Chain length dependence of solubility of monodisperse polypeptides

J. Phys. Chem. 1988, 92, 6161-6166

6161

Chain Length Dependence of Solubility of Monodisperse Polypeptides in Aqueous Solutions and the Stability of the &Structure Akio Nakaishi, Hiroshi Maeda, * Tetsuo Tomiyama, Shoichi Ikeda, Department of Chemistry, Faculty of Science, Nagoya University, Chikusa, Nagoya 464, Japan

Yuji Kobayashi, and Yoshimasa Kyogoku Institute for Protein Research, Osaka University, Suita, Osaka 565, Japan (Received: February 8, 1988)

Constant and reversible solubilities in water of uncharged poly(S-(carboxymethyl)-L-cysteine) were determined on monodisperse samples consisting of 10-12 repeating units. On the sample consisting of 13 repeating units, reversible and time-independent solubilities were obtained but they increased with the total concentration, which was regarded as manifestation of polymeric nature. Differences of free energy AGO, enthalpy AH",and entropy AS" between the solvated disordered state in solution and the @-structurein the precipitates were determined from the solubilities and their dependence on temperature. Contributions per uncharged residue to these thermodynamic quantities, AGOR, WR, and SoR, were evaluated on the basis of their chain length dependence. Linear dependence on the chain length was found in the case of AGO (at 25 "C), while it was assumed in evaluating A H O R and A S O R . The obtained values at 25 O C were as follows: AGOR = -3.8 kJ (-9.0 X lo2cal)/mol, WR = -6.2 kJ (-1.5 X lo3 cal)/mol, ASoR = -8 J K-' (-2 cal K-')/mol. Stabilization of the @-structuredue to aggregation was suggested to be largely of entropic nature. A lower bound for the cooperative unit (six residues) was estimated for the coil-fl-structure conversion of long polypeptide chains.

Introduction

Solubilities of long-chain molecules often increase with the total amount, which has been well-known as Bodenkorperregel.' Well-defined solubilities will not be obtained even with monodisperse preparations if chain lengths are long enough, since it usually takes a long time for the internal equilibrium of polymer solid to be reached. For a given kind of polymer, then, it is interesting to find the critical chain length range for well-defined solubilities to appear. In the present study, solubilities of monodisperse polymers of S-(carboxymethyl)-L-cysteine were examined for a range of chain length 10-13. Another relevance of the present study is related to the stability of the @-structure,an important secondary structure frequently found in proteins. Stability of the @-structure of polypeptides relative to random coils is still unexplored area. There are two approaches to the problem. Potentiometric titrations on long-chain polypeptides accompanying the coil-@-conversion yield free energy difference between these two states of uncharged However, most published titration data to date contain the contribution from the aggregation of @-sheetsin addition to that from conformational change, except poly(L-tyrosine) (PLT).4v5 This problem was analyzed and discussed previously.6 Another approach in terms of association equilibrium of extended short chains showed that the stability of these intermolecularly associated &structures strongly depended on chain length.7.8 In the case of poly(L-glutamic acid),7 the coexistence of the a-helix seems to interfere with evaluating correct chain length dependence of the stability of the @-structure. N o such unfavorable situation exists in the case of poly(S-(carboxymethyl)-L-cysteine) (p0ly[Cys(CH,C00H)]).~ However, no soluble /3-structure was formed in aqueous solutions with monodisperse samples, as shown in the present study. Therefore, only the stability of the @-structure in the solid state can be unam(1) Kruyt, H., Ed. Colloid Science; Elsevier: New York, 1949;Vol. 11, pp 76, 161, 243. (2) Pedersen, D.; Gabriel, D.; Hermans, J., Jr. Biopolymers 1971, 10,2133. (3) Maeda, H.;Ikeda, S . Biopolymers 1971, 10, 2525; 1975, 14, 1623. (4)McKnight, R. P.; Auer, H. E. Macromolecules 1976, 9, 939. (5) Auer, H. E.; McKnight, R. P. Biochemistry 1978, 17, 2798. (6) Maeda, H.Bull. Chem. SOC.Jpn. 1985,58,618;J. Phys. Chem. 1984, 88,5129. (7) Rinaudo, M.; Domard, A. J. Am. Chem. Soc. 1976,98, 6360. (8) Maeda, H.; Iwase, T.; Ikeda, S . Polym. J. 1984, 16, 471.

0022-3654/88/2092-6161$01.50/0

biguously evaluated from the solubilities of these monodisperse samples, which will be reported here. Experimental Section

Monodisperse samples were prepared from the polydisperse samples previously synthesized: by repeated fractionation on ion-exchange column (DEAE-Toyopearl, 650M d = 3 cm, 1 = 80 cm). Amino and carboxy terminals of the samples were blocked with acetyl andd ethylamide groups, respectively. Polypeptides were recovered as precipitates at low pH (2-2.5) at 4 "C. This procedure was effective to remove the contamination of salt but the yield of recovery of samples was not good. Final salt concentration was at most 50% (w/w) in the case of DP = 10, and smaller for other samples. Purity with respect to chain length distribution was examined with HPLC (column: TSK DEAESPW), eluted with 0.36 M NaCl 0.01 M phosphate buffer (pH 6.0) and monitored by absorption at 220-230 nm. However, the most sensitive criterion for the monodispersity was solubility itself, which will be shown in the text. Chain lengths of these samples, as determined on the basis of the ratio of the areas on 'H N M R spectra as described previo~sly,~ were 9.7 f 0.3, 11.0 f 0.2, 12.1 f 0.5, 12.9 f 0.2, and 15.0 f 0.3. Elution patterns of these samples on HPLC are shown in Figure 1. In the region of elution volume outside the indicated range in Figure 1, no peaks were found for all the five preparations. As is well-known, the logarithm of elution volume V, is linearly proportional to the number of charges or chain length n, which is expressed as9J0

+

n = K log V,

+C

(1)

In Figure 2 such relations at three different values of pH are shown on the present samples. The values of K and C were 11.4 and -1.72, 10.4 and -1.34, and 8.53 and -0.04 for pH 7.40, 6.70, and 6.00, respectively. The elution volumes of two reference substances carrying three negative charges, adenosine triphosphate and sodium citrate, also satisfied the relation of eq l . This consistency supported the assignments of chain lengths by 'H N M R . Solubility was measured in the following way. Polypeptides of a given chain length were dissolved in water at neutral pH (pH zz 7). By diluting the stock solution, solutions of different con(9) Maeda, H.;Ito, T.; Suzuki, H.; Hirata, s.;Kako, I.; Yoshino, M.; Ikeda, S.; Kobayashi, Y. Biopolymers 1983, 22, 2173. (10)Rinaudo, M.; Domard, A. Biopolymers 1975, 14, 2035.

0 1988 American Chemical Society

The Journal of Physical Chemistry, Vol. 92, No. 21, 1988

6162

I

a

I

e n

I

200

220

240

A/nm

Figure 3. Circular dichroism spectra of sample DP = 11 in solution with no added salt at various values of pH. Polymer concentration: C, = 1 X M. pH: (a) 6.54, (b) 4.36, (c) 3.52, and (d) 2.28.

log Ve

Figure 1. Elution patterns of samples on HPLC. Ionic strength, 0.36 M NaCl + 0.01 M phosphate buffer (pH 6.0). Monitored by absorption at 220-230 nm. Degrees of polymerization based on IH NMR spectra: (a) 10, (b) 11, (c) 12, (d) 13, and (e) 15. I

Nakaishi et al.

r

1

10

% M

,"

I

I

I

-4

'

I

01' -6

t--------"i

b

4 I

0 0

< -10 u

-14

L I

I

I

I

2

3

4

5

PH

Figure 4. Dependence of residue ellipticities at 230 nm (A) and 200 nm (B) on pH for four samples: (0)DP = 11, (a)DP = 12, ( 0 )DP = 13, (A) DP = 15. Concentrations: 1 X M for DP = 11, 12, and 13; 5 X lo4 M for DP = 15.

I 10

1.2

1.4

16

18

log Y e

Figure 2. Relationship between the logarithm of elution volume V, and the number of charges or chain length n of each sample eluted at different pH: (a) 7.40, (b) 6.70, (c) 6.00. The values of K and C in eq 1 were obtained from the fitted lines in the figure.

centrations were prepared in the tubes for centrifuge use (polycarbonate) followed by the addition of different amounts of HC1 to provide them a common pH (within f0.02). These solutions were incubated at 25 "C for about 12 h followed by centrifugation (7000 rpm, lOOOOg) under atmospheric pressure at 10 "C for 1 h and then incubated at 25.0 f 0.5 "C without stirring for 5-7 days. Intermittently, concentrations of supernatant solutions were determined from circular dichroism (CD) measurements. C D spectra of solutions were taken with a Jasco 540 A spectropolarimeter with cells of light paths 0.2-20 nm. Usually four scans were accumulated. The amino acid analyses of the samples were carried out with an amino acid analyzer HITACHI 835s after hydrolysis of the sample at 110 "C for 24 h by 6 N HC1 (using twice distilled water). S-(Carboxymethyl)-L-cysteinewas used as a reference for the optical density measurements. Results Determination of Polypeptide Concentration. CD spectra of a monodisperse sample DP = 11 are shown in Figure 3 at different values of pH. At neutral pH where all carboxyl groups were dissociated, the C D spectra (a) almost coincided with those of long-chain random coil polyions.11,12 At low pH, on the other hand, CD spectra of uncharged and disordered chains were ob(11) Maeda, H.; Gatto, Y.; Ikeda, S. Macromolecules 1984, 17, 2031. (12) Maeda, H.; Ooi, K. Biopolymers 1981, 20, 1549.

tained. The change of C D spectra with pH was associated not with the formation of the @-structurebut with some direct effect of charges on the chromophore.12 The change was characterized with an isodichroic point at 208 nm and could be described in terms of two species (charged and uncharged residues). As shown was indein Figure 4, the residue ellipticity at 230 nm [0]2s0 pendent of pH in the range below 4.0 and nearly the same for the chain lengths 10-15. Similar results were found for [e],, for the chain length 10-14. A deviation of [el2, of sample DP = 15 at pH range below 4 is ascribed to the formation of the @-structure. However, the maximum @-content(at pH 3.15) was about 0.06 and hence negligible. These results on monodisperse samples were in contrast with those on quasi-monodisperse samples:* the @-contentsunder the same conditions were 0.65-0.7 for (DP) 15-16 and 0.06-0.08 for (DP) zi 14. It was thus found that the @-structuresin solution consisting of short chains were less stable than the precipitates as the sample became more monodisperse. It is to be noted also in Figure 4 that optical activity of disordered chains exhibits little chain length dependence, a different situation from the reported dependence on poly(rly~ine).'~.'~ Since for the samples DP = 10-13 no complexity exists associated with possible aggregates (&structure) in solution, the concentration of the polypeptide, expressed in residue molarity, Cp,was calculated according to eq 2, where e230/l is expressed

=

(2) in deg cm-I. A value of -3800 deg cm2 dmol-I was determined for based on quasi-monodisperse preparations in the present c p

= 102(%o/1)/[e1230

(13) Yaron, Y.; Otey, M. C.; Sober, H. A.; Katchalski, E.; Ehrlich-Rogozinskia, S.; Berger, A. Polymers 1972, 1 1 , 607. (14) A more careful study is necessary to get any definite conclusion, since the range of chain length (10-15) is very narrow as compared with the reported case."

Solubility of Monodisperse Polypeptides

The Journal of Physical Chemistry, Vol, 92, No. 21. 1988 6163 I

I

I

1

I

1

1

10

100

I

CS/~O-~M

Figure 8. Dependence of residue ellipticitiesat 230 nm at 15 OC on the concentration of NaCl (Q. Sample DP = 11. 50

0

100

I

,

I

I

'

I

I

1

I

1

1

I

I

0"

1

1

3

4

5

6

7

8

I

I

{

I

200

t/ h

Figure 5. Temperature dependence of the time course of the supernatant concentration for sample DP = 11: (a) 35 O C , (b) 25 "C, and (c) 15 OC.

0

20

0

50

40

30

80

t / h

Figure 6. Time course of the residue ellipticities at 230 nm of sample DP = 11 at 25 OC after eauilibration of the solution at various temuer(0) 35 O C , (a) 2'5 OC, (0) 15 OC, and (A) 10 "C.

cp /10-3 M Figure 9. Solubilitiesat three different temperatures shown on sample DP = 11: (0) 15 OC, ( 0 )25 OC, (0) 35 OC.

I

'

I

1 I

I

I

0.2

0.4

0.6

I

I

0.8 1.0 C~/~O-~M

I

I

I

1.2

1.4

1.6

Figure 7. Time-independent solubilities of sample DP = 13 at 25 OC for different thermal histories. Solutions were incubated for 24 h at 35 OC (0) or 15 O C (0) before the measurement at 25 OC.

study. The solubility of sample DP = 15 was determined by means of amino acid analysis since it was too low to be reliably determined with CD. Attainment of Constant Solubility and Its Reversibility. In Figure 5 , the time course of the supernatant concentration is shown for sample DP = 11. Constant values were obtained within shorter periods at higher temperatures. To confirm that these constant values correspond to equilibrium solubilities, reversibility with respect to temperature change was examined, as shown in Figures 6 and 7. Figure 6 shows the time course of the supernatant concentration of sample D P = 11 at 25 OC. When the tubes containing both solution and precipitates were incubated at 35 "C for 24 h prior to equilibration at 25 O C , the equilibrium rate at 25 "C was rather rapid. On the other hand, when the tubes were incubated at 10 or 15 OC for 24 h, equilibration at 25 O C took place more slowly. However, final concentrations coincided with each other irrespective of the incubation temperatures prior to equilibration at 25 OC. The constant value was thus regarded as the equilibrium solubility. Time-independent solubilities of sample DP = 13 at 25 OC for different thermal histories are shown in Figure 7. Two series of solution + precipitates, incubated for

?w 24

'0

I

I

I

I

0.5

1.0

1.5

2.0

Cp/I@M

Figure 10. Solubilities at three different temperatures shown on sample DP = 13: (0) 15 OC, ( 0 )25 OC, (0) 35 OC.

24 h, one at 35 OC and the other at 15 OC prior to equilibrium at 25 OC, gave identical results. In this way, constant levels of supernatant concentrations attained for a period about 3-7 days after the preparation of solution were equilibrium solubilities. Equilibrium solubilities at 35 and 15 OC were also obtained within this time range. Since contamination of the salt was not completely avoided and its extent differed from sample to sample, one might suspect that the effect of ionic strength made it difficult to obtain the correct dependence on chain length. Effect of ionic strength was examined with sample DP = 11, as shown in Figure 8. The effect of ionic strength was negligible when it was lower than 2 X lo-* M. The highest salt concentration encountered in the present solubility M for a sample DP = 10 at C, = measurements was 1.5 X 9X M (the highest concentration examined). Therefore, all the solubilities reported in the present study were regarded as those

6164

Nakaishi et al.

The Journal of Physical Chemistry, Vol. 92, No. 21, 1988

TABLE I: Solubilities of Monodisperse Poly[Cys(CH2COOH)]in Water

T,

DP 10

*

25 15 25 35 15 25 35 15 25 35 25

11 12 13 15

solubility molarity (5.73 0.07) x 1 0 4 (5.82 f 0.02) x 10-5

residue molarity (5.73 0.07) x 10-3 (6.40 o . o i j x 10-4 (1.48 i 0.02) x 10-3

OC

mole fraction (1.04 f 0.01) X (1.05 o.oij x io” (2.42 f 0.03) X 10” (5.27 f 0.08) X 10” (1.7 f 0.1) x 10-7 (4.4 A 0.2) x 10-7 (1.07 f 0.01) X 10” (4.2 f 0.3) X (1.1 i 0.05) x 10-7 (2.9 f 0.1) x 10-7 (4.4-7.6) X IO-’’

(1.34 f 0.02) X IO4 (2.90 f 0.05) X lo4 (9.2 f 0.4) X 10” (2.4 0.1) x 10-5 (5.88 f 0.02) x 10-5 (2.3 f 0.2) X 10” (6.2 f 0.2) X 10” (1.6 f 0.1) x 10-5 (2.5-4.3) x 10-7

(3.20 f 0.05) X IO-’ (1.1 f 0.1) x 10-4 (2.9 f 0.1) X IO4 (7.05 f 0.02) x 10-4 (3.0 0.2) x 10-5 (8.0 f 0.3) X (2.1 i 0.1) x 10-4 (3.7-6.7) X 10”

“1

e

3 I

1

1

1

1

1

I

l

1

I

1

1.4 ~

0

0.2

0.4

e,

1 (10-3 Y )

3

5

Figure 11. Solubilities of two samples 25 O C : (A) DP = 10, (B) DP = 12.

1.3

in water. The effect of salt concentration C,on the solubility in Figure 8 was well described with a salting-out constant K as log S = -K log C,

+C

(3)

A value of 1.07 was obtained for K on uncharged sample DP = 11, which is identical with that for f i b r i n ~ g e n . ’ ~ Solubility and Its Temperature Dependence. In Figures 9-1 1, solubilities of uncharged polymers in water are shown for samples of DP = 10, 11, 12, and 13. In Figures 9 and 10, solubilities at three different temperatures are shown on samples DP = 11 and 13, respectively. The solubilities increased with increasing temperature for both polypeptides. Hence the heat of crystallization was negative. Since the solubilities of sample DP = 10 were high, the available amount of the sample prevented us from measuring the solubility at temperatures other than 25 O C . The solubilities of sample DP = 12 at two temperatures other than 25 O C were measured with a smaller number of points than at 25 “C because of shortage of the amount of the sample. the solubilities at 25 “ C of those two samples are shown in Figure 11. In Figures 9 and 11 three samples (DP = 10-12) show constant solubilities irrespective of the total concentration C,, indicating that the phase equilibria of the polypeptides take place like low molecular weight compounds up to the chain length 12. On the other hand, definite slopes were found on sample DP = 13 at all three temperatures examined, as shown in Figure 10. If this slope arose from polydispersity of the sample, the slope would decrease for the sample recovered from the precipitates. However, the slope was unchanged for those samples once and twice recovered from precipitates, as shown in Figure 12. It is likely that the slope (15) Edsall, J. T.; Wyman, J. Biophysical Chemistry; Academic: New York, 1958; Vol. 1, Chapter 5 , p 274.

i L

-

N

0

E. 1.2

c

0 ID

4

I

1.1

1.0

3.2

I

1

1

3.3

3.4

3.5

T-11 10-3

K-1

Figure 13. van’t Hoff plots on three samples: (a) DP = 13, (b) DP = 12, and (c) DP = 11.

represents some characteristics of polymeric nature. This point will be discussed later. Solubilities are summarized in Table I in different concentration scales. They are expressed in terms of the whole chain molecule basis as well as residue basis. In Table I, the solubility of sample DP = 15 at 25 O C is also given. The solubility S is related to the free energy difference between molecularly dispersed hydrated disordered chains p, and intermolecularly hydrogen-bonded /%sheets in three-dimensional aggregates pg as follows. pc(T,P,S) = p*,(T,P) + R T In S = pg(T,P) (4) AGO = pg - p*c = R T In S

(5)

The Journal of Physical Chemistry, Vol. 92, No. 21, 1988 6165

Solubility of Monodisperse Polypeptides

TABLE II: Summary of AGO, L W O , and AS' Associated with Crystallization and Contributions per Residue AG O R , UoR, and ASRO

DP

-AGo/kJ mol-I (cal mol-I)'

-AHo/kJ mol-l (cal mol-')b

-ASo/J K-I mo1-I

(cal K-'mol-I)"

10

28.4 f 0.3 (679 f 7) 11 32.1 f 0.4 60.0 f 2.9 9 4 f 10 (767 f 10) (1.43 f 0.7) X IO3 (2.29 f 0.2) X lo' 12 36.3 f 1.3 68.0 f 0.8 106 f 4 (1.63 f 0.2) X IO' (2.53 f 1.0) X lo3 (868 f 31) 13 39.7 f 0.5 72 f 2 107 f 7 (1.70 f 0.5) X lo3 (2.56 f 0.2) X lo3 (949 f 12) 15 46-48 (1.1-1.2) x 103 -AGoR/kJ mol-' -AHoR/kJ mol-' -ASRo/J K-' mo1-I (cal mol-') (cal mol-')c (cal K-I 3.8 f 0.1 (90 f 2) X 10

IO

11

12

14

13

DP

(DP).

12

13

DP

Figure 15. Dependence of AHo (0) and AS' ( 0 ) on the degree of

polymerization (DP). Contribution from the second virial coefficient to pc was evaluated assuming a rigid rod and it was found to be negligible for chain length encountered in the present study. In Figure 13, van't Hoff plots are shown for three samples, DP = 11, 12, and 13. The enthalpy change A W was evaluated from the slope, according to AHo = d(AGO/T)/d(l/T)

8 f 6 (2 f 1)

~ on the assumed 'Values at 25 OC. bFor 15-35 "C. C V a l u ebased linear dependence on DP.

15

Figure 14. Dependence of AGO at 25 OC on the degree of polymerization

11

6.2 f 2.4 (1.5 f 0.6) X lo3

(6)

In Figure 14, chain length dependence of AGO at 25 OC is shown. A straight line was obtained which can be expressed as (expressed in kJ mol-') AGO = 3.8n - 9.6 (7) From the slope we can evaluate the contribution per residue AGOR, which is free from the end effect and does not include the contribution of mixing entropy. This is the required quantity. Dependence of both AHo and ASoon chain length are also shown on chain length in Figure 15. Linear dependence of AH" and So was assumed to estimate the contributions per residue, AHOR or A S O R , as indicated by two straight lines in Figure 15, which were drawn according to the least-squares methods. A summary of thermodynamic quantities is given in Table 11. A value of -3.8 kJ/mol (-900 cal/mol) was obtained for AGO. It is greater than those obtained from titration of high molecular weight poly[Cys(CH2COOH)]accompanying pre~ipitation.~ This

indicates that three-dimensional aggregates consisting of short monodisperse chains are more stable than precipitates consisting of long chains. For the former case after the attainment of the initial equilibrium (within 5-7 days), well-ordered two-dimensional &sheets develop and further they stack with each other to produce three-dimensional crystals. In the latter case, on the other hand, such rearrangement in the precipitates will be difficult because many chain entanglements are supposed to occur for a given polymer chain. A value of -6.2 kJ/mol (-1500 cal/mol) was obtained as the enthalpy change, A H O R , based on linear approximation and this value is rather similar to that reported on PLT.4 It is to be noted that the enthalpy changes obtained in the present study are expected to be very close to those from calorimetry, although they were evaluated from van't Hoff plots. This is because there is no ambiguity associated with the problem of cooperative unit in solubility phenomena. According to this approximation, the present results enable us to estimate for the first time the size of the cooperative unit in the @-coil conversion in. the case of the long chains. The van't Hoff enthalpy change associated with the coil-/3 conversion of long chains has been obtained recently as -42 kJ (-10 kcal),16which gives a cooperative unit consisting of six residues when combined with the present data. However, this value should be taken as the lower bound, since a positive electric contribution was included in the value (-42 kJ).16

Discussion 1 . Dependence of the Solubility on the Total Polymer Concentration. In the present study the solubility of sample DP = 13 increased with the total polymer concentration, although the solid-solution phase equilibrium was confirmed to be reversible with respect to temperature change as shown in Figure 7. Generally, this behavior is ascribed to polydispersity of the sample with respect to chain lengths. In the present case, however, nearly identical solubility curves were obtained on the samples once and twice recovered from the precipitates (Figure 12). This indicates that Bodenkorperregel in the present case should be explained by a term other than polydispersity. As described in the Introduction, solid phases consisting of long-chain molecules are not always in the true internal equilibria or their Gibbs free energies are not always the minimum consistent with given thermodynamic variables. Precipitates formed in the solution of higher concentrations are likely to be less perfect than those formed at lower concentrations. Stabilities of two precipitates formed at concentrations differing by 10 times were estimated from Figure 10, assuming phase equilibria between solutions and metastable solid phase at both concentrations. A value of 0.28kT was obtained for the difference in the stabilities for a chain (16) Fukada, K.; Maeda, H.; Ikeda, S. Polymer 1987, 28, 1887.

6166 The Journal of Physical Chemistry, Vol. 92, No. 21, 1988 ( A i

Figure 16. Schematic representation of two processes of interest: (A) The unimolecular conversion for a long-chain polymer from random coil to the folded chain (unaggregated)p-structure. (B) Three-dimensional crystal formation consisting of short polymer chain in the 8-conformation from disordered chains in solution.

molecule (DP = 13). On the residue basis, this amounts to about 2.2 X 10-ZkT. In this way, the stability difference between the two precipitates formed at different total concentrations is very small, less than thermal energ,y kT. Nevertheless, the relaxation to the mast stable state cannot occur practically, probably because a kind of cooperative motion of many chains will be required for the relaxation to occur. Random motion of individual chains are not effective at all to drive the relaxation process. The phase equilibrium between the solution and the solid phase, on the other hand, is expected to be attained through random motion of individual chains. 2. Stability of the p-Structure. In the present study, a value of 3.8 kJ/mol (900 cal/mol) was obtained for -AGoR, corresponding to the free energy of transfer of a residue from the internal portion of a polymeric chain in random coil conformation in water to the interior of three-dimensional crystals consisting of polymer chain in the 8-conformation. For a long-chain poly[Cys(CH,COOH)] a value of 1.7 f 4 kJ/mol [(400 f 100) of cal/mol] was evaluated for the free energy change -AGO, the unimolecular conversion from random coil to the folded chain (unaggregated) @-structure." In Figure 16 two processes associated with the two quantities AGoR and AGO,@ are schematically drawn. They refer to rather similar processes and hence their comparison is meaningful. The difference, about -(2.1 f 4) J/mol (17) Kimura, M.; Maeda, H.; Ikeda, S. Biophys. Chem. 1988, 30, 185.

Nakaishi et al. [-(500 f loo)] cal/mol, is considered to be the stabilization due to aggregation or crystallization. The contribution from aggregation is large and comparable to or even slightly greater than that from conformational change. Similar results have been found on PLT; -2.1 kJ/mol (-500 cal/mol) for aggregated @-structureI8 and -1.3 kJ/mol (-310 cal/mol) for the folded-chain @-structure consisting of a single polypeptide chain.4 Stabilization due to aggregation is then expected to be general in the case of the @-structure. In contrast to AGOR, the values of AHoR are rather similar for the @-sheetof a single polypeptide chain of PLT (-5 to 6 kJ/mol) and for the present result (-6.2 kJ/mol) of three-dimensional aggregates consisting of monodisperse oligopeptides. These difon the effect of aggregation ferent dependences of AGOR and WR suggest that the contribution from aggregation to the stability is largely of entropic nature. The enthalpy changes associated with the a-helix-random coil conversion of uncharged polypeptides have been reported on poly@-glutamic acid) to be -7.1 to -4.1 kJ/mol (-1700 to -975 c a l / m 0 1 ) ' ~ -and ~ ~ on poly(L4ysine) to be -3.7 to -3.3 kJ/mol [-(885 to 790) ~ a l / m o l ] ' ~ ~These ~ ~ Jvalues ~ are rather similar to AHoRof the @-structuredescribed above. This is reasonable since the major contribution to AHOR is expected to be the hydrogen bond between peptide groups and hence to be independent, to the first approximation, of the kind of amino acid residue or of the secondary structure. Consequently, the major difference in AGOR between the a-helix and the @-structureis likely to be of entropic nature. In this context, it is to be noted that different values of AHOR have been reported in the case of the @-structureof poly-L-lysine: 3.64,2 0.54,25and -5.7 1 kJ/mo1.26 Slight enhancement of the stability (AHOR) of the @-structureover the a-helix is likely to arise from different environments of peptide groups in these two secondary structures: the dielectric constant will be lower in the interior of the @-sheetthan in the a-helix where peptide groups are exposed to aqueous solvent. Registry No. Poly[Cys(CH,CO,H)] (homopolymer), 29433-95-2; poly[Cys(CH2C0,H)] (sru), 3 1851-29-3. (18) Senior, M. B.; Gorrell, S. L. H.; Hamori, E. Biopolymers 1971, 10,

2387.

(19) Hermans, J., Jr. J. Phys. Chem. 1966, 70, 510. (20) Rialdi, G.; Hermans, J., Jr. J . Am. Chem. Soc. 1966, 85, 5719. (21) Olander, D. S.; Holtzer, A. J . Am. Chem. SOC.1968, 90, 4549. (22) Bychlcova, Y. E.; Ptitsyn, 0. B.; Barskaya, T. V. Biopolymers 1971, 10, 2161. (23) Maxfield, F. R.; Alter, J. E.; Taylor, G. T.; Scheraga, H. A. Macromolecules 1975, 8, 479. (24) Barskaya, T. V.; Ptitsyn, 0. B. Biopolymers 1971, 10, 2181. (25) Cosani, A.; Terbojevich, M.; Romanin-Jacur, L.; Peggion, E. Peptides and Proteins; Blout, E. R., et al., Eds.; Wiley: New York, 1974; p 166. (26) Chou, P. Y.; Scheraga, H. A. Biopolymers 1971, I O , 657.