Non-Canonical Ligand-Binding Events as Detected by NMR - Recent Developments in Biomolecular NMR - page 238

d ½ P t

dt ~{k PQ ½ P t ½ L t {k PR ½ P t ½ L t zk QP ½ Q t zk RP ½ R t
d ½ Q t

dt ~{k QS ½ Q t ½ L t {k QP ½ Q t {k QR ½ Q t zk PQ ½ P t ½ L t zk SQ ½ S t zk RQ ½ R t
d ½ R t

dt ~{k RS ½ R t ½ L t {k RP ½ R t {k RQ ½ R t zk PR ½ P t ½ L t zk SR ½ S t zk QR ½ Q t
d ½ S t

dt ~{k SQ ½ S t {k SR ½ S t {k QS ½ Q t ½ L t zk RS ½ R t ½ L t
d ½ L t
dt ~{k PQ ½ P t ½ L t {k PR ½ P t ½ L t {k QS ½ Q t ½ L t
{k RS ½ R t ½ L t zk QP ½ Q t zk RP ½ R t zk SQ ½ S t zk SR ½ S t

ð 10 : 33 Þ

The following starting conditions were used:

½ L t~0 ~ ½ L total
½ P t~0 ~ ½ P total
½ Q t~0 ~0
½ R t~0 ~0
½ S t~0 ~0

ð 10 : 34 Þ

The equilibrium concentrations were obtained after 1 s of integration.
We carried out the numerical integration for different starting conditions,
given in terms of the known total [P] and total [L], and obtained the species
concentrations after equilibrium was reached. The equilibrium association
constants in this particular calculation were K PQ 5 K RS 5 3 6 10 7

M 21

(primary binding site) K PR 5 K QS 5 10 4 M 21 (secondary binding site).
The results in Figure 10.12 show that this scheme allows for disappearance
of species Q, because it is being displaced by species S. Hence, the scheme can
account for the hallmark effect in the experimental titrations: the bound Cd 2+
signal loses intensity when excess Cd 2+ is present. In order to also account for
the progressive increase in linewidth of the 'bound' resonance, we must
translate the 'species' scheme of Figure 10.11 into a 'site' scheme for the Cd 2+
ion as shown in Figure 10.13. The scheme in Figure 10.11 corresponds to five-
site chemical exchange.
In the simulations, we make the reasonable assumption that no direct
exchange can occur between the internal and external sites d and e in species S
because the sites are occupied. Further, we assume that the internal site b
cannot directly interchange with the external site e and that the external site c
cannot directly interchange with the internal site d, because those processes
would require coordinated rearrangement and binding/release of two Cd 2+
ions. We do allow that external Cd 2+ in species R (site c) can become
internalised in species Q (site b). While this internalisation process does not