Journal of Magnetic Resonance Open (Dec 2019)
Dissolution dynamic nuclear polarization NMR studies of enzyme kinetics: Setting up differential equations for fitting to spectral time courses
Abstract
ABSTRACT: Dissolution dynamic nuclear polarization (dDNP) provides strikingly increased sensitivity for detecting NMR-receptive nuclei in molecules that are substrates of enzymes and membrane transport proteins. This paves the way for studying the kinetics of many such catalyzed reactions on previously unattainable short time scales (seconds). Remarkably, this can also be carried out not only in vitro, but in whole cells, tissues, and even in vivo. The information obtained from the emergent NMR time courses is a sequence of spectral-peak intensities (integrals) as a function of time. Typically, for 13C NMR studies, these consist of a series of spectra acquired every 1 s for a total time span of ∼3 min.The time evolution of an excited spin system in molecules that undergo chemical transformation via enzyme-catalyzed reactions is described by the Bloch-McConnell differential rate equations. For these equations, the estimation of the values of apparent first order rate constants, or enzyme kinetic parameters like the maximal velocity (Vmax), Michaelis constant (Km), and dissociation inhibition constant (Ki), is handled by nonlinear regression analysis. A twist (compared with previous methods) lies in the fact that modern software like Mathematica can perform regression analysis without the need for an analytical solution of these equations to fit to the time course data. Regression algorithms exist that can directly use the numerical solution of the differential equations in the fitting process. Therefore, the main challenge for data analysis now resides in formulating the correct set of differential equations that emulate the system being studied. It is this important aspect of dDNP enzyme-kinetic analysis that we address here.We show that formulating the array of simultaneous differential rate equations follows the same approach as more conventional chemical kinetics. Namely, this requires rigorous application of two key principles: the principle of conservation of mass; and the principle of balance of physical dimensions (dimensional veracity) on either side of the equal sign in all equations in a simultaneous array. This mathematical synthesis requires extra-careful thought when formulating equations that describe enzyme-catalyzed reactions that are probed by dDNP; potential pitfalls are discussed as well.
