Publications

Quantum mechanical equations of motion are strictly linear in density operators, but equations describing chemical kinetics and hydrodynamics may be nonlinear in concentrations. This incompatibility is fundamental, but special cases can be handledfor example, in magnetic resonance where nuclear spin interactions may be too weak influence concentration dynamics. For isolated spins and first-order reactions, this is a well-researched topic, but time evolution of complex nuclear spin systems in the presence of second-order kinetics, diffusion, and flow has so far remained intractable. In this communication, we report a numerically stable formalism for time-domain description of nuclear spin dynamics and relaxation in the simultaneous presence of diffusion, flow, and second-order chemical reactions. As an illustration, we use Diels-Alder cycloaddition of acrylonitrile to cyclopentadiene in the presence of diffusion and flow in a microfluidic NMR probe (a finite element model with thousands of Voronoi cells) with a spatially localized stripline radio frequency coil.

In pulsed dynamic nuclear polarization (DNP), enhancement of bulk nuclear polarization requires the repeated application of a microwave pulse sequence. So far, analysis of a one-time transfer of electron spin polarization to a dipolar-coupled nuclear spin has guided the design of DNP pulse sequences. This has obvious shortcomings, such as the inability to predict the optimal repetition time. In an actual pulsed DNP experiment, a balance is reached between the polarization arriving from the unpaired electrons and nuclear relaxation. In this article, we explore three algorithms to compute this stroboscopic steady state: (1) explicit time evolution by propagator squaring, (2) generation of an effective propagator using the matrix logarithm, and (3) direct calculation of the steady state with the Newton-Raphson method. Algorithm (2) is numerically unstable in dissipative DNP settings. Algorithms (1) and (3) are both stable; algorithm (3) is the most efficient. We compare the steady-state simulations to existing experimental results at 0.34 and 1.2 T and to the first experimental observation of X-inverse-X (XiX) DNP at 3.4 T. The agreement is good and improves further when electron-proton distance and electron Rabi frequency distributions are accounted for. We demonstrate that the trajectory of the spin system during one-time application of a microwave pulse sequence differs from the steady orbit. This has implications for DNP pulse sequence design.

NMR spectroscopy of biomolecules provides atomic level information into their structure, dynamics and interactions with their binding partners. However, signal attenuation from line broadening caused by fast relaxation and signal overlap often limits the application of NMR to large macromolecular systems. Here we leverage the slow relaxation properties of ¹³C nuclei attached to ¹⁹F in aromatic ¹⁹F¹³C spin pairs as well as the spinspin coupling between the fluorinated ¹³C nucleus and the hydrogen atom at the meta-position to record two-dimensional ¹H¹³C_F correlation spectra with transverse relaxation-optimized spectroscopy selection on ¹³C_F. To accomplish this, we synthesized [4-¹⁹F¹³C^ζ; 3,5-²H₂^ε] Phe, engineered for optimal relaxation properties, and adapted a residue-specific route to incorporate this residue globally into proteins and a site-specific 4-¹⁹F Phe encoding strategy. This approach resulted in narrow linewidths for proteins ranging from 30 kDa to 180 kDa, enabling interaction studies with small-molecule ligands without requiring specialized ¹⁹F-compatible probes. (Figure presented.)

Quantum optimal control methods, such as gradient ascent pulse engineering (GRAPE), are used for precise manipulation of quantum states. Many of those methods were pioneered in magnetic resonance spectroscopy, where instrumental distortions are often negligible. However, that is not the case elsewhere: the usual jumble of cables, resonators, modulators, splitters, amplifiers, and filters can and would distort control signals. Those distortions may be non-linear; their inverse functions may be ill-defined and unstable; they may even vary from one day to the next and across the sample. Here we introduce the response-aware gradient ascent pulse engineering framework, which accounts for any cascade of differentiable distortions within the GRAPE optimization loop, does not require filter function inversion, and produces control sequences that are resilient to user-specified distortion cascades with user-specified parameter ensembles. The framework is implemented into the optimal control module supplied with versions 2.10 and later of the open-source Spinach library; the user needs to provide function handles returning the actions by the distortions and, optionally, parameter ensembles for those actions.

The current standard method for amino acid signal identification in protein NMR spectra is sequential assignment using triple-resonance experiments. Good software and elaborate heuristics exist, but the process remains laboriously manual. Machine learning does help, but its training databases need millions of samples that cover all relevant physics and every kind of instrumental artifact. In this communication, we offer a solution to this problem. We propose polyadic decompositions to store millions of simulated three-dimensional NMR spectra, on-the-fly generation of artifacts during training, a probabilistic way to incorporate prior and posterior information, and integration with the industry standard CcpNmr software framework. The resulting neural nets take [¹H,¹³C] slices of mixed pyruvatelabeled HNCA spectra (different CA signal shapes for different residue types) and return an amino acid probability table. In combination with primary sequence information, backbones of common proteins (GB1, MBP, and INMT) are rapidly assigned from just the HNCA spectrum.

Fluorine substitution can have a profound impact on molecular conformation. Here, we present a detailed conformational analysis of how the 1,3-difluoropropylene motif (-CHF-CH₂-CHF-) determines the conformational profiles of 1,3-difluoropropane, anti- and syn-2,4-difluoropentane, and anti- and syn-3,5-difluoroheptane. It is shown that the 1,3-difluoropropylene motif strongly influences alkane chain conformation, with a significant dependence on the polarity of the medium. The conformational effect of 1,3-fluorination is magnified upon chain extension, which contrasts with vicinal difluorination. Experimental evidence was obtained from NMR analysis, where polynomial complexity scaling simulation algorithms were necessary to enable J-coupling extraction from the strong second-order spectra, particularly for the large 16-spin systems of the difluorinated heptanes. These results improve our understanding of the conformational control toolkit for aliphatic chains, yield simple rules for conformation population analysis, and demonstrate quantum mechanical time-domain NMR simulations for liquid state systems with large numbers of strongly coupled spins.

Solute translocation by membrane transport proteins is a vital biological process that can be tracked, on the sub-second timescale, using nuclear magnetic resonance (NMR). Fluorinated substrate analogues facilitate such studies because of high sensitivity of ¹⁹F NMR and absence of background signals. Accurate extraction of translocation rate constants requires precise quantification of NMR signal intensities. This becomes complicated in the presence of J-couplings, cross-correlations, and nuclear Overhauser effects (NOE) that alter signal integrals through mechanisms unrelated to translocation. Geminal difluorinated motifs introduce strong and hard-to-quantify contributions from non-exchange effects, the nuanced nature of which makes them hard to integrate into data analysis methodologies. With analytical expressions not being available, numerical least squares fitting of theoretical models to 2D spectra emerges as the preferred quantification approach. For large spin systems with simultaneous coherent evolution, cross-relaxation, cross-correlation, conformational exchange, and membrane translocation between compartments with different viscosities, the only available simulation framework is Spinach. In this study, we demonstrate GLUT-1 dependent membrane transport of two model sugars featuring CF₂ and CF₂CF₂ fluorination motifs, with precise determination of translocation rate constants enabled by numerical fitting of 2D EXSY spectra. For spin systems and kinetic networks of this complexity, this was not previously tractable.

We present a methodology for the design of optimal Raman beam-splitter pulses suitable for cold atom inertial sensors. The methodology, based on time-dependent perturbation theory, links optimal control and the sensitivity function formalism in the Bloch sphere picture, thus providing a geometric interpretation of the optimization problem. Optimized pulse waveforms are found to be more resilient than conventional beam-splitter pulses and ensure a near-flat superposition phase for a range of detunings approaching the Rabi frequency. As a practical application, we simulated the performance of an optimized Mach-Zehnder interferometer in terms of scale-factor error and bias induced by interpulse laser intensity variations. Our findings reveal enhancements compared to conventional interferometers operating with constant-power beam-splitter pulses.

Response functions of resonant circuits create ringing artefacts if their input changes rapidly. When physical limits of electromagnetic spectroscopies are explored, this creates two types of problems. Firstly, simulation: the system must be propagated accurately through every response transient, this may be computationally expensive. Secondly, optimal control: circuit response must be taken into account; it may be advantageous to design pulses that are resilient to such distortions. At the root of both problems is the popular piecewise-constant approximation for control sequences in the rotating frame; in magnetic resonance it has persisted since the earliest days and has become entrenched in the commercially available hardware. In this paper, we report an implementation and benchmarks of recent Lie-group methods that can efficiently simulate and optimise smooth control sequences.

This monograph is a fundamental reference for scientists and engineers who encounter spin processes in their work. The author, Ilya Kuprov, derives the concept of spin from basic symmetries and gives an overview of theoretical and computational aspects of spin dynamics: from Dirac equation and spin Hamiltonian, through coherent evolution and relaxation theories, to quantum optimal control, and all the way to practical implementation advice for parallel computers.

It is commonly believed that electromagnetic spectra of atoms and molecules can be fully described by interactions involving electric and magnetic multipoles. However, it has recently become clear that interactions between light and matter also involve toroidal multipolestoroidal absorption lines have been observed in electromagnetic metamaterials. Here, we show that a previously unexplored type of spectroscopy of the hitherto largely neglected toroidal dipolar interaction becomes feasible if, apart from the classical r × r × p toroidal dipole density term responsible for the toroidal transitions in metamaterials, the spin-dependent r × σ term (which only occurs in relativistic quantum mechanics) is taken into account. Toroidal dipole operators are odd under parity and time-reversal symmetries; toroidal dipole transitions can therefore be distinguished from electric multipole and magnetic dipole transitions.

Atom interferometers that employ atoms in superpositions of different electronic states are sensitive to any noise that affects these superposed states differently. Resilience to such noise results from using superpositions where the atomic states differ in momentum only, but implementation of such state-symmetric diffraction can lead to population loss into unwanted states and restricts the atomic velocity acceptance of the interferometer. In this paper, by varying the laser intensities and phases as functions of time, we present optimized pulses designed for use in state-symmetric interferometers that overcome these restrictions. We extend this optimization to multi-pulse sequences designed to increase the interferometer area and demonstrate significant improvements in the fringe visibility compared with sequences of π/2 and π pulses. We discuss the limits on the temperature of the atomic source required for efficient atomic diffraction and show how optimized pulse sequences enable efficient diffraction with considerably warmer clouds, hence reducing the need for velocity selection and increasing the measurement signal-to-noise ratio.

Membrane proteins are currently investigated after detergent extraction from native cellular membranes and reconstitution into artificial liposomes or nanodiscs, thereby removing them from their physiological environment. However, to truly understand the biophysical properties of membrane proteins in a physiological environment, they must be investigated within living cells. Here, we used a spin-labeled nanobody to interrogate the conformational cycle of the ABC transporter MsbA by double electron-electron resonance. Unexpectedly, the wide inward-open conformation of MsbA, commonly considered a nonphysiological state, was found to be prominently populated in Escherichia coli cells. Molecular dynamics simulations revealed that extensive lateral portal opening is essential to provide access of its large natural substrate core lipid A to the binding cavity. Our work paves the way to investigate the conformational landscape of membrane proteins in cells.

Dynamic nuclear polarization in the liquid state via Overhauser effect is enabled by the fluctuations of the electron-nuclear hyperfine interaction. Fermi contact (or scalar) hyperfine coupling can be modulated by molecular collisions on timescales of a few picoseconds and shorter, enabling an effective polarization transfer even at high magnetic fields. However, only a few studies have presented a theoretical analysis of the scalar mechanism. Here we report the current understanding of the scalar relaxation in liquid-state DNP and present different modeling strategies based on analytical relaxation theory and numerical calculations from molecular dynamics simulations. These approaches give consistent results in identifying the timescale of the fluctuations of the scalar interaction that drives ¹³C-DNP in the model system of CHCl₃ doped with nitroxide radical. Subpicosecond fluctuations arise not only from random molecular collisions but are also present when target molecule and polarizing agent form a transient complex that persists for tens of picoseconds. We expect that these kind of interactions, possibly based on hydrogen bond-like complexations, might be present in a large variety of compounds.

We have recently demonstrated supervised deep learning methods for rapid generation of radiofrequency pulses in magnetic resonance imaging (https://doi.org/10.1002/mrm.27740, https://doi.org/10.1002/mrm.28667). Unlike the previous iterative optimization approaches, deep learning methods generate a pulse using a fixed number of floating-point operations - this is important in MRI, where patient-specific pulses preferably must be produced in real time. However, deep learning requires vast training libraries, which must be generated using the traditional methods, e.g., iterative quantum optimal control methods. Those methods are usually variations of gradient descent, and the calculation of the gradient of the performance metric with respect to the pulse waveform can be the most numerically intensive step. In this communication, we explore various ways in which the calculation of gradients in quantum optimal control theory may be accelerated. Four optimization avenues are explored: truncated commutator series expansions at zeroth and first order, a novel midpoint truncation scheme at first order, and the exact complex-step method. For the spin systems relevant to MRI, the first-order midpoint truncation is found to be sufficiently accurate, but also significantly faster than the machine precision gradient. This makes the generation of training databases for the machine learning methods considerably more realistic.

Proline homopolymer motifs are found in many proteins; their peculiar conformational and dynamic properties are often directly involved in those proteins' functions. However, the dynamics of proline homopolymers is hard to study by NMR due to a lack of amide protons and small chemical shift dispersion. Exploiting the spectroscopic properties of fluorinated prolines opens interesting perspectives to address these issues. Fluorinated prolines are already widely used in protein structure engineering - they introduce conformational and dynamical biases - but their use as ¹⁹F NMR reporters of proline conformation has not yet been explored. In this work, we look at model peptides where Cγ-fluorinated prolines with opposite configurations of the chiral Cγ centre have been introduced at two positions in distinct polyproline segments. By looking at the effects of swapping these (4R)-fluoroproline and (4S)-fluoroproline within the polyproline segments, we were able to separate the intrinsic conformational properties of the polyproline sequence from the conformational alterations instilled by fluorination. We assess the fluoroproline ¹⁹F relaxation properties, and we exploit the latter in elucidating binding kinetics to the SH3 (Src homology 3) domain.

The sensitivity of atom interferometers depends on the fidelity of the light pulses used as beamsplitters and mirrors. Atom interferometers typically employ pulses that affect \u20ac/2 and πfractional Rabi oscillations, the fidelities of which are reduced when there are variations in atomic velocity and laser intensity. We have previously demonstrated the application of optimal control theory to design pulses more robust to such errors; however, if these variations exhibit a time dependence over periods on the order of the interferometer duration then phase shifts can be introduced in the final fringe that potentially reduce the sensitivity. In this paper, we explain why care must be taken when optimising interferometer pulse sequences to ensure that phase shifts arising from inter-pulse variations are not significantly increased. We show that these phase shifts can in fact be minimised by choosing an appropriate measure of individual pulse fidelity.

We report an experimental observation of ³¹P NMR resonances shifted by over 10 000 ppm (meaning percent range, and a new record for solutions), and similar ¹H chemical shifts, in an intermediate-spin square planar ferrous complex [^tBu(PNP)Fe-H], where PNP is a carbazole-based pincer ligand. Using a combination of electronic structure theory, nuclear magnetic resonance, magnetometry, and terahertz electron paramagnetic resonance, the influence of magnetic anisotropy and zero-field splitting on the paramagnetic shift and relaxation enhancement is investigated. Detailed spin dynamics simulations indicate that, even with relatively slow electron spin relaxation (T₁ ≈10⁻¹¹ s), it remains possible to observe NMR signals of directly metal-bonded atoms because pronounced rhombicity in the electron zero-field splitting reduces nuclear paramagnetic relaxation enhancement.

The lack of interpretability and trust is a much-criticized feature of deep neural networks. In fully connected nets, the signaling between inner layers is scrambled because backpropagation training does not require perceptrons to be arranged in any particular order. The result is a black box; this problem is particularly severe in scientific computing and digital signal processing (DSP), where neural nets perform abstract mathematical transformations that do not reduce to features or concepts. We present here a group-theoretical procedure that attempts to bring inner-layer signaling into a human-readable form, the assumption being that this form exists and has identifiable and quantifiable featuresfor example, smoothness or locality. We applied the proposed method to DEERNet (a DSP network used in electron spin resonance) and managed to descramble it. We found considerable internal sophistication: the network spontaneously invents a bandpass filter, a notch filter, a frequency axis rescaling transformation, frequency-division multiplexing, group embedding, spectral filtering regularization, and a map from harmonic functions into Chebyshev polynomialsin 10 min of unattended training from a random initial guess.

Molecular dynamics (MD) trajectories provide useful insights into molecular structure and dynamics. However, questions persist about the quantitative accuracy of those insights. Experimental NMR spin relaxation rates can be used as tests, but only if relaxation superoperators can be efficiently computed from MD trajectories no mean feat for the quantum Liouville space formalism where matrix dimensions quadruple with each added spin 1/2. Here we report a module for the Spinach software framework that computes Bloch-Redfield-Wangsness relaxation superoperators (including non-secular terms and cross-correlations) from MD trajectories. Predicted initial slopes of nuclear Overhauser effects for sucrose trajectories using advanced water models and a force field optimised for glycans are within 25% of experimental values.

Combining dynamic nuclear polarization with proton detection significantly enhances the sensitivity of magic-angle spinning NMR spectroscopy. Herein, the feasibility of proton-detected experiments with slow (10 kHz) magic angle spinning was demonstrated. The improvement in sensitivity permits the acquisition of indirectly detected ¹⁴N NMR spectra allowing biomolecular structures to be characterized without recourse to isotope labelling. This provides a new tool for the structural characterization of environmental and medical samples, in which isotope labelling is frequently intractable.

ConspectusComplexes of lanthanide(III) ions are being actively studied because of their unique ground and excited state properties and the associated optical and magnetic behavior. In particular, they are used as emissive probes in optical spectroscopy and microscopy and as contrast agents in magnetic resonance imaging (MRI). However, the design of new complexes with specific optical and magnetic properties requires a thorough understanding of the correlation between molecular structure and electric and magnetic susceptibilities, as well as their anisotropies. The traditional Judd-Ofelt-Mason theory has failed to offer useful guidelines for systematic design of emissive lanthanide optical probes. Similarly, Bleaney's theory of magnetic anisotropy and its modifications fail to provide accurate detail that permits new paramagnetic shift reagents to be designed rather than discovered.A key determinant of optical and magnetic behavior in f-element compounds is the ligand field, often considered as an electrostatic field at the lanthanide created by the ligands. The resulting energy level splitting is a sensitive function of several factors: The nature and polarizability of the whole ligand and its donor atoms; the geometric details of the coordination polyhedron; the presence and extent of solvent interactions; specific hydrogen bonding effects on donor atoms and the degree of supramolecular order in the system. The relative importance of these factors can vary widely for different lanthanide ions and ligands. For nuclear magnetic properties, it is both the ligand field splitting and the magnetic susceptibility tensor, notably its anisotropy, that determine paramagnetic shifts and nuclear relaxation enhancement.We review the factors that control the ligand field in lanthanide complexes and link these to aspects of their utility in magnetic resonance and optical emission spectroscopy and imaging. We examine recent progress in this area particularly in the theory of paramagnetic chemical shift and relaxation enhancement, where some long-neglected effects of zero-field splitting, magnetic susceptibility anisotropy, and spatial distribution of lanthanide tags have been accommodated in an elegant way.

We present designs for the augmentation "mirror"pulses of large-momentum-transfer atom interferometers that maintain their fidelity as the wave-packet momentum difference is increased. These biselective pulses, tailored using optimal control methods to the evolving bimodal momentum distribution, should allow greater interferometer areas and hence increased inertial measurement sensitivity, without requiring elevated Rabi frequencies or extended frequency chirps. Using an experimentally validated model, we have simulated the application of our pulse designs to large-momentum-transfer atom interferometry using stimulated Raman transitions in a laser-cooled atomic sample of Rb85 at 1 μK. After the wave packets have separated by 42 photon recoil momenta, our pulses maintain a fringe contrast of 90%, whereas, for adiabatic rapid passage and conventional π pulses, the contrast is less than 10%. Furthermore, we show how these pulses may be adapted to be robust to laser intensity variations between pulses and to suppress the detrimental off-resonant excitation that limits other broadband pulse schemes.

We present the theoretical design and experimental implementation of mirror and beamsplitter pulses that improve the fidelity of atom interferometry and increase its tolerance of systematic inhomogeneities. These pulses are designed using the GRAPE optimal control algorithm and demonstrated experimentally with a cold thermal sample of ⁸⁵Rb atoms. We first show a stimulated Raman inversion pulse design that achieves a ground hyperfine state transfer efficiency of 99.8(3)%, compared with a conventional π pulse efficiency of 75(3)%. This inversion pulse is robust to variations in laser intensity and detuning, maintaining a transfer efficiency of 90% at detunings for which the π pulse fidelity is below 20%, and is thus suitable for large momentum transfer interferometers using thermal atoms or operating in non-ideal environments. We then extend our optimization to all components of a Mach-Zehnder atom interferometer sequence and show that with a highly inhomogeneous atomic sample the fringe visibility is increased threefold over that using conventional π and π/2 pulses.

PSYCOSY is an f1 broadband homonuclear decoupled version of the COSY nuclear magnetic resonance pulse sequence. Here, we investigate by a combination of experimental measurements, spatially distributed spin dynamics simulations, and analytical predictions the coherence evolution delay necessary in PSYCOSY experiments to ensure intensity discrimination in favour of the correlations typically arising from short range (ⁿJ, n ≤ 3) ¹H¹H couplings and show that, in general, a coherence evolution delay of around 35 ms is optimum.

Interpreting dynamics in solid-state molecular systems requires characterization of the potentially heterogeneous environmental contexts of molecules. In particular, the analysis of solid-state nuclear magnetic resonance (ssNMR) data to elucidate molecular dynamics (MD) involves modeling the restriction to overall tumbling by neighbors, as well as the concentrations of water and buffer. In this exploration of the factors that influence motion, we utilize atomistic MD trajectories of peptide aggregates with varying hydration to mimic an amorphous solid-state environment and predict ssNMR relaxation rates. We also account for spin diffusion in multiply spin-labeled (up to 19 nuclei) residues, with several models of dipolar-coupling networks. The framework serves as a general approach to determine essential spin couplings affecting relaxation, benchmark MD force fields, and reveal the hydration dependence of dynamics in a crowded environment. We demonstrate the methodology on a previously characterized amphiphilic 14-residue lysine-leucine repeat peptide, LKα14 (Ac-LKKLLKLLKKLLKL-c), which has an α-helical secondary structure and putatively forms leucine-burying tetramers in the solid state. We measure the R₁ relaxation rates of uniformly ¹³C-labeled and site-specific ²H-labeled leucines in the hydrophobic core of LKα14 at multiple hydration levels. Studies of 9 and 18 tetramer bundles reveal the following: (a) for the incoherent component of ¹³C relaxation, the nearest-neighbor spin interactions dominate, while the ¹H-¹H interactions have minimal impact; (b) the AMBER ff14SB dihedral barriers for the leucine C_Î-C_Î bond ("methyl rotation barriers") must be lowered by a factor of 0.7 to better match the ²H data; (c) proton-driven spin diffusion explains some of the discrepancy between experimental and simulated rates for the C_β and C_α nuclei; and (d) ¹³C relaxation rates are mostly underestimated in the MD simulations at all hydrations, and the discrepancies identify likely motions missing in the 50 ns MD trajectories.

These are personal recollections and musings, written for the 50th Anniversary of the Journal of Magnetic Resonance. They are distilled from twenty years of writing simulation code, and filtered through hindsight and sarcastic intransigence. To me, the biggest recent achievements of the magnetic resonance community in the field of theory and simulation have been the successful war on the exponential scaling, the powerful and general simulation software, and the return to elegant notation. It appears that our future will be defined by computers. Three aspects are pertinent: simulation as the experiment is designed, optimal control as the experiment proceeds, and machine learning at the data processing stage.

We propose a solution to the matrix dimension problem in quantum mechanical simulations of MRI (magnetic resonance imaging) experiments on complex molecules. This problem is very old; it arises when Kronecker products of spin operators and spatial dynamics generators are takenthe resulting matrices are far too large for any current or future computer. However, spin and spatial operators individually have manageable dimensions, and we note here that the action by their Kronecker products on any vector may be computed without opening those products. This eliminates large matrices from the simulation process. MRI simulations for coupled spin systems of complex metabolites in three dimensions with diffusion, flow, chemical kinetics, and quantum mechanical treatment of spin relaxation are now possible. The methods described in this paper are implemented in versions 2.4 and later of the Spinach library.

Atomic-level information about the structure and dynamics of biomolecules is critical for an understanding of their function. Nuclear magnetic resonance (NMR) spectroscopy provides unique insights into the dynamic nature of biomolecules and their interactions, capturing transient conformers and their features. However, relaxation-induced line broadening and signal overlap make it challenging to apply NMR spectroscopy to large biological systems. Here we took advantage of the high sensitivity and broad chemical shift range of ¹⁹ F nuclei and leveraged the remarkable relaxation properties of the aromatic ¹⁹ F- ¹³ C spin pair to disperse ¹⁹ F resonances in a two-dimensional transverse relaxation-optimized spectroscopy spectrum. We demonstrate the application of ¹⁹ F- ¹³ C transverse relaxation-optimized spectroscopy to investigate proteins and nucleic acids. This experiment expands the scope of ¹⁹ F NMR in the study of the structure, dynamics, and function of large and complex biological systems and provides a powerful background-free NMR probe.

Fluorinated proline derivatives have found diverse applications in areas ranging from medicinal chemistry over structural biochemistry to organocatalysis. Depending on the stereochemistry of monofluorination at the proline 3- or 4-position, different effects on the conformational properties of proline (ring pucker, cis/trans isomerization) are introduced. With fluorination at both 3- and 4-positions, matching or mismatching effects can occur depending on the relative stereochemistry. Here we report, in full, the syntheses and conformational properties of three out of the four possible 3,4-difluoro-l-proline diastereoisomers. The yet unreported conformational properties are described for (3S,4S)- and (3R,4R)-difluoro-l-proline, which are shown to bias ring pucker and cis/trans ratios on the same order of magnitude as their respective monofluorinated progenitors, although with significantly faster amide cis/trans isomerization rates. The reported analogues thus expand the scope of available fluorinated proline analogues as tools to tailor proline's distinct conformational and dynamical properties, allowing for the interrogation of its role in, for instance, protein stability or folding.

Polarization transfer methods are widely adopted for the purpose of correlating different nuclear species as well as to achieve signal enhancement. Polarization transfer from ¹H to the ¹⁴N overtone transition (Δm = 2) can be achieved using cross polarization methods under magic-angle spinning conditions, where spin locks of the order of several milliseconds can be obtained on common bio-solids (α-glycine and N-acetylvaline). Signal enhancement factors up to 4.4 per scan, can be achieved under favorable conditions, despite MHz-sized quadrupolar interaction. Moreover, we present a detailed theoretical treatment and accurate numerical simulations which are in excellent agreement the unusual experimental matching conditions observed for cross-polarization to ¹⁴N overtone.

Magic-angle spinning solid-state NMR is increasingly utilized to study the naturally abundant, spin-1 nucleus ¹⁴N, providing insights into the structure and dynamics of biological and organic molecules. In particular, the characterisation of ¹⁴N sites using indirect detection has proven useful for complex molecules, where the 'spy' nucleus provides enhanced sensitivity and resolution. Here we exploit the sensitivity of proton detection, to indirectly characterise ¹⁴N sites using a moderate rf field to generate coherence between the ¹H and ¹⁴N at moderate and fast-magic-angle spinning frequencies. Efficient numerical simulations have been developed that have allowed us to quantitatively analyse the resulting ¹⁴N lineshapes to determine both the size and asymmetry of the quadrupolar interaction. Exploiting only naturally occurring abundant isotopes will aid the analysis of materials with the need to resort to isotope labelling, whilst providing additional insights into the structure and dynamics that the characterisation of the quadrupolar interaction affords.

In three structurally related series of nine-coordinate lanthanide(iii) complexes (Ln = Tb, Dy, Ho, Er, Tm and Yb) based on triazacyclononane, solution NMR studies and DFT/CASSCF calculations have provided key information on the magnetic susceptibility anisotropy. Both experimental and computational approaches have revealed a poor correlation to Bleaney's theory of magnetic anisotropy. CASSCF calculations suggested that the magnetic susceptibility is very sensitive to small geometric variations within the first coordination sphere, whereas DFT analyses indicate that it is the thermal accessibility of low energy vibrational modes that may lead to distortion. Parallel NMRD and EPR studies on the three Gd(iii) complexes revealed good correspondence in estimating the electronic relaxation time. The Gd(iii) tris-pyridinecarboxylate complex possesses a very long electronic relaxation time making it a promising starting point for responsive gadolinium EPR probe design.

Numerically optimised microwave pulses are used to increase excitation efficiency and modulation depth in electron spin resonance experiments performed on a spectrometer equipped with an arbitrary waveform generator. The optimisation procedure is sample-specific and reminiscent of the magnet shimming process used in the early days of nuclear magnetic resonance an objective function (for example, echo integral in a spin echo experiment) is defined and optimised numerically as a function of the pulse waveform vector using noise-resilient gradient-free methods. We found that the resulting shaped microwave pulses achieve higher excitation bandwidth and better echo modulation depth than the pulse shapes used as the initial guess. Although the method is theoretically less sophisticated than simulation based quantum optimal control techniques, it has the advantage of being free of the linear response approximation; rapid electron spin relaxation also means that the optimisation takes only a few seconds. This makes the procedure fast, convenient, and easy to use. An important application of this method is at the final stage of the implementation of theoretically designed pulse shapes: compensation of pulse distortions introduced by the instrument. The performance is illustrated using spin echo and out-of-phase electron spin echo envelope modulation experiments. Interface code between Bruker SpinJet arbitrary waveform generator and Matlab is included in versions 2.2 and later of the Spinach library.

We have developed a new approach, to our knowledge, to quantify the equilibrium exchange kinetics of carrier-mediated transmembrane transport of fluorinated substrates. The method is based on adapted kinetic theory that describes the concentration dependence of the transmembrane exchange rates of two competing, simultaneously transported species. Using the new approach, we quantified the kinetics of membrane transport of both anomers of three monofluorinated glucose analogs in human erythrocytes (red blood cells) using ¹⁹F NMR exchange spectroscopy. An inosine-based glucose-free medium was shown to promote survival and stable metabolism of red blood cells over the duration of the experiments (several hours). Earlier NMR studies only yielded the apparent rate constants and transmembrane fluxes of the anomeric species, whereas we could categorize the two anomers in terms of the catalytic activity (specificity constants) of the glucose transport protein GLUT1 toward them. Differences in the membrane permeability of the three glucose analogs were qualitatively interpreted in terms of local perturbations in the bonding of substrates to key amino acid residues in the active site of GLUT1. The methodology of this work will be applicable to studies of other carrier-mediated membrane transport processes, especially those with competition between simultaneously transported species. The GLUT1-specific results can be applied to the design of probes of glucose transport or inhibitors of glucose metabolism in cells, including those exhibiting the Warburg effect.

Recent developments in data sampling and processing techniques have made it possible to acquire 2-dimensional NMR spectra of small molecules at digital resolutions in both dimensions approaching the intrinsic limitations of the equipment and sample on a realistic timescale. These developments offer the possibility of enormously increased effective resolution (peak dispersion) and the ability to effectively study samples where peak overlap was previously a limiting factor. Examples of such spectra have been produced for a number of 2-dimensional techniques including TOCSY and HSQC. In this paper, we investigate some of the problems in applying such techniques to COSY spectra and suggest a modification to the classic experiment that alleviates some of these problems.

The established model-free methods for the processing of two-electron dipolar spectroscopy data [DEER (double electron-electron resonance), PELDOR (pulsed electron double resonance), DQ-EPR (double-quantum electron paramagnetic resonance), RIDME (relaxation-induced dipolar modulation enhancement), etc.] use regularized fitting. In this communication, we describe an attempt to process DEER data using artificial neural networks trained on large databases of simulated data. Accuracy and reliability of neural network outputs from real experimental data were found to be unexpectedly high. The networks are also able to reject exchange interactions and to return a measure of uncertainty in the resulting distance distributions. This paper describes the design of the training databases, discusses the training process, and rationalizes the observed performance. Neural networks produced in this work are incorporated as options into Spinach and DeerAnalysis packages.

Atom matterwave interferometry requires mirror and beam splitter pulses that are robust to inhomogeneities in field intensity, magnetic environment, atom velocity, and Zeeman substate. We present theoretical results which show that pulse shapes determined using quantum control methods can significantly improve interferometer performance by allowing broader atom distributions, larger interferometer areas, and higher contrast. We have applied gradient ascent pulse engineering (grape) to optimize the design of phase-modulated mirror pulses for a Mach-Zehnder light-pulse atom interferometer, with the aim of increasing fringe contrast when averaged over atoms with an experimentally relevant range of velocities, beam intensities, and Zeeman states. Pulses were found to be highly robust to variations in detuning and coupling strength and offer a clear improvement in robustness over the best established composite pulses. The peak mirror fidelity in a cloud of ∼80μKRb85 atoms is predicted to be improved by a factor of 2 compared with standard rectangular π pulses.

An approach to the indirect measurement of nuclear spin relaxation rates of low-magnetogyric ratio (γ) nuclei using the process of satellite exchange is described. The method does not require the observation of, or even the ability to provide radio-frequency pulses to, the low-γ nucleus, but requires this to be scalar coupled to an NMR observable nucleus, such as ³¹P or ¹H, making it especially attractive for the study of diamagnetic transition metals. In situations where spin relaxation is dominated by chemical shift anisotropy (CSA), the determination of the longitudinal spin relaxation time constant (T₁) of the metal becomes possible, as illustrated for ¹⁹⁵Pt and ^107/109Ag.

Recent developments in magic angle spinning (MAS) technology permit spinning frequencies of ≥100 kHz. We examine the effect of such fast MAS rates upon nuclear magnetic resonance proton line widths in the multi-spin system of β-Asp-Ala crystal. We perform powder pattern simulations employing Fokker-Plank approach with periodic boundary conditions and ¹H-chemical shift tensors calculated using the bond polarization theory. The theoretical predictions mirror well the experimental results. Both approaches demonstrate that homogeneous broadening has a linear-quadratic dependency on the inverse of the MAS spinning frequency and that, at the faster end of the spinning frequencies, the residual spectral line broadening becomes dominated by chemical shift distributions and susceptibility effects even for crystalline systems.

Liquid state nuclear magnetic resonance is the only class of magnetic resonance experiments for which the simulation problem is solved comprehensively for spin systems of any size. This paper contains a practical walkthrough for one of the many available simulation packages Spinach. Its unique feature is polynomial complexity scaling: the ability to simulate large spin systems quantum mechanically and with accurate account of relaxation, diffusion, chemical processes, and hydrodynamics. This paper is a gentle introduction written with a PhD student in mind.

The application of overtone nuclear magnetic resonance (OT NMR) to symmetric spin transitions of integer quadrupolar nuclei is of considerable interest since this transition is immune to the first-order quadrupolar interaction, thus resulting in narrow NMR lines. Owing to its roles in nature and its high natural abundance, ¹⁴N (I = 1) OT NMR has been explored, in which the indirect and/or direct acquisitions of ¹⁴N OT were experimentally demonstrated. However, other than ¹⁴N nucleus, no OT NMR observation of other integer quadrupolar nuclei has been reported in the literature. In this work, we extend the application of OT NMR to another integer quadrupolar nucleus, namely ¹⁰B (I = 3). However, this is not straightforward owing to the unfavorable characteristics of ¹⁰B isotope. Here, for the first time, we present the selective acquisition of ¹⁰B central (−1 ↔ +1) OT NMR via detection of ¹H nuclei on perborate monohydrate sample. Numerical calculations are in a good agreement with the experimental results. Both show that the optimal sensitivity is achieved when the carrier frequency is applied at the second OT spinning sideband, i.e. an offset of twice of the spinning frequency from the center band.

Monofluorination at the proline 4-position results in conformational effects, which is exploited for a range of applications. However, this conformational distortion is a hindrance when the natural proline conformation is important. Here we introduce (3S,4R)-3,4-difluoroproline, in which the individual fluorine atoms instil opposite conformational effects, as a suitable probe for fluorine NMR studies.

Lanthanide ions accelerate nuclear spin relaxation by two primary mechanisms: dipolar and Curie. Both are commonly assumed to depend on the length of the lanthanide-nucleus vector, but not on its direction. Here we show experimentally that this is wrong-careful proton relaxation data analysis in a series of isostructural lanthanide complexes (Ln = Tb, Dy, Ho, Er, Tm, Yb) reveals angular dependence in both Curie and dipolar relaxation. The reasons are: (a) that magnetic susceptibility anisotropy can be of the same order of magnitude as the isotropic part (contradicting the unstated assumption in Guéron's theory of the Curie relaxation process), and (b) that zero-field splitting can be much stronger than the electron Zeeman interaction (Bloembergen's original theory of the lanthanide-induced dipolar relaxation process makes the opposite assumption). These factors go beyond the well researched cross-correlation effects; they alter the relaxation theory treatment and make strong angular dependencies appear in the nuclear spin relaxation rates. Those dependencies are impossible to ignore-this is now demonstrated both theoretically and experimentally, and suggests that a major revision is needed of the way lanthanide-induced relaxation data are used in structural biology.

Two new heterometallic Zn₃Ln₃ (Ln³⁺=Dy, Tb) complexes, with a double triangular topology of the metal ions, have been assembled from the polytopic Mannich base ligand 6,6-{[2-(dimethylamino)ethylazanediyl]bis(methylene)}bis(2-methoxy-4-methylphenol) (H₂L) with the aid of an in situ generated carbonate ligand from atmospheric CO₂ fixation. Theoretical calculations indicate axial ground states for the Ln³⁺ ions in these complexes, with their local magnetic moments being almost coplanar and tangential to the Ln³⁺ atoms that define the equilateral triangle. Therefore, they can be considered as single-molecule toroics (SMTs) with almost zero total magnetic moment. Micro-SQUID measurements on the Dy³⁺ counterpart show hysteresis loops below 3 K that have an S-shape, with large coercive fields opening upon cooling. This behavior is typical of a single molecule magnet (SMM) with very slow zero-field relaxation. At around ±0.35 T, the loops have a broad step, which is due to a direct relaxation process and corresponds to an acceleration of the relaxation of the magnetization, also observed at this magnetic field from ac susceptibility measurements. Simulations suggest that the broad step corresponds to two level avoidance of crossing points where the spin chiral Kramers doublet meets excited states of the coupled manifold, whose position is defined by exchange and dipole interactions. The Tb³⁺ counterpart does not exhibit SMM behavior, which is due to the fact that the degeneracy of the ground state of the exchange coupled system is lifted at zero field, thus favoring quantum tunneling of magnetization (QTM).

Bleaney's long-standing theory of magnetic anisotropy has been employed with some success for many decades to explain paramagnetic NMR pseudocontact shifts, and has been the subject of many subsequent approximations. Here, we present a detailed experimental and theoretical investigation accounting for the anomalous solvent dependence of NMR shifts for a series of lanthanide(III) complexes, namely [LnL¹] (Ln = Eu, Tb, Dy, Ho, Er, Tm, and Yb; L¹: 1,4,7-tris[(6-carboxypyridin-2-yl)methyl]-1,4,7-triazacyclononane), taking into account the effect of subtle ligand flexibility on the electronic structure. We show that the anisotropy of the room temperature magnetic susceptibility tensor, which in turn affects the sign and magnitude of the pseudocontact chemical shift, is extremely sensitive to minimal structural changes in the first coordination sphere of L¹. We show that DFT structural optimizations do not give accurate structural models, as assessed by the experimental chemical shifts, and thus we determine a magnetostructural correlation and employ this to evaluate the accurate solution structure for each [LnL¹]. This approach allows us to explain the counterintuitive pseudocontact shift behavior, as well as a striking solvent dependence. These results have important consequences for the analysis and design of novel magnetic resonance shift and optical emission probes that are sensitive to the local solution environment and polarity.

A detailed analysis of paramagnetic NMR shifts in a series of isostructural lanthanide complexes relavant to PARASHIFT contrast agents reveals unexpected trends in the magnetic susceptibility anisotropy that cannot be explained by the commonly used Bleaney's theory. Ab initio calculations reveal that the primary assumption of Bleaney's theorythat thermal energy is larger than the ligand field splittingdoes not hold for the lanthanide complexes in question, and likely for a large fraction of lanthanide complexes in general. This makes the orientation of the magnetic susceptibility tensor differ significantly between complexes of different lanthanides with the same ligand: one of the most popular assumptions about isostructural lanthanide series is wrong.

Gd³⁺-based spin labels are useful as an alternative to nitroxides for intramolecular distance measurements at high fields in biological systems. However, double electron-electron resonance (DEER) measurements using model Gd³⁺ complexes featured a low modulation depth and an unexpected broadening of the distance distribution for short Gd³⁺-Gd³⁺ distances, when analysed using the software designed for S = 1/2 pairs. It appears that these effects result from the different spectroscopic characteristics of Gd³⁺ - the high spin, the zero field splitting (ZFS), and the flip-flop terms in the dipolar Hamiltonian that are often ignored for spin-1/2 systems. An understanding of the factors affecting the modulation frequency and amplitude is essential for the correct analysis of Gd³⁺-Gd³⁺ DEER data and for the educated choice of experimental settings, such as Gd³⁺ spin label type and the pulse parameters. This work uses time-domain simulations of Gd³⁺-Gd³⁺ DEER by explicit density matrix propagation to elucidate the factors shaping Gd³⁺ DEER traces. The simulations show that mixing between the |+, - and |-, + states of the two spins, caused by the flip-flop term in the dipolar Hamiltonian, leads to dampening of the dipolar modulation. This effect may be mitigated by a large ZFS or by pulse frequency settings allowing for a decreased contribution of the central transition and the one adjacent to it. The simulations reproduce both the experimental line shapes of the Fourier-transforms of the DEER time domain traces and the trends in the behaviour of the modulation depth, thus enabling a more systematic design and analysis of Gd³⁺ DEER experiments.

Ytterbium and yttrium complexes of octadentate ligands based on 1,4,7,10-tetraazacyclododecane with a coordinated pyridyl group and either tricarboxylate (L¹) or triphosphinate (L²) donors form twisted-square-antiprismatic structures. The former crystallizes in the centrosymmetric group P2₁/c, with the two molecules related by an inversion center, whereas the latter was found as an unusual kryptoracemate in the chiral space group P2₁. Pure shift NMR and EXSY spectroscopy allowed the dynamic exchange between the (RRR)-Δ-(δδδδ) and (RRR)-Δ-(λλλλ) TSAP diastereomers of the [Y.L²] complex to be detected. The rate-limiting step in the exchange between Δ and Δ isomers involves cooperative ligand arm rotation, which is much faster for [Ln.L¹] than for [Ln.L²]. Detailed analysis of NOESY, COSY, HSQC, and HMBC spectra confirms that the major conformer in solution is (RRR)-Δ-(λλλλ), consistent with crystal structure analysis and DFT calculations. The magnetic susceptibility tensors for [Yb.L¹] and [Yb.L²], obtained from a full pseudocontact chemical shift analysis, are very different, in agreement with a CASSCF calculation. The remarkably different pseudocontact shift behavior is explained by the change in the orientation of the pseudocontact shift field, as defined by the Euler angles of the susceptibility tensor.

The exchange interaction, J, between two spin centres is a convenient measure of through bond electronic communication. Here, we investigate quantum interference phenomena in a bis-copper six-porphyrin nanoring by electron paramagnetic resonance spectroscopy via measurement of the exchange coupling between the copper centres. Using an analytical expression accounting for both dipolar and exchange coupling to simulate the time traces obtained in a double electron electron resonance experiment, we demonstrate that J can be quantified to high precision even in the presence of significant through-space coupling. We show that the exchange coupling between two spin centres is increased by a factor of 4.5 in the ring structure with two parallel coupling paths as compared to an otherwise identical system with just one coupling path, which is a clear signature of constructive quantum interference.

The quadrupolar interaction experienced by the spin-1 ¹⁴N nucleus is known to be extremely sensitive to local structure and dynamics. Furthermore, the ¹⁴N isotope is 99.6% naturally abundant, making it an attractive target for characterisation of nitrogen-rich biological molecules by solid-state NMR. In this study, dynamic nuclear polarization (DNP) is used in conjunction with indirect ¹⁴N detected solid-state NMR experiments to simultaneously characterise the quadrupolar interaction at multiple ¹⁴N sites in the backbone of the microcrystalline protein, GB3. Considerable variation in the quadrupolar interaction (>700 kHz) is observed throughout the protein backbone. The distribution in quadrupolar interactions observed reports on the variation in local backbone conformation and subtle differences in hydrogen-bonding; demonstrating a new route to the structural and dynamic analysis of biomolecules.

A significant problem with paramagnetic tags attached to proteins and nucleic acids is their conformational mobility. Each tag is statistically distributed within a volume between 5 and 10 Angstroms across; structural biology conclusions from NMR and EPR work are necessarily diluted by this uncertainty. The problem is solved in electron spin resonance, but remains open in the other major branch of paramagnetic resonance-pseudocontact shift (PCS) NMR spectroscopy, where structural biologists have so far been reluctantly using the point paramagnetic centre approximation. Here we describe a new method for extracting probability densities of lanthanide tags from PCS data. The method relies on Tikhonov-regularised 3D reconstruction and opens a new window into biomolecular structure and dynamics because it explores a very different range of conditions from those accessible to double electron resonance work on paramagnetic tags: a room-temperature solution rather than a glass at cryogenic temperatures. The method is illustrated using four different Tm³⁺ DOTA-M8 tagged mutants of human carbonic anhydrase II; the results are in good agreement with rotamer library and DEER data. The wealth of high-quality pseudocontact shift data accumulated by the biological magnetic resonance community over the last 30 years, and so far only processed using point models, could now become a major source of useful information on conformational distributions of paramagnetic tags in biomolecules.

Linear π-conjugated porphyrin oligomers are of significant current interest due to their potential applications as molecular wires. In this study we investigate electronic communication in linear butadiyne-linked copper porphyrin oligomers by electron paramagnetic resonance (EPR) spectroscopy via measurement of the exchange interaction, J, between the copper(ii) centers. The contributions of dipolar and exchange interactions to the frozen solution continuous wave (cw) EPR spectra of the compounds with two or more copper porphyrin units were explicitly accounted for in numerical simulations using a spin Hamiltonian approach. It is demonstrated that a complete numerical simulation of the powder spectrum of a large spin system with a Hamiltonian dimension of 26244 and beyond can be made feasible by simulating the spectra in the time domain. The exchange coupling in the Cu₂ dimer (Cu⋯Cu distance 1.35 nm) is of the order of tens of MHz (Ĥ = -2JS₁·S₂) and is strongly modulated by low-energy molecular motions such as twisting of the molecule.

We show that the acquisition of 3D diffusion-ordered NMR spectroscopy (DOSY) experiments can be accelerated significantly with the use of spatial encoding (SPEN). The SPEN DOSY approach is discussed, analysed with numerical simulation, and illustrated on a mixture of small molecules.

Magnetic resonance spectroscopy and imaging experiments in which spatial dynamics (diffusion and flow) closely coexists with chemical and quantum dynamics (spin-spin couplings, exchange, cross-relaxation, etc.) have historically been very hard to simulate-Bloch-Torrey equations do not support complicated spin Hamiltonians, and the Liouville-von Neumann formalism does not support explicit spatial dynamics. In this paper, we formulate and implement a more advanced simulation framework based on the Fokker-Planck equation. The proposed methods can simulate, without significant approximations, any spatio-temporal magnetic resonance experiment, even in situations when spatial motion co-exists intimately with quantum spin dynamics, relaxation and chemical kinetics.

Paramagnetic metal ions accelerate nuclear spin relaxation; this effect is widely used for distance measurement and called paramagnetic relaxation enhancement (PRE). Theoretical predictions established that, under special circumstances, it is also possible to achieve a reduction in nuclear relaxation rates (negative PRE). This situation would occur if the mechanism of nuclear relaxation in the diamagnetic state is counterbalanced by a paramagnetic relaxation mechanism caused by the metal ion. Here we report the first experimental evidence for such a cross-correlation effect. Using a uniformly ¹⁵N-labeled mutant of calbindin D_9k loaded with either Tm³⁺ or Tb³⁺, reduced R₁ and R₂ relaxation rates of backbone ¹⁵N spins were observed compared with the diamagnetic reference (the same protein loaded with Y³⁺). The effect arises from the compensation of the chemical shift anisotropy tensor by the anisotropic dipolar shielding generated by the unpaired electron spin.

This paper presents an overview of the Fokker-Planck formalism for non-biological magnetic resonance simulations, describes its existing applications and proposes some novel ones. The most attractive feature of Fokker-Planck theory compared to the commonly used Liouville - von Neumann equation is that, for all relevant types of spatial dynamics (spinning, diffusion, stationary flow, etc.), the corresponding Fokker-Planck Hamiltonian is time-independent. Many difficult NMR, EPR and MRI simulation problems (multiple rotation NMR, ultrafast NMR, gradient-based zero-quantum filters, diffusion and flow NMR, off-resonance soft microwave pulses in EPR, spin-spin coupling effects in MRI, etc.) are simplified significantly in Fokker-Planck space. The paper also summarises the author's experiences with writing and using the corresponding modules of the Spinach library the methods described below have enabled a large variety of simulations previously considered too complicated for routine practical use.

We demonstrate that the induced toroidal dipole, represented by currents flowing on the surface of a torus, makes a distinct and indispensable contribution to circular dichroism. We show that toroidal circular dichroism supplements the well-known mechanism involving electric dipole and magnetic dipole transitions. We illustrate this with rigorous analysis of the experimentally measured polarization-sensitive transmission spectra of an artificial metamaterial, constructed from elements of toroidal symmetry. We argue that toroidal circular dichroism will be found in large biomolecules with elements of toroidal symmetry and should be taken into account in the interpretation of circular dichroism spectra of organics.

Quadratic convergence throughout the active space is achieved for the gradient ascent pulse engineering (GRAPE) family of quantum optimal control algorithms. We demonstrate in this communication that the Hessian of the GRAPE fidelity functional is unusually cheap, having the same asymptotic complexity scaling as the functional itself. This leads to the possibility of using very efficient numerical optimization techniques. In particular, the Newton-Raphson method with a rational function optimization (RFO) regularized Hessian is shown in this work to require fewer system trajectory evaluations than any other algorithm in the GRAPE family. This communication describes algebraic and numerical implementation aspects (matrix exponential recycling, Hessian regularization, etc.) for the RFO Newton-Raphson version of GRAPE and reports benchmarks for common spin state control problems in magnetic resonance spectroscopy.

Anomalous cross-peaks observed in the NOESY spectra of 2,4-disubstituted thiazolidines and oxazolidines that cannot be attributed to classical dipolar NOE or chemical exchange peaks have been investigated experimentally and computationally and have been shown to arise from scalar cross-relaxation of the first kind. This process is stimulated by the relatively slow modulation of scalar couplings and, for the systems studied, arises from slow on-off proton exchange of the amino nitrogen, a process influenced by solution temperature, acidity, and concentration. The mechanism is likely to be significant for many systems in which proton exchange occurs on the millisecond time scale, and misinterpretation of these cross-peaks may lead to erroneous conclusions should their true origins not be recognized.

This paper presents a detailed analysis of the pseudocontact shift (PCS) field induced by a mobile spin label that is viewed as a probability density distribution with an associated effective magnetic susceptibility anisotropy. It is demonstrated that non-spherically symmetric density can lead to significant deviations from the commonly used point dipole approximation for the PCS. Analytical and numerical solutions are presented for the general partial differential equation that describes the non-point case. It is also demonstrated that it is possible, with some reasonable approximations, to reconstruct paramagnetic centre probability distributions from the experimental PCS data.

To understand the properties and/or reactivity of an organic molecule, an understanding of its three-dimensional structure is necessary. Simultaneous determination of configuration and conformation often poses a daunting challenge. Thus, the more information accessible for a given molecule, the better. Additionally to ³J-couplings, two sources of information, quantitative NOE and more recently also RDCs, are used for conformational analysis by NMR spectroscopy. In this paper, we compare these sources of conformational information in two molecules: the configurationally well-characterized strychnine 1, and the only recently configurationally and conformationally characterized α-methylene-γ-butyrolactone 2. We discuss possible sources of error in the measurement and analysis process, and how to exclude them. By this means, we are able to bolster the previously proposed flexibility for these two molecules.

It is control that turns scientific knowledge into useful technology: in physics and engineering it provides a systematic way for driving a dynamical system from a given initial state into a desired target state with minimized expenditure of energy and resources. As one of the cornerstones for enabling quantum technologies, optimal quantum control keeps evolving and expanding into areas as diverse as quantum-enhanced sensing, manipulation of single spins, photons, or atoms, optical spectroscopy, photochemistry, magnetic resonance (spectroscopy as well as medical imaging), quantum information processing and quantum simulation. In this communication, state-of-the-art quantum control techniques are reviewed and put into perspective by a consortium of experts in optimal control theory and applications to spectroscopy, imaging, as well as quantum dynamics of closed and open systems. We address key challenges and sketch a roadmap for future developments.

Ultrafast (UF) NMR spectroscopy is an approach that yields 2D spectra in a single scan. This methodology has become a powerful analytical tool that is used in a large array of applications. However, UF NMR spectroscopy still suffers from an intrinsic low sensitivity, and from the need to compromise between sensitivity, spectral width, and resolution. In particular, the modulation of signal intensities by the spin-spin J-coupling interaction (J-modulation) impacts significantly on the intensities of the spectral peaks. This effect can lead to large sensitivity losses and even to missing spectral peaks, depending on the nature of the spin system. Herein, a general simulation package (Spinach) is used to describe J-modulation effects in UF experiments. The results from simulations match with experimental data and the results of product operator calculations. Several methods are proposed to optimize the sensitivity in UF COSY spectra. The potential and drawbacks of the different strategies are also discussed. These approaches provide a way to adjust the sensitivity of UF experiments for a large range of applications.

Solid-state NMR transitions involving outer energy levels of the spin-1 ¹⁴N nucleus are immune, to first order in perturbation theory, to the broadening caused by the nuclear quadrupole interaction. The corresponding overtone spectra, when acquired in conjunction with magic-angle sample spinning, result in lines, which are just a few kHz wide, permitting the direct detection of nitrogen compounds without the need for labeling. Despite the success of this technique, "overtone" resonances are still broadened due to indirect, second order effects arising from the large quadrupolar interaction. Here we demonstrate that another order of magnitude in spectral resolution may be gained by using double rotation. This brings the width of the ¹⁴N solid-state NMR lines much closer to the region commonly associated with high-resolution solid-state NMR spectroscopy of ¹⁵N and demonstrates the improvements in resolution that may be possible through the development of pulsed methodologies to suppress these second order effects.

Auxiliary matrix exponential method is used to derive simple and numerically efficient general expressions for the following, historically rather cumbersome, and hard to compute, theoretical methods: (1) average Hamiltonian theory following interaction representation transformations; (2) Bloch-Redfield-Wangsness theory of nuclear and electron relaxation; (3) gradient ascent pulse engineering version of quantum optimal control theory. In the context of spin dynamics, the auxiliary matrix exponential method is more efficient than methods based on matrix factorizations and also exhibits more favourable complexity scaling with the dimension of the Hamiltonian matrix.

Overtone ¹⁴N NMR spectroscopy is a promising route for the direct detection of ¹⁴N signals with good spectral resolution. Its application is currently limited, however, by the absence of efficient polarization techniques for overtone signal enhancement and the lack of efficient numerical simulation techniques to aid in both the development of new methods and the analysis and interpretation of experimental data. In this paper we report a novel method for the transfer of polarization from ¹H to the ¹⁴N overtone using symmetry-based R-sequences that overcome many of the limitations of adiabatic approaches that have worked successfully on static samples. Refinement of these sequences and the analysis of the resulting spectra have been facilitated through the development of an efficient simulation strategy for ¹⁴N overtone NMR spectroscopy of spinning samples, using effective Hamiltonians on top of Floquet and Fokker-Planck equations.

Anomalous NOESY cross-peaks that cannot be explained by dipolar cross-relaxation or chemical exchange are described for carbon-substituted aziridines. The origin of these is identified as scalar cross-relaxation of the first kind, as demonstrated by a complete theoretical description of this relaxation process and by computational simulation of the NOESY spectra. It is shown that this process relies on the stochastic modulation of J-coupling by conformational transitions, which in the case of aziridines arise from inversion at the nitrogen center. The observation of scalar cross-relaxation between protons does not appear to have been previously reported for NOESY spectra. Conventional analysis would have assigned the cross-peaks as being indicative of a chemical exchange process occurring between correlated spins, were it not for the fact that the pairs of nuclei displaying them cannot undergo such exchange. Scalar relaxation: Apparently anomalous cross-peaks in 2D ¹HNOESY spectra of carbon-substituted aziridines are shown to arise from scalar cross-relaxation of the first kind (SRFK; see picture), due to modulation of scalar coupling constants between protons. Such effects will likely be seen in NOESY spectra of other small molecules experiencing dynamic exchange modulation of scalar ¹H-¹H coupling constants.

Clusters of coupled nuclear spins may form long-lived nuclear spin states, which interact weakly with the environment, compared to ordinary nuclear magnetization. All experimental demonstrations of longlived states have so far involved spin systems which are close to the condition of magnetic equivalence, in which the network of spin-spin couplings is conserved under all pair exchanges of symmetry-related nuclei. We show that the four-spin system of trans-[2,3-¹³C₂]-but-2-enedioate exhibits a long-lived nuclear spin state, even though this spin system is very far from magnetic equivalence. The 4-spin longlived state is accessed by slightly asymmetric chemical substitutions of the centrosymmetric molecular core. The long-lived state is a consequence of the locally centrosymmetric molecular geometry for the trans isomer, and is absent for the cis isomer. A general group theoretical description of long-lived states is presented. It is shown that the symmetries of coherent and incoherent interactions are both important for the existence of long-lived states.

A comprehensive conformational analysis of both 2,3-difluorobutane diastereomers is presented based on density functional theory calculations in vacuum and in solution, as well as NMR experiments in solution. While for 1,2-difluoroethane the fluorine gauche effect is clearly the dominant effect determining its conformation, it was found that for 2,3-difluorobutane there is a complex interplay of several effects, which are of similar magnitude but often of opposite sign. As a result, unexpected deviations in dihedral angles, relative conformational energies and populations are observed which cannot be rationalised only by chemical intuition. Furthermore, it was found that it is important to consider the free energies of the various conformers, as these lead to qualitatively different results both in vacuum and in solvent, when compared to calculations based only on the electronic energies. In contrast to expectations, it was found that vicinal syn-difluoride introduction in the butane and by extension, longer hydrocarbon chains, is not expected to lead to an effective stabilisation of the linear conformation. Our findings have implications for the use of the vicinal difluoride motif for conformational control.

It is demonstrated that pseudocontact shift (PCS), viewed as a scalar or a tensor field in three dimensions, obeys an elliptic partial differential equation with a source term that depends on the Hessian of the unpaired electron probability density. The equation enables straightforward PCS prediction and analysis in systems with delocalized unpaired electrons, particularly for the nuclei located in their immediate vicinity. It is also shown that the probability density of the unpaired electron may be extracted, using a regularization procedure, from PCS data.

We introduce a new method, based on alternating optimization, for compact representation of spin Hamiltonians and solution of linear systems of algebraic equations in the tensor train format. We demonstrate the method's utility by simulating, without approximations, a N15 NMR spectrum of ubiquitin - a protein containing several hundred interacting nuclear spins. Existing simulation algorithms for the spin system and the NMR experiment in question either require significant approximations or scale exponentially with the spin system size. We compare the proposed method to the Spinach package that uses heuristic restricted state space techniques to achieve polynomial complexity scaling. When the spin system topology is close to a linear chain (e.g., for the backbone of a protein), the tensor train representation is more compact and can be computed faster than the sparse representation using restricted state spaces.

Nuclear magnetic resonance spectroscopy is one of the few remaining areas of physical chemistry for which polynomially scaling quantum mechanical simulation methods have not so far been available. In this communication we adapt the restricted state space approximation to protein NMR spectroscopy and illustrate its performance by simulating common 2D and 3D liquid state NMR experiments (including accurate description of relaxation processes using Bloch-Redfield-Wangsness theory) on isotopically enriched human ubiquitin - a protein containing over a thousand nuclear spins forming an irregular polycyclic three-dimensional coupling lattice. The algorithm uses careful tailoring of the density operator space to only include nuclear spin states that are populated to a significant extent. The reduced state space is generated by analysing spin connectivity and decoherence properties: rapidly relaxing states as well as correlations between topologically remote spins are dropped from the basis set.

We introduce a simple and general XML format for spin system description that is the result of extensive consultations within Magnetic Resonance community and unifies under one roof all major existing spin interaction specification conventions. The format is human-readable, easy to edit and easy to parse using standard XML libraries. We also describe a graphical user interface that was designed to facilitate construction and visualization of complicated spin systems. The interface is capable of generating input files for several popular spin dynamics simulation packages.

We present a solid-state NMR study of H₂ molecules confined inside the cavity of C₇₀ fullerene cages over a wide range of temperatures (300 K to 4 K). The proton NMR spectra are consistent with a model in which the dipole-dipole coupling between the ortho-H₂ protons is averaged over the rotational/translational states of the confined quantum rotor, with an additional chemical shift anisotropy δ^H _CSA=10.1 ppm induced by the carbon cage. The magnitude of the chemical shift anisotropy is consistent with DFT estimates of the chemical shielding tensor field within the cage. The experimental NMR data indicate that the ground state of endohedral ortho-H₂ in C₇₀ is doubly degenerate and polarized transverse to the principal axis of the cage. The NMR spectra indicate significant magnetic alignment of the C₇₀ long axes along the magnetic field, at temperatures below ∼10 K. Rattling the cage: A solid-state NMR study of H₂ molecules confined inside the cavity of C₇₀ fullerene cages over a wide range of temperatures (300 K to 4 K) is presented. The proton NMR spectra are consistent with a model in which the dipole-dipole coupling between the ortho-H₂ protons is averaged over the rotational/translational states of the confined quantum rotor.

Several methods are proposed for the analysis, visualization and interpretation of high-dimensional spin system trajectories produced by quantum mechanical simulations. It is noted that expectation values of specific observables in large spin systems often feature fast, complicated and hard-to-interpret time dynamics and suggested that populations of carefully selected subspaces of states are much easier to analyze and interpret. As an illustration of the utility of the proposed methods, it is demonstrated that the apparent ''noisy'' appearance of many optimal control pulses in NMR and EPR spectroscopy is an illusion - the underlying spin dynamics is shown to be smooth, orderly and very tightly controlled.

We demonstrate that Fokker-Planck equations in which spatial coordinates are treated on the same conceptual level as spin coordinates yield a convenient formalism for treating magic angle spinning NMR experiments. In particular, time dependence disappears from the background Hamiltonian (sample spinning is treated as an interaction), spherical quadrature grids are avoided completely (coordinate distributions are a part of the formalism) and relaxation theory with any linear diffusion operator is easily adopted from the Stochastic Liouville Equation theory. The proposed formalism contains Floquet theory as a special case. The elimination of the spherical averaging grid comes at the cost of increased matrix dimensions, but we show that this can be mitigated by the use of state space restriction and tensor train techniques. It is also demonstrated that low correlation order basis sets apparently give accurate answers in powder-averaged MAS simulations, meaning that polynomially scaling simulation algorithms do exist for a large class of solid state NMR experiments.

Molecules that support ¹³C singlet states with lifetimes of over 10 min in solution have been designed and synthesized. The ¹³C ₂ spin pairs in the asymmetric alkyne derivatives are close to magnetic equivalence, so the ¹³C long-lived singlet states are stable in high magnetic field and do not require maintenance by a radiofrequency spin-locking field. We suggest a model of singlet relaxation by fluctuating chemical shift anisotropy tensors combined with leakage associated with slightly broken magnetic equivalence. Theoretical estimates of singlet relaxation rates are compared with experimental values. Relaxation due to antisymmetric shielding tensor components is significant.

The physicochemical basis of probe design for ¹⁹F MRS and MRI applications is reviewed. Complexes that give a single major resonance in solution are described, in which the Ln³⁺ ion is about 6 É from the ¹⁹F label. Sensitivity improvements of 15-fold are reported in both imaging and spectroscopy based on longitudinal relaxation enhancement. The pseudocontact shift allows an amplification of chemical shift non-equivalence in responsive ¹⁹F probes, e.g. for monitoring pH in the range between 5 and 8. Sensitivity in ¹⁹F magnetic resonance spectroscopy and imaging is enhanced by placing a paramagnetic lanthanide within 7 E of the spin label. Faster relaxation allows more rapid data acquisition for systems generating one main resonance, and the proximate lanthanide ion amplifies the chemical shift non-equivalence in responsive probes.

An attempt is made to bypass spectral analysis and fit internal coordinates of radicals directly to experimental liquid- and solid-state electron spin resonance (ESR) spectra. We take advantage of the recently introduced large-scale spin dynamics simulation algorithms and of the fact that the accuracy of quantum mechanical calculations of ESR parameters has improved to the point of quantitative correctness. Partial solutions are offered to the local minimum problem in spectral fitting and to the problem of spin interaction parameters (hyperfine couplings, chemical shifts, etc.) being very sensitive to vibrational excursions from the equilibrium geometry.

A strategy is described for simulations of solid effect dynamic nuclear polarisation that reduces substantially the dimension of the quantum mechanical problem. Averaging the Hamiltonian in the doubly rotating frame is used to confine the active space to the zero quantum coherence subspace. A further restriction of the Liouville space is made by truncating higher spin order states, which are weakly populated due to the presence of relaxation processes. Based on a dissipative transport equation, which is used to estimate the transport of the magnetisation starting from single spin order to higher spin order states, a minimal spin order for the states is calculated that needs to be taken into account for the spin dynamics simulation. The strategy accelerates individual spin calculations by orders of magnitude, thus making it possible to simulate the polarisation dynamics of systems with up to 25 nuclear spins.

Several methods for density matrix propagation in parallel computing environments are proposed and evaluated. It is demonstrated that the large communication overhead associated with each propagation step (two-sided multiplication of the density matrix by an exponential propagator and its conjugate) may be avoided and the simulation recast in a form that requires virtually no inter-thread communication. Good scaling is demonstrated on a 128-core (16 nodes, 8 cores each) cluster.

We report some improvements to the gradient ascent pulse engineering (GRAPE) algorithm for optimal control of spin ensembles and other quantum systems. These include more accurate gradients, convergence acceleration using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton algorithm as well as faster control derivative calculation algorithms. In all test systems, the wall clock time and the convergence rates show a considerable improvement over the approximate gradient ascent.

Although originally designed for broadband inversion and decoupling in NMR spectroscopy, recent methodological developments have introduced adiabatic fast passage (AFP) pulses into the field of protein dynamics. AFP pulses employ a frequency sweep, and have not only superior inversion properties with respect to offset effects, but they are also easily implemented into a pulse sequence. As magnetization is dragged from the +z to the -z direction, Larmor precession is impeded since magnetization becomes spin-locked, which is a potentially useful feature for the investigation of microsecond to millisecond dynamics. A major drawback of these pulses as theoretical prediction is concerned, however, results from their time-dependent offset: simulations of spin density matrices under the influence of a time-dependent Hamiltonian with non-commuting elements are costly in terms of computational time, rendering data analysis impracticable. In this paper we suggest several ways to reduce the computational time without compromising accuracy with respect to effects such as cross-correlated relaxation and modulation of the chemical shift.

We present an algebraic foundation for the state space restriction approximation in spin dynamics simulations and derive applicability criteria as well as minimal basis set requirements for practically encountered simulation tasks. The results are illustrated with nuclear magnetic resonance (NMR), electron spin resonance (ESR), dynamic nuclear polarization (DNP), and spin chemistry simulations. It is demonstrated that state space restriction yields accurate results in systems where the time scale of spin relaxation processes approximately matches the time scale of the experiment. Rigorous error bounds and basis set requirements are derived.

A numerical procedure is presented for mapping the vicinity of the null-space of the spin relaxation superoperator. The states populating this space, i.e. those with near-zero eigenvalues, of which the two-spin singlet is a well-studied example, are long-lived compared to the conventional T ₁ and T ₂ spin-relaxation times. The analysis of larger spin systems described herein reveals the presence of a significant number of other slowly relaxing states. A study of coupling topologies for n-spin systems (4 ≤ n ≤ 8) suggests the symmetry requirements for maximising the number of long-lived states.

The enantioselective binding of the (SSS)-Δ isomer of an yttrium(iii) tetraazatriphenylene complex to 'drug-site II' of human serum albumin (HSA) was detected by the intensity differences of its STD ¹H NMR spectrum relative to the (RRR)-Λ isomer, by the effect of the competitive binder to that site, N-dansyl sarcosine, upon the STD spectrum of each isomer, in the presence of HSA and by 3D docking simulations.

We propose a novel avenue for state space reduction in time domain Liouville space spin dynamics simulations, using detectability as a selection criterion - only those states that evolve into or affect other detectable states are kept in the simulation. This basis reduction procedure (referred to as destination state screening) is formally exact and can be applied on top of the existing state space restriction techniques. As demonstrated below, in many cases this results in further reduction of matrix dimension, leading to considerable acceleration of many spin dynamics simulation types. Destination state screening is implemented in the latest version of the Spinach library (http://spindynamics.org).

The successful measurement of anisotropic NMR parameters like residual dipolar couplings (RDCs), residual quadrupolar couplings (RQCs), or residual chemical shift anisotropy (RCSA) involves the partial alignment of solute molecules in an alignment medium. To avoid any influence of the change of environment from the isotropic to the anisotropic sample, the measurement of both datasets with a single sample is highly desirable. Here, we introduce the scaling of alignment for mechanically stretched polymer gels by varying the angle of the director of alignment relative to the static magnetic field, which we call variable angle NMR spectroscopy (VA-NMR). The technique is closely related to variable angle sample spinning NMR spectroscopy (VASS-NMR) of liquid crystalline samples, but due to the mechanical fixation of the director of alignment no sample spinning is necessary. Also, in contrast to VASS-NMR, VA-NMR works for the full range of sample inclinations between 0° and 90°. Isotropic spectra are obtained at the magic angle. As a demonstration of the approach we measure ¹³C-RCSA values for strychnine in a stretched PDMS/CDCl₃ gel and show their usefulness for assignment purposes. In this context special care has been taken with respect to the exact calibration of chemical shift data, for which three approaches have been derived and tested.

The Liouville space spin relaxation theory equations are reformulated in such a way as to avoid the computationally expensive Hamiltonian diagonalization step, replacing it by numerical evaluation of the integrals in the generalized cumulant expansion. The resulting algorithm is particularly useful in the cases where the static part of the Hamiltonian is dominated by interactions other than Zeeman (e.g. in quadrupolar resonance, low-field EPR and Spin Chemistry). When used together with state space restriction tools, the algorithm reported is capable of computing full relaxation superoperators for NMR systems with more than 15 spins.

We introduce a software library incorporating our recent research into efficient simulation algorithms for large spin systems. Liouville space simulations (including symmetry, relaxation and chemical kinetics) of most liquid-state NMR experiments on 40+ spin systems can now be performed without effort on a desktop workstation. Much progress has also been made with improving the efficiency of ESR, solid state NMR and Spin Chemistry simulations. Spinach is available for download at http://spindynamics.org.

The electron-transfer (ET) reactions in photosystem I (PS I) of prokaryotes have been investigated in wild-type cells of the cyanobacterium Synechocystis sp. PCC 6803, and in two site-directed mutants in which the methionine residue of the reaction center subunits PsaA and PsaB, which acts as the axial ligand to the primary electron chlorophyll acceptor A₀, was substituted with histidine. Analysis by pulsed electron paramagnetic resonance spectroscopy at 100 K indicates the presence of two forms of the secondary spin-correlated radical pairs, which are assigned to [P₇₀₀⁺A _1A^-] and [P₇₀₀⁺A_1B ^-], where A_1A and A_1B are the phylloquinone molecules bound to the PsaA and the PsaB reaction center subunits, respectively. Each of the secondary radical pair forms is selectively observed in either the PsaA-M688H or the PsaB-M668H mutant, whereas both radical pairs are observed in the wild type following reduction of the iron-sulfur cluster F_X, the intermediate electron acceptor between A₁ and the terminal acceptors F_A and F_B. Analysis of the time and spectral dependence of the light-induced electron spin echo allows the resolution of structural differences between the [P₇₀₀⁺A_1A^-] and [P₇₀₀⁺A_1B^-] radical pairs. The interspin distance is 25.43 ± 0.01 Å for [P₇₀₀ ⁺A_1A^-] and 24.25 ± 0.01 Å for [P₇₀₀⁺A_1B^-]. Moreover, the relative orientation of the interspin vector is rotated by ∼60° with respect to the g-tensor of the P₇₀₀⁺ radical. These estimates are in agreement with the crystallographic structural model, indicating that the cofactors bound to both reaction center subunits of prokaryotic PS I are actively involved in electron transport. This work supports the model that bidirectionality is a general property of type I reaction centers from both prokaryotes and eukaryotes, and contrasts with the situation for photosystem II and other type II reaction centers, in which ET is strongly asymmetric. A revised model that explains qualitatively the heterogeneity of ET reactions at cryogenic temperatures is discussed.

A series of lanthanide complexes have been synthesized from 1,4,7,10-tetrakis(2-pyridylmethyl)-1,4,7,10-tetraazacyclododecane. Crystallographic studies indicate that, in the solid phase, all of the lanthanide ions are 9-coordinate and are bound to eight N atoms from the donor ligand, with the ninth site being filled by a counterion or solvent molecule. In solution, time-resolved luminescence studies indicate that the luminescence exhibits contributions from two species corresponding to the nonhydrated and hydrated forms. The NMR spectra in protic media show the presence of two dominant isomers on the NMR time scale; furthermore, the spectra are very different from those obtained for 1,4,7,10-tetraazacyclododecane-N, N,NT,NT-tetraacetic acid (DOTA) and its derivatives. The different forms of the complex undergo slow conformational and enantiomeric exchange in solution, which has been measured by NMR. The exchange path has been mapped out by density functional theory calculations and shows multiple metastable conformations (with respect to the dihedral angles of the cyclen ring). This contrasts with the established NMR behavior of DOTA complexes, which has been described by a two-state solution equilibrium.

We propose three basis screening methods for state space restriction in Liouville space simulations of large densely coupled spin systems encountered in electron paramagnetic resonance (EPR) spectroscopy and spin chemistry. The methods are based on conservation law analysis, symmetry factorization, and the analysis of state space connectivity graphs. A reduction in matrix dimensions by several orders of magnitude is demonstrated for common EPR and spin chemistry systems.

Systematic benchmarking of multi-dimensional protein NMR experiments is a critical prerequisite for optimal allocation of NMR resources for structural analysis of challenging proteins, e.g. large proteins with limited solubility or proteins prone to aggregation. We propose a set of benchmarking parameters for essential protein NMR experiments organized into a lightweight (single XML file) relational database (RDB), which includes all the necessary auxiliaries (waveforms, decoupling sequences, calibration tables, setup algorithms and an RDB management system). The database is interfaced to the Spinach library (http://spindynamics.org), which enables accurate simulation and benchmarking of NMR experiments on large spin systems. A key feature is the ability to use a single user-specified spin system to simulate the majority of deposited solution state NMR experiments, thus providing the (hitherto unavailable) unified framework for pulse sequence evaluation. This development enables predicting relative sensitivity of deposited implementations of NMR experiments, thus providing a basis for comparison, optimization and, eventually, automation of NMR analysis. The benchmarking is demonstrated with two proteins, of 170 amino acids I domain of αXβ2 Integrin and 440 amino acids NS3 helicase.

The synthesis and spectroscopic properties of a series of CF ₃-labelled lanthanide(III) complexes (Ln = Gd, Tb, Dy, Ho, Er, Tm) with amidesubstituted ligands based on 1,4,7,10tetraazacyclododecane are described. The theoretical contributions of the ¹⁹F magnetic relaxation processes in these systems are critically assessed and selected volumetric plots are presented. These plots allow an accurate estimation of the increase in the rates of longitudinal and transverse relaxation as a function of the distance between the Ln^III ion and the fluorine nucleus, the applied magnetic field, and the re-rotational correlation time of the complex, for a given Ln^III ion. Selected complexes exhibit pH-dependent chemical shift behaviour, and a pK_a, of 7.0 was determined in one example based on the holmium complex of an orthocyano DO3A-monoamide ligand, which allowed the pH to be assessed by measuring the difference in chemical shift (varying by over 14ppm) between two ¹⁹F resonances. Relaxation analyses of variable-temperature and variable-field ¹⁹F, ¹⁷O and ¹HNMR spectroscopy experiments are reported, aided by identification of salient lowenergy conformers by using density functional theory. The study of fluorine relaxation rates, over a field range of 4.7 to 16.5 T allowed precise computation of the distance between the Ln^III ion and the CF₃ reporter group by using global fitting methods. The sensitivity benefits of using such paramagnetic fluorinated probes in ¹⁹FNMR spectroscopic studies are quantified in preliminary spectroscopic and imaging experiments with respect to a diamagnetic yttrium(III) analogue.

We report analytical equations for the derivatives of spin dynamics simulations with respect to pulse sequence and spin system parameters. The methods described are significantly faster, more accurate, and more reliable than the finite difference approximations typically employed. The resulting derivatives may be used in fitting, optimization, performance evaluation, and stability analysis of spin dynamics simulations and experiments.

We extend the recently proposed state-space restriction (SSR) technique for quantum spin dynamics simulations [Kuprov et al., J. Magn. Reson. 189 (2007) 241-250] to include on-the-fly detection and elimination of unpopulated dimensions from the system density matrix. Further improvements in spin dynamics simulation speed, frequently by several orders of magnitude, are demonstrated. The proposed zero track elimination (ZTE) procedure is computationally inexpensive, reversible, numerically stable and easy to add to any existing simulation code. We demonstrate that it belongs to the same family of Krylov subspace techniques as the well-known Lanczos basis pruning procedure. The combined SSR + ZTE algorithm is recommended for simulations of NMR, EPR and Spin Chemistry experiments on systems containing between 10 and 10⁴ coupled spins.

Approximately 50 species, including birds, mammals, reptiles, amphibians, fish, crustaceans and insects, are known to use the Earth's magnetic field for orientation and navigation. Birds in particular have been intensively studied, but the biophysical mechanisms that underlie the avian magnetic compass are still poorly understood. One proposal, based on magnetically sensitive free radical reactions, is gaining support despite the fact that no chemical reaction in vitro has been shown to respond to magnetic fields as weak as the Earth's (∼50 μT) or to be sensitive to the direction of such a field. Here we use spectroscopic observation of a carotenoid-porphyrin-fullerene model system to demonstrate that the lifetime of a photochemically formed radical pair is changed by application of ≤50 μT magnetic fields, and to measure the anisotropic chemical response that is essential for its operation as a chemical compass sensor. These experiments establish the feasibility of chemical magnetoreception and give insight into the structural and dynamic design features required for optimal detection of the direction of the Earth's magnetic field.

Experimental measurements and theoretical analysis of magnetic properties, structural dynamics and acid-base equilibria for several lanthanide(III) complexes with tetraazacyclododecane derivatives as ¹⁹F NMR chemical shift pH probes are presented; pK_a values vary between 6.9 and 7.7, with 18 to 40 ppm chemical shift differences between the acidic and basic forms for Ho(III) complexes possessing T₁ values of 10 to 30 ms (4.7-9.4 T, 295 K).

We report progress with an old problem in magnetic resonance-that of the exponential scaling of simulation complexity with the number of spins. It is demonstrated below that a polynomially scaling algorithm can be obtained (and accurate simulations performed for over 200 coupled spins) if the dimension of the Liouville state space is reduced by excluding unimportant and unpopulated spin states. We found the class of such states to be surprisingly wide. It actually appears that a majority of states in large spin systems are not essential in magnetic resonance simulations and can safely be dropped from the state space. In restricted state spaces the spin dynamics simulations scale polynomially. In cases of favourable interaction topologies (sparse graphs, e.g. in protein NMR) the asymptotic scaling is linear, opening the way to direct fitting of molecular structures to experimental spectra.

We describe experimental results and theoretical models for nuclear and electron spin relaxation processes occurring during the evolution of ¹⁹F-labeled geminate radical pairs on a nanosecond time scale. In magnetic fields of over 10 T, electron-nucleus dipolar cross-relaxation and longitudinal ΔHFC-Δg (hyperfine coupling anisotropy - g-tensor anisotropy) cross-correlation are shown to be negligibly slow. The dominant relaxation process is transverse ΔHFC-Δg cross-correlation, which is shown to lead to an inversion in the geminate ¹⁹F chemically induced dynamic nuclear polarization (CIDNP) phase for sufficiently large rotational correlation times. This inversion has recently been observed experimentally and used as a probe of local mobility in partially denatured proteins (Khan, F.; et al. J. Am. Chem. Soc. 2006, 128, 10729-10737). The essential feature of the spin dynamics model employed here is the use of the complete spin state space and the complete relaxation superoperator. On the basis of the results reported, we recommend this approach for reliable treatment of magnetokinetic systems in which relaxation effects are important.

We describe a general method for the automated symbolic processing of Bloch-Redfield-Wangsness relaxation theory equations for liquid-phase spin dynamics in the algebraically challenging case of rotationally modulated interactions. The processing typically takes no more than a few seconds (on a contemporary single-processor workstation) and yields relaxation rate expressions that are completely general with respect to the spectral density functions, relative orientations, and magnitudes of the interaction tensors, with all cross-correlations accounted for. The algorithm easily deals with fully rhombic interaction tensors, and is able, with little if any modification, to treat a large variety of the relaxation mechanisms encountered in NMR, EPR, and spin dynamics in general.

Biosynthetic preparation and ¹⁹F NMR experiments on uniformly 3-fluorotyrosine-labeled green fluorescent protein (GFP) are described. The ¹⁹F NMR signals of all 10 fluorotyrosines are resolved in the protein spectrum with signals spread over 10 ppm. Each tyrosine in GFP was mutated in turn to phenylalanine. The spectra of the Tyr → Phe mutants, in conjunction with relaxation data and results from ¹⁹F photo-CIDNP (chemically induced dynamic nuclear polarization) experiments, yielded a full ¹⁹F NMR assignment. Two ¹⁹F-Tyr residues (Y92 and Y143) were found to yield pairs of signals originating from ring-flip conformers; these two residues must therefore be immobilized in the native structure and have ¹⁹F nuclei in two magnetically distinct positions depending on the orientation of the aromatic ring. Photo-CIDNP experiments were undertaken to probe further the structure of the native and denatured states. The observed NMR signal enhancements were found to be consistent with calculations of the HOMO (highest occupied molecular orbital) accessibilities of the tyrosine residues. The photo-CIDNP spectrum of native GFP shows four peaks corresponding to the four tyrosine residues that have solvent-exposed HOMOs. In contrast, the photo-CIDNP spectra of various denatured states of GFP show only two peaks corresponding to the ¹⁹F-labeled tyrosine side chains and the ¹⁹F-labeled Y66 of the chromophore. These data suggest that the pH-denatured and GdnDCl-denatured states are similar in terms of the chemical environments of the tyrosine residues. Further analysis of the sign and amplitude of the photo-CIDNP effect, however, provided strong evidence that the denatured state at pH 2.9 has significantly different properties and appears to be heterogeneous, containing subensembles with significantly different rotational correlation times.

The decay of the light-induced spin-correlated radical pair [P ₇₀₀⁺A₁^-] and the associated electron spin echo envelope modulation (ESEEM) have been studied in either thylakoid membranes, cellular membranes, or purified photosystem I prepared from the wild-type strains of Synechocystis sp. PCC 6803, Chlamydomonas reinhardtii, and Spinaceae oleracea. The decay of the spin-correlated radical pair is described in the wild-type membrane by two exponential components with lifetimes of 2-4 and 16-25 μs. The proportions of the two components can be altered by preillumination of the membranes in the presence of reductant at temperatures lower than 220 K, which leads to the complete reduction of the iron-sulfur electron acceptors F_A, F_B, and F_X and partial photoaccumulation of the reduced quinone electron acceptor A_1A ^-. The "out-of-phase" (OOP) ESEEM attributed to the [P ₇₀₀⁺A₁^-] radical pair has been investigated in the three species as a function of the preillumination treatment. Values of the dipolar (D) and the exchange (J) interactions were extracted by time-domain fitting of the OOP-ESEEM. The results obtained in the wild-type systems are compared with two site-directed mutants of C. reinhardtii [Santabarbara et al. (2005) Biochemistry 44, 2119-2128], in which the spin-polarized signal on either the PsaA- or PsaB-bound electron transfer pathway is suppressed so that the radical pair formed on each electron transfer branch could be monitored selectively. This comparison indicates that when all of the iron-sulfur centers are oxidized, only the echo modulation associated with the A branch [P₇₀₀⁺A_1A^-] radical pair is observed. The reduction of the iron-sulfur clusters and the quinone A₁ by preillumination treatment induces a shift in the ESEEM frequency. In all of the systems investigated this observation can be interpreted in terms of different proportions of the signal associated with the [P₇₀₀⁺A_1A^-] and [P ₇₀₀⁺A_1B^-] radical pairs, suggesting that bidirectionality of electron transfer in photosystem I is a common feature of all species rather than being confined to green algae.

The properties of a new class of pulse sequences for photo-CIDNP (photochemically induced dynamic nuclear polarization) are analysed in detail, and guidelines for their optimization and applicability are derived. Sensitivity is a central problem with time-resolved photo-CIDNP experiments. By using multiple laser flashes per acquisition and storing the polarizations temporarily in the spin system, a significant improvement is achieved. An alternative application is the reduction of the absorbed light needed to attain a given sensitivity. Compared to conventional signal averaging with the same number n of flashes, a maximum additional improvement by a factor of slightly more than can be achieved in both cases. By an analysis of the transfer pathways, it is shown that multiplet signals and CIDNP multiplet effects can also be investigated in this way, even for strongly coupled spin systems. Experimental examples are given.

Photoionization of N, N, N, N-tetramethyl-p-phenylenediamine (TMPD) in alcoholic solution produces the radical ion pair [TMPD⁺ e⁻]. However, the identity of the negatively charged counter-radical formed by photolysis of TMPD in DMSO (dimethylsulphoxide)/toluene mixtures, for which unusually large effects of weak applied magnetic fields have been observed, is unclear. Modulated MARY (Magnetically Affected Reaction Yield) experiments on solutions containing different isotopomers of TMPD, DMSO and toluene show that the counter-radical is likely to be the solvated electron. This result supports the idea that large effects of weak fields on radical recombination yields can be expected for radical pairs in which the electronnuclear hyperfine interactions are concentrated in one of the radicals, rather than being distributed more evenly between the two radicals.

The chromophore conformations of the red and far red light induced product states "Pfr" and "Pr" of the N-terminal photoreceptor domain Cph1-N515 from Synechocystis 6803 have been investigated by NMR spectroscopy, using specific ¹³C isotope substitutions in the chromophore. ¹³C-NMR spectroscopy in the Pfr and Pr states indicated reversible chemical shift differences predominantly of the C₄ carbon in ring A of the phycocyanobilin chromophore, in contrast to differences of C₁₅ and C₅, which were much less pronounced. Ab initio calculations of the isotropic shielding and optical transition energies identify a region for C₄-C₅-C₆-N₂ dihedral angle changes where deshielding of C₄ is correlated with red-shifted absorption. These could occur during thermal reactions on microsecond and millisecond timescales after excitation of Pr which are associated with red-shifted absorption. A reaction pathway involving a hula-twist at C₅ could satisfy the observed NMR and visible absorption changes. Alternatively, C ₁₅ Z-E photoisomerization, although expected to lead to a small change of the chemical shift of C₁₅, in addition to changes of the C₄-C₅-C₆-N₂ dihedral angle could be consistent with visible absorption changes and the chemical shift difference at C₄. NMR spectroscopy of a ¹³C-labeled chromopeptide provided indication for broadening due to conformational exchange reactions in the intact photoreceptor domain, which is more pronounced for the C- and D-rings of the chromophore. This broadening was also evident in the F2 hydrogen dimension from heteronuclear ¹H-¹³C HSQC spectroscopy, which did not detect resonances for the ¹³C₅-H, ¹³C₁₀-H, and ¹³C₁₅-H hydrogen atoms whereas strong signals were detected for the ¹³C-labeled chromopeptide. The most pronounced ¹³C-chemical shift difference between chromopeptide and intact receptor domain was that of the ¹³C₄-resonance, which could be consistent with an increased conformational energy of the C₄-C₅-C ₆-N₂ dihedral angle in the intact protein in the Pr state. Nuclear Overhauser effect spectroscopy experiments of the ¹³C- labeled chromopeptide, where chromophore-protein interactions are expected to be reduced, were consistent with a ZZZssa conformation, which has also been found for the biliverdin chromophore in the x-ray structure of a fragment of Deinococcus radiodurans bacteriophytochrome in the Pr form.

Pulse sequences have been developed that add up time-resolved photo-CIDNP signals from n successive laser flashes not in the acquisition computer of the NMR spectrometer but in the experiment itself, resulting in a greatly improved signal-to-noise ratio. For this accumulation, CIDNP is first stored in the transverse plane and then on the z axis, and finally superimposed on CIDNP produced by the next flash. These storage cycles also result in a very efficient background suppression. Because only one free induction decay is acquired for n flashes, the noise is digitized only once. The signal gain is demonstrated experimentally and analyzed theoretically. Losses are mostly due to nuclear spin relaxation, and to a small extent to instrument imperfections. With 10 laser flashes, a signal increase by a factor of about 7.5 was realized. As their main advantage compared to signal averaging in the usual way, these sequences yield the same signal-to-noise ratio with fewer laser flashes; the theoretical improvement is by a factor of √n.

We describe the design and operation of microsecond time-resolved photo-chemically induced dynamic nuclear polarization hardware, designed to go with a commercial 600 MHz nuclear magnetic resonance (NMR) spectrometer. A frequency-tripled neodymium-doped yttrium/aluminum garnet (Nd:YAG) laser is used as the light source, with a system of mirrors or prisms to route light to the NMR sample from above, removing the need for NMR probe modifications. The experiment has been designed for a shared NMR spectrometer and is straightforward and inexpensive to assemble and operate.

The spin-correlated radical pair [P₇₀₀⁺A ₁^-] gives rise to a characteristic "out-of- phase" electron spin-echo signal. The electron spin-echo envelope modulation (ESEEM) of these signals has been studied in thylakoids prepared from the wild-type strain of Chlamydomonas reinhardtii and in two site-directed mutants, in which the methionine residue which acts as the axial ligand to the chlorin electron acceptor A⁰ has been substituted with a histidine either on the PsaA (PsaA-M684H) or the PsaB (PsaB-M664H) reaction center subunits. The analysis of the time domain ESEEM provides information about the spin-spin interaction in the [P₇₀₀⁺A₁ ^-] radical pair, and the values of the dipolar (D) and the exchange (J) interaction can be extracted. From the distance dependence of the dipolar coupling term, the distance between the unpaired electron spin density clouds of the primary donor P₇₀₀⁺ and the phyllosemiquinone A ₁^- can be determined. The [P₇₀₀ ⁺A₁^-] ESEEM spectrum obtained in wild-type thylakoids can be reconstructed using a linear combination of the spectra measured in the PsaA and PsaB A₀ mutants, demonstrating that electron transfer resulting in charge separation is occurring on both the PsaA and PsaB branches. The [P₇₀₀⁺A_1B^-] distance in the point dipole approximation in the PsaA-M684H mutant is 24.27 ± 0.02 Å, and the [P₇₀₀⁺A_1A^-] distance in the PsaB-M664H mutant is 25.43 ± 0.01 Å An intermediate value of 25.01 ± 0.02 Å is obtained in the wild-type membranes which exhibit both spin-polarized pairs.

We demonstrate a simple, inexpensive method for in situ laser illumination of NMR samples using a stepwise tapered optical fibre to deliver light uniformly along the axis of a 5 mm NMR tube. The optical path length of the incident light inside the sample is about 3 mm, allowing efficient illumination of optically dense samples. The degradation in spectral resolution and the reduction in filling factor are both minimal. Probe modifications are not required.

Chemically induced dynamic nuclear polarisation (CIDNP) is explored as a source of nuclear hyperpolarisation in heteronuclear Overhauser effect experiments. A photochemical reaction proceeding through a radical pair intermediate is used to enhance ¹⁹F nuclear magnetisation in 3-fluorotyrosine by more than an order of magnitude with a corresponding increase in the amplitudes of ¹⁹F-¹H cross-relaxation and cross-correlation effects. The reactions employed are cyclic and leave the sample chemically unchanged. The potential for enhancing the sensitivity of heteronuclear NOEs in ¹⁹F-labelled proteins is discussed.