2.4 Analysis of PRE
The ratio of peak height of a paramagnetic sample to that of a diamagnetic sample, that is,I para/I dia, can be expressed as in Equation 2:22
\(\frac{I_{\text{para}}}{I_{\text{dia}}}=\frac{R_{2int}exp(-R_{2PRE}t)}{R_{2int}+R_{2PRE}}\)(2)
where R 2int and R 2PRErepresent the intrinsic transverse relaxation rate and PRE effects on the transverse relaxation rate for each amide proton, respectively; andt is the total INEPT evolution time of the HSQC (11.5 ms). TheR 2int for each amide proton was calculated from the half-height line-width of peaks of the diamagnetic sample. Using Equation 2, the R 2PRE was estimated by assigning the t andI para/I dia values. TheR 2PRE was then converted into distance using Equation 3:
\(r^{6}=\frac{K}{R_{2PRE}}\left(4\tau_{c}+\frac{3\tau_{c}}{1+\varpi_{H}^{2}\tau_{c}^{2}}\right)\)(3)
where r is the distance between the unpaired electron and nucleus, ω H is the Larmor frequency of protons,K is 1.23 ×10-32 cm6s-2,28 andτ c is the correlation time given as 1/τ c =1/τ r+1/τ e (τ r is the rotational correlation time of the electron–nucleus vector andτ e is the electron spin relaxation time). Sinceτ e (> 10-7 s) of the unpaired electron is much longer than τ r,τ c is approximately equal toτ r. The τ c value is approximately 12 ns for the folded conformation of OspA at 318 K and 0.1 MPa.29 The half-height line-widths (Δν(hertz) = R 2int/π) were approximately 20 Hz for amide proton signals in the protein. The distances from theI para/I dia were estimated with a τ r of 12 ns andR 2int (20 Hz). Back-calculation ofI para/I dia for structural models of OspA was performed with τ r = 12 ns andR 2int = 20 Hz.
Alternatively, estimation of PRE for the unfolded polypeptide chain is not as straightforward due to the fluctuations in both the orientation of the spins and its distance.30 Therefore, the distance r was characterized according to the probability distribution. Here, to simulate PREs of the OspA temperature-stabilized intermediate, we used the simplest restraining model, in which ris between the distance of closest approach for the two spheres (i.e. unpaired electron and nucleus). The r 6 in Eq.3 can be substituted withd 03L3 , assuming that the time scale of the distance fluctuation is slow on the time scale of τ rot.30d 0 and L are the distance of the minimum and the maximum separation, respectively. Further, the probability distribution of r was obtained from a trajectory of molecular dynamics simulation for heat unfolding of the protein (seeMolecular dynamics simulation ), and d 0 andL were estimated from the distance distribution. Theτ c of 12 ns, which is for native state OspA, was used for the intermediate conformations of the protein. More realistic restraining models were discussed in the literature.30