Optimisation of NMR dynamic models I. Minimisation algorithms and their performance within the model-free and Brownian rotational diffusion spaces

Abstract

The key to obtaining the model-free description of the dynamics of a macromolecule is the optimisation of the model-free and Brownian rotational diffusion parameters using the collected R 1 , R 2 and steady-state NOE relaxation data. The problem of optimising the chi-squared value is often assumed to be trivial, however, the long chain of dependencies required for its calculation complicates the model-free chi-squared space. Convolutions are induced by the Lorentzian form of the spectral density functions, the linear recombinations of certain spectral density values to obtain the relaxation rates, the calculation of the NOE using the ratio of two of these rates, and finally the quadratic form of the chi-squared equation itself. Two major topological features of the model-free space complicate optimisation. The first is a long, shallow valley which commences at infinite correlation times and gradually approaches the minimum. The most severe convolution occurs for motions on two timescales in which the minimum is often located at the end of a long, deep, curved tunnel or multidimensional valley through the space. A large number of optimisation algorithms will be investigated and their performance compared to determine which techniques are suitable for use in model-free analysis. Local optimisation algorithms will be shown to be sufficient for minimisation not only within the model-free space but also for the minimisation of the Brownian rotational diffusion tensor. In addition the performance of the programs Modelfree and Dasha are investigated. A number of model-free optimisation failures were identified: the inability to slide along the limits, the singular matrix failure of the Levenberg-Marquardt minimisation algorithm, the low precision of both programs, and a bug in Modelfree. Significantly, the singular matrix failure of the Levenberg-Marquardt algorithm occurs when internal correlation times are undefined and is greatly amplified in model-free analysis by both the grid search and constraint algorithms. The program relax (http://www.nmr-relax.com) is also presented as a new software package designed for the analysis of macromolecular dynamics through the use of NMR relaxation data and which alleviates all of the problems inherent within model-free analysis.

Dynamics of the Glycosidic Bond: Conformational Space of Lactose

“…was applied, in which D ijl is the dipolar coupling constant in Hertz in alignment condition i and m T jlc is the unit vector between spins j and l in conformation c. The formulae and analyses were implemented in the program relax [24] as described in the Supporting Information.…”

Bijvoet in Solution Reveals Unexpected Stereoselectivity in a Michael Addition

“…[31] The full ROE and RDC analysis was implemented in the program relax. [32,33] ORD analysis: The ten structures of the best ensemble of (R,R,R)-3 and (R,S,S)-3 were used as starting structures for DFT geometry optimization using Gaussian03 Revision C.02. [34] The optimizations were performed at the B3LYP/6-31G(d) level of theory.…”

Conservation of Flexible Residue Clusters among Structural and Functional Enzyme Homologues

“…The motional parameters on the faster ps-ns time scale were analyzed by fitting the NMR spin relaxation rates to the model-free formalism (reviewed in Ref. 37), using the methodology for the dual optimization of the model-free parameters and the global diffusion tensor proposed by d'Auvergne (38) and implemented in the program relax 2.1.0 (38,39). The crystallographic structure of free ECP was used as starting coordinates (PDB entry 1QMT), and hydrogen atoms were added in DS Visualizer 3.5 (Accelrys).…”

“…However, the average value of S 2 ϭ 0.740 Ϯ 0.052 indicates that ECP is a bit more flexible than the very rigid RNase A on the ps-ns time scale, which displays an average S 2 ϭ 0.910 Ϯ 0.051 (35). Interestingly, the catalytic residue Lys 38 in ECP, which is strictly conserved among pancreatic RNases, displays one of the highest order parameters in the protein (S 2 ϭ 0.848 Ϯ 0.026), along with its very rigid neighbor Asn 39 (S 2 ϭ 0.907 Ϯ 0.055). Lys 38 is thought to be involved in the stabilization of the excess negative charge on the phosphoryl oxygens in the transition state during RNA cleavage (15), which may require rigidity in ECP.…”