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268724 Parametric Sensitivities of Dynamic Systems with Linear Programs Embedded

Microorganisms are the main driving force in industrial processes such as biofuel production and pharmaceutical biotechnology. Hence, dynamic simulations of the microorganisms and their interactions with the bioreactor environment are important for design, optimization, and control of these processes. As a structured model for processes involving microorganisms, dynamic flux balance analysis (DFBA) combines the flux balance analysis model (Palsson, 2006) used in metabolic engineering with the dynamic mass balances of the reactor. The mathematical formulation is given by dynamic systems with linear programs (LPs) embedded. Sensitivity analysis for this class of systems is not only essential for simulation and optimization, but also of theoretical interest as they represent a class nonsmooth dynamic systems.

In this work, we develop first results on the nonsmooth parametric sensitivity analysis for dynamic systems with LPs embedded. Nonsmoothness is due to the nonsmooth dependence of the LP solution with respect to the problem data. In particular, the solution is a Lipschitz continuous function of the right-hand side of the equality constraints. Hence, the sensitivities are possibly set-valued, and this has to be accounted for during the integration and optimization.

The theoretical results are derived from parametric linear programming, sensitivity analysis of linear programs, and nonsmooth analysis. The dual problem of the embedded LP plays an important role since the dual solution set characterizes the subgradient of the LP solution map.

In the numerical implementation, the system is rewritten as a hybrid differential-algebraic equation (DAE) system such that the embedded LP only has to be solved at points along a trajectory where the optimal basis set, determined by the simplex algorithm, changes. A code, called DSL48LPR, has been developed which employs DAEPACK (Tolsma and Barton, 2000) component DSL48E, and is a numerical integrator for a dynamical system with LPs embedded, where the dynamic states determine the right-hand side of the equality constraints. The results are illustrated in a case study of ethanol production using genetically modified yeast and E. coli microorganisms.

Palsson, B., 2006. Systems Biology: Properties of Reconstructed Networks. Cambridge University Press, New York, NY.

Tolsma, J., Barton, P. I., 2000. DAEPACK: An open modeling environment for legacy models. Industrial & Engineering Chemistry Research 39 (6), 1826{1839.

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