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Formation of “kinetic” spectra from superimposed NMR resonances of different relaxation times
JOURNAL
OF MAGNETIC
RESONANCE
62, 556-560 (1985)
Formation of ‘Xinetic” Spectra from SuperimposedNMR Resonancesof Different Relaxation Times BRIAN
T. BULLIMAN
AND PHILIP
W. KUCHEL
Department of Biochemistry, University of Sydney, Sydney, New South Wales 2006, Australia Received June 12, 1984; revised January 22, 1985
We describe the application of a numerical procedure to NMR relaxation data that results in a spectrum which is not related to frequency but to relaxation rates. The background to our use of the method and its rationale are as follows. NMR-based analyses of systems in which there is exchange between distinct chemical species at different molecular sites usually involves the species having separate chemical shifts. Conventional strategies for calculating the exchange rates are determined largely by whether the exchange is “fast” or “slow” in the time scale of the high-energy nuclear magnetic state. Analysis of rapidly exchanging systems (millisecond time domain) involves spectral simulations and regression analysis (e.g., (1)) while slow exchange (1 s time domain) can be elegantly studied using saturation- and inversion-transfer spectroscopy (2, 3) or an appropriate 2-D experiment, e.g., 2-D NOESY (4). However, the rate of exchange of water and some low molecular weight solutes across the membranes of living cells can be measured in the absence of chemicalshift differences between the inside and outside of the cells; this is possible by virtue of different relaxation times of the solute in each compartment. Outside the cells relaxation is made fast by adding a paramagnetic ion such as Mnzf. The time course of relaxation of magnetization in inversion-recovery (T,) or spin-echo (T2) experiments is recorded (5-7). Data analysis involves estimating the pre-exponential coefficients and the exponents in double-, triple-, or multiple-exponential expressions which are consistent with the data; i.e., the total magnetization (M,,(t)) in the direction of the x axis in the rotating frame is described, for any time t, by M,,(t)
= i h4x~o)e-k”,
i=l
111
where M,(O) and ki are the zero-time magnetization and the decay-rate constant, respectively, of the ith compartment. The difficulties of obtaining a unique fit of Eq. [I] to real data, especially if i > 2, are well known; this is so even with modern nonlinear regression procedures (8) and the procedures require initial estimates, of the sought-after parameters, which are numerically “close” to the final fitted values. We have found the “kinetic spectrum analysis” method, recently introduced by Olson and Shuman (9) in an entirely different experimental context, valuable for obtaining initial estimates of the parameters in Eq. [l]. The analysis, appropriately 0022-2364185 $3.00 Copyright 0 1985 by Academic Press. Inc. All rights of reprcduction in any form reserved.
556
COMMUNICATIONS
557
implemented on a graphics computer (Tektronix 4052), more vividly indicates compartment sizes than the “exponential peeling” procedure that is in common ux (e.g., (5)). The basis of the analysis is that Eq. [l] can be written as a definite integral with th.e rate constant(s) considered to be the integration variable: M&t)
= [
F(k, t)e@dk,
PI
where F(k, t) is a function of the initial magnetizations and the rate constants. In other words it is a distribution function of the k:s, and it is the mathematical firm of&k, t) that is sought. Note that the expression for M,,(t) in Eq. [2] is simply the Laplace transform of F(k, t) so Laplace inversion of the equation yields the latter function; this can be done, albeit, approximately, numerically (IO) and the simplest approach is to use the real inversion formula of Post and Widder (11). The formula is based on approximating Eq. [2] by an “asymptotic” integral in which t is replaced by m/k, the upper limit of integration is m/k, and m - cc (II). We proceed, in th.e numerical analysis, by making the “bold” approximation that m = 2; this has, tat some extent, been vindicated with certain exponential functions (see Table 1 in Ref. (10)). The procedure then requires the first- and second-order numerical differentiation (using the central difference formula) of the data which consists of magnetization vs In t, and a plot of
a%, aw, (a(ln t)2 a(ln t)1
[31
vs In t yields the graph of the distribution function with respect to In t (9). Since the basic analysis was developed with t replaced by m/k, and m was set to 2, then In k = In 2 - In t; so, the graph is also the distribution function of the kts. Since the red cell water-exchange data of Fabry and Eisenstadt (5) is already well known we have chosen to demonstrate the analytical procedure with it, rather than ta introduce our own new data which require interpretations that may distract the reader from the main point of this Communication. Figure 1 is a reproduction of that of Fabry and Eisenstadt (Fig. 1, Ref. (5)) and shows the Ti-dependent decay of the ‘H NMR signal from water in a suspension of red cells doped with 2 mA4 M[nC12. The authors analyzed these data, according to a two-site exchange model, using “exponential peeling.” We photographically enlarged the diagram (4X) and digitized the data (crosses), taking means of replicated points, and regressed a double-exponential expression onto them (8). This yielded parameter estimates (II~~ = 11,100 f 260, Afx = 6 12 rt 242, kl = 0.083 + 0.004 ms-‘, k2 = 0.010 f 0.005 ms-‘) that can be compared with those obtained from the “kinetic spectrum analysis”; Fig. 2 shows the data set. The points in Fig. 3 are those of Fig. 2 but plotted vs In t and the line is the graph of the double-exponential expression obtained using the fitted values. Finally, Fig. 4 is the plot of Eq. [3] vs In k for (i) 200 points on the double-exponentially fitted curve (solid line) and (ii) the 15 data points originally used from which the derivatives for Eq. [3] were obtained numerically using linear interpolations; this is the simplest and most extreme of the interpolation procedures and was used here to present the “worst possible case.”
558
COMMUNICATIONS
102
50
100
150 200 Decay Time
250 r-(ms)
300
350
FIG. 1. Decay in an inversion-recovery experiment of ‘Hz0 NMR signal, plotted as log,, of the signal vs time, from two samples of red cells: (0) hematocrit 81.5%, 10 mkf MnC12; (X) hematocrit 18.52, 2 m&f MnQ. Magnetization is in arbitrary units. Reproduced from Fig. 1 of Fabry and Eisenstadt (5) by copyright permission of the Biophysical Society.
According to “the theory” the In k positions of the maximum ordinates yield the means of the rate constants that characterize the exchange between the compartments. While the two peaks from the smooth double-exponential data are not fully resolved good estimates of kl (0.06 ms-‘) and k2 (0.01 ms-‘) are easily obtained. Furthermore the ratio of areas of the two peaks is in reasonable accordance with the direct estimates obtained from the nonlinear least-squares analysis. While the points in 8. m
>, .
: Iz P
4.
2 n9 y2 z
. . .
l .*
0 0
P .
.
100
200 TIME
FIG. 2. The data of Fig. I
.
(X)
. 300
(ms)
digitized and reconverted to ‘H NMR signal vs time.
559
COMMUNICATIONS
0
1
0
In t
2
3
FIG. 3. Plot of the data of Fig. 2 vs In t. The solid line is from the two-exponential expression fitted to the original data and t is in ms.
Fig. 4 obtained from the original unsmoothed data are scattered about the smoothed fimction we note that it is still possible to discern two peaks with maxima at about kl = 0.08 ms-’ and k2 = 0.02 ms-‘. The derivatives of the two extreme points of the curve were not evaluated hence only 13 of the origimtl 15 data points are shown; also the resolution of the analysis, with only 13 points, is not sufficient to allow the claim that the point at 2.303 X In k = -1.1 in Fig. 4 lies in a trough between two resolved peaks. Note that this analysis is the “worst case” and that an implementation of the procedure, that does not involve nonlinear regression, takes the raw data and smooths it by use of a cubic spline from which a large number of
-1
-2
0
Inkx2.303 FIG. 4. The distribution function of kts (Eq. [3]) plotted vs In k. The points (0) were obtained by numerical differentiation and linear interpolation in the data of Fig. 3 while the line was obtained from 2(K) points given by the double-exponential fit to the original data of Fig. 2; the units of k are ms-‘.
560
COMMUNICATIONS
points can be taken. This smoothed function is then numerically differentiated to yield the figure corresponding to Fig. 4. In conclusion, “kinetic spectrum analysis” requires a plot of relaxation data vs In t without any preconceived notion of the number of exponentials involved. The difference between the second and first derivatives with respect to In t of the smoothed curve (cubic spline) through the data yields the distribution function of the k’s. The number of peaks gives the number of compartments and their relative areas gives the relative spin population in each compartment. It is conceptually feasible that 2-D kinetic spectra could be constructed in which the first dimension is the frequency spectrum while the second dimension is that of In k. Therefore spectra that are composed of signal contributions from chemical species in different compartments, that have different relaxation times in a heterogenous system, may be able to be “dissected apart.” This optimistic proposal awaits further study and may in fact rely, for its fulfillment, on a better means of numerical Laplace inversion than has hitherto been used. ACKNOWLEDGMENTS The work
was funded
by the Australian
NH
& MRC.
REFERENCES
1. G. BINSCH, 2. 3. 4. 5’. 6. 7.
T. G. P. M. P. P.
8. M. 9. D. 10. R. Il.
D.
in ‘Dynamic Nuclear Magnetic Resonance” (L. M. Jackman and F. A. Cotton, Eds.), Chap. 3, Academic Press, New York, 1975. R. BROWN ANI) S. OGAWA, Proc. Natl. Acad. Sci. USA 14, 3627 (1977). ROBINSON, B. E. CHAPMAN, AND P. W. KUCHEL, Eur. J. B&hem. 143,643 (1984). CARLICK AND C. J. TURNER, J. Magn. Reson. 51, 536 (1983). E. FABRY AND M. EISENSTADT, Biophys. J. 15, 1101 (1975). W. KUCHEL AND B. E. CHAPMAN, J. Theor. Biol. 105, 569 (1983). W. KUCHEL, G. R. BEILHARZ, B. E. CHAPMAN, AND C. R. MIDDLEHURST, in “Secretion: Mechanisms and Control” (R. M. Case, J. M. Lingard, and J. A. Young, Ed%), pp. l-10, Manchester Univ. Press, Manchester, 1984. OSBORNE, J. Aust. Math. Sot. B 19, 343 (1976). L. OLSON AND M. S. SHUMAN, Anal. Chem. 55, 1103 (1983). BELLMAN, R. E. KALABA, AND J. LOCKETT, “Numerical Inversion of the Laplace Transform,” Elsevier, New York, 1966. V. WIDDER, “The Laplace Transform,” Princeton Univ. Press, Princeton, N.J., 1941.
OF MAGNETIC
RESONANCE
62, 556-560 (1985)
Formation of ‘Xinetic” Spectra from SuperimposedNMR Resonancesof Different Relaxation Times BRIAN
T. BULLIMAN
AND PHILIP
W. KUCHEL
Department of Biochemistry, University of Sydney, Sydney, New South Wales 2006, Australia Received June 12, 1984; revised January 22, 1985
We describe the application of a numerical procedure to NMR relaxation data that results in a spectrum which is not related to frequency but to relaxation rates. The background to our use of the method and its rationale are as follows. NMR-based analyses of systems in which there is exchange between distinct chemical species at different molecular sites usually involves the species having separate chemical shifts. Conventional strategies for calculating the exchange rates are determined largely by whether the exchange is “fast” or “slow” in the time scale of the high-energy nuclear magnetic state. Analysis of rapidly exchanging systems (millisecond time domain) involves spectral simulations and regression analysis (e.g., (1)) while slow exchange (1 s time domain) can be elegantly studied using saturation- and inversion-transfer spectroscopy (2, 3) or an appropriate 2-D experiment, e.g., 2-D NOESY (4). However, the rate of exchange of water and some low molecular weight solutes across the membranes of living cells can be measured in the absence of chemicalshift differences between the inside and outside of the cells; this is possible by virtue of different relaxation times of the solute in each compartment. Outside the cells relaxation is made fast by adding a paramagnetic ion such as Mnzf. The time course of relaxation of magnetization in inversion-recovery (T,) or spin-echo (T2) experiments is recorded (5-7). Data analysis involves estimating the pre-exponential coefficients and the exponents in double-, triple-, or multiple-exponential expressions which are consistent with the data; i.e., the total magnetization (M,,(t)) in the direction of the x axis in the rotating frame is described, for any time t, by M,,(t)
= i h4x~o)e-k”,
i=l
111
where M,(O) and ki are the zero-time magnetization and the decay-rate constant, respectively, of the ith compartment. The difficulties of obtaining a unique fit of Eq. [I] to real data, especially if i > 2, are well known; this is so even with modern nonlinear regression procedures (8) and the procedures require initial estimates, of the sought-after parameters, which are numerically “close” to the final fitted values. We have found the “kinetic spectrum analysis” method, recently introduced by Olson and Shuman (9) in an entirely different experimental context, valuable for obtaining initial estimates of the parameters in Eq. [l]. The analysis, appropriately 0022-2364185 $3.00 Copyright 0 1985 by Academic Press. Inc. All rights of reprcduction in any form reserved.
556
COMMUNICATIONS
557
implemented on a graphics computer (Tektronix 4052), more vividly indicates compartment sizes than the “exponential peeling” procedure that is in common ux (e.g., (5)). The basis of the analysis is that Eq. [l] can be written as a definite integral with th.e rate constant(s) considered to be the integration variable: M&t)
= [
F(k, t)e@dk,
PI
where F(k, t) is a function of the initial magnetizations and the rate constants. In other words it is a distribution function of the k:s, and it is the mathematical firm of&k, t) that is sought. Note that the expression for M,,(t) in Eq. [2] is simply the Laplace transform of F(k, t) so Laplace inversion of the equation yields the latter function; this can be done, albeit, approximately, numerically (IO) and the simplest approach is to use the real inversion formula of Post and Widder (11). The formula is based on approximating Eq. [2] by an “asymptotic” integral in which t is replaced by m/k, the upper limit of integration is m/k, and m - cc (II). We proceed, in th.e numerical analysis, by making the “bold” approximation that m = 2; this has, tat some extent, been vindicated with certain exponential functions (see Table 1 in Ref. (10)). The procedure then requires the first- and second-order numerical differentiation (using the central difference formula) of the data which consists of magnetization vs In t, and a plot of
a%, aw, (a(ln t)2 a(ln t)1
[31
vs In t yields the graph of the distribution function with respect to In t (9). Since the basic analysis was developed with t replaced by m/k, and m was set to 2, then In k = In 2 - In t; so, the graph is also the distribution function of the kts. Since the red cell water-exchange data of Fabry and Eisenstadt (5) is already well known we have chosen to demonstrate the analytical procedure with it, rather than ta introduce our own new data which require interpretations that may distract the reader from the main point of this Communication. Figure 1 is a reproduction of that of Fabry and Eisenstadt (Fig. 1, Ref. (5)) and shows the Ti-dependent decay of the ‘H NMR signal from water in a suspension of red cells doped with 2 mA4 M[nC12. The authors analyzed these data, according to a two-site exchange model, using “exponential peeling.” We photographically enlarged the diagram (4X) and digitized the data (crosses), taking means of replicated points, and regressed a double-exponential expression onto them (8). This yielded parameter estimates (II~~ = 11,100 f 260, Afx = 6 12 rt 242, kl = 0.083 + 0.004 ms-‘, k2 = 0.010 f 0.005 ms-‘) that can be compared with those obtained from the “kinetic spectrum analysis”; Fig. 2 shows the data set. The points in Fig. 3 are those of Fig. 2 but plotted vs In t and the line is the graph of the double-exponential expression obtained using the fitted values. Finally, Fig. 4 is the plot of Eq. [3] vs In k for (i) 200 points on the double-exponentially fitted curve (solid line) and (ii) the 15 data points originally used from which the derivatives for Eq. [3] were obtained numerically using linear interpolations; this is the simplest and most extreme of the interpolation procedures and was used here to present the “worst possible case.”
558
COMMUNICATIONS
102
50
100
150 200 Decay Time
250 r-(ms)
300
350
FIG. 1. Decay in an inversion-recovery experiment of ‘Hz0 NMR signal, plotted as log,, of the signal vs time, from two samples of red cells: (0) hematocrit 81.5%, 10 mkf MnC12; (X) hematocrit 18.52, 2 m&f MnQ. Magnetization is in arbitrary units. Reproduced from Fig. 1 of Fabry and Eisenstadt (5) by copyright permission of the Biophysical Society.
According to “the theory” the In k positions of the maximum ordinates yield the means of the rate constants that characterize the exchange between the compartments. While the two peaks from the smooth double-exponential data are not fully resolved good estimates of kl (0.06 ms-‘) and k2 (0.01 ms-‘) are easily obtained. Furthermore the ratio of areas of the two peaks is in reasonable accordance with the direct estimates obtained from the nonlinear least-squares analysis. While the points in 8. m
>, .
: Iz P
4.
2 n9 y2 z
. . .
l .*
0 0
P .
.
100
200 TIME
FIG. 2. The data of Fig. I
.
(X)
. 300
(ms)
digitized and reconverted to ‘H NMR signal vs time.
559
COMMUNICATIONS
0
1
0
In t
2
3
FIG. 3. Plot of the data of Fig. 2 vs In t. The solid line is from the two-exponential expression fitted to the original data and t is in ms.
Fig. 4 obtained from the original unsmoothed data are scattered about the smoothed fimction we note that it is still possible to discern two peaks with maxima at about kl = 0.08 ms-’ and k2 = 0.02 ms-‘. The derivatives of the two extreme points of the curve were not evaluated hence only 13 of the origimtl 15 data points are shown; also the resolution of the analysis, with only 13 points, is not sufficient to allow the claim that the point at 2.303 X In k = -1.1 in Fig. 4 lies in a trough between two resolved peaks. Note that this analysis is the “worst case” and that an implementation of the procedure, that does not involve nonlinear regression, takes the raw data and smooths it by use of a cubic spline from which a large number of
-1
-2
0
Inkx2.303 FIG. 4. The distribution function of kts (Eq. [3]) plotted vs In k. The points (0) were obtained by numerical differentiation and linear interpolation in the data of Fig. 3 while the line was obtained from 2(K) points given by the double-exponential fit to the original data of Fig. 2; the units of k are ms-‘.
560
COMMUNICATIONS
points can be taken. This smoothed function is then numerically differentiated to yield the figure corresponding to Fig. 4. In conclusion, “kinetic spectrum analysis” requires a plot of relaxation data vs In t without any preconceived notion of the number of exponentials involved. The difference between the second and first derivatives with respect to In t of the smoothed curve (cubic spline) through the data yields the distribution function of the k’s. The number of peaks gives the number of compartments and their relative areas gives the relative spin population in each compartment. It is conceptually feasible that 2-D kinetic spectra could be constructed in which the first dimension is the frequency spectrum while the second dimension is that of In k. Therefore spectra that are composed of signal contributions from chemical species in different compartments, that have different relaxation times in a heterogenous system, may be able to be “dissected apart.” This optimistic proposal awaits further study and may in fact rely, for its fulfillment, on a better means of numerical Laplace inversion than has hitherto been used. ACKNOWLEDGMENTS The work
was funded
by the Australian
NH
& MRC.
REFERENCES
1. G. BINSCH, 2. 3. 4. 5’. 6. 7.
T. G. P. M. P. P.
8. M. 9. D. 10. R. Il.
D.
in ‘Dynamic Nuclear Magnetic Resonance” (L. M. Jackman and F. A. Cotton, Eds.), Chap. 3, Academic Press, New York, 1975. R. BROWN ANI) S. OGAWA, Proc. Natl. Acad. Sci. USA 14, 3627 (1977). ROBINSON, B. E. CHAPMAN, AND P. W. KUCHEL, Eur. J. B&hem. 143,643 (1984). CARLICK AND C. J. TURNER, J. Magn. Reson. 51, 536 (1983). E. FABRY AND M. EISENSTADT, Biophys. J. 15, 1101 (1975). W. KUCHEL AND B. E. CHAPMAN, J. Theor. Biol. 105, 569 (1983). W. KUCHEL, G. R. BEILHARZ, B. E. CHAPMAN, AND C. R. MIDDLEHURST, in “Secretion: Mechanisms and Control” (R. M. Case, J. M. Lingard, and J. A. Young, Ed%), pp. l-10, Manchester Univ. Press, Manchester, 1984. OSBORNE, J. Aust. Math. Sot. B 19, 343 (1976). L. OLSON AND M. S. SHUMAN, Anal. Chem. 55, 1103 (1983). BELLMAN, R. E. KALABA, AND J. LOCKETT, “Numerical Inversion of the Laplace Transform,” Elsevier, New York, 1966. V. WIDDER, “The Laplace Transform,” Princeton Univ. Press, Princeton, N.J., 1941.