5724
M. S. NARASIKGA R A OAND GERSONKEGELES
this technique to reacting systems already have been reported,8,16and the companion paper" de(16) R . Townend and s. N. Timasheff, 132nd National ACS Meeting, New York, N.Y., September, 1957, Abstracts, A. 31-C. (17) Ivl. S. Narasinga R a o and G. Kegeles, T w x s J O U R N A L , 80, 5724 (1958).
[CONTRIBUTION FROM
THE
VOl. 80
scribes a detailed study of such an application. Acknowledgment.--This study has been supported by u. s. Public Health Service Research Grant RG-3449. ~ ~ ' O R C E S T E RMASS. ,
DEPARTMENT O F CHEMISTRY O F CLARK UNIVERSITY]
An Ultracentrifuge Study of the Polymerization of a-Chymotrypsin BY M. S. NARASINGA RAO'AND GERSONKEGELES RECEIVEDMARCH3 1, 1958 T h e molecular weight of a-chymotrypsin has been determined a s it function of protein concentration in phosphate buffer solution of PH 6.2 and ionic strength 0.2, by the Archibald ultracentrifuge method. From molecular weight measurements alone, the data can be interpreted within experimental error either by assuming monomers and trimers t o be present or b!, assuming the simultaneous presence of monomers, dimers and trimers. The equilibrium constants and corresponding frec energies of depolymerization have been calculated for both assumptions. In spite of the prescnce of polymers larger than the dimer, only one peak is observed in velocity ultracentrifugation. In conjunction with the recent theory of Gilbert for velocity ultracentrifugation'of reacting systems, this fact would lend preference t o the assumption t h a t monomers, dimers and trimers are present together a t equilibrium. A straightforward extension has been made of Gilbert's theory t o the case where monomers, dimers and trimers are present simultaneousl\-, and this predicts, moreover, only a single peak for velocity ultracentrifugation of such a system. It is therefore possible t o deduce t h a t under t h e conditions of these experinlents, chymotrypsin is present as an equilibrium mixture containing monomers, dimers and trimers.
I n a separate communication2 we have justified the extension to the case of chemically reacting systerns of the suggestions of Xrchibald3for the determination of molecular weights with the ultracentrifuge. The conditions under which such determinations may be expected to be strictly valid for reacting systems also have been indicated. This communication reports the results obtained by the application of these suggestions to a system containing a-chymotrypsin, which has been reported to undergo a concentration-dependent polymerization reaction.3-7 Experimental
be 8.4 in good agreement with t h e reported v a l ~ e s . ~ ~ ' ~ Molecular weight determinations mere made with a Spinco Model E Ultracentrifuge, in a phosphate buffer solution of P H 6.2 and ionic strength 0.20 (0.029 Id disodium hydrogen phosphate 4- 0.114 AI sodium dihydrogen phosphate). T h e protein solutions were dialyzed in t h e cold, overnight against the buffer solution. At high protein concentrations it was observed t h a t slight pH changes took place when the protein was dissolved in the buffer solution. Dialysis against a large volume of buffer solution is essential t o restore the $H back t o the original value. T h e experimental technique and the method of molecular weight calculation described b y Klainer and Kegeles"*'z were followed, a11 experiments being performed with isoelectric protein t o obviate the effects of possible non-ideality of the solutions. Previous worklIJ* had demonstrated no observable concentration-dependence of t h e molecular T h e a-chymotrypsin was a \Torthington Biochemical weight of simple proteins a t their isoelectric points. T h e Corporation product, lot no. 577-82. original protein concentrations, C, iu refractive index units Electrophoretic mobility measurements !\-ere made with a were measured directly b y using the boundary-forming Tiselius Electrophoresis Apparatus (Perkin-Elmer Model cell.13 At low and high protein concentrations, extrapolated 38) fitted with a Longsworth scanning camera.8 T h e iso- values for C, were used, extrapolation being done with a electric point of a 1yoprotein solution in phosphate buffers of calibration curve of area versus concentration. For conionic strength 0.2 was found t o be PH 6.2. This value centrations below 5 g. per liter, a cell with 30 mm. optical differs markedly from the reported values of Y.19 and 8.3.1° path was used and appropriate corrections t o constant optical sensitivity lvere inncle ill such cases. All other measT h e cited investigations were made in buffer solutions of uniurenieiits were made ivith a standard 12 r n n i . cell. Carboil valent salts and of 0.1 ionic strength. T h e discrepancy can be explained on the basis t h a t this protein bitids phosphatc tetracliloritie \ w s used for the false bottom. ions strongly whereas i t binds, perhaps, very little of the The individual expcriniciits werc done :it various teinl)crtrunivalent buffer anions used by thesc investigators. T h a t tures between 20 and 2.5'. For t h e calculation of molecular weight :L value of 0.736 t h e phosphate ions are strongly bound b y this protei11 is further suggested b y the fact t h a t t h e isoelectric point in \vas used for the partial specific volume of a-chymotrypsin.6 phosphate buffer shifted t o pH 6.9 when the ionic strength Absolute protein concentrations were determined b y was decreased t o 0.1; also when the experiments were measuring the ultraviolet absorption a t 282 mp. The repeated in acetate and glycine-sodium hydroxide buffer specific absorption coefficient determined with a portion systems (ionic strength 0.1) the isoelectric $13 was found t o dried t o constant weight over phosphorus pentoxide n.as 2.07, in cxcellent agreement with the reported values of (1) Postdoctoral Fellow in Chemistry. 2.075. (2) G . Kegeles and M. S. Xarasinga Rao, T w r s J O U R N A L , 80, 5721 Buiicr salts v-ere of rcagent quality. fiH tneasuretncnts (1958). \yere made a t 25" with a Beckman pH-meter, model G . (3) W. J. Archibald, J . P h y s . Cdloid Chem., 5 1 , 1201 (1047). (4) G. W. Schwert, J . Bid. Chem., 179, 055 (1919). Results and Calculations ( 5 ) G. W. Schwert and S. K a u f m a n , ibid., 190, 807 (1921). Figure 1 gives a plot of the weight-average (6) R . F. S e i n e r , Avch. Biochem. Biophys., 5 3 , 4 5 7 (1954); I.Tinoco, molecular weight of a-chymotrypsin as a function ibid.. 68, 367 (1957). (7) V. Massey, W. F. Harrington and B. S. Hartley, Disc. F a r a d a y of protein concentration. I n this plot we have SOC., YO. ao, 24 (195;). used thc time-dependent values of C a t thc two (8) L. G. Longsworth, THISJOURNAL, 61, 529 (1939). (11) S. L I . Klniner and G. XIc;ele;, J . P h y r . Chewr., 59, 95'2 (19,331. (9) A. E. Anderson and R . A. Alberty, J . P h y r . Colloid C ~ ~ J5J2 , Z , ( 1 2 ) S. .\I. Klainer a n d G. Kegeles, Arch. E i o c h e w . I l i o p h y s . , 6S, 1315 (1945). 2.17 (1936). (10) 1 '. Kubacki, K. D. B r o w n and 11. Laskowski, J . U i o l . Chem , 180, 73 (1949). (13) G . K ~ ~ c l c '111rs b, Joi,nr*.,ir., 7 4 , X 3 9 ( 1 9 2 2 ) .
Nov. 5 , 1958
ULTRACENTRIFUGE STUDYOF POLYMERIZATION OF CY-CHYMOTRYPSIN
1
5725
menisci (air-solution and solution-carbon tetra6.0 chloride corresponding to the top and the bottom end, respectively, of the aqueous liquid column in 5.0 the cell), rather than the values of original concentrations. Centrifugation usually was done for 45 minutes and pictures were taken a t 15, 30 and 45 minutes after the attainment of the full operating speed. With the centrifugal fields employed in this study (3000-19,000 r.p.m.) and for such short I intervals of time i t was observed that marked 1.0 changes of concentration with time did not occur '1 1 1 l.--A a t either meniscus. For any single experiment we 1 0 . 0 20.0 30.0 40.0 have, therefore, averaged out the values of molecular weight and of concentration obtained at difConcn. of chymotrypsin (g./l). ferent intervals of time, the top and the bottom Fig. i.-Esperinicntal weight-average molecular weight values being averaged separately. I t is readily w y s u s concentration of chymotrypsin : 0, values from top seen (Fig. 1) that the values from the top and the meiiiscus; @?, values from bottom meniscus; @, 2 hr. values hotton meniscus fit into a single curve. from prolunged experiments (starting concn. 20.3 g./l.) ; Since in the case of other isoelectric proteins iii 0,3 hr. valucs from prolonged experiments (starting concn. this same range of concentration, no apparent con- 20.3 g , / l . ) ; 0 , 0 hr. values from prolonged experiments centration dependence of molecular weight is de- (starting concn. 14.2 g./l.): 0 , 18 hr. values from prolonged tectable,'* i t is justified to assume that in the experiments (starting concn. 14.2 g./l.). present case the concentration dependence repreinfinitely rapid, but is slow enough, the centrifugal sents a real physical change in the system. Thus by extrapolation of the curve to C = 0, field might be expected to deplete the solution of the molecular weight of chymotrypsin monomer, polymers a t the upper meniscus, and to concentrate the species favored by dilution, was obtained and polymers a t the lower meniscus, to an extent bewas found to be 2.3 X lo4. This is in good agree- yond that which would be expected at chemical ment with the value obtained by sedimentation- equilibrium. Reference to Fig. 1 shows that in diffusion studies" and also with the value (at C = fact the values from the top meniscus gradually de0) obtained by the light-scattering It crease below what they should be and those from may, hovever, he added that by the nature of the the bottom increase above what they should be. curve a t low concentration (Fig. 1) very precise Though the precision of our measurements is not high enough to derive any quantitative information, extrapolation is not possible. By knowing the molecular weight of the moiio- still the trend seems to he definite, indicating mer and applying the mass law equation, i t is pos- qualitatively that the rates of the association and sible to calculate from values of weight-average dissociation reactions are not very large. The molecular weight a t various chymotrypsin con- short-time experiments always were performed centrations the equilibrium constants for the as- after leaving the solutions overnight, and numerous sociation of monomers into polymers. However, checks were obtained, giving some confidence that before applying the mass law equation, i t is neces- the circles in Fig. 1 indicating the short-time exsary to know whether the system is truly in a con- periments represent equilibrium values. We can write the equations for the over-all disdition of reversible equilibrium. For this purpose in two sets of experiments ultracentrifugation was sociation of an n-mer into n niononiers as carried out for a long time (2-18 hr.) so that cona-mer +1~ monomers; KtL' = CM"/CP (1) siderable concentration changes took place at both where C? and C ~ \ are Z the weight-concentration of menisci. The values of molecular meeight a t the the wiiier and the iiiononicr. respectively. If in top meniscus should decrease with time gradually due to polymer dissociation accompanying the de- the equilibrium system only monoiners and n-mers crease in concentration, and similarly the values a t are p r e p t , then the weight-average molecular can be represented as the bottom meniscus should increase with time. weight M\"= ( C X M l ) CPJf,/C (2) Reference to Fig. 1 shows that, in these prolonged experiments, the molecular weights do change where M I and Mn are the molecular weight of the with time in general qualitative agreement with monoiiier and 11-mer, respectively, and C is the this hypothesis. Thus the association of a-chymo- total ~~eight.conceiitration.Since Cp = (C trypsin seems to be truly reversible. CJI) a i d :lI,L= T L J Ii~t can , be shown that For the case of truly reversible equilibrium, the n--l _(nJd, --__ - lV,)V K,&/= c (3) observed weight-average molecular weight when ( n - 1)Ml (aw - MI) plotted against the slowly decreasing concentration a t the upper meniscus, or against the slowly in- Then Krt' is obtained from the slope of a plot of veysus c ' 2 - 1. creasing concentration a t the lower meniscus, ( M , - M I ) '(niU1- Xn7)% First considering-the case of dimerization ( n = should follow the same curve as that obtained from - A f 1 ) / ( 2 M 1 - n/Iw)2were experiments of short duration a t low speeds. 2 ) , the values of (AITv However, if the chemical re-equilibration is not plotted against values of C. The plot (Fig. 2) showed a pronounced upward curvature a t high (14) N. Ivfichrel Green and H Xeurath. in "The Proteins," Ed. values of C. This fact, along with the observation Neurath and Bailey. Vol. 11, P a r t B, Acad Press, I n c , New York. S Y 1931, pp 1075 107b R that the experinleiitally c!etrrniincd weight-average
c
~
+
I--[
XI. S. NAKASIXGA Rso AND GEIISONKEGELES r--
I
120.0 110.0
100 1
I
so
Vol. -
/
I’
’
- - -
/
--1
,
I
i
Le& __ c‘
-
. _
10 0
__
2i) !I
c
Fig. 2--~iot of (*GW- .11~)/(2~f,\ ~ u c sr ( i ~ z C values; liiie ncdr horizontal axis has the liiniting slope.
molecular weights a t high concentrations are greater than that of the dimer, suggests that polymerization proceeded beyond dimerization. For purposes of comparative analysis, a value of Kz‘ = 2.07 was obtained from the slope of the line drawn to fit the data up to a concentration of 13 g./liter and a molecular weight of 42,000. It would be incongruous to obtain a least square value from the data o1er the entire concentration range, because _at Mw = 2M1 the slope would be infinite and a t M w > 2-!1 it would have no phyzical significance. 1%-iththis value of IC2’, values of M , as a function of C were calculated using equation 3 . It can be seen from Fig. 4 that this calculated curve does not fit the data either in the low or in the high concentration range. Thus it seems reasonable to conclude that the dimerization hypothesis does not adequately explain the experimental data. Hom ever, if it is assumed that a t low concentration dimerization is the predominant reaction then the constan for the dimerization reaction can be obtained from the limiting slope in Fig. 2, and this value of &’ is 652. As is to be expected, the curve calculated with this value of Kz’fits the data well a t low concentrations, but it completely fails to fit the data a t high concentrations. Similarly the case of trimerization ( n = 3 ) was c p i d e r e d next. The plot of (hf, - M1)/(3ldl M,)3 vs. C2 (Fig. 3) does not show a pronounced curvature even a t high concentration values, although there is considerable scatter in the points. Considering the extreme cases, 1% e have drawn lines in Fig. 3 giving entire weight to points a t low concentration or to points a t high concentration (dashed lines in Fig. 3) ; the solid line has the slope obtained by least squaring the data over the entire concentration range. The values of K3’ = 116.7 and 43.0 were obtained from the lowest and the highest slopes, respectively; the value of K3’ from the least square_slope is 56.2. With these various values of K3‘, M , as a function of C was calculated with equation 3 . These calculated curves are given in Fig. 4. I n view of the scatter in the experimental points it is reasonable to conclude that even the curves calculated with the extreme values of
t ~
200
000
1000 1400 10-1 I‘ig. 3 -Hot of (2Uw- Jf1)/\3M1- .UW)3vaiuesversus C2 values : , lines drawn giving entire weight t o d a t a at low concentration or high concentration only; ___ , line has the least square slope. (c’)
_________
r-
~~
I 1.0 [ .__-I
10.0
I
20.0
30.0
40.0
Concn. of c h y i o t r y p s i n (g./l.). Fig. 4.-Experimental points and theoretical curves for weight-average molecular weight uerszis concentration of chymotrypsin: -.- .-. , calculated for dimerization only Kz’= 2.07; calculated for trimerization only Ka‘ = 116.7; -’--, calculated for trimerization only Ks‘ = 56.2; , calculated for trimerization only Ks‘ = 45.0; --A-, calculated for dimerization and trimerization, Kz = 11.1 and K , = 4 . 5 ; -0-, experimental poirits.
Ks’ fit the data fairly wcll. From the above a i d ysis a possible picture of the polynierizatiou equilibrium is that only inononiers and trimers are present in the mixture. Both from the point of view of kinetics of the polymerization reaction and from the fact that equation 3 written for dimerization holds good a t low concentration, it may also be postulated that the equilibrium mixture contains all the three species, monomers, dimers and trimers. For the case when all the species are present (monomerJo n-mer) the weight-average molecular weight M , can be representedI5 in the present notation by (15) K. F. Steinrr, A r d i . Biuiirem. t r l O p h j 5 . , 39,333 (1952)
ULTRACENTRIFUGE STUDY OF POLYMERIZATION OF CY-CHI MOTRYPSIN
Nov. 5, 1958
5727
Discussion
_cn-lcM and C,
C,I, CO, CT . . . C,
are the weight Concentration of the monomer, dimer, trimer and n-mer, respectively. If we define x such that CM = X C then equation 4 can be rewritten as
+ 2( I / K ~ ) ( X C ) ' M +I 3(1/&&)(Xc)'*fi!fi . . . + n(l/KzK3 . . . Kn)(xC)"Mi (5)
C . a w = (SC)ML f
Following Steiner's15 treatment, we can write In x =
hc
-
{(e-:
l)/C)dC
where a = a W / M l . The values of x as a function of C can be obtained by means of graphical integration. With these values, we can plot x l l l ~- 1)xC as a function of xC. From the intercept and the slope, the values of KZand Ks,respectively, can be calculated. Any upward curvature in the plot would suggest the existence of polymers higher than trimers. The plot showed a slight upward curvature but the precision of our data is not high enough to place any emphasis on this curvature. By least squaring the data values of Kz = 11.1 and K3 = 4.5 were obtained. The value of Kz = 11.1 obtained by this method of calculation compares directly with the value of Kz' = 6.52 obtained above by considering the process a t low concentration as comprising the dissociation of dimers only, according to equation 3. The value for K3 obtained by Steiner's method is 4.5 and since K P = &'/&', the value of K S calculated from the values of Kz' and K3' obtained with equation 3 is 8.62. The agreement in the values of K z and K3 obtained by two methods can be considered fairly satisfactory. With the values of Kz = 11.1 and K3 = 4.5 in was calculated as a function of C. equation 5, It is seen from Fig. 4 that this curve is essentially the same as the one calculated by least squaring the data to equation 3 for the case of trimerization alone. Thus the interpretation that all the three species-monomer, dimers and trimers-are present in the equilibrium mixture seems t o be as valid as the one that only monomers and trimers are present. If in equation 4 the concentration is expressed in moles per liter and the successive dissociation constants are designated k*, k3, etc., i t readily can be shown that kz = K Z ( 2 / M 1 ) :ka = K3(3/2M1),etc. (6) The values of kz and k3 are 0.96 X and 0.29 X respectively. Thus, on this scale, the constant for the dissociation of dimers into monomers is nearly three times greater than the constant for the dissociation of trimers into dimers and monomers.
T o obtain some idea of the relative stability of dimers and trimers, assuming that monomers, dimers and trimers are all present, i t is desirable to examine the constants K Z and K 3 , on the grams/ liter concentration scale, and the corresponding standard free energies for dissociation. This concentration scale is useful for such a comparison, since the standard state reaction corresponds to polymer, a t a concentration of one gram per liter and unit activity coefficient, dissociating into products, each also a t one gram per liter concentration and unit activity coefficient. These concentrations and activity coefficients correspond to experimentally realizable physical conditions, whereas i t is quickly seen that the physical conditions corresponding to the standard state on the niole/liter scale are completely incongruous, because of the very large molecular weight of protein molecules. Failure to consider the physical realizability of the standard state can lead to confused deductions from the experimentally derived thermodynamic data. I n order t o eliminate the standard free energy of monomer, the reaction
{aw/
aW
dimer
+2 monomers;
AFz0 = -RT In K1 = -1411 cal./mole dimer
(7)
is multiplied by l / Z and subtracted from the reaction trimer
-+
monomer
+ dimer;
AF3O = -RT In K3 = -881 cal./mole trimer (8)
giving the reaction trimer
+3/2
dimer; A P = AFjo - l/2AFz0 = -176 cal./mole trimer
(9)
This implies that a solution containing 1 g./liter of chymotrypsin trimer would tend to dissociate spontaneously to chymotrypsin dimer, a t a concentration of 1g./liter, with a free energy change of AFO = - 176 cal./mole of chymotrypsin trimer. Equation 7 also implies that a t the same l gram/liter concentration level of protein species, dimers would dissociate spontaneously into monomers with a free energy change of - 1411 cal./niole of chymotrypsin dimer. It was desired to compare the moving boundary sedimentation behavior of this chemically reacting system with its predicted behavior from the recent theories of Gilbertl6 and of Gilbert and Jenkins" once a reasonable idea of the species present had been obtained by the presently employed non-hydrodynamic method. Three moving boundary ultracentrifuge experiments were performed a t 59,780 r.p.m. in phosphate buffer of PH 6.2 and ionic strength 0.2. The chymotrypsin concentrations were 15.0, 18.9 and 36.0 grams per liter. I n each case a single fairly symmetrical peak was observed. These observations are in qualitative agreement with the results of Schwert4under roughly comparable conditions. Reference to Table I provides an idea of the concentration of each species present a t a total chymotrypsin concentration of 18.9 g./liter ; the calculation has been made under two separate assumptions, first that only monomers and trimers (16) G. A. Gilbert, Disc. Faraday Soc.. 20,68 (1955). (17) G. A. Gilbert a n d R. C.L1. Jenkins, Nalurc, 1'77,853 (1956).
are present (coluiiiri 2 ) and secontl , t1i:i t I I ~ ~ dimers and trimers are all present ceoluinii 3'
I I ~ O I I ~ ~ E
TABLE I
SPECIESCOXCEXTRATIONS IS CHYMOTRYPSIN ( G.,'L.) Species concn , 6.A CTotal CMonomer
Eq. 3 (trimerization) Ka' = O l j 2
18 90
8 24
1s 90 7 lii
IO ti6
4 -70 7 24
CD>mer CTr I nl t r
Eq. 4 I = 1 1 . 1 ; k'i
4.,7
It is clear from Gilbert s treatment of the case (of infinitely rapid re-equilibration in a monomer trimer reaction that two peaks should be expected in velocity ultracentrifugation, provided that diffusion can be neglected and tiiiners arc no( ;die present. From the treatment of Gilbert and Jenkins for a rapid reaction -4 13 C it might perhaps be expected that three peaks u ould appedr in relocity ultracentrifugation if the predominant reactioii is the dissociation of trimers to dimers arid IIIOIIOmers, all three species being present. I-lomever the case treated by these authors does not really apply to the chymotrypsin system, since this treatment does not provide for interconrersion of the slowest species B into other species, once separation of this single species has been completed. On thc other hand, if the original tlieoryIF of Gilbert i i taken as correct, then the information that niono mers and trimers are necessarily present, :E judged from Fig. 2, and the information that only one peak is observed in velocity ultracentrifugation must lead to the conclusion that dimers nrc :ilso present a t equilibrium For otherwise Gilbert's theory would require two peaks in velocity ultracentrifugatiori which is contrary to experimental observation Furthermore, the assumption of finite dissociation and association rates could only be expected to leatl to better resolution in the T elocity ultracentrifugc experiments. According to Gilbert, in the theoi y of boundar?shapes in velocity ultracentrifugation oi a nionomer-polymer system which neglects diffusion, "no new principle is involved nhcn a group of coin plexes (each of different 1 2 ) is Iresciit, lout tiic :dgc bra is much inore complicated." Fortmic~tcl\,< L straightforward extension or Gilbert's origiiiLLl theory to the case of coexisting monomers, tliiiic r i and trimers has been found to lead to :I closed solu tion for the gradient of t o t d concentration. T h 1 \ extension is indicated briefly hcre, reference t o Gilbert's paperI6 being recomniended for a inore detailed discussion. The system is assumed to react SO rapidly that i t is always at chemical equilibrium The cell is assumed to be rectangular. The reLtive amounts of monomer, dimer and trimer arc then related through equation 1. If a frame o f reference is chosen which is stationary with respect to the monomer, and the velocities of dimer anrl trimer, Z'D and T I T ,are taken .is constnnt, tile f l J ~ h V-. ing equation applies
+
In these equations t is the time since the start of the experiment, and x is the distance from the center of rotation. .yo being the top surface of the liquid column. By taking advantage of the mathematical relationships between partial derivatives in equation IO and applying equation l to calculate concentration of dimer and trimer in terms of the concentration of monomer, i t is found that thc iiiaYiiniiin permissible value of 6 is given by
where Clr a t any gil-ci: total concentration ol chymotrypsin is obtained by Steiner's procedure, as outlined above. The total concentration of protein is given by
From equation 12, is found in ternis of 6, and insertion of this \ d u e into equation 1-1-results finally in
(lr,)
where
It is noted from equation 12 that (211 = 0 when 6 = 0 , and by evaluation of the resulting intermediate form in equations 1 4 and 15 it is found that (17)
The values of ED and UT which are required to evaluate equation 13 can be calculated from the sedimentation coefficients of the dimer and trimer, respectively. For the purposes of these calculations, tlic sc.tliment:itioii ccw!iicicnts of the chymotrypsin
Nov 5, 1958
ULTRACENTRIFUGE STUDY OF POLYMERIZATION OF CY-CHYMOTRYPSIN
5729
dimer and trimer were computed using the following relationship1*and using a value of 2.4 S for the sedimentation coefficient of the m ~ n o m e r , ~
so/s.
= { 2 M t / M , ] % = 2%;
ST/SII= { 3 M 1 / M j l % = 3%
where SM,SD and ST are the sedimentation coefficients of monomer, dimer and trimer, respectively, and M , is the monomer molecular weight. In using the above relationship, the basic assumption is that these molecules are spherical in shape. Since these values are only for purposes of sample computations of the gradient curves, there should not he much objection to the procedure adopted for getting SD and STvalues. The calculated SDand STvalues are 3.80 and 4.99S, respectively. Using the values of VD/ZIT = 0.762, &’ = 11.1, K1’= 50.0 in equation 15, the dC/& values as a function of 6 were calculated for a total concentration of 18.9 g./l. and for vT1 = 0.9159 cm. The monomer concentration that corresponds to this total concentration is 7.16 g./l. (equation 4), and this fixes the maximum value of 6 a t 0.756 (equation 12). In Fig. 5 8 , dC/dxvalues are plotted as a function of 6. It is seen that the schlieren pattern consists of a single peak. Similar calculations, for the same total concentration and v ~ value, t were made using the equations given by Gilbert16for the case when only monomers and trimers are present. The result is a double peak (Fig. 5A) with a valley a t 6 = 0.167. Comparison with an actual sedimentation velocity pattern, for the same total concentration of 18.9 g./L (Fig. 5C), leads to the conclusion that, under the experimental conditions, chymotrypsin exists as an equilibrium mixture of monomers; dimers and trimers. It has been pointed out earlier that from molecular weight data alone both the h y p o t h e s e s that monomers and trimers only are present or that all the three species are present-were equally valid. Thus with the aid of Gilbert‘s theory and values of the weight-average molecular weights. the sedimentation velocity experiments, a t any rate for this system, seem to give a decisive answer. Acknowledgment.-This study was made possible by U. S. Public Health Service Research Grant RG3449. (18) Svedberg and Pederseo, “The Ultracentrifuge," Oxford Uoiversity Press. 1940. p. 10.
i l A i b ,
- I B A,
-n
‘XUxd
0 0
IO
5 ,X.
xd
o.o 5
1.0
f
Fig. 5.-Theoretical and experimental sedimentation velocity patterns: A, theoretical pattern for the case of equilibrium mixture of chymotrypsin monomers and trimers, for a total concentration of 18.9 g./l.; B. theoretical pattern for t h e case of equilibrium mixture of chymotrypsin monomers. dimers a n d trimers for a total concentration of 18.9 g./L T h e position x = xo corresponds t o a value of 6 = -m/w; C, experimental sedimentation velocity pattern of chymotrypsin. Protein concentration = 18.9 g./l.; centrifugation for 120 min. a t 59,780 r.p.m. Sedimentation proceeds from left t o right. WORCESTRR, MASS.
