多维谱的简化探测

提出一种利用谱编辑的方法来简化二维谱的探测,使得一张二维谱可以被简化为N条一维谱(N等于二维谱中F1维上的谱线总数),从而达到缩短数据采集时间和减少数据贮存空间的目的。文中给出了运用此方法探测二维碳氢异核相关实验的实例,并分析了由于F1维的谱线不为δ函数而产生的编辑误差的大小,最后将此方法推广到三维的简化探测,即通过对N张二维谱的编辑获得三维谱的信息。     

Restricted linear least square treatment processing of heteronuclear spectra of biomolecules using the ANAFOR strategy

We explore the use of a processing procedure based on restricted least square minimization as a tool for reducing the time versus resolution dilemma often encountered for biomolecular multidimensional spectra. Using a 2D spectrum as a reference, we obtain the necessary input of frequency components and linewidths. Combined even with a limited time evolution in the indirect dimension, the amplitudes of the correlation peaks in all planes of the 3D spectra can be extracted, and can be used to reconstruct the interferograms in the third dimension. Parameters such as number of lines, threshold choice, resolution, lineshape, number of experimental data points and finally signal to noise ratio of the spectrum are examined starting from a triple-resonance HNCA spectrum of ubiquitin.     

Achievable accuracy of parameter estimation for multidimensional NMR experiments

A fundamental issue in NMR spectroscopy is the estimation of parameters such as the Larmor frequencies of nuclei, J coupling constants, and relaxation rates. The Cramer-Rao lower bound provides a method to assess the best achievable accuracy of parameter estimates resulting from an unbiased estimation procedure. We show how the Cramer-Rao lower bound can be calculated for data obtained from multidimensional NMR experiments. The Cramer-Rao lower bound is compared to the variance of parameter estimates for simulated data using a least-squares estimation procedure. It is also shown how our results on the Cramer-Rao lower bound can be used to analyze whether an experimental design can be improved to provide experimental data which can result in parameter estimates with higher accuracy. The concept of nonuniform averaging in the indirect dimension is introduced and studied in connection with nonuniform sampling of the data.     

3D NMR spectroscopy for resonance assignment and structure elucidation of proteins under MAS: novel pulse schemes and sensitivity considerations

Two types of 3D MAS NMR experiments are introduced, which combine standard (NC,CC) transfer schemes with (1H,1H) mixing to simultaneously detect connectivities and structural constraints of uniformly 15N,13C-labeled proteins with high spectral resolution. The homonuclear CCHHC and CCC experiments are recorded with one double-quantum evolution dimension in order to avoid a cubic diagonal in the spectrum. Depending on the second transfer step, spin systems or proton-proton contacts can be determined with reduced spectral overlap. The heteronuclear NHHCC experiment encodes NH-HC proton-proton interactions, which are indicative for the backbone conformation of the protein. The third dimension facilitates the identification of the amino acid spin system. Experimental results on U-[15N,13C]valine and U-[15N,13C]ubiquitin demonstrate their usefulness for resonance assignments and for the determination of structural constraints. Furthermore, we give a detailed analysis of alternative multidimensional sampling schemes and their effect on sensitivity and resolution.     

Spectral estimation of NMR relaxation

In this paper, spectral estimation of NMR relaxation is constructed as an extension of Fourier Transform (FT) theory as it is practiced in NMR or MRI, where multidimensional FT theory is used. nD NMR strives to separate overlapping resonances, so the treatment given here deals primarily with monoexponential decay. In the domain of real error, it is shown how optimal estimation based on prior knowledge can be derived. Assuming small Gaussian error, the estimation variance and bias are derived. Minimum bias and minimum variance are shown to be contradictory experimental design objectives. The analytical continuation of spectral estimation is constructed in an optimal manner. An important property of spectral estimation is that it is phase invariant. Hence, hypercomplex data storage is unnecessary. It is shown that, under reasonable assumptions, spectral estimation is unbiased in the context of complex error and its variance is reduced because the modulus of the whole signal is used. Because of phase invariance, the labor of phasing and any error due to imperfect phase can be avoided. A comparison of spectral estimation with nonlinear least squares (NLS) estimation is made analytically and with numerical examples. Compared to conventional sampling for NLS estimation, spectral estimation would typically provide estimation values of comparable precision in one-quarter to one-tenth of the spectrometer time when S/N is high. When S/N is low, the time saved can be used for signal averaging at the sampled points to give better precision. NLS typically provides one estimate at a time, whereas spectral estimation is inherently parallel. The frequency dimensions of conventional nD FT NMR may be denoted D₁, D₂, etc. As an extension of nD FT NMR, one can view spectral estimation of NMR relaxation as an extension into the zeroth dimension. In nD NMR, the information content of a spectrum can be extracted as a set of n-tuples (ω₁, … ω_n), corresponding to the peak maxima. Spectral estimation of NMR relaxation allows this information content to be extended to a set of (n + 1)-tuples (λ, ω₁, … ω_n), where λ is the relaxation rate.     

Universal Temporal Dependence of Survival Probability for Diffusing Particles in Multidimensional Media with Absorbing Traps in an Electric Field

The effect of electric-field-induced reduction of the “effective” dimension is detected in the multidimensional problem of diffusion in a medium with absorbing traps. An effective transition to the 1D case occurs due to directional motion of a particle when it begins to dominate over diffusion. A universal “1D” dependence of the particle survival probability in a multidimensional medium with absorbing traps over long times has been established.     

Randomization improves sparse sampling in multidimensional NMR

While a number of strategies have been developed to reduce data collection requirements for multidimensional NMR based on non-Fourier methods of spectrum analysis, there is an increasing awareness that the principal differences in the performance of these methods is attributable to the sampling strategies employed, and not the method of spectrum analysis per se. The ability of maximum entropy reconstruction to utilize essentially arbitrary sampling schemes makes it a useful platform for comparative analysis of sampling strategies. Here we use maximum entropy reconstruction to demonstrate that artifacts characteristic of sparse sampling result from regularity in the sampling pattern, and that they can be substantially reduced by introducing a degree of randomness to an otherwise regular sampling scheme, without requiring additional sampling.     

Signal identification in NMR spectra with coupled evolution periods

Novel multidimensional NMR experiments rely on modified time-domain sampling schemes to provide significant savings of experimental time. Several approaches are based on the coupling of evolution times resulting in a reduction of the dimensionality of the recorded spectra, and a concomitant saving of experimental time. We present a consistent and general tool, called EVOCOUP, for the analysis of these reduced dimensionality spectra. The approach is flexible in the sense that the input can consist of various forms of reduced dimensionality spectra, that any piece of information can be removed (provided enough information is left), e.g., signals undetectable due to poor signal-to-noise or covered by artifacts, and that it can be applied to spectra involving any number of nuclei. The use of a general optimization procedure and an appropriate target function provides for a robust approach with well-defined results and ensures optimal use of redundant information normally present in the input. Spectral overlap in the directly detected dimension is resolved in a fully automated manner, avoiding the assessment of signal quality and its use in combinatorial trials. The positions of all peaks in a corresponding full-dimensional spectrum are obtained without need for reconstruction of this spectrum. In a systematic analysis of a complete spectrum recorded for the 14 kDa protein azurin and involving five different nuclei, only four spin systems were missed and no false spins systems were detected.     

Signal enhancement for the sensitivity-limited solid state NMR experiments using a continuous, non-uniform acquisition scheme

We describe a sampling scheme for the two-dimensional (2D) solid state NMR experiments, which can be readily applied to the sensitivity-limited samples. The sampling scheme utilizes continuous, non-uniform sampling profile for the indirect dimension, i.e. the acquisition number decreases as a function of the evolution time (t1) in the indirect dimension. For a beta amyloid (Aβ) fibril sample, we observed overall 40-50% signal enhancement by measuring the cross peak volume, while the cross peak linewidths remained comparable to the linewidths obtained by regular sampling and processing strategies. Both the linear and Gaussian decay functions for the acquisition numbers result in similar percentage of increment in signal. In addition, we demonstrated that this sampling approach can be applied with different dipolar recoupling approaches such as radiofrequency assisted diffusion (RAD) and finite-pulse radio-frequency-driven recoupling (fpRFDR). This sampling scheme is especially suitable for the sensitivity-limited samples which require long signal averaging for each t1 point, for instance the biological membrane proteins where only a small fraction of the sample is isotopically labeled.     

The Multidimensional Filter Diagonalization Method: I. Theory and Numerical Implementation

The theory and numerical aspects of the recently developed multidimensional version of the filter diagonalization method (FDM) are described in detail. FDM can construct various “ersatz” or “hybrid” spectra from multidimensional time signals. Spectral resolution is not limited by the time-frequency uncertainty principle in each separate frequency dimension, but rather by the total joint information content of the signal, i.e., N_total = N₁ × N₂ × × N_D, where some of the interferometric dimensions do not have to be represented by more than a few (e.g., two) time increments. It is shown that FDM can be used to compute various reduced-dimensionality projections of a high-dimensional spectrum directly, i.e., avoiding construction of the latter. A subsequent paper (J. Magn. Reson. 144, 357–366 (2000)) is concerned with applications of the method to 2D, 3D, and 4D NMR experiments.     

基于近似l0范数最小化的NMR波谱稀疏重建算法

在核磁共振（NMR）波谱中,过长的数据采集时间会使很多化学以及分子生物学领域的高分辨率多维谱应用难以实现. 传统的解决办法是使用随机非均匀采样代替奈奎斯特采样,但这样会使谱图质量受损. 压缩传感的出现为此提供了更好的解决办法,合适的压缩传感重建算法可以通过很少的随机非均匀采样将谱图高质量的重建出来. 该文先介绍了一种可用于谱图重建的压缩传感重建算法,名为“平滑l₀范数最小化法”,然后针对该算法对采样噪声鲁棒性较差的缺点进行了改进. 通过将改进后的算法与原算法在一维实数域信号以及NMR波谱信号重建实验中进行对比后表明,改进后的算法对噪声的鲁棒性明显提高,并能获得更好的重建性能.     

Homogeneous FID signals in polypyrroles:T 1-estimate from a two-pulse sequence in homogeneous systems

Several experiments using pulse electron spin resonance (ESR) equipment reveal that homogeneous free induction decay (FID) signals are given by electron spins in polypyrrole (PPy). FID signals obtained from PPy are accurately fitted via single-exponential curves, thus indicating that PPy can be used as a standard sample for several experiments with pulse ESR. We particularly pay attention to the nutation phenomena resulting from two-pulse irradiation (θ-t−2θ-t) in the homogeneous systems. The analysis by the vector model suggests that the nutation curves are affected by spin-lattice relaxation. Such a tendency is actually observed for two types of PPy used as examples ofT ₁ >T ₂ andT ₁ ≈T ₂. Thus the fitting over the nutation curve can be utilized for estimatingT ₁. We particularly point out that such a procedure can be advantageously performed for electron spins with a short spin-lattice relaxation time.     

Centric scan SPRITE for spin density imaging of short relaxation time porous materials

The single-point ramped imaging with T1 enhancement (SPRITE) imaging technique has proven to be a very robust and flexible method for the study of a wide range of systems with short signal lifetimes. As a pure phase encoding technique, SPRITE is largely immune to image distortions generated by susceptibility variations, chemical shift and paramagnetic impurities. In addition, it avoids the line width restrictions on resolution common to time-based sampling, frequency encoding methods. The standard SPRITE technique is however a longitudinal steady-state imaging method; the image intensity is related to the longitudinal steady state, which not only decreases the signal-to-noise ratio, but also introduces many parameters into the image signal equation. A centric scan strategy for SPRITE removes the longitudinal steady state from the image intensity equation and increases the inherent image intensity. Two centric scan SPRITE methods, that is, Spiral-SPRITE and Conical-SPRITE, with fast acquisition and greatly reduced gradient duty cycle, are outlined. Multiple free induction decay (FID) points may be acquired during SPRITE sampling for signal averaging to increase signal-to-noise ratio or for T2* and spin density mapping without an increase in acquisition time. Experimental results show that most porous sedimentary rock and concrete samples have a single exponential T2* decay due to susceptibility difference-induced field distortion. Inhomogeneous broadening thus dominates, which suggests that spin density imaging can be easily obtained by SPRITE.     

Multidimensional Particle Size Distributions and Their Application to Nonspherical Particle Systems in Two Dimensions

Particle science and technology evolve toward ever increasing complexity with respect to the multidimensional particle properties of size, shape, surface, internal structure, and composition. In this study, the theoretical background is elaborated for multidimensional particle size distributions (PSDs) by transferring the concepts known from 1D size distributions to anisotropic particles comprising at least two different length dimensions, e.g., nanorods and platelets. After introducing 2D PSDs, the calculation of differently weighted probability density functions including their interconversion is presented. This is necessary in order to compare data resulting from different measurement techniques which probe different physical properties and thus provide differently weighted PSDs. In addition, it is shown how 1D distributions with reduced content of information can be deduced from 2D PSDs. As a proof‐of‐concept and for illustration purposes, this approach is applied to a 2D Gaussian size distribution. Furthermore, a generalized scheme is suggested which outlines the conversion of number, surface, and volume weighted densities within the 2D space. The application of these methods to the more general n‐dimensional case is straightforward.     

Q-shear transformation for MQMAS and STMAS NMR spectra

The multiple-quantum magic-angle spinning (MQMAS) and satellite-transition magic-angle spinning (STMAS) experiments refocus second-order quadrupolar broadening of half-integer quadrupolar spins in the form of two-dimensional experiments. Isotropic shearing is usually applied along the indirect dimension of the 2D spectra such that an isotropic projection free of anisotropic quadrupolar broadening can be obtained. An alternative shear transformation by a factor equal to the coherence level (quantum number) selected during the evolution period is proposed. Such a transformation eliminates chemical shift along the indirect dimension leaving only the second-order quadrupolar-induced shift and anisotropic broadening, and is expected to be particularly useful for disordered systems. This transformation, dubbed Q-shearing, can help avoid aliasing problems due to large chemical shift ranges and spinning sidebands. It can also be used as an intermediate step to the isotropic representation for expanding the spectral window of rotor-synchronized experiments.     

Fast multi-dimensional NMR by minimal sampling

A new scheme is proposed for very fast acquisition of three-dimensional NMR spectra based on minimal sampling, instead of the customary step-wise exploration of all of evolution space. The method relies on prior experiments to determine accurate values for the evolving frequencies and intensities from the two-dimensional 'first planes' recorded by setting t1=0 or t2=0. With this prior knowledge, the entire three-dimensional spectrum can be reconstructed by an additional measurement of the response at a single location (t1( *),t2( *)) where t1( *) and t2( *) are fixed values of the evolution times. A key feature is the ability to resolve problems of overlap in the acquisition dimension. Applied to a small protein, agitoxin, the three-dimensional HNCO spectrum is obtained 35 times faster than systematic Cartesian sampling of the evolution domain. The extension to multi-dimensional spectroscopy is outlined.     

Vector model of electron spin echo envelope modulation due to nuclear hyperfine and zeeman interactions

The transverse electron spin magnetization of a paramagnetic center with effective spinS=1/2 interacting with nonquadrupolar nuclei may be presented as a function of pairs of nuclei magnetization vectors which precess around the effective magnetic field directions. Each vector of the pair starts its precession perpendicular to both effective fields. The free induction decay (FID) signal is proportional to the scalar product of the vectors for nuclear spinI=1/2. The electron spin echo (ESE) signal can be described with two pairs of magnetization vectors. The ESE shape is not equal to two back-to-back FID signals except in the absence of ESE envelope modulation. A recursion relation is obtained which allows calculation of ESE signals for larger nuclear spins in the absence of nuclear quadrupole interaction. This relation can be used to calculate the time course of the ESE signal for arbitrary nuclear spins as a function of the nuclear magnetization vectors. While this formalism allows quantitative calculation of modulation from nuclei, it also provides a qualitative means of visualizing the modulation based on simple magnetization vectors.     

KDP晶体单点金刚石车削表面形貌分形分析

 分别使用2维和3维分形方法对单点金刚石车削加工的KDP晶体表面形貌进行了分析,并对表面的3维分形维数和3维粗糙度表征参数进行了比较,分析了二者对表面形貌表征的差异。使用2维轮廓分形方法计算了KDP晶体表面圆周各方向上的分形维数。通过分析得出：3维分形维数与表面粗糙度值成反比关系;使用单点金刚石车削方法加工KDP晶体会形成各向异性特征明显的已加工表面,在一定程度上容易形成小尺度波纹;已加工表面是否具有明显的小尺度波纹特征与表面粗糙度值并无直接关系,但与其表面轮廓分形状态分布密切相关;KDP晶体表面2维功率谱密度与其分形状态具有相近的方向性特征。     

The multidimensional filter diagonalization method

The theory and numerical aspects of the recently developed multidimensional version of the filter diagonalization method (FDM) are described in detail. FDM can construct various "ersatz" or "hybrid" spectra from multidimensional time signals. Spectral resolution is not limited by the time-frequency uncertainty principle in each separate frequency dimension, but rather by the total joint information content of the signal, i.e., N(total) = N(1) x N(2) x vertical ellipsis x N(D), where some of the interferometric dimensions do not have to be represented by more than a few (e.g., two) time increments. It is shown that FDM can be used to compute various reduced-dimensionality projections of a high-dimensional spectrum directly, i.e., avoiding construction of the latter. A subsequent paper (J. Magn. Reson. 144, 357-366 (2000)) is concerned with applications of the method to 2D, 3D, and 4D NMR experiments.     

Application of the slaving principle to quenching problems

The slaving principle theory is applied to quenching problems. It is shown that a system, which starts from the close vicinity of an unstable point and the evolution of which is described by a set ofn-dimensional equations, will approach and then follow the unstable manifold. In the presence of fluctuations, it is even possible that the system approaches the most unstable manifold. Thus, the actual dimension related to the problem can be considerably reduced.