Mathematica 8

All of the simulations carried out in this study assume two site exchange between a GS and ES. Synthetic carbon R1ρ relaxation data at 14.1 T magnetic field strength consisting of 150 data points with varying resonance offset (ΔΩ) and power (ω₁) (listed in Table S1) reflecting values typically used in our nucleic acid studies[33 (link), 37 ] were computed by solving the Bloch-McConnell equations (Equation 3) numerically using Mathematica 8.0 (Wolfram Research). To be consistent with the R1ρ experimental data analyzed in this study, we generated the synthetic Bloch-McConnell R1ρ data assuming the ¹³C selective 1D R1ρ experiment by Hansen et. al..[24 (link), 31 (link)] In this experiment an equilibration period exists prior to alignment with the effective field (ω_G or ω_eff for exchange on the slow and intermediate-fast timescale, respectively) for the spin lock period (Figure 1). The equilibration period allows for the build up of excited state magnetization that ends up aligned via a hard pulse with the effective field (ω_G or ω_eff, Figure 1). Simulations of the spin lock period with the Bloch-McConnell equations showed that this introduces negligible deviations relative to the conventional scheme for the asymmetric populations and intermediate/fast exchange rates examined here and in our previous dispersion studies of nucleic acids.[24 (link), 33 (link), 37 ] However, for relatively large p_E > 15% and slow exchange (k_ex/Δω < 0.5) the above scheme can result in oscillations due to precision of the initial ES magnetization that can be mitigated by simple modification of the pulse sequence as described by Kay[31 (link)] and more recently by Zhang[15 (link)]. R1ρ was computed by fitting the resultant projection of the net magnetization of the ground and excited states following the relaxation period along
${\omega }_{\mathit{\text{eff}}}={(\mathrm{\Delta }{\mathrm{\Omega }}^2+{\mathrm{\omega }}_1^2)}^{1/2}$ as a function of time to a monoexponential decay. For fast exchange (k_ex/Δω ≥ 2), R1ρ was calculated by initially aligning both the GS and ES magnetization along the average effective field (ω_eff) and by projecting the resultant GS and ES magnetization following the spin lock period about the average effective field (ω_eff) (Figure 1). For slow exchange (k_ex/Δω ≤ 1), R1ρ was calculated by initially aligning both the GS and ES magnetization along the GS effective field (ω_G) and by projecting the resultant GS and ES magnetization following the spin lock period about the average effective field (ω_eff) (Figure 1). We note that for systems in slow exchange, and when using the initial magnetization preparation scheme that aligns both the ES and GS along the GS effective field, the actual observed projection would be about the ground state effective field (ω_G).[24 (link)] Simulations using the Bloch-McConnell equations show that this can lead to small deviations relative to projections about the average effective field for a small subset of conditions examined here involving slow exchange, low spin lock fields of ω₁ < 1000 Hz and for p_E >15%. However, these deviations have negligible effects on the extracted exchange parameters. Similar exchange parameters are obtained when fitting 5% noise corrupted R1ρ data computed assuming projections along either the average (ω_eff) or ground state (ω_G) effective fields (data not shown). We also note that we carried out simulations for scenarios fast on the chemical shift timescale (k_ex/Δω ≥ 2) in which we calculated R1ρ by initially aligning both the GS and ES magnetization along GS effective field (ω_G) and obtained nearly identical uncertainties to those reported in Table 1 when fitting 5% noise corrupted data.
The synthetic R1ρ data was uniformly noise corrupted assuming 5% uncertainty by randomly selecting a value from a normal distribution centered at the R1ρ value with a standard deviation set equal to 5%. Standard simulations assumed Δω = 2.12 ppm (2,000 s⁻¹), p_E of varying asymmetry (5%, 15%, 30%), exchange rates (k_ex) ranging between 1,000 s⁻¹ and 30,000 s⁻¹ spanning k_ex/Δω values between 0.5 and 15. The R₂ and R₁ relaxation rate constants were equal to 11.0 s⁻¹ and 1.5 s⁻¹, respectively to be consistent with prior theoretical studies examining R1ρ and its associated algebraic expressions.[12 (link), 27 , 29 (link), 43 (link)] As shown below, increasing the value of R₂ but assuming a constant signal to noise ratio results in larger uncertainty in the extracted exchange parameters (p_E, Δω, k_ex) due to the increase in the relative contribution of R₂ to transverse relaxation as compared to R_ex.