Theory

Information

This page provides a brief overview of the equations, data, and theory used to correct the equilibrium constants. For a more rigorous description of the theory and assumptions involved in adjusting equilibrium constants, please consult the references.

For a step-by-step walkthrough of the adjustment calculations over the CK clamp, please consult this Jupyter notebook. This document contains descriptions of the process alongside Python code that implements the calculations.

Chemical Equations

The Chemical Equations for the hydrolysis of ATP, the Creatine Kinase reaction, and the Adenylate Kinase reaction are listed below

$$ K_{\text{ref ATP}} = \frac {[\text{ADP}^{3-}][\text{HPO}_4^{2-}][\text{H}^{+}]} {[\text{ATP}^{4-}][\text{H}_2\text{O}]} $$ $$ K_{\text{ref CK}} = \frac {[\text{ATP}^{4-}][\text{Cr}]} {[\text{ADP}^{3-}][\text{PCr}^{2-}][\text{H}^{+}]} $$ $$ K_{\text{ref AK}} = \frac {[\text{ATP}^{4-}][\text{AMP}^{2-}]} {[\text{ADP}^{3-}]^2} $$

Apparent Equilibrium Constants

The apparent equilibrium constant (K`) for the above chemical equations is related to the Reference Equilibrium Constant (Kref, which is measured under specific experimental conditions) through the following relationship:

$$ K^\prime_{ATP} = \frac {K_{\text{ref ATP}}}{[\text{H}{^+}]} \frac{ \left\{1 + \frac {[\text{H}{^+}]} {K_{\text{a}_{\text{ADP}}}} + K_{\text{b}_{\text{MgADP}}}[\text{Mg}^{2+}] + \frac {K_{\text{b}_{\text{MgHADP}}} [\text{H}{^+}][\text{Mg}^{2+}]} {K_{\text{a}_{\text{ADP}}}} \right\} \left\{ 1 + \frac {[\text{H}{^+}]} {K_{\text{a}_{\text{HPO}{_4}}}} + K_{\text{b}_{\text{MgHPO}{_4}}}[\text{Mg}^{2+}] \right\} } { \left\{1 + \frac {[\text{H}{^+}]} {K_{\text{a}_{\text{ATP}}}} + K_{\text{b}_{\text{MgATP}}}[\text{Mg}^{2+}] + \frac {K_{\text{b}_{\text{MgHATP}}} [\text{H}{^+}][\text{Mg}^{2+}]} {K_{\text{a}_{\text{ATP}}}} \right\} }$$ $$ K^\prime_{CK} = K_{\text{ref CK}} \frac { [\text{H}{^+}]\left\{1 + \frac {[\text{H}{^+}]} {K_{\text{a}_{\text{ATP}}}} + K_{\text{b}_{\text{MgATP}}}[\text{Mg}^{2+}] + \frac {K_{\text{b}_{\text{MgHATP}}} [\text{H}{^+}] [\text{Mg}^{2+}]} {K_{\text{a}_{\text{ATP}}}} \right\} } { \left\{1 + \frac {[\text{H}{^+}]} {K_{\text{a}_{\text{ADP}}}} + K_{\text{b}_{\text{MgADP}}}[\text{Mg}^{2+}] + \frac {K_{\text{b}_{\text{MgHADP}}} [\text{H}{^+}] [\text{Mg}^{2+}]} {K_{\text{a}_{\text{ADP}}}} \right\} \left\{1 + \frac {[\text{H}{^+}]} {K_{\text{a}_{\text{PCr}}}} + K_{\text{b}_{\text{MgPCr}}}[\text{Mg}^{2+}] \right\} }$$ $$ K^\prime_{ATP} = \frac {K_{\text{ref ATP}}}{[\text{H}{^+}]} \frac{ \left\{1 + \frac {[\text{H}{^+}]} {K_{\text{a}_{\text{ADP}}}} + K_{\text{b}_{\text{MgADP}}}[\text{Mg}^{2+}] + \frac {K_{\text{b}_{\text{MgHADP}}} [\text{H}{^+}][\text{Mg}^{2+}]} {K_{\text{a}_{\text{ADP}}}} \right\} \left\{ 1 + \frac {[\text{H}{^+}]} {K_{\text{a}_{\text{HPO}{_4}}}} + K_{\text{b}_{\text{MgHPO}{_4}}}[\text{Mg}^{2+}] \right\} } { \left\{1 + \frac {[\text{H}{^+}]} {K_{\text{a}_{\text{ATP}}}} + K_{\text{b}_{\text{MgATP}}}[\text{Mg}^{2+}] + \frac {K_{\text{b}_{\text{MgHATP}}} [\text{H}{^+}][\text{Mg}^{2+}]} {K_{\text{a}_{\text{ATP}}}} \right\} }$$

Each expression contained within {} represents one of the ions within the Kref expressed as a function of its its acid dissociation and magnesium binding properties. These are also equilibrium constants and must be adjusted for ionic strenght, pH, free magnesium, and temperature of the system. These variables along with their corresponding values are summarized in the table below

Vant Hoff Equation

Used to adjust the temperature of Kref

$$\ln\left(\frac{K_2}{K_1}\right) = -\frac{\Delta H ^\circ}{R}\left( \frac{1}{T_2} - \frac{1}{T_1} \right)$$

    \(R = 8.3144598 \frac{J}{K mol}\)
    \(K_1 = K_{ref_{I=0}}\)
    \(T_1 = \) Temperature of \(K_1\)
    \(K_2 = K_{ref_{I=0}}\) at new temperature \(T_2\)
    \(\Delta H^\circ = \) Change in Enthalpy at Ionic Strength 0

Debye–Hückel

The osmotic coefficient, \(A\), derived from literature measurements of the static dielectric constant and density of water (Clark and Glew, 1980)

$$A_m = 3 \left(-16.390 23 + \frac{261.337 1}{T} + 3.368 9633\ln T - 1.437 167\left(\frac{T}{100}\right) + 0.111 995 \left(\frac{T}{100}\right)^2 \right)$$

\(\gamma\) is the activity coefficient of each separate ionic species in the \(K_{ref}\) where:

$$\ln γ = \frac{-A_m \sqrt{I} z^2}{1 + B \sqrt{I}}$$

    \(I= \text{ionic strength} (\frac{mol}{L})\)
    \(B = 1.6 \frac{kg^{1/2}}{mol^{1/2}}\)
    \(z = \text{charge}\)
Using these equations, we can generate the activity coefficient of each separate ionic species within a chemical equilbrium and generate a constant \(\Gamma\):

$$\Gamma = \frac{\prod \gamma_{\text{products}_{I=\text{finite}}}} {\prod \gamma_{\text{reactants}_{I=\text{finite}}}}$$

The \(K_{ref}\) at Ionic Strength 0 is related by \(\Gamma\) to the measured \(K_{ref}\):

$$K_{\text{ref}_{I=O\text{, }T=\text{finite}}} = \Gamma K_{\text{ref}_{I=\text{finite, }T=\text{finite}}}$$

Constants for ATP Hydrolysis, Creatine Kinase, and Adenylate Kinase

Thermodynamic data required to adjust K` and Kref to temperature and ionic strength (Teague, Golding, Dobson 1996)
SymbolReactionEquilibrium Constant$$K_{\text{ref}}$$$$\Delta H{^\circ} (kJ mol^{-1})$$
$$K_{\text{a}_{\text{ATP}}}$$$$\text{HATP}^{3-} \leftrightarrow H{^+} + \text{ATP}^{4-}$$$$\frac{[H{^+}][\text{ATP}^{4-}]}{[\text{HATP}^{3-}]}$$2.512×10-8-6.30
$$K_{\text{b}_{\text{MgATP}}}$$$$\text{Mg}^{2+} + \text{ATP}^{4-} \leftrightarrow \text{MgATP}^{2-}$$$$\frac{[\text{MgATP}^{2-}]}{[\text{Mg}^{2+}][\text{ATP}^{4-}]}$$1.514×10622.90
$$K_{\text{b}_{\text{MgHATP}}}$$$$\text{Mg}^{2+} + \text{HATP}^{3-} \leftrightarrow \text{MgHATP}^{1-}$$$$\frac{[\text{MgHATP}^{1-}]}{[\text{Mg}^{2+}][\text{HATP}^{3-}]}$$4.266×10316.90
$$K_{\text{a}_{\text{ADP}}}$$$$\text{HADP}^{2-} \leftrightarrow H{^+} + \text{ADP}^{3-}$$$$\frac{[H{^+}][\text{ADP}^{3-}]}{[\text{HADP}^{2-}]}$$6.607×10-8-5.60
$$K_{\text{b}_{\text{MgADP}}}$$$$\text{Mg}^{2+} + \text{ADP}^{3-} \leftrightarrow \text{MgADP}^{1-}$$$$\frac{[\text{MgADP}^{1-}]}{[\text{Mg}^{2+}][\text{ADP}^{3-}]}$$4.466×10419.0
$$K_{\text{b}_{\text{MgHADP}}}$$$$\text{Mg}^{2+} + \text{HADP}^{2-} \leftrightarrow \text{MgHADP}$$$$\frac{[\text{MgHADP}]}{[\text{Mg}^{2+}][\text{HADP}^{2-}]}$$3.163×10212.50
$$K_{\text{a}_{\text{AMP}}}$$$$\text{HAMP}^{1-} \leftrightarrow H{^+} + \text{AMP}^{2-}$$$$\frac{[H{^+}][\text{AMP}^{2-}]}{[\text{HAMP}^{1-}]}$$1.862×10-7-5.40
$$K_{\text{b}_{\text{MgAMP}}}$$$$\text{Mg}^{2+} + \text{AMP}^{2-} \leftrightarrow \text{MgAMP}$$$$\frac{[\text{MgAMP}^{1-}]}{[\text{Mg}^{2+}][\text{AMP}^{2-}]}$$6.165×10211.30
$$K_{\text{a}_{\text{HPO}{_4}}}$$$$\text{H}{_2}\text{PO}{_4}^{1-} \leftrightarrow \text{H}{^+} + \text{HPO}{_4}^{2-}$$$$\frac {[\text{HPO}{_4}^{2-}][\text{H}^{+}]}{[\text{H}{_2}\text{PO}{_4}^{1-}]}$$6.026×10-83.60
$$K_{\text{b}_{\text{MgHPO}{_4}}}$$$$\text{Mg}^{2+} + \text{HPO}{_4}^{2-} \leftrightarrow \text{MgHPO}{_4}$$$$\frac{[\text{MgHPO}{_4}]}{[\text{Mg}^{2+}][\text{HPO}{_4}^{2-}]}$$5.128×10812.20
$$K_{\text{a}_{\text{PCr}}}$$$$\text{HPCr}^{1-} \leftrightarrow \text{H}{^+} + \text{PCr}^{2-}$$$$\frac{[\text{H}{^+}][\text{PCr}^{2-}]}{[\text{HPCr}^{1-}]}$$8.854×10-6 2.66
$$K_{\text{b}_{\text{MgPCr}}}$$$$\text{Mg}^{2+} + \text{PCr}^{2-} \leftrightarrow \text{MgPCr}$$$$\frac{[\text{MgPCr}]}{[\text{Mg}^{2+}][\text{PCr}^{2-}]}$$2.320×1028.19
$$K_{\text{eq}_{\text{Creatine Kinase}}}$$$$\text{PCr}^{2-} + \text{ADP}^{3-} + \text{H}^+ \leftrightarrow \text{ATP}^{4-} + \text{Cr} $$$$\frac{[\text{ATP}^{4-}][\text{Cr}]}{[\text{PCr}^{2-}][\text{ADP}^{3-}][\text{H}^+]}$$2.58×108-17.55
$$K_{\text{eq}_{\text{ATP Hydrolysis}}}$$$$\text{ATP}^{4-} + \text{H}_2\text{O} \leftrightarrow \text{ADP}^{3-} + \text{HPO}{_4}^{2-} + \text{H}^+$$$$\frac{[\text{ADP}^{3-}][\text{HPO}{_4}^{2-}][\text{H}^+]}{[\text{ATP}][\text{H}_2\text{O}]}$$2.946×10-1-20.50
$$K_{\text{eq}_{\text{Adenylate Kinase}}}$$$$2\text{ADP}^{3-} \leftrightarrow \text{ATP}^{4-} + \text{AMP}^{2-}$$$$\frac{[\text{ATP}^{4-}][\text{AMP}^{2-}]}{[\text{ADP}^{3-}]^2}$$2.248×10-1-1.50