~~~E~~ OF
C~IARGEON POLYMERIZING PRQTEIN
342
~ r ~ ~ p u tof( ~fhc~ .D,, ~ ~ p3o r n~ Values of I A , SA,p , and ~ t r a ~ g ~ i t algebraic ~ o r ~ ~manipulations ~ ~ r ~ ~ of eq 3fi,57, %, a i d 64 of ref 8 along with the definition of ,/-/-'b g-/.at a + give the following results
g+. I
p
f-3
arge in the Sedimentation Equilibrium of rotein Systems
6. J. Howlett, P. D. Jeffrey, and L. W. Nichol" Department of Physical Biochemistry, John Curtin School of Medical Research, Australian National Undaersity, Canberm, A.C.T., $601,Australia (Received M a y 8, fB7B)
Equations are developed which describe the distribution at sedimentation equilibrium of a chemically reacting syeteni involving polymeric species bearing net charges. The development is in terms of species defined as electrically neutral in the manner proposed by Casassa and Eisenberg. The differential equations cannot be directly integrated, since the molecular weights and partial specific volumes of the redefined species (as well as the solution density) are functions of total concentration. Expressions for these dependencies are developed from equilibrium dialysis considerations and are used in a numerical integration procedure employing the predictor -corrector method. The results show that when charge is conserved on polymerization, analysis of the distribution for both ideal and nonideal systems is possible in terms of equations derived for nonelectrolytes and, moreover, that nonsuperposition effects in plots of apparent weight-average molecular weight vs. total concentration cannot arise as a sole consequence of charge effects. The possibility that a volume change occurs on reaction, leading to nonsuperposition, is also considered. Brief comment is made on the sedimentation equilibrium of systems where charge is not conserved on polymerization and it is shown, for selected models, that the variation of the apparent equilibrium constant with radial distance is small.
Introd~ct~on The elucidation of polymerizing protein systems often involves studies performed at pH values away from the isodectric point and thus, in relation to sedimentation e ~ ~ ~ l ~ b experiments, rium involves the interpretation of results pertaining to macromolecular species bearing net charges. Frequently, the results are interpreted on thc hasis of equations formally derived for nonelectrolytes, Adam$ has provided some justification for this approach by examining the problem in t e r m of the dcfinition of an electrically neutral macromolecular component proposed by Casassa and Eisenberg.'L-6 Atlsms3 considered the possibility that the activity cocflicients of the monomeric and polymeric species were functions of total concentration; but neglected thc concentration dependence of the solution density arid of the molecular weights and partial specific ~ d u r n e sof the redefined species. Moreover, cases xvxe not considered where a volume change
accompanied polymerization7 or where charge was The purpose of not conserved on this work is to examine these poi& arid by numerical integration procedures to estimate their effect on sedimentation equilibrium distributions,
Theory Consider a three component syst'ern comprising water (component 1) a polyrnerizing an.d ionizing (1) E. T. Adams, Jr., and D. L. Filmer, Biochemistry, 5,2971 (1966). (2) G. J. Howlett, P. D. Jeffrey, and L. W. Nicbol, J . Phgs. Chem., 76, 777 (1972). (3) E . T. Adams, Jr., Biochemistry, 4, 1646 (1865). (4) E. F. Casassa and 13. Eisenberg, J . Phys. Chem., 64,753 (1960). (5) E. F. Casassa and H. Eisenberg, ihid., 65,427 (1961). (6) E. F. Casassa and H. Eisenberg, Advan. Protein Chem., 19, 287 (1964). (7) G. J. Howlett, P. D. Jeffrey, and L. W, Il'ichol, 9.Phys. Chem., 74,3607 (1970). (8) A. J. Sophianopoulos and K. E. Van Holde, J-. B i d . Chsm., 239, 2516 (1964). The Journal of Physical Chemistry, V d . 76, A'o. 23, 1972
G. J. HOWLETT, P. D. JEFFREY, AND L. W. NICHOL
3430 macromolecule (component 2), and a supporting uniunivalent electrolyte, BX (component 3). The polyC, where merization rextion is represented by nA n (>1) is the degree of polymerization. Initially, cases are considered where charge is conserved on polymerization, viz., p = np where p and 4 are the net charges borne by A and C, respectively. The activity of the monomeric species, U A , may be expressed as9
*
In
u.4 =
+
In (TVA/MA) In Y A VZB
where r is the radial distance, w the angular velocity, and BA* the partial weight molal volume. An expression for the right-hand side of eq 6 is available as follows. First, eq 3 is rewritten on a weight molal scale for species A such that a A * = YA*WA*, where ?A* is the activity coefficient on the same scale. Second, In Y A * is written as a power series in Wz* (=WA* WC*)and truncated after the first term, so that
+
+
yA* =
+
In (WBIMB) vzxIn ( W X I M X ) ( 1 )
where M denotes the molecular weight of a particular species, W the weight of the species per kilogram of solvent, and Y A is related to the mean ionic activity Coefficient on the same scale.g The ~ 2 ~(j' s= B or X) are the numbcrs of moles of diffusible species that are included pcr nole of monomer in the formulation of component 2 as an electrically neutral component. The choice of values of vZJ made in this work is that suggested by Casassa and Eisenberg,*--"who visualized an equilibrium dialysis experiment involving the three components and showed that equating the inner and outer salt concentradons led to the effcctive eliminalion of the folloning Fartial derivatives
(7)
eBA*MA*Wv,*
where BA* is a constant expressing nonideality. Third, the resultant expression for p A * is partially diff erentiated with respect to WA* and (separately) WC*. It follows that
where
LA* = (1 -
z?A*p)W2/2R5!'
(8b)
An entirely analogous procedure (defining YC* = eBc*Mc*Wz*) may be used to obtain an expression for Lc*Mc*, which may be added to eq 8a to give
d M A* Wz* (BA*WA* nBc*Wc*) (9) d(r2)
+
All quantitiles consistent with the definition are marked by asterisks. T h e chemical potential of component 2 per mole, pz* may be written as
where pzo* is the reference chemical potential. It follows from eq 2 and 3 that p2* is independent of the weight molality of component 3 and that a t equilibrium
The condition Mc* = nMA*, used in the formulation of eq 9, applies only for the cases under discussion where charge is conserved on polymerization. Rearrangement of eq 9 by division throughout by Wz*yields
+
LA*MA*WA* Lc*Mc*Wc* W z* (BA* WA* nBc*Wc*) dd4A */d (r2) d In Wz* = Wz*(l BA*MA*WA* Bc *Mc*Wc*) d(r2)
+
+
+
(10) Thus, at constant temperature, PA* is a function only of WA*,Wc*,and pressure, P .
)
wA*,wC*
dP dr
_ I
I n what follows it is assumed that BA* = Bc* = B*. Braswelll' has provided some justification for this assumption on the basis of the Debye-Huckel theory when charge is conserved. The molal volume change for the reaction may be defined as AV*
(5)
Substitution ir. eq 5 of dP/dr = r d p , ( d p ~ * / b P )= Z?A*MA* and the sedimentation equilibrium conditionlo ( d p ~ * / b ~ = ) , rclPM~*yields
=
Mc*fic* - nMA*BA*
(11)
and an apparent weight-average molecular weight as2
Mw*(r)sPP =
--
LA* d(y2)
~
Combination of eq 10, 11, and 12 gives (9) G. Scatchard, J.Amer. Chem. SOC.,68,2315 (1946). (10) H. Fujita in "Mathematical Theory of Sedimentation Analysis," Academic Press, New York, N. Y., 1962. (11) E. Braswell, J.Phys. Chem., 72,2477 (1968).
I'he Journal of Ph.wieaE Chemistr$/, Vol. 7 6 , A'o. 29, 1972
EFFECT OF C!aaan~ON YQLYMERIZING PROTEIN
3431
The last term in the numerator of eq 13 arises by virtue of the definitions of the activity coefficients in terms of M** and & I C * ? WhiCkl will e shown later to be funco n hence of r. Omission tions of total c o n c ~ n t r ~ t ~and of this term, which irj negligible in practice, leads to the formal. i d e n ~ ~ ~ fofi (eq~ ~13~with ~ ~ neq 16 of Howlett, et aL,7 which was deiived for a nonelectrolyte on the basis of a weight per unit volume scale. Moreover, the treatmcnC 1s then consistent with eq 36d of Ada,rns3 for cases where RV" = 10. Equation 83 w-hoclies a description of the nonsuperposition effect which may be seen as follows. An equilibrium constant on the neight molal scale may be defined as
X*
~
~
~
~
~
~
E *
W/ C (* / ( ~W AA* ) ~(14) ~ ~
sjn,, e**Mc*W,'/~R~?lM,*W,* = 1 \when Mc* nMA*, Let the w e i ~ h ~ - ~ ofr the a ~ redefined ~ ~ o ~ ~polymeric species be c ~ * == Wc*'/W2*. It follows that, Wn* = (1 a*)'CV2" and fmm eg 13
proves to be a function of total ~ o ~ ~ , e ~ ~ thrreby ~,rati~~~, preventing direct integration. Without loss of generality, the s i t ~ ~ ~ 15 a tconsid~o~ ered where the monomer bears R net charge of -+p and the polymer of -kq: the gegenions arrd X are taken as positive and negative, respectively. ~ ~ u ~ ~ dialysis of this system results in the ~~~~~~t~~~ of the diffusible ions, B and X, being equal 012 either bide of the membrane and this together with t,hci ~ ~ ~ ~ ~ of electroneutrality yields
+
m ~ ( m ~ pmcq mx(mx
- mAp
+ m ~ ) mz2 =
-
"CQ)
=
~3~
(1 8 )
(19)
where the symbol m denotes rnoldjlty znd I he subscript 3 refers to the solution outside &hemeribrane, It is implied in eq 18 and 19 that the rwan ionic ~ ~ t coefficients of the electrolyte inside and outside the membrane are equal (the ideal Donnan e ~ ~ ~ l ~ ~ r Equation 18 may be solved for mB and the square root of the discriminant expanded to 1 ield as ti fir4 approximation ~
A
"
~
~
I
(p/(I
**I"
x*
w
2
(20) or
*1?&--I
(15) Equation 13 niay als2 be written in terms of a* to give a!f,*(?"fnpp =: RIA" -4
[MC"
-- IlB'h" AV*p/(l - fla*p)]a*
(16)
uherc it has been assumed, for simplicity, that B* = 0. 'The effect of nonsuperposition occurs when different .~~, at the sanae value os Wz*, values of n / J ~ ~ ( r ) pertain, when drrived from experiments conducted with various values of initial. concentration and/or angular velocity. Provided M A * , Mc.C,gA*, and p are pressure independent, it folloT1.s from eq 15 and 16 that the effect will not arise if X * ( T ) 1s identical a t the same Wz* in the different exprnments. However, when AV* f 0 the values of X *(I-) (and hence a*(?))will not be identical when coimpan'ison:; are made a t the same Wz*, since (b Irt X*/dP) = -- AV*/RT, arid nonsuperposition wilP be observeid. Fsorn eq 11, the statement AV* f 0 implies that z?A+ ~ z f . bc' for the case of charge conservation (Mc* = n M A ' 9 . It is possible to compute numerical examples on the basis of eq 8, which with B* = 0 may be uritten for both species as
-
-m3
mAp
+ rncq mB
_-
=
Equation 21 corresponds to the definition of the membrane distribution parameter as used by Eisenberg and Casassa12 in their treatment of a single ionizing component. The number oi' moles of B t>obe included per mole of A i s termed vA,B* and in accordance with the chosen definition is VA,U*
I -
--(ma - m d p mAp
4- mcq
-
Similarly
It is noted that V C , B * / V A , B * = vc,x*/vA,x* = g / p . From these relations it is clear thak the v i , J * depends on the composition of the mixture which in the case of a The application of eq 17 in an integration procedure requires ccnsideration of the product, Li*Mi*, which
(12) €1. Eisenberg and E. F. Casassa, J . Polymer Sci,, 47, 29 (1960). The Jotirnal of Physical Chemistry, Vot. 75, Xo. 23, 197d
G. J. HOWLETT, P. D. JEFFREY, AND 1,.
3432 polymerizing system varies with total concentration. It i s however possible to calculate M A * and Mc* a t particular concentrations using eq 22-25 and
MA
MaF
+ VA,B*MB+ V A , X * M X
+ VC,B*MR+ vc,x*Mx
Mc* ==
(26) (27)
It could also be nlottd that eq 22-27 are consistent with the requirement that Mc* = % M A * when 9 = np. I n referring to sediutilizing eq 22-25 t o calculate the mentation ectrjlibriurn, it has been assumed that the equilibrium dialysis condition pertains even after solute redistribution , an nssumption assessed by Casassa and Eisenberg4 8s reasonable for experiments of usual design where the protein and other solute gradients are not exc~ssive It remains to discuss L,* or more explicitly the term (1 - 0,"~) The appropriate apparent specific volume is given by!' 4"
=
+ W/(l + 0
(284
(@
wherc t$ is the apparent specific volume determined by density rneasixreaients on undialyzed solutions, 03 i s the partial specific vohme of the salt in thc three component system, and
vZJ* has already beexi defined and v2J has the same significance a~ eq 1 but with a value chosen appropriate to the dcterminaticn of 4~ The apparent specific volumes, 4 and $*, are neight molal quantities if weight molal concentrations are used to evaluate them. Although B* = c$* 4 W2*(d+*/dW2*),it is assumed in what follows that tne concentration-dependence term is zcro and tFat t h Same relations apply to A and C, vparately . 'rhus
5," = (ai, -I- &i1)/(1
+ &) (i
=
A or C)
(29)
nhrre 4, is alm regarded as concentration independent. The mlution density in the (1 -- .j,*p) term should be evaluated at the isarne concentration as o,* and M,*. This is achieved by using the following relation
where p3 is the density of a solution of weight-molal concentration W3. The combined use of eq 22-27 and 28b -30 permits the calculation of L,*M,* a t a particular concentration for use m the numerical integration of eq 17. Attentiop is n o v directed to situations where charge is not eonscwrd an pol2merization, q Z np and Mc* # ~ L M A " .Equations 22-27 and 28b-30 apply and may be used to calculate L,*M,* at a series of total concentrations, which suggests that eq 17, based on eq 4, may again bc rmphyed in a numerical integration aimed at simulating an approximate sedimentation The Journal of PFysical Chemistry, Vol 76, KO
M ,lQrZ
equilibrium distribution. Since the prime purpose of this work is to examine the effects of charge, it is desirable to eliminate the possible additional effect of an intrinsic volume change on reaction, which has been considered in detail previously.*J Thus, in the following computations 4~ is set equal to for the case of charge conservation, to 22-25, 28b, and 29) and AB* = (eq 11). In contrast, when charge is not conserved, BA* f ire", even when +A = +e, because of the difference in i~ and (c.
Results and Discussion The numerical integration of eq 17 was performed using the predictor-corrector methodI3 in the following way. Consider first the simulation of an experiment conducted a t selected values of w and T with a system for which values are assigned to p , q, MA,M C (and hence r ~ ) ,M B , M x , Wa, i%, 4~ +e, W2*(rm), and X*(rm),rm referring to the meniscus. The latter two quantities define WA*(T,)and Wc*(r,), which may be ) mc(r,) by used to obtain first) estimates of ~ A ( T ~and respective division by and N c . Equations 22-27 were used to obtain csrresponding first estimates of v i , ) * ( r m ) and M,*(rm), the latter being employed as before, to obtain second estimates of the m1(rm)* The process was repeated until values of mL(rm)converged. The final values of v ~ , mere ~ * employed in eq 28b-30 to obtain the value of (1 - % * p ) at the meniscus. For this purpose, V A , X was equated with p (with V A , B = 0) and, similarly, YC,X was set equal to q (with YC,B = 0). This set of calculations provides a value of Ll'Mi* a t the meniscus, which may be used as the starting point in thc application of the predktor-corrector method in integrating eq 17. The complete distribution was generated by successively refining values of Mi*, Wi*, B,*, and p a t each increment of r 2 examine then possible utilizing eq 14 to find X* as a function of r . Column 1 of Table I shows the results of such ealculations performed for an example where y = np and &IC* = MA* (charge conserved), the values of the parameters being selected to resemble those describing the dimerizing lysozyme It is clear that X * is a constant, which is consistent with eq 17 written in the form d ln~Wc*/(Ct.'~*)"]/d(~~) = Lc*Mc* -nLa*Ma*
=
0 (31)
As noted previously, the constancy of X * ( r ) directly ) ~ ~ ~by, eq 12, must implies that values of M ~ * ( T defined be identical in different experiments when comparisons are made at the same value of Ws*. Thus, the effect of nonsuperposition cannot arise in the sedimentation (13) D. D. McCraken and W. S. Dorn in "Numerical Methods and Fortran Programming," Wiiey, New York, N. Y., 1984, p 330. (14) R. C. Deonier and J. W. Williams, Biochemistry, 9,4260 (1970).
EFFECTOF ~CT-IARCE ON POLYMERIZING PROTEIN
3433
extreme model is considered, involving the loss of all eqtii'iibriam of ideal systems as a consequence of charge charges un dimerization (column 31, the vuriatiov~in r4lect s tvhen charge- is conserved during the reaction :mi! the experiruaenf e, are conducted according to the X * ( T ) , d d e larger, i s only -10jo.15 'The svstem ~ i t h sy = 0 'ovas also used to calculate a ~ i s ~ l ~t h ~ ~ design suggested by Casassa and Eisenberg6 The efect aylny, OS C O I I ~ S C , arise when charge is conserved, W2*(r,) increased from 4.5 (Table Y) Lo 8.0, thereby simulating an experiment with a differed initial if diA Z $c('in8 f 3(-', AV* f Q ) . 2 , 3 AlthoughX*(r) is a constant in c ~ d i i r n r i1 OF Table I, the values of 7jA* = loading concentration. Equation 12 W B S trwd to dCc* = n M a", and p, nevertheless, varied with Wz* obtain Mw;"(rIapa values from the ti+s ~ i r d a i e d distributions, which ~ ~ e r n ~ ~c tot~etdp ~ ~ r i s113 o be n ~ made (and hence T ) in accordance with eq 22-30. The varialiens were' elight, &)eingO.OOITo for g ~ * ,Q.O030j, for at the same values of Wz*. The eifeet GI ~ ~ ~ n s ~ ~ p e ~ I W ~ and * ~ 0.13% for p, over the range of r reported in position was observed, but the D ~ ~ E X Y I I discrepancy LIU~ was only 02Voa ' r ~ b i i. c 'This ~ ~ ~ s supports e ~ the v neglect' a ~ ~of the ~ ~ term in eq 13 innvolv rig dMA*/d(r*), and indicates that I n several respects the findings in t h i s study are rein rasps where ,El* is also small that nonsuperposition assuring to the worker involved in the ~ ~ t e r ~ ~ofe t ~ ~ ~ o cffetts will bc ~ i e ~ as~ a ~consequence ~ i ~ ~ ofe charge s e d ~ ~ n ~e ~q ~~ ~ ~~results ~~ o ob~ ~r~ i ~R~ith~ ionizing ~ ~~ i ~ ~ d even when nonidesl systems are exanlined jg = np). c r o~~ i~o lje cnsolutes. ~ ~ ~ ~ , ~ In cases and p o ~ ~ ~ n ei a ~ where charge ix conserved on ~ ~ l it is ~ ~ reasonable t o utilize equations der.i~-e$for nonelectrolytes in the analysis of sediment ation ~ ~ ~ ~ l ~ b ~~b~~ I: Vrriatioa of ihe Apparent Weight Molal Equiiibiium Constant, X",ab a Function of Radial Distance, distributions. 'This follows from close similarity of T , in Sedimentation E q d i b r i u m Experiments" eq 13 with analogous expres;siir for the apparent weight-nveragc rnoleeulay weight, prwisusly d ~ r i v e d . ~ J Moreover, provided the nonitienlrl y coeflicient, i s nob large and charge is conserred,ala:$ observed non6.7000 0 ~03000 0.03000 0.03000 in plots of M W superposition 6.7223 0.03001 0.03003 0 83000 attributed not to a charge e 6.7816 0.03011 0 03000 0.03002 6.8403 0 03000 0.03003 0.03020 trinsic volume change on rea 6.8986 0 03000 0 03004 0.03028 more of t l i ~variety of other causei, ptevliously cited.2 0 03000 0.03036 6.9663 0.03006 The discussion of lack of charge conservation has been 0.03041 0,03007 6.9849 0 03000 restricted do ideal syst>ems for ~irnpliciby. Never0.03045 7.0136 0 03000 0.03008 theless, w e n when &* = BA* = 0,X" m s fourid to a All caiculations were performed employing the following vary wit11 T (Table I). The irnpoit:znt point, homver, values cf the required l~aramet~ers: MA = 14,400; n = 2; M B Is that the related nonsuperposilion cff ect observed with = 23; M x 3 5 " s ; d~ +C 0.726; E3 = 0.46; Wa = 0.15; the models selected is very xmdl It sce~nsto us Wz*(r,) = 4.5: X*(r,,) = 0.03: P
