ANALYTICAL
BIOCHEMISTRY
86,
34-49 (1978)
Isoelectric Focusing of Interacting Systems III. Carrier Ampholyte-Induced Macromolecular Association or Dissociation into Subunit9 JOHN R. CANN,* DONALD I. STIMPSON,* *Department Colorado
of Biophysics and Genetics, 80262, and the TDepartment Manhattan,
AND DAVID J.Coxt
University of Colorado Medical Center, Denver, of Biochemistry, Kansas State University, Kansas 66506
Received June 1, 1977; accepted November 5, 1977 A phenomenological theory of isoelectric focusing of interacting systems (Cann, J. R., and Stimpson, D. I. Biophys. Chem., 7, 103, (1977) has been extended to include carrier ampholyte-induced dimerization or dissociation of a subunit macromolecule. Equilibrium as well as transient focusing patterns can show two well-resolved peaks similar to the patterns for a mixture of two noninteracting macromolecules. Characteristically, one of the peaks in the transient pattern grows at the expense of the other, which under certain conditions may disappear completely as equilibrium is approached. The implications of these findings for conventional applications of isoelectric focusing and for the detection and characterization of macromolecular interactions are discussed.
During the past decade or so the method of isoelectric focusing (l-8) has become an established procedure for the separation and characterization of proteins and other amphoteric macromolecules with respect to isoelectric point (pZ). Briefly: A linear pH gradient is generated and maintained by electrolysis of a mixture of carrier ampholytes (most commonly, small aliphatic polyamino-polycarboxylic acids marketed by LKB, Stockholm, Sweden, under the tradename “Ampholine”), each initially distributed uniformly in a liquid column containing a sucrose density gradient to stabilize against convection or in a column or slab of solid support such as polyacrylamide gel. Electrolysis establishes a steady state in which the various carrier ampholytes are distributed along the column from anode to cathode in order of increasing pZ with each distributed approximately in a Gaussian fashion about its pZ, thus, generating the pH gradient. When a zone of isoelectrically homogeneous protein is inserted into the pH gradient, the protein molecules migrate under the influence of the electric field toward either the anode or cathode depending upon the pH (# pZ) at ’ Supported in part by Research Grants HL 13909-25 from the National Heart and Lung Institute (J.R.C.) and GM 22243-02 from the National Institute of General Medical Sciences (D.J.C.), United States Public Health Service, and by the Kansas Agricultural Experiment Station. This publication is No. 671 from the Department of Biophysics and Genetics, University of Colorado Medical Center, Denver, Colorado 80262, and No. 197-j from the Department of Biochemistry, Kansas Agricultural Experiment Station. 0003-2697/78/0861-0034$02.00/O Copyright Ail rights
0 1978 by Academic Press. Inc. of reproduction in any form reserved.
34
ISOELECTRIC
FOCUSING
OF
INTERACTING
SYSTEMS
35
the position of insertion until they reach that position in the column where the pH corresponds to the pZ. The protein focuses sharply at this position. In the case of an isoelectrically heterogeneous protein, each component focuses at its respective pZ so that the isoelectric focusing pattern (plot of concentration vs position) shows a corresponding number of peaks. This description applies to the commonly used procedure for isoelectric focusing in gels. Alternatively, the carrier ampholytes and the sample of protein can be premixed so that initially both are distributed uniformly throughout the gel or liquid column. Upon applying the electric field the pH gradient is generated relatively rapidly, after which each protein component converges from both ends of the column lipon its ~1. The foregoing considerations are for ideal situations uncomplicated by macromolecular interactions. Recently (9,lO) we have shown theoretically that rather different isoelectric focusing behavior is to be expected for systems undergoing either reversible, carrier ampholyte-induced macromolecular isomerization or pH-dependent conformational transitions both in the limit of instantaneous establishment of chemical equilibrium. Thus, for example, such interactions can give well-resolved, bimodal (sometimes multimodal) focusing patterns in which the peaks correspond to different equilibrium compositions and not to separated macromolecular species. Our calculations have now been extended to include the following rapidly equilibrating, macromolecular association-dissociation reactions in which Ju symbolizes macromonomer and &?. carrier ampholyte: The simple dimerization reaction the simple trimerization
3%/l% eAg,K2 the carrier ampholyte-induced
[III
dimerization
2.4 +4&d = .44,
Ks
and the carrier ampholyte-induced dissociation of a subunit romolecule into its isoelectrically identical half-mers J& + 4d e 2~fZ.d,,K,.
WI mac-
UVI
In each case the pKs of certain ionizable groups on the macromolecule are assumed to be altered due to the change in state of association, thereby leading to a change in ~1.~ [The contribution of reversibly bound carrier 2 The pKs may be altered via several possible mechanisms: Exposed groups may experience a change in either the local charge environment or the local. effective dielectric constant; the group(s) may be masked by becoming buried in an hydrophobic domain of the macromolecule or by participating in hydrogen-bonding interactions; etc. A change in net charge of one protonic unit at appropriate pH will typically shift the pI by about 0.1 pH unit irrespective ofthe total number of ionizable groups on the protein. Although it is not generally
36
CANN,
STIMPSON.
AND
COX
ampholyte per se to the change in pZ can be neglected since the focused system is in a medium of ampholytes which are essentially isoelectric (3).] It is evident, therefore, that hydrogen ions must be involved in each of the reactions. Consequently, the “equilibrium constant,” K,, must incorporate the pKs of macroreactant and macroproduct in such a way as to be a function of hydrogen ion concentration. We shall assume that the “equilibrium constant” is an insensitive function of pH and can be considered constant in the region of the column where the particular system focuses, given the shallow pH gradient and the relatively small difference in pls. Theoretical isoelectric focusing patterns have been computed by numerical solution of the simultaneous transport equations and mass action expression for each of Reactions I-IV as a function of time thereby constructing the approach to the equilibrium distribution of macromolecule along the isoelectric focusing co1umn,3 i.e., the equilibrium isoelectric focusing pattern in contradistinction to the transient patterns which pertain to the approach. Convergence to the equilibrium distribution was determined by comparison of the amplitudes and positions of the extrema in the transient patterns with those predicted for the equilibrium distribution by an analytical relationship. The results of these calculations are described below. THEORY4
Transient isoelectric focusing patterns were computed essentially as described previously for ampholyte-induced isomerization (9) except for use of the appropriate expressions for recalculation of chemical equilibrium after each time cycle of diffusion and driven transport. Consider, for example, Reaction III: Equation 7 of Cann and Stimpson (9) is replaced by possible to predict a priori the relationship of electrophoretic properties of an aggregated protein molecule to those of the monomer, it is clear that changes in state of association sometimes do influence significantly the electrophoretic mobility in free solution and the PI. Examples include the tetramerization of /3-lactoglobulin A (1 I- 13). the dissociation of the globulin arachin (14,151, and the multiple molecular weight isozymes of Neurospora mitochondrial malate dehydrogenase (16). In the case of malate dehydrogenase, the pl of the isozyme increases linearly with its molecular weight, suggesting that acidic groups are asymmetrically masked as the degree of association of the monomeric enzyme increases. 3 Since our model assumes that the macromolecule does not contribute to the conductance, the distribution of macromolecule in the final isoelectric focusing pattern is an equilibrium distribution given the particular environment created by the steady-state distribution of carrier ampholytes. ’ Aside from the introduction of a few new variables and functions defined herein, the notation is the same as used previously (9) except for the subscripts 1 and 2 which here denote Y and Y&,. respectively.
ISOELECTRIC
FOCUSING
OF INTERACTING
C(j,t,+,) = C*(jJ,+,) + Equation
8 is replaced by
Equation
9 by
and Equation
37
SYSTEMS
mL~n+J
11 by
K3 = ~‘z(j,t,)l[i-,O’,tn)lz[~~(j)l*. The apparent equilibrium constant K ‘3(j) = K3[CJ$( j)]” is normalized by assigning it the value, K’,(Z) = ~‘z(l,r,)l[i‘,(l,r,)]*, in segment I where the inflection point of the Gaussian distribution of constituent carrier ampholyte is located. As before it is assumed that the concentration of ampholytes is sufficiently greater than macromolecule that it is not perturbed significantly by the reaction, so that both the distribution of ampholytes and the pH gradient are invariant in time. The calculations are for a broad initial zone of macromolecule encompassing the pls of both JU and J&P& and can be placed into correspondence with the experimental procedure in which both the macromolecule and the ampholytes are distributed uniformly throughout the column initially. The correspondence is affected by making the limiting assumption that the steady-state distribution of ampholytes and the concomitant pH gradient are established so rapidly that, in effect, they obtain at the instant the macromolecule begins to migrate under the influence of the electric field. Of course, the equilibrium isoelectric focusing pattern is the same for all arbitrary initial distributions of a given amount of macromolecule. The equilibrium pattern itself can in principle be calculated in a manner similar to the procedure described previously (9): The continuity equation for constituent macromolecule is + 20,
a&x,t) ax
- V,(x@,(x,t)
- 2VZ(X)&X,l)
in which
&(x.t) = &
(-1 3
&(XJ> = ~3’(X,re,w>l’ K,‘(x 1 =K3[ 5(x)14.
+ [l + 8K,'(x)~(x,t)l"']
I
aJ(x,t) ax
E ~
38
CANN,
STIMPSON,
AND
COX
Equating 13~?(x,t)lat to zero, noting the physical requirement that at equilibrium Lim,, .Z(x,t) = 0, and integrating the resulting ordinary differential equation with respect to x gives the equilibrium distribution, but in terms of a nonlinear integral equation whose solution is a formidable mathematical problem. However, the ordinary differential equation readily admits a property of the family of equilibrium patterns given by the integral equation; namely, the amplitude of the extrema in the distributions, text, as an implicit function of their positions, xeXt. Equating ~3C(x,r)lax to zero in the explicit expression for Lim,,, .Z(x,t) Lim,,, = 0 and solving the resulting algebraic equation gives c’ext = &
[u” + 2u] 3
PI
- D2) ~dlnK3’ dxext
V2
+
V2
(D1 - D2) dlnK,’ V2
1
+ -2Vl + 1
+ 2V, ,+lr-%)111!
dxext
[lb]
in whichK’,, VI, and V, are now functions ofx,,t. These equations exhibit two interesting properties: (i) text = 0 for V, = 0 and (ii) a singularity, I c!xt I + m as V, + 0. Both can be understood in terms of the law of mass action as it applies to Reaction III: (i) The equilibrium pattern can show a peak centered precisely at the pZ of & only in the limit of infinite dilution of total macromolecule where the reaction is shifted completely in the direction of reactant. (ii) The pattern can show a peak centered precisely at the pZ of &,d4 only in the other limit of infinitely high concentration of macromolecule where the reaction is driven completely to product. At finite concentrations the two species must coexist wherever there is macromolecule in the isoelectric focusing column, so that the peak(s) in the pattern will be centered somewhere between the pls. Note that while text as a function of xeXt is a property of the family of all equilibrium patterns admitted by Reaction III, it is expedient to consider subfamilies thereof. For example, one subfamily corresponds to specified Kf3(x), diffusion coefficients and velocity gradients; members of this subfamily are generated by varying the total amount of macromolecule applied to the isoelectric focusing column. Another subfamily is characterized by varying values of K’3(x) holding diffusion coefficients, velocity gradients, and total material constant. Other subfamilies can be defined in an analogous fashion. Relationships similar to Eqs. la and lb, and with analogous properties, have also been derived for Reactions I and IV. In the case of Reaction IV use is made of the simplification that for given distribution of ampholyte, diffusion coefficients, and velocity gradients, f?e‘,,t/K’norm as a function of x,,~ corresponds to a subfamily of
ISOELECTRIC
FOCUSING
OF INTERACTING
SYSTEMS
39
equilibrium patterns whose members are generated by varying K’,,,,, at constant total amount of macromolecule. K’,,,, is the normalized value of K’4(~) as assigned at the inflection point of the distribution of ampholyte. Computations were made on the University of Colorado’s CDC 6400 computer. With the following exceptions the values of the several parameters are the same as assigned previously (9): Ax = 0.007 cm; At = 10 set; diffusion coefficients of macromonomer and macrodimer in Reactions I, III, and IV, 5.1 x lo-’ and 3.6 x lo-’ cm2 set-‘, respectively; diffusion coefficients of macromonomer and macrotrimer in Reaction II, 6.2 x low7 and 3.6 x lo-’ cm2 set-I, respectively; pI of macromonomer, pH 7.25, located at x = 0.9950 cm; p1 of macrodimer and macrotrimer, pH 7.00, located at x = 0.4950 cm. The steady-state parameters of the isoelectric focusing column and the macromolecular velocity gradients are presented in Fig. 1. Material balance was excellent, to better than 10P6%. Computed isoelectric focusing patterns are displayed as plots of constituent concentration of macromolecule, C = C, vs X; the overbar is also dropped in plots of r?‘,,t vs x,,~ and of cd vs x; the circumflex is dropped when (?“, and c2 are plotted; the two vertical arrows in
0
0.4
0.8
1.2
1.6
X (cm) FIG. 1. Parameters of the calculations. (A) Environment created by the steady-state distribution of carrier ampholytes: a, sharp distribution of carrier ampholyte used to calculate the isoelectric focusing patterns presented in Fig. 4A; b, broad distribution of ampholyte pertaining to Figs. 4B, 5, and 6; c, sharp distribution pertaining to Figs. 7 and 9. (B) Velocity gradients of macromolecular species: V,, macromonomer pertaining to Figs. 2 and 4-8: V2. macrodimer pertaining to Figs. 2A and 4-8 and macrotrimer pertaining to Fig. 2B. These several parameters and the calculated focusing patterns can be applied to any other region of the isoelectric focusing column merely by translating the abscissa to the desired pH range.
40
CANN,
STIMPSON,
AND
COX
each of the figures indicate the positions of the pls. The constituent concentration of macromolecule in the initial zone is designated as Co. RESULTS
The control calculations on the simple dimerization Reaction I serve a twofold purpose: They allow assessment of two criteria for convergence of the transient isoelectric focusing patterns to the equilibrium pattern and, together with the bimodal patterns calculated previously for a mixture of two noninteracting macromolecules (Fig. 1A of Ref. 9), they provide a framework within which to consider carrier ampholyte-induced association and dissociation reactions. Focusing patterns have been computed for three different values of K, at constant Co. In each case the transient patterns, while skewed in shape, remain strictly unimodal during the approach to equilibrium, which, for all practical purposes, is attained in 1.6 x lo5 set as judged by the weak5, criterion that the constituent concentration of macromolecule at all positions in the pattern remains constant to seven places between 1.6 x lo5 and 2 x lo5 sec. The patterns for 2 x lo5 set are displayed in Fig. 2A. As might be anticipated from mass action considerations, the position of the maximum in the pattern shifts progressively from the vicinity of the pZ of Ju to the vicinity of the pZ of .Mz as K, is increased; i.e., the higher K,, the more highly dimerized the system and, thus, the closer in value to the pZ of dimer is the average pZ. We now apply the rigorous criterion for convergence by comparing the amplitude and position of the maximum with the values predicted analytically by equations analogous to Eqs. la and lb. The comparison is made graphically in Fig. 3A. Clearly, equilibrium has been virtually reached in all three cases. Turning to the control calculations on the trimerization Reaction II, we find that here too the transient patterns are at all times unimodal, albeit markedly skewed due to the strong concentration dependence of the reaction, Fig. 2B. Equilibrium is almost attained in 1 x IO5 set with the concentrations changing only in the fifth place during the last 2 x lo4 set of focusing. The calculations for carrier ampholyte-induced reactions assume either specific binding of one of the several ampholytes that may be positioned in the region of the pZs of the macromolecular species or nonspecific binding of all of the ampholytes of which one (or a closely spaced family with 5 Weak, because in the case of ampholyte-induced isomerization (9) and pH-dependent conformational transitions (IO) the approach to equilibrium is impeded under certain conditions by a diffusion barrier located midway between the pls where the constituent velocity of the macromolecule vanishes. Such behavior has not been observed for ampholyte-induced association and dissociation even though there is a potential diffusion barrier, evidently because its location oscillates in a damped fashion with time so as to be coincident with a strong gradient of constituent macromolecule during intermediate times of focusing. This appears to be a characteristic feature of concentration-dependent reactions.
ISOELECTRIC
Oh!.,
Q2
FOCUSING
.
0.4
OF INTERACTING
,
0.6
.
,
QE
.
,
1.0
.
SYSTEMS
41
,
1.2
FIG. 2. Control calculation of isoelectric focusing patterns. (A) Simple macromolecular dimerization (Reaction I): a, initial zone of macromolecule, Co = 5 x 10e6 M; b, focusing pattern for K, = 1 x 103M-l; c, pattern forK, = 1 X 105M-'; d, K, = 1 X 106M-'; t = 2 x 105 sec. (B) Simple trimerization (Reaction II): a, initial zone. Co = 1.83 x 10m5M: b, pattern for K, = 3.33 x lo9 M-* and t = 1 x lo5 sec.
Gaussian envelope) overwhelmingly dominates the region. Both sharp and broad distributions of ampholyte (Fig. lA), which approximate the extreme widths of the discrete, steady-state zones of “Ampholines” visualized experimentally (17), have been considered. In Fig. 4 the shapes of the transient focusing patterns computed for dimerization (Reaction III) induced by interaction with a sharp distribution of ampholyte (curve a in Fig. 1A) are compared with those for a broad distribution (curve b in Fig. 1A). Values ofK’,(Z) and Co were chosen such that the macromolecule would be only slightly dimerized in the equilibrium focusing pattern. For the sharp distribution of ampholyte the transient patterns (Fig. 4A) show two well-resolved, intense peaks during much of the time course of approach to equilibrium, the bimodality being reminiscent of a mixture of two noninteracting macromolecules. But there the resemblance stops, since one of the peaks grows at the expense of the other as the transient patterns converge upon the symmetrical, unimodal equilibrium pattern which is focused close to the pZ of monomer. When the interaction is with the broad distribution of ampholyte the bimodality of the transient patterns is much less pronounced (Fig. 4B), but the general features of approach to the unimodal equilibrium pattern are similar. Were
42
CANN, STIMPSON, 10
I
A
a
b
I
AND COX
4-
1 IB D
8f; a
1
c
3-
6-
I
0
2-
x
"g 441 2-
l-
a
.'....I
c
L, Ok,
,
b ,
. ,
,
,
0230.6 0.7 a8 0.0 II)
Xext (cm) FIG. 3. Analytical criterion for convergence of transient isoelectric focusing patterns to the equilibrium pattern; -, amplitude vs position of extrema in subfamilies of equilibrium patterns as given by Eqs. la and lb or their analogs in the case of Reaction I. (A) Simple dimerization (Reaction I), Co = 5 X lo-@ M: a, K, = 1X 106M-l; b, 1 X lo5 M-l; c, 1 X 103M-l. 0, amplitude and position of maximum shown by transient isoelectric focusing pattern for corresponding K, at t = 2 x IO5 set (Fig. 2A). (B) Carrier ampholyte-induced dimerization (Reaction III), broad distribution of ampholyte given by curve b in Fig. IA, K'JI) = 1 x 105 M-I: A, amplitude and position of maximum shown by transient isoelectric focusing patterns at t = 2 x lo5 set for Co = 1 x 10mBM (practically the same as r = 1 x 105 set, pattern d in Fig. 4B); 0, extrema in bimodal transient pattern att = 2 x IO5set for Co = 5 x lOA M (practically the same as f = 1 x IO5 set, pattern d in Fig. 5) with path to equilibrium, . . . , shown by curves a, b, and c, where b denotes the minimum; 0, extrema in pattern at t = 2 x lo5 set for Co = 1 x 10m5M(pattern C in Fig. 6).
the interaction to involve the binding of a larger number of ampholyte molecules, resolution of the transient patterns would be enhanced (9,lO). The differences in the focusing behavior between ampholyte-induced and simple dimerization become even more striking when the value of Co is increased, other things held constant, for then the bimodality of the transient patterns shown by the induced reaction is preserved in the equilibrium pattern (Figs. 5 and 6). For all practical purposes, by f = 2 x IO5 set, the transient patterns shown in Figs. 4B, 5, and 6 constitute a subfamily of equilibrium patterns generated by varying Co, as judged by both criteria for convergence, in particular the comparison drawn in Fig. 3B between the amplitudes and positions of their extrema with analytically predicted values. [Small deviations of the points from the curve are to be expected because of the truncation error inherent to the discrete transport calculations (9).] As illustrated in Fig. 6, the bimodal equilibrium patterns shown at the higher Co are reaction zones in which the two peaks correspond to different chemical equilibrium compositions each enriched
ISOELECTRIC
0.2
FOCUSING
0.4
OF INTERACTING
0.6
0.8
1.0
SYSTEMS
43
1.2
x (cm)
FIG. 4. Theoretical transient isoelectric focusing patterns for carrier ampholyte-induced macromolecular dimerization (Reaction III). (A) Sharp distribution of ampholyte given by curve a in Fig. IA. K’3(1) = I x IO5 M-', Co = 5 x IOe6 M: a, 3.75 x IO3set; b, I x IO’sec; c, 3 x IO4set; d, 6 x IO’ set; e, I .8 x lo5 sec. (B) Broad distribution ofampholyte (curve b in Fig. IA), K’,(l) = 1 x IO5 M-', Co = I x 10m6M: a, 3.75 X IO3set; b, 1 X lo* set; c, 2.5 x IO4set; d, I x IO5 set; the amplitude of the maximum in the pattern increased by only 0.03% between 1 x IO5 and 2 x IO5 sec. The breadth of the initial zone in this figure and in Figs. 5-8 is the same as in Fig. 2A.
in the macromolecular species of proximal pZ, and not to separated .& and &A,. If Co were to be increased still further, the peak enriched in .A@& would grow in size until at sufficiently high concentration the equilibrium pattern would show a single peak focused close to the pZ of dimer. Thus, as predicted by Eqs. la and lb, the shapes of the equilibrium patterns for ampholyte-induced and simple dimerization become indistinguishable in the limits of very low or exceedingly high Co. Rather similar results have been obtained for carrier ampholyte-induced dissociation of a subunit macromolecule into its isoelectrically identical half-mers (Reaction IV). Here too the characteristic bimodality of the transient patterns (e.g., Fig. 7) is, under certain conditions, preserved in the equilibrium pattern. (In all cases examined one peak grows at the expense of the other during the approach to equilibrium.) Transient patterns, obtained after 2 x IO5 set of transport, are displayed in Fig. 8 to illustrate how the shape of the pattern changes with increasing K’,(Z) at constant Co. For small Kr4(Z) the pattern (curve a) shows a major peak positioned close to the pZ of the subunit macromolecule and a small one
CANN, STIMPSON,
AND COX
12
10
I Oh
’
I 0.8
1.0
I 1.2
x (cm)
FIG. 5. Transient isoelectric focusing patterns for ampholyte-induced dimerization (Reaction III). Broad distribution of ampholyte (curve b in Fig. IA), K’,(I) = 1 x lo5 M-i, Co=5 x 10-B~:a,3.75 x 103sec;b,l x 104sec;c,2.5 x 104sec;d,l x 105sec;amp1itudesof the maxima in the pattern increased by only -0.03% between 1 x lo5 and 2 x lo5 sec.
x(m)
FIG. 6. Transient isoelectric focusing pattern for ampholyte-induced dimerization (Reaction III) showing distributions of monomer (C,) and dimer (2C,) as well as constituent macromolecule (C): broad distribution of ampholyte (curve b in Fig. IA), K’,(I) = 1 X lo5 M-I, Co = 1 x 1O-5 M, t = 2 x IO5sec. The resolution of earlier patterns (3.75 x 103-2.5 X lo4 set) into two peaks is much more pronounced, comparable to pattern b in Fig. 5. The amplitudes of the maxima in the pattern increased by -2% between 1 x IO5 and 2 x lo5 sec.
ISOELECTRIC
0.2
FOCUSING
0.4
OF INTERACTING
0.6
0.8
1.0
SYSTEMS
45
2
X (cm1
FIG. 7. Transient isoelectric focusing patterns for ampholyte-induced dissociation of a subunit macromolecule into its isoelectrically identical half-mers (Reaction IV). Sharp distribution of ampholyte (curve c in Fig. IA), K’,(I) = 1 x 10m5 M, Co = 1 x 10e5 M: a, 3.75 x lo3 set; b. 1 x lo* set; c, 3 x IO* set; d, 1 x IO5 set; amplitudes of the maxima in the pattern increased by -3.5% between 1 x lOsand 2 x lo5 set (see pattern b in Fig. 8 for 2 x lo5 sec.).
near the pZ of the half-mers; i.e., overall there is very little dissociation throughout the pattern. For K’,(Z) an order of magnitude larger in value the pattern (curve b) consists of two intense peaks; while for two more orders of magnitude increase in Kf4(Z) the pattern (curve c) now shows a major peak near the pZ of half-mers and a small one near the pZ of subunit macromolecule, since overall there is little intact subunit macromolecule left. The amplitudes and positions of the maxima in these patterns are compared graphically with analytically predicted values in Fig. 9. The patterns for the two smaller values of K’,(Z) are practically equilibrium patterns (Fig. 9A), but the pattern for the largest value still has a distance to go (Fig. 9B). Whereas the amplitude and position of its major peak falls close to the analytical curve, the small peak falls in the forbidden region to the left of the curve, and the minimum falls far from the curve. Since the amplitude of the small peak decreased appreciably between 1.8 x lo5 and 2 x lo5 set (with concomitant growth of the major one), it might be surmised that the equilibrium pattern consists of a single peak focused close to the pZ of the half-mers. In any case we note that out of the total of ten sets of transient patterns computed for Reactions I, III, and IV, nine sets virtually converge to their equilibrium pattern as judged by the analytical criterion. Thus, iterative solution of the simultaneous transport equations and mass action expression is one way of solving the nonlinear
46
CANN, STIMPSON,
AND COX
I
6-
a
2 x (cm)
FIG. 8. Transient isoelectric focusing patterns for ampholyte-induced dissociation of a subunit macromolecule into its isoelectrically identical half-mers (Reaction IV). Sharp distribution of ampholyte (curve c in Fig. 1A). Co = 1 x 1O-5 M, t = 2 x IO5 set: a, K’,(f) = 1 X lo-* M; b, K’#) = I X lO-5 M; c, K’,(I) = I X 1O-3 hf.
integral equation describing the equilibrium distribution of constituent macromolecule along the isoelectric focusing column. Although the calculations presented here are for a broad initial zone of macromolecule, the transient focusing behavior described for the ampholyte-induced interactions also applies, with only two exceptions, to other initial conditions in which a narrow zone is inserted into the pH gradient at some arbitrary position. The exceptions arise when the equilibrium pattern shows a single peak focused close to the pZ of one of the macromolecular species. In that event, the transient patterns will show a single peak when (i) the peak in the equilibrium pattern is close to the pZ of the more acidic species and the initial zone is positioned anodically thereto,or (ii) the peak is close to the pZ of the less acidic species and the initial zone is cathodic thereto. Bimodal transient patterns can be expected for other conditions. [See previous calculations (9,lO) for peculiar transient focusing behavior alluded to in Footnote 5, which may obtain in the case of ampholyte-induced isomerization and pH-dependent conformational transitions.] DISCUSSION
The new insights provided by these calculations have important implications for conventional analytical and preparative applications of isoelectric focusing. Thus, equilibrium as well as transient focusing patterns given by carrier ampholyte-induced association or dissociation can show two peaks similar to the patterns for a mixture of two
ISOELECTRIC
FOCUSING
).5 0.6 0.7 Cl8 0.9
OF INTERACTING
1.0
0.5 Q6 a7
SYSTEMS
0.8
0.9
47
1.0
FIG. 9. Analytical criterion for convergence of transient isoelectric focusing patterns to the equilibrium pattern for ampholyte-induced dissociation of a subunit macromolecule into its isoelectrically identical half-mers (Reaction IV): -, ratio of amplitude to normalized value of the apparent equilibrium constant vs position of extrema in equilibrium patterns; sharp distribution ofampholyte (curve c in Fig. IA); analytical expressions analogous to Eqs. la and lb. (A) A, transient focusing pattern at 2 x IO5set for Co = I x 10e5 M and K’,(I) = 1 x 10e6 M (pattern a in Fig. 8); 0, K’XI) = I x 10e5M (pattern b in Fig. 8). (B) 0, K’,(I) = 1 x 10e3 M (pattern c in Fig. 8).
noninteracting macromolecules, and the same can be said for ampholyteinduced isomerization (9) and pH-dependent conformational transitions (10). Moreover, a molecule undergoing sequential conformational transitions can give multiple peaks. In practice such patterns could easily be misinterpreted in terms of inherent heterogeneity with respect to isoelectric point, but with due precautions fractionation provides an unambiguous method for distinguishing between interactions and heterogeneity. In the fractionation test the protein in each of the peaks is isolated. The resulting fractions are reconstituted to the concentration of the unfractionated material, as dictated by the concentration dependence of the patterns for association-dissociation reactions, and resubjected to isoelectric focusing taking care to position the initial zone as in the original separation. For interaction each fraction will behave like the unfractionated material and show two or more peaks, while for heterogeneity a single peak focused at a reproducible isoelectric point will be obtained. Rigorous completion of the analysis requires that the unfractionated material and its fractions be focused from initial zones located at different positions along the isoelectric focusing column.
48
CANN, STIMPSON,
AND COX
The fractionation test has been recommended previously by Ressler (18) and Wrigley (19) for distinguishing between artifactual and real multipeaked patterns; our calculations provide a theoretical rationale for the test. The investigation of Talbot (20) on the pH-dependent conformational transition of the 12s subunit protein of the capsid of foot-and-mouth disease virus is exemplary in the way fractionation experiments were combined with focusing procedural variations and ultracentrifugation to establish the interaction. It is characteristic of interacting systems that during the approach to equilibrium one of the peaks in the transient focusing pattern grows at the expense of the other, which under certain conditions may disappear completely (Fig. 4A). This property may well prove to be diagnostic for interactions. Examples are afforded by a-glycerol phosphate dehydrogenase and pyruvate kinase (21). When focused from an initially uniform distribution throughout a density-gradient isoelectric focusing column, both give transient patterns showing two peaks migrating from either end of the column. As the two peaks converge upon the isoelectric region they either coalesce or, possibly, one of the peaks grows at the expense of the other as judged from the few transient patterns presented. Clearly, these results are suggestive of an interaction, and it may be pertinent that both enzymes have a subunit structure. It must be emphasized, however, that several classes of interaction exhibit certain common features on isoelectric focusing. Accordingly, elucidation of the mechanism of interaction requires the combined application of several physicochemical methods such as ultracentrifugation and circular dichroism. Finally, the results of our calculations add to the store of fundamental information required for application of isoelectric focusing to the detection and eventual characterization of interacting systems. For example, it is anticipated that the results reported herein may be useful in the application of the method to the study of subunit enzymes, enzyme complexes, and other associating-dissociating systems. Note added in proof. Subsequent to submission of our manuscript M. Dishon and G. H. Weiss (1977, Anal. Biochem. 81, 1) reported a theoretical study of isoelectric focusing in a nonlinear pH gradient. Their calculations, which are for a broad initial zone of noninteracting macromolecule, predict two transient peaks arising at the ends of the initial concentration profile, migrating towards each other as time goes on and finally merging at the PI. It should be possible to distinguish this effect of a nonlinear pH gradient from macromolecular interactions by comparison of transient and equilibrium patterns given by a broad initial zone with those from a narrow zone located at different positions along the isoelectric focusing column.
REFERENCES I. Svensson, H. (1961) Acra Chem. Stand. 15, 325. 2. Svensson, H. (1962) Acta Chem. Stand. 16, 456. 3. Vesterberg, 0.. and Svensson, H. (1966) Acta Chem.
Stand.
20, 820.
ISOELECTRIC 4. 5. 6. 7. 8. 9. 10. 1 I. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
FOCUSING
OF INTERACTING
SYSTEMS
49
Vesterberg, 0. (1969) Acra Chem. Stand. 23, 2653. Haglund, H. (1971) Methods Biochem. Anal. 19, I. Catsimpoolas, N. (ed.) (1973) Ann. N. Y. Acad. Sci. 209, l-529. Righetti. P. G., and Drysdale, J. W. (1974) J. Chromatogr. 98, 271. Catsimpoolas, N. (ed.) (1976) Isoelectric Focusing, Academic Press, New York. Cann, J. R.. and Stimpson. D. I. (1977) Biophys. Chem., 7, 103. Stimpson, D. I.. and Cann. J. R. (1977) Biophys. Chem.. 7, 115. Tombs, M. P. (1957) Biochem. J. 67, 517. Brown, R. A., and Timasheff, S. N. (1959) in Electrophoresis Theory, Methods and Practice (Bier, M., ed.), Chap. 8, Academic Press, New York. Timasheff, S. N., and Townend, R. (1960) J. Amer. Chem. Sot. 82, 3157. Johnson, P.. and Shooter, E. M. (1950) Biochem. Biophys. Acta 5, 361. Johnson, P., Shooter, E. M., and Rideal, E. K. (1950) Biochem. Biophys. Acfa 5, 376. Benveniste, K. B., and Munkres, K. D. (1975) in Isozymes I. Molecular Structure (Marker?. C. L.. ed.). p. 561, Academic Press, New York. Felgenhauer, K., and Pak, S. J. (1973) Ann. N. Y. Acad. Sci. 209, 147. Ressler, N. (1973) Anal. Biochem. 51, 604. Wrigley, C. W. (1976)in Isoelectric Focusing(Catsimpoolas, N.. ed.), Chap. 4, Academic Press, New York. Talbot, P. (1975) in Isoelectric Focusing (Arbuthnott, J. P.. Beeley, J. A., eds.), p. 270, Butterworths, London. Behnke, J. N., Dagher. S. M., Massey, T. H., and Deal, W. C., Jr. (1975)AnaL Biochem. 69, 1.
BIOCHEMISTRY
86,
34-49 (1978)
Isoelectric Focusing of Interacting Systems III. Carrier Ampholyte-Induced Macromolecular Association or Dissociation into Subunit9 JOHN R. CANN,* DONALD I. STIMPSON,* *Department Colorado
of Biophysics and Genetics, 80262, and the TDepartment Manhattan,
AND DAVID J.Coxt
University of Colorado Medical Center, Denver, of Biochemistry, Kansas State University, Kansas 66506
Received June 1, 1977; accepted November 5, 1977 A phenomenological theory of isoelectric focusing of interacting systems (Cann, J. R., and Stimpson, D. I. Biophys. Chem., 7, 103, (1977) has been extended to include carrier ampholyte-induced dimerization or dissociation of a subunit macromolecule. Equilibrium as well as transient focusing patterns can show two well-resolved peaks similar to the patterns for a mixture of two noninteracting macromolecules. Characteristically, one of the peaks in the transient pattern grows at the expense of the other, which under certain conditions may disappear completely as equilibrium is approached. The implications of these findings for conventional applications of isoelectric focusing and for the detection and characterization of macromolecular interactions are discussed.
During the past decade or so the method of isoelectric focusing (l-8) has become an established procedure for the separation and characterization of proteins and other amphoteric macromolecules with respect to isoelectric point (pZ). Briefly: A linear pH gradient is generated and maintained by electrolysis of a mixture of carrier ampholytes (most commonly, small aliphatic polyamino-polycarboxylic acids marketed by LKB, Stockholm, Sweden, under the tradename “Ampholine”), each initially distributed uniformly in a liquid column containing a sucrose density gradient to stabilize against convection or in a column or slab of solid support such as polyacrylamide gel. Electrolysis establishes a steady state in which the various carrier ampholytes are distributed along the column from anode to cathode in order of increasing pZ with each distributed approximately in a Gaussian fashion about its pZ, thus, generating the pH gradient. When a zone of isoelectrically homogeneous protein is inserted into the pH gradient, the protein molecules migrate under the influence of the electric field toward either the anode or cathode depending upon the pH (# pZ) at ’ Supported in part by Research Grants HL 13909-25 from the National Heart and Lung Institute (J.R.C.) and GM 22243-02 from the National Institute of General Medical Sciences (D.J.C.), United States Public Health Service, and by the Kansas Agricultural Experiment Station. This publication is No. 671 from the Department of Biophysics and Genetics, University of Colorado Medical Center, Denver, Colorado 80262, and No. 197-j from the Department of Biochemistry, Kansas Agricultural Experiment Station. 0003-2697/78/0861-0034$02.00/O Copyright Ail rights
0 1978 by Academic Press. Inc. of reproduction in any form reserved.
34
ISOELECTRIC
FOCUSING
OF
INTERACTING
SYSTEMS
35
the position of insertion until they reach that position in the column where the pH corresponds to the pZ. The protein focuses sharply at this position. In the case of an isoelectrically heterogeneous protein, each component focuses at its respective pZ so that the isoelectric focusing pattern (plot of concentration vs position) shows a corresponding number of peaks. This description applies to the commonly used procedure for isoelectric focusing in gels. Alternatively, the carrier ampholytes and the sample of protein can be premixed so that initially both are distributed uniformly throughout the gel or liquid column. Upon applying the electric field the pH gradient is generated relatively rapidly, after which each protein component converges from both ends of the column lipon its ~1. The foregoing considerations are for ideal situations uncomplicated by macromolecular interactions. Recently (9,lO) we have shown theoretically that rather different isoelectric focusing behavior is to be expected for systems undergoing either reversible, carrier ampholyte-induced macromolecular isomerization or pH-dependent conformational transitions both in the limit of instantaneous establishment of chemical equilibrium. Thus, for example, such interactions can give well-resolved, bimodal (sometimes multimodal) focusing patterns in which the peaks correspond to different equilibrium compositions and not to separated macromolecular species. Our calculations have now been extended to include the following rapidly equilibrating, macromolecular association-dissociation reactions in which Ju symbolizes macromonomer and &?. carrier ampholyte: The simple dimerization reaction the simple trimerization
3%/l% eAg,K2 the carrier ampholyte-induced
[III
dimerization
2.4 +4&d = .44,
Ks
and the carrier ampholyte-induced dissociation of a subunit romolecule into its isoelectrically identical half-mers J& + 4d e 2~fZ.d,,K,.
WI mac-
UVI
In each case the pKs of certain ionizable groups on the macromolecule are assumed to be altered due to the change in state of association, thereby leading to a change in ~1.~ [The contribution of reversibly bound carrier 2 The pKs may be altered via several possible mechanisms: Exposed groups may experience a change in either the local charge environment or the local. effective dielectric constant; the group(s) may be masked by becoming buried in an hydrophobic domain of the macromolecule or by participating in hydrogen-bonding interactions; etc. A change in net charge of one protonic unit at appropriate pH will typically shift the pI by about 0.1 pH unit irrespective ofthe total number of ionizable groups on the protein. Although it is not generally
36
CANN,
STIMPSON.
AND
COX
ampholyte per se to the change in pZ can be neglected since the focused system is in a medium of ampholytes which are essentially isoelectric (3).] It is evident, therefore, that hydrogen ions must be involved in each of the reactions. Consequently, the “equilibrium constant,” K,, must incorporate the pKs of macroreactant and macroproduct in such a way as to be a function of hydrogen ion concentration. We shall assume that the “equilibrium constant” is an insensitive function of pH and can be considered constant in the region of the column where the particular system focuses, given the shallow pH gradient and the relatively small difference in pls. Theoretical isoelectric focusing patterns have been computed by numerical solution of the simultaneous transport equations and mass action expression for each of Reactions I-IV as a function of time thereby constructing the approach to the equilibrium distribution of macromolecule along the isoelectric focusing co1umn,3 i.e., the equilibrium isoelectric focusing pattern in contradistinction to the transient patterns which pertain to the approach. Convergence to the equilibrium distribution was determined by comparison of the amplitudes and positions of the extrema in the transient patterns with those predicted for the equilibrium distribution by an analytical relationship. The results of these calculations are described below. THEORY4
Transient isoelectric focusing patterns were computed essentially as described previously for ampholyte-induced isomerization (9) except for use of the appropriate expressions for recalculation of chemical equilibrium after each time cycle of diffusion and driven transport. Consider, for example, Reaction III: Equation 7 of Cann and Stimpson (9) is replaced by possible to predict a priori the relationship of electrophoretic properties of an aggregated protein molecule to those of the monomer, it is clear that changes in state of association sometimes do influence significantly the electrophoretic mobility in free solution and the PI. Examples include the tetramerization of /3-lactoglobulin A (1 I- 13). the dissociation of the globulin arachin (14,151, and the multiple molecular weight isozymes of Neurospora mitochondrial malate dehydrogenase (16). In the case of malate dehydrogenase, the pl of the isozyme increases linearly with its molecular weight, suggesting that acidic groups are asymmetrically masked as the degree of association of the monomeric enzyme increases. 3 Since our model assumes that the macromolecule does not contribute to the conductance, the distribution of macromolecule in the final isoelectric focusing pattern is an equilibrium distribution given the particular environment created by the steady-state distribution of carrier ampholytes. ’ Aside from the introduction of a few new variables and functions defined herein, the notation is the same as used previously (9) except for the subscripts 1 and 2 which here denote Y and Y&,. respectively.
ISOELECTRIC
FOCUSING
OF INTERACTING
C(j,t,+,) = C*(jJ,+,) + Equation
8 is replaced by
Equation
9 by
and Equation
37
SYSTEMS
mL~n+J
11 by
K3 = ~‘z(j,t,)l[i-,O’,tn)lz[~~(j)l*. The apparent equilibrium constant K ‘3(j) = K3[CJ$( j)]” is normalized by assigning it the value, K’,(Z) = ~‘z(l,r,)l[i‘,(l,r,)]*, in segment I where the inflection point of the Gaussian distribution of constituent carrier ampholyte is located. As before it is assumed that the concentration of ampholytes is sufficiently greater than macromolecule that it is not perturbed significantly by the reaction, so that both the distribution of ampholytes and the pH gradient are invariant in time. The calculations are for a broad initial zone of macromolecule encompassing the pls of both JU and J&P& and can be placed into correspondence with the experimental procedure in which both the macromolecule and the ampholytes are distributed uniformly throughout the column initially. The correspondence is affected by making the limiting assumption that the steady-state distribution of ampholytes and the concomitant pH gradient are established so rapidly that, in effect, they obtain at the instant the macromolecule begins to migrate under the influence of the electric field. Of course, the equilibrium isoelectric focusing pattern is the same for all arbitrary initial distributions of a given amount of macromolecule. The equilibrium pattern itself can in principle be calculated in a manner similar to the procedure described previously (9): The continuity equation for constituent macromolecule is + 20,
a&x,t) ax
- V,(x@,(x,t)
- 2VZ(X)&X,l)
in which
&(x.t) = &
(-1 3
&(XJ> = ~3’(X,re,w>l’ K,‘(x 1 =K3[ 5(x)14.
+ [l + 8K,'(x)~(x,t)l"']
I
aJ(x,t) ax
E ~
38
CANN,
STIMPSON,
AND
COX
Equating 13~?(x,t)lat to zero, noting the physical requirement that at equilibrium Lim,, .Z(x,t) = 0, and integrating the resulting ordinary differential equation with respect to x gives the equilibrium distribution, but in terms of a nonlinear integral equation whose solution is a formidable mathematical problem. However, the ordinary differential equation readily admits a property of the family of equilibrium patterns given by the integral equation; namely, the amplitude of the extrema in the distributions, text, as an implicit function of their positions, xeXt. Equating ~3C(x,r)lax to zero in the explicit expression for Lim,,, .Z(x,t) Lim,,, = 0 and solving the resulting algebraic equation gives c’ext = &
[u” + 2u] 3
PI
- D2) ~dlnK3’ dxext
V2
+
V2
(D1 - D2) dlnK,’ V2
1
+ -2Vl + 1
+ 2V, ,+lr-%)111!
dxext
[lb]
in whichK’,, VI, and V, are now functions ofx,,t. These equations exhibit two interesting properties: (i) text = 0 for V, = 0 and (ii) a singularity, I c!xt I + m as V, + 0. Both can be understood in terms of the law of mass action as it applies to Reaction III: (i) The equilibrium pattern can show a peak centered precisely at the pZ of & only in the limit of infinite dilution of total macromolecule where the reaction is shifted completely in the direction of reactant. (ii) The pattern can show a peak centered precisely at the pZ of &,d4 only in the other limit of infinitely high concentration of macromolecule where the reaction is driven completely to product. At finite concentrations the two species must coexist wherever there is macromolecule in the isoelectric focusing column, so that the peak(s) in the pattern will be centered somewhere between the pls. Note that while text as a function of xeXt is a property of the family of all equilibrium patterns admitted by Reaction III, it is expedient to consider subfamilies thereof. For example, one subfamily corresponds to specified Kf3(x), diffusion coefficients and velocity gradients; members of this subfamily are generated by varying the total amount of macromolecule applied to the isoelectric focusing column. Another subfamily is characterized by varying values of K’3(x) holding diffusion coefficients, velocity gradients, and total material constant. Other subfamilies can be defined in an analogous fashion. Relationships similar to Eqs. la and lb, and with analogous properties, have also been derived for Reactions I and IV. In the case of Reaction IV use is made of the simplification that for given distribution of ampholyte, diffusion coefficients, and velocity gradients, f?e‘,,t/K’norm as a function of x,,~ corresponds to a subfamily of
ISOELECTRIC
FOCUSING
OF INTERACTING
SYSTEMS
39
equilibrium patterns whose members are generated by varying K’,,,,, at constant total amount of macromolecule. K’,,,, is the normalized value of K’4(~) as assigned at the inflection point of the distribution of ampholyte. Computations were made on the University of Colorado’s CDC 6400 computer. With the following exceptions the values of the several parameters are the same as assigned previously (9): Ax = 0.007 cm; At = 10 set; diffusion coefficients of macromonomer and macrodimer in Reactions I, III, and IV, 5.1 x lo-’ and 3.6 x lo-’ cm2 set-‘, respectively; diffusion coefficients of macromonomer and macrotrimer in Reaction II, 6.2 x low7 and 3.6 x lo-’ cm2 set-I, respectively; pI of macromonomer, pH 7.25, located at x = 0.9950 cm; p1 of macrodimer and macrotrimer, pH 7.00, located at x = 0.4950 cm. The steady-state parameters of the isoelectric focusing column and the macromolecular velocity gradients are presented in Fig. 1. Material balance was excellent, to better than 10P6%. Computed isoelectric focusing patterns are displayed as plots of constituent concentration of macromolecule, C = C, vs X; the overbar is also dropped in plots of r?‘,,t vs x,,~ and of cd vs x; the circumflex is dropped when (?“, and c2 are plotted; the two vertical arrows in
0
0.4
0.8
1.2
1.6
X (cm) FIG. 1. Parameters of the calculations. (A) Environment created by the steady-state distribution of carrier ampholytes: a, sharp distribution of carrier ampholyte used to calculate the isoelectric focusing patterns presented in Fig. 4A; b, broad distribution of ampholyte pertaining to Figs. 4B, 5, and 6; c, sharp distribution pertaining to Figs. 7 and 9. (B) Velocity gradients of macromolecular species: V,, macromonomer pertaining to Figs. 2 and 4-8: V2. macrodimer pertaining to Figs. 2A and 4-8 and macrotrimer pertaining to Fig. 2B. These several parameters and the calculated focusing patterns can be applied to any other region of the isoelectric focusing column merely by translating the abscissa to the desired pH range.
40
CANN,
STIMPSON,
AND
COX
each of the figures indicate the positions of the pls. The constituent concentration of macromolecule in the initial zone is designated as Co. RESULTS
The control calculations on the simple dimerization Reaction I serve a twofold purpose: They allow assessment of two criteria for convergence of the transient isoelectric focusing patterns to the equilibrium pattern and, together with the bimodal patterns calculated previously for a mixture of two noninteracting macromolecules (Fig. 1A of Ref. 9), they provide a framework within which to consider carrier ampholyte-induced association and dissociation reactions. Focusing patterns have been computed for three different values of K, at constant Co. In each case the transient patterns, while skewed in shape, remain strictly unimodal during the approach to equilibrium, which, for all practical purposes, is attained in 1.6 x lo5 set as judged by the weak5, criterion that the constituent concentration of macromolecule at all positions in the pattern remains constant to seven places between 1.6 x lo5 and 2 x lo5 sec. The patterns for 2 x lo5 set are displayed in Fig. 2A. As might be anticipated from mass action considerations, the position of the maximum in the pattern shifts progressively from the vicinity of the pZ of Ju to the vicinity of the pZ of .Mz as K, is increased; i.e., the higher K,, the more highly dimerized the system and, thus, the closer in value to the pZ of dimer is the average pZ. We now apply the rigorous criterion for convergence by comparing the amplitude and position of the maximum with the values predicted analytically by equations analogous to Eqs. la and lb. The comparison is made graphically in Fig. 3A. Clearly, equilibrium has been virtually reached in all three cases. Turning to the control calculations on the trimerization Reaction II, we find that here too the transient patterns are at all times unimodal, albeit markedly skewed due to the strong concentration dependence of the reaction, Fig. 2B. Equilibrium is almost attained in 1 x IO5 set with the concentrations changing only in the fifth place during the last 2 x lo4 set of focusing. The calculations for carrier ampholyte-induced reactions assume either specific binding of one of the several ampholytes that may be positioned in the region of the pZs of the macromolecular species or nonspecific binding of all of the ampholytes of which one (or a closely spaced family with 5 Weak, because in the case of ampholyte-induced isomerization (9) and pH-dependent conformational transitions (IO) the approach to equilibrium is impeded under certain conditions by a diffusion barrier located midway between the pls where the constituent velocity of the macromolecule vanishes. Such behavior has not been observed for ampholyte-induced association and dissociation even though there is a potential diffusion barrier, evidently because its location oscillates in a damped fashion with time so as to be coincident with a strong gradient of constituent macromolecule during intermediate times of focusing. This appears to be a characteristic feature of concentration-dependent reactions.
ISOELECTRIC
Oh!.,
Q2
FOCUSING
.
0.4
OF INTERACTING
,
0.6
.
,
QE
.
,
1.0
.
SYSTEMS
41
,
1.2
FIG. 2. Control calculation of isoelectric focusing patterns. (A) Simple macromolecular dimerization (Reaction I): a, initial zone of macromolecule, Co = 5 x 10e6 M; b, focusing pattern for K, = 1 x 103M-l; c, pattern forK, = 1 X 105M-'; d, K, = 1 X 106M-'; t = 2 x 105 sec. (B) Simple trimerization (Reaction II): a, initial zone. Co = 1.83 x 10m5M: b, pattern for K, = 3.33 x lo9 M-* and t = 1 x lo5 sec.
Gaussian envelope) overwhelmingly dominates the region. Both sharp and broad distributions of ampholyte (Fig. lA), which approximate the extreme widths of the discrete, steady-state zones of “Ampholines” visualized experimentally (17), have been considered. In Fig. 4 the shapes of the transient focusing patterns computed for dimerization (Reaction III) induced by interaction with a sharp distribution of ampholyte (curve a in Fig. 1A) are compared with those for a broad distribution (curve b in Fig. 1A). Values ofK’,(Z) and Co were chosen such that the macromolecule would be only slightly dimerized in the equilibrium focusing pattern. For the sharp distribution of ampholyte the transient patterns (Fig. 4A) show two well-resolved, intense peaks during much of the time course of approach to equilibrium, the bimodality being reminiscent of a mixture of two noninteracting macromolecules. But there the resemblance stops, since one of the peaks grows at the expense of the other as the transient patterns converge upon the symmetrical, unimodal equilibrium pattern which is focused close to the pZ of monomer. When the interaction is with the broad distribution of ampholyte the bimodality of the transient patterns is much less pronounced (Fig. 4B), but the general features of approach to the unimodal equilibrium pattern are similar. Were
42
CANN, STIMPSON, 10
I
A
a
b
I
AND COX
4-
1 IB D
8f; a
1
c
3-
6-
I
0
2-
x
"g 441 2-
l-
a
.'....I
c
L, Ok,
,
b ,
. ,
,
,
0230.6 0.7 a8 0.0 II)
Xext (cm) FIG. 3. Analytical criterion for convergence of transient isoelectric focusing patterns to the equilibrium pattern; -, amplitude vs position of extrema in subfamilies of equilibrium patterns as given by Eqs. la and lb or their analogs in the case of Reaction I. (A) Simple dimerization (Reaction I), Co = 5 X lo-@ M: a, K, = 1X 106M-l; b, 1 X lo5 M-l; c, 1 X 103M-l. 0, amplitude and position of maximum shown by transient isoelectric focusing pattern for corresponding K, at t = 2 x IO5 set (Fig. 2A). (B) Carrier ampholyte-induced dimerization (Reaction III), broad distribution of ampholyte given by curve b in Fig. IA, K'JI) = 1 x 105 M-I: A, amplitude and position of maximum shown by transient isoelectric focusing patterns at t = 2 x lo5 set for Co = 1 x 10mBM (practically the same as r = 1 x 105 set, pattern d in Fig. 4B); 0, extrema in bimodal transient pattern att = 2 x IO5set for Co = 5 x lOA M (practically the same as f = 1 x IO5 set, pattern d in Fig. 5) with path to equilibrium, . . . , shown by curves a, b, and c, where b denotes the minimum; 0, extrema in pattern at t = 2 x lo5 set for Co = 1 x 10m5M(pattern C in Fig. 6).
the interaction to involve the binding of a larger number of ampholyte molecules, resolution of the transient patterns would be enhanced (9,lO). The differences in the focusing behavior between ampholyte-induced and simple dimerization become even more striking when the value of Co is increased, other things held constant, for then the bimodality of the transient patterns shown by the induced reaction is preserved in the equilibrium pattern (Figs. 5 and 6). For all practical purposes, by f = 2 x IO5 set, the transient patterns shown in Figs. 4B, 5, and 6 constitute a subfamily of equilibrium patterns generated by varying Co, as judged by both criteria for convergence, in particular the comparison drawn in Fig. 3B between the amplitudes and positions of their extrema with analytically predicted values. [Small deviations of the points from the curve are to be expected because of the truncation error inherent to the discrete transport calculations (9).] As illustrated in Fig. 6, the bimodal equilibrium patterns shown at the higher Co are reaction zones in which the two peaks correspond to different chemical equilibrium compositions each enriched
ISOELECTRIC
0.2
FOCUSING
0.4
OF INTERACTING
0.6
0.8
1.0
SYSTEMS
43
1.2
x (cm)
FIG. 4. Theoretical transient isoelectric focusing patterns for carrier ampholyte-induced macromolecular dimerization (Reaction III). (A) Sharp distribution of ampholyte given by curve a in Fig. IA. K’3(1) = I x IO5 M-', Co = 5 x IOe6 M: a, 3.75 x IO3set; b, I x IO’sec; c, 3 x IO4set; d, 6 x IO’ set; e, I .8 x lo5 sec. (B) Broad distribution ofampholyte (curve b in Fig. IA), K’,(l) = 1 x IO5 M-', Co = I x 10m6M: a, 3.75 X IO3set; b, 1 X lo* set; c, 2.5 x IO4set; d, I x IO5 set; the amplitude of the maximum in the pattern increased by only 0.03% between 1 x IO5 and 2 x IO5 sec. The breadth of the initial zone in this figure and in Figs. 5-8 is the same as in Fig. 2A.
in the macromolecular species of proximal pZ, and not to separated .& and &A,. If Co were to be increased still further, the peak enriched in .A@& would grow in size until at sufficiently high concentration the equilibrium pattern would show a single peak focused close to the pZ of dimer. Thus, as predicted by Eqs. la and lb, the shapes of the equilibrium patterns for ampholyte-induced and simple dimerization become indistinguishable in the limits of very low or exceedingly high Co. Rather similar results have been obtained for carrier ampholyte-induced dissociation of a subunit macromolecule into its isoelectrically identical half-mers (Reaction IV). Here too the characteristic bimodality of the transient patterns (e.g., Fig. 7) is, under certain conditions, preserved in the equilibrium pattern. (In all cases examined one peak grows at the expense of the other during the approach to equilibrium.) Transient patterns, obtained after 2 x IO5 set of transport, are displayed in Fig. 8 to illustrate how the shape of the pattern changes with increasing K’,(Z) at constant Co. For small Kr4(Z) the pattern (curve a) shows a major peak positioned close to the pZ of the subunit macromolecule and a small one
CANN, STIMPSON,
AND COX
12
10
I Oh
’
I 0.8
1.0
I 1.2
x (cm)
FIG. 5. Transient isoelectric focusing patterns for ampholyte-induced dimerization (Reaction III). Broad distribution of ampholyte (curve b in Fig. IA), K’,(I) = 1 x lo5 M-i, Co=5 x 10-B~:a,3.75 x 103sec;b,l x 104sec;c,2.5 x 104sec;d,l x 105sec;amp1itudesof the maxima in the pattern increased by only -0.03% between 1 x lo5 and 2 x lo5 sec.
x(m)
FIG. 6. Transient isoelectric focusing pattern for ampholyte-induced dimerization (Reaction III) showing distributions of monomer (C,) and dimer (2C,) as well as constituent macromolecule (C): broad distribution of ampholyte (curve b in Fig. IA), K’,(I) = 1 X lo5 M-I, Co = 1 x 1O-5 M, t = 2 x IO5sec. The resolution of earlier patterns (3.75 x 103-2.5 X lo4 set) into two peaks is much more pronounced, comparable to pattern b in Fig. 5. The amplitudes of the maxima in the pattern increased by -2% between 1 x IO5 and 2 x lo5 sec.
ISOELECTRIC
0.2
FOCUSING
0.4
OF INTERACTING
0.6
0.8
1.0
SYSTEMS
45
2
X (cm1
FIG. 7. Transient isoelectric focusing patterns for ampholyte-induced dissociation of a subunit macromolecule into its isoelectrically identical half-mers (Reaction IV). Sharp distribution of ampholyte (curve c in Fig. IA), K’,(I) = 1 x 10m5 M, Co = 1 x 10e5 M: a, 3.75 x lo3 set; b. 1 x lo* set; c, 3 x IO* set; d, 1 x IO5 set; amplitudes of the maxima in the pattern increased by -3.5% between 1 x lOsand 2 x lo5 set (see pattern b in Fig. 8 for 2 x lo5 sec.).
near the pZ of the half-mers; i.e., overall there is very little dissociation throughout the pattern. For K’,(Z) an order of magnitude larger in value the pattern (curve b) consists of two intense peaks; while for two more orders of magnitude increase in Kf4(Z) the pattern (curve c) now shows a major peak near the pZ of half-mers and a small one near the pZ of subunit macromolecule, since overall there is little intact subunit macromolecule left. The amplitudes and positions of the maxima in these patterns are compared graphically with analytically predicted values in Fig. 9. The patterns for the two smaller values of K’,(Z) are practically equilibrium patterns (Fig. 9A), but the pattern for the largest value still has a distance to go (Fig. 9B). Whereas the amplitude and position of its major peak falls close to the analytical curve, the small peak falls in the forbidden region to the left of the curve, and the minimum falls far from the curve. Since the amplitude of the small peak decreased appreciably between 1.8 x lo5 and 2 x lo5 set (with concomitant growth of the major one), it might be surmised that the equilibrium pattern consists of a single peak focused close to the pZ of the half-mers. In any case we note that out of the total of ten sets of transient patterns computed for Reactions I, III, and IV, nine sets virtually converge to their equilibrium pattern as judged by the analytical criterion. Thus, iterative solution of the simultaneous transport equations and mass action expression is one way of solving the nonlinear
46
CANN, STIMPSON,
AND COX
I
6-
a
2 x (cm)
FIG. 8. Transient isoelectric focusing patterns for ampholyte-induced dissociation of a subunit macromolecule into its isoelectrically identical half-mers (Reaction IV). Sharp distribution of ampholyte (curve c in Fig. 1A). Co = 1 x 1O-5 M, t = 2 x IO5 set: a, K’,(f) = 1 X lo-* M; b, K’#) = I X lO-5 M; c, K’,(I) = I X 1O-3 hf.
integral equation describing the equilibrium distribution of constituent macromolecule along the isoelectric focusing column. Although the calculations presented here are for a broad initial zone of macromolecule, the transient focusing behavior described for the ampholyte-induced interactions also applies, with only two exceptions, to other initial conditions in which a narrow zone is inserted into the pH gradient at some arbitrary position. The exceptions arise when the equilibrium pattern shows a single peak focused close to the pZ of one of the macromolecular species. In that event, the transient patterns will show a single peak when (i) the peak in the equilibrium pattern is close to the pZ of the more acidic species and the initial zone is positioned anodically thereto,or (ii) the peak is close to the pZ of the less acidic species and the initial zone is cathodic thereto. Bimodal transient patterns can be expected for other conditions. [See previous calculations (9,lO) for peculiar transient focusing behavior alluded to in Footnote 5, which may obtain in the case of ampholyte-induced isomerization and pH-dependent conformational transitions.] DISCUSSION
The new insights provided by these calculations have important implications for conventional analytical and preparative applications of isoelectric focusing. Thus, equilibrium as well as transient focusing patterns given by carrier ampholyte-induced association or dissociation can show two peaks similar to the patterns for a mixture of two
ISOELECTRIC
FOCUSING
).5 0.6 0.7 Cl8 0.9
OF INTERACTING
1.0
0.5 Q6 a7
SYSTEMS
0.8
0.9
47
1.0
FIG. 9. Analytical criterion for convergence of transient isoelectric focusing patterns to the equilibrium pattern for ampholyte-induced dissociation of a subunit macromolecule into its isoelectrically identical half-mers (Reaction IV): -, ratio of amplitude to normalized value of the apparent equilibrium constant vs position of extrema in equilibrium patterns; sharp distribution ofampholyte (curve c in Fig. IA); analytical expressions analogous to Eqs. la and lb. (A) A, transient focusing pattern at 2 x IO5set for Co = I x 10e5 M and K’,(I) = 1 x 10e6 M (pattern a in Fig. 8); 0, K’XI) = I x 10e5M (pattern b in Fig. 8). (B) 0, K’,(I) = 1 x 10e3 M (pattern c in Fig. 8).
noninteracting macromolecules, and the same can be said for ampholyteinduced isomerization (9) and pH-dependent conformational transitions (10). Moreover, a molecule undergoing sequential conformational transitions can give multiple peaks. In practice such patterns could easily be misinterpreted in terms of inherent heterogeneity with respect to isoelectric point, but with due precautions fractionation provides an unambiguous method for distinguishing between interactions and heterogeneity. In the fractionation test the protein in each of the peaks is isolated. The resulting fractions are reconstituted to the concentration of the unfractionated material, as dictated by the concentration dependence of the patterns for association-dissociation reactions, and resubjected to isoelectric focusing taking care to position the initial zone as in the original separation. For interaction each fraction will behave like the unfractionated material and show two or more peaks, while for heterogeneity a single peak focused at a reproducible isoelectric point will be obtained. Rigorous completion of the analysis requires that the unfractionated material and its fractions be focused from initial zones located at different positions along the isoelectric focusing column.
48
CANN, STIMPSON,
AND COX
The fractionation test has been recommended previously by Ressler (18) and Wrigley (19) for distinguishing between artifactual and real multipeaked patterns; our calculations provide a theoretical rationale for the test. The investigation of Talbot (20) on the pH-dependent conformational transition of the 12s subunit protein of the capsid of foot-and-mouth disease virus is exemplary in the way fractionation experiments were combined with focusing procedural variations and ultracentrifugation to establish the interaction. It is characteristic of interacting systems that during the approach to equilibrium one of the peaks in the transient focusing pattern grows at the expense of the other, which under certain conditions may disappear completely (Fig. 4A). This property may well prove to be diagnostic for interactions. Examples are afforded by a-glycerol phosphate dehydrogenase and pyruvate kinase (21). When focused from an initially uniform distribution throughout a density-gradient isoelectric focusing column, both give transient patterns showing two peaks migrating from either end of the column. As the two peaks converge upon the isoelectric region they either coalesce or, possibly, one of the peaks grows at the expense of the other as judged from the few transient patterns presented. Clearly, these results are suggestive of an interaction, and it may be pertinent that both enzymes have a subunit structure. It must be emphasized, however, that several classes of interaction exhibit certain common features on isoelectric focusing. Accordingly, elucidation of the mechanism of interaction requires the combined application of several physicochemical methods such as ultracentrifugation and circular dichroism. Finally, the results of our calculations add to the store of fundamental information required for application of isoelectric focusing to the detection and eventual characterization of interacting systems. For example, it is anticipated that the results reported herein may be useful in the application of the method to the study of subunit enzymes, enzyme complexes, and other associating-dissociating systems. Note added in proof. Subsequent to submission of our manuscript M. Dishon and G. H. Weiss (1977, Anal. Biochem. 81, 1) reported a theoretical study of isoelectric focusing in a nonlinear pH gradient. Their calculations, which are for a broad initial zone of noninteracting macromolecule, predict two transient peaks arising at the ends of the initial concentration profile, migrating towards each other as time goes on and finally merging at the PI. It should be possible to distinguish this effect of a nonlinear pH gradient from macromolecular interactions by comparison of transient and equilibrium patterns given by a broad initial zone with those from a narrow zone located at different positions along the isoelectric focusing column.
REFERENCES I. Svensson, H. (1961) Acra Chem. Stand. 15, 325. 2. Svensson, H. (1962) Acta Chem. Stand. 16, 456. 3. Vesterberg, 0.. and Svensson, H. (1966) Acta Chem.
Stand.
20, 820.
ISOELECTRIC 4. 5. 6. 7. 8. 9. 10. 1 I. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
FOCUSING
OF INTERACTING
SYSTEMS
49
Vesterberg, 0. (1969) Acra Chem. Stand. 23, 2653. Haglund, H. (1971) Methods Biochem. Anal. 19, I. Catsimpoolas, N. (ed.) (1973) Ann. N. Y. Acad. Sci. 209, l-529. Righetti. P. G., and Drysdale, J. W. (1974) J. Chromatogr. 98, 271. Catsimpoolas, N. (ed.) (1976) Isoelectric Focusing, Academic Press, New York. Cann, J. R.. and Stimpson. D. I. (1977) Biophys. Chem., 7, 103. Stimpson, D. I.. and Cann. J. R. (1977) Biophys. Chem.. 7, 115. Tombs, M. P. (1957) Biochem. J. 67, 517. Brown, R. A., and Timasheff, S. N. (1959) in Electrophoresis Theory, Methods and Practice (Bier, M., ed.), Chap. 8, Academic Press, New York. Timasheff, S. N., and Townend, R. (1960) J. Amer. Chem. Sot. 82, 3157. Johnson, P.. and Shooter, E. M. (1950) Biochem. Biophys. Acta 5, 361. Johnson, P., Shooter, E. M., and Rideal, E. K. (1950) Biochem. Biophys. Acfa 5, 376. Benveniste, K. B., and Munkres, K. D. (1975) in Isozymes I. Molecular Structure (Marker?. C. L.. ed.). p. 561, Academic Press, New York. Felgenhauer, K., and Pak, S. J. (1973) Ann. N. Y. Acad. Sci. 209, 147. Ressler, N. (1973) Anal. Biochem. 51, 604. Wrigley, C. W. (1976)in Isoelectric Focusing(Catsimpoolas, N.. ed.), Chap. 4, Academic Press, New York. Talbot, P. (1975) in Isoelectric Focusing (Arbuthnott, J. P.. Beeley, J. A., eds.), p. 270, Butterworths, London. Behnke, J. N., Dagher. S. M., Massey, T. H., and Deal, W. C., Jr. (1975)AnaL Biochem. 69, 1.
