VOL. 15, 757-783 (1976)

BIOPOLY MERS

Buoyant and Potentiometric Titrations of Synthetic Polypeptides. 11. Five Copolypeptides and Two Nonionizable Homopolypeptides in CsCl Solutions* DAN S. SHARP,** ROBERT ALMASSY,*** LAWRENCE G. LUM,? KATHLEEN KINZIE, JOHN S. V. ZIL,?? and JAMES B. IFFT,+t+ Department of Chemistry, University of Redlands, Redlands, California 92373 Synopsis The buoyant density titrations of five ionizable copolypeptides in concentrated CsCl solutions have been determined. The results are used to formulate models for predicting the buoyant density titration behavior of copolypeptides and proteins using the previously reported homopolypeptide buoyant density titratiop curves. I t was determined for these copolypeptides that the best predictive model must include not only the buoyant densities of the constituent amino acid residues and the relative composition, but also hydration and salt binding. Hydrations determined for the homopolypeptides are used in the copolypeptide predictive model. The hydrations of the neutral homopolypeptides were readily calculable since their buoyant densities are thermodynamically defined in terms of their partial specific volumes and hydrations. For the case of a charged macromolecule, an expression for the buoyant density as a function of the number and nature of the bound ions, its partial specific volume, and its relative hydration has also been available for some time. This heretofore intuitive relationship is now derived from thermodynamic principles and allows calculations of hydrations to charged macromolecules which bind either cations, anions, or both. The potentiometric titrations of three of the five copolypeptides in concentrated CsCl solutions were determined in order to study the effect of residue interaction and solvation effects on their ionization behavior. The potentiometric results are also combined directly with the buoyant density titration results to determine the correlation of the buoyant density with the degree of ionization. As in the cases of poly(Glu) and poly(His), the buoyant density of the copolypeptides changed linearily with the degree of ionization. The buoyant density titrations of two nonionizahle homopolypeptides, poly(G1y) and poly(Ala), were determined in concentrated CsCl solutions. The buoyant density was

* Part I: Almassy et al.' * * Present address: California College of Medicine, University of California, Irvine, Ca.

* * * Present address: Department of Chemistry, California Institute of Technology, Pasadena, Ca. Present address: Department of Pediatrics, University of California, San Francisco, Ca. t t Present address: Department of Psychiatry, Yale University, New Haven, Conn. + t tTo whom reprint requests should be addressed. 757

0 1976 by John Wiley & Sons, Inc.

758

SHARP ET AL.

found to increase with increasing pH, despite the fact that side chains do not contain ionizable groups. This is the first evidence from homopolypeptide or copolypeptide data that buoyant density changes can be observed from effects other than side-chain ionizations.

INTRODUCTION The technique of sedimentation equilibrium in density gradients in the analytical ultracentrifuge has been employed to study the properties of proteins and polypeptide^.'-^ Initial studies4 were directed toward the study of the behavior of a single protein, bovine serum mercaptalbumin, in CsCl solutions at the isoelectric point. More recently, buoyant titrations, the measurement of the buoyant density of a polymer as a function of pH, of several protein^^.^ have been measured. Because of the complexities involved in interpreting these data, studies have been made' of the buoyant behavior of homopolypeptides. The buoyant titrations of six ionizable homopolypeptides in CsCl solutions have been reported and these results correlated with potentiometric data. The next logical step in the utilization of this homopolypeptide data in the interpretation and prediction of the buoyant behavior of proteins was to measure the buoyant titrations of a number of copolypeptides which contained two residues which had already been characterized in CsCl gradients. Because the molar compositions of these copolymers were known, a variety of predictive methods could be employed to determine which was the most effective in predicting the behavior of the copolymers. The best method then in turn could be used to interpret more fully the buoyant titration data for proteins. The goals of this interpretation are the determination of the number of anions and cations bound, the hydration, the number of ionizable residues of each class of amino acids and their approximate pK,'s, and the relation between the amino acid composition of a protein and its buoyant density. Because one of the copolymers examined in the present study contained alanine, it was necessary to measure the buoyant titration of poly(A1a). The results of this study were so surprising that the buoyant titration of poly(G1y) was also measured.

EXPERIMENTAL A description of the polypeptides used in this study is provided in Table I. All materials, apparata, and procedures for both buoyant density and potentiometric titrations are the same as described in Part I,l with the following exceptions andlor additions. Amino acid compositions of poly(Ala), poly(Gly), and p ~ l y ( A l a ~ ~ . ~ Glu9.') were determined with a Spinco model 118 amino acid analyzer. Photographic plates from buoyant density experiments were mea-

POLYPEPTIDE TITRATIONS. I1

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sured using a 2*/4 X 3l/4 in. Beseler 23C enlarger in the case of soluble bands, and a Nikon 6C comparator for precipitated bands. For the polypeptides not soluble at any pH, a suspension of the relative polypeptide in deionized water was effected with the gentle use of a mortar and pestle. pH shifts for buoyant density points are represented by an arrow as PHinit, pHfinalon the buoyant density titration plots. Rotor speeds employed varied between 47,660 rpm and 56,100 rpm, and equilibrium was attained in 24 hr or less in all cases, as determined by successive time measurements of the band center.

-

TABLE I Description of Polypeptides ReMolecular ported Measured mole % mole % weight

Polypeptide

Source

Control number

Poly(A1a) POlY(GlY1 Poly(Ala, Glu)

Miles-Y eda Miles-Yeda Pilot Chemicals

AL-52 G-4 1 M-82-D

Poly(Glu, Tyr)

Miles-Yeda

107-A

Poly(Glu, Tyr)

Miles-Yeda

138

21,850

Poly(Glu, Lys)

Pilot Chemicals

C-35

90,000

Poly(Lys, Tyr)

Miles-Yeda

LYTY3

41,480

2,675 a a b

100 100 91 9 90 10 50 50 60 40 50 50

100 100 90.9 9.1 93.8 6.2 54.5 45.5 60.9 39.1 51.4 48.6

a Indeterminable. h Unknown.

All copolypeptide and homopolypeptide buoyant densities were calculated using the expression of Ifft5 for the isoconcentration position, instead of using extrapolations or interpolations of the isoconcentration ratio method.6 The required slopes were obtained from those calculated in Part 1.l All results of the potentiometric titrations are described by a versus pH plots, a being the degree of deprotonation. In addition, a plot of pH - log[a/(l - a ) ]versus a is given for the Glu-Lys copolypeptide. A pH Stat experiment was conducted on poly(Ala90~9Glu9~1) to determine if the carboxyl groups of the Glu residues were buried, and thus did not titrate until higher pH values were reached. A suspension of the copolypeptide in concentrated CsCl solution was adjusted to an initial pH of 3 with 12.ON HCl. It was then rapidly titrated with 1.ON NaOH t o pH 8 using the Radiometer Copenhagen titration system. The pH Stat function then measured the volume of NaOH needed to maintain the pH of the suspension at 8.00.

760

SHARP ET AL.

CALCULATIONS Hydration Calculations Using Eq. (l),Almassy et al.’ calculated a total hydration for the nonionized forms of the six ionizable homopolypeptides they ~ t u d i e d . ~ J PO =

+ r’ + r’D1

1

o3

PO is the measured buoyant density, I” is the hydration of the neutral homopolypeptide in g HzOIg homopolypeptide, D 3 is the partial specific volume of the homopolypeptide, and 01 is the partial specific volume of water. Using Eq. (2) and necessary assumptions they also calculated a total hydration for the ionized forms of the homopolypeptides.

PO =

+ + r:

1 zi

o3 + Z

+ r:D1

~ D ~

I’: is the total hydration of the homopolypeptide-ion complex in g HaO/g anhydrous, charged, saltfree homopolypeptide, zi is the weight fraction of bound ion in g ionlg anhydrous charged, saltfree homopolypeptide, and D i is the partial specific volume of the bound ion. Equation (2) was originally presented as an extension of Eq. (1)to describe the BMA-salt complex at its isoelectric point? In the first paper of this series, Almassy et a1.l have extended this model to homopolypeptide-ion complexes a t all pH’s. A thermodynamic derivation of this equation is given in Appendix A. The interaction terms in Eq. (2), zi and r:, can be redefined in terms of amino acid residues by the relationships

Mp

N

nMr

(3)

M , is the molecular weight of the amino acid residue. The partial specific volume of the homopolypeptide can be approximated by the partial specific volume of the amino acid residue by the relationship

P p and P, are the partial molar volumes of the homopolypeptide and the amino acid residue, respectively. These redefinitions mean it is not necessary to know the molecular weight of the homopolypeptide in order to redefine the solvation param-

POLYPEPTIDE TITRATIONS. I1

761

eters, r: and zi on a mole basis, which is necessary for their use in predictive Eq. (10) (uide infra). The same approximations hold for redefining Eq. (1) on a residue basis. Objections may be raised to the form of Eq. (2) in that the solvation parameters are in terms of the charged polypeptide. Equation (27) in Appendix A can be manipulated in such a manner as to lead to solvation parameters in terms of the neutral homopolypeptide-ion complex by dividing the numerator and denominator of the equation by M p n M x . However, it is still necessary to approximate the specific volume of the homopolypeptide-ion complex in terms of the constituents, U P (or or) and D;. This necessity defeats the purpose of expressing the solvation parameters in terms of the neutral homopolypeptide-ion complex. When specific volumes of macroion-salt complexes are available, the use of Eq. (2) will no longer be necessary. For the case of certain copolypeptides and for most proteins, the macroion-salt complex consists of the macroion (P) plus both cations (X) and anions (Y). The derivation in Appendix A has been extended to include this possibility, and is presented in Appendix B. The derivationesults in an expression for the buoyant density in terms of water, anion, and cation binding:

+

PO =

u3

1 + z x + z y + rTl + zXvx + zYoy + rTul

(7)

This equation is identical to the one used by Williams and Ifft.7 The same limitations applicable to the use of Eq. (2) are applicable to Eq. (7)kthese being the use of the specific volumes of charged species comprising the macroion-ion complex. In calculating the hydration from the algebraic equivalent of either Eq. (1) or (2) certain assumptions must be made about the terms comprising these equations. 0 3 is the partial specific volume of the amino acid residue, and is taken from data compiled by Cohn and Edsall.8 These are calculated and not experimentally measured. I t is assumed that the specific volume of the amino acid residue does not change with ionization and ion binding. The lack of data relating the specific volume of the amino acid residue with pH necessitates this assumption. The partial specific volume of the hydration term is assumed to be 1.000 ml/g. This is an assumption because the electrostriction effects associated with the ionized side chains may result in a decreased partial specific volume of the bound water. The partial specific volumes of the bound ions were interpolated from the published data.g Ion binding of one oppositely charged ion for each charged amino acid residue is assumed. This assumption is necessary due to the lack of precise ion binding data for these polypeptides in CsCl solutions. Some investigations, however, indicate that there may also be salt binding to the polypeptide backbone.1°-12

762

SHARP ET AL.

Predictive Methods for Buoyant Densities One of the objects in studying the buoyant density behavior of the copolypeptides was to provide a standard with which to compare different predictive techniques using buoyant density data of the constituent homopolypeptides. Three types of predictive methods were tested. Equation (8)-( 10) illustrate these methods.

Pp

-wa

+-w b

Pa

Pb

The subscripts a and b refer to the two amino acid residues comprising the copolypeptide. pp is the predicted buoyant density. p a and p b are the respective measured buoyant densities of the homopolypeptides a and b a t a given pH. na and n b are the respective mole fractions of the amino acid residues of the copolypeptide. wa and W b are the respective weight fractions. They reflect the mole fractions of the respective components in the copolypeptide and the molecular weights of the amino acid residues, bound water, and bound ions if applicable. The use of Eqs. (8) and (9) gave poor predictive results for all copolypeptides. Since they have no physical significance, their use was not pursued. Equation (10) can be called the additive volumes relationship. It is obtained by assuming that the density of the whole is equal to the sum of the component masses divided by the sum of the component volumes. The predictive method of Eq. (10) requires a knowledge of the density and weight fractions of the component amino acid residues. The densities are accurately obtained from the densities of the homopolypeptides.' The weight fraction terms must include not only amino acid composition data, but also hydration and salt binding data for each individual amino acid residue. The weight fraction terms are further complicated by the fact that the hydration and salt binding terms are dependent on the degree of ionization. The salt binding is obtained by assuming that one ion of opposite charge binds to each charged residue in the copolymer. Therefore, in the flat regions of the homopolypeptide buoyant density titration curves, there is either zero or one ion bound, depending on whether the homopolypeptide is either completely neutral or completely charged. The hydration is also dependent on the degree of titration. Again, in the flat regions of the buoyant density titration curves of the homopolypeptides, the hydrations are assumed to be constant and equal the hydrations of the neutral or ionized forms. The salt binding and hydrations in regions of buoyant density changes are more difficult to calculate. For the homopolypeptide in

POLYPEPTIDE TITRATIONS. I1

763

this transition region, the extent of ion binding is assumed to be dependent on the degree of titration of the ionizable side chains of the amino acid residues. This degree of ion binding can be described by the following equations-a form of the additive volume relationship. PO

Pu

Pion

T = (1 - ( Y ) (+~ r:)

+

f

f

(

+ r;on+ z~") ~

(12)

po is the measured buoyant density of the homopolypeptide a t a given pH. pu is the measured buoyant density of the unionized homopolypeptide at a pH where no ion binding is assumed. pion is the measured buoyant density of the homopoIypeptide a t a pH where 100% ion binding is assumed. M is the molecular weight of the amino acid residue. i'I is the calculated hydration (Eq. (1))of the homopolypeptide at the pH where no ion binding is assumed. is the calculated hydration (Eq. (2)) of the homopolypeptide at the pH where 100% ion binding is assumed. 2,: is the ion binding to the ionized form of the homopolypeptide. (Y is the degree of ionization, and also represents the fraction of ion binding. All interaction parameters are in terms of g;/mole residue. Solving for (Y yields,

(13) (Y values computed according to Eq. (13) were employed in the following calculations because the homopolypeptide potentiometric data are incomplete. (It would be preferable to use the potentiometric data.) The term (Y is used to calculate weighted hydration and ion binding terms.

2, =

ffz;

These weighted hydration and ion binding terms are used in the expressions defining w, and wb of Eq. (10). In summary, all of the parameters comprising Eq. (10) are either measured (pa, Pb, and mole fractions of amino acid residues), calculated (e.g., molecular weights and hydrations of residues), or assumed (e.g., ion binding). A predicted buoyant density can be calculated and compared with the measured buoyant density of the copolypeptide a t a given pH. Several predictive methods have been studied, based on different models of amino acid hydration and salt binding. Two particular methods correlate with the experimental data closely and are described below. In method A, ion binding is assumed to be as described above (i.e., one oppositely charged ion is bound per charged residue). In this

764

SHARP ET AL.

model, the hydration is calculated as the net hydration of the neutral homopolypeptide (Eq. (l)), and is assumed not to change as the amino acid residue is ionized. Although this assumption does not appear to have physical significance, a constant hydration has been observed7 for bovine serum mercaptalbumin from pH 5.5 to pH 12. In method B, the ion binding is treated in the same manner as in method A. However, the hydration is calculated somewhat differently. For the amino acid residues in their neutral form, the hydration is still calculated from Eq. (1). However, for the ionized form of the residue, an ion is bound, and the hydration is calculated from Eq. (2). This hydration represents the total hydration of the amino acid residue-ion complex, assuming one ion is bound. The calculation of this hydration is further complicated since i ~ iin Eq. (2) is a function of the solution d e n ~ i t yand , ~ must be evaluated a t pp. This value is obtained by calculating an initial pp a t the particular pH using Eq. (8), which depends only on the mole percentage of the residues and the measured PO’S of the homopolypeptides. This density is then used to interpolate an ionic specific volume, bi, from the data of Ifft and W i l l i a m ~ . ~ This new bi is used in Eq. (2) to calculate a new buoyant density, ph, for the homopolypeptide. This new homopolypeptide buoyant density results from a change in the specific volume of the ion in going from the solution density of the homopolypeptide to the predicted density of the copolymer. All other parameters are invariant. The hydration used in Eq. (2) to calculate this new buoyant density, p i , is the hydration originally calculated from Eq. (2) and the experimentally determined po of the homopolypeptide. In addition to the hydration, the salt binding and the partial specific volume of the amino acid residue were assumed to remain constant. This newly calculated p; is used in place of the appropriate pa or P b in Eq. (10) to calculate pp. This new pp is compared with the previous pp and if the difference between the two is less than 0.0001 g/ml then the last pp is called the predicted buoyant density. Otherwise, the last pp is used to interpolate a new ionic specific volume, and the procedure is repeated to produce a third, fourth, etc., pp. This iteration continues until the aforementioned criterion is met. An IBM Fortran computer and a Hewlett-Packard 2000B basic time sharing computer were used in calculating pp’s a t pH intervals of 0.5. The results of the calculations are presented with the measured buoyant densities of the copolypeptides and constituent homopolypeptides.

RESULTS AND DISCUSSION As was noted for the six ionizable homopolypeptides studied by Almassy et al.,l there is a marked increase in buoyant density with increasing pH for the copolypeptides. Again, this increase can be interpreted in terms of the change in ion binding and hydration, with Cs+

POLYPEPTIDE TITRATIONS. I1

765

TABLE I1 Solubility Ranges of Homopolypeptides and Copolypeptides in CsCl Polymer

Buoyant density

Po ly (Ala ) b b POlY(G1Y) Poly(Tyr)C 12.2-14.0 P o l y ( G l ~ ~ ~ . ~ )L y s ~ ~ . ' 0.0-14.0 Poly(G I u ~ ~5 . . ) ~ T ~ ~ ~ 10.0-1 4 .O Poly(Glu93.8Tyr6.2) 6.0-14.0 Po ly ( Lys5 4Tyr48.) 11.8-14.0 P ~ l y ( A l a ~ ~ . ~ )G l u ~ . ' b

Potentiometrica

10.7-14.0 0.0-14.0 10.5-14.0 6.0-14.0 11.8-14.0

a Potentiometric titration solutions have solution densities approximating the average densities of the buoyant experiments. Polymer was precipitated at all pH's. C The potentiometric solubility range for poly(Tyr ) was incorrectly stated in the first paper of this series.'

binding to side chains with a negative charge, and C1- binding to side chains with a positive charge. T h e increase in buoyant density with a loss of the positive charge, as in the case of Lys residues, is interpreted in terms of the loss of C1- ions and the hydration associated with those anions. This loss results in a n increased buoyant density because the density of the hydrated C1- ion is less than the density of the amino acid residues. The increase in buoyant density with the gain of negative charge, as in the case of Glu residues, is interpreted in terms of Cs+ binding because the density of the hydrated Gs+ ion is greater than the polymer density. As is expected, all buoyant density titration curves of the copolymers are between the buoyant density curves of the constituent homopolypeptide curves. The soluble p H ranges for the various homopolypeptides and copolypeptides in the density gradient experiments correspond closely t o those found in the potentiometric titrations as demonstrated in Table 11.

Experimental and Predicted Buoyant Densities

T h e buoyant densities for the copolymers poly( G l ~ ~ ~ . ~ L ypolys~~.'), poly( G l ~ ~ ~ . ~ T ypoly( r ~ .G~ l )~, ~ ~ . ~ A l and a ~ . ~poly), (Ly~".~Tyr~ have ~ . ~been ) predicted by methods A and B, as described previously. T h e predicted buoyant density titration curve for p o l y ( G 1 ~ ~ ~ ~ ~ L y s ~ ~ ~ ~ is presented in Figure 1 along with the experimentally determined curves for this copolymer and the corresponding homopolymers. Predictive method A is consistently higher than the experimentally determined buoyant densities, although the error is not too great in any region. In contrast, method B predicts buoyant densities less than the observed ones a t low pH and greater than the observed ones a t high pH.

SHARP ET AL.

766 I.9

I

I

I

I

I

I

1.8 -

1.7

poly(Glu)

-

-

1.6 -

Predictive hlethod B /-------

>
a > 0.90 as manifested by the flatness of the curves throughout this range. At the extremes of a the curve deviates significantly from ideality. While this deviation may result from uncertainties in the data a t low or high a,it also may indicate that only at the extremes in the titration range are residue interactions important in altering the titration behavior. The buoyant density and potentiometric titration data for poly111) indicate that the Glu residues of (G1~~~.~T (Fig. y r 7~ and ~ . ~Table ) the copolypeptide generally titrate at a higher pH than those of poly(Glu) while the Tyr residues titrate a t a lower pH relative to poly(Tyr). A similar observation can be made for the titration of the Glu residues of poly(la90~9Glu9~1) (Table 111). Examination of the buoyant density and potentiometric titrations of p~ly(Lys~~.~T (Figs. y r ~5~and . ~ )8) is complicated by the overlapping titrations of the two residues. The Tyr residues of the copolypeptide generally titrate at a higher pH than those of poly(Tyr) in the potentiometric titration (Fig. 8). Also, the buffer capacity displayed by the titration curve a t 1.0 < a < 2.0 (roughly corresponding to titration of Tyr residues) is greater than the ideal case. Comparison of the buoyant density titration behavior of the Glu residues of p o l y ( G l ~ ~ ~ . * T y(Fig. r ~ . 3) ~ ) and that of poly(G1u) is difficult due to the high mole ratio of Glu residues in the copolypeptide. Because of the low mole ratio of Tyr, comparison between Tyr residues in this compound and poly(Tyr) is not possible.

POLYPEPTIDE TITRATIONS. I1

775

poly ( G I y )

1.1

tI

0

I 2

I 4

I

I

6

8

I 10

I

1

12

14

I

PH Fig. 10. The buoyant density titrations of poly(G1y) and poly(A1a) in CsC1.

Buoyant Density Titrations of Poly(G1y) and Poly(A1a) The buoyant titrations of poly(G1y) and poly(A1a) were measured and are displayed in Figure 10. Two features of these curves are of interest. The buoyant densities of poly(G1y) at all pH's are considerably larger than those of poly(A1a). This is to be expected from the higher methylene content of the poly(A1a). The analysis of the contribution of a methylene group to the buoyant densities of poly(Lys) and poly(0rnj in the first paper in this series indicates that poly(G1y) should have a density about 0.1 g/ml higher than poly(Ala).l This is much smaller than the measured Ap of about 0.3 g/ml observed here. The recent data of Von Hippel and co-workers11,12may provide the correct explanation. They measured salt binding by recycling chromatography to polystyrene, polyacrylamide, and various small amides. They concluded that there is significant salt binding to the amide group which is appreciably reduced by the presence of nearby methyl groups. Thus there should be considerably higher salt binding to the peptide bond dipoles in poly(G1y) than in poly(A1a). The second feature of interest is that both curves display a definite inflection point at approximately pH 7. This pH corresponds rather closely to the pK of N-terminal amino groups. However, the molecular weights of these polymers are about 3,000. Thus, unless these polymers were rather highly branched, end effects should be negligible and ionization properties cannot account for this inflection. In the absence of such structural changes, changes in the solvent structure might be expected to affect the density of the hydrated polymer.19 However, changes in the hydronium and hydroxide ion concen-

SHARP ET AL.

776

trations in the mid-pH range are so small that this is unlikely. We conclude that some alteration in the conformation of the polymer in the gel-like precipitates measured in the CsCl gradients must occur as the pH increases and this alteration is responsible for decreased hydration or increased salt-binding. The lack of crystalline or fiber structure as well as no evidence of alpha helical or beta pleated sheet structures of the solid amorphous polymer20 gives no clues as to what conformations can be considered.

Buoyant Density as a Function of Degree of Deprotonation The data from the buoyant density titrations and the potentiometric titrations were combined to determine the buoyant density as a function of ionization. This gives information about the residue’s titration only in the transition regions of the buoyant density and potentiometric titration curves. For both residues of p o l y ( G l ~ ~ ~ . ~ Land y s the ~ ~ .glu~) tamic residues of p o l y ( G l ~ ~ ~ . ~ Tthere y r ~ ~is. sufficient ~), buoyant density data in this transition region to accurately construct such curves (Figs. 11 and 12). As in the case of homopolypeptidesl the data can be fit to a straight line. However, in the case of these copolypeptides, the linear relationship is continuous throughout the transition, and not divided into separate regions of different slope. In this case where the function is linear and continuous throughout the transition, the slope should be identical to the difference between the density of the copolymer prior to and after the buoyant density ti-

%

1.2

7 0.2 0.4 0.6 0.8 1.0

0

a Fig. 11. The buoyant density of Glu and Lys residues of p o l y ( G l ~ ~ ~ . ~asL ay funcs~~~~) tion of the degree of deprotonation in CsC1.

777

POLYPEPTIDE TITRATIONS. I1 I

I

I

I

I

I 0.0

€

1

-2

0

Q

I

I

I

0.2

0.4

0.6

1.0

a Fig. 12. The buoyant density of Glu residues of p o l y ( G l ~ ~ ~ . ~ as T ya rfunction ~ ~ . ~ )of the degree of deprotonation in CsC1.

tration. Since this is the difference between the flat portions of the buoyant density titration curves, it can be determined very accurately. The results of p o versus LY plots are summarized in Table IV, along with the differences in densities of the copolymers, as observed in Figures 1 and 2. Excellent agreement is observed for both residues of the p o l y ( G 1 ~ ~ ~ ~ ~The L y srather ~ ~ ~ poor ~ ) . agreement for the glutamic acid residues of Glu-Tyr is probably due to the rather large scatter of data points in the a versus p o plot. The density changes associated with the titration of glutamic residues in different polypeptides can be compared using data from poly(Glu), p o l y ( G l ~ ~ ~ ~ ~p L o lyys( ~G~l ~ ~~ ~) ., ~ Tand y r ~ovalbumin. ~.~), Comparisons using poly(G1u) are complicated because of a phase transition and the values of 0.73 g/ml per unit ionization and 0.073 g/ml per unit ionization' span the values obtained for the other polypeptides. The slope, TABLE IV Slopes of Buoyant Density as a Function of Degree of Deprotonation Compared to Changes in Buoyant Density

P o l y ( G 1 ~ ~ ~ ~ ~)L y s ' ~ . ' Glu residues Lys residues Poly(Glus4~ 5Tyr45 . ) Glu residues

Slope of po vs. 01

AP

0.150 g/ml 0.052 g/ml

0.150 g/ml 0.055 g/ml

0.1 36 g/ml

0.193 g/ml

SHARP ET AL.

778

0.150 g/ml per unit ionization for glutamic residues in polycan be normalized by dividing by the fraction of glutamic residues to give a value that can be compared to normalized values from other polypeptides of different composition. For the glutamic residues of p o l y ( G 1 ~ ~ ~ ~ ~ L y s ~ ~ ~ ~ ) : 0.150 g/ml = 0.246 g/ml/unit ionization 60.9/(60.9 39.1)

+

A similar calculation yields a value of 0.249 g/ml/unit ionization for the glutamic residues of p o l y ( G l ~ ~ ~ . ~ T y r ~ ~ . ~ ) . Finally, it has been determined that there is a change in density of 0.00054 g/ml per ionization of each of the 90 -COOH groups in ovalbumin.21 This can be converted to a change in density as described above for the copolypeptides. 90 X 0.00054 g/ml = 0.214 g/ml/unit ionization 90/397

This compares favorably with the ionization behavior of glutamic residues in poly (G1u60.9Lys39.1)and poly Quantitative comparisons of the titration of lysine residues in poly(G1u60.9Lys39.1)and ovalbumin cannot be done because of the experimental difficulties in precisely measuring the ovalbumin buoyant density titration curve. However, in both cases, the change in density upon ionization is relatively small.

CONCLUSIONS The buoyant density titration curves of five ionizable copolypeptides have been determined. As expected, their buoyant densities fall within the densities of the respective homopolypeptides. In order to calculate quantitatively where these copolypeptide buoyant densities should occur, utilizing the densities of the homopolypeptides, several predictive methods were examined. The only methods which gave consistently good results were methods A and B. These methods, described in detail previously, predict buoyant densities from the densities and the weight fractions of the amino acid residues. The densities are obtained from the homopolypeptide densities. The weight fractions are more difficult to obtain since they must include not only the amino acid residues, but also the associated hydrations and salt binding. Despite the superiority of these two methods, there are several sources of uncertainty. With future experiments designed to provide additional data to eliminate these uncertainties, the predictive methods should improve considerably. One source of uncertainty concerns calculations of hydrations. A derivation of the necessary equation for this calculation is presented in Appendix A. This equation allows the calculation of the “total” hydra-

POLYPEPTIDE TITRATIONS. I1

779

tion of a residue from the homopolypeptide buoyant density, provided that ion binding and partial specific volumes are known. A t present, we were only able to estimate the ion binding from electroneutrality principles and use calculated, rather than experimental, partial specific volumes. Another explanation of the discrepancies in the predicted results arises from changes in hydration and ion binding in going from the homopolypeptide to the copolypeptide. In most cases, the CsCl concentration changes considerably in going from the homopolypeptide density to the copolypeptide density. This could have significant effects. In addition, there are many cases where the precipitated homopolypeptide is used to predict the density of a soluble copolypeptide and vice versa. This change in phase of the polypeptide will almost certainly effect the hydration and salt binding. As a result, the calculated values from the homopolypeptides might differ from the actual hydration and ion binding to the copolypeptides. The copolypeptide potentiometric results obtained in CsCl solutions were used to help understand the relationship between the buoyant density and ionization behavior. Plots of the buoyant density as a function of the degree of deprotonation were linear. The slopes, the change in buoyant density per unit ionization, compared favorably for glutamic residues in poly( Glu), poly( G1u60.9Lys39.1),poly(G 1 ~ ~ ~ . ~ T and yr~~.~), ovalbumin. The buoyant density titration curves for the nonionizable homopolypeptides poly(G1y) and poly(A1a) are presented. The densities are not independent of pH, but rather have an inflection point around pH 7. The differences in densities between these two polymers a t identical pH’s are greater than would be expected by adding a methylene group to each glycine residue. These observations suggest that further experiments are needed to look for changes in partial specific volumes of the residues or changes in hydration and ion binding.

APPENDIX A The equation describing the equilibrium of electrically charged components in a gravitational field is expressed as

All variables and coefficients are the same as defined by The three equations for the three components considered, P for macroion, X for cation, and Y for anion, plus the electroneutrality condition stated by

yield four simultaneous equations for the four unknowns, d m p l d r , d m x l d r , d m y l d r , and d+ldr. Application of Cramer’s rule yields a determinant expression for dmpldr.

SHARP ET AL.

780

dmp= dr

AP Ax AY 0 CP

PPY PXY

ZP

ZX

CY ZY

ZY 0 ZP

PPX

cx PYX PPX

PPY

PXP

cx

PXY

PYP

PYX

ZP

zx

CY ZY

zx

zx ZY 0

Redefining the components as 2 = X,+Y,- and 3 = P X , and simplification of Eq. (18) in the manner described by Hearst yields Hearst's Eq. (22).

R and T are the usual thermodynamic parameters, the molality. The net solvation is defined as

p

is the chemical potential, and m is

(22) rz=---- 6m3

mz

and is in units of moles 2lmole 3. Water will be designated as component 1. Because the above equations are applicable to any neutral species, Eq. (19) can be rewritten in terms of component 1. Using the new notation, dmsldr becomes zero at band center and transforms Eq. (19) into,

A3 + r i A i = 0

(21)

Redefining A3 into its components yields,

As = Ap

+ nAx

(22)

and n is defined as

The quantity n represents the number of negatively charged sites on a homopolymer as well as the number of amino acid residues per homopolypeptide. This definition of n is not affected by considering anion binding to a homopolypeptide of the form PY,, where n = -Zp/Zy and is still a positive number. The validity of the derivation is not affected by considering anion or cation binding, as Hearstzz has stated. A p and A x are defined as,

MP Ap = - (1- Dppo)uzr RT

and A 1 is defined as

POLYPEPTIDE TITRATIONS. I1

781

M is the molecular weight of the respective component, ir is the partial specific volume of the component, w is the angular velocity, r is the distance from the center of rotation, and p o is the buoyant density (i.e., band center density).

Redefining Ag into its original components necessitates knowledge of the separate partial specific volumes of the charged homopolypeptides, irp, and the bound ion, ox. Although this cannot be done rigorously, partial specific volumes of the cesium and chloride ions can be approximated by methods described in the text. By assuming that ion binding to the charged homopolypeptide minimizes charge repulsion as well as co-volume effects associated with the charged side groups, specific volumes of the charged homopolymers can be approximated by Cohn and Edsall’s* calculated amino acid residue partial specific volumes. Substitution of Eqs. (22), (24), (25), and (26) into Eq. (21) and algebraic manipulation yields an expression for PO.

Division of both numerator and denominator by Mp yields,

nMx i+-+-rl MP

MI MP

vu -

nMx u p + -fix MP

MI +rlul MP

The third term in the numerator of Eq. (28) redefines the hydration parameter rl on a weight basis, and is denoted as r:. The subscript asterisk indicates hydration to a macroion-salt ion complex. The second term of the numerator can be defined as the grams of ion bound per gram anhydrous, saltfree homopolypeptide and is denoted as zx. These definitions result in: PO =

i+zx+r:

up + zxux

+ r:ul

z x and D X are analogous to zL and ir,, respectively, of Eq. (2).

APPENDIX B The buoyant density of the solvated macroion PX,Y, can be derived from the theory of the buoyant density equilibrium of the macroion in an ionic solution. Following the notation of Hearst22and the steps presented in Eqs. (16)-(18) yields the determinant of Eq. (18). (Eq. (18) being also applicable to the macroion-ion complex PX,Y,.) The charge of the polypeptide can be divided into the positive charge and the negative charge components.

zp = zp++ zp-

(30)

where Zp+ is the total charge of positive sites, all assumed to be accessible to anion binding, and Zp- is the total charge of the negative sites, all assumed to be accessible to cation binding. The following quantities are introduced in order to reduce the determinants of Eq. (18) to a 2 X 2 determinant.

-zy = zx

u+ = u-

(31) (32)

782

SHARP ET AL. m = --zP+ ZY

IAp + nAx

dmp dr

-

(34)

+ mAy}

+ V--BPY + m(v+@yx+ v-CY)J Iv+(u+Cx + v - B X Y )

lY+PPX

+ n(v + C X + u - 0 ~ ~ )

b + A x + V-AYI

f

ICP + nPpx + m o w + n@xp + nCx + mPxy) + m ( P w + nPyx + mCyN {v+(Pxp+ nCx

+ mPxY). + U - ( ~ Y P+ nPyx +mCy)l

p-(v+@Yxf

IU+PPX

.

v-cY)l

+ U-PPY

(36)

+ n(v+Cx + u-PXY)

+ m(v+Pyx + ~-CY)J Iv+(v+Cx + v - P X Y ) + V - ( U + P Y X + V-CYN

Substitution of Eqs. (37) and (38) and the defining expressions for Ci and & j Z 2 into Eq. (36) yields the determinant equivalent to Eq. (19). The steps followed in Eqs. (20) and (21) are applicable to the present consideration. A3 is redefined into its components, yielding

A3 = Ap + nAx

+ mAy

(39)

All parameters are as defined either previously in this text or by HearstZ2 Substitution of Eq. (39) and the defining expressions for Ap, A x , A y , and A1 into Eq. (19) and appropriate manipulation in the manner previously described for PX, yields,

POLYPEPTIDE TITRATIONS. I1

783

Division of Eq. (40) by M p in both numerator and denominator and redefining the interaction terms on a weight basis in the same manner as was done for PX, in Appendix A yields Eq. (7). This work was supported in part by research grants HE 11512 from the National Heart Institute and GM 18871 from the National Institute of General Medical Sciences. D.S.S., R.A., L.G.L., and J.S.V.Z. were supported by Undergraduate Research Participation Grants Numbers GY-4600, GY-5867, GY-7324, GY-8951, and GY-9791. D.S.S. was supported in part by the Health Professions Scholarship Program, Department of the Army. The authors are indebted to Dr. Ib Svendsen, Assistant Director, Carlsberg Laboratorium, Copenhagen, Denmark, for.discussions on salt binding to dipoles and peptide backbones, and Prof. John Hearst, University of California, Berkeley, for discussions of the thermodynamic theory of density gradient centrifugation.

References 1. Almassy, R., Zil, J. S. V., Lum, L. G. & Ifft, J. B. (1973) Biopolymers, 12,2713-2729. 2. Hearst, J. E. & Vinograd, J . (1961) Proc. Natl. Acad. Sci., 47,999-1004. 3. Williams, J. W., van Holde, K. E., Baldwin, R. L. & Fujita, H. (1958) Chem. Reu., 58,715-806. 4. Ifft, J. B. & Vinograd, J. (1966) J . Phys. Chem., 70,2814-2822. 5. Ifft, J. B. (1971) C. R. Trau. Lab. Carlsberg, 38,315-338. 6. Ifft, J. B., Voet, D. H. & Vinograd, J. (1961) J. Phys. Chem., 65, 1138-1145. 7 . Williams, A. E. & Ifft, J. B. (1969) Biochim. Biophys. Acta, 181,311-318. 8. Cohn, E. J. 81 Edsall, J. T. (1943) Proteins, Amino Acids, and Peptides as Ions and Dipolar Zons, Reinhold, New York, p. 370. 9. Ifft, J. B. & Williams, A. E. (1967) Biochim. Biophys. A c ~ Q136,151-153. , 10. Schleich, T. & von Hippel, P. H. (1969) Biopolymers, 7,861-877. 11. von Hippel, P. H., Peticolas, V., Schack, L. & Karlson, L. (1973) Biochem., 12, 1256-1264. 12. Hamabata, A., Chang, S. & von Hippel, P. H. (1973) Biochem., 12,1271-1278. 13. Gill 111, T. J., Ladoulis, C. T., Kunz, H. W. & King, M. F. (1972) Biochem., 11, 2644-2653. 14. Steinhardt, J. & Stocker, N. (1973) Biochem., 12,2798-2802. 15. Steinhardt, J . & Stocker, N. (1973) Biochem., 12,1789-1797. 16. Avruch, J., Reynolds, J. A. & Reynolds, J . H. (1969) Biochem., 8,1855-1859. 17. Foster, J. F. (1960) T h e Plasma Proteins, Putman, F. W., Ed., Academic Press, New York, p. 179. 18. Gratzer, W. B. & Beaven, G. H. (1972) Biopolymers, 11,689-705. 19. von Hippel, P. H. & Schleich, T. (1969) Structure and Stability of Biological Macromolecules, Vol. 2, Timasheff, S.& Fasman, G., Eds., Marcel Dekker, Inc., New York, p. 422. 20. Finegold, L. & Cude, J. L. (1972) Biopolymers, 11,683-687. 21. Ifft, J . B. & Lum, L. G. (1971) C. R . Trau. Lab. Carlsberg, 38,339-349. 22. Hearst, J. E. (1965) Biopolymers, 3,57-68.

Received January 1,1975 Accepted September 17,1975