Under the assumptions made, the derived equations indicate several kinetically different cases corresponding to different possible mechanisms in terms of the p H dependence of the kinetic parameters, the maximum velocity and Michaelis constant. These differences may be summarized as follows: if both ionized forms of substrate are utilizable, V p and the ratio V F ( ~ ( H + ) / K s H ) / K F would be expected to vary with pH in a manner more complex than would be expected from the ionization of one or two groups in the active site of the enzyme molecule. On the other hand, when only one ionized form of substrate is utilizable, the two possibilities that the lion-utilizable form of substrate be either a “good” or “poor” competitive inhibitor are kinetically distinguishable. For both these possibilities, l i ~ ( 1 (H+)/KsH)/KF os. pH is a function only of the ionizable groups in the enzymatically active site which are responsible for activity and the pH dependence may be described easily by an equation similar to equation II,3. For the case of poor competitive inhibition, the niaxirnum velocity repre-
+
+
[CONTRIBUTION FROM
THE
sents only the ionization of groups of the active site of enzyme to which substrate is bound. However, for the case of good competitive inhibition, the maximum velocity will be influenced by the degree of ionization of the substrate and will therefore be more complex than would be expected. I n order that the equations derived in this paper be applicable, a large amount of kinetic data must be obtained. These data must be uncomplicated by interference due to substrate inhibition, substrate activation, irreversible enzyme denaturation, buffer effects due to different concentrations of anions or cations a t different pH values and so forth. Thus, care must be exercised in the reinterpretation of data already presented in the literature. For example, the two enzymes arginase and enolase which have been discussed both involve metal ions for enzymatic activity. Although constant metal ion concentration was used, the mechanism of interaction of enzyme with the metal ions is not completely understood. SAINT LOUIS,MISSOURI
DEPARTMENT O F CHEMISTRY,
CORKELL
UKIVERSII Y]
The Elastic Properties of Elastin1S2 BY
c. A. 3. HOEVE3 AND P. J. FLoRY3 RECEIVED JUNE 2, 1958
Force-temperature measurements have been carried out on elastin (ox ligamentum nuchae) held a t fixed elongation and immersed in glycol-water (3:7) mixture. The equilibrium degree of swelling of elastin in this mixture is independent of temperature, and the retractive force is directly proportional to the absolute temperature. It follows that ( d E / d L ) ~ v= 0 for elastin and hence that the internal energy of the elastin chain is independent of its conformation. Contrary t o previous studies on elastin, in which the influence of changes of swelling with temperature were overlooked, the thermoelastic behavior offers no indication whatever for crystallization on stretching a t any elongation. The shape of the stress-strain curve is explained in terms of the morphology of native elastin; the abrupt rise in stress a t high elongations is attributed t o straightening out of the initially curled fibers of collagen which are associated with the native elastin.
Introduction Elastiii is ai1 important constituent of various elastic tissues including ligaments, blood vessel walls and skin. I t possesses a high extensibility combined with a low modulus not unlike that of rubber. Moreover the stress-strain curve for elastin, like that of rubber, swings upward sharply a t high extensions5; the rise of the stress-strain curve is however more abrupt, and i t occurs a t somewhat lower elongations, as compared with vulcanized rubber. Thermoelastic s t ~ d i e s ~on- ~elastin have yielded large positive stress-temperature coefficients even a t low extensions. It has been inferred from this alleged deviation from ideal rubber elasticity that crystallization occurs on stretching. On the contrary, however, Astbury7 found only an (1) Support of the Xational Science Foundation is gratefully ackilo wledged. ( 2 ) Presented in September 1957 a t the 132nd American Chemical Society Meeting, New York, N. Y . (3) Mellon Institute, Pittsburgh, Pennsylvania. (4) K. H. hleyer and C. Ferri, Pjlugtr’s Arch. EFT. Phyr., 238, i R (l!W). ( 5 ) E. Wohlisch, fioiloiii Z., 89, 2.19 (1039j. (6) E. Wohlisch, H. Weitnauer, W. Grtining and R . Kohrbach. ibid., 104, 14 (1943). (7) W. T. Asthury, J . l i t t c u 7 z . Soc. Leather T r a i i e ~ ’Chem., 24, 69 ( 1 940).
amorphous halo in the X-ray diff ractiori pattern of stretched elastin in which the collagen component had been destroyed. The principles underlying rubber-like elasticity of amorphous polymers are of course well known. Progress recently has been made in the analysis of the thermoelastic behavior of partly crystalline polymers.8 In extension of studies in this area, and especially those relating to fibrous proteins, i t became of interest to examine elastin, and in particular to endeavor to resolve the apparent contradiction between the thermoelastic results and those of X-ray diffraction. Theoretical Insight into the nature of the molecular processes itivolved in elastic deformation may be gained by analysis of experimentally determined stress-straintemperature results according to the thermodynamic equation of state for elastic deformation. This equation may be expressed as follows for a system subject to elongation a t constant pressure
f
-
=
+ T(df/bl’)pL
(dE/dL)p~
(1)
( 8 ) J. F. M . Oth, E. T. Dumitru, 0. K. Spurr and P J. Flory, THIS JOURNAL, 79, 3288 (1957); J. F. M . 0 t h and P. J. Flory, i b i d . , SO, 1297 (1958).
C. A. J. HOEVB AND P.J. FLORY
(i524
where f is the refractive force, E the internal energy,
L the length, T the absolute temperature and P the pressure on the sample. volume I/
(bE/dL’,Tl,
= I(dPidI)vL -
I’
= -1’
uiitlcr the foregoitig cundiiions, :md hciice that (bE/bL),p
=
(bE/bL)TV
approximation that
For systems a t constant
The advantages of experimentation a t constant pressure weigh heavily in favor of eq. 1, by means of which ( b E / d L ) mmay be evaluated from observations on the temperature coefficient of the force a t constant pressure. Unfortunately, however, it is the constant volume coefficient (dE/dL) V T that is representative of the energy change due solely to changes in molecular conformation brought about by the deformation. The constant pressure coefficient includes a contribution from dilation of the sample. Direct evaluation of the force-temperature coefficient a t constant volume (eq. 2) being generally impractical, it is customary to determine ( d j ’ / b T ) p ~ from measurements a t constant pressure and subsequently to apply suitable corrections for volume changes with t e r n p e r a t ~ r e . ~Through use of eq. 2 it is then possible to deduce the desired coefficient (3E/dL)v~, which affords an index of the mo1ect;lar processes accompanying deformation. Manifestation of high elasticity in fibrous proteins generally requires the presence of a swelling agent, usually water. If the sample is in equilibrium with the swelling agent available in large excess, as for example if the sample is immersed in the swelling medium, then the quantity of diluent in the sample will, in general, depend on L and T . Equations 1 and 2 are rertdily shown to apply to such an open thermodynamic system comprising the swollen polymer in equilibrium with the external diluent phase, provided however that the partial derivatives are understood to be restricted to states in which swelling equilibrium prevails. Then E may be considered to represent the sum of the internal energies of both phases; P,V and L refer to the polymer phase specifically. Direct evaluation of (djhi?) V L in general would require the experimentally awliward feat of applying pressures to the polymer phase in order to maintain its volume constant. This difficulty may be avoided by choice of a diluent in which the volume L* of the swollen polymer is independent of temperature, ie., for which ( d I . ’ / b T ) p , ~= 0. If this condition holds for the unstrained network, then i t may be assumed to hold also for extended states, to a close approximation a t least (;.e., neglecting the effects of anisotropv, which should be trivial). It follows from well-known thermodynamic relations that
+ (bE/dV)7I(bV/bL)PP = (3EidL)mr
- P(bT’/bL)TP
Except For high pressures, we h a l e to
L:
very good
(9) P J P l o r ~ . Piincipleq of Polymer Lhemislry,” Cornel1 Uai vcrsity Press I t h d c d 4’ Y 1953, pp 41O--llJ
Vol. 80
( d E / b L ) ~ p= ( ~ E / B L ) T I ‘
( 31
Hence we may use the experimentally convenient eq. 1 to evaluate the theoretically desired derivative (dE/dL)v~ of eq. 2 , provided, of course, that the condition ( b V / b T ) p ~ = 0. I t is obvious also ~ then identical with (bf/dTjvr.. that (df / b T ) p is The procedure indicated has been adopted in the present study. A mixed diluent was used; possible disproportionation of the two components between the phases is of no consequence, provided only that the diluent phase is present in large excess. Through the use of statistical mechanical pro. cedures, the following equations have been derived’O for an amorphous polymer uuder a tcnsile force while in swelling equilibrium ( vvI/Iva V u ) [02(LiO/L) - ?‘2/2]=
-
[In(1 -
2’2)
If the retr Ztive force is zcro, ell.
+ + ~ ~ r ’ ~ (*4;) 212
bec(Jtlles
(uvl/S,’i.~o)[oZv2’:l -- uz/’?j =
- [1tl(l
- Vz)
-+ +
‘iJ? xlZ’z2]
(5)
where u is the
total number of chains in the network structure is the molar volume of the diluent LV, is Avogadro’s number Li, and VUare, respectively, the isotropic length and the volume of the polymeric sample in the absence of a diluent 2 ) ~is the volume fraction of polymer in the swollen network o is the dilatation factor in the absence of diluent and is given by VI
”2
=
rio2/?7
where 2 is the mean square end-to-end distance of the polymer chains in the isotropic amorphous sample in the absence of diluent and is the corresponding quantity for unconstrained free chains. x1 is the interaction parameter for solvent-polymer interaction. In general the degree of swelling v2-l depends on the temperature coefficients of < a > 0 2 and X I . The temperature coefficient of x: is proportional to the heat of swelling which, if positive, causes the swelling to increase on raising the temperature, o2 being considered constant. On the other hand, depending on the nature of the energy barrier restricting bond rotation, 2 and hence < c r > 0 2 , may change with temperature. (We neglect for the moment the relatively small effects of the bulk thermal expansion.) For amorphous polyethylene d< a> 02,/ d T has been found1I to be positive, hence d g / ’ d T is negative, as a result, no doubt, of rotational hindrance about the C-C bonds of the chain. Consequently measurement of the temperature coefficient of swelling alone is not sufficient for determining d< a> ,,“dT. However, by combining swellingtemperature measurements with force-temperature
3
(IO) P.J . Flory, THISJ O U R N A L , 7 8 , 5 2 2 1 (1956). (11) I?. J. Flory. C. A. J. Hoeve and A. Ciferii. paper presented before the IUPAC Conference on hlacromolecular Chemistry, Nottinpham, 21-24 July, 1958.
Dec. 20, 1958
THE
ELASTIC PROPERTIES OF ELASTIN
measurements a t constant length separate evaluation of dxl/dT and of d02/dT is in principle possible through use of eq. 4 and 5 . The relation for the force as a function of the length, temperature and degree of swelling is given bylo f = vkT(02/Lio2)L(1- LiO3/a2L3)
(6)
where v2 is in general a function of T and L as determined by eq. 4. At large extensions, the second term of eq. 6 is small and therefore the force-temperature coefficient a t constant length can be related - directly to changes in < a> o2/LiO2 and hence in yo3. For small L the changes in v2 with temperature will in general contribute to the force-temperature coefficient. But, if a solvent is chosen in which the volume of the swollen polymer (proportional to L3io/'vZ) a t zero force remains constant with temperature, the increase in volume from thermal expansion is exactly counterbalanced by de-swelling, and the force-temperature coefficient a t constant length can be related without appreciable error to changes in yo2over the whole range of extensions. This deduction from eq. 6 is in accord with the foregoing thermodynamic considerations. Thus, a t constant length and a t swelling equilibrium in a medium in which the condition (bV/bT)tp = 0 holds, we have
[bw)]vL = -d In ?/dT
=
-(bE/aL)z,v/Tf
6523
larly curled thinner fibers between them. Staining techniques13 showed the thicker fibers to consist of elastin and the thinner ones of collagen. Decomposition of the elastin component with elastase left fibers which, though coherent, accounted for only about 20% of the initial (dry) specimen. The lengths of these residual fibers, when straightened out under tension, were about twice those of the relaxed native sample. Identification of the residual fibers with the irregularly curled component seen under the microscope is thus indicated. These fibers displayed low elongation and high modulus; they underwent shrinkage (in water) a t 6.5' to about one-fourth of their length. These observations leave little doubt that the residual fibers are collagen. The fact that macroscopic collagen fibers survive the enzymatic treatment is clear proof of the co-existence in ox ligamentum nuchae of separate fabrics comprising elastin and collagen fibers. The results of the force-temperature measurements are given in Fig. 1. The points were taken in r--
CROSS-SECTION
OF
ELASTIN 0048
ern'
1
(7)
as has been shown for single phase polymer systems."
Experimental Fiber bundles of ca. 0.05 cm.' in cross section and 4 cm. in length were obtained from unpurified ox ligamentzkm nuchae. Length and diameter were measured with a cathetometer. Measurements in water showed that the degree of swelling decreased in water on elevation of the temperature. However, the dimensions remained essentially constant from 0 to 50' in a 30% by volume glycol-water mixture. Force measurements were performed in this medium. The fiber bundle was suspended between clamps a t both ends. The lower clamp was fixed, while the upper clamp was suspended from a strain gauge which could be adjusted vertically. The strain gauge (Statham Instrument Company, Transducer Model G-1) had a capacity of 750 g. and a linear response of 0.05 mv. per g. under a supplied e.m.f. of 12 v. The output of the strain gauge was connected to a Leeds and Northrup recorder giving a full scale deflection of 10 mv. The instrument was calibrated by addition of known weights before and after each experiment. Reversible measurements were made in the following way. The fiber bundle was extended at the highest temperature (50.5') and allowed to remain in this condition for half an hour for dissipation of stress relaxation. The temperature then was lowered to 0.5', and again relaxation was allowed to occur for 30 minutes, whereupon the force was recorded. After returning the sample t o 50.5', the previous value for the force was restored; hence attainment of equilibrium forces was assured. Thereafter the elongation of the sample was changed and the cycle repeated. Fiber bundles were decomposed enzymatically with an impure preparation of elastase extracted from pancreas according to the procedure of Balb and Banga.l*
Results and Discussion As observed in the microscopela the elastin samples consisted of many separate fibers with irregu(12) J. Ba16 and I. Banga, Biockcm. J . . 46, 384 (1950). (13) A. A. Marimow and W. Bloom, "A Textbook of Histology," W. B.Sunders Co., Philadelphi., 1952, p. 70.
I
I
40
50
60
70
L e n g t h i n rnm.
Fig. 1.-Stress-strain curves of an elastin bundle a t different temperatures in 30% glycol-water mixture.
random order. Deviations from the lines are small, in further support of the conclusion that equilibrium was closely approached. The coefficients ( b f / b T ) ~were ~ obtained from the slopes, and (&E/ ~ L ) v values T were calculated according to eq. 2. For extensions up to 5Oy0 and temperatures ranging does not deviate from from 0 to 50.5', (bE/dL)v~ zero by more than 1% of the total force, which is well within the experimental error, estimated to be about 2%. Compelling evidence is thus afforded for the conclusion that elastin does not crystallize on stretching. The failure of elastin to crystallize is rather unexpected, since most proteins are a t least partly crystalline in their natural environment even without the assistance of elongation. On application of eq. 6 to these results i t is seen that < a>O2/LiO2,and therefore G2,is independent of temperature. Thus the different conformations of the protein chain must possess equal energy. This result is by no means a t variance with current view^^^^'^ on impedance of rotation about the C-N '(14) R . B . Corey and L. Pauling, Proc. Roy. SOC.(London), B141, 10 (1953). (15) S. hfizushima, "The Structure of hlolecule and Internal Rotation," Academic Press, Iuc., New York, N.Y.,1954. (16) S. M. Partridge and H. F. Davis. Uiorhem. J . , 61, 2 1 ( l ( I 5 3 .
withiii expcriiiiciit;il u r o r over tlie raiige c)i extcii-. sions given in Fig. 1 is regarded as fortuitous. At extensioiis higher than cu. TOYd the collagcll libers evidently become taut and hence cause thc (J stress to rise abruptly with further extension. C o h cloublc bond chdracter. In fact, this plmaritp of' p i , beiiig a crL-stallinepolymer where elastic ~ n o d u the amide group should be expected to suppress lus and tensile strength are much higher tlian tliosc steric hindrance to rotation about the other bonds in oi elastin, prevents rupture of the fiber hulidlc ;tt tlie protein chain, thereby reducing the change of higher stresses. 111 contrast with rubber. tlic energy with chain conformation. Furthermore. steep rise in stress at. elongations near the iriasielastin is known16 to contain few polar amino inurn attainable leligth is not precipitated by crysacids and the electrical forces should not therefore tallization! but by- a permanently crystalline conibe expected to contribute appreciably to the con- Iionent (collagenj iiiterwol-en with the deforniable formational character of the chain. jelastinj component. Although the elastin bundle conforms to the conThe conclusions vie has-e reached depart from thc = 0 for ideal rubber elasticity dition ( b E b L ) v ~ interpretations of Ileyer and Ferri4 and of LYohlisch the linear stress-strain relation presents a striking and co-workers.5 Tliese investigators concluded contrast to the isotherm reported for rubber. This from stress- -temperature measurements that elastili feature finds qualitative explanation in the fact crystallizes on stretching. W6hlisch went even SO that elastin consists of separate fibers, curled to far :is to calculate the heat of crystallization o f clasdifferent degrees. Coiisequently some of the fibers tiii from the strcss--temper:tturerelatioli. Thc ~ : i l u ~ . do not contribute a t all to the force a t low exteii- ioulld was surprisingly low. Failure to take acsioiis. .Is the extension iticreaies the proportioii of couiit of de-swelling of elastin in water with elevathe fibers subjected to extension increases; hence tion of temperature led these investigators to tltc more of them contribute to the force. The stress- erroiieous inferences. The internal energy changes strain relation which would obtain for a homogenc- with length as deduced by them are to be attributed o m polymer is therefore modified to the extent that rather to the eiiergy change associated with the the initial negative curvature which would other- riioreirientioric,d de-sw~ellinji with rise in ienilwraturc.. wise he obserwd is eliminatctl 'The 1iiic:irity noted
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