Enzyme cost landscape of a metabolic pathway

Given a pathway flux profile and a kinetic model of the pathway, one can predict the enzyme demand by assuming that cells minimize the enzyme cost in that pathway. A reaction rate v = E ⋅ r(c) depends on enzyme level E and metabolite concentrations c_i through the enzymatic rate law, r(c). If the metabolite levels were known, we could directly compute enzyme demands E = v/r(c) from fluxes, and similarly calculate the flux-specific enzyme demand E/v = 1/r(c). However, metabolite levels are often unknown and vary between experimental conditions. Therefore, there can be many solutions for E and c realizing one flux distribution. To select one of them, we employ an optimality principle: we define an enzyme cost function (for instance, total enzyme mass) and choose the enzyme profile with the lowest cost while restricting the metabolite levels to physiological ranges and imposing thermodynamic constraints. As we shall see below, the optimal solution is in many cases unique. Let us demonstrate this with a simple example (Fig 2a). In the pathway X ⇌ A ⇌ B ⇌ Y, the external metabolite levels [X] and [Y] are fixed and given, while the intermediate levels [A] and [B] need to be found. As rate laws for all three reactions, we use reversible Michaelis-Menten (MM) kinetics (1) with enzyme level E, substrate and product levels s and p, turnover rates and , and Michaelis constants K_S and K_P. In kinetic modeling, steady-state concentrations would usually be obtained from given enzyme levels and initial conditions through numerical integration. Here, instead, we fix a desired pathway flux v and compute the enzyme demand as a function of metabolite levels: (2)

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Fig 2. Enzyme demand in a metabolic pathway.

(a) Pathway with reversible Michaelis-Menten kinetics (equilibrium constants, catalytic constants, and K_M values are set to values of 1, [A] and [B] denote the variable concentrations of intermediates A and B in mM). The external metabolite levels [X] and [Y] are fixed. Plots (b)-(d) show the enzyme demand of reactions 1, 2, and 3 at given flux v = 1 according to Eq (2). Grey regions represent infeasible metabolite profiles. At the edges of the feasible region (where A and B are close to chemical equilibrium), the thermodynamic driving force goes to zero. Since small forces must be compensated by high enzyme levels, edges of the feasible region are always dark blue. For example, in reaction 1 (panel (b)), enzyme demand increases with the level of A (x-axis) and goes to infinity as the mass-action ratio [A]/[X] approaches the equilibrium constant (where the driving force vanishes). (e) Total enzyme demand, obtained by summing all enzyme levels. The metabolite polytope—the intersection of feasible regions for all reactions—is a triangle, and enzyme demand is a convex function on this triangle. The point of minimum total enzyme demand defines the optimal metabolite levels and optimal enzyme levels. (f) As the k_cat value of the first reaction is lowered by a factor of 5, states close to the triangle edge of reaction 1 become more expensive and the optimum point is shifted away from the edge. (g) The same model with a physiological upper bound on the concentration [A]. The bound defines a new triangle edge. Since this edge is not caused by thermodynamics, it can contain an optimum point, in which driving forces are far from zero and enzyme costs are kept low.

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Fig 2 shows how the enzyme demand in each reaction depends on the logarithmic reactant concentrations. To obtain a positive flux, substrate levels s and product levels p must be restricted: for example, to allow for a positive flux in reaction 2, the rate law numerator must be positive. This implies that [B]/[A] < K_eq where the reaction’s equilibrium constant K_eq is determined by the Haldane relationship, . With all model parameters set to 1, we obtain the constraint [B]/[A] < 1, i.e., ln[B] − ln[A] < 0, putting a linear boundary on the feasible region (Fig 2(c)). Close to chemical equilibrium ([B]/[A] ≈ K_eq), the enzyme demand E₂ approaches infinity. Beyond the boundary ([B]/[A] > K_eq) no positive flux can be achieved (grey region). Such a threshold exists for each reaction (see Fig 2b–2d). The remaining feasible metabolite profiles form a triangle in log-concentration space, which we call metabolite polytope (Fig 2e), and Eq (2) yields the total enzyme demand E_tot = E₁ + E₂ + E₃, as a function on the metabolite polytope. The demand increases steeply towards the edges and becomes minimal in the center. The minimum point marks the optimal metabolite profile, and via Eq (2) we obtain the resulting optimal enzyme profile.

The metabolite polytope and the large enzyme demand at its boundaries follow directly from thermodynamics. To see this, we consider the unitless thermodynamic driving force Θ = −Δ_rG′/RT [38] derived from the reaction Gibbs free energy Δ_rG′. For a given mass-action ratio Q = [B]/[A], the thermodynamic force can also be written as Θ = ln(K_eq/Q), i.e., the driving force is positive whenever Q < K_eq, and it vanishes if Q = K_eq. How is this force related to enzyme cost? A reaction’s net flux is given by the difference v = v⁺ − v⁻ of forward and backward fluxes, and the ratio v⁺/v⁻ depends on the driving force as v⁺/v⁻ = e^Θ. Thus, only a fraction v/v⁺ = 1 − e^−Θ of the forward flux acts as a net flux, while the remaining forward flux is canceled by the backward flux (Figure A in S1 Text). Close to chemical equilibrium, where the mass-action ratio approaches the equilibrium constant, i.e. Q → K_eq, the driving force goes to zero, the reaction’s backward flux increases, and the flux per unit enzyme level drops. This is what happens at the triangle edges in Fig 2. Exactly on the edge, the driving force vanishes and no enzyme level, no matter how large, can support a positive flux. The quantitative cost depends on model parameters: for example, lowering a k_cat value increases the cost of each enzyme unit, making the polytope boundary steeper and thus the optimum is shifted away from the boundary (see Fig 2f and Figure B in S1 Text).

Enzyme cost as a function of metabolite profiles

The prediction of optimal metabolite and enzyme levels can be extended to models with general rate laws and complex network structures. In general, enzyme demand depends not only on driving forces and k_cat values, but also on the kinetic rate law, which includes Michaelis-Menten constants (K_M) and allosteric regulation. Thus, one must model these factors using the available kinetic information [39, 40], or approximate them when the information is not available. For some of these parameters, genome-scale prediction methods exist [41, 42]. The rate of a reaction depends on enzyme level E, forward catalytic constant (i.e. the maximal possible forward rate per unit of enzyme, in s⁻¹), driving force (i.e., the ratio of forward and backward fluxes), and on kinetic effects such as substrate saturation or allosteric regulation. If all active fluxes are positive, reversible rate laws like the Michaelis-Menten kinetics in Eq (1) can be factorized as [22] (3)

Negative fluxes, which would complicate this formula, can be avoided by orienting all reactions in the direction of fluxes. The reversible Michaelis-Menten rate law Eq (1), for example, can be written in this separable form [22]: (4) and similar factorizations exist for reactions of any stoichiometry (see S1 Text section 2.2). The term describes the maximal reaction velocity, which is reduced, depending on metabolite levels, by condition-specific factors η^rev and η^kin (see Fig 1b), accounting for backward fluxes, incomplete substrate saturation, or saturation with product (see Table 1 for a summary of all mathematical symbols used throughout this paper). The reversibility factor η^rev can be expressed in terms of the driving force Θ ≡ −Δ_rG′/RT by the general formula η^rev = 1 − e^−Θ, which also applies to reactions with multiple substrates and products [22]. The factor η^kin depends on the rate law and thus on the enzyme mechanism considered (see S1 Text section 2.2). In some cases, it could be convenient to subdivide Eq (3) even further: the value can be decomposed into a product , where denotes the catalytic constant of a hypothetical, infinitely fast enzyme whose rate is only limited by substrate diffusion. The enzyme-specific, constant factor is a unitless number between 0 and 1. A realistic value of s⁻¹ can be obtained by considering a very fast enzymatic reaction, the breakdown of water structure around a polymer [43]. Furthermore, with some rate laws, η^kin can be further decomposed into η^kin = η^sat ⋅ η^reg, where η^reg refers to certain types of allosteric regulation (see example in Methods).

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Table 1. Mathematical symbols used.

The fitness unit Darwin (D) is a proxy for the different fitness units used in cell models. Reaction must be orientated in such a way that all fluxes are positive. To define metabolite log-concentrations, we use the standard concentration c_σ = 1 mM. For a more comprehensive list of mathematical symbols used in ECM, see Table C in S1 Text.

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The factorization in Eq 3, and any finer subdivision into factors, will lead to a subdivision of enzyme demands. Enzyme demand can be quantified as a concentration (e.g., enzyme molecules per volume) or mass concentration (where enzyme molecules are weighted by their molecular weights). If rate laws, fluxes, and metabolite levels are known, the enzyme demand of a single reaction l follows from Eq 3 as (5)

To determine the enzyme demand of an entire pathway, we sum over all reactions: E_tot = ∑_l E_l. Based on its enzyme demands E_l, we can associate each metabolic flux with an enzyme cost , describing the effort of maintaining the enzymes. The burdens of different enzymes represent, e.g., differences in molecular mass, post-translation modifications, enzyme maintenance, overhead costs for ribosomes, as well as effects of misfolding and non-specific catalysis. The enzyme burdens can be chosen heuristically, for example, depending on enzyme sizes, amino acid composition, and lifetimes (see S1 Text section 2.1). Setting (protein mass in Daltons), q will be in mg protein per liter. Considering the specific amino acid composition of enzymes, we can also assign specific costs to the different amino acids. Alternatively, an empirical cost per protein molecule can be established by the level of growth impairment that an artificial induction of protein would cause [44, 45]. Thus, each reaction flux v_l is associated with an enzyme cost q_l, which can be written as a function of flux and metabolite concentrations. From now on, we refer to log-scale metabolite concentrations x_i = ln c_i in order to obtain simple optimality problems below. From the separable rate law Eq 5, we obtain the enzyme cost function (6) for a given pathway flux v. If the fluxes are fixed and given, our enzyme cost becomes, at least formally, a function of the metabolite levels. We call it enzyme-based metabolic cost (EMC) to emphasize this fact. The cost function is defined on the metabolite polytope , a convex polytope in log-concentration space containing the feasible metabolite profiles. Like the triangle in Fig 2, the polytope is defined by physiological and thermodynamic constraints. It can be bounded by two types of faces: On “E-faces”, one reaction is in equilibrium, and enzyme cost goes to infinity; “P-faces” stem from physiological metabolite bounds. The shape of the cost function depends on rate laws, rate constants, and enzyme burdens, and its minimum points can be inside the polytope or on a P-face (see Fig 2f).

Enzyme cost minimization

The cost function q(x, v) reflects a trade-off between fluxes to be realized and enzyme expression to be minimized, where the relation between fluxes and enzyme levels is not fixed, but depends on metabolite log-concentrations x. Wherever trade-offs exists in biology, it is common to assume that evolution converges to Pareto-optimal solutions [1], i.e. cases where there are no other solution with both a higher flux and a lower cost. Therefore, we can now use this principle to predict metabolite and enzyme concentrations in cells. As with our simple model in Fig 1, the metabolite profile that minimizes the enzyme cost for a given flux, and the corresponding enzyme profile (computed using Eq 5) could be good predictions for the abundance of metabolites and enzymes in naturally evolved organisms.

The resulting method, which we call enzyme cost minimization (ECM), is a convex optimization problem and can be solved with local optimizers. Enzyme demand and enzyme cost functions, for single reactions or pathways, are differentiable, convex functions on the metabolite polytope. This convexity holds for a variety of rate laws, including rate laws describing polymerization reactions [46], and even for the more complicated problem of preemptive enzyme expression, i.e., a cost-optimal choice of enzyme levels that allows the cell to deal with a number of future conditions (see S1 Text section 3.7). If a model contains non-enzymatic reactions, this changes the shape of the metabolite polytope, but not the enzyme cost function, and the polytope remains convex, e.g., if the non-enzymatic reactions are irreversible with mass-action rate laws (see Methods). Obviously, metabolite and enzyme levels may be subject to various other constraints that are not reflected by our pathway model. To assess how easily the metabolic state can be adapted to external requirements, we can study the cost of deviations from the optimal metabolite levels. If the cost function q(x) has a broad optimum as in Fig 2, cells may flexibly realize metabolite profiles around the optimal point, and the choice of metabolite levels may vary from cell to cell. We can quantify the tolerable variations by relaxing the optimality assumptions and computing a tolerance range for each metabolite level (see Methods). To apply ECM in practice, we developed a workflow in which a kinetic model is constructed, all necessary enzyme parameters are determined by a method called parameter balancing, and optimal metabolite and enzyme levels are predicted along with their tolerance ranges. In parameter balancing [47, 48], a complete, consistent set of enzyme parameters is determined from measured values by employing prior distributions, parameter dependencies arising from thermodynamic laws, and Bayesian statistics (for details, see Methods). Different kinds of EMC functions and constraints (e.g., defining concentration ranges for specific metabolites) can be chosen. Missing data (e.g., K_M values), can thus be handled in two ways: either, by using a simplified EMC function that does not require this parameter, or by relying on parameter values chosen by the workflow.

Which factors shape the optimal enzyme profile and how?

Beyond minimizing the total enzyme cost, one can also use ECM to analyze the individual enzyme demands. When the metabolite levels are known, the demand can be directly calculated and each efficiency factor in Eq (6) reflects a different part of the cost (see Methods). Alternatively, by omitting some factors or replacing them with constant numbers 0 < η ≤ 1, simplified enzyme cost functions with fewer parameters can be obtained. For example, η^rev = 1 would imply an infinite driving force Θ → ∞ and a vanishing backward flux, η^kin = 1 implies full substrate saturation, as well as full allosteric activation and no allosteric inhibition (or no allosteric regulation at all). In these limiting cases, enzyme activity will not be reduced, and enzyme demand will be given by the capacity-based estimate , a lower bound on the actual demand. Such simplifications are practical if rate constants are unknown.

Depending on the data available (e.g., k_cat values, equilibrium constants, or even K_M values), one may choose between different cost functions with different data requirements: EMC0 (“sum-of-fluxes-based” same prefactors for all enzymes), EMC1 (“capacity-based”, setting all η = 1 and thus replacing reaction rates by the maximal velocities), EMC2 (“reversibility-based”; considering driving forces, and setting η^kin = 1), EMC3 (“saturation-based”, assuming simple rate laws depending on products of substrate or product concentrations, and including the driving forces), and EMC4 functions (“kinetics-based”; with dependence on individual metabolite levels). Details of the simplified EMC functions are given in Table 2 and Table A in S1 Text. Each EMC function is a lower bound on the subsequent functions; i.e., even if only a simplified cost function can be used, it will always yield a lower bound on the cost computed using the full EMC4 model.

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Table 2. Simplified enzyme cost functions.

By omitting some terms in Eq (5), we obtain a number of cost functions with simple dependencies on enzyme parameters and metabolite levels. Terms marked by ✓ appear explicitly in the rate and cost formulae, while other terms are omitted or set to constant values. The EMC0 function yields the sum of fluxes, EMC1 functions contain enzyme-specific flux burdens based on k_cat and h values (i.e., replacing reaction rates by their maximal velocities). EMC2 depends on metabolite levels only via the driving forces. EMC3 functions are based on simplified rate laws, and EMC4 functions capture all rate laws, possibly including allosteric regulation. The rate law denominators D^S, D^SP, D^1S, and D^1SP, and the EMC functions themselves are described in Table A in S1 Text.

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Let us consider the various simplifications in more detail. If fluxes are the only data available, we may assign identical catalytic constants and burdens to all enzymes and assume that all reactions run at their maximal velocities. Then, enzyme levels and fluxes will be proportional for all reactions, the cost function in Eq (6) will be EMC0, and the cost will be proportional to the sum of fluxes. However, catalytic constants span many orders of magnitude [25], as do molecular masses of enzymes, suggesting that EMC0 is an oversimplification. If individual and values are known, we can define an individual flux burden for each enzyme, independent of metabolite levels. Then we obtain an EMC1 cost function , which is the same as the cost weights used in FBA with flux minimization [17] or molecular crowding [19]. When k_cat values are unknown, they can be estimated [42], replaced by “typical” values [25], or bounded by the value 1/s for a very fast, but diffusion-limited enzyme. The enzyme burdens h_E can include factors like protein size, protein lifetime, covalent modifications, or space restrictions (see [20] and S1 Text section 2.1).

However, by assuming that enzymes work at their maximal rate and setting η^rev = η^kin = 1, we may obtain unrealistic results. First, the simplifying assumption η^rev = η^kin = 1 implies uncontrollable metabolic states. In a kinetic model with completely irreversible and substrate-saturated enzymes, the reaction rates would be independent of metabolite levels and the steady-state fluxes and metabolite levels would depend on finely tuned enzyme levels [15]. Random variation in enzyme levels would lead to non-steady states, with fast accumulation or depletion of intermediate metabolites. Such states are extremely fragile and thus uncontrollable. When assuming efficiencies η^rev or η^kin smaller than 1, we accept an increased cost and thereby acknowledge that control must be paid for by enzyme investments. Second, EMC1 functions underestimate all enzyme costs, and for reactions close to chemical equilibrium the errors may be quite large. For a reaction Gibbs energy of Δ_rG′ = −0.1RT, the efficiency of the catalyzing enzyme is reduced by a factor of η^rev = 1 − e^0.1 ≈ 0.1, and the demand for enzyme increases by a factor of 1/η^rev ≈ 10. To account for this decreased efficiency, we can use EMC2 functions, which include the reversibility factor . The driving forces are expressed in terms of metabolite log-concentrations Θ_l(x) and equilibrium constants, which need to be known. This factor approaches infinity as reactions reach equilibrium (i.e. where Θ_l → 0), which is what forces reactions away from equilibrium during cost minimization (see, for example, Fig 2).

The advantage of reversibility-based cost functions (EMC2) is that they are based on k_cat and equilibrium constants only. Several in-silico methods exist to estimate K_eq for virtually any biochemical reaction [41, 49] and the values can be easily obtained at http://equilibrator.weizmann.ac.il/ [50]. As in the case of EMC1, k_cat values can be estimated or set to a default constant value. Methods like MDF [15] and mTOW [23] have been developed to address exactly this situation, where detailed kinetic information is hard to obtain. We discuss the relation between EMC2 and MDF in section 4 of the S1 Text. Aside from the EMC2 function, there are other reversibility-based estimates of the enzyme cost. For instance, the enzyme demand in Fig 2 (an EMC3-function with kinetic constants, fluxes, and enzyme burdens set to 1) has the reversibility-based cost as a lower bound. Since 1 − e^−x ≤ x for all positive x, an even lower estimate is ∑_l Θ(c)⁻¹ (Figure B and Figure C in S1 Text). Some variants of FBA relate fluxes to metabolite profiles, which are then required to be thermodynamically feasible, i.e., within the metabolite polytope. ECM constrains the metabolite profiles even further: as shown in Fig 2, profiles close to an E-face are very costly and can never be optimal. This holds for EMC2 functions and for the more realistic enzyme costs, which will even be higher. Thus, regions close to E-faces can be excluded from the polytope. At P-faces, defined by physiological bounds, there will be no such increase, so the optimum may lie on a P-face (see Fig 2f). To exclude regions near E-faces, we simply define lower bounds for all driving forces (see S1 Text section 7.1). These bounds can be used both in ECM or in thermodynamic FBA to reduce the search space.

The next logical step is to relax the assumption that η^kin = 1. Just like the reversibility factor η^rev, the kinetic factors η^sat and η^reg can be used to define tighter constraints on metabolite levels. However, unlike η^rev, the kinetic terms may take various forms and contain many kinetic parameters. To obtain simple, but reasonable formulae in EMC3, we first consider rate laws in which enzyme molecules exist only in three possible states: unbound, bound to all substrate molecules, or bound to all product molecules. Metabolites affect the rate only through the mass-action terms S = ∏_i(s_i/K_Mi) (for substrates) and P = ∏_j p_i/K_Mj (for products), and the degree of saturation is determined by η^sat = S/(1 + S + P), where the formula effectively has two Michaelis-Menten constants: one for substrates and one for products (which are equivalent to the product of all K_Mi and all K_Mj values). EMC3 represents a balance between complexity and requirement for kinetic parameters, and is a practical cost function if simple, realistic rate laws are desired. The EMC4 functions, finally, represent general rate laws and η^kin can take many different forms depending on mechanism and order of enzyme-substrate binding. Again, for simplicity, we resort to analyzing only a small set of relatively general templates for EMC4, known as convenience kinetics [51] or modular rate laws [21]. Nevertheless, our formalism allows a much wider range of rate laws, and we consider EMC4 a wild-card cost function that covers almost any reasonable rate law (see S1 Text section 2.2 for more details).

Enzyme and metabolite levels in E. coli central metabolism

To benchmark our optimality-based prediction of metabolite, we applied ECM to a model of E. coli central metabolism, containing three major pathways: glycolysis, the pentose phosphate pathway, and the TCA cycle (see Fig 3a, and Methods for modeling details). Fig 3b–3d compares predicted enzyme profiles to measured protein levels [53]. The absolute values of predicted enzyme levels arise directly from the model, using the fluxes reported in [52], while cellular protein concentrations were obtained from proteomics data (measured in similar conditions [53]) and assuming an average cell volume of ~ 1 fL (10⁻¹⁵ liters) [54]. EMC4 predicts values that are of the right order of magnitude and reflect differences in enzyme levels along the pathways. The prediction error of 0.42 for enzyme levels (RMSE: root mean square error on a log₁₀-scale) corresponds to a typical fold error of 10^RMSE = 2.6. In line with the measured protein levels, the predicted enzyme levels tend to be larger in glycolysis than in TCA and pentose phosphate pathway, reflecting the larger fluxes and less-favorable thermodynamics. All predictions including metabolite concentrations, thermodynamic forces and c/K_M ratios can be found online at the accompanying website www.metabolic-economics.de/enzyme-cost-minimization/.

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Fig 3. Predicted enzyme levels in E. coli central metabolism.

(a) Network model with pathways marked by colors. Flux magnitudes are represented by the arrows’ thickness. (b) The ratio flux/ (EMC1) as a predictor for enzyme levels. Points on the dashed line would represent precise predictions. (c) Enzyme levels predicted by the reversibility-based EMC2(S) function. Vertical bars indicate tolerance ranges obtained from a relaxed optimality condition (allowing for a one percent increase in total enzyme cost). (d) Enzyme levels predicted with EMC3 function representing fast substrate or product binding. (e) Enzyme levels predicted with EMC4 function based on the common modular rate law [21]. In all sub-figures (b-e), RMSE is the root mean squared error (in log₁₀-scale) of our predictions compared to the measured enzyme levels, and r stands for the Pearson correlation coefficient. Predictions are based on fluxes from [52], and K_M values from BRENDA [40], and compared to protein data from [53]. For metabolite predictions, see Figure E in S1 Text.

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We note that predicted enzyme levels become more accurate as more complex cost functions are used, with a prediction error decreasing monotonically from 1.35 with EMC0 to 0.42 with EMC4. The capacity-based enzyme cost (EMC1) assumes that enzymes operate at full capacity () and therefore underestimates all enzyme levels (Fig 3b). In reality, many reactions in central metabolism are reversible and many substrates do not reach saturating concentrations. When taking these effects into account, predictions come closer to measured enzyme levels (Fig 3c–3e). For instance, FUM (fumarase, fumA) and MDH (malate dehydrogenase) have a much higher predicted level in EMC2-4 than in EMC1 as the reversibility-based costs account for their low driving forces. Similarly, the predicted levels of two pentose-phosphate enzymes (ribulose-5-phosphate epimerase RPE and ribose phosphate isomerase RPI) are much higher in EMC3 and EMC4 because of their low affinity for the substrate ribulose-5-phosphate (Ru5P). In some cases, however, the more complex EMC4 fails to improve the prediction over the simpler methods. For instance, the 6-phosphogluconolactonase (PGL) and phosphoglycerate kinase (PGK) reactions are underestimated by all EMC functions, perhaps due to regulation mechanisms that reduce activity such as allosteric inhibition. In very few cases, EMC4 overestimates the level of an enzyme that has a more precise prediction in EMC1-3, e.g. phosphofructokinase (PFK). Overall, the EMC4 function performs substantially better on average than the simpler cost functions even though it relies on a larger set of parameters, many of which are known with low certainty. Moreover, EMC4 predicts well the total of all enzyme levels (0.64 mM, compared to the measured value—0.62 mM), while the other EMC function underestimate this value (0.17, 0.24 and 0.43 mM for EMC1, EMC2 and EMC3 respectively). To test the sensitivity of our results to the choice of parameters, we performed random sampling of kinetic constants, fluxes and fixed metabolite levels, and analyzed the effect on the enzyme level predictions (see Methods and Fig 4a). We further tested the sensitivity to our choice of proteomic data, by repeating the entire analysis using measured enzyme concentrations from [55] and reached essentially the same findings (see Figure F in S1 Text).

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Fig 4. Prediction uncertainties and evidence for cost optimality.

(a) Uncertainty of predicted enzyme levels due to uncertain model parameters. A hundred sets of kinetic model parameters were generated by Monte Carlo sampling. Due to the multivariate distribution used for sampling, each parameter set satisfies the Haldane relationships. At the same time, fluxes were sampled according to their experimental error bars (typically around 15% of the measured flux), and the fixed metabolite concentrations were randomly varied in a ± 5% range. The resulting predicted enzyme levels, computed using the EMC4cm score, are shown by small gray dots. Solid blue circles show medians, and error bars show 25% and 75% quantiles; empty red circles show the original ECM4cm prediction, i.e. without sampling. (b) The enzyme levels in E. coli appear to be cost-optimized. We compared the ECM solution (with ECM4cm score) to enzyme profiles obtained from metabolite profiles randomly sampled in the metabolite polytope. The ECM solution (red) or metabolite profiles sampled in a close neighborhood (pink) yield significantly better enzyme predictions (quantified by RMSE, compare Fig 3) than metabolite profiles sampled in the entire polytope (light blue). The total enzyme cost (on x-axis) represents the sum of weighted enzyme concentrations (in mM); the weight of an enzyme is given by its amino chain length, divided by the median chain length of all enzymes considered.

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Finally, we tested whether our kinetic model can also predict enzyme levels without the assumption of cost optimality: to do so, we randomly sampled feasible metabolite profiles from the metabolite polytope, computed the resulting enzyme profiles, and compared them to proteomic data. It turned out that the cost-optimal metabolite profile, or similar profiles, yielded significantly better predictions than metabolite profiles sampled from a broader range (see Methods and Fig 4b). This supports the hypothesis that cost-optimality shapes the metabolic state in E. coli.

Although ECM puts enzymes on a pedestal due to their relatively high cost, the metabolite concentrations are key to minimizing that cost. One would thus expect to find good correspondence between the predicted metabolite profile and concentrations measured in vivo, especially when predictions of enzyme levels are good. Since some EMC functions leave metabolite levels underdetermined, we penalized very high or low metabolite concentrations by adding a second, concentration-dependent objective to the optimization problem. In particular for EMC0 and EMC1, this regularization term is the only term—aside from global constraints—that determines the metabolite concentrations as they do not affect enzyme cost whatsoever. In all other cases, the term mostly influences metabolites that have a minimal effect on the cost. Comparing the EMC metabolite prediction with in-vivo experimental data, as shown in Figure E in S1 Text, the predicted metabolite levels are in the correct scale. Similar to enzyme level predictions, EMC4cm has the smallest prediction error—about 0.62 (corresponding to a typical fold error of 4.1).

We can now use EMC analysis to rationalize cellular enzyme levels. Fig 5 (like the scheme in Fig 1b) shows the specific contributions to enzyme demand for each reaction. The reversibility cost terms provided by EMC2s (purple bars in Fig 5a) improve the enzyme demand predictions in most cases, compared to the basic capacity-based costs. However, the EMC4cm predictions show that saturation-based costs (orange bars in Fig 5b) are often larger than the reversibility costs, and they improve the predictions even more. For practical cost estimates, for example when computing flux burdens for FBA, we can conclude that multiplying the experimentally determined k_cat values by reversibility factors will likely improve the fidelity of FBA predictions. For more details, see S1 Text section 4.

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Fig 5. Enzyme demand in central metabolism.

(a) Measured fluxes for all reactions (black dots on top) lead to an enzyme demand (bottom). The enzyme demand, predicted by using the reversibility-based EMC2s cost function, can be split into factors representing enzyme capacity and thermodynamics (see Methods). Bars show predicted enzyme levels in mM for individual enzymes on logarithmic scale. Yellow dots denote measured enzyme levels (in μM). Note that the bars do not represent additive costs, but multiplicative cost terms on logarithmic scale; therefore, the relevant feature of the blue bars is not their absolute lengths, but their differences between enzymes. (b) The kinetics-based EMC4cm cost function includes saturation terms and yields more accurate predictions. Starting from the capacity cost (in blue), the reversibility (purple) and saturation (red) terms increase the enzyme demands and decrease the variability between enzymes (on log-scale). Note that flux data (circles) and protein data (yellow dots) are identical in both plots.

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Metabolite polytope and enzyme cost functions

A metabolic network with given flux directions, equilibrium constants, and metabolite bounds defines the metabolite polytope. This convex polytope in the space of log-concentrations x_i = ln c_i represents the set of feasible metabolite profiles. The flux profile used can be stationary (e.g. determined by FBA or ¹³C MFA) or non-stationary (e.g. from dynamic ¹³C labeling experiments [66]). If the provided flux directions are thermodynamically infeasible, the metabolite polytope will be an empty set, . The faces of the metabolite polytope arise from two types of inequality constraints. First, the physical ranges of metabolite levels define a box-shaped polytope (bounded by P-faces). Some metabolite levels may even be constrained to fixed values. Second, each reaction must dissipate Gibbs free energy, and to make this possible, driving forces and fluxes must have the same signs (Θ_l ⋅ v_l > 0), and thus . The resulting constraints define E-faces of the metabolite polytope (representing equilibrium states, Θ_l = 0). Close to these faces, enzyme cost goes to infinity.

Separable rate laws and enzyme cost functions

According to Eq (3), reversible rate laws can be factorized into four terms: the enzyme level E, its forward catalytic constants , and two efficiency factors [22]. In Fig 6 we add a non-competitive allosteric inhibitor x. While the enzyme level and are not directly affected by the concentration of metabolites (although can vary with conditions such as pH, ionic strength, or molecular crowding in cells), the efficiency factors are concentration-dependent, unitless, and can vary between 0 and 1. The reversibility factor η^rev depends on the driving force (and thus, indirectly, on metabolite levels), and the equilibrium constant is required for its calculation. The saturation factor η^sat depends directly on metabolite levels and contains the K_M values as parameters. Allosteric regulation yields additive or multiplicative terms in the rate law denominator, which in our example can be captured by a separate factor η^reg. In general, η^sat and η^reg can be combined into one kinetic factor η^kin, as depicted in Eq 6.

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Fig 6. Rate law and enzyme demand of reversible Meichalis-Menten reactions.

For a reaction S ⇌ P with reversible Michaelis-Menten kinetics, a driving force θ = −Δ_rG′/RT, and a prefactor for non-competitive allosteric inhibition, the rate law can be written as with inhibitor concentration x. In the example, with non-competitive allosteric inhibition, the kinetic factor η^kin could even be split into a product η^sat ⋅ η^reg. The first two terms in our example, , represent the maximal velocity (the rate at full substrate-saturation, no backward flux, full allosteric activation), while the following factors decrease this velocity for different reasons: the factor η^rev describes a decrease due to backward fluxes (see Figure A in S1 Text) and the factor η^kin describes a further decrease due to incomplete substrate saturation and allosteric regulation (see Fig 1b). The inverse of all these terms appear in the equation for enzyme demand, q, which is given by the enzyme level multiplied by the burden of that enzyme, h_E.

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The second equation in Fig 6 describes the enzyme cost for a flux v, and contains the terms from the rate law in inverse form multiplied by the enzyme burden h_E. The left-hand part of the equation, , defines a minimum enzyme cost, which is then increased by the following efficiency factors. Again, 1/η^kin can be split into 1/η^sat ⋅ 1/η^reg. By omitting some of these factors, one can construct simplified enzyme cost functions with higher specific rates, or lower enzyme demands (compare Fig 1b). Since both rate and enzyme demand are a product of several terms, it is convenient to depict them as a sum on a logarithmic scale (Fig 7), where the simplified functions are seen as upper/lower bounds on the more complex rate/demand functions.

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Fig 7. The conversion between fluxes and enzyme levels, in both directions.

(a) Starting from the logarithmic enzyme level (dashed line on top), we add the terms , log η^rev, and log η^kin, and obtain better and better approximation of the rate. In the example shown, has a numerical value smaller than 1. The more precise approximations (with more terms) yield smaller rates. The EMC4 arrows refer to other possible rate laws with additional terms in the denominator. (b) Enzyme demand is shaped by the same factors (see Eq (5)). Starting from a desired flux (bottom line), the predicted demand increases as more terms are considered.

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Enzyme cost minimization can be formulated as a convex optimality problem for metabolite levels

Enzyme cost minimization (ECM) uses a metabolic network, a flux profile v, kinetic rate laws, enzyme burdens, and bounds on metabolite levels to predict optimal metabolite and enzyme concentrations. The enzyme cost of reactions or pathways is a convex function on the metabolite polytope (proof in S1 Text section 3.2), that is, a log-scale metabolite vector x, linearly interpolated between vectors x_a and x_b, cannot have a higher cost than the interpolated cost of x_a and x_b. Convexity also holds for cost functions h(E) that are non-linear, but convex over E. Some EMC functions are strictly convex (i.e., Eq. (S18) holds with a < sign instead of ≤), while others are not (e.g. EMC2). The most simplified EMC functions are actually constant (as in EMC0 and EMC1). To find an optimal state, we choose an EMC function and minimize the total enzyme cost within the metabolite polytope. Optimal metabolite profiles, enzyme profiles, and enzyme costs are obtained by solving the enzyme cost minimization (ECM) problem (7)

The total cost q(x, v) (defined in Eq (6)) is the sum of enzyme costs given by EMC functions. Since q(x) and the metabolite polytope itself are convex, ECM is a convex optimization problem. The optimal enzyme levels depend on external conditions and have to be recalculated after any change in external metabolite levels. There are cases where q(x) is convex, but not strictly convex, and therefore Eq (7) will have a continuum of optimal metabolite solutions. This holds, in particular, for EMC1 scores, which are independent of metabolite levels, and for EMC2 scores, which only depend on reaction Gibbs free energies, i.e., on some linear combinations of the logarithmic metabolite levels. In such cases, to enforce a unique solution one may add a strictly convex side objective that scores the log-metabolite levels, e.g., a quadratic function favoring metabolite levels close to some typical concentration vector , where λ is a small, heuristically chosen weighing factor (see S1 Text section 3.3). Such extra objectives can be justified biologically, e.g. by assuming that intermediate metabolite levels give cells more flexibility to adapt to perturbations. Strict convexity not only simplifies numerical calculations, but it also guarantees that the optimization problem has a unique solution. In fact, metabolite polytope and cost functions remain convex even under various modifications of the problem. When adding constraints on the total metabolite level, on weighted sums of metabolite levels, or on weighted sums of enzyme levels, the metabolite polytope is intersected with curved manifolds (since we are dealing with concentrations in logarithmic scale) but remains convex (S1 Text section 3.4). Finally, we can consider the more complicated problem of preemptive enzyme expression, where a fixed enzyme profile and allosteric inhibition must allow a cell to realize different flux distributions under different conditions. Also this problem is convex (S1 Text section 3.7). If a model contains non-enzymatic reactions (or non-enzymatic processes such as metabolite diffusion out of the cell or dilution in growing cells), each such reaction leads to an extra constraint on the metabolite polytope (S1 Text section 3.8). A known flux in an irreversible diffusion or dilution reaction fixes the concentration of one metabolite. In the presence of irreversible non-enzymatic reactions with mass-action rate laws, the polytope is intersected by a subspace. In both cases, the resulting sub-polytope may be empty, i.e., the given flux distribution will not be realizable.

Non-stationary states and the importance of boundary metabolites

Flux balance analysis and kinetic models rely on the assumption that certain metabolites are mass-balanced: in FBA, this assumption, together with stationarity, defines the set of steady-state fluxes; in kinetic models, the mass-balanced metabolites are the ones whose dynamics is described by the system equations. ECM, in contrast, assumes fluxes to be given and makes no assumption about mass balances. If the fluxes in our pathway model lead to a mass imbalance in a metabolite, we may still assume that the entire cell is in stationary state, but that mass balances are reached with the help of other pathways that are not part of our model. Alternatively, we may assume that the metabolite is actually not mass-balanced and that we are describing a non-stationary, transient state. In both cases, ECM is fully applicable as long as metabolic fluxes are predefined and loop-less (in order to be realizable by a thermodynamically consistent state [67]).

A key point in ECM is the choice of metabolite levels on the model boundary. If we predefine all these metabolite levels, our pathway will be “isolated” from the rest of the network, and any information about the surrounding network can be safely ignored. In our E. coli model, ATP is one such important boundary metabolite: if we allowed for a lower ATP level, ATP could be produced at a lower enzyme cost because of the more favorable driving forces. If we do not fix the ATP concentration, but define an allowed range, ECM would choose the lowest possible ATP level; thus, if the allowed ATP range is too broad, no meaningful predictions can be expected. In a model of ATP-consuming biosynthesis pathways, the situation would be exactly the opposite: here, it is a high ATP level that would lead to higher driving forces and to lower enzyme requirements. Whole-cell models contain both types of pathways—ATP-producing and ATP-consuming ones. In such a model, ECM could predict some meaningful compromise, i.e. an intermediate ATP level that minimizes the enzyme cost of ATP production plus the enzyme cost of biosynthesis. Since in our central metabolism model there are only 3 reactions that produce or consume ATP, it is unlikely that so few reactions would be representative of the cost tradeoff between the dozens of enzymes that use ATP in the full metabolic network. Therefore, we chose to fix the ATP level to its measured value (∼3 mM).

Our use of small-scale pathway models, which ignore most of the metabolic network, is therefore justified: as long as we predefine all metabolic fluxes and all metabolite levels at the pathway boundary, pathways can be modeled separately and the models can later be combined without any adjustment. This makes the ECM approach fully modular. All the input parameters (kinetic parameters, fluxes and boundary concentrations) are directly obtained from measurements without any further tuning. By setting all relevant fluxes and boundary metabolite concentrations to their measured values, we isolate our submodel from any effects that the surrounding network might have on predicted enzyme cost. Finally, there may be metabolites that are “free” in a model, but that affect enzymes that are not in the model (e.g. pyruvate, which affects 30 other enzymes). By neglecting these enzymes, we ignore some of the complex compromises between them, and the predicted metabolite concentration may be wrong. In our specific case of central metabolism, whose enzymes comprise a very large fraction of E. coli’s proteome, this effect is probably not so severe. In any case, this problem can be easily fixed by imposing constraints or fixed concentrations for central metabolites such as pyruvate (similar to how we deal with ATP and other co-factors).

Tolerance ranges for nearly optimal solutions

Evolution could tolerate non-optimal enzyme costs; this tolerance depends on population dynamics and can sometimes be quite significant, e.g. in small isolated communities. To compute realistic tolerance ranges for the ECM problem, we start from the optimum (total cost q) and choose a tolerable cost q^tol (e.g., one percent higher than the optimal cost). This defines a tolerable region in . A tolerance range for each metabolite is defined by the minimal and maximal values the metabolite can show within . Tolerance ranges for enzyme levels are defined in a similar way. Alternatively, tolerance ranges and nearly optimal solutions can be estimated from the Hessian matrix (see S1 Text section 7.3).

Sensitivity analysis with respect to kinetic constants

The predicted enzyme and metabolite levels depend on the kinetic model chosen, and in particular on the kinetic constants (k_cat and K_M values). Errors or uncertainties in these constants will cause errors or uncertainties in the predicted enzyme profiles. To estimate these uncertainties, we considered a joint distribution of all model parameters, describing both the uncertainties of individual parameters and the correlations between dependent parameters. This probability distribution was directly obtained from parameter balancing (S1 Text section 5.2). We sampled the kinetic parameters from this distribution, sampled metabolic fluxes according to their experimental mean values and standard deviations, and varied the fixed metabolite levels in a ± 5% range around their standard values. Then we applied ECM on each of the sampled parameter sets, and gathered statistics for the optimal enzyme and metabolite levels. Fig 4(a) shows the distributions of the predicted enzyme levels. For narrow parameter distributions, the mean values, variances, and covariances of the predicted enzyme levels can even be computed, approximately, from the ECM solution with standard parameters (see S1 Text section 3.5). Enzyme uncertainties caused by parameter uncertainties should not be confused with the tolerance ranges described before. The tolerance ranges are always associated with sub-optimal solutions, i.e., enzyme profiles with a higher total cost; enzyme variations caused by parameter variation, in contrast, can go both ways and may sometimes decrease the cost.

Testing the hypothesis of cost-optimal metabolite profiles

Our enzyme level predictions rely on two main assumptions: a mechanistic model that defines a quantitative relation between metabolite levels, enzyme levels, and fluxes, and an optimality assumption stating that metabolite levels are optimized for a minimal total enzyme cost. To test whether such cost optimality holds in reality, we used the same mechanistic model and predicted enzyme levels based on feasible, randomly sampled metabolite profiles. We first sampled metabolite profiles around the ECM optimum by adding normally distributed random numbers (standard deviation 0.05, for metabolite levels on natural log scale); then we sampled metabolite profiles in a much wider range, by sampling convex combinations of extreme points in the metabolite polytope (i.e., points realizing minimal or maximal values of individual metabolite concentrations). As shown in Fig 4(b), the metabolite profiles close to the ECM optimum yield significantly better enzyme level predictions than broadly sampled metabolite profiles. The fact that predictions from the same kinetic model, without the optimality assumption, become much worse provides strong support for cost-optimality as a principle in living cells.

Workflow for model building and enzyme prediction

To predict enzyme and metabolite levels in metabolic pathways we developed an automated workflow (Fig 8). In a consistent model, all parameters must satisfy Wegscheider conditions for equilibrium constants [68] and Haldane relationships between equilibrium constants and rate constants [69]. The kinetic constants used in rate laws should represent effective parameters, which may differ from “ideal” parameters, e.g., by crowding effects. However, since measured parameter values are usually incomplete and inconsistent, parameter balancing [47] is used to translate measured kinetic constants into consistent model parameters. Based on a network and given fluxes, the software extracts relevant data from a database (thermodynamic constants, rate constants, fluxes, and protein sizes; metabolite and protein levels for validation), determines a consistent set of model parameters, builds a kinetic model, and optimizes enzyme and metabolite profiles for the EMC function chosen. To assess the effects of parameter variation, parameter sets can be sampled from the posterior distribution as described above. The workflow has been implemented in MATLAB and uses Systems Biology Markup Language (SBML) for model structures and the SBtab table format for numerical data [70].

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Fig 8. Data integration in the ECM-based modeling workflow.

After collecting all available kinetic and thermodynamic data and mapping them onto the network model, we use parameter balancing to obtain a consistent, complete set of kinetic constants. For a fully parameterized kinetic model, the metabolite and enzyme levels must be determined. We compute them by enzyme cost minimization with predefined metabolic fluxes (obtained from experiments or computationally). Finally, the predicted values are validated with measured metabolite and protein concentrations.

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E. coli model

The model shown in Fig 3 was built automatically from a list of chemical reactions in E. coli central metabolism (for details, see S1 Text section 6). Equilibrium constants were estimated using the component contribution method [41], kinetic constants ( and K_M values) were obtained from the BRENDA database (after which each value was curated manually), and a complete, globally consistent parameter set was determined by parameter balancing. During ECM, all metabolite levels were limited to predefined ranges, and the levels of cofactors and some other metabolites were fixed at experimentally known values. To compute tolerances for predicted metabolite and enzyme levels, we defined an acceptable enzyme cost, one percent higher than the minimal value, and determined ranges for metabolite levels that agree with this cost limit. The enzyme cost function accounts for protein composition, giving different costs to different amino acids. However, models with equal cost weights for all proteins, or with size-dependent protein costs yielded similar results (results are provided on the website). Data, model, and MATLAB code for ECM can be obtained from www.metabolic-economics.de/enzyme-cost-minimization/.