# ðŸ“’ Alberty 2006

Relations between biochemical thermodynamics and biochemical kinetics

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## Intro

Close connection between the biochemical thermodynamics and biochemical kinetics of an enzyme-catalyzed reaction.

Difference between chemical & biochemocal thermodynamics is the specification of the

**pH as an independent variable**in biochemical thermodynamicsGibbs energy:

`G = H-TS`

- T : intensive variable (not changed by number of particles)
- S : extensive variable (changed by number of particles)
- @ equilibrium: G is @ minimum

specification of pH as an independent variable in biochemical thermodynamics has another advantage: ease of binding polynomial (adenosine triphosphate can be treated as ATP, rather than the species ATP4âˆ’,HATP3âˆ’, and H2ATP2âˆ’)

- The apparent equilibrium constant Kâ€² are also represented by mathematical functions of T, pH, and ionic strength
- The assumption is that the
**pH is held constant**during the reaction,

The apparent equilibrium constant for the catalyzed reaction and the equilibrium composition do not depend in any way on the properties of the enzyme.

In biochemical thermodynamics it is more convenient to take the

**apparent equilibrium constant**and transformed thermodynamic properties of reactants like ATP to be functions of ionic strength in addition to temperature and pH.The

**Haldane relations**connect chemical and biochemical thermodynamics at equilibrium.

## Consideration of the effect of ionic strength on a simple mechanism

### Simple mechanism

$$ E + S \rightleftharpoons EX \rightleftharpoons E + P $$

the complete steady-state rate equation is $$ \begin{aligned} v &= \frac{V_f \phi_S - V_r \phi_P}{1 + \phi_S + \phi_P} \cr \phi_S &= [S] / K_S \cr \phi_P &= [P] / K_P \cr \end{aligned} $$

The Haldane relation: $$ K_{eq}^{app} = \frac{[P]_{eq}}{[S]_{eq}} = \frac{V_fK_P}{V_rK_S} = \frac{k_{1}k_{2}}{k_{-1}k_{-2}} $$

Apparrent equilibrium constants are functions of temperature, pH, and ionic strength.

### Multistep mechanism

$$ E + S \rightleftharpoons ES \rightleftharpoons EP \rightleftharpoons E + P $$ The Haldane relation: $$ K_{eq}^{app} = \frac{[P]_{eq}}{[S]_{eq}} = K_1 K_2 K_3 = \frac{k_{1}k_{2}k_{3}}{k_{-1}k_{-2}k_{-3}} $$

## Consideration of the effect of pH on a simple mechanism

$$ \begin{aligned} K_{1S} &= [S^-][H^+] / [HS] \cr K_{1H2S} &= [HES^-][H^+] / [H2ES] \cr K_{H2ES} &= [HS][HE] / [H2ES] \cr \ [E]_t &= [E^-] + [HE] + [H2E^+] + [HES^-] + [H2ES] + [H3ES^+] \cr \ [S] &= [S^-] + [HS] + [H2S] \cr \end{aligned} $$

Then:

$$ \begin{aligned} f_H([H^+], K_1, K_2) &:= 1 + K_{1} / [H+] + [H+] / K_{2} \cr \ [E]_t &= [HE] / f_H([H^+], K_{1E}, K_{2E})+ [H_2ES]/ f_H([H+], K_{1ES}, K_{2ES}) \cr \ [E]_t &= f([H_2ES]) = … \cr \ [S] &= [HS] / f_H([H^+], K_{1S}, K_{2S}) \end{aligned} $$

Finally: $$ \begin{aligned} v &= V_{f} \frac{ [S] }{ [S] + K_{S} } \cr V_{f} &= k_f [E]_{t} / f_{H} ([H^+], K_{1ES}, K_{2ES}) \cr K_{S} &= K_{H2ES} \cdot f_{H} ([H^+], K_{1E}, K_{2E}) \cdot f_{H}([H^+], K_{1S}, K_{2S}) / f_{H}([H^+], K_{1ES}, K_{2ES}) \cr \end{aligned} $$

## pH-dependent reverse reaction

- Using the King-Altman method, and Roberts has derived it using determinants.
- Similar relations for P

The Haldane relation: $$ K_{eq}^{app} = \frac{[P]_{eq}}{[S]_{eq}} = \frac{V_f K_P}{V_r K_S} = \frac{k_fK_{H2EP}f_H([H^+], K_{1P}, K_{2P})}{k_rK_{H2ES}f_H([H^+], K_{1S}, K_{2S})} $$

## Expression of the apparent equilibrium constant for A + B = P + Q

in terms of the rate constants of the steps in the forward and reverse direction

4400 and 4700 respectively, in good agreement. They depend only on the properties of the reactants in the reaction above; we don’t have to consider ionic strength.

## Discussion

The apparent equilibrium constant (Kâ€²) for an enzyme-catalyzed reaction can be calculated using parameters from the complete

**rate equation**or**rate constants**for the steps in the mechanism- knowledge of K â€² makes it possible to calculate the equilibrium composition of reactants for given initial concentrations of the reactants. Go left or go right.
- values of Kâ€² can be used to calculate Î”rGâ€²Â° using -RTlnKâ€². partial derivatives of Î”rGâ€²Â° can be used to calculate Î”rHâ€²Â°, Î”rSâ€²Â°.
- values of Kâ€² or Î”rGâ€²Â° can be used to calculate values of Î”fGÂ° of species of a new reactant if the Î”fGÂ° are known for species of all the other reactant

the most efficient way to store information on the thermodynamics of enzyme-catalyzed reactions is to store

**standard Gibbs energies**of formation and**standard enthalpies**of formation of speciesWhen

**magnesium ions**or other divalent cations are present, their effects on the thermodynamics of enzyme-catalyzed reactions can be**handled in the same way as effects of hydrogen ions**. Just introduce p Mg as an independent variable of the transformed Gibbs energy in addition to pH.

Alberty RA. Relations between biochemical thermodynamics and biochemical kinetics. Biophys. Chem. 2006;124(1):11-17. doi:10.1016/j.bpc.2006.05.024. ↩︎