Thermodynamics and Kinetics of Glycolytic Reactions. Part I: Kinetic Modeling Based on Irreversible Thermodynamics and Validation by Calorimetry

Abstract

In systems biology, material balances, kinetic models, and thermodynamic boundary conditions are increasingly used for metabolic network analysis. It is remarkable that the reversibility of enzyme-catalyzed reactions and the influence of cytosolic conditions are often neglected in kinetic models. In fact, enzyme-catalyzed reactions in numerous metabolic pathways such as in glycolysis are often reversible, i.e., they only proceed until an equilibrium state is reached and not until the substrate is completely consumed. Here, we propose the use of irreversible thermodynamics to describe the kinetic approximation to the equilibrium state in a consistent way with very few adjustable parameters. Using a flux-force approach allowed describing the influence of cytosolic conditions on the kinetics by only one single parameter. The approach was applied to reaction steps 2 and 9 of glycolysis (i.e., the phosphoglucose isomerase reaction from glucose 6-phosphate to fructose 6-phosphate and the enolase-catalyzed reaction from 2-phosphoglycerate to phosphoenolpyruvate and water). The temperature dependence of the kinetic parameter fulfills the Arrhenius relation and the derived activation energies are plausible. All the data obtained in this work were measured efficiently and accurately by means of isothermal titration calorimetry (ITC). The combination of calorimetric monitoring with simple flux-force relations has the potential for adequate consideration of cytosolic conditions in a simple manner.

Keywords: biothermodynamics; glycolysis; isothermal titration calorimetry; systems biology.

Figure 1

Scheme of the investigations within the present work. (A) Shows exemplarily the validity range of the various kinetic models used to describe the reaction progress, (B) shows the number of adjustable parameters required by the models, and (C) shows the most important cytosolic conditions, highlighting the temperature as the one used to test the models.

Figure 2

Heat flow diagrams from the isothermal titration calorimetry (ITC) measurement of reactions 2 and 9 at 310.15 K. The arrows mark the time of the two injections (black = 1st injection, blue = 2nd injection). At the end of the reaction, when the equilibrium is reached, the signal returns to the baseline. The concentrations were 14.3 nmol kg⁻¹ PGI and 4.8 mmol kg⁻¹ F6P for reaction 2 and 2 µmol kg⁻¹ enolase and 74.8 mmol kg⁻¹ PEP for reaction 9. (A,C) Show the heat flow curves of the total reaction signal (green), the reference measurement without the enzyme (red), and the reference measurement without the substrate (black). The heat of dilution causes the positive peak of the red and green curve. (B,D) Display the net reaction (subtraction of the reference signals from the total reaction signal). The integrated heat of the net signal (Q) is shown in gray.

Figure 3

Plot of reaction rate against the substrate concentration for both reactions at 310.15 K. Both panels show a single example reaction. (A) Displays reaction 2 with a concave curvature but no plateau at high substrate concentrations (14.3 nmol kg⁻¹ PGI and 4.8 mmol kg⁻¹ F6P). (B) Reaction 9 with a convex curvature, also not showing a plateau at high substrate concentrations (2 µmol kg⁻¹ enolase and 74.8 mmol kg⁻¹ PEP).

Figure 4

Fitting of the kinetic data (scatter) with the Noor model (red solid line). Both panels show a single example reaction. (A) Shows the data of reaction 2 with concentrations of 14.3 nmol kg⁻¹ PGI and 4.8 mmol kg⁻¹ F6P (R² = 0.99972). (B) Shows the data of reaction 9 with concentrations of 2 µmol kg⁻¹ enolase and 74.8 mmol kg⁻¹ PEP (R² = 0.99903). Both were measured at 310.15 K. The adjustable parameters are $r_{max}$ and $K_{F6P}$ in A and $\Lambda$ and $K_{2PG}$ in B.

Figure 5

Verification of the suitability of the flux-force model to describe the reaction kinetics at 310.15 K. Both panels show a single example reaction. (A) Shows the result of one chosen condition of reaction 2 (14.3 nmol kg⁻¹ PGI and 4.8 mmol kg⁻¹ F6P) with R² = 0.99921 and (B) of reaction 9 with concentrations of 2 µmol kg⁻¹ enolase and 74.8 mmol kg⁻¹ PEP (R² = 0.99892).

Figure 6

The Arrhenius plot of the kinetic parameter L (flux-force model) for both reactions. (A) Shows results for reaction 2 (14.3 nmol kg⁻¹ PGI and 4.8 mmol kg⁻¹ F6P) with a p-value of the slope of 0.01389 (slope is significantly different from 0) and R² = 0.99952 and (B) for reaction 9 with concentrations of 2 µmol kg⁻¹ enolase and 74.8 mmol kg⁻¹ PEP (p-value = 0.01922 (slope is significantly different from 0), R² = 0.99909). The error bars correspond to the standard deviations of the triple determinations. The activation energy $E_a$ was calculated from the slope of the fit. For reaction 2, it is $E_a$ = 55.4 ± 1.2 kJ mol⁻¹ and for reaction 9, it is $E_a$ = 44.8 ± 1.4 kJ mol⁻¹.