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Homonuclear proton cross relaxation in the presence of paramagnetic metal ions
JOURNAL
OF MAGNETIC
RESONANCE
88,60-7
1 ( 1990)
Homonuclear Proton Cross Relaxation in the Presenceof ParamagneticMetal Ions ANTHONY J. DUBEN Computer Science Department, Southeast Missouri State University, I University Plaza, Cape Girardeau, Missouri 63701 AND
WILLIAM C. HUTTON * Monsanto Corporate Research, Monsanto Company, 700 Chesterfield Village Parkway, St. Louis, Missouri 63198 Received June 13, 1989; revised October 17, 1989 Homonuclear cross relaxation for a proton pair with external relaxation is evaluated theoretically. Results for nuclear Overhauser effect experiments and for two-dimensional NOESY spectra are given. The effects of internal motion are treated. The calculations give insight into experimental results obtained from cross-relaxation studies of proteins containing paramagnetic metals. When external relaxation is present, a much longer correlation time is required for the spins to be in the pure spin-diffusion limit. The value of the NOE enhancement becomes dependent on the interproton distance when external relaxation is significant. The initial slope of cross-peak evolution in two-dimensional spectra is not significantly changed by external relaxation. The slope is, however, very sensitive to the rotational correlation time. The implications ofthe theoretical analysis for comparing interproton distances for pairs of protons near and distant from the paramagnetic metal are summarized. 0 1990 Academic Press, 1~.
The use of homonuclear, proton cross relaxation is an established technique for determining molecular conformation. In one-dimensional spectroscopy, cross relaxation is determined experimentally by the nuclear Overhauser effect enhancement, q. In the extreme-narrowing limit 7 can become vanishingly small when external relaxation is present ( I ) . On the other hand, several groups have observed cross relaxation between protons close to paramagnetic ions in proteins (2-11) . The measured range of enhancements falls between - 1 and -75%. In the context of relaxation theory, these experimental results are interesting for two reasons: (i) the unpaired electron in the metal is a strong source of external relaxation (ye-/y ‘H x 600) that could overwhelm the intramolecular cross relaxation between the protons and (ii) in very large proteins, where 7, is close to the pure spin-diffusion limit, the maximum q of - 1 is not observed. * To whom correspondence should be addressed. 0022-2364190 $3.00 Copyright 0 1990 by Academic Press, Inc. All rights of reproduction in any form reserved.
60
PROTON
CROSS RELAXATION
AND PARAMAGNETIC
61
IONS
I
.o -31.0 -21.0 4.0
o’.o
II.0
2’.oG’.o
’ .O
log (0 %> FIG. 1. Motional dependence of 9 for a spin pair attached to the surface of the sphere or when /3 is near zero. Werbelow’s model ( 14) for calculating cross relaxation in an isolated, two-spin system with autocorrelated, mixed molecular motion is used with the addition of external relaxation, R,. R, is set to valuesof O(O), 3.0 X 1O-4(O), 3.0 X IO-'(A), 3.0 X lo-‘(+), 3.0 X 10-l (X),and3.0(0).
For an isolated two-spin system, the magnitude of the NOE depends on the rotational diffusion rate of the molecule as described by the rotational correlation time, T,. If T, is very short (extreme-narrowing limit), 7 takes the values of 0.5 regardless of the rate and geometry of motional averaging. When T, is very long (pure spindiffusion limit), v approaches - 1, again without regard to the rate or geometry of motional averaging. For a given spectrometer frequency, w. , there is a small range of 7, values where it is possible to measure 7 values that are not at these extreme values. When a molecule is very large and rigid, or the viscosity of the medium is very high, it is not possible to distinguish among the protons as they all give rise to an r] of - 1. This is depicted graphically in Fig. 1. Although values for 7 calculated in Fig. 1 are based on the one-dimensional, presaturation technique, similar results hold for the exchange of longitudinal magnetization through cross relaxation as measured in homonuclear, two-dimensional experiments ( 12, 13). Simple substitution of the formulae in Ref. ( 13) demonstrates that in twodimensional spectra the ratio of cross-peak to diagonal-peak intensity at the optimal mixing time is - 7 as calculated for the one-dimensional spectra. For this reason these remarks hold for both one-dimensional and two-dimensional experiments.
62
DUBEN
AND HUTTON
FIG. 2. Model system for cross relaxation in the presence of mixed motion. The overall molecular tumbling rate, TV,is isotropic and the internal motional averaging of the proton pair is characterized by 7,.
A theoretical examination of cross relaxation between a proton pair in the presence of strong external relaxation predicts that large 0 values can be measured. In addition when 11is greater (more positive) than - 1, neglect of external relaxation can result in an estimate of 7Cthat is in error by as much as two orders of magnitude. We begin with Werbelow’s ( 14) treatment of two-spin cross relaxation augmented by the inclusion of an external relaxation term R,. The electron-nuclear distance is assumed to be fixed and the motion of the proton pair and the electron is assumed to be uncorrelated. - Jo(O)/3
2J2(2%) ’
Jill+
= [2J2(2~0)-
ill
Jo(0)/3l+R,
with Jo
= (6a/5)(y4h2r6)((-1)k/167r)[(3 + 12 Sil12/3
COS2/3F,(D,
COST@Dint,
l>2Fo(D,Dint, W)
+
3 SiI14@F2(D,
W) Dint,
fd)]
[2]
and
60 + t12Dint
Fn(D, Dint, W)= (60 + n2Dint)’ + CO’.
131
In Eq. [3] D = l/67,, Dint = l/7,, T, is the correlation time for global molecular reorientation, 7, is the correlation time for internal motion, and /3 is the angle between the internuclear vector and the axis of internal rotation (see Fig. 2). When there is no internal rotation or when ,L3is zero all of the F,, terms become equal and the Bdependent term in Eq. [ 21 reduces to 47,( 1 + 7:~~))‘. Substitution of these expressions into Eq. [ l] with the use of w = mwo, where m = 0, 1,2, yields
PROTON
CROSS
RELAXATION
AND
6 rl=
PARAMAGNETIC
-1
IONS
63
.
1 + 47&J; [ 1 + 4&lJ; 6 + 1+ 37:o; +1 I +R,
3
T&? in which q is the collection of values q = ( l/ lO)(r&h */r6) and Y is the interproton distance. In Fig. 1, Eq. [ 41 was plotted using six different values of R, assuming an instrument frequency, wo, of 500 MHz. The value of q was selected to be 3.33 X lo7 s-* which corresponds to a ‘H-‘H separation of 3.5 A. If external relaxation is not important, the interproton distance has no effect on 7. When external relaxation becomes significant, the value for 1 will vary with Y. Shorter ‘H- ‘H distances increase q and decrease the importance of the R, term. A ‘H- ‘H separation of 2.4 A increases q by one order of magnitude. A longer ‘H- ‘H distance, 5.0 A, decreases q by one order of magnitude. There is the usual parametric dependence of r] on o but note that with respect to the protons the external relaxation term, RJ~T,, is frequency independent. There are two limiting cases based on the product of woT,. When oo7,e 1,~ reduces to 5
II=
lo+&
151 Tc9
In this case, when R, = 0 then r~ = 0.5. When paramagnetic ions are present, R,/qT, can assume values larger than 10 and n will become vanishingly small. The other limiting case is the pure spin-diffusion
limit where
OoT,
% 1 and
If R, = 0, n = - 1 and n becomes equal for all of the protons. When R,/qT, must be accounted for, Eq. [ 61 should be applied with care. Consider a macromolecule with a rigid paramagnetic metal binding site. For proton pairs near to the metal, R, will be nonzero. Since 7, is to be large, the value of R,/qTc will be significant and 7 will take values more positive than - 1 (see Fig. 1). The measurement of 0 will be useful in conformational studies as n will depend on q which contains r. The reduced enhancements will differentiate protons near the metal from those far from the metal which will have full enhancements. A proton pair with a small interproton distance can be more di$icult to study compared to a pair that is further apart. The product of qTc will decrease and, as q approaches - 1, it will be difficult to differentiate between proton pairs near and far from the paramagnetic metal. By contrast, a proton pair that is further apart, but near to the metal, will be conspicuous by their smaller enhancement and will be more useful in conformational studies. Inspection of Fig. 1 reveals another interesting feature of external relaxation. Counter to the normal case, when UT, is tl the measured peak intensity actually increases as the rotational correlation time increases. In this case one would actually
O’Z
0’1
0’0
O’l(‘z co ) 801
O’Z-
O’E-
0
-2 -ri
9 0
I
0 - c-4
--P
L r-
0 - 4
9
I
‘;” 0 -P-i
--T
O.-b-
p=36
-4.0
-3.0
-2.0
-1.0
0.0
1% (0 k) FIG.
3-Continued 65
1.0
2.0
3.0
4.0
d
-2.0
-1.0
0.0 1%
(0
1.0 -k)
FIG. S-Continued 66
-2.0
0.0
-1.0
log (0 w FIG. 3-Continued 67
1.0
68
DUBEN
AND
HUTTON
9m I $
-4.0
-3.0
-2.0
0.0
-1.0 1% FIG.
(0
1.0
2.0
k)
3-Continued
These differences are particularly noticeable at values of log( w,,T,) < 0 with the greatest difference between - 1 < log( ~~7,) < 0. Regardless of the angle /3 (including the magic angle), for extreme values of log( w~T,)( -4), the curves converge to the same values for a given R,. In this case the internal rotation is slow enough so that the only important rotation becomes the overall molecular rotation. In a more limited sense, a similar convergence of curves for R, = 0 and /3 not equal to the magic angle seems to hold for negative values of log( woT,)( - -4). Physically, log( war,) = -4 corresponds to the internal rotational correlation time of an unhindered methyl group ( - lOpi3 s). At angles other than the magic angle, the rapid averaging of the protons produces an environment that has lost its distinct angular character. The extent of deviation at negative values of log( OoT,) increases with proximity of /I to the magic angle. When an experiment is performed, the relaxation time scale samples only the rapidly moving protons and an averaged conformation is obtained. When R, does not equal zero, the convergence of the curves for negative values of log( ~~7,) no longer holds. For a given log( WoT,), the profile of NOE responses across a range of log(wo~,) values depends on /3 as illustrated in Figs. 3e-3h. The averaged character of the response discussed above is reduced and the angle the rotating protons make with the axis about which they spin remains, in principle, determinable in the NOE profile over Tc. The secondary and tertiary structure of proteins is determined by using ‘H- ‘H distance constraints from cross-relaxation studies in conjunction with molecular
PROTON
CROSS RELAXATION
AND PARAMAGNETIC
69
IONS
I I
b.00
0:02
0:04
0.06
0108
c 10
mixing time, t, FIG. 4. Cross-peak evolution ofa proton pair R, is0 (-----), and R, = 1 (-). In both, w,,~~equals 5.6 (O), 11.2(O), 112(A),and 1120(+)andqisequalto3.33X lO’.Thespectrometerfrequencyis5OOMHz.
modeling. The upper boundary for the distance constraints is calculated using time dependence of 7 in transient NOE experiments (15) or from the time evolution of cross-relaxation cross peaks in a two-dimensional spectrum ( 16). The time dependence of the integrated intensity of the cross peaks, I&t,,,), in an isolated two-spin system is given by Macura and Ernst (23) from the following equation in which the Wfunctions have been revised to be consistent with Werbelow’s usage of the spectral density functions, Z*L3(4n)
= -(MO/41
(2Jz _,(2J2
J”‘3) Jo,3),
( 1 -exp(-I?&))(
-exp( -RI&)),
[7]
where t, is the mixing time; R, = 2 I (2 J2 - JO/ 3) 1, the cross-relaxation term; and R, = R, - J, + Jo/ 3 + 2 J2, the leakage term (expanded to include external relaxation). Each J, is a product of q and a function depending only on angle. Differentiating [ 71 with respect to t, yields the slope. At short mixing times, the exponentials may be represented by linear expressions in t, . slope z -(Mo/4)[(2Jz
- Jo/3)( 1 - 2R,t,) + 2(2J, - Jo/3)(J1
+ Jo/3 + 2J,)].
[8]
Only the first term of Eq. [ 81 depends on R,. As can be seen in Fig. 4., the initial
70
DUBEN
AND
HUTTON
slope is barely changed when R, = 1. However, when r is calculated, it is necessary to take the sixth root of the fraction ( 1 - 2R,t,). Even if this fraction were 0.90, the sixth root would be 0.983. The effect on predicted r would be quite small, even for large R, and a long mixing time. Figure 4 depicts the calculated cross-peak evolution with (R, = 1) and without external relaxation using the same conditions as in Fig. 1. The angle p is assumed to be zero. The initial slope of the cross-peak intensity curve is used to calculate the internuclear distance between the pair of protons. The initial slopes are essentially identical with and without the presence of external relaxation. The rotational correlation time has a dramatic effect on the slope, even in the limit oft, = 0. When external relaxation is present, correct determination of internuclear distances requires either accurate information on correlation times or the use of an internal standard of known geometry. The usual assumption that all of the protons have the same effective correlation time is particularly risky when paramagnetic metals are present. Using a proton pair with a known distance to calibrate distance constraints will be productive only when all of the proton pairs of interest are equidistant from the metal. The implications of our calculations for the study of macromolecules where significant pathways for external relaxation must be considered are summarized below. 1. The wo7, dependence of n indicates that segments of a macromolecule that are well below the critical correlation time (war, $ 1) will in fact have small ( < 10%) enhancements when external relaxation becomes significant. Increasing the correlation time by increasing the viscosity of the solution is recommended for initial studies. 2. With regard to ‘H- ‘H relaxation the external relaxation term, R,, is frequency independent. Comparison of single NOE measurements at different field strengths would permit R, to be determined. 3. For very large macromolecules, distance constraints are difficult to determine when R, = 0. The mixing times required to fulfill the initial rate approximation used to determine rare very short ( lo-30 ms) making cross-peak quantification difficult. When RJqr, is large, the cross-peak maxima will decrease making quantification even more demanding. 4. When paramagnetic metal ions are present, qualitative and quantitative determinations of interproton distances, the relative flexibility of segments of the molecule, and metal-proton distances based on measurements of n will be in error unless the effect of the external relaxation terms is considered. 5. When comparing n values for systems with and without paramagnetic metals to determine metal binding sites, the full form of Eq. [4] must be used. The closer the protons are, the further away the metal will appear if the interproton distance is not considered when comparing v values. 6. The method of using qualitative cross-peak classifications (weak, medium, and strong cross peaks) in distance geometry programs should be applied with caution. While this paper does not specifically address multispin relaxation, some insight can be gained into the observations that protons near paramagnetic metals do not participate in spin-diffusion networks ( IO, 17). Figures 1 and 3 clearly show that proton pairs close to the metal must have very long correlation times in order to be in the
PROTON
CROSS
RELAXATION
AND
PARAMAGNETIC
IONS
71
pure spin-diffusion limit. While the actual correlation times for protons close to the metal may be as long as those further away, R, has essentially isolated the protons near the metal by rendering J,, less dominant when compared to those where R, is negligible. R, makes the summed spectral densities responsible for the relaxation of the two types of proton pairs different and spin diffusion between them becomes inefficient. REFERENCES
1. J. H. NIGGLE
AND R. R. SCHIRMER, “The Nuclear Overhauser Effect: Chemical Applications,” Academic Press, New York, 197 1. 2. R. M. KELLER AND K. W~THRICH, Biochem. Biophys. Rex Commun. 83,1132 ( 1978). 3. J. TREWHELLA, P. E. WRIGHT, ANDC. A. APPLEBY, Nature (London) 280,87 ( 1979). 4. R. M. KELLER ANDK. W~THRICH, Biochim. Biophys. Acta 621,204 ( 1980). 5. R. D. JOHNSON, S. A. RAMAPRASAD, ANDG. N. LAMAR, J. Am. Chem. Sot. 105,7205 ( 1983). 6. R. D. JOHNSON, S. RAMAPRASAD, ANDG. N. LA MAR, J. Am. Chem. Sot. 106,533O ( 1984). 7. M. BARBUSH AND D. W. DIXON, Biochem. Biophys. Rex Commun. 129,70 ( 1985). 8. J. D. SATERLEE, J. E. ERMAN, ANDJ. S. DEROOP, J. Biol. Chem. 262,11,578 (1987). 9. J. D. SATERLEE, S. J. MOENCH, ANDD. AVIZONS, Biochim. Biophys. Acta 952,317 ( 1988). 10. V. THANABAL, J. S. DE ROPP, ANDG. N. LA MAR, J. Am. Chem. Sot. 110,3027 ( 1988). 11. S. D. EMERSON, J. T. J. LECOMTE, ANDG. N. LA MAR, J. Am. Chem. Sot. 110,4176 ( 1988). 12. J. JEENER, B. H. MEIER, P. BACHMANN, ANDR. R. ERNST, J. Chem. Phys. 71,4546 ( 1979). 13. S. MACURAANDR. R. ERNST, Mol. Phys. 41,95 ( 1980). 14. L. G. WERBELOW, J. Am. Chem. Sot. 96,4747 ( 1974). 1.5. S. L. GORDONANDK. WOTHRICH, J. Am. Chem. Sot. 100,7095 ( 1978). 16. A. KUMAR, G. WAGNER, R. R. ERNST, ANDK. WUTHRICH, J. Am. Chem. Sot. 103,3654 ( 1981). 17. V. THANABAL, G. N. LA MAR, AND J. S. DE ROPP, Biochemistry 27,540O ( 1988).
OF MAGNETIC
RESONANCE
88,60-7
1 ( 1990)
Homonuclear Proton Cross Relaxation in the Presenceof ParamagneticMetal Ions ANTHONY J. DUBEN Computer Science Department, Southeast Missouri State University, I University Plaza, Cape Girardeau, Missouri 63701 AND
WILLIAM C. HUTTON * Monsanto Corporate Research, Monsanto Company, 700 Chesterfield Village Parkway, St. Louis, Missouri 63198 Received June 13, 1989; revised October 17, 1989 Homonuclear cross relaxation for a proton pair with external relaxation is evaluated theoretically. Results for nuclear Overhauser effect experiments and for two-dimensional NOESY spectra are given. The effects of internal motion are treated. The calculations give insight into experimental results obtained from cross-relaxation studies of proteins containing paramagnetic metals. When external relaxation is present, a much longer correlation time is required for the spins to be in the pure spin-diffusion limit. The value of the NOE enhancement becomes dependent on the interproton distance when external relaxation is significant. The initial slope of cross-peak evolution in two-dimensional spectra is not significantly changed by external relaxation. The slope is, however, very sensitive to the rotational correlation time. The implications ofthe theoretical analysis for comparing interproton distances for pairs of protons near and distant from the paramagnetic metal are summarized. 0 1990 Academic Press, 1~.
The use of homonuclear, proton cross relaxation is an established technique for determining molecular conformation. In one-dimensional spectroscopy, cross relaxation is determined experimentally by the nuclear Overhauser effect enhancement, q. In the extreme-narrowing limit 7 can become vanishingly small when external relaxation is present ( I ) . On the other hand, several groups have observed cross relaxation between protons close to paramagnetic ions in proteins (2-11) . The measured range of enhancements falls between - 1 and -75%. In the context of relaxation theory, these experimental results are interesting for two reasons: (i) the unpaired electron in the metal is a strong source of external relaxation (ye-/y ‘H x 600) that could overwhelm the intramolecular cross relaxation between the protons and (ii) in very large proteins, where 7, is close to the pure spin-diffusion limit, the maximum q of - 1 is not observed. * To whom correspondence should be addressed. 0022-2364190 $3.00 Copyright 0 1990 by Academic Press, Inc. All rights of reproduction in any form reserved.
60
PROTON
CROSS RELAXATION
AND PARAMAGNETIC
61
IONS
I
.o -31.0 -21.0 4.0
o’.o
II.0
2’.oG’.o
’ .O
log (0 %> FIG. 1. Motional dependence of 9 for a spin pair attached to the surface of the sphere or when /3 is near zero. Werbelow’s model ( 14) for calculating cross relaxation in an isolated, two-spin system with autocorrelated, mixed molecular motion is used with the addition of external relaxation, R,. R, is set to valuesof O(O), 3.0 X 1O-4(O), 3.0 X IO-'(A), 3.0 X lo-‘(+), 3.0 X 10-l (X),and3.0(0).
For an isolated two-spin system, the magnitude of the NOE depends on the rotational diffusion rate of the molecule as described by the rotational correlation time, T,. If T, is very short (extreme-narrowing limit), 7 takes the values of 0.5 regardless of the rate and geometry of motional averaging. When T, is very long (pure spindiffusion limit), v approaches - 1, again without regard to the rate or geometry of motional averaging. For a given spectrometer frequency, w. , there is a small range of 7, values where it is possible to measure 7 values that are not at these extreme values. When a molecule is very large and rigid, or the viscosity of the medium is very high, it is not possible to distinguish among the protons as they all give rise to an r] of - 1. This is depicted graphically in Fig. 1. Although values for 7 calculated in Fig. 1 are based on the one-dimensional, presaturation technique, similar results hold for the exchange of longitudinal magnetization through cross relaxation as measured in homonuclear, two-dimensional experiments ( 12, 13). Simple substitution of the formulae in Ref. ( 13) demonstrates that in twodimensional spectra the ratio of cross-peak to diagonal-peak intensity at the optimal mixing time is - 7 as calculated for the one-dimensional spectra. For this reason these remarks hold for both one-dimensional and two-dimensional experiments.
62
DUBEN
AND HUTTON
FIG. 2. Model system for cross relaxation in the presence of mixed motion. The overall molecular tumbling rate, TV,is isotropic and the internal motional averaging of the proton pair is characterized by 7,.
A theoretical examination of cross relaxation between a proton pair in the presence of strong external relaxation predicts that large 0 values can be measured. In addition when 11is greater (more positive) than - 1, neglect of external relaxation can result in an estimate of 7Cthat is in error by as much as two orders of magnitude. We begin with Werbelow’s ( 14) treatment of two-spin cross relaxation augmented by the inclusion of an external relaxation term R,. The electron-nuclear distance is assumed to be fixed and the motion of the proton pair and the electron is assumed to be uncorrelated. - Jo(O)/3
2J2(2%) ’
Jill+
= [2J2(2~0)-
ill
Jo(0)/3l+R,
with Jo
= (6a/5)(y4h2r6)((-1)k/167r)[(3 + 12 Sil12/3
COS2/3F,(D,
COST@Dint,
l>2Fo(D,Dint, W)
+
3 SiI14@F2(D,
W) Dint,
fd)]
[2]
and
60 + t12Dint
Fn(D, Dint, W)= (60 + n2Dint)’ + CO’.
131
In Eq. [3] D = l/67,, Dint = l/7,, T, is the correlation time for global molecular reorientation, 7, is the correlation time for internal motion, and /3 is the angle between the internuclear vector and the axis of internal rotation (see Fig. 2). When there is no internal rotation or when ,L3is zero all of the F,, terms become equal and the Bdependent term in Eq. [ 21 reduces to 47,( 1 + 7:~~))‘. Substitution of these expressions into Eq. [ l] with the use of w = mwo, where m = 0, 1,2, yields
PROTON
CROSS
RELAXATION
AND
6 rl=
PARAMAGNETIC
-1
IONS
63
.
1 + 47&J; [ 1 + 4&lJ; 6 + 1+ 37:o; +1 I +R,
3
T&? in which q is the collection of values q = ( l/ lO)(r&h */r6) and Y is the interproton distance. In Fig. 1, Eq. [ 41 was plotted using six different values of R, assuming an instrument frequency, wo, of 500 MHz. The value of q was selected to be 3.33 X lo7 s-* which corresponds to a ‘H-‘H separation of 3.5 A. If external relaxation is not important, the interproton distance has no effect on 7. When external relaxation becomes significant, the value for 1 will vary with Y. Shorter ‘H- ‘H distances increase q and decrease the importance of the R, term. A ‘H- ‘H separation of 2.4 A increases q by one order of magnitude. A longer ‘H- ‘H distance, 5.0 A, decreases q by one order of magnitude. There is the usual parametric dependence of r] on o but note that with respect to the protons the external relaxation term, RJ~T,, is frequency independent. There are two limiting cases based on the product of woT,. When oo7,e 1,~ reduces to 5
II=
lo+&
151 Tc9
In this case, when R, = 0 then r~ = 0.5. When paramagnetic ions are present, R,/qT, can assume values larger than 10 and n will become vanishingly small. The other limiting case is the pure spin-diffusion
limit where
OoT,
% 1 and
If R, = 0, n = - 1 and n becomes equal for all of the protons. When R,/qT, must be accounted for, Eq. [ 61 should be applied with care. Consider a macromolecule with a rigid paramagnetic metal binding site. For proton pairs near to the metal, R, will be nonzero. Since 7, is to be large, the value of R,/qTc will be significant and 7 will take values more positive than - 1 (see Fig. 1). The measurement of 0 will be useful in conformational studies as n will depend on q which contains r. The reduced enhancements will differentiate protons near the metal from those far from the metal which will have full enhancements. A proton pair with a small interproton distance can be more di$icult to study compared to a pair that is further apart. The product of qTc will decrease and, as q approaches - 1, it will be difficult to differentiate between proton pairs near and far from the paramagnetic metal. By contrast, a proton pair that is further apart, but near to the metal, will be conspicuous by their smaller enhancement and will be more useful in conformational studies. Inspection of Fig. 1 reveals another interesting feature of external relaxation. Counter to the normal case, when UT, is tl the measured peak intensity actually increases as the rotational correlation time increases. In this case one would actually
O’Z
0’1
0’0
O’l(‘z co ) 801
O’Z-
O’E-
0
-2 -ri
9 0
I
0 - c-4
--P
L r-
0 - 4
9
I
‘;” 0 -P-i
--T
O.-b-
p=36
-4.0
-3.0
-2.0
-1.0
0.0
1% (0 k) FIG.
3-Continued 65
1.0
2.0
3.0
4.0
d
-2.0
-1.0
0.0 1%
(0
1.0 -k)
FIG. S-Continued 66
-2.0
0.0
-1.0
log (0 w FIG. 3-Continued 67
1.0
68
DUBEN
AND
HUTTON
9m I $
-4.0
-3.0
-2.0
0.0
-1.0 1% FIG.
(0
1.0
2.0
k)
3-Continued
These differences are particularly noticeable at values of log( w,,T,) < 0 with the greatest difference between - 1 < log( ~~7,) < 0. Regardless of the angle /3 (including the magic angle), for extreme values of log( w~T,)( -4), the curves converge to the same values for a given R,. In this case the internal rotation is slow enough so that the only important rotation becomes the overall molecular rotation. In a more limited sense, a similar convergence of curves for R, = 0 and /3 not equal to the magic angle seems to hold for negative values of log( woT,)( - -4). Physically, log( war,) = -4 corresponds to the internal rotational correlation time of an unhindered methyl group ( - lOpi3 s). At angles other than the magic angle, the rapid averaging of the protons produces an environment that has lost its distinct angular character. The extent of deviation at negative values of log( OoT,) increases with proximity of /I to the magic angle. When an experiment is performed, the relaxation time scale samples only the rapidly moving protons and an averaged conformation is obtained. When R, does not equal zero, the convergence of the curves for negative values of log( ~~7,) no longer holds. For a given log( WoT,), the profile of NOE responses across a range of log(wo~,) values depends on /3 as illustrated in Figs. 3e-3h. The averaged character of the response discussed above is reduced and the angle the rotating protons make with the axis about which they spin remains, in principle, determinable in the NOE profile over Tc. The secondary and tertiary structure of proteins is determined by using ‘H- ‘H distance constraints from cross-relaxation studies in conjunction with molecular
PROTON
CROSS RELAXATION
AND PARAMAGNETIC
69
IONS
I I
b.00
0:02
0:04
0.06
0108
c 10
mixing time, t, FIG. 4. Cross-peak evolution ofa proton pair R, is0 (-----), and R, = 1 (-). In both, w,,~~equals 5.6 (O), 11.2(O), 112(A),and 1120(+)andqisequalto3.33X lO’.Thespectrometerfrequencyis5OOMHz.
modeling. The upper boundary for the distance constraints is calculated using time dependence of 7 in transient NOE experiments (15) or from the time evolution of cross-relaxation cross peaks in a two-dimensional spectrum ( 16). The time dependence of the integrated intensity of the cross peaks, I&t,,,), in an isolated two-spin system is given by Macura and Ernst (23) from the following equation in which the Wfunctions have been revised to be consistent with Werbelow’s usage of the spectral density functions, Z*L3(4n)
= -(MO/41
(2Jz _,(2J2
J”‘3) Jo,3),
( 1 -exp(-I?&))(
-exp( -RI&)),
[7]
where t, is the mixing time; R, = 2 I (2 J2 - JO/ 3) 1, the cross-relaxation term; and R, = R, - J, + Jo/ 3 + 2 J2, the leakage term (expanded to include external relaxation). Each J, is a product of q and a function depending only on angle. Differentiating [ 71 with respect to t, yields the slope. At short mixing times, the exponentials may be represented by linear expressions in t, . slope z -(Mo/4)[(2Jz
- Jo/3)( 1 - 2R,t,) + 2(2J, - Jo/3)(J1
+ Jo/3 + 2J,)].
[8]
Only the first term of Eq. [ 81 depends on R,. As can be seen in Fig. 4., the initial
70
DUBEN
AND
HUTTON
slope is barely changed when R, = 1. However, when r is calculated, it is necessary to take the sixth root of the fraction ( 1 - 2R,t,). Even if this fraction were 0.90, the sixth root would be 0.983. The effect on predicted r would be quite small, even for large R, and a long mixing time. Figure 4 depicts the calculated cross-peak evolution with (R, = 1) and without external relaxation using the same conditions as in Fig. 1. The angle p is assumed to be zero. The initial slope of the cross-peak intensity curve is used to calculate the internuclear distance between the pair of protons. The initial slopes are essentially identical with and without the presence of external relaxation. The rotational correlation time has a dramatic effect on the slope, even in the limit oft, = 0. When external relaxation is present, correct determination of internuclear distances requires either accurate information on correlation times or the use of an internal standard of known geometry. The usual assumption that all of the protons have the same effective correlation time is particularly risky when paramagnetic metals are present. Using a proton pair with a known distance to calibrate distance constraints will be productive only when all of the proton pairs of interest are equidistant from the metal. The implications of our calculations for the study of macromolecules where significant pathways for external relaxation must be considered are summarized below. 1. The wo7, dependence of n indicates that segments of a macromolecule that are well below the critical correlation time (war, $ 1) will in fact have small ( < 10%) enhancements when external relaxation becomes significant. Increasing the correlation time by increasing the viscosity of the solution is recommended for initial studies. 2. With regard to ‘H- ‘H relaxation the external relaxation term, R,, is frequency independent. Comparison of single NOE measurements at different field strengths would permit R, to be determined. 3. For very large macromolecules, distance constraints are difficult to determine when R, = 0. The mixing times required to fulfill the initial rate approximation used to determine rare very short ( lo-30 ms) making cross-peak quantification difficult. When RJqr, is large, the cross-peak maxima will decrease making quantification even more demanding. 4. When paramagnetic metal ions are present, qualitative and quantitative determinations of interproton distances, the relative flexibility of segments of the molecule, and metal-proton distances based on measurements of n will be in error unless the effect of the external relaxation terms is considered. 5. When comparing n values for systems with and without paramagnetic metals to determine metal binding sites, the full form of Eq. [4] must be used. The closer the protons are, the further away the metal will appear if the interproton distance is not considered when comparing v values. 6. The method of using qualitative cross-peak classifications (weak, medium, and strong cross peaks) in distance geometry programs should be applied with caution. While this paper does not specifically address multispin relaxation, some insight can be gained into the observations that protons near paramagnetic metals do not participate in spin-diffusion networks ( IO, 17). Figures 1 and 3 clearly show that proton pairs close to the metal must have very long correlation times in order to be in the
PROTON
CROSS
RELAXATION
AND
PARAMAGNETIC
IONS
71
pure spin-diffusion limit. While the actual correlation times for protons close to the metal may be as long as those further away, R, has essentially isolated the protons near the metal by rendering J,, less dominant when compared to those where R, is negligible. R, makes the summed spectral densities responsible for the relaxation of the two types of proton pairs different and spin diffusion between them becomes inefficient. REFERENCES
1. J. H. NIGGLE
AND R. R. SCHIRMER, “The Nuclear Overhauser Effect: Chemical Applications,” Academic Press, New York, 197 1. 2. R. M. KELLER AND K. W~THRICH, Biochem. Biophys. Rex Commun. 83,1132 ( 1978). 3. J. TREWHELLA, P. E. WRIGHT, ANDC. A. APPLEBY, Nature (London) 280,87 ( 1979). 4. R. M. KELLER ANDK. W~THRICH, Biochim. Biophys. Acta 621,204 ( 1980). 5. R. D. JOHNSON, S. A. RAMAPRASAD, ANDG. N. LAMAR, J. Am. Chem. Sot. 105,7205 ( 1983). 6. R. D. JOHNSON, S. RAMAPRASAD, ANDG. N. LA MAR, J. Am. Chem. Sot. 106,533O ( 1984). 7. M. BARBUSH AND D. W. DIXON, Biochem. Biophys. Rex Commun. 129,70 ( 1985). 8. J. D. SATERLEE, J. E. ERMAN, ANDJ. S. DEROOP, J. Biol. Chem. 262,11,578 (1987). 9. J. D. SATERLEE, S. J. MOENCH, ANDD. AVIZONS, Biochim. Biophys. Acta 952,317 ( 1988). 10. V. THANABAL, J. S. DE ROPP, ANDG. N. LA MAR, J. Am. Chem. Sot. 110,3027 ( 1988). 11. S. D. EMERSON, J. T. J. LECOMTE, ANDG. N. LA MAR, J. Am. Chem. Sot. 110,4176 ( 1988). 12. J. JEENER, B. H. MEIER, P. BACHMANN, ANDR. R. ERNST, J. Chem. Phys. 71,4546 ( 1979). 13. S. MACURAANDR. R. ERNST, Mol. Phys. 41,95 ( 1980). 14. L. G. WERBELOW, J. Am. Chem. Sot. 96,4747 ( 1974). 1.5. S. L. GORDONANDK. WOTHRICH, J. Am. Chem. Sot. 100,7095 ( 1978). 16. A. KUMAR, G. WAGNER, R. R. ERNST, ANDK. WUTHRICH, J. Am. Chem. Sot. 103,3654 ( 1981). 17. V. THANABAL, G. N. LA MAR, AND J. S. DE ROPP, Biochemistry 27,540O ( 1988).