Anal. Chem. 1982, 5 4 ,
Sediment trap samples were generally too small to allow duplicate analyses for U, Th, and Pa. However, sample sizes from PARFLUX Site E (21) were sufficiently large to allow duplicate analysis of one sample as part of this work. Material from 389 m at Site E was chosen since it had the lowest specific activitieEi and therefore it was considered to be the most difficult sample for which to obtain reproducible results. Results of duplicate analyses are given in Table IV along with other results from sediment trap samples. Uncertainties are larger for the second analysis because a smaller sample size was used. Results of the two analyses are in good agreement. Complete results of this work are discussed in detail elsewhere (22). Analyses of PARFLUX Site P (21)sediment trap samples for eoSr, 137Cs,*41Am,and 55Fewere carried out by personnel in the laboratory of V. T. Bowen (WHOI) and these results will also be reported elsewhere (23). Selected results for particulate material sampled with sediment traps and by in situ fitration are presented in Tables IV and V, respectively. Uncertainties resulting ffrom counting statistics for samples and standards as well an blank and background corrections have been propagated to the final reported values. There are few reported data with which to compare these results. Brewer et al. (19) measured concentrations of several isotopes in the site E sediment trap samples for which results are presented here, but by slightly different methods. The agreement between their results and those in Table IV is very good.
ACKNOWLEDGMENT We are grateful to S. Honjo, D. W. Spencer, and P. G. Brewer for providing the sediment trap samples, and to P. L. Sachs for assistance in the assembly and deployment of the in situ filtration systems. The inclusion of the fallout isotopes in this method resulted from many discussions with H. D. Livingston. M. P. Bacon and C. W. Sill provided helpful comments on the manuwript. Calibrated 242Puand 243Am tracers and 239Puand 241Amcalibrated 01 sources were kindly provided from the laboratory of V. T. Bowen.
1147
1147-1152
LITERATURE CITED Ku, T. L. Ph.D. Dissertation, Columbia University, New York, 1966. Rosholt, J. N.; Szabo, B. J. I n “Modern Trends in Activation Analysis”; DeVoe, J. F., LaFleur, P. D., Eds.; National Bureau of Standards: Washlngton, DC, 1969; Spec. Publ. 312, Vol. 1, pp 327-333. Kraemer, T. F. Ph.D. Dissertation, University of Miami, Coral Gables, FL, 1975. SIII, C. W.; Puphai, K. W.; Hindman, F. D. Anal. Chem. 1974, 4 6 , 1725-1 737. SIII, C. W. Anal. Chem. 1978, 50 1559-1571. Sill, C. W.; Williams, R. L. Anal. Chem. 1981, 53,412-415. Krishnaswami, S.;Sarin, M. M.Anal. Chlm. Acta 1978, 83,143-156. Lal, D.; Schink D. R. Rev. Scl. Instrum. 1980, 37,395. Sill, C. W. Anal. Chem. 1974, 4 6 , 1426-1431. Spencer, D. W.; Sachs, P. L. Mar. Geol. 1970, 9 , 117-136. Krishnaswami, S.; Lal, D.; Somayajulu, B. L. K.; Weiss, R. F.; Craig, H. Earth Planer. Scl. Lett. 1978, 32,420-429. Keller, C. Angew. Chem., Int. Ed. Engl. 1988, 5, 23-35. Sill, C. W. Health fhys. 1975, 2 9 , 619-626. Wong, K. M.; Noshkvn V. E.; Bowen, V. T. I n “Reference Methods for Marine Radioactivity Studies”; International Atomic Energy Agency: Vlenna, 1970; Tech. Rep. Ser.-IAEA, No. 118, pp 119-127. Labeyrie, L. D.; Livingston, H. D.; Gordon; A. G. Nucl. Insfrum. Methods 1075, 728,575-580. Livingston, H. D.; Mann, D. R.; Bowen, V. T. I n “Analytical Methods in Oceanography”; Gibb, Thomas R. P., Jr., Ed.; American Chemical Soclety: Washington, DC, 1975. “Reference Methods for Marine Radioactivity Studies 11. Iodine, Ruthenium, Silver, Zirconium, and the Transuranic Elements”; International Atomic Energy Agency: Vienna, 1975; Tech. Rep. Ser.-IAEA, No. 69. D 5Sil, C. W.; Olson, D G. Anal. Chem. 1970, 4 2 , 1596-1607. Brewer, P. G.; Nozaki, Y.; Spencer, D. W.; Fleer, A. P. J. Mar. Res. 1980. 38. 703-728. Rosholt, J. N.; Prijana; Noble, D. C. €con. Geol. 1971, 6 6 , 1061-1 069. Honjo, S.J. Mar. Res. 1980, 38,53-97. Anderson, R. F. Ph.D. Dissertation, Massachusetts Institute of Technology-Woods Hole Oceanographic Institution Joint Program in Oceanography; Woods Hole, MA, 1981. Livingston, H. D.; Anderson, R. F., unpublished results.
--.
FCECEIVEDfor review August 24, 1981. Resubmitted and accepted March 9, 198%. Financial support for this work was plrovided in part by the National Science Foundation under Chants OCE 7826318, OCE 7825724, and OCE 7727004, the Department of Energy under Contract EY-764-02-3566, and a fellowship to R.F.A. from the WHOI Education Office.
Titration of Individual Components in a Mixture with Resolution of Difference Spectra, pKs, and Redox Transitions Richard
I. Shrager”
Laboratory of Applled Studies, Divlsion of Computer Research and Technology, National Institutes of Health, Bethesda, Maryland 20205
Richard W. Hendler Laboratory of Cell Biology, IUational Heart, Lung, and Blood Institute, National Institutes of Health, Bethesda, Maryland 20205
A method Is presented far analyzing a set of spectra from a tltratlon, to determine the transitions of components from one state to another (e.g., pKs or mldpolnt potentials) and to determlne the individual difference spectra of those components. The use of singular value decomposition (SVD), a completely automatic procedure, replaces the modellng of the original spectra, e.g., by sums of Gausslans. SVD is partlcularly useful for complex spectra wlth overlapping features. Techniques of nolse suppression are dlscussed. A concise description of the method is glven, along with some examples uslng laboratory data.
A spectrum of a complex biological system such as a cell membrane will contain all of the individual spectra of the
components. If the membrane contains the respiratory chain of the organism, spectral contributions from a variety of dehydrogenase enzymes, cytochromes,iron-sulfur proteins, and cofactors will be included. Each of these components has a different spectrum for its reduced and oxidized conditions and the spectrum presented for any particular redox state of the medium will be determined by the relative concentrations of these two components as specified by the Nernst equation
RT [oxidized species] E = Eo + - In nF [reduced species] We have recently developed and described techniques for generating a large number of such complex spectra, each at a different oxidizing potential of the suspending medium (1). Contained in this large amount of data is information on the
This article not subject to U S . Copyright. Publlshed 1982 by the American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 54, NO. 7, JUNE 1982
number of individual redox components their unique spectra, their redox potentials, and the numbers of electrons each species takes up and donates. Extracting this kind of information is the problem we wish to solve. By convention we will use absorbance (a) vs. wavelength (w)as our notation for a spectrum although this work clearly applies to other types of spectra as well. By titration, we mean the stepwise addition of titrants, either physically or by electrodic conversion of mediator titrants. Associated with each step in a titration is a value which we call the control condition, e.g., pH or voltage. We will sometimes use voltage (e) as a generic term for control condition. Often, when a complex substance or mixture is exposed to a series of voltages, the analysis requires a complete spectrum a(w) over a fixed range of w for each value of e (1). Given such a collection of spectra, this paper presents a method for (1)detecting and characterizing the transitions of sample components from one state to another, e.g., from the oxidized to the reduced state, and (2) describing the difference spectrum of each titrating component. Further, the process does not require any modeling of the original spectra, e.g., by sums of Gaussians. For this method to be applicable, the absolute spectra must behave in a linear fashion. That is, the spectrum of a mixture must be the sum of the spectra of the individual components, and the spectrum of a single substance must be proportional to the concentration of that substance. Also, one must have some class of models in mind for the transitions, e.g., Henderson-Hasselbach, Nernst, or Adair (see the Examples section). In previous work involving collections of spectra ( I ) ,we have had to select models (e.g., sums of Gaussian peaks) for each spectrum as a function of wavelength and then model the variation of each peak height as a function of voltage: a two-phase modeling process. In this paper, we propose to replace the fiist phase, modeling the spectra, by singular value decomposition (SVD), a completely automatic procedure, leaving the investigator to model the titrations of automatically chosen components. The organization of the paper is as follows. The mathematics is explained in the Theory section, but for those who wish to apply the method directly, it is given in step-by-step form in the Procedure section. Some results are presented in the Examples section, and finally, some merits and drawbacks are given in the Discussion section. As to notation, we will use upper case letters to denote matrices and vectors. The corresponding lower case letters, subscripted, will denote the matrix elements. One possible point of confusion is that we use ej to denote voltage and the matrix E (epsilon) to denote noise, but the distinction should be clear from context. The ith row of A will be denoted A row i, and the j t h column of A will be denoted A col j. Superscript T will denote matrix transpose.
THEORY We will assume that the reader is familiar with the elementary concepts of matrix algebra as explained, e.g., in ref 2 and 3. Let the m by n matrix A be a collection of spectra, with each A col j being a complete difference spectrum at some fixed e. That is, if Y ( e j )is the j t h absolute spectrum, then ( A col j ) = Y(ej)- 7,where 7 is a reference spectrum. Our model of the titration process is as follows. Let c be the number of titrating components. Then
A=DF+E where D = m by (c + 1)matrix. D cols 1to c are difference spectra, final state minus initial state, of each titrating component. D col ( c + 1)is a base curve, i.e., the spectrum that would appear in A if all components were in the initial state. F = n by (c + 1) matrix of rank (c + 1). F cols 1 to c are
transition curves for each titrating component, i.e., f i , is the fraction of component j in the final state in the ith spectrum of A. F col (c + 1)is all ones, because the base curve applies to all curves in A . E = m by n matrix of random error, which we will assume normal and independent with zero mean and nearly uniform variance. Our goal is: given A , find D and F. We can do this only because we invoke physical principles to select appropriate curves for the columns of F,e.g., Henderson-Hasselbach, Nernstian, or Adair-type curves (see the Examples section). By fitting sums of such curves to, say, selected rows of A , we can determine parameters, e.g., midpoint potentials or pKs, that help us to decide how many distinct transitions there are. Alternately, we can model the spectra by sums of Gaussian peaks or their derivatives ( 4 ) ,e.g., determine peak heights by curve fitting every column of A and then fit the sums of transition curves to the peak heights plotted vs. e. In either case, the process is only as good as our judgments in choosing appropriate rows of A or in choosing appropriate peaks to model the columns of A. Wrong choices can be minor, or they can be ruinous. In fortunate cases, when we have the columns of F deduced, D can be found directly by matrix algebra
D = AF(PF)-l
(2) In contrast to the methods just described, the proposed singular value decomposition (SVD)procedure, defined below, accepts the entire matrix A , unfiltered by our judgments or curve fits, and produces much smaller matrices that contain all of the information we need to determine F and D. Any real matrix A can be written as the product of three matrices called the SVD of A
A = USV (3) where f l u = I , U is m by n, I is the n x n identity, V'V = 2 ... V p = I , V is n by n, and S is n by n diagonal, 1 0. S is unique, while U and V are sometimes nonunique, but the method discussed here does not depend on the choice of U and V in nonunique cases. The theory of this process is discussed in ref 5 and an Algol program is given in (6). A FORTRAN version of SVD is available from G. H. Golub, Computer Science Department, Stanford University, Stanford, CA. These papers, and many others in the field, see SVD mainly as a means to an end in solving least-squares problems or in smoothing data. Our interest is mainly in the direct outputs U, S, and V as aids to the understanding of spectral events. As far as we know, the first use of SVD to analyze a multicomponent system was in the context of excitationemission matrices of fluorescence where the purpose was to estimate concentrations, not transitions or spectra. Consider first the noiseless case E = 0. Let r be the rank of D, i.e., the maximum number of columns of D which are not linear combinations of each other. Then r Ic + 1,and the rank of A is also r. SVD reveals the rank at once, because for any rank r matrix only the first r singular values sL,L are positive; the rest are zero. By design, m and n are much larger than r , and since sL,& = 0 means that U col i and V col i contribute nothing to A , we can discard these numbers, retaining the minimum representation of A
(a,
A = usv (4) where 9 is the upper left r by r submatrix of S, and U and p are the first r columns of U and V, respectively. It can be proven that every transition in F is exhibited by at least one
p col j, so that no information is lost. The fundamental relationship of the method is A = B S P = DF from which one can deduce
P=FF
(5)
ANALYTICAL CHEMISTRY, VOL. 54, NO. 7, JUNE 1982
where
IT
= S l i F D = PF(F'F)-l
(7)
i.e., of all matrices of rank I r , B is closest to A . If we know what the noise level is, then the choice of r that yields n
n
C .si,: Imna2 < CS,.;~
and finally
D = DSH
(8)
One extrads Hand F from P by curve fitting. For example, suppose in a pH titration that the model transition curves (derived from physical considerations) are all of the Henderson-Hasselbach typo
+ 10(PKi-PH~))
(9)
+ 1) + j=1 Chijf(pH, pKj)
(10)
f i j = f(pHi,pKj) = 1/(1 Then we perform r curve fits C
V C O ~i = hi,c+l(FC O ~c
in which the f(pH, pKj) are column vectors, one element for every pH, and the parameters are the hs and pKs. Usually the resulting pKs will not match exactly from one column of P to the next, so it often becomes a matter of judgment as to whether a pK from one column of P is practically equal to a pK from another, i.e., how many distinct pKs there really are. Each distinct pK forms its own column of F by eq 9. The noiseless process can now be completely summarized. (1) By SVD
A = USV
(11)
(2) Eliminating si,i = 0
A = USP (3) By curve-fitting relation (10) to columns of 7 vs. voltage, find F and H such that p = F € P (4) D = DSH (13) We now consider the case where the noise E in eq 1 is nonzero. Since the noise in one spectrum is independent of the noise in the others, the rank of A is almost always n or n - 1,depending on the reference spectrum, even when D = 0, i.e., when A is all noise. Consequently, nearly all the singular values s,,~are positive, so that selecting 0, and p will be more complex than step (12) in the noiseless case. Let a,J 2 be the variance of a,, with the average variance in A being a2. Let each a,, be the sum of a true reading b,, and an error eLJ. Them the Euclidian norm of the error is
s,
m n
( A- BIZ = ~ ~ e L= Jmna2 2
(14)
FlJ=1
The quantity JA- BI is also called the distance between A and E. Since, by design, m and n are much larger than r, our object is to find a matrix B which is of much lower rank than A , yet which has a high likelihood, as defined in (8), of being equal to A. Minimizing distance and maximizing likelihood are equivalent only if the O,,S are equal. In our experience, max(u)/min(u) 5 3 seems workable. We will show, in this section how SVD enables us to find B that minimizes ( A- B(. When the a,s, are reasonably uniform, then B, as specified below will be a suitable candidate to replace A as a noisesuppressed matrix. SVD provides such candidates with the help of the following two theorems (5): Let B = UST, r = rank ( B ) = number of columns in 0, 9, and P, r < n. (1) n
IA - BI2 =
i=r+l
s,,;
C of rank r or less ( A- B( IIA - CI
(15)
(2) For any matrix
(16)
1149
i=r+l
(17)
i=r
also yields B of minimum rank that differs from A by no more than is attributable to noise. However, because of the variance in si,, and in the estimate of u2, and because of some nonuniformity in the noise in A, we may be led by this technique to set r too large (i.e., to include columns of U and V that are mostly noise) or too small (i.e., to reject columns of U and V that represent signal). The above estimate of r is best used as a starting estimate, to be refined by examining the columns of U and V. A better clue to the significance of the columns of U and V is the shape of those columns when plotted against their respective abscissas (wavelength and voltage respectively). Assuming that the spectra are taken at wavelength increments small enough to have several points on each peak or trough, the number of sign changes in these curves should be considerably less than m. Even the magnitudes of successive points will tend to be alike. Some columns of U , representing signal, will be very smooth. Rapid fluctuations in sign or magnitude will be observed only in those columns of U in which noise dominates, usually the later columns. A simple statistic for detecting such noise is the autocorrelation coefficient of order 1,which, in our case, can be closely estimated by m-1
Typically, one should observe Cuco, well above +0.5 for small j , larger values indicating greater smoothness. The value +0.5 seems to be a good cutoff point given small wavelength increments as described above. Similarly, if voltage increments are small enough to have several e values in the neighborhood of each transition, then n-1
v'
COl J
=
CuiJui+lJ
i=l
(19)
can be used in a manner entirely analogous to CumlJ. One may wish to use either or both depending on the density of samples in w and e. We have now described three tools, one for each of the matrices S, U , and V. Combined, these three should be sufficient in most cases to select the effective rank of A. The matrix
B = USP
(20)
will then represent a noise-suppressed version of A . PROCEDURE (1) Prepare t h e I n p u t Data. Let the vector W contain
the wavelengths at which a spectrum will be taken. Fix the voltage e at some value el and place the corresponding spectrum in A col 1. Fix the voltage at a new value e2, and place the correspondingspectrum in A col2, and so on. Each A col j is a complete spectrum at fixed e,, and each A row i is a titration curve at fixed w,. (2) Estimate Noise Level. This step is done automatically by smoothing the data, and determining the total variance as the sum of squared deviations from the smoothed values, which are used only in this step and then discarded. Call the total variance NT. (3) Compute SVD(A ) and Discard Noisy Components. The SVD part is automatic; most large scale computers have SVD routines available. Three vectors help decide what is "too noisy". These are (1) distance, = ( ~ , = , " s J J 2 ) (2) /N~, autocorrelations of columns of U , and (3) autocorrelations of
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ANALYTICAL CHEMISTRY, VOL. 54, NO. 7, JUNE 1982 WI
1.2
0.0M
.9
0
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.6 -
01 350
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; 600
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Figure 1. Acid titration of a mixture of three pH Indlcators: (a) difference spectra; (b) columns of U ; (c) columns of V , first three with fitted curves: (d) reconstructed spectra compared to pure spectra (smooth curves).
columns of V, where U , S, and V are defined in formula 3 of the Theory section. Our usual practice has been to discard U col i through n, and the corresponding parts of S and V if distance; I 1 or either autocorrelation is less than 0.5. (4) Obtain “Fits” for the Transitions in Matrix V. Again, most large scale computing systems have nonlinear regression programs available. Each remaining column of V should be plotted against voltage or pH or other control condition. In the case of a redox titration, the Nernst equation will be used. For a pH titration, which we use here for illustration, the Henderson-Hasselbach (H-H) model is employed, i.e., a sum of terms of the form aik,H/(l + kiH) where H = [H+]. Assume that there are two transitions and two remaining columns in V. Then the function fit to each V col i will have two H-H terms and the results may look like this (where pKi = log ( k i ) ) a1
Vcol 1 Vcol 2
1.1 3.3
PK,
a2
5.9 6.2
-4.4
2.2
PK, 8.1 7.9
(5) Compute a Difference Spectrum for Each Transition. Using results from the example of the previous step, we may be convinced that pK1 is really the same transition in both fits (V col1 and V col2) and similarly for pK,. The difference spectra are then estimated by PK1: 2 1 = l.l(Sl,l)(U COl 1) 3 . 3 ( S 2 , 2 ) ( U COl 2)
+
PK2: 2 2 = 2.2(S,,l)( U COl 1) - 4.4(S2,2)( U COl 2) The vectors Z1 and Z2plotted vs. W will be the estimated difference spectra. These may be optionally fitted by, say, sums of Gaussians to determine absorbance bands. Further details on any phase of this procedure are available from the authors.
EXAMPLES Three examples of the use of the method are given in this section. The results shown in Figures 1-3 use some mathematical forms which are defined here: (1)Henderson-Hasselbach (H-H) term hf(pH, pK) = h / ( l
+ 10pK-pH)
(2) Nernstian term
hN(e, em, n) = h / ( l
+ 10n(em-e)/59)
where e, = midpoint potential in millivolts and n = number of electrons passed. (3) Gaussian peak
hdw, w,,
S)
= h expkln (2)[(w- ~ m ) / s l ~ l
where w, = peak center position in nanometers and s = half-width (center to half-height) in nanometers. In the first example, Figure 1, three pH indicators were mixed and acid titrated. The three indicators were meta cresol purple and chlorphenol red (Fisher Scientific Co. Pittsburgh, PA) and phenol red (Sigma Chemical Co., St. Louis, MO). Suitable colorless buffers were present in all cases to increment pH by about 0.25 units when several microliters of 1 N HC1 were added. Figure l a shows the data with a central spectrum as the reference. The method of the Procedure section was used to derive the three pKs and to reconstruct the spectra of the indicators. Figure l b shows the first four columns of U plotted vs. W and shifted upward by multiples of 0.3 for graphic separation. The column labeled SVs are the corresponding singular values, and the column labeled ACs are the corresponding first order autocorrelations. Figure ICshows the first four columns of V vs. pH shifted upward by multiples of 0.5 and labeled as in Figure lb. The fourth autocorrelation
ANALYTICAL CHEMISTRY, VOL. 54, NO. 7, JUNE 1982
1151
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.14
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.12
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.10
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.08
2.0
a
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.04 .02 ' 50
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600
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,060
-I
620 640 660 Wavelengfh(nrn)
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I
C
13
! -.01 5 580
600
620 640 660 Wavslength(nrn)
I 680
150
200
250
300 E(mV)
350
400
Flgure 2. Potentiometric titration of E . colicytochromes: (a)difference spectra; (b) columns of U ; (c)columns of V , first five with fitted curves: (d) reconstructed 8E (lower) and 4E (upper spectra), each fit by Gaussians with base line (smooth curves).
(0.515) was chosen as the cutoff value, so that only the first three columns of V were processed further. These were each fit to a s u m of three H-H curves, which are the smooth curves superimposed on the P vs. pH curves in Figure IC. The derived pKs (with previously published "range" values (9) in parentheses) are meta cresol purple, 8.0 (7.4-9.0), chlorphenol red, 5.8 (5.0-6.6), and phenol red, 7.5 (6.&8.4), showing good agreement. Finally, the amplitudes of the H-H terms from the curve fits, the singulslr values, and the columns of U were used as shown in the final step of the Procedure section to estimate the individual difference spectra. In contrast to the example in the Procedure section, there were three spectra with three terms oach in this case. These spectra are shown in Figure Id, shifted upward by 0.2, 0.3, and 0.5 for clarity. For verification of these spectra, each indicator was acid titrated separately, and the SVD method was used to separate the individual spectra from noise and minor impurities. The resulting smooth spectra are shown in Figure Id superimposed on the spectra derived from the mixture. Example 1was difficult on two counts. First, the spectra of all three indicators are qualitatively similar, having troughs in the low 400 nm range and peaks in the high 500 nm range, so the distinctions to be extracted are subtle compared to overall spectral change. Second, the upper two pKs are only 0.5 pH unit apart, making it difficult to resolve the sum of two nearby H-H curves. The point here is to show that the method can extract features in a complex case, and also to show the errors, as in Figure Id, that one encounters. The second example (Figure 2) uses data from ref 1,Figure 6, page 11291, the spectra-voltage surface for cytochromes d. As in example 1, Figure 2a is the original data, this time referenced to the initial spectrum. Figure 2b shows columns of U vs. W , shifted upward by multiples of 0.5 and labeled
with SVs and ACs as in Figure 1. Figure 2c shows columns of V vs. e shifted and labeled as in Figure 2b. The first five columns (i.e., were fit to sums of Nerstian terms as shown by the smooth curves superimposed on 7in Figure 2c. The coefficients of these terms, the singular values, and the columns of U were used, as in the last step of the Procedure section, to produce the difference spectra shown in Figure 2d. The lower curve corresponds to a species titrating at 323 mV with N = 8. The upper curve is derived from a titration at 312 mV with N = 4 Sums of Gaussians with base lines (superimposed smooth curves in Figure 2d) were fit to the estimated spectra to determine absorbance bands. Results are shown above each curve in Figure 2d. The third example (Figure 3) is an independent repeat of the second. The results of this experiment were used in ref 1, Tables I, 11, and 111, but the spectra were not shown in graphic form. Example 3 and Figure 3 are entirely analogous to example 2 and Figure 2. Examples 2 and 3 involved samples of E . coli membranes with some variability between preparations, as reflected in the noticeable disparity between Figures 2b and 3b, and likewise between Figures 2c and 3c. Nevertheless the final reconstructed spectra of the BE (Le., n = 8 in the Nernstian term)and 4E components are clearly similar from experiment to experiment. These results, while mostly supportive of the conclusions in ref 1,suggest a refined model of spectral changes around 640 nm and 325 mV. The reconstructed " B E components in Figures 2 and 3 support the conclusion expressed in ref 1that a peak near 650 nm is disappearing, while a peak near 630 nm is coming up on reduction. However, for ease of resolution, the "4E" component near 320 mV was originally modeled using the same two Gaussians as in the " 8 E case, but the SVD results suggest that a new, broader peak (mid-
1152
ANALYTICAL CHEMISTRY, VOL. 54, NO. 7,JUNE 1982 .10
1
h
i80
600
620 640 660 Wovelength(nm)
680
g80
>
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'
Figure 3.
620 640 660 Wavelength(nm)
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I w
-.040
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680
0 50
100
150
200
250
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'
0 191
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E(mV)
Repeat with new data of the experiment in Figure 2: (a),(b), (c), and (d) as in Figure 2.
point = 644, half-width E 18.6) is rising on reduction. This observation would be difficult to make in any other manner because of the many overlaps in both the wavelength and the voltage domains.
DISCUSSION The SVD technique has been shown in the previous sections to be useful for resolving overlapping spectral features and overlapping titrations. One can avoid the modeling of spectra entirely as in the first example, or one can be assisted in such modeling as in examples 2 and 3. SVD can be expensive for large matrices and the curve fitting of V is not easy, especially when the singular values are near the noise level. However, the benefits are well worth the inconvenience, and in many cases, the alternatives hold out the false hope of simplicity. For example, when features overlap considerably, what one perceives as,say, the shoulder of one peak may in fact be composed of several peaks. A two-point difference plotted against voltage will exhibit a titration pattern just as complex as a column of V. As a final example, when one is modeling the transition of a single substance from one state to another (e.g., the pH denaturation of a protein or the oxygenation of hemoglobin), it is common practice to use only one or two wavelengths. But this practice may be deceiving if there is more than one substance undergoing a transition, or if one substance has more than two states. The assumption of simple two-state behavior may be checked by SVD (the A matrix must be practically of rank l),and if there is another state or substance present, SVD
can help to detect its midpoint potential and its spectrum. Since one is interested in only one of the two transitions, some corrective action is in order: either remove the offending substance or monitor a wavelength unaffected by its derived spectrum or resolve it mathematically by a more complex transition model (e.g., two Nernst components instead of one). For such cases, SVD is a prerequisite screening device and an experimental design technique. Once a system is thus analyzed, it may be possible to use a one- or two-point absorbance monitor with some assurance that the numerical results are not being distorted by spurious transitions.
LITERATURE CITED Hendler, R. W.; Shrager, R . I . J . Biol. Chem. 1979, 2 5 4 , 11288-1 1299. Ayres, F., Jr. "Theory and Problems of Matrices"; Schaum Outline Series; Schaum: New York, 1962, Chapters 1-7. Noble, B. "Applied Linear Algebra"; Prentice-Hall: Englewood Cliffs, NJ, 1969; Chapters 1-5. Shipp, W. S. Arch. Biochem. Biophys. 1972, 150, 459-472. Golub, G.; Kahan, W. J . SIAM Numer. Anal., Ser. 6 1965, 2 , 205-223. Golub, G. H.; Reinsch, C. Numer. Math. 1970, 14, 403-420. Ho, C.-N.; Christian, G. D.; Davldson, E . R. Anal. Chem. 1978. 5 0 , 1108. Kendall, M. G.; Stuart, A. "The Advanced Theory of Statistics", 3rd ed.; Hafner Press: New York, 1973; Vol. 2, p 78. Sober, H. A,, Ed. "Handbook of Blochemistry", 2nd ed.; Chemical Rubber Co.: Cleveland, OH, 1970; J241-J242. Knott, G. Comput. Programs Biomed. 1979, IO, 271-280.
RECEIVED for review June 17, 1981. Resubmitted December 16, 1981. Accepted March 11, 1982.
