L. W. Nichol a n d
460
D.J. Winzor
Difference Gel Chromatography of Kinetically Controlled Systems. Irreversible Polymerization L. W. Nichol* Department of Physical Biochemistry, John Curtln School of Medical Research, Australian National University, Canberra, A. C. T., 2601, Australia
and D. J. Winror Department of Biochemistry, University of Queensland, St. Lucia, Queensland, 4067, Australia
(Received September 27, 7973)
A set of partial differential equations expressing mass conservation during migration of a system undergoing a kinetically controlled and irreversible association to form a single higher polymer has been solved to yield expressions for concentrations of species as a function of distance. These expressions have been adapted to describe elution profiles obtainable in difference gel chromatographic experiments. Characteristic features of these profiles are illustrated with numerical examples and are compared with those previously predicted for a kinetically controlled irreversible isomerization or dissociation. The potential use of a difference chromatography experiment in which a solute in one solvent system (a)is introduced onto a gel column preequilibrated with another ( p ) has been explored in relation to the examination of rapidly polymerizing systems1 and irreversibly isomerizing systems subject to kinetic control.2 The latter treatment yielded equations which described elution profiles and provided analytical expressions for the evaluation of the relevant first-order rate constant from these profiles. In practice a change of solvent environment may also result in an irreversible and kinetically controlled polymerization as illustrated3 by the time-dependent increase in weight-average molecular weight observed on the transfer of trypsin from pH 2 to 4.8. The purpose of this article is to present the theoretical expressions describing elution profiles pertaining to the difference chromatography of systems of the type nA
C
(n
>
Equations l a and the sum of eq l a and l b (written in terms of the constituent concentration, E @ ) have been solved by integrating the corresponding Lagrange system of ordinary differential equations, the resulting general solution being examined as described previously.2 Sets of solutions are obtainable for both cases UA' > UC' and u ~ ' < uc', but as the latter is the realistic relative order of magnitude of species velocities in gel chromatography (polymers with larger Stokes radii than those of monomers being excluded more completely from the gel phase) only the following set is relevant in the present context. In the region 0 Q x v A ' t
e
11
lz being the nth-order rate constant.
The treatment is commenced by considering migration in a single-phase system with subsequent adaptation of the resulting expressions to the chromatographic situation. The statement of mass conservation during migration ignoring diffusional spreading4 (for distance, x > 0 and time, t > 0) is v ~ ( a c , @ / a x ) , (dc,$/at), = - k ( C A d P (la)
+
u((acc@/ax),
+ (ac,@/at), = k(C.4")"
Ob)
where x is measured relative to an origin moving with the velocity of the solvent front (cup boundary), UA' and UC' are the respective velocities of monomer and polymer relc, 0 = A,C) is ative to the same frame of reference, and O the weight concentration in the p region. In these terms and for a loading concentration Fa the boundary conditions are c,@ r C" (n 0,t > 0) E
?'
E
CA'
=o + c$
(t=Qx>O) = ?*
(% =
Ojt
= 0 ( t = 0,x
> 0) > 0)
The Journal of Physical Chemistry. Voi. 78, No. 4, 1974
(3b) From eq 3b it follows that at x = vc't, E @ ( x , t ) = 0, the onset of the fl-solvent plateau region ahead of all solute. When x = u A ' t , eq 2b and 3b describe respectively the upper and lower limits of a step function of E @ , behavior entirely analogous with the corresponding isomerization situation.2 As required by mass conservation, the sum of the definite integrals of eq 2b and 3b between the limits indicated yields ?auA't. It is also clear from the form of these equations that the isomerization case (n = 1) required separate treatmentaZ Equations 2 and 3 were adapted to describe elution profiles by the series of steps detailed previouslyZ (including conversion to a fixed frame of reference) and with the ad-
Difference Gel Chromatography of Kinetically Controlled Systems a
I
1
b
'
volume
I
C
461
d
I
I
(mt)
Volume
Figure 1. Computed elution profiles for solutes undergoing kinet-
ically controlled polymerization. Patterns were calculated from eq 4 using the following common parameters: Vb = 35 ml, V = 0.5 ml/min, Vs = 33.8 ml, VA = 21.5 ml, Ea = 5 g/l. (a) IJ = 2, Vc = 16.9 rnl, k = 0.0033 I . g - ' rnin-'; (b) n = 3, VC = 14.3 ml, k = 0.001 g-* m i n - l ; (c) n = 4 , VC = 12.9 ml, k = 0.00031 g-3 min-l. ditional relationship = taVA/Vb, where ?a denotes the concentration of A averaged over the mobile and stationis the elution volume of A, and vb is the ary phases, bed volume of the column; this relationship follows from eq 5 and 16 of ref 5 . The expressions describing a chromatographic experiment are
v~
(ml)
Figure 2. Computed elution profiles for a solute undergoing kinetically controlled trimerization. Patterns were calculated from eq 4 using the foilowing parameters: Vb = 35 ml, V = 0.5 ml/min, VS = 33.8 ml, VA = 21.5 ml, VC = 14.3 ml, k = 0.001 i , * g P 2 min-' and values of 2 g/l. (-), 5 g/l. and 10 g/l. (---I for E m . (---e-),
and (b) the existence of a distinct step in solute concentration at VA. Since the profiles in Figure 1 all refer to systems with equivalent rates of polymer formation (on a weight basis), it is also evident that this step in concentration is relatively larger the larger the value of n. A feapolymer conversions ture that distinguishes monomer from those involving isomerization is concentration dependence of the elution profile. This is illustrated for a trimer system in Figure 2, which shows that monomer higher values of lead to decreased magnitudes of the discontinuity in ce@at VA. In the earlier treatment of kinetically controlled isomerization2 it was suggested that integration of the elution profile between the limits VAand VCcould provide a possible method of evaluating k . From eq 4b, the relevant expression for integration in the present study, it is evident that the same procedure could be adopted with systems undergoing monomer polymer conversion, but that different expressions for the experimentally determined area Q pertain depending on the value of n (eq 5).
-
+
-+
in which E,@ is the constituent concentration of A in the 4 region of the elution profile, V,, Vc, and Vs are the respective elution volumes of A, C, and the solvent front, and V is the volume rate of flow of the column. In eq 4, P is the applied concentration and also the plateau concentration in the a region of the elution profile. In the numerical examples of elution profiles that follow, V, and Vc have been assigned magnitudes in keeping with the gel chromatographic behavior of a polymerizing protein with monomer molecular weight 25,000 on a 35-ml (V,) column of Sephadex G-100. Vs was taken as the elution volume of sucrose, while V was set at 0.5 ml/min. Values of the nth order rate constants were selected such that half of the solute (on a weight basis) converted to polymer on exposure t o p solvent for 60 min. Theoretical elution profiles for monomer dimer, monomer trimer, and monomer tetramer systems with P = 5 g/l. are shown in Figure 1. Comparison with the corresponding isomerization case (Figure 2b of ref 2 ) reveals qualitative similarities with respect to (a) the terminating points of the reaction boundaries (Vc and Vs)
-
-
-
(VS =
(vs -
[
(1 ~ " ( VS
v
- vA>(v,A. - vc)vk?
-
X
VcXn 2)kv~(i;*V~/V$' hVA(n l)(?vA/Vb)n-' 'nr2)"n-') V
+
-
v,)(vA - vc)/(vs- vc)
1
(n
>
>+
2) (5b)
Although these expressions may also be solved for K , diffusional spreading would render the experimental measurement of Q subject to overestimation in the event of there being a pronounoed concentration step at VA.However, it is evident from Figure 2 that this error may be diminished by suitable choice of the applied solute concentration Ea in order to decrease the magnitude of the step. As in the case of kinetically controlled isomerization, a slower flow rate af the column (V)would also lead to an improved experimental estimate of the required area. The Journal of Physical Chemistry. Vol. 78. No. 4. 7974
462
Communications to
In summary, the previous treatment of the difference gel chromatographic behavior of irreversibly isomerizing systems has been extended to include solutes undergoing a kinetically controlled association in the solvent system used to preequilibrate the column. It is clear from eq 1 that consideration has only been given to the formation of a single type of polymer (C), treatment of alternate reaction schemes involving appreciable amounts of polymers of size intermediate between A and C, each characterized by a different velocity, being exceedingly difficult. It is noteworthy, however, that the behavior of a system of the nB, where dissociation occurs in the solvent systype A tem used to preequilibrate the column, has been implicity described in the earlier work2 since eq 1 of that reference expresses mass conservation during migration for either irreversibly isomerizing or dissociating systems. This ob+
the Editor
servation together with the expressions derived herein for associating systems suggests that the method should find wider experimental application than previously visualized for isomerizing systems because of greater differences between the elution volumes of monomer and polymer species compared with those likely to apply to isomeric solute forms. References and Notes (1) P. A . Baghurst, L. W. Nichol, R. J . Richards, and D. J. Winzor, Nature (London), 234,299 (1971). (2) S. M. A . Meggitt, L. W . Nichol, and D. J. Winzor, J. Phys. Chern., 77, 352 (1973). (3) A. D'Albis, Biochirn. Biophys. Acta, 200, 40 (1970). (4) G. A . Gilbert and R. C. LI. Jenkins, Proc. Roy. SOC., Ser. A, 253, 420 (1959). ( 5 ) L. W. Nichol, A . G. Ogston, and D. J. Winzor, J. Phys. Chern., 71, 726 (1967)
COMMUNICATIONS TO THE EDITOR
Disproportionationand Recombination of Cyclopentyl RadicaIs
TABLE I: G Values of Cyclopentene and Dicyclopentyl for Molten and Dissolved Cyclopentane Samples G(c-CbHio)
Pubiication costs assisted by the University of Louvain
Sir: When annealed hydrocarbons are irradiated at low temperature, the radical concentration remains constant in the matrix as long as the sample is maintained below a temperature characteristic of the solid where the decay occurs. Recent results1 have shown that these radicals can be quantitatively scavenged by dissolving the irradiated samples in a solvent that contains several specific scavengers, provided the dissolution takes place below the transition temperature. Previously reported works2 are concerned with the measurement of the contribution of alkyl radical combinations to the formation of dimers in the case of solid undecane and solid n-pentane irradiated at low temperature. This technique is now extended to the determination of the ratio of the rate constants for the disproportionation and the combination of cyclopentyl radicals in solid cyclopentane at 120°K. The cyclopentyl radical is the only paramagnetic species observed by esr spectroscopy in solid cyclopentane samples irradiated with y rays at 77"K.334 In the case of annealed samples, the intensity of the esr signal remains constant when the temperature is raised from 77 to l2O"K.b A rapid radical decay occurs near the transition point (122"K);5 at this temperatgre, cyclopentyl radicals react to produce cyclopentane and dicyclopentyl. These compounds are found to be the main radiolysis products of cyclopentane.6
Molten Dissolved AG
2.25
1.09 1.16
(u =
0.24)
= 0.25) ( u = 0.35) (U
G(CioHis)
0.69 ( u
= 0.05) 0 . 3 1 (u = 0.05) 0 . 3 8 ( u = 0.07)
The yields of these products are measured for several annealed samples irradiated at a dose of 4 Mrads. All these samples are bleached with visible light. Some of them are melted, others are dissolved at 1lO"K in a solution of oxygen in propane following a procedure described in ref 1. In this case, cyclopentyl radicals are scavenged and reactions 1 and 2 do not occur. The results of these experiments are summarized in Table I. The differences of the G factors between the two sets of experiments measure the contributions of reactions 1 and 2 to the production of cyclopentene and dicyclopentyl, respectively. They are related to the disproportionationcombination rate constants ratio in the following way
G, c ~ H -~ Gc~C , H ,~, ~ ~ ~~' ~ ~ dt ~ hd =-= -' Shd(R.)* G~ l H smolten - G C I J H s J"h,(R*)*dt k, 1.16 - - - 3.05 (a = 1.1) 0.38 It is worthwhile to note that these determinations do not need any assumption about the mechanism (the scavenger effect is specific) nor any approximation in the calculations.
'"-
Acknowledgment. The authors are much indebted to the Fonds de la Recherche Fondamentale et Collective for financial assistance. The Journal of Physical Chemistry. Voi 78. No. 4 . 7974
