User`s manual

18 Chapter 3: Theory of Operation
Calorimetry Sciences Corp.
CSC 5300 N-ITC III 19
User’s Manual
which can then be substituted into Equation 3-1 or Equation 3-2. Unfortunately, Equation
3-6 is given in terms of the concentration of free ligand, [X], whereas the quantity we
control in the experiment is the total concentration of ligand, [X]
tot
. We thus need to
express Equation 3-6 in terms of the total ligand concentration.
Again, rearranging Equation 3-3 and substituting, the total concentration of ligand is
given as:
Combining Equation 3-7 with Equation 3-4 gives a quadratic equation which can be
solved to express [X] in terms of the total concentrations of ligand and macromolecule:
This can then be substituted into Equation 3-6 to give an analytical expression for q
i
or
Q
i
for each injection. If we have multiple sites which are identical and independent we
simple multiply [M]
tot
(Equation 3-4) by N, the number of sites.
Notice that the value of N is equal to the exponent of the concentration of the free
ligand in Equation 3-3. In a system where the binding ratio is one to one (i.e., N=1) the
exponent is implied, but expressing the binding reaction in general terms for any value of
N gives:
Hence, the equilibrium constant for the overall reaction is:
When N=1, the binding constant, K, is equal to the equilibrium constant, K
eq
. Owing to
this relationship it should be noted that even if the known stoichiometry is one binding
[ ] [ ] [ ] [ ] [ ]( )
MK1XMXXX
tot
+=+=
Equation 3-7
[ ]
[ ] [ ]
( )
[ ] [ ]
( )( )
[ ]
2K
X4KXMK1XMK1
X
tot
2
tottottot
tot
+−++−−−
=
Equation 3-8
N
MX NX M =+
Equation 3-9
[ ]
[ ][ ]
N
N
eq
XM
MX
K =
Equation 3-10