Kinetic Theory Modeling and Efficient Numerical Simulation of Gene Regulatory Networks Based on Qualitative Descriptions

Using mathematical modeling to address large scale problems in the world of biological regulatory networks has become increasingly necessary given the sheer quantity of data made available by improved technology. In the most general sense, modeling approaches can be thought of as being either quantitative or qualitative. Quantitative methods such as ordinary differential equations or the chemical master equation are widespread in the literature; when the model is well developed, the detail therein can be incredibly informative. However, they require an in depth knowledge of the reaction kinetics and generally fail as the problem size grows. The alternative approach, qualitative models, does not possess the same amount of detail but captures the essential dynamics of the system. Gene regulation, as a sub-genre of biological regulatory networks, is characterized by large numbers of interconnected species whose influences depend on passing some threshold, thus, largely sigmoidal behaviors. The application of qualitative methods to these systems can be highly advantageous to the modeler. As just mentioned realistic models in gene regulation are immense and highly interconnected, such that the simply enumeration of the possible states of the resulting system creates a combinatorial explosion. There are some questions for which one must access the underlying probability distribution associated with the Markov transitions of the qualitative model, as for example a qualitative and intuitive analysis of the system as a whole. The most pervasive methods have historically been simulation-based. Here, we propose a method to solve the system by treating the Markov equations of a Process Hitting model with numerical techniques. Proper Generalized Decomposition (PGD) can be used to overcome the curse of dimensionality, providing fast and accurate solutions to an otherwise intractable problem. Moreover PGD allows considering unknown parameters as a model extra-coordinate to obtain a parametric solution.