Long-lived states in solution NMR: selection rules for intramolecular dipolar relaxation in low magnetic fields

The Dynamical Ensemble of the Posner Molecule Is Not Symmetric

The Posner molecule, Ca₉(PO₄)₆, has long been recognized to have biochemical relevance in various physiological processes. It has found recent attention for its possible role as a biological quantum information processor, whereby the molecule purportedly maintains long-lived nuclear spin coherences among its ³¹P nuclei (presumed to be symmetrically arranged), allowing it to function as a room temperature qubit. The structure of the molecule has been of much dispute in the literature, although the S₆ point group symmetry has often been assumed and exploited in calculations. Using a variety of simulation techniques (including ab initio molecular dynamics and structural relaxation), rigorous data analysis tools, and by exploring thousands of individual configurations, we establish that the molecule predominantly assumes low-symmetry structures (C_s and C_i) at room temperature, as opposed to the higher-symmetry configurations explored previously. Our findings have important implications for the viability of this molecule as a qubit.

Nuclear singlet relaxation by chemical exchange

The population imbalance between nuclear singlet states and triplet states of strongly coupled spin-1/2 pairs, also known as nuclear singlet order, is well protected against several common relaxation mechanisms. We study the nuclear singlet relaxation of ¹³C pairs in aqueous solutions of 1,2-¹³C₂ squarate over a range of pH values. The ¹³C singlet order is accessed by introducing ¹⁸O nuclei in order to break the chemical equivalence. The squarate dianion is in chemical equilibrium with hydrogen-squarate (SqH^-) and squaric acid (SqH₂) characterized by the dissociation constants pK₁ = 1.5 and pK₂ = 3.4. Surprisingly, we observe a striking increase in the singlet decay time constants T_S when the pH of the solution exceeds ∼10, which is far above the acid-base equilibrium points. We derive general rate expressions for chemical-exchange-induced nuclear singlet relaxation and provide a qualitative explanation of the T_S behavior of the squarate dianion. We identify a kinetic contribution to the singlet relaxation rate constant, which explicitly depends on kinetic rate constants. Qualitative agreement is achieved between the theory and the experimental data. This study shows that infrequent chemical events may have a strong effect on the relaxation of nuclear singlet order.

Centralizer theory for long-lived spin states

Nuclear long-lived spin states represent spin density operator configurations that are exceptionally well protected against spin relaxation phenomena. Their long-lived character is exploited in a variety of Nuclear Magnetic Resonance (NMR) techniques. Despite the growing importance of long-lived spin states in modern NMR, strategies for their identification have changed little over the last decade. The standard approach heavily relies on a chain of group theoretical arguments. In this paper, we present a more streamlined method for the calculation of such configurations. Instead of focusing on the symmetry properties of the relaxation superoperator, we focus on its corresponding relaxation algebra. This enables us to analyze long-lived spin states with Lie algebraic methods rather than group theoretical arguments. We show that the centralizer of the relaxation algebra forms a basis for the set of long-lived spin states. The characterization of the centralizer, on the other hand, does not rely on any special symmetry arguments, and its calculation is straightforward. We outline a basic algorithm and illustrate advantages by considering long-lived spin states for some spin-1/2 pairs and rapidly rotating methyl groups.

Magnetization Lifetimes Prediction and Measurements Using Long-Lived Spin States in Endogenous Molecules

Nuclear magnetization storage in biologically-relevant molecules opens new possibilities for the investigation of metabolic pathways, provided the lifetimes of magnetization are sufficiently long. Dissolution-dynamic nuclear polarization-based spin-order enhancement, sustained by long-lived states can measure the ratios between concentrations of endogenous molecules on a cellular pathway. These ratios can be used as meters of enzyme function. Biological states featuring intracellular amino-acid concentrations that are depleted or replenished in the course of in-cell or in-vivo tests of drugs or radiation treatments can be revealed. Progressing from already-established long-lived states, we investigated related spin order in the case of amino acids and other metabolites featuring networks of coupled spins counting up to eight nuclei. We detail a new integrated theoretical approach between quantum chemistry simulations, chemical shifts, J-couplings information from databanks, and spin dynamics calculations to deduce a priori magnetization lifetimes in biomarkers. The lifetimes of long-lived states for several amino acids were also measured experimentally in order to ascertain the approach. Experimental values were in fair agreement with the computed ones and prior data in the literature.

Rotational-permutational dual-pairing and long-lived spin order

Quantum systems in contact with a thermal environment experience coherent and incoherent dynamics. These drive the system back toward thermal equilibrium after an initial perturbation. The relaxation process involves the reorganization of spin state populations and the decay of spin state coherences. In general, individual populations and coherences may exhibit different relaxation time constants. Particular spin configurations may exhibit exceptionally long relaxation time constants. Such spin configurations are known as long-lived spin order. The existence of long-lived spin order is a direct consequence of the symmetries of the system. For nuclear spin systems, rotational and permutational symmetries are of fundamental importance. Based on the Schur-Weyl duality theorem, we describe a theoretical framework for the study of rotational and permutational dual-symmetries in the context of long-lived spin order. Making use of the proposed formalism, we derive refined bounds on the number on long-lived spin populations and coherences for systems exhibiting rotational-permutational dual-symmetries.

Singlet excitation in the intermediate magnetic equivalence regime and field-dependent study of singlet–triplet leakage

Measuring field-dependence of singlet lifetimes in the intermediate magnetic equivalence regime.

Effect of convection and B1 inhomogeneity on singlet relaxation experiments

Nuclear spin singlet lifetimes can often exceed the T₁ length scales by a large factor. This property makes them suitable for polarization storage. The measurement of such long lifetimes itself can become challenging due to the influence of even very weak relaxation mechanisms. Here we show that a judicious choice of the singlet-to-triplet conversion method is highly important in order to achieve reliable singlet relaxation measurements. In particular, we identify thermal convection, in connection with B₁ field gradients, asa significant apparent decay mechanism, which limits the ability to measure the true singlet state lifetimes. Highly B₁-compensated broadband singlet excitation/detection sequences are shown to minimize the influence of macroscopic molecular motion and B₁ inhomogeneity.

NMR Spin-Lock Induced Crossing (SLIC) dispersion and long-lived spin states of gaseous propane at low magnetic field (0.05T)

When parahydrogen reacts with propylene in low magnetic fields (e.g., 0.05T), the reaction product propane develops an overpopulation of pseudo-singlet nuclear spin states. We studied how the Spin-Lock Induced Crossing (SLIC) technique can be used to convert these pseudo-singlet spin states of hyperpolarized gaseous propane into observable magnetization and to detect ¹H NMR signal directly at 0.05T. The theoretical simulation and experimental study of the NMR signal dependence on B₁ power (SLIC amplitude) exhibits a well-resolved dispersion, which is induced by the spin-spin couplings in the eight-proton spin system of propane. We also measured the exponential decay time constants (T_LLSS or T_S) of these pseudo-singlet long-lived spin states (LLSS) by varying the time between hyperpolarized propane production and SLIC detection. We have found that, on average, T_S is approximately 3 times longer than the corresponding T₁ value under the same conditions in the range of pressures studied (up to 7.6atm). Moreover, T_S may exceed 13s at pressures above 7atm in the gas phase. These results are in agreement with the previous reports, and they corroborate a great potential of long-lived hyperpolarized propane as an inhalable gaseous contrast agent for lung imaging and as a molecular tracer to study porous media using low-field NMR and MRI.

Singlet NMR methodology in two-spin-1/2 systems

This paper discusses methodology developed over the past 12years in order to access and manipulate singlet order in systems comprising two coupled spin-1/2 nuclei in liquid-state nuclear magnetic resonance. Pulse sequences that are valid for different regimes are discussed, and fully analytical proofs are given using different spin dynamics techniques that include product operator methods, the single transition operator formalism, and average Hamiltonian theory. Methods used to filter singlet order from byproducts of pulse sequences are also listed and discussed analytically. The theoretical maximum amplitudes of the transformations achieved by these techniques are reported, together with the results of numerical simulations performed using custom-built simulation code.

Prediction of low-field nuclear singlet lifetimes with molecular dynamics and quantum-chemical property surface

Molecular dynamics and quantum chemistry methods are implemented to quantify nuclear spin-1/2 pair singlet-state relaxation rates. Illustrated is the relevant spin-internal-motion mechanism (SIM).

NMR SLIC Sensing of Hydrogenation Reactions Using Parahydrogen in Low Magnetic Fields

Parahydrogen-induced polarization (PHIP) is an NMR hyperpolarization technique that increases nuclear spin polarization by orders of magnitude, and it is particularly well-suited to study hydrogenation reactions. However, the use of high-field NMR spectroscopy is not always possible, especially in the context of potential industrial-scale reactor applications. On the other hand, the direct low-field NMR detection of reaction products with enhanced nuclear spin polarization is challenging due to near complete signal cancellation from nascent parahydrogen protons. We show that hydrogenation products prepared by PHIP can be irradiated with weak (on the order of spin-spin couplings of a few hertz) alternating magnetic field (called Spin-Lock Induced Crossing or SLIC) and consequently efficiently detected at low magnetic field (e.g., 0.05 T used here) using examples of several types of organic molecules containing a vinyl moiety. The detected hyperpolarized signals from several reaction products at tens of millimolar concentrations were enhanced by 10000-fold, producing NMR signals an order of magnitude greater than the background signal from protonated solvents.

The role of level anti-crossings in nuclear spin hyperpolarization

Nuclear spin hyperpolarization is an important resource for increasing the sensitivity of NMR spectroscopy and MRI. Signal enhancements can be as large as 3-4 orders of magnitude. In hyperpolarization experiments, it is often desirable to transfer the initial polarization to other nuclei of choice, either protons or insensitive nuclei such as (13)C and (15)N. This situation arises primarily in Chemically Induced Dynamic Nuclear Polarization (CIDNP), Para-Hydrogen Induced Polarization (PHIP), and the related Signal Amplification By Reversible Exchange (SABRE). Here we review the recent literature on polarization transfer mechanisms, in particular focusing on the role of Level Anti-Crossings (LACs) therein. So-called "spontaneous" polarization transfer may occur both at low and high magnetic fields. In addition, transfer of spin polarization can be accomplished by using especially designed pulse sequences. It is now clear that at low field spontaneous polarization transfer is primarily due to coherent spin-state mixing under strong coupling conditions. However, thus far the important role of LACs in this process has not received much attention. At high magnetic field, polarization may be transferred by cross-relaxation effects. Another promising high-field technique is to generate the strong coupling condition by spin locking using strong radio-frequency fields. Here, an analysis of polarization transfer in terms of LACs in the rotating frame is very useful to predict which spin orders are transferred depending on the strength and frequency of the B1 field. Finally, we will examine the role of strong coupling and LACs in magnetic-field dependent nuclear spin relaxation and the related topic of long-lived spin-states.

Accessing long-lived nuclear singlet states between chemically equivalent spins without breaking symmetry

Long-lived nuclear spin states could greatly enhance the applicability of hyperpolarized nuclear magnetic resonance. Using singlet states between inequivalent spin pairs has been shown to extend the signal lifetime by more than an order of magnitude compared to the spin lattice relaxation time (T1), but they have to be prevented from evolving into other states. In the most interesting case the singlet is between chemically equivalent spins, as it can then be inherently an eigenstate. However this presents major challenges in the conversion from bulk magnetization to singlet. In the only case demonstrated so far, a reversible chemical reaction to break symmetry was required. Here we present a pulse sequence technique that interconverts between singlet spin order and bulk magnetization without breaking the symmetry of the spin system. This technique is independent of field strength and is applicable to a broad range of molecules.

Long-lived 1H singlet spin states originating from para-hydrogen in Cs-symmetric molecules stored for minutes in high magnetic fields

Nuclear magnetic resonance (NMR) is a very powerful tool in physics, chemistry, and life sciences, although limited by low sensitivity. This problem can be overcome by hyperpolarization techniques dramatically enhancing the NMR signal. However, this approach is restricted to relatively short time scales depending on the nuclear spin-lattice relaxation time T(1) in the range of seconds. This makes long-lived singlet states very useful as a way to extend the hyperpolarization lifetimes. Para-hydrogen induced polarization (PHIP) is particularly suitable, because para-H(2) possesses singlet symmetry. Most PHIP experiments, however, are performed on asymmetric molecules, and the initial singlet state is directly converted to a NMR observable triplet state decaying with T(1), in the order of seconds. We demonstrate that in symmetric molecules, a long-lived singlet state created by PHIP can be stored for several minutes on protons in high magnetic fields. Subsequently, it is converted into observable high nonthermal magnetization by controlled singlet-triplet conversion via level anticrossing.

Dependence of nuclear spin singlet lifetimes on RF spin-locking power

We measure the lifetime of long-lived nuclear spin singlet states as a function of the strength of the RF spin-locking field and present a simple theoretical model that agrees well with our measurements, including the low-RF-power regime. We also measure the lifetime of a long-lived coherence between singlet and triplet states that does not require a spin-locking field for preservation. Our results indicate that for many molecules, singlet states can be created using weak RF spin-locking fields: more than two orders of magnitude lower RF power than in previous studies. Our findings suggest that for many endogenous biomolecules, singlets and related states with enhanced lifetimes might be achievable in vivo with safe levels of RF power.

Storage of nuclear magnetization as long-lived singlet order in low magnetic field

Hyperpolarized nuclear states provide NMR signals enhanced by many orders of magnitude, with numerous potential applications to analytical NMR, in vivo NMR, and NMR imaging. However, the lifetime of hyperpolarized magnetization is normally limited by the relaxation time constant T(1), which lies in the range of milliseconds to minutes, apart from in exceptional cases. In many cases, the lifetime of the hyperpolarized state may be enhanced by converting the magnetization into nuclear singlet order, where it is protected against many common relaxation mechanisms. However, all current methods for converting magnetization into singlet order require the use of a high-field, high-homogeneity NMR magnet, which is incompatible with most hyperpolarization procedures. We demonstrate a new method for converting magnetization into singlet order and back again. The new technique is suitable for magnetically inequivalent spin-pair systems in weak and inhomogeneous magnetic fields, and is compatible with known hyperpolarization technology. The method involves audio-frequency pulsed irradiation at the low-field nuclear Larmor frequency, employing coupling-synchronized trains of 180° pulses to induce singlet-triplet transitions. The echo trains are used as building blocks for a pulse sequence called M2S that transforms longitudinal magnetization into long-lived singlet order. The time-reverse of the pulse sequence, called S2M, converts singlet order back into longitudinal magnetization. The method is demonstrated on a solution of (15)N-labeled nitrous oxide. The magnetization is stored in low magnetic field for over 30 min, even though the T(1) is less than 3 min under the same conditions.

Density matrix tomography of singlet states

First direct and quantitative study of singlet states using density matrix tomography is reported. A robust scheme for the tomography of a general density matrix of two spin 1/2 nuclei is introduced for this purpose. The study is carried out at different spin-lock conditions and the results are compared.

Long-lived coherences for homogeneous line narrowing in spectroscopy

Line broadening, which can arise from inhomogeneities or homogeneous relaxation effects that lead to finite lifetimes of quantum states, is the Achilles' heel of many forms of spectroscopy. We show that line broadening may be considerably reduced by exploiting long lifetimes associated with superpositions of quantum states with different symmetry, termed long-lived coherences. In proton NMR of arbitrary molecules (including proteins) in isotropic solution, the slow oscillatory decays of long-lived coherences can yield spectra with very high resolution. This improvement opens the way to high-field magnetic resonance of molecular assemblies that are almost an order of magnitude larger than could be hitherto studied. Coherences between states of different symmetry may be useful in other forms of spectroscopy to cancel unwanted line broadening effects.

Long-lived states to sustain hyperpolarized magnetization

Major breakthroughs have recently been reported that can help overcome two inherent drawbacks of NMR: the lack of sensitivity and the limited memory of longitudinal magnetization. Dynamic nuclear polarization (DNP) couples nuclear spins to the large reservoir of electrons, thus making it possible to detect dilute endogenous substances in magnetic resonance spectroscopy (MRS) and magnetic resonance imaging (MRI). We have designed a method to preserve enhanced ("hyperpolarized") magnetization by conversion into long-lived states (LLS). It is shown that these enhanced long-lived states can be generated for proton spins, which afford sensitive detection. Even in complex molecules such as peptides, long-lived proton states can be sustained effectively over time intervals on the order of tens of seconds, thus allowing hyperpolarized substrates to reach target areas and affording access to slow metabolic pathways. The natural abundance carbon-13 polarization has been enhanced ex situ by almost four orders of magnitude in the dipeptide Ala-Gly. The sample was transferred by the dissolution process to a high-resolution magnet where the carbon-13 polarization was converted into a long-lived state associated with a pair of protons. In Ala-Gly, the lifetime T(LLS) associated with the two nonequivalent H(alpha) glycine protons, sustained by suitable radio-frequency irradiation, was found to be seven times longer than their spin-lattice relaxation time constant (T(LLS)/T(1) = 7). At desired intervals, small fractions of the populations of long-lived states were converted into observable magnetization. This opens the way to observing slow chemical reactions and slow transport phenomena such as diffusion by enhanced magnetic resonance.

Extremely low-frequency spectroscopy in low-field nuclear magnetic resonance

We demonstrate a new phenomenon in nuclear magnetic resonance spectroscopy, in which nuclear spin transitions are induced by radio frequency irradiation at extremely low frequencies (of the order of a few Hz). Slow Rabi oscillations are observed between spin states of different exchange symmetry. These "forbidden" transitions are rendered weakly allowed by differential electronic shielding effects on the radio frequency field. We generate coherence between the singlet and triplet states of 15N-labeled nitrous oxide in solution, and estimate the scalar coupling between the two 15N nuclei with a precision of a few mHz.

Producing over 100 ml of highly concentrated hyperpolarized solution by means of dissolution DNP

New low-temperature inserts compatible with an existing hyperpolarizer were developed to dynamically polarize nuclei in large samples. The performance of the system was tested on 8 ml glassy frozen solutions containing 13C-labeled molecules and doped with nitroxyl free radicals. The obtained 13C low-temperature polarization was comparable to the one measured on 20 times smaller sample volume with only 3-4 times higher microwave power. By using a dissolution insert that fits to the new design, it was possible to obtain about 120 ml of room-temperature hyperpolarized solution. The polarization as well as the molecule concentration was comparable to the values obtained in standard size hyperpolarized samples. Such large samples are interesting for future studies on larger animals and possibly for potential clinical applications.

Hyperpolarized long-lived states in solution NMR: three-spin case study in low field

Recent work has shown that singlet states in two-spin systems can possess lifetimes exceeding the T(1) relaxation time, provided that the system is kept under conditions that minimize the effects of the chemical shift Hamiltonian (for instance under low magnetic field or RF irradiation). Similar observations have been made in hyperpolarized states of multi-spin systems prepared via parahydrogen-induced polarization (PHIP). However, lifetime prolongation mechanisms in multi-spin systems are still under investigation. Here we present experimental observations of a long-lived state in a three-spin system prepared by PHIP and stored at low field. The observed lifetime of the long-lived state is 144s, about twice as long as the longest T(1) measured in the system at high field. The results are analyzed using a recently proposed theory of lifetime prolongation in multi-spin systems in low field. It is shown that quantum mechanical selection rules governing intramolecular dipolar relaxation in low field account for the enhanced lifetime and spectral features of this state.

The long-lived nuclear singlet state of 15N-nitrous oxide in solution

A 15N nuclear singlet lifetime of over 26 min has been observed in a solution of 15N2O, by using a field-cycling NMR pulse sequence. This observation suggests applications of hyperpolarized 15N2O in medical imaging and for flow and diffusion studies.

Long-lived states in solution NMR: theoretical examples in three- and four-spin systems

Long-lived spin states have been observed in a variety of systems. Although the dynamics underlying the long lifetimes of these states are well understood in the case of two-spin systems, the corresponding dynamics in systems containing more spins appear to be more complex. Recently it has been shown that a selection rule for transitions mediated by intramolecular dipolar relaxation may play a role in determining the lifetimes of long-lived states in systems containing arbitrary numbers of spins. Here we present a theory of long-lived states in systems containing three and four spins and demonstrate how it can be used to identify states that have little or no intramolecular dipolar relaxation.

Isotropic filtering using polyhedral phase cycles: application to singlet state NMR

A technique is described for filtering out the components of an NMR signal that have passed through an isotropic spin order term. The method involves a coordinated cycle of three radiofrequency phase angles, where two of the phases correspond to the polar angles describing the vertices of a regular polyhedron, and the third angle is stepped around a circle. The most economical isotropic filtering scheme involves a 12-step phase cycle based on tetrahedral symmetry. The method is used to filter out NMR signals that have passed through singlet populations in a solution NMR experiment.

Extending the scope of singlet-state spectroscopy

Different decoupling sequences are tested-using various shaped radio-frequency (RF) pulses-to achieve the longest possible lifetimes of singlet-state populations over the widest possible bandwidths, that is, ranges of offsets and relative chemical shifts of the nuclei involved in the singlet states. The use of sinc or refocusing broadband universal rotation pulses (RE-BURP) for decoupling during the intervals where singlet-state populations are preserved allows one to extend the useful bandwidth with respect to prior state-of-the-art methods based on composite-pulse WALTZ decoupling. The improved sinc decoupling sequences afford a more reliable and sensitive measure of the lifetimes of singlet states in pairs of spins that have widely different chemical shifts, such as the two aromatic protons H(5) and H(6) in uracil. Similar advantages are expected for nucleotides in RNA and DNA. Alternative approaches, in particular frequency-modulated decoupling sequences, also appear to be effective in preserving singlet-state populations, even though the profiles of the apparent relaxation rate constants as a function of the offset are somewhat perturbed. The best decoupling sequences prove their utility in sustaining longer lifetimes of singlet states than previously achieved for the side-chain tyrosine protons in bovine pancreatic trypsin inhibitor (BPTI) at 600 MHz (14.1 T), where the differences of chemical shifts between coupled protons are a challenge.