%. Depending on the sampled probes the error can be larger. In this case, however, we cannot afford to start with a high measurement error, as already for eM = 1% the estimates exceed 60% of error (Table 6.5). Only parameters q5 and q18 indicate high accuracy of estimation. Parameters q1 Estimates error 58% q5 2.1% q9 67% q12 58% q13 15% q14 q18 39% 2.8% Table 6.5: Parameter estimation error for heuristic design; measurement error eM = 1%. Problem 6.9 To refine the estimates using the idea of sequential design, we have to find the best sampling protocol. 6.4. OPTIMAL EXPERIMENTAL DESIGN 111 We apply the sequential design, and therefore, in the following iterations we minimize the covariance matrix Cov derived in the direction of estimated parameters. Numerical solutions show improvement in the estimates in the subsequent iterations evaluated by succesive optimization procedure for T race (T-optimality), Det (D-optimality), Eig (Eoptimality) and Ratio (C-optimality). Table 6.6 displays the sampling points found in the subsequent steps of optimization for the considered optimality criteria based on the covariance matrix Cov starting with heuristic design ξ 0 = {t1 , ..., t7 }. iteration 1 time t8 Trace 2800 Det 6000 Eig 2100 2100 Ratio iteration 2 time t9 28400 50000 27900 27900 iteration 3 time t10 13400 13400 13400 13400 iteration 4 time t11 4200 50000 3200 3200 iteration 5 time t12 32800 5300 12300 12300 Table 6.6: Sampling points in subsequent iterations for criteria: T race, Det, Eig, Ratio; measurement error eM = 1%. We investigate the progressive refinement of the estimates for five iterations. We notice that Eig and Ratio criteria consist of the same sampling design in every step and they differ from T race and Det except third iteration. T race and Det criteria also represent various sampling schemes. Such variety of possible sampling schemes leads to the situation where the experimentalist can provide measurements for his own preferred sampling scheme. For this reason the procedure of data collection can be less expensive, i.e., the experiment can be stopped earlier. The Eig and Ratio criteria meet the shortest time duration with tlast = t9 = 27900. In this case we can benefit from Eig and Ratio optimization and stop experiment on tlast . An interesting fact emerges in the case of Det, where during optimization process the same probe time is computed twice. This accounts for the repeated measurement for the same time point. The question is which protocol is the best for parameter identification considering costs and efficiency of the experimental procedure. The enhancement of the parameter estimates is depicted in Table 6.7, 6.8 and 6.9. 112 CHAPTER 6. PARAMETER ESTIMATION AND EXPERIMENTAL DESIGN heuristic points 58% 2.1% 67% 58% 15% 39% 2.8% Trace criterion q1 q5 q9 q12 q13 q14 q18 iter. 1 20% 2% 2.7% 2.1% 14% 13% 0.5% iter. 2 11% 1.9% 2.6% 2.1% 7% 9% 0.5% iter. 3 5.6% 0.8% 2.5% 2.1% 4.4% 5% 0.4% iter. 4 5.4% 0.8% 1.5% 1.2% 4.3% 4.3% 0.4% iter. 5 5% 0.8% 1.5% 1.2% 3.9% 3.8% 0.4% Table 6.7: Parameter estimation error in subsequent optimization iterations for Trace criterion; measurement error eM = 1%. Det criterion q1 q5 q9 q12 q13 q14 q18 heuristic points 58% 2.1% 67% 58% 15% 39% 2.8% iter. 1 21% 2% 3.3% 2.2% 14% 13% 0.51% iter. 2 12% 1.9% 3.1% 2.1% 6.5% 9% 0.5% iter. 3 5.8% 0.8% 1.9% 1.5% 4.2% 4.3% 0.44% iter. 4 5.4% 0.8% 1.8% 1.5% 3.9% 4.1% 0.4% iter. 5 5.3% 0.8% 1.4% 1.1% 3.9% 3.7% 0.4% Table 6.8: Parameter estimation error in subsequent optimization iterations for Det criterion; measurement error eM = 1%. Eig, Ratio criteria q1 q5 q9 q12 q13 q14 q18 heuristic points 58% 2.1% 67% 58% 15% 39% 2.8% iter. 1 20% 2% 3.7% 3.1% 14% 13% 0.53% iter. 2 11% 1.9% 3.6% 3% 6.9% 9.2% 0.5% iter. 3 6.1% 0.9% 3.6% 3% 4.5% 6.1% 0.4% iter. 4 5.6% 0.8% 1.9% 1.6% 4.4% 4.6% 0.4% iter. 5 5.2% 0.7% 1.9% 1.6% 4.3% 4.4% 0.4% Table 6.9: Parameter estimation error in subsequent optimization iterations for Eig, Ratio criteria; measurement error eM = 1%. 6.4. OPTIMAL EXPERIMENTAL DESIGN 113 Already, after the first iteration the error of the estimates is strongly reduced for all criteria. The increase in accuracy of parameter estimates accompanies within subsequent steps. Numerical solutions indicate that the difference between third and fourth step is small. This implies that only three optimization cycles are essential to obtain reasonable estimates. However, in further investigations we consider all five steps, as the result for five iterations is the best. The well observable fact is that for different parameters the optimization has different impact. For instance, examine parameter q1 and q12 , where the heuristic design results in an estimation error of 58%. For parameter q1 the reduction is only almost three times, whereas for parameter q12 the reduction is substantial of thirty times after first iteration for T race criteria. In the case of other parameters, we notice also the diversity in parameter estimates refinement. To clearly depict the process of identification according to the iterations and given criterion, we present results in Figure 6.37 to Figure 6.40. Each parameter is separately considered. Parameter q estimation error Parameter q estimation error 1 5 60 2.2 Trace Det Eig, Ratio 50 Trace Det Eig, Ratio 2 1.8 error [%] error [%] 40 30 1.6 1.4 1.2 20 1 10 0.8 0 0 1 2 3 iterations 4 5 0 1 2 3 4 iterations Figure 6.37: Comparison of optimality criteria of parameters q1 and q5 estimation error, five optimization iterations; measurement error eM = 1%. 5 CHAPTER 6. PARAMETER ESTIMATION AND EXPERIMENTAL DESIGN 114 Parameter q estimation error Parameter q 9 12 70 estimation error 60 Trace Det Eig, Ratio 60 Trace Det Eig, Ratio 50 50 error [%] error [%] 40 40 30 30 20 20 10 10 0 0 1 2 3 4 0 0 5 1 iterations 2 3 4 5 iterations Figure 6.38: Comparison of optimality criteria of parameters q9 and q12 estimation error, five optimization iterations; measurement error eM = 1%. Parameter q13 estimation error Parameter q14 estimation error 16 40 Trace Det Eig, Ratio 14 Trace Det Eig, Ratio 35 30 error [%] error [%] 12 10 8 25 20 15 6 10 4 2 0 5 1 2 3 iterations 4 5 0 0 1 2 3 4 iterations Figure 6.39: Comparison of optimality criteria of parameters q13 and q14 estimation error, five optimization iterations; measurement error eM = 1%. 5 6.4. OPTIMAL EXPERIMENTAL DESIGN Parameter q 18 115 estimation error 3 Trace Det Eig, Ratio 2.5 error [%] 2 1.5 1 0.5 0 0 1 2 3 4 5 iterations Figure 6.40: Comparison of optimality criteria of parameter q18 estimation error, five optimization iterations; measurement error eM = 1%. Comparison of optimality criteria for 5 steps of sequential design Problem 6.10 To obtain the best parameter estimates in the context of optimal experimental design using sequential design the best optimality criteria must be established. By analyzing the criteria in the context of error for parameter estimates, we can state that Eig and Ratio are worse criteria for our model, (see Table 6.7, 6.8 and 6.9). In the case of Det and T race some estimations are better and some are worse. After first iteration the Det criterion seems to be worse, but in the fifth iteration it performs comparably to T race. Essentially, using all criteria estimation finishes with reasonable results after five iterations. To recall, for the heuristic design we make a sweep of the additional measurement time point from t1 to tl,l=50000 and we find the minimum to obtain the best time point in each iteration. In Figure 6.41 to Figure 6.44 we illustrate the behavior of particular property of covariance matrix Cov represented by the criteria T race, Det, Eig and Ratio in the time interval (t1 , t50000 ). CHAPTER 6. PARAMETER ESTIMATION AND EXPERIMENTAL DESIGN 116 [A], Trace of Cov in 1th iteration −3 0.25 1.25 x 10 [B], Trace of Cov in 5th iteration 1.2 0.2 1.15 Trace Trace 0.15 1.1 0.1 1.05 0.05 1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 time [s] 5 0.95 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 time [s] 4 x 10 5 4 x 10 Figure 6.41: Trace of Cov in the process of sampling scheme optimization, first [A] and last [B] iteration. th [A], Det of Cov in 1 iteration −32 2.5 x 10 [B], Det of Cov in 5th iteration −38 1.7 x 10 1.6 2 1.5 1.4 1.5 Det Det 1.3 1 1.2 1.1 1 0.5 0.9 0.8 0 0 0.5 1 1.5 2 2.5 time [s] 3 3.5 4 4.5 5 4 x 10 0 0.5 1 1.5 2 2.5 3 3.5 4 time [s] Figure 6.42: Determinant of Cov in the process of sampling scheme optimization, first [A] and last [B] iteration. For each criterion we depict the first and last iteration. Values of particular property (T race, Det, Eig, Ratio) are much higher in the first than the last iteration, for all criteria. It implies that reduction in values of T race, Det, Eig and Ratio entails reduction in error of parameter estimates for subsequent iterations. We achieve a satisfying error not exceeding 5% for every parameter qi,i={1,5,9,12,13,14,18} for measurement error eM = 1%. In the case of concentrations, the measurement error of the experimental probes can be (and in laboratorie’s reality is found to be) bigger than eM = 1%. Hence, we carry forward our investigation. We introduce measurement errors eM = 5% and eM = 10% and give the results are in Table 6.10, 6.11 and 6.12 for all 4.5 5 4 x 10 6.4. OPTIMAL EXPERIMENTAL DESIGN 117 [A], Eig of Cov in 1th iteration [B], Eig of Cov in 5th iteration −4 0.25 9 x 10 8.8 0.2 8.6 8.4 Eig Eig 0.15 0.1 8.2 8 7.8 0.05 7.6 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 time [s] 7.4 0 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 time [s] 4 x 10 5 4 x 10 Figure 6.43: Maximal eigenvalue of Cov in the process of sampling scheme optimization, first [A] and last [B] iteration. 7 9 x 10 [A], Ratio of Cov in 1th iteration 5 3.8 x 10 [B], Ratio of Cov in 5th iteration 8 3.6 7 3.4 Ratio Ratio 6 5 4 3 3.2 3 2 2.8 1 0 0 0.5 1 1.5 2 2.5 time [s] 3 3.5 4 4.5 5 4 x 10 2.6 0 0.5 1 1.5 2 2.5 3 3.5 4 time [s] Figure 6.44: Ratio between the maximum eigenvalue and minimum eigenvalue of Cov in the process of sampling scheme optimization, first [A] and last [B] iteration. optimality criteria. 4.5 5 4 x 10 118 CHAPTER 6. PARAMETER ESTIMATION AND EXPERIMENTAL DESIGN Trace criterion q1 q5 q9 q12 q13 q14 q18 1% 5% 0.8% 1.5% 1.2% 3.9% 3.8% 0.4% 5% 25% 3.8% 7.5% 6.1% 19% 19% 1.9% 10% 50% 8% 15% 12% 39% 38% 3.8% Table 6.10: Parameter estimation error for measurement error 1%, 5%, 10%. Trace criterion. Eig, Ratio criteria q1 q5 q9 q12 q13 q14 q18 1% 5.2% 0.7% 1.9% 1.6% 4.3% 4.4% 0.4% Det criterion q1 q5 q9 q12 q13 q14 q18 1% 5% 5.3% 27% 0.8% 3.9% 1.4% 7% 1.1% 5.4% 3.9% 20% 3.7% 19% 0.4% 1.9% 10% 53% 7.7% 14% 11% 39% 37% 3.8% Table 6.11: Parameter estimation error for measurement error 1%, 5%, 10%. Det criterion. 5% 27% 3.6% 9.3% 7.7% 19% 22% 1.9% 10% 53% 7.3% 19% 15% 43% 44% 3.9% Table 6.12: Parameter estimation error for measurement error 1%, 5%, 10%. Eig, Ratio criteria. As for eM = 10% the parameters are still estimable, the resultant error for some estimates is quite large and is even of 53% for Det, Eig and Ratio criteria. However, four out of seven parameters, q5 , q9 , q12 and q18 , are obtained with a surprisingly good accuracy, below 20%. For eM = 5% we have very good estimates for those parameters after five iterations for all optimality criteria, where the best is T race and Det. Short Summary 6.4 This subsection described strategies in the parameter estimation process regarding the optimal experimental design. We introduced the optimality criteria that support better estimation results. We faced the situation, where measurement error impacts strongly on the parameters accuracy. Finally, we showed that with eM = 5% the estimates were achieved with good precision and that the best results were obtained with T race and Det optimality criteria. The drawback of deterministic estimation is that in our case the subset of admissible estimates Qad is small (see Remark 6.4). 6.4. OPTIMAL EXPERIMENTAL DESIGN 119 Summary of Chapter 6 The parameter estimation problem appears to be intricate under many aspects. The represented nonlinear Gauss-Newton method and linear approximation of confidence regions highlighted difficulties and details of parameter estimation. With the optimal experimental design, we refined estimates and showed which of the optimality criteria is the best and leads to the reduction in estimation error. In our model, however, we could find only seven from nineteen parameters, which are well estimable (Table 6.10, Table 6.11 and Table 6.12): q1 , q5 , q9 , q12 , q13 , q14 , q18 . Consequently, even in assistance of optimized measurement points, we could not estimate more than these. 120 CHAPTER 6. PARAMETER ESTIMATION AND EXPERIMENTAL DESIGN CHAPTER 7 Summary Understanding biological processes which are altered in medulloblastoma is important for building a relevant mathematical model. There are four distinct molecular variants of medulloblastoma [49]. These four variants are characterized by different chromosome aberrations, diverse even in a singular medulloblastoma subgroup [55]. Furthermore, prognosis radically varies depending on the molecular diagnosis and major differences are also found between the adult and pediatric case. In this work we focus on the aberration of chromosome 6q in the pediatric case only. Monosomy 6 (6q loss) is linked to good prognosis and trisomy 6 (6q gain) to poor prognosis. Both types involve a distortion in the expression of the target genes of the Wnt/βcatenin signaling pathway. This may suggest changes at the embryonal or developmental level and may explain the larger incidence of medulloblastoma among children than among grown-ups. We observe the correlation between the SGK1 deregulation and prognosis, i.e., increase of SGK1 seems to favor a negative influence on the patient’s prognosis. Conversely, the SGK1 downregulation is more frequent in good prognosis. We build a novel mathematical model in the form of a system of nonlinear ODEs to investigate the different prognosis in the two types of medulloblastoma. We model interactions between particular genes involving biological processes such as transcription, translation, phosphorylation, degradation and transport between the nucleus and cytoplasm. Because of the complexity of the biological system we choose genes which seem to play a crucial role in the system, hence we model dynamics of the loop SGK1-GSK3βMYC. Numerical simulations indicate the importance of SGK1. We simulate different scenarios for the monosomy 6 and trisomy 6 patient samples based on the microarray data from the clinics. Consistently, we reveal that high SGK1 mRNA production strongly 121 122 CHAPTER 7. SUMMARY correlates with the patient poor prognosis. It is obtained due to the comparison between the patient data and nuclear cMyc level obtained in the simulations. We elucidate the discrepancy between prognosis of the two types of medulloblastoma due to the SGK1 protein concentration. Based on the patient data we formulate hypothesis that gene SGK1 is an essential factor in medulloblastoma treatment. We propose a pharmacological inhibition of the SGK1 protein as a novel way of the patient treatment. We find that inhibition of the SGK1 protein yields the best effect as a pharmacological intervention for the modeled system and leads to decrease of the nuclear cMyc and the GSK3β stabilization. Our results are in line with recent experimental results obtained in the laboratory of the group of Prof. Dr. med. Stefan Pfister at the Division of Pediatric Neuro-oncology Research Group of the German Cancer Research Center (DKFZ) showing that SGK1 is an important gene in the investigated tumor and its inhibition entails remission of malignancy [56]. We can "translate" our mathematical study to an application in medicine, what was the main goal of this work. In the analytical part we show well-possedness of the proposed model in the sense of classical theory of ODEs. We show global existence and uniqueness of solutions. At next, we perform parameter estimation process in the framework of optimal experimental design. Using sensitivity analysis and a tool of confidence ellipses combined with Monte Carlo simulations we are able to refine seven estimates for the model. This result shows difficulties in the process of deterministic estimation of the nonlinear problem. Importantly, we indicate also that the most sensitive model parameter, for the species of the nuclear cMyc concentration, is responsible for the degradation of the SGK1 protein. Consequently, we underline the importance of SGK1 also in the sensitivity analysis. The question that still arises is whether the effects of SGK1 deregulation are executed just through the MYC dynamics or if there are other target genes that are significant for the disease development. It means that further studies could expose on which proteins SGK1 has further negative impact. The processes in tumor development are complex and one should follow many tests in order to deeply comprehend the biological dynamics. We have a well-posed model and a qualitative solution based on the input in the form of microarray data (mRNAs), but what about quantitative solutions on the protein level? Todays laboratory techniques do not assure the necessary data. However, we may estimate parameters using surrogate data. 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Kaplan Meier curves of overall survival probabilities for medulloblastoma patients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Fold change and log2 -ratio of SGK1 in medulloblastoma . . . . . . . . . 2.7 Scheme of the signaling pathway from a ligand-receptor to the nucleus . . 2.8 Scheme of the cell fate dependent on the multiple extracellular signals . . 2.9 Scheme of the Wnt signaling pathway without Wnt ligands . . . . . . . . 2.10 Scheme of the Wnt signaling pathway with Wnt ligands . . . . . . . . . . 2.11 Scheme of the Wnt signaling pathway with β-catenin mutation . . . . . . 2.12 Schematic diagram of the modeled biological dynamics . . . . . . . . . . 8 9 10 11 12 13 14 16 3.1 3.2 Scheme of GSK3β phosphorylation process . . . . . . . . . . . . . . . . Scheme of cMyc phosphorylation process . . . . . . . . . . . . . . . . . 21 21 4.1 4.2 4.3 Phase portrait for the state variables u5 and u8 - healthy sample . . . . . . Phase portrait for the state variables u5 and u8 - trisomy sample . . . . . . Time change of concentrations for perturbations in initial conditions for SGK1 mRNA, cytoplasmic SGK1, MYC mRNA . . . . . . . . . . . . . . Time change of concentrations for perturbations in initial conditions for cytoplasmic cMyc, nuclear cMyc, cytoplasmic GSK3β . . . . . . . . . . Time change of concentrations for perturbations in initial conditions for nuclear GSK3β, phosphorylating GSK3β, phosphorylating SGK1 . . . . 36 36 4.4 4.5 131 5 6 7 7 37 38 39 132 LIST OF FIGURES 5.1 Graphical user interface - time dependence graph . . . . . . . . . . . . . 46 5.2 Graphical user interface - phase portrait . . . . . . . . . . . . . . . . . . 47 5.3 Time change of concentrations of SGK1 mRNA, cytoplasmic SGK1, MYC mRNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Time change of concentrations of cytoplasmic cMyc, nuclear cMyc, cytoplasmic GSK3β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.4 5.5 Time change of concentrations of nuclear GSK3β, phosphorylating GSK3β, phosphorylating SGK1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.6 Trisomy 6. The nuclear cMyc level versus death status . . . . . . . . . . 54 5.7 Monosomy 6. The nuclear cMyc level versus death status . . . . . . . . . 55 5.8 Time change of concentrations in samples: (SGK1 : 0.54, M Y C : 28.59)monosomy6 (SGK1 : 4.16, M Y C : 29.78)trisomy6 . . . . . . . . . . . . . . . . . . . 57 Time change of concentrations in samples: (SGK1 : 0.32, M Y C : 12.33)monosomy6 (SGK1 : 7.92, M Y C : 12.02)trisomy6 . . . . . . . . . . . . . . . . . . . 57 5.10 Time change of concentrations in samples: (SGK1 : 0.19, M Y C : 17.18)monosomy6 (SGK1 : 0.47, M Y C : 17.20)monosomy6 . . . . . . . . . . . . . . . . . . 59 5.11 Time change of concentrations in samples: (SGK1 : 7.92, M Y C : 12.02)trisomy6 (SGK1 : 1.12, M Y C : 13.59)trisomy6 . . . . . . . . . . . . . . . . . . . 59 5.12 Time change of concentrations in samples: (SGK1 : 0.54, M Y C : 28.59)monosomy6 (SGK1 : 7.92, M Y C : 12.02)trisomy6 . . . . . . . . . . . . . . . . . . . 61 5.13 Time change of concentrations in samples: (SGK1 : 0.65, M Y C : 11.76)monosomy6 (SGK1 : 5.70, M Y C : 5.22)trisomy6 . . . . . . . . . . . . . . . . . . . . 61 5.14 Time change of concentrations in samples: (SGK1 : 0.27, M Y C : 38.11)monosomy6 (SGK1 : 0.27, M Y C : 31.06)monosomy6 . . . . . . . . . . . . . . . . . . 63 5.15 Time change of concentrations in samples: (SGK1 : 7.92, M Y C : 12.02)trisomy6 (SGK1 : 7.86, M Y C : 3.05)trisomy6 . . . . . . . . . . . . . . . . . . . . 63 5.16 Influence of GSK3β on the nuclear cMyc in trisomy 6 and monosomy 6 . 64 5.17 Relationship between the cMyc protein in the nucleus, MYC mRNA and SGK1 mRNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.9 LIST OF FIGURES 133 5.18 Correlation between the level of the cMyc protein in the nucleus and SGK1 mRNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.19 Trisomy 6. Influence of protein inhibitors on the nuclear cMyc . . . . . . 68 5.20 Monosomy 6. Influence of SGK1 inhibiton on the nuclear cMyc . . . . . 69 5.21 Trisomy 6. Influence of cMyc inhibition on GSK3β and nuclear cMyc . . 70 5.22 Trisomy 6. Influence of SGK1 inhibition on GSK3β and nuclear cMyc . . 70 6.1 Graphical representation of the parameter estimation problem . . . . . . . 80 6.2 95% confidence ellipses - general examples . . . . . . . . . . . . . . . . 81 6.3 Sensitivity with time points for species C5 and parameter q5 . . . . . . . . 86 6.4 Sensitivity with time points for species C5 and parameter q13 . . . . . . . 86 6.5 Sensitivity with time points for species C5 and parameter q14 . . . . . . . 87 6.6 95% confidence regions with various time points design, parameters q5 and q13 , species C5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 95% confidence regions with various time points design, parameters q5 and q14 , species C5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 95% confidence regions with various time points design, parameters q13 and q14 , species C5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 95% confidence regions and Monte Carlo simulation sampling in nonlinearity detection, parameters q2 , q5 and q8 , species C5 and C8 . . . . . . . 91 6.10 95% confidence regions and Monte Carlo simulation sampling in nonlinearity detection, parameters q2 , q5 and q18 , species C5 and C8 . . . . . . . 91 6.11 95% confidence regions and Monte Carlo simulation sampling in nonlinearity detection, parameters q5 , q8 and q18 , species C5 and C8 . . . . . . . 92 6.12 95% confidence regions for different linearizations, parameters q5 , q9 and q12 , species C5 and C8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.13 95% confidence regions for different linearizations, parameters q5 , q13 and q14 , species C5 and C8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.14 95% confidence regions for different linearizations, parameters q9 , q12 and q13 , species C5 and C8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.15 95% confidence regions for different linearizations, parameters q12 , q13 and q14 , species C5 and C8 . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.16 95% confidence regions for different linearizations, parameters q9 , q13 and q14 , species C5 and C8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.17 95% confidence region and Monte Carlo numerical results for estimation of parameters q5 and q9 for C5 and C8 with 1% and 10% of measurement error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.7 6.8 6.9 134 LIST OF FIGURES 6.18 95% confidence region and Monte Carlo numerical results for estimation of parameters q5 and q12 for C5 and C8 with 1% and 10% of measurement error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.19 95% confidence region and Monte Carlo numerical results for estimation of parameters q5 and q13 for C5 and C8 with 1% and 10% of measurement error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.20 95% confidence region and Monte Carlo numerical results for estimation of parameters q5 and q14 for C5 and C8 with 1% and 10% of measurement error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.21 95% confidence region and Monte Carlo numerical results for estimation of parameters q9 and q12 for C5 and C8 with 1% and 10% of measurement error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.22 95% confidence region and Monte Carlo numerical results for estimation of parameters q9 and q13 for C5 and C8 with 1% and 10% of measurement error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.23 95% confidence region and Monte Carlo numerical results for estimation of parameters q9 and q14 for C5 and C8 with 1% and 10% of measurement error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.24 95% confidence region and Monte Carlo numerical results for estimation of parameters q12 and q13 for C5 and C8 with 1% and 10% of measurement error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.25 95% confidence region and Monte Carlo numerical results for estimation of parameters q12 and q14 for C5 and C8 with 1% and 10% of measurement error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.26 95% confidence region and Monte Carlo numerical results for estimation of parameters q13 and q14 for C5 and C8 with 1% and 10% of measurement error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.27 95% confidence regions for variation in fixed parameters from 1% up to 10%, parameters q5 , q9 . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.28 95% confidence regions for variation in fixed parameters from 1% up to 10%, parameters q5 , q12 . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.29 95% confidence regions for variation in fixed parameters from 1% up to 10%, parameters q5 , q13 . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.30 95% confidence regions for variation in fixed parameters from 1% up to 10%, parameters q5 , q14 . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.31 95% confidence regions for variation in fixed parameters from 1% up to 10%, parameters q9 , q12 . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.32 95% confidence regions for variation in fixed parameters from 1% up to 10%, parameters q9 , q13 . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.33 95% confidence regions for variation in fixed parameters from 1% up to 10%, parameters q9 , q14 . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.34 95% confidence regions for variation in fixed parameters from 1% up to 10%, parameters q12 , q13 . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6.35 95% confidence regions for variation in fixed parameters from 1% up to 10%, parameters q12 , q14 . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6.36 95% confidence regions for variation in fixed parameters from 1% up to 10%, parameters q13 , q14 . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.37 Comparison of optimality criteria of parameters q1 and q5 estimation error 113 6.38 Comparison of optimality criteria of parameters q9 and q12 estimation error 114 6.39 Comparison of optimality criteria of parameters q13 and q14 estimation error114 6.40 Comparison of optimality criteria of parameter q18 estimation error . . . . 115 6.41 Trace of Cov in the process of sampling scheme optimization. First and last iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.42 Determinant of Cov in the process of sampling scheme optimization. First and last iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.43 Maximal eigenvalue of Cov in the process of sampling scheme optimization. First and last iteration . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.44 Ratio between the maximum eigenvalue and minimum eigenvalue of Cov in the process of sampling scheme optimization. First and last iteration . . 117

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