Theory of the trNOESY Experiment

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At spectrometer frequencies of 500 to 600 MHz (Hv), small molecules (molecular weight < 1500) in the free state yield small, positive NOEs, whereas large protein molecules (molecular weight > 10,000) yield large, negative NOEs. In the fast-exchange regime (binding constant between 10-3 and 10-7 M), the ligand acquires the NOE characteristics of the large molecule during the reversible binding and shows large negative NOEs (Fig. 1). These characteristics are transferred from the bound state of the ligand to its free state, and, therefore, the ligand signals are still sharp owing to the rapid total rotational correlation time, Tc, of the free ligand. The ligand-binding event is thus identified by both the change in sign and the intensity buildup rate of its intramolecular NOEs.

The theory behind the mechanism of trNOE has been well developed over the years (24-29). The dynamics of the NOE is governed by the three species equilibrium processes given by k

koff in which [P], [L], and [PL] are the molar concentrations of the protein, the ligand, and the complex, respectively; and kon and koff are the association and dissociation constants, respectively. In the trNOE experiment, the exchange of the ligand between the free and bound states alters the relaxation dynamics of the ligand more significantly than of the protein. Under equilibrium conditions, the binding constant, KD, is the ratio of koff to kon. The exchange rate that is relevant to the NMR experiments, kex, depends on the relative populations of the protein and ligand as well as the binding constant and is defined as kex = kon[P] + koff = koff/(1 - Lb) (2)

in which Lb is the bound ligand fraction. For a single ligand-binding site, Lb is given by

Lb = |(Pr + Lt + Kd) - [V(PT + Lt + Kd)2 - 4PrLr]j/2Lr (3)

in which PT and LT are the total protein and ligand concentrations, respectively; and Kd is the binding constant. In the performance of trNOE experiments, it is useful to have the protein-binding site at least half-saturated (Lt ~ Kd). This is accomplished by using a large molar excess of the ligand of approx 5 to 50 times that of the protein. Exchange of the magnetization between the protein-bound form and the free form of the ligand produces an averaged NOE.

The exchange-averaged NOE depends on the rate of exchange and the magnitude of the NOE between the free (NOEy) and bound (NOEb) forms. The exchange rate, kex, is considered fast, intermediate, or slow if kex » [NOEy-NOEb], kex - [NOEy - NOEb], and kex « [NOEy- NOEb], respectively. In the fastexchange regime (kex » [NOEy - NOEb]), the trNOE experiments are extremely useful because the observed NOE is then a population weighted average.

The dynamic interplay between the ligand intramolecular NOEs and the ligand/protein intermolecular exchange NOEs can be described by combining Solomon's equations and chemical exchange equations (30-32). Using this approach, Clore and Gronenborn (25,26) described the observed effect of trNOE by using matrix notation. The combined matrix includes the pairwise interactions in a multiple-spin system undergoing exchange, which accounts for the spin diffusion effects as well (29,33,34). The evolution of the intensity in a two-dimensional (2D) trNOE experiment (trNOESY) is given by (30,32)

in which Tm is the mixing time, and the elements of the matrix V(Tm) are the measured peak volumes of the crosspeaks in the trNOESY spectrum, which are described in terms of the exchange-relaxation matrix r. The exchange rates (kex) as well as the self- (py) and cross-relaxation (Gy) rates of the various proton pairs are included in r. When the exchange is fast relative to the relaxation rates, the effective rate constants are molar fraction weighted averages of the rate constants of the free and bound forms. Thus, if i and j are the ligand spins, the effective cross-relaxation rate Gijavg is v

In this fast-exchange regime, as pointed out by Landy and Rao (35), the relaxation + exchange matrix, r, can be symmetrized, and, thus, the rate equation for the m-spin ligand and «-spin protein simplifies to an (n + m) differential equation (36). The relative concentrations of the ligand and protein, following Zabell and Post (37) and Eq. 2, can be written as

in which rb and rf represent the symmetrical n x n relaxation matrices of the ligand in the bound and free forms, respectively; and rpb and rpf have analogous definitions for the protein. The relative concentrations are defined by as follows:

Mbi = [PL]/([PL] + [L]) Mfi = [L]/([PL] + [L]) Mbp = [PL]/([PL] + [P]) f = [P]/([PL] + [P])

These equations are generally used in cases in which quantification of ligand crosspeaks in the trNOESY spectra is needed to determine the structure of the bound form of the ligand. This information is especially useful when combined with computational approaches in order to optimize models of protein/ligand complexes, which, in turn, are used to develop new ligands with higher affinity and specificity for the target site on the protein.

Fig. 3. trNOESY pulse sequence. Bars represent 90° radiofrequency pulses and FID stands for free induction decay. The pulses and the receiver are phase cycled to select the NOE and perform phase-sensitive detection along tx. WET, water suppression enhanced through T effects; DPFGE, double pulsed field gradient echo.

Fig. 3. trNOESY pulse sequence. Bars represent 90° radiofrequency pulses and FID stands for free induction decay. The pulses and the receiver are phase cycled to select the NOE and perform phase-sensitive detection along tx. WET, water suppression enhanced through T effects; DPFGE, double pulsed field gradient echo.

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