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Standard Gibbs energy of metabolic reactions: I. Hexokinase reaction Florian Meurer, Maria Bobrownik, Gabriele Sadowski, and Christoph Held Biochemistry, Just Accepted Manuscript • DOI: 10.1021/acs.biochem.6b00471 • Publication Date (Web): 21 Sep 2016 Downloaded from http://pubs.acs.org on September 22, 2016
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Biochemistry
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Standard Gibbs energy of metabolic reactions: I. Hexokinase
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reaction
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Florian Meurer, Maria Bobrownik, Gabriele Sadowski, Christoph Held*
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Laboratory of Thermodynamics, Department of Biochemical and Chemical Engineering,
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Technische Universität Dortmund, Emil-Figge-Str. 70, 44227 Dortmund, Germany
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* corresponding author:
[email protected] 21
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Symbols
23
Greek letters Symbol r
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Property
Unit
relative dielectric permittivity
-
association-energy parameter
∗,
activity coefficient of component i on molality base
-
rational activity coefficient of component i on molality base
-
activity coefficient of component i at infinite dilution on molality base
-
association-volume parameter
-
stoichiometric coefficient of component i
-
segment diameter of component i
Å
osmotic coefficient of aqueous solution
-
,
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K
Latin letters Symbol
Property
Unit
res
residual Helmholtz energy
J
hc
hard-chain contribution to Helmholtz energy
J
disp
dispersive contribution to Helmholtz energy
J
association contribution to Helmholtz energy
J
ionic contribution to Helmholtz energy
J
f-function of component i
-
chemical standard Gibbs energy of chemical reaction
J·mol
biochemical standard Gibbs energy of reaction
J·mol
Boltzmann constant
J·K
chemical thermodynamic equilibrium constant
-
biochemical thermodynamic equilibrium constant
-
A
A A
assoc
A
A
ion
∆
∆ ′ kB
-1
-1
-1
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Biochemistry
∗
chemical activity-coefficient ratio
-
biochemical activity-coefficient ratio
-
chemical molality-based equilibrium constant
-
biochemical molality-based equilibrium constant
-
molality of component i
mol·kg
molality of component i at thermodynamic equilibrium
mol·kg
segment number
-
molar mass of component i
g·mol
number of association sites of component i
-
∗
mi
seg
assoc
Ni
!
osmolality, ! =
-1
-1
-1
-1
osmol kg -1
-1
-1
-1
R
ideal gas constant, 8.31446 J·mol ·K
J·mol ·K
T
temperature
K
dispersion-energy parameter of component i
K
ui/kB
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Abstract
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The standard Gibbs energy of reaction enables calculation of the driving force of a
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(bio)chemical reaction. Gibbs energies of reaction are required in thermodynamic
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approaches to determine fluxes as well as single reaction conversions of metabolic
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bioreactions. The hexokinase reaction (phosphorylation of glucose) is the entrance step
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of glycolysis, and thus its standard Gibbs energy of reaction (∆ ) is of great impact.
∆ is accessible from equilibrium measurements, and the very small concentrations of
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the reacting agents cause usually high error bars in data reduction steps. Even worse,
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works from literature do not account for the non-ideal behavior of the reacting agents
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(activity coefficients were assumed to be unity), thus published ∆ values are not standard data. Consistent treatment of activity coefficients of reacting agents is crucial
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for
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measurements.
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In this work equilibrium molalities of hexokinase reaction were measured with an
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enzyme kit. These results were combined with reacting agents’ activity coefficients
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obtained with the thermodynamic model ePC-SAFT. Pure-component parameters for
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adenosine triphosphate (ATP) and adenosine diphosphate (ADP) were fitted to
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the
accurate
determination
of
standard
Gibbs
energy
from
equilibrium
experimental osmotic coefficients (water+Na2ATP, water+NaADP). ∆ of the
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hexokinase reaction at 298.15 K and pH 7 was found to be -17.83 ± 0.52 kJ·mol-1. This
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value was compared with experimental literature data; very good agreement between
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the different ∆ values was obtained by accounting for pH, pMg, and the activity coefficients of the reacting agents.
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Introduction
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The activation of glucose by phosphorylation to glucose-6-phosphate (G6P) is the entry
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step of glycolysis in biological cells. This reaction is catalyzed by the enzyme
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hexokinase, and it contains adenosine triphosphate (ATP) as reactant that is reduced to
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adenosine diphosphate (ADP) according to: Glucose + ATP ↔ G6P + ADP
(1)
Glucose + ATP3- ↔ G6P- + ADP2-
(2)
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Eq. (1) shows the biochemical reaction while Eq. (2) displays the chemical expression
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that contains the highest ionized species of each reacting agent present at biological
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standard conditions (298.15 K and pH 7). Due to its importance for metabolism the
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hexokinase reaction was subject to different investigations in the last decades. While
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Fromm et al. in 1964 (1) and later Blázquez et al. (2) worked on enzyme kinetics and
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inhibition of the hexokinase reaction using different hexokinases, Bianconi et al. (3)
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studied the kinetics as well as reaction enthalpy of reaction (1), and they used two
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isozymes of hexokinase for that.
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The thermodynamic equilibrium of this hexokinase reaction has been subject to some
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investigations (4, 5). The reaction equilibrium was found to be far on the product side,
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and the reaction therefore has usually been presented as an irreversible reaction.
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Nevertheless, hexokinase reaction was proven to be reversible. Studies on (standard)
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enthalpy of reaction by calorimeter experiments were published by different authors (3,
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5, 6). They applied titration calorimetry to determine enthalpies of reaction at different
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conditions considering the influence of pH, pMg (6) and the influence of some inert
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sugars (5). The conditions under investigation in those works are non-ideal from the
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thermodynamic point-of-view with respect to deviations from the biological standard ACS Paragon Plus Environment
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state, which is a one molal hypothetically ideal solution. Deviations from this standard
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state are accounted for by the activity coefficients of the reacting agents. However, all
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previous works on hexokinase have assumed that the activity coefficients of the reacting
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agents are unity. As demonstrated by Hoffmann et al. (7, 8) the equilibrium of biological
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reactions depend on the reacting agents’ concentrations, even if the molalities of
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reacting agents are very low (in the mmolal range). All data available on literature on the
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hexokinase reaction were measured at conditions that yielded equilibrium molalities of
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the reacting agents up to 47.21 mmol·kg-1. One exception are the conditions applied by
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Robbins and Boyer (4) who determined equilibrium-composition data at very low
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equilibrium molalities (max. 7.6 mmol·kg-1) of the reacting agents. Only at these highly
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diluted conditions activity coefficients might be close to unity, whereas activity
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coefficients are required for investigations at higher molalities.
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Activity coefficients might be accessible from measurements (e.g. osmometry).
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However, measuring activity coefficients of the reacting agents of the hexokinase
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reaction depending on a broad variety of conditions (reacting agents’ molalities, ionic
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strength, pH, temperature, presence of additives and cofactors) is not reasonable.
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Application of a thermodynamic model is more convenient, especially since
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thermodynamic models for biomolecules improved significantly. To give some examples,
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gE-models like the Pitzer equation (9) as well as equations of state (EOS) like Peng-
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Robinson EOS and (electrolyte) Perturbed-Chain Statistical Associating Fluid Theory
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(ePC-SAFT) (10, 11) were applied to describe thermodynamic properties of biological
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solutions. Using ePC-SAFT, excellent results were obtained on modeling activity
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coefficients in solutions containing urea, amino acids, or adenine (12-14). Further, ePC-
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SAFT was successfully applied to model thermodynamic equilibrium of bioreactions (7, ACS Paragon Plus Environment
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8, 15, 16). ePC-SAFT requires pure-component parameters and binary interaction
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parameters. In (7, 8, 15, 16), the pure-component ePC-SAFT parameters for the
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reacting agents were fitted to reaction-independent data e.g., thermodynamic water
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activity data (7). This yielded good prediction results of activity coefficients in biological
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multi-component systems. In contrast, the prediction of activity coefficients in multi-
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component mixtures failed using classical models like (Pitzer)-Debye-Hückel (17). The
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reason for this is the complex interaction behavior among the components present in the
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mixtures. This yields a strongly non-ideal solution with activity coefficients distinctly
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different from unity, which can only be predicted by state-of-the-art models (e.g. ePC-
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SAFT) that account for short-range forces and long-range interactions.
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The availability of equilibrium molalities and activity coefficients of the reacting agents
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allows determining the standard Gibbs energy of the hexokinase reaction ∆ . Since
the entrance in the glycolytic cycle is an important step in catabolic activity of the cell, a thermodynamically-consistent ∆ value is of great impact for other scientists working on strand and production optimization in terms of metabolic engineering and
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thermodynamic metabolic flux analysis (18, 19). All previous equilibrium studies on
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hexokinase have assumed that the activity coefficients of the reacting agents are unity
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despite that fact that the reactions were carried out at non-standard conditions. Thus,
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∆
values calculated from these results are just observed values but not consistent
∆ values at standard state. Thus, in this work, the reaction equilibrium of the
hexokinase reaction was measured, activity coefficients of the reacting agents were
modeled, and ∆ of the hexokinase reaction at 298.15 K and pH 7 is derived as the
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main result. These values will further be compared to experimental reaction data from
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literature in the very last section of the results chapter. This will show the importance of ACS Paragon Plus Environment
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consistent treatment of activity coefficients of reacting agents for the accurate
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determination of Gibbs energy at standard state.
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Thermodynamic formalism for hexokinase reaction
The Gibbs energy of reaction ∆ g is a fundamental property for the characterization of
reactions. It allows predicting whether or not a reaction will occur at defined conditions. At thermodynamic equilibrium ∆ g will be zero. Therefore at equilibrium Eq. (3) will be
fulfilled.
∆ = −RT ln* +
(3)
Here, R is the ideal gas constant with value of 8.31446 J·mol-1·K-1 and T is the absolute temperature and is the thermodynamic equilibrium constant. Many compounds in biological reactions are electrolytes. They dissociate to a certain extent depending on solution conditions and may even form complexes with metal cations e.g., with Mg2+.
The chemical definition of is formulated in terms of species and would therefore
require single species activities. Thus, the equilibrium constant belongs to Eq. (2). In contrast, the biochemical equilibrium constant refers to Eq. (1). In this work, was
investigated, yielding the biochemical standard Gibbs energy of reaction ∆ ′ ∆ ′ = −RT ln * ′+
(4)
The thermodynamic equilibrium constant ′ for the hexokinase reaction (Eq. (1)) is thus defined via the sum of all species activities Σaeq at reaction equilibrium according to: ∑ -./0 ∙ ∑ -20 ′ = ∑ -345678 ∙ ∑ -90
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(5)
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As the aeq values are defined as product of molality and activity coefficient on molality basis ∗, , ′ can be expressed as product of
and ∗ via
′ = ′ ∙
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∗
∑ ./0 ∙ ∑ 20
∗
∗
./0 ∙ 20 = ∙ ∗
∗
∙ 90 ∑ 345678 ∙ ∑ 90 345678
(6)
In Eq. (6) ∗, values have to be calculated at equilibrium molalities . The
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assumption behind Eq. (6) is that the activity coefficients of all species of one
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component are equal. The reference state in biological systems is the one molal
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hypothetically ideal solution, in which the activity coefficients ∗, are assumed to be
one. At conditions other than the biological standard state, ∗, differ from unity. The rational activity coefficients can be obtained according to: ∗,
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= :,
(7)
where and , are the molality-based activity coefficients of component i at any
given molality and at infinite dilution in the reaction medium. With this definition the formulation of the thermodynamically-consistent calculation of ∆ ′ is expressed as: + ∆ ′ = −RT ln*
− RT ln;∗
