Discussion and Conclusion

Super-cooled RD could be used to obtain site specific kinetics for two residues whose experimentally determined lifetimes can be attributed to the time constant for conformer

interconversion (Figure 4 and 5). The temperature dependence of their exchange lifetimes places the conformer interconversion to be fast at 309 K with a lifetime of 10 9 s (Figure 6). This experimental observation also agrees with a long MD trajectory of BPTI where backbone fluctuations also coincided with the microsecond timescale [117]. Additionally, through the use of ensembles that capture the motional variances from this timescale calculated motional

amplitudes were the largest for the same two residues across all RDC-derived ensembles (Figure 7). An independent experimental approach was also utilized to verify the kinetics that had been detected.

The site specifically resolved kinetics for ubiquitin were also tested using an independent method that does require extrapolation of the extracted lifetimes and can be probed directly at 309 K. The employed technique was dielectric relaxation (DR) spectroscopy. DR is sensitive to motions that originate from changes of the electric dipole moment of solutes, solvent, and ions in solution [118]. Generally in DR, motions are broken down into different regimes ,  and  which correspond to motions from conductivity of small ions in solution, dipole relaxation processes, and bond librations as well as the rearrangement of water dipoles, respectively. DR was measured on free ubiquitin in solution and at 309 K the  peak correctly corresponded to the

35

c for ubiquitin. Once the effect of relaxation due to ionic charge transport is removed from the measured permittivity, the dielectric loss spectrum presented a new peak [112]. Deemed the

sub-, its resonant frequency was less than  peak. The mean value for this peak centers around 1 s at 309 K providing independent validation for the super-cooled RD based Arrhenius extrapolation [112]. This peak was also reproduced at different temperatures and with different choices of ions in solution [112].

Kinetics from the supra-c range, which spans four orders of magnitude, has remained elusive until now. We could narrow down this range and identify that for ubiquitin the lifetime for interconversion is around 10 s at physiological temperatures. Via super-cooled RD we could obtain site specific kinetics whose amplitudes were also verified by ensembles that report on the motional variance of ubiquitin within the same timescale. In addition, NMR relaxation dispersion before has not been used to provide kinetic characterization of ground-state fluctuations from a protein. This motion was also identified by an independent experimental technique, namely solution DR. These studies also allow for this motion to be studied under solution conditions without having to make chemical modifications to the system itself. The combination of super-cooled RD, RDC-derived ensembles, and solution DR spectroscopy will open the doors for future investigations that should be applicable to a wide range of systems in order to elucidate motions from the supra-c range.

36 2.5 Materials and Methods

Super-cooled off-resonance R_1NMR samples for measurements below the freezing point of water were conducted by filling 1 mm capillaries in which twelve could then be placed into one 5 mm NMR sample tube. ¹⁵N labeled ubiquitin at a concentration of 6 mM was in a buffer

composed of 100 mM NaCl, 50 mM sodium phosphate at pH 6.5. Dynamic light scattering was used to confirm that ubiquitin still existed as a single monomeric species at the employed

concentrations. Off-resonance R1 experiments were conducted at 265, 269, 273, and 277 K with a TROSY based sequence similar to Kempf et al [110,119]. A TROSY [120] block was used not only to prevent sample heating during decoupling applied in the direct dimension, but also to account for the decrease in sensitivity due to the slower tumbling of ubiquitin at lower temperatures. Water handling was also optimized for the employed gradient based

Echo-Antiecho readout used for frequency discrimination. Each spectrum was recorded with 512 and 128 complex points in the direct (t2) and indirect (t1) dimensions, respectively, with 24 transients per t1 increment. The t1,max and t2,max were 81.4 and 65.5 ms, respectively. At each temperature eight spin-lock field strengths (1) were used and varied between 265 to 3050 Hz. The spin-lock carrier frequency was set outside of the spectral range to 134 ppm. For each 1, a relaxation series was conducted by changing the length of the applied spin-lock between 20 to 240 ms. All spectra were acquired on a Bruker Avance III spectrometer operating at a ¹H Larmor frequency of 600 MHz. All data were processed with the NMRPipe sotware package [121] and visualized with the CARA program [122].

R1 rates were determined by fitting each relaxation series to the function I0=exp(-R1T).

Assessment of conformational exchange with respect to the effective fields (eff) that were used were ascertained by fits to a fast exchange model of

37

longitudinal relaxation rate, intrinsic inphase transverse relaxation rate, the conformational amplitude, and the exchange time constant, respectively. All data fitting routines were carried out using Mathematica (Wolfram Research). Errors in the fitted parameters were determined by Monte-Carlo simulations run with 500 iterations that used the base-plane noise as the standard deviation for a given intensity value. All parameters that describe exchange can be found in Appendix Table 1. Temperature dependent time constants were fit to an Arrhenius type exchange

of the form    activation energy, respectively. Errors in A and E_a were determined from error propagation.

ensemble calculation The RDC-derived ensembles are non-canonical and therefore each

conformer has an equal probability of existing and that all conformers interconvert with the same rate (k). The time dependent change in the populations is given by

   

The equation above is a first-order rate equation, but in NMR observables are made at

equilibrium therefore K P 0. Where the kinetic matrix (K) follows the formalism for an N-site jump model [123] and takes the form:

r k

38 where ^{r }^{N 1}^k and the P matrix is a column vector with the form of P^T p1,p2, ...p_N. The conditional probabilities are given by

 ^{, | , 0} ¹ _ij ^{1 N kt} correlation function takes the form

       

where  is the precession frequency for a nucleus in a given state. After substitution of equation 2.5.3, into equation 2.5.4 yields

 

Separation of the time-invariant, which do not cause relaxation, and time variant terms gives

  2 2

 

The first term in equation 2.5.6 is time independent and ineffective in causing relaxation.

Focusing on the time-variant component we can expand the summation into two components in which

39 Equation 2.5.7 is a series that is the expanded form for the square difference between states i and j and simplifies to

Equation 2.5.8 is reduced by a factor of two because the summation is performed for jumps between states i to j and j to i, but whose squared difference is equal in magnitude. The NMR observable form of equation 2.5.8 comes after Fourier transform which in the case of exchange in the fast limit becomes

where  = (Nkt)^-1, eff is the experimentally employed frequency offset and spin-lock field. The term preceding the Lorentzian is the amplitude of the motion or structural variances reported by the ensembles

40

Exceeding the kinetic limit for dynamic