S. AINSWORTH
1968
mately to any real solution. It is our hope that in the near future there will exist experimental data on simple systems in sufficient abundance to provide thorough tests of the present equations and to suggest, perhaps, new equations of more general validity. Appendix A Glossary of Notation for the Diffusion Coefficients.To aid the reader, we here prevent an index of notation together with our interpretation of the meaning of each of the symbols.
BEARMAN D . . . . . . . . . . . . . . . . .Mutual diffusion coefficient in the volume frame
i-01. ti5
Di, i = 1, 2, A , 5 . .Self-diffusion (tracer diffusion) coefficient in the mixture EYRINQ D1.. . . . . . . . . . . . . ..Mutual diffusion coefficient in the volume frame D, DLo............. Self-diffusion coefficient in the mixture HARTLEY-CRANK Dv.. . . . . . . . . . . . ..Mutual diffusion coefficient in the volume frame BA,BB... . . . . . . . ..Mutual diffusion coefficients in the mass frame (intrinsic diffusion coefficients) in their section seven. Mutual diffusion coefficients in the volume frame in their section eight DA, D B . .. . . . . . . . . .Mutual diffusion coefficients at infinite dilution (equal to the corresponding self-diffusion coefficients)
SPECTROPHOTOMETRIC ANALYSIS OF REACTION MIXTURES' BY
s. AINSWORTH
Department of Biochemistry, University of Shefield, Shefield, England Received April 1 1 , 1961
A method is described to give the number of absorbing species in a reaction mixture by employing spectrophotometric data in matrix form. Application of the method to various types of reactions is discussed and several test systems are examined.
Introduction Sternberg, Stillo and Schwendeman2 have described a least squares method in matrix form to fit the spectrum of a known mixture by the spectra of the pure components, the concentrations of which act as variables. They applied the method to determine the percentage composition of mixtures arising in the irradiation of ergosterol. Later, Reid and Pratt3 employed the same method in the analysis of ribonucleic acid. I n a recent paper, Weber4 has described another application of matrix analysis. The fluorescence of a mixture is excited at a number of wave lengths, n, and the emission observed a t a further wave lengths, rn. The nm observations are set out in a matrix, the rank of which gives the number of fluorescent components in the mixture. This paper will describe a method, similar to that of Weber, to enumerate the absorbing species in a reaction mixture using spectrophotometric data. Prior knowledge of the spectra of the pure components is not required. Enumeration of Components.-Assuming that the optical density, ds,x,a t wave length X of a mixture s, containing k components, is a function of its composition alone, we have
+
d8.k = c ~ l C Y x 1
Cs2CYAz
+ .......
CikcYkk
(1)
where c is the concentration of component k and CY is its extinction coefficient a t wave length A. For n such mixtures and m wave lengths, the nm optical density readings may be set out as an absorbance matrix, As,x (1) This work was supported in part b y t h e Atomic Energy Commission. ('2) J. C. Sternherg, H. 8. Stilt0 and R. H. Schwendeman, Anal. Chem., 3 2 , 84 (1960). (3) J. C . Reid and A. W. P r a t t , Biochem. Biophys. Research Conanun., 3 , :337 (1960). (4) Q. Weber, Nolure, 190, 27 (1961).
1
A,,.+ =
I ~
dll dz1 . ' . . ....................d n l dlrn dL, .... d u m
(2)
~
I
The &,A matrix is the product, taken row by column, of two subsidiary matrices 'a11 a12 . ' . CYlk 'cll CZl * ' ' ' c.1'
. . . . . . . . ., / . . . . . . . . . . . . . . . . . I*.&. . . . . . . . . .... . '
A.,x=
a&
CXmk
'Clk
c2k
(3)
C*l
'
* '
In the same way, difference spectrum measurements may be used to set up a D,,x matrix
- &I dnl .......................
dll dl,
. . * e
- dOm
' *
' * dnm
&I1
=
- domi
- COl . . . . - COIj 1 1 .................... a l c l k - C0k Cnk - &k 1 1 c11
CDl
' ' ' '
(4)
where subscript 0 denotes some reference state of the system: the a matrix remains unchanged. Or, more shortly 'All
D,,x =
A21
....
An1
~
j
................ a IAlk A$ .... Ank 1
(5)
where Ank represents the change in concentration of component k between the state n and the chosen reference state. By definition, a matrix is of order F if it contains a t least one non-zero minor of order F , while all 1 are zero. In general, deminors of order F terminants derived from the CY matrix are non-zero, irrespective of order, since, for absorbing species, each element has an arbitrary value. The possibility of a zero solution occurring by chance is small, and can be tested for by measurements a t a further number of wave lengths. The ranks of the concentration matrices, therefore, determine the ranks of the corresponding -4,~ and D,,x matrices, and operations performed on the one are equivalent to operations on the other. Further, it is a standard theorem that if the matrix corresponding, for example, to the forms
+
SPECTROPHOTOMETRIC ANALYSIS OF RUCTIONMIXTURES
Kov., 19Gl
1969
4. Conversion of a Closed System to an Open System:-The conversion of a closed system to an open system, by permitting the independent enumeration of the absorbing species, could, in principle, provide additional information concerning the posP A FA = c (6) sible inter-relationships among the components PD F D c of the closed system. (depending on whether absolute or difference Application of Matrix Analysis to Reaction Sysspectrum measurements are employed in the esti- tems.-It is suggested that the main use of the mation of rank), where C enumerates the compo- methods described above will be in the spectronents in the system and P the relationships among photometric investigation of reaction systems, them. Since the values of P and F depend on the i.e., closed systems in the above sense. type of system examined, these may new be deIf the composition of the mixture varies as a fined. of time, the matrix can be set up by 1. Closed Systems.-In a closed system, the function extinction measurements made after successive total concentration of the several components re- intervals. It is necessary, of course, to measure mains constant. Two main types of closed sys- the several optical densities in a time short in comtem may be distinguished. parison with the rate a t which the system changes (a) A ---f B.-In this system, a complementary in composition; if this is not possible, measurements relationship exists between A and B, that is, -AI* a t different wave lengths can be made in successive = +ALE, and so on. The rank of the D,,Amatrix runs. is therefore one For equilibrium systems, the composition of the I I ‘ A I * A>*’ mixture may be varied by alteration of such physi(7) cal conditions as temperature or pressure or by control of the reactant and product concentrations. indicating the presence of a single inter-relation- Similarly, photostationary states may be varied ship, PD = 1. Kevertheless, the absolute con- by alteration of the incident light intensity. centrations of A and B remain arbitrary and the The conversion of a closed to an open system may rank of the A,,Amatrix, FA,is two, directly enumer- be possible, in favorable cases, by stopping the ating the components in the system. Similarly, reaction in several states followed by addition of the system A+B-+C has ranks, FA and FD, of components in arbitrary amounts. 3 and 2, respectively. Determination of Rank.-The rank of a matrix * * *.. * + PP qQ (b) UA bB is determined as one less than the order of the In this svstem there is, in addition to the com- lowest minor giving a zero solution. Estimation plemental-y relation discussed under (a), the of the order of the first zero minor is subject to possibility of further direct relations between two, uncertainty arising from errors of measurement. or more, components. The effect of this is to Computational errors also may arise in the evaluareduce F D , relative to C, by one unit for each such tion of the determinant since, in this process, relationship. In the present example, -AIA/a = numbers of the same order of magnitude are sub-AIB/b := +Alp/p = +AIQ/q, etc., and F D = 1 tracted from one another. The consequent loss (column or row multipliers do not affect the proper- of leading significant figures may result in a soluties of a matrix). tion less accurate than the given data. Assuming The value of FAdepends on the initial state of that computational error is guarded against (by the system. If the initial concentrations of the carrying the calculation to more figures than the components are arbitrary, FA = C. However, data), it is of interest to consider the effect of if [Alo/[B10 = u / b and [P10/I&lo= p/q, PA = 2 errors of measurement on a determinant as a and there is a corresponding reduction in the value function of its order, so that an objective test for of FA. a zero solution may be advanced. 2. Open Systems.-In an open system, the Let S, be the value of a determinant of order n concentrations of the components are varied with elements aij, and 8,’ the value of the corarbitrarily. KO inter-relationships are present responding determinant in which the elements are between the different components and FA = F D = subject to an error with standard deviation d. c. 8,’ is the sum, having regard to sign, of n! terms, 3. Systems Containing Colorless Components.each being the product of n elements from dif(i.e. components that do not absorb a t any of the m ferent columns of the determinant. Expanding wave lengths employed to set up the matrix).-For i ((ali d)(anj f d ) . . . . . ( a n x d)] (8) 8.‘ = the closed system, A-+B+C, FA = 3 and F D = 2. i#j However, if one of the components is colorless, the rank of both matrices is 2 . The presence of the determinant and neglecting powers of d greater the colorless component has produced, essentially, than one, we have a two-component open system. Considered as a closed system, the situation is represented formally 8,’= C i (ali * * . anx) i#j by inclusion of a column of zeros in the a matrix, with the consequent appearance of only two concentration terms in the & , A matrix. The appearance of additional colorless components in a The signs in the second term are determined by the error, and therefore have equal probability. closed system has no further effect on the rank.
d , , ~is of rank F , then tlhese forms contain F linearly independent variables, any variables in excess of F being linearly dependent. Thus, for a given system, we have
+ +
+
+
+
+
*
*
+
S.AIXSWORTH
1970
But, as the probability of positive and negative signs as determined by the transposition ,rules is also equal, equation 9 may be rewritten as
Vol. 65 Sn ‘ - f- n --
3 ’,
p,n-z
un
2
f nd
(17)
where t,he equation is applied to the mean of a number o j determinants of order n. Since S,‘ represents all possible minors of Sn’
(ali . . . .
The second term is the sum of n2 minors of order nz = (n - l), having random sign. If the values, Smi,are randomly distributed about their average, the second term is equivalent to a random walk in one dimension of n2 steps whose length is the root mean square of the min0rs.j Hence, equation 11 becomes
s,‘
=
S,
ndd\/S,2
However, if the selection of minors of order m is limited to those obtained in the expansion of S,’ the additional terms in equation 17 may be employed empirically as corrections, the subscripts defining the selection of elements included in the two expansions. Equation 17 may be used as a test for a zero solution, thus ( a ) If Sn‘/SIn’ and, S, = O
(la)
Assuming that \Then S , # 0, S, =S,’ and employing the numerical average of the minors as an approximation to the root mean square, we have
the factor 2 being introduced to define the level of significance. Application of Matrix Analysis to Model Systems. 1. Introduction.-A matrix of random numbers was used to test the validity of equations This equation may be used a_s a test for a zero 16 and 17 in the absence of errors of measurement. solution for, when S, = 0, Sn‘/Sm‘ = nd. The equations then were tested using artificial However, to avoid chance effects, it is desirable mixtures of dyes. Two real systems also were to apply equation 13 to a significant sample of the examined. minors of order n that can be obtained from the 2. Experimental Methods and Results. Ranmatrix of experimental results. In these circum- dom Numbers.-A sequence of 32 random numbers stances, eraluation of all the possible minors of less than 11 were obtained from statistical tables order m, required by equation 13, becomes too and arranged in a 4 X 8 matrix. laborious: the selection of minors of order m is Dye Mixtures.-Three series of dye solutions limited. tberefore, to those arising in the expan- were prepared containing 2, 3 and 4 components, sions of S,’. For this reason, an approximate solu- respectively. Individual dyes were made up in tion of the ratio S,I S m r for non-zero determinants, alcohol (so as to prevent dimerization6)and the conin terms of the elements employed in the respective centrations adjusted to give approximately equal expansions is useful as a comparison to the value optical densities a t their absorption maxima. of nd. Mixtures of the dyes were prepared by replacing We have a given volume of one dye solution by an equal volume of the second, and so on. The two component system contained thioniiie and methylene blue. The 3-component system contained phenol where pn is the average numerical value of all the red, acridine orange and di-iodo (R)fluorescein, elements in S , (negative rows or columns may be with the addition in the 4component system of changed in sign without altering the properties rhodamine B. Optical densities at several wave of the determinant) and ( p n q I J ) = alJ. lengths and for various proportions of the difWhen this equation is expanded, terms in ferent components were measured and set out in F~~ and pnn-’ disappear. The remaining terms matrices. Table I gives the results obtained with have real sums, but, for n < 4 and ~ / p > O . 5 , only the 4-component system. those in pnn--?need be considered. If the variance, Cytochrome Oxidase Reaction-A miom-soluu * , is employed as an approximation to the mean hilized preparation of cytochrome oxidase was value of the products of residuals taken two at a reduced by dimercaptopropanol in the presence of time, me have catalytic amounts of cytochrome c. Optical n Z ( n -2) density changes a t different wave lengths occurring during the reaction of reduced cytochrome oxidase s, = i ( p a - ’ u,?), (15) (2 x M after mixing, pH 9, room tempera2=1 ture) with oxygen (3 x 111 after mixing) Equation 15 represents a random walk and hence were measured in a stopped flow apparatus. These results, which were kindly made available by Gibson and G r e e n ~ o o d , are ~ given in Table 11. Substitution of equation 16 into equation 13 gives (6) E. Rabino-itch and L. r. Cpstein, J Am. C‘hern Soc., 63, 69
+
5
(5) L. Brillouin. “Science and Information Theory,” Academic Press. Inc., New York, N. Y . , lY56, pp. 129-132.
(1941).
(7) &. H. Gibson and C. Greenwood, private commnnication.
SPECTROPHOTOMETRIC ANALYSIS OF REACTION ~IIXTCRES
s o v . , 1961
1971
TABLE I OPTICALDENSITIES X 10 FOR MIXTURES OF FOUR DYES
-
Wave length, m p
1
2
3
4
5
Mixture
6
7
8
9
10
427 490 525 543
1.47 3.56 4.60 4.73
3.48 5.02 2.01 1.71
5.75 4.15 0.83 0.42
1.62 4.50 4.29 3.31
0.48 2.84 5.62 6.26
2.48 4.50 3.66 2.48
4.06 4.17 2.03 1.71
5.97 4.91 0.33 0.12
1.52 3.72 4.73 4.20
0.40 2.36 6.61 6.85
TABLE I1 DEKSITYCHAXGES x 100 MEASURED AT TIMEt AFTER h l I X l K G SOLUTIONS O F REDCCED CYTOCHROME AND OXYGEN(GIBSOK A X D GREENWOOD?)
OPTICAL
The natural order of the table has been reduced so as to increase tetrad differences Wave length, mp t , mec.
415
435
23 3 33
2.0 6.3 1.4
13
3.6
-0.8 -3.0 -0.4 -1.8
445
425
405
430
395
440
450
5.1 -10.2 - 3.7 - 7.3
4.3 7.2 3.8 5.6
2.0 6.1 1.5 3.7
18 2.6 1.6 2.0
1.4 3.3 0.9 2 0
-2 5 -7 8 -1 4 -4.6
-4.6 -8.5 -3 8 -6 0
-
420
2 6 2 4
8 6 0 1
OXIDASE
-400 1 4 1 2
4 6 0 8
the use of equation 16. The predictions are less good for the dye systems, particularly for the 4component system. A similar comparison betwe-en the calculated and observed values of the ratio Sn’/Sm’ gives better results, and application of the proposed test for a zero solution permits, in every example, an unequivocal estimation of rank. The reduction in rank on passing from nbsorption spectra to difference spectra is illustrated in the three dye systems. Cytochrome Oxidase Reaction.-The data in Table IV show that F D = 2. This implies that a t least three components contribute t o the spectral changes occurring during the oxidation of reduced TABLE I11 cytochrome oxidase. The simplest interpretation OPTICALDENSITIES X 10 FOR M I X T ~ R EOF S HEMOGLOBINof this finding is that the system involves successive reactions of the type A+-B+C. However, there AND OXYHEMOGLOBIN is good evidence8 that cytochrome oxidase comThe optical densities of mixtures 3, 4 and 5 mere used in calprises two pigments, cytochromes a and u3, both culating the results given in Table IV. of which participate in the transfer of electrons to -Wace length, m p oxygen Mix596 678 572 561 546 536 454 505
Hemoglobin Mixtures.-Mixtures of oxyhemoglobin and reduced hemoglobin were prepared by the progressive deoxygenation of a dilute solution of oxyhemoglobin (pH 9, borate buffer) contained in a closed glass cuvette, in a side arm of which was placed a solution of reducing agent (anthraquinone-P-sulfonic acid and sodium dithionite in alkaline solution). Since reduction took place by diffusion of oxygen through the intervening gaseous phase, the rate of reaction was slow, and readings a t a succession of wave lengths were easily made in a time during which no change in the composition of the solution occurred. Table I11 gives the results obtained.
---
ture
1 2 3 4 5
160-
bestic point
1.00 1.83 2.18 2.55 2 37
6 5 5 4 4
70 60 27 56 92
5.62 5.21 5 24 5 08 5 16
3.87 4.83 5.30 5 86 5 55
6 22 5.70 5 63 5 29 5 50
5.76 4 88 4 60 4 16 4 38
6 38 2 36 71 2 39 37 2 37
5 5 4 5
89 2 33 20 2 34
3. Evaluation of Determinants.-N X n determinants were expanded by minors. In every case, severd such determinants were evaluated by moving crabwise along the matrix. The mean of the values thus obtained, together with mean values of l,he minors, are recorded in Table IV, where they are compared with values predicted by equations 16 and 17. The error term, 2,was calculated on the assumption that the extinction measurements were subject to a 2% error, t o which was added a further 2% making up error in the case of the dye mixtures. Measurements of the cytochrome oxidase reaction were assumed subject to a 5% error. 4. Discussion. Test Systems.-Table IV shows thal, for a random system, the value of S,, may be predicted within reasonable limits by
where the superscripts indicate the oxidation states of the two pigments. This system contains two independent variables for -&as3+ # -Anaz- and but - A,,aa3+ = A n a 2 + and
# Ana3+
- Ana2+ =
&as+
and is, therefore, consistent with the finding that F D = 2. Hemoglobin Mixtures.-Although it appears that the combination of hemoglobin with ligands, such as oxygen or carbon monoxide, takes place in four distinct stages, there is no spectral evidence to show the existence of the several intermediates.1O Many investigations, such as those of Iiahas” ( 8 ) D. Kellin and E. F. Hartree, Proc. Roy. Soc. ( L o n d o n ) , Bl27, 167 (1939). (9) T.Yonetani, J . Biol. Cnem., 235, 845 (1960); 235, 3138 (1960). (10) Q. H.Gibson a n d F. J. W.Roughton, Proc. Roy. Soc. ( L o n d o n ) , B146, 206 (1957). (11) G. G . Nahas, Science,lis, 723 (1951).
1972
1. 2. 3. 4. 0.
(j.
7. 8. 9. 10.
0. K. RICE
System Random numbers 4 dyes-A.,h 4 dyes-D,.h 3 dyes--A,,A 3 dyes-D,.A 2 dyes--A,.h 2 dyes--D,,h Cytochrome oxidase-D,, Hb/OaHb-A..X Hb/O*Hb-Da,h
h
Matrix 4 x 8 4 x 10 4 x 10 3 x 4 3 x 4 4 x 5 4 x 5 4 x 11 3 x 7 3 x 7
-s-
1'01. 65
TABLEIV OF R A N K DETERMINATION
Obsd. Calcd. 27.1 12.3 2.4 6.3 2.0 8.4 12.6 4.5 8.2 2.7 4.6 1.4 0.14 0.91 3.1 4.5 1.5 1.5 0.11 0.33
S -nObsd.
Calcd. 184 140 12.6 48.8 7.8 55.1 47.9 49.2 3.7 47.4 0.50 17.8 0.01 1.0 101 0.12 21.8
...
..... and Harboe,12have shown that the light absorption of a partially saturated hemoglobin solution at a given wave length is a linear function of its percentage saturation. It seemed worthwhile to check this result by the present method since it is independent of determinations of percentage saturation. Table IV shows that F A = 2, thus confirming that the four haems of the hemoglobin molecule act independently in the absorption of light. Chlorella Difference Spectra.-Coleman nnd RabinowitchlYhave presented difference spectra in (12) h1. Ifarboe, Scand. J . Clin. and Lab. Inoest., 11, 66 (1959). (13) J W. Coleman a n d E. Rabinowitch, J . Phya. Chem., 6 3 , 30 ( 1959 1.
-s4-
Obsd. 2000 21.1 5.1
..... .....
0.06 ,032 .82
--&/Sa7-Ss/Sr Calcd. Obsd. Calcd. nd Obsd. 2060 1 0 . 9 14.7 0.8 462 1.7 9 . 5 0 53 5 . 1 1050 0.65 19.0 0.53 3.9 3.8 0.45
..
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . . 810
0.S2
8.3
..
..
0.80
-
Calcd. na Rank 11.4 .. 7 . 5 0 . 4 2 Q-l 6.6 .42 :3 10.9 .46 4 3 17.3 .40 1 1 .I1 12.5 .30 1" . . . . .. . 5 l 22.5 .GO 2 2 .OS 1 5 . 0 . 2 8 I?
..... . . . . . . . ....... . . . . . . . .. which the light absorption of a suspension of Chlorella kept in the dark is compared with that of an identical suspension exposed to one of a range of light intensities. These spectra have been examined by the present method. Unfortunately, the available data are not extensive enough to reach reliable conclusions. However, matrix analysis, in principle, should permit enumeration of the species contributing to the various bands of the difference spectra. Acknowledgment.-The author gratefully ncknowledges the advice and criticism of Dr. G. Weber.
ON THE RELATION BETWEEN AX' EQUILIBRIUM CONSTANT AND THE XONEQUILIBR1UR.I: RATE COXSTANTS OF DIRECT AND REVERSE REACTIONS1 BY 0. K. RICE Department o j Chemistry, University of North Carolina, Chapel Hill, A'orth Carolina Recewed April 86, 1961
It is kr own that the rat2 of dissociation of a small niolecule activated by an inert substanre is affected by the fact that some of the excited states with energies close to the dissociation energy have less than their equilibrium population. It is shown that, in spite of this, the actual observed rates of dissociation and association will bear the same relation to the equilibrium constant as if equilibrium in all intermediate states were completely maintained. A similar conclusion can be extended to other types of reaction.
Recent calculations* have indicated that the rate of dissociation of small molecules activated by inert
molecules may be affected, probably not by a very large factor but still appreciably, due to t,he fact that the energy states leading up t)odissociation do not have their equilibrium population. A complementary effect JTould of course occur in the reverse association of the fragments in the presence of the same third body. Since the rates which are thus observed are not true equilibrium rates, it has been sugqested by Nikit'in and Sokolov,2 by Pritchardl2 by Widoiq2and, by implication, by Ross and Mazur3 that it is not correct to set kobs/ka,oba
= K
(1)
where kd,oba is the observed rate constant for the dissociation, say, (1) Work assisted by the National Science Foundation. (2) E. E. Nikitin and N. D. Sokolov, J . Chern. Phya., 3 1 , 1.571 (1959); J . C. Polanpi. ibid., 31, 1338 (1959); H. 0. Pritohard. J . P h o n . Chem., 66, 504 (1061): B. Widorn, 139th meeting, Am. Chern. Sac.. Paper No. 3, Div. of I'hys. Chern. (March, 1961). (3) J . 1 b s s and P. Rlazur, J . Chem. Phys., 35, 19 (1961).
AB
+ M --+ A + B + BI
with no A or B present, k a , o b s is the observed rate constant for the association A
+ 13 + PII +AB + $1
with no AB present, and K is the equilibrium constant for A B Z A + B
This is of some practical importance in connection with experiments on the association of atoms and the dissociation of diatomic molecules. For the most part experiments a t low temperatures (usually using flash photolysis) give the rate of the association reaction, while experiments a t high temperatures (usually using shock waves) give the rate of the dissociation reaction. It has been suggested that these experiments are not strictly comparable, because it has been supposed that, one cannot, calculate one rate from the other via the equilibrium constant. In order to decide whether such a conclusion is
