NMR determined Rotational correlation time

Brownian rotation diffusion of a particle in solution has a characteristic time constant called rotational correlation time (τ_c). It is the time it takes the particle to rotate by one radian and it depends on the particle size. For globular proteins a spherical approximation can be used and the rotational correlation time is given by Stoke's law

<math>\tau_c=\frac{4\pi\eta r^3}{3kT}</math>,

where η is the viscosity of the solvent, r is the effective hydrodynamic radius of the protein molecule, k is the Boltzmann constant and T is the temperature. The hydrodynamic radius can be estimated from the molecular weight of the protein (M) as

<math>r\approx\sqrt[3]{\frac{3M}{4\pi\rho N_a}}+r_w</math>,

where ρ is the average density for proteins (1.37 g/cm3), N_a is the Avogadro's number and r_w is the hydration radius (1.6 to 3.2 A). As a general rule of thumb, the τ_c of a monomeric protein in solution in nanoseconds is approximately 0.6 times its molecular weight in kDa.

For rigid protein molecules, in the limit of slow molecular motion (τ_c >> 0.5 ns) and high magnetic field (500 MHz or greater), a closed-form solution for τ_c as a function of the ratio of the longitudinal (T₁) and transverse (T₂) ¹⁵N relaxation times exists:

<math>\tau_c\approx\frac{1}{4\pi\nu_N}\sqrt{6\frac{T_1}{T_2}-7}</math>,

where ν_N is the ¹⁵N resonance frequency (in Hz). This equation is derived from Eq. 8 in ref. by considering only J(0) and J(ωN) spectral density terms and neglecting higher frequency terms.

Average ¹⁵N T₁ and T₂ relaxation times for a given protein can be measured using 1D ¹⁵N-edited relaxation experiments . To minimize contributions from unfolded segments each 1D spectum is integrated over the downfield backbone amide ¹H region (typically 10.5 to 8.5 ppm) and the results are used to fit an exponential decay as a function of delay time. One then computes the correlation time using Eq. 3, and compares it to a standard curve of τ_c vs. protein molecular weight (MW) obtained at the same temperature on a series of known monomeric proteins of varying size. The T₁/T₂ method is suitable for proteins with molecular weight of up to MW ≈ 25 kDa. Accurate measurement of the diminishing ¹⁵N T₂ becomes difficult for larger proteins and cross-correlated relaxation rates are measured instead.

Below is a table of rotational correlation time values compiled for known monomeric NESG targets (J. Aramini; June 2010).  All data was recorded on a Bruker 600 NMR instrument at 298 K.  The molecular weight for each target takes into account isotopic enrichment and the presence of affinity purification tags (if any).

NESG target (isotope labeling)	MW (kDa)	15N T1 (ms)	15N T2 (ms)	τc (ns)
PsR76A  (NC5)	7.2	478.0	128.0	5.10
VfR117 (NC)	11.2	605.0	119.0	6.30
SyR11 (NC5)	12.4	630.0	104.0	7.10
ER541-37-162 (NC5)	15.8	729.0	66.5	10.0
ER540 (NC5)	18.8	909.0	66.5	11.3
SoR190 (NC)	13.8	697.5	100.9	7.70
TR80 (NC5)	10.5	612.8	102.9	7.00
Ubiquitin (NC)	9.0	441.8	144.6	4.40
HR2873B (NC)	10.7	492.0	115.0	5.70
B-domain (NC)	7.2	423.5	153.3	4.05
BcR97A (NC)	13.1	705.8	80.6	8.80
PfR193A (NC)	13.6	733.9	80.9	9.00
MvR76 (NC)	20.2	1015.0	64.5	12.2
DvR115G (NC)	10.9	608.7	115.6	6.50
MrR110B (NC5)	11.8	707.0	99.2	7.80
VpR247 (NC5)	12.5	661.2	88.3	8.05
BcR147A (NC)	11.9	645.0	104.0	7.20
WR73 (NC5)	21.9	1261.0 *	41.3 *	13.0
NsR431C (NC5)	16.8	855.5	71.2	10.6
StR82 (NC)	9.2	537.3	100.4	6.6

• data obtained on an 800 MHz Bruker spectrometer at 298 K.

Protocols for Bruker and Varian NMR Instruments

• Bruker protocol
• Varian protocol