SEDIMENTATION AND ELECTROPHORESIS OF INTERACTING SUBSTANCES
NaK alloys with compositions close to 78% K and with low oxygen contents behave normally with regard to flow and viscosity. Even at temperatures as low aa -6.1°, they flow readily and have viscosities that are only moderately higher than those at room temperature. The effect of oxide content is not clear, and some investigation is required before a positive statement can be made. It does appear, however, that a small amount of oxide contaminant does reduce the viscosity. A comparison of the data from this study with that of Ewingl is shown in the Andrade plot9 given in Figure 4. On this plot In 9 ~ ’ ’is~ plotted against 1000/vT, where 7 is the viscosity, Y is the specific volume, and T is the absolute temperature. Our viscosity data are in
3785
agreement with the trend discussed by Ewing. He noted that the data below 100’ showed significant deviations from the Andriule equation based on the data above looo. Both our data and his show lower values than those predicted by the equation based on data above 100’. An Andrade plot of the data from this study is reasonably linear over a considerable range of temperature. Acknowledgment. The authors wish to acknowledge the support of this work by Atomics International, Canoga Park, Calif., and Messrs. M. Perlow and R. Keene of that organization for their technical guidance. ~
(9) E. N. daC. Andrade, Phil. Mag., 17, 698 (1934).
Sedimentation and Electrophoresis of Interacting Substances. IV. Theory of the Analysis of Interacting Systems by Differential Boundary Experiments
by R. C. L. Jenkins Department of Mathematics, Portsmouth College of Technology, Portsmouth, Hunts, England (Received May 9, 1966)
The type of boundary formed in differential boundary sedimentation, electrophoresis, or chromatography experiment8 on reversibly interacting systems is investigated mathematically. With the simplifying assumption that the transport velocities of the components are independent of total concentration, it is shown that the boundary splits up into a number of separate parts, the velocities and “areas” of which are simply related to the free molar concentrations of the components and their transport velocities. This is in distinct contrast with conventional boundary experiments on reversibly interacting systems.
Introduction The detailed mathematical analysis of conventional sedimentation, electrophoresis, or chromatography experiments on reversibly interacting systems’ is difficult with the exception of a few important but comparatively simple caaes.2,8 With the possible exception of these caaes, differential boundary experiments,4,6 in which a boundary is formed between two solutions ofslightly differing constituent concentrations, seem to
offer distinct advantages.6 It is shown in this paper that if the simplifying assumption is made that the (1) L. W. NichoI, J. L. Bethune, G. Kegeles, and E. L. Hess in “The Proteins,” 2nd Ed., H. Neurath, Ed., Academic Press, New York, N. Y., 1964, Chapter 9. (2) G. A. Gilbert Proc. Roy. SOC.(London), A250, 377 (1959). (3) G. A. Gilbert and R. C. L. Jenkins, ibid., A253, 420 (1959). (4) L . G. Longsworth in “Electrophoresis,” M. Bier, Ed., Academic Press, New York, N. Y., 1959, Chapter 3.
Volume 69,Number 11 November 1966
R. C.L.JENKINS
3786
transport velocities of the reacting species are independent of concentration, the initial boundary breaks up into several separate boundaries, each traveling with a characteristic velocity, and the mathematical analysis required to find reaction constants, etc., then consists of solving a set of linear (algebraic) equations. The case where the concentration dependence of the transport velocities cannot be neglected (as will usually be the case in practice) will be considered in a subsequent paper where it will be shown that the initial boundary again breaks up into several separate parts but the set of equations that have to be solved to determine reaction constants, etc., are no longer entirely linear. In addition to the assumption of concentration independent transport velocities, it will be assumed that the transport of the reacting species occurs in a rectangular or cylindrical cell, their reactions are governed by the simple form of the law of mass action, and that they are subject to a uniform force field parallel to the axis of the cell. Finally, the effects of diffusion and finite time of reaction will be neglected. The neglect of such effectswill not be as serious in the present context as it might appear as they will only tend to broaden the various separate parts of the boundary without altering their characteristic velocities or “areas.”
Theory Consider a system in which n reactants A I , Az, . . . A , form m complexes C1, Cz, , , . C, reversibly in solution according to the m linearly independent equations
2 Atvtr =+
C5
i = l
distance x down the axis of the “cell” at a time t must equal the net rate at which it flows into this region. Since the effects of diffusion are being neglected this implies that
For a boundary that is sharp at a time t = 0, the molar concentrations a, and c5 can be functions only of v = x / t where x is measured from the initial position of the boundary. This follows on dimensional grounds as only constants of the dimensions of molar concentration and velocity occur in the problem. Equation 4 therefore becomes’
As the assumption is being made that the reversible reactions 1 are so rapid that the components are always in chemical equilibrium, it follows from eq. 2 that
If the
are now eliminated from eq. 5 by means of eq. 6, it is found that
(1)
where the vtj are, of course, integral or fractional constants. These reactions are governed by’ the simple form of the law of mass action, i.e.
(7) where rn
n
n i = l
aiYij
= k5~5
(2)
whete at and c5 are the free molar concentrations of At and C,, respectively, and kl, kz, . . . k, are dissociation constants. The constituent molar concentrations %I, 7?22, , . . m,,of the A , are defined by
+c m
%, = a,
j = l
VtfC5
(3)
Let the velocities of A t and C, be v, and u5,respectively, which are by assumption independent of concentration. The constituent molar flux of A , is vg,
2 vtjujc5. The net rate of
j-1
+
accumulation of A i in
its free and combined states within a small region at a The Journal of Physical Chemistry
and lifi=j at,
=
Oifi#j
Thus if all the d/dv(log at) are not to be identically zero, the determinant of the coefficients of the set of n linear eq. 7 must vanish, i.e. (5) R. T. Hersh and H. K.Schachman, J. Am. Chem. SOC.,77, 5528 (1955); J . Phys. Chem., 62, 170 (1968). (6) G . A. Gilbert and R. C. L. Jenkins, Nature, 199, 688 (1983). (7) If f(v) is a function only of v where v = z/t then
SEDIMENTATION AND ELECTROPHORESIS OF INTERACTINQ SUBSTANCES
I
qnl
*..
-.
... .
*-. . Vpnl QnZ - Vpn2
Qnn
3787
If the quantities fffk are now defined to be the solutions of the equationsg
- Vpnn
The determinant is a polynomial of degree n in v , ~ and it can be shown that all the roots, say VI, V2, . . . and V,, of this eq. 9 must be real (see Appendix). Hence all n n the d/dv(log a,) are zero unless v is equal to one sf the MfPtjffjic= 1 V,. i = l j = l Now suppose a differential boundary experiment is it follows from eq. 10 and 13that the are given by carried out in which a solution of the A , at constituent molar concentrations iZLis layered over a similar solution with the A , at slightly differing constituent molar concentrations Ki A G to form an initially sharp From the eq. 3 the changes in the constituent molar differential boundary. Let the corresponding free concentrations of the A t across the kth part of the molar concentrations of the A , and C j in these solutions boundary are given by be at, cj, and ai 4- Aut, cj Acj, respectively. Since m the over-all changes in the free molar concentrations = vfjAcj'k) (17) across the boundary, Aut and Acj, are small, it follows j = 1 that throughout the whole boundary region the free By eq. 12,16, and 8 these equations (17) reduce to molar concentrations of the A , and C j differ little from the molar concentrations a, and cj. Therefore, eq. 7 with these values of the free molar concentrations are valid throughout the boundary region. Therefore, However, the total changes in the constituent molar as the d/dv(log at) are all zero unless v is equal to VI, concentrations across the whole boundary region Aml, V z , . . . or Vn, it follows that the boundary consists Am2, . . . Amn are just the s u m of the changes across of n separate parts, the kth part traveling with a each part of the boundary and so by eq. 18 velocity V,. Let the changes in the free molar concentrations n n of the A , and C j across the kth part of the boundary Ami = Am,(k) = atgAWk (19) k = 1 k = l by Aa,(k) and Aci(k). It then follows from eq. 7 that where n (qti - vkPtj)A(log = 0 (10)
c c
+
+
+
j = 1
If the molecular weights of the A , are M,, the molecn
ular weights of the C j will be
M,v,,.
Therefore,
i = l
the total changes in concentration (w./v.) across the kth part of the boundary AWkwill be given by
However, from eq. 2 or 6 Agog c,)'~) =
2 vtj A(log ai)(')
i = l
and if the Acj(k) = cjA('og eq. 11it is found that
cj)
(k)
are
(12)
from
(8) It is clear that the determinant cannot be a polynomial in v of degree greater than n. The coefficient of I" is the determinant (pij( but as the quadratic form
is evidently positive definite it follows from a well-known theorem that lpijl must be positive. Thus the determinant must be a polynomial degree n in v. (9) It can be shown that if the constituent molar concentrations mi are chosen so that eq. 14 and 15 happen to be inconsistent for some value of k, then the corresponding total change in concentration (w./v.) AWh will be identically zero no matter how the A% are varied. Such an experiment i s ill suited to the type of analyak proposed in this paper 80 the possibitity of eq. 14 and 15 being inconsistent will be neglected.
Volume 69,N u d e r 11 November 1966
R. C. L. JENKINS
3788
As the ff5k satisfy the eq. 15, it follows that the ufk are related to one another by the linear relations
where the Y f 5are given by
n
Mfufk = 1 j-1
In an actual experiment, the Amf will be known and the AWk measured. By repeating the experiment for different values of 'the Amt, the values of the coefficients utk can be determined using eq. 19 with the help of the eq. 21.'0 The V kare just the velocities with which the various parts of the boundary are observed to travel. The velocities of the reactants ut can be found by preliminary experiments. The object of such a series of experiments is of course to determine the dissociation constants k, together with the velocities of the complexes u5. In order to derive equations that allow one to deduce these unknown quantities, consider the eq. A10 derived in the Appendix
Now the determinant laIkl # 0" and so the eq. 25 and 28 imply that all the XIj and Yi, are zero. Hence by eq. 26 and 29
c n
Qt5
=
k = l
If eq. 30 are now written out in full, with the help 2) of eq. 8, together with eq. 3 the following n(n equations are obtained
+
ai
+
m
'75khka5Z
j=1
=
6kZ
(23)
If these equations (23) are multiplied in turn by the at%and summed over k it is found that
mi
n
< j ) (31) = (i < j ) These n(n + 2) equations are linear in the 2(n +m)unVfkVjkCk
k = l
n
=
VIkCk
k = l
m
+c vfafsfl + @fj
where it is shown that all the hk must be positive. These equations, with the help of eq. 20 (remembering that p5t= pf,), can be written in the form
UtkU5kVk~k
=
m
Ufku5dk
k = l
(i
n
~fkvfk~kCk
UfkUjkVkhk
k = 1
k = l
known quantities at, c,, u,q, and X k . Having determined the ut, U f k ) and v k experimentally, eq. 30 can be solved provided m 5 n2/2 and the free molar concentrations ut and c5 together with u5cjand Xk found. From these the dissociation constants kj and the velocities of the complexes u j can be immediately determined.
Example with the help of eq. 20. These equations (24) can therefore be written in the form n
(25)
To provide a numerical example of the analysis of a differential boundary experiment, suppose reactants A1 and A2form complexes C1 and Cz reversibly in solution according to the equations
xfj = Pf5 - k2= lufkujkhk Again, if the eq. 23 are multiplied in turn by and summed over k it is found that n
C2
(26) UfkVk
n UfkUjkVkhkajl
k=lj-1
=
-
+ A2 Ci 2A1 + A2 =+= AI
where the X t 5are given by
'JflVZ=
Supposing an ultracentrifuge study is being made and that the reactants A1 and AZ roughly resemble pepsin and serum albumin, it will be assumed that M1 = 35,000, M z = 70,000, and, on the basis of the relative masses of the four components, that VI = 1, vz = 2'/' = (10) It should be noted that the linear relations between AWh and A%. are valid only if the Aiiii are sufficiently small. (11) By a well-known theorem on products of determinants and eq. 22
with the help of eq. 20 and 14. Equations 27 can therefore be written in the form and aa lpiil be nonzero.
The J o u r d of Physieol Chemislry
>0
and finite, it follows immediately that I q j l must
SEDIMENTATION AND ELECTROPHORESIS OF INTERACTING SUBSTANCES
1.587, u1 = 3"' = 2.080, u2 = 4"' = 2.520, where the sedimentation velocity of AI has been chosen as a convenient unit of velocity. To correspond to a study of the complexes between pepsin and serum albumin by Cann and Klapper,12 it will be assumed that fhl = 0.849 X loF4mole/l., fhz = 2.190 X loM4 mole/l., and kl = 0.650 X mole/l. Finally, it will be assumed that k2 = 4k12. It is worth noting that from their study, Cann and Klapper had reason to believe that complexes containing more than one molecule of pepsin were being formed. From the above data the free molar concentration of the components can be found and hence the p f l and qtr. From eq. 9 it is then found that V1 = 1.540 and V z = 1.948. The at, can then be determined byeq. 14 and 15 and by eq. 20 it is found that ull = -0.421 X arid uZ2= uZ1= 1.639 X 10-5, u12 = 0.902 X 0.977 X Therefore, if a differential boundary experiment is performed on such a system, the velocities of the two parts of the boundary would be found to be 1.540 and 1.948 while the relations between the Aml, A% and the Awl, AW2 would be found, by such experiments, to be
+ 0.902 X 10-5Aw2 Am2 = 1.639 X 10-5AW1 + 0.977 X lO-'AW2
Am1
=
-0.421 X lO-'AW1
Therefore, the set of equations that the experimenter would have to solve is
+ + 2c2 = 0.849 X a2 + c1 + 2.190 X a1 + + 4C2 0.177 X 10-loX1 + 0.815 X 1O-'OX2 a2 + + 2.687 X 10-loX1 + 0.955 X 10-'OX2 + 2 ~ 2 -0.690 X 10-loX1 + 0.882 X 10-'0X2 + + 4UzC2 = 0.273 X 1O-'OX1 + 1.586 X 1O-'OXz 1.587~2+ + 4.137 X + a1
CI
~2
=
=
~1
~1
~2
=
=
~1
UIC~
U~CI
~
2
=~
2
1O-'OX1
3789
This example illustrates the relatively simple manner in which unique values for the unknown parameters of a reversibly interacting system can be determined from the results of differential boundary experiments provided of course that m 5 n2/2. This should be compared with the problem of determining the unknown parameters in conventional boundary experiments where, with the exception of certain important cases,* the nonlinear ordinary differential equations that govern the form of the boundary at sufficiently large times must be solved numerically with some reasonable set of values for the unknown parameters involved and this procedure repeated until sufficiently good agreement is obtained with the observed experimental boundary. There is of course no guarantee in general that the answers obtained in this way are unique; i.e., other values of the unknown parameters may happen to give equally good agreement with the experimental boundary. Acknowledgment. The author is grateful to Dr. G. A. Gilbert for many helpful discussions during the preparation of this paper.
Appendix Consider the equations
where V I ,is one of the roots of eq. 9. As the determinant of the coefficients of the eq. A1 therefore vanishes, one of these n equations at least is a linear combination of the others. Thus the atk are not all identically zero but are defined (at most) to within a multiplicative constant (dependent on k) by these equations. Note that until it has been proved that the roots V k of eq. 9 are all real, it must not be assumed that the aikare real. Denoting the complex conjugate of c y t l by atl* it is found on multiplying eq. A1 in turn by the cyi1* and summing over i that
1.860 X 10-'0X2 ~
1
+ 2 ~ 2 ~=2 -1.063 ~
1
X 10-loX1
+ 1.718 X 10-'oX~
Actually, these equations when solved give a1 = 0.219 X mole/l., az = 1.605 X mole/l., c1 = 0.540 X mole/l., c2 = 0.0454 X mole/l., and ulcl = 1.123 X lo-* mole/l., uzcz= 0.114 X lo-' mole/l. (while = 0.439 x lo6 and Xz = 1.058 X 106). Hence, it follows that U I = 2.080, uz = 2.518, and kl = alu2/c1 = 0.650 X 10-4mole/l., k2 = u ~ ~ u = ~/c 1.689 X lo-* mole2/L2. These "unknown" parameters are of course in agreement (to sufficient accuracy) with their initially assumed values.
On interchanging the i and j (and remembering that p,( = pcl and q5( = qU) these equations (A2) can be rewritten in t'heform
cc w
w
i - l j - 1
(Qtl
-
Vk,pi,)aika,l*
=
0
(A3)
Taking the complex conjugate of eq. A2, it is found that (12) J. R. Cann and J. A. Klapper, Jr., J. Biol. Chem., 236, 2446 (1961).
Volume 69,Number 11 November 1966
R. C. L. JENKINS
3790
and so it fol1ows from the eq. A7 that v k * = Vk; ie., the V kmust all be real. It then follows from the eq. A1 and 15 that all the q k must be real also. Equations A6 can therefore be written in the form
Interchanging the k and I, eq. A4 become n
n
c c (Pi, - VZ*Pdafka,t*
i = l j - 1
= 0
(A51
and if eq. A5 are now subtracted from eq, A3, it is found that
Now consider eq. A6 in the case where 1 = k , ie. n
Now in any experiment the constituent molar concentrations f i i of the A iwould be chosen so that the various parts of the boundary are well separated; i e . , no two of the vk would be equal. Equations A9 then imply that
n
c c
(Vk* - Vk) i = l j - l Prf"tka,k*
=0
(A71
By eq. 8 1 3
where, by eq. A8, all the X k must be positive. m n
=
c ai
i = l
m
laikI2
+ c c,J c j - 1
(13) The terms I a i k / and
n
i = l
m
tended to represent the modulus of and not determinants.
The Journal of Physical Chemiatry
that occur in eq. A8 are in-
Yt,(Yfkj2 aik
and Z
j - 1
vgjajk,
respectively,
