(599e) Optimization of Dynamic Flux Balance Analysis Systems

Gomez, J. A. - Presenter, Massachusetts Institute of Technology
Barton, P. I., Massachusetts Institute of Technology
Optimization of Dynamic Flux Balance Analysis Systems

Jose A. Gomeza and Paul I. Bartona

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.



A. Varma and B. Ø. Palsson, "Stoichiometric flux balance models quantitatively predict growth and metabolic by-product secretion in wild-type Escherichia coli W3110," Applied and Environmental Microbiology, vol. 60, no. 10, pp. 3724-3731, 1994.


J. D. Orth, I. Thiele and B. Ø. Palsson, "What is flux balance analysis?," Nature Biotechnology, vol. 28, pp. 245-248, 2010.


R. Mahadevan, J. Edwards and F. I. Doyle, "Dynamic flux balance analysis of diauxic growth in Escherichia coli.," Biophysical Journal, vol. 83, no. 3, pp. 1331-40, 2002.


K. Höffner, S. M. Harwood and P. I. Barton, "A reliable simulator for dynamic flux balance analysis," Biotechnology and Bioengineering, vol. 110, no. 3, pp. 792-802, 2013.


J. A. Gomez, K. Höffner and P. I. Barton, "DFBAlab: A fast and reliable MATLAB code for Dynamic Flux Balance Analysis," BMC Bioinformatics, vol. 15, p. 409, 2014.


J. A. Gomez, K. Höffner and P. I. Barton, "From sugars to biodiesel using microalgae and yeast," Green Chemistry, vol. 18, no. 2, pp. 461-475, 2016.


J. Chen, J. Gomez, K. Höffner, P. Barton and M. Henson, "Metabolic modeling of synthesis gas fermentation in bubble column reactors," Biotechnology for Biofuels, vol. 8, no. 89, 2015.


J. Chen, J. Gomez, K. Höffner, P. Phalak, P. Barton and M. Henson, "Spatiotemporal modeling of microbial metabolism," BMC Systems Biology, vol. 10, no. 21, 2016.


A. U. Raghunathan, J. R. Pérez-Correa, E. Agosin and L. T. Biegler, "Parameter estimation in metabolic flux balance models for batch fermentation - formulation and solution using differential variational inequalities," Ann Oper Res, vol. 148, pp. 251-270, 2006.


U. Kaplan, M. Türkay, L. Biegler and B. Karasözen, "Modeling and simulation of metabolic networks for estimation of biomass accumulation parameters," Discrete Applied Mathematics, vol. 157, no. 10, pp. 2483-2493, 2009.


S. Harwood, K. Höffner and P. Barton, "Efficient solution of ordinary differential equations with a parametric lexicographic linear program embedded," Numerische Mathematik, vol. 133, no. 4, pp. 623-653, 2016.


D. Bertsimas and J. N. Tsitsiklis, Introduction to Linear Optimization, Nashua, NH: Athena Scientific, 1997.


F. H. Clarke, Optimization and Nonsmooth Analysis, Philadelphia: Society for Industrial and Applied Mathematics, 1990.


K. A. Khan and P. Barton, "A vector forward mode of automatic differentiation for generalized derivative evaluation," Optimization Methods & Software, vol. 30, no. 6, pp. 1185-1212, 2015.


K. A. Khan and P. I. Barton, "Generalized Derivatives for Solutions of Parametric Ordinary Differential Equations with Non-differentiable Right-Hand Sides," Journal of Optimization Theory and Applications, vol. 163, no. 2, pp. 355-386, 2014.