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weight percentage. The % C1 de te rmined in all the Aro- clot samples was in close agreement with the nominal values expected, as well as the values ob ta ined f rom the po ten t iomet r ic method. An interest ing aspect of these results is t ha t the % C1 could be de te rmined wi thout prior identification of the individual PCB congeners, again indicating t ha t the C1 emission observed was indepen- den t of the s t ruc ture for each respect ive PCB congener p resen t in the Aroclor samples. A p r imary concern was t h a t several PCB congener peaks would overlap with the in ternal s tandards; however, no significant peaks were observed a t the re tent ion t imes of the in ternal s tandards (~15.6 min for T C B and ~24.4 min for H C B under the presen t exper imenta l conditions).
Compar i sons to ch romatograms obta ined with the use of a flame ionization detec tor indicated t h a t there was no discernible loss of resolut ion or peak qual i ty th rough the hea ted interface between the column and HDD. One advan tage no ted was the absence of detectable column bleed a t the high GC column t empera tu re s (280-310°C) necessary to elute the highly chlor inated P C B congeners. Even a t the highest permissable gain set t ing (4 nA full scale) a t t empe ra tu r e s of 300°C over a 30 min period, no greater t han a 5 % full-scale increase in the baseline was observed. This observat ion is in direct cont ras t to results f rom convent ional nonselect ive GC detectors, in which the background signal can significantly increase over the course of a ch romatogram, especially at low a t tenua t ions and high column tempera tu res . Such prob lems can resul t in in tegrat ion errors for peak area de te rmina t ions or in the need for greater signal a t t enua t ion to compensa te for excessive background shifts.
ACKNOWLEDGMENTS This work was supported by an ACS-PRF Grant (#17417-GB3) and
a Thomas F. and Kate Miller Jeffress Memorial Research Grant (J- 99). This work was presented in part at the 1987 Federation of Ana- lytical Chemistry and Spectroscopy Societies Conference, Detroit, MI.
1. J. H. Koeman, M. C. Debrouw, and R. H. DeVos, Nature (London) 221, 1126 (1969).
2. L. Fishbein, J. Chromatogr. 68, 345 (1972). 3. M. Cooke, G. Nickless, A. M. Prescott, and D. J. Roberts, J. Chro-
matogr. 156, 293 (1978). 4. A. L. Robbins and C. R. Wilhite, Bull. Environ. Contam. Toxicol.
21, 428 (1979). 5. J. M. Lin and S. S. Que Hee, Anal. Chem. 57, 2130 (1985). 6. R. E. Lea, R. Branston-Cook, and J. Tschida, Anal. Chem. 55, 626
(1983). 7. W. J. Dunn, D. L. Stalling, T. R. Schwartz, J. W. Hogan, J. D.
Petty, E. Johansson, and S. Wold, Anal. Chem. 56, 1308 (1984). 8. A. L. Alford-Stevens, T. A. Bellar, J. W. Eichelberger, and W. L.
Budde, Anal. Chem. 58, 2014 (1986). 9. A. Poland and E. Glover, Mol. Pharmacol. 13, 924 (1977).
10. J. Goldstein, P. Hickman, H. Bergman, J. McKinney, and M. Walk- er, Chem. Biol. Interact. 17, 69 (1977).
11. J. W. Rote and P. G. Murphy, Bull. Environ. Contam. Toxicol. 6, 377 (1971).
12. M. Mullin, C. M. Pochini, S. McCrindle, M. Ronkes, S. Safe, and L. Safe, Environ. Sci. Technol. 18, 648 (1984).
13. S. D. Cooper, M. A. Moseley, and E. D. Pellizzari, Anal. Chem. 57, 2469 (1985).
14. T. Cairns and E. S. Siegmond, Anal. Chem. 53, 1183A (1981). 15. G. W. Rice, A. P. D'Silva, and V. A. Fassel, Spectrochim. Acta 40B,
1573 (1985). 16. G. W. Rice, Quarterly Report, Ames Laboratory, Ames, IA, 4/84-
6/84. 17. K. Ballschmitter and M. Zell, Fresenius Z. Anal. Chem. 302, 20
(1980). 18. M. L. Ware, M. D. Argentine, and G. W. Rice, Anal. Chem., 60,
383 (1988).
Spectroscopic Quantitative Analysis of Strongly Interacting Systems: Human Plasma Protein Mixtures
M A R C R. N Y D E N , * G L E N N P. F O R N E Y , and K R I S H N A N C H I T T U R Center for Fire Research, National Bureau of Standards, Gaithersburg, Maryland 20899 (M.R.N., G.P.F.); and National Center for Biomedical Infrared Spectroscopy, Battelle Laboratories, Columbus, Ohio (K.C.)
Blood plasm~L protein infrared spectra, while qualitatively very similar, display subtle differences in the frequencies and intensities of absorption bands. These small differences are sufficient to permit an accurate quan- titative analysis of mixtures of these proteins. In this paper we examine the performance of some alternative methods of spectroscopic quanti- tative analysis in determining the concentrations of proteins in aqueous solutions. T~e widely-used K matrix method, using sloping baselines and intercept functions, was found to be inadequate for these determi- nations. In contrast, a method based on the little-known Q matrix ap- proach, augmented by a robust equation solver, yielded results with a sufficient degree of accuracy to make it a viable tool for use in the study of proteins at solid interfaces and for more general applications in the field of protein chemistry. Index Headings: Infrared; Spectroscopic techniques.
Received 7 December 1987. * Author to whom correspondence should be sent.
I N T R O D U C T I O N
Techniques based on Fourier t r ans fo rm infrared with a t t enua ted to ta l in ternal reflection ( F T - I R / A T R ) have been used to probe the dynamics of the interact ions be- tween synthet ic po lymer surfaces and blood proteins. 1~ In te res t in this area has been mot iva t ed by the real izat ion t ha t the fo rmat ion and behavior of ha rmfu l pla te le t and white cell t h romb i on the surfaces of synthe t ic polymer ic surfaces are de te rmined to a large ex ten t by the na ture of the adsorbed pro te in layer t h a t forms on such surfaces. Recen t papers have highl ighted the kinds of in format ion t ha t can be ob ta ined by s tudying cer tain prote ins such as a lbumin, immunoglobul in G, and fibrinogen on a num- ber of polymeric surfaces of clinical interest . Pa s t efforts were focused on the descr ipt ion of the kinetics and c o n -
588 Volume 42, Number 4, 1988 ooo3-7o28/ss/42o4-o~ss$zoo/o APPLIED SPECTROSCOPY © 1988 Society for Applied Spectroscopy
formational changes occurring as these individual pro- teins adsorb to the polymer surfaces. These studies, while revealing some important characteristics of the blood- material interactions, are nevertheless limited in their ability to predict blood compatibility.
In an effort to better simulate the conditions of real- life blood-material interactions, we have begun working with mixtures of these proteins. The kinds of information that we hope to obtain are:
1. The kinetics of adsorption of individual proteins from whole blood.
2. The conformations of individual proteins during ad- sorption from whole blood.
Since whole blood is a mixture of many proteins, it is clear that accurate quantitative techniques need to be established before the dynamics of protein adsorption from mixtures can be understood.
Despite the considerable progress made in the field of quantitative infrared analysis, most of the investigations hitherto reported in the literature have been restricted to the analysis of simple mixtures involving structurally dissimilar, noninteracting components. In this paper we will break from this tradition by examining the effec- tiveness of mathematical curve-fitting techniques in ex- tracting concentration information from the FT-IR spec- tra of multicomponent protein solutions. This is a challenging problem characterized by the following dif- ficulties:
1. The proteins of interest must be studied in aqueous media at dilute concentrations. As a result, the spec- tral features of the proteins are almost completely obscured by the overwhelming infrared absorbance of water.
2. The spectra of the proteins overlap strongly and are nearly indistinguishable. The subtle differences which do exist tend to have much lower intensity than do the spectral similarities.
3. Proteins are interactive molecules which readily change conformation as a function of concentration and tem- perature. As a consequence, the spectra of protein mixtures exhibit strong deviations from Beer's law. The nonlinearity which characterizes the spectra of protein solutions has an adverse effect on the accuracy of all methods of quantitative analysis.
We shall demonstrate below that, although the quanti- tation of proteins in solution is a formidable task, it is in fact possible to obtain accurate determinations of pro- tein concentrations. This is accomplished without sub- jective data reduction schemes. Rather, accurate quan- titation is achieved by using a large number of calibration spectra in conjunction with a numerically robust Q ma- trix algorithm.
THEORY
The objective of quantitative analysis is to predict the concentrations of specific analytes in unknown mixtures. It is assumed that some information about these analytes is available to the analyst in the form of standard mix- tures. Thus, in determining the concentration of a spe- cific analyte in an unknown mixture, the analyst makes
use of a set of calibration spectra which were obtained from mixtures containing known amounts of this analyte. As a minimum, it is necessary that all the components present in the unknown mixtures be represented in the calibration set. However, effective quantitations using one well-established method, the K matrix method, re- quire a more complete characterization of the calibration mixtures. In conventional implementations of the K ma- trix method, the analyst must supply the concentrations of every component in each calibration spectra. This de- tailed information is not required if alternative methods of analysis are employed (vide infra).
The basis of quantitative spectroscopy is Beer's law. This law holds that the spectrum of a dilute mixture is approximately a linear combination of the spectra of its components. Mathematically, this can be written as fol- lows:
Ai" = ~ Ki~Cj" + R~ (i = 1, 2 , . . . , N) (1) j~l
where Ai u is the absorbance of the n component mixture at frequency, ~, K~i is the absorbance of the j t h com- ponent at this frequency, and Ci" is the concentration of this component. The residual, Ri, is the deviation from Beer's law which arises from molecular interactions and measurement errors. To the extent that this residual is systematic, Beer's law can be generalized to:
A," = ~_~ K, jCi" (i2 > n) (i = 1, 2 , . . . , AT). (2) j=l
In this equation the additional I~ - n values of Ki i are spectral representations of the effects, such as molecular interactions, which give rise to the residual in Eq. 1.
There are many approaches to spectroscopic quanti- tative analysis. All of these approaches involve the use of Beer's law in one form or another. The most widely used is the so-called K matrix method2 ,~ This is a two- step process. In the calibration step, an approximate K matrix is determined by least-squares analysis as follows:
= AC'(CC')- ' (3)
where A is an (N x M) matrix containing the spectral data (N absorbencies from M mixtures), and C is the corresponding (n x M) matrix, which contains the con- centration of each of the n components present in the M calibration mixtures. Primes are used to indicate the transpose of a matrix, and the superscript ( - 1 ) is used to denote matrix inversion. The prediction of the con- centrations in the unknown is achieved by solving Eq. 2 for the Ci" after substitution of the K matrix and the spectral absorbencies of the unknown mixture.
In its conventional implementation, the K matrix method is subject to errors which result from the rank reduction involved in the calibration step. The source of the problem is that the coefficients, C u (j > n), of the spectral representations of the deviations from Beer's law are usually not available. The magnitude of the error which results when these deviations are neglected de- pends on the intensities of the spectral representations of these deviations and also on the degree of overlap between the component spectra. For similar reasons, the K matrix method is also inconvenient to employ in the analysis of mixtures containing strongly absorbing ira-
APPLIED SPECTROSCOPY 589
purities. That is, unless the concentrations are known for each component in the calibration mixtures, errors will be propagated in the prediction step. The problem is exacerbated if there is appreciable spectral overlap between the components, since this leads to ill-condi- tioning of E',q. 2.
Another widely used method for quantitative spectro- scopic analysis is the P matrix method: ,s Unlike the K matrix method, the implementation of this method usu- ally does not involve the use of potentially inaccurate approximations. However, the P matrix method is some- times inconvenient to use. The source of the problem is that the P :matrix is determined by fitting the concen- trations of the calibration mixtures to linear combina- tions of the corresponding spectra. This means that the elements of the P matrix are not uniquely determined unless there are at least as many calibration mixtures as there are points in the reference spectra. Unfortunately, the number of calibration mixtures is often limited by practical considerations, so that the analyst is forced to use narrow spectral windows and poorly resolved spectra.
Conventionally, the P matrix is evaluated from the following formula, derived from least-squares:
P = CA'(AA') -1. (4)
The concentrations of the components in the analyte are then predicted with the use of the formula
C- = PA-. (5)
The P matrix is actually the pseudo-inverse of the K matrix. This is expressed in the following formula:
(
(P, IK:-)=/1 ( i = j ) (6) otherwise
where the bracket is used to indicate that the scalar product between the vectors is taken. Thus, the concen- tration of each component is obtained by projecting the contributions of the interfering components out of the mixture spectrum.
Finally, a third formulation is possible. This is the Q matrix method, which involves using the reference mix- ture spectra directly in the fit of the spectrum of the unknown mixture. The essence of this method has been described in a paper by McClure e t al .; 9 however, until now no practical solutions have been demonstrated whereby this method has been able to solve problems intractable to the more established methods of quanti- tative infrared analysis.
The basis of the Q matrix method is the following formula, w]hich is a corollary of Eq. 2:
A" = AQ. (7)
The M-component row vector Q is related to the com- ponent concentrations in the analyte by
C" = CQ. (8) In contrast to the P matrix method, there are no con- ditions on the calibration spectra, other than the con- straint that the number of spectral absorbencies (N) exceed the number of calibration spectra (M). However, since the accuracy of the results does depend on the numerical ,conditioning of the A matrix, it is wise to use as many data points as possible.
An undesirable property of the Q matrix method is that sizeable errors in the predicted concentrations will result from linear dependencies, or even near-linear de- pendencies, in the calibration spectra. Unfortunately, such dependencies are frequently encountered in the analyses of protein mixture spectra. The source of the problem is the spectral similarities between the com- ponents and the fact that many calibration spectra, span- ning a small range of compositions, must be used to model the deviations from Beer's law. For this reason, the use of a numerically robust algorithm to solve the linear equations which constitute the Q matrix method is essential.
Our response was to adapt a technique called QR de- composition, which involves transforming the A matrix to upper triangular form by a series of Householder re- flections. 1° Linear dependencies in the calibration spectra are detected in the process of transforming the spectra which make up the columns of the A matrix. Specifically, a spectrum is considered to be a linear combination of the previous spectra if the average of the spectral inten- sity corresponding to data points which fall below the principle diagonal of the A matrix is less than a prede- termined threshold. The value of this threshold is de- termined from the estimated signal-to-noise ratio of the calibration spectra. Thus, for example, a signal-to-noise of 1000:1 implies that the threshold should be approxi- mately 10 -3 times the average spectral intensity of the calibration spectra. One of the attributes of this algo- rithm is that the "normal" equations, which would be even more poorly conditioned than is Eq. 7, are circum- vented.
EXPERIMENTAL
Proteins were used as received from the Sigma Chem- ical Companyt (St. Louis, MO). They were: (1) albumin (A1), human, crystallized, and lyophilized, globulin free (Cat. No. A-8763); (2) immunoglobulin G (IgG), human, lyophilized, salt free (Cat. No. P4506); and (3) fibrinogen (Fg), human, 95% clottable (Cat. No. F-4883). Solutions of A1 and IgG were made by dissolving the protein in isotonic saline (9.5 g/L) and were used without further purification or characterization. Fg solutions prepared in Tris buffer were dialyzed overnight against buffer. This step was necessary in order to remove L-glycine, an ami- no acid used during the protein purification from human plasma. The pH of all protein solutions was adjusted to 7.4 with dilute HC1 or NaOH. Mixtures of A1, IgG, and Fg were also prepared at different concentrations from stock solutions of the single proteins. All protein solu- tions were prepared on the day of use.
Transmission spectra were acquired with a Model FTS- 15 Fourier transform infrared spectrometer system (Dig- ilab, Inc., Cambridge, MA) equipped with fast-scan capabilities; a Hycomp 32 data array processor; and a
t Certain commercial equipment, instruments, materials, or companies are identified in this paper in order to adequately specify the exper- imental procedure. In no case does such identification imply rec- ommendation of, or endorsement by, the National Bureau of Stan- dards, nor does it imply that the equipment or material identified is necessarily the best available for the purpose.
590 Volume 42, Number 4, 1988
I
0 . 0 2 0
0 . 0 ] 5
0 . 0 1 0
I I I I I I [
0 . 0 0 5
i i i i i i 2 0 0 0 t 9 0 0 1 6 0 0 1 7 0 0 1 6 0 0 1 6 0 0 1 4 0 0 1 3 0 0 1 2 0 0 i t 0 0 1 0 0 0
~ a v e n u m b e r
FIG. 1. The spectrum of IgG was obtained by subtracting the spectrum of the buffer from the spectrum of a 10 mg/mL solution of IgG.
high-sensit ivity, narrow-range, l iquid-ni t rogen-cooled mercury-cadmium-te l lur ide (MCT) detector. The trans- mission cell (Harrick Scientific, Ossining, NY) was used with CaF2 windows and 6-/~m polyethylene spacers. For each sample, 2000 interferogram scans were coadded in order to obtain spectra at a resolution of 8 cm -1. The interferograms were then apodized with a tr iangular function, zero-filled, and Fourier t ransformed to produce spectra from 1000 to 2000 cm -1 with a digital resolution of 4 cm-L
One of the problems in the measurement of the protein spectra is the variability in cell pathlengths due to changes in the polyethylene spacer during cell assembly/disas- sembly by the analyst. For example, it is well known tha t actual cell pathlengths can vary by as much as 10-20 % between experiments. This variat ion would introduce an unacceptable degree of imprecision into the measured spectra and would result in a degradat ion in the accuracy of the component quanti tat ions. In an a t t empt to correct this inconsistency, we used the OH stretching band, which occurs at 1640 cm -~ in the spectrum of water, as an in- ternal reference for the comparison of band intensities in the spectra of protein solutions measured at different times. All spectra used in this s tudy were normalized so
0 . 0 2 5
0 , 0 2 0
0.030
I I J I I I I I I
0.015
0 . 0 1 0
0 . 0 0 5
i i 2 0 0 0 1 9 0 0 ] 0 0 0 t 7 0 0 ] 6 0 0 1 5 0 0 1 4 0 0 1 3 0 0 t 2 0 0 1 1 0 0 t 0 0 0
Fro. 2. The spectrum of A1 was obtained by subtracting the spectrum of the buffer from the spectrum of a 10 mg/mL solution of A1.
I [
0 . 0 2 5
0 . 0 2 0
0 , 0 i 5
O . O l O
0 . 0 0 5
I I 2 0 0 0 l g O 0 i 8 0 0 1 7 0 0
I I I I I
i i 1 I I I 1 6 0 0 1 5 0 0 1400 i 3 0 0 t 2 0 0 1 1 0 0 1000
W a v e n u m b e r
FIG. 3. The spectrum of Fg was obtained by subtracting the spectrum of the buffer from the spectrum of a 16.75 mg/mL solution of Fg.
tha t the absorbance due to water was 0.700 absorbance units. The details of this procedure have been repor ted elsewhere.11,12
The K matr ix results were generated with the use of QUANT, a program for mul t icomponent analysis, dis- t r ibuted by Mat tson Ins t ruments Inc. as par t of their E X P E R T S Y S T E M software. The reliability of this pro- gram was tes ted on a series of synthet ic mixture spectra which were constructed from random linear combina- tions of l ibrary spectra. No discrepancies between the predicted and actual component concentrat ions were ob- served.
The Q matr ix calculations were run on a program de- veloped at the Center for Fire Research for the purpose of quant i ta t ing target compounds in fire atmospheres. I t has been repeatedly tes ted for reliability, with the use of techniques similar to the one described above.
R E S U L T S AND D I S C U S S I O N
The spectra of IgG (10 mg/mL), A1 (10 mg/mL), and Fg (16.5 mg/mL) are displayed in Figs. 1-3, respectively. Each protein spectrum was obta ined by digital subtrac- t ion of the spectrum of the buffer from the corresponding
I I I I I I I I
O , O t S
0 . 0 1 4
0 . 0 1 2
0 . 0 l O
o . o o e
o oo
0 . 0 0 4
0 . 0 0 2
0 . 0 0 0
i t I 2 0 0 0 1 9 0 0 ] 8 0 0 1 7 0 0 1 6 0 0 1 5 0 0 1 4 0 0 1 3 0 0 ] 2 0 0 1 1 0 0 1 0 0 0
Wavenumber "
FIG. 4. The difference spectrum obtained by subtracting the spectra displayed in Figs. 1 and 2.
APPLIED SPECTROSCOPY 591
A
b
S
0
r
b
a 0.30
n
C 0.20
e o.io
5.2 5.4 5.6 I I I
0.70
0,60
0.50
0.40
Microns 60 6.5 7,0 7.5 B.O 8.5 g.o I0.0
I I I I I [ / ~ ~ ~uf fer
1900 1800 1700 |FoO0 JSO0 1400 |300 1200 1100 |000
Navenumber
Fro. 5. The spectrum of the buffer is compared to the spectrum dis- played in Fig. 1.
protein solution spectrum. The small differences be- tween these spectra are emphasized in Fig. 4, which was obtained by digital subtraction of the A1 and IgG spectra. The most prominent feature in this difference spectrum, an intense peak at 1640 cm -1, is not distinguishing in- formation a'L all, but rather an artifact which results from inconsistencies in the water subtraction. It is obvious that the spectral interference of water can have an ad- verse effect on the accuracy of the protein concentration determinations. A further illustration of the problem is presented in Fig. 5, which compares the relative inten- sities of the spectral features of the protein and buffer.
The strong deviations from Beer's law which charac- terize protein solutions constitute another potential source of error. In our experience, the nonlinearities in the mixture spectra, which result from optical aberra- tions and molecular interactions, are comparable in mag- nitude to the spectral differences between the proteins of interest. An assessment of this situation is facilitated by comparing Figs. 4 and 6. The latter figure is the dif- ference spectrum generated by subtracting 0.8 times the IgG spectrum and 0.2 times the A1 spectrum from the spectrum of a solution consisting of 8 mg/mL IgG and 2 mg/mL of A1. This spectrum is representative of the
0.0060
A 0 . 0 0 5 0 b S 0 0 , 0 0 4 0 r
b
a 0 . 0 0 3 0 n ¢
e 0 , 0 0 2 0
0.0010
0.0000
- 0 , 0 0 t 0
I I I [ I I I I I
I T I i I I I I I 2000 :tgo0 1000 t 7 0 0 t 6 0 0 1500 t 4 0 0 t 3 0 0 1200 1100 | 0 0 0
Wavenumber
Fro. 6. The difference spectrum obtained by subtracting the com- ponent spectra from a mixture spectrum.
4 0 . 0
.J S
3 2 . 0 •
o E O • ~ 2 4 . 0 •
¢,-
u 1 6 . 0 • ¢ -
• ~ 0 . 0
.l-i w
0 . 0
/ /
• 4 / ~ / O , / " / 4
o s:o tl;.o 2~.o s~.o
Actual Concentration. mg/mL
. 0
FIG. 7. Q matrix estimates of the concentration of A1 in ternary mix- tures, plotted with the actual concentration. Data points on the solid line mean the prediction is exact, while the dashed line was determined from linear regression.
difference spectra obtained from many solutions con- taining various amounts of the three proteins.
On the basis of the previous discussion, it is clear that, if a method is to be successful in obtaining accurate protein concentrations, it must be sensitive enough to detect subtle spectral differences. Furthermore, the method of choice must be capable of extracting mean- ingful information from the large number of calibration spectra which are necessary to adequately model the systematic deviations from Beer's law. Interferences due to unrepresented nonlinearities will propagate as large concentration errors because of the ill-conditioning in Eq. 2. The presence of numerical ill-conditioning in this problem is an inevitable result of the small differences between the component spectra.
In Table I we summarize results obtained by using both the K and Q matrix methods in a subspace con- sisting of 131 absorbencies in the spectral region between 1200 and 1700 cm -1. These data are also reported as three X,Y plots, where the estimated concentrations of the three proteins are plotted as a function of the actual concentration (Figs. 7 to 9). The first set of concentra- tions presented in Table I corresponds to a mixture which was in the calibration set. The remaining concentrations correspond to 20 mixtures which were not used for cal- ibration. The spectral window between 1200 and 1700 cm -1 encompasses the amide I, II, and III bands which characterize protein absorbance spectra. The spectral domains between 1000 and 1200 cm -~ and between 1700 and 2000 cm -z were omitted because, in our judgement, they did not contain any discriminating information.
A set of 50 reference spectra were initially chosen as the calibration set. However, we detected linear depen- dencies in this data in the process of transforming the A matrix to an upper triangular form. Of the original 50 calibration spectra, only 19 were independent. The re- maining 31 spectra could be expressed as linear combi- nations of these 19. By discarding the linear dependen- cies, we were able to obtain solutions to Eqs. 7 and 8, with a minimum of round-off error.
It is clear, from the magnitude of the errors in the concentration predictions obtained from standard K ma-
592 Volume 42, Number 4, 1988
3 0 . 0
.J E
" ~ 2 4 . 0
g • ~ i 8 . 0
L
g u ~ 2 . 0
• o S . O
0 . 0
/
o:o e'.o t~,.o tk.o 2~.o 30.0
Actual Concentration. mg/mL
FIG. 8. Q matrix estimates of the concentration of Fg in ternary mix- tures, plotted with the actual concentration. Data points on the solid line mean the prediction is exact, while the dashed line was determined from linear regression.
trix calculations, that this method has limited usefulness for the problem at hand. The total standard error of analysis 1~ in the K matrix predictions of the protein con- centrations using the 21 mixtures reported in Table I is 9.7 mg/mL. The corresponding value for the Q matrix concentration predictions is only 1.7 mg/mL. The source of the large errors in the K matrix calculations is the rank reduction approximation involved in solving Eq. 3. This approximation is an integral part of the K matrix method. It is necessary because values for the coefficients of the spectral representations of the deviations from B e e r ' s l a w a r e n o t g e n e r a l l y a v a i l a b l e . C o n s e q u e n t l y , a n ( n x M ) C m a t r i x r a t h e r t h a n t h e e x a c t (~ x M ) C m a t r i x is u s e d t o d e t e r m i n e I( . T h i s s t r a t e g y l e a d s t o a c c e p t a b l e r e s u l t s o n l y w h e n i t is a p p l i e d t o m i x t u r e s w h i c h d o n o t d e v i a t e s t r o n g l y f r o m B e e r ' s l aw. I t is c l e a r l y n o t s u i t a b l e f o r o u r p u r p o s e s - - t h e q u a n t i t a t i o n o f p r o t e i n s i n s o l u - t i o n .
R e c e n t l y , H a a l a n d h a s d e m o n s t r a t e d a v a r i a t i o n o f t h e K m a t r i x m e t h o d w h i c h i n v o l v e s u s i n g t h e r e s i d u a l s f r o m t h e c a l i b r a t i o n s t e p i n t h e l e a s t - s q u a r e s f i t t o t h e s p e c -
20.0
.J E
"~- l S . O •
1= o
" ~ t 2 . 0
¢._
g u 8 . 0
. . ~ 4 . 0
0 . 0
, , i , • / / yO o.o 4:0 e:o t~,.o sb.o 20.0
Actua l Concentration. mg/mL
FIG. 9. Q matrix estimates of the concentration of IgG in ternary mixtures, plotted with the actual concentration. Data points on the solid line mean the prediction is exact, while the dashed line was de- termined from linear regression.
TABLE I. Comparison of K and Q matrix methods. ",b
IgG A1 Fg
-1 .5 14.8 21.7 0.0 20.0 12.3 0.0 20.0 12.3 5.3 11.2 -5 .6 0.4 10.1 -0 .5 0.0 10.0 0.0 1.1 0.3 17.2
-0 .3 0.0 17.9 0.0 0.0 18.5
24.3 7.4 - 24.8 10.0 0.2 1.3 10.0 0.0 0.0 11.6 14.5 5.0 14.2 15.5 1.0 15.0 15.0 0.0 9.7 17.2 5.8
12.1 18.4 6.9 10.0 20.0 6.2 4.3 8.1 23.2 8.1 15.4 10.0 6.0 12.0 11.1
11.2 28.4 -7 .1 8.5 22.5 5.9
10.0 20.0 5.6 4.5 8.5 24.3 8.5 16.2 10.5 6.0 12.0 10.0 2.3 29.8 23.0 6.2 39.2 5.0 6.7 40.0 2.1
11.1 11.9 10.5 15.8 16.6 -1 .0 15.0 15.0 0.0 17.5 7.3 5.6 22.0 14.1 -2 .4 20.0 10.0 0.0 18.0 - 7.0 27.8 28.2 2.6 -2 .3 25.0 5.0 0.0 28.2 -2 .5 7.8 27.6 2.2 -3 .2 27.0 3.0 0.0
9.8 8.0 -6 .2 2.9 8.4 0.1 2.0 8.0 0.0
25.3 -3 .7 9.1 30.4 2.4 0.3 30.0 0.0 0.0
2.1 28.3 1.7 4.7 30.2 -2 .9 3.0 27.0 0.0
-1 .2 21.1 17.5 5.3 25.2 - 1.6 5.0 25.0 0.0
17.8 16.3 -30.6 4.1 7.3 -0 .3 5.0 5.0 0.0 4.2 10.6 20.1 8.1 15.7 9.6 7.5 15.0 9.2
17.7 1.3 -6 .6 9.6 3.1 -1 .6 8.0 2.0 0.0
. The calculations included all absorbencies in the region 1200 to 1700 c m - 1 .
b The concentrations (given in units of mg/mL) are listed in the order K matrix, Q matrix, and actual concentration proceeding down any column.
A P P L I E D S P E C T R O S C O P Y 5 9 3
trum of the analyte, it In the same paper, Haaland also reportedresults obtained from calculations with an aug- mented K matrix which includes multiplicative products of the component spectra. These product spectra are used to model Beer's law deviations which result from molecular interactions. The incorporation of these mod- ifications into our K matrix calculations would undoubt- edly reduce the magnitude of the errors we observed; however, it i:3 not likely that this approach to the problem would offer a consistent improvement in accuracy over the Q matrix method. Only modifications in the proce- dures used to select the calibration spectra and improve- ments in the numerical stability of the algorithms can be productive in this respect. In contrast to the conven- tional implementations of the K matrix method, the Q matrix method extracts all of the information which is present in the calibration spectra. This property is evi- dent from the results of the analysis of the first mixture listed in Table I. The Q matrix method always gives the exact concentrations of the analytes in the calibration mixtures. T:he error of calibration in the Q matrix meth- od is exactly zero.
CONCLUSIONS
The feasibility of using quantitative FT-IR spectros- copy to determine the concentrations of proteins in aqueous solution was examined. The K matrix method with sloping baseline and intercept functions was found to be inadequate, and a numerically robust version of the Q matrix method which gave acceptable results was derived. This method was used to estimate the concen- trations of 21 ternary protein solutions with a total stan- dard error of analysis of 1.7 mg/mL. This level of accu- racy is sufficient to make the Q matrix method a viable tool for general use in the field of protein chemistry. The specific application of this method to the study of blood- protein adsorption on synthetic surfaces should yield important information about the changes in blood pro- tein composition which accompany this process. Fur- thermore, s
