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An automated method to find the proper modulation phase in continuous wave EPR spectroscopy
JOURNAL
OF MAGNETIC
RESONANCE
80,493-50
1 ( 1988)
An Automated Method to Find the Proper Modulation Phase in Continuous Wave EPR Spectroscopy F. P. AUTERI,*
A. H. BETH,?
AND B. H. ROBINSON*$
*Department ofchemistry University of Washington, Seattle, Washington 98195, and TDepartment of Molecular Physiology and Biophysics, Vanderbilt University Medical Center, Nashville, Tennessee 3 7232 Received January 11, 1988; revised March 8,1988 Modulation of the dc magnetic field is universally employed in the acquisition of continuous wave electron paramagnetic resonance spectra, which includes linear and saturation transfer (ST) EPR spectra. Modem two-channel lock-in analyzers produce two components which are at angles $ and 4 + 90’ with respect to the modulation reference. Experimentally it is often desired to choose I$ such that the quadrature signal is “minimized,” so that the in-phase signal exhibits the maximum signal-to-noise ratio. This is called phase nulling. In this paper we provide a quantitative definition to the concept of a “minimized quadrature signal,” using a least-squares criterion, and derive a formula which enables computation of 4 which is then used to generate proper in-phase and quadrature signals. Furthermore, an expression is derived for the error in Q which is of particular interest when acquiring STEPR spectra. The formulas are noniterative and can be easily programmed to rapidly and automatically phase null signals alter acquisition. @ 1988 Academic Press, Inc.
INTRODUCTION
To improve the signal-to-noise ratio of cw EPR spectroscopy, the dc magnetic field is modulated at frequency V, , usually 100 kHz, which is called Zeeman modulation. The signal is then fed into a lock-in amplifier which removes the modulation carrier and produces a time-independent signal. For linear cw EPR, the choice of a proper phase between the driving modulation of the dc magnetic field is easily obtained visually. Once set, at 100 kHz, it rarely requires adjustment. However, with the advent of STEPR, the choice of phase becomes critical. Typical STEPR spectra are quadrature spectra under saturation conditions. Quadrature signals nearly vanish when the power is low enough that the response is linear, but grow rapidly when the microwave radiation incident on the sample causes partial saturation of the spin system (1-4). Even under partially saturating conditions, however, the in-phase signal is much larger than the quadrature signal, thereby necessitating accurate determination of 4 to avoid mixing of the large in-phase signal with the desired quadrature signal. Furthermore, phase settings change as one changes modulation frequencies and harmonics, thereby making it desirable to define an automated, accurate method for determining 4. $ To whom correspondence should be addressed. 493
0022-2364188 $3.00 Copyright Q 1988 by Academic F’ress, Inc. All rights of npmdwtion in any form reserved.
494
AUTERI,
BETH,
AND
ROBINSON
The difficulty of phasing STEPR spectra has led to suggesting that different combinations of the in-phase and quadrature signals could be used for STEPR studies. Hemminga and co-workers suggested using a magnitude spectrum which is completely independent of the phase (5). Vistnes suggested using the arctangent of the ratio of the quadrature to in-phase signals (6); this signal is insensitive to the phase up to a baseline correction. These approaches, however, result in a loss of information since those STEPR components most sensitive to motion are the pure quadrature components (3,4). Several criteria have been suggested to determine the true phase. One is that the phase be chosen so that the total integral of the in-phase signal is zero (7). Another is that the phase be adjusted so that the zero crossings (or nodes) of the in-phase signal (which have a weak dependence on the rotational tumbling rate of the spin probe) exist at predefined field positions (8). Both suggestions have the drawback that the in-phase signal is used to determine the correct phase. The most popular method is the phase null method. This is based on the approximation that, in the limit of low microwave power (no saturation) and low modulation amplitude, the out-of-phase components are essentially zero (9). The proper application of this method to STEPR requires one to adjust the phase so that the signal has effectively vanished, at the harmonic of choice and under low power. The signals of interest in STEPR are generally wide line signals, and, relative to in-phase signals under linear conditions, are quite small. However, for a single Lorentzian line the ratio of the quadrature to in-phase signal is Tzeu,, not zero (ZO, II). This holds for wide line signals as well and suggests that the phase null criterion must be used with care. Experimentally, the EPR signal is generated by microwave radiation detected by a biased crystal (or mixer). The resulting signal, M(t), would be time independent except that the dc magnetic field (or Zeeman level) is modulated by a small RF, v,, in the range 1 to 200 kHz. The signal is then processed by a lock-in amplifier which effectively performs the integrals (3) Z(4) = s,l/” cos(nw, t + 4)M(t)dt
[la1
and
Q(4) = JO[“‘” sin(nw,t + c$)M(t)dt. The above integrations correspond to finding the Fourier components at the nth harmonic of M(t): Z(4) is the in-phase response, and Q(4) is the phase quadrature signal w,,, = 27~~. Regardless of the value of 4, Z(4) and Q(4) are phased by 90” with respect to each other. The reference phase angle, 4, is independent of the magnetic field. Application of the appropriate trigonometric identities applied to [l] yields the relationships Z(4) = cos(4NO) - sin(@)Q(O>
Pal
and eC4> = W4MO)
+ cos(+>QW.
Pbl
AUTOMATED
MODULATION
PHASE
495
Hence, one can generate the set of in-phase and quadrature spectra for any phase, 4, from the true set of spectra Z(0) and Q(0). As an extension of Eqs. [2] one can generate the signals for all possible values of 4 from the set at any single value of $J.However, in so doing, only the relative phasing is known and the absolute phasing still remains to be determined. This summarizes the vector picture of STEPR spectra (12). The traditional method of phase nulling required one to record a single component of the phased output and manually adjust the phase knob on the instrument so that the signal was as near to the baseline across the entire spectrum as possible. Gaffney suggested back and forth phasing and interpolation giving a signal phased to within 0. lo (14). What follows is a computational algorithm which computes both 4 and the standard error in I#Jand carries out the signal rotation. This mathematical method using least squares to determine the phase angle 4 reproduces the results of the manual “self-null” method. THEORY
Assume that the in-phase and quadrature signals at all relevant field positions, 1, denoted by 4, and Q,, respectively, have been acquired, but that the signals are not phased properly. The signals can then be phased with a suitable choice of 4, after the manner of Eqs. [2], according to the operations Z; = cos($)Zl - sin(4)Q/
Pal
and Q; = sin($)Z, + cos(@)Q,. The null condition is reached when $Jis chosen such that Q; = 0. A least-squares approach to calculating can be taken by defining
WI [41
,i
I=1
[51
N
and minimizing a2 with respect to C#J and c, where c is an adjustable baseline parameter, independent of field position. Then the two criteria are
and
acr2 2N -=O=,,:(Q;-cl,,.
aQ;
a4
[6bl
From the first criterion, Eq. [6a], and Eq. [3b] one finds that the best baseline value, c, is given, in a least-squares sense by c = Isin where r= ,XE, Z,/N, and Q = CE, Q//N.
+ Q cos(4),
[71
496
AUTERI,
BETH, AND ROBINSON
Then evaluating the second criterion, Eq. [6b], leads to 0 = sin(4)cos(4)( VI - VP) + (cos2(+)- sin2(4)} VIQ,
PI
where the following definitions and identities have been used: bQ
= $ (b - &Q, - G) = 2 VI-
I=1
/=I
~QI = ; Z/(Ql- 0)
I=1
v, = ; (II- I>’ = ; Z[(Z[- I> I=1 I=1 vQ
=
i$ I=1
CQ/-
0)’
=
Q/CQ/-
;
[9al WI [9cl
0).
I=1
Rearranging Eq. [ 81 to solve for 4 then yields
As a practical matter, the arctangent is principal valued over the range from -7r/2 to 7r/2. The evaluation of 4 should be done by an arctangent routine that will accept two arguments: the numerator and denominator exactly as shown in [lo]. This extends the range of 4 to include the interval from -x to ?r. The resulting signals are then ambiguous only by ?F.4 is substituted into [3] and used with Z, and QI to generate the properly phase-nulled signals, Z; and Q;. By a standard least-squares error analysis the uncertainty in 4, Q, , can be calculated from the standard propagation of error formula,
[Ill where or, and uQ, are the uncertainties in the acquired in-phase and quadrature signals. We may assume that the errors in Z and Q are independent, so that the covariante, &,a, vanishes. The g2 from Eq. [5] may be used as an estimate of the individual variances in both Z! and QI (4, and Q&, respectively). Alternatively, since there is always a residual signal, even when phase nulled, one may estimate a2 from a region of the data which contains no signal. After some algebra the resulting expression for O$is 2= u’
(6 + (VI -
VQ)”
vQ)2
+ (2vIQ)2
*
One can show that u+ is independent of 4. Therefore the error in the phase angle should be independent of the lock-in setting. Of particular importance for STEPR spectra, u4 provides a criterion for a satisfactory null. EXPERIMENTAL
The experiments were performed on an EPR spectrometer, described elsewhere (Z3). The modulation was generated by a Wavetek function generator 182A (an oscil-
AUTOMATED
MODULATION
PHASE
497
lator), and the phase-sensitive detection of the modulation was performed by an Ithaca Dynatrak 3 lock-in analyzer, which generates, at separate channels, both the in-phase and the quadrature signals for either the first or the second harmonics. Signals were converted to digital form by a Data Translation DT2781 and stored and processed in a DEC PDP 11/23 microcomputer. Modern lock-in amplifiers (or phasesensitive detectors) can simultaneously produce Z(4) and e<$), which can both be digitally recorded simultaneously by a laboratory microcomputer (13). If one is limited by instrumentation (including all commercial spectrometers) and cannot simultaneously record both signals, alternatives are possible. The simplest solution is to record the spectrum twice, with a 90” phase shifter used during the second sweep. PADS (peroxyl amine disulfonate) was obtained from Aldrich and prepared in a carbonate buffer according to the prescription of Jones (15). NMP TCNQ was prepared according to Acker et al. (Z6). The [ 15N,2H1,]-maleimide spin-labeled GAPDH ($~~;ddehyde-3-phosphate dehydrogenase) was prepared according to Beth et
.
. RESULTS
In-phase and quadrature first harmonic absorption spectra of PADS were simultaneously recorded at different phase angles (set on the lock-in) and subsequently rotated to null according to Eq. [lo]. Figure 1 shows representative absorption spectra of PADS for the in-phase and quadrature signals at the first harmonic at 100 kHz before and alter rotation to null. The resulting phase-nulled quadrature signal had the same spectral shape and relative amplitude as that reported by Beth et al. (18). This null position is 8.5” away from the true quadrature position. The reason for the discrepancy is that the quadrature signal is (u,Tze) times as large as the in-phase signal; for this sample q,,Tze - 0.13. The position of 8.5” occurs due to the partial and maximal cancellation of the two signals. Figure 2a plots the phase angle 4, computed according to Eq. [lo] vs the phase setting at the lock-in amplifier. The phase on the lock-in is arbitrary by a constant.
FIG. 1. Experimental spectra of the first harmonic, in-phase (left) and quadrature (right) absorption signals, of the mi = 0 manifold of PADS before and after rotation to null. Relative peak to peak amplitudes are given in the upper right comer of each tracing. The spectra, obtained from the mi = 0 resonance line of a 1 mM PADS in standard buffer, are shown before rotation and then after rotation to null, by 4 = 5.47”. The amplitude of the 100 kHz modulation field was 0.1 G (peak to peak) and H, = 0.01 G in the rotating frame. Samples were placed in a flat cell in a TE 102 cavity.
498
AUTERI,
BETH, AND ROBINSON
25 -
-50
0
50
LOCK-IN
bi.3
I
I
100
PHASE
I
1
I l
1.2 -
0
l
I
Oo
0
0
9 08
to ‘*I
00
0
b” 1.0 00 0.9 -. a0
e
0
d
‘,A l
0.8
I
-50
I
I
0
I
1
50
I
100
FIG. 2. (a) Same sample and conditions as those given in Fig. 1. The value of 4 (a), determined by Eq. [lo], as a function of the phase setting on the lock-in (an arbitrary value) is plotted for the first harmonic signal. The ideal relation is shown by the solid line. (b) Same sample and conditions as those given in Fig. 1. The relative error in 4, ~+/a~, determined by Eq. [ 121, is plotted as a function of the phase setting, @, computed according to Eq. [lo]. The Q/U,, for both the first (0) and the second (0) harmonic signals is shown. The experiment was performed by changing the phase on the lock-in and 4 was computed for each setting. When the experiment is performed on a single data set by rotating numerically by a given amount and then rotating to null, ~~/a,, = 1, always, verifying that Eq. [ 121 is independent of phase.
The resulting straight line relation of the computed phase to the set phase convinces one that the phase null method finds the same position (and hence the same spectra) regardless of phase setting on the lock-in. Figure 2b shows the relative errors in the phase angle as a function of the phase angle, 4. As can be seen there is approximately a 30% change in the estimated error of 4 as the relative phase is changed on the lock-in through 180”. Similar results, not shown, were obtained with samples of NMP TCNQ. a, was shown to be independent of $ analytically and confirmed by rotating numerically a given data set through different angles and then phase nulling. Figure 3 shows the three-step process used to obtain a properly phased STEPR spectrum. In the first step (top spectra) both linear EPR spectra (at the second harmonic and in absorption mode, with a 5 G p/p modulation amplitude and 50 kHz modulation frequency) are recorded and rotated to null; the value of (b to perform
AUTOMATED
MODULATION
PHASE
499
+------
H 20 G
FIG. 3. The in-phase and quadrature signals of ‘5N-perdeuterated maleimide spin-labeled GAPDH in 77% glycerol in water at 2°C are shown. All spectra are. taken in absorption mode at the second harmonic of the Zeeman modulation at 5 G p/p amplitude and 50 kHz frequency. In each case the quadrature signal is plotted on the same scale as and to the right of the corresponding in-phase signal. Top: linear in-phase and quadrature signal after rotating to null, plotted on the same scale; $I = 28.14”. Middle: the in-phase and quadrature signal under saturation conditions (Hi = 0.2 G in the rotating frame) unrotated. Bottom: in-phase and quadrature (STEPR) spectra rotated by $J= 28.14”, as determined by phase nulling the linear spectra (see top). The bottom spectra were generated from the middle spectra by the numerical rotation according to Eqs. [3] of the text.
this rotation is noted. In step 2 (middle spectra) the spectra are recorded under a partially saturating microwave field (the amplitude, H1 = 0.2 Gin the rotating frame). In step 3 the recorded spectra are numerically rotated by 4 (see the bottom spectra). At this point the quadrature signal is a properly phase-nulled STEPR spectrum. These spectra are of a sample of spin-labeled GAPDH in 70% glycerol at 2°C and compare very favorably with previously reported spectra (I 7, 18). STEPR spectra were also acquired at 5 and 100 kHz Zeeman modulation as well as at 50 kHz, as described above. The phase angle, 4, was different for different frequencies. The resultant STEPR spectra are shown in Fig. 4 and are compared to theoretically calculated STEPR spectra, at the same modulation frequencies, for a molecule undergoing isotropic Brownian rotational motion with a characteristic correlation time of 2 I.LS(17, 19). CONCLUSIONS
Phase nulling can be automated. The results are in excellent agreement with literature spectra for both linear EPR and STEPR spectroscopy. When acquiring STEPR spectra this procedure provides a considerable saving in time and a method for easily reproducing phase settings. The value of $Jcorresponds well to the offset on the lockin; the phase rotation is always to the same signal (up to an ambiguity in sign). The values of 4 and Q+can be determined easily, without any adjustment of spectrometer settings. The variation in ud as a function of 4 was unexpected. We conclude that the
500
AUTERI,
BETH, AND ROBINSON
II
-Experimental --Computed
A
100kHz FIG. 4. STEPR (quadrature spectra) obtained at Zeeman modulation frequencies of 550, and 100 kHz are shown (solid lines). Each spectrum was obtained from the sample of and by the procedure described in Fig. 3. The resultant phases, saturated (HI = 0.2 G), second harmonic, quadrature signals (STEPR spectra) are compared with computed spectra (dashed lines). The rotational correlation time was determined to be 20 g. The simulated spectra were computed with input parameters of 9.45 GHz microwave frequency; A,= 10.62G,A,= 10.38G;A,= 50.15G,g,=2.0091;g,=2.0061;gzz= 2.0022;H, =0.2G,r,=20 ps; T,. = 50,25, and 15 gs; r,, = 60,40, and 40 ns for the top, middle, and bottom spectra, respectively, according to the formulas given by Robinson (20). Each computed spectrum was postconvoluted with a Gaussian function of 0.9 G width.
variation in a, for different settings is due to the competing amplitudes of the signals relative to the noise and the relative noise of the amplifiers in the lock-in, providing yet another reason to rotate numerically: set the lock-in’s phase for minimum noise and do the rotations computationally. ACKNOWLEDGMENTS This work was partially supported by Grants PCM-82-16762 (B.H.R.) and DMB-87-06175 (B.H.R.) from the National Science Foundation; by a grant from IBM; and by Grant HL34737 (A.H.B.) from the National Institutes of Health. REFERENCES 1. J. S. HYDE AND L. DALTON, Chem. Phys. Lett. 16,568 (1972). 2. D. D. THOMAS, L. R. DALTON, AND J. S. HYDE, J. Chem. Phys. 65,3006 (1976). 3. L. R. DALTON, B. H. ROBINSON, L. A. DALTON, AND P. COFFEY, in “Advances in Magnetic Resonance” (J. S. Waugh, Ed.), Vol. 8, p. 149, Academic Press, New York, 1976. 4. B. H. ROBINSON, H. THOMANN, A. H. BETH, P. FAJER, AND L. R. DALTON, “EPR and Advanced
AUTOMATED
5.
6. 7.
8. 9. IO.
MODULATION
PHASE
501
EPR Studies of Biological Systems” (L. R. Dalton, Ed.), Chap. 2, CRC Press, Boca Raton, Florida, 1985. M. A. HEMMINGAANDP. A. DE JAGER, J. Magn. Resort. 43,324 (1981). A. I. VISTNES, Biophys. .I. 43,3 1 ( 198 1). C.A. EVANS, J.Magn. Reson.44,109(1981). B. H. ROBINSON, Biophys. J. 41,399 (1983). D. D. THOMAS AND H. M. MCCONNELL, Chem. Phys. Lett. 25,470 (1974). A.M. PoRTIs,P~~~. Rev. 100,1219(1955).
Il. A. M. PORTIS, Technical Note No. 1, Sara Mellon Scaife Radiation Laboratory, University of Pittsburgh, Pittsburgh, Pennsylvania, unpublished. 12. Y. SHIMYAMAAND H. WATARI, J. Chem. Phys. 84,3688 (1986). 13. C. MAILER, J. D. S. DANIELSON, ANDB. H. ROBINSON, Rev. Sci. Instrum. S&l917 (1985). 14. B. J. GAFFNEY, C. H. ELBRECHT, AND J. P. A. SCIBILIA, J. Magn. Reson. 44,436 ( 198 1). 15. M. T. JONES, J. Chem. Phys. 38,2892 (1963). 16. D. S. ACKER AND W. R. HERTLER, J. Am. Chem. Sot. 84,337O (1962). 17. A. H. BETH, K. BALASUBRAMANIAN, R. T. WILDER, S. D. VENKATARAMU, B. H. ROBINSON, L. R. DALTON, D. E. PEARSON, AND J. H. PARK, Proc. Natl. Acad. Sci. USA 78,4955 (198 I). 18. A. H. C. 19. (a) A. A.
BETH, R. WILDER, L. S. WILKERSON, R. C. PERKINS, B. P. MERIWETHER, R. PARK, AND J. H. PARK, J. Chem. Phys. 71,2074 (1979). H. BETH, B. H. ROBINSON, C. E. COBB, AND T. E. CONTURO, Biophys. J. H. BETH AND B. H. ROBINSON, in “Spin Labeling III: Theory and Applications”
Ed.), Academic Press, San Diego, 1988. 20. B. H. ROBINSON, J. Chem. Phys. 78,2268 (1983).
L. R. DALTON,
51, 75a (1987); (b) (L. J. Bcrlinger,
OF MAGNETIC
RESONANCE
80,493-50
1 ( 1988)
An Automated Method to Find the Proper Modulation Phase in Continuous Wave EPR Spectroscopy F. P. AUTERI,*
A. H. BETH,?
AND B. H. ROBINSON*$
*Department ofchemistry University of Washington, Seattle, Washington 98195, and TDepartment of Molecular Physiology and Biophysics, Vanderbilt University Medical Center, Nashville, Tennessee 3 7232 Received January 11, 1988; revised March 8,1988 Modulation of the dc magnetic field is universally employed in the acquisition of continuous wave electron paramagnetic resonance spectra, which includes linear and saturation transfer (ST) EPR spectra. Modem two-channel lock-in analyzers produce two components which are at angles $ and 4 + 90’ with respect to the modulation reference. Experimentally it is often desired to choose I$ such that the quadrature signal is “minimized,” so that the in-phase signal exhibits the maximum signal-to-noise ratio. This is called phase nulling. In this paper we provide a quantitative definition to the concept of a “minimized quadrature signal,” using a least-squares criterion, and derive a formula which enables computation of 4 which is then used to generate proper in-phase and quadrature signals. Furthermore, an expression is derived for the error in Q which is of particular interest when acquiring STEPR spectra. The formulas are noniterative and can be easily programmed to rapidly and automatically phase null signals alter acquisition. @ 1988 Academic Press, Inc.
INTRODUCTION
To improve the signal-to-noise ratio of cw EPR spectroscopy, the dc magnetic field is modulated at frequency V, , usually 100 kHz, which is called Zeeman modulation. The signal is then fed into a lock-in amplifier which removes the modulation carrier and produces a time-independent signal. For linear cw EPR, the choice of a proper phase between the driving modulation of the dc magnetic field is easily obtained visually. Once set, at 100 kHz, it rarely requires adjustment. However, with the advent of STEPR, the choice of phase becomes critical. Typical STEPR spectra are quadrature spectra under saturation conditions. Quadrature signals nearly vanish when the power is low enough that the response is linear, but grow rapidly when the microwave radiation incident on the sample causes partial saturation of the spin system (1-4). Even under partially saturating conditions, however, the in-phase signal is much larger than the quadrature signal, thereby necessitating accurate determination of 4 to avoid mixing of the large in-phase signal with the desired quadrature signal. Furthermore, phase settings change as one changes modulation frequencies and harmonics, thereby making it desirable to define an automated, accurate method for determining 4. $ To whom correspondence should be addressed. 493
0022-2364188 $3.00 Copyright Q 1988 by Academic F’ress, Inc. All rights of npmdwtion in any form reserved.
494
AUTERI,
BETH,
AND
ROBINSON
The difficulty of phasing STEPR spectra has led to suggesting that different combinations of the in-phase and quadrature signals could be used for STEPR studies. Hemminga and co-workers suggested using a magnitude spectrum which is completely independent of the phase (5). Vistnes suggested using the arctangent of the ratio of the quadrature to in-phase signals (6); this signal is insensitive to the phase up to a baseline correction. These approaches, however, result in a loss of information since those STEPR components most sensitive to motion are the pure quadrature components (3,4). Several criteria have been suggested to determine the true phase. One is that the phase be chosen so that the total integral of the in-phase signal is zero (7). Another is that the phase be adjusted so that the zero crossings (or nodes) of the in-phase signal (which have a weak dependence on the rotational tumbling rate of the spin probe) exist at predefined field positions (8). Both suggestions have the drawback that the in-phase signal is used to determine the correct phase. The most popular method is the phase null method. This is based on the approximation that, in the limit of low microwave power (no saturation) and low modulation amplitude, the out-of-phase components are essentially zero (9). The proper application of this method to STEPR requires one to adjust the phase so that the signal has effectively vanished, at the harmonic of choice and under low power. The signals of interest in STEPR are generally wide line signals, and, relative to in-phase signals under linear conditions, are quite small. However, for a single Lorentzian line the ratio of the quadrature to in-phase signal is Tzeu,, not zero (ZO, II). This holds for wide line signals as well and suggests that the phase null criterion must be used with care. Experimentally, the EPR signal is generated by microwave radiation detected by a biased crystal (or mixer). The resulting signal, M(t), would be time independent except that the dc magnetic field (or Zeeman level) is modulated by a small RF, v,, in the range 1 to 200 kHz. The signal is then processed by a lock-in amplifier which effectively performs the integrals (3) Z(4) = s,l/” cos(nw, t + 4)M(t)dt
[la1
and
Q(4) = JO[“‘” sin(nw,t + c$)M(t)dt. The above integrations correspond to finding the Fourier components at the nth harmonic of M(t): Z(4) is the in-phase response, and Q(4) is the phase quadrature signal w,,, = 27~~. Regardless of the value of 4, Z(4) and Q(4) are phased by 90” with respect to each other. The reference phase angle, 4, is independent of the magnetic field. Application of the appropriate trigonometric identities applied to [l] yields the relationships Z(4) = cos(4NO) - sin(@)Q(O>
Pal
and eC4> = W4MO)
+ cos(+>QW.
Pbl
AUTOMATED
MODULATION
PHASE
495
Hence, one can generate the set of in-phase and quadrature spectra for any phase, 4, from the true set of spectra Z(0) and Q(0). As an extension of Eqs. [2] one can generate the signals for all possible values of 4 from the set at any single value of $J.However, in so doing, only the relative phasing is known and the absolute phasing still remains to be determined. This summarizes the vector picture of STEPR spectra (12). The traditional method of phase nulling required one to record a single component of the phased output and manually adjust the phase knob on the instrument so that the signal was as near to the baseline across the entire spectrum as possible. Gaffney suggested back and forth phasing and interpolation giving a signal phased to within 0. lo (14). What follows is a computational algorithm which computes both 4 and the standard error in I#Jand carries out the signal rotation. This mathematical method using least squares to determine the phase angle 4 reproduces the results of the manual “self-null” method. THEORY
Assume that the in-phase and quadrature signals at all relevant field positions, 1, denoted by 4, and Q,, respectively, have been acquired, but that the signals are not phased properly. The signals can then be phased with a suitable choice of 4, after the manner of Eqs. [2], according to the operations Z; = cos($)Zl - sin(4)Q/
Pal
and Q; = sin($)Z, + cos(@)Q,. The null condition is reached when $Jis chosen such that Q; = 0. A least-squares approach to calculating can be taken by defining
WI [41
,i
I=1
[51
N
and minimizing a2 with respect to C#J and c, where c is an adjustable baseline parameter, independent of field position. Then the two criteria are
and
acr2 2N -=O=,,:(Q;-cl,,.
aQ;
a4
[6bl
From the first criterion, Eq. [6a], and Eq. [3b] one finds that the best baseline value, c, is given, in a least-squares sense by c = Isin where r= ,XE, Z,/N, and Q = CE, Q//N.
+ Q cos(4),
[71
496
AUTERI,
BETH, AND ROBINSON
Then evaluating the second criterion, Eq. [6b], leads to 0 = sin(4)cos(4)( VI - VP) + (cos2(+)- sin2(4)} VIQ,
PI
where the following definitions and identities have been used: bQ
= $ (b - &Q, - G) = 2 VI-
I=1
/=I
~QI = ; Z/(Ql- 0)
I=1
v, = ; (II- I>’ = ; Z[(Z[- I> I=1 I=1 vQ
=
i$ I=1
CQ/-
0)’
=
Q/CQ/-
;
[9al WI [9cl
0).
I=1
Rearranging Eq. [ 81 to solve for 4 then yields
As a practical matter, the arctangent is principal valued over the range from -7r/2 to 7r/2. The evaluation of 4 should be done by an arctangent routine that will accept two arguments: the numerator and denominator exactly as shown in [lo]. This extends the range of 4 to include the interval from -x to ?r. The resulting signals are then ambiguous only by ?F.4 is substituted into [3] and used with Z, and QI to generate the properly phase-nulled signals, Z; and Q;. By a standard least-squares error analysis the uncertainty in 4, Q, , can be calculated from the standard propagation of error formula,
[Ill where or, and uQ, are the uncertainties in the acquired in-phase and quadrature signals. We may assume that the errors in Z and Q are independent, so that the covariante, &,a, vanishes. The g2 from Eq. [5] may be used as an estimate of the individual variances in both Z! and QI (4, and Q&, respectively). Alternatively, since there is always a residual signal, even when phase nulled, one may estimate a2 from a region of the data which contains no signal. After some algebra the resulting expression for O$is 2= u’
(6 + (VI -
VQ)”
vQ)2
+ (2vIQ)2
*
One can show that u+ is independent of 4. Therefore the error in the phase angle should be independent of the lock-in setting. Of particular importance for STEPR spectra, u4 provides a criterion for a satisfactory null. EXPERIMENTAL
The experiments were performed on an EPR spectrometer, described elsewhere (Z3). The modulation was generated by a Wavetek function generator 182A (an oscil-
AUTOMATED
MODULATION
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497
lator), and the phase-sensitive detection of the modulation was performed by an Ithaca Dynatrak 3 lock-in analyzer, which generates, at separate channels, both the in-phase and the quadrature signals for either the first or the second harmonics. Signals were converted to digital form by a Data Translation DT2781 and stored and processed in a DEC PDP 11/23 microcomputer. Modern lock-in amplifiers (or phasesensitive detectors) can simultaneously produce Z(4) and e<$), which can both be digitally recorded simultaneously by a laboratory microcomputer (13). If one is limited by instrumentation (including all commercial spectrometers) and cannot simultaneously record both signals, alternatives are possible. The simplest solution is to record the spectrum twice, with a 90” phase shifter used during the second sweep. PADS (peroxyl amine disulfonate) was obtained from Aldrich and prepared in a carbonate buffer according to the prescription of Jones (15). NMP TCNQ was prepared according to Acker et al. (Z6). The [ 15N,2H1,]-maleimide spin-labeled GAPDH ($~~;ddehyde-3-phosphate dehydrogenase) was prepared according to Beth et
.
. RESULTS
In-phase and quadrature first harmonic absorption spectra of PADS were simultaneously recorded at different phase angles (set on the lock-in) and subsequently rotated to null according to Eq. [lo]. Figure 1 shows representative absorption spectra of PADS for the in-phase and quadrature signals at the first harmonic at 100 kHz before and alter rotation to null. The resulting phase-nulled quadrature signal had the same spectral shape and relative amplitude as that reported by Beth et al. (18). This null position is 8.5” away from the true quadrature position. The reason for the discrepancy is that the quadrature signal is (u,Tze) times as large as the in-phase signal; for this sample q,,Tze - 0.13. The position of 8.5” occurs due to the partial and maximal cancellation of the two signals. Figure 2a plots the phase angle 4, computed according to Eq. [lo] vs the phase setting at the lock-in amplifier. The phase on the lock-in is arbitrary by a constant.
FIG. 1. Experimental spectra of the first harmonic, in-phase (left) and quadrature (right) absorption signals, of the mi = 0 manifold of PADS before and after rotation to null. Relative peak to peak amplitudes are given in the upper right comer of each tracing. The spectra, obtained from the mi = 0 resonance line of a 1 mM PADS in standard buffer, are shown before rotation and then after rotation to null, by 4 = 5.47”. The amplitude of the 100 kHz modulation field was 0.1 G (peak to peak) and H, = 0.01 G in the rotating frame. Samples were placed in a flat cell in a TE 102 cavity.
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FIG. 2. (a) Same sample and conditions as those given in Fig. 1. The value of 4 (a), determined by Eq. [lo], as a function of the phase setting on the lock-in (an arbitrary value) is plotted for the first harmonic signal. The ideal relation is shown by the solid line. (b) Same sample and conditions as those given in Fig. 1. The relative error in 4, ~+/a~, determined by Eq. [ 121, is plotted as a function of the phase setting, @, computed according to Eq. [lo]. The Q/U,, for both the first (0) and the second (0) harmonic signals is shown. The experiment was performed by changing the phase on the lock-in and 4 was computed for each setting. When the experiment is performed on a single data set by rotating numerically by a given amount and then rotating to null, ~~/a,, = 1, always, verifying that Eq. [ 121 is independent of phase.
The resulting straight line relation of the computed phase to the set phase convinces one that the phase null method finds the same position (and hence the same spectra) regardless of phase setting on the lock-in. Figure 2b shows the relative errors in the phase angle as a function of the phase angle, 4. As can be seen there is approximately a 30% change in the estimated error of 4 as the relative phase is changed on the lock-in through 180”. Similar results, not shown, were obtained with samples of NMP TCNQ. a, was shown to be independent of $ analytically and confirmed by rotating numerically a given data set through different angles and then phase nulling. Figure 3 shows the three-step process used to obtain a properly phased STEPR spectrum. In the first step (top spectra) both linear EPR spectra (at the second harmonic and in absorption mode, with a 5 G p/p modulation amplitude and 50 kHz modulation frequency) are recorded and rotated to null; the value of (b to perform
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FIG. 3. The in-phase and quadrature signals of ‘5N-perdeuterated maleimide spin-labeled GAPDH in 77% glycerol in water at 2°C are shown. All spectra are. taken in absorption mode at the second harmonic of the Zeeman modulation at 5 G p/p amplitude and 50 kHz frequency. In each case the quadrature signal is plotted on the same scale as and to the right of the corresponding in-phase signal. Top: linear in-phase and quadrature signal after rotating to null, plotted on the same scale; $I = 28.14”. Middle: the in-phase and quadrature signal under saturation conditions (Hi = 0.2 G in the rotating frame) unrotated. Bottom: in-phase and quadrature (STEPR) spectra rotated by $J= 28.14”, as determined by phase nulling the linear spectra (see top). The bottom spectra were generated from the middle spectra by the numerical rotation according to Eqs. [3] of the text.
this rotation is noted. In step 2 (middle spectra) the spectra are recorded under a partially saturating microwave field (the amplitude, H1 = 0.2 Gin the rotating frame). In step 3 the recorded spectra are numerically rotated by 4 (see the bottom spectra). At this point the quadrature signal is a properly phase-nulled STEPR spectrum. These spectra are of a sample of spin-labeled GAPDH in 70% glycerol at 2°C and compare very favorably with previously reported spectra (I 7, 18). STEPR spectra were also acquired at 5 and 100 kHz Zeeman modulation as well as at 50 kHz, as described above. The phase angle, 4, was different for different frequencies. The resultant STEPR spectra are shown in Fig. 4 and are compared to theoretically calculated STEPR spectra, at the same modulation frequencies, for a molecule undergoing isotropic Brownian rotational motion with a characteristic correlation time of 2 I.LS(17, 19). CONCLUSIONS
Phase nulling can be automated. The results are in excellent agreement with literature spectra for both linear EPR and STEPR spectroscopy. When acquiring STEPR spectra this procedure provides a considerable saving in time and a method for easily reproducing phase settings. The value of $Jcorresponds well to the offset on the lockin; the phase rotation is always to the same signal (up to an ambiguity in sign). The values of 4 and Q+can be determined easily, without any adjustment of spectrometer settings. The variation in ud as a function of 4 was unexpected. We conclude that the
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100kHz FIG. 4. STEPR (quadrature spectra) obtained at Zeeman modulation frequencies of 550, and 100 kHz are shown (solid lines). Each spectrum was obtained from the sample of and by the procedure described in Fig. 3. The resultant phases, saturated (HI = 0.2 G), second harmonic, quadrature signals (STEPR spectra) are compared with computed spectra (dashed lines). The rotational correlation time was determined to be 20 g. The simulated spectra were computed with input parameters of 9.45 GHz microwave frequency; A,= 10.62G,A,= 10.38G;A,= 50.15G,g,=2.0091;g,=2.0061;gzz= 2.0022;H, =0.2G,r,=20 ps; T,. = 50,25, and 15 gs; r,, = 60,40, and 40 ns for the top, middle, and bottom spectra, respectively, according to the formulas given by Robinson (20). Each computed spectrum was postconvoluted with a Gaussian function of 0.9 G width.
variation in a, for different settings is due to the competing amplitudes of the signals relative to the noise and the relative noise of the amplifiers in the lock-in, providing yet another reason to rotate numerically: set the lock-in’s phase for minimum noise and do the rotations computationally. ACKNOWLEDGMENTS This work was partially supported by Grants PCM-82-16762 (B.H.R.) and DMB-87-06175 (B.H.R.) from the National Science Foundation; by a grant from IBM; and by Grant HL34737 (A.H.B.) from the National Institutes of Health. REFERENCES 1. J. S. HYDE AND L. DALTON, Chem. Phys. Lett. 16,568 (1972). 2. D. D. THOMAS, L. R. DALTON, AND J. S. HYDE, J. Chem. Phys. 65,3006 (1976). 3. L. R. DALTON, B. H. ROBINSON, L. A. DALTON, AND P. COFFEY, in “Advances in Magnetic Resonance” (J. S. Waugh, Ed.), Vol. 8, p. 149, Academic Press, New York, 1976. 4. B. H. ROBINSON, H. THOMANN, A. H. BETH, P. FAJER, AND L. R. DALTON, “EPR and Advanced
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EPR Studies of Biological Systems” (L. R. Dalton, Ed.), Chap. 2, CRC Press, Boca Raton, Florida, 1985. M. A. HEMMINGAANDP. A. DE JAGER, J. Magn. Resort. 43,324 (1981). A. I. VISTNES, Biophys. .I. 43,3 1 ( 198 1). C.A. EVANS, J.Magn. Reson.44,109(1981). B. H. ROBINSON, Biophys. J. 41,399 (1983). D. D. THOMAS AND H. M. MCCONNELL, Chem. Phys. Lett. 25,470 (1974). A.M. PoRTIs,P~~~. Rev. 100,1219(1955).
Il. A. M. PORTIS, Technical Note No. 1, Sara Mellon Scaife Radiation Laboratory, University of Pittsburgh, Pittsburgh, Pennsylvania, unpublished. 12. Y. SHIMYAMAAND H. WATARI, J. Chem. Phys. 84,3688 (1986). 13. C. MAILER, J. D. S. DANIELSON, ANDB. H. ROBINSON, Rev. Sci. Instrum. S&l917 (1985). 14. B. J. GAFFNEY, C. H. ELBRECHT, AND J. P. A. SCIBILIA, J. Magn. Reson. 44,436 ( 198 1). 15. M. T. JONES, J. Chem. Phys. 38,2892 (1963). 16. D. S. ACKER AND W. R. HERTLER, J. Am. Chem. Sot. 84,337O (1962). 17. A. H. BETH, K. BALASUBRAMANIAN, R. T. WILDER, S. D. VENKATARAMU, B. H. ROBINSON, L. R. DALTON, D. E. PEARSON, AND J. H. PARK, Proc. Natl. Acad. Sci. USA 78,4955 (198 I). 18. A. H. C. 19. (a) A. A.
BETH, R. WILDER, L. S. WILKERSON, R. C. PERKINS, B. P. MERIWETHER, R. PARK, AND J. H. PARK, J. Chem. Phys. 71,2074 (1979). H. BETH, B. H. ROBINSON, C. E. COBB, AND T. E. CONTURO, Biophys. J. H. BETH AND B. H. ROBINSON, in “Spin Labeling III: Theory and Applications”
Ed.), Academic Press, San Diego, 1988. 20. B. H. ROBINSON, J. Chem. Phys. 78,2268 (1983).
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51, 75a (1987); (b) (L. J. Bcrlinger,