Singular Value Decomposition Method to Determine Distance Distributions in Pulsed Dipolar Electron Spin Resonance

Abstract

Regularization is often utilized to elicit the desired physical results from experimental data. The recent development of a denoising procedure yielding about 2 orders of magnitude in improvement in SNR obviates the need for regularization, which achieves a compromise between canceling effects of noise and obtaining an estimate of the desired physical results. We show how singular value decomposition (SVD) can be employed directly on the denoised data, using pulse dipolar electron spin resonance experiments as an example. Such experiments are useful in measuring distances and their distributions, P(r) between spin labels on proteins. In noise-free model cases exact results are obtained, but even a small amount of noise (e.g., SNR = 850 after denoising) corrupts the solution. We develop criteria that precisely determine an optimum approximate solution, which can readily be automated. This method is applicable to any signal that is currently processed with regularization of its SVD analysis.

Figure 1

Model Data: The exact solution using the SVD method; P(r) vs number of singular value contributions (SVCs). Unimodal Distance Distribution: (A1) Noise-free dipolar signal. Comparison of model distribution with P(r) generated from (B1) fewer SVCs i = 3, σ_i = 8.04; (C1) exact solution obtained for i = 83, σ_i = 4.8 × 10⁻⁶; (D1) more SVCs i = 85, σ_i = 10⁻⁹; (E1) Piccard plot of ${\mathit{log}}_{10}({\mathrm{\sum }}_i{\mid u_i^{T}S/{\sigma }_i\mid }^2)$ vs the number of singular values from i = 1 to 200 starting from largest value; it shows the contributions of singular values that lead to stable and unstable distributions. Bimodal Distance Distribution: (A2) noise-free dipolar signal. Comparison of model distribution with P(r) generated from (B2) fewer SVCs i = 3, σ_i = 8.04; (C2) exact solution obtained for i = 82, σ_i = 5.1 × 10⁻⁵; (D2) more SVCs i = 84, σ_i = 10⁻⁷; (E2) Piccard plot of ${\mathit{log}}_{10}({\mathrm{\sum }}_i{\mid u_i^{T}S/{\sigma }_i\mid }^2)$ vs the number of singular values from i = 1 to 200 starting from largest value; it shows the contributions of singular values that lead to stable and unstable distributions.

Figure 2

Piccard plots of ${\mathit{log}}_{10}({\mathrm{\sum }}_i{\mid u_i^{T}S/{\sigma }_i\mid }^2)$ comparing noise-free and some noise (SNR = 850) cases for different distance segments. Case 1 is unimodal and Case 2 is the bimodal model. (A) Piccard plots for noise-free vs some noise. (B) Comparison of distance (r) dependent Piccard plots at 3.2 nm, 5 nm, and 8.1 nm for noise-free models. (C) Same as B, except for models with some noise. Comparison of distance (r) dependent Piccard plots for noise-free vs some noise cases for (D) 3.2 nm, (E) 5 nm, and (F) 8.1 nm.

Figure 3

Reconstruction of distance distribution for noise-free model data and noisy model data (SNR ≈ 850) using the new SVD method. (A) Model dipolar signal; (B) model dipolar signal with added noise (see added noise in red plot); (C) singular value cutoff at each distance (nm) for the model dipolar signal; (D) singular value cutoff at each distance (nm) for the model dipolar signal with added noise; and (E) Distance distribution reconstructed from the model dipolar signal and model dipolar signal with noise using the singular value cut-offs shown in C and D, respectively. Note that the added noise is so small that A and B still appear identical, but convergence to the virtually identical final results requires segmentation in the latter case.

Figure 4

Reconstruction of distance distribution using Tikhonov regularization + Maximum entropy method (TIKR+MEM) and new SVD method for noisy and denoised (WavPDS) experimental dipolar signal. (A1) Noisy experimental dipolar signal from spin labeled IgE cross-linked with DNA-DNP ligand in PBS buffer solution. The dipolar signal was collected after 18 h of signal averaging and has SNR = 3.8; (B1) distance distribution obtained from noisy dipolar signal using TIKR+MEM; (C1) distance distribution obtained from noisy dipolar signal using new SVD method; (A2) Denoised experimental dipolar signal using WavPDS; (B2) distance distribution obtained from denoised dipolar signal using TIKR+MEM; (C2) distance distribution obtained from denoised dipolar signal using new SVD method.

Scheme 1

Block Diagram Showing the Determination of Distance Distributions from Pulsed Dipolar Spectroscopy Using (A) Tikhonov Regularization and (B) the New Singular Value Decomposition Method

Scheme 2

Block Diagram Showing Generation of Probability Values at Each Distance Measurement^{a a}For practical analyses, it is sufficient to consider just 3 or 4 ranges (cf. Figures 3 and S8.