[29] Nuclear relaxation measurements of water protons and other ligands

654

RESONANCE TECHNIQUES

[29]

Nuclear Relaxation Measurements Protons and Other Ligands I

[29]

of Water

B y ALBERT S. MILDVAN and JAMES L. ENGLE

The measurement of the relaxation rates of magnetic nuclei is a specialized branch of nuclear magnetic resonance spectroscopy which, especially when carried out with paramagnetic probes, can provide thermodynamic, structural, and kinetic information on enzyme complexes. Specifically, the stoichiometry and dissociation constants of binary -~-* and ternary "~,6 complexes of enzymes with paramagnetic ions, p a r a m a g netic substrate analogs, 7,s and diamagnetic substrates m a y be measured. Coordination schemes and interatomic distances between enzyme-bound paramagnetic metal ions or substrate analogs, and the substrate molecules have been determined. Exchange rates of substrates into paramagnetic and diamagnetic environments on enzymes have been estimated. A review of the principles and applications of nuclear magnetic relaxation to enzyme mechanisms in the presence of paramagnetic probes has recently been published2 An earlier review TM provides a comprehensive account of the principles and applications of nuclear relaxation in diamagnetic systems, i.e., in the absence of unpaired electrons. Here we will discuss in detail how nuclear relaxation rates are measured and consider only those aspects of the theory which elucidate the methods and their applications.

Basic Principles of Nuclear Magnetic Relaxation Definition o] Nuclear Relaxation Rates

Any atomic nucleus with an odd number of protons or neutrons has a net magnetic moment and its magnetic vector will tend to orient in ~This work was supported by National Science Foundation Grant GB-8579, United States Public Health Service Grants AM-13351, CA-06927, and RR-05539 from the National Institutes of Health, and American Cancer Society Grant IN-49, and by an appropriation from the Commonwealth of Pennsylvania. oA. S. Mildvan and M. Cohn, Biochemistry 2, 910 (1963). 3A. S. Mildvan and M. Cohn, J. Biol. Chem. 240, 238 (1965). 4j. Reuben and M. Cohn, J. Biol. Chem. 245, 6539 (1970). A. S. Mildvan and M. Cohn, J. Biol. Chem. PAl, 1178 (1966). s W. J. O'Sullivan and M. Cohn, J. Biol. Chem. PAl, 3104 (1966). ~A. S. Mildvan and It. Weiner, Biochemistry 8, 552 (1969). 8A. S. Mildvan and H. Weiner, J. Biol. Chem. 244, 2465 (1969). 9A. S. Mildvan and M. Cohn, Advan. Enzymol. 33, 1 (1970).

[29]

NUCLEAR RELAXATION MEASUREMENTS

655

a magnetic field. The longitudinal relaxation time (T1) is the first-order time constant for orientation of a population of magnetic vectors in a magnetic field. Its reciprocal, 1/TI is the first-order rate constant for this process called the longitudinal relaxation rate. Nuclear magnetism results from net nuclear spin. When a torque is applied to a spinning particle it will tend to precess. By analogy, when a population of magnetic nuclei arc placed in a magnetic field, their magnetic vectors experience a torque and precess about the direction of the field. Energy may be applied to this system to align the magnetic vectors of the nuclei to precess in phase with each other. Since each nucleus experiences a different magnetic microenvironment, each will precess at a different rate, and after a time, phase coherence will be lost with a characteristic time constant (T2) the transverse relaxation time. Its reciprocal, 1/T2, is the transverse relaxation rate. Reorientation of magnetic vectors (longitudinal relaxation) disrupts phase relationships among magnetic vectors and is therefore always accompanied by transverse relaxation but the converse is not true. Hence the transverse relaxation rate is always greater than or equal to the longitudinal relaxation rate: 1/T2 >_ 1/T1 (1) D i a m a g n e t i c Effects on R e l a x a t i o n R a t e s

Magnetic nuclei undergo relaxation by interacting with and exchanging magnetic energy with their magnetic environment. In an aqueous solution of diamagnetic salts and buffers, the predominant magnetic environment of a magnetic nucleus (e.g., a proton) consists of the other water protons. For transfers of magnetic energy to take place which cause relaxation, the magnetic environment or "lattice" must be capable of absorbing magnetic energy which fluctuates at the appropriate frequency, i.e., the precession (Larmor) frequency of the proton (100 MHz or 6.28 × l0 s sec-1 at 23.5 Kgauss). The fluctuating character of the magnetic interaction between the proton and its magnetic environment is caused by its periodic interruption (modulation) by molecular tumbling, a high-frequency process for small molecules (~1011 sec-1). Because protons are weak magnets, and because of the large difference in the tumbling frequency of small molecules and the Larmor precession frequency of protons, the transfer of magnetic energy occurs at a low rate. Hence, the relaxation rates of the protons of water and of other small molecules in aqueous solution are low ( ~ 1 sec-~). When such small molecules are immobilized on macromolecules their interaction with the ~oo. Jardetzky, Advan. Chem. Phys. 7, 499 (1964).

656

RESONANCE TECHNIQUES

[29]

lattice is modulated by the slower tumbling of the macromolecule (,~108 sec-1) or by the rate of exchange into a different magnetic environment (~109 sec-1), processes with lower frequencies closer to that of the precession frequency of protons. Hence, more efficient transfer of magnetic energy is possible and much greater relaxation rates (103 sec -1) may be observed. Such diamagnetic effects on relaxation rates have been used to study the interaction of antibiotics with serum albumin 1°,11 inhibitors with enzymes, 12 and water with macromolecules. 13 A theoretical treatment of such diamagnetic effects has been given. TM Paramagnetic Effects on Relaxation Rates

Unpaired electrons, with magnetic moments which are three orders of magnitude greater than protons, are much stronger magnets. Hence paramagnetic ions and radicals are exceedingly effective in increasing the relaxation rates of ligands, e.g., ~ 1 0 ~ see -~ for a coordinated H~O in Mn(H_~O)6 ~÷. The paramagnetic contributions to the relaxation rates (1/T~p, 1/T~_p) are defined as

l/Tip = (l/T1) - (l/T1)0 1/T2, = ( l / T 2 ) - (1/T2)0

(2) (3)

where (1/T~) and (l/T2) are the measured relaxation rates in the presence of the paramagnetic species and (1/T~)o and (1/T2)o are the measured relaxation rates in absence of the paramagnetic species. To compare different systems the paramagnetic contribution to the relaxation rates may be divided by the concentration of the paramagnetic species to yield the molar relaxivity, or by the factor p = [paramagnetic species]/[ligand]. For 10=~ M Mn 2÷ in water at 25 ° and 24.3 MHz 1/T~p = 0.84 sec-~; the molar relaxivity 1/Tlp[Mn] = 8.4 × 103M -1 sec-~; and 1/pT~p = 1 / ( [ M n ] / [ H 2 0 ] ) T1, = 4.7 × 102 sec -~ where the concentration of water is 55.5 M. The equations relating the experimentally measured parameters (1/pT~,, 1/pT2,) to theoretical parameters were derived from the Bloch equations by Swift and Connick, 14 and by Luz and Meiboom, ~5 to be: 1

q

pT1, 1

pT~,

T1M -5 TM _

q

r~ \

+

1

(4)

To.~. +

(~/T2M?~ ~

+

1

+ -A~-~ / + (To.~.~

,10. Jardetzky and N. G. Wade-Jardetzky, Mol. Pharmacol. 1, 214 (1965). ,o.B. D. Sykes, J. Amer. Chem. Soc. 91, 949 (1969). "S. H. Koenig and W. E. Schillinger, J. Biol. Chem. 244, 3283 (1969). '~T. J. Swift and R. E. Connick, J. Chem. Phys. 37, 307 (1962). I~Z. Luz and S. Meiboom, J. Chem. Phys. 40, 2686 (1964).

(5)

[29]

NUCLEAR RELAXATION MEASUREMENTS

657

where 1~To.8. is the outer sphere contribution to the relaxation rate due to ligand molecules beyond the inner coordination sphere, q is the coordination number, and ~M the residence time of the ligand in the inner coordination sphere, T1M and T~M are the relaxation times of a coordinated ligand and A~M is the chemical shift difference between free and coordinated ligands. For the interaction of Mn 2+ and Cu 2÷ with ligands Ao~r~ ~ 1/T2M and Eq. (5) simplifies to 1/pT2p = T2M q4- rM + ~ To.~.

(5a)

The relaxation rates of coordinated ligands (l/TIM and 1/T2M) depend on the rate of exchange of magnetic energy between ligands and the u~paired electrons. Magnetic energy can be exchanged between protons and unpaired electrons through space (dipolar interaction) and through chemical bonds (hyperfine interaction). The dipolar interaction is modulated by the sum of the three processes: tumbling (~101~ see-l), ligand exchange (~109 sec-~), or electron spin relaxation ( ~ 1 0 s sec-1). The hyperfine interaction is modulated by the sum of the two processes, ligand exchange and electron spin relaxation. For those paramagnetic species with slow electron spin relaxation rates, as manifested by visible E P R spectra at room temperature (e.g., Mn 2+, Cu 2÷, Cr 3÷, nitroxide radicals, flavin radicals), tumbling modulates the dipolar process in small complexes and determines the relaxation rate. When tumbling is slowed by immobilization of the paramagnetic species on macromolecules, slower processes with frequencies closer to the Larmor frequency such as hindered rotation, electron spin relaxation, or ligand exchange now modulate the electron-nuclear interaction and enhanced relaxation rates are observed (e.g., 1/pT~p = 106.6 sec-~ for H20 in pyrurate kinase-Mn(H~O)3, which is a 25-fold enhancement). The enhancement factor 16 is the ratio of the paramagnetic contribution to the relaxation rate of the ligand in the presence of the macromolecule to that in its absence, and is separately defined for longitudinal (~1) and transverse (e2) relaxation. ~ = (1/T~p)*/1/V~p = (1/pT~p)*/1/pT~p ~2 = (1/T2p)*/1/T2p = (1/pT2p)*/1/pT2p

(6) (7)

where the asterisk denotes the presence of the macromolecule. The definition of enhancement may be generalized for all ligands under diverse conditions by inserting the normalizing factor p = [paramagnetic species] / [relaxing ligand]. lej. Eisinger, R. G. Shulman, and B. M. Szymanski, J. Chem. Phys. 36, 1721 (1962).

658

[29]

RESONANCE TECHNIQUES

From the above discussion, the relaxation rates of ligands in the coordination sphere of Mn ~÷ (1/TiM, 1/T2M) are directly related to the correlation times or time constants of those processes ~which modulate the dipolar electron nuclear interaction (rc) and the hyperfine interaction (r~). Expressed mathematically 1

1 q_ 1 q_ 1

Tc

Tr

Ts

1

1

Ts

TM

(8)

TM

and 1

-

Te

= -- + - -

(9)

where rr is the time constant for rotation of the complex, vs is the longitudinal relaxation time of the electron spin, and rM is the residence time of the ligand in the coordination sphere. The quantitative relationships between the inner sphere relaxation rates (1/TiM and 1/T~M) and the structural parameters of the complex are given by the SolomonBloembergen equations, ires

1

2 S(S q-

T1M = 15

1)3,~g2~2/' 3T¢ q_ 7r¢ r6 ~1 --~ WI2rc2 1 -~- ws2rc2/

2 S(S + 1)A2 ( -[- 3

1

1 s(s + 1)~g~/4to +

T~M = 15

r~

~,

h2

3to

+

1 + o~i2v~2 h2

) (10)

13~o 1 ~r~

1S(S + I)A2 (T

-t- 3

Te

1 "~ ~s2re 2

2]

r~

~ q- 1 q-~2ro:

)

(11)

where S is the electron spin quantum number; ~/~ is the nuclear magnetogyric ratio; r is the ion-proton internuclear distance; g is the electronic "g" factor; fl is the Bohr magneton; ~ and ¢o~ are the Larmor angular precession frequencies for the nuclear and electron spins, respectively; and A is the hyperfine coupling constant. In equations (10) and (11) the first term represents the dipolar contribution, and the second term the hyperffne contribution, to the relaxation rates. Since these equations were derived for rotational motion, an additional term is required in the dipolar contribution to 1/TiM and 1/T.~Mwhen Tc contains a contribution from TM, a process which involves linear motion. 19 A further refinement of the Solomon-Bloembergen equations has been 17I. Solomon, Phys. Rev. 99, 559 (1955). ~N. Bloembergen, J. Chem. Phys. 27, 572 (1957). 19A. McLaughlin, personal communication, 1971.

[29]

NUCLEAR RELAXATION MEASUREMENTS

659

made by Reuben et al., ~° who pointed out that the re values in the nuclear (~) and electronic (~o~) terms of the dipolar contribution may not be identical, but would be of the same order of magnitude. Hence, in most cases, this small difference may be ignored. For macromolecular complexes and certain small complexes of Mn 2÷, Cu 2÷, Cr 3÷, nitroxide and flavin radicals, the value of re is sufficiently large such that . , J r ~ > > 1, and therefore the hyperfine contribution to T1M is negligible. The term ~ro for the water protons in Mn(H20)62+ is approximately 1 at a frequency of approximately 40 MHz, but in enhanced complexes (~ ~ 10) then ~Jrc2>> 1, and the Solomon-Bloembergen equations simplify to 1 = 2 S ( S -4- 1)Ti2g2B~ 3r¢ ) 1 -~- 0~I2Tc2 T1M 15 r~ 1

T2M

1 s(s + 1) i g2 (4 o + 15

r6

(12)

i s(s + 1)A 1 + 0JI2r¢~] + 3

h2

re

(13)

The values of the parameters of the Solomon-Bloembergen equation and their units are given in Table I. ~1 M e t h o d s of Measuring Relaxation Rates

There are three general methods for measuring nuclear relaxation rates: pulsed methods requiring a pulsed N M R spectrometer; continuous wave methods which can be done on most conventional N M R spectrometers; and the Fourier transform method, a combination of the pulsed and continuous wave methods requiring a combined instrument. All three types of instrument are commercially available. Pulsed methods are highly accurate to better than + 2 % for 1/T1 and to ___5% for 1/T2, but are low in sensitivity, generally requiring 0.05 to 0.10 ml samples with concentrations of the nuclei under investigation greater than 5 M . Pulsed methods are therefore ideal for measuring the relaxation rates of solvents such as water, which is 55.5M or l l l N in protons. Continuous wave methods conversely are less accurate (___40% errors in l/T1 and 10% errors in 1/T2 may be found), but are more sensitive and discriminating than the pulsed methods since the relaxation rates of any nucleus that can be observed in an N M R spectrum can be measured. Typically, for protons at 100 MHz, 0.3 to 0.4 ml of a 0.01 M solution can be well resolved in a single spectral scan and an n-fold improvement in signal-to-noise ratio can be achieved by the summation of n 2 scans. Continuous wave methods 2oj. Reuben, G. H. Reed, and M. Cohn, J. Chem. Phys. 52, 1617 (1970). -~IA. S. Mildvan, J. S. Leigh, Jr., and M. Cohn, Biochemistry 6, 1805 (1967).

660

RESONANCE TECHNIQUES

[29]

TABLE I DEFINITIONS AND VALUES OF SYMBOLS USED IN THE SOLOMON-BLOEMBERGEN EQUATION a

Symbol

Definition

Units (cgs)

Numerical value

TIM Longitudinal relaxation time Second of ligand in the first coordination sphere of metal ion S Electronic spin quantum -5/2 (for Mn 2+) number "n Nuclear gyromagnetic ratio Rad. sec-1. gauss-1 2.675 X 104 (for IH) h Planck's constant/27r Erg. second 1.054 × 10-17 g Electronic "g" factor -2.00 (for Mn 2+) t~ Bohr magneton Rad. sec-1. gauss -1 8.795 X 10e r Average electron-nuclear disCentimeter tance To Dipolar correlation time Second ~3.0 X 10-11 (Mn-H20) b re Hyperfine correlation time Second ~ 1 X 10-s (Mn-H20) ~ ~i Nuclear resonance frequency Rad. sec-1 6.28 X 108 (for ~H at 23,487 gauss) o~, Electron resonance frequency Rad. sec-~ 4.13 X 10~ (for 23,487 gauss) A/h Isotropic hyperfine coupling Rad. see-1 3.9 X 10~ for Mn-H in constant/Planck's Constant Mn-H~O ¢ , Modified from A. S. Mildvan, J. S. Leigh, Jr., and M. Cohn, Biochemistry 6, 1805 (1967). b N. Bloembergen and L. O. Morgan, J. Chem. Phys. 34, 842 (1961). c From Z. Luz and R. G. Shulman, J. Chem. Phys. 43, 3750 (1965). a r e t h e r e f o r e s u i t a b l e for m e a s u r i n g t h e r e l a x a t i o n r a t e s of s u b s t r a t e s in f a i r l y d i l u t e solutions (>_1 m M ) . T h e F o u r i e r t r a n s f o r m m e t h o d ~2,2~ which should, in principle, combine t h e a d v a n t a g e s of p u l s e d a n d c o n t i n u o u s w a v e m e t h o d s , is s u i t a b l e for m e a s u r i n g 1/T1 of each r e s o n a n c e in an N M R s p e c t r u m . T h i s m e t h o d is t h e r e f o r e a p p l i c a b l e to c o m p l i c a t e d molecules i n c l u d i n g enzymes. I t will be discussed in P a r t D of " E n z y m e S t r u c t u r e . "

Cart-Purcell Pulsed M e t h o d ]or T1 24 Principle. A t e q u i l i b r i u m , i n d i v i d u a l n u c l e a r spins in the s a m p l e precess a t the L a r m o r f r e q u e n c y a b o u t t h e lines of force of t h e l a b o r a t o r y R. L. Void, J. S. Waugh, M. P. Klein, and D. E. Phelps, J. Chem. Phys. 48, 3831 (1968). ~' A. Allerhand, D. Doddrell, V. Glushko, D. W. Cochran, E. Wenkert, P. J. Lawson, and F. R. N. Gurd, J. Amer. Chem. Soe. 93, 544 (1971). H. Y. Carr and E. M. Purcell, Phys. Rev. 94, 630 (1954).

[~9]

661

NUCLEAR RELAXATION MEASUREMENTS y'

y

X~J

Z

c y

.7'

X

B

X I

D

Fro. I. Precession of the magnetization vector. (A) Equilibrium conditions. M0 is parallel to the magnet's lines of force. (B) After a 90 ° pulse the magnetization is in the x-y plane. (C) The magnetization vectors are fanning out due to field inhomogeneities. (D) After a 180 ° rotation about the y' ("rotating frame") axis, the vectors are reconverging.

magnet, i.e., around the z direction. The population of spins with magnetic moments oriented with the magnetic field slightly exceeds the population with magnetic vectors oriented against the field (Fig. 1A). Hence the total resultant magnetic moment is a vector M, which is in the direction of the laboratory magnetic field and has a magnitude Mo (Fig. 1A). If an intense radio frequency (rf) magnetic field H1 at the Larmor precession frequency is directed perpendicular to the lines of forces of the laboratory magnet, the magnetization vector M is made to tilt or nutate away from its original direction. If the radio frequency is turned off, the magnitude of M is unchanged but the angle that M makes with the z axis depends upon the duration and intensity of the rf burst. In the example in Fig. 1B, the magnetic moment has been nutated by 90 ° so that we say that we have delivered a 90 ° (or 7r/2) pulse to the sample. This particular angle produces the maximum rf signal in the receiver coil because the coil responds only to the component of M that is in the x - y plane. On the other hand, M in the relaxed orientation produces no receiver signal because it has no component in the x - y plane. The basic principle (Fig. 2) used in measuring the longitudinal relaxation time T1 is first to deliver a 180 ° excitation pulse which nutates

662

RESONANCE TECHNIQUES

[29]

I v

I 180 ° pulse

90 °

r Time

I 180 °

I 90 °

I

L

180 °

90 °

1 180 °

k_ 90 °

FIG. 2. T1 measurement. A 180° pulse reorients the nuclear magnetization against the magnet field. A 90° pulse at time (t) monitors the magnitude (V) of the magnetization. (A) t < t,,.H, (B) t = t,,,,~, (C) and (D) t > t,,.H. M until it is parallel to the z axis but oriented against the magnetic field. As the individual spins randomly change energy state, M remains parallel with the z axis but its magnitude decreases through zero and then increases until it has reached the equilibrium magnitude Mo. The magnitude of the relaxing vector follows an exponential curve with a time constant T1. The magnitude and polarity of the relaxing vector can be ascertained at any time by delivering a sampling 90 ° pulse to nutate M into the x-y plane, where it can be detected by the receiver for a short time before it disappears owing to dephasing (see section Instrument, below). At the half-time of the exponential decay the magnitude of the vector goes through a null, as ascertained by the delivery of a 90 ° pulse. T1 is related to t,u~ by the formula: T1 = t,u~/ln 2, where tnu~ is the time between the 180 ° and the 90 ° pulses. One must be sure to wait until the spins relax to equilibrium before applying the next 180 ° pulse. A waiting time of 8 × t , , ~ is sufficient; it introduces an error of around 0.5%. A waiting time of 4 × t,lu~, introduces a measured error of around 11%. Instrument. In a typical pulsed N M R system (Fig. 3) the source of radio frequency is a stable crystal-controlled oscillator. The prog r a m m e r timing is also precisely controlled by a crystal oscillator and allows a wide range of settings of duration and initiation times of the

[29]

NUCLEAR RELAXATION MEASUREMENTS

RF

I

Oscillator

J

Programmer 180 o

663

--[-[-

Power ,

Phase

control

omplifer

I Transmitter signal

NII--~'7 ,,

I

I.......

1

!

Probe

=

Receiver

]

Reference

~I

Phase

Input

. Oscilloscope

control

FIG. 3. Pulsed N M R spectrometer system. The programmer controls the passage of the oscillator power to the probe. The receiver is sensitive to the phase and

amplitude of the NMR signal. The phase shift is used only for the measurement of T2. rectangular waves appearing at its outputs. These outputs will gate the 180 ° and 90 ° rf pulses in the transmitter signal. In the example, the phase of the rf in the 90 ° pulse is fixed, while that of the 180 ° pulse is variable. If the sample's relaxation times are longer than 1 msec, then the transmitter peak power need be no greater than about 2 W. At this power, the 180 ° pulse length is about 200/~sec the precise value depending on the geometry of the probe. For shorter relaxation times, stronger pulses of shorter duration are required. The receiver input should be low noise to allow working with weak samples. Ideally the first stage of the receiver should be in the form of a preamplifier located at the probe. The receiver includes a phase sensitive detector which responds to both the amplitude of the received NMR signal and its phase relative to a reference rf signal. The reference phase is adjusted to maximize the signal which one wishes to observe, e.g., the free induction decay signal following the 90 ° pulse. This type of detection has the virtue that the sign of the detected signal changes as

664

RESONANCE TECHNIQUES

[29]

the 90 ° pulse delay is made equal to tnu11, so that tnuI1 is much easier to find. The receiver output is viewed on an oscilloscope whose timing and vertical sensitivity can be chosen to make the almost-nulled signal most visible. It is very helpful to be able to trigger the oscilloscope once just preceding the 180 ° excitation pulse (to show some base line) and once again at the time of the 90 ° sampling pulse. The demands on the magnet for measuring T1 of H20 are not great when compared to those in a high resolution system. The only adverse effect of inhomogeneity and minor drift on measuring T1 is that the free induction decay signal following the 90 ° pulse will dephase more rapidly and therefore be harder to see on the oscilloscope. The relaxation along the z axis will be essentially unchanged. With a modern 12-inch electromagnet, the field may be made strong enough to give a resonant frequency of more than 60 MHz for protons. A flux stabilizer is used to correct short-term variations in field strength. The probe is positioned until the 90 ° pulse (by itself) produces the longest free induction decay. The programmer may be designed to make the waiting period before the next 180 ° pulse automatically equal to a constant (e.g., 8) times the interval between 180 ° and 90 ° pulses in order for the spins to return to equilibrium as explained above. Sample. The optimum sample size depends in part on the geometry of the coils in the probe insert. For measurements of T1 and T2 of water protons a crossed-coil probe insert is preferable. With the Varian V4331 crossed-coil probe insert at 24.3 MHz, 0.1-ml samples are optimal since serious losses in signal are observed with smaller samples. With the NMR Specialties PR-04 crossed-coil probe at 24.3 MHz, sample volumes between 0.02 and 0.1 ml are suitable. For subsequent EPR studies, to determine the concentration of free Mn 2÷~ or free [nitroxide radical], 7 a sample of at least 0.03 ml is required. After measurements of the relaxation rates are complete, the sample is drawn up into a quartz capillary (1 mm I.D., 2 mm O.D.-Fig. 4B) for the EPR measurements. 2,7 Smaller samples decrease the signal level and larger samples may distort the appearance of a true null due to inhomogeneities in the 180 ° and 90 ° pulses. Standard NMR tubes have the appropriate geometry but are inconvenient for titrations. Small cellulose centrifuge tubes (e.g., Beckman Spinco No. 305528, 3//16 inch × 15/~ inch) are most suitable. A long Teflon rod split as shown (Fig. 4A) serves to close the tube and to permit rapid retrieval of the sample. Method. The nonspinning sample tube is positioned in the probe to optimize the amplitude of the signal following a single 90 ° pulse. After thermal equilibration, a 180 ° pulse followed after a time t by a 90 °

[29]

NUCLEAR RELAXATION MF_~SUREMENTS

665

Iv ~Teflon rod ~-J"O"ring

SeT

B

~ /Teflon cap ~( 1,-J .~ to fit EPR, ]~l~

~ 5 mm Cap plastti:bc:ntrifuge

!

--Sample (0.05 m l ) ~ --~

C " ~

5 mm

I ~

TMS reference

~-[-- ~'q]:~,-~Sample (0.4 ml) b u j ~ I ~ : J ~ l ~ 2mm

5 mm i

T-

10mm J__

Quartz tube (I mm I.D.,2ram O.D.) Sample(0.025 ml) Plug . olyethyleneluaing lion seal

/Screw thread ~Teflon holder

Sample (0.1ml)

Fro. 4. Sample tubes for relaxation rate and EPR measurements. (A) Celhdose acetate centrifuge tube (Beckman-Spinco No. 305528) for 1/7'1 and 1/% of water. (B) Quartz capillary for determining free [Mn] *÷ or free [nitroxide radical] by EPR. (C) Coaxial tube (Wilmad No. 520, No. 516-0) for relaxation rates by continuous wave methods. TMS, tet.ramethylsilane. (D) Microcell (Wilmad No. 529E) for relaxation rates by continuous wave methods.

pulse is applied. After at least 8t another 180°-90 ° sequence is applied with a different interval. This process is repeated until an interval is found such that the 90 ° pulse is followed by a null signal (Fig. 2). This interval, tn~l, can be read from the oscilloscope or from a timer which times the interval between the 180 ° and the 90 ° pulse. The longitudinal relaxation time, T1 = t,u~/in 2 = 1.443 tnull. In carrying out titrations, the tuning of the field to resonance should be examined after each addition of titrant. Since T1 can be determined more rapidly than T2, the former is a more convenient parameter to measure during titrations. Check o/Instrument. At 24.3 M H z and at 24 °, redistilled water and diamagnetic buffers equilibrated with air yield a I/T~ value of ~0.39 sec -~. A freshly prepared 10 -3 M aqueous solution (19.8 rag/100 ml) of reagent grade MnCI~.4H20 yields a 1/T~ value of 8.3 sec-1. The values of the relaxation rates of MnCl~ solutions at other frequencies are given by Nolle and Morgan. 2~ = A. W. Nolle and L. O. Morgan, I. Chem. Phys. 26, 642 (1957).

666

RESONANCE TECHNIQUES

[291

Carr-PurceU Pulsed Method ]or T2 24 Principle. The transverse relaxation time T2 is the time constant for the exponential decay of a magnetic moment preeessing in the x-y plane assuming a perfectly homogeneous magnetic field. The factors contributing to the true T._, decay are those intrinsic to the sample. In principle all that one should have to do would be to deliver a 90 ° pulse to the sample and observe the free induction decay. A well shimmed magnet capable of producing a continuous wave N M R spectral line width of 0.2 Hz would give an apparent T: of 1.6 sec for a sample possessing an infinitely long intrinsic T~. The reason for the excessively rapid free induction decay following a 90 ° pulse is that different parts of the sample experience slightly different field strengths so that the individual nuclear magnetization vectors precess at different frequencies. Therefore a single vector in the x-y plane soon breaks up into a "fan" of vectors in the x-y plane (Fig. 1C) and then cancels out and becomes undetectable by the receiver. In the Carr Purcell method as modified by Meiboom and Gill 2G the vectors are rebunched after they have fanned out. This is done bv delivering a 180 ° pulse whose rf phase has been delayed 90 ° with res ,ect to the original 90 ° pulse so that the fan of vectors is in effect flipped over on its back. It is therefore still in the x-y plane, but now the faster precessing vectors are placed behind the slower ones so that they can catch up (Fig. 1D). If the 180 ° pulse occurs r seconds after the 90 ° pulse, then the vectors will be rebunched at time 2r and an "echo" will be detected by the receiver. The form of the echo is that of two free induction decays placed back-to-back. Following the echo the vectors will again fan out, but may be rebunched by another 180 ° pulse. In Fig. 5 is shown a Carr-Purcell train of pulses in which the 90 ° pulse is followed by 180 ° pulses at times r, 3r, 5r, etc. The envelope of the echo heights decays exponentially with a time constant T2 unless diffusion or convection effects are too great. Diffusion

Sign°LK-I-'~.-k-x:-t--~_-L-~ 90 °

180 °

180 °

180 °

180 °

180 °

.

0

r

2r

.

.

.

3v ......

Time

FIG. 5. T: m e a s u r e m e n t with a Carr-Purcell pulse train. T h e height of the echoes following the 180 ° pulses d e t e r m i n e s an envelope decaying with a time c o n s t a n t T_,. ~ S . M e i b o o m and D. Gill, Rev. Sci. Instrum. 29, 688 (1958).

[29]

NUCLEAR RELAXATION MEASUREMENTS

667

can disturb the rebunching of the vectors in the x-y plane because a given precessing nucleus as it moves throughout the sample volume will experience a different field and therefore have a different precezsion frequency. The effect can be minimized by making the 180 ° pulses more frequent, or can be utilized to measure rates of water proton diffusion.24 With an unshimmed 12-inch magnet and a sample with very long T~ (benzene), we have measured an instrument-limited T._,of 2.8 seconds at 24.3 MHz. Instrument. The instrument is essentially unchanged from the configuration of the T1 experiment (Fig. 3), except that the programmer now puts out the gating wave forms for the Carr-Purcell train, i.e., a 90 ° pulse followed by a sequence of 180 ° pulses. To prevent drift of the field strength, it is mandatory to use an auxiliary ("piggyback") NMR probe which automatically adjusts the laboratory magnet field strength via the flux stabilizer to resonate its sample so that the analytical sample also automatically stays on resonance. Sample. The sampling is as described for 1/T1 measurements. Method. After tuning the field as described for T1, a 90 ° pulse followed by a series of 180 ° pulses is delivered to the sample. The receiver output can be photographed on Polaroid film from the oscilloscope. Enough 180 ° pulses should be included in the train for the decay of the envelope to fall to less than one-third of its maximum. The spacing of the 180 ° pulses should not be so great that the envelope of the echoes is poorly defined. Also, if diffusion makes the apparent T2 too short, the 180 ° pulse spacing should be shortened at least until the measured T2 reaches a maximum. The 180 ° pulse phasing should be adjusted to prevent even-numbered echoes being unequal to odd-numbered ones. The programmer can be in a single-sequence mode so that one can wait for the spins to relax (10 T_~ after the last 180 ° pulse is a conservative period). The camera shutter is then opened and the pulse train is initiated for the photograph. The oscilloscope should be triggered so as to show some baseline at the beginning of the trace. Calculation of T2 from Data. On the photograph of the train of echoes (Fig. 5), draw a smooth curve touching the tops of the echoes and draw a straight base line at 0 voltage. The horizontal axis of the oscilloscope must be properly calibrated. The calibration can be checked against the timer in the T1 measuring mode. Transfer the curve to semilog paper and fit a straight line to the points. Find any two ordinates having a ratio of e (= 2.718). The time interval between them is T2. Check o] Instrument. At 24.3 MHz and at 24 °, redistilled water and diamagnetic buffers yield a 1/T2 value of 0.55 sec-1. A freshly prepared 10-~M aqueous solution (19.8 rag/100 ml) of reagent grade MnCI~.4H.~O

668

RESONANCE TECHNIQUES

[29]

yields a 1/T2 value of 43.0 see-1. The values of the relaxation rates of MnC12 solutions at other frequencies are given by Nolle and MorganY 5

Continuous Wave Methods METHOD 1. DIRECT METHOD FOR T1

Principle. At high enough rf power an N M R resonance disappears due to equalization of the populations of nuclei with magnetic moments oriented against the laboratory magnetic field (excited state) and with the magnetic field (ground state). Hence no net absorption can occur. When the rf power is then diminished to a value well below saturation, the signal reappears at a rate which is equal to the rate of repopulation of the ground state. Hence the rate constant of this process is 1/TI. Instrumentation. A commercial N M R spectrometer equipped with field frequency lock is required, to remain at the center of the resonance for a time longer than T1. To monitor signal height as a function of time, an external recorder is most convenient, but the oscilloscope of the instrument and a camera may be used. Sample. Regular 5-mm N M R sample tubes are used. For external locking, a separate piggyback probe is most convenient. If this is not available, locking on a separate sample in a coaxial tube is suitable (e.g., Wilmad No. 520 inner tubes and No. 516-0 outer tubes). The reference liquid for locking (e.g., tetramethylsilane) is placed in the inner tube (Fig. 4C). Evaporation of the tetramethylsilane may be prevented by sealing the inner tube with epoxide. With these coaxial tubes, 0.4-ml sample volumes of fairly concentrated solutions (>_0.05 M protons) are required for 1/T~ measurements because one is observing with the rf field well below saturation. The sample volume may be reduced by 4 and the signal to noise only halved by using a cylindrical microcell (Fig. 4D) (e.g., Wilmad No. 529E). 27 However, this method requires an internal standard for locking. Keeping the field sweep fixed at the center of the resonance to be measured, one raises the rf power beyond saturation until the signal disappears. Recording externally at an appropriate rate, one quickly decreases the rf power to a value of 10 dB below saturation and records the time course of reappearance of the N M R signal. For narrow lines one may repetitively sweep through the resonance before and after saturation and desaturation, using the internal sweep control of the spectrometer, and monitor the reappearance of the N M R signal with an external recorder. A semilogarithmic plot of (h~ - h) against time after desaturation is then made, where h is the 2~j. j. Villafranca, unpublished observation, 1971.

[29]

NUCLEAR RELAXATION MEASUREMENTS

669

signal height at time t, and h~ is the signal height asymptotically approached at infinite time. A straight line is fitted to the 1)oints and T1 is the time elapsed between the occurrence of a value ( h ~ - h) and a value (h~ - h)/2.718. The direct method is satisfactory for T1 values ranging from 0.1 seconds to 10 seconds. METHOD 2. PROGRESSIVE SATURATION METHOD FOR T1

Principle. From the steady-state solution of the Bloch equations, under conditions of slow passage through the resonance frequency (vo), the signal height is given by: (o~1)T2 Signal height a 1 + 47r2T22(~0 -- v)2 + (¢ol)~T1T2

(14)

In Eq. (14) T1 and T~ have their usual meanings, and 0)1 is the frequency of the applied rf fi'eld (yH'H1): the height of the peak at the center of the resonance (v = vo) is the following function of ,o1: wiT2

(signal height)voa 1 + ~2T~T:

(15)

The peak height increases linearly with 0)1 when o)12~ 1/T, T2, reaches a maximum when 0)1~ = 1~TIT2 and increases linearly with 1/(~t when 0)12>> 1~TIT2. The determination of 1/T~ is done in three stages: (1) calibration of the attenuator of the N M R spectrometer in terms of 0)1; (2) determination of the value of 0)1 at which the signal is a maximum [i.e., 0)1" (at saturation) = 1~TIT..,]; (3) determination of 1/T., (e.g., from the line width at rf power -->5 dB below saturation). Hence, 1/T~ = 0) 2 (at saturation).T._,. Instrumentation. Most commercial N M R spectrometers are suitable for power saturation studies. Attenuation of the rf field 0)~ with an error --<0.6 dB is required. The attenuators supplied with certain instruments (e.g., the Varian XL-100 and the Jeolco C-60H) are not accurately calibrated, but can be replaced. ,Sample. The sampling techniques as described for the direct method are satisfactory. Method. (i) CALIBRATIONOF THE RADIO FREQUENCY (0),) FIELD. Obviously, from the relationship 0)12 (at saturation) = 1/T1T,, the value of 0)1 can be determined by measuring the rf power at saturation of a sample of known T1 and T2.~1 A direct method of measuring the intensity of the rf field has been described. 2s If an intense rf field is suddenly applied to a sample having a single narrow line, the nuclear magnetization will nutate away from "~J. S. Leigh, Jr., Rev. Sci. Instrum. 39, 1594 (1968).

670

RESONANCE TECHNIQUES

[29]

the field direction of the laboratory magnet, exactly as in a pulsed N M R system. For as long as the rf field is present, the magnetization will continue to nutate so that the angle it makes with the z axis starts from 0 °, goes through 90 °, 180 °, 270 °, 0 °, etc. The result is a fluctuating signal ("ringing") in the receiver as the component of magnetization in the x - y plane varies. The frequency of this ringing, which is typically in the subaudio range ( ~ 2 Hz) is, to a good approximation, ~1/27r at the given setting (dB) of the attenuator. The value of ~'1 at any other attenuator setting (dB') is described by the formula 20 loglo oJ1/o~'1 = dB -- dB'

(16)

The practical application of this principle depends on the spectrometer accurately remaining on a narrow resonance peak for a period of several seconds even though a powerful rf field is present which may disturb the circuits used to stabilize the magnet field. One lock system which is insensitive to this disturbance is that using an auxiliary, "piggyback" probe for its field locking signal. Another good alternative is heteronuclear internal lock where the stabilizing N M R signal is derived from the same sample, but a different nuclear species (e.g., deuterium). If one is forced to use homonuclear internal lock then the rf power used in the locking channel may approach that used in the observing channel. Ordinary tetramethylsilane (TMS) may present difficulties for locking in this case because it will saturate at high rf levels. If such difficulties are encountered, locking should be done on a line which is much broader (5-10 Hz) than the line which is observed, such as a sample of water broadened by the addition of Fe(NO:~).~ (0.5-1.0 mM) in 0 . 2 M HNO~ using coaxial tubes (Fig. 4C). Using such tubes one can then observe the ringing of any narrow resonance such as that of acetone or benzene. If coaxial tubes are not used, one may mix two samples such as t-butyl alcohol and water containing Fe(NO:e):~ as above. The resulting subaudio signal may be put onto a chart recorder or recorded by camera from an oscilloscope trace. If an oscilloscope is used, a convenient time calibration is 60 Hz line frequency recorded as a separate trace on the same film. (ii) DETERMINATION OF ¢o'1, THE RADIO FREQUENCY AT WHICH THE SIGNAL IS MAXIMAL. The resonance is scanned slowly (no ringing; sweep rate. TiT2 ~ 0.06) 29 and repeatedly at rf powers varying the attenuation by 5 dB until the approximate dB value at saturation is located. This region is examined more closely (e.g., every 1 or 2 dB). The line heights are then measured and plotted against dB attentuation on 29R. R. Ernst and W. A. Anderson, Rev. Sci. Instrum. 36, 1696 (1965).

[29]

NUCLEAR RELAXATION MEASUREMENTS

671

semilog paper. The peak location is determined by fitting the theoretical curve of Eq. (15) to the data with a plastic template 9,3° or by a computer. The value of the attentuation at the peak (dB') is converted to ~'~ by Eq. (16). For least-squares computer fitting or visual fitting, a linear form of Eq. (15) has been derived31: o:1 = aT~(~l) 2 + a/T2 (signal height) ~o

(17)

Hence after determining ~1 for each dB setting using Eq. (16), a plot of ~l/(signal height) against ~12 is made and the intercept on the ~2 axis is ~,2. (iii) DETERMINATION OF T~. Since the data have already been collected, the most convenient way of determining T.~ is from the line widths at rf powers at least 5 dB below saturation (see next section). METHOD 3. LINE-WIDTH METHOD FOR T2 Principle. From Eq. (14) the full width of a resonance line at half height is

(Width) = (1 + ~2TIT2) 1/2 7rT~

(18)

Hence at rf powers well below saturation (~1'-'~ 1~TIT2), v(width) = 1/T~_ while at saturation ( ~ = 1/T~T2), v(width) = ~/'2/T2. Instrumentation. Most commercial high resolution NMR spectrometers capable of resolving resonances of ~0.5 Hz are suitable for measurements of line width. Sampling techniques are as described above for T~. Method. Passage through the resonance should be slow enough to avoid ringing. Even if one is measuring only line widths, a power saturation study should be carried out to make certain one is well below saturation. Although rf powers of one-tenth (i.e., 10 dB below) saturation are recommended for line width measurements, we see little effect on the line widths of protons and improvements in signal-to-noise ratios at power levels of one-third (5 dB below) saturation. Spectra should therefore be run at many rf powers from 20 dB below to 10 dB above saturation and the lines measured at rf power levels ~ 5 dB below saturation are utilized to measure the line width, provided no systematic broadening can be detected between 5 and 10 dB attenuation. If measured at saturation, the line width should be corrected by the factor V/2. ,o j . S. Leigh, Jr., Ph.D. Dissertation, University of Pennsylvania, Philadelphia, 1970. ,Ij. Reuben, D. Fiat, and M. Folman, J. Chem. Phys. 45, 311 (1966).

672

RESONANCE TECHNIQUES

[29]

To measure a paramagnetic contribution to a line width, one subtracts 1/T2 observed in the presence of all diamagnetic components from 1/T2 observed in the presence of all components including the paramagnetic species. OTHER METHODS Other continuous wave methods suitable for measuring 1/T~ include the reversal of polarization 32 and spin locking or adiabatic half passage? ~ The latter method r-' and the method of adiabatic rapid passage 33 have been used for measuring 1/T~. These methods have been reviewed elsewhere2 Use of Nuclear Relaxation D a t a

Water Relaxation Rates The enhancement factors of the paramagnetic contribution to relaxation rates of water are defined by Eqs. (6) and (7). As previously shown for macromolecular complexes of manganese(II) 2,5 or nitroxide radicals, 7 the observed enhancement of the relaxation rate of water is the weighted average of the enhancements of all complexes of the paramagnetic species: ~* = ~xi~i

(19)

where X~ is the fraction of the ith complex of the paramagnetic species and ** is its enhancement. Although the following equations are expressed in terms of Mn, they are applicable to other paramagnetic ions and to paramagnetic substrate analogs.

Determination o] Enhancement, Stoichiometry, and Dissociation Constants o] Binary Complexes of Enzymes with Paramagnetic Species. For a binary complex of Mn with an enzyme, Eq. (19) becomes .

[Mn]f

[Mn]b

(20)

= [Mn]------~~ + [-[-M--~t~b

where the subscripts f, b, and t refer to free, bound, and total Mn, respectively. The enhancement of free Mn (~f) is 1 by definition, and [Mn]b = [ M n ] t - [Mn]~. With these substitutions, Eq. (20) can be solved for [Mn]f and [Mn]b in terms of [Mn]t, ~'*, and the enhancement Eb of the Mn bound in the binary complex. 3:L. E. Drain, Proc. Phys. Soc. (London), Set. A, 6~, 301 (1949). 33E. E. Salpeter, Proc. Phys. Soc. (London), Set. A, 63, 337 (1960).

[29]

NUCLEAR R E L A X A T I O N MEASUREMENTS

673

[Mn]~--- \ ~ b -

1] [Mnlt

(21)

[Mn]b ---- \ e b -

1] [Mn]t

(22)

The dissociation constant K . of the binary Mn-enzyme complex is defined as KD --

[Mn]f[Sites]f [Mn]f(n[E]t- [Mn]b) [Mn]s = [Mn]~

(23)

where n is the number of sites per enzyme molecule with dissociation constant Ko and [E]t is the molar concentration of the enzyme. From Eqs. (21)-(23) we m a y write: KD = (~b -- e*)(n[E]t(~b -- 1) -[- [Mn]t(1 -- ~*))

(24)

(~*- 1)(~b- 1) When a single class of binding sites exists, titration data of ~ as a function of [E]t and [Mn]t m a y be used directly with Eq. (24) to solve analytically for n, .Cb, and K~. With multiple classes of binding sites a stepwise analysis is necessary. = A computer program for such data processing which utilizes each point has been written. 34 The following graphical and analytical methods m a y be used. (i) DETERMINATIONOf Cb. A good approximation of ~b for the tightest binding sites can be found by titration of the metal with enzyme and extrapolation of a plot of 1/~ ~ vs. 1 / [ E ] t to infinite enzyme concentration (type 1 titration).2 Cb for several types of sites m a y be approximated by titration of the enzyme with the metal and extrapolation of a plot of 1/c ~ against [Mn]t to zero metal. This gives a lower limit to ~b (type 2 titration).2 The simplest and most direct method for evaluating ~b is to measure [ M n ] J [ M n ] t independently by E P R at each point of type 2 titration. 2 With Mn metalloenzymes (i.e., K . ~ 10-T M) ~b may be directly determined since ~ = Oh.a5-87 However, a small diamagnetic correction to I / T , should be made due to the effect of the protein. The apoenzyme or the enzyme complex of a diamagnetic metal provides the appropriate correction. (ii) DETERMINATIONOF n AND KD. When ~b is known or has been approximated it may be used to determine [Mn]f and [Mn]b at each point ~4G. H. Reed, M. Cohn, and W. J. O'Sullivan, J. Biol. Chem. 245, 6547 (1970). UA. S. Mildvan, M. C. Serutton, and M. F. Utter, J. Biol. Chem. 241, 3488 (1966). W. J. Ray, Jr., and A. S. Mildvan, Biochemistry 9, 3886 (1970). ~ A. S. Mildvan, R. D. Kobes, and W. J. Rutter, Biochemistry 10, 1191 (1971).

674

RESONANCE TECHNIQUES

[29]

of a type 2 titration curve using Eqs. (21) and (22). The data may then be plotted as a Hughes-Klotz plot of [ E ] t / [ M n ] b vs. 1 / [ M n ] f or preferably 3s as a Scatchard 3'~ plot of [Mn]~/[E]T[Mn]~ vs. [ M n ] b / [ E ] t to determine n and KD graphically. T y p e 1 titration curves are generally unsuitable for such analyses. Difficulties arise when there are more than one class of binding site or when cb changes due to site interaction as several sites with the same affinity become occupied. For such cases independent determinations of [Mn]~ by another method (e.g., E P R ) are essentiaF ,~° and iterative procedures to fit the data are required. 2,4° Determination o] Enhancement Factors and Dissociation Constants of Ternary Complexes. In the presence of an enzyme E and a substrate S the observed enhancement of the effect of Mn on the relaxation rate of water protons from Eq. (19) is: . [Mn]f [MnS] [EMn] [EMnS] c = [Mn]-----~+ ~ ca + ~ cb + [Mnl----Tcr

(25)

where Ca and cv are the enhancements of the binary Mn-substrate complex, and the ternary enzyme-Mn-substrate complex, respectively? In addition to the dissociation constant (Ko) for the binary E - M n complex, defined by Eq. (23), five additional dissociation constants must be defined to describe the system: K1 K2 K3 K'a

= = = =

[Mn][S]/[MnS] [E][MnS]/[EMnS] [EMn][S]/[EMnS] [ES][Mn]/[EMnS]

K s = [E][S]/[ES]

(26) (27) (28) (29) (30)

From the definitions of these constants it can be shown that K1K2 = K3KD = K ' A K s

(31)

Substituting the dissociation constants into Eq. (25), one obtains Eq. (32) relating the observed enhancement d to the concentrations of free enzyme and substrate: K3KD =

KsKD

K2

K3 K2

K~

(32)

[E][S----] + [-~ + ]-~ + 1 B y separate experiments, as described above, the dissociation constants ~D. A. Deranleau, J. Amer. Chem. Soc. 91, 4044, 4050 (1969). G. Scatchard, Ann. N.Y. Acad. Sci. 51, 660 (1949). 4oj. j. Villafranca and A. S. Mildvan, J. Biol. Chem. 246, 5791 (1971).

[29]

NUCLEAR RELAXATION MEASUREMENTS

675

(K1, Ko) and enhancements (*a, ~b) of the binary Mn-S and E-Mn complexes are found. To determine the dissociation constants (K2, K3) and enhancement (~T) of the ternary complex, titrations are carried out measuring c'* at constant [Mn] but with variable [E] and IS]. At least 5 concentrations of [E] and [S] are required, most of which should exceed [Mn] in concentration.5 From these titration data, K2, K3, and ~Tmay be roughly approximated by graphical methods based on simplifications of Eq. (32) at high [E] and [S].° In cases involving E-S-M complexes in which ~ > ~b 6 and in certain E-M-S complexes when KI <_Ko,K.~, the graphical methods have yielded reasonable approximations of the dissociation constants but unsatisfactory approximations of cT.34 For all cases a computer fit of the titration data to the theoretical curve which has been devised is essential (Fig. 6).34 In some cases in which *T < eb and K3 < K1 graphical analyses of data have yielded dissociation constants which are in good agreement with constants obtained by kinetic studies ~ or by other binding studies, 41 and c, values can often be measured directly and reliably as end points in the titration curve2 ,4~ However, in all cases the graphical procedures are useful mainly in providing initial values for K2, K3, and cT for further iteration by the computer. Three graphical procedures which are useful for cases in which cT < cj, have been described in detail2 Here they are summarized. In procedure I, the value of c at a fixed enzyme concentration and infinite substrate concentration, denoted by ~ , is obtained by extrapolating to infinite substrate concentration in a plot of ~ against 1/[S]T. A secondary plot of 1/C~c against l / [ E ] t yields the value of cT at infinite enzyme concentration and K2 from the slope of the plot. Because K~ and Ko are known quantities, K.~ may be evaluated from Eq. (31) above. In procedure II, the PRR titration data are analyzed by reversing the order of extrapolation. The value of ~* at infinite enzyme concentration is obtained by plotting 1/~~ against 1/[E]t; this value is denoted by *¢*d.The values of c~ at different substrate concentrations are then extrapolated in a secondary plot to infinite substrate concentration to obtain eT. K~ is approximated by the slope of the secondary plot. Procedure III leads only to an evaluation of K.~ which may be equated with the value of the concentration of free substrate which produces half-maximal change in enhancement at saturating enzyme concentrations. The value obtained at several enzyme concentrations may, if neces4~R. S. Miller, A. S. Mildvan, I-I. C. Chang, R. L. Easterday, H. Maruyama, and M. D. Lane, J. Biol. Chem. 243, 6030 (1968). 42T. Nowak and A. S. Mildvan, J. Biol. Chem. 245, 6057 (1970).

676

RESONANCE TECHNIQUES

STEP 1

[29]

Read in all known equilibrium constants and enhancement factors. Read in all experimental data -- observed enhancements as functions of f o r m a l concentrations of components

STEP 2

Make an initial guess at K ?

STEP 3

With this trial K? and other known equilibrium constants solve numerically simultaneous equilibria to obtain concentration of all species

STEP 4

Insert these concentrations (for each data point) and known enhancement factors into the observation ZMi equation, e* = × el, and calculate and e ? i MT

STEP 5

Calculate the average ~ ? and % relative standard deviation of e ?'s

STEP 6

Adjust K? and repeat steps 3-5 until minimum % relative standard deviation of ?'s is obtained

FIO. 6. General scheme for computer analysis of P R R titration data for an unknown equilibrium constant, K?, and enhancement factor, ~?, generously provided by Dr. George Reed. See also G. H. Reed, M. Cohn, and W. J. O'Sullivan, J. Biol. C h e m . 245, 6547 (1970). sary, be e x t r a p o l a t e d to infinite e n z y m e c o n c e n t r a t i o n . W h e n a p p r o x i m a t e v a l u e s for K2, K~, a n d ~T h a v e been o b t a i n e d , t h e y should be i t e r a t e d to y i e l d m o r e precise v a l u e s b y a c o m p u t e r p r o g r a m of t h e t y p e o u t l i n e d in Fig. 6. 8~ A n e n l a r g e d c o m p u t e r p r o g r a m to deal w i t h t e r n a r y an d higher complexes a t m u l t i p l e sites has r e c e n t l y been w r i t t e n . 42~ 42. j. p. Slater, I. Tamir, L. A. Loeb, and A. S. Mildvan, J. Biol. C h e m . , submitted.

[29]

NUCLEAR

RELAXATION

677

MEASUREMENTS

Provisional Determination o] Coordination Scheme o] Ternary Complexes. As pointed out previously 9,4'~ for a 1:1:1 complex of an enzyme, metal, and substrate, four coordination schemes are possible: M E--S--M

E--M--S

E

M--E--S

\ s

These are the substrate bridge complex, the metal bridge complex (simple and cyclic), and the enzyme bridge complex. Water relaxation enhancement data provide a useful provisional means for determining the coordination scheme. 44 In addition, the dissociation constants from enhancement data provide an independent provisional method. Table II summarizes the expected enhancement and thermodynamic behavior for each of the coordination schemes. As pointed out elsewhere, occasional exceptions to the relationships in Table II are found. 9,37,4'~ Hence a confirmatory method such as the measurement of substrate relaxation rates is required to establish the coordination scheme. Water Proton Exchange Rates on Paramagnetic Metals. From Eqs. (1), (4), and (5), since 1/To.,. is usually small, the exchange rate of water protons (q/rM) is related to the relaxation rates as follows:

q/r~ >_ 1/pT~., >_ 1/pT1,

(33)

TABLE II CORRELATION OF ENHANCEMENT AND THERMODYNAMIC BEHAVIOR OF TERNARY COMPLEX OF ENZYME ( E ) M E T A L (M) AND SUBSTRATE (S) WITH THE COORDINATION SCHEME a

Coordination scheme

Substrate bridge

Metal bridge

/ F~--S--M

Enzyme bridge M

E--M---S E

M--E---S

\ s

Enhancement behavior Thermodynamic data

eb ~ eT

eb ~> eT

K l ' ~ 10-3M

K s - - ~¢ Ks < ~

K.. ~ Ks

Ks ;> Ka

eb ~-- er

Ks < K3 = Ks

a Adapted from A. S. Mildvan, in "The Enzymes" (P. D. Boyer, ed.), 3rd ed., Vol. II, p. 445. Academic Press, New York. 4SA. S. Mildvan, in "The Enzymes" (P. D. Boycr, cd.), 3rd ed., Vol. II, p. 445. Academic Press, New York, 1970. ~ M. Cohn and J. S. Leith, Jr., Nat~re (London) 193, 1037 (1.962).

678

RESONANCE TECHNIQUES

[29]

Hence the measured paramagnetic contributions to the relaxation rates set lower limits on the water proton exchange rates. Frequently the water exchange rate is slow enough such t h a t the transverse or even the longitudinal relaxation rate measures q/TM. AS pointed out elsewhere, such a relationship m a y be established by measuring the temperature and frequency dependence of the relaxation rate." If the relaxation rate is limited by chemical exchange it should typically have a positive temperature coefficient with an energy of activation > 3 k c a l / m o l e and should be independent of frequency. The q/rM for protons need not be a measure of the exchange rate of the entire water molecule. Although the q/rM for protons in M n ( H 2 0 ) 6 -~÷ measures the exchange rate of entire water molecules, TM this is not generally true. Thus in Fe(H..,O)6 :~+45 and in metmyoglobin 4s the proton exchange rate is much faster than the water exchange rate. In such cases measurements of the relaxation rates of ~O in H._,O m a y be used to measure the ligand exchange rate. 14

Determination of the Coordination Number (q) ]or Water on ~nzyme-Bound Paramagnetic Metals. When water exchange (q/rM) is rapid compared to the longitudinal relaxation rate of coordinated water ligands (q/T~r~) and the outer sphere contribution to the relaxation rate is small, Eqs. (4) and (12) simplify to: 1/pT~p = q/T~M = ]-~ q

re

_1 + ~2r¢~]

(34)

Equation (34) contains the measured quantity 1/pT,, and three "unknown" quantities q, r~, and r. For Mn(H~O) 2÷ and F e ( H 2 0 ) ~+ complexes the metal to proton distances, r, can be calculated from crystallographic data to be 2.87 __. 0.05/~* and 2.45 +_ 0.08 £, respectively. .7 I f ~ can be estimated from frequency dependence *,*s or by temperature dependence 47 the value of q can be determined.

Relaxation Rates of Substrates and Inhibitors Confirmation of Coordination Scheme.s. An enhanced paramagnetic effect of an enzyme-bound metal ion such as M n -~÷ on the relaxation rates of a substrate m a y be used to establish an E - M - S coordination scheme. 2~,.3 As pointed out below, distance calculations should be made 45Z. Luz and R. G. Shulman, J. Chem. Phys. 43, 3750 (1965). 46A. S. Mildvan, N. M. t~umen, and B. Chance, in "Probes of Structure and Function of Macromoleeules and Membranes," Vol. 2 (B. Chance et al., eds.), Part 2, p. 205. Academic Press, New York, 1971. 4,j. j. Villafranc~ and A. S. Mildvan, J. Biol. Chem. 246, 772 (1971). *SA. R. Peacocke, R. E. Richards, and B. Sheard, Mol. Phys. 16, 177 (1969).

[29]

NUCLEAR RELAXATION MEASUREMENTS

679

since enhancements can occur even without direct coordination, due to large increases in re. Conversely a deenhanced or negligible paramagnetic effect of an enzyme-bound metal ion on a substrate or ifihibitor in the presence of independent evidence of ternary complex formation indicates an M-E-S complexY1,4~ An exception to this relationship can occur when the structure of the M-S complex is very different on the enzyme than in free solution. EPR, but not nuclear relaxation, may be used to establish a substrate bridge complex (E-S-M). 44 Distances in Binary and Ternary Complexes. The dipolar contribution to the longitudinal (1/T1M) or transverse (1/T2~) relaxation rate of an atom of a coordinated ligand can be used to calculate the distance r between the paramagnetic center and the atom2 ,-~1,49 For metal-ligand interactions the dipolar contribution to the longitudinal relaxation rate is easier to obtain since the hyperfine term in Eq. (10) is small. Hence the measured value of 1/pT~p may be used to determine distance when it is equal to q/TIM or 1/T~M for a 1:1 complex. When 1/pT~p is significantly less than 1/pT,~ the equality of 1/pTlp and I/Ti~ is a valid assumption. When 1/pTl, = 1/pT._,p two alternative possibilities exist. Both relaxation rates may be equal to the ligand exchange rate (1/TM) or to the outer sphere term (1/T .... ), and a temperature study is required to distinguish between these possibilities. In the former case 1/pTlp increases and in the latter case it decreases with increasing temperature. When 1/pT~p measures the rate of ligand exchange, it is less than 1/T~M and may be used to calculate an upper limit to r2 When 1/pT~p is dominated by outer sphere relaxation, as evidenced by a small inverse temperature dependence (Ea ~ 3 kcal/mole) and equality to 1/pT2p, it cannot he used to calculate inner sphere distances, but may be used to calculate distances of closest approach of noncoordinated ligands.l~, 46 A general equation for the calculation of r from a paramagnetic center to a magnetic nucleus from T~M of that nucleus, obtained from the dipolar term of Eq. (10), is r(in A) = C T1M i -]- (3.94) (1013)~2rc~ + 1 -t- (1.71-~-10~9)~rcVJ

(35) In Eq. (35) ,~ is expressed in NMR frequency units in Megahertz (e.g., 24.3 MHz, 60 MHz, 100 MHz, etc.); the correlation time ro and the coefficient C depend on the nature of the paramagnetic center. The coefficient C also depends on the nature of the magnetic nucleus under,9A. S. Mildvan and M. C. Scrutton, Biochemistry 6, 2978 (1967).

680

RESONANCE TECHNIQUES

[29]

TABLE III C VALUES AND CORRELATION TIME 1"c FOR DISTANCE CALCULATIONS USING

EQ. (35) Range of ~o values Paramagnetic center

No. of C value unpaired for electrons protons ~

Nitroxide radicals Neutral flavin radical Cu 2+ Ni 2+ (high spin) Co ~+

1 1 1 2 3

539 539 539 635 705

Fe ~+ (high spin) Fe 3+ (high spin) Mn 2+ (high spin)

4 5 5

763 812 812

T~ (sec) 10-8 10-s 10-8 _<_3 X 10-12 5 X 10-13 to 10-11 1.5 X 10-12 10-11 to 10-l° 3 X 10-g

r~ b (sec) 10-12 to 10 -H to 10-11 to 10 -11 to 10-11 to

10-8 10-10 10-s 10-8 10-s

10-11 to 10-s 10-11 to 10-s 10-11 to 10 -8

References

c-f g,h i-k 1

m,n j o p

a For 19F, zip, and 13C, the C value should be multiplied by 0.980, 0.740, and 0.631, respectively. b Lower limit (which may be 10-12 second for nonrigid ligands) is tumbling time in absence of macromolecule. Upper limit is value typical for a ligand rigidly bound to a macromolecule of M W ~105 or highest measured value for re. c A. S. Mildvan and H. Weiner, Biochemistry 8, 552 (1969). A. S. Mildvan and H. Weiner, J. Biol. Chem. 244, 2465 (1969). • j. S. Leigh, Jr., Ph.D. Dissertation, University of Pennsylvania, Philadelphia, 1970. f M. Cohn, Quart. Rev. Biophys. 3, 61 (1970). g A. Ehrenberg, Ark. Kemi 19, 97 (1962). h G. Palmer and A. S. Mildvan, Struct. Funet. Oxidation-Reduction Enzymes Proc. Int. Wenner-Gren Syrup. 1971. i T. J. Swift and R. E. Connick, J. Chem. Phys. 37, 307 (1962). i j. Eisinger, R. G. Shulman, and B. M. Szymanski, J. Chem. Phys. 36, 1721 (1962). k B. P. Gaber, W. E. Schillinger, S. H. Koenig, and P. Aisen, J. Biol. Chem. 245, 4251 (1970). Z. Luz and S. Meiboom, J. Chem. Phys.'40, 1066 (1964). '~ Z. Luz and S. Meiboom, J. Chem. Phys. 40, 1058 (1964). " M . E. R. Fabry, S. H. Koenig, and W. E. Schillinger, J. Biol. Chem. 245, 4256 (1970). o A. Wishnia, J. Chem. Phys. 32, 871 (1960). M. Tinkham, R. Weinstein, and A. F. Kip, Phys. Rev. 84, 848 (1951). g o i n g r e l a x a t i o n . T a b l e I I I g i v e s v a l u e s of C a n d Tc for v a r i o u s p a r a m a g n e t i c c e n t e r s a n d n u c l e i . T h e v a l u e s of rc a r e seen to v a r y o v e r o n e t o f o u r o r d e r s of m a g n i t u d e . F o r s m a l l c o m p l e x e s of p a r a m a g n e t i c m e t a l s one m a y t a k e as re t h e v a l u e o b t a i n e d for t h e M - H 2 0 i n t e r a c t i o n in t h e s a m e c o m p l e x . T h e re v a l u e of w a t e r p r o t o n s in t h e s a m e c o m p l e x m a y b e c a l c u l a t e d f r o m 1 / p T l p of w a t e r p r o t o n s w i t h a n a p p r o p r i a t e v a l u e of t h e M - p r o t o n d i s t a n c e (2.87/~ for M n 2+, 2.45 A f o r F e "~+) a n d a v a l u e of 2 - 4 for t h e n u m b e r of c o o r d i n a t e d w a t e r l i g a n d s . F o r e x a m p l e , in

[29]

NUCLEAR RELAXATION MEASUREMENTS

681

the case of manganese complexes with small nonrigid ligands, To(MnH20) ~ 3 × 10-11 second. Fe 2÷, Co 2÷, and Ni 2÷ have very short electron spin relaxation times (Ts ~ 10-12 second) which serve as the correlation time (Table I I I ) . In enzyme systems, as discussed above, Tc increases. If rc has not been determined independently, a reasonable alternative is to set upper and lower limits on To. Since the calculated distance is proportional to (re)l/+, the value of r is not too sensitive to the value of r~, and one can assume a generous range of values. The value of re (M-ligand) in the absence of the enzyme obtained as described above provides, a lower limit to T¢(Mligand) on the enzyme. An upper limit of r¢ for the enzyme-bound Mligand interaction is estimated by using T¢ of water protons in the same complex which may be calculated from 1/pTlp of the water protons with an appropriate value of the M-proton distance and for the number of coordinated water ligands. The value of T~ can be obtained directly by studies of the frequency dependence of Tim4,+s or in certain cases with Mn 2÷, Cu 2÷, or radicals, its lower limit can be estimated by the line width of the E P R spectrum of the bound paramagnetic species. 9,5° For nitroxide radical interactions with water or with substrates on enzymes, since there is no chemical bonding, the hyperfi'ne coupling constant A may be assumed to be ~ 0 . s Hence there is only a dipolar contribution to both 1~Tim and 1/T+_m, and 1/pT2p, if not limited by chemical exchange, may also be used to calculate r in Eq. (36).

I(

r(in A) = C , ,T2M 2r~ ~- 1 + 3.94(10")v,'v¢ 2 + 1 + l ~ ~ x 2 r o s / J

11, ' (36)

where C and re are as given in Table III. When T1, and T2p have been determined, and are neither limited by chemical exchange nor contain a T~ ~"

(1.96)(10 -7) .]T__~M 7 vI

~T2M

6

(37)

hyperfine contribution, their ratio m a y be used to evaluate ~¢ from the relationship, where v~ is expressed in Megahertz. Determination o] the Hyperfine Coupling Constant. For metal corn= plexes in which re ~ 1/v~, the values of 1/pT2, and 1/pTip m a y be used to estimate (A/h) the hyperfine coupling constant (in frequency units) from the relationship 16,+9,5~ M. Cohn, Quart. Rev. Biophys. 3, 61 (1970). M. C. Scrutton and A. S. Mildvan, Arch. Biochem. Biophys. 140, 131 (1970).

682

RESONANCE TECHNIQUES

1

pT2p

7

6pT1, ~- ll4re(A/h)2

[30]

(38)

where re, the correlation time for hyperfine interaction, is in the range given for rc in Table III. While the detection of superhyperfine splitting in the EPR spectrum or the detection of an NMR contact shift provides a more accurate method for determining the value of A/h, the qualitative detection of hyperfine coupling by nuclear relaxation is useful since it provides independent evidence for direct coordination of a ligand by a paramagnetic metal.

[30] Magnetic Susceptibility By TETSUTARO IIZUKA and TAKASHI YONETANI The magnetic susceptibility of a substance x is defined as the ratio of the magnetization of the substance M to the magnetizing force or the strength of the external magnetic field H to which it is subjected:

x = M/H

(1)

Principles and methods of magnetic susceptibility measurements have been described in a number of physics books. Reference books by Selwood1 and Bates 2 give a comprehensive survey of the subject. Two major methods have been applied in the measurement of magnetic susceptibility: (1) Force Method and (2) Induction Method. The former measures the ponderomotive force exerted by a magnet on a substance brought into the magnetic field gradient. The latter method measures the change of the electromotive force, which is induced in an alternating magnetic field. Magnetometoric studies of biological substances have been almost exclusively confined to the measurements of paramagnetic susceptibility. Free radicals and transition-metal ions are the origin of paramagnetism in biological substances. Since the content of paramagnetic centers per unit mass is;yery low in biological compounds and preparations of pure biological compounds in large quantities are generally difficult, highly sensitive magnetometers are required for the paramagnetic susceptibility measurement of biological substances. The force methods 1p. W. Selwood, "Magnetochemistry." Wiley (Interscience), New York, 1956. ~L. F. Bates, "Modern Magnetism." Cambridge Univ. Press, London and New York, 1951.