Using optimal control to understand complex metabolic pathways

Abstract

Background: Optimality principles have been used to explain the structure and behavior of living matter at different levels of organization, from basic phenomena at the molecular level, up to complex dynamics in whole populations. Most of these studies have assumed a single-criteria approach. Such optimality principles have been justified from an evolutionary perspective. In the context of the cell, previous studies have shown how dynamics of gene expression in small metabolic models can be explained assuming that cells have developed optimal adaptation strategies. Most of these works have considered rather simplified representations, such as small linear pathways, or reduced networks with a single branching point, and a single objective for the optimality criteria.

Results: Here we consider the extension of this approach to more realistic scenarios, i.e. biochemical pathways of arbitrary size and structure. We first show that exploiting optimality principles for these networks poses great challenges due to the complexity of the associated optimal control problems. Second, in order to surmount such challenges, we present a computational framework which has been designed with scalability and efficiency in mind, including mechanisms to avoid the most common pitfalls. Third, we illustrate its performance with several case studies considering the central carbon metabolism of S. cerevisiae and B. subtilis. In particular, we consider metabolic dynamics during nutrient shift experiments.

Conclusions: We show how multi-objective optimal control can be used to predict temporal profiles of enzyme activation and metabolite concentrations in complex metabolic pathways. Further, we also show how to consider general cost/benefit trade-offs. In this study we have considered metabolic pathways, but this computational framework can also be applied to analyze the dynamics of other complex pathways, such as signal transduction or gene regulatory networks.

Keywords: Dynamic modeling; Multi-criteria optimization; Optimal control; Pareto optimality.

Fig. 1

Classification of solution strategies for nonlinear optimal control problems. Figure adapted from [146]

Fig. 2

General workflow of our approach

Fig. 3

Network representation for the LPN3B case study

Fig. 4

Case study LPN3B: Pareto front computed with three different approaches

Fig. 5

Case study LPN3B: Optimal controls for different points (A, B and C) on the Pareto front. The solutions presented here were computed with a multistart of ICLOCS (msICLOCS)

Fig. 6

Network representation of the SC case study

Fig. 7

Case study SC: Pareto front computed with the three different approaches. Point A and C correspond to the extreme points of the three approaches while B and K are in the vicinity of the knee point

Fig. 8

Case study SC: comparison of the solution histograms corresponding to the strategies AMIGO2_DO+ICLOCS (on the left) and msICLOCS (on the right), for point K on the Pareto front shown in Fig. 7

Fig. 9

Case study SC: an illustration of the impact of the ATP critical value path constraint on the Pareto front previously shown in Fig. 7 is presented in subfigure I. Subfigure II shows the impact of this path constraint on the optimal state trajectories corresponding to the extreme point of maximum $t_f$

Fig. 10

Network representation of case study BSUB

Fig. 11

Case study BSUB: subfigure I shows the Pareto front obtained with the three different strategies. Subfigure II compares the corresponding optimal state trajectories for point B of the Pareto (subfigure I)