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Constrained Allocation Flux Balance Analysis

Matteo Mori, Terence Hwa, Olivier C. Martin, Andrea De Martino, Enzo Marinari

arXiv:1607.00128v1q-bio.MNcond-mat.dis-nnphysics.bio-phq-bio.QM

TL;DR

Genome-scale models have limited ability to represent the costs of gene expression and protein synthesis in metabolism. The paper introduces CAFBA, which adds a single proteome-allocation constraint to FBA and uses empirical growth laws to optimize growth and fluxes. In E. coli, CAFBA predicts a shift from respiratory, yield-maximizing states at slow growth to fermentative carbon overflow at fast growth, with quantitatively accurate acetate-excretion and growth-yield predictions from few parameters.

  • Problem

    Genome-scale metabolic approaches have limited molecular-level treatment of gene-expression and protein-synthesis costs, while more detailed alternatives are more involved than FBA.

  • Method

    CAFBA adds a single global flux constraint encoding growth-dependent proteome-sector allocation and optimizes growth under stoichiometric and proteomic constraints.

  • Results

    CAFBA predicts a transition from respiratory, growth-yield-maximizing states at slow growth to fermentative carbon-overflow states at fast growth, including quantitatively accurate acetate secretion.

  • Takeaways & Limitations

    E. coli overflow metabolism can be predicted from a small set of empirical growth-law parameters while linking metabolic fluxes with proteome allocation.

  • Takeaways & Limitations

    Reaction-specific protein-cost weights remain poorly characterized because proteomic coverage and abundance accuracy are limited, so the study uses simplifying parameterizations.

Abstract

from arXiv · show

New experimental results on bacterial growth inspire a novel top-down approach to study cell metabolism, combining mass balance and proteomic constraints to extend and complement Flux Balance Analysis. We introduce here Constrained Allocation Flux Balance Analysis, CAFBA, in which the biosynthetic costs associated to growth are accounted for in an effective way through a single additional genome-wide constraint. Its roots lie in the experimentally observed pattern of proteome allocation for metabolic functions, allowing to bridge regulation and metabolism in a transparent way under the principle of growth-rate maximization. We provide a simple method to solve CAFBA efficiently and propose an "ensemble averaging" procedure to account for unknown protein costs. Applying this approach to modeling E. coli metabolism, we find that, as the growth rate increases, CAFBA solutions cross over from respiratory, growth-yield maximizing states (preferred at slow growth) to fermentative states with carbon overflow (preferred at fast growth). In addition, CAFBA allows for quantitatively accurate predictions on the rate of acetate excretion and growth yield based on only 3 parameters determined by empirical growth laws.

Author Summary

CAFBA extends genome-scale flux balance analysis by incorporating proteome-allocation costs through a single global constraint, addressing the limited molecular detail of existing genome-scale approaches. Applied to E. coli, it links growth-rate optimization with sector allocation and predicts overflow metabolism quantitatively using few empirical inputs.

  • Approach: CAFBA extends FBA with a single global flux constraint encoding growth-dependent adjustment of ribosomal, biosynthetic, and carbon-intake proteome sectors.The approach models regulation effectively without the biochemical detail of ME-models or RBA.
  • Predicted metabolism: CAFBA captures the growth-yield trade-off underlying overflow metabolism, predicting respiratory, yield-maximizing states at slow growth and fermentative acetate secretion at fast growth.The model targets quantitative overflow features that standard FBA has generally captured only qualitatively.
  • Proteome sectors: CAFBA models bacterial proteomes as ribosomal, biosynthetic-enzyme, carbon-intake, and housekeeping sectors whose fractions sum to one.The growth-dependent sectors are represented through experimentally motivated allocation rules.
  • Proteome sectors: The ribosomal sector increases linearly with growth rate, while the carbon sector depends linearly on carbon intake and includes co-expressed nutrient-scavenging proteins.The carbon-sector prescription includes basal proteins and proteins induced during carbon limitation, not only glucose transporters.
  • Model inputs: CAFBA uses a simple carbon-intake control parameter based on enzyme kinetics to represent changes in extracellular sugar concentration.The parameter wC can be increased to model reduced extracellular sugar levels.

Proteome-wide constraint

CAFBA recasts proteome-sector allocation as a genome-wide linear constraint coupled to stoichiometric flux balance and biomass growth. In E. coli, the resulting solutions describe growth-dependent shifts between respiration and fermentation, while unknown reaction costs motivate homogeneous and heterogeneous parameter treatments.

  • Constraint formulation: The proteome-wide constraint balances weighted non-catabolic fluxes against growth-dependent ribosomal allocation within the accessible proteome fraction.As growth increases, ribosomal allocation rises and enzymatic and carbon-intake allocations must adjust.
  • Optimization: CAFBA optimizes fluxes subject to stoichiometric mass balance, flux bounds, and a biomass-defined growth rate.The formulation retains the core constraint structure of FBA while adding proteome allocation.
  • Experimental scenarios: For E. coli glucose-limited growth, carbon limitation is modeled by increasing wC from its saturating-glucose minimum, while translational limitation increases wR.The study uses the iJR904 GSM/GPR reconstruction and considers both growth-independent and growth-dependent biomass composition.
  • Results: Figure 1 reports a carbon-limitation transition from fermentation to respiration around 0.7–0.9/h and a corresponding translational-limitation regime in which ribosomal allocation expands.Under translational limitation, carbon metabolism and biosynthesis receive smaller proteome shares.
  • Parameterization: Because reaction-specific protein costs are poorly characterized, CAFBA evaluates both uniform weights and randomly sampled heterogeneous weights.The homogeneous weight is calibrated to match the empirical maximum growth rate, while the heterogeneous case represents uncertainty in individual costs.

Homogeneous weights and patterns of flux

CAFBA links growth-rate optimization to proteome allocation through global sector-cost parameters, producing growth-dependent shifts between respiratory and fermentative metabolism. Across carbon sources and perturbations, the model reproduces several experimentally observed flux patterns and captures average behavior despite heterogeneous enzyme costs.

  • Parameterization: wE = 8.3 × 10−4 gh/mmol yields λmax = 1/h for glucose, while 1/wC and 1/wR correspond to nutritional and translation capacities.CAFBA recovers the phenomenological growth-rate relation within a genome-scale model.
  • Proteome allocation: As carbon becomes limiting, the C-protein fraction increases while E- and R-sector fractions change with growth rate.The predicted C-sector response is consistent with experimentally observed catabolic-protein and PTS expression.
  • Central carbon flux: Respiration dominates at low growth, whereas fermentation, acetate secretion, and ED-pathway use increase at high growth; the acetate onset is within 10% of experiment.The model places the transition near the experimentally observed onset for NCM3722 across carbon sources.
  • Translational limitation: Increasing wR to model translational limitation extends acetate secretion to slower growth rates while respiratory flux becomes negligible.The ribosomal fraction increases and the C- and E-sector fractions shrink approximately linearly under stronger inhibition.
  • Flux transitions: Optimal flux configurations can change discontinuously with growth because maximizing growth responds to continuous changes in the C-sector weight wC.The rearrangements reflect the geometry of the constraint-based optimum rather than changes in a directly controlled flux.
  • Carbon-source patterns: Across glycolytic carbon sources, acetate excretion decreases nearly linearly with growth under limitation, while TCA and glyoxylate-shunt fluxes peak near λac ≃0.79/h.CO2 secretion also changes regime around λac, distinguishing slower- and faster-growth behaviors.
  • Carbon-source patterns: Carbon sources entering glycolysis as glucose-6P show similar average fluxes, whereas fructose-6P substrates favor processing through fructose bisphosphate.These substrate groups support distinct but internally consistent pathway-use patterns.
  • Carbon-source patterns: For lower-glycolysis and TCA-cycle substrates, CAFBA predicts multiple strategies sharing increased CO2 production at faster growth.The passage identifies a common outcome while allowing pathway strategies to vary.

Comparison between CAFBA and FBA solutions

CAFBA is compared with FBA by matching growth rates and examining flux similarity, growth yield, and acetate secretion. The comparison shows that CAFBA aligns more closely with yield-maximizing behavior at slow growth, while faster growth is associated with carbon overflow and parameter-dependent acetate secretion.

  • Comparison procedure: CAFBA solutions are compared with pFBA solutions at matched growth rates and glucose influxes.pFBA solutions were generated by varying glucose-intake bounds, then interpolated at CAFBA growth rates.
  • Acetate secretion: Acetate secretion above λac is approximated by vac = s × (λ − λac), with s = 45 mmol/gDW and λac = 0.79/h in the representative case.This linear approximation applies for λ ≥ λac.
  • Flux similarity: The overlap between CAFBA and FBA fluxes is generally high at low growth rates and decreases as growth rate increases.The overlap is evaluated across all reactions, glycolysis/gluconeogenesis, and the TCA cycle.
  • Parameter dependence: Changing βATP alters the acetate-secretion slope and onset growth rate, while tuning ⟨w⟩ to fix λmax = 1/h yields λac ≃0.8/h across tested values.The same parameter changes also affect TCA flux, growth yield, and maximum growth rate before λmax is normalized.
  • Model structure: CAFBA uses one additional global proteome-allocation constraint while retaining the computational complexity of standard FBA.The constraint represents resource allocation across ribosomal, transport, and biosynthetic protein sectors.
  • Growth-regime comparison: CAFBA transitions from growth-yield maximizing states at slow growth to carbon-overflow states at faster growth.The transition is reported as continuous, with acetate secretion emerging as growth increases.

Optimization problem

CAFBA is formulated as a proteome-constrained optimization problem whose absolute-value terms can be eliminated, reducing the model from MILP to LP while retaining growth-rate maximization.

  • Formulation: CAFBA combines stoichiometric constraints, flux bounds, growth rate, and a proteome allocation constraint involving weighted absolute fluxes.The weighted sum excludes biomass, ATP maintenance, transport, exchange, and carbon-intake pathways.
  • Formulation: Splitting each flux into non-negative forward and backward components initially makes CAFBA a mixed-integer linear program.The integer structure arises from absolute values in the proteome constraint.
  • Optimization: Growth maximization forces one directional component to vanish for each flux, so CAFBA becomes a simple linear-programming problem.This follows from the tight connection between CAFBA and flux minimization.
  • Optimization: CAFBA solution degeneracies arise from futile loops or alternative pathways with identical conversions and proteome costs.Heterogeneous random weights make exact cost ties between equivalent configurations unlikely.
  • Implementation: The implementation uses E. coli genome-scale reconstructions, including iJR904, with Matlab and COBRA-compatible solvers.The iJR904 model contains 761 metabolites and 1075 reactions; related reconstructions produce similar results.

Supporting Text

The listed authors are M. Mori, T. Hwa, O.C. Martin, A. De Martino, and E. Marinari.

  • M. Mori is listed as an author.
  • T. Hwa is listed as an author.
  • O.C. Martin, A. De Martino, and E. Marinari are listed as authors.

Note A. The linear enzyme–flux relation

The linear enzyme–flux relation links proteome allocation to the flux required by each reaction, with kinetic assumptions explaining when this relation is appropriate.

  • Core relation: CAFBA assumes that larger reaction fluxes require larger enzyme proteome shares to sustain them.This constitutive relation connects enzyme fractions and fluxes within the model.
  • Kinetic basis: For Michaelis–Menten kinetics, the derivation accounts for substrate and enzyme concentrations while neglecting metabolite dilution flux in steady-state calculations.The resulting substrate factor is f([s], [p]) = [s]/([s] + KM).
  • Core relation: Under saturating kinetics, enzyme levels become linearly related to flux, corresponding to φi,0 = 0 and wi = 1/κcat,i.The saturating case is recognized as an idealization because reaction saturation varies across growth conditions.
  • Kinetic basis: If substrate concentration is proportional to flux, as described for some glycolytic enzymes, enzyme levels again depend linearly on flux.The same framework connects this relation to substrate-driven feedforward activation.
  • Baseline expression: Finite substrate concentrations introduce baseline enzyme expression, represented by a positive constant offset φi,0 > 0.The baseline is supported by quantitative proteomics data.

Note B. The choice of the control parameter

CAFBA uses the proteome cost of carbon uptake, rather than glucose uptake alone, to control growth and determine the optimal glucose intake. The choice of control parameter changes predicted high-growth flux patterns, including the appearance of discontinuities when carbon limitation is varied.

  • Control parameter: Standard FBA tunes growth by imposing an upper bound on carbon intake, whereas CAFBA tunes growth through the carbon-scavenging proteome weight wC.In CAFBA, glucose intake remains a free optimization variable.
  • Control parameter: The weight wC increases as extracellular glucose decreases, so higher wC represents stronger carbon limitation.Because wC and glucose concentration are invertibly related, wC can replace glucose concentration as the control parameter.
  • Flux consequences: Varying wC produces jump discontinuities and large-scale flux rearrangements, unlike the smoother behavior generally associated with varying glucose intake.Carbon-overflow solutions can become growth-optimal below the maximum achievable growth rate.
  • Figures: Figure N1 compares growth against glucose uptake across four wC values, distinguishing directly constrained glucose fluxes from unconstrained CAFBA solutions.The plotted weights are wC = 0, 10−3, 10−2, and 10−1 gDWh/mmol.
  • Figures: Figure N2 contrasts fluxes controlled by wC with fluxes obtained by imposing an upper glucose-influx bound at fixed wC = 0.Panel C magnifies the high-growth region of the glucose-bound case.

Note C. Extension to different growth media and/or bacterial species

CAFBA can be adapted across media and species by calibrating a small set of proteome-allocation parameters and nutrient-specific weights. The framework supports homogeneous and heterogeneous enzyme-cost models while retaining a compact parameterization.

  • Implementation: CAFBA setup requires defining the growth medium and assigning exchange-flux bounds to available and unavailable substrates.Available substrates receive large negative intake bounds, whereas unavailable metabolites receive zero lower bounds.
  • Implementation: Nutrient limitation is modeled by increasing the corresponding nutrient-specific weight above its empirical lower bound.The same procedure can represent carbon, phosphate, nitrogen, or oxygen limitation.
  • Parameter calibration: For MG1655, CAFBA uses wE = 1.55 × 10^-3 gDWh/mmol, wR = 0.169/h, and φmax = 48.4%, giving λmax ≈ 0.7/h when nutrient weights vanish.These parameters calibrate the strain-specific proteome allocation constraint.
  • Parameter calibration: Applying CAFBA to another strain or species requires a constraint-based metabolic model plus values for wR, φmax, and the E-sector weights.The E-sector includes intracellular metabolic reactions except those associated with carbon uptake.
  • Homogeneous and heterogeneous costs: In the homogeneous model, all E-sector weights equal wE, calibrated so saturating-carbon growth extrapolates to 1/h; heterogeneous models sample weights independently.The homogeneous and heterogeneous cases have three and four parameters, respectively, apart from nutrient-specific lower bounds and one arbitrary scale choice.
  • Homogeneous and heterogeneous costs: Carbon-specific uptake reactions can be assigned to either the C-sector or E-sector without affecting results beyond a rescaling of wC.The paper places empirically carbon-upregulated uptake systems in the C-sector.

Note D. Translational inhibition and protein over–expression in CAFBA solutions

This note examines how translational capacity and unnecessary-protein overexpression alter CAFBA growth and flux predictions. These perturbations mainly rescale growth and flux magnitudes, while the qualitative flux patterns remain approximately unchanged.

  • CAFBA parameters: The homogeneous proteome-allocation constraint contains wC, wE, wR, and φmax, but calibration and scale invariance leave wR and φmax as the principal free parameters.wE is fixed to give λmax = 1/h at wC = 0.
  • Growth-rate dependence: CAFBA predicts growth as a Michaelis–Menten function of translational capacity and nutritional quality, with growth decreasing linearly as φQ increases.The approximation is accurate for λ ≳ 0.1/h.
  • Overflow transition: Acetate begins to be excreted near kc,ac ∼ 200 mmol/gDWh, corresponding to λac ∼ 0.8/h.The fit is reported as accurate from very low growth rates up to the acetate switch.
  • Overflow transition: When carbon overflow begins, the cell switches to a higher-maximum-growth, lower-affinity state, whereas low-quality nutrients favor efficient, high-yield growth.The paper links the trade-off to different preferred strategies across nutrient quality.
  • Flux patterns: Changing wR or φmax approximately rescales growth and fluxes with λ without significantly modifying flux patterns.This covers translational limitation and unnecessary-protein overexpression modeled through increasing φQ.

Note E. Case of inhomogeneous proteome costs

The inhomogeneous CAFBA extension treats enzyme costs as random variables and averages solutions across model replicas. Increasing cost heterogeneity smooths pathway transitions but lowers yield and increases flux variability.

  • Ensemble construction: Inhomogeneous CAFBA samples each E-sector weight independently and averages fluxes across many model replicas.The weights are quenched random variables, so each replica has a distinct realization of protein costs.
  • Ensemble construction: The weight distribution is log-uniform, with δ giving the number of decades spanned by the sampled costs.The density satisfies p(wi) ∝ 1/wi over the interval set by the mean scale and δ.
  • Robustness of averaging: Different weight distributions give quantitatively similar results when their mean and standard deviation are similar.The paper specifically reports similar outcomes for log-uniform and lognormal choices when logarithmic variance is matched.
  • Effects of heterogeneity: For δ ∼ 1, average acetate excretion grows approximately linearly with growth rate from λ ∼ 0.7/h; larger δ lowers yield and increases glucose uptake and byproduct excretion.The added excretions are mainly dihydroxyacetone and formate.
  • Effects of heterogeneity: Flux heterogeneity increases with both δ and growth rate, as measured by the standard deviation across model replicas.Fluxes generally scale with growth rate, contributing to the growth-rate dependence of variability.
  • Effects of heterogeneity: The study uses δ close to one because empirical evidence is better reproduced at that fluctuation width.The authors adopt this value for the remainder of the analysis.

Note F. Growth-dependent biomass composition

CAFBA can incorporate growth-dependent biomass composition by enforcing empirical biomass constraints and iteratively solving fixed-composition linear programs. The procedure converges rapidly, but the direct optimization problem is no longer linear.

  • Biomass representation: The model represents biomass through growth-dependent coefficients for amino acids, DNA, RNA, and lipids, while retaining small cofactors from iJR904.The remaining components account for roughly 2.5% of total dry weight.
  • Biomass constraints: The biomass functions satisfy total-mass normalization and the empirical RNA/protein relation R(λ) = r0 + λ/κt.The reported parameters are r0 = 0.087 ± 0.009 and κt = 4.5 ± 0.2/h.
  • Problem formulation: Growth-dependent biomass composition makes the CAFBA optimization problem nonlinear and prevents direct solution with standard linear programming.The paper therefore treats the problem as a sequence of LPs with temporarily fixed biomass coefficients.
  • Iterative solution: The iterative algorithm initializes λ0, recomputes biomass coefficients, resolves CAFBA, and stops when successive growth rates differ by less than 10^-4/h.Each iteration uses the coefficients evaluated at the preceding growth rate.
  • Iterative solution: The iteration converges rapidly because |λk − λk−1| decreases exponentially with the number of steps.The procedure is applied separately for each carbon-limitation value while other parameters remain fixed.
  • Model outputs: Allowing biomass composition to vary produces representative acetate, CO2, TCA, glyoxylate-shunt, and growth-yield flux predictions across ATP-maintenance settings.The supplied figure descriptions compare constant and variable biomass composition for λ-dependent and λ-independent ATP hydrolysis.

Supplementary Tables

The supplementary tables define CAFBA parameter settings and report how maximum growth rates are determined across carbon sources and metabolic models.

  • Maximum growth rates: CAFBA extrapolates maximum growth rates across carbon sources using wC = 0 and wE = 8.3×10−4 gDWh/mmol.Under these conditions, growth is limited by the E- and R-sectors.
  • Maximum growth rates: λmax is the x-axis intercept of the C-line, with non-phosphorylated carbon sources yielding values between 0.984 and 1.01/h.Glucose-6P and mannose-6P support larger maximum growth rates because they generate extra ATP per flux unit.
  • Parameter settings: The reported CAFBA calculations use uniform E-sector weights wi = wE = 8.3 × 10−4 gDWh/mmol, wR = 0.169 h, and φmax = 48.4%.These settings specify the common parameterization used for the carbon-source comparison.
  • ATP requirements: The ATP hydrolysis flux is modeled as vATP = vATP M + βATP λ across the iJR904, iAF1260, and iJO1366 E. coli models.Table C compares the maintenance and growth-dependent ATP hydrolysis parameters for these three models.

Supplementary Figures

The supplementary figures examine CAFBA predictions across weight distributions, substrates, growth constraints, biomass compositions, and metabolic pathways, comparing model behavior with experiments where available.

  • Growth constraints: CAFBA growth rates vary continuously with nutritional quality kc and translational capacity kr, while acetate excretion changes discontinuously across kc values.The growth-rate surface is approximated by a Michaelis–Menten function of kc and kr.
  • Flux comparisons: Flux differences in glucose minimal medium are measured relative to fluxes scaled linearly with growth rate from the maximum-growth reference.Positive differences indicate fluxes above that reference, and carbon limitation suggests proportional upregulation for some pathways.
  • Redox and respiration: CAFBA activates the energy-dependent THD2 transhydrogenase at high growth rates, while ubiquinol oxidase usage shifts from CYTBO3 at low growth to comparable CYTBD and CYTBO3 fluxes at high growth.These averages use 1000 independent weight realizations for each wC value in glucose minimal medium.
  • Carbon sources: Across seven glycolytic carbon sources, acetate excretion remains consistent, with slight deviations for phosphorylated substrates, while ED-pathway fluxes are more heterogeneous.The calculations use ⟨w⟩ = 8.8 × 10−4 gDWh/mmol and δ = 1, averaging 500 samples per wC value.
  • Carbon sources: For five TCA carbon sources, carbon dioxide production is much higher, whereas acetate excretion persists above λac = 0.79/h.The compared substrates are α-ketoglutarate, fumarate, malate, succinate, and pyruvate.
  • Biomass composition: Varying ATP hydrolysis rates and biomass composition changes predicted acetate, carbon dioxide, TCA, and growth-yield profiles while tuning ⟨w⟩ to keep the average maximum growth rate at 1/h.The tested growth-dependent ATP hydrolysis rates are 35, 45.5608, and 60 mmol ATP/gDW.
  • Weight sensitivity: Changing individual enzyme weights tests how αKG dehydrogenase, malate synthase, and oxidative-phosphorylation reactions affect acetate, respiratory, glyoxylate-shunt, and yield predictions.The perturbations multiply selected weights by five or divide oxidative-phosphorylation weights by five.
  • Experimental comparison: CAFBA predictions agree with most experimental E. coli MG1655 fluxes, capture glyoxylate-shunt behavior, and show their largest discrepancy in the ED pathway.The ED flux depends strongly on the substrate under the single-carbon-source optimization assumption.
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