Nuclear magnetic resonance line-shape analysis and determination of exchange rates

The fact that chemical exchange processes occur at rates that cover a broad range and produce readily detectable effects on the spectrum is one of the attractive features of high-resolution NMR. The description of these line shapes in the presence of spin-spin coupling requires the density matrix theory which is rather complex. Analysis of the line shapes usually needs computer simulations and is capable of providing reliable information on the exchange rates as well as spectral parameters in the absence of exchange. Simplified procedures, ignoring spin-spin coupling, often result in deviations in these exchange and spectral parameters determined. A step-by-step procedure is detailed in this chapter for setting up the matrices required for computing the line shapes of exchanges involving weakly coupled spin systems on the basis of the density matrix theory without the need for a detailed understanding of the theory. A knowledge of the energy level structure and allowed transitions in the NMR spectra of the individual weakly coupled spin systems is all that is required. The procedure is amenable to numerical computation. The group of illustrative examples chosen to demonstrate the development of the computational tools cover some of the commonly encountered cases of exchange from simple systems to rather complex ones. Such exchanges occur frequently in biological molecules, especially those involving enzyme-substrate complexes. In cases where the experimental line shapes are obtained with respectable precision, and the relevant exchange processes are unambiguously identifiable, the computer simulation method of line-shape analysis is capable of providing useful and incisive information. The example of the 31P exchanges in the adenylate kinase is illustrative of this point. Not only has the line-shape analysis clearly indicated the role of the interchange process, but it has also produced evidence that the rates of interchange of the ADP molecules bound to the enzyme become relevant to the kinetics and mechanism of catalysis by this enzyme. It is probably difficult to obtain this information by any other experimental method or by any other method of analysis.