Slow relaxation of longitudinal multispin orders in weakly and strongly coupled two-spin systems

Abstract

Longitudinal multispin order (LOMO) corresponds to a nonequilibrium population distribution in spin systems that exhibit scalar (J), dipolar, or quadrupolar coupling. We investigated the relaxation of longitudinal two-spin order (2-LOMO) in systems that had either weakly or strongly J-coupled spins. Our results indicated longer relaxation times for the 2-LOMO state compared with the corresponding longitudinal single-spin state (1-LOMO). Accessing nuclear spin states that have relaxation times longer than T₁, without the use of external contrast agents, is potentially useful for in vivo imaging and also for studying systems using dynamically hyperpolarized nuclear spins where longer life times are sought to increase the time available to study (bio)chemical events. Copyright © 2012 John Wiley & Sons, Ltd.

Introduction

The available time span of all nuclear magnetic resonance (NMR) pulse sequences is restricted by the spin–lattice relaxation time, T₁, which determines the time required for the nuclear spins to reach equilibrium with the surroundings. Thus, the generation of a spin state having a longer “effective T₁” would be valuable for following slow molecular processes such as diffusion, chemical exchange and rearrangements of structural domains in macromolecules. Achieving longer lifetimes of spin states within different molecules is of great interest for a wide range of applications in NMR imaging using hyperpolarized spin populations.[1] Recently, several groups have reported the potential for increasing relaxation times by the creation of long-lived states[2, 3]. These are zero-quantum states involving populations that have lifetimes that are much greater than T₁. It has been reported recently that such states can be obtained by creating spin orders that are immune to dipolar relaxation, by using special radiofrequency (RF) pulse sequences. In this context, there is great interest in understanding the relaxation behavior of longitudinal orders.

We have previously shown[4, 5] that by using the “frequency-cycling approach” (for a definition, see section on Theory), longitudinal order can be prepared without the presence of zero-quantum coherence, double-quantum coherence, I_z and S_z magnetization. In addition, this approach permits the editing of different longitudinal orders within a spin cluster. In the present work, we investigated the relaxation behavior of two-spin longitudinal order generated by using the previously described frequency-cycling procedure.[4] In this earlier work, such states were called longitudinal multispin orders (LOMOs).[4] A LOMO state corresponds to a nonequilibrium population distribution that can be created in spin systems that exhibit scalar (J) coupling, dipolar coupling, or quadrupolar coupling.[6, 7] The longitudinal orders are also called magnetization modes, which can be used to analyze longitudinal relaxation in coupled spin systems.[8-10] Several RF pulse sequences incorporate the generation of LOMO as an intermediate state for editing spectra of different metabolites, such as lactate and γ-aminobutyrate.[11-13] LOMOs can also be created via cross-correlated relaxation between different relaxation pathways that are present for a given spin system.[14] The buildup of longitudinal two-spin order can be observed via an experiment that combines double-quantum filtration with two-dimensional nuclear Overhauser effect spectroscopy.[15] The time evolution of the two-spin order created in this way can convey important information on macromolecular structure. Similarly, in a three-spin system, cross-correlated relaxation between different dipolar vectors creates longitudinal three-spin order. This can be observed selectively by using a triple-quantum filter in conjunction with two-dimensional exchange experiments. The resulting triple-quantum filtered nuclear Overhauser effect spectroscopy method provides information on angles subtended by internuclear vectors.[16] In heteronuclear spin systems, it is possible to selectively observe the conversion of single-spin order to two-spin longitudinal order via cross correlation between chemical shift anisotropy and dipolar interactions. This is carried out by converting longitudinal order into antiphase magnetization by a small angle pulse acting on both spins or by a selective 90° pulse acting on one of them.[17] Longitudinal orders can also serve as an intermediate state in many polarization transfer[18] and exchange experiments.[19, 20]

The experimental and theoretical basis of the creation of LOMO states with different approaches has been reported elsewhere.[4] The creation of 2-LOMO states has been investigated for weakly and strongly coupled spin systems using a frequency-cycling method.[5, 21] Furthermore, the relaxation behavior of LOMO states generated via interference of various relaxation mechanisms has been well explored,[14, 22] but to our knowledge, the relaxation behavior of LOMO states generated by frequency cycling has not been previously reported. Hence, the aim of the present work was to investigate the longitudinal relaxation behavior of 2-LOMO states in both weakly and strongly J-coupled spin systems, generated via the frequency-cycling approach. The observed experimental relaxation enhancements in LOMO states were then compared with the theoretically estimated enhancements based on the previous work.[23] The relaxation study given here will be of importance in the design of NMR and MRI sequences that generate and/or select LOMO states.

Theory

Scalar-coupled spin systems can be classified as weakly and strongly coupled, depending on the relative magnitude of the coupling constant, J, and the difference between the chemical shifts of the resonances of the two (or more) coupled spins. The 2-LOMO states of both weakly and strongly coupled spin systems[21] can be prepared independently of the value of J by a frequency-cycling procedure. A pulse sequence that is used to edit LOMO states has been studied previously,[4] which involves frequency-selective inversion of transitions by a shaped (tailored) pulse of appropriate duration and spectral bandwidth, followed by a high power (“hard”) read pulse of fixed phase and flip angle. The phase of the receiver is altered for each transient as the selective inversion frequency is incremented. In the present study, we created 2-LOMO states by executing a two-step frequency cycling. For a system of two spin-1/2 nuclei, each component of the doublet of one spin is selectively inverted in each transient with successive changes in the receiver phase of 90^o and −180^o. Table 1 in the previous work[4, 5, 21] shows the generation of a 2-LOMO state using this approach and how its effectiveness is completely independent of the value of the coupling constant, J, between the two spins.

Table 1. Longitudinal relaxation time of 1-LOMO and 2-LOMO states of an AX (para amino-benzoic acid) and AB (sodium citrate) spin system

Compound	T1z (s)	T1zz (s)
PABA	3.69 ± 0.01	4.96 ± 0.08
NaC	0.833 ± 0.001	2.012 ± 0.001

The pulse sequence that is used to measure the longitudinal relaxation time of a 2-LOMO state, which has been selected by the frequency-cycling procedure, is shown in Fig. 1. The inversion-recovery pulse sequence was implemented within the frequency-cycling protocol for relaxation analysis. It incorporates a transition-selective inversion of magnetization followed by a 180° hard pulse. The selective inversion of each line of a multiplet in successive transients, combined with a suitable receiver add/subtract cycle, is called frequency cycling. The frequency of the selective inversion pulse is switched to the frequency of an appropriate transition of a multiplet, along with the receiver phase during each transient. Using this procedure, the 2-LOMO state can be distinguished by recording two transients (for which the frequency-selective inversion is performed on two transitions in the AX or AB spin system).

Figure 1. Pulse sequence used in the longitudinal relaxation analysis of the 2-LOMO state of AX and AB systems. The first transition-selective 180° gaussian pulse was applied for a duration of 200 ms. The last pulse was the optimal reconversion pulse with a flip angle of 45°.

The variable delay after the hard pulse is followed by a read pulse of flip angle θ. A 45° read pulse was used here to maximize the conversion of the 2-LOMO state to observable magnetization.[4, 5]

Longitudinal relaxation analysis of a two-spin system

Longitudinal relaxation can be represented by the time-dependent recovery of the diagonal elements of the density matrix to their equilibrium values. This relaxation is characterized by the first block of the “relaxation kite”, and the diagonal elements are in general coupled and their time evolution is given by[24]

• (1)

where α and β are the eigenstates of the Hamiltonian and σ_αα and R_ααββ are the elements of the density matrix and Redfield relaxation matrix, respectively. Fig. 2 shows the energy-level diagram of the two-spin system together with the corresponding transition probabilities between the levels. Since the diagonal elements of the density matrix represent populations, Eqn (1) can also be written in terms of each population and the relevant transition probabilities. The identical rate equation for a two-spin system that describes the recovery of the population distribution between the various energy levels, relative to their equilibrium values, can be expressed as[25]

• (2)

Figure 2. Energy level diagram for a two-spin system together with the transition probability for each relaxation connectivity.

The relationship between the populations denoted by the vector N and various magnetization modes, M, is given by

• (3)

where V is the transformation matrix connecting the populations to the modes. Thus, the relationship between these modes and the populations for an AX spin system can be written as

• (4)

and on expanding Eqn (4), we obtain

• (5)

Hence, the rate of change of the spin modes, expressed in terms of the populations, is

• (6)

The rate at which each of these populations returns to equilibrium is given as (after rearranging Eqns (2) and (5)):

• (7)

where

• (8)

Substituting Eqn (7) into Eqn (6) gives the relationship between the rate of change of spin modes to the transition probabilities and the populations. Thus, in the resulting equation, replacing the transition probability elements by the relaxation matrix elements and populations by spin modes, we obtain

• (9)

where,

• (10)

and

• (11)

where ρ_A and ρ_X are the direct relaxation rate constant 1/T_1A and 1/T_1X, respectively, and σ_AX is the cross-relaxation rate constant.

For an AB spin system, because of strong coupling and no inherent symmetry between the energy levels, a simple set of magnetization modes cannot be defined. Therefore, for such cases one has to retain the matrix of the complete longitudinal multiple-quantum modes.[26]. Hence, for an AB system, Eqn (4) can be written as

• (12)

Following similar algebraic manipulation to that used to obtain the expression for the longitudinal relaxation term for an AX spin system, we obtain

• (13)

where .

In this case, the expressions for ρ_A and ρ_B cannot be derived because of the absence of pure <A_z> and <B_z> magnetization modes.[26-28] However, from the previously mentioned analysis, it is clear that the relaxation rate constants for the 2-LOMO states, ρ_AX (for 2A_zX_z) and ρ_AB (for 2A_zB_z), depend on the first order-transition probabilities of the two spins and can be prepared independently of the strong coupling parameter using the frequency-cycling procedure.

Experimental

4-Aminobenzoic acid (para-amino benzoic acid; PABA) and tri-sodium citrate (NaC) were used as an AX and AB spin systems, respectively (Figs 3A and 4A). They were obtained from Sigma-Aldrich in the pure “Analytical Reagent” form. The sodium salt of PABA was prepared by dissolving 25 mg of it and 20 mg of sodium phosphate in 750 µl of D₂O. The NaC was prepared by dissolving 25 mg of it in 500 µl of D₂O. Both samples were sealed after three freeze-vacuum pump–thaw cycles to remove dissolved oxygen that acts as a source of enhanced nuclear magnetic relaxation.

Figure 3. (A) Chemical structure of NaC and (B) 400.13-MHz 1-D ¹H NMR spectrum acquired at 298 K. To create the 2- LOMO state, the frequency cycle was applied to the high-frequency multiplet. (C) 2-LOMO spectrum of NaC at different variable delays.

Figure 4. (A) Chemical structure of PABA and (B) 400.13-MHz 1-D ¹H NMR spectrum acquired at 298 K. To create the 2- LOMO state, the frequency cycle was applied to the high-frequency multiplet. (C) 2-LOMO spectrum of PABA at different variable delays.

All spectra were recorded at 298 K on a Bruker 400-MHz DRX spectrometer using a Bruker, inverse “broadband observe” probe. The pulse sequence used for the inversion recovery of the 2-LOMO state is shown in Fig. 1. A transition-selective Gaussian pulse of 200 ms duration was used for selective inversion. The frequency-cycling approach was used for the creation of two-spin orders by changing the receiver phase in each transient, followed by a 180° hard pulse of 24 µs duration. The inversion-recovery experiment was performed for 28 variable delays ranging from 10 ms to 15 s. A 45° read pulse of 6.0 µs duration was used to record the LOMO spectra at the end of each variable delay. For each variable delay, 32 averages were recorded with a repetition delay of 10 s. All spectra were processed using XWINNMR 3.5 (Bruker) and MestReNova 6.1.0 (Mestrelab Research S.L.)

Results and Discussion

Fig. 3A and 4A show the chemical structures, and Figs 3B and 4B show the one-dimensional (1-D) ¹H NMR spectra of NaC and PABA, respectively. Although both of these compounds have a four-spin system (AA′BB′ and AA′XX′), their NMR behavior can be considered to be two independent pairs of two-spin systems, as described previously.[29] This standpoint is based on the unresolved interpair J-coupling in the 1-D NMR spectrum. The intrapair J-couplings of 15.5 Hz (J_AB) and 8.0 Hz (J_AX) were measured from the spectra. The 2-LOMO states in both AB and AX spin systems were created by running the frequency cycle sequence on the high-frequency multiplet. Fig. 3C and 4C show the longitudinal relaxation of the 2-LOMO state in NaC and PABA, respectively. The longitudinal relaxation time of the 2-LOMO state (T_1zz) was estimated from the biexponential decay of the peak area with respect to the variable delay. The longitudinal relaxation times (T_1z) of the 1-LOMO state of these systems were measured independently using the same experimental parameters but with the absence of a transition-selective pulse. The normalized exponential curves of the 2-LOMO state and the 1-LOMO state for the measurement of T_1zz and T_1z for the AX and AB spin systems are shown in Figs 5 and 6, respectively. Relaxation times were estimated by nonlinear fitting of a monoexponential for 1-LOMO and biexponential for 2-LOMO function. Table 1 lists the estimates of the longitudinal relaxation times for both AB and AX spin systems.

Figure 5. Exponential recovery and decay curves of the 1-LOMO (half circles) and the 2-LOMO (full circles) states of PABA.

Figure 6. Exponential recovery and decay curves of the 1-LOMO (half circles) and the 2-LOMO (full circles) states of NaC, respectively.

The estimate of T_1z for PABA was 3.69 ± 0.01 s. After creating the 2-LOMO state, the estimated relaxation time was larger, being 4.96 ± 0.08 s. The effect of the 2-LOMO state on the overall relaxation of magnetization of the spin system in NaC (AB system) was also investigated to explore the effect of strong coupling. The estimates of T_1z were 0.833 ± 0.001 s for NaC and 2.012 ± 0.001 s for 2-LOMO. Under extreme narrowing conditions (ωτ_c < <1) and within the dipolar relaxation framework, the ratio of W₀:2W₁:W₂ for an AX system is 1:3:6.[30] On substituting this ratio into Eqn (10), it is readily seen that

• 

or

• (14)

Whereas in the case of AB system, the ratio (derived using the spectral density function[14, 23] of W₀ : ¹W_1A : W₂ is and W₀ : ¹W_1B : W₂ is , where _.

On substituting these ratios into Eqn (13), we obtain

• 

or

• (15)

The estimated larger relaxation time of the longitudinal two-spin-order state, for both AX and AB systems, was in agreement with the theoretical estimates based on the assumption of pure dipolar relaxation. The longer relaxation time was concluded to be because the W₀ and W₂ transition probabilities are not involved in the relaxation of the 2-LOMO state as described by Eqn (10) for dipolar relaxation. Longer relaxation times for the singlet state of NaC and PABA[29] and the zero quantum state of NaC[31] have been previously reported.

Our results clearly showed that the 2-LOMO states had longer T₁ values compared with the 1-LOMO states for both weakly and strongly coupled spin systems. The prolonged relaxation of the 2-LOMO state could be exploited in NMR or MRI (e.g., imaging of hyperpolarized nuclei) experiments where longer T₁ values are of benefit when investigating, for example, the metabolism of hyperpolarized [¹³C] pyruvate via a series of enzyme catalyzed reactions that lead into and involve the Krebs cycle.[32] The frequency cycle–based relaxation measurements can be readily extended to other higher-order spin systems. However, there are concerns regarding the amount of RF power that might be deposited in tissues when applying this technique in vivo. It is relevant to note that in combination with hyperpolarized nuclei, it may be possible to exploit any spin states that have prolonged relaxation times. In addition, the detection of antiphase magnetization can lead to signal losses under line broadening conditions, for example, in the in vivo situation or with large line widths brought about by magnetic field inhomogeneity.

Conclusions

We quantified the relaxation behavior of 2-LOMO states in weakly and strongly coupled spin systems. We showed that the relaxation time of the 2-LOMO state was greater than the corresponding 1-LOMO state for both weakly and strongly coupled systems. It should be possible to use LOMO relaxation within an imaging environment for generating new types of contrast, compared with conventional T₁- or T₂-dependent methods. The longitudinal spin order might also be used to study slow motional processes such as translational molecular diffusion, slow conformational changes in macromolecules, and slow chemical exchange,[6] or membrane transport under conditions of dynamic equilibrium, over longer time intervals than is possible by using spin states that undergo conventional longitudinal relaxation.

Acknowledgement

SBIC awarded intramural research funds for this work.