Installations

All simulations were performed with Julia version 1.11.5 (https://julialang.org/). You can install the latest Julia version and then, install the required packages by coping the contents of the file Project.toml to your file Project.toml. Then, in Julia REPL, enter the package management mode by typing the key "]" and run the command "instantiate". This operation will resolve a set of feasible packages from the copied file that are compatible with the target Julia version, and will install or update them if suitable.

The package IPOPT (https://coin-or.github.io/Ipopt/index.html) is used for solving large-scale nonlinear optimization problems. IPOPT supports a number of linear solvers to improve performance. You need to obtain an HSL (free academic) license from https://licences.stfc.ac.uk/products/Software/HSL/LibHSL (https://github.com/jump-dev/Ipopt.jl?tab=readme-ov-file).

We used Julia in VS Code (https://www.julia-vscode.org/docs/dev/).

Files

FBA&pFBA_Reformulations.ipynb simulates the aerobic growth of Escherichia coli metabolic models from BiGG database (http://bigg.ucsd.edu/) on glucose with Flux Balance Analysis (FBA) and parsimonious Flux Balance Analysis (pFBA). FBA and pFBA are reformulated using duality theory and Karush-Kuhn-Tucker optimality conditions.

Bioreactors-ODEs.ipynb simulates the aerobic growth of Escherichia coli on glucose in three common types of bioreactors (batch, fed-batch, and continuous stirred tank reactor) with a system of Ordinary Differential Equations (ODEs) solved by Julia ODE solver RadauIIA3() and orthogonal collocation on fixed and moving finite elements at Legendre-Gauss–Radau collocation points.

Bioreactors-dFBA.ipynb simulates the aerobic growth of Escherichia coli metabolic models from BiGG database (http://bigg.ucsd.edu/) on glucose in three common types of bioreactors (batch, fed-batch, and continuous stirred tank reactor) with dynamic Flux Balance Analysis (dFBA). DFBA is solved by a sequential approach using FBA solver and Julia ODE solver Tsit5(), and simultaneous approaches using selected FBA and pFBA reformulations and orthogonal collocation on moving finite elements at Legendre-Gauss–Radau collocation points.