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A novel approach for the determination of fast exchange rates
JOURNAL
OF MAGNETIC
RESONANCE
95, 309-3 19 ( 199 1)
A Novel Approach for the Determination of Fast ExchangeRates
Laboratory of Cardiac Energetics, National Institutes of Health, Bethesda, Maryland 20892 Received February 1, 199 1; revised May 28, I99 1 A novel technique is described to monitor fast exchange rates using NMR. This approach relies on the effects of partial saturation of the longitudinal magnetization with a saturation RF field at various o&et frequencies on the observed resonance lines. A complete quantum mechanical approach for the two-site chemical exchange of nuclei is presented, which yields access to rate constants from slow to very fast exchange rates using the described technique. The model system evaluated using this approach was the hydroxyl proton exchange in water-ethanol solutions. Exchange rate constants from the intermediate range (about 170 liters/s) at pH 7.5 to very fast rate constants up IO lo6 liters/s at pH 5 were measured. 0 1991 Academic Press. Inc.
Nuclear magnetic resonance spectroscopy is a well-established technique for the determination of chemical exchange rates over a wide range, with minimum rates in the order of the inverse spin-lattice relaxation time 1 / T, ( I ) . Slow-exchange reactions, which occur in a time comparable to T, , can be studied by double-resonance techniques originally suggested by For&r and Hoffmann (2). Other workers have extensively exploited this approach (3, 4). For more complex reactions, two-dimensional spectroscopic techniques have also been shown to be very useful (5, 6). For intermediate exchange, where the rate constant is in the order of the chemicalshift differences Au = VA - uB of both resonance lines VAand vs, lineshape analysis is a very sensitive method ( 7-9). However, this technique yields less accurate measurements of faster and slower reaction rates. For faster reactions &r-Purcell sequences have been utilized. At sufficient short interpulse delays ( = lop4 s) rate constants of the order of lo4 liters/s have been determined ( 10). However, the accuracy of adjusting the 180” pulses required for this approach drops for interpulse delays less than 10e4 s. An alternative method for fast exchange rates relies upon measurements of the spin-lattice relaxation time, T rp, in the presence of an RF field (II, 12). Herein we describe a new NMR method for the determination of intermediate to very fast chemical exchange rates where the individual resonance lines of the exchange partners are not resolvable. The method relies upon the determination of the frequency dependence of an off-resonance irradiation on the longitudinal magnetization of the participating resonances. To analyze these data a complete quantum mechanical treatment is presented. This approach allowed us to determine the dependence of hydroxyl proton exchange rates in water-ethanol solutions on mole fraction and pH. 309
0022-2364191 $3.00 Copyright 0 1991 by Academic Press, Inc. All tights of reproduction in any form reserved.
310
KNiiTTEL
AND
BALABAN
THEORY
The density-matrix formalism was used with the following assumptions. As the simplest case, the two-site exchange A
k.4 -B
ill
b
was evaluated, where A and B each contain one nucleus or a group of magnetically equivalent nuclei. Any effect due to scalar coupling is neglected. The exchange rate constants kA and kB can indicate either the real rate constants for intramolecular exchange or the pseudo-rate constants for intermolecular exchange ( 13). In the equilibrium condition the ratio of the rates is given by
k,lb = Wl/[Al,
t21
where [A] and [B] represent the populations at .the two sites. According to Kaplan and Fraenkel ( 14), the complete density-matrix the rotating frame has the form p = 0 = -ih-‘[&‘,
p] t- l? p + Ep,
where the first derivative of the density-matrix Hamiltonian operator is given by
equation in [31
operator p vanishes. The transformed
&” = h 2 {(u,, - va) fi + IQ@; + i,-)/2}.
[41
The subscript s indicates either the nucleus A or the nucleus B, so that vAo and em represent the corresponding Larmor frequencies. The rotating coordinate system revolves with the angular frequency v. of the saturation field, and the field strength is expressed in frequency units vl = yB,/( 27r). i’ and i- are the raising and lowering operators, respecti-vely. The operators R and l? in Eq. [ 3 ] are designated relaxation and exchange operators ( 14)) respectively. The phenomenologically introduced relaxation operator for spinlattice relaxation T, and spin-spin relaxation T;! is written Pm;-
gp = _ 2 s
(
PO-+ Poyiag IS
2s
.
[51
1
For a two-site exchanging system, the corresponding contribution to the density-matrix equation is given by E pA = kBpB - kApA E pB = kApA - kBpB,
161
where the sub- and superscripts A and B refer to the nuclei A and B. At equal populations, the rate constants kA and kB are equal according Eq. [ 21. Even for a twosite exchange, the intermolecular case is far more difficult to describe mathematically than the intramolecular case. However, it was shown by Alexander ( 15) and others
FAST
EXCHANGE
RATES
311
( 1, 16) that Eq. [ 61 describes both cases in the high-temperature limit, and this is valid for most applications. Almost the entire equilibrium z magnetization is maintained as long as the perturbation caused by the RF irradiation is small. In this case, Eq. [ 31 need be solved only for the off-diagonal elements which represent the transitions. For the lineshape analysis the slow- and fast-exchange limits have been derived analytically, and for the intermediate state numerical solutions are appropriate ( 14). In this work we apply a strong perturbation with the RF saturation field and therefore the complete quadratic density matrix of dimension eight according to Eq. [ 31 has been solved. The z magnetization M, of both nuclei is partially saturated immediately after the saturation pulse is switched off and is calculated as a function of the offset frequencies vAO- vo, vm - vo and the parameters TIA, TIB, TZA, TzB, v1. The diagonal matrix elements yield the total z magnetization Mz - P?, - P% + d1 - A!.
171
For the general case of the partially saturated z magnetization, the density-matrix equation is solved numerically, because analytical approaches are extremely cumbersome. The computer creates an inverse matrix with the help of the Gaussian elimination procedure ( 17). EXPERIMENTS
A two-site exchanging system is irradiated using the second channel at the offset frequencies VA0- v. and VBO- vo. The resonance lines VA0and YBoof the nuclei A and B are separately resolvable in the slow-exchange limit. After irradiating for a time of 5 T, , the system is at steady state. The second channel is gated off, and a spoil gradient is applied in order to destroy residual transverse steady-state magnetization. This provides excellent suppression in a single scan and yields more accurate measurements than pure phase cycling without a spoil gradient. About 5 ms after gating off the second channel, a 90” RF pulse enables the acquisition of both resonance lines. The integral of both lines in the Fourier-transformed spectrum represents the remaining z magnetization (of both nuclei), which is a function of the offset frequency and RF field strength of the saturation field. The measurements were carried out on a GE spectrometer at a magnetic field strength of 4.7 T. A 5 mm NMR tube containing the sample was placed in a saddle coil of 2 cm diameter tuned to the proton Larmor frequency of 200 MHz. The homebuilt gradient set of 5 cm diameter produced a one-dimensional spoil gradient of 0.15 mT / cm. The density-matrix calculations were done on a Sun computer 3 / 370 using IDL software (Research Systems, Inc.). In order to verify the theory, water-ethanol solutions have been chosen with two different proton molar equivalent fractions ( pmf). Since water has two exchangeable magnetically equivalent protons, pmf was used instead of mole fractions. Results for pmf values of 1/ 1 water/ethanol (mole fraction of 0.5 / 1) and 2 / 1 (mole fraction of 1 / 1) were evaluated. The intermolecular equilibrium reaction can be written
312
KNiiTTEL
R-OH;
+ HZ0
AND
BALABAN
+
R.-OH + H30+, 1
[81
where R-OH represents ethanol. The forward and backward rate constants are called k, and k-, , respectively. The pseudo-first-order NMR rate constants, kA and k,, for this system are expressed by
k/i Hz0 e
R-OH.
b
By comparison with Eq. [I] the protons of the water molecule are related to the nuclei A and the proton of the OH group is related to 13. Some parameters must be determined in advance in order to measure the pseudofirst-order rate constants. The RF field strength B, was calibrated by adjusting the pulse width of an inversion pulse. The following values were determined: B, = lop6 T and B1 = 2 X lop6 T. These were expressed in frequency units as Y, = 42 Hz and v1 = 84 Hz, respectively. The spin-lattice relaxation times TIA of pure water and TIB of the OH group of pure ethanol were determined by the inversion-recovery method (18). A least-squares fit procedure yielded the relaxation times T,, = 2.75 s and TIB = 2.00 s, respectively. The accuracy of the above cited values was estimated to be 5%. It is difficult to get an exact measurement of the spin-spin relaxation times T2. One method uses the application of the CPMG sequence (19), but this requires very careful adjustment of the RF pulses. Another method, suggested by Bain et al. (20), was chosen, where the partially saturated z magnetization was measured as a function of the offset frequency. On the basis of the Bloch equations, the known saturation field strength B1 and spin-lattice relaxation times T IA and TIB yield a TZA of 2.25 s for pure water and a T,, of 1.4 s for the OH group of ethanol. The chemical-shift difference Au = r$o - VA0is sometimes not easy to determine. Fortunately at water-ethanol solutions with a pmf of 1/ 1 and 2/ 1 the changes in Au are very small in the pH range between 8 and 5 and have been neglected. The difference in frequency has been determined to be Au = 150 Hz (4.7 T) in the slow-exchange limit at pH 8. We prepared solutions with either mole fraction and adjusted the pH to be between 5.0 and 8.0 in steps of 0.5 units. In order to keep the pH stable, 1% of a 1 mol stock buffer solution was added and titrated to the exact pH value. The buffer solution consisted of three 0.33 mol concentrations of MES, Hepes, and Tricine (Sigma). The low-power spectra of the water-ethanol solution with equal proton populations (pmf of l/ 1) are shown for pH 7.5 in Fig. la and for pH 6 in Fig. lb. As a pHindependent frequency standard, the line of the CH3 group of ethanol was always set to -690 Hz (4.7 T). At equal population and pH 8 (not shown) the water line and OH line possess the resonance frequencies VA0 - u. = 0 Hz and um - u. = 150 Hz, respectively. According to Fig. 1 in the pH range from 7 to 7.5 both lines merge, and this is referred
FAST EXCHANGE
313
RATES
a
800
800
400
200 Frequency
0
-200
-400-800
-800
(Hertz)
b
800
800
400
200 Frequency
0 -200 -400 -800 -800 (Hertz)
FIG. 1. Proton spectra of a water-ethanol solution with an equal population of water and OH protons or proton mole equivalent fraction (pmf) of I/ 1 (mole fraction, 0.5 / 1) . The lines at the chemical-shift frequencies of -690 and -200 Hz, respectively, represent the CH3 group and the CHr group of ethanol. (a) At pH 7.5 the already broadened lines of the OH groups of water and ethanol, respectively, are visible at 10 and 140 Hz. (At pH 8.0 these lines are visible at exactly 0 and 150 Hz). The calculated exchange rate constant is k = 170 liters/s. (b) At pH 6 the lines of the OH groups of water and ethanol merge and are visible at 75 Hz. The calculated rate constant is k = 12.0 X 10 3 liters/s.
to as the intermediate state. Case b shows the fast-exchange limit with one line at the mean frequency of 75 Hz, which is consistent with the above-mentioned data. Figure 2 depicts the theoretical and measured values of the partially saturated z magnetization as a function of the offset frequency. The y axis of each plot is scaled to the unperturbed equilibrium magnetization MO of both nuclei. The sample and the frequency calibration were the same as those for the spectra in Fig. 1. Since the populations are equal (pmf is 1/ 1)) the rate constants are equal according to Eq. [ 21 and therefore called k. Different saturation field strengths v, (expressed in frequency units) were applied. Figure 2a shows the curve at pH 6.5 with v1 = 42 Hz, and the fit yielded k = 10.0 X lo3 liters/s. The curves in Figs. 2b ( v1 = 42 Hz) and 2c (v, = 84 Hz) were obtained at pH 6 and a rate constant of k = 12.0 X lo3 liters/s has been calculated for both cases. In Figs. 2d and 2e the plots recorded
WTTEL
-0.2t.
I..
-400
-200 Offset
0 Frequency
AND BALABAN
I.
.I
200
-0.2L
400
1,.
-400
-200 Offset
(Hertz)
1.0
1.0
0.8:
0.8
0.6:
0.6
0.2
0.2
0.O. cl 2 -0.2 0.4;s -400
cH -0.204. 1.;...-400
I..
0 Frequency
200
I
400
(Hertz)
i
r"
‘.’ -200 Offset
0 Frequency
200
400
d)
(Hertz)
-0.2/------g& -400
-200 Offset
200
400
(Hertz)
200
-200 Offset
0 Frequency
Frequency
(Hertz)
FIG. 2. Partially saturated z magnetization (IV,/&) as function of the irradiation offset frequency in hertz, measured at a water-ethanol solution with a pmf of 1/ 1 (see Fig. 1). The y axis is scaled to the equilibrium magnetization MO of both nuclei. The parameters are the pH and the saturation field strength v1 (expressed in frequency units). The asterisks represent the measured values and the solid lines are the best fits. Measured at pH 6.5 (k = 10.0 X lo3 liters/s) with V, = 42 Hz; (a) at pH 6 (k = 12.0 X 10’ liters/ s) with V, = 42 Hz(b) and V, = 84 Hz(c); and at pH 5.5 (k = 14.0 X 10“ liters/s)(d) and pH 5 (k = 50.0 X lo4 liters/s) (e), both times with v, = 84 Hz.
with v1 = 84 Hz at pH 5.5 and pH 5, respectively, represent rate constants of k = 14.0 X lo4 and 50.0 X lo4 liters/s. The mean quadratic error of the fit was always better than 0.0 1.
FAST EXCHANGE
RATES
315
The spectra of the water-ethanol solution with an unequal populations ( pmf of 2 / 1) are visible in Fig. 3a at pH 7.5 and in Fig. 3b at pH 6. Analogous to Fig. 1, case a represents the intermediate state and case b the fast-exchange limit. In Fig. 4 the rate constants k and kA for pmf of 1 / 1 and 2 / 1, respectively are plotted versus pH. Cases a and b, respectively, differ in the population ratios and have pmf values of l/ 1 and 2/ 1, respectively. In the intermediate range (pH 7.5 to 7) this method, which is based on partial saturation, attains the accuracy of the lineshape analysis for a well-optimized saturation field strength (compare Fig. 5 ). In this range the lineshape analysis served as control. For both population sizes, the measured rate constants fitted fairly well to a straight line and a linear regression analysis yielded a slopeof24forapmfof l/l and 16forapmfof2/1. The accuracy of the rate constants kcAj is dependent on the right choice of the saturation field strength IQ. The optimized saturation field strength vlOPtas a function of the rate constants k has been calculated by minimizing the uncertainty of k for a
a
Frequency (Hertz)
600 600
400
200 0 -200 -400 -600 -600 Frequency (Hertz)
FIG. 3. Proton spectra of a water-ethanol solution with a proton mole equivalent fraction of 2/ 1 (mole fraction, 1/ 1) . The lines at the chemical-shift frequencies of -690 and -200 Hz, respectively, represent the CHx group and the CHr group of ethanol. The lines of the OH groups are related as explained in the legend to Fig. 1, (a) Measured at pH 7.5 with the calculated exchange rate constant kA = 170 liters/s. (b) Measured at pH 6 with the calculated rate constant kA = 12.0 X 10’ liters/s.
316
KNiiTTEL
AND BALABAN
10s 11’~““‘1’1”““‘11”“‘1”1~“~“~“1~“‘~~”’
a
4
5
7
6
8
9
8
9
PH 1oe
1”“““‘I”“““‘I”“““‘I”““”
b 106
-2 2 104 Ai’
A (_
A
A
102
100 4
\
6
7 PH
FIG. 4. Exchange rate constants k(,, plotted versus pH, measured at a water-ethanol solution. (a) At pmf 1/ 1 the slope indicates a rate multiplication factor of 24 per d’ecreased pH unit. (b) At pmf 2/ 1 the multiplication factor is 16.
given mean quadratic error. This is shown in Fig. 5 for a solution of equal population sizes. The plot of the unequal population sizes ( pmf of 2 / 1) looks similar and is therefore not shown. The error bars represent the uncertainty of k, if a mean quadratic
FAST EXCHANGE
RATES
317
k (l/s) FIG. 5. Optimized saturation field strength v,,,~ (expressed in frequency units) versus the rate constants k calculated for a two-site exchanging system with equal population sites and the parameter of the measured mixture. The error bars represent the uncertainty of k if a mean quadratic error of 0.01 is assumed. The highest accuracy of k exists around k = 3.2 X lo3 liters/s and k = 10.0 X IO3 liters/s with 11% uncertainty.
error of 0.0 1 is assumed. As already mentioned, this value represents the upper limit of the accuracy of the reported measurements. The lowest mean uncertainty of about 11% is associated with rate constants between 3.2 and 10.0 X 10 3 liters/s. For the extreme cases shown in this plot, the error exceeds 20%. DISCUSSION
We have shown that the total z magnetization of the two sites participating in a fast exchange reaction is a function of the offset frequency of an irradiation field. To determine the rate of exchange, a complete quantum mechanical density-matrix equation for a two-site exchange model was solved numerically. Experimental data on the hydroxyl proton exchange between water and ethanol fitted very well to this model. Optimum values for saturation field strength as a function of the reaction rate constants were also calculated. The pseudo-first-order rate constants for the water-ethanol exchange increased linearly with decreasing pH from 7.5 to 5. The multiplication factor for a unity pH step is 24 for a proton mole equivalent fraction of 1/ 1 and 16 for pmf of 2 / 1. At a chemicalshift difference of Au = 150 Hz between the OH groups of water and ethanol, the highest measured rate constant was k = 0.5 X lo6 liters/s (for a pmf of 1/ 1 at PH 5). The accuracy of the measurements depends on the spin-lattice and spin-spin relaxation times T, and T2 of both nuclei and the saturation field strength v1 (expressed
318
KNiiTTEL
AND BALABAN
in frequency units). Furthermore the difference in frequency Au must be known from the slow-exchange limit. Alternatively, one can step through the saturation field strength u, in order to get the difference frequency even at fast exchange rates. This is necessary if Au changes dramatically with pH, temperature., etc. A corresponding test for two different v1 values is shown in Figs. 2b and 2c, because only at this particular difference frequency is the rate constant the same for different vl. A problem with this alternative is that the measurements are time consuming, and the ability to achieve an optimum saturation field strength vloDtconfines the range in which v1 can be used with reasonably accurate results. For the slow-exchange limit the present method covers the For&n and Hoffman approach (2). Provided that the saturation field strength is adjusted properly, this method is also suitable for the intermediate range., when both lines begin to merge. When extended to faster rate constants, the accuracy reaches a maximum and finally drops until the optimum saturation field strength. vlopt becomes independent of the rate constant. At these very fast exchange rates, there is only one resonance line in the spectrum which has an effective spin-spin relaxation time, given by (18) 1 -=$+;(Av)~;, T 2eff
[lOI
where a two-site exchange with equal populations is assumed. In this case the average spin-spin relaxation time of both sites is 1/ T2 = 0.5 / T2* + OS/ T2B. If both spinlattice relaxation times T, are equal or at least very similar, the measured data can be fitted to the well-known Bloch equation (20) using the effective spin-spin relaxation time (Eq. [lo] ) . Generally the upper limit is expected to be in the order of k = $(Av)ZT;.
1111
For the actual difference frequency Au = 150 Hz and an average spin-spin relaxation time T2 = 1.8 s the limit would be k x 0.2 X lo6 liters/s, but with sufficient accuracy the measurement of rate constants up to 10 6 liters/s is feasible. The present method has the advantage of being very simple experimentally, because it is insensitive to lineshape distortions and it covers the whole exchange range with the same instrumental arrangement. ACKNOWLEDGMENT It is a pleasure to thank Dr. A. Bax for valuable discussions. A. Knuettel thanks the NIH council for his postdoctoral fellowship. REFERENCES 1. C. S. JOHNSON, in “Advances in Magnetic Resonance” (.J. S. Waugh, Ed.), Vol. 1, p. 33, Academic Press, San Diego, 1965. 2. R. A. HOFFMANN AND S. FOR&N, J. Chem. Phys. 39,28!)2 ( 1963). 3. S. D. WOLFF AND R. S. BALABAN, J. Magn. Reson. 86, 164 ( 1990). 4. S. D. WOLFF AND R. S. BALABAN, Magn. Reson. Med. 10, 135 ( 1989). 5. J. JEENER, B. H. MEIER, P. BACHMANN, AND R. R. ERNST, J. Chem. Phys. 71,4546 (1979).
FAST EXCHANGE 6. 7. 8. 9. 10. II.
12. 13. 14. IS.
16. 17.
18. 19. 20.
RATES
319
S. MACURA AND R. R. ERNST, Mol. Phys. 41,95 ( 1980). A. ALLERHAND, H. S. GUTOWSKY, J. JONAS, AND R. A. MEINZER, J. Am. Chem. Sot. 88,3 185 ( 1966). F. A. L. ANET AND A. J. R. BOURNE, J. Am. Chem. Sot. 89,760 (1966). P. T. INGLEFIELD, E. KRAKOWER, L. W. REEVES, AND R. STEWARD, Mol. Phys. 15,65 ( 1968). A. ALLERHAND, F. CHEN, AND H. S. GUTOWSKY, J. Chem. Phys. 42,304O ( 1965). C. DEVERELL, R. E. MORGAN, AND J. H. STRANGE, Mol. Phys. 18,553 ( 1970). P. STILBS AND M. E. MOSELEY, J. Magn. Reson. 31, 55 ( 1978). J. SANDSTROM, “Dynamic NMR Spectroscopy,” Academic Press, London/New York, 1982. “NMR of Chemically Exchanging Systems,” Academic Press, New J. KAPLAN AND G. FRAENKEL, York, 1980. S. ALEXANDER, J. Chem. Phys. 37,967 ( 1962). P. D. BUCKLEY, K. W. JOLLEY, AND D. N. PINDER, Prog. NMR Spectrosc. 10, 1 ( 1975 ). I. N. BRONSTEIN AND K. A. SEMENDJAJEW, “Taschenbuch der Mathematik,” Han-i Deutsch, Thun/ Frankfurt/Main, 1981. R. K. HARRIS, “Nuclear Magnetic Resonance Spectroscopy,” Longman Scientific &Technical, Harlow, England, 1983. R. FREEMAN AND H. D. W. HILL, “Dynamic Nuclear Magnetic Resonance Spectroscopy,” Academic Press, New York, 1975. A. D. BAIN, W. P. Y. Ho, AND J. S. MARTIN, J. Magn. Reson. 43,328 ( 1981).
OF MAGNETIC
RESONANCE
95, 309-3 19 ( 199 1)
A Novel Approach for the Determination of Fast ExchangeRates
Laboratory of Cardiac Energetics, National Institutes of Health, Bethesda, Maryland 20892 Received February 1, 199 1; revised May 28, I99 1 A novel technique is described to monitor fast exchange rates using NMR. This approach relies on the effects of partial saturation of the longitudinal magnetization with a saturation RF field at various o&et frequencies on the observed resonance lines. A complete quantum mechanical approach for the two-site chemical exchange of nuclei is presented, which yields access to rate constants from slow to very fast exchange rates using the described technique. The model system evaluated using this approach was the hydroxyl proton exchange in water-ethanol solutions. Exchange rate constants from the intermediate range (about 170 liters/s) at pH 7.5 to very fast rate constants up IO lo6 liters/s at pH 5 were measured. 0 1991 Academic Press. Inc.
Nuclear magnetic resonance spectroscopy is a well-established technique for the determination of chemical exchange rates over a wide range, with minimum rates in the order of the inverse spin-lattice relaxation time 1 / T, ( I ) . Slow-exchange reactions, which occur in a time comparable to T, , can be studied by double-resonance techniques originally suggested by For&r and Hoffmann (2). Other workers have extensively exploited this approach (3, 4). For more complex reactions, two-dimensional spectroscopic techniques have also been shown to be very useful (5, 6). For intermediate exchange, where the rate constant is in the order of the chemicalshift differences Au = VA - uB of both resonance lines VAand vs, lineshape analysis is a very sensitive method ( 7-9). However, this technique yields less accurate measurements of faster and slower reaction rates. For faster reactions &r-Purcell sequences have been utilized. At sufficient short interpulse delays ( = lop4 s) rate constants of the order of lo4 liters/s have been determined ( 10). However, the accuracy of adjusting the 180” pulses required for this approach drops for interpulse delays less than 10e4 s. An alternative method for fast exchange rates relies upon measurements of the spin-lattice relaxation time, T rp, in the presence of an RF field (II, 12). Herein we describe a new NMR method for the determination of intermediate to very fast chemical exchange rates where the individual resonance lines of the exchange partners are not resolvable. The method relies upon the determination of the frequency dependence of an off-resonance irradiation on the longitudinal magnetization of the participating resonances. To analyze these data a complete quantum mechanical treatment is presented. This approach allowed us to determine the dependence of hydroxyl proton exchange rates in water-ethanol solutions on mole fraction and pH. 309
0022-2364191 $3.00 Copyright 0 1991 by Academic Press, Inc. All tights of reproduction in any form reserved.
310
KNiiTTEL
AND
BALABAN
THEORY
The density-matrix formalism was used with the following assumptions. As the simplest case, the two-site exchange A
k.4 -B
ill
b
was evaluated, where A and B each contain one nucleus or a group of magnetically equivalent nuclei. Any effect due to scalar coupling is neglected. The exchange rate constants kA and kB can indicate either the real rate constants for intramolecular exchange or the pseudo-rate constants for intermolecular exchange ( 13). In the equilibrium condition the ratio of the rates is given by
k,lb = Wl/[Al,
t21
where [A] and [B] represent the populations at .the two sites. According to Kaplan and Fraenkel ( 14), the complete density-matrix the rotating frame has the form p = 0 = -ih-‘[&‘,
p] t- l? p + Ep,
where the first derivative of the density-matrix Hamiltonian operator is given by
equation in [31
operator p vanishes. The transformed
&” = h 2 {(u,, - va) fi + IQ@; + i,-)/2}.
[41
The subscript s indicates either the nucleus A or the nucleus B, so that vAo and em represent the corresponding Larmor frequencies. The rotating coordinate system revolves with the angular frequency v. of the saturation field, and the field strength is expressed in frequency units vl = yB,/( 27r). i’ and i- are the raising and lowering operators, respecti-vely. The operators R and l? in Eq. [ 3 ] are designated relaxation and exchange operators ( 14)) respectively. The phenomenologically introduced relaxation operator for spinlattice relaxation T, and spin-spin relaxation T;! is written Pm;-
gp = _ 2 s
(
PO-+ Poyiag IS
2s
.
[51
1
For a two-site exchanging system, the corresponding contribution to the density-matrix equation is given by E pA = kBpB - kApA E pB = kApA - kBpB,
161
where the sub- and superscripts A and B refer to the nuclei A and B. At equal populations, the rate constants kA and kB are equal according Eq. [ 21. Even for a twosite exchange, the intermolecular case is far more difficult to describe mathematically than the intramolecular case. However, it was shown by Alexander ( 15) and others
FAST
EXCHANGE
RATES
311
( 1, 16) that Eq. [ 61 describes both cases in the high-temperature limit, and this is valid for most applications. Almost the entire equilibrium z magnetization is maintained as long as the perturbation caused by the RF irradiation is small. In this case, Eq. [ 31 need be solved only for the off-diagonal elements which represent the transitions. For the lineshape analysis the slow- and fast-exchange limits have been derived analytically, and for the intermediate state numerical solutions are appropriate ( 14). In this work we apply a strong perturbation with the RF saturation field and therefore the complete quadratic density matrix of dimension eight according to Eq. [ 31 has been solved. The z magnetization M, of both nuclei is partially saturated immediately after the saturation pulse is switched off and is calculated as a function of the offset frequencies vAO- vo, vm - vo and the parameters TIA, TIB, TZA, TzB, v1. The diagonal matrix elements yield the total z magnetization Mz - P?, - P% + d1 - A!.
171
For the general case of the partially saturated z magnetization, the density-matrix equation is solved numerically, because analytical approaches are extremely cumbersome. The computer creates an inverse matrix with the help of the Gaussian elimination procedure ( 17). EXPERIMENTS
A two-site exchanging system is irradiated using the second channel at the offset frequencies VA0- v. and VBO- vo. The resonance lines VA0and YBoof the nuclei A and B are separately resolvable in the slow-exchange limit. After irradiating for a time of 5 T, , the system is at steady state. The second channel is gated off, and a spoil gradient is applied in order to destroy residual transverse steady-state magnetization. This provides excellent suppression in a single scan and yields more accurate measurements than pure phase cycling without a spoil gradient. About 5 ms after gating off the second channel, a 90” RF pulse enables the acquisition of both resonance lines. The integral of both lines in the Fourier-transformed spectrum represents the remaining z magnetization (of both nuclei), which is a function of the offset frequency and RF field strength of the saturation field. The measurements were carried out on a GE spectrometer at a magnetic field strength of 4.7 T. A 5 mm NMR tube containing the sample was placed in a saddle coil of 2 cm diameter tuned to the proton Larmor frequency of 200 MHz. The homebuilt gradient set of 5 cm diameter produced a one-dimensional spoil gradient of 0.15 mT / cm. The density-matrix calculations were done on a Sun computer 3 / 370 using IDL software (Research Systems, Inc.). In order to verify the theory, water-ethanol solutions have been chosen with two different proton molar equivalent fractions ( pmf). Since water has two exchangeable magnetically equivalent protons, pmf was used instead of mole fractions. Results for pmf values of 1/ 1 water/ethanol (mole fraction of 0.5 / 1) and 2 / 1 (mole fraction of 1 / 1) were evaluated. The intermolecular equilibrium reaction can be written
312
KNiiTTEL
R-OH;
+ HZ0
AND
BALABAN
+
R.-OH + H30+, 1
[81
where R-OH represents ethanol. The forward and backward rate constants are called k, and k-, , respectively. The pseudo-first-order NMR rate constants, kA and k,, for this system are expressed by
k/i Hz0 e
R-OH.
b
By comparison with Eq. [I] the protons of the water molecule are related to the nuclei A and the proton of the OH group is related to 13. Some parameters must be determined in advance in order to measure the pseudofirst-order rate constants. The RF field strength B, was calibrated by adjusting the pulse width of an inversion pulse. The following values were determined: B, = lop6 T and B1 = 2 X lop6 T. These were expressed in frequency units as Y, = 42 Hz and v1 = 84 Hz, respectively. The spin-lattice relaxation times TIA of pure water and TIB of the OH group of pure ethanol were determined by the inversion-recovery method (18). A least-squares fit procedure yielded the relaxation times T,, = 2.75 s and TIB = 2.00 s, respectively. The accuracy of the above cited values was estimated to be 5%. It is difficult to get an exact measurement of the spin-spin relaxation times T2. One method uses the application of the CPMG sequence (19), but this requires very careful adjustment of the RF pulses. Another method, suggested by Bain et al. (20), was chosen, where the partially saturated z magnetization was measured as a function of the offset frequency. On the basis of the Bloch equations, the known saturation field strength B1 and spin-lattice relaxation times T IA and TIB yield a TZA of 2.25 s for pure water and a T,, of 1.4 s for the OH group of ethanol. The chemical-shift difference Au = r$o - VA0is sometimes not easy to determine. Fortunately at water-ethanol solutions with a pmf of 1/ 1 and 2/ 1 the changes in Au are very small in the pH range between 8 and 5 and have been neglected. The difference in frequency has been determined to be Au = 150 Hz (4.7 T) in the slow-exchange limit at pH 8. We prepared solutions with either mole fraction and adjusted the pH to be between 5.0 and 8.0 in steps of 0.5 units. In order to keep the pH stable, 1% of a 1 mol stock buffer solution was added and titrated to the exact pH value. The buffer solution consisted of three 0.33 mol concentrations of MES, Hepes, and Tricine (Sigma). The low-power spectra of the water-ethanol solution with equal proton populations (pmf of l/ 1) are shown for pH 7.5 in Fig. la and for pH 6 in Fig. lb. As a pHindependent frequency standard, the line of the CH3 group of ethanol was always set to -690 Hz (4.7 T). At equal population and pH 8 (not shown) the water line and OH line possess the resonance frequencies VA0 - u. = 0 Hz and um - u. = 150 Hz, respectively. According to Fig. 1 in the pH range from 7 to 7.5 both lines merge, and this is referred
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313
RATES
a
800
800
400
200 Frequency
0
-200
-400-800
-800
(Hertz)
b
800
800
400
200 Frequency
0 -200 -400 -800 -800 (Hertz)
FIG. 1. Proton spectra of a water-ethanol solution with an equal population of water and OH protons or proton mole equivalent fraction (pmf) of I/ 1 (mole fraction, 0.5 / 1) . The lines at the chemical-shift frequencies of -690 and -200 Hz, respectively, represent the CH3 group and the CHr group of ethanol. (a) At pH 7.5 the already broadened lines of the OH groups of water and ethanol, respectively, are visible at 10 and 140 Hz. (At pH 8.0 these lines are visible at exactly 0 and 150 Hz). The calculated exchange rate constant is k = 170 liters/s. (b) At pH 6 the lines of the OH groups of water and ethanol merge and are visible at 75 Hz. The calculated rate constant is k = 12.0 X 10 3 liters/s.
to as the intermediate state. Case b shows the fast-exchange limit with one line at the mean frequency of 75 Hz, which is consistent with the above-mentioned data. Figure 2 depicts the theoretical and measured values of the partially saturated z magnetization as a function of the offset frequency. The y axis of each plot is scaled to the unperturbed equilibrium magnetization MO of both nuclei. The sample and the frequency calibration were the same as those for the spectra in Fig. 1. Since the populations are equal (pmf is 1/ 1)) the rate constants are equal according to Eq. [ 21 and therefore called k. Different saturation field strengths v, (expressed in frequency units) were applied. Figure 2a shows the curve at pH 6.5 with v1 = 42 Hz, and the fit yielded k = 10.0 X lo3 liters/s. The curves in Figs. 2b ( v1 = 42 Hz) and 2c (v, = 84 Hz) were obtained at pH 6 and a rate constant of k = 12.0 X lo3 liters/s has been calculated for both cases. In Figs. 2d and 2e the plots recorded
WTTEL
-0.2t.
I..
-400
-200 Offset
0 Frequency
AND BALABAN
I.
.I
200
-0.2L
400
1,.
-400
-200 Offset
(Hertz)
1.0
1.0
0.8:
0.8
0.6:
0.6
0.2
0.2
0.O. cl 2 -0.2 0.4;s -400
cH -0.204. 1.;...-400
I..
0 Frequency
200
I
400
(Hertz)
i
r"
‘.’ -200 Offset
0 Frequency
200
400
d)
(Hertz)
-0.2/------g& -400
-200 Offset
200
400
(Hertz)
200
-200 Offset
0 Frequency
Frequency
(Hertz)
FIG. 2. Partially saturated z magnetization (IV,/&) as function of the irradiation offset frequency in hertz, measured at a water-ethanol solution with a pmf of 1/ 1 (see Fig. 1). The y axis is scaled to the equilibrium magnetization MO of both nuclei. The parameters are the pH and the saturation field strength v1 (expressed in frequency units). The asterisks represent the measured values and the solid lines are the best fits. Measured at pH 6.5 (k = 10.0 X lo3 liters/s) with V, = 42 Hz; (a) at pH 6 (k = 12.0 X 10’ liters/ s) with V, = 42 Hz(b) and V, = 84 Hz(c); and at pH 5.5 (k = 14.0 X 10“ liters/s)(d) and pH 5 (k = 50.0 X lo4 liters/s) (e), both times with v, = 84 Hz.
with v1 = 84 Hz at pH 5.5 and pH 5, respectively, represent rate constants of k = 14.0 X lo4 and 50.0 X lo4 liters/s. The mean quadratic error of the fit was always better than 0.0 1.
FAST EXCHANGE
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315
The spectra of the water-ethanol solution with an unequal populations ( pmf of 2 / 1) are visible in Fig. 3a at pH 7.5 and in Fig. 3b at pH 6. Analogous to Fig. 1, case a represents the intermediate state and case b the fast-exchange limit. In Fig. 4 the rate constants k and kA for pmf of 1 / 1 and 2 / 1, respectively are plotted versus pH. Cases a and b, respectively, differ in the population ratios and have pmf values of l/ 1 and 2/ 1, respectively. In the intermediate range (pH 7.5 to 7) this method, which is based on partial saturation, attains the accuracy of the lineshape analysis for a well-optimized saturation field strength (compare Fig. 5 ). In this range the lineshape analysis served as control. For both population sizes, the measured rate constants fitted fairly well to a straight line and a linear regression analysis yielded a slopeof24forapmfof l/l and 16forapmfof2/1. The accuracy of the rate constants kcAj is dependent on the right choice of the saturation field strength IQ. The optimized saturation field strength vlOPtas a function of the rate constants k has been calculated by minimizing the uncertainty of k for a
a
Frequency (Hertz)
600 600
400
200 0 -200 -400 -600 -600 Frequency (Hertz)
FIG. 3. Proton spectra of a water-ethanol solution with a proton mole equivalent fraction of 2/ 1 (mole fraction, 1/ 1) . The lines at the chemical-shift frequencies of -690 and -200 Hz, respectively, represent the CHx group and the CHr group of ethanol. The lines of the OH groups are related as explained in the legend to Fig. 1, (a) Measured at pH 7.5 with the calculated exchange rate constant kA = 170 liters/s. (b) Measured at pH 6 with the calculated rate constant kA = 12.0 X 10’ liters/s.
316
KNiiTTEL
AND BALABAN
10s 11’~““‘1’1”““‘11”“‘1”1~“~“~“1~“‘~~”’
a
4
5
7
6
8
9
8
9
PH 1oe
1”“““‘I”“““‘I”“““‘I”““”
b 106
-2 2 104 Ai’
A (_
A
A
102
100 4
\
6
7 PH
FIG. 4. Exchange rate constants k(,, plotted versus pH, measured at a water-ethanol solution. (a) At pmf 1/ 1 the slope indicates a rate multiplication factor of 24 per d’ecreased pH unit. (b) At pmf 2/ 1 the multiplication factor is 16.
given mean quadratic error. This is shown in Fig. 5 for a solution of equal population sizes. The plot of the unequal population sizes ( pmf of 2 / 1) looks similar and is therefore not shown. The error bars represent the uncertainty of k, if a mean quadratic
FAST EXCHANGE
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317
k (l/s) FIG. 5. Optimized saturation field strength v,,,~ (expressed in frequency units) versus the rate constants k calculated for a two-site exchanging system with equal population sites and the parameter of the measured mixture. The error bars represent the uncertainty of k if a mean quadratic error of 0.01 is assumed. The highest accuracy of k exists around k = 3.2 X lo3 liters/s and k = 10.0 X IO3 liters/s with 11% uncertainty.
error of 0.0 1 is assumed. As already mentioned, this value represents the upper limit of the accuracy of the reported measurements. The lowest mean uncertainty of about 11% is associated with rate constants between 3.2 and 10.0 X 10 3 liters/s. For the extreme cases shown in this plot, the error exceeds 20%. DISCUSSION
We have shown that the total z magnetization of the two sites participating in a fast exchange reaction is a function of the offset frequency of an irradiation field. To determine the rate of exchange, a complete quantum mechanical density-matrix equation for a two-site exchange model was solved numerically. Experimental data on the hydroxyl proton exchange between water and ethanol fitted very well to this model. Optimum values for saturation field strength as a function of the reaction rate constants were also calculated. The pseudo-first-order rate constants for the water-ethanol exchange increased linearly with decreasing pH from 7.5 to 5. The multiplication factor for a unity pH step is 24 for a proton mole equivalent fraction of 1/ 1 and 16 for pmf of 2 / 1. At a chemicalshift difference of Au = 150 Hz between the OH groups of water and ethanol, the highest measured rate constant was k = 0.5 X lo6 liters/s (for a pmf of 1/ 1 at PH 5). The accuracy of the measurements depends on the spin-lattice and spin-spin relaxation times T, and T2 of both nuclei and the saturation field strength v1 (expressed
318
KNiiTTEL
AND BALABAN
in frequency units). Furthermore the difference in frequency Au must be known from the slow-exchange limit. Alternatively, one can step through the saturation field strength u, in order to get the difference frequency even at fast exchange rates. This is necessary if Au changes dramatically with pH, temperature., etc. A corresponding test for two different v1 values is shown in Figs. 2b and 2c, because only at this particular difference frequency is the rate constant the same for different vl. A problem with this alternative is that the measurements are time consuming, and the ability to achieve an optimum saturation field strength vloDtconfines the range in which v1 can be used with reasonably accurate results. For the slow-exchange limit the present method covers the For&n and Hoffman approach (2). Provided that the saturation field strength is adjusted properly, this method is also suitable for the intermediate range., when both lines begin to merge. When extended to faster rate constants, the accuracy reaches a maximum and finally drops until the optimum saturation field strength. vlopt becomes independent of the rate constant. At these very fast exchange rates, there is only one resonance line in the spectrum which has an effective spin-spin relaxation time, given by (18) 1 -=$+;(Av)~;, T 2eff
[lOI
where a two-site exchange with equal populations is assumed. In this case the average spin-spin relaxation time of both sites is 1/ T2 = 0.5 / T2* + OS/ T2B. If both spinlattice relaxation times T, are equal or at least very similar, the measured data can be fitted to the well-known Bloch equation (20) using the effective spin-spin relaxation time (Eq. [lo] ) . Generally the upper limit is expected to be in the order of k = $(Av)ZT;.
1111
For the actual difference frequency Au = 150 Hz and an average spin-spin relaxation time T2 = 1.8 s the limit would be k x 0.2 X lo6 liters/s, but with sufficient accuracy the measurement of rate constants up to 10 6 liters/s is feasible. The present method has the advantage of being very simple experimentally, because it is insensitive to lineshape distortions and it covers the whole exchange range with the same instrumental arrangement. ACKNOWLEDGMENT It is a pleasure to thank Dr. A. Bax for valuable discussions. A. Knuettel thanks the NIH council for his postdoctoral fellowship. REFERENCES 1. C. S. JOHNSON, in “Advances in Magnetic Resonance” (.J. S. Waugh, Ed.), Vol. 1, p. 33, Academic Press, San Diego, 1965. 2. R. A. HOFFMANN AND S. FOR&N, J. Chem. Phys. 39,28!)2 ( 1963). 3. S. D. WOLFF AND R. S. BALABAN, J. Magn. Reson. 86, 164 ( 1990). 4. S. D. WOLFF AND R. S. BALABAN, Magn. Reson. Med. 10, 135 ( 1989). 5. J. JEENER, B. H. MEIER, P. BACHMANN, AND R. R. ERNST, J. Chem. Phys. 71,4546 (1979).
FAST EXCHANGE 6. 7. 8. 9. 10. II.
12. 13. 14. IS.
16. 17.
18. 19. 20.
RATES
319
S. MACURA AND R. R. ERNST, Mol. Phys. 41,95 ( 1980). A. ALLERHAND, H. S. GUTOWSKY, J. JONAS, AND R. A. MEINZER, J. Am. Chem. Sot. 88,3 185 ( 1966). F. A. L. ANET AND A. J. R. BOURNE, J. Am. Chem. Sot. 89,760 (1966). P. T. INGLEFIELD, E. KRAKOWER, L. W. REEVES, AND R. STEWARD, Mol. Phys. 15,65 ( 1968). A. ALLERHAND, F. CHEN, AND H. S. GUTOWSKY, J. Chem. Phys. 42,304O ( 1965). C. DEVERELL, R. E. MORGAN, AND J. H. STRANGE, Mol. Phys. 18,553 ( 1970). P. STILBS AND M. E. MOSELEY, J. Magn. Reson. 31, 55 ( 1978). J. SANDSTROM, “Dynamic NMR Spectroscopy,” Academic Press, London/New York, 1982. “NMR of Chemically Exchanging Systems,” Academic Press, New J. KAPLAN AND G. FRAENKEL, York, 1980. S. ALEXANDER, J. Chem. Phys. 37,967 ( 1962). P. D. BUCKLEY, K. W. JOLLEY, AND D. N. PINDER, Prog. NMR Spectrosc. 10, 1 ( 1975 ). I. N. BRONSTEIN AND K. A. SEMENDJAJEW, “Taschenbuch der Mathematik,” Han-i Deutsch, Thun/ Frankfurt/Main, 1981. R. K. HARRIS, “Nuclear Magnetic Resonance Spectroscopy,” Longman Scientific &Technical, Harlow, England, 1983. R. FREEMAN AND H. D. W. HILL, “Dynamic Nuclear Magnetic Resonance Spectroscopy,” Academic Press, New York, 1975. A. D. BAIN, W. P. Y. Ho, AND J. S. MARTIN, J. Magn. Reson. 43,328 ( 1981).