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.

Fig. 1

The results of the optimisation of both the R_ex Grid (RG) and the DMG using Modelfree4, Dasha and relax presented as difference surfaces. For each grid point the programs were used to optimise either model m4 or m5 for the RG and DMG, respectively. If the minimum has been found for each point the difference between the optimised and true parameter values should be zero. Positive and negative differences correspond to over and underestimation respectively. As a surface has been draped over the discrete differences, perfect optimisation should result in a flat surface of zero height. The top three plots correspond to the RG τ_e difference surfaces for the subset of the grid whereby S² = 0.952. The bottom three plots correspond to the DMG difference surfaces for the subset of the grid whereby The optimisation methods used are: Modelfree4, the Levenberg–Marquardt algorithm; Dasha, the combined Newton–Raphson/conjugate gradient algorithm; relax, Newton optimisation together with the backtracking line search, GMW Hessian modification, and the Method of Multipliers constraint algorithm

Fig. 2

The failure of the constraints algorithm in Modelfree4 demonstrated by minimisation terminating at the upper limit of the τ_e parameter of 10 ns. The figure is a map of the chi-squared space of model m4 which is composed of the parameters {S², τ_e, R_ex}. Different χ² values are demonstrated by the four isosurfaces which, from outermost to innermost, possess the values of 1, 0.5, 0.3 and 0.05. The relaxation data was generated by back calculation from the model-free parameter values of S² = 0.970, τ_e = 2048 ps and R_ex = 0.149 s⁻¹. As no noise was added the minimum for this model is located at this position. The four spheres in the plot correspond to the results as found by Modelfree4 using the Levenberg–Marquardt algorithm (black sphere), Dasha using either Levenberg–Marquardt or Newton–Raphson–CG minimisation (grey spheres), and relax using Newton minimisation together with the backtracking line search and the GMW Hessian modification (white sphere). The exact coordinates of the spheres are listed in Table S6 of the supplementary material

Fig. 3

The effect of optimisation precision on the final model-free parameter values. The deep and curved tunnel of the model-free space is illustrated by the four isosurfaces which correspond, from outermost to innermost, to chi-squared values of 50, 20, 5 and 0.5, respectively. The χ² space belongs to the grid point and τ_s = 32 ps. The white sphere corresponds to the true parameter values as well as the results of the program relax, the black sphere corresponds to the results of Modelfree4, and the two grey spheres are the results from Dasha using the two available optimisation algorithms. The parameter and chi-squared values of these positions are given in Table S7 of the supplementary material

Fig. 4

An example of optimisation failure in Modelfree4 caused by the bug in the Levenberg–Marquardt algorithm. The chi-squared space belongs to the grid point where the true parameter values are S² = 0.388, τ_e = 128 ps and R_ex = 0.223 s⁻¹. From outermost to innermost, the five isosurfaces illustrating the curvature of the space correspond to chi-squared values of 1371.79, 500, 100, 20 and 7. The true parameter values which were found by both relax and Dasha are indicated by the white sphere whereas the black sphere corresponds to the final parameter values found by Modelfree4 (S² = 0.263, τ_e = 526.316 ps and R_ex = 1.053 s⁻¹). The failure occurred in the first iteration hence the final parameter values are those of the grid point with the lowest chi-squared value (1371.79)

Fig. 5

A rare example of the double minima phenomenon existent in a space of a simplistic approximation to the true model. In this case the true model is tm5 where τ_m = 7 ns, and τ_s = 0.8 ns whereas the space in (a) is a map of model tm2. The back calculated relaxation data used to generate this plot consisted of data at 500 and 600 MHz. The four isosurfaces shown correspond, from outermost to innermost, to χ² values of 30, 22, 17 and 5 respectively. The resolution of the plot is 100 data points per dimension. The three chi-squared distributions shown in (b), corresponding to the two minima of model tm2 and the single minimum of model tm5, were generated by 500 randomisations of the original noise-free data assuming Gaussian noise. The chi-squared values were binned to increments of 2. These distributions demonstrate that experimental noise will cause most instances of model tm2 in which two minima are present to be eliminated by AIC model selection as model tm5 is selected instead

Fig. 6

Map of the chi-squared space of the spheroid diffusion tensor parameters of cytochrome c₂. This figure demonstrates the multiple minima of the space due to both the prolate and oblate approximations to the true ellipsoid diffusion tensor together with the glide reflection symmetry of the space. The glide reflection is most evident in the top right image where the subspace between is a duplication and mirror image reflection about θ of the subspace between To map this space all S² parameters and τ_m were fixed to the values of the minimised prolate spheroid. The χ² values of the four isosurfaces from outermost to innermost are 300, 200, 100 and 70, respectively (dark grey to white). The resolution of the plot is 100 data points per dimension