3.3 Three State
Binding Reactions
Any binding mechanism beyond one-step becomes very complex
very rapidly, and data from multistep mechanisms are challenging
to analyze. Microscopic rate constants from two different steps
(Schemes 2 and 3) will couple when they are of similar magnitude,
which happens at lower concentrations of B. Moreover, the magnitude of the kinetic amplitudes may be very different such that one is
small in relation to the other. Both of these scenarios can produce
ambiguous looking exponentials (Fig. 1c). For example, we consider a two-step binding (Schemes 2 and 3), which theoretically will
result in two kinetic phases where one increases linearly and the
other hyperbolically with [B] according to Eq. 4 (Scheme 2) or
Eq. 5 (Scheme 3), under pseudo-first-order conditions.
k obs1,2 ¼
k 1 B
½ þ k À1 þ k 2 þ k À2 Æ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k 1 B
½ þ k À1 þ k 2 þ k À2
ð
Þ
2 À 4 k 1 B
½ k 2 þ k À1 k À2 þ k 1 B
½ k À2
ð
Þ
q
2
ð4Þ
k obs1,2 ¼
k 1 þ k À1 þ k 2 B
½ þ k À2 Æ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
k 1 þ k À1 þ k 2 B
½ þ k À2
ð
Þ
2 À 4 k 1 k 2 B
½ þ k À1 k À2 þ k 1 k À2
ð
Þ
q
2
ð5Þ
What often happens in a real experiment is that in the time
window where we can measure kinetics, the kinetic phases are on
similar scales and one of the associated amplitudes might dominate
over the other. The resulting experimental trace will not be perfectly represented by a single exponential. Neither is the data good
enough to justify the use of a double exponential. A fit to a single
exponential will then yield k obs values intermediate between the two
real ones given by the respective theoretical kinetic phase. Generally, k obs values that differ by a factor of 3–4 or less result in
exponential transients, which may be very difficult to resolve independently unless there are favorable kinetic amplitudes. Therefore,
giving general advice on “borderline” curve fitting cases is particularly difficult. The best thing to do in such cases is to perform
experiments over a range of protein concentrations, fitting the
data for both single and double exponentials and plot k obs and
amplitudes versus [B] and, if possible, versus [A] as well (see Subheading 3.6.2 below). If the dependences of k obs and amplitudes
from the double exponential fit are in accordance with what can be
expected from a two-step binding, the fitted parameters are likely
sound. It may also be useful to change experimental conditions
such that a suspected intermediate is stabilized, for example, by
increasing the ionic strength or using sodium sulfate, 2,2,2trifluoroethanol or trimethylamine N-oxide (see Note 11).
3.4 Displacement
Experiments
to Determine k off
It is common that k off is not well determined in binding experiments by linear extrapolation to the y-axis (zero concentration of
B). This occurs when k off is small in relation to the lowest measured
k obs . The k off should then be determined separately in a
116
Elin Karlsson and Per Jemth
Binding Reactions
Any binding mechanism beyond one-step becomes very complex
very rapidly, and data from multistep mechanisms are challenging
to analyze. Microscopic rate constants from two different steps
(Schemes 2 and 3) will couple when they are of similar magnitude,
which happens at lower concentrations of B. Moreover, the magnitude of the kinetic amplitudes may be very different such that one is
small in relation to the other. Both of these scenarios can produce
ambiguous looking exponentials (Fig. 1c). For example, we consider a two-step binding (Schemes 2 and 3), which theoretically will
result in two kinetic phases where one increases linearly and the
other hyperbolically with [B] according to Eq. 4 (Scheme 2) or
Eq. 5 (Scheme 3), under pseudo-first-order conditions.
k obs1,2 ¼
k 1 B
½ þ k À1 þ k 2 þ k À2 Æ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
k 1 B
½ þ k À1 þ k 2 þ k À2
ð
Þ
2 À 4 k 1 B
½ k 2 þ k À1 k À2 þ k 1 B
½ k À2
ð
Þ
q
2
ð4Þ
k obs1,2 ¼
k 1 þ k À1 þ k 2 B
½ þ k À2 Æ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
k 1 þ k À1 þ k 2 B
½ þ k À2
ð
Þ
2 À 4 k 1 k 2 B
½ þ k À1 k À2 þ k 1 k À2
ð
Þ
q
2
ð5Þ
What often happens in a real experiment is that in the time
window where we can measure kinetics, the kinetic phases are on
similar scales and one of the associated amplitudes might dominate
over the other. The resulting experimental trace will not be perfectly represented by a single exponential. Neither is the data good
enough to justify the use of a double exponential. A fit to a single
exponential will then yield k obs values intermediate between the two
real ones given by the respective theoretical kinetic phase. Generally, k obs values that differ by a factor of 3–4 or less result in
exponential transients, which may be very difficult to resolve independently unless there are favorable kinetic amplitudes. Therefore,
giving general advice on “borderline” curve fitting cases is particularly difficult. The best thing to do in such cases is to perform
experiments over a range of protein concentrations, fitting the
data for both single and double exponentials and plot k obs and
amplitudes versus [B] and, if possible, versus [A] as well (see Subheading 3.6.2 below). If the dependences of k obs and amplitudes
from the double exponential fit are in accordance with what can be
expected from a two-step binding, the fitted parameters are likely
sound. It may also be useful to change experimental conditions
such that a suspected intermediate is stabilized, for example, by
increasing the ionic strength or using sodium sulfate, 2,2,2trifluoroethanol or trimethylamine N-oxide (see Note 11).
3.4 Displacement
Experiments
to Determine k off
It is common that k off is not well determined in binding experiments by linear extrapolation to the y-axis (zero concentration of
B). This occurs when k off is small in relation to the lowest measured
k obs . The k off should then be determined separately in a
116
Elin Karlsson and Per Jemth