AMPK Methods and Protocols

it is then straightforward to calculate Gibbs free energy (ΔG) and the
entropy (ΔS). The following relationships are required to determine
these parameters. For a single binding site, the binding constant,Ka,
is given by:

Ka¼

Θ

ð 1 ΘÞ½XŠ

ð 1 Þ

whereΘis the fraction of sites occupied by ligandXand[X]is the
free concentration of ligand. The total concentration of ligandXtis
represented by:

Xt¼½ŠþX nΘPt ð 2 Þ

wherePtis the total concentration of protein in the cell. The total
heat content (Q) of the volume of solution contained in the cell
(V 0 ) at fractional saturation is defined by the following equation:

Q¼nΘPtΔHV 0 ð 3 Þ

Combining eqs. (1) and (2) results in a quadratic equation that
can be solved and substituted into eq. (3) to give the following:

Q¼

nPtΔHV 0

2

1 þ

Xt

nPt

þ

1

nKaPt



ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þnPXttþnK^1 aPt

 2

 4 Xt

nPt

vu

u

t

2

6

6

4

3

7

7

5

ð 4 Þ
The value ofQcan now be calculated as a function ofKa,n, and
ΔHas the other parameters are all known. However, this value of
Qonly applies forV 0 (initial volume), and therefore a correction
(Vi) must be made after each injection. The change in heat after
each injection is now defined as:

ΔQiðÞ¼QiðÞþ

dVi

Vo

QiðÞþQiðÞ 1

2

QiðÞ 1



ð 5 Þ

Equation (5) can now be used to obtain best fit values forKa,n,
andΔHby standard Marquardt methods, using the commercial
software of the instrument, until no further improvement of fit
occurs. The dissociation constant,Kd, is obtained from the inverse
ofKa. The Gibbs free energy (ΔG^0 ) can be obtained from the
following thermodynamic equation:

ΔG^0 ¼ΔH^0 TΔS^0 ¼RTlnKd ð 6 Þ

where R is the gas constant (1.987 cal.K^1 .mol^1 ) and T is
temperature.

A range of NMR experiments are available for monitoring
protein conformational change on a range of timescales [7]. Such
experiments can also probe the kinetics of ligand binding. Usually

Carbohydrate Binding Kinetics of theβ-Subunit CBM 89