ANALOGS OF MOVING BOUNDARY ELECTROPHORESIS AND SEDIMENTATION
Oct., 1961
1761
COUXTERCURRENT DISTRIBUTION OF CHEMICALLY REACTIK G SYSTEMS. 111. ANALOGS OF MOVING BOUNDARY ELECTROPHORESIS AND SEDIMENTATION BYJ. L. BETHUNE AND GERSON KEGELES (7matributionfrom the Departmat of Chemistry of Clark Unwereity, Worcesler, Mass. Recaird March 9, 1961
Countercurrent distribution as usually carried out experimentally is a discontinuous zone separation process. If, however, the same arnouat of solute is first placed in every tube, each of which contains both phases, and the distribution is carried out as usual, introducing fresh upper phase into tube zero at each transfer, a discontinuous moving boundary method of analysis results. An analogous relation between calculated results from this process and results obtained from moving bounda electrophoresis or sedimentationis established. Then, using the approach described in Part I of this series, computer c.&lations are carried out for polymerizimg system in the ultracentrifuge. The resulting patterns show the effects of diffusion as well as transport and chemical reaction.
Introduction Gilbert*-' and Gilbert and Jenkins4J have published a series of papers showing, theoretically, the effects of chemical reactions upon the schlieren patterns obtained from experiments using moving boundary methods of analysis. I n all of their papers the effects of diffusion upon the schlieren patterns a t times not infinitely long have been neglected. Countercurrent distribution, as practiced in the laboratory, is a zone separation method. The equatione relating concentration C,,, to position T in the apparatus after n transfers is
where Ca is the initial concentration of an independently partitioning solute of partition coeffcient K. For suffcientl y large numbers of transfers, when the solute has a partition ratio of approximately 1.0, equation 1 may be transformede,' into equation 2
tion of x and t. Therefore if C,,t in equation 3 is replaced by ( 2 ~ / b zthe ) ~ resulting equation should be that describing the schlieren pattern in electrophoresis or sedimentation, as indeed it is.8 Therefore, if a front is established in the countercurrent distribution apparatus, the derivative of this front should exhibit a pattern which is completely analogous to that experimentally obtained from electrophoresis or sedimentation in a rectangular cell, if the correct values are chosen for the parameters between which analogies are drawn. Since countercurrent distribution is a discon tinuous process, this derivative curve must be approximated hy taking first differences betveen the amounts of solute in adjacent tubes of the appnratus. This may be verified by a step by step calculation similar to that carried out for a zone distribution.6 If this is done, for example for four transfers, assuming one unit weight of solute in each of the tubes a t the beginning, the weights of solute in each of thc tubes are Tube number
__
__ K + l r
'jikr+-Di
time, 1
+-+
0
reservoir)
t )
0
velocity, u
AI
1
(Solvent
If, then, the following analogies are drawn n
Tots
4 -
1
E-Tv 4K
t3 Diffusion
+ 1)'
iK
+--+ position, x 1
coefficient, D
6K'
equation 2 may be re-written as equation 3
(k'+l)' 2
In continuous processes, the equation describing the concentration of the zone as a function of x and t is identical with that describing the concentration gradient of the moving boundary as a func(1) G. A. Gilbert, D~scussionaForadaV Soc., 13, 239 (1953). (2) G. A. Gilbert, rbid.. 20, 68 (1955). (3) G. A. Gilbert Proc. Row SOC.(London). M60,377 (1959). (4) G. A. Gilbert and R. C. L1. Jenkins. Nafure, 177, 853 (1950). (5) G. A. Gilbert and R. C. LI. Jenkim, Proc. Roy. SOC.(London), h 6 3 , 420 (1959). (0) L. C. Craig and D. Craig in A . Wekberger. "Technique of Or-
ganic Chemistry," VOL 111. Interscienoe Publ.., New York, N.
4K'
3
4
1
+ 4K + GK* + 4K* (K
I
+
W.Bock.
.7. Am. Chem. Soc., 72, 4269 (1950).
+ 1)'
114
K'
+ 4 K + GI(' + 4Ka + h" (Z +
m+-1)-4
114
and the first differences are indeed the usual TnsF as calculated from equation 1; (7'n.r = Cn,r/Co). Y., Therefore the first differences between tubes are
1950. (7) R.
(K
(8) L. J. Gosting, ;bid., 74, 1548 (1952).
J. L. BETHUNE AND GERSON KEGELES
176-3
Vol. 65
venient to consider the analog of an ascending froilt in moving boundary electrophoresis. A countercurrent distribution train of tubes, each filled originally with unit volume of only pure lower phase solvent, may be imagined to be fed at each transfer with a unit volume of upper phase containing the amount K / ( K I) of a solute having a partition coefficient K . After n transfers, the total amount of solute in the apparatus is nK,' (I( I), distributed over tubes numbered r = 0 to r = n - 1. This may be expressed by
+
+
(K
+ 1)'(1 + K)n-* + . . . . + ( K +
(1
+ K)o] (4)
/'T.
i
/
'
Here each term in the expression on the right.-hand side of equation 4 is individually equal to K,' (K l ) , and there are n terms in all. The s-mibo1 Sn,,represents the total contents of the rth tube after n transfers, as a result, of the n successive additions of solute to the apparatus. To find the value of the general terni S,,,, it is necessary to combine all t'erms from the right-hand side of equation 4 which contain K r multiplied by thc K ) , ie.. successivelv increasing powers of (1 (1 K)O, (I K ) l , ( I K ) 2 ,etc. Enumeration of the amounts per tube for each tube after each transfer in a small number of transfers serves t o verify that only such terms contribute t'o the contents of the rth tube. The extradon of all such terms leads to the general expression
+
+
+
/
-
i /'
i
t
0.0
b.
+
+
Fig. l.--ThPorettcal sedimentation velocity patterns calculated from analog: upper, theoretical pattern for the case of equilibrium mixture of chymotrypsin monomers and trimers; lower, theoretical pattern for the case of equilibrium mixture of chymotrypsin monomers, dimers and trimers.
indeed those values of the function which delineates the zone. When more than one solute is distributed, and a reaction may occur among them, the actual calculation procedure is much more difficult. I n Parts I9 and II'O of this series, it has been shown that patterns may be calculated for countercurrent distribution of zones when a chemical reaction may occur, using material balance equations which are solved repeatedly by a computer. From the preceding discussion, it is apparent that the only changes necessary in this method of calculation, for it to be applicable to the present case, are first, that solute must initially be placed in each tube of the apparatus and second, that first differences between the contents of adjacent tubes must be calculated. This involves only obvious minor modifications of the computer program shown in Part I. Theory It is now desired to establish in somewhat more general terms the identity between the 3 r d differences in tube contents of adjacent tubes for the case of a moving front, and the corresponding tube contents for a moving zone. For the purpose of this computation it is con(9) J. L Bcthune and G . Kegeles, J. Phus. Chcm.. 61, 433 (1961). (IO) J i, Bethune and G. Kegelea t b t d . 66, 17.55 (1961).
Rearrangement of this equa.tion leads directly t o
+ 1 ]r?& - 1 - r)! ( K + l ) n - l+
K S",, = -__
K
( n - l)!
\
(n -
r!(n - 2 or Sn,r =
K T-
k + l
____ K'
a)! - r ) ! ( K + I)--*
+
+
(Z'%-.l,r T,-?., Tn--.i
+
.
...
+ T,,,) (7)
where each T,- I,, represents the binomial coefficient expression for the contents of the rth tube after n - 1 transfers in the corresponding fundamental zone process. The first difference in tube content,s for adjacent tubes is now given by Sn.r-I
K I i ( T n - 1 , r-I - T n - 1 . r ) - Tn-Lr) + . . . . + (TW-1 - T,,,)
- S".r =
(T,-*. r-1
+
+
+ T,-I, ,-,I ( 8)
The conservation of matt'er now requires that in thec orresponding fundamental process, the accumulation in the rth tube during t.he (n l)st transfer, is equal to the amount transferred into the rth tube from the (r - l)sttube during the nth transfer less the amount transferred out of the rth tube to the (r l)stt,iibeduring the same transfer, i.e.
+
+
Oct., 1961
ANALOGS OF MOVING BOUNDARY ELECTROPHORESIS A N D SEDIMENTATION 17G3
K K+1 (T.
.r-,
-
T. .)
=
T,,.,
-
T...
(9)
Here the right-hand side of the equation is simply the amount present in the rth tube after n 1 transfers less the corresponding amount after n transfers, or the accumulation during the (n transfer. Application of the left side of equatiori 9 to each pair of T-values enclosed in parentheses in the right-hand side of equation 8 then leads to S,,,,-I - Sw,r = (T",, - T--I,,) + (T"-I,? - Tm-4 (!r.z,, - T " - d ... (Tr+*.7- T.+,.J K (T,+l., - 1 . .I f - - - - T , _ , .,-, (IO) K+1
+ +
+
+
+
6.
+
, I
and all values except Tn,, add out, the final trrrn ( K / ( K l ) ]T, - I - bring equal to T,,, because only the former amourit can contrihut,c to the colitents of the rth tube aftcr r transfers. Hence, the general result described above follows
+
,.
Sn.,-i - Sw = Tn.r
(11)
To illustrate the method by which the parameters u (the velocity), D (the diffusion coefficient) and t (the time) are transformed into the analogous parameters n and K , aSsuinc that a substanre M, charact,erizcd by the parameters u~ and Dnl iindergoes sediment,at,ion for a time t. At the cnd of this time, the . sprcad of the schlieren curve is pro..-. portional to . \ / 2 D ~ t ( L e . , the square root of the second moment), and t.hc displacement undergone by the maximum of the ciirvc is u ~ l . In the countercunrml. distribution apparatus, after n transfers the maximum of the curve has beendisplaced by nK/(K 1) ~tubrsaiid thcpattcrn spread is proportional to d n K / ( K 1)2. The ratio of spread to movement in the iiltracentrifugc is .\/%&&~t and in the countercurrent apparatus is l/.\/nK. If these two ratios are equated for given values of l ) ~t ,and I I M , and the number of trmsfers, n, which will give adequate detail to tho curve is assignid, the valuc of K which will give the required ratio of sprcad to displaccmmt is automatically fixrd. For a fixed n, this calculation procedure corresponds t,ii division of the ultraccnt,rifuge cell into n compartments, determination of the arnount of solutc in each i:nmpartmcnt and extraction of diffrrmccs betwren these to obtain the analog of the schlieren pattrrn. Results The approach dcscrihcd nhovc has I)POII used to refine previous calculations dnnc for t,he polymerization nf a-chymntrgpsin," iising Gilbert's methods.'.? The same v ~ l i i c sfnr the sediinentation constants, cqiiilibririni constants arid time arc chosen as in Fig. 5 nf rcfen:iice 1 1 . A value of 10.2 X IO-' was used for t,he diffusion eoeffirient of the monomer,I2 and from this diffusion corfficients of dimcr and trimcr were calculated, assurning t.he polymrrs t,o he sphrriral in shape. The raloiilat,rd valurs arc 8.1 X lo-' and 7.1 X lo-' for dimer and trimer, ri!spccfivcly. The rquilihrium constant for t,he reactinn 1111 M. S. N. Rno and G. Kegelcs. . I . Am. Chcn. Sor.. 80, .572
+
(1958). (12) G . W. Scbwerl and (19511.
S. Kaufrnan.
~~
+
. I . Bid. Chsm., 190. 807
-. Is'ig. Z.-Theorctical scdimcirtation velocity pattcrns calculated irom Gilbert's equations: upper, thrumtical p:ittern for the cnso oi cquilihrium mixtun: of chymchrypsin monomers and trimcrr; Iowcr, thcorcticnl pattern for the P ~ S Cof equilihriurn mixtiirr of chymotrypsin monnmcrg: dimers and trimers.
I
Fig. 3.--Eaperimcnt:il chymotrypsin.
sdimmt;itii,ii
w h ~ i~ l~ ~i i i l v ri il li
3 Mariomer % Trimer
was taken to be 50 (K3' of reference 1 1 ) and those of the reactions 2 Monomer
Dimer
% Dimer
+ Monomer
Trimer
were taken to he 11.4 and 4.5, respectively (&and IC8 of ref. 11). From these, at n = 340, K M = 0.04, K D = 0.13 and KT = 0.24. where K M : K D and KT are the
1764
J. L. BETHUNE AND GERSONKEGELES
partition coefficients of monomer, dimer and trimer, respectively. The results of this calculation are shown in Fig. 1, where the upper curve represents the pattern obtained when only monomer and trimer are present, and the lower curve that when monomers, dimers and trimers are present. These are to be compared with the Gilbert plots, for the same situation, shown in Fig. 2, and the experimental pattern of Fig. 3. None of the theoretical patterns is symmetrical, but the lower curve of Fig. 1 shows the closest approach t,o symmetry. The inclusion of kinetic terms in the chemicd reaction could only broaden the curve, and enhance resolution, judging from the one published study including kinetic effect,s,Ia aiid it is difficult’ to see how the inclusion of higher polymers in the system could remove the asymmetry which exist,s in the region of lowest concentrations. The most reasonable explanation of the lack of agreemcnt is that in none of the calculations are the sedimentation and diffusion coefficients concent,rati.on dependent. If this concentration dependence could be included in the calculations it, mould t;prrd up the monomer in the dilute region and retard the polymers in the regions of higher concentrationt to give a more symmetrical curve. (13; J. R. Cann, J. G . ICirkwood and R. A . Brown: Arch. Bioclem. Riophys.. ‘12, 37 !1957i. (14) The model used for the present calculation ia ale0 in error because the slowest moving constituent (monomer) is w i g n e d the sinallest diffuaion coefficient. whereas in the actual case the reverse is ? r u e , blurring tile tentienry for resolution of a trailing peak.
Vol. 65
Acknowledgments.-This work was supported by U. S. Public Health Service liesearch Grant number A3508. All computations were carried out at the M.I.T. Computation Center as problem number N1064, under the Cooperating Colleges of New England scheme. DISCUSSION J. R. CANN(University of Colorado Medical School).The important computations of Drs. Bethune and Kegeles afford a forreful demonstration of the hazards of classical interpretation of zone and moving boundary patterns of interacting systems while, at the same time, pointing up the power of these methods for studying such systems. Recently we have solved the system of differential equations deacrihing the electrophoretic transport of two-component ka
B, by numerical solution of the
isomerizing sytems, A ki
correspondin- difference equations (in press, Arch. Biochnz. Uiophys.). novel feature of the resulta is that under certain conditi ms of rates of reaction and time of electrophoresis the sch: eren patterns of two-component isomerizing systems may show three peaks. This is another example of how our intuition often fails us in thinking about the transport of interacting system of macromolecules.
!.
G. KEGELES.-Ih. Cann has kindly provided information from unpublished materid mentioned in his discussion above whirh permitted Dr. Bethune to compute the countercurrent distribution analog. It is gratifying that with this rather different approach to the computation, three peaks also were found under certain conditions. It has just been called to my attention by Dr. A. Eilinkenberg that he has treated the chromatography problem for slow isomerization reactions [“(ha Chromatography,” Edinburgh, June, 1960, 1i.P IT.Scott, rd. and Chem. Eng. Sci., in press (1961)].
