Introduction to Fast Field Cycling (FFC) relaxometry

Molecular dynamics are generally rotational or translational motions that are modeled as small angle or small step translational jumps which occur randomly in time. The random functions of time are usually characterized by a time correlation function that characterizes the conditional probability that if there is a particular orientation or position at time t, what is the probability at time (t + τ) later? For Brownian rotational diffusion, the correlation function decays exponentially:

Where B2 is the strength of the coupling modulated and τc is the correlation time for the molecule to diffuse approximately one radian. Nuclei are relaxed by the transition induced by these fluctuating fields at the Larmor frequency (single quantum) and twice the Larmor frequency (double quantum) spin flips. What motions are important depends on the magnetic field strength experienced by the nuclei. The more intense the fluctuations are near the Larmor frequency, the faster the spins relax. The frequency dependence of the fluctuation intensity, i.e., the power spectrum or the spectral density, is the Fourier transform of the time correlation function. For rotation this becomes,

Which to within a constant has the form:

Thus, the Nuclear Magnetic Resonance Dispersion (known as NMRD or MRD) maps the Larmor frequency dependence of the spectral density function that is related directly, by the Fourier transform, to the time correlation function characterizing molecular motions. Any motion that causes changes in the local fields contributes to the relaxation process including rotation, translation and chemical exchange among different chemical or physical environments.

A critical feature of NMRD is that it is possible to map spectral densities as low as 5-10 kHz which corresponds to a time regime in the vicinity of 30 microseconds, i.e., well into the range of chemical exchange events. Variations of the method permit exploration to lower frequencies.

ACKNOWLEDGEMENT:
Thanks to Prof. Robert G. Bryant (University of Virginia, USA) for providing some of the text for this section.

FIG 02: schematic representation of the field cycling technique.

FFC NMR relaxometry requires a small amount of a solid or liquid sample (enough to fill a standard 10mm NMR tube to a volume of around 1 cm3) with no other form of preparation required.

The Stelar wide-bore magnet is able to accommodate larger samples such as rock cores or even small animals. The Stelar relaxometer works by fast electronic switching of the magnetic field from an initial polarizing magnetic field (BPOL), where the equilibrium of nuclear magnetization is attained in about 4T1, to a field of interest (relaxation field; BRELAX) at which the nuclear spins relax to the new equilibrium state with a characteristic relaxation time constant T1.

After a delay time, τ, the BRELAX is switched to the field of acquisition (BACQ) and the NMR signal is detected after a π/2 RF pulse (fig. 2).

FIG. 3: NMRD profile for bovine serum albumin in water at 200 mg/mL.
Picture provided by Prof. Robert G. Bryant (University of Virginia, USA).

FIG 4a & 4b: The relaxation dispersion profiles for cross-linked bovine serum albumin samples at different compositions shown in semilogarithmic (a) and log-log (b) presentations.
Picture provided by Prof. Robert G. Bryant
(University of Virginia, USA)

FIG 5a: 1H NMRD profiles of three different weight distributions of polyethylene glycol (PEG) melts. Data produced from in-house studies at Stelar. Samples kindly provided by Dr. M. Fleury, IFPEN, Paris.

FIG 5b: NMRD profiles of BTPI as a function of salt concentration.

FIG 5c: NMRD profiles of starch samples with differing water content.

A FFC relaxometer, such as the Stelar SPINMASTER or SMARtracer, is a specialized system with a low field magnet which is able to electronically switch the magnetic field strength required, from a few kHz up to the maximum magnetic field strength allowed by the magnet (up to 42 MHz 1H Larmor frequency with the 1 Tesla magnet) and which is able to overcome the limits of the NMR signal-to-noise ratio at low magnetic field strengths.

A standard fixed field magnet measures the relaxation rate constant 1/T1 with the limitation that it is unable to change the magnetic field strength of operation and thus is only able to measure 1/T1 at a single magnetic field strength. Fixed field time domain instruments generally operate in the range of 2 – 60 MHz. The FFC relaxometer is able to measure 1/T1 at very low magnetic field strengths (down to a few kHz) which is of particular advantage as many molecular processes occurring in the range between nano-seconds and milliseconds (slow molecular dynamics) are very difficult to measure at higher magnetic field strengths.

In the following we show three examples. The NMRD profiles in fig.5A, measured with a Stelar FFC NMR relaxometer, show three different weight distributions of polyethylene glycol (PEG) melts (blue squares 8,000 Da, black squares 20,000 Da, red squares 35,000 Da). This demonstrates how the ability to measure nuclear spin relaxation over a wide range of magnetic field strengths, and especially at very low magnetic fields, can be of advantage. It would be difficult to distinguish between these three different polymers at magnetic field strengths higher than 0.1 MHz. In fig. 5b it is shown how protein aggregation can be investigated using FFC. NMRD profiles allowed to study BTPI (Bovine Pancreatic Trypsin Inhibitor) self-association as a function of pH, salt type, salt concentration and temperature.

FFC method sensitively detects stable oligomers without being affected by other intercactions. Looking at the NMRD profiles in fig. 5B, it is evident that it would be difficult to distinguish between the different samples at magnetic field strengths higher than 5 MHz.

In fig. 5C it is shown that applying FFC to starch samples it is possible possible to distinguish between samples with different water content. The difference between samples is not detectable for field strengths over 10MHz.