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Emergence of Diauxie As an Optimal Growth Strategy Under Resource Allocation Constraints in Cellular Metabolism

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Emergence of Diauxie As an Optimal Growth Strategy Under Resource Allocation Constraints in Cellular Metabolism Emergence of diauxie as an optimal growth strategy under resource allocation constraints in cellular metabolism Pierre Salvya and Vassily Hatzimanikatisa,1 aLaboratory of Computational Systems Biotechnology, Ecole´ Polytechnique Fed´ erale´ de Lausanne, CH-1015 Lausanne, Switzerland Edited by Jens Nielsen, BioInnovation Institute, Copenhagen, Denmark, and approved January 15, 2021 (received for review July 23, 2020) Diauxie, or the sequential consumption of carbohydrates in bac- maximization assumptions, FBA models predict the simultane- teria such as Escherichia coli, has been hypothesized to be an ous consumption of two or more carbon sources to achieve evolutionary strategy which allows the organism to maximize its the maximum possible growth (3). However, this contradicts instantaneous specific growth—giving the bacterium a competi- Monod’s observation of distinct, sequential phases of carbon tive advantage. Currently, the computational techniques used in consumption and suggests that diauxie does not come from industrial biotechnology fall short of explaining the intracellular stoichiometric constraints. dynamics underlying diauxic behavior. In particular, the under- To account for diauxie beyond stoichiometric modeling, we standing of the proteome dynamics in diauxie can be improved. looked into other biological features. Because a cell has a We developed a robust iterative dynamic method based on physiological constraint on the total amount of enzymes it can expression- and thermodynamically enabled flux models to sim- house, which we will call a proteome allocation constraint, it is ulate the kinetic evolution of carbohydrate consumption and likely that the cell will preferentially distribute its limited cat- cellular growth. With minimal modeling assumptions, we cou- alytic capacity toward pathways that utilize the most efficient ple kinetic uptakes, gene expression, and metabolic networks, substrate/enzyme combination (2, 10, 11). Therefore, models at the genome scale, to produce dynamic simulations of cell cul- that account for proteome limitation in cells may be able to tures. The method successfully predicts the preferential uptake account for diauxie. Toward this end, the role of protein lim- of glucose over lactose in E. coli cultures grown on a mixture itation in diauxie was demonstrated by Beg et al. (12) with BIOPHYSICS AND COMPUTATIONAL BIOLOGY of carbohydrates, a manifestation of diauxie. The simulated cel- their formulation of flux balance analysis with molecular crowd- lular states also show the reprogramming in the content of the ing (FBAwMC). Their method correctly predicts the uptake proteome in response to fluctuations in the availability of car- order of five different carbon sources in a batch reactor, using bon sources, and it captures the associated time lag during the a proteome allocation constraint. In a push toward more global diauxie phenotype. Our models suggest that the diauxic behavior models, models of metabolism and expression (ME models) of cells is the result of the evolutionary objective of maximiza- (13, 14) include proteome allocation but also gene expression tion of the specific growth of the cell. We propose that genetic mechanisms, a modeling paradigm that is ideal for studying regulatory networks, such as the lac operon in E. coli, are the diauxie at the proteome level. ME models also fully describe SYSTEMS BIOLOGY biological implementation of a robust control system to ensure the requirements of enzyme synthesis, degradation, and dilu- optimal growth. tion effects, as well as messenger RNA (mRNA) and enzyme concentrations. diauxie j dynamic FBA j resource allocation j ME models j iterative Significance n his pioneering work on the growth of bacterial cultures, the IFrench biologist Jacques Monod (1) observed that the growth of Escherichia coli in a mixture of carbohydrates followed two dis- When several sugars are at its disposition, the bacterium Escherichia coli tinct exponential curves separated by a plateau—a phenomenon consumes them in a specific order—a behav- he called diauxie. Hypothesized to allow optimal growth of the ior called diauxie. We developed a framework combining culture (2), this cellular behavior corresponds to the sequential dynamic methods and models of metabolism and gene expres- consumption of sugars: one sugar is preferentially consumed, sion to show that diauxie and its associated lag in cell growth and the second is only consumed after depletion of the first. can be explained simply as an optimal behavior under con- Although current optimality-based computational models can straints on the protein amount and renewal in a cell. We predict diauxie, these lack a detailed description of the protein validate our model by reproducing experimental results and dynamics during the phenomenon (3). Diauxie is an evolved, successfully predict a diauxic behavior on a growth medium complex behavior, and its occurrence is controlled by the regula- containing two types of sugar, glucose and lactose. Finally, we lac tion network of the lac operon in E. coli (4, 5). The emergence claim that the regulation mechanism inducing diauxie (the of such a control mechanism is the product of evolutionary pres- operon) is a control system to implement growth optimality sure, and being able to fully elucidate its raison d’ˆetre in terms at the cellular level. of cell physiology is an important milestone to understand and Author contributions: P.S. and V.H. designed research; P.S. performed research; P.S. con- better engineer the intracellular dynamics of bacterial growth. tributed new reagents/analytic tools; P.S. and V.H. analyzed data; and P.S. and V.H. wrote There is thus a need for a formulation describing diauxie at the the paper.y proteome level. The authors declare no competing interest.y Genome-scale models of metabolism (GEMs) combine This article is a PNAS Direct Submission.y constraint-based modeling and optimization techniques to study This open access article is distributed under Creative Commons Attribution-NonCommercial- cell cultures (6–8). A key method for studying GEMs is flux NoDerivatives License 4.0 (CC BY-NC-ND).y balance analysis (FBA) (9), which formulates a linear optimiza- 1 To whom correspondence may be addressed. Email: vassily.hatzimanikatis@epfl.ch.y tion problem that employs stochiometric constraints through the This article contains supporting information online at https://www.pnas.org/lookup/suppl/ mass conservation of metabolites given their synthesis and degra- doi:10.1073/pnas.2013836118/-/DCSupplemental.y dation reactions. Under the typical steady-state and growth rate Published February 18, 2021. PNAS 2021 Vol. 118 No. 8 e2013836118 https://doi.org/10.1073/pnas.2013836118 j 1 of 11 Downloaded by guest on September 26, 2021 Since diauxie is also a time-dependent phenomenon, we chose dard mixed integer linear programming (MILP) solvers (19). We to complement ME models with a dynamic modeling approach. herein leverage ETFL for dynamic analysis, in a method called Dynamic flux balance analysis (dFBA) (15) is a generalization of dETFL. It includes a method based on Chebyshev centering to FBA for modeling cell cultures in time-dependent environments. robustly select a representative solution from the feasible space In its original static optimization approach formulation, the time at each time step. The representative solution captures phe- is discretized into time steps, and an FBA problem is solved at notypic and genotypic differences between cells precultured in each step. At each iteration, kinetic laws and the FBA solution different media. The ETFL method, on which dETFL is built, are used to update the boundary fluxes, extracellular concentra- does not need dedicated quadprecision solvers, unlike previous tions, and cell concentration, based on the amount of substrate ME model formulations (13, 14, 16, 17, 20). It also does not use consumed, by-products secreted, and biomass produced by the strict equality coupling between flux rates and enzyme concen- cells. We expected that the combination of a dFBA and ME tration, unlike the previous state-of-the-art ME model methods models would yield a formulation that can describe diauxie at (16, 17). This strict coupling was instrumental in improving the the proteome level. solving performance of these formulations at the cost of reduc- However, we identified three major challenges in the con- ing the predictive capacity of the methods, in particular with ception of dynamic ME models. First, while dFBA studies of respect to the prediction of the lag phase (discussion is in SI metabolic networks can be solved by common linear solvers, Appendix, Note S1). Instead, (d-)ETFL relies on a combination ME models are nonlinear by nature and significantly more com- of scaling methods and MILP formulation, which allows models plex. The species and reactions introduced and considerations to be solved efficiently. As a result, whole-proteome reconfigura- of the interactions between enzyme expression and metabolism tion during sugar consumption can be simulated with reasonable result in nonlinear problems that are often one to two orders solving times, which enabled the modeling of the lag phase in of magnitude bigger in terms of constraints and variables than diauxie. the corresponding linear (d-)FBA problem. The increase in com- Herein, we model the emergence and dynamics of diauxie aris- plexity is compounded when iteratively solving an optimization ing at the proteome level. We first propose a small conceptual problem. As
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