data on dynamics of the model components, like for example time courses of metabolites, will contribute to minimization of parameter uncertainty: only parameter sets allowing for the successful simulation of the time-course will be approved. The parameter estimation of nonlinear dynamic modeling approaches can be classified as a nonlinear programming problem being subject to nonlinear differential-algebraic constraints . In general, this mathematical problem can be formulated as follows: (2) #STA-4783 order keyword# where Z represents the cost function to be minimized, yexp contains experimentally determined state variables (for example metabolite concentrations), ypred(p,t) is the model prediction of state variables depending on estimated parameters p and time t, and W(t) is the weighting matrix containing information about the level of importance of single state variables and determining their influence on the cost function. This optimization problem of minimization
of Z is subject to the differential/algebraic Inhibitors,research,lifescience,medical equality constraints describing the systems dynamics and additional requirements for system performance. Additionally, the estimation of model parameter p is subject to lower (plow) and upper (pup) bounds: (3) Due to nonlinearities Inhibitors,research,lifescience,medical in objective function and constraints, solving these optimization problems frequently means having to cope with multimodality, i.e., the potential existence of multiple local solutions [37,38]. This implies the application of algorithms, which are able to overcome local minima to ultimately yield the best solution, Inhibitors,research,lifescience,medical i.e., the global optimum. Gradient-based local optimization methods fail to reliably determine the global optimum in multimodal problems because of nonconvexity arising from the previously mentioned nonlinearities. A graphical representation of this problem
is shown in . In a simple example it was demonstrated that even with Inhibitors,research,lifescience,medical only two decision variables, e.g., unknown kinetic model parameters, multimodal surfaces may result from optimization problems, i.e., surfaces of the cost function with multiple peaks and valleys, which do not allow for the determination of one unique optimal solution by local optimization methods. Solving such multimodal problems is the goal of global optimization , which was discussed and reviewed in the only context of parameter estimation in biochemical pathways . One example for a global optimization method is the particle swarm pattern search method for bound constrained global optimization . This algorithm was shown to be highly competitive with other global optimization methods and is a demonstrative example of how possible nonconvexity of the objective function can be globally explored. The basic idea behind this approach is to construct a hybrid of a pattern search method and a particle swarm search .