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The protein cost of metabolic fluxes: prediction from enzymatic rate laws and cost minimization
Elad Noor, Avi Flamholz, Arren Bar-Even, Dan Davidi, Ron Milo, Wolfram Liebermeister
TL;DR
The paper addresses the difficulty of predicting enzyme levels and protein costs from metabolic fluxes when enzyme demand also depends on metabolite concentrations and nonlinear kinetics. It develops enzyme cost minimization as a convex, scalable optimization over metabolite levels and validates the approach against E. coli central-metabolism measurements. The method provides a kinetics-based way to estimate enzyme levels and protein costs while incorporating enzyme kinetics into constraint-based metabolic models.
Problem
Existing metabolic models predict fluxes more readily than the enzyme levels and protein costs needed to sustain them, despite enzyme demand depending on metabolite concentrations and nonlinear kinetics.
Method
Enzyme cost minimization treats enzyme cost as a function of metabolite levels and formulates the resulting problem as a convex, numerically tractable optimization with tiered model detail.
Results
ECM predicts enzyme levels and protein costs while accounting for substrate, product, and allosteric saturation effects and connecting enzyme cost with thermodynamics.
Takeaways & Limitations
ECM offers a physically plausible and computationally tractable route for including enzyme kinetics in constraint-based models of natural and engineered pathways.
Takeaways & Limitations
The assumption that enzyme levels are continuously cost-optimized is debatable, and the enzyme-cost profile can be strongly shaped by large variation in catalytic constants and molecular masses.
Abstract
from arXiv · showhide
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 3.8 and 2.7, respectively, for the two kinds of data. ECM can be used to predict enzyme levels and protein cost in natural and engineered pathways, 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.
1 Introduction
Metabolic pathways must balance flux benefits against enzyme-resource demands, but existing constraint-based approaches predict feasible fluxes more readily than the enzyme levels and costs required to realize them. The paper motivates a kinetics-based enzyme-cost framework that connects fluxes, metabolite concentrations, thermodynamics, and protein investment.
- Existing gap: Constraint-based methods define feasible fluxes and assess benefits, but predict necessary enzyme levels and protein-maintenance costs less effectively.This limits quantitative analysis of protein demand for desired reaction or pathway fluxes.
- Protein investment: Protein cost also varies with enzyme-specific properties such as molecular size and effective lifetime, not solely with flux or enzyme amount.These differences motivate cost-weighted analysis of enzyme distributions and alternative pathways.
- Existing gap: Enzyme demand depends on metabolite levels as well as flux, because metabolite concentrations shape nonlinear enzyme kinetics.Thus, clarifying the relation between enzyme amounts and flux requires more than assigning numerical flux-cost weights.
- Kinetic determinants: Maximal-rate estimates divide flux by kcat but underestimate enzyme demand because enzymes experience backward fluxes, incomplete saturation, and regulation.Lower thermodynamic driving forces increase backward flux, reduce efficiency, and raise enzyme demand.
- Proposed direction: The proposed approach uses kinetics to translate fluxes into enzyme demand and relates heuristic constraints such as avoiding small driving forces to enzyme economy.It also connects enzyme-cost minimization with cost-benefit approaches and recovers relationships involving enzyme costs and metabolic control coefficients.
2 Results
The paper formulates enzyme demand for a specified flux as a function of metabolite levels, then selects metabolite and enzyme profiles minimizing protein cost. The resulting ECM framework is convex, supports tiered kinetic detail, and predicts measured E. coli metabolite and enzyme levels with typical fold errors of 3.8 and 2.7.
- Enzyme cost minimization: ECM treats enzyme cost as a function of metabolite levels and solves the resulting optimization problem as a convex, scalable method.Its tiered EMC functions range from capacity-only costs to models incorporating thermodynamics, simplified kinetics, full rate laws, and possible allosteric regulation.
- Enzyme demand: For a fixed pathway flux, enzyme demand can be computed from kinetic rate laws as a function of metabolite concentrations.The method reverses the usual integration problem by fixing flux and calculating the enzyme amount required at each metabolite profile.
- Enzyme demand landscape: Thermodynamic feasibility restricts metabolite profiles: enzyme demand diverges near equilibrium, while profiles beyond equilibrium cannot support positive flux.For the example pathway, feasible profiles form a triangular metabolite polytope in log-concentration space.
- Enzyme cost minimization: The minimum of total enzyme demand identifies the optimal metabolite profile and determines the corresponding optimal enzyme levels.The demand rises steeply toward the polytope edges and is minimized toward its center, unless kinetic or physiological constraints shift the optimum.
- Factors shaping enzyme profiles: Thermodynamic driving force and enzyme saturation increase predicted enzyme demand beyond capacity-only estimates, improving agreement with measured protein levels.At ΔrG′ = −0.1 RT, the thermodynamic factor is approximately 0.1 and enzyme demand increases by about 10-fold; incomplete saturation has an especially large effect in many enzymes.
- Validation: In E. coli central metabolism, predicted metabolite concentrations have a typical fold error of 3.8, while predicted enzyme levels have a typical fold error of 2.7.More complex cost functions improve enzyme-level predictions, with error decreasing monotonically from 1.34 to 0.43; EMC4 performs better on average than simpler functions.
3 Discussion
The discussion presents ECM as a flexible framework that makes enzyme-cost optimization tractable in metabolite space while connecting kinetics, thermodynamics, and pathway design. Its tiered models support different data conditions, but continuous cost optimality remains an important biological assumption.
- Solving the enzyme optimality problem in metabolite space: ECM reformulates enzyme optimization in flux and metabolite space, avoiding difficult direct searches over enzyme profiles that can be non-convex.The metabolite polytope provides a feasible search space, while log-concentrations allow systematic screening of metabolic states.
- Convexity: Convexity makes ECM tractable and scalable, with an efficiently findable unique optimum when strict convexity is ensured.The cost function is convex in metabolite log-concentrations, and regularization can produce strict convexity.
- Separable rate laws disentangle individual enzyme cost effects: ECM’s tiered rate-law formulations trade prediction precision against parameter demands, allowing model detail to match available data.Including more physical terms improves predictions, while stripped-down versions can remain useful under conditions such as batch-fed E. coli.
- Relationship to other optimality approaches: ECM accounts for substrate and product saturation and allosteric effects while retaining a convex optimization problem, extending earlier approaches.The method uses factorized enzyme-cost functions to represent distinct kinetic and thermodynamic contributions.
- Enzyme cost is related to thermodynamics: ECM links thermodynamic feasibility and enzyme cost by using the metabolite polytope and enzyme-cost function together to summarize flux cost.Thermodynamic efficiency emerges as a quantitative condition related to enzyme economy, and costs can exclude high-cost regions of metabolite space.
- Improved parameters for flux analysis: ECM can support pathway selection and flux analysis by predicting enzyme demands, comparing pathway structures, and identifying deviations from resource-optimality.Its predictions can focus attention on enzymes or pathways that may reflect optimization or adaptations beyond simple resource optimality.
4 Methods
ECM represents enzyme cost as a function of metabolite levels and minimizes that cost over a thermodynamically feasible metabolite polytope. The formulation remains convex while incorporating kinetic constraints, enzyme burdens, parameter balancing, and uncertainty analysis.
- Optimization problem: ECM minimizes total enzyme cost over metabolite profiles constrained by concentration bounds, flux directions, and thermodynamic feasibility.Driving forces and fluxes must have matching signs; infeasible flux directions can yield an empty feasible polytope.
- Optimization problem: The enzyme cost function and metabolite polytope are convex, making ECM a numerically tractable convex optimization problem.Convexity can persist under several added constraints and problem modifications.
- Model construction: ECM predicts optimal metabolite profiles, enzyme profiles, and enzyme costs from a metabolic network, flux profile, kinetic rate laws, enzyme burdens, and metabolite bounds.The workflow builds a consistent kinetic model before optimizing profiles for the selected enzyme metabolic cost function.
- Uncertainty and tolerance: Nearly optimal metabolite and enzyme ranges are obtained by restricting solutions to a tolerable cost above the optimum.A tolerable region is defined by q(s) ≤ qtol, after which minimum and maximum levels are calculated within that region.
- Model construction: Parameter balancing reconciles incomplete or inconsistent kinetic measurements with thermodynamic and kinetic consistency requirements.The workflow enforces Wegscheider conditions and Haldane relationships, then can sample posterior parameter sets to assess prediction variation.
S1 Kinetic rate laws
The kinetic framework links reversible reaction rates to thermodynamic driving forces and molecularities while allowing rate laws of varying mechanistic detail. Simplified rate laws are computationally convenient but can underestimate enzyme demand relative to richer formulations.
- Rate-law construction: General reversible rate laws use substrate and product concentrations, molecularities, binding-state terms, and optional allosteric activation or inhibition terms.The denominator is constructed from enzyme binding states, with exponents encoding bound reactant numbers and prefactors encoding binding energies.
- Thermodynamic consistency: Thermodynamic driving force is defined from negative reaction Gibbs energy, and forward-to-backward flux ratios increase exponentially with that force.At fixed net flux, forward and backward fluxes rise as the reaction approaches chemical equilibrium.
- Rate-law construction: Driving forces must be defined from molecularities rather than nominal stoichiometric coefficients when the two differ.Reaction-specific scaling factors or Hill-like coefficients account for these differences in thermodynamic expressions.
- Thermodynamic consistency: Equilibrium constants and rate constants must satisfy Haldane relationships, while equilibrium constants must also satisfy Wegscheider conditions across thermodynamic cycles.These constraints ensure that reaction rates vanish at chemical equilibrium and that the kinetic model has a consistent equilibrium state.
- Rate-law alternatives: Simplified rate laws with fewer denominator terms produce higher predicted rates and therefore tend to underestimate enzyme demand and cost.Averaging rate-law denominators yields corresponding geometric or harmonic means of enzyme costs.
- Rate-law alternatives: Direct-binding modular and common modular rate laws provide alternative levels of mechanistic detail for reversible Michaelis-Menten kinetics.The common modular law contains more denominator terms, whereas realistic rate laws may lie between these simplified extremes.
S2 Enzyme cost functions
ECM represents enzyme cost with functions that combine enzyme demand, kinetic efficiency, and enzyme-specific burdens. Costs can reflect protein size, degradation, composition, complexes, modification state, physical constraints, and nonlinear interactions, but biological cost effects remain incompletely captured.
- Cost formulation: ECM assumes cells realize fluxes at minimal enzyme cost and models total cost as a weighted function of enzyme concentrations.The default linear weights can be proportional to protein size, such as amino-acid length or molecular mass.
- Scope and limitations: The model's simplified cost functions omit many context-dependent effects, including promiscuous activity and detailed physicochemical properties that require extensive biological information.These omissions motivate a cost function that is simple to calculate while capturing selected biological aspects.
- Cost formulation: Enzyme-specific cost weights can incorporate protein degradation, with h_El proportional to protein size divided by enzyme lifetime.When growth is much faster than degradation, degradation has negligible effects because enzyme lifetimes become similar.
- Biological cost factors: Amino-acid composition can be included through amino-acid production costs, although neglecting composition produces similar enzyme-level predictions.The paper reports using composition-dependent costs in its calculations while finding similar predictions with simpler cost functions.
- Biological cost factors: For multisubunit enzymes, catalytic constants refer to catalytic sites whereas measured protein levels refer to subunits, requiring explicit accounting for subunit and site numbers.The numbers of subunits and catalytic sites enter reaction-rate, enzyme-demand, and enzyme-cost formulas.
- Biological cost factors: Posttranslational modification increases effective cost when only a fraction of total enzyme is in the active modification state.If the active fraction is ρ_l, the cost weight is multiplied by 1/ρ_l.
- Biological cost factors: ECM can represent physical enzyme constraints with upper bounds on enzyme-level sums or penalty terms for high levels.These constraints can apply to cells, compartments, or membranes, including restrictions on respiration complexes.
S3 Enzyme cost minimization
ECM represents steady states using metabolite profiles and derives the enzyme levels required for a specified flux. Under its rate-law assumptions, enzyme cost is convex over the feasible metabolite polytope, enabling tractable optimization.
- Metabolic-state representation: ECM replaces exhaustive scans of enzyme levels and initial conditions with metabolite profiles that parameterize steady states for a given flux.The enzyme levels are recovered from inverted kinetic rate laws.
- Metabolic-state representation: For every feasible metabolite profile, the required enzyme levels are unique and given by E_l(ln c) = v_l/r_l(c).The enzyme-level function is differentiable on the metabolite polytope.
- Metabolic-state representation: Thermodynamically feasible metabolite profiles can be realized at steady state, and their feasible set depends on equilibrium constants rather than other enzyme-specific parameters.The set of metabolic states can therefore be parameterized by the metabolite polytope.
- Convex optimization: The enzyme cost functions are convex in logarithmic metabolite concentrations, because energetic and saturation terms are convex and total cost sums reaction costs.This convexity supports numerical minimization under metabolite and additional resource constraints.
- Catalytic constants and optima: In the two-reaction example, enzyme demand diverges near chemical equilibrium, while the optimum metabolite concentration lies away from those boundaries.Changing the forward catalytic constant in reaction 1 shifts the optimum toward higher intermediate concentration.
S3.7 The enzyme cost profile obtained by ECM is a linear combination of metabolic control profiles
ECM links enzyme cost to metabolic control coefficients. The relationship generalizes the proportionality between enzyme levels and flux control coefficients under simpler network and boundary conditions.
- Control-coefficient relationship: ECM relates enzyme costs to a linear combination of metabolic control coefficients associated with stationary flux modes and bounded or fixed internal metabolites.The relevant coefficients represent both flux control and metabolite control contributions.
- Control-coefficient relationship: With one stationary flux mode, enzyme costs are proportional to the corresponding flux control coefficients.When all enzyme cost weights are equal, this also yields proportionality between enzyme levels and flux control coefficients.
- Control-coefficient relationship: If fixed metabolites are external and no internal metabolite reaches a bound, concentration-control terms disappear from the cost relationship.Under these assumptions, enzyme costs are directly proportional to flux control coefficients.
S4 Energy-based cost functions and limits on driving forces
ECM-derived cost estimates connect enzyme demand to thermodynamic driving forces and provide stricter metabolite constraints than sign-based thermodynamic FBA. These constraints exclude regions requiring implausibly high enzyme costs.
- Driving-force constraints: ECM can derive stricter driving-force constraints that exclude costly regions from the metabolite polytope and can also be used in thermodynamics-based FBA.The bounds depend on fluxes, catalytic constants, and enzyme cost weights.
- Driving-force constraints: Near polytope E-faces, enzyme costs rise rapidly, so optimal metabolite profiles avoid these costly regions.The enzyme-cost minimum may lie inside the metabolite polytope or on a boundary face.
- Driving-force constraints: Unlike sign-based thermodynamic FBA, the stricter formulation requires driving forces large enough to realize fluxes at plausible enzyme costs.This prevents fluxes from being supported by infinitesimal thermodynamic forces.
- Additional metabolite constraints: Upper bounds on enzyme cost yield lower bounds on saturation efficiency and substrate levels, with analogous constraints for multisubstrate reactions and allosteric regulators.These constraints can be expressed as linear inequalities in log-concentration space for multisubstrate reactions.
- Driving-force constraints: Lower-bound cost functions based on driving forces support MDF-like optimization, whose optimum avoids polytope faces and excessive enzyme costs.MDF maximizes the smallest pathway driving force and corresponds to minimizing a lower bound on enzyme cost.
S5 Workflow for model building and metabolic optimization
The ECM workflow integrates kinetic-data balancing with convex optimization of metabolite profiles, then computes enzyme costs and validates predictions. It accommodates incomplete data and several pathway-model extensions.
- Workflow: The workflow has a kinetics phase for collecting and adjusting parameters and an optimization phase for realizing the desired flux with optimal enzyme and metabolite profiles.The optimization phase is followed by calculation and validation of predicted levels.
- Optimization phase: ECM sets metabolite bounds, selects a feasible log-concentration starting point, minimizes an EMC function under model constraints, and computes enzyme levels and cost.Starting points can come from polytope geometry, quadratic programming, or MDF optimization.
- Validation: Predicted metabolite and enzyme levels are validated against experimental data after tolerances are computed around the optimal cost.The workflow can repeat optimization from different starting points as a convergence check.
- Kinetics phase: Parameter balancing converts incomplete or contradictory kinetic and thermodynamic measurements into a complete, consistent parameter set.Bayesian priors provide plausible estimates when data are sparse, while the posterior captures parameter uncertainty and correlations.
- Extensions and scope: ECM supports nonstationary flux profiles, inactive and non-enzymatic reactions, compartment models, and bounds on total metabolite or enzyme levels.Substrate channeling is not modeled directly and requires approximate effective rate constants.
S6 Model of central metabolism in E. coli
The E. coli central-metabolism model uses KEGG reactions and identifiers, literature kinetic parameters, and an enzyme cost function that distinguishes protein composition. Predicted metabolite levels are compared with measured data, while alternative protein-cost weightings give similar results.
- The central-metabolism model was built from KEGG chemical reactions, with compounds and reactions identified by KEGG identifiers.
- The enzyme cost function assigns different costs to amino acids through protein-composition weighting.
- Equal protein cost weights and size-dependent protein costs produced similar results to composition-based weighting.
- Figure S6 compares predicted and measured metabolite levels using tolerance ranges and measurement uncertainties.
S7 Proofs and derivations
The derivations establish thermodynamic bounds, treatment of lumped reactions, convex tolerance regions, and constrained enzyme-cost optimality conditions. Together, these results formalize how ECM represents kinetics, costs, and active constraints.
- An upper limit on individual enzyme cost yields a lower bound on thermodynamic driving forces.
- For lumped reactions, the effective enzyme concentration sums component enzymes, implying smaller effective kcat values after parameter rescaling.
- Lumped-enzyme parameters can be chosen using geometric means of catalytic constants or arithmetic means of enzyme concentrations while preserving pathway flux.
- Strict convexity and a positive-definite Hessian define tolerance regions around the global minimum, with bounding boxes available when ellipsoids are inconvenient.
- ECM minimizes enzyme cost subject to stationary-flux and metabolite-bound constraints, with Lagrange multipliers describing the two constraint classes.
- For linear cost functions, enzyme-cost slopes equal the corresponding cost weights and enter optimality relations expressed through control coefficients.
S8 Mathematical symbols
Table S5 defines ECM's mathematical symbols, including fitness units, reaction orientations, and flux units.
- Table S5 defines Darwin as a hypothetical fitness unit and specifies positive reaction orientations and concentration-per-time flux units.Amounts-per-time units are also allowed and are more practical for transport reactions.