S.M. A. Meggitt, L. W. Nichol, and D.J. Winzor
352
Chromatographic Behavior of Irreversibly stems Subject to Kinetic Control
I]
A , Meggitt, L. W. Nichol,"' and D. J. Winzor The Department of Applied Mafhematics, School of General Sftidies, and The Department of Physical Biochemistry, John Curtin School of Medicai Research, Australian Nationai University, Canberra, A.C. T., 2601, Australia, and The Department of Biochemistry, University of Queensland, St. Lucia, Queensiand, 4067, Atistraiia (Received July IO, 1972)
Analytical expressions are obtained describing the migration behavior of a solute introduced by frontal gel chromatography into a new solvent where it undergoes a kinetically controlled and irreversible isomerization. The expressions are derived for migration in a one-phase system and adapted to the chromatographic situation involving two phases. Numerical examples are presented of plots of concentration us. distance (for free migration) and of concentration us. time (elution profiles) for the possibilities where the reactant is characterized by a velocity greater or less than that of the product. Features of the patterns are discussed in relation to their use in the detection of kinetic conversion and its characterization iq terms of the rate constant.
~
~
t
~
~
~
u
c
~
i
~
~
~
In a previous study,2a a differential technique of frontal gel chromatography was described in which a solution of a macromolecule was applied to a gel column preequilibrated with a different solvent system. This design permitted the comparison of the constituent velocities2b of the macromolecule ahead of' and behind the migrating solvent front. With. a suitable selection of gel type, these velocities will differ if the isomeric or polymeric state of the macromolecule is different in the two solvent environments. In the systems studied,2a the conformational change of bovine serum albumin and the change in the extent of association of /%lactoglobulin A were rapid, which facilitated the interpretation of the elution profiles in terms of th'e Johnston-Ogston equation3 and two timeindependent constituent velocities. Several v ~ o r k e r s ~have - ~ drawn attention to the complicated reactior, boundaries which may arise in the mass migration of systems subject to kinetic control,8 where constituent velocities are no longer time independent.2b hereas these workers considered systems initially at equilibrium and in a fixed environment, the present theoretical treatment is concerned with the kinetic conversion between states of a solute inherent on its introduction into a new environment m a chromatographic experiment of differential design. An irreversible transition between two states of' the solute is explored and it is suggested that the analytical solutions describing the forms of elution profiles may find appkication in the kinetic study of such phenomena as protein renaturationgJO and denaturationll when these interactions approximate to the selected two-state model. The0ry
The experiment is commenced by layering a large volume of solute A in one solvent ( a region) over a large volume of a different solvent (/? region). At time t the C@ boundary will have migrated to a position X = ust, where v g is the constant velocity of the solvent front. In the a region solute A is stable and moves with velocity U A . Provid7he Journal of Physical Chemistry, Voi. 77, No. 3, 7 973
ed UA > U S , some A migrates across the crp boundary into the p region, where it undergoes a time-dependent conversion to the species B, governed by the rate constant h, i.e. k
A - B
Species A and B move independently in the /3 region with velocities U A and U B , respectively, the latter also being considered to be greater than us. The situation visualized, in fact, pertains to migration in a single-phase system; later, the theory will be adapted to meet the specific requirements of chromatography involving both mobile and stationary phases. Migration in. a Single-phase System. Since the plateau of A alone at its original concentration ( e a ) extends to the = ust, i t is convenient to employ a frame of position reference moving with the velocity of the C.U@ boundary, US. Velocities of A and B relative to this frame of reference are denoted by uA' and us', respectively, with dis. boundary tances denoted by x; clearly, x = X - u ~ t The conditions become
x
CAP
-8
c = CAP
>o
= Fa
x = 0, t
=o
t=o,x>o
+
CgP
= Fff
=o
x = 0, t > o t=O,x>O
(1) Address correspondence to Department of Physical Biochemistry, The John Curtin School of Medical Research, Australian National University, Canberra City, A.C.T. 2601, Australia. (2) (a) P. A. Baghurst, L. W. Nichol, R. J. Richards, and D. J. Winzor, Nature (London), 234, 299 (1971); (b) L. W. Nichol and A. G. Ogston, ProC. SOC.Ser. E., 163,343 (1965). ), (3) L. W. Nichol and A. G .Ogston, J. Phys. Chem., 69,1754 (1965). (4) J. R. Cann and H. R. Bailey, Arch. Biochem. Biophys., 93, 576 (1961). (5) P. C. Scholten, Arch. Biochem. Biophys., 93,568 (1961). (6) K. E. Van Holde, J. Chem. Phys., 37,1922 (1962). (7) G. G. Belford and R. L. Belford, J. Chem. Phys., 37,1926 (1962). (8) L. G. Longsworth and D. A. Maclnnes, J . Gen. Physiol., 25, 507 (1942). (9) W. F. Harrington and P. H. Von Hippel, Arch. Biochem.'Biophys., 92,100 (1961). (10) P. H. Von Hippel and K-Y. Wong, Biochemistry, 1 , 664 (1062). (11) H. A. McKenzie and G. B. Ralston, Experieniia, 27,617 (1977).
353
Migration of Kinetically Controlled Isomerizing Systems The statemenl of miiss conservation during migration (for x > 0, t > 0) is giveii by L‘,’(L%:,~
/t?X ;lt
/ at),
-k
(la)
= -kCAd.
+
U ~ ( & , ; ~ / &(dcBp : ) ~ /at),
=
(Ib)
kc$
which naglec’ts diffusional e f f e c t ~ . ~, Ib3 ,Since ~ ~ A and B move independently, we first consider the distribution of A, which is obtained by solution of eq la using the corresponding Lagrange system of ordinary differential equations or Laplace trarisfol-ms.14
clis(x, t ) := cA5(x- ua’t, 0 ) exp(-kt) = 0; x C,’(X,
t ) = cf(0, t
-=
x / u L ) exp(-kx/v,’)
cL‘exp(-kx/u,’);
> uA’t
> UB’t
(74
x < uLt (3)
( v i - u,l)(dc,”/x),
(4)
The function @ is de1,ermined by use of the appropriate boundary condi‘ions Consider first the case UA’ > U B I . For x > uA’t and x c: ~ ~ the ’ t values , of @ are determined by setting t = 0 and x = 0, respectively, in eq 5 . In the reniaiiiing region, bB’t < x < L A ’ t , the value of c A B ( x , t ) is given by eq 3, the relevant value., of x and t being (x uB’t)u4’/(uA’ - uB’) and (x - uH’t)/(uA’ - vg’), respectively, the lattep debnote intersections of the line x = vA‘t with lines parallel to x = uB’t, along which @ has a constant value. The solutions of eq 5 become
> UA’t
x
=
The corresponding Lagrange system of ordinary differential equations rnay readily be written and integrated to yield
c”(x,t>= 0; x
C P ( X , t ) = 0;
(2)
Equation 2 describes the solvent plateau or a region containing B alone ahead of the sharp boundary of A arising a t vAt, while eq 3 expresses the exponential decay of A as a function of the distance travelled by A in the fl region. In gas chromatcgmphic studies of surface-catalyzed react i 0 n s , ~ ~ Jincorporation 5 of the catalytic microreactor into a conventional :malytical column permits analysis of the emergent pulse in ternis of reactant concentration, whereupon eq 3 allows evaluation of k . The success of this design of experi.ment relies on ability to freeze the reaction a t a specified time b y removal of the reaction mixture from the catalyst.. In contrast,, the reactions presently being considered progress with Lime both on and off the column and thus it is necessary to obtain the distribution of total solute concentration in the region, E D , which may be the only concentration available to the experimenter. Addition of eq l a and l.b9and subsequent rearrangement, gives uB’(aP31ax:, -i- (ac@/at), =
readily be shown that these surfaces intersect along the line x = u g ’ t . Thus, a t constant time in the (CD,x) plane, the curve exhibits a discontinuity of slope a t x = uB’t. Also, in this plane the area under the curve is given by the sum of the integrals of eq 6b and 6c, which equals E“uAft,the total amount of original solute to have crossed the crp boundary in time t. In the alternative situation, ug‘ > u ~ ’ ,the solutions analogous to eq 6 are
(64
Equations 6b and 6c describe surfaces in the space defined by the cortesian coordinates ($,x,t) and it may
Migration in a Two-Phase System. In accordance with the preceding sectipn, we again assign the reaction a single rate constant k , which implies identity of reaction rates in the mobile and stationary phases. Equations 6 and 7 describe concentration-distance distributions a t a given time, whereas construction of an elution profile requires knowledge of the concentration-time profile at the exit plane located a t a fixed distance X F from the entry plane. The rewriting of eq 6 and 7 in appropriate form requires not only a conversion to a fixed frame of reference ( X = x f ust, V A = v A f us, and u~ = uw’ u s ) , but also consideration of the partitioning of solute between the mobile and stationary phases on the coiumn.16 Thus, the total solute concentrations designated or 3 in the previous discussion must now be summed over both phases and are termed 201 or 83.In the same connection, the velocities of the individual species and that of the cup boundary must be represented by oA, uH, and oh to denote that they too are weighted averages over both phases. The procedure for converting eq 6 and 7 to forms describing elution profiles need only be illustrated for one relation and we choose eq 6c and 7c since it pertains to cases where A is either the slower or faster moving species and to regions where A and B coexist. I t becomes
+
+
and pertains to the region XF/L?S9 t > X F / u H if 0 4 > Dp, or to the region XF/& > t > X F / D A if > irA. As noted earlier,l6 concentrations of solutes in the effluent from a column are identical with those in the mobile solution reaching the exit plane a t that time, the relevant expression of mass conservation in the /3 region being
where V is the volume rate of flow of the column and Q is the total concentration of A and B in the p region in the eluate. Use of eq 9 requires specification of E 4 a ( X ~ , tand ) (12) G. A . Gilbert, Proc. Roy. SOC.,Ser. A , 250, 377 (1959). (13) 6. A . Gilbert and R. C. LI. Jenkins, Proc. Roy. SOC.. Ser. A, 253, 420 (19591. (14) S. H. Langer, J. Y. Yurchak, and J. E. Patton, lnd. Eng. Chem., 61(4), 10 (1969). (15) D. W. Bassett and H. W. Habgood, J. Phys. Chem., 64, 769 (1960). (16) L. W. Nichol, A. 6. Ogston, and D. J. Winzor. J , Phys. Chem., 71, 726 (1967).
The Journal of Physical Chemistry, Vol. 77, No. 3, 1973
S. M. A. Meggitt, I.W. Nichol, and D. J. Winzor
354
cBO(XF,t); the former is obtained directly from eq 3 and the latter by subtracting EAO(XF,t) from i.s(X,,t) given in eq 8. In these terms eq 9 becomes
-- Bs)Ca/(B, - Os)] [ D ~ ( D , -- V B ) ~ ~ / ( D-BOS)] exp[-k(X,
v c e a = [c&,
-
2 .o
1.5
- vst)/(~A- Q] (lo;,
Two more operations are required. The first involves the substitution i n eq 10 of 5. = VCU/D,which expresses conservation of mass in the DL region cf. eq 9). The second involves a change of variable from velocity to elution volume, effected by noting that uA = XF/tAwhere t~ = VA/ v, similar expressions defining the elution volumes VR and Vs. Equation 10 written in these terms is
where the variable V = Vt. Since eq 6b and 7b may be transformed in a similar manner, thle general features of elution profiles may now be summarized. For VA C VB ( U A > UB), the solvent used to preequilibrate emerges initially at elution volumes less than VA, a t which value A begins to emerge. Thereafter between Vi, and VB both species coexist, the total concentration being given essentially by eq 6b, which in its transformed chromatographic form is
At V = VR eq 3 % reverts to eq 11, the latter describing the total emerging concentration until the solvent front (cup boundary) is eluted from the column at Vs; indeed at Vs eq 11 simplifies to = F ,this being the onset of the emergence of a plateau of the originally applied solution. For VH C VA ( Q H > u4), the solvent plateau elutes first until H begins t o emerge a t VB. Between Vu and V4 only B exists, the concentration being given by the transformed eq 7b
A t V = VA, A begins to emerge and thus a t this elution volume the total concentration i s a step function to be illustrated later, the lower limit being given by eq 13 and the upper by eq 11. Equation 11 continues to describe the profile until the emergence of the original solution with the solvent front at k: Numerical Illustrations a n d Discussion In the use of eq 6 and 7 to compute concentration distributions a t fixed t m e s for migration in a one-phase system, it is helpful to introduce certain reduced variables in order l o give greater generality to numerical illustrations. Thus, the ordinate will be expressed relative to the applied concentration (?B/?a), while the abscissa (distance) will be in terms of the solvent velocity U S and the rate constant; timc is also expressed relative to h . Illustrations will first be presented for cases where species A travels faster than species B The Journal of Physical Chemistry, Vol. 77, No. 3, 7973
0.51-
ELUTIONVOLUME
(VI
Figure 1. Computed concentration distributions for a system where reactant A travels faster than product B, both moving ahead of the solvent front S: (a) migration in a one-phase system with velocities V A = Z V B = 4vs; broken line t = l / k , solid line t = 2 / k , time t being expressed in terms of.rate constant k: (b) the corresponding elution profiles f r q m two-phase chromatography, VS = 2 v B = 4V.4; broken line V (flow rate) = k V s / 2 , solid line (slower flow rate) V = k V s / 8 .
Figure l a presents concentration distributions at two fixed times, t = l / h and 2 / h , for a system in which the relative magnitudes of the velocities were arbitrarily se. interesting features in Figlected as U A = 2uu = 4 ~ s The ure l a are that there is a maximal value of the ordinate (at the point of intersection uBt) and that it increases with increasing time. Figure l b refers to the corresponding sit) chromatography calculated uation ( v s = 2vB = 4 v ~ in on the basis of eq 11 and 12. In this instance, the two patterns refer to different flow rates, V, expressed in terms of Vs and h. These elution profiles are qualitatively similar to the concentration distributions shown in Figure l a since a point of discontinuity occurs a t VA and the slope is discontinuous at VS and VB. Again, tlieve is no evidence of a plateau in the fl region, a maximum value of the ordinate, though less pronounced, being observed at VR.The lack of a plateau in the p region may prove to be valuable in distinguishing kinetically controlled systems from those a); for in the differenin which conversion is rapid (12 tial chromatography of the latter systems (equilibrium or irreversible) a plateau in the fl region would be observed in accordance with the Johnston-Ogston equation.2as3 The advantage (in making such a distinction) of performing experiments at different flow rates is evident. Figures 2a and 2b refer to situations in free migration and chromatography, respectively, where species B travels faster than species A (uB > U A or V , < VA). Since both Figures 2a and 2b exhibit qualitatively similar features,
-
355
Migration of Kinetically Controlled Isomerizing Systems I
I
a
DISTANCE ( X I
V§/2
VS
(V) Figure 2. Computed concentration distributions for a system where reactant A travels slower than product B, but faster than the solvent front: (a) migration in a one-phase system with VB = Z V A = 4 ~ s brciken ; line t = l / k , solid line t = 2/k: (b) the corresponding elution profiles from column chromatography, with V s = 2 v A = 4 v B ; broken line v = kVs/2, solid line V = kVs/ ELUTION
VOLUME
8.
attention is directed to the chromatographic situation, Figure 2b, calculated on the basis of eq 11 and 13 with VS = 2VA = 4VB and flow rates as for Figure l b . In contrast to the behavior observed in Figure l b , the total concentration in the 6 region never exceeds the original concentration Ea. Again, the lack of a plateau in the 6 region may prove to be a valuable diagnostic test for kinetic conversion. However, i3 may be seen from Figure 2b that as the flow rate is decreased, the step discontinuity a t VR becomes progressively smaller until the pattern gives the appearance of two plateaux separated by a concentration gradient near Vs, behavior also found in the study of a rapidly reacting syBtem.2a Therefore, it would be advantageous to study the system at different flow rates, but in contrast t,o the earlier situation (Figure l b ) the difference between kinetic control and rapid conversion is accentuated a t fast flow rates. In this case (broken line of Figure 2b), a derivative plot of dc,p/dV us. V would exhibit trimodality with a sharp central rtegion. Complicated reaction boundaries of this type have been discussed in relation to reversible kinetically controlled isomerizations examined in migration experiments of different It is noted that the main value of earlier theoretical studies4-7 on the behavior of kinetically controlled sys-
tems in mass migration was also the recognition of such systems. Thus, quantitative measurement of thermodynamic and/or kinetic parameters describing the systems may well be more easily performed by means other than mass migration. For example, the rate constant governing the irreversible isomerization presently under discussion might be readily obtained by rapidly changing (say) the pH or ionic strength of the solution studied in situ and following the time dependence of a spectral or optical rotatory property. On the other hand, if the study is to involve a solvent in which the solid solute cannot rapidly be dissolved, the chromatographic method of differential design which effects a complete transfer of environment in the solution phase, may offer a unique method of elucidating the kinetics. For this purpose an analytical expression must be found involving h and experimentally determinable parameters. The values of V.4, V R , and V , may be found in separate experiments while V is readily measurable. For the case VA < VB, the simplest analytical expression involving these parameters and k is eq 11, the use of which would involve measurement from the elution profile of c ~ P / cat~ V R . However, the value of iQ!/Ca at VR will be affected by diffusional spreading, of which eq 11 takes no account. An experimental measurement less affected by diffusion would appear to be the area under the pattern from the solvent plateau to VR, which corresponds to the amount of solute in the eluate at that stage. If this experimentally determined area is termed Q , integration of eq 12 between the limits V ,and VR leads to
Q
=
[V(V,
V A ) ( V-~VA)f/kVA(V, - V B ) {I ] exP[-kVA(Vs -
v~)/+(vs - VA)]!
(14)
which may readily be solved for k . Two points require comment in relation to the choice of the limits of integration. First, the experimental measurement of Q should include the small amount of A which has diffused ahead of VA to be consistent with the choice of b", as the lower limit in the intergration of eq 12, as is apparent from Figure l b . Second, it is recalled that integration from VA to VS leads to ca(Vs - VA), the condition for conservation of mass, and thus provides no information on h. In the event that V R < V A ,the value of Q obtained by integration of eq 13 between the limits Vn and V , is
The experimental measurement of Q in this case is subject to overestimation because of diffusional spreading if there is a pronounced step at VA; in this regard, adjustment of the flow rate would be desirable to obtain a pattern such as shown by the solid line in Figure 2b. In summary, it is hoped that the analytical expressions presented may prove useful in the detection from elution profiles of irreversibly isomerizing systems and be of use in the elucidation of kinetic parameters when more direct methods of study are precluded.
The Journal of Physical Chemistry, Vol. 77, No. 3, 1973
