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Configuration, simulation and visualization of simple biochemical reaction-diffusion systems in 3D
(2004)
Background In biological systems, molecules of different species diffuse within the reaction compartments and interact with each other, ultimately giving rise to such complex structures like living cells. In order to investigate the formation of subcellular structures and patterns (e.g. signal transduction) or spatial effects in metabolic processes, it would be helpful to use simulations of such reaction-diffusion systems. Pattern formation has been extensively studied in two dimensions. However, the extension to three-dimensional reaction-diffusion systems poses some challenges to the visualization of the processes being simulated. Scope of the Thesis The aim of this thesis is the specification and development of algorithms and methods for the three-dimensional configuration, simulation and visualization of biochemical reaction-diffusion systems consisting of a small number of molecules and reactions. After an initial review of existing literature about 2D/3D reaction-diffusion systems, a 3D simulation algorithm (PDE solver), based on an existing 2D-simulation algorithm for reaction-diffusion systems written by Prof. Herbert Sauro, has to be developed. In a succeeding step, this algorithm has to be optimized for high performance. A prototypic 3D configuration tool for the initial state of the system has to be developed. This basic tool should enable the user to define and store the location of molecules, membranes and channels within the reaction space of user-defined size. A suitable data structure has to be defined for the representation of the reaction space. The main focus of this thesis is the specification and prototypic implementation of a suitable reaction space visualization component for the display of the simulation results. In particular, the possibility of 3D visualization during course of the simulation has to be investigated. During the development phase, the quality and usability of the visualizations has to be evaluated in user tests. The simulation, configuration and visualization prototypes should be compliant with the Systems Biology Workbench to ensure compatibility with software from other authors. The thesis is carried out in close cooperation with Prof. Herbert Sauro at the Keck Graduate Institute, Claremont, CA, USA. Due to this international cooperation the thesis will be written in English.
The paper proposes a variation of simulation for checking and proving contextual equivalence in a non-deterministic call-by-need lambda-calculus with constructors, case, seq, and a letrec with cyclic dependencies. It also proposes a novel method to prove its correctness. The calculus' semantics is based on a small-step rewrite semantics and on may-convergence. The cyclic nature of letrec bindings, as well as non-determinism, makes known approaches to prove that simulation implies contextual equivalence, such as Howe's proof technique, inapplicable in this setting. The basic technique for the simulation as well as the correctness proof is called pre-evaluation, which computes a set of answers for every closed expression. If simulation succeeds in finite computation depth, then it is guaranteed to show contextual preorder of expressions.