(599e) Optimization of Dynamic Flux Balance Analysis Systems

Jose A. Gomez^a and Paul I. Barton^a

^a Process Systems Engineering Laboratory, Massachusetts Institute of Technology, Cambridge MA 02139, USA

Bioprocesses involving microbial communities have widespread applications in the pharmaceutical, food and biofuels industries. These complex bioprocesses can be modeled accurately using dynamic flux balance analysis (DFBA) [1], [2], [3], which combines genome-scale metabolic network reconstructions with dynamic process models. DFBA models result in dynamic systems with linear programs (LPs) embedded [4]. These LPs are embedded because their right-hand side depends on the dynamic states and the dynamic states depend on the solution vector of the LP. Newly available simulators [4], [5] have made the reliable and efficient implementation of DFBA possible. This modeling framework has opened a new set of possibilities with regards to the discovery of novel bioprocesses [6], [7], [8] and the optimal design and control of bioprocesses. The optimal design and control of bioprocesses requires optimization algorithms for DFBA systems.

Optimization algorithms that rely on transforming the DFBA system into a Mathematical Program with Complementarity Constraints [9] or possibly high-index differential-algebraic equations system [10] have been published in literature. However, numerical difficulties and shortcomings of these approaches have also been reported [11]. Here, we use the simulation framework reported in [11], [4], [5] that transforms the LP embedded into a lexicographic LP (LLP) to address complications associated with nonunique and infeasible LP solutions.

The objectives of a LLP in standard form as a function of its right-hand side are piecewise linear functions [12], and therefore, nonsmooth. This source of nonsmoothness can be propagated to the parametric dependence of the final states of the dynamic system. Therefore, there exist some parameter values for which the Jacobian of the dynamic system may not exist. Computing elements of Clarke’s generalized Jacobian for complex nonsmooth functions is challenging [13], but can be done efficiently for piecewise differentiable functions, such as LLPs parameterized by their right-hand side, with lexicographic-directional (LD) derivatives [14].

This paper presents an algorithm for the optimization of DFBA systems. This algorithm relies on LD-derivatives to compute sensitivity information. It then uses bundle methods to solve the resulting nonsmooth optimization problem. An example illustrating the application of this algorithm is presented.

Keywords: Linear programming, generalized Jacobian, lexicographic differentiation, LD-derivative, lexicographic optimization, nonsmooth sensitivities, nonsmooth equation solving, flux balance analysis, dynamic flux balance analysis.

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