BIOPOLYMERS VOL. 4 (1966)

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Equilibrium and Kinetic Studies of the Cooperative I S I1 Transition in Poly- L-proline

The synthetic polymer poly-L-proline exists in two conformational forms, I and 11, the structures of which have been elucidated by x-ray crystallography.'-3 Form I is a right-handed helix with all peptide bonds in the cis configuration, and is stable in pyridine and aliphatic alcohols. Form I1 is a left-handed helix having all peptide bonds in the trans configuration, and is stable in water, acetic acid, formic acid, and benzyl alcoh01.~~~

The reversible transformation I ~ I1 may be induced by appropriate changes of the solvent system.4 Measurement of the considerable change of optical rotation accom- panying the transition affords a convenient means of studying it. The rate of the transi- tion is independent of the polymer concentration, indicating that the transformation is intramolecular. trans isomerization of the peptide bond.4 One of the arguments in favor of that comes from the fact that the theoretically predicted energy of activation of about 21 kcal./mole6 for the rotation about this partial double bond is in good agreement with the experimentally observed values.'J The nar- row range of solvent composition in which the transition occurs indicates that it is coop erative, i.e., that the equilibrium and rate constants for the isomerization of a given pep- tide unit depend on the conformation of its neighbors.

The most simple model for approximating such a process is that in which only nearest- neighbor interactions are considered. A statistical thermodynamical theory which is based on such a model has been developed by Zimm and Braggo for the cooperative helix

It is proposed in this paper, that a generalized form of the Zimm-Bragg model can also be applied to the I I1 transition in poly-L-proline. Therefore, the equilibrium theory of Zimm and Bragg and the kinetic theory of Schwarz,'O.11 which are both based on the Zimm-Bragg model, must be capable of describ- ing the transition of poly-L-proline, a t least to a first approximation.

The propagation step, in which a helix I unit is added to an existing helix I segment, is assigned an equilibrium constant s, according to the Zimm-Bragg theory. The equilib- rium constant for the nucleation step, in which a helix I unit is created between helix I1 units, is us. Studies with spacefilling models verified that both the nucleation step (e.g., a trans + cis isomerization somewhere in the middle of a I1 segment) and the p r o p agation step (e.g., a trans + cis isomerization at the boundary between a section of I and a section of I1 helix) are sterically possible. This shows that the above definitions are meaningful. It also emerged qualitatively from these studies that the nucleation should be more difficult than the propagation step. In the former, the segments adja- cent to the site of nucleation are forced to assume a very special (antiparallel) position, whereas the transformation of one kind of unit into the other at a boundary seems to add little or no restriction to the degree of freedom of the molecule. It is therefore expected that the nucleation parameter U, which is small if the difficulty of nucleation is large, is much smaller than 1 in the case of the I I1 transition in poly-L-proline. The param- eter u is a characteristic constant for each system, whereas the equilibrium constant s depends on external parameters. For the case of poly-Lproline it changes with a change of solvent composition, thereby inducing the transition. From the results of the Zimm- Bragg theory it can be concluded that the sharpness of the transition for a given system decreases with decreasing number of residues per polymer chain. Moreover, its mid- point is shifted toward higher s values. That these predictions hold for the ply-L-pro- line system is shown by the results displayed in Figures 1 and 2.

The elementary process is the cis

coil transition of polypeptides.

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946 COMMUNICATIONS TO THE EDITORS

(D *) .I

‘I

u cl I

1000 $-O-

800 - M = 3,300

600 -

400 -

‘ \ 200 -

\ 0 ’ ” ’ . ” ” ’ 0 50 100

100 0 - n-BUTANOL

PERCENT (v./v.)

BENZYL ALCOHOL Fig. 1. Equilibrium values of the specific optical rotation a t 436 mp as a function of

solvent composition in the system benzyl alcohol/n-butanol for poly-L-proline: (0 ) molecular weight 3300; (0) molecular weight 21,000. Polymer concentration, 0.25- 0.50 g./100 ml.; T = 70°C. The straight line in the lower part of the diagram shows the dependence of specific optical rotation of polyproline I on solvent composition. This was measured by fast dilution of a solution of form I polymer in 90% n-butanol with varying amounts of benzyl alcohol. The dilu- tion was carried out a t 20°C., since a t this temperature the conversion into form I1 is ex- tremely slow. The optical rotation of polyproline I1 did not depend of the composition in this solvent system (not shown in the figure).

All values are corrected for refractive index.

In Figure 1 are plotted equilibrium values of the specific optical rotation as a function of solvent composition for two samples of poly-cproline (synthesized by a Leuchs anhy- dride polymerization12) having molecular weights 3300 and 21,000, respectively, as de- termined by the Yphantis midpoint method.13 The degree of conversion into polypro- line I, eI, has been calculated from these data and is plotted in Figure 2 versus solvent composition. The evaluation of u from equilibrium data is difficult because the dependence of s on the ratio of benzyl alcohol to n-butanol is not known.

Information about the kinetics of the transition can be obtained from chemical relaxa- tion experiments. Schwarz’s theorylO.ll predicts that for infinitely long chains (number of residues n + m ) the mean reciprocal relaxation time I * should have a maximum at 01 = 0.5. Its value T * ~ ~ ~ is ‘ / 4 a k ~ , where k F is the rate constant for the elementary propagation step, which is the cis + trans isomerization of the peptide bond. For finite chain lengths, I* decreases with decreasing n.14 Experimentally I* may be obtained from the relation

The equilibrium constant s runs from right to left a t the abscissa.

I / ~ * = (i/aer)(der/di),-o In this equation (deI/dt)t=o is the initial change of eI, and is measured after a sudden

jump of solvent composition which causes a Ath. An equilibrated poly-cproline solution of a certain solvent composition (e.g., initially 60% n-butanol) was “disturbed” by the addition of a solvent mixture which shifted the composition to a new value (e.g., to 70% n-butanol). The dilution in polymer concentration was always twofold. The ratio of n-butanol to benzyl alcohol in the mixture used to upset the equilibrium was calculated such that the jump in solvent composition was always from X to X + A per cent n-bu- tanol. A constant stepwidth A was employed. Twofold dilution by a solvent with equal

COMMUNICATIONS TO THE EDITORS 941

0 PERCENT (v.lv.1 - BENZYL ALCOHOL

100

n-BUTANOL 81

was calculated from the formula er = ([a111 - ( [ a ] ) / ( [a111 - [ a ] ~ ) . Values for the specific optical rotation of form I, [(Y]I, corresponding to the solvent composition for which eI had to be calculated were taken from Figure 1. The specific optical rotation of form 11, [(Y]II, was independent of solvent composition and values of - 1040 and -980' were chosen for molecular weights 21,000 and 3300, respectively.

Fig. 2. Degree of conversion into polyproline I , eI, versus solvent composition.

1000

500

zoo

v) u c 100 3 Z

I * 50 P

-

20

10 J 0.2 0.4 0.6 0.8 1.0

8, Fig. 3. Mean reciprocal relaxation time, T * , as a function of eI, at the equilibrium es-

Polymer concentration, 0.25-0.50 tablished after the change in solvent composition. g./IOO ml.; T = 70°C.

948 COMMUNICATIONS TO THE EDITORS

composition was shown to have no effect on the specific optical rotation. Furthermore, variation of A from A = 3 to 10 had very little effect on T * . The change of eI was fol- lowed by measurement of optical rotation. Valiies of A& and (&/dt),-o were read from el versus time plots."

From the observed 25-fold increase of the maximum mean relaxation time rmaX* for a change from n = 34 to 215 a u of an order of magnitude of 10- is estimated from the theory.14 Using this value of u, a k~ = lo-' sec.-l may be calculated, which together with the activation energy of the elementaryprocess (about 21 kcal./mole, see above), yields a frequencyfac- tor of about lox2 see.-'. The agreement of this value with the frequency factor of 6 X 1 0 I 2 set.-' obtained by I R spectroscopy for the cis S trans isomerization of the CO-NH bond in N-rnethyla~etamide~~ gives some confidence to the significance of the results ob- tained by applying Schwarz's theory to the I

Further experimental work and a detailed theoretical analysis to check the more special predictions of this theory are in progress.I6 At this stage it may be stated with confidence that the I S I1 transition in poly-Lproline is a process possessing a high de- gree of cooperativity, as indicated by the observed molecular weight dependence of the transition curves and the dependence of the mean relaxation time on molecular weight and degree of conversion.

The results for the two molecular weights are plotted versus eI in Figure 3.

I1 transition in pdy-Lproline.

I wish to thank Dr. G. Schwarz and Dr. M. Eigen for helpful discussions and, in par- ticular, Dr. G. Schwarz for making available to me his unpublished results.

References

1. P. bl. Cowari and S. McGavin, Nature, 176,1062 (1955). 2. V. Sasisekharan, Acta Cryst., 12, 897 (1959). 3. W. Traub and U. Shmueli, Nature, 198. 1165 (1963). 4. I. Z. Steinberg, W. F. Harrington, A. Berger, M. Sela, and E. Katchalski, J . Am.

5. E. Katchalski, A. Berger, and J. Kurtz, Aspects of Protein Structure, Academic

L. Pauling, The Nature of the Chemical Bond, 3rd ed., Cornell University Press,

Chem. SOC., 82, 5263 (1960).

Press, New York, 1963, p. 205.

Ithaca, 1960, p. 281. 6.

7. A. R. Downie and A. A. Randall, Trans. Faraday Soc., 55,2132 (1959). 8. F. Gornick, L. Mandelkern, A. F. Diorio, and D. E. Roberts, J . Am. Chem. SOC.,

9. B. H. Zimm and J. K. Bragg, J . Chem. Phys., 31, 526 (1959). 86,2549 (1964).

10. G. Schwarz, Ber. Bunsenges Physik. Chem., 68, 843 (1964). 11. G. Schwarz, J . Mol. Biol., 11, 64 (1965). 12. A. Berger, J. Kurtz, and E. Katchalski, J . Am. Chem. SOC., 76, 5552 (1954). 13. D. A. Yphantis, Ann. iV. Y . Acad. Sci., 88,586 (1960). 14. G. Schwarx, unpublished results. 15. T. Miyazawa, Bull. Chem. SOC. Japan, 34,691 (1961); in Polyamino Acids, Polypep-

tides and Proteins, M. A. Stahmann, Ed., Univ. of Wisconsin Press, Madison, 1962, p. 201. 16. J. Engel and G. Schwarz, in preparation.

J ~ R G E N ENGEL*

Max-Planck-Institnt fur Eiweiss- nnd Lederforsrhiing, A f ~ r - * Permanent address: nich, Germany.

Max-Planck-Institut fur Physikalische Chemie Gottingen, Germany

Received April 25, 1966 Prod. No. 224