The advantage of this method is that, by rapid visual comparison scanning of complex sample spectra, elements are quickly identified and quantitatively determined to be present above chosen minimum concentrations or to be present in the important concentration levels of interest in connection with the subject problems with which the samples are associated. This avoids computations that would otherwise be required
to ascertain whether elements are present in sufficient concentrations to be important. Two years' experience with this method has shown that the average time expenditure in a diverse sample program is about 2.5 hours total per sample in duplicate for 26 element's, and about 1.5 hours per sample in duplicate when a few major need be determined.
LITERATURE CITED
(1) Hedge, E. s,, B ~ Q-.~ K,, ~ ~, ~ Spectr. 10, 150 (1956). (2) Mittledorf, A. J., Landon, D. 0.: "9 l2 (1956). (3) Task Group VI, Subcommittee 11, ASTM E-2, A S T ~~ ~~ i 216, l . p. 29 (SeDtember 1956). . -
P. ZSCHEILE,
the Mid-America Spectroscopy Symposium, Chicago, Ill., May 3, 1962.
Jr. and HAZEL C. MURRAY
Department of Agronomy, University of California, Davis, Calif.
G. A. BAKER and R. G. PEDDICORD Deparfmenf of Mafhematics, University of California, Davis, Calif.
b Linear relationship among the absorption curves of multicomponent mixtures may lead to an extremely unstable system of equations. The stability may be estimated by solving the system many times, each time changing the coefficients slightly. The method can also be used to measure the effectiveness of various choices of wavelengths. This method was applied in this study to a four-component system of RNA constituents (adenylic, cytidylic, guanylic, and uridylic acids); poor results were obtained because of the extreme linear relation between the absorption curves of adenylic and uridylic acids. When uridylic was removed, the results improved markedly. Various partitions of the wavelengths were selected, and the corresponding stabilities were computed. Best results were obtained when all available wavelengths were used.
S
analysis of unknown mixtures has often been confined to the simpler one- and twocomponent systems. Earlier methods (7, 19, 17, 18) were used on systems for which analysis of more than two components was not required. Some workers, by use of fortunate isolated absorption bands characteristic of one component only (high absorption for one component with negligible absorption of other components), have analyzed or studied systems with more than two components by stepwise analysis for single components ( 2 , 5, 6, 13). In some methods, approximations precede PECTROPHOTOMETRIC
1776 *
ANALYTICAL CHEMISTRY
the final solution (4,9, 13). It has been assumed, not always tacitly (1, 4, 8, 10, 19), that three or more components, each with different but appreciable absorptivity values a t the wavelength concerned, could be analyzed with equal effectiveness simply by the use of more wavelengths and the solution of more simultaneous equations. This assumption is correct mathematically, provided the absorption curves are not linearly related, but in practice certain disappointing (even ridiculous) results have been obtained by various experimenters. Stearns (14) mentioned that the application of simultaneous equations to a three-component system sometimes gives absurd calculated values-Le., negative values for components known to be present. In some cases, these discrepancies have been attributed t o deviations from Beer's law at certain rvavelengths, caused (perhaps) by interaction of the components or by impurities in extracts which absorb appreciably a t the shorter wavelengths. While these reasons may often apply, the major source of difficulty, linear relationship among the absorption spectra of the components, has seldom been mentioned. Recently Sternberg, Stillo, and Schmendeman (16) in a study of a five-component system by the method of least squares, stated the problem correctly but only hinted a t the cause. They explain that ([Past attempts for obtaining the composition of the irradiation mixture, based upon the direct application of the Beer-Lambert law to the ultraviolet absorption spectra of the mixtures,
i
~ ~ ~ ~ ~ o ~ ~ ~ ~ e f ~ p ~ S ~ ~ a y ~ ,
Instability of Linear Systems Derived from Spectrophotometric Analysis of Multicomponent Systems F.
~
failed because the system of five simultaneously linear equations obtained lacked sufficient independence. This failure has generally been attributed to the lack of accuracy with which the spectra of the components were known." The absorption curves of related compounds (isomers, for example) are often linearly related in the ultraviolet region (they may have their maxima and minima a t approximately the same wavelengths, for example). In such cases extreme accuracy may be required of the data to obtain only reasonable accuracy in the solution of the equations. This problem is less serious in the infrared; by careful choice of wavelengths to minimize absorption by more than one component, and by study of known systems, mixtures of eight components were successfully analyzed ( 3 ) . These considerations lead to the notion of the stability of a linear systemLe., the susceptibility to change in the solution of the linear system due to changes in its coefficients. This paper presents a method for determining the stability of the linear systems encountered in spectroscopic work and thus for estimating the precision which can be expected of the solution, given the precision of the data (the distinction between accuracy and precision will be made clear later). Various factors influencing stability are also discussed, along with some rules of thumb for obtaining better results. FACTORS INFLUENCING STABILITY
For ribonucleic acid constituents, many environmental factors surround-
ing the RSA molecules, such as pH, ionic strength, and concentrations of salts and other solutes, may influence the shape of the absorption curve and hence its linear relationship to other curves. The absorption spectra of two components are linearly related if the absorptivity of one component is a multiple of the absorptivity of the other a t each wavelength used in the analysis. The deviation of the ratio of their absorptivities from constancy is then a measure of the linear relationship. We are indebted to the referee for suggesting the following correlation index. Let T be the mean of the absorptivity ratios taken over the wavelengths used. Let d be the mean deviation of the ratios. Then the correlation index is (T - d ) / r . For more than two components, the correlation index between pairs cannot detect a linear relationship involving three or more curves. One rule of thumb is to adjust the p H for separation of the maxima and minima of the curves. When the pH and other environmental details have been decided, the absorptivities (and the absorbance values of the mixture to be analyzed) are obtained a t as many wavelengths as possible. Certain of these wavelengths are chosen to form the linear system. In general, the more wavelengths the better, but as mentioned previously, certain portions of the spectrum may deviate from Beer's law, and extremely low wavelengths should be omitted if impurities expected in natural and/or unknown extracts absorb appreciably. If many samples are to be run, it is of course desirable to select only as many wavelengths as are needed to maintain the desired accuracy, DERIVATION OF THE LINEAR SYSTEM OF EQUATIONS A N D ESTIMATES OF ITS STABILITY
The molar absorptivities of each of the and the absorbance values of the mixture itself are obtained a t k wavelengths A,, . . . , Ab, where IC 3 n. Let a,, be the molar absorptivity of the j t h compound a t the ith wavelength, A t , and let bi be the absorbance of the mixture a t that wavelength. Using Beer's law, we then form the overdetermined system n, ('ompounds
whose least-squares solution, if it exists, will yield estimates for the respective concentrations xi (in moles per liter) of the j t h component, j = 1, . . ,
.
n.
Using matrix notation we may ren ~ i t Equation e 1 as
Ax
= b
The precision of a set of observations refers to the spread about the mean, and is usually taken as the standard deviation, For our purposes it is convenient to normalize the standard deviation by dividing it by the absolute value of the mean; this ratio, multiplied by 100, is called the relative standard deviation (RSD). The accuracy of a set of observations, on the other hand, refers to the difference between the mean and true values, and expresses the bias of the measurement. Unless the true value is known the accuracy cannot be estimated; the precision, however, can be estimated, simply by making several observations. Suppose we repeat the spectroscopic analysis several times, each time obtaining new coefficients of A and b and corresponding solutions x, with components 21. Suppose also that the RSD of the zi's is considerablygreater than the RSD of the coefficients of A and b. I n this case, the effect of the linear system is to amplify the original error in the observations; this amplification fac-
where A is a k times n matrix and b is a k times 1 column vector. In general, Equation 2 will not have an exact solution since it is overdetermined] but we may form a related system by premultiplying Equation 2 by A transpose (denoted A') to obtain A'Ax
=
A%
(3)
ALA is a square matrix, and if the rank of A = n,A ' A will have an inverse, so we may premultiply Equation 3 by (A") -l to obtain 5
= (A'A)-' A %
(4)
This solution z has the property that it minimizes the length of the residual vector Ax - b. For a discussion of this property see Lanzcos (11). This method is the so-called least squares technique for solving an overdetermined linear system. The coefficients which appear in A and b are not exact numbers; each coefficient represents a particular observation of a theoretically fixed quantity.
Table 1.
No. of
Wavelength,
wavelength
mM
I 2 3 4 5 6
220 222 224 226 228 230 232 234 236 238 240 242 244 246 248 250 2 52 254 256 258 260 262 264 266 268 270 272 274 276 278 280 282 284 286 288 290
7
b9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
Spectrometric Data for Four-Component System
Molar absorptivities, AbSOd~nce ( x 10-3) Obsd. Calcd. Adenylic Cytidylic Guanylic Uridylic 0.368 0.305 0.253 0.210 0.184 0.170 0.169 0.178 0.198 0.224 0.260 0.297 0.339 0.385 0.434 0.470 0.515 0.555 0.585 0 . GO1 0.607 0.607 0.602 0,593 0.571 0.553 0.517 0.487 0.458 0.430 0.399 0.366 0.332 0.299 0.263 0.228
0.370 0.311 0.261 0.223 0.193 0.181 0.183 0.191 0.213 0,236 0.269 0.308 0.349 0.397 0.445 0.489 0.531 0.568 0,592 0.609 0.616 0.615 0.606 0.593 0.571 0.547 0.519 0,490 0.459 0.429 0.396 0.363 0.330 0.298 0,262 0.225
'
7.00 5.50 4.40 3.70 3.22 3.30 3.60 3.88 4.45 5.10 6.00 7.08 8.10 9.26 10.45 11.40 12.35 13.30 13.68 13.70 13.40 12.95 12.38 11.60 10.47 9.10 7.60 6.20 4.90 3.84 2.95 2.25 1.70 1.20 0.80 0.50
8.70 7.94 7.00 5.95 4.85 3 96 3.26 2.65 2.30 2.00 1 90 1 90 2.02 2.30 2 70 3.20 3.80 4.45 5.15 5.98 6.90 7.90 8.65 9.50 10.36 11.12 11 80 12.30 12.55 12.73 12.59 12.20 11.60 10.90 9.88 8.65
4.50 3.57 2.90 2.60 2.50 2.65 3.06 3.65 4.45 5.18 6.05 7.00 7.90 8.90 9.75 10.45 11.00 11.36 11.42 11.35 11.00 10.45 9.90 9.33 8.79 8.40 8.15 8.00 7.92 7.79 7.60 7.30 7.00 6.57 6.03 5.38
4 30 3.55 2.92 2.50 2.20 2.10 2.25 2.50 3.00 3.45 4.00 4.62 5.30 6.07 6.85 7.60 8.30 8.83 9.25 9.55 9.70 9.62 9.37 8.97 8.27 7.58 6.70 5.80 4.80 3.80 2.85 2.03 1.30 0.80 0.40 0.18
Molar concentrations Adenylic 1 . 4 4 X 10-6M Cytidylic 1 . 5 5 x 10-6M Guanylic 1.51 X 10-6M Uridylic 1.54 x 10-6M Total 6.04 X 10-6M
(2) VOL 34, NO. 13, DECEMBER 1962
1777
/
s
Figure 1.
Table
II.
c
'
I
3
220 222 224 226
228 ---
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
30 31
230 232 234 236 238 240 242
-244 -_
Spectrometric Data for Three-Component System
n- . 428 _-_
260 262 264 266 268 270 272 274 276 278 280 282 284 286 290 292 294 296
0,364 0.311 0.255 0,199
2.58 ---
288 ---
38 39
0.499 0.414 0.344 0.289 0. ~-~ 2.i2 0.233 0.228 0.238 0.259 0.289 0,328 0.374 0.481 0.5-12 0.598 0.648 0.695 0.732 7.in 0.755 0 . 756 0,750 0.744 0.726 0.704 0.680 0,657 0.632 0.603 0.577 0.545 0,508 0.464
246 248 250 252 254 256
n
417
Molar absorptivities,
x
ANALYTICAL CHEMISTRY
10-3
Adenylic
Cytidylic
Guanylic
0.757 0 . 753 0.744 0. 728 0.706 0.681
7.49 5.66 4.38 3.74 3.23 3.19 3.46 3.86 4.51 5.33 6 32 7.18 8.53 9.75 11.00 12.00 12.88 13.86 14.46 14.42 _14_ 07 . 13.61 13.02 12 25 11.09 9.61 8.07
0.630 0.603 0,576 0.544 0.506 0.464 0.416 0.362 0.309 0.253 0.197
5.i 7 4.03 3.13 2 37 1.74 1.25 0.86 0.52 0.31 0.18 0.10
8.35 7.53 6.59 5.61 4.64 3.77 3.04 2.48 2.08 1.86 1.74 1.76 1.89 2.16 2.47 3.02 3.58 4.24 4.99 5.81 G 67 7.63 8.53 9.47 10.34 11.18 11.85 12.44 12.79 12.95 12.89 12.56 11.95 11.14 10.13 8.93 7.69 6.32 4.91
4.50 3.56 2.86 2.42 2.29 2.45 2.87 3.43 4.13 4.91 5.78 6.66 7.61 8.53 9.38 10.13 10.72 11.07 11.30 11.21 11.00 10.44 9.85 9.20 8.70 8.33 8.08 7.90 7.79 7.64 7.46 7.20 6.88 6.47 5.92 5.25 4.56 3.81 3.01
0.495 0.409 0.339 0.289 0.248 0.229 0.226 0.234 0.254 0.286 0.326 0.367 0.. 424 -~~ 0.481 0.538 0.593 0.642 0.690 0.741 n 747
n 7.66
n- .6.5.5 ___
6 49
Molar concentrations Adenylic. 2.373 x io-5-if Cytidylic 2.580 x 1 0 - 5 ~ Guanylic 2.273 X 10-6i1.1 Total 7.226 X 10-6iTf
1778 *
I
Absorption spectra of nucleotides at pH 1.1
Absorbance No. of Wavelength, wavelength mp Observed Calculated 1 2
I
tor is a measure of the stability of the linear system. Various techniques have been devised for measuring the stability (sometimes called condition) of the linear systems (9, 16), but so far most of these measures have been of more theoretical than practical interest. The crude way, and the one we have adopted because it converges to the desired estimate of the RSD, is t o solve Equation 2 many, many times, each time changing the coefficients of .4 and b by small amounts, and to observe the distribution of the 2 ' s . The more times we solve the equation, the closer we will be to the limiting estimate. This scheme, although sound, requires a n electronic computer and considerable time. It is reasonable to assume that all coefficients are equally precise, that is, that the RSD, E , of each a,, and b, is the same. Let u be a normally distributed random variable with mean 0 and standard deviation 1. (The machine generates new values of u each tune u is required; the machine - generated u is not strictly random but for all practical purposes may be considered eo.) Let A and b denote the origins1 coefficient matrices. The computing scheme then consists of the following steps: 1. Solve the system A z = b for x. Store x. 2. Form nev A ' and b' by setting
3. Solve A'z' = b' for 2'. Store 2'. 4. Repeat steps 2 and 3 many times (say 50) and then go t o step 5 5 . Compute the mean, standard deviapion, and RSD of the distribution of each Z,.]= 1, . . . , n.
I n the tabulated results the mean nil1 be called the mean value. EXPERIMENTAL
Standard solutions were inade from weights of about 25 mg. dissolved in 100 ml. of aqueous HC1 (pH 2.2). Solutions were stored at 2" C.; dilutions were made a t 25" C. Volumes of 4 or 10 ml. were diluted to 100 ml. of HC1 (final pH 1.1). Spectra were determined niaiiually at room temperature (near 25" C.) on a Becknian Spectrophotometer, Model DU, using carefully matched cells 1 cm. in thickness. Corrections Ivere applied when necessary for cell differences of 0.001 or more absorbance units. Absorbance values (read to the third decimal place) were restricted to the range between 0.220 and 0.600 over the wavelength range 220 to 296 mp for cytidylic and guanylic acids. For adenylic and uridylic acids this range was 220 to 280 mp; for wavelengths above 280 nip, absorbances were less than 0.220. Plotting of data per-
Table 111.
Four-component system Three-component system
P1
P2
P3
P4
P5
P6
1-36 1-39
1-19 1-19
2W36 20-39
19, 21, 30, 34, 36 19, 21, 30, 34, 36
18-21 18-21
22-25 22-25
mitted some choice of absorbance values a t overlapping wavelengths among the three concentrations of each standard. Mixtures of three and four coniponentq were made by mixing appropriate volumes of standards. Spectra were determined as with the pure standards.
low pII. The molar nbsorptivitics w e presented in Tables I and 11. They were determined carefully on purified standards [grade A, 3' (2') isomers, supplied by California Corp. for Biochemical Research, Los Angeles, Calif.], which were shown to obey Beer's law in the concentration ranges involved. Repetition of all weights, dilutions, and measurements gave nearly identical results. Seven different sets of n-avelengths (which we call partitions) were chosen
RESULTS A N D DISCUSSION
The ultraviolet abborption spectra of the more common RSX nucleotides are presented in Figure 1 for solutions a t
Table IV.
All iiiolar concentration values
x
10-5
Partition
True value
P1
Calcd. value Rel. error Xean value RPD
P2
TCI Calcd. value Rel. error Mean value RPI)
TCI P3
P4
C a l d value Itel. error Mean value RSD TCI Calcd. value
Rel. error
Mea11 value
RSD TCI P3
and thc resulting equations 15 L're analyzed. The numbers of the wavelengths for each partition are displayed in Table I11 (for the corresponding wavelengths, see Tables I and 11). The four-component systcm equations were solved 100 times each, using an assumed RSD of 2% and the three-component system equations 50 times each, using an assumed RSD of 1%. Table IV presents the results. The calculated values are the solutions of the original equations (with unchanged coefficients), and the
Analysis of Stability of Three- and Four-Component Systems
Adenylic
Four-comnonent si-stcm Cytidylic Guanylic
Three-comnonont svstein
Cytidylic
Guan>-%
1.55
1.51
1.54
2 37
2.58
2.27
0.95 34.0 0.90 26.2 1.30
1.56 0.64 1.56 2.29 0.314
1.49 1.3% 1.49 5.60 0.687
2.12 37.7 2.22 15.2 0.786
2.36 0.42 2 38 3 . '28 0.373
2.56 0.78 2 57 2.04 0 272
2.32 2.20 2.29 4.78 0.563
2.92 103.0 1.86 54.5 1.93
1.27 18.1 1.33 7.57 0.393
0.48 68.2 0.58 75.3 1.60
0.67 3 . 5 2.10 '76.8 1.85
2.l.i 9.28 2 07 30 1 0 975
2.67 3.49 2.68 5 86 0.258
2.55 12.33 2.64 25.5 0.828
1.93 34.0 1.52 56.7 0 902
1.78 14.8 1.68 15.9 0.607
1.14 24.5 1.31 36.8 0.i48
1.03 33.1 1.48 61 . 0 0,654
2.32 2.11 2.31 7.05 0.107
2,55 1.16 2.52 7.60 0.572
2.33 3.52 2.40 15.4 0.657
0.40 72.2 1.07 777.0 0.596
1 22 21 3 1.50 210.0 0 319
"15 42.4 1.63 360.0 0.395
2.40 53.8 1.88 376.0 0.519
2.26 4.64 2.37 20.4 0.082
2.48 3.88 2.57 16.1 0.292
2.47 8.81 2.30 35.8 0.373
1.44
I
Vridylic
.klenylic
20 0 1190.0 -2.54 2555.0 2.23
2~. 0 32.5 -6.i3 999.0 2.61
-50.0 3347.0 10.25 1436.0 2.64
2.65 11.8 2.56 245.0 1.71
2.32 10.1 2.54 54.5 1.44
2.06 9.25 2.09 375.0 1.10
Calcti. value Itel. error
-30.0 2183.0 -3.64 1208.0 2.63
-2.00 229.0 0.60 1441.0 2.17
9.00 496.0 -6.62 1304.0 2.67
30.0 1848.0 17.5 804.0 2.67
2.51 5.91 1.14 325,O 1.67
2.59 0.39 2.53 25.3 1.40
2.08 8.37 3.90 133.0 1.67
0.31 78.7 0.66 '730.0 1.01
1.60 3.23 1.51 113.0 1.28
1.54 1.99 1.68 170.0 1.29
2.36 8.53 2.29 i4.3 1.23
2 68 ._ 18.1 2.82 105.0 1.25
2.24 55.5 2.10 93.2 1.22
1.90 22.6 1.86 211.0 0.511
RSD
TCI
P8
10, 25, 28, 34 10, 25, 28, 34
"0.0 1289.0 4.05 1350.0 2.65
Mean value
P7
P8
P7 29-36 29-36
Calcd. value Rel. error Mean value
RSD TCI P6
Partitions of Wavelengths by Number
C'alcd. v d u e Rel. error
Mean value RSD TCI ('alcd. value Rel. error Mean value RSD TCI
0.89 41.0 0.95 77.9 0.827
2.49 61.7 2.18 194.0 0,539
2.34 1.27 2.29 18 1 0.538
0.72 53.2 0.88 2331.0 0.797
2.33 2.10 2.32 5.94 0.423
2.53 1.55 2.53 3.76 0.476
~
2.39 4.36 2.40 8.40 0.777
VOL. 34, NO. 13, DECEMBER 1962
1779
Table V. Correlation Indices for Four-Component System Adenylic Cytidylic Adenylic Adenylic Cytidylic Guanylic US.
us.
us.
US.
us.
0s.
Partition
cytidylic
guanylic
guanylic
uridylic
uridylic
uridylic
P1
0.280 0.464 0.232 -0.015 0.857 0.842 0.520 0.278
0.465 0.241 0.739 0.561 0.844 0.852 0.986 0.657
0.682 0.869 0.449 0.283 0,988 0.984 0.529 0.611
0.872 0.978 0.804 0.718 0.971 0.979 0.682 0.879
-0.141 0.346 -0.086 -0.061 0.884 .. _ _ _ 0.860 0.197 -0.050
0.079 0,881 -0.016 -0.006 0.959 ~.... 0.988 0.188 0.149
P2
P3
P4 P5 P6
P7 P8
Table Partition P1 P2
P3
P4
P5 P6 P7 P8
VI.
Correlation Indices for Three-Component System Adenylic Cytidylic Adenylic US.
cytidylic 0.202 0.450 0.106 -0.024 0.853 0.834 0.512 0.247
us.
us.
0.481 0.236 0.749 0,540 0.846 0.837 0.986 0.644
0.576 0.879 0.310 0.285 0.991 0.986 0.525 0.602
guanylic guanylic
.
relative errors are their deviations from the true values expressed in per cent of the corresponding true value. The mean values and relative standard deviations are described in the computing scheme. It must be remembered that the RSD and not the actual relative error is the true index of the stability; the RSD may be interpreted as a measure of the precision of the solution. The relationship between the calculated and mean values is not known. For the three-component system the RSD tends to be greater than the actual error, especially for the partitions containing fewer wavelengths. The only explanation we can offer is that in these regions the data were more precise than 1%. The absorption curves of the standard components (Figure 1) (the curves for the three- and four-component systems differ only slightly because of different standard preparations), together with the correlation indices of Tables V and VI, give a good idea as to how the curves are linearly related (in pairs) for the various partitions. To facilitate comparison with the RSD, we have defined the total correlation index (TCI) of each component to be the sum of the squares of its correlation index with each of the other components. The TCI values are listed in Table IV. Partition P1 contains all available wavelengths, and yields the most stable system. The curve for cytidylic is displaced considerably to the right of the other three curves, indicating that it is not linearly related to them. As the
1780
ANALYTICAL CHEMISTRY
results show, cytidylic has the lowest RSD. The curves for adenylic and uridylic show a high correlation index and have high RSD’s. When uridylic is removed-Le., the three-component system-the results improve considerably. Part of this improvement, as we have mentioned, is due t o the increased precision of the data. Partitions P2 and P 3 represent, respectively, the lower and upper regions of the spectrum. With the exception of adenylic in the four-component system, the T C I predicts how the RSD will change passing from P2 to P3. Since there are three-way linear relationships possible (which are undetected by the correlation index) in the four-component system, the TCI is a weaker predicter, relative to the three-component system. Partition P4 is essentially partition P3 with many wavelengths removed. We retained only the minimum number of wavelengths which permitted a French curve fit to the data. The TCI’s of all components substantially decreased, passing from P3 to P4, yet the RSD’s of all components increased. This illustrates the effect of removing wavelengths from the analysis, even though the linear relationships are reduced. The three-component system did not suffer as much from this loss of information (a twofold increase in instability) as did the four-component system (a tenfold increase). Partitions P5 and P6 represent the minimum partitions (each has four consecutive wavelengths), and they occupy adjacent regions of the spectrum. The TCI predicts the change in stability passing from P5 to P6 in two out of three components (adenylic is the exception) for the three-component system, and in three out of four components (uridylic is the exception) for the fourcomponent system. Partition P 7 was selected because the curves for adenylic and uridylic practically coincide for the higher wavelengths. We conjectured that the fourcomponent system could be analyzed as a three-component system in this region. The correlation index between adenylic and uridylic is not as large as me had
expected, and the RSD’s of cytidylic and guanylic are higher than we had expected. The last partition, P8, was selected by the referee on the basis that the correlation indices were low (for using only four wavelengths). The results were better than those of the other fourwavelength partitions ( P 5 and P6), indicating that the correlation index is a useful measure of stability. If a specified accuracy must be obtained, and if the system is too unstable to provide this accuracy, then an alternate method of analysis must be used. The use of adsorption column or other methods of fractionation into pairs of constituents, followed by spectroscopic analysis of the resultant binary systems, is suggested. A simple but effective method for these RNA constituents will be described elsewhere (do), LITERATURE CITED
(1) Allen, E., Rieman, W., 111, ANAL. CHEM.25,1325 (1953). (2) Avery, W-.H., Morrison, J. R., J. A p p l . Phys. 18, 960 (1947). (3) Barnett, H. A., Bartoli, A, ANAL. CHEM.32, 1153 (1960). (4) Brattain, R. R., Rasmussen, R. S., Cravath. A. PII.. J . A .. v d Ph:US. 14. 418 (1943). (5) Brode, W. R., Pattern, J. W., Brown,
J. B., Frankel, J., IND.ENG. CHEM., ANAL.ED.16, 77 (1944). (6) Clark, C., “Physical Techniques in Biological Research, Vol. I, Optical Techniques” chapter on Infrared SpectroDhotometrv G. Oster and A. W. ster, eds“., Academic Press, New York:, 1955. (7) c omar, C. L., Zscheile, F. P., Plant
Physiol. 16, 6 51 (1941). ( 8 ) Ibid., 17, 198 (1942). (9) Forsythe, Ci . E., Bull. Am. Xath. SOC.59, 299 ( 1953). 110) Frev. D. L., Nusbaum, R. E., ‘ Randall, H. M., J . A p p l . Phys. 17, 150 (1946).
(11) Lanzcos, C., “Applied Analysis,” p. 164, Prentice Hall, Englewood Cliff, N. J. (1956). (12) Miller, E. S.,“Quantitative Biologi-
cal Spectroscopy,”Burgess, Minneapolis
i19391. \ _ _ _ _
(13) Niklsen, J. R., Smith, D. C., IND. ENQ.CHEM.,ANAL.ED. 15, 609 (1943). (14) Stearns, E. I., Ibid., 25, 1004 (1953). (15) Sternberg, J. C., Stillo, H. S., Schwendeman. R. W., Ibid... 32.. 84 (1960). (16) Todd, J., Proc. Cambridge Phil. SOC.43., 116 (1949). (17) White, J. W., Jr., Brunson, A. hI., Zscheile, F. P., IND.ENG.CHEM.,ANAL. ED. 14, 798 (1942). (18) Zscheile, F. P., Jr., J . Phys. Chem. 38,95 (1934). (19) Zscheile, F. P., Jr., Cold Spring Harbor Symp., Quant. Biol. 3, 108 (1935). (20) Zscheile, F. P., Jr., ?lurray, Hazel, University of California, Davis, unpublished data, 1962. RECEIVEDfor review April 17, 1962. iiccepted October 15,. 1962. The computing and programming time was sup-
ported by the National Science Foundation grant, NSF-G14637. The spectroscopic and chemical work was supported in part by a grant from the Herman Frasch Foundation.
