Modeling metabolic networks including gene expression and uncertainties

Abstract

Constraint based methods, such as the Flux Balance Analysis, are widely used to model cellular growth processes without relying on extensive information on the regulatory features. The regulation is instead substituted by an optimization problem usually aiming at maximal biomass accumulation. A recent extension to these methods called the dynamic enzyme-cost Flux Balance Analysis (deFBA) is a fully dynamic modeling method allowing for the prediction of necessary enzyme levels under changing environmental conditions. However, this method was designed for deterministic settings in which all dynamics, parameters, etc. are exactly known. In this work, we present a theoretical framework extending the deFBA to handle uncertainties and provide a robust solution. We use the ideas from multi-stage nonlinear Model Predictive Control (MPC) and its feature to represent the evolution of uncertainties by an exponentially growing scenario tree. While this representation is able to construct a deterministic optimization problem in the presence of uncertainties, the computational cost also increases exponentially. We counter this by using a receding prediction horizon and reshape the standard deFBA to the short-time deFBA (sdeFBA). This leads us, along with further simplification of the scenario tree, to the robust deFBA (rdeFBA). This framework is capable of handling the uncertainties in the model itself as well as uncertainties experienced by the modeled system. We applied these algorithms to two case-studies: a minimal enzymatic nutrient uptake network, and the abstraction of the core metabolic process in bacteria.

Optimality in cellular storage via the Pontryagin Maximum Principle

“…The short time scale is relevant for optimization based models of cellular metabolism, which have previously been studied in Waldherr et al (2015) and Lindhorst et al (2016). The key criterion for optimality is which growth mode is instantaneously faster for biomass accumulation, represented by the value of b E k E for exponential growth and b M k M for linear growth.…”

“…Based on previous numerical analyses (Waldherr et al, 2015;Lindhorst et al, 2016), we hypothesize that optimal solutions will be composed by up to three different phases:…”