CHARLES TANFORD
1628
sample diluted with water t o 25 nil. in a volumetric flask. Ornithine was determined by the method of Chinard.'6 Thiosulfate and glycine were found not t o interfere with the analysis. Values of 2.9, 3.1 and 2.9 moles ornithine/ mole ferrichrome were found in triplicate analyses. Analysis for L-Ornithine.-The assay was conducted with E . coli mutant 160-37 which was kindly supplied by H. J. Vogel.0 Glycine and D-ornithine were found not t o interfere. Triplicate analyses of the H I hydrolysate described above gave 2.8, 3.2 and 2.8 moles L-ornithine/mole ferrichrome. Syntheses of 1.-The preparation of the w-N-hydroxy derivatives of DL-ornithine and DL-lysine will be the subject of a succeeding communication.11 The following procedure relates to the preparation of DL-compound I in sufficient quantities for comparison with the natural product. A solution containing 500 mg. (2.26 mmoles) of DL-5-7bromopropylhydantoin,'O 250 mg. (3.62 mmoles) of NaNOl and 250 mg. of urea in 5 ml. of dimethylformamide was agitated with a magnetic stirrer for 2 hr. at room temperature as described by Kornblum, et al.'I The solvent mas then removed and the residue extracted with acetone. The acetone extracts were filtered t o yield a clear yellow solution. The acetone was evaporated and the residue dissolved in 10 ml. of ethanol. After addition of 3 ml. of water and 500 mg. of NHrC1, the solution was heated t o near boiling and 500 mg. of zinc dust added in portions over a period of several minutes. The solution was then filtered, acidified with dilute HC1 and evaporated t o dryness. The residue was dissolved in 2-3 ml. of water and applied t o a 1 X 5 cm. column of Dowex 50H. The column was washed with water and the tetrazolium positive material eluted with 0.2 M XHpOH. The tetrazolium-positive fractions were pooled, acidified with dilute HCl and evaporated t o dryness. The residue was dissolved in 1 ml. of G N HCl and heated in an autoclave at 135' for 4 hr. After removal of excess HC1, this preparation on electrophoretic analysis (see above) showed the presence of approximately equal portions of ninhydrin positive materials with the mobility chsracteristics of neutral amino acid(s), compound I and ornithine. The entire preparation was placed on a 1.5 X 50 cm. (16) F. P. Chinard, J . Biol. Chem., 199, 91 (1952). (17) N. Kornblum, H. 0. Larson, R. K. Blackwood, D. D. Mocberry, E. P. Oliveto and G. E. Graham, J. A m . Chem. Sac., 78, 1494 (1956).
[CONTRIBUTION FROM
THE
Vol. 83
column of 200-400 mesh Dowex 50.H X 4 equilibrated with 2.5 N HC1. The column was operated at a flow rate of one fraction (4 ml.) per 12 miti. with 2.5 N HCl as developing solvent. The neutral amino acid(s) as well as an unidentified tetrazolium-reducing peak emerged within the first 50 fractions. Fractions 50-60 gave a positive spot test on paper with tetrazolium. These fractions were pooled, evaporated t o dryness and the total residue applied in a narrow band across the entire 15 cm. width of a sheet of Whatman 3 MM filter paper. A spot of authentic ornithine was placed on the origin at one side of the paper. The buffer and operating conditions were the same as those used ahove. After approximately 1 hr. the paper was air dried and a strip cut off along the edge bearing the authentic ornithine spot. The ninhydrin spray revealed the presence of only two components, the authentic ornithine (15.7 cm.) arid compound I (11.1 cm.). h second strip was sprayed with ninhydrin and this showed only a band at 11.1 cm. Finally, a third strip sprayed with tetrazolium reagent gave one band at the 11.1 cm. position. The 11.1 cm. band from the remainder of the paper was eluted and reduced with HI as was done with the natural product (see above). Paper electrophoretic analysis of the reduced preparation revealed the presence of ornithitie as the sole amino acid. Synthetic and natural I were indistinguishable by ionic mobility on paper and by their behavior toward the ninhydrin and tetrazolium reagents. In the experiment described above, the ionic mobility ratio of synthetic I/ornithine was 11.1/15.7 = 0.708. I n a separate but sinilar analysis the corresponding ratio for natural I/ornithine was 10.1/14.3 = 0.m.
Elementary Analyses. (a) Ferrichrome.-Calcd. for CnHa&OlzFe: C, 43.79; H, 5.72; Tu', 17.02; Fe, 7.51. Found: C,41.02; H,5.90; N, 16.55; Fe, 7.54. (b) Ferrichrome A.-Calcd. for C41Hb8NB02DFe.4Hz0: C, 43.78; H, 5.91; N, 11.21. Found: C, 43.62; H , 5.80; N, 11.18. A thrice recrystallized sample was dried a t 100" under reduced pressure over P2Oa for 24 hr. The specimen was then allowed to reach constant weight under atmospheric conditions. The dry sample, 26.5 mq. (25.17 Wmoles) reached constant weight at 28.3 mg. The water absorbed, 1.8 mg. (100 pmoles) was 99.3y0. Anal. Calcd. for CIIHbsNoOIoFe: Fe, 5.3. Found: Fe, 5.3.
DEPARTMENT O F CHEMISTRY, STATE UNIVERSITY O F IOWA,IOWA CITY, IOTVA]
Ionization-linked Changes in Protein Conformation. I. Theory BY CHARLES TANFORD* RECEIVEDAUGUST15, 1960 If a protein molecule can exist in two conformations, CY and p , and if the equilibrium constant @)/(CY) depends on pH, then a necessary consequence of the laws of thermodynamics is that at least one titratable group has a different pK in the two conformations. The objective of this paper is t o give exact equations for the relation between the pH-dependence of ( p ) / (CY)and the course of titration of the anomalous titratable groups, in a form suitable primarily in situations where electrostatic interaction does not play a n important role.
If a protein molecule in solution can exist in two different conformations, CY and p, and if the equilibrium distribution between these depends on PH, then the titration curves characteristic of the two conformations are necessarily different. This propa t PHI difosition is readily proved. If @)/(cy) fers from the same ratio a t pH2, then P(S,pH11 - F0 (CY,PHI) Z (D, PHz) - Fo (CY, PHz) where Fo is the free energy in a suitable standard state. It follows a t once that F0 (D, pH2) - P '3, pH11 f F ' (CY, PHz) - F0 (CY, PHI) which is equivalent to saying that the titration
*
Department of Birochemistry, Duke University, Durham,
N.C.
curve of p between pH1 and PH2 differs from the titration curve of CY between the same limits. The purpose of this paper is to derive the equations which connect these two observable events: the effect of pH on conformation and the accompanying changes in titration parameters. The titration curve depends on two properties of the protein molecule's side chains,'-* their (1) K. L i n d e r s t r g m - h o g , C o m p f . rend. frau. lab. Carlsbcrg, 16, No. 7 (1924). (2) G. Scatchard, A n n . N. Y.A c Q ~Sci.. . 61, 660 (1949). (3) C. Tanford, "Electrochemistry in Biology and Medicine," T. Shedlovsky, editor, John Wiley and Sons, Inc., New York, N. Y.,
1955. (4) C. Tanford and J. G. Kirkwood, J . A m . Chcm. Sac., 79, 5333
(1957).
PROTEIN CONFORMATION THEORY
April 5, 1961
intrinsic free energies of ionization (;.e., the free energies of ionization appropriate to a hypothetical discharged protein molecule) and the electrostatic interaction between them. This paper will consider primarily those situations in which the intrinsic free energy of one or more side chains differs in two conformations, but electrostatic interactions between charged side chains remain the same. The results will thus be most important for situations involving “buried” acidic or basic groups or pK anomalies which arise from hydrogen bonding, intramolecular ion-pair formation, etc. They are in principle applicable to situations in which reduced electrostatic interaction is the driving force (e.g., molecular unfolding a t extreme PH’s), but the results will not be in convenient form for application to that situation. Conformation Changes Associated with a Single Titratable Group.-We suppose that a protein molecule can exist in two conformations a and p and that the transition between them is associated with the state of a single acidic group AH,S such that a is more stable when the group is in its acidic form and P is more stable when i t is in its basic form. Where (aAH) and (PAH) represent the concentrations of molecules with undissociated AH groups in the two conformations and (EA) and (PA) the corresponding concentrations of molecules with dissociated A groups, we may write the following equilibrium constants for the transition a -+ p.
We also have the expressions for the acid dissociation constants ( K ) of group AH in the two configurations K=
(aA)(H+)/(aAH)
Kg
= (PA)(H+)/(BAH)
(2)
where (H f) will ordinarily represent hydrogen ion activity. By considering the two paths along which one can proceed from aAH to PA, one obtains two expressions for (PA)(H+)/(aAH), from which i t follows a t once that Kg/Ka = kl/ko (3 1 Le., if p is stabilized by dissociation of AH, then DAH is necessarily more acidic than aAH. The assumption that AH is the only group affecting the transition is by equation 3 equivalent to saying that the dissociation constants of other groups are the same in a and P. It also implies, of course, that there is no specific interaction between AH and other titratable groups. Nonspecific electrostatic interaction may be incorporated into the model as shown below. Two parameters which are subject to experimental measurement are of interest: the extent of dissociation of AH, which we shall call X A , ;.e.
and the extent of conversion of a to B , which we shall call fs, L e . , where ( a ) is the concentration of all forms of the protein molecule in conformation ( 5 ) We need not distinguish between dissociations of the type A H C + A 4- H t and AH + A H + , and charges are therefore omitted in the symbols for AH and A.
+
1629
CY and (/3) the corresponding quantity in conformation P f 5 = @)/[(a)f @)I (5) It is to be noted, however, that no method is ordinarily available to determine whether all molecules in a given protein solution have the same conformation, i.e., KO and kl are not directly measurable. I n the present situation experimental parameters which depend on conformation will be invariant before titration of AH begins, will change during titration of AH, and will be invariant again when the titration is complete. It will not be possible to distinguish between situations in which the transition proceeds from essentially 1 0 0 ~ oQ! to essentially 100% P (equivalent to K O 0,kl w ) , and situations in arhich the transition is from a mixture (say, 80% CY,20T0 0) to another mixture (say, 20% a, 80% P). It is thus expedient to define an apparent extent of conversion, yo, as
- -
which will correspond to the observable data regardless of the values of ko and kl. Using equations 1, 2 and 3, equation 4 becomes (7)
where
Our first important conclusion is then that the link between the titration of AH and the conformation change does not change the f o r m of the titration curve of AH, but simply affects the pK. In the same way we obtain from equation 6 that yg is also equal to [K*/(H+)]/[l K*/(H+)],
+
%.e. Y@ = XA
(9)
and the second important conclusion is that with the present assumptions the apparent course of the transition is invariably identical with the course of titration of AH. Effect of Electrostatic Interaction.-Electrostatic interaction between the charged groups of a protein molecule is usually considered in terms of the approximate theory of Linderstr#m-Lang, in which the electrostatic free energy W - a t an average net charge 2 is proportional to Z2. The corresponding effect on the titration curve is to replace the dissociation constant K by the product of an intrinsic constant Kint and an exponential chargedependent factor K = Ki,,&wl
(10)
The relation-between W and w may be written as
W = 2wRTZ2. It is assumed in this section that electrostatic interaction per se does not affect the transition a + P. Outside the region of titration of AH, Wo - W , must therefore be independent of charge, which requires that wa = wp. With this restriction, normal electrostatic interaction between titratable groups may be taken into account by replacing the K’s of the preceding equations by equation 10. This will make K,, K5
CHARLES TAKFORU
1630
VOl. 83
and K * pH-dependent, flattening the titration group or strong vicinal electrostatic effects can curve in the usual way.2 The ratio kl ‘ko (equation change p K values by more than about one pi7 unit. 3 ) will also be slightly pH-dependent, because, Conformation Changes Associated with Two a t any pH in the region of the transition, Z will Titratable Groups.-We suppose again that the have a slightly different value in the two conforma- protein molecule can exist in two conformations tions. However, kl,/ko will be primarily a measure a and /3 but that the transition is associated with of (Kint)p! ( K i n t ) a . two acidic groups AH and BH, such that a is more It is to be noted that the principal conclusions stable when both of these are in the acid form, and of this section, as expressed by equations 7 and 9, /3 more stable when both are in the basic form. are independent of any assumption concerning the We shall use aAHBH to designate the species with actual magnitude of electrostatic interactions. two undissociated protons, aABH the species with Titratable Group Buried in a Hydrophobic the proton dissociated from AH only, aAHB Region.--,4 titratable group may be buried in the species with the proton dissociated from BH the hydrophobic interior of a globular protein only, aXB the species with both protons dismolecule. Ordinarily i t must be uiacharged when sociated, all of these being in conformation a. so buried because the electrostatic self-energy of a The corresponding molecules in conformation @ are charged site in a medium of low dielectric constant designated PXHBH, etc. The equilibrium for the is prohibitively large.6 We shall therefore assume transition CY -+ /3 is now described by the equithat if a change in conformation is associated with librium constants titration of a buried group, only the uncharged form will be buried. The conformation containing the group in its charged form will be such as to leave the group in contact with free solvent, such that its pK is that normally expected of the group.’ If the buried group is an acidic group (AH un- and the assumption concerning the relation between charged) we have K P equal t o the normal K for conformation and titration may be expressed as ko the group, and k l - a . We have (froni equation 8) < 1, ks > 1, kl and kz intermediate. Acid dissociation constants are again designated by K . Three K * = koKp/(l ko) (11) subscripts will be used: the first to indicate whether Thus ko is determinable from the difference be- the proton dissociated is the first or second one, the tween pK* and the normal pK for the group. next to indicate conformation, the last to indicate Similarly, if the buried group is basic (A un- whether the proton is dissociated from AH or BH, charged) we have K , equal to the normal K and 1.c. K ~ P= A (BABH)(H+)/(pAHBH) KO 0, giving
+
-
K* = ( 1
+ kl)Ka
(12)
so that k l becomes determinable. It is to be noted that we cannot determine whether the buried group is hydrogen-bonded or not. The value of ko or kl reflects the total relative stability of the two conformations, no matter what the factors which determine the stability. It should also be noted that the foregoing considerations provide a third possible important mechanism whereby the intrinsic pK of a titratable group, in the normal treatment of titration curves, may be anomalous. Two other mechanisms have been suggested previously : pure hydrogen bonding8 and strong near-neighbor electrostatic interaction~.~ What the present paper shows is that burying of a group in a hydrophobic region, with rearrangement accompanying ionization, will produce a similar eRect. Moreover, this mechanism may be used to explain relatively large shifts in p K , because there is no limit on the magnitudes of k o and k l . There is no evidence a t present to suggest that pure hydrogen bonding of a single (0) Ref. 4, footnote 14. (7) This does not mean t h a t other sitnations are impossible. A charged group could be buried as part of an ion pair, for instance. However, we clearly cannot consider all possible examples in a single paper. ( 8 ) M. Laskowski a n d H. A . Scheraga, J . A m . Chem. SOL.,76, 6305 (1954). We use t h e term “pure hydrogen bonding” t o describe t h e situation envisaged b y these authors because t h e calculations they made ascribe t h e free energy of formation solely t o t h e heat of formation of t h e hydrogen bond and t o t h e loss in entropy resulting from loss of rotational freedom of the donor group. (9) C. Tanford, {bid., 79, 5340 (1957).
Kzp.4 = @AB)(H‘)/(PAHB) kip^ = (PXHB)(H’,/(3AHBH) K ~ B= B (6AB)(H+)/(oA4BH) (14)
and a similar set for conformation a. These constants are not all independent, being related by the equations K I ~ A K ~=PKBI ~ B K zK~,AKz,B ~A, = K ~ , B K ~ , ACorresponding . to equation 3 we have four relations (three of them independent)
We shall confine the examples chosen for analysis to situations in which the K ’ s are “normal” in one of the two conformations. In the derivations it will be assumed that the conformation a t high p H (;.e., p) has the normal K’s. The manner in which the equations must be transposed if the opposite situation prevails will be indicated a t the end. The assumptions that the K’s in @ are normal includes the assumption that AH and BH are independent in @. Thus K l p ~ = I G ~ B= K P B , Kip* = Kzpa= K P ~ As . the corresponding groups in a may not be independent, these equalities do not reduce the number of independent relations in equations 15. Defining ZA by equation 4, X B by a similar relation involving BH rather than AH, and yp as in equations 5 and 6, we obtain
PROTEIN CONFORMATION THEORY
April 5 , 1961
1631
1.0
0.5
where
ko(1
ki(1
+ k,) K I ~ B ko(1 + k i ) KOAKPB(19) + ko) (H+)
k m & )
+
mTz
0
This result is clearly not as simple as that given by equations 7 to 9. However, i t reduces to simple form whenever the conformational change proceeds from essentially lOO% a to essentially 1OO%p, i.e., when ko 0,ka 0 3 , provided that both k1 and ke are considerably larger than ko.'O %-e obtain, whenever K ~ A K ~ B
- -
K"
=
koKpAKBB
(21)
I t is to be noted that the course of reaction follows an equation for the simultaneous dissociation of two protons. If, with ko 0, k~ K ~ >> A I&, then AH and BH are titrated separately
- -
j
6
12
10
8
PH.
and the interesting result is obtained that the transition a -+ ,/3 is also split into two parts
Fig. 1.-The course of titration and conformational ka > 100. change with KOA= 10-4, K ~ = B 10-lo, ko = The value of kl is 0.25 in ( a ) and 5.0 in (b). In each case curve 1 represents (xa X B ) / 2 ; curve 2 represents yp.
+
as illustrated for two values of k , in Fig. 1. The criterion for use of equations 20 and 21, derived on the basis KOA K ~ isBthat KPA.'&B < 1/100ko. The criterion for use of equations 22 t o A KPB,is KBA/ 24, derived on the basis of K ~ >> K ~ B> 100/ko. Thus deviations from the two limiting situations here given will occur primarily when K ~ A I K ~ Bl l k o . If, for instance, AH is 0 2 4 0 2 4 a carboxyl group and BH a phenolic group we (a) (b) can expect that K ~ A ! K ~ Blo5. With ko =: pH (relative). we would then be in the intermediate region to which the simple expressions above are inapplicable. Fig. 2.-The course of titration and conformational Calculations for this case, with kl = 5 and k, = change with KBA/KPB= lo6 and ko = The value 0.25 are shown in Fig. 2. With kl = 3 the result of kl is 5 in (a) and 0.25 in (b), and ka has been assumed is close to that obtained when K ~ % A K ~ Bbut , the >loo. In each case curve 1 represents ( x a xg)/2, curve
-
-
-
+
(10) I t is difficult t o envisage a non-trivial situation in which the condition concerning kl and k2 does not apply. If kl Y kz -c ko a t the same time as ka kc, this means b y equation 15 t h a t Klaa = Kpn, K i a B = KBB, whereas K z a ~ (ko/ka)Kp.4 Kla.4, Kzap 2 T h e condition ko 0 , k3 -' (ko/ka)XpB KlaB. requires t h a t k d k 8 he certainly no larger t h a n lo-', and this means then t h a t titration of either of the two groups in conformation a changes the free energy of ionization of the other group by more t h a n 5000 calories (without change in conformation). At t h e present time we know of no interaction between two titratahle groups which would produce so large a n effect in t h e absence of conformational change. On the other ko, while kz ko, but this is a hand. it is of course possible t h a t KI trivial exception t o t h e rule, for it corresponds t o t h e situation in which only one group (in this case BH) is associated with t h e conformational change. i.e., it reduces t o the situation treated in the previous section,
>>
-
1. change when either ko or l / k t is not very small, Le., when the (However, k3 will be even larger.) If ko 0 and conformational change is between a pure conformation KO* K ~ Bequations , 20 and 21 will apply, for ( a or 8 ) and a mixture of the two. In ( a ) we have ko = these equations were obtained without specific 0.25, kl = kl = 5, k, = lo3. I n ( b ) we have ko = assumption concerning an upper limit for k, or kl = kt = 0.1, kr = 5. In both cases KEA= K p . Curve 1 k z . The value of ko can be obtained i r o n the obrepresents X A = XB, curve 2 represents yo, and the dashed served value of I> K B Dsuch , that AJI and BH cases the curve for X A = X B does not coincide exactly with the curve for ya. Moreover the curves are are titrated separately, equations 22 to 24 will be all intermediate between one-proton and two-proton obeyed. However, with kl large, k’~* = k o k ’ p A , dissociation curves, whereas a two-proton curve is KB* = Kps, y8 = X‘A, i.e., the reaction would be invariably obtained for two identical groups (equa- that characteristic of a single buried group. If a carboxyl group is buried and k , is not too small tion 20) when ko 0, ka m. lo+), we would be unable to tell whether In conclusion we give the equations applicable (KO when the groups AH and BH are “normal” in the phenolic groups were buried in the same region. If the two buried groups are basic, equations 20 acidic form (conformation a) rather than the basic ~0 K ~ A K ~ B and , form. I n this situation K l a ~= K Z ~ = A Ka.4 and 29 would apply when ka and KlaB = K 2 a = ~ K,B become the reference ka could be determined from the data. If KOA>> , with k1 now necessarily very small, the pK’s in terms of which the relations are most con- K o ~then, veniently expressed. In place of equations 16 to situation would again reduce to that Characteristic of a single buried group, in this case the less acidic 19 we get one (BH). Extension to n Titratable Groups. Conformation Changes Due to Electrostatic Repulsion.When this treatment is extended to consider conformational changes associated with any number ( n ) of titratable groups, i t is necessary to abandon the “microscopic” dissociation constants used in the earlier portion of this paper and to substitute, as one does in the treatment of titration curves, dissociation constants which are appropriate averages of many i‘microscopic” constants. One method is to group together all molecules with the same number of protons, dissociated from the 71 The condition for applicability of equation 20 groups considered. Thus (PPHn) and (aPH,) may be used to designate concentrations of rnolenow becomes ko 0, ka m K,A/K,B < k3/100, cules in the two conformations which have all 72 and to replace equation 21 we have acidic groups intact, and (PI”,-,) and (aPH,-,) I(* = ,w a ~100ka and to replace equation 23 we have
-
- -
(11) If t h e protein molecule contains several interior regions which can be independently unfolded, then both groups must be in t h e same region. Otherwise, we should be dealing with two independent conformation changes.
PROTEIN CONFORMATION THEORY
April 5 , 1961
may designate concentrations of molecules with v of the n protons dissociated. The state of titration of groups not connected with the transition from a to 6 is, as before, not explicitly considered. We may now define the dissociation constants
1633
tion stable a t high pH requires that that k, = kop'.) We may make use of the identity
5
x w ! / ( n-
Y)!v! 3
(1
p
>
1; also
+ x)"
(38)
u p 0
and x times its derivative with respect to v
2
with 1 5 v 5 n and the equilibrium constants for the transition
vxvfi!/(n
-
v)!9!
= nx(1 + X I " - 1
(39)
v - 0
and define an auxiliary function with 0 5 v we now get
5 n.
For the relation between these
kv/ku-1 = K ~ v I K a v
(33)
and, again taking 0 as the conformation stable a t high @H,this means that a t least some of the Kpv (in the absence of unusual cooperative effects all of the KO")must be larger than the corresponding Kav. We obtain, for the titration curve of the n groups
5
+ (1 + kv)KaiKat. - .Kav/(H+)'
~ ( 1 k,)KaiKaz.. .Kav/(H+)v
i ; = u = O
n
5
+
+ l / k v ) K g i K g z . . .ICpv/(H+)v
(1
(34b)
A single calculation using equations 40 to 43 is shown in Fig. 4. We have used n = 6, ko =
" P O
while for the extent of conversion to
2
0
kvKalKa2.. .K a v / ( H + ) v
f p = v = O 11
(1 " P O
-
5
(42
which gives for the experimental parameter y~ (equation 6 )
~ ( 1 1/k,)KpIKp2...Kp./(H+P
v = o
+
where V , = [nK,/(H+)I/[l K,/(H+)] gives the titration curve of conformation a alone and :p is the corresponding expression for conformation p . For fs we obtain
(344
v=.o
-
which varies from 1 to p between the beginning and the end of the transition. With these relations we get
+ kv)KaiKaz. . .Kav/(H+)u
(354
K,?iKpz.. . K p v / ( H + ) ~
v - 0
n
(1 " - 0
+ 1lkv)KpiKpz.. .Kpv/(H+)v
(35b)
We shall consider in detail only the simple situation in which all groups are identical in both conformations and independent of each other except for non-specific electrostatic interaction. In that case the various K,, and Kpuare related to the corresponding constants K , and Kp, for any single group, such that12 Kov/Kp = Kav/Ka =
(fi
-
Y
+ l)/v
(36)
Combining this relation with equation 33, we have, independent of v Kpv/Ka,
kv/ku-i
=
Kp/Ka =
p
(37)
where p is a constant defined by the equation. Electrostatic interaction may be incorporated into this equation by expressing Kp and K , as products of an intrinsic K and an electrostatic factor, as in equation 10. This will make p PH-dependent, even if w, = wp, since the two conformations will differ in the relation between charge and pH. (It is to be noted that defining 0 as the conforma(12) I. M. Klotz, in H. Neurath and K . Bailey, "The Proteins," Vul. I, Academic Press, Inc., New York, N . Y., 1953, Chapter 8.
o!
.//:
-2
-1
4 0
I
I
1
1
2
3
PH-PKp. Fig. 4.-Calculatiuns for n = 6, ko = 0 001, ke = 1000, with electrostatic effects assumed negligible. Curve 1 represents y p , curve 2 represents the degree of dissociation, x = i / 6 , curves 3 and 4 are the titration curves in con, represent xp = formations and C Y , respectively, i . ~ .they i p / 6 and x, = Ca/6.
ks = loa. We have neglected electrostatic interaction so that p = constant = 10. We see that both the transition curve and the titration curve are much steeper than the curves which would correspond to dissociation of a single proton : the curve for yp approximately follows a relation of the form K * / ( H+)"' (44 1 " 1 K*/(H+)"
+
163i
-
CHARLES TA~TORD A N D VALERIEG. TAGGART
-
with nz 3 . The curve for 2 = ?In approximately follows a similar curve with rn 2. While these results apply only to the particular situation treated, we may expect that their qualitative aspects are typical, i . e . , we shall often expect close correspondence between the titration curve and the transition curve and shall usually find both to be steep, such that rn of equation 44 lies somewhere between 1 and n. For n titratable groups buried in the interior of a protein molecule we can expect p to be large, which leads to equation 44 with rn + n for both y p and ?. The equations of this section may be applied to the expansion of a compact conformation to a more loosely coiled one under the influence of electrostatic repulsion. In this case the change in conformation is associated with all titratable groups. However, the eCfect of electrostatic interaction on K , and Kp, and, hence, on p and u , now plays the major role in the process, in contrast to the examples discussed explicitly in this paper. We shall postpone consideration of this situation to a later paper. Discussion example of an ionization-linked change in conformation is the N + R transition in P-lactoglobulin, which is discussed in the following paper.I3 I t will be seen that this transition is associated with a single titratable group. 9 n example of a transition associated with several titratable groups is the acid denaturation of h e m ~ g l o b i n . ~This ~ transition obeys equation 44 with rn 5 and it (13) C. Tanford and
V. G. Taggart. J . A m . Chem. Soc., 83, 1634
I l!l(il).
(14) S. Beychok and J. Steinhardt, ibid., 81, 5679 (1959); references t o m a n y earlier papers on this suhject are given in this paper.
Vol.
s3
is accompanied by the titration of a large number of anomalous groups. However, both intrinsic and electrostatic factors are involved, so that the present treatment is not directly applicable. I t should be observed that the general approach we have formulated is not restricted to changes in conformation. For example, a could represent a free protein molecule and P a protein molecule to which a small molecule has been bound. The parameter yp would represent the extent of binding, and the conclusion would be t h a t if the extent of binding depends on pH, then there must be titratable groups which change their pK values when binding occurs. This particular situation has been discussed in considerable detail in connection with the binding of 0, by h e m ~ g l o b i n . Another ~~~~~ possibility is that p represents a polymer of a , rather than a new conformation. In this case, if the degree of polymerization depends on pH, then the titration curve of the polymer must differ from that of the monomer. il well-known example is the polymerization of fibrin monomer. I n conclusion, it must be noted that the general approach of this paper is not original. It should properly be regarded as an extension of the principles already formulated by Wyman'j in connection with the hemoglobin-oxygen reaction. Acknowledgments.-This investigation was supported by research grants G-5829 from the Xational Science Foundation and -1-3114 from the National Institute of -lrthritis and Metabolic Diseases, U. S. Public Health Service. 17218
(15) J. \\'yman, Advences in PYotein Chem., 4 , 407 (1948). (16) R. A. Alberty, J . A m . Chem. Soc., 77, 4522 (1955). (17) H. A. Scheraga and M. Laskowski, Jr., Advaizces in Protein Ckcm., 12, 1 (1957). (18) E. Alihalyi, J . Btol. Cizetn., 209, 783 (1954).
[ COXTRIBUTION FROM THE DEPARTMENT O F CHEMISTRY, STATE UNIVERSITY
OF
IOWA, IOWACITY, I O W A ]
Ionization-linked Changes in Protein Conformation. 11. The N p-Lactoglo bulin
R Transition in
BY CHARLESTANFORD* AND VALERIEG. TAGGART RECEIVED -4UGCST 15, 1960 Sear pH T 5 native p-lactoglobulin ( S )undergoes a transition to a new conformation (R), as previously described. The reaction satisfies the criteria for a conformational change involving a single titratable group. This group is a carboxyl group which appears to be buried in the hydrophobic interior of the protein molecule in conformation N. Our earlier study of this reactiori showed two rather than one anomalous carboxyl group per molecule, so that there must be two locations in the protein molecule where the N + R transition occurs independently. There is presumably one location in each of p-lactoglobulin's two polypeptide chains. The thermodynamic parameters of the transition, a t 2 5 O , at each of the two locations before titration of the buried groups, are ko = (R)/(N) = 0.0025, AFO = 3500 cal./mole, AHo = 7000 cal /mole, A S o = 12 cal./deg. mole. These estimates depend on assignment of normal values t o the thermodynamic parameters for dissociation of the carboxyl group in conformation R, so that AFO has an uncertainty of perhaps 1000 cal./mole. The uncertainty in AH0 is somewhat greater than this, while A S o has an uncertainty of the order of A7 cal./deg. mole.
h previous paper from this Laboratory' has described a reversible change in conformation which /3-lactoglobulin undergoes near #H 7.5. I t was shown that this transition is accompanied by a change in specific rotation (but not in the dispersion parameter bo2), by a small change in Department of Biochemistry, Duke University, Durham, . ' h C. (1) C. Tanford, I,. G. Bunville and Y.Nozaki, J . Am. Chem. Soc.. 81, 1032 (1959). (2) C. Tanford, P. K. De and V. G T a g g a r t , ibid., 82, 6028 (1960).
sedimentation coefficient,3 (but not in molecular weight4) and by the titration of two anomalous carboxyl groups. The curves describing the changes in optical rotation and in sedimentation coefficient are essentially superimposable, and both coincide within experimental error with the titration curve of the two anomalous g r o ~ p s . ~ (3) K. 0. Pedersen, Biochein J . , 30, 961 (1936). (4) S. i T.Timasheff, personal communication. (5) Figure 1 of ref 1 shows the titration curve to be so.newhat *teeper.
