Qualitative simulation of genetic regulatory networks using piecewise-linear models (original) (raw)
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Methods developed for the qualitative simulation of dynamical systems have turned out to be powerful tools for studying genetic regulatory networks. We present a generalization of a simulation method based on piecewise-linear differential equation models that is able to deal with discontinuities. The method is sound and has been implemented in a computer tool called GNA.
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Microarray chips generate large amounts of data about a cell's state. In our work we want to analyze these data in order to describe the regulation processes within a cell. Therefore, we build a model which is capable of capturing the most relevant regulating interactions and present an approach how to calculate the parameters for the model from time-series data. This approach uses the discrete approximation method of least squares to solve a data fitting modeling problem. Furthermore, we analyze the features of our proposed system, i.e., which kinds of dynamical behaviors the system is able to show.
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Piecewise-linear Models of Genetic Regulatory Networks: Equilibria and their Stability
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A formalism based on piecewise-linear (PL) differential equations, originally due to Glass and Kauffman, has been shown to be well-suited to modelling genetic regulatory networks. However, the discontinuous vector field inherent in the PL models raises some mathematical problems in defining solutions on the surfaces of discontinuity. To overcome these difficulties we use the approach of Filippov, which extends the vector field to a differential inclusion. We study the stability of equilibria (called singular equilibrium sets) that lie on the surfaces of discontinuity. We prove several theorems that characterize the stability of these singular equilibria directly from the state transition graph, which is a qualitative representation of the dynamics of the system. We also formulate a stronger conjecture on the stability of these singular equilibrium sets.
Simulating Genetic Regulatory Networks Version 1.2
If we assume that the process modelled is stable over time, then we can represent the causal structure of the series with a time series graph that includes the smallest fragment of the series that repeats. The number of temporal slices in the time series graph is the longest lag of direct influence plus one. For example, the time series graph in Figure 2, 1 which represents the series in Figure 1, needs three temporal slices to represent a repeating sequence, because G2 has a direct effect on G3 with a temporal lag of two. time= i
2005
Methods developed for the qualitative simulation of dynamical systems have turned out to be powerful tools for studying genetic regulatory networks. A bottleneck in the application of these methods is the analysis of the simulation results. In this paper, we propose a combination of qualitative simulation and model-checking techniques to perform this task systematically and efficiently. We apply our approach to the analysis of the complex network controlling the nutritional stress response in the bacterium Escherichia coli.