System theory applications in biology: From stochastic chemical kinetics to deterministic model invalidation

Ideas from System Theory lie behind many of the new powerful methods being developed in the burgeoning field of Systems Biology. In this paper, we show two examples of this: one in the area of stochastic chemical kinetics, and the other in biological model invalidation. Stochastic chemical kinetics has gained a lot of attention in the last few years. In order to capture certain important dynamics in the subcellular environment, it is necessary to model molecular interactions at the gene level as discrete stochastic events. The dynamics of such processes is typically described by probability distributions, which evolve according to the set of linear ordinary differential equations known as the chemical master equation (CME). Until recently, it has been believed that the CME could not be solved analytically except in the most trivial of problems, and the CME has been analyzed almost exclusively with Monte Carlo (MC) algorithms. However, concepts from linear systems theory have enabled the Finite State Projection (FSP) approach and have significantly enhanced our ability to solve the CME without resorting to MC simulations. In this paper we review the FSP approach as well as a variety of systems theory based modifications to the FSP algorithm that dramatically improve the computational efficiency of the algorithm and expand the class of solvable problems. Notions such as observability, controllability and minimal realizations enable large reductions in the order of models and increase efficiency with little to no loss in accuracy. Model reduction techniques based upon linear perturbation theory allow for the systematic projection of multiple time scale dynamics onto a slowly varying manifold. Our second example shows the application of systems ideas in the area of biological model invalidation. As a specific case study, we use a dynamic model of the bacterial heat-shock response to demonstrate the approach. Using recent sum-of-squares techniques we show that the heat-shock model, when stripped from a certain protein-protein interaction that implements a certain feedback loop, cannot account for the input-output data regardless of the parameter choice for the model. In essence, such a deficient model is invalidated. Such conclusions are essential for pointing out the likelihood of missing components or interactions, thereby guiding new biological experiments.