Here we address the problem of parameter estimation (inverse problem) of nonlinear dynamic biochemical pathways. This problem is stated as a nonlinear programming (NLP) problem subject to nonlinear differential-algebraic constraints. These problems are known to be frequently ill-conditioned and multimodal. Thus, traditional (gradient-based) local optimization methods fail to arrive at satisfactory solutions. To surmount this limitation, the use of several state-of-the-art deterministic and stochastic global optimization methods is explored. A case study considering the estimation of 36 parameters of a nonlinear biochemical dynamic model is taken as a benchmark. Only a certain type of stochastic algorithm, evolution strategies (ES), is able to solve this problem successfully. Although these stochastic methods cannot guarantee global optimality with certainty, their robustness, plus the fact that in inverse problems they have a known lower bound for the cost function, make them the best available candidates.Mathematical optimization can be used as a computational engine to arrive at the best solution for a given problem in a systematic and efficient way. In the context of biochemical systems, coupling optimization with suitable simulation modules opens a whole new avenue of possibilities. Mendes and Kell (1998) highlight two types of important applications:1. Design problems: How to rationally design improved metabolic pathways to maximize the flux of interesting products and minimize the production of undesired by-products (metabolic engineering and biochemical evolution studies); 2. Parameter estimation: Given a set of experimental data, calibrate the model so as to reproduce the experimental results in the best possible way.This contribution considers the latter case, that is, the socalled inverse problem. The correct solution of inverse problems plays a key role in the development of dynamic models, which, in turn, can promote functional understanding at the systems level, as shown by, for example, Swameye et al. (2003) and Cho et al. (2003) for signaling pathways.The paper is structured as follows: In the next section, we state the mathematical problem, highlighting its main characteristics, and very especially its challenging nature for traditional local optimization methods. Next, global optimization (GO) is presented as an alternative to surmount those difficulties. A brief review of GO methods is given, and a selection of the presently most promising alternatives is presented. The following section outlines a case study considering the estimation of 36 parameters of a three-step pathway, which will be used as a benchmark to compare the different GO methods selected. A Results and Discussion section follows, ending with a set of Conclusions.
METHODS
Statement of the Inverse ProblemParameter estimation problems of nonlinear dynamic systems are stated as minimizing a cost function that measures the goodness of the fit of the model with respect to a given experimental data set, subject to the dynamics of the ...
