(533f) Discovering Cellular Objectives through Differentiable Optimization

From microbial to mammalian cells, metabolic models have been widely used in synthetic and systems biology. They are a useful tool to understand the main metabolic pathways of cells, the way energy is allocated and mechanisms that enable protein secretion. Despite their utility, the challenge of measuring intracellular fluxes poses obstacles to their use; the model becomes an underdetermined system of equations that has many potential solutions. One approach to identify a solution is to utilize optimization methods, such as Flux Balance Analysis (FBA) [1]. Typical objectives include maximization of growth, maximization of protein production or minimization of energy. Although growth maximization yields satisfactory results for microbial cells, for mammalian cells this objective usually falls short of accurately predicting net experimental growth rate. As a result, alternative methods that do not assume an objective have gained popularity, such as flux sampling, whereby the metabolic model solutions are sampled to understand their distribution [2]. Although useful, flux distributions may be challenging to interpret and use in further metabolism studies and experimental activities.

Given the cellular objective is not known, there have been many efforts towards performing inverse FBA (iFBA). Here, one becomes interested in finding the most likely cellular objective, given a set of measured fluxes, via bilevel optimization [3]. The upper level identifies a parameterization of the objective function to recover the experimental data, while the lower level is provided by FBA with an objective defined by the upper level. Classic iFBA assumes the FBA lower level is a linear program with a linear objective function. However, often the recovered cell objective does not replicate the observed flux measurements well. Additionally, the existing approaches are challenged by the use of multiple sets of partial flux measurements, and typically identify an objective for each dataset as a result. This is because the number of constraints in the mathematical program grows with the number of datapoints, resulting in increased computational times for solution [4].

We propose a methodology to abstract convex nonlinear objectives for cell metabolism given experimental data. Our framework utilizes Karush-Kuhn-Tucker (KKT)-based sensitivities from the lower-level FBA problem and backpropagation to solve the upper-level problem using first order gradient information. The approach is agnostic to the lower-level solver, which allows for biologically inspired convex formulations and solution methods. Further, the methodology is amenable to the use of batch Quasi-Newton and stochastic gradient updates, which promises to mitigate the computational effects of many sets of flux measurements in the iFBA procedure.

We demonstrate the applicability of our framework in two case studies: firstly, we demonstrate the methodology in a case study previously presented in [5]; and secondly, we use an experimental dataset of Chinese Hamster Ovary (CHO) cells fluxes in different cell culture phases. The method is benchmarked against existing iFBA approaches ([3] and [6]).

In summary, we propose a methodology to learn a convex parametric nonlinear objective function for a cell, through differentiable convex optimization. This framework is amenable to use of many partial and noisy flux measurements, which is often what is available in practice. We believe the framework is suitable for wider use of metabolic models, particularly in mammalian cell systems where FBA is often unable to well describe observed cell behaviour.

References:

[1] Orth, J., Thiele, I. & Palsson, B. (2010). What is flux balance analysis?. Nature Biotechnology 28, 245–248. https://doi.org/10.1038/nbt.1614

[2] Herrmann, H.A., Dyson, B.C., Vass, L. et al. (2019). Flux sampling is a powerful tool to study metabolism under changing environmental conditions. npj Systems Biology and Applications, 5, 32 https://doi.org/10.1038/s41540-019-0109-0

[3] Zhao, Q., Stettner, A.I., Reznik, E. et al. (2016). Mapping the landscape of metabolic goals of a cell. Genome Biology 17, 109 https://doi.org/10.1186/s13059-016-0968-2

[4] Zhao, Q., Stettner, A., Reznik, E., Segrè, D., & Paschalidis, I. C. (2015). Learning cellular objectives from fluxes by inverse optimization. In 2015 54th IEEE Conference on Decision and Control (CDC) (pp. 1271-1276). IEEE.

[5] Smallbone, K., & Simeonidis, E. (2009). Flux balance analysis: a geometric perspective. Journal of theoretical biology, 258(2), 311–315. https://doi.org/10.1016/j.jtbi.2009.01.027

[6] Burgard, A. P., & Maranas, C. D. (2003). Optimization‐based framework for inferring and testing hypothesized metabolic objective functions. Biotechnology and bioengineering, 82(6), 670-677.