Note for roughly estimating the protein aggregation state from its diffusion coefficient measured by NMR method

1. Use the (modified) NH-diffusion pulse sequence from biopack to measure a series of 1D spectra with different gradient values for the N¹⁵, N¹⁵/C¹³, or N¹⁵/C¹³/D labeled protein in 90%/10% H₂O/D₂O buffer. (For nsp10~12kDa and ubiquitin~8.5kDa, array “gzlvl2 from 2000 to 20000 with interval =3000, and total 7 values are enough here. Note: nt = N*16, N is 1, 2, 3…)

2. Use vnmrj to process the 1D spectra, find a strongest peak and use “peak” command to get the intensity values under each gradient value.

3. Calculate D based on the next equation:

A(g²) = A(0)exp[-γ²δ²g²D(Δ-δ/3)],

Where γ is the gyromagnetic ratio for protons, Δ is the diffusion period time, and δ is the duration time of the gzlvl2 gradient pulse. (Δ and δ values can be gotten from the pulse sequence.)

After the equation transforming:

ln(A(g²)) = –[γ²δ²(Δ – δ/3)*D]*(g²) + ln(A(0))  → y = -a*x + b

Use excel to do the linear fitting, and get the slope “a”, so D = a / [γ²δ²(Δ-δ/3)].

 

4. Based on Stokes-Einstein equation, D = kT/6πηR_s,

Where k is the Boltzman constant, T is the absolute temperature (e.g. 278K here), η is the viscosity of the solvent (for water, η = 0.01gcm^-1s^-1), R_s is the Stokes hydrodynamic radius of the spherical particles, so

R_s = (2.2/D) x 10^-6 nm.

5. Since the proteins have the same approximate specific weight (density), 1.37g/cm³, the simple estimation for the relationship between the molecular weight M(Da)and its spherical radius R(nm) (assume the molecule is spherical shape) is as the following:

M (g)/6.023*10²³ = 1.37(g/cm³) *V(cm³)

V = 1/(1.37*6.023)*10^-23 (cm³) = 1.212*10^-3 * M (nm³)

On the other hand, assuming the protein molecule is spherical shape,

V = (4π/3)*R³,

So, R = [3V/(4π)]^1/3 = 0.066*M^1/3 (nm)

For a protein (assuming it’s spherical) with its known MW, its radius should be at least has the R value, its Stokes hydrodynamic radius R_s should be more accurate, and bigger than its R.

For example here, for nsp10 (MW~12.5kDa), after measurement, D = 1.30*10^-6 cm²/s; so its R_s = 1.69 nm and R = 1.53 nm.

For ubiquitin (MW~8.5kDa), D = 1.57 *10^-6 cm²/s; so its R_s = 1.40 nm and R =1.35 nm.

Based on the known data and the measured D, we can estimate if a new protein exists as a monomer or oligomer…

Useful references:
1. JMR 2004, 166, 129-133.
2. Biological Procedures Online, 2009, 11(1), 32-51.

Hongwei edited on 12/22/2014