Abstract
We present a novel surrogate modeling method that can be used to accelerate the solution of uncertainty quantification (UQ) problems arising in nonlinear and non-smooth models of biological systems. In particular, we focus on dynamic flux balance analysis (DFBA) models that couple intracellular fluxes, found from the solution of a constrained metabolic network model of the cellular metabolism, to the time-varying nature of the extracellular substrate and product concentrations. DFBA models are generally computationally expensive and present unique challenges to UQ, as they entail dynamic simulations with discrete events that correspond to switches in the active set of the solution of the constrained intracellular model. The proposed non-smooth polynomial chaos expansion (nsPCE) method is an extension of traditional PCE that can effectively capture singularities in the DFBA model response due to the occurrence of these discrete events. The key idea in nsPCE is to use a model of the singularity time to partition the parameter space into two elements on which the model response behaves smoothly. Separate PCE models are then fit in both elements using a basis-adaptive sparse regression approach that is known to scale well with respect to the number of uncertain parameters. We demonstrate the effectiveness of nsPCE on a DFBA model of an E. coli monoculture that consists of 1075 reactions and 761 metabolites. We first illustrate how traditional PCE is unable to handle problems of this level of complexity. We demonstrate that over 800-fold savings in computational cost of uncertainty propagation and Bayesian estimation of parameters in the substrate uptake kinetics can be achieved by using the nsPCE surrogates in place of the full DFBA model simulations. We then investigate the scalability of the nsPCE method by utilizing it for global sensitivity analysis and maximum a posteriori estimation in a synthetic metabolic network problem with a larger number of parameters related to both intracellular and extracellular quantities.
Highlights
The utility of mathematical modeling in biology is on the rise due to computational advancements and the increasing availability of data provided by high-throughput experimental techniques [1]
The method is applied to infer extracellular kinetic parameters in a batch fermentation reactor consisting of diauxic growth of E. coli on a glucose/xylose mixed media as well as a larger synthetic metabolic network problem
We propose an extension to polynomial chaos expansions (PCEs), referred to as non-smooth PCE, that can adequately capture the non-smooth behavior exhibited by dynamic flux balance analysis (DFBA) models
Summary
The utility of mathematical modeling in biology is on the rise due to computational advancements and the increasing availability of data provided by high-throughput experimental techniques [1]. Given a constrained metabolic network, FBA assumes the intracellular fluxes are regulated by the cell to optimize a predefined cellular objective function (e.g., maximizing the biomass growth rate [4]) subject to mass balances of the intracellular metabolites and other feasibility constraints (e.g., bounds on the substrate uptake and product secretion rates). In DFBA models, the intracellular fluxes are given by the solution of a FBA model, which is coupled to a set of dynamic equations that describes the time-varying nature of the extracellular substrate and product concentrations as a function of the extracellular environment [10]. The key assumption in DFBA is that the intracellular fluxes equilibrate instantaneously. This “quasi steady-state” assumption is valid as long as the intracellular dynamics are significantly faster than the extracellular dynamics
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