Calculation of the Aqueous Thermodynamic Properties of Citric Acid Cycle Intermediates and Precursors and the Estimation of High Temperature and Pressure Equation of State Parameters

The citric acid cycle (CAC) is the central pathway of energy transfer for many organisms, and understanding the origin of this pathway may provide insight into the origins of metabolism. In order to assess the thermodynamics of this key pathway for microorganisms that inhabit a wide variety of environments, especially those found in high temperature environments, we have calculated the properties and parameters for the revised Helgeson-Kirkham-Flowers equation of state for the major components of the CAC. While a significant amount of data is not available for many of the constituents of this fundamental pathway, methods exist that allow estimation of these missing data.


Introduction
The citric acid cycle (CAC), also known as the Krebs or tricarboxylic acid cycle, is a fundamental pathway in intermediary metabolism among all the domains of life. Organisms that use the CAC (or especially the reverse or reductive CAC) are represented among the most deeply-rooted autotrophic hyperthermophilic Archaea and Bacteria and the most derived of organotrophic evolutionary lineages [1][2][3][4]. The key role of iron-sulfur proteins, thioester intermediates, and the reductive use of the CAC in hyperthermophilic autotrophs link the CAC to prebiotic theories of energy metabolism and abiotic carbon fixation at deep-sea hydrothermal vents [5][6][7]. Because of the fundamental metabolic roles and evolutionary importance of the CAC and the reverse CAC, it is of interest to understand the conditions constraining the many reactions of which it may be composed under the considerable range of physical and chemical environments where it functions. Most thermodynamic data available for the substrates used in the CAC are incomplete and have only been determined at the 25 °C and 0.1 MPa standard state. Therefore it is difficult to predict accurate reaction thermodynamics well beyond those conditions, including those likely to have hosted the emergence of life [7][8][9][10]. For examination of CAC reactions under the relatively extreme high-temperature and high-pressure conditions where life can thrive and may have originated, and to determine the geochemical environments where prebiotic conditions may have been favorable, the high pressure and temperature thermodynamic parameters of the substrates must be determined.
To evaluate the standard Gibbs free energy ( ) for each substance at nonstandard temperature and pressure must be calculated. Most thermodynamic data available for aqueous organic molecules beyond the 25 °C and 0.1 MPa reference conditions are generally for compounds that are either potential growth substrates or metabolic by-products. Only in a few works are there data available for compounds that are intermediates in the ubiquitous biochemical pathways found in nature. The general lack of empirical data for aqueous organic compounds at elevated temperature and pressure has lead to the development of various equations of state to describe the behavior of various thermodynamic properties at non-standard temperature and pressure [11,12]. The revised Helgeson-Kirkham-Flowers equations of state [11,13], along with methods for the estimation of high pressure and temperature thermodynamic properties [14][15][16][17][18][19][20], can be used to predict accurately reaction thermochemical properties for aqueous organic compounds well within the thermal and pressure range of the possible environments where these reactions may be expected to occur in biological processes. In addition to aqueous organic compounds [13,18,21,22], the revised HKF equations of state have been used to calculate accurately the thermodynamic reaction properties, without the benefit of high pressure and temperature data, for aqueous inorganic [17] and organic ions [13,23], and inorganic electrolytes [19] at subcritical temperatures and pressures up to 500 MPa. Such versatility makes it one of the more useful tools for evaluating geochemical and biochemical reaction properties in a wide variety of environments, as the revised HKF allows calculation of species and reaction properties among minerals, gases and aqueous species. This provides a framework for understanding many interdependent metabolic and geochemical processes [24].  [25] Data for reactions depicted in Figure 1 are used as a suggestion for a prebiotic reductive CAC. However, estimation of the thermochemical properties of the thioester intermediates may also function as proxies in lieu of high pressure and temperature reaction data for coenzyme A or other thioester intermediates found in Archaea.

Strategy used for the estimation of missing reference state values
The approach taken to estimate reference state values herein is to utilize methods using state variables and reaction data, where available, to calculate missing values. In absence of state or reaction data, the group contribution method is used, with emphasis on attaining data for the closest structural analogues as a base structure. In doing so, the fewest group values are used to modify the base structure as a means of decreasing the probability of error accumulation. The selection of methods used to estimate reference state data are summarized in Figure 2 using the neutral species as examples. thioester acid-anion pair. The aq G f 0  (>C=O) group value was calculated from the difference in the yintercepts for regression equations between n-alkanones and (n-1) n-alkanes (both taken from reference [18]). The aq G f 0  (-S-) was calculated similarly, from the difference in y-intercepts from dialkylsulfides [34] and n-alkanes [18], with the same number of carbon atoms. Data for other components of the citric acid cycle (ethyl thiol and acetic acid; see Figure 1) are available in the literature [16,23].

Figure 2.
Group contribution method used to estimate (a) the standard partial molal entropy (S°) and volume (V°) and (b) the heat capacity (C p°) and free energy (G f°) of aqueous organic species considered in this study.  Table 1. b -group values from [35]. c -the carbonyl group value was estimated from the difference in 0 V between succinic acid [23] and -ketoglutarate [36]. d -estimated using group values from [31]. e -malonic acid from [23]. f -group aq S f 0 value for >C=O assumed to be the difference in aq S f 0 between -ketoglutaric acid [37] and succinic acid [23]. g -calculated form pKa values from [33] and aq G f 0  for ions from Table 1 (as described in text). h -value for aq G f 0  was taken from [38], the aq H f 0  was from [33], and elements S f 0 were the CODATA values from [39]. i -estimated from 0 V of H-fumarate -1 ( Figure 2) assuming the same difference in 0 V between the acid and ion as between succinic acid and H-succinate -1 from [23]. jestimated by subtracting the difference in y-intercept values for 0 P C between n-alkanes and n-alkenes (taken from [18]) from that of succinic acid (taken from [23]). k -calculated from pKa and aq G f 0  values from [33] for ions (as described in text) , aq H f 0  was from [37)] and elements S f 0 were the CODATA values from [39]. l -the 0 P C for succinic acid was taken from [23]. The group value for the >C=O was taken from [31] . m -from [18]. n -group value estimated from difference in aq S f 0 values between ethyl sulfide (as described in Appendix) and n-butane (taken from [18]). o -group value from [40]. p -group value estimated from difference in 0 P C values between ethyl sulfide [34] and n-butane (taken from [18]. q -from [23]. r -group value estimated from difference in yintercepts of aq S f 0 between n-alkanones and n-1 n-alkanes taken from [18]. s -group value from [30]. t -group value estimated from difference in y-intercepts of 0 P C between n-alkanones and n-1 n-alkanes taken from [18]. u -group value estimated from difference in y-intercepts of aq G f 0  between n-alkanones and n-1 n-alkanes taken from [18]. v -group value estimated from difference in aq  -kJ mol -1 . b -J mol -1 K -1 . c -cm 3 mol -1 . d -J mol -1 bar -1 . e -J mol -1 . f -J K mol -1 . g -J mol -1 K -1 . h -value as described in Table 1. i -from [33]. j -calculated from pKa values from [33] and aq G f 0  of oxaloacetate -2 from above. k -from [38]. l -from [37]. mfrom [41]. n -estimated as described in Figure 2 . o -estimated as was succinyl thioester in Figure 2, except using pentanoate, taken from [23], as the base structure. p -estimated as was oxaloacetic acid in Figure 2, except using Hmalonate -1 , taken from [23], as the base structure. q -estimated as was oxaloacetic acid in Figure 2, except using malonate -2 , taken from [23], as the base structure. r -calculated from aq S f 0 of malic acid in above table using the S ion  from [33]. sestimated from aq S f 0 of a-ketoglutaric acid from above table assuming the same S ion  as between succinic acid and its respective ions in [23]. t -estimated from 0 V of pyruvic acid in above table assuming the same V ion  as between lactic acid and lactate, in [23]. u -estimated as was oxaloacetic acid in Figure 2, but using H-malonate -1 , from [23] as the base structure. v -estimated as was oxaloacetic acid in Figure 2, but using malonate -2 , from [23] as the base structure . w -from [42]. x -estimated by assuming the same V ion  as between succinic acid, taken from [35] and its respective ions, taken from [23]. y -from [43]. z -from [36]. aa -from [44]. bb -estimated from 0 P C of pyruvic acid in above table assuming the same 0 P ion C  as between lactic acid and lactate in [23] . cc -estimated as described in Figure 2 for oxaloacetic acid but using respective 0 P C of H-malonate -1 or malonate -2 . dd -estimated form the acid in the above table assuming the same 0 P ion C  as between succinic acid and the respective -1 or -2 ion from [23]. ee -estimated using Equation (6) ff -estimated using Equation (7) . ggestimated using Equation (8) . hh -calculated from the a 2 parameter in above table as described by [23]. ii -calculated using Equation (10) . jj -calculated using Equation (11). kk -calculated using Equation (12) . ll -calculated from aq S f 0 in above table using Equations (30-32) from [18]. mm -estimated as described in text.
The estimation of high temperature and pressure thermochemical properties for cis-aconitate and isocitrate, for the reactions between citrate and α-ketoglutarate, were not calculated due to deficient reference state data. Since citric acid and its ions are the only feasible tricarboxylated base structures available to use for the estimation, this approach was rejected to avoid circularity. Furthermore, the primary goal of estimating the high pressure and temperature parameters for CAC reactions calculated herein is primarily driven to examine these reactions under the near equilibrium conditions that are present in anoxic systems. Extant microbes that use the CAC reductively to fix carbon generally only contain a partial cycle where the pathway is used to supply precursors for biosynthesis via succinyl-Coenzyme A or α-ketoglutarate [2][3][4]45].

Calculation of the reference state entropy ( aq S 0 ) and estimation of missing values
The aq S 0 values selected are shown in Table 1. If aq S 0 values were not available, third-law entropies were calculated from the aqueous enthalpy and Gibbs free energy values along with the sum of entropies of the elemental constituents using Equation (1): The aq S 0 for some anions in Table 1 were calculated from the aq S 0 of the acid using S ion  for the ionization reactions from Miller and Smith-Magowan [33]: The aqueous formation entropies for H-α-ketoglutarte -1 and α-ketoglutarte -2 were estimated by assuming similar S ion  from succinic acid and its mono-and divalent anions (from Shock [23]). The carbonyl group aq S 0 contribution value (54.3 J mol -1 ) used for oxaloacetic acid and its respective anions was calculated from the difference in aq S 0 between α-ketoglutaric acid and succinic acid from Shock [23]: To calculate the aq S 0 needed for acetyl thioester the value of aq S 0 (-S-) was calculated from the difference in aqueous entropies between ethyl sulfide (see Appendix A, Equation 21), and n-butane [18].
The aq S 0 (>C=O) value used for succinyl thioester and its anion was estimated from the difference (68.15 J mol -1 K -1 ) in y-intercepts between n-alkanones and (n-1) n-alkanes, both from Shock and Helgeson [18]. The aq S 0 (>C=O) value used here was chosen since the value derived from an nalkanone may be more likely to represent the value of a subterminal carbonyl adjacent to a sulfur atom, as opposed to the α-carbonyl adjacent to a carboxylic acid, as was used above for oxaloacetic acid.

Calculation of the reference state partial molal volume (V°aq) and estimation of missing values
The chosen partial molal volume values are shown in Table 1. The group value 0 V (>C=O) used for the oxaloacetate series was estimated by the difference in 0 V between α-ketoglutaric acid [36] and succinic acid [35] such that: This 0 V (>C=O) was used for pyruvic acid and pyruvate as well since it may rather well represent the volume of an α-carbonyl ( 0 V~5 cm 3 mol -1 ) adjacent to a carboxylic acid, rather than n-alkanone's carbonyl with a much larger volume of (~14-15 cm 3 mol -1 [30,31]).
The 0 V for fumarate -2 was used to estimate the 0 V of fumaric acid and the H-fumarate -1 anion by assuming the same V ion  relationship for fumaric acid and its ions as for succinic acid from Criss and Wood [35], and H-succinate -1 or succinate -2 anion, from Shock [23], as demonstrated in Figure 2. The same relationship, using the V ion  (13.13 cm 3 mol -1 ) between lactic acid and lactate from Shock [23] was used to estimate the 0 V for pyruvate from its acid. The partial molal volume of the (-S-) group for a thioester was estimated by the addition of the value of 0 V (-S-) from a value published by Lepori and Gianni [40] to the 0 V of 2-butanone taken from Shock and Helgeson [18]. The 0 V of succinyl thioester and its anion were estimated by the addition of 0 V (-S-) [40] and 0 V (>C=O) to the 0 V of pentanoic acid and pentanoate, respectively (both from Shock [23]). The 0 V (>C=O) value was estimated to be the difference between the 0 V of n-alkanones and (n-1) n-alkanes (both taken from Shock and Helgeson [18]).

Calculation of the reference state heat capacity (C°p aq) and estimation of missing values
The standard molal isobaric heat capacities at the reference temperature and pressure are shown in Table 1 and estimation procedures in Figure 2. The 0 P C of pyruvate ion was assumed to have the same difference in 0 P ion C  from its acid, as did the lactic acid-lactate pair [23], of 167.4 J mol -1 K -1 . The 0 P C for oxaloacetic acid was estimated by adding the 0 P C (>C=O) value (-52.0 J mol -1 K -1 ) from that reported by Cabani et al. [31] to that of malonic acid [23]. For the estimation of 0 P C values for both Hoxaloacetate -1 and oxaloacetate -2 ions, it was assumed that the difference from the acid was the same as between succinic acid and it respective anions from Shock [23]. This assumption was also used to estimate 0 P C values of the H-mono-and divalent anions of malate, fumarate, and α-ketoglutarate. The 0 P C of fumaric acid was estimated through subtracting the values for the difference (70.7 J mol -1 K -1 ) in y-intercepts between n-alkanes from n-alkenes (both values being from Shock and Helgeson [18]) from that of succinic acid. For α-ketoglutaric acid, the 0 P C (>C=O) value (-52.0 J mol -1 K -1 ) reported by Cabani et al. [31] was added to that of succninc acid. The 0 P C for succinyl thioester was estimated from addition of the 0 P C (>C=O), and the 0 P C (-S-) (-81.2 J mol -1 K -1 ), both calculated in this work, to the 0 P C of pentanoic acid from Shock [23]. The 0 P C (-S-) for succinyl thioester was calculated from the difference in y-intercepts of C 2 ,C 4 ,C 6 n-diaklysulfides [34] and C 2 ,C 4 ,C 6 n-alkanes [23]. The 0 P C (>C=O) value (-135.2 J mol -1 K -1 ) was calculated to be the difference between 2-pentanone and n-butane, both from Shock and Helgeson [18]. This value for 0 P C (>C=O) was chosen for succinyl thioester because a carbonyl from an n-alkanone may be more likely to represent the 0 P C (>C=O) in a thioester, as opposed to that from a α-carbonyl adjacent to a carboxylic acid. The resulting value of 216.1 J mol -1 K -1 is in close agreement with the sum of group values provided by Cabani et al. [31], using the (-S-) calculated here, of 216.4 J mol -1 K -1 . The 0 P C of the succinyl thioester anion was assumed to have the same difference from its acid as that between pentanoic acid and pentanoate [23]. The 0 P C of acetyl thioester was estimated by adddition 0 P C (-S-), as described above, to that of 2-butanone from Shock and Helgeson [18].

Extrapolation of reference state data to high pressures and temperatures
Starting with 25°C, 0.1 MPa reference state data, calculating the at the elevated P and T involves the integration of Equation (5) as: The revised-HKF model allows for the incorporation of the changes in the partial molal volume and heat capacity as described in Appendix B.

Estimation of the temperature and pressure effects on the partial molal volume of aqueous organic species: the non-solvation contribution
The partial molal volume of a substance in the revised HKF model is defined by Equation (24). At temperatures of ~150°C, at the water-saturation vapor pressure (Psat), the solute-dependent contribution (the non-solvation volume, parameters have been generated from empirical data gathered at high pressure and temperature conditions for a variety of compounds [11,[15][16][17][18][19][20]46,47]. The a 1 variable is correlated to a high degree with the 0 n V  of a wide range of neutral and charged aqueous organic compounds with a variety of functional groups ( Figure 3) and can therefore be used to estimate the values of a 1 for compounds with structural homology to those with high pressure and temperature data for which there are no volumetric data beyond the reference state. For the range of compounds regressed in Figure 3, the line is defined by: where a 1 is in J mol -1 K -1 and 0 n V  in cm 3 mol -1 . This regression equation was used to generate the a 1 parameter for all neutral and charged compounds in this work.
The a 2 parameter is somewhat more dependent on the functional group characteristics of a particular molecule. In Figure 4 where a 2 is also in J mol -1 K -1 . For n-alkanones and n-alcohols ( Figure 4) the slope is somewhat shallower and the y-intercept lower so these values are fit better by the line   (25) using the partial molal volumes and effective Born coefficients of (C 3 -C 5 ) carboxylic acids, (C 2 -C 6 ) carboxylate anions, (C 3 -C 5 ) hydroxy acids, ( C 3 -C 6 ) hydroxylate anions, (C 2 -C 6 ) dicarboxy acids, -1, and -2 anions [23], and aldehydes [15]. (b) Upper line: Regression plot generated from points in upper figure. Lower line: Regression of non-solvation volumes calculated with Equation (25) using the partial molal volumes and effective Born coefficients of (C 3 -C 5 ) nalkanones, and (C 2 -C 5 ) n-alcohols [23].

(a) (b)
The a 2 parameter for all dicarboxylic acids, and dicarboxylate anions, citric acid and its anions, pyruvic acid and pyruvate, and succinyl thioester and its anion were estimated with Equation (7). Since the only apparently large departures from the line in Equation (7) are for shorter-chained carboxylate and hydroxylate anions, a fit for hydroxylates was considered separately to calculate the a 2 parameter for pyruvate. The y-intercept for hydroxylates (Equation (9)) is slightly lower with a steeper slope than Equation (8): However, if the y-intercept value for n-alcohols vs. n-alkanones can be considered comparable to that of the hydroxy and carbonyl groups in hydroxy-and  keto-acids, respectively, then pyruvate's yintercept would shift toward the line in Equation (7). Therefore the a 2 value for pyruvic acid using Equation (7) was retained. The a 2 parameter for acetyl thioester was estimated with the line defined by Equation (8) for n-alcohols and n-alkanones.
All a 4 parameters were generated, as suggested by Shock and Helgeson and Shock [18,23], using the correlation with the a 2 fitting parameter. The a 3 parameter was then calculated by solving the rearranged non-solvation molal volume term (

Estimation of temperature and pressure effects on the isobaric heat capacity of aqueous organic species: The non-solvation contribution
The non-solvation and solvation heat capacity contributions to the partial molal heat capacity are defined in Equation (27). The non-solvation contribution term of Equation (27) is expanded in Equation (28) and combines the influence of pressure on the 0 ,n P C  with incorporation of a substance's a 3 and a 4 fitting parameters, from above, and the influence of temperature by the use of two heat capacity fitting parameters, c 1 and c 2 . The c 2 parameter correlates closely with the reference state 0 P C for compounds with similar functional groups ( Figure 5). Therefore, from this correlation high temperature and pressure data from structurally similar compounds can be used to predict the c 2 parameter of molecules for which no high temperature heat capacity data are available. Figure 5a shows the plot of c 2 vs. 0 P C values, taken from Shock [23], for organic acid anions. The regression of these values for the range of compounds shown is described by the line in Equation (10): (10) where c 2 is in J mol -1 . This regression equation was used to estimate the c 2 parameters for all the dicarboxylate and carboxylate anions from the reference state 0 P C used in this study. Equation (11) describes the upper line in the plot of c 2 vs. 0 P C values from organic acids (taken from Shock [23]) in Figure 5b: (11) which was used to estimate the c 2 parameter for all organic acids. The lower line in Figure 5(b) is a plot of the c 2 vs. 0 P C of short-chained n-alcohols and n-alkanones [18] and is described by the line in Equation (12), which was used to estimate the c 2 parameter for acetyl thioester: c 2 = 0.283 0 P C -16.970 (12) Figure 5. Regression plots of the non-solvation molal heat capacity variable c 2 against the reference state heat capacity of neutral and ionic organic species from the literature [18,23]. (a) Plot of organic acid anions: (C 2 -C 5 ) hydroxylates, (C 2 -C 5 ) carboxylates and (C 3 -C 6 ) H-dicarboxylate -1 and dicarboxylate -2 ions (all from reference [23]) used to generate Equation (10). (b) Upper line: Plot of neutral acids: (C 4 -C 5 ) carboxylic acids, (C 3 -C 5 ) hydroxy acids, and (C 4 -C 6 ) dicarboxylic acids from reference [23] used to generate Equation (11). Lower line: Plot of (C 3 -C 5 ) n-alkanones and (C 2 -C 5 ) n-alcohols from reference [18] used to generate Equation (12).

Estimation of temperature and pressure effects on the partial molal properties of aqueous organic species: The solvation contribution
At temperatures of ~150°C, at the water-saturation vapor pressure (Psat), the solvent-dependent contribution to the partial molal volume (Equation (26)) and heat capacity (Equation (29)) terms begin to dominate each partial molal function. The conventional (ω) and effective (ω e ) Born coefficients, respectively, are used to describe the substance-specific solvation properties of an ionic species or electrolytes, and neutral species (see Appendix B).

Calculation of the conventional Born coefficient () for ionic species
The ω of ionic species has been demonstrated to have a strong correlation with the aq S f 0  [17] and can therefore be calculated from this quantity. In Figure 6 ω is plotted against the aq S f 0  for a variety of mono-, di-and trivalent inorganic and organic anions (taken from Shock and Helgeson (1988) [17]) using relationships in Equations (30)(31)(32) to calculate ω, which was also used to calculate the ω for all anions in this work (data also shown in Figure 6).  [17]), organic anions (taken from Shock [23]) and compounds calculated in this work.

Calculation of the effective Born coefficient ( e ) for neutral species
The effective Born coefficient used to describe the solvation contribution of neutral species in the revised-HKF model has been calculated from aq S f 0 [13,22,23] and the Gibbs free energy of hydration ( G hyd 0  ) (Appendix C) in the absence of high temperature 0 P C and 0 V data. It has been demonstrated from empirical data that ω e generally has a negative value for low-molecular weight neutral organic compounds. The negative value is the result of the inflection of 0 P C and 0 V values towards positiveinfinity near the critical point of water. As the properties of a set of molecules become increasingly polar, the solute-solvent interaction increases. In the case of some neutral inorganic polyhydroxyl compounds such as aqueous silica, of which the hydrated form is thought to be H 4 SiO 4 , and boric acid (H 3 BO 3 ), both demonstrate electrolyte-like behavior and thus positive ω e values. The mechanism associated with this phenomenon is thought to arise from water-solute versus water-water competition near the critical point [48], where solutes that are associated with more solvent molecules than is the solvent itself, and will have 0 P C and 0 V values that approach -∞. The volatility of a substance in comparison to water is also thought to have influence over this process as well. This issue is discussed in detail by Amend and Plyasunov [49], where predictions were made concerning the near-critical point behavior of carbohydrates. It is clear, through their discussion and others, that the relationship between e  and solvation at higher temperatures is not an obvious one. For instance, the nonelectrolyte amino acid proline displays low-temperature solvation behavior that is best described with a negative value for ω e . As the temperature increases, however, proline's solvation is best fit with a positive ω e . However, this type of solvation behavior may be particular to the proline zwitterion due to its particularly asymmetrical dipole and spatial arrangement of hydrophobic and hydrophilic moieties [22]. If the high density of polar functional groups, as in polyhydroxy compounds, is to be viewed as an indicator of near-critical behavior of compounds with hydroxyl functionalities, then for dicarboxylic acids, oxalic acid with an ω e that displays neutral behavior may be the closest analogue.
In Table 1, different values for e  are displayed as calculated from the correlations with the G hyd 0  (Equation (44)) and entropy (Equation (45)), as has been done previously. Also included is a systematic estimation of ω e using values provided by Shock and Shock and Helgeson [18,23] with results shown in Figure 7 in comparison to other series of neutral organic compounds. The values of these estimations were used for ω e in all the neutral compounds in this work. The ω e for pyruvic acid was estimated by taking the ω e from lactic acid and subtracting the difference between n-propanol and acetone: ω e lactic acid -(ω e n-propanol -ω e acetone) = ω e pyruvic acid (13) The ω e for oxaloacetic acid was estimated from the value for malonic acid with addition of the carbonyl group value such that: ω e malonic acid + (ω e 2-pentanone -ω e n-butane) = ω e oxaloacetic acid (14) Malic acid's ω e value was estimated by from succinic acid from: ω e succinic acid + (ω e hydroxybutanoic acid -ω e butanoic acid) = ω e malic acid (15) to add the value of the hydroxyl group. For fumaric acid the difference in n-alkanes and n-alkenes were used to modify succinic acid as: ω e succinic acid + (ω e n-butene -ω e n-butane) = ω e fumaric acid (16) The ω e for α-ketoglutaric acid also used succinic acid and carbonyl value for oxaloacetic acid: e  succinic acid + ( e  2-pentanonee  n-butane) = e  α-ketoglutaric acid (17) Since there are currently no analogous compounds (tricarboxyl) for citric acid, its ω e was estimated to be the same as a C 6 compound using the slope and y-intercept for dicarboxylic acids. The value for succinyl thioester was estimated by: ω e hexanoic acid + (ω e 2-pentanone -ω e n-butane) + (ω e diethyl sulfide -ω e n-butane) = ω e succinyl thioester (18) using the value from [22] for diethyl sulfide. The ω e estimation for acetyl thioester also used the same value for diethyl sulfide: ω e diethyl sulfide + (ω e 2-pentanone -ω e n-butane) -(ω e alkane -ω e alkane (n-1) ) = ω e acetyl thioester .  Table 1. Datum points are the values of e  for the neutral compounds calculated as described in the text.
In consideration of consistency with previous works and the pragmatic aspects of the temperature ranges to be expected to be relevant for these compounds, negative values for ω e were chosen. In addition, for practical purposes, since the pKa of the strongest acid among these compounds is 2.49 (pyruvic acid from reference [33]), at pH above 3-4 the acids of these compounds will not usually be germane in writing reactions.

Analysis of the possible error associated with methods used in estimating reference and standard state parameters
Since the motivation for this work is to estimate the high temperature and pressure thermochemical parameters for compounds for which there was no data, the credibility of the resulting values need be examined. The effects of errors on the reference state data, as well as for values calculated beyond the reference state, can be analyzed by the comparison of expected calculations, using the estimated values, with calculations from values in which a sensible error is incorporated. Figure 8a demonstrates the influence of the over-and underestimation of e  on the partial molal volume of propanoic acid along P sat calculated using Equations (25) and (26) with revised HKF parameters from Shock [23]. Although the reference state 0 r r T P V is unaffected by ω e , the error in the calculated 0 PT V increases with temperature as the partial molal volume is influenced increasingly by the solvation term, although at higher pressures the error is diminished (Figure 8b). However, as we are most concerned with providing data for reaction thermodynamics at elevated pressure and temperature, the effects of inaccurate estimation of the 0 ,T P G  are of primary concern. As can be seen in Figure 8c, the revised-HKF method is quite insensitive to even a 2-fold under-or over estimation of ω e , with a maximum of ~0.3% relative error at 350 0 C. The insensitivity to errors from the estimation methods can be further tested by swapping the HKF parameters of one compound for another, while using the original . To demonstrate this, for compounds in Figure 8d the original aq G f 0  values were retained while the remaining values ( aq S 0 , a 1 -a 4 , c 1 , c 2 , and Born coefficients) were mutually swapped from another compound: pyruvic acid and propanoic acid, H-oxaloacetate -1 and H-α-ketoglutarate -1 , and oxaloacetate -2 and α-ketoglutarate -2 (all values are from Table 2 or Shock [23] for propanoic acid) and used to solve Equation (39). The largest error (~4% relative at 350 °C) is encountered when the equation of state parameters from propanoic acid are replaced with those of pyruvic acid. Considering that there is a relative difference (compared to pyruvic acid) of 15%, 24%, and 12% between the aq S 0 , 0 V , and 0 P C of propanoic acid, respectively, this level of error is quite tolerable, as group additivity estimations generally give relative errors of ~5% for these classes of organic compounds [50]. At 150 0 C the error is approximately half that at 350 0 C and is likely to fall below the analytical errors encountered when quantifying the concentrations of these substances.  Table   1) for the labeled acid or ion, the remaining values ( aq a 1 , a 2 , a 3 , a 4 , c 1 , c 2 , and e  ) were swapped: propanoic acid (taken from [23] for pyruvic acid, and the respective ketoglutarate anion for H-oxaloacetate -1 and oxaloacetate -2 ions (Table 1), and vice versa. a -calculated as described in text and Figure 6. b -calculated from aqueous entropies from Table 4 using Equation (45). c -calculated from the hydration Gibbs free energy using Equation (44).

Estimation of equilibrium constants at high temperatures and pressures
As an example of the usefulness of these data, values of the logarithm of the equilibrium constant (log K) for acid dissociation reactions were calculated at different temperatures from the 0 ,T P G  values at P sat using Equations (38) and (39) and: with the partial molal properties and equation of state parameters in Table 1 (Figure 9). These values allow us to evaluate the potential for each of these reactions to occur under different geochemical conditions. The plots in Figure 8 allow investigation of the pH dependence of the speciation among CAC components. Similarly, the data and parameters, along with the revised HKF equation of state, allow evaluation at wide ranges of temperatures and pressures of reaction energetics among species in the various steps of the CAC (see Figure 1) to determine the thermodynamic viability of these reactions for a variety of conditions. If life did indeed begin under hydrothermal geochemical conditions, these calculations can help identify the conditions necessary for this development.

Concluding Remarks
Using the thermodynamic data that have either been measured experimentally or estimated through methods described and provided in this paper for the constituents of the citric acid cycle, we can begin to place the fundamental biological process of energy transfer into a geochemical context. With the data and parameters presented in this paper, we can for the first time calculate thermodynamic reaction properties for the citric acid cycle under hydrothermal conditions, whence life may have emerged. Furthermore, we can use calculations such as the ones described above to evaluate the energy cycles of microorganisms that live at elevated temperatures and pressures and thus gain insight into the conditions necessary for the initiation of these cycles. We are now also able to evaluate quantitatively the steps in the reverse or reductive citric acid cycle, which may have preceded the more modern oxidative citric acid cycle as the primary energy transfer mechanism for life. Such calculations are facilitated through use of the computer program SUPCRT92 [51]. Figure 9. Plot of the logarithms of equilibrium constants for dissociation reactions (as indicated) involving organic species from this work as a function of temperature at P sat .
In addition, we can gain some insight into the conditions (including factors such as pH, temperature, pressure and concentrations of the chemical components) on the early Earth that may have facilitated the initiation of the central metabolic pathways such as the reductive and oxidative citric acid cycles in biological energy systems by evaluating quantitatively the energy gained through various metabolic reactions.
For example, careful examination of Figure 9 reveals that the logarithms of the equilibrium constants of many of the deprotonation reactions involving components of the citric acid cycle vary by up to an order of magnitude over the known temperature range for life (currently up to 122°C). Because these reactions are functions of pH, changes in the equilibrium constants over the temperature range will change the range of pH at which they would be thermodynamically favorable. A full evaluation of the geochemical parameters attending any environment would be required to determine the effect each would have on reaction favorability and the viability of the citric acid cycle.
The revised HKF eos combines substance specific structural or non-solvation (  The non-solvation contribution to the standard molal term is a summation of the intrinsic property of a substance and its affect the solvent structure in its local vicinity and is the dominant contribution to the partial molal quantity at temperatures of ~ 150 °C. The solvent contribution to the standard molal term is a factor of the intrinsic properties of water and the Born transfer properties of solvation [11]. In effect, for a given structural moiety of a compound the greater the degree or hydrophilicity (i.e. compounds that form hydrogen bonds and/or ions), will reflect in the collapse of the local solvent structure, whereas more hydrophobic moieties will necessitate the formation of cavities within the solvent. The solvation contribution to a standard molal property has the greatest affect when the solvent (H 2 O) is undergoing the greatest degree of change in permitivity, volume, and heat capacity as these functions are influenced by high temperature and pressure. The non-solvation and solvation contribution to the partial molal volume for ions, electrolytes, and neutral aqueous organic species is defined by: where the non-solvation term is defined by:  (25) and the solvation term for ions and electrolytes is: For neutral organic species, the non-solvation term partial derivative function   T P   is taken to be zero, simplifying the term to - e Q since the effective Born coefficient is used.
The combination of non-solvation and solvation function for integration of the heat capacity function is: where the non-solvation contribution is defined by: and the solvation term for ions and electrolytes is: and reduces to  e TX for neutral organic species, where  T   P and  2  P 2   P terms become zero.
The a 1 , a 2 , a 3 , a 4 , c 1 , and c 2 coefficients define the substance-specific non-solvation parameters and  and  e parameters are the conventional and effective Born coefficients for ionic species or electrolytes and neutral species, respectively. The Q, X, and Y coefficients are the solvent, and P-T dependent Born functions from [11].  and  are the solvent parameters corresponding to 2600 bar and 228 K, respectively. T and P are the temperature and pressure of interest, respectively, and the T r and P r terms are those of the reference state temperature of 298. 15 T  T   T  T  T  T  T  T  c   P   P  a  P  P  a