Thermodynamics of a model biological reaction: A comprehensive combined experimental and theoretical study

Accepted Manuscript Thermodynamics of a model biological reaction: a comprehensive combined experimental and theoretical study Vladimir N. Emel´yanenko, Andrei V. Yermalayeu, Matthias Voges, Christoph Held, Gabriele Sadowski, Sergey P. Verevkin PII:

S0378-3812(16)30034-6

DOI:

10.1016/j.fluid.2016.01.035

Reference:

FLUID 10972

To appear in:

Fluid Phase Equilibria

Received Date: 11 November 2015 Revised Date:

13 January 2016

Accepted Date: 19 January 2016

Please cite this article as: V.N. Emel´yanenko, A.V. Yermalayeu, M. Voges, C. Held, G. Sadowski, S.P. Verevkin, Thermodynamics of a model biological reaction: a comprehensive combined experimental and theoretical study, Fluid Phase Equilibria (2016), doi: 10.1016/j.fluid.2016.01.035. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Thermodynamics of a model biological reaction: a comprehensive combined experimental and theoretical study

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Vladimir N. Emel´yanenko,a Andrei V. Yermalayeu,a Matthias Voges, b Christoph Held,b,* Gabriele Sadowskib, Sergey P. Verevkin,a,** a Department of Physical Chemistry and Department „Science and Technology of Life, Light and Matter“, University of Rostock, Dr-Lorenz-Weg 1, D-18059, Rostock, Germany b Department BCI, Laboratory of Thermodynamics, Technische Universität Dortmund, EmilFigge-Str 70, D-44227 Dortmund, Germany

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In this work we applied experimental and theoretical thermodynamics to methyl ferulate hydrolysis, a model biological reaction in order to calculate the equilibrium constant and reaction enthalpy. In the first step, reaction data was collected. Temperature-dependent equilibrium concentrations of methyl ferulate hydrolysis have been measured. These were combined with activity coefficients predicted with electrolyte PC-SAFT in order to derive thermodynamic equilibriums constants Ka as a function of temperature. In a second step, thermochemical properties of the highly pure reaction participants methyl ferulate and ferulic acid were measured by complementary thermochemical methods including combustion and differential scanning calorimetry. Vapor pressures and sublimation enthalpies of these compounds were measured by transpiration and TGA methods over a broad temperature range. Thermodynamic data on methyl ferulate and ferulic acid available in the literature were evaluated and combined with our own experimental results. Further, the standard molar enthalpy of methyl ferulate hydrolysis reaction calculated according to the Hess´s Law applied to the reaction participants was found to be in agreement with the experimental reaction enthalpy from the equilibrium study. In a final step, the gas-phase equilibrium constant of methyl ferulate hydrolysis at 298.15 K was calculated with the G3MP2 method. This value was adjusted to the liquid phase using the experimental vapor pressures of the reaction participants. As a result, the liquid phase Ka value calculated by quantum chemistry with additional data on the pure reaction participants was in good agreement with the experimental Ka reported in the literature for the aqueous phase. The thermodynamic procedure based on the quantum-chemical calculations is found to be a valuable option for assessment of thermodynamic properties of biologically relevant chemical reactions. Key words: equilibrium constant, ePC-SAFT, thermochemistry, enthalpy of formation, quantum-chemical calculations *To whom correspondence concerning reaction-equilibrium studies and ePC-SAFT modeling should be addressed, E-mail address: [email protected] (C. Held) **To whom correspondence concerning the thermochemical measurements (Rostock) and data evaluation and discussion should be addressed, E-mail address: [email protected] (S.P. Verevkin) 1

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1. Introduction Thermodynamics plays an important role in biochemical reactions. Key reaction properties such as Gibbs energies of reactions or reaction enthalpies are accessible by thermodynamic tools. Such tools can be experimental considerations, theoretical approaches

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at different levels of molecular size or time-scale, or combined approaches making use of experiments and theory. This work presents a comprehensive thermodynamic study of a biological model reaction, aiming at application of combined experimental and theoretical state-of-the-art methods to the methyl ferulate hydrolysis as an archetype of a biological

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reaction. The hydrolysis of methyl ferulate presents an appropriate model reaction for the breakdown of hemicellulose from plants and thus for investigations on the enzyme-catalyzed ester hydrolysis of hemicellulose [1], which is an important step in the production route of

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biofuel from biomass. In this work the methyl ferulate hydrolysis in the presence of the feruloyl esterase is investigated. Methyl ferulate hydrolysis is presented by the following

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overall biochemical reaction (1a) and the reference chemical reaction (1b):


ℎ   +  ⇋    +
ℎ

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ℎ   () +  () ⇋     () +   () +
ℎ()

(1a) (1b)

In addition, methyl ferulate hydrolysis is a meaningful model reaction from thermodynamic point-of-view, since the hydrolysis of methyl ferulate is limited by thermodynamic equilibrium. Such limitations can be overcome by manipulating the reaction conditions (e.g. T, pH, reactant concentration, initial reactant ratio) in a way that allows shifting the equilibrium position towards the side of products. From a thermodynamic point, it is most reasonable to change reaction temperature, as this does not influence the composition of the reaction medium (which would be necessary for changing pH, reactant concentration or reactant ratios, respectively). Thus, it is convenient in reaction engineering to optimize reaction temperature as a first step in order to shift the equilibrium position towards the 2

ACCEPTED MANUSCRIPT product side for increasing yield. Varying reaction temperature requires that the enzyme is still active in the temperature window under consideration. Note, that temperature-dependent enzyme activity was not of particular interest of the current work.

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Thermodynamic approach used in this study

A general sketch which comprises the experimental and theoretical methods involved in this work is presented in Fig. 1. The detailed explanation of this figure will be given along the text, but the main idea behind this study is a development of a procedure suitable for the

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reliable prediction of thermodynamic properties, as well as reasonable assessment of equilibrium constants of biochemical reactions. Using the model reaction of methyl ferulate

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hydrolysis Eq. (1) we intend to perform mutual validation of experimental and theoretical methods presented in Fig. 1 in order to extend this thermodynamic procedure to more

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complex reactions.

Fig. 1. Sketch of thermodynamic calculations performed in this work. The question sign shows the points of mutual validation of the experimental results. TK – titration calorimetry, QC – quantum-chemistry, CC – combustion calorimetry, T – transpiration method, TGA – thermogravimetry, DSC – differencial scanning calorimetry, SC – solution calorimetry

The key property that allows evaluating thermodynamic limitations and reaction yield is the thermodynamic equilibrium constant Ka. From the van`t Hoff relation 3

ACCEPTED MANUSCRIPT d ln Ka/dT = ∆ r H mo / RT 2

(2)

the slope of a plot R(ln Ka) vs 1/T provides the standard molar enthalpy of reaction ∆ r H mo . The derivation of reaction enthalpy from the temperature-dependence of the equilibrium constant is commonly referred to the Second Law Method. This method is well established for chemical reactions in the liquid [2,3] or in the gas phase [4]. However, this method becomes

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very demanding and time consuming especially for biologically relevant or enzyme catalyzed reactions, which mostly proceed in aqueous medium. In contrast to common chemical reactions, apparent equilibrium constants of the overall chemical reaction (1a) K´m (where m means that K´m is molality based) are mostly used instead of the thermodynamic equilibrium

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constant of a reference chemical reaction Ka (where a means that Ka is activity based). The apparent equilibrium constant K´m is usually calculated from equilibrium molalities of the

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reaction participants of the overall biochemical reaction (see Equation 1a). The apparent equilibrium constant K´m provides very useful information for biochemical reactions, but it crucially depends on the composition of the reaction medium. Admittedly, only the calculations based on thermodynamic equilibrium constant Ka can lead to standard thermodynamic properties. Usually, Km values are determined from measured equilibrium molalities of the reaction participants of the reference chemical reaction (see Equation 1b).

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Knowledge on activity coefficients γi or γi* (depending on the reference state of the reaction participant) of all reaction participants allows converting Km values into the Ka value using Ka=KmKγ, where Kγ is the ratio of product activity coefficients to reactant activity coefficients. One of the possibilities to obtain activity coefficients is thermodynamic modeling using an

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equation of state as, e.g. ePC-SAFT [5]. This method is partly based on model parameters adjusted to the thermodynamic properties of pure species. The applicability of ePC-SAFT

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towards biological reactions and validity of the calculated thermodynamic properties have been in focus of our recent studies [5,6]. Further, these works showed that neglecting Kγ for the conversion of Km to Ka values might cause strong thermodynamic inconsistencies. It was the surprising results of our first works on bioreactions [5,6] that Km values were found to be strongly concentration dependent, even at low metabolite concentrations. This was ascribed to metabolite activity coefficients. In this context, the methyl ferulate hydrolysis reaction Eq. (1) seems to be a good model of an enzyme catalyzed chemical reaction, where reaction participants are well defined, as well as commercially available. Moreover, for this reaction some previous valuable thermodynamic studies have been already performed just recently [1,5]. Goldberg et al. [1] measured reaction enthalpy of methyl ferulate hydrolysis in a citrate buffer using a micro-calorimeter as well as K´m values also in citrate buffer. Hoffmann et al. 4

ACCEPTED MANUSCRIPT [5] published a thermodynamic constant Ka at 298.15 K that was based on experimental biochemical K´m values extrapolated to infinite dilution of methyl ferulate and ferulic acid. To the best of our knowledge, temperature-dependent Ka values independently leading to the “true” reaction enthalpy ∆ r H mo from the “Second Law” method have not been published yet. In this context, it was reasonable to study the model reaction Eq. (1) at several

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temperatures (as well as at different conditions from Goldberg´s conditions [1]) and apply ePC-SAFT to derive ∆ r H mo (298.15 K)-value. This method is denoted as “Second Law” thermodynamic procedure is presented in the left part of Fig. 1, and it allows for comparison and mutual validation of experimental and theoretical results. Thus, in this work the

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equilibrium molalities of the methyl ferulate hydrolysis were measured at 293 K, 298 K, and 303 K in a 0.5 mol·kg-1 sodium phosphate buffer at pH 6.40. This allows calculating 

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values at these temperatures. The corresponding activity coefficients of reactants and products were modelled with ePC-SAFT using the procedure developed in our previous work [5]. As the result of this equilibrium study, the ∆ r H mo (298.15 K)-value was derived for comparison with the calorimetric result by Goldberg et al. [1], which is presented in the left part of Fig. 1. Determinations of the key properties Km and activity coefficients of reactants and products of biochemical reactions are thwarted with experimental as well as with theoretical

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complications [5], being partly due to low absolute metabolite concentrations. Thus, development of an alternative to the “Second Law” thermodynamic procedure is crucial for the validation of the result from equilibrium study. Thus, in this work a method based on

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Hess´s Law is applied, denoted as First Law of thermodynamics. It is applied to the standard molar enthalpies of formation, ∆ f H mo , of reaction participants, and is illustrated in the right

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part of Fig. 1. One of the advantages to use reaction (1) as model reaction is that the reaction participants methyl ferulate and ferulic acid are commercially available and can be easily purified for thermochemical measurements, which require a purity of chemicals of 99.99%. The combustion calorimetry is a classic method for measuring standard molar enthalpies of formation ∆ f H mo in the solid or in the liquid state. In order to relate the ∆ f H mo (cr)-value of methyl ferulate and ferulic acid to the liquid state, their fusion enthalpies were measured with DSC. Applying the Hess´s Law to ∆ f H mo (liq) of all reaction (1) participants (values for methanol and water are well known), the experimental ∆ r H mo (liq, 298.15 K) according to the “First Law” was derived as independent result for validation of those derived from the “Second Law”. This is illustrated in the middle part of Fig. 1. 5

ACCEPTED MANUSCRIPT Having established the mutual consistency of the experimental reaction enthalpies according to “Second Law” and “First Law” principles for this model reaction, it was the idea to apply modern quantum chemical calculations in order to reduce the experimental efforts as well as facilitate evaluation and handling of thermodynamic information. In our recent work we have demonstrated that the high-level composite quantum-chemical (QC)

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methods from the G*-family are reliable for estimation of thermodynamic properties of small and middle-size organic molecules [8-9]. However, it should be noticed that methyl ferulate and ferulic acid belong already to rather large molecules in QC, where accuracy of results might be questionable. Nevertheless, a complex of thermochemical methods including

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combustion calorimetry, DSC, transpiration and TGA methods applied to methyl ferulate and ferulic acid were used within the current work in order to derive the gas phase standard molar enthalpies of formation ∆ f H mo (g, 298.15 K) and validate the theoretical values calculated

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with QC-method. This is presented in the middle part of Fig. 1. Moreover, knowledge on theoretical ∆ f H mo (g, 298.15 K)-values of methyl ferulate and ferulic acid allow deriving

∆ f H mo (liq, 298.15 K)-values, if experimental standard molar sublimation enthalpies, ∆gcr H mo (298.15 K) and experimental standard molar fusion enthalpies ∆lcr H mo (298.15 K) are

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available. ∆ f H mo (liq, 298.15 K) values by using Hess´s Law provide an additional and again independent experimental ∆ r H mo (liq, 298.15 K) value, which is expected to be in agreement with the result based on the combustion calorimetry (as presented in the right part in Fig. 1).

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Besides the reaction enthalpy of the model reaction, this work intends to prove how accurate QC-based methods are for prediction the thermodynamic constant Ka in the liquid phase. This could be very valuable for evaluation of the thermodynamically dominant

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metabolic pathways in living cells. As a matter of fact, calculation of thermodynamic properties of pure compounds is incorporated in the methods provided by the Gaussian package [10]. Thus, the thermodynamic equilibrium constant in the gas phase, Kp(g), can be estimated for any reaction from the QC-results for reaction participants (as it shown in the middle part in Fig. 1). Furthermore, for the model reaction (1), the equilibrium constant Ka (liq) at any desired temperature can be calculated by using the vapor pressures p of pure reaction participants (at appropriate temperature):

K a (liq) = K p ( g ) ⋅

p (methyl ferulate) ⋅ p ( water) p (ferulic acid) ⋅ p (methanol)

6

(3)

ACCEPTED MANUSCRIPT Values of vapor pressures p for the model reaction (1) were measured in this work or taken from the literature. Provided that the QC-based thermodynamic equilibrium constant matches at least the order of the experimental value, we could extend application of this QC-based procedure for additional and more complex biochemical reactions.

2. Materials and methods Materials and solution preparations

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2.1.

All the substances used in this work (see Table 1) were of commercial origin. They were used for equilibrium studies as obtained without further purification. For the reactionequilibrium measurements, the recombinant feruloyl esterase (EC 3.1.1.73) from a rumen

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microorganism (Megazyme International, Bray, Ireland) was used. Millipore water was used for all aqueous solutions. All solutions were prepared gravimetrically by using a Sartorius

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CPA324S balance (Sartorius, Göttingen, Germany) with an accuracy of ±1·10-4 g. The pH of the samples was adjusted with phosphate buffer, and pH values were measured with a pHmeter GMH 3531 (GHM Messtechnik GmbH, Regenstauf, Germany) with an accuracy of ±1·10-2. Samples of methyl ferulate (methyl 4-hydroxy-3-methoxycinnamate) and ferulic acid (4-hydroxy-3-methoxycinnamic acid) used for thermochemical measurements (combustion calorimetry, transpiration, TGA and DSC) were additionally purified by fractional

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sublimation in vacuum. No impurities (greater than 0.001 mass fractions) were detected in these samples as analysed by GC and DSC. Table 1

Provenance and purity of the material: Chemical Abstracts Service (CAS) registry numbers,

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empirical formulae, suppliers (A = Alfa Aesar GmbH & Co KG, M = Merck KGaA, S = Sigma-Aldrich Chemie GmbH), and approximate mass-fraction purities as given by the

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suppliers.

Purification

Substance

CAS-No.

Formula

Supplier

Purity

Acetonitrile

75-05-8

C2H3N

A

>0.999

None

Ferulic acid

1135-24-6

C10H10O4

S

>0.99a

a,b

67-56-1

CH4O

M

>0.999

None

Methyl ferulate

2309-07-1

C11H12O4

A

>0.99a

a,b

Sodium dihydrogen phosphate

7558-80-7

NaH2PO4

S

>0.99

None

Disodium hydrogen phosphate

7558-79-4

Na2HPO4

M

>0.995

None

Methanol

method

a

Purity of 0.999 mass fraction after fractional sublimation, used for thermochemical measurements.

b

No purification for equilibrium study. 7

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2.2.

Chemical equilibrium study in the aqueous phase The equilibrium reactions were carried out in sealed glass vessels having a reaction

solution volume of 5 mL. The equilibrium of the methyl ferulate hydrolysis was investigated in the range 293-303 K. Preliminary results showed that equilibrium at 310 K was not reached

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due to loss of catalytic activity of the enzyme. Thus,  values were calculated from the measured equilibrium concentrations at 293.1 K, 298.1 K and 303.1 K. The temperature was mantained using a C12 CP Lauda thermostat (Lauda, Lauda-Königshofen, Germany) and controlled with a temperature sensor (Pt 100; ±0.1 K) that was placed inside the glass vessel

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(directly in the reaction mixture). The reaction vessels were continuously stirred (~ 300 rpm) to ensure homogenous solution. The resulting  values were found to be independent of the stirring speed (data not shown). A sodium phosphate buffer (0.5 mol· kg-1) was used to adjust

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the reactant and enzyme stock solutions to pH 6.43. Each initial reactant solution contained 5 mol· kg-1 methanol and 0.010 mol·(kg-1 methyl ferulate. The reaction was initiated by adding the enzyme solution to the initial reactant solution, so that an enzyme concentration of 0.334 U/ml was adjusted for every reaction batch. The pH of the reaction mixture was controlled prior to sampling after a reaction time of four hours (for the validation of reaction equilibrium

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see Hoffmann et al. [5]) and was found to be pH 6.40 at equilibrium for all initial reactant molalities under investigation. The equilibrium molalities of reactants and products were analyzed by HPLC with an UV-detector (HPLC method is described by Hoffmann et al. [5]). Prior to chromatographic analysis, the reaction was terminated by centrifugation to prevent

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any equilibrium shift during storage before HPLC analyses. All samples were centrifuged for 20 minutes at reaction temperature in a Hettich ‘Universal 32R’ centrifuge (Hettich,

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Tuttlingen, Germany) at 16000 U⋅min-1 using 10 kDa ultrafiltration units to separate the esterase from the reaction mixture. The measured molalities of reactants and products and the pH were used to calculate the equilibrium constant  (see section 3.1).

2.3. ePC-SAFT modelling of activity coefficients The activity coefficients of the reactants and product at equilibrium were modeled with ePC-SAFT [5]. The modeling assumption, the strategy, and the pure-component parameters and binary parameters were all inherited from the previous work [5], except for two kij values, which were published incorrectly in the previous work [5]. The corrected parameters are listed in Table 2. 8

ACCEPTED MANUSCRIPT Table 2 ePC-SAFT pure-component parameters for all components present in the experiments. Their binary interaction parameters with water and methanol are also shown.  !



Component

" 

# ⁄

%$'(% ⁄

%$)(%

$ * (water)

$ * (methanol)

10.2982

3.5124

414.71

1936.6

0.000100

-

-0.080

Ferulic acid [5]

15.9236

3.9985

360.15

617.2

0.022700

-0.180

-0.001 b

Water [39]

1.2047

estimateda

353.94

2425.7

0.045099

-

-0.05 b

Methanol [40]

1.5255

3.2300

188.90

2899.5

0.035176

-0.05

-

Na+ [41]

1.0000

2.4122

646.05

-

-

-

-

H2PO4- [41]

1.0000

3.7026

-

-

-

-

-

1.0000

4.4608

-

-

-

-

-

HPO4 [41]

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2-

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Methyl ferulate [5]

Estimated σwater = 2.7927 + 10.11 exp(−0.01775 T) − 1.417 exp(−0.01146 T)

b

Value of kij was corrected in this work, since the kij in the literature [5] was incorrect. In the future work, the kijs

re-calculated here have to be used.

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a

2.4. Combustion calorimetry. Enthalpies of formation of methyl ferulate and ferulic acid

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The standard molar enthalpies of combustion of the methyl ferulate and ferulic acid were measured with a static bomb in an isoperibol combustion calorimeter. In a typical experiment a pellet of the sample was placed in a platinum crucible and it was burned in oxygen at a pressure p=3.04 MPa. We used small pieces of polyethylene to achieve a completeness of

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combustion. The detailed procedure has been described previously [11]. Calibration of the calorimeter was performed with benzoic acid (sample SRM 39j, from NIST). Conventional

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procedures [12] were used for the reduction of the data to the standard conditions using the auxiliary data are collected in Table S1. Results for combustion experiments are summarized in Tables S2 and S3.

2.5. Transpiration Method. Enthalpy of vaporisation of methyl ferulate The standard molar vaporization enthalpy of methyl ferulate was derived from the temperature dependence of vapor pressures. Absolute vapor pressures of methyl ferulate at different temperatures in the range from 340 to 376 K were measured with the transpiration method [13,14]. A nitrogen stream was passed through the thermostatted (± 0.1 K) U-shaped 9

ACCEPTED MANUSCRIPT saturator filled with small glass beads mixed with the fine powdered sample. At a series of constant temperatures, the transported substance, at equilibrium conditions, was collected in a cold trap. The amount of condensed sample was determined by GC analysis using n-tridecane as an external standard. The absolute vapor pressure p at each temperature Ti was calculated from the amount of the product collected within a definite period of time. Absolute values of

p = m·R·Ta / V·M ;

V= VN2 + Vi;

(VN2 » Vi)

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vapor pressures p were calculated with equation: (4)

where R is the universal gas constant; m is the mass of the transported compound i, M is the molar mass of the compound, and Vi, is volume contribution of the compound (was neglected)

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to the total volume V. VN2 is the volume of the carrier gas and Ta is the temperature of the soap bubble meter used for measurement of the gas flow. The volume of the carrier gas VN2 was

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determined from the flow rate and the time measurement.

2.6. Isothermal TGA method. Enthalpy of sublimation of ferulic acid The standard molar sublimation enthalpy of ferulic acid was derived from the temperature dependence of mass loss rates measured with a Perkin Elmer Pyris 6 TGA. The

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sample of about 70 mg was placed in a plane platinum crucible and it was heated under the nitrogen stream. The mass loss rates dm/dt were measured in the temperature range 385-442 K at a nitrogen flow rate of 200 ml·min-1. The measurements were done by using both increasing and decreasing temperature steps. The detailed procedure has been described

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elsewhere [15]. The primary experimental results are given in Table S4.

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2.7. DSC. Enthalpy of fusion measurements The thermal behavior of ferulic acid was studied with a DSC Mettler Toledo 822. The samples were hermetically sealed in 50 µL pans supplied by Perkin Elmer and heated with a rate of 10 K·min-1. The DSC measurements were repeated in triplicate and values agreed within the experimental uncertainties u( ∆lcr H mo ) = 0.2 kJ·mol-1 for the enthalpy of fusion and u(T) = 0.5 K for the melting temperature. No phase transitions in the solid state of ferulic acid were observed below the melting temperature. The same observation is valid for methyl ferulate [16].

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2.8. Quantum chemical calculations We used the Gaussian 09 Revision C.01 series software [10] for quantum chemical calculations. Optimisation of the molecules structures was performed usin the MMFF94 method [17]. Energies of optimised structures were calculated using the G3MP2 method [18].

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General computational details were reported elsewhere [19]. The thermodynamic properties of each molecule were calculated according to standard procedures [20].

3. Results and Discussion

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3.1. Equilibrium constants ( ) and “Second Law” reaction enthalpy in the

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aqueous phase

For the reference chemical reaction 1b of the methyl ferulate hydrolysis, the thermodynamic equilibrium constant  is defined as ∗,-7

∗,-7

 =

-7

,-./012 2134 · 9 : · -;<=>0 ∗,-7 -7 -;
∗,-7

-7

(5),

-7

∗, 
,-./012 2134 · A,-./012 213 4 ·  : ·
-;<=>0 · A-;<=>0 9 -7 -7 ∗, 
-;
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=





where the  value is calculated from the equilibrium molalities
1 , the activity coefficients -7

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A1 or A1∗, of the reacting agent  at equilibrium (calculated as shown by Hoffmann et al. [5,6]) and 9 : (calculated from the experimentally-determined pH values at reaction

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equilibrium). In this work, the molality-based activity coefficients A1 and A1∗, are used, together with equilibrium molality
1 (moles of component  per kg water) as concentration unit. The standard state for A1 is the pure component and the standard state for A1∗, is the hypothetical ideal solution of reacting agent , which is defined as a one molal solution of the reacting agent  exhibiting the same interactions as at infinite dilution of component  in the same medium [21]. These standard states have been chosen according to [5] as ferulic acid (aq) and methyl ferulate (aq) were present in very small concentrations in the reaction medium; this was due to the fact that the reacting agents water (liq) and methanol (aq) were used in high excess in the reaction experiments. The experimental molalities of reactants and products at equilibrium were obtained at 293.15 K, 298.15 K, and 303.15 K for initial aqueous reaction solutions containing 5 moles 11

ACCEPTED MANUSCRIPT methanol and 0.01 moles methyl ferulate per kg water, respectively. The resulting equilibrium molalities are listed in Table 3. At these conditions, more than 99% of the initial 0.01 moles methyl ferulate per kg water were converted at equilibrium; that is, the equilibrium position is clearly on the product side. The corresponding activity coefficients of reactants and products at equilibrium were

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modeled with ePC-SAFT; they are listed in Table 3. It can be observed from Table 3 that activity coefficients of reactants and products are neither unity, nor temperature-independent. This requires involving the concentration and temperature dependence of activity coefficients in order to finally obtain the temperature-dependent  values according to Eq. 5. However,

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the  value of this work differs from the  value published by Hoffmann et al. [5] since

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latter is based on the overall biochemical reaction 1 and thus, based on the ′ value.

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Table 3

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Experimental equilibrium molalities and ePC-SAFT modeled activity coefficients of the reactants and products in the methyl ferulate hydrolysis in -1 -1 aqueous solution at different temperatures (at pH 6.40 in sodium phosphate buffer:
C : = 0.568 mol·kg ,
9 EF 4 = 0.432 mol·kg , D G -1 a
9 D4 = 0.068 mol·kg ), as well as the activities of the protons and the resulting  values of reaction 1’. EF G

K

mol·kg-1

293.15

4.9182±0.001

9.91±0.001

0.077±0.0001

55.5093

298.15

4.9392±0.001

10.35±0.037

0.076±0.0025

55.5093

303.15

4.9318±0.001

10.26±0.003

0.067±0.0041

55.5093

mmol·kg-1

∗, ∗,  A,-./012 213 4 A-;0

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-7
-;<=>0

mol·kg-1

∗,-7

 A@;-.

9 :



0.05819

1.88034

0.98123

0.01932

3.98·10-7

2.59·10-5

0.05693

1.86337

0.98123

0.01931

3.98·10-7

2.67·10-5

0.05563

1.83562

0.98123

0.01931

3.98·10-7

2.90·10-5

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a

-7
Σ-7,-./012 213 #
Σ-7-;
T

Standard uncertainties u are u(meqmethanol)= 0.001 mol·kg-1, u(meqferulic acid)= 0.01035 mmol·kg-1, u(meqmethyl ferulate)= 0.00166 mmol·kg-1, u(meqwater)= 0.001 mol·kg-1, u(T)= 0.1 K, ∗,-7

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u(pH)= 0.01, u(9 : )= 9.2·10-9, and the combined expanded uncertainty Uc is Uc( )= 3.28·10-7 (0.95 level of confidence). # Σferulic acid= ferulic acid0 + ferulic acid-+ ferulic acid2-. Note that at pH 6.40 and for the temperature range under investigation Σferulic acid = ferulic acid-.

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values see [1])

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Σmethyl ferulate= methyl ferulate0 + methyl ferulate -. Note that at pH 6.40 and for the temperature range under investigation Σmethyl ferulate = methyl ferulate0. (for pKa

13

ACCEPTED MANUSCRIPT The resulting Ka of reaction 1b calculated with Eq. (5) using the values in Table 3 are also listed in Table 3 and illustrated in Fig. 2 for the ln ( ) values as a function of

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temperature.

Fig. 2. KL (MN ) as a function of (O/M)Q of the enzyme-catalyzed methyl ferulate hydrolysis at pH 6.4 in 0.5 mol·kg-1 sodium phosphate buffer. Symbols represent experimental data of this work; dashed line represents the linear fit of the experimental data. It can be seen in Fig. 2 that ln ( ) of reaction 1b is decreasing with increasing R S .

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Thus, hydrolysis of methyl ferulate is an endothermic reaction at the conditions under investigation. According to the van’t Hoff equation Eq. (2), ∆ r H mo was determined from the slope of the linear fit of the experimental data in Fig. 2, yielding ∆ r H mo = 8.3±0.6 kJ·mol-1 for

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reaction1b. The enthalpy of reaction determined in this work is in reasonable agreement with the enthalpy of reaction published by Goldberg et al. [1] ( ∆ r H mo =7.3 ± 1.7 kJ·mol-1).

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Having these two consistent ∆ r H mo values of the biologically relevant chemical reaction we pose now a challenge to obtain a comparable result using a reasonable combination of the quantum-chemical and empirical methods. This requires data on the pure reaction participants (sections 3.2 - 3.5), and the different methods are then compared in section 3.6.

3.2.

Combustion calorimetry and enthalpies of formation of methyl ferulate

and ferulic acid The results of the combustion experiments on methyl ferulate (cr) and ferulic acid (cr) are given in Tables S2 and S3. Value of the standard specific energies of combustion ∆cu° were 14

ACCEPTED MANUSCRIPT used to derive standard molar enthalpies of combustion ∆ c H mo and standard molar enthalpies of formation in the crystalline state ∆ f H mo (cr) = -(677.3±3.2) kJ·mol-1 for methyl ferulate and

∆ f H mo (cr) = -(708.3±2.6) kJ mol-1 for ferulic acid (see Table 4). Uncertainties of combustion

Table 4

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experiments were calculated according to the guidelines presented in [12,22].

Thermochemical results for methyl ferulate and ferulic acid at T = 298.15 K (p° = 0.1 MPa) in kJ·mol-1

∆ f H m° (g)d

compounds

∆ c H m° (cr) a

∆ f H m° (cr) a

∆ gcr H m b

methyl ferulate

-5366.3±2.9

-677.3±3.2

123.3±1.2

-554.0±3.4

-555.0

ferulic acid

-4656.1±2.2 e

-708.2±2.6

135.7±1.0

-572.5±2.8

-574.4

SC

∆ f H m° (g)c

Uncertainties correspond to expanded uncertainties of the mean (0.95 confidence level)

b

From Table 6. Uncertainties correspond to the standard deviation.

c

Calculated as the sum ∆ f H m° (g) = ∆ f H m° (cr) + ∆ gcr H m

d

Calculated by the G3MP2 method using the atomization reaction with an assessed

uncertainty of ±3.6 kJ·mol-1 [7,8].

For comparison ∆ c H m° (cr) = (-4666.0±3.8) kJ⋅mol-1 [23] (see text).

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e

M AN U

a

Values of ∆cu° and ∆ c H mo for methyl ferulate (cr) and ferulic acid (cr) refer to the respective

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reactions:

C11H12O4 (cr) + 12 O2 (g) = 11 CO2 (g) + 6 H2O (liq)

(6)

C10H10O4 (cr) + 10.5 O2 (g) = 10 CO2 (g) + 5 H2O (liq)

(7)

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Enthalpies of formation ∆ f H mo (cr) of methyl ferulate and ferulic acid were calculated according to the Hess´s Law applied to Eqs. (6) and (7) using standard molar enthalpies of formation of H2O (l) and CO2 (g) as assigned by CODATA [24]. The enthalpy of formation ∆ f H mo (cr) of methyl ferulate is now reported for the first time. A value of the enthalpy of formation ∆ f H mo (cr) = -(698.2±4.0) kJ·mol-1 of ferulic acid measured by using the combustion calorimetry was recently reported [23]. This result is in disagreement with our value (see Table 4). This may be because the sample in the work [23] was used “as purchased” and without additional purification except for drying under vacuum at 320 K. Nevertheless, the accuracy of our result for ∆ f H mo (cr) of ferulic acid is supported 15

ACCEPTED MANUSCRIPT (see section 3.5) by the consistency of experimental and theoretical results for the gas phase enthalpies of formation, ∆ f H mo (g). The ∆ f H mo (g) experimental values for ferulic acid and methyl ferulate were derived by summation of ∆ f H mo (cr) measured with the combustion calorimetry and experimental sublimation enthalpies ∆gcr H mo for these compounds as

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described below.

3.3. Vapor pressure and vaporization enthalpy of methyl ferulate from the

SC

transpiration method

The volatility of the methyl ferulate is too low to be measured accurately in the temperature range below the melting point 335.7 K with the experimental techniques

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available in our lab. However, the measured vapor pressures from 340 K to 376 K, where the pressures become more accurately measurable. The temperature dependence of absolute vapor pressures p measured for methyl ferulate (see Table 5) was fitted with the equation [14]:

R ⋅ ln p = a +

T  b + ∆gl C op ,m ⋅ ln  T  T0 

(8)

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where a and b are adjustable parameters. The value ∆glC po ,m is the difference of the molar heat capacities of the gaseous and the liquid phase respectively. The value T0 is an arbitrarily chosen reference temperature (usually chosen to be 298.15 K) and R is the molar gas constant. The combined uncertainties u(p) of vapor pressures (as described in detail in ref. [25]) are

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generally on the level (1 to 3) %.

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The vaporization enthalpy of methyl ferulate was calculated by using equation:

∆gl H mo (T ) = −b + ∆gl C po ,m ⋅ T

(9)

A value of ∆gl C po ,m was calculated (see Table S5) according to the procedure developed by Chickos and Acree [26] based on the isobaric molar heat capacity C po , m (liq, 298.15 K) for estimated by group-additivity [27]. Sublimation entropies at temperature T were also derived from the temperature dependence of vapor pressures using Eq. (10):

∆gl S mo (T ) = ∆gl S mo / T + R ln( p / p o )

(10)

Experimental vapor pressures measured by the transpiration method, coefficients a and b of Eq. (9), as well as values of ∆gl H mo (T) and ∆gl S mo (T) are given in Table 5. Uncertainty of 16

ACCEPTED MANUSCRIPT temperature adjustment of vaporization enthalpy to the reference temperature T = 298.15 K was calculated using an assumption that a standard deviation of ±16 J·mol-1·K-1 of the liquid phase heat capacity, C op , m (liq) [28]. The combined uncertainty of the vaporization enthalpy was developed as described elsewhere [25]. They include uncertainties of the transpiration experimental conditions, uncertainties of vapor pressure, and uncertainties of temperature

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adjustment to 298.15 K.

Table 5

Thermodynamic parameters of vaporization of methyl ferulate: absolute vapor pressures p,

SC

sublimation enthalpies, ∆gl H mo , and sublimation entropies, ∆gl S mo obtained by the transpiration method. m/

V(N2)c /

Ta

Flow/

Ka

mgb

dm3

Kd

p/

U(p)/

M AN U

T/

dm3·h-1

Pae

%f

∆gl H mo /

∆gl Smo /

kJ⋅mol-1

J⋅K-1⋅mol-1

∆gl H mo (298.15 K) = (99.8±0.6) kJ.mol-1 384.8 132597.2 110.1  T/K  − − ln  R R ⋅ (T/K) R  298.15

ln(p/p°) = 64.92

293.2

343.2

0.78

71.31

293.2

348.2

1.10

61.13

353.1

1.04

358.2

4.46

0.097

0.007

95.15

164.6

4.46

0.128

0.008

94.82

163.5

293.2

4.46

0.210

0.010

94.27

162.1

37.74

293.2

4.46

0.324

0.013

93.72

160.3

0.72

16.34

293.2

4.46

0.513

0.018

93.16

158.8

363.2

0.88

13.37

293.2

4.46

0.769

0.024

92.61

157.1

368.1

0.84

8.394

293.2

4.46

1.17

0.03

92.07

155.8

370.2

1.07

8.914

293.2

4.46

1.40

0.04

91.84

155.2

373.2

1.08

7.057

293.2

4.46

1.79

0.05

91.51

154.3

376.2

1.09

5.571

293.2

4.46

2.29

0.06

91.18

153.6

EP

TE D

0.54

AC C

a

340.2

Saturation temperature (u(T) = 0.1 K).

b

Mass of transferred sample condensed at T = 243 K.

3

nitrogen (u(V) = 0.005 dm ) used to transfer m (u(m) = 0.0001 g) of the sample. soap bubble meter used for measurement of the gas flow.

e

d

c

Volume of

Ta is the temperature of the

Vapor pressure at temperature T, calculated by

iteration from the m and the residual vapor pressure at T = 243 K. f Expanded uncertainty in p (0.95 level of confidence) calculated with: U(p/Pa) = 0.005 +0.025(p/Pa).

The sublimation enthalpy ∆gcr H mo (298.15 K) of methyl ferulate required for calculation of the ∆ f H mo (g)-value was obtained as the sum of ∆gl H mo (298.15 K) from Table 5 and the 17

ACCEPTED MANUSCRIPT enthalpy of fusion ∆lcr H mo reported for this compound [16] and adjusted (see Table 6) to the reference temperature as described in the supporting information.

Table 6 kJ⋅mol-1 Compounds

∆lcr H mo

Tfus, K

∆lcr H mo

a

at Tfus

a

25.8±0.9[16]

23.5±1.1

435.3

(17.9±0.1)[29]

(9.8±2.4)

448

32.7±1.0[30]

24.6±2.6

444.6

33.3±1.0 [31]

25.2±2.6

444.9

31.9±0.9 [32]

23.8±2.6

445.1

33.5±0.5 [23]

25.4±2.5

445.9 b

34.7±0.2 b

26.6±2.4

445.9

d

99.8±0.6 b

34.4±0.2

∆gcr H mo

c

123.3±1.2 c

SC

ferulic acid

335.7

∆gl H mo

M AN U

methyl ferulate

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Thermodynamic parameters of of phase transitions of methyl ferulate and ferulic acid,

25.2±1.1 d

110.5±1.5 e

135.7±1.0 b

The experimental enthalpies of fusion ∆lcr H mo measured at Tfus and adjusted to 298.15 K (see

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ESI). The uncertainties in the temperature adjustment of fusion enthalpies from Tfus to the reference temperature are estimates and amount to 30 % of the total adjustment [28]. Our study (see Table 5).

c

Calculated as the sum (column 4 and 5 in this table).

d

Average value was calculated using the uncertainty of the experiment as a weighing factor.

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b

Value in brackets were disregarded by the averaging. Recommended values are given in bold. Calculated as the difference between (column 6 and 4 in this table).

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e

3.4. Sublimation enthalpy of ferulic acid from the TGA method Vapor pressures and the sublimation enthalpy of the ferulic acid reported in the literature are in disarray (see Fig. S1 and Table 7). Temperature dependence of vapor pressures measured by Chen et al. [33] using the Knudsen cell hung on the Cahn microbalance have the similar slope with the results obtained conventional Knudsen technique [23]. However, the absolute values of vapor pressures are significantly different (see Fig. S1). This disagreement partly could be explained due to some uncertainty of the temperature measurements specific for the Knudsen method with the hung cell. Moreover, vapor pressures and sublimation enthalpy derived from the correlation gas chromatography (CGC) method [34] are in dramatic 18

ACCEPTED MANUSCRIPT disagreement with the other results and they are disregarded. One of the inherent problems of the CGC method is that the data treatment is based on a reference compound, but in many cases this simple assumption is not correct. In order to resolve contradiction of available data, in this work, we derived the sublimation enthalpy of the ferulic acid according to the Clausius-Clapeyron equation by using the mass loss rate dm/dt measured by the TGA (instead ∆ cr H m (T0 ) − ∆ cr C p mT0  dm  ln  T  = A'− R  dt  g

o

g

o

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of the absolute pressure) [15]: g  1 1  ∆ cr C p m  T   −  + ln   R  T T0   T0  o

(11)

with a constant A´ which is essentially unknown and which includes parameters specific for

SC

our setup but is independent from the substance studied. T0 appearing in Eq. (11) is an arbitrarily chosen reference temperature (which we have chosen to be 298.15 K). Due to the unknown constant A´ in Eq. 11, the absolute vapor pressures of ferulic acid were not derived

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in this study, but the experimental value ∆gcr H mo (298.15 K) = 137.4±1.9 kJ·mol-1 was in agreement within the combined standard uncertainties with the sublimation enthalpies measured with the conventional techniques. In order to provide more confidence, we have averaged the available experimental sublimation enthalpies (except for value from CGC) for the ferulic acid and the value ∆gcr H mo (cr, 298.15 K, cr) = 135.7±1.0 kJ·mol-1 is adopted and

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used for calculation of the gas-phase enthalpy of formation of this compound.

Table 7

Thermodynamics of vaporisation and sublimation processes on methyl ferulate and ferulic

EP

acid.

Method

AC C

Compounds

methyl ferulate (liq)

T

a

T-range

∆gcr H mo /

∆gcr H mo /

Tav

298.15 Kb kJ⋅mol-1

K 340.2-376.2

93.2±0.4 c

methyl ferulate (cr)

ferulic acid (cr)

a

Ref

99.8±0.6 c

this study

123.3±1.2 d

this study

ME

394-522

132.2±1.3

135.1±1.4

33

CGC

338-403

121.4±2.2

(124.0±2.3)

34

ME

379.7-394.6

129.2±4.0

132.4±4.1

16

TGA

385-442

133.2±1.8

137.4±1.9

this study

135.7±1.0 e

mean

Methods: ME = mass loss Knudsen effusion method; T = transpiration method; CGC =

correlation gas-chromatography; TGA = thermogravimetric analysis. 19

ACCEPTED MANUSCRIPT b

Uncertainties of vaporization enthalpies are expressed in this table as standard deviations.

Vapor pressure available in the literature were treated using Eqs. (8) and (9) in order to evaluate enthalpy of sublimation at 298.15 K in the same way as our own results. Measured above melting point and referred to the vaporization enthalpy ∆gl H mo .

d

Calculated as the sum ∆gcr H mo = ∆gl H mo + ∆lcr H mo (see Table 6)

e

Average value was calculated using the uncertainty of the experiment as a weighing factor.

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c

Value in brackets were disregarded by the averaging. Recommended values are given in bold.

formation of methyl ferulate and ferulic acid

SC

3.5. Mutual validation of experimental and theoretical gas phase enthalpies of

Results from combustion experiments on methyl ferulate and ferulic acid (Table 4)

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combined with their sublimation enthalpies derived in this work (Tables 5 and 6) have been used for further calculation of the experimental gaseous standard molar enthalpy of formation,

∆ f H mo (g) at 298.15 K for these compounds. The resulting values are given in Table 4, column 5. Now they can be applied for the validation of the high-level quantum-chemical calculations. In this work we used the composite method G3MP2 [18] and the atomization

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procedure for calculation of the theoretical gaseous enthalpy of formation of both compounds for the comparison with the experimental data. Demonstration of agreement or disagreement between the independent experimental and computed results can indicate a possible error or provide a strong validation for both results. An excellent agreement (see Table 4) between

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theoretical and experimental values of ∆ f H mo (g, 298.15 K) was found, and serves as an evidence of the mutual consistency of the thermochemical data sets for methyl ferulate and

3.6.

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ferulic acid.

Reaction enthalpy of methyl ferulate hydrolysis

3.6.1. Theoretical reaction enthalpy in the gas phase. According to the Hess´s Law the enthalpy of chemical reaction, ∆ r H mo , is defined as the stoichiometric difference of the enthalpies of formation of the reactants and products in the pure states. A crucial advantage of the quantum chemical methods in this context is that reaction enthalpy of any reactions can be calculated as the stoichiometric difference of H298(g)-values directly calculated by composite methods included in the Gaussian 09 package. In this way the standard reaction enthalpy,

∆ r H mo (g)G3MP2 = 22.6 kJ⋅mol-1, of the reaction 1 in the ideal gaseous state at 298.15 K was 20

ACCEPTED MANUSCRIPT calculated (see Table 8) using the G3MP2 method. This theoretical value is in very good agreement with the experimental reaction enthalpy ∆ r H mo (g) = 21.8±4.4 kJ⋅mol-1 (see Table 9) which was calculated using the experimental gas phase enthalpies of formation of the reaction participants methyl ferulate, water, methanol, and ferulic acid listed in Tables 4 and

Table 8

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9.

Thermodynamics of methyl ferulate hydrolysis reaction in the gas phase at 298.15 K as calculated by the G3MP2 method

kJ⋅mol-1

J.K-1.mol-1

22.6 a

13.4

For comparison

∆ r H mo

∆ r Gmo

SC

∆ r Smo

Kp(g)

kJ⋅mol-1 18.5

0.00056

M AN U

a

∆ r H mo

(g) = 21.8±4.4 kJ⋅mol-1 (see Table 9) was calculated using the experimental gas phase

enthalpies of formation of the reaction participants (see Table 9).

3.6.2. “First Law” reaction enthalpy in the liquid phase. The theoretical value of ∆ r H mo (g)

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calculated by the G3MP2 is related to the reaction enthalpy, ∆ r H mo (liq), in the liquid state by the following equation [25]:

∆ r H mo (liq) = ∆ r H mo ( g )G 3MP2 − ∑ν i ∆gl H mo ,i

(12)

i

where ∆ l H m,i are the molar enthalpies of vaporization of the pure reaction 1 participants i at o

EP

g

the reference temperature 298.15 K. Vaporization enthalpies of compounds involved in the

AC C

reaction 1 are given in Tables 3,6, and 9). Enthalpy of reaction ∆ r H mo (liq)G3MP2 = 18.6±1.6 kJ⋅mol-1 (see Table 9) estimated using Eq. (12) is in a good agreement (within the boundaries of experimental uncertainties) with those ∆ r H mo (liq) = 17.1±4.4 kJ⋅mol-1 (see Table 9) derived from the experimental enthalpies of formation according to the Hess´s Law. In our opinion, such a combination of quantum-chemical methods with the experimental (or estimated) vaporization enthalpies seems to be a very fruitful way to get access to the energetics of chemical reactions in the condensed phases. It should be underlined, that the values of the

∆ r H mo (liq)G3MP2 for reaction 1 was calculated directly using enthalpies H298 (available from the Gaussian output) of the reaction participants. In this way it is possible to overcome ambiguities inherent to quantum-chemical methods towards calculations leading to the 21

ACCEPTED MANUSCRIPT enthalpies of formation, ∆ f H mo (g), using conventional isodesmic or homodesmic reactions [7,8].

Table 9

(p° = 0.1 MPa), kJ·mol-1.

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Results for the reaction enthalpy ∆ r H mo of methyl ferulate hydrolysis reaction at T = 298.15 K

Experimental enthalpies of formation in the gas state water (g)

ferulic acid (g)

methanol (g)

-554.0±3.4

-241.8±0.1[24]

-572.5±2.8

-201.5±0.3 [35]

SC

methyl ferulate (g)

∆ r H mo (g) a = 21.8±4.4 kJ⋅mol-1

M AN U

Experimental enthalpies of formation in the liquid state methyl ferulate (liq)

water (liq)

ferulic acid (liq)

methanol (liq)

-653.8±3.4

-285.8±0.1[24]

-683.0±2.8

-239.5±0.2[35]

∆ r H mo (liq) b = 17.1±4.4 kJ⋅mol-1

Experimental enthalpies of formation in the aqueous state water (liq)

ferulic acid0 (aq)

methanol0 (aq)

-642.3±4.4

-285.8±0.1[24]

-676.6±4.1

-246.4±0.3[35]

TE D

methyl ferulate0 (aq)

∆ r H mo (aq) = 5.1±6.0 kJ⋅mol-1

For comparison: the theoretical value ∆ r H mo (g) G3MP2 = 22.6 kJ⋅mol-1 was calculated by the

G3MP2 method. b

EP

a

For comparison: ∆ r H mo (liq) G3MP2 = 18.6±1.6 kJ⋅mol-1 was calculated from the theoretical

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value ∆ r H mo (g)G3MP2 = 22.6 kJ⋅mol-1 (see Table 8) using enthalpies of vaporization of water

∆gl H mo (298.15 K) = 44.0±0.1 kJ⋅mol-1 [24], methanol ∆gl H mo (298.15 K) = 37.3±0.1 kJ⋅mol-1 [36], methyl ferulate (Table 3) and ferulic acid (Table 6)

3.6.3. “First Law” reaction enthalpy in the aqueous phase. Theoretical values ∆ r H mo (g)G3MP2 in the gas phase as well as ∆ r H mo (liq) G3MP2 in the liquid phase (and their comparison with the experimental values given in Table 9) are very important indicators of the mutual consistency of the experimental and theoretical methods applied in this work. However, these values are still not relevant to the real biological reactions which are mostly referred to the aqueous solutions. Admittedly, the reaction 22

ACCEPTED MANUSCRIPT enthalpy ∆ r H mo (aq) in the aqueous solution is usually calculated according to the Hess´s Law from the aqueous standard molar enthalpies of formation ∆ f H mo (aq) of the reaction participants. This value for methanol is well established (see Table 9). The required ∆ f H mo (aq)-values for methyl ferulate and ferulic acid are easily calculated by the summation of the

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appropriate ∆ f H mo (cr)-values (see Table 4) with the standard molar enthalpies of solution,

∆ sol H mo as follows: ∆ f H mo (aq) = ∆ f H mo (cr) + ∆ sol H mo

(13)

The latter values for methyl ferulate and ferulic acid are given in Table S6 and used for

SC

calculation of the aqueous enthalpies of formation. Thus, the reaction enthalpy of the methyl ferulate hydrolysis ∆ r H mo (aq) = 5.1±6.0 kJ⋅mol-1 was calculated according to the “First Law”.

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Table 10

Calculation of standard molar aqueous enthalpies of formation, ∆ f H mo (aq), of methyl ferulate and ferulic acid at T = 298.15 K (p° = 0.1 MPa), kJ·mol-1

compounds

∆ f H mo (cr) a ∆lcr H mo

b

∆ f H mo (liq) c ∆ sol H mo d ∆ f H mo (aq) e

ferulic acid a

TE D

methyl ferulate -677.3±3.2 23.5±1.1 -653.8±3.4 35.0±3.0 -642.3±4.4 -708.2±2.6 25.2±1.1 -683.0±2.8 31.6±3.2 -676.6±4.1

From Table 4. Uncertainties correspond to expanded uncertainties of the mean (0.95

confidence level)

From Table 6. Uncertainties correspond to the standard deviation.

c

Calculated as the sum ∆ f H mo (liq) = ∆ f H mo (cr) + ∆lcr H mo

d

From Table S6.

e

Calculated as the sum ∆ f H mo (aq) = ∆ f H mo (cr) + ∆ sol H mo

AC C

3.7.

EP

b

Comparison of “First Law” and “Second Law” reaction enthalpies in the

aqueous phase. Strongly speaking the aqueous reaction enthalpy ∆ r H mo (aq) = 5.1±6.0 kJ⋅mol-1 of the methyl ferulate hydrolysis independently derived according to the “First Law” is referred to the reference chemical reaction:
ℎ  () +  () ⇄   () +
ℎ() 23

(14)

ACCEPTED MANUSCRIPT It is well established [1] that the ferulic acid undergoes dissociation in the aqueous phase (which results in reference chemical reaction (1b)): (15)

   () ⇋     () +   ()

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We used the enthalpy of this reaction (15) ∆ r H mo = 4.1 kJ⋅mol-1 [1] at 298.15 K to derive the standard molar enthalpy of formation of the aqueous ferulate anion ∆ f H mo (ferulic acid-, aq) = -(672.5±4.1) kJ⋅mol-1, which is required for the reaction according to Eq. (1b). And now

SC

we have all necessary information to calculate the enthalpy ∆ r H mo (aq) = 9.2±6.0 kJ⋅mol-1 of the methyl ferulate hydrolysis reaction (1b), which can be now compared with the result by Goldberg et al. [1] ∆ r H mo (aq) = 7.3±1.7 kJ⋅mol-1 because both values are referred now to the

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same reaction and to identical experimental conditions. The absolute values of reaction enthalpies derived by totally different methods are in very good agreement providing confidence in the whole scope of thermochemical methods leading to from the theoretical enthalpies in the gas phase to the biologically relevant aqueous conditions. It should be mentioned that the standard molar reaction enthalpy reported by Goldberg et al. [1] was based

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on a directly measured reaction enthalpy by using titration microcalorimetry with errors assumed to be small. In contrast, the reaction enthalpy of the reaction 1b was derived in this work that involved numerous steps and thus the larger expanded uncertainty of ±6.0 kJ⋅mol-1. However, in the future it is possible to significantly reduce the uncertainties at least at the step

EP

of the experimental determination of molar enthalpy of solution ∆ sol H mo . In the current work we used the available literature data on the water solubility in order to estimate ∆ sol H mo –

AC C

values. It is well known that these values often suffer from insufficient accuracy. At the same time, a most accurate technique for this thermodynamic quantity is the classic solution calorimetry which can be used to measure standard molar enthalpy of solution with uncertainties of ~ ±0.1 kJ⋅mol-1. This technique is established in our lab and it will be applied for more accurate determination of ∆ sol H mo (298.15 K) of the molecular and ionic species. Moreover, comparison of our new result at 298.15 K ∆ r H mo (aq) = 9.2±6.0 kJ⋅mol-1 (referred

to reaction 1b, at ionic strength I =0) with the “Second Law” value ∆ r H mo = 8.3±0.6

kJ·mol-1 (see section 3.2) measured in this work, as well as with the literature calorimeterbased value ∆ r H mo = 7.3±1.7 kJ·mol-1 reported by Goldberg et al. [1], leads to a promising 24

ACCEPTED MANUSCRIPT conclusion, that the thermodynamic procedure suggested in this work is able to provide reasonable energetics of biologically relevant chemical reactions with the accuracy sufficient to be applied for the measurement of many metabolic pathways and biochemical reactions. For the latter purpose, the reliable appraisal of the equilibrium constant is even more important and challenging. The next section shows if the combination of quantum chemistry

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with some empirical data will allow predicting equilibrium constants using methods totally independent from experimental reaction data.

3.8

Equilibrium constant of the methyl ferulate hydrolysis from quantum-

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chemical calculations.

The gas phase thermodynamic equilibrium constant, Kp, at 298.15 K calculated (see

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Table 9) for the reaction (1) is related to the thermodynamic constant Ka(liq) of this reaction in the liquid phase by Eq. (3). Saturated vapor pressures for reaction participants were available in the literature (see Table 11) or they were measured in the present work. Using the theoretical KP calculated with help of the G3MP2-method and presented in Table 9, the thermodynamic constant Ka (liq) = 5⋅10-4 in the liquid phase was calculated (see Table 11). Surprisingly, this theoretical Ka for reaction (1) is very good comparable with the

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experimental constant Ka = 1.9⋅10-4 in aqueous solution reported by Goldberg et al. [1]. Such agreement can be considered as highly sufficient taking into account the extremely low vapour pressures of methyl ferulate and ferulic acid at 298.15 K. Note that both constants are related to the practically different reference conditions (pure liquid and aqueous solution). In

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any case, from this study of the model reaction 1 we learned that the level of the thermodynamic equilibrium constant for biological reactions can be reasonably assessed by

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using QC methods complemented with the restricted number of additional thermochemical data. However, a lot of additional work is still required in order to ascertain this conclusion.

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ACCEPTED MANUSCRIPT Table 11 Adjustment of the theoretical thermodynamic equilibrium constant Kp to the liquid phase Vapour pressure in the liquid state at 298.15 K, Pa methyl ferulate (liq)

water (liq)

ferulic acid (liq)

methanol (liq)

0.00072

3179.4 [37]

0.00015 [23]

16940.0 [38]

0 .00072 ⋅ 3166 .7 ⋅ K p ( g ) = 0 .9 ⋅ 0 .00056 = 0 .0005 0 .00015 ⋅ 16938 .5

a

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K a ( liq ) =

From the practical point of view, the most interesting thermodynamic value is the equilibrium constant Ka (liq). A qualitatively correct order of magnitude of this constant

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provides preliminary information about feasibility of the chemical reaction according to equation:

(16)

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∆ r Gmo (liq) = -RT ⋅ ln Ka(liq)

as well as about expected theoretical concentrations of reaction participants. Moreover, Ka values can be assessed by equilibrium molalities and using any suitable method (e.g. ePCSAFT [5]) able to provide activity coefficients of biological components required in Eq. (5). For the model reaction 1, we used Ka (liq) = 5⋅10-4 at 298.15 K (see Table 11) to calculate

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with Eq. (16) ∆ r Gmo (liq) = 18.8±5.0 kJ⋅mol-1 and this value is in fair agreement with the value

∆ r Gmo (aq) = 21.3±0.1 kJ⋅mol-1 reported by Goldberg et al. [1]. At this point it is not possible to quantitatively compare Ka (liq) and the Ka value shown in table 3 (“Second law”), since the reference states of these Ka values differ from each other. However, the dependence of γ and

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γ* on the temperature are assumed to be equal and this allows to compare the value for ∆ r H mo from the “Second law” with values from the “First law” and the literature. Thus, the QC-

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based thermodynamic procedure developed in this work could be further tested for assessment of thermodynamically dominant pathways of more complex biological processes.

4. Conclusion

Using the example of methyl ferulate hydrolysis, this work shows thermodynamic procedures for an access to reaction data of enzymatic reactions. In a first step, equilibrium molalities of reacting agents have been measured, and combined with activity coefficients predicted with ePC-SAFT to derive the true thermodynamic equilibrium constant Ka. The Ka– values were determined at several temperatures, and the standard enthalpy of reaction was derived from that information. In a second step, thermochemical properties of the pure 26

ACCEPTED MANUSCRIPT reaction participants have been measured, finally providing the enthalpy of formation of the reaction participants in the liquid phase. Both, reaction enthalpy from Ka values and from enthalpies of formation were shown to be mutually consistent. Finally, high-level quantum chemical calculations were applied yielding Ka values in the liquid phase. Qualitatively correct agreement between Ka from apparent equilibrium constants and from QC calculation

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were observed. This is a highly promising result, and the method might serve to reduce the experimental efforts as well as facilitate evaluation and handling of thermodynamic

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information.

Acknowledgment

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The authors acknowledge financial support from Leibniz award to G. Sadowski given by the German Science Foundation DFG. The authors also acknowledge for the valuable advices of the reviewers.

Appendix A. Supplementary data

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Supplementary data related to this article can be found at…

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ACCEPTED MANUSCRIPT List of symbols activity of reacting agent i (standard state = pure component) activity of reacting agent i (standard state = hypothetical ideal solution) at reaction equilibrium Saturated vapor pressure, Pa Standard pressure = 105 Pa Arbitrary chosen reference temperature for data treatment, K

∆glGmo (θ)

standard molar Gibbs energy of vaporization at the reference temperature, kJ·mol-1

∆gl H mo (θ)

standard molar enthalpy of vaporization at the reference temperature, kJ·mol-1

∆gl Smo (θ)

standard molar entropy of vaporization at the reference temperature, J·K-1·mol-1

∆lcr H mo (θ)

standard molar enthalpy of fusion at the reference temperature, kJ·mol-1

∆gcr H mo (θ)

standard molar enthalpy of sublimation at the reference temperature, kJ·mol-1

Cop,m

standard molar heat capacity, J·K-1·mol-1

∆glC op,m

molar heat capacity difference at the reference temperature, J·K-1·mol-1

∆gcrCop,m

molar heat capacity difference at the reference temperature, J·K-1·mol-1

∆ f H mo

standard molar enthalpy of formation, kJ·mol-1

∆ r H mo

standard molar entropy of reaction, J·K-1·mol-1

∆ r Gmo

standard molar Gibbs energy of reaction, kJ·mol-1

∆ sol H mo

standard molar enthalpy of solution, kJ·mol-1

Σ

sum of all ion species



activity-based equilibrium constant



molality-based equilibrium constant

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apparent molality-based equilibrium constant of the overall biochemical reaction

mi

molality of component i (mole component i per kg water), mol⋅kg-1 molality based activity coefficient of reacting agent i (standard state = hypothetical ideal solution), kg⋅mol-1 molality based activity coefficient of reacting agent i (standard state = pure component), kg⋅mol-1 electrolyte Perturbed-Chain Statistical Associating Fluid Theory

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θ

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ai ai* Eq pi po

A1∗, A1

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ePC-SAFT

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activity-coefficient ratio

V W1X1 /YX κAiBi

association-energy parameter of component i, K

kB

Boltzmann constant, J⋅K-1

kij

association-volume parameter of component i

binary interaction parameter between components i and j

miseg

segment number of component i

R

ideal gas constant, J/(molK)

T

temperature, K

σi

segment diameter of component i, Å

ui/kB

dispersion-energy parameter of component i, K

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Graphical abstract:

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