ETFL: A formulation for flux balance models accounting for expression, thermodynamics, and resource allocation constraints

Since the introduction of metabolic models and flux balance analysis (FBA) in systems biology, several attempts have been made to supplement these formulations with expression information to form metabolic and expression models (ME-models). However, directly accounting for enzyme and mRNA production in the mathematical programming formulation of the problem is challenging because of the dilution of the macromolecules, which introduces a bilinear term in the mass-balance equations, making the problem non-linear and harder to solve than linear formulations like FBA. Furthermore, no attempts have been made at including thermodynamic constraints in these formulations, which would yield an even more complex mixed-integer non-linear problem. We propose here a new framework, called Expression and Thermodynamics Flux (ETFL), as a new ME-model implementation. ETFL is a top-down model formulation, from metabolism to RNA synthesis, that simulates thermodynamic-compliant intracellular fluxes as well as enzyme and mRNA concentration levels. The formulation results in a mixed-integer linear problem (MILP) that enables both relative and absolute metabolite, protein, and mRNA concentration integration. The proposed formulation is compatible with mainstream MILP solvers and does not require a non-linear solver. It also leverages the MILP formulation to account for growth-dependent parameters, such as relative protein or mRNA content. We present here the formulation of ETFL along with its validation using results obtained from a well-characterized E. coli model. We show that ETFL is able to reproduce proteome-limited growth, which FBA cannot. We also subject it to different analyses, including the prediction of feasible mRNA and enzyme concentrations in the cell, and propose ETFL-based adaptations of other common FBA-based procedures. The software is available on our public repository at https://github.com/EPFL-LCSB/etfl. Author summary Metabolic modeling is a useful tool for biochemists who want to tweak biological networks for the direct expression of key products, such as biofuels, specialty chemicals, or drug candidates. To provide more accurate models, several attempts have been made to account for protein expression and growth-dependent parameters, key components of biological networks, though this is computationally challenging, especially when also attempting to include thermodynamics. To the best of our knowledge, there is no published methods integrating these three types of constraints in one model. We propose here a transparent mathematical formulation to model both expression and metabolism of a cell, along with a reformulation that allows a computationally tractable inclusion of growth-dependent parameters and thermodynamics. We demonstrate good performance using community-standard software, and propose ways to adapt classical modeling studies to expression-enabled models. The incorporation of thermodynamics and growth-dependent variables provide a finer modeling of expression because they eliminate thermodynamically unfeasible solutions and consider phenotypic differences in different growth regimens, which are key for accurate modeling.