Dissertation Joanna Kawka

Dissertation Joanna Kawka
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
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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 . . . .
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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 . . . . . . . . . . .
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CONTENTS
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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
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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].
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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)
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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.
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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.
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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).
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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. During investigation, we see that the deterministic
methods used here failed to estimate all parameters for our nonlinear problem. Therefore,
with the help of stochastic methods, which are more sophisticated, we could try to obtain
estimates for more model parameters. The model with refined parameters may be a powerful tool for more accurate studies (e.g., individual patient dose) on the adjuvant therapy
reagarding the SGK1 inhibition in near future.
Bibliography
[1] Angelika E Altmann-Dieses, Johannes P Schlöder, Hans G Bock, and Otto Richter.
Optimal experimental design for parameter estimation in column outflow experiments. Water Resources Research, 38(10):4–1, 2002.
[2] Celina Ang, David Hauerstock, Marie-Christine Guiot, Goulnar Kasymjanova,
David Roberge, Petr Kavan, and Thierry Muanza. Characteristics and outcomes
of medulloblastoma in adults. Pediatric blood & cancer, 51(5):603–607, 2008.
[3] Maria F Arteaga, Diego Alvarez de la Rosa, Jose A Alvarez, and Cecilia M
Canessa. Multiple translational isoforms give functional specificity to serum-and
glucocorticoid-induced kinase 1. Molecular biology of the cell, 18(6):2072–2080,
2007.
[4] Samuel Bandara, Johannes P Schlöder, Roland Eils, Hans G Bock, and Tobias
Meyer. Optimal experimental design for parameter estimation of a cell signaling
model. PLoS computational biology, 5(11):e1000558, 2009.
[5] Julio R Banga. Optimization in computational systems biology. BMC systems biology, 2(1):47, 2008.
[6] Roland Becker, Malte Braack, and Boris Vexler. Numerical parameter estimation
for chemical models in multidimensional reactive flows. Combustion Theory and
Modelling, 8(4):661–682, 2004.
[7] Larissa Belova, Sanjay Sharma, Deanna R Brickley, Jeremy R Nicolarsen, Cam
Patterson, and Suzanne D Conzen. Ubiquitin–proteasome degradation of serum-and
glucocorticoid-regulated kinase-1 (sgk-1) is mediated by the chaperone-dependent
e3 ligase chip. Biochemical Journal, 400(Pt 2):235, 2006.
123
124
BIBLIOGRAPHY
[8] Hans G Bock, Thomas Carraro, Willi Jäger, Stefan Körkel, Rolf Rannacher, and
Johannes P Schlöder. Model based parameter estimation. Theory and applications.
Springer, 2013.
[9] Hans Georg Bock, Ekaterina Kostina, and Johannes P Schlöder. Numerical methods for parameter estimation in nonlinear differential algebraic equations. GAMMMitteilungen, 30(2):376–408, 2007.
[10] Carmen Chicone. Ordinary differential equations with applications, volume 34.
Springer, 2000.
[11] Kwang-Hyun Cho, Sung-Young Shin, Walter Kolch, and Olaf Wolkenhauer. Experimental design in systems biology, based on parameter sensitivity analysis using a
monte carlo method: A case study for the tnfα-mediated nf-κ b signal transduction
pathway. Simulation, 79(12):726–739, 2003.
[12] Y-J. Cho, A. Tsherniak, P. Tamayo, S. Santagata, A. Ligon, H. Greulich,
R. Berhoukim, V. Amani, L. Goumnerova, C.G. Eberhart, et al. Integrative genomic analysis of medulloblastoma identifies a molecular subgroup that drives poor
clinical outcome. Journal of Clinical Oncology, 29(11):1424–1430, 2011.
[13] Hans Clevers. Wnt/β-catenin signaling in development and disease.
127(3):469–480, 2006.
Cell,
[14] Chi V Dang. c-myc target genes involved in cell growth, apoptosis, and metabolism.
Molecular and cellular biology, 19(1):1–11, 1999.
[15] Chi V Dang, Linda Resar, Eileen Emison, Sunkyu Kim, Qing Li, Julia E Prescott,
Diane Wonsey, and Karen Zeller. Function of the c-myc oncogenic transcription
factor. Experimental cell research, 253(1):63–77, 1999.
[16] Manuel Dehner, Michel Hadjihannas, Jörg Weiske, Otmar Huber, and Jürgen
Behrens. Wnt signaling inhibits forkhead box o3a-induced transcription and apoptosis through up-regulation of serum-and glucocorticoid-inducible kinase 1. Journal
of Biological Chemistry, 283(28):19201–19210, 2008.
[17] Jacques Delforge, Andre Syrota, and Bernard M Mazoyer. Experimental design
optimisation: theory and application to estimation of receptor model parameters
using dynamic positron emission tomography. Physics in medicine and biology,
34(4):419, 1989.
[18] Bradley W Doble and James R Woodgett. Gsk-3: tricks of the trade for a multitasking kinase. Journal of cell science, 116(7):1175–1186, 2003.
BIBLIOGRAPHY
125
[19] David W Ellison, Olabisi E Onilude, Janet C Lindsey, Meryl E Lusher, Claire L Weston, Roger E Taylor, Andrew D Pearson, and Steven C Clifford. β-catenin status
predicts a favorable outcome in childhood medulloblastoma: the united kingdom
children’s cancer study group brain tumour committee. Journal of Clinical Oncology, 23(31):7951–7957, 2005.
[20] D Faller, U Klingmüller, and J Timmer. Simulation methods for optimal experimental design in systems biology. Simulation, 79(12):717–725, 2003.
[21] Gary L Firestone, Jennifer R Giampaolo, and Bridget A OKeeffe. Stimulusdependent regulation of serum and glucocorticoid inducible protein kinase (sgk)
transcription, subcellular localization and enzymatic activity. Cellular Physiology
and Biochemistry, 13(1):1–12, 2003.
[22] Riccardo Fodde and Thomas Brabletz. Wnt/β-catenin signaling in cancer stemness
and malignant behavior. Current opinion in cell biology, 19(2):150–158, 2007.
[23] Sheelagh Frame and Philip Cohen. Gsk3 takes centre stage more than 20 years after
its discovery. Biochemical Journal, 359(Pt 1):1, 2001.
[24] Scott F Gilbert. Developmental biology: The anatomical tradition. Sinauer Associates, 2000.
[25] Michael D Gordon and Roel Nusse. Wnt signaling: multiple pathways, multiple receptors, and multiple transcription factors. Journal of Biological Chemistry,
281(32):22429–22433, 2006.
[26] Mark A Gregory, Ying Qi, and Stephen R Hann. Phosphorylation by glycogen
synthase kinase-3 controls c-myc proteolysis and subnuclear localization. Journal
of Biological Chemistry, 278(51):51606–51612, 2003.
[27] Narendra Gupta and Raman Mehra. Computational aspects of maximum likelihood
estimation and reduction in sensitivity function calculations. Automatic Control,
IEEE Transactions on, 19(6):774–783, 1974.
[28] Eldad Haber, Lior Horesh, and Luis Tenorio. Numerical methods for experimental design of large-scale linear ill-posed inverse problems. Inverse Problems,
24(5):055012, 2008.
[29] Philip Hartman. Ordinary differential equations. John Wiley & Sons, Inc. New
York-London-Sydney, 1964.
126
BIBLIOGRAPHY
[30] Stefan Hoppler and Claire L Kavanagh. Wnt signalling: variety at the core. Journal
of cell science, 120(3):385–393, 2007.
[31] Hans A Kestler and Michael Kühl. From individual wnt pathways towards a wnt
signalling network. Philosophical Transactions of the Royal Society B: Biological
Sciences, 363(1495):1333–1347, 2008.
[32] Ehab M Khalil. Treatment results of adults and children with medulloblastoma
nci, cairo university experience. Journal of the Egyptian National Cancer Institute,
20(2):175, 2008.
[33] David M King and Chris Perera. Sensitivity analysis for evaluating importance of
variables used in an urban water supply planning model. In Proceedings of the
International Congress on Modelling and Simulation (MODSIM’07), pages 2768–
2774, 2007.
[34] Peter K Kitanidis and Robert W Lane. Maximum likelihood parameter estimation
of hydrologic spatial processes by the gauss-newton method. Journal of Hydrology,
79(1):53–71, 1985.
[35] Alexandra Klaus and Walter Birchmeier. Wnt signalling and its impact on development and cancer. Nature Reviews Cancer, 8(5):387–398, 2008.
[36] Frank T Kolligs, Guido Bommer, and Burkhard Göke. Wnt/beta-catenin/tcf signaling: a critical pathway in gastrointestinal tumorigenesis. Digestion, 66(3):131–144,
2002.
[37] M. Kool, J. Koster, J. Bunt, N.E. Hasselt, A. Lakeman, P. van Sluis, D. Troost,
N. Schouten-van Meeteren, H.N. Caron, J. Cloos, et al. Integrated genomics identifies five medulloblastoma subtypes with distinct genetic profiles, pathway signatures
and clinicopathological features. PloS one, 3(8):e3088, 2008.
[38] Andrey Korshunov, Axel Benner, Marc Remke, Peter Lichter, Andreas von Deimling, and Stefan Pfister. Accumulation of genomic aberrations during clinical progression of medulloblastoma. Acta neuropathologica, 116(4):383–390, 2008.
[39] Andrey Korshunov, Marc Remke, Wiebke Werft, Axel Benner, Marina Ryzhova,
Hendrik Witt, Dominik Sturm, Andrea Wittmann, Anna Schöttler, Jörg Felsberg,
et al. Adult and pediatric medulloblastomas are genetically distinct and require
different algorithms for molecular risk stratification. Journal of Clinical Oncology,
28(18):3054–3060, 2010.
BIBLIOGRAPHY
127
[40] Roland Kruger and Reinhart Heinrich. Model reduction and analysis of robustness
for the wnt/beta-catenin signal transduction pathway. Genome Informatics Series,
pages 138–148, 2004.
[41] Julia Kzhyshkowska, Anna Marciniak-Czochra, and Alexei Gratchev. Perspectives
of mathematical modelling for understanding of intracellular signalling and vesicular trafficking in macrophages. Immunobiology, 212(9):813–825, 2008.
[42] Ethan Lee, Adrian Salic, Roland Krüger, Reinhart Heinrich, and Marc W Kirschner.
The roles of apc and axin derived from experimental and theoretical analysis of the
wnt pathway. PLoS biology, 1(1):e10, 2003.
[43] Han C Lee, Miran Kim, Jack R Wands, et al. Wnt/frizzled signaling in hepatocellular
carcinoma. Front Biosci, 11(5):1901–1915, 2006.
[44] Tomasz Lipniacki, Pawel Paszek, Allan R Brasier, Bruce Luxon, and Marek Kimmel. Mathematical model of nf-κ b regulatory module. Journal of theoretical biology, 228(2):195–215, 2004.
[45] Catriona Y Logan and Roel Nusse. The wnt signaling pathway in development and
disease. Annu. Rev. Cell Dev. Biol., 20:781–810, 2004.
[46] Franklin C McLean. Application of the law of chemical equilibrium (law of mass
action) to biological problems. Physiol. Rev, 18:495, 1938.
[47] Christopher Z Mooney. Monte carlo simulation, volume 116. SAGE Publications,
Incorporated, 1997.
[48] Jorge Nocedal and Stephen J Wright. Numerical optimization. Springer Science+
Business Media, 2006.
[49] Paul A Northcott, Andrey Korshunov, Hendrik Witt, Thomas Hielscher, Charles G
Eberhart, Stephen Mack, Eric Bouffet, Steven C Clifford, Cynthia E Hawkins, Pim
French, et al. Medulloblastoma comprises four distinct molecular variants. Journal
of Clinical Oncology, 29(11):1408–1414, 2011.
[50] Alex B Novikoff.
A transplantable rat liver tumor induced by 4dimethylaminoazobenzene. Cancer Research, 17(10):1010–1027, 1957.
[51] Fabio Orlandi, Enrico Saggiorato, Giovanni Pivano, Barbara Puligheddu, Angela
Termine, Susanna Cappia, Paolo De Giuli, and Alberto Angeli. Galectin-3 is a
presurgical marker of human thyroid carcinoma. Cancer research, 58(14):3015–
3020, 1998.
128
BIBLIOGRAPHY
[52] Andrzej Palczewski. Równania różniczkowe zwyczajne: teoria i metody numeryczne
z wykorzystaniem komputerowego systemu obliczeń symbolicznych. Wydawnictwa
Naukowo-Techniczne, 1999.
[53] Stella Pelengaris, Mike Khan, and Gerard Evan. c-myc: more than just a matter of
life and death. Nature Reviews Cancer, 2(10):764–776, 2002.
[54] Stefan Pfister, Christian Hartmann, and Andrey Korshunov. Histology and molecular
pathology of pediatric brain tumors. Journal of child neurology, 24(11):1375–1386,
2009.
[55] Stefan Pfister, Marc Remke, Axel Benner, Frank Mendrzyk, Grischa Toedt, Jörg
Felsberg, Andrea Wittmann, Frauke Devens, Nicolas U Gerber, Stefan Joos, et al.
Outcome prediction in pediatric medulloblastoma based on dna copy-number aberrations of chromosomes 6q and 17q and the myc and mycn loci. Journal of Clinical
Oncology, 27(10):1627–1636, 2009.
[56] Sabrina V Pleier, Dominik Sturm, Stefan Pfister, Marcel Kool, Anna MarciniakCzochra, Joanna Kawka, et al. Serum- and glucocorticoid-regulated kinase 1 is a
druggable regulator of cell migration and apoptosis in high-risk medulloblastoma.
In Review.
[57] Paul Polakis. Wnt signaling and cancer. Genes & development, 14(15):1837–1851,
2000.
[58] Scott L Pomeroy and Lisa M Sturla. Molecular biology of medulloblastoma therapy.
Pediatric neurosurgery, 39(6):299–304, 2004.
[59] Luc Pronzato. Optimal experimental design and some related control problems.
Automatica, 44(2):303–325, 2008.
[60] M. Remke, T. Hielscher, A. Korshunov, P.A. Northcott, S. Bender, M. Kool, F. Westermann, A. Benner, H. Cin, M. Ryzhova, et al. Fstl5 is a marker of poor prognosis
in non-wnt/non-shh medulloblastoma. Journal of Clinical Oncology, 29(29):3852–
3861, 2011.
[61] Laurent Riffaud, Stephan Saikali, Emmanuelle Leray, Abderrahmane Hamlat, Claire
Haegelen, Elodie Vauleon, and Thierry Lesimple. Survival and prognostic factors in
a series of adults with medulloblastomas: Clinical article. Journal of neurosurgery,
111(3):478–487, 2009.
BIBLIOGRAPHY
129
[62] Alessandra Rossi, Valentina Caracciolo, Giuseppe Russo, Krzysztof Reiss, and Antonio Giordano. Medulloblastoma: from molecular pathology to therapy. Clinical
cancer research, 14(4):971–976, 2008.
[63] Kevin M Ryan and George D Birnie. Myc oncogenes: the enigmatic family. Biochemical Journal, 314(Pt 3):713, 1996.
[64] Einat Sadot, Maralice Conacci-Sorrell, Jacob Zhurinsky, Dalia Shnizer, Zeev Lando,
Dorit Zharhary, Zvi Kam, Avri Ben-Ze’ev, and Benjamin Geiger. Regulation of
s33/s37 phosphorylated β-catenin in normal and transformed cells. Journal of cell
science, 115(13):2771–2780, 2002.
[65] HR Schlesinger, JM Gerson, PS Moorhead, H Maguire, and K Hummeler. Establishment and characterization of human neuroblastoma cell lines. Cancer research,
36(9 Part 1):3094–3100, 1976.
[66] George AF Seber and Chris J Wild. Nonlinear regression. John Wiley & Sons, New
York, New York, USA, 2003.
[67] Lawrence F Shampine and Mark W Reichelt. The matlab ode suite. SIAM journal
on scientific computing, 18(1):1–22, 1997.
[68] Perikles Simon, Michaela Schneck, Tabea Hochstetter, Evgenia Koutsouki, Michel
Mittelbronn, Axel S Merseburger, Cora Weigert, Andreas M Niess, and Florian
Lang. Differential regulation of serum-and glucocorticoid-inducible kinase 1 (sgk1)
splice variants based on alternative initiation of transcription. Cellular Physiology
and Biochemistry, 20(6):715–728, 2007.
[69] Gordon D Smith. Numerical solution of partial differential equations: finite difference methods. Oxford University Press, 1985.
[70] Michael D Taylor, Paul A Northcott, Andrey Korshunov, Marc Remke, Yoon-Jae
Cho, Steven C Clifford, Charles G Eberhart, D Williams Parsons, Stefan Rutkowski,
Amar Gajjar, et al. Molecular subgroups of medulloblastoma: the current consensus.
Acta neuropathologica, 123(4):465–472, 2012.
[71] Esther M Verheyen and Cara J Gottardi. Regulation of wnt/β-catenin signaling by
protein kinases. Developmental Dynamics, 239(1):34–44, 2010.
[72] Ferdinand Verhulst. Nonlinear differential equations and dynamical systems.
Springer verlag, 1996.
130
BIBLIOGRAPHY
[73] Karina J Versyck, Kristel Bernaerts, Annemie H Geeraerd, Jan F Van Impe, et al.
Introducing optimal experimental design in predictive modeling: a motivating example. International Journal of Food Microbiology, 51(1):39, 1999.
[74] Brian J Wagner and Judson W Harvey. Experimental design for estimating parameters of rate-limited mass transfer: Analysis of stream tracer studies. Water Resources
Research, 33(7):1731–1741, 1997.
[75] Hong-Xing Wang, Francis R Tekpetey, and Gerald M Kidder. Identification of
wnt/β-catenin signaling pathway components in human cumulus cells. Molecular
human reproduction, 15(1):11–17, 2009.
[76] Jianbo Wang and Anthony Wynshaw-Boris. The canonical wnt pathway in early
mammalian embryogenesis and stem cell maintenance/differentiation. Current opinion in genetics & development, 14(5):533–539, 2004.
[77] Kan Wang, Shuchen Gu, Omaima Nasir, Michael Föller, Teresa F Ackermann, Karin
Klingel, Reinhard Kandolf, Dietmar Kuhl, Christos Stournaras, and Florian Lang.
Sgk1-dependent intestinal tumor growth in apc-deficient mice. Cellular Physiology
and Biochemistry, 25(2-3):271–278, 2010.
[78] Russell H Zurawel, Sharon A Chiappa, Cory Allen, and Corey Raffel. Sporadic medulloblastomas contain oncogenic β-catenin mutations. Cancer research,
58(5):896–899, 1998.
List of Figures
2.1
2.2
2.3
2.4
2.5
Computer tomography brain scan showing a medulloblastoma tumor . . .
Overall survival probabilities in medulloblastoma mutations . . . . . . .
Diagram of chromosome structure . . . . . . . . . . . . . . . . . . . . .
A genomic hybridization profiles of chromosome 6 . . . . . . . . . . . .
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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