DynSyst_Special_Topics: Polynomial Dynamical s Systems Over Finite Fields: From Structure to Dynamics

Virginia Polytechnic Institute And State University, Blacksburg VA

Investigators

Abstract

Progress in our understanding of increasingly complex engineered and natural systems depends crucially on the use of mathematical models as the basis for computational analysis and prediction. Such models allow the simulation on a computer of dynamic processes that are driven by several different interacting forces. This is true in particular for systems studied in the life sciences, which form the basis for advances in biomedicine and bioengineering. Molecular networks inside human cells which process external signals and drive cellular metabolism provide important examples of such processes. Relatively little is currently known about the design principles of such networks. One approach to gaining increased understanding is to study properties of the mathematical models that capture their key features. An understanding of the relationship between structural features of the models and the constraints these features put on model dynamics will allow the formulation of hypotheses about design features of biological networks based on observed dynamics. These hypotheses can then be tested in the laboratory. The goal of this project is to study the relationship between structure and dynamics for a type of model that has proven to be very useful in capturing key features of a variety of intracellular molecular networks. Beyond molecular networks, aspects of this model type have been used in electrical engineering and computer science, so this project might have an impact beyond the life sciences. Time-discrete dynamical systems models are ubiquitous not only in engineering but also the life sciences. Especially during the last decade finite dynamical systems, that is, time-discrete dynamical systems with a finite state space, have been used increasingly in systems biology to model a variety of biochemical networks, such as gene regulatory networks and signal transduction networks. In many cases, the available data quantity and quality is not sufficient to build detailed quantitative models such as systems of ordinary differential equations, which require many parameters that are frequently unknown. In addition, discrete models tend to be more intuitive and more easily accessible to life scientists. The premise of this project is that polynomial dynamical systems over finite fields form a unified, mathematically rich class of dynamical systems that are increasingly used in systems biology. The goal of the project is to establish results that relate their structure to their dynamics. To obtain strong results it is necessary to focus on specific families of such systems. The choice for this project is the class of Boolean networks constructed from so-called nested canalyzing functions, and their multi-state generalizations. Many regulatory mechanisms in molecular biology can be described by such functions, and the resulting networks have good dynamic properties.