899
EQUILIBRIUM CONSTANTS OF ASSOCIATING PROTEIN SYSTEMS Another possibility, however, would be that CzHz is formed in the singlet state, either from a spin-forbidden predissociation or from the vibrationally excited singlet ground state of ethylene. The second mechanism has been recently advocated by Hunxiker,13who has introduced a third intermediate state (C&f) in addition to the other two already postulated by Callear and Cvetanovi6.1° This new state would decompose into acetylene and hydrogen and would be produced from the triplet ethylidene by an htersystem crossing to some vibrationally excited singlet state. What would be the geometrical configuration of this new intermediate state, postulated by Hunxiker? According to our calculations there are three possibilities corresponding to the three possible low-energy configurations of CzH4 in its singlet ground state: (1)
the planar configuration, (2) the asymmetrical ethylidene, and (3) the asymmetrical bridge proposed by W h a l l e ~ . Concerning ~ the latter, an independent observation by Hunxiker in the isotopic scrambling might provide some possible evidence for the occurrence of the third possibility.
Acknowledgment. The author wishes to thank Dr. G. R. De Mar6 for suggesting the problem, and for a critical reading of the manuscript. She is also indebted to Professor R. B. Cundall for a fruitful correspondence. It is a pleasure to thank Professor L. D’Or for his interest in this work. The financial support of the Fonds de la Recherche Fondamentale Collective and of the Fonds National de la Recherche Scientifique of Belgium is gratefully acknowledged. (13) H. E. Hunaiker, J. Chem. Phys., 50, 1288 (1969).
Determination of the Equilibrium Constants of Associating Protein Systems. V.
Simplified Sedimentation Equilibrium
Boundary Analysis for Mixed Associations by P. W. Chun and S. J. Kim Department of Biochemistry, College of Medicine, University of Florida, Gainesville, Florida (Received August 86,1969)
5.9601
+
A simplified procedure for the determination of equilibrium constants for mixed associations of the type i A j B F! AtBj or nA mB F! AtBj AhBl, where i -I- h = n, and j 1 = m, are described. The procedure is applied to a thermodynamically ideal situation, but its application in the anal) sis of reaction boundaries of any mixed association in a biological system is also considered. The equations outlined are based on concentration as a function of radial distance at sedimentation equilibrium.
+
+
In recent years, the theoretical treatment of assomB i? C has ciating protein systems of the type nA been based on the interpretation of data obtained by various physical techniques. 1--6 Nichol and Ogstons and Adams’ have described procedures for analyzing mixed associations of this type in an ideal system from sedimentation equilibrium boundary experiments. Adams, et d,* describe the equilibrium constants and nonideal term B,, evaluated from mixed associations using osmometric measurements
+
(@napp).
The principal drawback of these earlier procedures for determination of the apparent equilibrium constant and composition of the complex is the cumbersome manipulation of the data involved.
+
This communication describes a greatly simplified procedure for quantitative evaluation of the equilibrium constants of any mixed type of association in an ideal (1) R. F. Bteiner, Arch. Biochem. Bhphys., 49,71 (1954). (2) G. A. Gilbert, Proc. Roy. Soc., A250,377 (1959). (3) J. L. Bethune and G. Kegeles, J. Phys. Chem., 6 5 , 1755 (1961). (4) G. A. Gilbert and R. C. L1. Jenkins in “Ultracentrifugal Analysis in Theory and Experiment,” J. W. Williams, Ed., Academic Press, New York, N. Y., 1963, p 59. (5) L. W. Nichol, A. G. Ogston, and D. J. Winaor, Arch. Bwchem. Bwphys., 121, 517 (1967). (6) L. W. Nichol and A. G. Ogston, J.Phys. Chem., 69,4365 (1965). (7) E. T. Adams, Jr., New York Academy of Science Conference on Advances in Ultracentrifugal Analysis, Feb 15, 1968. (8) E. T. Adams, Jr., A. H. Pekar, D. A. Soucek, L. H. Tang, and G . Barlow, Biopolymers, 7 , 5 (1969). Volume 74,Number 4 February 19, 1970
900
situation. Application of column matrix transformation is used in determining concentration as a function of radial distance at sedimentation equilibrium. Associations of the following three types are considered
nA
A + B ~ A B iA jB AiBl
+
+ mB
A&
6) (ii)
+ AIBI, i
+ h = n, j + I = m
(iii)
Column Matrix Transformation for Evaluation of the Equilibrium Constant and Composition of the Complex
Basic Equations and Assumptions. The assumptions that are used in the analysis of mixed associations of all types based on sedimentation equilibrium experiments are (1) that the partial specific volumes of all species are the same, (2) that the refractive index increments of all species are equal, (3) that the system is ideal in that the activity coefficient of each solute species is unity, and (4)it is assumed that the system undergoes no volume change on chemical reaction. As the result of these assumptions, the differential equation for sedimentation equilibriumg is given by
c = ci CGl j
(5)
Equations 1, 4,and 5 are fundamental expression8 in the evaluation of equilibrium constants and composition of the complex for mixed association. Both Nichol and Qgstone and Adams7 determined the apparent weight, average molecular weight, and concentration at each radial distance ( r ) . Knowing these quantities, it is possible to evaluate CA,CB, and K for a mixed association of the type A B C (see Appendix); however, the necessary calculations are burdensome. In contrast, the simplified procedure described here provides all necessary data for evaluation of equilibrium constants and composition using only the determination of c = C(,).
+
Case I. A
+ B + AB
Method I. The total concentration at r is given by
+
C, = C A ( ~ ) CB(,)
+ CAB(,)
(6)
From eq 1 and 4
CAB(?)= K C A ( ~ ) C B ( ~ )
(7b)
Substitution of eq 7a and 7b into 6 gives
c, = C A ( & where Ct3 is the concentration of the component AiBj ~ M B the , partial differentia1,IO and Aft, = E MA. bp/bCi, at constant concentration of all other species equals (1 - q p ) . Equation 1 is integrated between any two limits, a and r, provided that rm 6 a and r 6 rb where rm and rb refer to meniscus and bottom of cell, respectively.
+
In[%]
=
LMlj(r2- a2) =
L & I A ( T Z - U*)
+
+ CB(a)eL1vB(re-~2) ?d(MA+ MB)(‘i‘’-a*) TcCA(a)CB(a)e
(8)
In order to solve eq 8 and evaluate CA(,)> CB(r),and cqr) = CAB(^),^^-^^ the curve of C us. r2 obtained from either
Schlierenz0or Rayleigh fringe data”-I2 must be constructed. In general, the C us. r 2 1 3 curve for mixed association will not be linear. When three species are involved in equilibrium, the curve is bisected by three r2 lines, each equidistant from the next. When four species coexist, four r2 lines are plotted, and four simultaneous equations set up, etc. (9) T . Svedberg and K. 0. Pedersen in “Ultracentrifuge,” Oxford University Press, London, 1940. (LO) E. F. Casassa and H. Eisenberg, Advan. Protein Chem., 19, 287 (1964).
(11) E. G. Richard and H. K. Schachman, J . Phys.Chem., 63, 1578 (1969). (12) D. A, Yphantis, Biochemistry, 3 , 297 (1964). (13) S. M. Klainer and G. Kegeles, J. Phys. Chem., 5 0 , 952 (1955). (14) L. D. Harris, “Numerical Methods using FORTRAN,’’ Charles E. Merrill Books, Columbus, Ohio, 1964, p 149. (15) E. T. Adams, Jr., Biochemistry, 4, 1646 (1965). (16) G. K. Ackers and T. E. Thompson, Proc. Nut. Acad. Sei. 53, 342 (1965).
Equation 3b can be written as Caj = K$jCAiCB’
(4)
The equilibrium constant, K i d ,for reactions of the type j B Ft AtBj is given by eq 4 and the total concentration at r as
iA
+
The Journal of Physical Chemistry
U.S.,
(17) P. W. Chun, S. J. Kim, C. A. Stanley, and G. K. Ackers, Biochemistry, 8, 1625 (1969). (18) P. W. Chun and 9.J. Kim, ibid.,8, 1633 (1969). (19) E. T. Adams, Jr., {bid., 4, 1655 (1965). (20) D.J. Winzor, J. P. Lolce, and L. W. Nichol, J. Phys. Chem., 71, 4492 (1967). (21) 0. Bryugdahl and S. Ljunggren, ibid., 64,1264 (1960).
EQUILIBRIUM CONSTANTS OF ASSOCIATING PROTEIN SYSTEMS
1
C l
90 1
analysis and ease of application to various model systems. By determinant^,'^ the solution to eq 9 is given by eq loa. Note that a matrix array is distinguished from some other array by the brackets as shown here. Here Q! = eLMAAz,P = eLMBAz, and by Cramer's rule, eq 10b may be set up. Thus the concentration term CA,~,CBro) and Cero becomes
Having shown how to obtain the quantities CAro1 CBro,and CC,,, we evaluate the equilibrium constant from
The composition of the reacting boundary of the complex is determined by the expressions
CAT
CAroeLMA(T2 - To,)
C B =~ CnloeLMB(Ta - 7 0 , ) L(MA+MB)(TZ - T o 2 ) CC, = CcToe
The evaluations which follow are arrayed in matrix form, due to the adaptability of matrices to computer
P2
The value of the equilibrium constant may be reconfirmed by selecting three new equidistant points r12,r22, and ra2 and repeating the procedure. The resulting value should be identical with the first. Method 2. A modification of associations of the j B @ AtB,. When i and j are known, eq type iA 8 takes the form
+
c, = CAroeL M ~ ( r % - r o *+) c B r o e L M ~ W - r o * )+
x ~ t cj eL(iM~+jM~)(~~-~r~~) Aro Bro (13)
P P2 Pa P P2 P8 C Y 1
Cl, Cra P '02 P3 Volume 74,Number 4 Febrzcary 10,1QYO
P. W. CHUNAND S. J. KIM
902
*
0, i = 2, the result of analysis is similar to 2P1 Pz. This method can be readily applied to a monomer-r+ mer self-associating system. 16-18,20
*
+
+
Case 11. nA mB AiBj AkBz; i + h = n,j+l = m When four species are involved in chemical equilibrium, the following simultaneous equations similar to equation set 8 may be set up
Cr
(LMA(v2-ro9)) +
CAroe
=
c
+ +
e(LM~(r2-rol)J
Bro
KijCAroC Bro,(L(Z'MA+I-~B)(rz-1.O2))
Kk ZChAroCzBroe( L ( h M A + I M B ) (re-To2)
1
(17)
Here rm< ro< rb. Choosingequidistant points rl, r2, and r3, rZ2- ro2 = 2(r12 - ro2)and r32- ro2 = 3(rI2 ro2). Letting (r12 - ro2) = AZ, Ctjra = KijC'AroC'gro and Cklro = K~zCA7~CB7011 four simultaneous equations result
-
cro = C A r o + C B r o + KijC'AroC'Bro f K k l C k A r o CzBro cr, = C A T 8L M ~ ( r i-2 roz) + c ~ ~ ~ ~ -Lroe)J +~ B ( ~ ~
+
L(%MA+~MB (r1* ) -roe)
KtjC'AroC'Broe
) Kk lCkBroC'Brae L ( ~ M A + Z M B(riz-roz)
crz
+ cBroeLM~W-ro9
LMAZ(ri2 -roe)
L(iM.4 f l M B ) 2 ( T l 2 - - T O Z )
KijC'AroC'Broe
cre = c
+ +
(18)
A -70%) Kk zCkAPoC'Brae L ( ~ M+ZM~)2(riz Aro
+ +
eLMA(rl*-roz) + c ~ ~ ~ ~ L M B ~ ( ~ I ~ - ~ o ~ ) L(iMA+JMB)3(r11 -To%)
K&' A~oC' B roe
L ( ~ M+ZM8)3 A (TI,
KazCkAroCzBroe
-To,)
Equation 18 becomes
CTO=
Cr, =
cr, Cra
CAroa
+
CAroff2
=
CAro
CA70ff3
+
+
CBr8
CBroP2
+
CRro
+
CBr@
+
+
[$I [
a!
and the composition of the reaction boundary is determined from the expression CAT
c
eLM~(~z-roZ)
= JAro
C B r = CBroe
L.44B(Tz - T o z )
L(~MA+~MB)(T~-VOS)
Ccr = Ccroe
In solving for K, a number of values for i and j are assumed until the resulting K value remains constant. Note that when i = j = 1, equation set 14 becomes identical with equation set 9 of method 1; and when j = The Journal of PhgsiCal Chemistry
a2
=
ff3
+ +
CkZro
Cijroa!tP'
+
chzroffkPz
[email protected] C h Z r o f f 2 k 0 2 z
+
1
1
p 82 P3
a!ip
+
(19)
c t j 7 0 ~ ~ ' P ~ ' CkIroffahf13'
where a! = eLMAAz, P = e L M B A Z solution to equation set 19 is 1
ctjro
(a!y9)3
,
By determinants, the
][;%I
1
(20)
(ffkPZ)2
CkZro
(a!'P')3
Application of Cramer's rule gives Values for CArO) C A ~C, B r , C{jr, C h z r , K i d ,and Kkzare evaluated as before. When m = j = I = 0, then three species qP1 & SP, + rPh coexist at chemical equilibrium, where h > i > I, for the self-associating ~ y s t e r n . ' ~ -In ~~ addition to the mixed associations described here, this method could be extended to indefinite mixed associamB IC $ tions or associations of the type nA
c~~~~ Cijro,and C h / r O J and the quantities
+
+
EQUILIBRIUM CONSTANTS OF ASSOCIATING PROTEIN SYSTEMS A,BmC1. Thus it is widely applicable to the association of biological systems which exhibit multiple equilibria through the construction of the C = C(,) curve.
903
2. Adams M e t h ~ d . ~
Acknowledgment. This work was supported by National Institutes of Health Research Grant N I H FR 05362-06, 07. Dividing eq 3' by e+B [which is In [CB~/CB~,] = LIMB(r2 r d 1
-
Appendix The abbreviated outline of procedures for determination of equilibrium constants developed by Nichol and Ogston6 and Adam$ is given below for contrast with our simplified method. 1. Nichol and Ogston Method.6 Cro(Mc -
Mwro)
eLM~(r2-ro2)
- Cr(Mc - J l w r )
LMA(rz-ro2)
( M , - MA)CAT,[~
-
=
eLMB(ra-ro9
CrdMc - Mwr& LMB(rz-ro2) - Cr(Mc- M,,) = (M~ MB)CBro[eLMB(r2-rD2) - e L MA(r 2-YOz)
1
(1')
3
(2')
From eq 1' and 2', C A ~and , CBro may be evaluated. CC,,is determined from the equation cC70
=
Cro -
CAro
where
Similarly
- CBra
knowing CA,,, CB,~,and Cero, we may determine the composition of the boundary complex from =
CAroeL I + ~ A (- rT$~)
CBr =
CBraeLMB(r1 -r$)
CAT
CcT = Cc,,e LMc(r2-ro2) The equilibrium constant is then evaluated from K = CCr/CArCBr*
plottingyB us. erm@A+(n-l)+B1yields CB,,[(MB/MA)- 11 as the hypothetical YBintercept where the exponential function is zero and K C m ~CnBro ,, [(Mc/~?A)- 11is the Slope. A value for Cdrois determined from a plot of YA us. e[(nz-l)'A+n'B1 at the intercept. Then knowing C A r o , CBfo,and CC,,, we may evaluate the composition of the complex and the equilibrium constant.
Volume 74?Number 4 February 10,1970
