# Algebraic differentiation for fast sensitivity analysis of optimal flux modes in metabolic models

## TL;DR

This paper introduces a new method for analyzing how changes in parameters affect optimal metabolic pathways, using algebraic differentiation to improve sensitivity analysis in metabolic models.

## Contribution

The paper introduces optimal flux modes (OFMs) and proves that algebraic differentiation can efficiently calculate their sensitivities, enabling detailed sensitivity analysis of optimal metabolic solutions.

## Key findings

- Optimal flux modes (OFMs) allow for fine-grained sensitivity analysis of optimal solutions in metabolic models.
- DifferentiableMetabolism.jl enables efficient differentiation of constraint-based model solutions.
- OFM sensitivities predict the impact of genetic knockouts on waste product accumulation in E. coli.

## Abstract

Sensitivity analysis is a useful tool to identify key parameters in metabolic models. It is typically only applied to the growth rate, disregarding the sensitivity of other solution variables to parameters. Further, sensitivity analysis of elementary flux modes could provide low-dimensional insights into optimal solutions, but they are not defined when a model is subject to inhomogeneous flux constraints, such as the frequently used ATP maintenance reaction.

We introduce optimal flux modes (OFMs), an analogue to elementary flux modes (EFMs), but specifically applied to optimal solutions of constraint-based models. Further, we prove that implicit differentiation can always be used to efficiently calculate the sensitivities of both whole-model solutions and OFM-based solutions to model parameters. This allows for fine-grained sensitivity analysis of the optimal solution, and investigation of how these parameters exert control on the optimal composition of OFMs. This novel framework is implemented in DifferentiableMetabolism.jl, a software package designed to efficiently differentiate solutions of constraint-based models. To demonstrate scalability, we differentiate solutions of 342 yeast models; additionally we show that sensitivities of specific subsystems can guide metabolic engineering. Applying our scheme to an Escherichia coli model, we find that OFM sensitivities predict the effect of knockout experiments on waste product accumulation. Sensitivity analysis of OFMs also provides key insights into metabolic changes resulting from parameter perturbations.

Software introduced here is available as open-source Julia packages DifferentiableMetabolism.jl (https://github.com/stelmo/DifferentiableMetabolism.jl) and ElementaryFluxModes.jl (https://github.com/HettieC/ElementaryFluxModes.jl), which both work on all major operating systems and computer architectures. Code to reproduce all results is available from https://github.com/HettieC/DifferentiableOFMPaper, and as an archive from https://doi.org/10.5281/zenodo.15183208.