Paramagnetic relaxation enhancement (PRE)

A paramagnetic centre, such as a spin label, a metal tag or a gadolinium cosolvent, speeds up the relaxation of nearby nuclei. Use pre2d to measure the ¹H transverse enhancement Γ₂ from spectra with and without the paramagnetic agent, and lpre2d to measure the longitudinal enhancement Γ₁ as well, from spectra at a series of recycle delays.

Both fit every peak across all of its spectra at once. Each peak has one diamagnetic amplitude and diamagnetic linewidths, and the enhancement then sets its linewidths and height in every spectrum:

• Γ₂ broadens the ¹H linewidth, and in an HMQC the multiple-quantum linewidth in the indirect dimension too.
• Γ₂ also lowers the peak height by $\exp(-\Gamma_2 T_\text{relax})$ during the coherence transfer delays.
• For lpre2d, the height recovers between scans as $1 - \exp(-(R_1 + \Gamma_1) \tau)$, where $\tau$ is the recovery time.

For a single paramagnetic spectrum this gives the familiar intensity ratio (Battiste & Wagner, Biochemistry 39, 5355 (2000)):

\[\frac{I_\text{para}}{I_\text{dia}} = \frac{R_2 \exp(-\Gamma_2 T_\text{relax})}{R_2 + \Gamma_2}\]

Measure Γ₂ from a diamagnetic and a paramagnetic sample

Give the two spectra and the sequence type. Γ₂ is reported in s⁻¹.

using NMRAnalysis

pre2d("diamagnetic/pdata/1", "paramagnetic/pdata/1"; type=:hsqc)

# experiment numbers work too
pre2d(11, 12; type=:hmqc)

Measure Γ₂ from a concentration series

For a paramagnetic cosolvent, give the spectra and the concentration of the agent in each, with 0 for the diamagnetic reference. Γ₂ is then reported per unit of concentration, in whatever unit you give, for example s⁻¹ mM⁻¹.

pre2d([10, 11]; concentrations=[0, 2], type=:hmqc)            # one Gd concentration
pre2d([10, 11, 12, 13]; concentrations=[0, 1, 2, 5], type=:hsqc)

Measure Γ₁ and Γ₂ from a series of recycle delays

With spectra at several recycle delays, lpre2d fits the diamagnetic R₁ and both enhancements together, so Γ₂ is free of the bias that incomplete recovery gives at short delays. Give the diamagnetic and paramagnetic spectra as two lists, or one list with its concentrations.

# spin-labelled protein: nine delays without and nine with the label
lpre2d(10:2:26, 11:2:27; type=:hmqc)

# 2 mM Gd: the nine diamagnetic experiments first, then the nine with Gd
lpre2d([10:18; 103:111]; concentrations=[fill(0, 9); fill(2, 9)], type=:hmqc)

The recovery time of each spectrum is its d1 plus its acquisition time, read from the acquisition parameters. To set the recovery times yourself, pass recycledelays: the values are used exactly as given, with nothing added. For two lists, you can give one list of delays that applies to both.

R₁ and Γ₁ describe the recovery in the sequence you used. A selective or BEST-type sequence recovers faster than a non-selective one, so compare them only between experiments recorded the same way.

Parameters

The :auto transfer time assumes a plain HSQC or HMQC, where ¹H is transverse for two delays of 1/(2J). In a sensitivity-enhanced HSQC the time is longer and differs between coherence pathways (Iwahara, Tang & Clore, J. Magn. Reson. 184, 185 (2007)), so give Trelax explicitly, or use a sequence without sensitivity enhancement.

Check the fits

Each peak's plot shows its fit:

• For a pair, cross-sections through both spectra, the diamagnetic and paramagnetic in different colours, so you can see both the broadening and the loss of height.
• For a concentration series, the peak height relative to its diamagnetic amplitude, against concentration, with the fitted curve.
• For lpre2d, the peak height against recovery time, a series and fitted curve for each concentration.

Output

Clicking Save to folder writes the usual files (see Peak Lists and Output Files). results.csv has these columns:

Rates carry the unit s⁻¹ for a diamagnetic and paramagnetic pair, and no unit for a concentration series, since the concentration unit isn't known. In series.csv, amp is the height measured in each spectrum at the linewidths the fit gives it, and amp_fit the height the model predicts.

summary.pdf stacks panels against residue number: the fitted enhancements, each peak's height in every paramagnetic spectrum relative to the diamagnetic reference (for lpre2d, at the same recycle delay), and the diamagnetic R₂ and, for lpre2d, R₁. To plot a single parameter instead, pass it to summaryplot:

expt = lpre2d(10:2:26, 11:2:27; type=:hmqc)
summaryplot(expt; param=:Gamma1)