TY  - GEN
N2  - This thesis comprises the modelling of and parameter estimation in dynamical systems, with a focus on applications in systems biology. In an interdisciplinary research project on the systems biology of cancer, we develop a predictive mathematical model of an intracellular crosstalk in cytokine signalling. Expected and unexpected predictions are confirmed in experiments and lead to new biological insights. For model calibration with measurement data, we apply well established methods for parameter estimation in ordinary differential equation models. Extending these to stochastic differential equations, we develop, analyse, and implement a new method for parameter estimation in dynamical processes with noise, and demonstrate its performance in several selected examples from systems biology and mathematical finance.

Many processes, especially in biology, obey deterministic ground rules (e.g. metabolic processes or signal transduction pathways), but may be heavily in?uenced by ?uctuations and stochasticity inherent to the system that change its behaviour both qualitatively and quantitatively. Therefore, frequently, a deterministic description is not constructive.

A large class of such systems can be adequately described by nonlinear multi-dimensional stochastic differential equations (SDEs). Classical estimation techniques for SDEs, relying on (approximations of) transition densities, are all too often not applicable to these problems, due, inter alia, to their high computational costs and prerequisites on the measurements. The proposed new method is based on the method of multiple shooting, using piecewise deterministic solutions of ordinary differential equations (ODEs) to approximate the SDE realization that corresponds to the studied process from which measurements have been taken.

The generally discontinuous concatenation of ODE trajectories mimics the consequences of stochastic effects, and, further, allows to formulate the parameter estimation problem as a deterministic nonlinear optimization problem that can be solved with efficient derivative-based solution methods. In this thesis, a generalized GAUSS-NEWTON method is deployed.

Main results and contributions of this thesis are summarized in the following:

? We propose a new method for parameter estimation in nonlinear multi-dimensional SDEs, based on a piecewise approximation by solutions of ODEs. Discontinuities (jumps) occurring at the interval borders are used for regularization. Unknown parameters and initial states are estimated by a generalized weighted least squares method from data that can originate from direct complete or partial state measurements or from indirectly observed quantities. Measurement data may be afflicted with errors and arbitrarily sampled. Non-linear parameter and point constraints may be formulated as equality and inequality constraints. The resulting nonlinear constrained optimization problems are highly structured and efficiently solved using a generalized GAUSS-NEWTON method.

? We give a proof that the discontinuities at the interval borders asymptotically tend to zero if the number of equidistantly distributed shooting nodes goes to infinity.

? We show in a numerical analysis that the resulting equation systems are sparse, that the number of nonzero elements depends only linearly on the number of shooting nodes, and give sharp upper bounds. Moreover, we prove that the sparsity is maintained if an appropriate (stable) decomposition is a applied.

? It is demonstrated in comparative simulation studies that the estimates are robust w.r.t. the exact choice of jump regularization weights. Moreover, the effects of jump regularization on estimates and approximated trajectories are investigated and described.

? A lifting approach with per-interval parameter sets, coupled by additional equality constraints, is developed and its numerical properties are analysed. Moreover, we propose a homotopy method for the treatment of hard problems.

? We demonstrate the performance of the new estimation technique in examples from systems biology, each shedding some light on different aspects. Especially, we show that the method can also be used for hidden state estimation and for trajectory reconstruction in time spans without observations. Further, we derive a criterion for local grid refinement.

? We show for an ORNSTEIN-UHLENBECK process driven by a LEVY jump process, that, in addition to mean reversion level and mean reversion rate, also the diffusion constant may be estimated by analysing the jump residuals.

? The software package :sfit is an efficient implementation of the proposed method, offering easy symbolic problem formulation to the user, from which the stochastic parameter estimation problem can be automatically built and solved.


Parts of this work emerged in the interdisciplinary research project SBCancer of the Helmholtz Alliance on Systems Biology. In close collaboration with expert biologists, we developed an mathematical model for a crosstalk of two cytokines in human skin cells that interfere in a signalling pathway frequently found aberrantly activated in cancer. After an extensive analysis of the deployed measurement data processing, the model proposed by the author of this thesis has been calibrated from experimental data. Its counter-intuitive predictions have been verified in wet lab experiments and lead to new biological insights.

Main novelties and contributions in this thesis are:

? Development of a mathematical model of the crosstalk of two cytokines in human keratinocytes (HaCaT cell line). The predicted and hitherto unknown nonlinear moderating effects of GM-CSF on the IL-6-induced JAK-STAT signalling pathway has been verified in vitro.

? An extensive mathematical analysis of the frequently utilized quantitative WESTERN blotting measurement procedure shows that established data normalization methods, relying on housekeeping proteins or manually added calibrator proteins, are prone to signal deteriorating statistical artefacts. Moreover, we show that the frequently declared assumption of normally distributed measurement errors cannot be maintained if these normalization techniques are applied.

? We propose as a remedy a normalization technique based on the calculation of amplification factors, and develop criteria for (approximate) normally distributed errors. These criteria can be easily checked using solely the raw measurement data. Moreover, we demonstrate the advantages of the proposed amplification factors method in a large comparative simulation study.
A1  - Sommer, Andreas
UR  - https://archiv.ub.uni-heidelberg.de/volltextserver/22589/
ID  - heidok22589
KW  - Parameter Estimation
KW  -  Nonlinear Stochastic Differential Equation
KW  -  Nonlinear Ordinary Differential Equation
KW  -  Systems Biology
KW  -  Mathematical Modelling
KW  -  Western Blot
KW  -  IL-6
KW  -  GM-CSF
KW  -  JAK/STAT
KW  -  HaCaT Keratinocytes
KW  -  Signal Transduction
KW  -  Cytokine
AV  - public
TI  - Numerical Methods for Parameter Estimation in Dynamical Systems with Noise with Applications in Systems Biology
Y1  - 2017///
ER  -