Parameter estimation in rational models of molecular biological systems.

Based on statistical thermodynamics or Michaelis -Menten kinetics, molecular biological systems can be modeled by a system of nonlinear differential equations. The nonlinearity in the model stems from rational reaction rates whose numerator and denominator are linear in parameters. It is a nonlinear problem to estimate the parameters in such rational models of molecular biological systems. In principle, any nonlinear optimization methods such as Newton-Gauss method and its variants can be used to estimate parameters in the rational models. However, these methods may converge to a local minimum and be sensitive to the initial values. In this study, we propose a new method to estimate the parameters in the rational models of molecular biological systems. In the proposed method, the cost function in all parameters is first reduced to a cost function only in the parameters in the denominator by a separable theorem. Then the parameters in the denominator are estimated by minimizing this cost function using our proposed new iteration method. Finally, the parameters in the numerator are estimated by a well defined linear least squares formula. A simple gene regulatory system is used as an example to illustrate the performance of the proposed method. Simulation results show that the proposed method performs better than the general nonlinear optimization methods in terms of the running time, robustness (insensitivity) to the initial values, and the accuracy of estimates.