Gradients for measuring diffusion

Translational diffusion, also known as Brownian motion, is the random motion of particles among the other components of a solution. The rate of diffusion is related to the size of the molecule so measuring this rate can provide an estimate of molecular weight and information on aggregation, or binding, at close to physiological conditions. NMR experiments have been devised to measure translational diffusion using a pair of gradient pulses separated by a delay. These Diffusion Ordered SpectroscopY, or DOSY¹, experiments, are typically shown as 2D spectra with a horizontal ¹H dimension and a vertical diffusion rate dimension.

Gradient pulses create a position dependent alteration in the magnetic field of an NMR sample. In most NMR experiments gradients are used in pairs so that the second gradient reverses the effects of the first to recover the magnetization. Maximum recovery of the magnetization depends on the molecules not moving from their position when the first gradient was applied. If the molecules do move between pulses then the second gradient will not properly compensate for the first, the magnetization will not refocus properly, and the signal will be reduced. This is the basis for DOSY experiments. The delay between pulses is made large enough that the molecules can diffuse from their original positions by the time of the second gradient, causing signal attenuation. By recording a series of spectra with different delays, the signal intensity can be plotted against the delay to give a roughly exponential decay. Fitting an equation to the decay curve allows the diffusion rate to be determined.

The figure below shows a series of spectra recorded on a sample of ¹⁵N urea and ¹³C methanol in DMSO-d₆. The large pair of peaks around 5.5 ppm are from the ¹⁵N urea. The peaks at 4.1, 3.3 and 3.0 ppm are from the ¹³C methanol and the peak at 2.5 ppm is from the DMSO. The spectra are ordered to show the decay of the signals.

Fitting the decay of the signals gives a diffusion rate for each peak. The figure below shows an expansion of the DOSY spectrum obtained by extracting diffusion rates from the data above. The horizontal axis corresponds to the standard 1D ¹H spectrum. The vertical axis is the diffusion rate. The three components of the sample have been readily separated by their diffusion constants as indicated by the red lines. Extracting rows at these positions would give 1D ¹H spectra of the pure components.

DOSY experiments can be used to estimate molecular weight², but obtaining accurate results requires careful control of the experimental conditions and the data analysis³. I have found them more qualitative than quantitative. Still, useful information can be obtained, particularly if information under physiological conditions is required.

References
1. K.F. Morris and C.S. Johnson Jr
"Diffusion-ordered two-dimensional nuclear magnetic resonance spectroscopy"
J. Am. Chem. Soc. 1992 114(8):3139–314

2. R. Evans, Z. Deng, A.K. Rogerson, A.S. McLachlan, J.J. Richards, M. Nilsson and G.A. Morris
"Quantitative Interpretation of Diffusion-Ordered NMR Spectra: Can We Rationalize Small Molecule Diffusion Coefficients?"
Angew. Chem. Int. Ed., 2013 52:3199–3202

3. J. Lapham, J.P. Rife, P.B. Moore and D.M. Crothers
"Measurement of diffusion constants for nucleic acids by NMR"
J Biomol NMR, 1997 10:255