Evaluating the uncertainty in exchange parameters determined from off-resonance R1ρ relaxation dispersion for systems in fast exchange

Abstract

Spin relaxation in the rotating frame (R1ρ) is a powerful NMR technique for characterizing fast microsecond timescale exchange processes directed toward short-lived excited states in biomolecules. At the limit of fast exchange, only k(ex)=k(1)+k(-1) and Φex=p(G)p(E)(Δω)(2) can be determined from R1ρ data limiting the ability to characterize the structure and energetics of the excited state conformation. Here, we use simulations to examine the uncertainty with which exchange parameters can be determined for two state systems in intermediate-to-fast exchange using off-resonance R1ρ relaxation dispersion. R1ρ data computed by solving the Bloch-McConnell equations reveals small but significant asymmetry with respect to offset (R1ρ (ΔΩ)≠R1ρ (-ΔΩ)), which is a hallmark of slow-to-intermediate exchange, even under conditions of fast exchange for free precession chemical exchange line broadening (k(ex)/Δω>10). A grid search analysis combined with bootstrap and Monte-Carlo based statistical approaches for estimating uncertainty in exchange parameters reveals that both the sign and magnitude of Δω can be determined at a useful level of uncertainty for systems in fast exchange (k(ex)/Δω<10) but that this depends on the uncertainty in the R1ρ data and requires a thorough examination of the multidimensional variation of χ(2) as a function of exchange parameters. Results from simulations are complemented by analysis of experimental R1ρ data measured in three nucleic acid systems with exchange processes occurring on the slow (k(ex)/Δω=0.2; pE=∼0.7%), fast (k(ex)/Δω=∼10-16; p(E)=∼13%) and very fast (k(ex)=39,000 s(-1)) chemical shift timescales.

Keywords: DNA; Dynamics; Excited state; Fast exchange; RNA; Rotating-frame relaxation dispersion.

Figure 1

Resonance offsets and effective fields in R1ρ relaxation dispersion.

Figure 2

Difference in for R1ρ between positive and negative offsets for the spin lock powers and offsets listed in Table S1. Each data point was sorted by decreasing difference in R1ρ between a matching spin lock power and its positive-negative offsets. Data is shown for the following exchange scenarios: black, p_E = 5%, k_ex = 4,000 s⁻¹; blue, p_E = 5%, k_ex = 8,000 s⁻¹; cyan, p_E = 5%, k_ex = 12,000 s⁻¹; green, p_E = 5%, k_ex = 16,000 s⁻¹; red, p_E = 5%, k_ex = 20,000 s⁻¹. The Δω used to generate the data sets was 2.12 ppm (¹³C at 14.1 T).

Figure 3

Grid search analysis of R1ρ relaxation dispersion data. Shown is the value of $e^{(-{\chi }^2/2{\chi }_{\mathrm{\text{min}}}^2)}$ where ${\chi }^2=\sum _{i=1}^N{\left(\frac{R_{1\rho (i)}^{\mathit{\text{Calc}}}-R_{1\rho (i)}^{\mathit{\text{Exp}}}}{{\sigma }_{\mathit{\text{Exp}}(i)}}\right)}^2$ as a function of exchange parameter. The Δω used to generate the data sets was 2.12 ppm (¹³C at 14.1 T). Data is shown for different exchange scenarios a) p_E = 5%, k_ex = 1,000 s⁻¹, and k_ex/Δω =0.5 b) p_E = 5%, k_ex = 4,000 s⁻¹, and k_ex/Δω =2 c) p_E = 5%, k_ex = 20,000 s⁻¹, and k_ex/Δω =10.

Figure 4

Evaluating uncertainty in exchange parameters derived for R1ρ data using bootstrap and Monte-Carlo approaches for various exchange scenarios. Error bars represent the standard deviation between 3 independent runs. The Δω used to generate the data sets was 2.12 ppm (¹³C at 14.1 T). Note that although the absolute uncertainty in population generally increases with input p_E, the fractional error decreases or stays constant.

Figure 5

Example bootstrap fits for various exchange scenarios. Shown are density plots of exchange parameters obtained upon fitting 1,000 bootstrap R1ρ data sets with 5% noise corruption. The Δω used to generate the data sets was 2.12 ppm (¹³C at 14.1 T). Results are shown for various exchange scenarios a) p_E = 5%, k_ex = 4,000 s⁻¹, and k_ex/Δω =2 b) p_E = 5%, k_ex = 20,000 s⁻¹, and k_ex/Δω =10.

Figure 6

Grid search analysis of exchange parameters for experimental R1ρ data describing a slow process (k_ex/Δω = 0.2) involving Watson-Crick to Hoogsteen transitions in DNA.[37] Shown is representative example data for G10-C1' and G10-C8. Similar results were obtained from analysis of other dispersion R1ρ data in DNA describing Watson-Crick to Hoogsteen transitions (data not shown). a) The DNA duplex studied by NMR with G10 highlighted in red. b) The Watson-Crick to Hoogsteen transition. c) Shown is value of $e^{(-{\chi }^2/2{\chi }_{\mathrm{\text{min}}}^2)}$ where ${\chi }^2=\sum _{i=1}^N{\left(\frac{R_{1\rho (i)}^{\mathit{\text{Calc}}}-R_{1\rho (i)}^{\mathit{\text{Exp}}}}{{\sigma }_{\mathit{\text{Exp}}(i)}}\right)}^2$ as a function of exchange parameter. The extracted exchange parameters and uncertainties are listed in Table 3.

Figure 7

Grid search analysis of exchange parameters for experimental R1ρ data describing a fast transition (k_ex/Δω ≥ 10) involving localized changes in base-pairing with the HIV-1 TAR RNA apical loop.[33] a) The TAR sequence studied by NMR with exchanging residues highlighted in red b) The proposed ground to excited state transition with exchanging residues highlighted in red. c) Shown is value of $e^{(-{\chi }^2/2{\chi }_{\mathrm{\text{min}}}^2)}$ where ${\chi }^2=\sum _{i=1}^N{\left(\frac{R_{1\rho (i)}^{\mathit{\text{Calc}}}-R_{1\rho (i)}^{\mathit{\text{Exp}}}}{{\sigma }_{\mathit{\text{Exp}}(i)}}\right)}^2$ as a function of exchange parameter for three representative residues. The extracted exchange parameters and uncertainties are listed in Table 4.