Supporting Information
S1. 1H-15NHSQC spectra of Syn protein under the following conditions (A) pH 7.4, 15 °C; (B) pH 6.1, 15 °C; (C) pH 7.4 35 °C; and (D) pH 6.1, 35 °C. Sample solutions were prepared in 10 mM sodium phosphate buffer, 137 mM NaCl, 2 mM KCl, and 90%/10% H2O/D2O.
S2. A kinetic model for hydrogen exchange effect on the 15NR2 relaxation.
In H/H exchange, the reactant (NH) and the product (NH’) are distinguished as shown in Scheme 1.The relevant rate equations forthe HX scheme are,
(1’)
(2’)
The integral forms of these equations with I0 (the initial concentration) are,
(3’)
(4’)
When the NMR observable (I) is proportional to the sum of [NH]and [NH’], the signal decay is,
(5’)
When fanti1, Q’ Q and equation 5’ is approximated as equation 6’ shown in equation 5 in the paper.
(6’)
S3.The R2CPMGrelaxation data were acquired with the SS pulse and fit with the singleexponential decay function. The differences between R2CPMG values at pH 7.4 (red circles) pH 6.2 (blue rhombi) are reduced compared to Fig. 1 (single exponential fit without the SS pulse) but larger than Fig. 4 (multi-exponential fit with the SS pulse). The resulted average 15NR2CPMG values at pH 7.4 and pH 6.2 are 3.73 ± 0.49 Hz and 3.29 ± 0.44 Hz.(b) Correlation between R2CPMGvalues at pH 7.4 and pH 6.2.
S4. Comparison of the proposed HX model (equation 1) in this paper and the generalized chemical exchange model (Bloch-McConnell equation)
(1) The generalized chemical exchange model (Bloch-McConnell equation) (Hansen and Led 2003; McConnell 1958)
The chemical exchange between two states (A and B) is described as,
The state A is NxHE and the state B is NxDE under HX. By definition, kHX = kf + kb where kf is the forward exchange rate, kb is the backward exchange rate, and kHX is the nominal HX rate. At equilibrium, pAkf = pBkb where pA is the population of A and pB is the population of B. Since pA and pB can be determined by mixed sample solvents, pA = Q and pB = (1-Q) where Q is the H2O content and (1-Q) is the D2O content in the sample, so Q = kb/kHX. The magnetization of the state A (MA) and that of the state B (MB) are subject to undergoing their own R2CPMG relaxation rates (R2A and R2B). Because of CPMG refocusing pulses, the chemical shift evolution of the transverse magnetizations are not considered. The intensity change of magnetizations, MA(t) and MB(t), during the relaxation time (t) are given by this differential equation including the chemical exchange.
Equation S4.1
The solution of this equation is
Equation S4.2
(2) Comparison of the proposed HX model (equation 1 in the main text) in this paper and the generalized chemical exchange model (Bloch-McConnell equation)
The observable magnetization MA(t) in equation S4.2 is identical to equation 1 (our proposed model) by substituting these parameters; I(t) = MA(t), I0 = MA0, R2CPMG = R2A = R2B, kHX = kf + kb, kf = (1-Q)kHX, kb = QkHX, and MB0 = 0. Because of out-and-back characteristics of HSQC, the initial magnetization of MB (MB0) is zero. As long as R2AR2B, equation 1 and equation S4.2 are equivalent regardless of the range of kHX. Generally, R2A and R2B are different from each other but it is difficult to get these separately from a single data set of a multi-exponential decay. Thus, the main difference between equation1 and equation S4.2 is due to the difference of R2A and R2B.
(3) Difference of R2A and R2B
R2A (R2NH, R2 of the protonated nitrogen) and R2B (R2ND, R2 of the deuterated nitrogen) were calculated as a function of the molecular tumbling time (m). At the magnetic field strength of 800 MHz (18.79 T), it is assumed that only gyromagnetic ratios of proton and deuterium count in the difference of R2A and R2B and the molecular dynamics is based on the isotropic rigid body (S2 = 1) (Farrow et al. 1994; Palmer III 2004). The R2B was estimated as about 44.6% of R2A (shown in the figure below).
(4) Effect of the difference of R2A and R2B
As shown in the figures below, by using R2A = 5 Hz and R2B = 2.25 Hz, MA(t) and MB(t) (black lines with symbols) were calculated based on equation S4.2 and MA(t) was compared to I(t) of equation 1 (red lines) with R2CPMG = R2A. Two representatives plots with kHX = 5 Hz and 50 Hz are shown. The difference of the two equations becomes obvious as kHX increases gradually while Q = 0.9 is kept constant for both equations. With increasing kHX, the onset of the deviation happens at the early relaxation time as well.
To estimate the impact on R2CPMG by ignoring the difference of R2A and R2B, data (with R2A = 5 Hz, R2B = 2.25 Hz, and kHX = 1 – 55 Hz) generated by the Bloch-McConnell equation were fit with equation 1. As shown in the figure below, the relative errors of R2CPMG (fit by equation 1) from R2A (calculated from the Bloch-McConnell equation) were plotted as function of the ratio of kHX and R2A. At kHX / R2A > 10 (the extreme fast exchange limit), the R2CPMG value will approach the population-weighted average of R2A and R2B, which is Q R2A + (1-Q) R2B = 4.725 Hz and the resulting relative error, (R2A – R2CPMG) x 100 / R2A, is 5.5 %. Under our experimental conditions (kHX / R2A ≤ about 10), the expected maximum error of R2CPMG is about 4.5 % by using equation 1 instead of using the Bloch-McConnell equation. Since it is difficult to get R2A and R2B from experimental data, equation 1 will be used as a good approximation of equation S4.2 because the H/D exchange obeys the pseudo-first order kinetics before reaching the equilibrium.
(5) Impact on corrected R2CPMG
To check whether corrected R2CPMG data presented in the paper are influenced by ignoring the difference of R2A and R2B, the difference of corrected R2CPMG data between pH 7.4 and pH 6.2 (Figure 4 in the paper) were compared to the ratio of kHX and R2CPMG at pH 7.4 (Figure 3 in the paper). As shown in the figure below, the impact on corrected R2CPMG by the difference of R2A and R2B is not significant since the difference of corrected R2CPMG data between pH 7.4 and pH 6.2 (figure below) did not show a dependence on kHX values, and the range of the difference was larger than the maximum error (0.15 Hz) by equation 1 which can be calculated as 4.5 % of the average R2CPMG (3.27 Hz) at pH 7.4.
1