(444b) Fast Uncertainty Quantification in Dynamic Flux Balance Analysis Models Using Sparse Multi-Element Polynomial Chaos

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]. Dynamic flux balance analysis (DFBA) is one such modeling approach that combines genome-scale metabolic network reconstructions with dynamic process models. Complex bioprocesses involving microbial communities, including various unit operations in the pharmaceutical, food, and biofuel industries, can be accurately modeled using DFBA [2,3]. However, the predictions of these types of biological models are typically subject to various sources of uncertainty including unknown model parameters, unknown model structure, and experimental uncertainty such as measurement errors. The accurate characterization of these uncertainties as well as their impact on the model predictions are crucial for the application of these models in decision-support or optimization tasks. This has motivated the development of a broad range of tools for uncertainty quantification (UQ) that involves both propagating uncertainties onto the model outputs and calibration of the model with experimental data. However, the majority of UQ methods quickly become intractable for large-scale computationally expensive models, which has severely limited their application in DFBA [4].

In this work, we develop a novel surrogate modeling method that enables UQ approaches to be executed significantly faster on genome-scale DFBA models [5]. We focus on polynomial chaos expansions (PCEs) as it has been demonstrated to perform well on a broad range of engineering applications using limited computational resources [6-9]. An important assumption in PCE is that the model response depends smoothly on the model parameters so that any particular quantity of interest can be accurately represented by a collection of polynomial functions. Since DFBA models result in dynamic systems with linear programs (LPs) embedded [10], they are inherently non-smooth models, meaning that standard PCE methods may converge very slowly or even fail to converge altogether in these problems [11]. The main contribution of this work is the development of a multi-element version of PCE that systematically decomposes the parameter space into a collection of non-overlapping regions on which the model response behaves smoothly. In this way, separate PCEs can be fit in each of the elements such that the overall DFBA model response is approximated by a piecewise polynomial function. Additionally, we present a procedure for selecting these elements that takes advantage of the fact that singularities can be detected during integration of the DFBA model. To ensure the approach is applicable to large-scale problems with many parameters, we estimate the coefficients of the PCEs using tailored sparse regression methods that are capable of locating the terms in the expansion that have the largest impact on the model response. Since the proposed method directly accounts for both sparsity and discontinuities, the resulting surrogate model is very fast to evaluate.

We demonstrate the effectiveness of the proposed sparse multi-element PCE framework by applying it for acceleration of Bayesian inference of parameters in the substrate uptake kinetic expressions of diauxic growth of a batch monoculture of Escherichia coli on a glucose/xylose mixed media. The metabolic network reconstruction used for E. coli is iJ904, which contains 1075 reactions and 761 metabolites [12]. The six uncertain parameters are estimated from measurements of the extracellular metabolite and biomass concentrations taken throughout the batch. The posterior parameter distribution is estimated using a sequential Monte Carlo approach. We demonstrate a 200-fold savings in computational cost when using the proposed surrogate model as opposed to the original DFBA model, while observing a negligible loss in accuracy. We also discuss how the resulting posterior provides significant physical insights when compared to trial-and-error procedures used to estimate the kinetic parameters in previous work [13].

References

[1] M. W. Covert et al. “Integrating high-throughput and computational data elucidates bacterial networks,” Nature, vol. 429, pp. 92–96, 2004.

[2] J. L. Hjersted and M. A. Henson, “Optimization of fed-batch Saccharomyces cerevisiae fermentation using dynamic flux balance models,” Biotechnology Progress, vol. 22, pp. 1239–1248, 2006.

[3] K. P. Lisha and D. Sarkar, “Dynamic flux balance analysis of batch fermentation: Effect of genetic manipulations on ethanol production,” Bioprocess and Biosystems Engineering, vol. 37, pp. 617–627, 2014.

[4] F. Scott, P. Wilson, R. Conejeros, and V. S. Vassiliadis, “Simulation and optimization of dynamic flux balance analysis models using an interior point method reformulation,” Computers & Chemical Engineering, vol. 119, pp. 152–170, 2018.

[5] J. A. Paulson, M. Martin-Casas, and A. Mesbah, “Black-box surrogate modeling of biological systems with non-smooth behavior: Applications in Bayesian parameter inference for dynamic flux balance analysis models,” PLOS Computational Biology, 2019 (under review).

[6] D. Xiu, “Efficient collocational approach for parametric uncertainty analysis,” Communications in Computational Physics, vol. 2, pp. 293–309, 2007.

[7] G. Blatman and B. Sudret, “Adaptive sparse polynomial chaos expansion based on least angle regression,” Journal of Computational Physics, vol. 230, pp. 2345–2367, 2011.

[8] J. A. Paulson and A. Mesbah, “Nonlinear Model Predictive Control with Explicit Backoffs for Stochastic Systems under Arbitrary Uncertainty,” IFAC-PapersOnLine, vol. 51, pp. 523–534, 2018.

[9] J. A. Paulson and A. Mesbah, “An efficient method for stochastic optimal control with joint chance constraints for nonlinear systems,” International Journal of Robust and Nonlinear Control, 1-21, 2017.

[10] 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, pp. 409, 2018.

[11] X. Wan and G. E. Karniadakis, “Multi-element generalized polynomial chaos for arbitrary probability measures,” SIAM Journal on Scientific Computing, vol. 28, pp. 901–928, 2006.

[12] J. L. Reed, T. D. Vo, C. H. Schilling, and B. O. Palsson, “An expanded genome-scale model of Escherichia coli K-12 (i JR904 GSM/GPR),” Genome Biology, vol. 4, pp. R54, 2003.

[13] T. J. Hanly and M. A. Henson, “Dynamic flux balance modeling of microbial co-cultures for efficient batch fermentation of glucose and xylose mixtures,” Biotechnology and Bioengineering, vol. 108, pp. 376–385, 2011.