Geometric Approximation: A New Computational Approach To Characterize Protein Dynamics from NMR Adiabatic Relaxation Dispersion Experiments

Abstract

A new computational strategy is reported that provides a fast approximation of numerical solutions of differential equations in general. The method is demonstrated with the analysis of NMR adiabatic relaxation dispersion experiments to reveal biomolecular dynamics. When an analytical solution to the theoretical equations describing a physical process is not available, the new approach can significantly accelerate the computational speed of the conventional numerical integration up to 10(5) times. NMR adiabatic relaxation dispersion experiments enhanced with optimized proton-decoupled pulse sequences, although extremely powerful, have previously been refractory to quantitative analysis. Both simulations and experimental validation demonstrate detectable "slow" (microsecond to millisecond) conformational exchange rates from 10(2) to 10(5) s(-1). This greatly expanded time-scale range enables the characterization of a wide array of conformational fluctuations for individual residues, which correlate with biomolecular function and were previously inaccessible. Moreover, the new computational method can be potentially generalized for analysis of new types of relaxation dispersion experiments to characterize the various dynamics of biomolecular systems.

General expressions for R1 relaxation for N-site chemical exchange and the special case of linear chains

“…Exact numerical solutions for R 1 ρ can be obtained by numerical integration of the Bloch-McConnell differential equation or by calculation of the largest (least negative) real eigenvalue of the Bloch-McConnell evolution matrix and elegant fast numerical algorithms have been described [9]. However, analytic expressions for R 1 ρ facilitate understanding of the relationship between model parameters and the phenomenological relaxation rate constant and can lead to new methodological advances.…”

“…The general solution obtained from this approach is valid only if one of the N populations is highly dominant. While most treatments of R 1 ρ have assumed a constant-amplitude rf field, the theoretical and experimental results have been extended to time-varying adiabatic sweeps of the rf field [9, 15–18]. …”

Application of geometric approximation to the CPMG experiment: Two- and three-site exchange

“…Recently, a new computational strategy, geometric approximation 17 , was developed to analyze the sophisticated data from adiabatic relaxation dispersion experiments 18 . The strategy is composed of several steps (Fig.…”

“…1): 1) to formulate a computable mathematical problem based on a well-established physical theory, 2) to analyze the geometric features of the solution surface, under the assumption of continuity, using a high-performance computing system, and to save the geometric features in a library, and 3) to approximate the solution surface, based on the library, during data analysis in personal computers. This new strategy allows accurate data analysis in any system, within a given domain, with minimal computational power 17 . The generality of the new computational approach implies the potential broader application of this technique to other problems in data analysis.…”

“…The concepts of the geometric approximation methodology have been proposed and validated previously for the analysis of adiabatic relaxation dispersion experiments 17 . Herein, we explore the geometric features of the solution surfaces for the CPMG experiment acting on both the simple 2-site exchange model and a linear 3-site exchange model.…”