Protein dynamics and estimation of additional relaxation from ¹³C to measured ¹⁵N relaxation, diffusion tensor and...

Studies of protein dynamics are mainly based on ¹⁵N relaxation data measured from a sample labeled by ¹⁵N only. For ¹⁵N/¹³C-labeled samples, contribution of additional relaxation from ¹³C to the ¹⁵N relaxation complicates studies of protein dynamics and model-free analysis. Here we present an optimization approach via correction of ¹⁵N-¹³C^α and ¹⁵N-¹³CO dipole-dipole interactions for ¹⁵N/¹³C-labeled NACI to determine correlation time and diffusion tensor and to correct measured R₁ and R₂ values for residues in which internal motion is restricted and chemical exchange is absent, by fitting the measured and theoretical R₂/R₁ ratios. The additional increased longitudinal and transverse relaxation rates via ¹⁵N-¹³C relaxation are determined as well by calculating ¹⁵N-¹³C^α and ¹⁵N-¹³CO dipole-dipole interactions. The estimated additional increased longitudinal and transverse relaxation rates via ¹⁵N-¹³C relaxation in the ¹⁵N/¹³C-labeled NACI are not constant over the protein, but different from residue to residue, and range between 0.124 s^-1 and 0.132 s^-1 for the additional increased transverse relaxation rate, corresponding to around 2 % increase of the transverse relaxation rate in the absence of ¹³C, and between 0.057 s^-1 and 0.062 s^-1 for the additional increased longitudinal relaxation rate, corresponding to around 3 % increase of the longitudinal relaxation rate in the absence of ¹³C. For the ¹⁵N/¹³C-labeled NACI with determined correlation time 5.0 ns, the measured R₁ and R₂, because of the increased relaxation in R₂/R₁ ratio close to 1, can be used directly as R₂/R₁ input into program 'r2r1_diffusion'. For verification of our approach, the measured R₁ and R₂ from the ¹⁵N/¹³C-labeled NACI were directly inputted into our approach and also into the program 'r2r1_diffusion' for comparison. The diffusion tensor obtained from the calculations of our approach is consistent with results derived from the program 'r2r1_diffusion', revealing that it is feasible to correct the effect of the additional relaxation on R₁ and R₂ values and to determine the correlation time and the diffusion tensor for the doubly ¹⁵N/¹³C-labeled NACI containing an additional long peptide from expression plasmid. When correlation time (from 3.5 to 30 ns) or magnetic field (from 500 to 800 MHz) increases, the contributions from the ¹³C^α and ¹³CO relative to ¹⁵N R₁ and R₂ without ¹³C labeling change only slightly and the changes in the additional increased percentages are very small; i.e., the additional increased percentages remain around 3 % increase in R₁ and around 2 % increase in R₂, and hence the additional increased percentages from the contribution of ¹³C to ¹⁵N could be considered approximately independent of both the correlation time (from 3.5 to 30 ns) and the magnetic field (from 500 to 800 MHz). For the diffusion parameters, such as correlation time and diffusion tensor, derived from using R₂/R₁ ratio, the contribution from carbon labeling is relatively very small and does not influence the fitting of the diffusion parameters. The reason is that the R₂/R₁ ratio is used, in which the ¹³C contribution effects are cancelled out each other via (1 + 3 %)/(1 + 2 %) in the R₂/R₁ ratio. But because the 3 % increase in R₁ and the 2 % increase in R₂, even though relatively small, do affect S² values that are derived from the program 'Fast-Modelfree' using R₁ and R₂ individually, rather than using the R₂/R₁ ratio, and do make the S² values higher by 0.2 % to 5.9 % (more rigid), the contribution from carbon labeling is hence not negligible for parameters fitted using individual R₁ and R₂ inputs. Overall, parameters fitted from R₂/R₁ ratio, in case of similar increased percentages, such as 3 % and 2 %, will not be influenced, whereas parameters, such as S², fitted from using individual R₁ and R₂ will be influenced. Here, the corrections of the contributions from ¹³C to measured ¹⁵N relaxation are limited to those residues in which internal motion is restricted and chemical exchange is absent, and can be taken directly as R₂^NC,measured/R₁^NC,measured for input in the program 'r2r1_diffusion' or quantified as R₁^NC,measured/1.03 and R₂^NC,measured/1.02 for input in the program 'Fast-Modelfree'. In addition, relaxation data show significantly increased R₂ values and R₂/R₁ ratios and hence suggest conformational exchange for assigned residues S13, C38, G39, G40 and N41 which are mainly confined to a region surrounding C14-C38 disulfide bond. S13 in the NACI experimentally shows temperature-dependent conformational exchange with reduced R₁ value, significantly increased R₂ value and R₂/R₁ ratio, and larger deviation of NOE, whereas P13 in BPTI lacks amide proton.