Spin Diffusion

Spin diffusion is a coherent process that promotes spontaneous exchange of magnetisa- tion between spins. If this magnetisation transfer happens sufficiently fast compared to relaxation times during relaxation rate measurements, the measured rates are averaged over several sites, meaning that they are no longer site specific.68–71 Thus it is important that the effects of spin diffusion are effectively suppressed in relaxation measurements. It is particularly challenging to do so in R1 measurements.

During protein 13C and 15N R1 measurements the most efficient spin diffusion

is assisted by the dipolar couplings to protons. This is known as proton-driven spin diffusion (PDSD) and its rate is dependent on the cross terms involving 1H-X and X-X dipolar couplings, where X = 13C or 15N. 15N measurements tend to be less affected by PDSD than 13C due to their lower gyromagnetic ratio and therefore smaller dipolar couplings, as well as being sparsely spread throughout the protein. The rate of PDSD can be reduced by fast MAS, dilution of the 1H network or application of RF pulses.

Although, the latter technique is rarely used in practice due to the requirement of high RF pulses that may damage the equipment or sample.

Suppressing PDSD is huge problem for aliphatic 13C nuclei in particular, due to their strong one-bond1H-13C and13C-13C dipolar couplings. As previously discussed, if the coupled nuclei have a small chemical shift difference, relative to the dipolar coupling constant, higher-order terms will become relevant. This is the case for some13Cα 13_Cβ

nuclei that are close in chemical shift, and therefore fast MAS will only eliminate some of the dipolar interaction terms.

In addition to fast spinning, partial 13C labelling or deuteration can be used to remove some of these pathways for PDSD. This combination of techniques allows for accurate, site-specific R1 measurements on aliphatic 13C nuclei, but there are various

disadvantages to using these isotopic labelling schemes. In this light, Chapter 4 focuses on methods of eliminating the effects of proton-driven spin diffusion from site-specific

13_Cα _R

1 measurements on fully-protonated, uniformly13C labelled proteins. Calculating the Effect of Spin Diffusion on R1

In the case of a two-spin system coupled by spin diffusion, the spin diffusion can be described by the following equation:69

∂ ∂t " I S # =−(K+R). " I S # − " M0 M0 # ! (2.80) Where K =   −σ σ σ −σ  , R =   RI 1 0 0 R₁S 

 and M0 is the magnetisation at thermal

equilibrium. In this system, two spins are considered, I and S, with spin-lattice relaxation ratesR1I and R1S, magnetisationsI and S (at time t) and an exchange rate ofσ (due to PDSD).

The evolution of the above system can be described by Equations 2.81 and 2.82: ∂I ∂t =−R I 1(I−M0)−σ(I−S) (2.81) ∂S ∂t =−R S 1(S−M0)−σ(S−I) (2.82)

Equations 2.81 and 2.82 were solved (using Matlab) allowing the effect of PDSD on the true R1 to be calculated. Calculating the time dependence of the magnetisation

of spins I and S will simulate the relaxation of this nucleus over time when coupled to the other spin, and fitting an exponential curve to this allows for calculation of the effectiveR1. The solution to Equations 2.81 and 2.82, and Matlabscripts used for these

calculations are included in Appendix A.2.

The simulations presented in Figure 2.26 highlight just how effective PDSD is at altering the measured relaxation rates from their true values. If the exchange rate is faster than either of the relaxation rates it will completely dominate the system and the

measured relaxation rates will no longer contain any site-specific information. Even a PDSD exchange rate an order of magnitude smaller than the true relaxation rates can still significantly alter the measured rates, so it is critical to reduce the PDSD exchange rate as much as possible during relaxation measurements.

There are two limiting cases for magnetisation exchange due to PDSD: 2σ >> |RI₁−RS₁|and 2σ <<|RI₁−RS₁|.72 In the former case the PDSD rate will clearly dominate the measured relaxation rates, causing the magnetisation to decay bi-exponentially for both spins at an averaged rate. In this situation it is not possible to determine reliable dynamics data. The latter case, where the difference in relaxation rates is much greater than the rate of spin diffusion, will produce independent mono-exponential decays for the spins I and S with the rates ofRI₁+σ and RS₁ +σ, respectively. In this case ifσ is sufficiently small compared to the relaxation rates it will be possible to obtain reliable dynamics data.

Further simulations revealed that the system that causes the largest deviation in the measured R1 of spin I is one where spin I relaxes much faster and is much less

polarised than spin S. This, in the majority of cases, matches the situation between the adjacent 13Cα _and 13_Cβ _{nuclei in the relaxation measurements throughout Chapter}

4: 13Cβ _R

1 is typically far greater than 13Cα R1 due to the increased dynamics of a

sidechain, however the13Cα _{peaks tend to be far more intense. This suggests why it can}

be so challenging to suppress PDSD at during these 13Cα _{relaxation measurements.}