| In silico models of signal transduction pathways have been highly successful in describing, quantitatively, how complex protein networks govern overall cell function. Understanding these signaling pathways helps us not only in understanding biology at its roots, but provides insight into how we can constructively manipulate biological functions, i.e., the developement of treatments of human diseases. However, the complexity of these signaling pathways or networks, characterized by feedback loops, cross-talk, redundancy, hinders the generation of new knowledge, strategies and breakthroughs for the regulation of cellular machinery. Sensitivity analysis, as one of the most effective approaches for studying mathematical models of biochemical systems, has the ability to identify dominant parameters, simplify models and answer "what if" questions. In this study, a stiff Rosenbrock integrator has been developed for sensitivity analysis using a direct sensitivity approach. Automated sparse Jacobian and Hessian calculations of the coupled system (the original model equations and the sensitivity equations) have been implemented in the freely available software package CellSim. The accuracy and efficiency of this newly developed R/DM method (Rosenbrock with direct method) are tested extensively on the complex MAPK (mitogen-activated protein kinase) pathway model of Bhalla et al. Both time-dependent concentration and parameter based sensitivity coefficients are measured using several integration schemes. The method is shown to perform sensitivity analysis in a manner that is both cost effective and accurate. It is several magnitudes faster than traditional integrators, such as adaptive Runge-Kutta, etc. The error control strategies between the DDM (decoupled direct method) and the R/DM are discussed and their computational accuracies are compared. The method is used to analyze the positive feedback loop within the MAPK signal transduction pathway.; As systems biology models move from purely kinetic to spatio-temporal models, important analysis approaches such as sensitivity analysis must be appropriately expanded to fit this change. We have developed a fast integrator for the sensitivity analysis of spatio-temporal reaction-diffusion PDE systems. The method is an extension of the previously developed Rosenbrock integration method aimed for pure reaction systems. The expanded spatio-temporal sensitivity analysis method is successfully applied to the canonical Gray-Scott reaction-diffusion system. The mixture of this new integrator and the simulation together provide an efficient way to analyze the localization of a nonlinear system response at different times and locations as well as the pattern transitions between adjacent patterns. |
