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TitleStochastic Oscillations in Living Cells
File Size3.9 MB
Total Pages142
Table of Contents
1. An excitable p53 model
	1.1. Introduction to p53 and the DNA damage response system
		1.1.1. The guardian of the genome
		1.1.2. DNA double strand breaks
		1.1.3. Established p53 dynamics
	1.2. Experimental findings not covered by oscillatory dynamics
		1.2.1. Introducing IPI distributions
		1.2.2. p53 data analysis
	1.3. Modeling theory
		1.3.1. Preface - dynamical systems theory and biochemical reaction networks
		1.3.2. Negative feedback oscillators
		1.3.3. Introduction to excitability - case study FitzHugh Nagumo
	1.4. The ATM-Wip1 switch building block
		1.4.1. ATM as Signalling Switch
		1.4.2. Incorporating the Phosphatase Wip1
	1.5. Including the core negative feedback loop - the full p53 model
		1.5.1. The effectual modeled p53 network for the DSB response
		1.5.2. Bifurcation analysis and deterministic dynamics
	1.6. Driving the p53 model with a stochastic DSB process
		1.6.1. Constructing a stochastic process for the DSB dynamics
		1.6.2. Stochastic forcing of the excitable p53 model
	1.7. Reanalysis of inhibitor experiments
	1.8. Discussing the p53 modeling approach
2. Hierarchic stochastic modelling of intracellular Ca2+
	2.1. Introduction to intracellular Ca2+ signaling
	2.2. An analytical approach to hierarchic stochastic modelling
		2.2.1. What is HSM ?
		2.2.2. Semi-Markov processes
		2.2.3. Non-Markovian master equations and first passage times
		2.2.4. A simple semi-Markovian system
		2.2.5. Explicit solutions for the tetrahedron Ca2+ model
	2.3. Exploiting time scale separation - The generic Ca2+ model
		2.3.1. Model Construction
		2.3.2. Results of the generic model
		2.3.3. Error analysis
	2.4. Numerical Analysis of the HSM Ca2+ model
		2.4.1. The DSSA algorithm
		2.4.2. An exact semi-Markovian simulation algorithm
	2.5. Encoding Stimulus intensities in random spike trains
		2.5.1. Theoretical predictions
		2.5.2. Experiments supporting the fold change encoding hypothesis
	2.6. Discussion of the stochastic Ca2+ modeling
3. Concluding remarks
A. Appendix p53
	A.1. Peak detection with wavelets
	A.2. Table of Parameters for the p53 model
	A.3. Sensitivity of pulse shapes and the excitation threshold on parameter variations
	A.4. Codimension-2 bifurcation diagrams
	A.5. Period of oscillations in the p53 model
B. Appendix Ca2+
	B.1. Laplace transformations of Ψo and Ψc
C. List of abbreviations
Document Text Contents
Page 71

2.2. An analytical approach to hierarchic stochastic modelling

stochastic process emerging by clustering of IP3 receptors.
An approach to model such complex stochastic systems was coined hierarchic

stochastic modeling (HSM), its theoretical developments started already some
years ago [61, 79]. The successful application to Ca2+ dynamics was achieved
recently [103] and further analytical insights were gained even more recently
[72] and shall be presented in the following.

2.2. An analytical approach to hierarchic stochastic

In this section the general theoretical framework involved in hierarchic stochastic
modeling (HSM) shall be developed. At first, the general idea to refrain from
a pure Markovian description shall be motivated. The formal consequences
including semi-Markovian processes, the correspondent non-Markovian Master
equations and the concept of probability fluxes will be discussed subsequently.
Finally, a specific Ca2+ model developed in ref. [103] shall be analytically solved
in the context of a first passage time problem.

2.2.1. What is HSM ?

The main goal of HSM is a state space reduction without fully neglecting the
microscopic dynamics of the system. As to be seen in the following, this effectively
implies a semi-Markovian description. However, the greater theoretical challenge
is paid off well by a substantial reduction in the number of free parameters in
the model. Additionally, it makes the theory readily applicable to experiments,
which especially in molecular biology rarely observe microscopic state changes
directly. This is due to the enormous complexity of actual molecular interactions
often found for elementary cellular processes such as transcription, translation or
Ca2+ signaling. The many cooperative interactions translate via combinatorics
to high dimensional state spaces which in turn make the application of standard
methods like the chemical master equation often intractable. The integration
of many microscopic states into one mesoscopic observable state is therefore a
naturally choice for the description of intracellular processes and is the core idea
of HSM.

When describing a receptor channel molecule, often the main question of
interest is if the channel is open or closed. The receptor molecule might have
many internal states where only one corresponds to the channel being open
[22, 93]. However, in a standard Markovian description, all internal state
transitions have to be described, also the ones which are not leading to an
opening event. The classical master equation reads


i (t) =



l (t) − qilP

i (t)

. (2.1)

Here, N is the number of system states and P ji (t) = P (i, t|j, 0) is the conditional
probability that the system is in state i at time t conditioned by being in state


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