[Paper Review] Fast computation of gradient and sentitivity in 13C metabolic flux analysis instationary experiments using the adjoint method

Stéphane Mottelet|arXiv (Cornell University)|Jun 22, 2012

Microbial Metabolic Engineering and Bioproduction9 references3 citations

TL;DR

This paper presents an efficient adjoint method for computing gradients and sensitivities in 13C metabolic flux analysis (MFA) during non-stationary experiments. By leveraging the implicit trapezoidal rule for time discretization and solving adjoint equations backward in time, the method enables fast and accurate computation of gradients with respect to fluxes and pool sizes, significantly reducing computational cost compared to traditional sensitivity analysis approaches.

ABSTRACT

Metabolic flux analysis using 13C labeled substrates is an important tool for metabolic engineering. Although it has now been evolving for more than ten years, metabolic flux analysis has still not reached the limits of its application. First and foremost, there is only one reference software for the analysis and identification of metabolic fluxes using stationnary carbon labeling experiments, which is closed-source. Moreover, this software lacks connections with the new standards of systems biology community, for example the Systems Biology Markup Language, which allows to describe arbitrary large metabolic networks. The first part of this paper, after recalling all the mathematics involved in the mathematical problem of flux identification in the case of multiple experiments (state equations, regularized cost function, explicit computation of the gradient) concentrates on the problem of specific automatic generation of scripts (Matlab or Scilab) implementing the numerical resolution. To this purpose, we will describe the architecture of the software chain implementing the transformation from the XML file describing the metaboling network and the carbon transitions to the final collection of scripts computing, for example, the exact gradient of the regularized least-squares cost function and the output sensitivities. In the unstationnary case the adjoint state method is used to speed up computations.

Motivation & Objective

• To address the high computational cost of gradient and sensitivity computation in 13C metabolic flux analysis during non-stationary experiments.
• To develop a scalable and automated approach for computing gradients and sensitivities in isotopomer labeling systems.
• To enable efficient optimization in metabolic flux identification by reducing the computational burden of sensitivity analysis.
• To integrate the method into a workflow using SBML and XML-based code generation for model specification and implementation in Scilab.
• To ensure consistency between discrete state and discrete adjoint equations through proper time discretization, particularly using the implicit trapezoidal rule.

Proposed method

• Formulates the isotopomer labeling system as a cascade of linear ODEs with time-varying coefficients, where each cumomer weight level depends on lower-weighted cumomers.
• Applies the implicit trapezoidal rule for time discretization to ensure that the discrete adjoint equation is the adjoint of the discrete state equation, preserving numerical consistency.
• Derives the discrete adjoint equations by taking the Lagrangian of the cost function and state equations, leading to a backward-in-time system of equations for the adjoint variables.
• Computes the gradient of the cost function with respect to fluxes and pool sizes using the adjoint variables and the Jacobians of the right-hand side of the ODE system.
• Uses SBML as the model description format and applies XSL transformations to generate Scilab code for numerical solution and optimization.
• Implements the method in Scilab with XML-based GUI generation via XMLlab, enabling user-friendly input and optimization execution.

Experimental results

Research questions

• RQ1 How can gradients and sensitivities be computed efficiently in 13C metabolic flux analysis during non-stationary experiments?
• RQ2 What is the most effective way to discretize the adjoint system so that it remains the adjoint of the discrete state system?
• RQ3 Can the adjoint method reduce computational cost compared to direct sensitivity analysis in isotopomer labeling experiments?
• RQ4 How can the method be integrated into a reusable, automated pipeline using standard formats like SBML and XML?
• RQ5 What is the impact of using high-order integration schemes on the accuracy and consistency of the discrete adjoint method?

Key findings

• The adjoint method enables gradient computation with a computational cost comparable to solving one additional ODE, making it significantly more efficient than forward sensitivity analysis.
• The use of the implicit trapezoidal rule ensures that the discrete adjoint is consistent with the discrete state equation, preserving accuracy in stiff systems.
• The method achieves accurate gradient computation for both fluxes and pool sizes, with the gradient of the cost function expressed as a sum over time steps involving adjoint variables and Jacobians.
• The implementation using SBML and XSL-based code generation allows automatic, reusable, and efficient code generation for any given metabolic network.
• The approach supports both stationary and non-stationary cases, with the non-stationary case requiring solution of the full state and adjoint systems over time.
• The method is validated on a small branching network, demonstrating feasibility and scalability for larger networks through automated code generation.