The Influence of Crowding Conditions on the Thermodynamic Feasibility of Metabolic Pathways

Abstract

Intracellular reactions are carried out in a crowded medium where the macromolecules occupy ∼40% of the total volume. This decrease in the available volume affects the activity of the reactants. Scaled particle theory is used for the estimation of the activity coefficients of the metabolites, and thereby for the assessment of the impact of the presence of background molecules, on the estimation of the Gibbs free energy change (Δ_rG) of the reactions. The lactic acid pathway and the central carbon metabolism of Actinobacillus succinogenes for the production of succinic acid from glycerol have been used as illustrative case studies. Results suggest the importance of maintaining intracellular crowded regions to favor the feasibility of a pathway that in other circumstances would be infeasible. Moreover, the crowding conditions may change the directionality of reactions and can modify the feasible range of fluxes estimated for a metabolic system compared with those obtained at standard biological conditions.

Introduction

Stoichiometric-based models are valuable tools for the estimation of the likely range of fluxes in a metabolic network under certain environmental conditions (1, 2, 3, 4).

The use of thermodynamic constraints allows the calculation of fluxes in the direction of the Gibbs free energy drop (5). Thermodynamic analysis indicates the feasibility, directionality, and reversibility of the reactions involved in the network by the estimation of the corresponding change in Gibbs free energy Δ_rG, which depends on intracellular conditions such as pH, ionic strength (I), temperature (T), activity of the metabolites, and also the intracellular crowding conditions (6).

The cytoplasm contains several species including macromolecules (e.g., the enzymes) and solutes whose concentrations are not considered high but all together may occupy ∼40% of the cellular space (7). Due to the impenetrability of the molecules, this reduces the available volume for the motion and reaction of the species (excluded volume effect). Hence, the macromolecular crowding may affect the equilibrium and rate of the reactions involving changes in the available volume (7).

The thermodynamic activity of a metabolite a′ (equivalent to the number of moles per available volume) is related to its concentration C (moles per total volume) through the activity coefficient of metabolite, γ, a′ = γC/C^st. Here γ is the ratio between the total volume and the available volume, while C^st is the standard concentration.

The activity coefficient, γ, encompasses the deviations from the ideal behavior of a mixture caused by all kinds of interactions among the molecules, e.g., interactions due to crowding and electrostatics. Under crowding conditions such interactions are also due to steric repulsion (7). This article focuses on the crowding effect on the thermodynamics of biochemical reactions.

Assuming that all the species in a solution are volumeless particles, then the available volume is equal to the total volume of the solution, and therefore γ = 1. However, in a more realistic scenario where the available volume is reduced due to the presence of background molecules of different size (and therefore γ > 1), the activity of the reactants is increased. This could shift an infeasible reaction (Δ_rG > 0 for a positive net reaction flux) to a feasible one (Δ_rG < 0) even at low reactant concentrations.

Several attempts have been made to integrate the thermodynamic information to flux balance analysis or to other stoichiometric methods (8, 9, 10, 11, 12, 13, 14, 15, 16), but none of them has developed an explicit relationship between the crowding conditions and the thermodynamic analysis.

Nevertheless, the crowding effect has been incorporated to flux balance analysis (17) by restricting the maximum number of macromolecules or the concentration of cytoplasmic enzymes. Through this restriction and the use of kinetic parameters, the maximum metabolic flux attainable for a reaction is estimated. However, the influence of crowding conditions on the thermodynamic feasibility of a pathway and its corresponding flux distribution has not been analyzed.

Scaled particle theory (SPT) (18, 19) provides the theoretical framework to calculate the activity coefficients of species in a hard-sphere mixture as a function of the concentration and the radii of the metabolites. SPT has been widely used to examine the influence of macromolecular crowding on the stability (20) and thermodynamic activity (21) of globular proteins, solvation (22), and the estimation of the osmotic pressure (23).

The aim of this article is to illustrate the qualitative effect of the macromolecular crowding on thermodynamic analysis of metabolic pathways, and its further impact on the prediction of ranges of fluxes. To achieve this, we used SPT for the estimation of the activity coefficients and we subsequently coupled the crowding effect with thermodynamically constrained flux variability analysis, formulated as a nonlinear optimization problem.

Our illustrative case studies to demonstrate the excluded volume effects on the prediction of Gibbs free energies and ranges of fluxes of a metabolic system are the lactic acid metabolic pathway, and the central metabolism of Actinobacillus succinogenes for the production of succinic acid from glycerol, which is an industrially relevant biochemical system with important implications in biorefinery engineering for the valorization of the by-product crude glycerol of the biodiesel manufacture (24, 25, 26).

Materials and Methods

The simultaneous solution of the mass conservation equation and the thermodynamic constraints is fundamental for the calculation of metabolic flux distributions whose components do not violate the second law of thermodynamics.

Mass balance constraints

As with all stoichiometric models, this methodology is based on the assumption that the intracellular reactions are at steady state, and they are carried out in a homogeneous medium hence

$$\mathbf{S}\times \mathbf{v}=0,$$ (1)

where the m × (n + N) stoichiometric matrix S contains the information of m intracellular metabolites, and the n + N reactions involved in the metabolic network (n intracellular reactions and N reactions for the transport of metabolites outside/inside the cell), while v is the (n+N) × 1 flux vector.

Incorporation of the crowding conditions on thermodynamic constraints

The feasibility of a reaction in a certain direction is assessed by its Δ_rG value. The Δ_rG of a metabolic reaction can be split into the intracellular biochemical reaction contribution (Δ_rG_br) and the (coupled) transport reaction of some metabolites across the cell membrane (Δ_rG_tr),

$${\Delta }_{r}G={\Delta }_r{G}_{br}+{\Delta }_r{G}_{tr},$$ (2)

where

$${\Delta }_r{G}_{br}=\sum _{i=1}^{\text{Products}}{\eta }_i\left[{\Delta }_f{G}_i^{0\prime }+RT\text{ln}{a}_i^{\prime }\right]-\sum _{i=1}^{\text{Reactants}}{\eta }_i\left[{\Delta }_f{G}_i^{0\prime }+RT\text{ln}{a}_i^{\prime }\right],$$ (3)

$${\Delta }_r{G}_{tr}={\eta }_{H^{+}}RT\text{ln}\left({10}^{-p{H}_c+p{H}_e}\right)+{\eta }_{H^{+}}F\Delta \varphi .$$ (4)

Here, η_i is the stoichiometric coefficient of the metabolite i involved in the intracellular reactions, and a_i′ its thermodynamic activity. Δ_fG_i⁰′ indicates the transformed Gibbs free energy of formation of metabolite i (at a defined constant pH value, and ionic strength I) (27). Because Δ_fG_i⁰′ is calculated at the specified (constant) pH, the hydrogen ion H⁺ is not included in the stoichiometry of the intracellular reactions (27). In this article each metabolite is represented by its pseudoisomer group.

The pseudoisomer group i consists of all the protonated species j formed by the dissociation of metabolite i in aqueous solutions (27). The equations used to estimate Δ_fG_i⁰′ are summarized in Appendix A.

An example of intracellular reactions coupled to the transport of metabolites, e.g., the transport of hydrogen ion H⁺ across the membrane, is the production of succinic acid (succ) from fumarate (fum) by the fumarate reductase (28):

$$\begin{array}{c}fu{m}_{\text{c}}+{\text{NADH}}_{\text{c}}+{\text{H}}_{\text{c}}^{+}\leftrightharpoons suc{c}_{\text{c}}+{\text{NAD}}_{\text{c}}^{+}\\ 2{\text{H}}_{\text{c}}^{+}\leftrightharpoons 2{\text{H}}_{\text{e}}^{+}.\end{array}$$ (5)

In Eq. 5, the fumarate reductase has been lumped to the NADH dehydrogenase.

Equation 4 only works for the H⁺ transport from outside the cell into the cell, but a more general transport equation can be found in Jol et al. (29). Here, η_H+ is used to represent the stoichiometric coefficient or number of H⁺ crossing the membrane, and F is the Faraday constant (equivalent to 2.306 × 10⁻² kcal mV⁻¹ mol⁻¹).

The Δ_rG_tr value is a function of the H⁺ concentration difference (or pH) between both sides of the cell membrane (first terms of the right-hand side of Eq. 4).

The second term of the right-hand side of Eq. 4 is the electrical potential difference of the membrane. The difference between the internal potential of the membrane and the external one (Δϕ) is calculated as (5)

$$\Delta \varphi =33.33\left(p{H}_c-p{H}_e\right)-143.33,$$ (6)

where pH_c and pH_e are the intracellular and extracellular medium pH, respectively.

As shown in Eqs. 2‒4, Δ_rG is a function of the metabolites’ activity a_i′, defined as

$$a_i^{\prime }={\gamma }_i\frac{C_i}{C_i^{st}},$$ (7)

where C_i is the molar concentration of the metabolite i (i.e., moles per total volume), and C^st_i is its standard concentration equal to 1 M.

The activity coefficient γ_i represents the nonideal deviations raised by molecular interactions. For example, when an electrolyte is dissociated into its component ions, γ_i can be estimated using the Debye-Hückel equation (in fact, this expression is used to calculate Δ_fG_i⁰′ at different I, from its standard Gibbs free energy of formation Δ_fG_i⁰ at the biological standard conditions pH = 7, I = 0 M, and T = 298.15 K).

Assuming hard-core molecules, and no molecular interactions other than steric repulsion, γ_i can be estimated using SPT.

The activity of the solvent, water in this case, a_water′ can be calculated from the difference between its chemical potential in the solution (μ_water,sol) and the chemical potential of pure water (μ_{pure_water}) (30):

$$k_{\text{B}}T\text{ln}{a}_{\text{water}}^{\prime }={\mu }_{\text{water,sol}}-{\mu }_{\text{pure}\_\text{water}}.$$ (8)

According to SPT, the chemical potential of species i, μ_i, is defined as (18, 19)

$${\mu }_i=k_{\text{B}}T\text{ln}\left(\frac{{\rho }_i{h}^3}{{\left(2\pi m_i{k}_{\text{B}}T\right)}^{\frac{3}{2}}}\right)+k_{\text{B}}T\text{ln}{\gamma }_i,$$ (9)

where h is Planck’s constant (equivalent to 6.6260 × 10⁻²⁷ g cm⁻² s⁻¹), k_B is Boltzmann’s constant (equivalent to 1.3806 × 10⁻²⁷ g cm² s⁻² K⁻¹), ρ_i is the density number (molecules per cm³), and m_i is the mass of one molecule i (g per molecule).

Using Eq. 9 to calculate μ_water,sol and μ_{pure_water} and substituting them in Eq. 8, we get

$$a_{\text{water}}^{\prime }=\frac{{\rho }_{\text{water,sol}}{\gamma }_{\text{water,sol}}}{{\rho }_{\text{pure}\_\text{water}}{\gamma }_{\text{pure}\_\text{water}}},$$ (10)

where ρ_{pure_water} is proportional to water concentration 55.34 M, i.e., ρ_{pure_water} = 3.3327 × 10⁻²² molecules per cm³.

Scaled particle theory

The mathematical derivation of SPT (18, 19) is based on the calculation of the probability to find an empty cavity in an arbitrary location of the mixture or solution analyzed, where a molecule of certain size and shape can fit without overlapping with other molecules.

SPT allows the analysis of mixtures of molecules of different size, but with the same shape. In this article we assume that all the metabolites are hard spheres whose radii are proportional to the cubic root of their molecular weight. Therefore, the activity coefficient of the metabolite i (γ_i) can be expressed as (18, 19)

$$\text{ln}{\gamma }_i=-\text{ln}\left(1-S_3\right)+\left[\frac{6S_2}{1-S_3}\right]r_i+\left[\frac{12S_1}{1-S_3}+\frac{18S_2^2}{{\left(1-S_3\right)}^2}\right]r_i^2+\left[\frac{8S_0}{1-S_3}+\frac{24S_1{S}_2}{{\left(1-S_3\right)}^2}+\frac{24S_2^3}{{\left(1-S_3\right)}^3}\right]r_i^3,$$ (11)

where S_x (0 ≤ x ≤ 3) is given by

$$S_x=\frac{\pi }{6}\sum _{i=1}^m{\rho }_i{\left(2r_i\right)}^x$$ (12)

$${\rho }_i=\frac{C_i{N}_A}{1000}$$ (13)

$$r_i={\left(\frac{M_i{\upsilon }_i}{\frac{4\pi }{3}{N}_A}\right)}^{\frac{1}{3}},$$ (14)

where ρ_i represents the density number (molecules per cm³), r_i the radius (cm), υ_i the specific volume (cm³ g⁻¹), C_i the concentration (mol L⁻¹), and M_i is the molecular weight (g mol⁻¹) of metabolite i, while N_A is Avogadro’s number. Note that the denominator in Eq. 13 is a conversion factor from 1 L to 1000 cm³.

As can be seen in Eqs. 11–14, γ_i depends on the density number (or its equivalent concentration) of all species (metabolites and macromolecules) present in the cell or the system analyzed, regardless if they are involved or not in a reaction. Hence, the impact of inert molecules (or crowders) can be easily simulated.

To do that, additional information related to the crowder molecules has to be provided such as the concentration (or its equivalent the density number), and radii (or the specific volume and molecular weight).

In this article, the coupling of both constraints (mass balance and thermodynamics) is formulated as a nonlinear optimization (nonlinear programming) problem. This methodology (expressed in Eqs. 15–22 below) allows the prediction of thermodynamically feasible ranges of fluxes, intracellular metabolites’ concentrations, and Gibbs free energy change (Δ_rG) of the reactions involved in a metabolic network under crowding conditions:

$$\begin{array}{l}\text{maximize}/\text{minimize}func\hfill \\ s.t.,\hfill \end{array}$$ (15)

$$\mathbf{S}\times \mathbf{v}=0$$ (16)

$$v_{\text{min},k}\le v_k\le v_{\text{max},k}$$ (17)

$$\begin{array}{l}{C}_{\text{min},i}\le C_i\le C_{\max ,i}\text{if}\text{for}\text{any}\text{reaction}y,y=1,\dots ,nr\_of\_reactions\left\{\begin{array}{c}{\eta }_{i,y}{v}_y>0\\ \begin{array}{c}\text{or}\\ \begin{array}{ccc}{v}_y=0& \text{and}& {\Delta }_r{G}_y=0\end{array}\end{array}\end{array}\right\}\hfill \\ C_i=0\text{otherwise},\hfill \end{array}$$ (18)

$$v_k\times {\Delta }_r{G}_k\le 0$$ (19)

$$abs\left(v_k\right)+\frac{{\Delta }_r{G}_k}{tol}{v}_k\le 0$$ (20)

$${\Delta }_r{G}_k={\Delta }_r{G}_{br,k}+{\Delta }_r{G}_{tr,k}$$ (21)

$$C_{\text{crowder}},M_{\text{crowder}},{\upsilon }_{\text{crowder}}.$$ (22)

In the above scheme, the objective function, func (Eq. 15), represents the variable of the system, the range of which we want to estimate by computing its maximum and minimum values. It can be the net flux of the reaction k (v_k), the concentration of the metabolite i (C_i), or the Gibbs free energy of the reaction k (Δ_rG_k).

The inputs for the nonlinear programming formulations (Eqs. 15–22) are Δ_fG⁰′ of all metabolites at the pH and I conditions of the system, the specific volume (υ) and the molecular weight M of the system molecules (including υ_crowder and M_crowder), and the tolerance value for the Gibbs free energy change (tol), that is the value beyond which Δ_rG_k is considered as nonzero; in this article we use tol = 10⁻⁴ kcal mol⁻¹.

Equations 17 and 18 represent the lower/upper bounds of the variables v_k and C_i, respectively, given by either experimental measurements or physiological limits. It should be noted here that if the values of one or more fluxes and/or concentrations (v_k or C_i) are known, their corresponding lower/upper bounds can be modified appropriately to restrict the computed range of the other unknown variables.

Here we assume that a reaction is blocked (i.e., its enzyme is absent or inactive) when v_k = 0 and Δ_rG_k ≠ 0, hence the concentration of the metabolites produced from this reaction is equal to zero. A metabolite has nonzero concentration when 1) it is produced by at least one reaction y, with nonzero flux, y = 1,…,nr_of_reactions, hence η_i,yv_y > 0, or 2) when reaction y is at equilibrium (v_y = 0 and Δ_rG_y = 0), otherwise C_i = 0 (Eq. 18).

Equation 19 restricts the flux of reaction k, v_k of the metabolic network to the direction of the corresponding Gibbs free energy drop, Δ_rG_k; hence v_k and Δ_rG_k have opposite sign. The prediction of a negative flux value means that the reaction goes in the opposite direction to that indicated in S.

Equation 20 is always valid when a reaction is not at (or even far from) thermodynamic equilibrium, hence abs(Δ_rG_k) ≥ tol. Then it follows that $abs\left(v_k\right)\le \left({\Delta }_r{G}_k/tol\right)v_k$. If Δ_rG_k is estimated as zero (thermodynamic equilibrium), i.e., when abs(Δ_fG_k) < tol, then Eq. 20 can only be satisfied when v_k = 0, in practice when abs(v_k) < tol_opt and tol_opt being the tolerance for convergence of the optimizer, here set to 10⁻⁶.

Unlike previous thermodynamically based stoichiometric models, in this formulation there is an explicit relationship between the crowding conditions (macromolecular crowders and small molecules such as metabolites) and the thermodynamic constraint. This relationship is given by the calculation of Δ_rG_k (Eq. 21), which is a function of the activity of the metabolites (a_i) (Eq. 2‒4, 7, 10 and 11), i.e., their concentrations (C_i) and activity coefficients (γ_i). Here γ_i (Eq. 11) takes also into account the presence of background molecules. Equation 22 indicates the parameters related with the properties of these molecules, i.e., concentration, molecular weight, and specific volume.

In this article, the thermodynamic analysis is focused on intracellular reactions and not on the transport of the metabolites across the membrane, except for those transport reactions coupled with biochemical ones, such as the transport of the hydrogen ion H⁺ (Eq. 5).

Due to the nonlinear nature of the problem expressed in Eqs. 15‒22, several local optima can be computed. To increase our chances to approximate the global optimum, we use the following initial guesses according to the type of objective function (func) to optimize:

• 1)func = v_f, f ≤ n (i.e., intracellular fluxes). The initial guesses composed by all the fluxes v_k (k ≠ f), metabolite concentrations ln(C_i), and Gibbs free energy changes Δ_rG_k are given by the solution of the linear optimization problems: (i) the corresponding mass balance (Eqs. 15–17), where func = v_f; and (ii) the maximization/minimization of Δ_rG_f using the linear Eqs. 2‒4, 7, 15, 18, and 21 assuming γ_i = 1, whose vector solution includes Δ_rG and ln(C).
• 2)func = v_f, n < f < N (i.e., extracellular fluxes). The initial guesses for the fluxes are estimated as in Item 1. The initial guess for Δ_rG is a unit vector whose elements have the opposite sign of the corresponding intracellular flux previously calculated. The initial guesses for ln(C) are (ln(C_min)) and (ln(C_max)).
• 3)func = ln(C_l), l < nr_of_species. The initial guesses for the fluxes are the upper bound (v_max) and the lower bound (v_min) while Δ_rG is estimated from the solution of the linear optimization problems of maximizing/minimizing func constrained by Eqs. 18 and 21 with γ_i = 1.
• 4)func = Δ_rG_f, f ≤ n. The initial guesses are computed as in item 1.

Results and Discussion

Thermodynamic feasibility of lactic acid pathway and glycolysis

The importance of the environmental conditions I, T, pH, and pMg (the negative logarithm of Magnesium ions) and of the concentrations of metabolites on Δ_rG of the reactions and therefore their feasibility is unquestionable (27). Vojinović and Stockar (6) have shown how these intracellular factors can turn an infeasible pathway into a feasible one. However, how important is the presence of background molecules on this type of thermodynamic analysis has not yet been clarified.

We take as an example the lactic acid metabolic pathway and the Δ_fG_i⁰′ values reported in Vojinović and Stockar (6) (Table 1) to analyze the impact of crowding on the feasibility of the pathway (the detailed information for the 11 reactions and 18 metabolites can be found in Tables S1–S3 in the Supporting Material). Here, the standard state is defined as I = 0 M, pH = 7, and T = 298.15 K. The solvent is considered as a continuum and therefore neglected.

Table 1.

Lactic acid pathway, transformed standard reaction Gibbs free energy, and metabolite concentration ranges

Reactions	ΔrG0′ (6) (kcal mol−1)	Metabolite Concentration Ranges (6) (mM)
(1) glc + ATP ⇋ g6p + ADP	−4.00	13dpg	0.06–0.25
(2) g6p ⇋ f6p	0.39	2pg	0.0295–0.295
(3) f6p + ATP ⇋ fdp + ADP	−3.40	3pg	0.118–1.18
(4) fdp ⇋ dhap + g3p	5.70	ADP	0.075–0.84
(5) dhap ⇋ g3p	1.79	ATP	0.31–1.85
(6) g3p + NAD + pi ⇋ 13dpg + NADH	1.50	dhap	0.138–1.38
(7) 13dpg + ADP ⇋ 3pg + ATP	−4.50	f6p	0.014–0.48
(8) 3pg ⇋ 2pg	1.09	fdp	0.031–3.29
(9) 2pg ⇋ pep	0.39	g3p	0.0185–0.185
(10) pep + ADP ⇋ pyr + ATP	−7.50	g6p	0.083–1.21
(11) pyr + NADH ⇋ lac + NAD	−5.99	glc	5–5
lac	0.014–5
NAD	0.65–3.55
NADH	0.038–0.145
pep	0.023–0.6
pi	0.5–1
pyr	0.051–0.4

Because SPT considers no other particle interactions than the excluded volume, we use Δ_fG_i⁰′ at I = 0 M, instead of the physiological I = 0.2 M (which involves electrostatic interactions) to carry out the thermodynamic analysis.

The optimization problem given by Eqs. 2–4, 7, 11, 15, 18, 21, and 22 was solved using the function fmincon in MATLAB R2011a (The MathWorks, Natick, MA), where the variable func (Eq. 15) can be any Δ_rG_k. If a solution with Δ_rG < 0 is found then the pathway is determined as thermodynamically feasible.

To evaluate the effects of crowding on the feasibility of the lactic pathway, we use the same concentration bounds for all metabolites by setting a fixed concentration lower bound of 10⁻² mM and varying the concentration upper bound from 2 to 3.5 mM. Using these concentration bounds, we estimate the Δ_rG values of the reactions of Table 1 in the presence of different concentrations of a single type of background molecules (crowder) of 72 kDa (31). The purpose of this is to determine if the system is thermodynamically feasible under a range of crowding conditions.

Assuming the size of all species (metabolites and macromolecules) is proportional to the cubic root of the molecular weight of their neutral form with specific volume υ_i = 0.73 cm³ g⁻¹ (17), their radii can be estimated using Eq. 14 (Scheme r ∝ M^1/3).

The r values estimated for some solutes such as glucose, water, ethanol, and glycerol using υ_i = 0.73 cm³ g⁻¹ are within the experimental/calculated radii compiled by Tang and Bloomfield (22).

Fig. 1 shows the feasibility diagram where the square-line indicates the minimum crowder concentration C_crowder that makes the lactic pathway feasible (Δ_rG < 0, regions I + II) for each concentration upper bound tested. In the absence of crowders (C_crowder = 0 mM) and when the metabolites’ concentration bounds are very narrow (in this case, using a concentration upper bound < 3.5 mM), then the pathway is predicted as not feasible. As the concentration bounds are allowed to be wider (i.e., with a concentration upper bound ≥ 3.5 mM), the thermodynamic constraints are relaxed so the pathway becomes feasible. However, when there is an increase in the crowder concentration the activity coefficients of the metabolites (Eq. 11) also increase. Therefore their activities, which affect the estimation of Δ_rG of the reactions, make the system thermodynamically feasible even if the metabolites’ concentration upper bound is <3.5 mM.

Figure 1.

Feasibility diagram of the lactic acid pathway using different metabolite concentration upper bounds (the concentration lower bound was set as 10⁻² mM) and crowder concentration (M_crowder = 72 kDa). (Square line) The scheme r ∝ M_i^1/3, according to Eq. 14. (Diamond line) The scheme r = 0, where the radii of all the metabolites are neglected. In both cases the feasible region is the area above each line. For the scheme r ∝ M_i^1/3 the feasible region is given by region I+II, while for scheme r = 0 it corresponds to region I.

To simplify the problem, we assume that small molecules with M < 72 kDa (including all metabolites) are volumeless particles (r = 0). Then Eq. 11 reduces to ln γ_i = −ln(1 – S₃), where S₃ represents the volume fraction occupied by molecules, in this case only by macromolecules. Then, the reactions affected by the crowding are those where the number of reactants are different from the number of products, so that the nonideal term implicit in Eq. 3 (obtained when Eq. 7 is substituted in Eq. 3) is $RT\text{ln}\left({\prod }_{i=1}^{\text{Products}}{\gamma }_i^{{\eta }_i}/{\prod }_{i=1}^{\text{Reactants}}{\gamma }_i^{{\eta }_i}\right)\ne 0$. Note that if this nonideal term is equal to zero, then Δ_rG is a function of the metabolites’ concentration as it happens in dilute solutions. However, because the system is feasible only when all reactions have Δ_rG < 0, then these crowding-sensitive reactions can indirectly affect the maximum/minimum Δ_rG value attainable by other reactions. Fig. 1 shows the feasibility diagram for metabolites with negligible radii.

For both schemes r = 0 and r ∝ M^1/3, the decrease of the available volume due to the crowder’s presence push the pathway toward the feasible region for a fixed concentration upper bound.

However, when the radii of the metabolites are considered (scheme r ∝ M^1/3), the pathway becomes feasible at lower macromolecules concentrations. This is because the available volume to the center of mass of the metabolites decreases with the molecules’ size, which is taken into account by the second, third, and fourth terms of Eq. 11.

The radius of the metabolite impacts directly the calculation of the activity coefficient (Eq. 11). In fact, for small molecules (e.g., r < 200 Da), the first term of the right-hand side of Eq. 11 becomes the most dominant term, while for large molecules (e.g., macromolecules), the fourth term is most dominant one.

The importance of the parameter r on the feasibility of a pathway is also evident when a comparison is made using a vector r_random of random radii taken from the range r ± 0.1r for all the metabolites (where r is proportional to the cubic root of M and υ = 0.73 cm³ g⁻¹ as mentioned before) but keeping constant the original r_crowder = 2.75 × 10⁻⁷ cm. The range r ± 0.1r covers most of the radii values reported in Tang and Bloomfield (22) for each metabolite (glucose, water, ethanol, and glycerol).

Fig. 2 A shows the general tendency to achieve feasibility at lower C_crowder as the bounds of metabolite concentration are relaxed (i.e., when the concentration upper bound is increased). Nevertheless, the specific C_crowder value at which the lactic acid pathway turns feasible depends on the molecular radii of the metabolites.

Figure 2.

Feasibility diagram of the lactic acid pathway using a range of metabolite concentration upper bounds (the concentration lower bound was set as 10⁻² mM) and crowder concentrations. (A) Three different random metabolite radii vectors r_random were tested, while the radius of the crowder is equal to 2.75 × 10⁻⁷ cm for all cases. (B) Three different crowder radii, 2.48 × 10⁻⁷, 2.75 × 10⁻⁷, and 3.03 × 10⁻⁷ cm were tested; the metabolites’ radii were kept constant and proportional to the cubic root of their molecular weight. The feasible region is the area above each line.

Analyzing the effect of metabolites’ radius change on the term RT ln(a_i′) of Eq. 3, we found that for a species i of 200 Da (r_i = 3.87 × 10⁻⁸ cm) with C_i = 0.01 mM, in the presence of crowders of 72 kDa (r_crowder = 2.75 × 10⁻⁷ cm, C_crowder = 10.5 mM), a radius change of 0.1r_i (T = 298.15 K) causes an average change of 0.0604 kcal mol⁻¹ in RT ln(a_i′), while for a species i of 600 Da (r_i = 5.58 × 10⁻⁸ cm) the average RT ln(a_i′) change is 0.1112 kcal mol⁻¹. This indicates that for a larger molecule, a change in its radius has more impact on the RT ln(a_i′) value and therefore on Δ_rG of the reaction (Eq. 3) in which metabolite i is involved.

In particular, NAD and NADH are the largest metabolites in the lactic acid pathway. In Fig. 2 A, vector r_random3 is equal to r_random1, except for the elements corresponding to the radii of NAD and NADH, which are equal to those given in r_random2. The comparison of the minimum C_crowder required to make the lactic acid pathway feasible using metabolites of radii r_random3 (triangle line), r_random1 (diamond line), and r_random2 (square line) indicates that the particular radii of NAD and NADH contribute significantly to the difference shown between the feasibility diagram obtained for r_random1 and r_random2 (Fig. 2 A). This suggests that reactions where large-size metabolites are involved are more sensitive to changes in the crowding conditions. The values of r_random1, r_random2, and r_random3 are given in Table S1.

Fig. 2 B shows the impact that changes in the crowder’s radius (within the range r_crowder ± 0.1 r_crowder) have on the feasibility diagram of the lactic acid pathway. For this, the radius of each metabolite i was kept constant (calculated from Eq. 14, υ_i = 0.73 cm³ g⁻¹). As it can be seen, the C_crowder required to make the pathway feasible (for a specific metabolite concentration upper bound) is higher for smaller crowder molecules. Nevertheless, the volume fraction occupied by the crowder is similar for the three r_crowder values tested. For example, using a metabolite concentration upper bound of 2.5 mM, the volume fraction occupied by the crowder is 0.634 (r_crowder = 2.48 × 10⁻⁷ cm), 0.646 (r_crowder = 2.75 × 10⁻⁷ cm), and 0.656 (r_crowder = 3.03 × 10⁻⁷ cm).

Although the intracellular crowding conditions may turn a positive Δ_rG to a negative value, they cannot overcome by themselves the influence of uncertainties in the measurements of Δ_fG_i⁰′ and of the concentration of the metabolites.

For example, using the physiological concentration ranges reported as Range 2 in Vojinović and Stockar (6) (see Table 1), the lactic pathway could not be predicted as feasible (scheme r ∝ M^1/3) even at high background molecules concentration. In fact for this particular set of concentrations, Vojinović and Stockar (6) found that the feasibility of the pathway could be reached only under unrealistic I, pH, and pMg.

On the other hand, slight variations were experimentally detected in the intracellular concentration of pep and dhap in Escherichia coli growing on either glycerol or acetate (32), suggesting that Δ_rG⁰′ of the reactions involved in lower glycolysis (from dhap to pep) must be near equilibrium, so that these small variations cause the activation of glycolysis in the direction required depending on the carbon source used, i.e., dhap → pep for glycerol, and pep → dhap for acetate.

Furthermore, the crowding conditions could help to change the direction of the reactions in glycolysis. For example, lumping reactions 8 and 9 of Table 1 into the single reaction

$$3\text{pg}\to \text{pep}$$ (23)

gives Δ_rG⁰′ (I = 0 M, pH = 7, T = 298.15 K) = 1.48 kcal mol⁻¹, with the concentration limits reported by Bennett et al. (32) for the growth on glycerol: 1.04 mM < C_pep < 1.73 mM, and 3.36 mM < C3pg < 4.95 mM.

The Δ_rG estimated for this reaction in absence of crowders (C_crowder = 0 M) indicates the tendency toward the formation of 3pg (Δ_rG > 0), i.e., the likely glycolysis pathway is active in the direction pep → g6p, as was expected for growing on acetate. However, under the presence of 16.9 mM (or higher) of crowders of 72 kDa (scheme r ∝ M^1/3), which occupy 89% of the total volume, the likely direction of reaction of Eq. 23 is (3pg → pep) as indicated by a computed negative Δ_rG.

While the estimated volume fraction occupied by molecules is more than twice that expected in the cell (∼40%) (7), this example shows qualitatively the change of reaction direction due to the excluded volume effects.

Nevertheless, other intracellular conditions like I, pH, T, and pMg (not considered in this example) may affect the Δ_rG⁰′ value making it to approach zero, as was suggested by Bennett et al. (32).Then the crowding condition needed to change the reaction direction could be within the physiological 40%.

Moreover, the cell consists of a heterogeneous medium where regions more crowded than others may occur, favoring a metabolic pathway that in other circumstances or even other regions would be infeasible.

Prediction of thermodynamically feasible flux distributions: the metabolic network of A. succinogenes

To study the influence of crowding conditions on the flux distribution, thermodynamically constrained flux variability analysis (33) was carried out on the central carbon pathways of A. succinogenes (Fig. 3; the detailed information of the 53 reactions and 41 metabolites is in Tables S4–S6. The list of metabolites and reactions is in Appendix B.

Figure 3.

Central carbon pathways of Actinobacillus succinogenes for the production of succinic acid from glycerol. (Unidirectional arrow) The flux through the reaction is a priori considered unidirectional. (Bidirectional arrow) The reaction is a priori considered reversible. (Larger arrowhead) How the reaction was written in the stoichiometric matrix. (Dashed arrows) Input/output fluxes to/from the cell, which were fixed to experimental flux values. (Shaded arrows) Biomass formation. (Italic numbers) Flux estimated for the corresponding reaction, such flux value was found to be independent of the environmental condition tested. Fluxes have units of mmol g_DW⁻¹ h⁻¹.

In this section we use the values of standard Gibbs free energy of formation of all metabolites (Δ_fG⁰) estimated by the expanded group contribution method proposed by Jankowski et al. (34).

Both input and output fluxes to/from the cell were fixed to the experimental data reported by Vlysidis et al. (24) for the batch production of succinic acid from glycerol by A. succinogenes. The aim here is to determine the intracellular and unmeasured fluxes.

The uptake flux of glycerol was set to 5.6524 mmol g_DW⁻¹ h⁻¹, while the excreted products were set to the following fluxes: acetate to 0.5965, formate to 0.4735, and succinate to 3.8711 mmol g_DW⁻¹ h⁻¹. Biomass production was equal to 1.8504 mmol g_DW⁻¹ h⁻¹ while the cell composition is assumed as CH₂O_0.5N_0.2 (35). Because lactate, ethanol, fumarate, pyruvate, and oxaloacetate were not experimentally detected in the medium (24) their production fluxes were set to zero.

Because there are no available values for the metabolite concentrations for this system, the concentration bounds of all the metabolites were set to C_max = 10 mM and C_min = 10⁻² mM (12), while the crowding concentration was set to the fixed value C_crowder1 = 6.5 mM for macromolecules of M_crowder1 = 72 kDa and C_crowder2 = 300 mM for other small molecules of M_crowder2 = 200 Da. The bounds of intracellular fluxes were v_max = 100 mmol g_DW⁻¹ h⁻¹ and v_min = −100 mmol g_DW⁻¹ h⁻¹ (5). The water flux exchange to/from the cell was unconstrained.

If water is assumed as a volumeless particle, then the number of molecules that can fit into the system (i.e., the water density number) is proportional to the space not occupied by other molecules and the density number of pure water ρ_water,sol = (1 – S₃) ρ_{pure_water}, while the activity coefficient (Eq. 11) is reduced to γ_water,sol = 1/(1 – S₃) and γ_pure,water = 1. Therefore, the water activity (Eq. 10) is equal to unity.

The optimization problem (constrained by thermodynamics and mass balances) given by Eqs. 2‒4, 7, 11, and 15‒22 was also solved using the function fmincon (MATLAB R2011a; The MathWorks). To increase the possibility of finding a minimum/maximum value close to the global one, four different initial guesses were tested for each variable x (as discussed above). These values were then compared with the solution obtained when the objective function is another variable y (with its own four initial guesses), which is equivalent to increasing the number of effective initial guesses per variable.

Table 2 shows the environmental conditions (I, pH_c, pH_e, and macromolecular crowding) and the molecules’ volume consideration (r = 0 for volumeless particles, and r ∝ M^1/3 when the radii of the molecules are proportional to cubic root of the molecular weight, the radii of molecules with M < 20 Da are neglected) employed for the estimation of the maximum and minimum flux values predicted in this example.

Table 2.

Sets of environmental conditions tested for the FVA of A. succinogenes

Condition 1a	Condition 2b	Condition 3c	Condition 4d	Condition 5e	Condition 6f
pHc = 7	pHc = 6.7	pHc = 7	pHc = 7	pHc = 7	pHc = 7
pHe = 6.2	pHe = 6.2	pHe = 7.4	pHe = 6.2	pHe = 6.2	pHe = 6.2
T = 298.15 K	T = 298.15 K	T = 298.15 K	T = 298.15 K	T = 298.15 K	T = 298.15 K
I = 0 M	I = 0 M	I = 0 M	I = 0.2 M	I = 0 M	I = 0 M
r = 0	r = 0	r = 0	r = 0	ri = 0 for Mi < 20 Da	ri = 0 for Mi < 72 kDa
ri ∝ Mi1/3 for Mi ≥ 20 Da	ri ∝ Mi1/3 for Mi ≥ 72 kDa

^aBiological standard conditions, volumeless molecules.

^bChange in pH_c, volumeless molecules.

^cChange in pH_e, volumeless molecules.

^dChange in I, volumeless molecules.

^eMolecules’ volume.

^fMacromolecules’ volume.

Conditions 2‒4 express one change at a time in the biological standard values I, pH_c, or pH_e toward physiological conditions. The aim of this is to compare the crowding effect (condition 5 and 6) on flux distribution regarding other environmental factors.

Although the crowders’ concentration is the same in all the conditions tested (C_crowder1 = 6.5 mM for molecules of M_crowder1 = 72 kDa and C_crowder2 = 300 mM for molecules of C_crowder2 = 200 Da), the assumption that the radii of all the molecules is zero (conditions 1‒4) causes their activity coefficient to be equal to one, therefore the presence of background molecules does not have an effect on the thermodynamics and flux distribution of the metabolic pathway.

As shown before, the crowding conditions can change the directionality of some reactions. For this metabolic network (Fig. 3), and neglecting the (group contribution) uncertainty in Δ_rG⁰′, a greater number of reactions were predicted as reversible for condition 5 compared with those under conditions 1–4 (Fig. 4 A). A reversible reaction is one that can go in both forward and backward directions; hence, the feasible range of flux predicted covers both positive and negative values, beyond a tolerance of 10⁻⁴ mmol g_DW⁻¹ h⁻¹.

Figure 4.

(A) Comparison of the number of reactions predicted as reversible under conditions 1–6. (B) Number of reactions whose range of fluxes (under conditions 2‒6) differs from those predicted for condition 1. The group contribution uncertainty in Δ_rG^0′ is not considered.

The fact that more reactions go in both backward and forward directions affects the attainable flux of other reactions, therefore there exist more thermodynamically feasible flux distributions for the same output fluxes (that in this example were fixed to experimental values). FVA reveals that the range of fluxes predicted under crowding conditions (condition 5) is either equal to or wider than those from conditions 1–4 (colored lines in Fig. 5).

Figure 5.

Comparison of the direction and range of fluxes predicted for some reactions under conditions 1–6. (Colored lines) Fluxes when the uncertainty in Δ_rG^0′ is not taken into account; (shaded lines) behind them is the flux range predicted considering the uncertainty. To see this figure in color, go online.

Fig. 3 shows the fluxes estimated for those reactions whose flux values do not change under the environmental conditions 1–6.

A comparison of the number of reactions (under conditions 2–5) whose range of fluxes differ from those predicted under biological standard conditions (condition 1) suggests that for this pathway the flux distributions are more sensitive to crowding conditions primarily, and to ionic strength secondarily (Fig. 4 B).

The impact of pH_c and pH_e was less significant on the flux ranges (Fig. 4 B). That was expected because no drastic changes were tested inasmuch as A. succinogenes regulates the intracellular pH_c within a narrow range (36), and pH_e was maintained between 6.2 and 7.4 during the fermentation (24).

In this study we focus on the individual effect of I, pH_c, pH_e, and crowders’ concentration, but certainly the environmental conditions exert a combined effect on the feasibility of metabolic networks. Hence, an appropriate model which combines the effect of electrostatic and excluded volume interactions (the two parameters with more influence on the flux distributions), together with reliable thermodynamic data and metabolite concentrations, would give a deep insight into the complex intracellular processes.

To simplify the calculation of the activity coefficients, we assume that all metabolites and the crowder of type 2 (with M_crowder2 = 200 Da) have negligible volume (condition 6).

Because the radii are equal to zero, all terms of the right-hand side of Eq. 8 vanish except for the first one, which represents the volume fraction occupied by molecules. In this case, this is determined by a constant concentration of 72 kDa macromolecules equivalent to 34% of the total volume.

A comparison between the ranges of fluxes predicted using the full Eq. 8 (condition 5) and its simplified version (condition 6) shows no difference (colored lines in Fig. 5). This suggests that a good approximation of the crowding effect can be easily incorporated to any linear thermodynamically stoichiometric model (e.g., Henry et al. (5) and Hoppe et al. (12)) by the addition of parameter RT ln(γ_i) = RT/(1 – S₃) to Δ_fG⁰_i′.

However, there are slight differences in the range of Δ_rG estimated for both approximations (conditions 5 and 6) (colored lines in Fig. 6). Although for this example these Δ_rG differences have no influence on the range of fluxes, they could be more important for reactions near equilibrium because small changes are needed to change the directionality or feasibility of a reaction.

Figure 6.

Comparison of Δ_rG estimated for some reactions under conditions 1–6. (Colored lines) Range of Δ_rG when the uncertainty in Δ_rG^0′ is not taken into account. (Shaded lines) Behind them is the Δ_rG predicted considering the uncertainty. To see this figure in color, go online.

Up to this point the uncertainty SE associated with the transformed standard Gibbs free energy of the reaction, i.e., ${\Delta }_r{G}^{0\prime }={\sum }_{i=1}^{\text{Products}}{\eta }_i{\Delta }_f{G}_i^{0\prime }-{\sum }_{i=1}^{\text{Reactants}}{\eta }_i{\Delta }_f{G}_i^{0\prime }$, estimated by the group contribution method was not taken into account.

To incorporate the influence of the uncertainty on the thermodynamic analysis, the error variable E is added to the Eq. 3, giving

$${\Delta }_r{G}_{br}=\sum _{i=1}^{\text{Products}}{\eta }_i\left[{\Delta }_f{G}_i^{0\prime }+RT\text{ln}\left(a_i^{\prime }\right)\right]-\sum _{i=1}^{\text{Reactants}}{\eta }_i\left[{\Delta }_f{G}_i^{0\prime }+RT\text{ln}\left(a_i^{\prime }\right)\right]+E.$$ (24)

The variable E can take any value within the limits ±SE of each reaction as calculated by the group contribution method (34).

The new optimization problem that takes into account the uncertainty in Δ_rG⁰′ (Eqs. 2, 4, 7, 11, 15‒22, and 24) predicts a range of Δ_rG values wider than those where this is not considered (or neglected) (gray lines in Fig. 6).

This causes some reactions to change their directionality, so that in general the range of fluxes is also estimated to be wider for all conditions tested (gray lines in Fig. 5). For this example, the uncertainty given by the group contribution method masks the effects of I, pH_c, pH_e, and crowding conditions on the estimation of the range of fluxes (Fig. 5). This reflects the importance of using more accurate thermodynamic data.

Finally, to determine the impact of crowding conditions on the maximum attainable biomass flux, the (thermodynamically constrained) flux balance analysis (Eqs. 2‒4, 7, 11, and 15‒22) was applied to the central carbon pathways of A. succinogenes (Fig. 3). The uncertainty in Δ_rG⁰′ is not taken into account.

Experimental observations indicate the relation between the CO₂ availability in the medium and the growth and succinate production of A. succinogenes. The following uptake fluxes were set: glycerol 5.6524 mmol g_DW⁻¹ h⁻¹, CO₂ 1.7602 mmol g_DW⁻¹ h⁻¹, and phosphate 0.1598 mmol g_DW⁻¹ h⁻¹. The remaining extracellular fluxes were left unrestricted. The concentration and intracellular flux bounds mentioned before were unchanged.

The maximum biomass production rate estimated was 1.8495 mmol g_DW⁻¹ h⁻¹ under both crowding conditions and the biological standard condition (conditions 5 and 1, respectively, see Table 2), which agrees with the experimental results (24).

Only small differences were found for the maximum succinate flux: (condition 5) 4.100 mmol g_DW⁻¹ h⁻¹ and (condition 1) 3.5302 mmol g_DW⁻¹ h⁻¹. Both values are within the range v_{succinate,exp} ± 9%. This confirms that, for the system analyzed, crowding conditions acting on the robustness of the metabolic network, aid in reaching the same output flux but with different flux distributions.

It is noteworthy that the concentration bounds used here have an important role in this conclusion, because the use of other very restricted bounds/limits may turn complete pathways to be infeasible (as shown in the lactic acid pathway example). However, where the crowding conditions may favor the flux through them, this will affect the estimation of some output fluxes.

Conclusions

Along with other environmental conditions such as I, T, pH_c, and pH_e, the presence of background molecules may also play an important role in the feasibility and directionality of reactions.

In this article, SPT was used to incorporate the crowding effects on the thermodynamic analysis. It was found that changes in the crowding conditions can push a pathway to one direction or another. This suggests the importance of heterogeneity in crowding (i.e., some regions within the cell being more crowded than others) or the compartmentalization (in eukaryotes) of the intracellular space, which could favor the feasibility of certain pathways.

Moreover, a crowded medium may affect the flux distribution in a metabolic system. In fact, FVA estimates a wider range of fluxes when crowding is taken into account compared with the standard conditions. Because more thermodynamically feasible flux distributions are possible, the microorganism could cope more easily with environmental or genetic changes that prevent certain flux distributions. Maybe, a rearrangement in the spatiotemporal distribution of the cellular components (37), increasing/decreasing the crowding conditions of a particular region, would also help to maintain the feasibility of the system despite some environmental/genetic perturbations.

Nevertheless, the enhancement in the metabolite’s activity caused by the crowding conditions is accompanied by a decrease in the diffusion and the degree of mixing of the molecules, hence reaction between two reactants is less likely (38, 39), which was not considered in this article.

Due to the high influence of the excluded volume and electrostatic interactions (related with the crowding conditions and ionic strength) on the feasibility and flux distribution of a metabolic network, the use of a model that incorporates the combined effect of these two interactions would give a better understanding of environmental conditions’ control on the metabolism.

Author Contributions

L.A.-M. developed the methodology, performed all the computations, and drafted the article; and C.T. supervised the method development and the work on the case studies, and reviewed and edited the article.

Editor: Daniel Beard.

Appendix A: Calculation of Δ_fG_i⁰′

In this section the methodology for the estimation of transformed standard Gibbs free energy of formation of the metabolite i (Δ_fG_i⁰′) is presented, which for the purpose of this article is represented by its pseudoisomer group i, at different values of I and pH.

A pseudoisomer group i is made up of all the protonated species j formed by the dissociation of the metabolite i in aqueous solutions (27), whose proportions depend on the logarithm of the acid dissociation constant pK_a. So, for example, for the reaction $HA\underset{\text{pKa}}{\to }{A}^{-}$, if the change in standard Gibbs free energy of the species HA (ΔfG⁰_HA) is known, then Δ_fG⁰_A− for the species A⁻ can be calculated by

$${\Delta }_f{G}_{A^{-}}^0={\Delta }_f{G}_{HA}^0+p{K}_{a}RTln\left(10\right).$$ (A1)

The Δ_fG_j⁰ values of species j (at standard biological conditions pH = 7, I = 0 M, and T = 298.15 K) of several metabolites can be found in databases (e.g., Alberty (27) and Li et al. (40)) or estimated by any group contribution method and other predictive method, for example the one proposed by Jankowski et al. (34), or by Noor et al. (41).

In this article we use the Calculator Plugins, MarvinSketch ver. 5.5.1.0 (ChemAxon, http://www.chemaxon.com) for the estimation of the pK_a values of all the metabolites (the molecular structure is required).

Once the Δ_fG_j⁰ values for all the species of a metabolite are found, then if the environmental conditions I and pH are different from the standard biological conditions, the transformed Gibbs free energy change Δ_fG_j⁰′ (when pH is considered constant) can be calculated as (27)

$${\Delta }_f{G}_j^{0\prime }={\Delta }_f{G}_j^0-\frac{A{z}_j^2{I}^{\frac{1}{2}}}{1+B{I}^{\frac{1}{2}}}RT\text{ln}\left(10\right)-N_{H,j}{\Delta }_f{G}_H,$$ (A2)

where

$${\Delta }_f{G}_H={\Delta }_f{G}_H^0-\frac{A{I}^{\frac{1}{2}}}{1+B{I}^{\frac{1}{2}}}RT\text{ln}\left(10\right)-RT\text{ln}\left({10}^{-\text{pH}+\text{pH}ref}\right).$$ (A3)

The constants of the extended Debye-Hückel equation are given as A = 0.51 and B = 1.6. Here, z_j is the electric charge of species j; N_H,j is the number of H atoms in species j; and pH_ref = 7 is the reference pH at the standard biological conditions.

Finally, assuming that all species j of the metabolite i are in equilibrium, the Δ_fG_i⁰′ of the pseudoisomer group i can be estimated as (27)

$${\Delta }_f{G}_i^{0\prime }=-R\times T\times \text{ln}\left\{\sum _{j=1}^{N_{\text{species}}}{e}^{-\left(\frac{{\Delta }_f{G}_j^{0\prime }}{R\times T}\right)}\right\}.$$ (A4)

Appendix B: List of Metabolites and Reactions

• 13dpg, 3-phospho-D-glyceroyl phosphate

• 2pg, D-glycerate 2-phosphate

• 3pg, 3-phospho-D-glycerate

• 6pgc, 6-phospho-D-gluconate

• 6pgl, 6-phospho-D-glucono-1,5-lactone

• Ac, acetate

• acald, acetaldehyde

• accoa, acetyl-CoA

• actp, acetyl phosphate

• ADP, adenosine diphosphate

• ATP, adenosine triphosphate

• CO₂, carbon dioxide

• Coa, coenzyme A

• dhap, dihydroxyacetone phosphate

• e4p, erythrose 4-phosphate

• etoh, ethanol

• f6p, fructose 6-phosphate

• fdp, fructose 1,6-bisphosphate

• for, formate

• fum, fumarate

• g3p, glyceraldehyde 3-phosphate

• g6p, glucose 6-phosphate

• glyc, glycerol

• glyc3p, glycerol 3-phosphate

• H₂O, water

• lac, lactate

• mal, malate

• NAD, nicotinamide adenine dinucleotide

• NADH, nicotinamide adenine dinucleotide - reduced

• NADP, nicotinamide adenine dinucleotide phosphate

• NADPH, nicotinamide adenine dinucleotide phosphate - reduced

• Oaa, oxaloacetate

• pep, phosphoenolpyruvate

• pi, phosphate

• pyr, pyruvate

• r5p, ribose 5-phosphate

• ru5p, ribulose 5-phosphate

• s7p, sedoheptulose 7-phosphate

• succ, succinate

• xu5p, xylulose 5-phosphate

• ACALD, acetaldehyde dehydrogenase

• ACKr, acetate kinase

• ACYP, acetyl phosphate phosphohydrolase

• ALCD2x, alcohol dehydrogenase

• ATPm, ATP requirements for other reactions

• ENO, enolase

• EX_ac(e), acetate exchange

• EX_CO₂(e), CO₂ exchange

• EX_coa(e), coenzyme A exchange

• EX_etoh(e), ethanol exchange

• EX_for(e), formate exchange

• EX_fum(e), fumarate exchange

• EX glyc(e), glycerol exchange

• EX_H₂O(e), water exchange

• EX_lac(e), lactate exchange

• EX_oaa(e), oxaloacetate exchange

• EX_pi(e), phosphate exchange

• EX_pyr(e), pyruvate exchange

• EX_succ(e), succinate exchange

• FBAr, fructose-bisphosphate aldolase

• FBP, fructose-1,6-bisphosphatase

• FDH3, formate dehydrogenase

• FRD, fumarate reductase

• FUM, fumarase

• G3PD2, glycerol 3-phosphate dehydrogenase

• G6PDH2r, glucose 6-phosphate dehydrogenase

• GAPD, 3-phosphate glyceraldehyde dehydrogenase

• GLYK, glycerol kinase

• GND, phosphogluconate dehydrogenase

• LDH, lactate dehydrogenase

• MDH, malate dehydrogenase

• ME2, malic enzyme

• NADHm, NADH requirements for other reactions

• NADPHm, NADPH requirements for other reactions

• OAD, oxaloacetate decarboxylase

• PDH, pyruvate dehydrogenase complex

• PGI, glucose-6-phosphate isomerase

• PGK, 3-phosphoglycerate kinase

• PGL, 6-phosphogluconolactonase

• PGM, phosphoglycerate mutase

• PFK, 6-phosphofructokinase

• PFL, pyruvate formate lyase

• PPCK, phosphoenolpyruvate carboxykinase

• PTAr, acetate phosphotransferase

• PYK, pyruvate kinase

• RPE, ribulose 5-phosphate 3-epimerase

• RPI, ibose-5-phosphate isomerase

• TALA, transaldolase

• TKT1, transketolase

• TKT2, transketolase

• THD2, NAD(P) transhydrogenase

• TPI, triose-phosphate isomerase

Supporting Citations

References (42, 43) appear in the Supporting Material.