[45]
FOURTH
DERIVATIVE
SPECTRA
501
fluorescence increase, consistent with the knowledge that it stimulates energy-linked reactions in mitochondrial fragments. Energy-linked ANS fluorescence increases can be obtained with N A D H (0.1-1.0 mM) or ascorbate (2 mM plus phenazine methosulfate) (0.1 mM), which are reversed, respectively, by rotenone (2/zM) and KCN (1 mM). When N A D H is employed as substrate, the wavelengths to be selected for excitation and emission are 436 and 560 nm, respectively, in order to minimize the interference of N A D H fluorescence on the ANS signal. The fluorescence decrease of ANS in mitochondrial fragments reflects an energy-linked process and can be utilized down to protein concentration of the order of 0.05 mg/ml.
[45] F o u r t h D e r i v a t i v e S p e c t r a By WARREN L. BUTLER Higher derivative analysis provides a powerful and useful technique to aid in the resolution of complex spectra. The efficacy of the analysis can be readily demonstrated with digitized spectral data. Derivatives are obtained by the simple procedure of computing a difference spectrum between the original curve, A(h), and the same curve shifted a finite wavelength interval, A(h + Ak), with the difference value being assigned the wavelength corresponding to the midpoint of Ah. Higher derivatives are obtained by repeating the Ah differentiation the desired number of times. The primary thrust of this chapter is to elucidate and justify the use of fourth derivatives for the analysis of spectral data. Fourth Derivative Analysis of an Experimental Spectrum The absorption spectrum of beef heart mitochondria measured at liquid nitrogen temperature (Fig. l) is used to demonstrate the kinds of information that can be extracted from the data by a fourth derivative operation. This spectrum was measured with a single-beam spectrophotometer on line, via a 12-bit A-D converter, with a small computeH with data points being taken every 0. l nm. The spectrum is a complex mixture of the absorption bands of several b- and c-type cytochromes. Some of the absorption bands are apparent only as small shoulders which cannot be assigned precise wavelength maxima. The absorption bands of other components known to be present are completely buried in the cornl W. L. B u t l e r , Vol. 24, Part B, p. 3.
METHODS IN ENZYMOLOGY, VOL. LVI
Copyright © 1979 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-181956-6
502
[45]
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Wavelength, nm FIG. I. Absorption spectrum (dithionite-reduced minus oxidized difference spectrum) of mitochondria (7.5 mg protein per milliliter). See Davis e t al. z for details. The fourth derivative curve in the upper part of the figure was taken with Ah differentiating intervals of 1.0, 1.1, 1.2 and 1.4 nm.
[45]
FOURTH DERIVATIVE SPECTRA
503
plexity of the spectrum. The constituent absorption bands are resolved to a much greater degree in the fourth derivative of the absorption spectrum obtained with Ah differentiating intervals of 1.0, 1.1, 1.2 and 1.4 nm, shown in the upper part of Fig. 1. All of the fourth derivative peaks which are marked for their wavelength maxima correspond to absorption bands with essentially the same wavelength maxima in the original spectrum. In the study from which the data in Fig. 1 was taken, 2 absorption spectra of all of the individual cytochromes at - 196° were obtained by separating the mitochondria into various submitochondrial complexes and measuring difference spectra between differential redox treatments. In this manner, the maxima observed in the fourth derivative spectrum of the mitochondria at - 196° could be compared against the absorption maxima of the individual cytochrome components at - 196°. The shoulder at about 562 nm in the absorption spectrum which is resolved as a sharp peak at 562.5 nm in the fourth derivative spectrum is due to cytochrome bx. This cytochrome at -196 ° has a sharp ai band at 562.5 nm and a smaller a2 band at 554 nm. The absorption maximum at 558.5 nm is due to the combination of two cytochromes; one, cytochrome bK, has a single band at 559.5 nm at - 196°C and the other, the cytochrome b of succinic dehydrogenase (complex II), has an al band at 557.5 nm and an a~ band at 550 nm. The a bands of these two cytochromes are resolved in the fourth derivative spectrum, although the 557.5 nm band appears to be shifted to 557 nm. The fourth derivative band at 553.5 nm is due to cytochrome cl which has a maximum at 553 nm with probably some contribution from the 554 nm a2 band of cytochrome b+. The sharp fourth derivative bands at 547.5 and 544 nm are due to the al and a2 bands of cytochrome c, which is a soluble component that can be washed free from the membrane fragments. Most of the/3 bands (in the spectral region between 500 and 540 nm) can also be assigned to particular cytochromes. It is worth noting that the sample of mitochondria used for the spectral measurements shown in Fig. 1 was by no means ideal from the standpoint of making photometric measurements. The mitochondria were suspended in a buffer medium and frozen to -196 ° resulting in a sample which was about 3 mm thick, highly scattering, and optically quite dense. Even so, the measurements were made with sufficiently low noise that noise was not a problem in the fourth derivative curve. One of the purposes of this chapter is to explore the effects of noise on fourth derivative spectra and to explore techniques to minimize those effects. 2 K. A. Davis, Y. Hatefi, K. L. Poff, and W. L. Butler, Biochim. Biophys. Acta 325, 341 (1973).
504
SPECIALIZED TECHNIQUES
[45]
Simulated Spectra The rationale for using higher derivatives to resolve complex spectra was established in a previous study2 Spectra of known content were simulated with a small computer by summing together analytical expressions for bands which were specified by their wavelength maximum, amplitude, half-width and band shape (i.e., specified mixtures of Gaussian and Lorentzian bands). The spectra were constructed so that some of the individual bands were not apparent in the sum, and the ability of second and fourth derivative curves to resolve those bands was examined by summing together the analytical expressions for the second and fourth derivatives of the individual bands (the sum of the derivatives is identical to the derivative of the sum). Even numbered derivatives have the advantage over the odd numbered ones in that they have maxima at the same position as the maxima of the original bands, which makes the correspondence between the derivative bands and the bands in the original spectrum more apparent. (Strictly speaking, derivatives which are odd numbered multiples of two, such as the second and the sixth, give minima at the position of the band maxima, but the negative of these curves can be plotted so that the minima appear as maxima.) The results of the previous study confirmed that the second and fourth derivative curves could reveal the presence of constituent bands which were not apparent in the simulated spectrum, that the bands were resolved better by the fourth derivative than by the second derivative, and that the wavelength maxima in the fourth derivative curves agreed closely to the wavelength maxima of the original bands. The enhanced resolution results because the derivative bands are narrower than the original bands, e.g., a Lorentzian band of 20 nm half-width has a half-width of 5 nm in the fourth derivative. Thus, the fourth derivative of a spectrum comprised of overlapping bands may show well-defined peaks at the wavelength maxima of the bands even though the bands blend together in the spectrum to obscure the individual maxima. The fourth derivative of an individual band has negative minima on both sides of the central maximum, and these minima can interact with the fourth derivative curves of adjacent bands to create a potential for artifacts. Small band shifts may result from such interactions and bands may even be obliterated. It was also pointed out previously~ that false bands may be generated by the interaction between the side wings of adjacent bands. For instance, with two Gaussian bands of 20 nm halfwidth spaced 40 nm apart, the negative side wings come together to form what appears to be a third band halfway between the two bands. The :~W. L. Butler and D. W. Hopkins,Photochem. Photobiol. 12, 439 (1970).
[45]
FOURTH DERIVATIVE SPECTRA
505
unmarked band near 540 nm in the fourth derivative spectrum in Fig. 1 may be an example of such a false band. None of the spectra of the individual cytochromes showed a 540 nm absorption band which would be correlated with that fourth derivative band. If it is a false band it is one of the few that have appeared to date in the fourth derivative spectra of experimental data. While the potential for such artifactual bands should be kept in mind, in actual practice it has not proved to be a serious problem. ~,~4A/~h4 versus d 4 A M h 4 In the previous study, the fourth derivatives of the simulated spectra were obtained mathematically from the analytical expressions for the Gaussian and Lorentzian bands. Such analytically derived fourth derivative curves showed the maximum resolution which could be obtained theoretically by a fourth derivative. However, the question of how the resolving power of a fourth derivative obtained with four finite ~;~ intervals depended on the size of those intervals was not addressed because of the limitations set by the 12-bit resolution of the computer used in those studies. In the present study sufficient accuracy was available (a computer with 16-bit resolution was used) to make those calculations and that question is examined in Fig. 2. (All of the simulated spectra presented in this chapter are comprised of 1000 points spanning a simulated spectral region from 500 to 600 nm, except in one case, Fig. 3, where the spectral range is from 450 to 650.) The sum of two Gaussian bands of equal amplitude and 20 nm half-width with wavelength maxima at 544 and 556 nm is shown in Fig. 2A. The mathematical fourth derivative is presented as curve A w. Fourth derivative curves taken by the Ah method are also shown. With four ~h intervals of 2 nm each, the resolution is equal to that obtained by the mathematical differentiation. With intervals of 4 nm, the two bands are still resolved but not quite as well and with intervals of 8 nm the resolution is lost. The previous study3 using the mathematical differention showed that Lorentzian bands could be resolved much better than Gaussian bands. While Gaussian bands of 20 nm half-width could not be resolved by the fourth derivative if the bands were closer together than 12 nm, Lorentzian bands of 20 nm half-width could be resolved even when the bands were 6 nm apart. The ability of the fourth derivative to resolve a spectrum which is the sum of two such Lorentzian bands with maxima at 547 and 553 nm is shown in Fig. 2B. In this case in order to achieve a resolution with four 5h differentiations which approaches that of the mathematical fourth derivative, A Iv, the intervals have to be 1 nm or less. The resolution is somewhat
506
SPECIALIZED TECHNIQUES A
[45]
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Wavelength(nm) FIG. 2. Computer-generated spectra. (A) Two Gaussian bands with 20 nm half-widths and wavelength maxima at 544 and 556 nm were summed together. (B) Two Lorentzian bands with 20 nm half-widths and wavelength maxima at 547 and 553 nm were summed together. The mathematical fourth derivative calculated from the analytical expressions for the Gaussian and Lorentzian bands are shown as the M v curves. The fourth derivative by four Ah differentiations are identified by the four Ah intervals used. The numbers outside the parentheses indicate the multiplication factor by which the curves were multiplied for plotting. less with intervals of 2 n m and is lost with intervals of 4 nm. Thus, the A)~ method o f differentiation confirms that the resolvability of Lorentzian bands is inherently greater than that o f Gaussian bands but, as should be expected, smaller Ah intervals are needed to achieve that resolution. T h e n u m b e r s outside the p a r e n t h e s e s in Fig. 2 indicate the amplification factors b y which the curves w e r e multiplied to m a k e t h e m equal in amplitude to the m a t h e m a t i c a l fourth derivative. I f the fourth derivative curves obtained by the Ah differentiations w e r e true fourth derivatives, the amplitudes should be proportional to Ah 4, i.e., doubling the Ah interval should increase the amplitude 16 times. T h e fact that the amplitude ratios due to doubling Ah are less than 16 for the curves in Fig. 2 is due to the deviations f r o m the true fourth derivative c u r v e which o c c u r as the size the Ah intervals is increased. B a n d Width Just as the amplitude o f the fourth derivative band has a strong dependence on the size o f the Ah intervals used for the differentiations (ap-
[45]
FOURTH DERIVATIVE SPECTRA
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FIG. 3. Computer-generated spectrum from three Gaussian bands with wavelength maxima at 540, 560, and 570 nm with amplitude ratios of 5:10:1 and half-widths of 20, 40 and 10 nm, respectively. Fourth derivative curves taken with four equal Ah intervals of 2, 8, and 16 nm are shown. Vertical lines indicate wavelengths of 540 and 570 nm.
proaching Ah4 for sufficiently small ah) so is the amplitude of the fourth derivative strongly dependent on the half-width of the original band, approaching an inverse fourth power dependence. Thus, narrow bands are highly selected for in a fourth derivative spectrum. Figure 3 shows the sum of three Gaussian bands at 540,560, and 570 nm in amplitude ratios of 5:10:1 and half-widths of 20, 40, and 10 nm, respectively. The fourth derivative curve of the simulated spectrum obtained with four Ah's of 2 nm shows the 540 nm band, but the presence of the large broad 560 nm band is almost obliterated by the fourth derivative signal of the small narrow band at 570 nm. One might suppose that using larger Ah intervals would enhance broad bands more than narrow bands so that it might be possible to preferentially select for the broad bands in the spectrum by increasing the size of the ah intervals. However, using four Ah intervals of 8 or 16 nm does not enhance the broad 560 nm band relative to the narrow 570 nm band. In fact, the broad 560 nm band is suppressed even more in the spectrum taken with the broader Ah intervals because the dominating influence of the narrow 570 nm band extends over a wider wavelength region.
508
SPECIALIZED TECHNIQUES
[45]
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Wavelength (nm)
Fro. 4. Computer-generated spectra. The solid curve is the sum of two 50% Gaussian50% Lorentzian bands of equal amplitude with 20 nm half-widths and wavelength maxima at 547 and 553 rim. The dashed curve (superimposed on the solid curve) is a single band, 55% Gaussian, with a 22 nm half-width and a wavelength maximum at 550 nm. Fourth derivative curves of these two spectra, solid and dashed, were taken with four Ah intervals of 1 nm. Vertical lines indicate wavelengths at 547 and 553 nm. Information Retrieval An e x a m p l e demonstrating the capacity of the fourth derivative analysis to reveal information that is not readily apparent in the original spectrum is presented in Fig. 4. T h e solid c u r v e in Fig. 4 is the sum of two bands o f equal amplitude which are 50% Gaussian and 50% Lorentzian with half-widths o f 20 nm and wavelength m a x i m a at 547 and 553 nm. S u p e r i m p o s e d on the solid c u r v e is a curve drawn in dashes which is a single band, 55% Gaussian, with a 22 nm half-width, a 550 nm wavelength m a x i m u m and an amplitude chosen to m a t c h that of the solid curve. The two simulated spectra are almost indistinguishable by visual inspection and the question is asked w h e t h e r the fourth derivative obtained by four Ah differentiations o f l n m each can distinguish the different origins of these two curves. Indeed the fourth derivative of the solid curve shows that that c u r v e is c o m p r i s e d of two c o m p o n e n t s with m a x i m a at 547 and 553 nm, while the fourth derivative of the dashed curves shows a single band with the m a x i m u m at 550 nm. It should be e m p h a s i z e d that this Ah method o f obtaining the fourth derivative is entirely independent of the composition o f the s p e c t r u m and is the same method that we use for experimental spectra where the shapes and the widths of the bands are generally unknown. It should be apparent from the results of Fig. 4 that a
[45]
FOURTH DERIVATIVE SPECTRA
509
spectral curve may contain information that is not readily available in the original data but which may be processed into recognizable forms by the higher derivative analysis. Convolution Functions and Even Higher Derivatives It was shown previously 4 that higher derivative operations obtained by successive Ah differentiations are equivalent to convolution functions. The convolution function for a fourth derivative obtained with four equal Ah intervals can be represented as a wavelength bar 4 Ah long with factors of 1 at the end points, a factor of 6 at the midpoint, and factors of - 4 at the one-quarter and three-quarters points. The factors for any particular derivative can be obtained by expanding (a - b) n, where n is the order of the derivative or by taking the factors from the appropriate row of Pascal's triangle. The convolution function steps sequentially along the curve averaging at each digitization wavelength the readings on the curve according to the factors on the wavelength bar. A convolution function giving a sixth derivative can be represented by a wavelength bar 6Ah long with factors - 1 , 6, - 1 5 , 20, - 1 5 , 6, - 1 spaced each Ah interval, while the convolution function for an eighth derivative would have the factors l, - 8 , 28, - 5 6 , 70, - 5 6 , 28, - 8 , 1 spaced at Ah intervals along the bar. The use of these convolution functions for higher derivatives is demonstrated in Fig. 5 on a spectrum comprised of two 20 nm half-width Lorentzian bands, 4 nm apart, with maxima at 548 and 552 nm. This spectrum is not resolved by the convolution function for the fourth derivative which is equivalent to four Ah differentiations of 1 nm each, nor would it be resolved by using smaller Ah intervals. The mathematical differentiation of these bands shows that this spectrum is not resolvable by a fourth derivative operation. However, these bands are just resolved in the sixth derivative and are well resolved in the eighth derivative, both being obtained with convolution functions which are equivalent to a series of equal Ak differentiating intervals of 1 nm. However, some fluctuations are apparent in the eighth derivative curve which may be due to limitations set by the 16-bit resolution of the computer. Band Shape The effect which small changes of band shape may have on the resolvability of a spectrum is demonstrated in Fig. 6A. The simulated spectrum is comprised of five bands of equal amplitude with wavelength maxima at 544, 547,550, 553, and 556 nm. All of the bands have a 5 nm half-width 4 W. L. Butler and D. W. Hopkins, Photochem. Photobiol. 12, 451 (1970).
510
SPECIALIZED TECHNIQUES
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(1500) th (600)
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550 600 Wavelength (nrn)
FIG. 5. Computer-generated spectrum fi'om two Lorentzian bands with 20 nm halfwidths and wavelength maxima at 548 and 552 nm. Fourth, sixth, and eighth derivative curves taken with convolution functions equivalent to Ah differentiations of 1 rim. Numbers in parentheses indicate factors by which the curves were multiplied for plotting.
and a shape which is 90% Gaussian and 10% Lorentzian. The fourth derivative of the spectrum taken with four Ah intervals of 1.0 nm (curve a) resolves all of the bands. It is apparent that the magnitude of bands in the fourth derivative curve bears no direct relationship to the magnitude of the corresponding bands in the original spectrum. The heights of the three fourth derivative bands in the middle of curve a are decreased by negative wings of the fourth derivative bands on each end. However, the wavelength maxima of the fourth derivative bands agree quite closely to the wavelength maxima of the original bands. The resolution shown in curve a, however, is markedly dependent on the 10% Lorentzian character of the bands. Curve b shows the fourth derivative curve obtained in the same manner from a simulated spectrum comprised of the same bands with the single exception that each band was 100% Gaussian. The difference in shape of the simulated spectrum resulting from the change of the bands from 90% Gaussian to 100% Gaussian was scarcely detectable. However, in the latter case, the fourth derivative bands of the 547 and 553 nm components are obliterated by the negative wings of the other bands. Using smaller Ah intervals would sharpen up the three fourth derivative bands that are present but would not restore the two bands which are missing. If any adjacent pair of the 100% Gaussian bands were analyzed
[45]
FOURTH DERIVATIVE SPECTRA
511
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Wavelength (nrn) Fie. 6. (A) Computer-generated spectrum from five 90% Gaussian, 10% Lorentzian bands of equal amplitude with 5 nm half-widths and wavelength maxima at 544,547,550,553 and 556 nm. Curve a, fourth derivative curve taken with four Ah intervals of 1 nm. Curve b, fourth derivative of a similar spectrum comprised of the same five bands which were 100% Gaussian. (B) Fourth derivative curves of the spectrum shown in (A) with the addition of random noise with a peak-to-peak amplitude equal to 1/100 of the maximum amplitude of the spectrum. Curve a, fourth derivative taken with four Ah intervals of i.5 nm without the addition of the noise. Curve b, same as curve A with noise added to the spectrum. Curve c, curve b after a 10-point digital filtering. Curve d, fourth derivative of spectrum plus noise taken with 2~h intervals of 1.3, 1.4, 1.5, and 1.7 nm. Curve e, curve d after a 10-point digital filtering. Curve f, curve d after a 30-point digital filtering. Curve g, fourth derivative of spectrum plus noise taken with Ah intervals of 1.0, 1.1, 1.2, and 1.4 nm after a 10-point digital filtering. Curve h, same as curve g except that a 20-point digital filtering was used. Vertical lines indicate the wavelength position at 544, 547,550, 553, and 556 nm. separately, the two peaks would be resolved by a fourth derivative comp r i s e d o f f o u r 1 n m Ah i n t e r v a l s s i n c e , in t h a t c a s e , t h e r a t i o s b e t w e e n b a n d w i d t h , b a n d s e p a r a t i o n , a n d t h e Ah d i f f e r e n t i a t i n g i n t e r v a l (5 : 3 : 1) is t h e s a m e as f o r t h e p a i r o f b a n d s in Fig. 1A d i f f e r e n t i a t e d b y 4 n m i n t e r v a l s ( 2 0 : 1 2 : 4 ) . H o w e v e r , w h e n all five G a u s s i a n b a n d s a r e p r e s e n t e d together some destructive interference occurs. Noise T h e i n f l u e n c e o f n o i s e on t h e f o u r t h d e r i v a t i v e s p e c t r a w a s e x a m i n e d b y a d d i n g a k n o w n l e v e l o f r a n d o m n o i s e to t h e s i m u l a t e d s p e c t r u m s h o w n in Fig. 6 A ( c o m p r i s e d o f t h e 90% G a u s s i a n b a n d s ) . R a n d o m n o i s e w i t h a p e a k - t o - p e a k a m p l i t u d e e q u a l t o 1/100 o f t h e m a x i m u m a m p l i t u d e o f t h e s p e c t r u m w a s g e n e r a t e d b y a r a n d o m n o i s e p r o g r a m in t h e c o m p u t e r . T h e d i f f e r e n t i a t i n g i n t e r v a l s w e r e i n c r e a s e d to 1.5 n m f o r c u r v e s a,
512
SPECIALIZED TECHNIQUES
[45]
b, and c in Fig. 6B which increased the magnitude of the fourth derivative bands about fivefold. (The fourth derivative curves in Fig. 6A taken with h~ intervals of 1.0 were multiplied by a factor of 5 for plotting while those in Fig. 6B taken with intervals of 1.5 nm were plotted without multiplication.) Curve a in Fig. 6B, which was obtained without addition of the noise to the spectrum, shows some loss of resolution, in comparison with curve a in Fig. 6A, because of the larger Ah intervals. Addition of the moderate level of noise to the simulated spectrum (SIN = 100) generates an appreciable amount of noise in the fourth derivative spectrum (curve b of Fig. 6B). Consider the random noise apart from the absorbance signal and assume that the noise at each digitized wavelength is expressed by an amplitude between 0 and 1. The first derivative of the noise by the AX method will give values between 1 and - 1 which will be independent of the size of Ah. A similar doubling of noise occurs at each differentiation until by the fourth derivative the noise will assume values between 8 and - 8 . Thus, noise increases as 2 n, where n is the order of the derivative. Now consider a spectrum with random noise which is to be resolved into its individual components by a fourth derivative operation. If the size of hh is decreased to increase the resolution, the amplitude of the fourth derivative will decrease (approaching A~,4 dependence) while the noise will remain constant at 16 times the noise in the original data. Thus, the signal-tonoise ratio of the fourth derivative spectrum will be strongly dependent on the size of the hh differentiating intervals. Suppression of Noise by Digital Filtering Noise can be suppressed by using a convolution function to give a running average of a specified number of adjacent readings. Such a convolution function can be represented by a wavelength bar with a specified number of factors of 1 spaced each digitization interval along the bar. The convolution function steps along the digitized curve summing that number of adjacent readings and plotting the average value at the wavelength corresponding to the midpoint of the wavelength bar. The ability of the digital filtering technique to suppress random noise is demonstrated in curve c of Fig. 6B. Curve c (Fig. 6B) was derived from curve b (Fig. 6B) by a digital filtering convolution function which averaged 10 points.
Suppressing Noise by Using Different Ah Intervals It was shown previously 4 that the noise in the fourth derivative curves can be reduced substantially without loss of resolution by using four Ah
[45]
FOURTH DERIVATIVE SPECTRA
513
values which are slightly different rather than all being identical. That improvement is apparent in Fig. 6B by comparing the fourth derivative curve obtained with Ah intervals of 1.3, 1.4, 1.5, and 1.7 nm (curve d in Fig. 6B) versus that obtained with four equal Ah intervals of 1.5 nm (curve b in Fig. 6B). The reason for the lower noise when the four different Ah values were used is apparent if one considers the equivalent convolution functions. When all four Ah's are the same, the factor at the midpoint of convolution function is six, while the factor at the one-quarter and threequarters points is - 4 . In that case each fourth derivative value is an average in which one measurement and the associated noise are multiplied by 6 and two are multiplied by 4. When the four Ah's are used which are slightly different, the factor of 6 is replaced by six separate factors of 1 separated by 0.1 nm (the digitizing interval) clustered about the midpoint of the convolution function, and the factors of - 4 are each replaced by four separate factors of - 1 clustered about the one-quarter and threequarters points. In the latter case, each fourth derivative value is the average of 16 individual measurements so that the noise of the 16 separate readings is averaged together. This greater averaging of noise when four different Ah's are used results in a lower noise level in the fourth derivative curve. It is important, however, in order that the information not be degraded by using different A~, intervals that the differences between the Ah intervals be small compared to the size of the intervals. This is one reason why it is advantageous to digitize the spectral data at smaller wavelength intervals (e.g., 0.1 nm) than the resolution of the spectrophotometer would appear to require. The Fourth Derivative of Random Noise Close inspection of curves c and d in Fig. 6B reveals a phenomenon which is often observed in fourth derivative curves of experimental spectra; namely, that the fourth derivative random noise is not necessarily random. Instead a periodic fluctuation with a wavelength of approximately 2Ah is often found in spectral regions devoid of absorption bands. Such a periodic fluctuation is apparent in curves c and d. The oscillation is also present in curve b, but there it is largely buried in the noise. The source of the oscillation is inherent in the fourth derivative operation. As noted previously, the convolution function for the fourth derivative has a factor of 6 at (or clustered about) the midpoint, factors of - 4 at (or clustered about) the one-quarter and three-quarters points, and factors of 1 at each end. The wavelength interval between the negative values at the one-quarter and three-quarters points is 2Ah. Likewise the wavelength interval between the positive values of 1 on each end and the 6 in the
514
SPECIALIZED TECHNIQUES
[45]
middle is 2Ah. In essence, the convolution function for the fourth derivative is a tuned filter which tends to select and emphasize features in the random noise which reinforce the 2Ah period of the convolution function. Noise can be suppressed in the fourth derivative by using both the digital filtering and the unequal Ah intervals. The order in which the two processes are taken is immaterial. The results obtained by carrying out the digital filtering on the original spectrum followed by the fourth derivative are identical, point by point, with the results obtained by taking the fourth derivative first followed by the digital filtering. Curve e (in Fig. 6B) shows the results of a 10-point digital filtering on the fourth derivative curve obtained with A;~ intervals of 1.3, 1.4, 1.5, and 1.7 nm. The random noise component is suppressed but the oscillation remains. In this case the improvement due to digital filtering is scarcely significant, since it is the magnitude of the oscillation which limits the credibility of the fourth derivative bands. The oscillation can be suppressed by using a convolution function for digital filtering which is comparable in length to the wavelength of the oscillation. The wavelength of the oscillation in curve d is approximately 3 nm. The results obtained after a 30 point convolution function, spanning 2.9 nm (curve f in Fig. 6B), show that the oscillation is markedly suppressed. However, in this case, as is often the case with complex spectra, the separation between the absorption maxima is comparable to the wavelength of the oscillation so that the oscillation cannot be suppressed by digital filtering without sacrificing information. An attempt is shown in curves g and h (Fig. 6B) to use smaller Ah intervals to increase resolution and to make the wavelength of the oscillation smaller than the 3 nm band separation. The wavelength of the oscillation with Ah intervals of 1.0, 1. I, 1.2, and 1.4 nm will be approximately 2.4 nm. The amplitude of the fourth derivative bands is less with the smaller Ah intervals so that these spectra are multiplied by a factor of 2 for plotting. The oscillation remains after the 10-point digital filtering (curve g in Fig. 6B) and is more pronounced with the smaller Ah intervals because of the amplification factor. However, with 20-point digital filtering (curve h in Fig. 6B), the oscillation is suppressed more than the fourth derivative bands. Nevertheless, the results of curve h do not represent an appreciable improvement over those in curve e (Fig. 6B) obtained with the larger Ah intervals. It is apparent that even with the noise suppression techniques the resolvability of a spectrum is limited by the level of noise in the data. In examining any of the fourth derivative curves in Fig. 6B it would be difficult to decide whether the 547 and 553 nm bands were real or were part of the oscillation inherent in the fourth derivative operation. How-
[46]
MEMBRANE
SURFACE
POTENTIAL
MEASUREMENTS
515
ever, with independent spectral measurements the peaks in the oscillation will appear at random, whereas the position of real bands will be fixed. Therefore, the validity of the 547 and 553 nm bands could be confirmed by their reproducibility. Or if the fourth derivative curves from several independent spectral measurements were added together, the oscillations would tend to average out while the bands would reinforce. Conclusion The question can be raised as to how or why the fourth derivative can extract information that is not apparent in the original spectrum. How can two spectral curves, as similar as the two in Fig. 4, be so different in their fourth derivatives or how can so much detail be present in the fiat top of the spectral curve of Fig. 6A. The answer lies in the width of the pen line. In general, the width of the penline will be in the order of 1/100 to 1/500 of the maximum amplitude of the curve. The information content of a 16-bit word in the computer is essentially 65,000. Thus, with the spectral curves generated by the computer there may be more information buried within the width of the pen line than is revealed in the amplitude values of the curve. With experimental measurements the limitation becomes the signal-to-noise ratio. Signal-to-noise ratios of several thousand can be achieved readily in many photometric types of measurement. The signalto-noise ratio in the measurements presented in Fig. 1 was probably about 1000. Thus, there is more information in the spectral data than is apparent in the spectra curve and that additional information can be extracted by the fourth derivative. Acknowledgment: The work of Ms. S. Swarbrickin programmingthe computer is gratefullyacknowledged. The work was supported by grants from the National Science Foundation,GB-43512, and the United States Public Health Service, GM-20648.
[46] M e m b r a n e
Surface Potential Measurements with Amphiphilic Spin Labels
By R. J. MEHLHORN and L. PACKER The electrical potential at a membrane surface governs the distribution of ions within the aqueous interface, and hence this potential may be implicated in control mechanisms of transmembrane ion fluxes. Therefore, it is important to measure surface potentials accurately.
METHODS IN ENZYMOLOGY, VOL. LVI
Copyright (~) 1979 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-181956-6
FOURTH
DERIVATIVE
SPECTRA
501
fluorescence increase, consistent with the knowledge that it stimulates energy-linked reactions in mitochondrial fragments. Energy-linked ANS fluorescence increases can be obtained with N A D H (0.1-1.0 mM) or ascorbate (2 mM plus phenazine methosulfate) (0.1 mM), which are reversed, respectively, by rotenone (2/zM) and KCN (1 mM). When N A D H is employed as substrate, the wavelengths to be selected for excitation and emission are 436 and 560 nm, respectively, in order to minimize the interference of N A D H fluorescence on the ANS signal. The fluorescence decrease of ANS in mitochondrial fragments reflects an energy-linked process and can be utilized down to protein concentration of the order of 0.05 mg/ml.
[45] F o u r t h D e r i v a t i v e S p e c t r a By WARREN L. BUTLER Higher derivative analysis provides a powerful and useful technique to aid in the resolution of complex spectra. The efficacy of the analysis can be readily demonstrated with digitized spectral data. Derivatives are obtained by the simple procedure of computing a difference spectrum between the original curve, A(h), and the same curve shifted a finite wavelength interval, A(h + Ak), with the difference value being assigned the wavelength corresponding to the midpoint of Ah. Higher derivatives are obtained by repeating the Ah differentiation the desired number of times. The primary thrust of this chapter is to elucidate and justify the use of fourth derivatives for the analysis of spectral data. Fourth Derivative Analysis of an Experimental Spectrum The absorption spectrum of beef heart mitochondria measured at liquid nitrogen temperature (Fig. l) is used to demonstrate the kinds of information that can be extracted from the data by a fourth derivative operation. This spectrum was measured with a single-beam spectrophotometer on line, via a 12-bit A-D converter, with a small computeH with data points being taken every 0. l nm. The spectrum is a complex mixture of the absorption bands of several b- and c-type cytochromes. Some of the absorption bands are apparent only as small shoulders which cannot be assigned precise wavelength maxima. The absorption bands of other components known to be present are completely buried in the cornl W. L. B u t l e r , Vol. 24, Part B, p. 3.
METHODS IN ENZYMOLOGY, VOL. LVI
Copyright © 1979 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-181956-6
502
[45]
SPECIALIZED TECHNIQUES
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Wavelength, nm FIG. I. Absorption spectrum (dithionite-reduced minus oxidized difference spectrum) of mitochondria (7.5 mg protein per milliliter). See Davis e t al. z for details. The fourth derivative curve in the upper part of the figure was taken with Ah differentiating intervals of 1.0, 1.1, 1.2 and 1.4 nm.
[45]
FOURTH DERIVATIVE SPECTRA
503
plexity of the spectrum. The constituent absorption bands are resolved to a much greater degree in the fourth derivative of the absorption spectrum obtained with Ah differentiating intervals of 1.0, 1.1, 1.2 and 1.4 nm, shown in the upper part of Fig. 1. All of the fourth derivative peaks which are marked for their wavelength maxima correspond to absorption bands with essentially the same wavelength maxima in the original spectrum. In the study from which the data in Fig. 1 was taken, 2 absorption spectra of all of the individual cytochromes at - 196° were obtained by separating the mitochondria into various submitochondrial complexes and measuring difference spectra between differential redox treatments. In this manner, the maxima observed in the fourth derivative spectrum of the mitochondria at - 196° could be compared against the absorption maxima of the individual cytochrome components at - 196°. The shoulder at about 562 nm in the absorption spectrum which is resolved as a sharp peak at 562.5 nm in the fourth derivative spectrum is due to cytochrome bx. This cytochrome at -196 ° has a sharp ai band at 562.5 nm and a smaller a2 band at 554 nm. The absorption maximum at 558.5 nm is due to the combination of two cytochromes; one, cytochrome bK, has a single band at 559.5 nm at - 196°C and the other, the cytochrome b of succinic dehydrogenase (complex II), has an al band at 557.5 nm and an a~ band at 550 nm. The a bands of these two cytochromes are resolved in the fourth derivative spectrum, although the 557.5 nm band appears to be shifted to 557 nm. The fourth derivative band at 553.5 nm is due to cytochrome cl which has a maximum at 553 nm with probably some contribution from the 554 nm a2 band of cytochrome b+. The sharp fourth derivative bands at 547.5 and 544 nm are due to the al and a2 bands of cytochrome c, which is a soluble component that can be washed free from the membrane fragments. Most of the/3 bands (in the spectral region between 500 and 540 nm) can also be assigned to particular cytochromes. It is worth noting that the sample of mitochondria used for the spectral measurements shown in Fig. 1 was by no means ideal from the standpoint of making photometric measurements. The mitochondria were suspended in a buffer medium and frozen to -196 ° resulting in a sample which was about 3 mm thick, highly scattering, and optically quite dense. Even so, the measurements were made with sufficiently low noise that noise was not a problem in the fourth derivative curve. One of the purposes of this chapter is to explore the effects of noise on fourth derivative spectra and to explore techniques to minimize those effects. 2 K. A. Davis, Y. Hatefi, K. L. Poff, and W. L. Butler, Biochim. Biophys. Acta 325, 341 (1973).
504
SPECIALIZED TECHNIQUES
[45]
Simulated Spectra The rationale for using higher derivatives to resolve complex spectra was established in a previous study2 Spectra of known content were simulated with a small computer by summing together analytical expressions for bands which were specified by their wavelength maximum, amplitude, half-width and band shape (i.e., specified mixtures of Gaussian and Lorentzian bands). The spectra were constructed so that some of the individual bands were not apparent in the sum, and the ability of second and fourth derivative curves to resolve those bands was examined by summing together the analytical expressions for the second and fourth derivatives of the individual bands (the sum of the derivatives is identical to the derivative of the sum). Even numbered derivatives have the advantage over the odd numbered ones in that they have maxima at the same position as the maxima of the original bands, which makes the correspondence between the derivative bands and the bands in the original spectrum more apparent. (Strictly speaking, derivatives which are odd numbered multiples of two, such as the second and the sixth, give minima at the position of the band maxima, but the negative of these curves can be plotted so that the minima appear as maxima.) The results of the previous study confirmed that the second and fourth derivative curves could reveal the presence of constituent bands which were not apparent in the simulated spectrum, that the bands were resolved better by the fourth derivative than by the second derivative, and that the wavelength maxima in the fourth derivative curves agreed closely to the wavelength maxima of the original bands. The enhanced resolution results because the derivative bands are narrower than the original bands, e.g., a Lorentzian band of 20 nm half-width has a half-width of 5 nm in the fourth derivative. Thus, the fourth derivative of a spectrum comprised of overlapping bands may show well-defined peaks at the wavelength maxima of the bands even though the bands blend together in the spectrum to obscure the individual maxima. The fourth derivative of an individual band has negative minima on both sides of the central maximum, and these minima can interact with the fourth derivative curves of adjacent bands to create a potential for artifacts. Small band shifts may result from such interactions and bands may even be obliterated. It was also pointed out previously~ that false bands may be generated by the interaction between the side wings of adjacent bands. For instance, with two Gaussian bands of 20 nm halfwidth spaced 40 nm apart, the negative side wings come together to form what appears to be a third band halfway between the two bands. The :~W. L. Butler and D. W. Hopkins,Photochem. Photobiol. 12, 439 (1970).
[45]
FOURTH DERIVATIVE SPECTRA
505
unmarked band near 540 nm in the fourth derivative spectrum in Fig. 1 may be an example of such a false band. None of the spectra of the individual cytochromes showed a 540 nm absorption band which would be correlated with that fourth derivative band. If it is a false band it is one of the few that have appeared to date in the fourth derivative spectra of experimental data. While the potential for such artifactual bands should be kept in mind, in actual practice it has not proved to be a serious problem. ~,~4A/~h4 versus d 4 A M h 4 In the previous study, the fourth derivatives of the simulated spectra were obtained mathematically from the analytical expressions for the Gaussian and Lorentzian bands. Such analytically derived fourth derivative curves showed the maximum resolution which could be obtained theoretically by a fourth derivative. However, the question of how the resolving power of a fourth derivative obtained with four finite ~;~ intervals depended on the size of those intervals was not addressed because of the limitations set by the 12-bit resolution of the computer used in those studies. In the present study sufficient accuracy was available (a computer with 16-bit resolution was used) to make those calculations and that question is examined in Fig. 2. (All of the simulated spectra presented in this chapter are comprised of 1000 points spanning a simulated spectral region from 500 to 600 nm, except in one case, Fig. 3, where the spectral range is from 450 to 650.) The sum of two Gaussian bands of equal amplitude and 20 nm half-width with wavelength maxima at 544 and 556 nm is shown in Fig. 2A. The mathematical fourth derivative is presented as curve A w. Fourth derivative curves taken by the Ah method are also shown. With four ~h intervals of 2 nm each, the resolution is equal to that obtained by the mathematical differentiation. With intervals of 4 nm, the two bands are still resolved but not quite as well and with intervals of 8 nm the resolution is lost. The previous study3 using the mathematical differention showed that Lorentzian bands could be resolved much better than Gaussian bands. While Gaussian bands of 20 nm half-width could not be resolved by the fourth derivative if the bands were closer together than 12 nm, Lorentzian bands of 20 nm half-width could be resolved even when the bands were 6 nm apart. The ability of the fourth derivative to resolve a spectrum which is the sum of two such Lorentzian bands with maxima at 547 and 553 nm is shown in Fig. 2B. In this case in order to achieve a resolution with four 5h differentiations which approaches that of the mathematical fourth derivative, A Iv, the intervals have to be 1 nm or less. The resolution is somewhat
506
SPECIALIZED TECHNIQUES A
[45]
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Wavelength(nm) FIG. 2. Computer-generated spectra. (A) Two Gaussian bands with 20 nm half-widths and wavelength maxima at 544 and 556 nm were summed together. (B) Two Lorentzian bands with 20 nm half-widths and wavelength maxima at 547 and 553 nm were summed together. The mathematical fourth derivative calculated from the analytical expressions for the Gaussian and Lorentzian bands are shown as the M v curves. The fourth derivative by four Ah differentiations are identified by the four Ah intervals used. The numbers outside the parentheses indicate the multiplication factor by which the curves were multiplied for plotting. less with intervals of 2 n m and is lost with intervals of 4 nm. Thus, the A)~ method o f differentiation confirms that the resolvability of Lorentzian bands is inherently greater than that o f Gaussian bands but, as should be expected, smaller Ah intervals are needed to achieve that resolution. T h e n u m b e r s outside the p a r e n t h e s e s in Fig. 2 indicate the amplification factors b y which the curves w e r e multiplied to m a k e t h e m equal in amplitude to the m a t h e m a t i c a l fourth derivative. I f the fourth derivative curves obtained by the Ah differentiations w e r e true fourth derivatives, the amplitudes should be proportional to Ah 4, i.e., doubling the Ah interval should increase the amplitude 16 times. T h e fact that the amplitude ratios due to doubling Ah are less than 16 for the curves in Fig. 2 is due to the deviations f r o m the true fourth derivative c u r v e which o c c u r as the size the Ah intervals is increased. B a n d Width Just as the amplitude o f the fourth derivative band has a strong dependence on the size o f the Ah intervals used for the differentiations (ap-
[45]
FOURTH DERIVATIVE SPECTRA
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FIG. 3. Computer-generated spectrum from three Gaussian bands with wavelength maxima at 540, 560, and 570 nm with amplitude ratios of 5:10:1 and half-widths of 20, 40 and 10 nm, respectively. Fourth derivative curves taken with four equal Ah intervals of 2, 8, and 16 nm are shown. Vertical lines indicate wavelengths of 540 and 570 nm.
proaching Ah4 for sufficiently small ah) so is the amplitude of the fourth derivative strongly dependent on the half-width of the original band, approaching an inverse fourth power dependence. Thus, narrow bands are highly selected for in a fourth derivative spectrum. Figure 3 shows the sum of three Gaussian bands at 540,560, and 570 nm in amplitude ratios of 5:10:1 and half-widths of 20, 40, and 10 nm, respectively. The fourth derivative curve of the simulated spectrum obtained with four Ah's of 2 nm shows the 540 nm band, but the presence of the large broad 560 nm band is almost obliterated by the fourth derivative signal of the small narrow band at 570 nm. One might suppose that using larger Ah intervals would enhance broad bands more than narrow bands so that it might be possible to preferentially select for the broad bands in the spectrum by increasing the size of the ah intervals. However, using four Ah intervals of 8 or 16 nm does not enhance the broad 560 nm band relative to the narrow 570 nm band. In fact, the broad 560 nm band is suppressed even more in the spectrum taken with the broader Ah intervals because the dominating influence of the narrow 570 nm band extends over a wider wavelength region.
508
SPECIALIZED TECHNIQUES
[45]
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Wavelength (nm)
Fro. 4. Computer-generated spectra. The solid curve is the sum of two 50% Gaussian50% Lorentzian bands of equal amplitude with 20 nm half-widths and wavelength maxima at 547 and 553 rim. The dashed curve (superimposed on the solid curve) is a single band, 55% Gaussian, with a 22 nm half-width and a wavelength maximum at 550 nm. Fourth derivative curves of these two spectra, solid and dashed, were taken with four Ah intervals of 1 nm. Vertical lines indicate wavelengths at 547 and 553 nm. Information Retrieval An e x a m p l e demonstrating the capacity of the fourth derivative analysis to reveal information that is not readily apparent in the original spectrum is presented in Fig. 4. T h e solid c u r v e in Fig. 4 is the sum of two bands o f equal amplitude which are 50% Gaussian and 50% Lorentzian with half-widths o f 20 nm and wavelength m a x i m a at 547 and 553 nm. S u p e r i m p o s e d on the solid c u r v e is a curve drawn in dashes which is a single band, 55% Gaussian, with a 22 nm half-width, a 550 nm wavelength m a x i m u m and an amplitude chosen to m a t c h that of the solid curve. The two simulated spectra are almost indistinguishable by visual inspection and the question is asked w h e t h e r the fourth derivative obtained by four Ah differentiations o f l n m each can distinguish the different origins of these two curves. Indeed the fourth derivative of the solid curve shows that that c u r v e is c o m p r i s e d of two c o m p o n e n t s with m a x i m a at 547 and 553 nm, while the fourth derivative of the dashed curves shows a single band with the m a x i m u m at 550 nm. It should be e m p h a s i z e d that this Ah method o f obtaining the fourth derivative is entirely independent of the composition o f the s p e c t r u m and is the same method that we use for experimental spectra where the shapes and the widths of the bands are generally unknown. It should be apparent from the results of Fig. 4 that a
[45]
FOURTH DERIVATIVE SPECTRA
509
spectral curve may contain information that is not readily available in the original data but which may be processed into recognizable forms by the higher derivative analysis. Convolution Functions and Even Higher Derivatives It was shown previously 4 that higher derivative operations obtained by successive Ah differentiations are equivalent to convolution functions. The convolution function for a fourth derivative obtained with four equal Ah intervals can be represented as a wavelength bar 4 Ah long with factors of 1 at the end points, a factor of 6 at the midpoint, and factors of - 4 at the one-quarter and three-quarters points. The factors for any particular derivative can be obtained by expanding (a - b) n, where n is the order of the derivative or by taking the factors from the appropriate row of Pascal's triangle. The convolution function steps sequentially along the curve averaging at each digitization wavelength the readings on the curve according to the factors on the wavelength bar. A convolution function giving a sixth derivative can be represented by a wavelength bar 6Ah long with factors - 1 , 6, - 1 5 , 20, - 1 5 , 6, - 1 spaced each Ah interval, while the convolution function for an eighth derivative would have the factors l, - 8 , 28, - 5 6 , 70, - 5 6 , 28, - 8 , 1 spaced at Ah intervals along the bar. The use of these convolution functions for higher derivatives is demonstrated in Fig. 5 on a spectrum comprised of two 20 nm half-width Lorentzian bands, 4 nm apart, with maxima at 548 and 552 nm. This spectrum is not resolved by the convolution function for the fourth derivative which is equivalent to four Ah differentiations of 1 nm each, nor would it be resolved by using smaller Ah intervals. The mathematical differentiation of these bands shows that this spectrum is not resolvable by a fourth derivative operation. However, these bands are just resolved in the sixth derivative and are well resolved in the eighth derivative, both being obtained with convolution functions which are equivalent to a series of equal Ak differentiating intervals of 1 nm. However, some fluctuations are apparent in the eighth derivative curve which may be due to limitations set by the 16-bit resolution of the computer. Band Shape The effect which small changes of band shape may have on the resolvability of a spectrum is demonstrated in Fig. 6A. The simulated spectrum is comprised of five bands of equal amplitude with wavelength maxima at 544, 547,550, 553, and 556 nm. All of the bands have a 5 nm half-width 4 W. L. Butler and D. W. Hopkins, Photochem. Photobiol. 12, 451 (1970).
510
SPECIALIZED TECHNIQUES
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[45]
II
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(1500) th (600)
4th (150)
550 600 Wavelength (nrn)
FIG. 5. Computer-generated spectrum fi'om two Lorentzian bands with 20 nm halfwidths and wavelength maxima at 548 and 552 nm. Fourth, sixth, and eighth derivative curves taken with convolution functions equivalent to Ah differentiations of 1 rim. Numbers in parentheses indicate factors by which the curves were multiplied for plotting.
and a shape which is 90% Gaussian and 10% Lorentzian. The fourth derivative of the spectrum taken with four Ah intervals of 1.0 nm (curve a) resolves all of the bands. It is apparent that the magnitude of bands in the fourth derivative curve bears no direct relationship to the magnitude of the corresponding bands in the original spectrum. The heights of the three fourth derivative bands in the middle of curve a are decreased by negative wings of the fourth derivative bands on each end. However, the wavelength maxima of the fourth derivative bands agree quite closely to the wavelength maxima of the original bands. The resolution shown in curve a, however, is markedly dependent on the 10% Lorentzian character of the bands. Curve b shows the fourth derivative curve obtained in the same manner from a simulated spectrum comprised of the same bands with the single exception that each band was 100% Gaussian. The difference in shape of the simulated spectrum resulting from the change of the bands from 90% Gaussian to 100% Gaussian was scarcely detectable. However, in the latter case, the fourth derivative bands of the 547 and 553 nm components are obliterated by the negative wings of the other bands. Using smaller Ah intervals would sharpen up the three fourth derivative bands that are present but would not restore the two bands which are missing. If any adjacent pair of the 100% Gaussian bands were analyzed
[45]
FOURTH DERIVATIVE SPECTRA
511
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Wavelength (nrn) Fie. 6. (A) Computer-generated spectrum from five 90% Gaussian, 10% Lorentzian bands of equal amplitude with 5 nm half-widths and wavelength maxima at 544,547,550,553 and 556 nm. Curve a, fourth derivative curve taken with four Ah intervals of 1 nm. Curve b, fourth derivative of a similar spectrum comprised of the same five bands which were 100% Gaussian. (B) Fourth derivative curves of the spectrum shown in (A) with the addition of random noise with a peak-to-peak amplitude equal to 1/100 of the maximum amplitude of the spectrum. Curve a, fourth derivative taken with four Ah intervals of i.5 nm without the addition of the noise. Curve b, same as curve A with noise added to the spectrum. Curve c, curve b after a 10-point digital filtering. Curve d, fourth derivative of spectrum plus noise taken with 2~h intervals of 1.3, 1.4, 1.5, and 1.7 nm. Curve e, curve d after a 10-point digital filtering. Curve f, curve d after a 30-point digital filtering. Curve g, fourth derivative of spectrum plus noise taken with Ah intervals of 1.0, 1.1, 1.2, and 1.4 nm after a 10-point digital filtering. Curve h, same as curve g except that a 20-point digital filtering was used. Vertical lines indicate the wavelength position at 544, 547,550, 553, and 556 nm. separately, the two peaks would be resolved by a fourth derivative comp r i s e d o f f o u r 1 n m Ah i n t e r v a l s s i n c e , in t h a t c a s e , t h e r a t i o s b e t w e e n b a n d w i d t h , b a n d s e p a r a t i o n , a n d t h e Ah d i f f e r e n t i a t i n g i n t e r v a l (5 : 3 : 1) is t h e s a m e as f o r t h e p a i r o f b a n d s in Fig. 1A d i f f e r e n t i a t e d b y 4 n m i n t e r v a l s ( 2 0 : 1 2 : 4 ) . H o w e v e r , w h e n all five G a u s s i a n b a n d s a r e p r e s e n t e d together some destructive interference occurs. Noise T h e i n f l u e n c e o f n o i s e on t h e f o u r t h d e r i v a t i v e s p e c t r a w a s e x a m i n e d b y a d d i n g a k n o w n l e v e l o f r a n d o m n o i s e to t h e s i m u l a t e d s p e c t r u m s h o w n in Fig. 6 A ( c o m p r i s e d o f t h e 90% G a u s s i a n b a n d s ) . R a n d o m n o i s e w i t h a p e a k - t o - p e a k a m p l i t u d e e q u a l t o 1/100 o f t h e m a x i m u m a m p l i t u d e o f t h e s p e c t r u m w a s g e n e r a t e d b y a r a n d o m n o i s e p r o g r a m in t h e c o m p u t e r . T h e d i f f e r e n t i a t i n g i n t e r v a l s w e r e i n c r e a s e d to 1.5 n m f o r c u r v e s a,
512
SPECIALIZED TECHNIQUES
[45]
b, and c in Fig. 6B which increased the magnitude of the fourth derivative bands about fivefold. (The fourth derivative curves in Fig. 6A taken with h~ intervals of 1.0 were multiplied by a factor of 5 for plotting while those in Fig. 6B taken with intervals of 1.5 nm were plotted without multiplication.) Curve a in Fig. 6B, which was obtained without addition of the noise to the spectrum, shows some loss of resolution, in comparison with curve a in Fig. 6A, because of the larger Ah intervals. Addition of the moderate level of noise to the simulated spectrum (SIN = 100) generates an appreciable amount of noise in the fourth derivative spectrum (curve b of Fig. 6B). Consider the random noise apart from the absorbance signal and assume that the noise at each digitized wavelength is expressed by an amplitude between 0 and 1. The first derivative of the noise by the AX method will give values between 1 and - 1 which will be independent of the size of Ah. A similar doubling of noise occurs at each differentiation until by the fourth derivative the noise will assume values between 8 and - 8 . Thus, noise increases as 2 n, where n is the order of the derivative. Now consider a spectrum with random noise which is to be resolved into its individual components by a fourth derivative operation. If the size of hh is decreased to increase the resolution, the amplitude of the fourth derivative will decrease (approaching A~,4 dependence) while the noise will remain constant at 16 times the noise in the original data. Thus, the signal-tonoise ratio of the fourth derivative spectrum will be strongly dependent on the size of the hh differentiating intervals. Suppression of Noise by Digital Filtering Noise can be suppressed by using a convolution function to give a running average of a specified number of adjacent readings. Such a convolution function can be represented by a wavelength bar with a specified number of factors of 1 spaced each digitization interval along the bar. The convolution function steps along the digitized curve summing that number of adjacent readings and plotting the average value at the wavelength corresponding to the midpoint of the wavelength bar. The ability of the digital filtering technique to suppress random noise is demonstrated in curve c of Fig. 6B. Curve c (Fig. 6B) was derived from curve b (Fig. 6B) by a digital filtering convolution function which averaged 10 points.
Suppressing Noise by Using Different Ah Intervals It was shown previously 4 that the noise in the fourth derivative curves can be reduced substantially without loss of resolution by using four Ah
[45]
FOURTH DERIVATIVE SPECTRA
513
values which are slightly different rather than all being identical. That improvement is apparent in Fig. 6B by comparing the fourth derivative curve obtained with Ah intervals of 1.3, 1.4, 1.5, and 1.7 nm (curve d in Fig. 6B) versus that obtained with four equal Ah intervals of 1.5 nm (curve b in Fig. 6B). The reason for the lower noise when the four different Ah values were used is apparent if one considers the equivalent convolution functions. When all four Ah's are the same, the factor at the midpoint of convolution function is six, while the factor at the one-quarter and threequarters points is - 4 . In that case each fourth derivative value is an average in which one measurement and the associated noise are multiplied by 6 and two are multiplied by 4. When the four Ah's are used which are slightly different, the factor of 6 is replaced by six separate factors of 1 separated by 0.1 nm (the digitizing interval) clustered about the midpoint of the convolution function, and the factors of - 4 are each replaced by four separate factors of - 1 clustered about the one-quarter and threequarters points. In the latter case, each fourth derivative value is the average of 16 individual measurements so that the noise of the 16 separate readings is averaged together. This greater averaging of noise when four different Ah's are used results in a lower noise level in the fourth derivative curve. It is important, however, in order that the information not be degraded by using different A~, intervals that the differences between the Ah intervals be small compared to the size of the intervals. This is one reason why it is advantageous to digitize the spectral data at smaller wavelength intervals (e.g., 0.1 nm) than the resolution of the spectrophotometer would appear to require. The Fourth Derivative of Random Noise Close inspection of curves c and d in Fig. 6B reveals a phenomenon which is often observed in fourth derivative curves of experimental spectra; namely, that the fourth derivative random noise is not necessarily random. Instead a periodic fluctuation with a wavelength of approximately 2Ah is often found in spectral regions devoid of absorption bands. Such a periodic fluctuation is apparent in curves c and d. The oscillation is also present in curve b, but there it is largely buried in the noise. The source of the oscillation is inherent in the fourth derivative operation. As noted previously, the convolution function for the fourth derivative has a factor of 6 at (or clustered about) the midpoint, factors of - 4 at (or clustered about) the one-quarter and three-quarters points, and factors of 1 at each end. The wavelength interval between the negative values at the one-quarter and three-quarters points is 2Ah. Likewise the wavelength interval between the positive values of 1 on each end and the 6 in the
514
SPECIALIZED TECHNIQUES
[45]
middle is 2Ah. In essence, the convolution function for the fourth derivative is a tuned filter which tends to select and emphasize features in the random noise which reinforce the 2Ah period of the convolution function. Noise can be suppressed in the fourth derivative by using both the digital filtering and the unequal Ah intervals. The order in which the two processes are taken is immaterial. The results obtained by carrying out the digital filtering on the original spectrum followed by the fourth derivative are identical, point by point, with the results obtained by taking the fourth derivative first followed by the digital filtering. Curve e (in Fig. 6B) shows the results of a 10-point digital filtering on the fourth derivative curve obtained with A;~ intervals of 1.3, 1.4, 1.5, and 1.7 nm. The random noise component is suppressed but the oscillation remains. In this case the improvement due to digital filtering is scarcely significant, since it is the magnitude of the oscillation which limits the credibility of the fourth derivative bands. The oscillation can be suppressed by using a convolution function for digital filtering which is comparable in length to the wavelength of the oscillation. The wavelength of the oscillation in curve d is approximately 3 nm. The results obtained after a 30 point convolution function, spanning 2.9 nm (curve f in Fig. 6B), show that the oscillation is markedly suppressed. However, in this case, as is often the case with complex spectra, the separation between the absorption maxima is comparable to the wavelength of the oscillation so that the oscillation cannot be suppressed by digital filtering without sacrificing information. An attempt is shown in curves g and h (Fig. 6B) to use smaller Ah intervals to increase resolution and to make the wavelength of the oscillation smaller than the 3 nm band separation. The wavelength of the oscillation with Ah intervals of 1.0, 1. I, 1.2, and 1.4 nm will be approximately 2.4 nm. The amplitude of the fourth derivative bands is less with the smaller Ah intervals so that these spectra are multiplied by a factor of 2 for plotting. The oscillation remains after the 10-point digital filtering (curve g in Fig. 6B) and is more pronounced with the smaller Ah intervals because of the amplification factor. However, with 20-point digital filtering (curve h in Fig. 6B), the oscillation is suppressed more than the fourth derivative bands. Nevertheless, the results of curve h do not represent an appreciable improvement over those in curve e (Fig. 6B) obtained with the larger Ah intervals. It is apparent that even with the noise suppression techniques the resolvability of a spectrum is limited by the level of noise in the data. In examining any of the fourth derivative curves in Fig. 6B it would be difficult to decide whether the 547 and 553 nm bands were real or were part of the oscillation inherent in the fourth derivative operation. How-
[46]
MEMBRANE
SURFACE
POTENTIAL
MEASUREMENTS
515
ever, with independent spectral measurements the peaks in the oscillation will appear at random, whereas the position of real bands will be fixed. Therefore, the validity of the 547 and 553 nm bands could be confirmed by their reproducibility. Or if the fourth derivative curves from several independent spectral measurements were added together, the oscillations would tend to average out while the bands would reinforce. Conclusion The question can be raised as to how or why the fourth derivative can extract information that is not apparent in the original spectrum. How can two spectral curves, as similar as the two in Fig. 4, be so different in their fourth derivatives or how can so much detail be present in the fiat top of the spectral curve of Fig. 6A. The answer lies in the width of the pen line. In general, the width of the penline will be in the order of 1/100 to 1/500 of the maximum amplitude of the curve. The information content of a 16-bit word in the computer is essentially 65,000. Thus, with the spectral curves generated by the computer there may be more information buried within the width of the pen line than is revealed in the amplitude values of the curve. With experimental measurements the limitation becomes the signal-to-noise ratio. Signal-to-noise ratios of several thousand can be achieved readily in many photometric types of measurement. The signalto-noise ratio in the measurements presented in Fig. 1 was probably about 1000. Thus, there is more information in the spectral data than is apparent in the spectra curve and that additional information can be extracted by the fourth derivative. Acknowledgment: The work of Ms. S. Swarbrickin programmingthe computer is gratefullyacknowledged. The work was supported by grants from the National Science Foundation,GB-43512, and the United States Public Health Service, GM-20648.
[46] M e m b r a n e
Surface Potential Measurements with Amphiphilic Spin Labels
By R. J. MEHLHORN and L. PACKER The electrical potential at a membrane surface governs the distribution of ions within the aqueous interface, and hence this potential may be implicated in control mechanisms of transmembrane ion fluxes. Therefore, it is important to measure surface potentials accurately.
METHODS IN ENZYMOLOGY, VOL. LVI
Copyright (~) 1979 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-181956-6
