Microbial pathway thermodynamics: structural models unveil anabolic and catabolic processes

The biotechnological exploitation of microorganisms enables the use of metabolism for the production of economically valuable substances, such as drugs or food. It is, thus, unsurprising that the investigation of microbial metabolism and its regulation has been an active research field for many decades. As a result, several theories and techniques were developed that allow the prediction of metabolic fluxes and yields as biotechnologically relevant output parameters. One important approach is to derive macrochemical equations that describe the overall metabolic conversion of an organism and basically treat microbial metabolism as a black box. The opposite approach is to include all known metabolic reactions of an organism to assemble a genomescale metabolic model. Interestingly, both approaches are rather successful to characterise and predict the expected product yield. Over the years, especially macrochemical equations have been extensively characterised in terms of their thermodynamic properties. However, a common challenge when characterising microbial metabolism by a single equation is to split this equation into two, describing the two modes of metabolism, anabolism and catabolism. Here, we present strategies to systematically identify separate equations for anabolism and catabolism. Based on metabolic models, we systematically identify all theoretically possible catabolic routes and determine their thermodynamic efficiency. We then show how anabolic routes can be derived, and use these to approximate biomass yield. Finally, we challenge the view of metabolism as a linear energy converter, in which the free energy gradient of catabolism drives the anabolic reactions.


Introduction
The macrochemical equation can be understood as catabolism (the breakdown of nutrients to gain free energy in the form of ATP) and anabolism (the formation of new biomass from the nutrients) [46].In this picture, microbial growth is described as a thermodynamic energy converter, where the catabolic reactions provide the required free energy to drive anabolism (see Fig. 1).Here, the negative free energies of reaction of the catabolic and anabolic halfreactions, denoted by −∆ cat G and −∆ ana G, respectively, are generalised thermodynamic forces, and the respective reaction rates, J cat and J ana , are generalised thermodynamic fluxes.The relation between these generalised forces and fluxes is often assumed to be linear [40,54,50,47], following Onsager's theory [25] for non-equilibrium thermodynamics.Onsager has shown that the linearity holds in general for systems close to equilibrium.Despite the attractiveness of the linear converter theory, it is not fully clear, to what extent this approximation is actually adequate for microbial growth.Regardless of these uncertainties, this simplified view of microbial growth as two coupled processes is insightful and allows estimating some principle thermodynamic limitations, such as maximally possible yields.
Describing macrochemical, catabolic and anabolic equations experimentally requires a precise measurements of all chemical substances that are consumed and produced by the growing microbes.These measurements are possible in controlled chemostat cultures, in which microbes grow on a defined growth medium.However, many mircobes cannot easily be cultured in chemically defined media, which prevents an experimental determination of macrochemical growth equations.However, with the recent scientific advancement in genome sequencing, genomic data became available for a huge number of organisms, including those which are difficult to culture [22].This information greatly facilitates building a stoichiometric (or structural) metabolic models, manually or semi-automatically [11,18,20,35].With a structural model, metabolic capabilities can be systematically assessed and optimal flux distributions optimizing some objective function (such as biomass production) can be easily calculated [27].Thus, such models can support strategies to improve the product yield in biotechnological applications.Elementary flux modes (EFM) are a systematic way to quantify the metabolic capabilities of an organism [36].
EFMs describe all possible pathways between substrate and products.However, due to combinatorial explosion [17], it is still challenging to calculate all EFMs for larger structural metabolic models, also with modern computational facilities.To overcome this, and because for many investigations only the conversion between substrate and product is relevant, elementary conversion modes (ECM) were introduced by Urbanczik and Wagner [44].ECMs ignore all intracellular processes and only focus on the results of metabolic pathways.ECMs describe a minimal set of 97 pathways that generates all steady-state substrate-to-  Gibbs free energies of reactions with a higher energy gradient than 50 kJ/C-mol (around 17 ECMs).The pathways with the largest absolute Gibbs free energy of reaction are the combustion of glucose (∆ cat G 0 ≈ −488 kJ C-mol −1 ), and the production of formate from glucose (∆ cat G 0 ≈ −325 kJ C-mol −1 ).In particular, oxygen-using ECMs belong to the group with the largest absolute ∆ cat G 0 (compare red bars in Fig. 2).
As shown in Fig. 2, usage of nitrogen does not appear to be an indicator of whether the respective ECM has a high or low energy gradient.
For each catabolic route, we use the metabolic model to calculate the maximal ATP yield.For this, the exchange reactions were constrained to the stoichiometries of the respective ECM, and subsequently the flux through ATP hydrolysis was maximized (see Meth- Additionally, oxygen is allowed to be a substrate in the calculation of the elementary conversion modes.The efficiency is based on a typical value of 46.5 kJ/mol for production of ATP in E. coli [42].
where c ATP is the maximal ATP yield per carbon mol,

Deriving anabolic information from a metabolic network
Likewise, also anabolic pathways can be investigated in separation by employing genome-scale metabolic models.The theoretical limit how much carbon in the nutrient can be converted to biomass carbon is given by the degree of reduction.If biomass is more reduced than substrate, a fraction of the substrate carbons need to be oxidised in order to ensure the overall redox balance.Specifically, if γ S and γ X are the degrees of reduction of substrate and biomass, respectively, then the theoretical maximal yield is given by [32] Based on the elemental composition of the biomass ).If γ S ≤ γ X the stoichiometry

Is the linear energy converter a good model for microbial growth?
In the original publication that we draw our data from for E. coli [15], also higher dilution rates were investigated.For these, however, the carbon recovery rates were significantly below 90%, indicating that not all metabolic products were measured.The incomplete carbon recovery prevents a reliable calculation of catabolic stoichiometries.We therefore omitted these data, and will in the following focus on the energetic analysis of the metabolism of S. cerevisiae.
Microbial growth is often thermodynamically interpreted in the context of a linear energy converter model [50,47], which assumes that the anabolic and catabolic fluxes linearly depend on the catabolic and anabolic forces, i. e. that where the anabolic flux equals the growth rate, which   and 3).
With the usual definition of the power as the product of flux and force, we can quantify the catabolic and anabolic powers P cat = −J cat • ∆ cat G and P ana = −D • ∆ ana G, as well as the power of ATP production shows that the powers increase approximately linearly with growth rate, despite the fact that metabolism changes considerably for fast growth rates.Moreover, E. coli and S. cerevisiae behave rather similarly.For large growth rates, the powers of E. coli appear to be somewhat lower, but this could be a result from incomplete carbon recovery in the experiments, because besides acetate no other fermentation products were measured.This could lead to an underestimation of the Gibbs free energy gradients and consequently of into account, the highest efficiency, which is observed for the fermentation pathways, is very close to the 50% that is predicted to yield the highest ATP production rates by simple linear thermodynamic energy converter models [52].It is remarkable that the actually realised catabolic pathways in chemostat cultures (see red circles in Fig. 3) provide a higher efficiency than elementary pathways with a similar free energy gradient.This observation stresses the important role of the pure respiration and fermentation pathways of catabolism.Because of their high efficiencies, operating them in combination always provides a higher thermodynamic efficiency than any single elementary conversion mode.
While a linear energy converter model seems adequate to predict optimal thermodynamic efficiencies of ATP producing pathways with a reasonable accuracy [52], our interpretation of experimental data shows that this is not the case when microbial growth is considered.Our results clearly demonstrate that the fluxforce relationship is not linear, and that in fact an-

Calculating elementary conversion modes
Elementary conversion modes (ECMs, [44]) are a fast way to describe metabolic capabilities of an organism.
We and thus the full catabolic potential of the core network could be described.

Estimating Gibbs free energy of reaction
To approximate the standard Gibbs free energy of reaction (∆ cat G 0 ) for each obtained ECM, we used the Python API of the eQuilibrator tool [24].We extracted the Gibbs free energies of formation (∆ f G • ) for all external metabolites involved in an ECM.Next, we normalized the ECMs with respect to the carbon atoms of the carbon source (C-mol) and applied a Laplace transformation, adapting for temperature (298.15K), pH (7.4), pMg (3.0), and ionic strength (0.25 M).We used Hess's law to calculate the standard Gibbs free energy of reaction for each ECM, where ν i and ∆ f G Every carbon that is converted to biomass will have

Figure 1 .
Figure 1.The view of microbial metabolism as a thermodynamic energy converter.Catabolic reactions have a large negative free energy gradient, driving anabolic reactions.

98
product conversions [6].Using ECMs instead of EFMs 99 reduces the necessary computational power drasti-100 cally.Additionally, modern software, such as ecmtool 101 that allows parallelization of the computation, helps 102 to obtain an exhaustive list of all metabolic capabili-103 ties of an organism in the form of ECMs [5].104 As suggested in [6], we view ECMs as building blocks 105 of macrochemical equations.We show how genome-106 scale metabolic models can be used to systemati-107 cally enumerate all possible catabolic pathways.With 108 thermodynamic data, in particular energies of for-109 mation of substrates and products obtained from 110 the eQuilibrator tool [2, 24], we characterise the 111 catabolic pathways by their energy gradient.Using 112 the network models, we further estimate the maximal 113 ATP production capacity for each catabolic pathway, 114 and thus determine their thermodynamic efficiencies.115 We then analyze experimental data for Saccharomyces 116 cerevisiae [31] and Escherichia coli [15], grown un-117 der controlled chemostat conditions in defined me-118 dia, to separate the macrochemical equation into the 119 catabolic and anabolic parts and characterise their 120 thermodynamic properties.Interpreting our findings

Figure 2 .
Figure 2. Standard Gibbs free energies of all catabolic pathways, normalised to carbon mole.The catabolic pathways were derived using elementary conversion modes (ECMs) calculated from the E. coli core network.Red symbolises ECMs that use oxygen, while blue denotes ECMs not using oxygen.Filled bars belong to ECMs that include no compounds with the element nitrogen while empty ones include nitrogen containing metabolites.The black crosses indicate the maximal yield of ATP per carbon mole nutrient for each ECM (right axis).standard Gibbs free energy of reaction, based on the standard energies of formation estimated by eQuilibrator [2].The standard energies of catabolism ∆ cat G 0 , normalised to one carbon mole of consumed substrate, are displayed in Fig. 2. Most of the catabolic pathways have relatively low energy gradients between approximately 20 and 50 kJ/Cmol.To this large group of pathways belong key catabolic routes, such as the fermentation of glucose to lactate or ethanol.Few catabolic routes exhibit

Figure 3 .
Figure 3. Thermodynamic characterisation of catabolic routes in S. cerevisiae genome-scale model (iND750) for α-ketoglutarate, glucose, xylose, and pyruvate as carbon source.Additionally, oxygen is allowed to be a substrate in the calculation of the elementary conversion modes.The efficiency is based on a typical value of 46.5 kJ/mol for production of ATP in E. coli[42].

∆ r G
ATPase the energy of reaction for ATP synthesis from ADP and inorganic phosphate, and ∆ cat G the energy of reaction of the respective catabolic pathway.We assume the typical value of ∆ r G ATPase = 46.5 kJ mol −1[42].Further, we approximate the Gibbs free energy of catabolism by the corresponding standard Gibbs free energy of reaction, because changes in substrate and product concentrations in the medium are likely to have only a minor effect on the quantity.The determined efficiencies η are depicted in the right panel of Fig.3.Interestingly, the pathways with the highest energy gradient are not the most efficient.For instance, under full respiration of glucose, only 28% of the released free energy is converted to chemical work producing ATP.In contrast, fermentation of glucose to lactate exhibits an efficiency of 43%.The fermentation of glucose to lactate is one of the most efficient reactions, with higher efficiencies only found in fermentation processes involving ethanol production.

339
The determined catabolic coefficients are summarised 340 in Fig. 4. It can clearly be seen that the onset of over-341 flow metabolism, when glucose is partly fermented 342 even in the presence of sufficient oxygen, occurs at 343 growth rates of around 0.3 h −1 for S. cerevisiae and 344 around 0.4 h −1 for E. coli.With the catabolic co-345 efficients, we calculate the standard Gibbs free en-346 ergy of reaction of the overall catabolic conversion, 347 where we obtained the standard Gibbs free energies 348 of formation, required for this calculation, from the 349 equilibrator tool [2].It can be observed (blue lines 350 in Fig. 4) that with the onset of overflow metabolism, 351 also the Gibbs free energy gradients are reduced sig-352 nificantly.

Figure 4 .
Figure 4.The catabolic stoichiometric coefficients and thermodynamic driving forces determined for chemostat growth of E. coli [31] and S. cerevisiae [15] at different dilution rates.All coefficients are given in mol/C-mol substrate, except for ethanol and acetate, which are given in C-mol/C-mol substrate.For E. coli, CO2 production is identical to O2 consumption.The thermodynamic driving force is given as standard energy of reaction of the overall catabolic conversion, normalised to one carbon mole of substrate.
in turn is set by the dilution rate of the chemostat, J ana = D.In this model, the matrix L is the Onsager matrix of the phenomenolocigal coefficients [25].

Figure 5 . 386 ∆
Figure 5. Metabolic fluxes as function of the catabolic driving force.Shown are the catabolic (blue), anabolic (red) and total (green) glucose consumption rates in dependence of the catabolic driving force, −∆catG 0 .On the x-axis on the top, the force ratio x = ∆catG 0 /∆anaG 0 is given.Considering the large energy gradients, we approx-374

Figure 6 .
Figure 6.Catabolic and anabolic powers, as well as power of ATP synthesis as a function of growth rate.Catabolic power is depicted in blue, anabolic power in red, and ATP synthase power in black.Circles present results for S. cerevisiae, crosses for E. coli.

16 )
• i are the stoichometric coefficient and the Gibbs free energy of formation of the i th external compound in the ECM, respectively.For the calculation of the maximal ATP production for an ECM, we constrained all external fluxes to the values of the respective ECMs while maximizing ATP hydrolysis (excluding ATP maintenance):maximize v ATPM , such that N • v = 0 v i,ex = ν i for i ∈ ECM, v j,ex = 0 for j / ∈ ECM (14)where N is the stoichiometric matrix of the metabolic 642 model, v ATPM is the flux through the reaction643 ATP + H 2 O −−⇀ ↽−− ADP + Pi,(15)and v i,ex are fluxes through the reaction exchanging 644 metabolite i, which is constrained to the stoichiomet-645 ric coefficient ν i obtained by the respective ECM.The 646 stoichiometric coefficients are normalised to one car-647 bon mole substrate.648 The thermodynamic efficiency of ATP production cal-649 culates as 650 η = c ATP • ∆ r G ATPase |∆ cat G| .(For the Gibbs free energy of ATP synthesis, we 651 used a typical value for E.coli of ∆ r G ATPase = 652 46.5 kJ mol −1 [42].653 Calculating the stoichiometry of anabolism 654 We assume that the substrate [S] has the nor-655 malised sum formula CH x O y and the biomass [X] has 656 CH a O b N c , and that the biomass is more reduced than 657 the substrate, i.e. 658 γ S = 4 + x − 2y ≤ γ X = 4 + a − 2b − 3c.(17) We assume an overall stoichiometry of 659 b 1 [S] + b 2 NH 3 −−→ [X] + b 3 CO 2 + b 4 H 2 O. (18)

660 1 = 1 ( 21 )
to be reduced by γ X − γ S .From the overall redox bal-661 ance it follows that for each carbon that is converted662 into biomass, 663 b 3 = γ X − γ S γ S (19)carbons have to be oxidised to CO 2 .From the carbon 664 balance of (18) it follows that b b and b 4 = b 1 x + 3b 2 − a 2 .It is straight-forward to generalise these calculations to include sulfur and phosphorus into the 669 biomass.hhu/thermodynamics-task-force/2023-energy-750 metabolism-of-microorganisms.
↽−− 1 [X] eco + 0.0530 CO 2 + 0.5635 H 2 O + 0.2205 H + , 2 (see Methods).256Thisequation allows for the calculation of the stan-257 dard energy of reaction of anabolism, ∆ ana G 0 .To es-258 timate the Gibbs free energy of formation of biomass, 259 which is required to determine the energy of reaction, 260 we employ the empirical method proposed by Batt-261 ley [1].262Constraint-based models can be employed to inves-263 tigate to what extent such an ideal anabolic reaction 264 can be realised by a microorganism's metabolism.We 269 optimisation, in which the minimal nutrient uptake 270 is fixed and the required reverse ATP hydrolysis is 271 minimised, allows determining the minimal ATP re-272 quirement per carbon mole biomass formed.For the 273 iJR904 model of E. coli metabolism, we obtain the 274 following optimal anabolic stoichiometry for growth 283 95.0% is slightly lower than expected by Eq. (2).This 284 is explained by the fact that also small amounts of 285 oxygen are required for the pure anabolic biomass 286 formation.In iJR904 this is caused by a minimal 287 required flux through a cytochrome oxidase which re-288 quires molecular oxygen as substrate.289Asubsequent optimisation reveals a minimal require-290 ment of 1.766 mol ATP per carbon biomass pro-291 duced.292For the iND750 metabolic model of S. cerevisiae, we −−⇀ ↽−− 1 [X] sce + 0.0718 CO 2 + 0.4035 H 2 O + 0.1404 H + , 296 of 297 γ X,sce = 4.080, ∆ f G 0 X,sce = −128.76kJC-mol −1 .(9)Here, the descrepancy between the computationally 298 determined maximal yield of Y max X/S = 1 1.0718 = 93.3%299 and the 98.0% expected from Eq. (2) is even larger.300 The minimal requirement of ATP to produce biomass 301 is predicted to be slightly larger than for E. coli with 302 2.031 mol ATP per C-mol biomass.303 We repeated the calculations for different carbon 304 sources.The results are summarized in Table 1.In 317 in fact, nitrogen) yield, but a larger free energy gra-318 dient.319 Separating catabolism from anabolism based on 320 chemostat data 321 In a controlled continuous microbial cultivation sys-322 tem, such as a chemostat [21, 14], it is possible to grow 323 microbial cultures at a steady state with pre-defined 324 growth rates.Measuring nutrient and gas exchange 325 rates as well as nutrient and product concentrations in 326 the reactor allows experimental determination of the 327 overall growth stoichiometries [12, 13, 48, 49].328 In the following we employ experimentally determined 329 macrochemical equations for growth of S. cerevisiae 330 [31] and E. coli [15] in chemostats at different dilution 331 rates to calculate catabolic stoichiometries, ATP pro-332 duction potential, and thermodynamic efficiencies for 333 each condition.The catabolic stoichiometry is calcu-334 lated by first identifying the ideal anabolic stoichiom-335 etry based on the degrees of reduction of substrate 336 and biomass, and then subtracting this anabolic sto-337 ichiometry from the macrochemical equation (for de-338 tails, see Methods).

Table 1 .
Thermodynamic properties of anabolic pathways.The theoretical maximal yield Y max X/S is calculated according to Eq. (2).The maximal yield Y max,model X/S predicted by the metabolic model was determined using the linear program (22).The minimal anabolic ATP requirement per carbon mole biomass, aATP,min was determined using the linear program (23).The standard energies of reaction for anabolism, ∆anaG 0 , were determined from overall anabolic stoichiometries like given for glucose in Eqs.(4) and (7).ATP,min (C-mol C-mol −1 ) ∆anaG 0 (kJ C-mol −1 ) calculated ECMs to thermodynamically charac-The output of ecmtool is a matrix, in which the rows are the respective elementary conversion modes, and the columns are all external metabolites that were not hidden.For the E. coli core network[26], no metabolite had to be hidden,