The Protein Cost of Metabolic Fluxes: Prediction from Enzymatic Rate Laws and Cost Minimization

Abstract

Bacterial growth depends crucially on metabolic fluxes, which are limited by the cell’s capacity to maintain metabolic enzymes. The necessary enzyme amount per unit flux is a major determinant of metabolic strategies both in evolution and bioengineering. It depends on enzyme parameters (such as kcat and KM constants), but also on metabolite concentrations. Moreover, similar amounts of different enzymes might incur different costs for the cell, depending on enzyme-specific properties such as protein size and half-life. Here, we developed enzyme cost minimization (ECM), a scalable method for computing enzyme amounts that support a given metabolic flux at a minimal protein cost. The complex interplay of enzyme and metabolite concentrations, e.g. through thermodynamic driving forces and enzyme saturation, would make it hard to solve this optimization problem directly. By treating enzyme cost as a function of metabolite levels, we formulated ECM as a numerically tractable, convex optimization problem. Its tiered approach allows for building models at different levels of detail, depending on the amount of available data. Validating our method with measured metabolite and protein levels in E. coli central metabolism, we found typical prediction fold errors of 4.1 and 2.6, respectively, for the two kinds of data. This result from the cost-optimized metabolic state is significantly better than randomly sampled metabolite profiles, supporting the hypothesis that enzyme cost is important for the fitness of E. coli. ECM can be used to predict enzyme levels and protein cost in natural and engineered pathways, and could be a valuable computational tool to assist metabolic engineering projects. Furthermore, it establishes a direct connection between protein cost and thermodynamics, and provides a physically plausible and computationally tractable way to include enzyme kinetics into constraint-based metabolic models, where kinetics have usually been ignored or oversimplified.

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“…This results in a non-convex optimization problem which is handled by approximation. There are two not uncommon categories of deterministic substitute problems: (i) expected total cost minimization [19][20][21]; and (ii) reliability-based optimization [22][23][24].…”

“…Three kinds of optimization algorithms, evolutionary [8][9][10][11][12][13][14][15][16][17], stochastic [18][19][20][21][22][23][24][25][26][27][28][29] and combinatorial optimization [30][31][32][33][34][35][36][37][38] will be addressed. For machine learning algorithms, the discussion is based on un-supervised learning [39][40][41][42][43][44][45][46][47][48][49], supervised learning and semi-supervised learning [71][72][73][74][75][76][77][78]…”

Expanding Metabolic Capabilities Using Novel Pathway Designs: Computational Tools and Case Studies

“…Fundamental metabolic trade‐offs between metabolic rates and yield can be systematically explored by considering the associated protein cost of the underpinning pathway and/or phenotype . Indeed, the inclusion of enzyme cost as the objective function for predicting metabolic network states using GSMMs and ME models (metabolic and expression models) has greatly increased their prediction fidelity and our general understanding of metabolism.…”

“…Indeed, the inclusion of enzyme cost as the objective function for predicting metabolic network states using GSMMs and ME models (metabolic and expression models) has greatly increased their prediction fidelity and our general understanding of metabolism. In the context of pathway prediction, application of this criterion has also been attempted for predicting alternative, “biologically convenient” pathways with specific metabolic functions . For this purpose, the enzyme cost minimization (ECM) formulation has been proposed as a way of measuring the (minimum) enzyme load—here represented by the aggregated or total enzyme mass allocated—required for running a particular metabolic pathway .…”

“…It is worth mentioning that, in the context of constraint‐based modeling methods, the total sum of fluxes (minFlux) has been employed as an approximation for estimating flux solutions with minimum enzyme cost. To perform this estimation, ECM requires a fully parameterized kinetic model based on a thermodynamically consistent kinetic formalism such as mass action or reversible Michaelis–Menten kinetics, among others, which renders a harder (though convex) nonlinear optimization program . If one considers a fixed flux through the metabolic pathway, then minimization of the enzyme mass allocated yields a biologically convenient solution.…”