optical rotatory dispersion and sedimentation in the study of association-dissociation: bovine...

BIOCHIMICA ET BIOPHYSICA ACTA 73

BBA 35119

OPTICAL ROTATORY D I S P E R S I O N AND SEDIMENTATION IN T H E

STUDY OF ASSOCIATION-DISSOCIATION: BOVINE fl-LACTOGLOBULINS

NEAR pH 5

H. A. McKENZIE, W. H. SAWYER AND M. B. SMITH*

Department of Physical Biochemistry, Institute of Advanced Studies, Australian National University and C.S.I.R.O. Physical Chemistry Unit (Division of Food Preservation), Biochemistry Department, University of Sydney, Sydney (Australia)

(Received March 29th, i967)

SUMMARY

A comparative s tudy is made of the sedimentation and optical rotatory dispersion behaviour of genetic variants of bovine fl-lactoglobulin in sodium aceta te- acetic acid buffers near pH 5-

Sedimentation velocity patterns of the A variant at pH 4.7 and low temperature are in agreement with the observations of others that the dimer unit of molecular weight 35000 polymerizes rapidly to form predominantly an octamer (for a review see ref. I). The dimer of the B variant shows only a small tendency to undergo polymerization.

The relationship of sedimentation coefficient to protein concentration is em- ployed as a useful index of polymerization in several cases wherein association is sufficiently weak that it is not readily apparent from the sedimentation patterns.

The weight average molecular weight of the A and B variants is measured at pH 4.7 in the range 4 - 3 2o using the approach to sedimentation equilibrium principle of ARCHIBALD. A method is presented for treating the data to determine the equilibrium constant of the dimer-octamer reaction in the A variant. Values of relevant thermodynamic functions are determined.

Sedimentation equilibrium measurements are made at low concentration for the A, B and C variants at pH 5.4 and 20 °. There is some evidence of dissociation of the dimer to the monomer under these conditions contrary to the conclusions of others.

The effect of the dimer-octamer reaction on the optical rotatory dispersion properties of the A variant is studied extensively. The reaction is accompanied by a considerable change in [~j~ and ao, but only a small change in bo. I t is shown how the former two changes may be used to calculate the equilibrium constant and thermodynamic functions of the reaction. The results obtained are in good agreement with those obtained by other weight average methods. The method is a valuable one and applicable generally in favourable cases of associating-dissociating systems.

Possible conformational changes in the dimer of the A variant to facilitate octamerization are discussed.

* ]Present address: C.S.I.R.O. Division of Food Preservation, North Ryde, N.S.\¥., Australia.

Bioehim. Biophys. Acta, 147 (i967) 73-92

74 H. A. MCKENZIE, W. H. SAWYER, M. B. SMITH

INTRODUCTION

Early studies of fl-lactoglobolin by sedimentation-diffusion, X-ray and osmotic pressure methods indicated it had a molecular weight of approx. 36 ooo. (for a review see ref. I). OGSTON AND TILLEY 2 found that some fl-lactoglobulin samples prepared from milk of individual cows showed a tendency to associate at pH 4.7 and low temperature (see also ref. 3). Considerable light was thrown on these observations by two important discoveries. Firstly, ASCHAFFENBURG AND DREWRY 4,5 showed that bovine fl-lactoglobulin exhibited genetic polymorphism, there being two variants (A and B). Secondly, GILBERT 6 developed a theory to predict the kind of patterns obtained in transport experiments for rapidly associating-dissociating systems. OGSTON AND TOMBS 7-9 examined the heterogeneity of the A and B proteins and their behaviour in transport experiments in the light of GILBERT'S theory. They concluded that their samples of each variant contained about ~o % of a minor component, and that the A and B variants associate at pH 4.7 and low temperature to form at least a trimer of the 36000 molecular weight particle. MCKENzIE AND S~iTHI°, 11 concluded from sedimentation measurements that the A variant polymerized to form a te t ramer of the 36000 unit, but the B variant showed little tendency for this association. A similar conclusion was reached by TIMASHEFF and his cc, llaborators 12, la from sedilnen- tation and light scattering measurements. I t is now known ~ that this ' te t ramer ' is in fact an octamer of the. minimum molecular weight unit of 18ooo.

In I958 two of us (H.McK., M.B.S.) were carrying out a study of the effects of pH, temperature and organic solutes oil the conformation and state of association of various proteins. In studies of urea denaturation we assumed that large changes in laevorotation were associated with appreciable conformational changes and that polymerization produced only negligible effect on the laevorotation. However, in the absence of urea, it was found that there were abnormalities in the laevorotation vs.

temperature curves, where polymerization was occurring and where large changes in conformation were unlikely. I t became apparent that optical rotation measurements could be a useful aid in the s tudy of association-dissociation reactions. Thus concurrent studies were made of the association of bovine fl-lactoglobulins A and B, at pH 4.6 over the temperature range 4-3 °0 , by making approach to sedimentation-equilibrium and optical rotation measurements. This work was extended subsequently in Canberra to include optical rotatory dispersion measurements. Also studies by two of us (H.McK., W.H.S.) of fi-lactoglobulins A and B and of the recently discovered C variant14,15 indicated that the 36 ooo particle weight unit ("dimer") could dissociate into tile single chain 18 ooo unit ("monomer") over a wider pH range than had been realised generally, following the important dissociation studies at low pH of TOWNEND AND TIMASHEFF 16,17. We have now made an extensive study of the behaviour of the fl-lactoglobu]ins over the pH range 2-1o. In the present paper we report our results for the association-dissociation behaviour of bovine fl-lactoglobulins in the pH range 4.6-6.0 (preliminary accounts have been given elsewhere 1°, 11,1a). TIMASHEFF and his collaborators ls-2° have made an independent s tudy of the association of bovine fl-lactoglobulins A and B near pH 4.6. Their results are discussed also in this paper.

Biochim. Biophys. Acta, 147 (1967) 73-92

~-LACTOGLOBULIN ASSOCIATION-DISSOCIATION 75

MATERIALS AND METHODS

Materials In early experiments bovine fldactoglobulins A and B were prepared by a

modification of the method of ASCHAFFENBURG AND I)REWRY 21. Later preparations were made by the method of ARMSTRONG, MCKENZIE AND SAWYER 22. The C variant was prepared by the method of BELL AND MCKENZlE 23.

Methods Sedimentation Sedimentation velocity measurements were carried out as described previous-

ly 23. The weight average molecular weight measurements for the "octamerizat ion" reaction were made by the method of KEGELES AND NARASINGA RAO 24,~5 using the EHRENBERG 26 technique. The methods of calculation were different from those of these authors and are discussed in the next section.

Weight average molecular weights at low concentrations were measured and calculated by the method of YPHANTIS 27. Experiments were performed at pH 5.4 in acetate buffer (o.Io M sodium acetate, o.o16 M acetic acid; I o.I). Estimates of the equilibrium time were made using the criteria of VAN HOLDE AND BALDWIN ~s. In practice it was found that displacement distributions at 9 and IO h were identical within experimental error. Low speed exposures at the beginning of the experiments were used as blanks. Measurement of fringe displacements in solutions of pure solvent were found to be independent of rotor speed.

optical rotatory dispersion Early measurements of optical rotation were made with a Schmidt and Haensch

polarimeter (Schmidt and Haensch, Berlin, Germany). Measurements of optical rotatory dispersion were made with a Perkin Elmer Model 141 photoelectric polarimeter (Bodenseewerk, Perkin Elmer, Uberlingen, Germany). Most of the measurements were made using medium pressure mercury arc light sources with filters to cover the range 313-578 m/~. Some measurements were made using a quartz-halogen light source and Zeiss double monochromator. Standard Perkin Elmer water-jacketed cells were used. Condensation of water on these cells at low temperature was prevented by circulating dry N,, through the cell compartment. The temperature was controlled to o.o5-0.1o ° over the temperature range 1-45 ° , and the temperatures in the cell were measured using a nichrome- constantin thermocouple.

Partial specific volume (v) The determination of ~, involved accurate measurements of density over a

range of protein concentration. These were made by the density gradient tube method of LINDERSTROM-LANG AND LANZ 29. I t was orginally planned to measure the density of the ~-lactoglobulins in a o.i M sodium acetate-o.o88 M acetic acid (pH 4.65) buffer. However it was found that drops of this buffer (0.00o2-0.0005 ml) introduced into the column, after falling initially, drifted steadily upwards. The effect is probably due to the transfer of acetic acid from the drop to the gradient liquid (brombenzene- kerosene) saturated with H20, the drop becoming less dense. Since it was impossible to saturate the column with acetic acid it was necessary to use another solvent. Accordingly the measurements were made in 0.05 M NaC1, the pH of the protein solutions being approx. 5.2. Protein concentrations in this and other parts of the

Biochim. Biophys. Acta, 147 (1967) 73-9z

7 6 H . A . MCKENZIE, W. H. SAWYER, M. B. SMITH

present work, were determined by the Kjeldahl nitrogen method of MCKE~ZIE AND WALLACE 30. KC1 solutions of known density were used to calibrate the column. Apparent specific volumes were calculated from the experimental density values using the relationship given by DAYHOFF, PERLMANN AND MACIN~,'ES 31.

RESULTS AND CALCULATIONS

Partial specific volumes (~) The apparent specific volume for each variant was independent of concentration

over the range of density measurements, o.5-1. 9 g protein per ioo g solution, within the accuracy of the measurements. Thus the partial specific volume was taken as equal to the apparent specific volume. The value at 3 °0 of ~ for fl-lactoglobulin A was found to be 0.747 -k 0.005 mug and that for the B variant to be 0.745 :~ 0.005 ml/g. The mean value of 0.746 ml/g was used for both variants at 3 o°. The value of ~ for fl-lactoglobulin C was assumed to be the same as that for the A and B variants. The value of 0.746 may be compared with PEDERSEN'S value 32 of o.751 for pooled fl-lactoglobulin. I t was also assumed in our work that ~ for the monomer and octamer are the same as that of the dimer. There is little information in the literature on the variation of ~ with temperature. LUNDGREN AND WARD 33 state that ~ increases by 0.0o05 mug per degree "for many proteins", while ADAIR AND ADAIR 3~ give the value of 0.00038 ml/g per degree for haemoglobin. Thus for calculation of ~ at a temperature, t °, we used the formula

vt = 0 . 7 4 6 - 0.0004 (3 ° - - t ) (i)

Sedimentation measurements Sedimentation velocity Sedimentation-velocity measurements were made for fl-lactoglobulins A and B

at a concentration of approx. 20 g/1 in the OGSToN--TILLEY 2 acetate buffer (o.I M sodium acetate-o.o88 M acetic acid (pH 4.65) at 20°). I t can be seen from Fig. IA that the sedimentation pat tern of the A variant is strongly dependent on temperature as was found for pooled fl-lactoglobulin by TIMASHEFF and co-workers ~, 12. The pat tern at 3 ° is of the typical bimodal type predicted by GILBERT for rapidly associating- dissociating systems (nA ~ An where n > 2). (The short t ime for at ta inment of equilibrium of this system was established originally by TIMASHEFF'S group using light-scattering measurements13.) The s2o,w value for the fast-moving material was 5-9 S and for the slow material, 3.6 S. (It is important to note that the sedimentation coefficients determined under the actual conditions of the experiment have been converted to s2o,w values for convenience in comparison. Such coefficients do not apply for systems of this type if the experiment is carried out at a temperature different from the original experimental temperature.) At 8.7 ° there is less of the fast-moving material, the s2o,w values of the slow and fast material being 3.1 S and 5.1 S, respec- t ively (Fig. IB). At 20 ° a single broad peak of s2o,w of ~ 3.7 S (Fig. IC) and at 29 ° a single symmetrical peak of s~0 ' w 3.0 S (Fig. Id) are obtained. In the lower temperature range, where association is appreciable, the proportion of fast-moving material at a given temperature depends on the protein concentration and increases with increasing concentration. In contrast fl-lactoglobulin B, in this concentration range, at all temperatures studied, gave a single peak, typical examples being shown in Fig. i,

13iochim. Biophys. Acta, 147 (1967) 73-92

fl-LACTOGLOBULIN ASSOCIATION-DISSOCIATION 77

for a concentration of approx. 20 g/ l , giving an s~o,w value at all temperatures of approx. 2.83-2. 9 S.

BELL AND McKENZIE 2a obtained similar patterns to those of the B variant for the C variant when it was first isolated.

I i

A .3 " A .9 "

A .20 ° A . 29 °

B. 2" B. 7 °

Fig. I. The effect of t e m p e r a t u r e on the s e d i m e n t a t i o n p a t t e r n s of bovine f l - lactoglobulins A and B a t p H 4.65 (sodium ace t a t e - ace t i c acid, I o.1). Ro to r speed: 59 780 rev /min . Phase pla te angle, 75 °. A, 3 °, p ro te in concent ra t ion , 19.5 g/I; io8 min (after r each ing full speed); s~o,w fas t = 5.9 S; Szo,w slow = 3.6 S. A, 9 °, I9.5 g/l; 90 rain; S2o,w fas t = 5.I S; s2o,w slow = 3.r S. A, zo °, I9.5 g/l; io 7 m in ; s2o,w ~ 3.7 S. A, 29 °, 19.5 g/l; 79 min ; s20,w = 3.0 S. B, 2 °, i8.2 g/l; i24 min ; s2o,w = 3.o S. B, 7 ° , 18.2 g/l; i24 min ; s2o,w = 2.9 S.

At low temperature and pH 4.65 fl-lactoglobulin A continues to show its characteristic bimodal pat tern with a smaller amount of fast-moving material as the concentration is lowered. The s20,w values for the fast and slow material at 3.0 g/1 are 4-5 S and 3.I S, respectively. However, as the concentration is lowered further only a single broad peak is observed eventually, for example, at a concentration of 1.5 g/1 the se0,w value of this peak is 3.2 S.

In addition to the strong association of the A variant observed at pH 4.7 and low temperature, it is of interest to consider the effect on the s vs. concentration curve of association which is sufficiently weak that the slow and fast peaks are not resolved. Calculated ~ vs. concentration curves (see DISCUSSION) are shown in Fig. 2A for fl- lactoglobulin A for conditions when no association occurs ("monomer" (dimer) only present) and when there is a slight tendency towards rapid association to form a

B i o c h i m . B i o p h y s . A c t a , 147 (1967) 73-92

7 8 H. A. MCKENZIE, W. H. S A W Y E R , M. B. SMITH

" t e t r a m e r " (octamer). These curves may be compared qualitatively with experimental data obtained under various conditions.

The A variant undergoes weak association at pH 4.7 and 2o °, but the peaks are not resolved as has already been seen from Fig. I. A plot of s2o,w vs. concentration for this variant, under these conditions, is shown in Fig. 2B. Although it has been considered, usually on the basis of the appearance of single sedimentation patterns, that association of the A variant at pH 5.o is negligible, it can be seen from the s2o,w vs.

concentration plot, in Fig. 2B, that association may occur under these conditions even at 20 ° .

The plots of the S2o,w vs. concentration data for the B variant at pH 4.7 both at 3 ° and 3 °o are shown in Fig. 2C. BELL AND MCKE;','ZlE ~a and W. H. SAWYER (private communication) have obtained a similar type of curve for the C variant under similar conditions. Their data are discussed further by BELL AND McKE~'ZlE 23.

30 &

25

($)

2O

B 3.2

S2°w 3o{- (s)

2

C. 281

S20,W 26 (s)

2.4

" ' ' .44 " 4

& pH a 7 20 ° / & ,

© I I I I I

o ~ pH 5O 20 °

, t I [ I

& o

3-"

o ~ ~ o A~A " ' " -

O 300

t I I I I l I [ I l ,

10 20 30 40 50

CONCENTRATION (g/L)

Fig . 2. A. T h e o r e t i c a l c u r v e s for t i le c o n c e n t r a t i o n d e p e n d e n c e of s e d i m e n t a t i o n of b o v i n e /~- l a c t o g l o b u l i n A u n d e r c o n d i t i o n s of w e a k a s s o c i a t i o n (based o n GILBERT40). B. E x p e r i m e n t a l c u r v e s for t h e c o n c e n t r a t i o n d e p e n d e n c e of s e d i m e n t a t i o n f o r / 3 - 1 a c t o g l o b u l i n A a t p H ' s 4 .7 a n d 5 .0 for 2o °. C. E x p e r i m e n t a l c u r v e s for t h e c o n c e n t r a t i o n d e p e n d e n c e of s e d i m e n t a t i o n f o r / ~ - l a c t o g l o b u l i n B a t p H 4.65 for 3 ° a n d 3 °o .

Biochim. Biophys. dcta, I47 ( I967) 73 -92

J~-LACTOGLOBULIN ASSOCIATION-DISSOCIATION 79

Molecular weight and equi l ibr ium constant measurements at high concentration and

I~H 4.6 by the method af KEGELES AND NARASINGA RAO Apparent weight average molecular weight measurements were made on fl-

lactoglobulins A and B at pH 4.65, using the principle of ARCHIBALD aS,~. KEGELES AND NARASINGA RAO 2a,25 showed that, for an associating system, the apparent weight average molecular weight (]~rw) is given by

M w = RT(~c/Ox) m (2) (I - - ~p) ~2XmCT

where xm is the distance from the axis of rotation to the meniscus; CT is the total weight concentration of protein at the meniscus at any time, p is the density and co the angular velocity.

Now for the monomer-n-mer system, the weight concentration dissociation constant (Ka') is given by

C m n Ka' - (3)

cp

where CD and Cm are weight concentrations (g/l) of n-mer and monomer, respectively. Now assuming the absence of intermediate polymers in appreciable concen-

trat ion M w = CmM1 4- CpMp (4)

CT

where M1 and Mp are the molecular weights of monomer and n-mer respectively and CT is the weight concentration of total protein (g/l).

KEGELES AND NARASINGA RAO showed that (converted to our symbols}

CT .n-1 (nM1 - - Mw) n K a ' = {~ ---~/~} (Mw- M,) (5>

They obtained K d ' from the slope of a plot of (~rw --M1)/(nM1 --217rw)n vs. C n-1.

This plot is not particularly satisfactory, especially at higher values of n as this results in crowding of the experimental points mainly into one region. Thus we have preferred to proceed in a different way.

Now the stoicheiometric dissociation constant on the base mole scale (Kd) is related to K d ' by

n Ka = K a " - - - - (6)

Mn-1

Hence,

Thus,

i

Ka = n (nM1 - - M1) n-1 (M~w - - M,)

log Ka = log n + (n - - I ) log (CT/M1) - - (n - - I ) log (nM1 - - M1) +

n log (riM1 - - Mw) - - log (Mw - - M1)

(7)

(8)

Now if the right hand side of Eqn. 8 is plotted against CT and n has been chosen correctly, assuming little departure from ideality, the least squares line of best fit should have zero slope and the intercept should give log Kd. The association constant (Ka) can be derived from the relationship,

log Ka = --log Ka (9)


8 0 H. A. MCKENZIE, W. H. SAWYER, M. B. SMITH

The molecular weight experiments were carried out at sufficiently high concentration to avoid appreciable dissociation of the dimer into the monomer. At the same time the concentrations were chosen so that they were not too high to be examined conveniently with the centrifuge's optical system or that departure from ideality would be appreciable (fortunately for fl-lactoglobulin this appears to be small in the concentration range studied). In calculating Ka it has been assumed that the equilibrium can be treated as between the dimer and higher polymers. Thus the dimer was regarded as a "monomer" and the oetamer as "tetramer". It was also assumed that the "dimer" and "trimer" were only present in insignificant quantities. At the time the experiments were carried out there was no major evidence to support this, but the subsequent ingenious small angle X-ray scattering studies of WITZ, TIMASHEFF AND LUZZAT137 support, but do not demand this assumption (see also BISCUSSION).

TABLE I

log Ka FOR THE DIMER--OCTAMI~R ASSOCIATION OF fl-LACTOGLOBULIN A (pH 4.65)

L i g h t - s c a t t e r i n g d a t a of TOWNEND AND TIMASHEFF 13. K a is g i v e n in I s" b a s e mo les -3.

Temp. log Ka from (°C)

ARCHIBALD [0¢]578 Optical Light Mu, × Io -3 (o) rotatory dis- scattering

persion (ao)

I.O - -13 . 3 12.6 2.2 3.0 - -12 . 3 12.O 4.0 I I . I 5.0 - -11 .8 11.6 6.1 7.0 - -11 . 4 11.2

IO.O - -10 .8 10. 7 10.5 I I .O 9,9 13.O - -10 .2 IO.I

15.5 16.O - - 9.8 9.8 20.O - - 9.2 9.1 20.1 9.4 25 .o 30.O

11. 7

I I . I

10.5

9.8

9.2 8.5

7.9

Results of the determination of weight average molecular weight measurements for fl-lactoglobulin A at pH 4.65 and 4.0 ° are given in Fig. 3-

Values of the right hand side (RHS) of Eqn. 8 are plotted in Fig. 4a against CT assuming n - - 4 and n = 5- The difference between these two plots is striking. Thus the correct value for n is 4.

Results are shown in Fig. 3 for measurements at 11.7 °, 2o.1 ° and 32°. The calculated values of log Ka for n = 4 are given in Table I. Relevant thermodynamic parameters are given in Table II.

Molecular weight measurements for fl-lactoglobulin B at 4.6 ° and 32o are summarized in Fig. 3. They indicate there is slight association of this protein beyond the 36000 molecular weight under the conditions of the experiments.

Biochim. Biophys. Acta, 147 (1967) 73 -92

fl-LACTOGLOBULIN ASSOCIATION-DISSOCIATION 81

Molecular weight measurements at low concentrations by the method of YPHANTIS In view of the s vs. concentration curves found above, it was decided to measure

the weight average molecular weight for/~-lactoglobulin A at low concentration and pH 5-4; I o.I. The sedimentation equilibrium method of YPHANTIS 17 seemed well suited to this measurement. A plot of log fringe displacement versus half the square of the distance from the centre of rotation showed some departure from linearity.

Values of the weight average molecular weight ()llw) for the three variants

T A B L E II

THERMODYNAMIC PARAMETERS OF THE DIMER--OCTAMER ASSOCIATION OF /~-LACTOGLOBULIN

Light-scat ter ing da ta of TOWNEND AND TIMASHEFF 13.

Temp. ARCHIBALD tool. wt. Optical rotatory dispersion (ao) Light scattering (°co (AH = --4 ° Kcal/mole)

(A H = --64 Kcal[mole) _ A F ° - - A S ° (Kcal/mole) (e.u.)

(AH = --53 Kcal/mole)

__AF ° __AS ° __AF ° - -AS ° (Kcal]mole) (e.u.) (Kcal/mole) (e.u.)

I.O 3.0 4.0 14.o 96 4.5 5.0 7.0 IO.O

II.O 12.9 IOO 13.o 16.o 20.0

2o.1 12.6 96

15.8 178 15.2 179

14.8 179 14.4 179 13.9 179

13.3 179 12.9 178 12.2 178

14.4 138

120

110 e ~ ~

100 o / 90

80

A~ % ~ , / " K~

I'S" 60 Au 7'~[

50 A20.f~ ~" dO ~I**

3fl 10 20 30

CONCENTRATION (9/l)

Fig. 3. Concentra t ion dependence of the apparen t weight average molecular weight (Mw) of bovine fl-lactoglobulins A and B at p H 4.65 (sodium ace ta te-ace t ic acid, I o. i) . Detai ls of var iant , t empera tu re and initial concent ra t ion: A, 4°: ~ , 12. 3 g/l; &, 24.6 g/l; A, 3 ° g/l; 0 , 5 ° g/1. A, I I .7° : @, 12. 3 g/l. A, co°: *, 24.6 g/1. A, 32°: O, 5.7 g/l; (D, e4.6 g[1. B, 4.6°: + , 25.2 g/1. B, 32°: ~,, 6.2 and 25.2 g/1.

Biochim. Biophys. Acta, 147 (1967) 73-92:

~2 H.A. IVICKENZIE, W. H. SAWYER, M. B. SMITH

obtained from these experiments are shown in Table I n . I t can be seen that -Mw decreases with decreasing concentration and that each of the variants dissociates slightly under conditions of low concentration and to about the same extent.

At I 0.5, the displacement versus concentration plot became essentially a straight line, the slope corresponding to a weight average molecular weight of 315oo.

-RHS 13

12

\

. ~ O • @ O O O •

" - 0 0 ' •

10 20 30 CONCENTRATION (g/L)

12.0

-RHS 11.5

11 i0

o o 7 . / ~

. / o /

/ . /•

o o / ' / o / j .

• e /

ee •

,b ~o CONCENTRATION {g /L )

Fig. 4. a. R igh t h a n d side of Eqn . 8 (RHS) p lo t ted aga ins t pro te in concen t ra t ion for bovine fl-lactoglobul in A (pH 4.65) (sodium ace ta te -ace t i c acid, I o.I) a t 4.0 °. ~ , calculated, a s s u m i n g d i m e r - oc t amer associat ion; O, calculated, a s s u m i n g d i m e r - d e c a m e r associat ion, b. The dependence of the r igh t h a n d side (RHS) of Eqn. 8 on prote in concen t ra t ion for the d i m e r - o c t a m e r associat ion of bovine f l - lactoglobulin A . - - - - , a s s u m i n g IOO % associa t ing mater ia l ; . . . . , a s s u m i n g 5 % non-assoc ia t ing mater ia l p resent ; . . . . . , a s s u m i n g io % non-assoc ia t ing mater ia l present . To avoid confusion in the graph , the po in t s for the 5 % line are omit ted•

T A B L E I I I

S VEIGHT A V E R A G E M O L E C U L A R ~,VEIGHTS (x~u , ) OF f l - L A C T O G L O B U L I N S A , B A N D C AT p H 5 . 4

fl-Lactoglobulin ]VI w Concn. varia~t (g/l)

A 27 200 0.02 33 ooo 0.43

B 26 ooo 0.05 32 700 0.68

C 25 700 0.o 5 33 ooo 0.43

The effect of non-associating com~onents in ~-lactoglobulin A OGSTON AND TOMBS 7 a n d TIMASHEFF AND TOWNEND 17 reached the conclusion

that their samples of fl-lactoglobulin A contained about IO % of material unable to associate on the basis of phase solubility studies and the proportion of fast-moving material in sedimentation velocity measurements. This possibility was examined in relation to our Mw measurements. If it is assumed that x is the fraction of the mixture which does not take part in the equilibrium system then, the

Biochim. Biophys. Acta, I47 (1967) 73-92

~-LACTOGLOBULIN ASSOCIATION--DISSOCIATION 83

value of the weight average moleculal weight (-~MP) for the reactants is given by

MMP- MMV xM1 I - - X

where ]~MV is the measured value of ]~rw. The total concentration of reactants becomes, CM + Cp = (I - - x) CT. These relations were used to plot the right hand side of Eqn. 8 for the sedimentation data, assuming various proportions of non-associating material. If x were taken as 0.20, o.15 or o.io the right hand side of Eqn. 8 was far from constant. With x = 0.05, the value was more nearly constant. Typical plots with x = o, 0.05 and o.io are shown in Fig. 4 b. Lines of best fit were found by the method of least squares. The regression coefficients of the lines for x ~ o and 0.05 were examined by the Student t test and found to be significantly different at the o/ 5 /o level. Thus our protein would appear to contain negligible material incapable of undergoing octamerization.

Determination of sedimentation coeftcient of the "tetramer" NICHOL AND BETHUNE 3s have described a simple procedure for determining the

sedimentation coefficient of the "n-mer" from the sedimentation patterns for rapidly equilibrating polymerizillg systems. L. W. NICHOL (private communication) has kindly examined some of our sedimentation patterns for fl-lactoglobulin A at pH 4.7 by their procedure. This calculation was carried out assuming n = 4, 6, 8, IO. The values of S!0olymer so obtained were compared with those computed from a simple model. Only for n = 4, were the two values in close agreement. For all the other values of n there was considerable discrepancy between the two values.

Optical rotation measurements The effect of temperature on the optical rotation at 589 mr* was examined for

solutions of ~-lactoglobulin A and of/3-1actoglobulin B in the acetate buffer (pH 4.65, I o.1) over the temperature range 1-3 o°. It was found that the laevorotation (the numerical value of E~5s9 denoted hereafter as /EXlss9/) increased with increasing temperature for both variants. However, there was a marked variation in the slope of the /[~15s9/vs. temperature curves between the variants. The curve for the B variant showed a slight, almost linear, increase with increasing temperature. The A variant had an increase of almost the same slope over the range 25-300 , but below 25 ° the slope increased considerably as the temperature was lowered. A comparison of the temperature variation of/E~]sso / was also examined for both variants at pH's 6. 9 and 2.1. At pH 2.1 the curves were similar in shape to those of the B variant at pH 4.65. These observations led to the conclusion that the optical rotation vs. temperature change for the A variant at pH 4.65 reflected its tendency to associate. Experiments carried out at various concentrations confirmed this conclusion. Subsequently these experiments were extended to include the new variant/3-1actoglobulin C by BELL AND 1V[CKENZIE 23 and H. A. McKENZIE AND W. H. SAWYER (unpublished observations). Its behaviour resembled that of the B variant.

The marked effect of protein concentration on the shape of the /[a157s/ vs. temperature curve for fl-lactoglobulin A at pH 4-7 is shown in Fig. 5 A. The effect of protein concentration at the lower temperatures is even more striking when/[cc)57s / is plotted against protein concentration at a given temperature, as shown in Fig. 6.


~4 H . A . MCKENZIE, W. H. SAWYER, M. B. SMITH

The results for the A variant may be contrasted with the lack of effect of concentration on the /~]57s/vs. temperature curve for the B and C variants at pH 4.7, shown in Fig. 5 B (for the C variant see also ref. 23). Similar kinds of relationship were found when ~] was measured at other wavelengths in the range 313-57 8 m/,.

The data for the optical rotatory dispersion were examined using the phenom- enological equation of MOFFITT AND YANG ag, namely

M.R. \ 'V. 3 ao~o 2 bo)lo 4

[ m ' ] ) . = [(X]). IOO n 2 -~- 2 - - ~ 2 _ _ ~ O ~ + (Z2 _ _ ~ O 2 ) 2 ( IO)

where Ire'l), is the reduced mean residue rotation, M.R.W. is the mean residue weight (112 for fl-lactoglobulin), n is the refractive index of the solvent at the wavelength and ao and bo are constants. ~o was taken as 212 mr,. Plots of

~2 _ _ Ao 2 ,~o 2 [m'] - - - - - - vs. - - - - - -

~o e 2e - - Ao 2

for fl-lactoglobulin A at a concentration of Io.8 g/1 over the temperature range 1-45 ° showed slight departure from linearity at lower wavelengths (see DISCUSSION). (13 such isotherms were determined.) The method of least squares was used to obtain the intercept, ao, and the slope, be. The resultant small variation in be with temperature for the A variant (at lO.8 g/l) is shown in Fig. 7C. The marked variation of a 0 and I=]57s with temperature, due to the octamerization reaction, are shown ill Fig. 7 B. It is not surprising that these are closely parallel as the variation of bo is small. Like-

I

A

: 5

3°Io ~ _ DJ_MEEb_T_¢5 °_ ...... ~5 o

F3 ~ 10 °

OCTAMER AT 0 °

. . . . ~ 210 31 ,~ 510 I0 20 3O 4O

IEMPERA~ URE CONCENTRATION (g/h

Fig. 5. The effect of temperature on the specific rotation of the/~-|actoglobulins at pH 4.65 (sodium ace ta t e -ace t i c acid, I o.I) and a t several concent ra t ions . A. f l -Lactoglobul in A. F igures refer to concen t r a t ions in g/1. E x p e r i m e n t a l po in t s are omi t t ed to avoid confusion. B. f l -Lactoglobul ins B a n d C. O , fl-C, 11.2 g/l; O, fl-C 22.0 g/l; y , fl-B, 21.2 g/l; A , fl-B, 50.0 g/1.

Fig. 6. The effect of concen t ra t ion on the specific ro ta t ion of/~-lactoglobulin A a t p H 4.65 (sodium ace t a t e - ace t i c acid, I o.i) . The b roken lines represen t the dependence of the pure d imer and pure oc tamer .

Biochim. Biophys. Acta, 147 (I967) 73-92

~-LACTOGLOBULIN ASSOCIATION-DISSOCIATION 85

wise W. H. SAWYER (private communication) has shown that the variation of A, the constant of the single Drude equation, parallels that of ao. The variation in ao and bo for fl-lactoglobulin B with temperature at pH 4.7 was found to be small. This is in accord with the small tendency for the B variant to polymerize.

A

e / _. / . / ~-"

m e/e - - " -

B ~so / o / r ~ 4 ,~°

IdO ~ ° / ° -ao , ,~ . . -,-~o

C 50"

-bo 50

4o

,; 2; ~'o ,; TEMPERATURE

Fig. 7. The dependence of optical rotation on temperature for/Sqactoglobulin A (lO.8 g/l) a t pH 4.65. The broken lines represent the dependence for the pure dimer and pure octamer. A. Specific rotation at 578 m/~. B. The ao constant of the MOFFITT--YANG equation. C. The bo constant of the 1V[OFFITT--YANG equation.

The determination of Ka from the optical rotatory measurements We made then an attempt to determine the association constant for the

"tetramerization" (octamerization) reaction from the optical rotatory data. If it is assumed once again that "monomer" and "n-mer" are only present and

there are no major interaction effects between "monomer" and "n-mer", then the observed rotation ~obs, is equal to the sun) of a rotatio~ ~p, normally due to "n-mer" and a rotation, ~rn, normally due to "monomer" if either were alone present. Thus

~obs -- O~p ~- ~Xm ( I I a )

Thus [~!, the observed "specific" rotation, is given by

CT ( l i b )

CT{[~] -- [~]p} C m -- (12)

E~x] m -- [0c] p


86 H . A . MCKENZIE, W. H. SAWYER, M. B. SMITH

Now since

CT = Cm -}- Cp (13)

log Ka" = n log Cm - - log (CT - - Cm) (14)

Hence log K a ' ~ (n - - I ) log CT - - (n - - I ) l o g {[0¢] rn - - [0¢]p} -!

n log {[~] - - [0¢]p} - - log {[~]m - - [0¢]} ( I 5 )

If n is chosen correc t ly a plot of the r ight hand side of Eqn. 15 vs. C T should give log K a ' as in te rcep t wi th zero slope. (Al te rna t ive ly a plot of log C T ( x - - {[~" - - [=Jv/ [aim - - [=]p}) vs. log {CT([R] - - [C~V/[=]m - - [=~p} should give a slope of n and an in te rcep t of K d ' . ) Hence K a and Ka m a y be ob ta ined using the usual re la t ionships wi th K d ' .

The opt ica l r o t a t o r y dispers ion measurements indica te tha t the value of ao is r e la ted to the ex ten t of associat ion. If i t is assumed tha t the observed value, do, is a weight average p r o p e r t y we have,

- C m a o , m @ Cpao ,p ao = (16)

CT

where ao,m and ao,p are the charac te r i s t i c values of ao for monomer and n-mer, respect ively . Hence,

Cw(ao - - do,p) C~ (17)

(a . . . . - - ao,p)

C . ( a o , m - - ao) Cp = CT - - C~ -- (18)

(ao,m - - ao,p)

log K a ' = (n - - I) log CT - - ( n - - I) log (ao,m - - ao,p) +

n log (ao - - ao,p) - - log (ao,m ao) (19)

A plot of the r ight hand side of Eqn. 19 aga ins t CT is made as above. In app ly ing the first of the above me thods i t is necessary to know [aim and

[~]p. I t was assumed, in view of the sed imen ta t ion measurements , t ha t the ex ten t of associat ion of f l - lactoglobulin A a t p H 4.65 and 35 ° was negligible and the curve was e x t r a p o l a t e d back to o °, assuming the slope of the curve for f l- lactoglobulin C (for which associa t ion is negligible over the range 1-35 °) appl ied to the f l- lactoglobulin A " m o n o m e r " (dimer). The values so ob ta ined gave [0~]m over the range 0-45 °. A value of [a~p for o ° was ob ta ined from the [c~] vs. t curve of f l - lactoglobul in A at a concent ra t ion of 50 g/1 where i t was assumed tha t it was mos t l y in the form of " t e t r a m e r " (octamer) a t o °. A line was d rawn from this po in t to 45 ° assuming a slope equiva len t to t ha t of the "monomer" . Thus [a]p was ob ta ined for the range 0-45 °. A s imi lar procedure was a dop t ed wi th the ao vs. t curve to ob ta in ao,m and ao,p for the second method of analys is of the da ta .

Values of log K a ob ta ined from the opt ica l r o t a t o r y measurements are compared in Table I wi th our values ob ta ined from ARCHIBALD measurements , and the values deduced b y TOWNEND AND TIMASHEFF 13 from l igh t - sca t te r ing measurements .

When log K a was p lo t t ed agains t I / T the value of 2 H ° could be ca lcula ted from the slope. This and values of AS ° a n d / I F ° are given in Table I I .


~-LACTOGLOBULIN ASSOCIATION--DISSOCIATION 8 7

DISCUSSION

We have seen from the present sedimentation and optical rota tory dispersion measurements that bovine fl-lactoglobulin A associates over a wide range of concentration in acetate buffers at pH 4.65 and low temperatures. Our results for this reaction are in accordance with the earlier sedimentation velocity observations of OGSTON AND TOMBS ~ and the sedimentation velocity and light scattering studies of TIMASHEFF and co-workers. On the other hand fl-lactoglobulin B shows much less tendency for this association. The recent measurements of BELL AND IV[cKENzIE 23 and W. H. SAWVER (private communication) on fl-lactoglobulin C indicate lack of any appreciable tenden* cy for this association even up to concentrations of 50 g/1. Contrary to previous studies, we have shown that there is some tendency for the dimer of all three variants to dissociate into the monomer unit in the pH region 4.6-6.0.

S e d i m e n t a t i o n v e l o c i t y

In a strongly polymerizing system the sedimentation velocity increases with increasing concentration, reaches a maximum value, and then decreases with further increasing concentration. This behaviour is due to the opposition of two factors: the increase in overall sedimentation velocity as the average particle size increases with increasing concentration and the usual decrease in velocity caused by changes in the viscosity and density of the solution and back flow of solvent. GILBERT 4° calculated the s vs . concentration curves for the leading and trailing sedimentation peaks of fl-lactoglobulin A at pH 4.7 and 2 °. He obtained satisfactory agreement with the experimental data of TIMASHEFF AND TOWNEND for the leading peak.

We have made similar assumptions in our weight average measurements of the association of fl-lactoglobulin A to those of GILBERT 4° in his theoretical calculations for the s vs . concentration curves. The justification for these assumptions is based on the satisfactory t reatment of a variety of weight average data (ARCHIBALD and optical rotatory dispersion in this paper and light scattering by TOWNEND AND TIMASHEFF13). As stated earlier, the outstanding small angle X-ray scattering work of WITZ, TIMASHEFF AND LUZZAT137 indicates that the " te t ramer" (octamer) is a compact cyclc structure. GILBERT 4° assumed such a structure stabilized considerably the " te t ramer" over the intermediate species. In considering the possibility of intermediatesTowNEND AND TIMASHEFF 13 imply that their absence is supported by the bimodality of the schlieren pat tern and furthermore that this bimodality precludes "n-mers" higher than the " te t ramer" . These conclusions are not demanded by GILBERT 6 in his theory (for a discussion see ref. I).

GILBERT 4° has pointed out that, at higher temperatures, where the association of B-lactoglobulin A becomes weak, the peaks of the bimodal schlieren pat tern move closer together and they cease to be distinguishable because of blurring by diffusion. Nevertheless the shape of the s vs . concentration curve should enable polymerization to be detected. Typical theoretical curves calculated by GILBERT for the B-lactoglobulin dimer-octamer reaction are shown in Fig. 2. Our experimental curves for the A variant at pH values of 4.7 and 5.0 at 20 ° have been seen already to be in qualitative agreement with GILBERT'S curves (no quantitat ive comparison is of course possible as our curves were obtained for different conditions).

I t can be seen, from the A curve at pH 5.0 and the B curves for 3 ° and 20 ° at


~ H. A. MCKENZIE, W. H. SAWYER, M. B. SMITH

pH 4.7, that the shape of the s vs . concentration curve is a sensitive criterion for de- tecting an associating-dissociating system. I t should be stressed that, while the general factors outlined above influence the shape of the s vs . concentration curve for the /~-lactoglobulins, the situation is made more complex by the tendency for dissociation of the dimer to the monomer at low concentration. The slope of the "linear" (higher concentration) part of our s vs . concentration curve for B-lactoglobulin B at pH 4 . 7 ,

is less than the calculated limiting slope. The difference is greater at 3 ° than 3 °o . This may be contrasted with the results of BELL AND lVlcKENZlE 23 and W. H. SAWYER (private communication) for the C variant where the slope of this part of the s vs .

concentration curve is the same as the calculated limiting slope. Thus it would appear that, in the case of the B variant, there are one or more rapid association-dissociation reactions occurring and one of these is probably the dimer-octamer reaction. Never- theless the tendency for the latter reaction would appear to be less than that implied by the recent light-scattering measurements of KusiostxsKI A~'D TIMASHEFF 41 who reported a value of 3.5" lO8 13"moles-3 for Ka for the B variant at pH 4.7 and 2 °. This value would suggest that approx, o/ 43 ,o of the protein would be present as octamer at a total protein concentration of 5 ° g/1 and a bimodal pat tern would be expected in sedimentation velocity experiments. Such a pat tern has not been observed in the present work at this or any other concentration studied.

Our 2~w measurements at low concentration by the method of YPHANTIS 27 justify the assumption that the extent of dissociation of the dimer ("monomer") unit is negligible under the conditions used in the determination of Ka for tile dimer- octamer ("monomer- te t ramer") by the sedimentation and optical rotatory dispersion methods. Nevertheless the extent of dissociation is appreciable at very low concentration. Thus, in the pH 4.5-6.0 region, the dissociation of each of the three variants into the 18 ooo monomer unit must be considered in studying their behaviour undet conditions of very low concentration (compare results at pH 7.5 in ref. 43)-

Furthermore in considering the shape of the schlieren pattern obtained in sedimentation velocity studies of the dimer-octamer reaction of fl-lactoglobulin A it is apparent from our work that dissociation into monomer may be appreciable at the low concentrations prevailing on the trailing side of the slow "peak". This would render the simple application of the GILBERT 6 treatment for estimating the shape of tile slow peaks somewhat suspect.

YPHANTIS ~7 obtained a value of 365oo ~ ilOO for the molecular weight of a sample of mixed ~-lactoglobulin in o.5 M ammonium acetate buffer (pH 5.5) at 25 °. He records that/~-lactoglobulin appears to be homogeneous under these conditions. However, an examination of his plot of fringe displacement v e r s u s concentration shows that deviation from the straight line exists at low concentrations, even at fringe displacements greater than IOO / z.

In considering the " t e t r a m e r i z a t i o n " reaction other workers have considered o/ that the area distribution of their sedimentation velocity patterns indicates IO ,o of

non-associating material in /~-lactoglobulin A. I t is not our experience that such material is present in significant quantities. Zone electrophoresis and column chrom- atography studies in this laboratory have not given evidence of this impurity. The present 21Iw measurements indicate that there is appreciably less than 5 % of any such material.

When our original work on 217w measurements was undertaken to determine Ka,

13iochim. Biophys. Acta, 147 (I967) 73-92

~-LACTOGLOBULIN ASSOCIATION--DISSOCIATION 8 9

short column equilibrium methods had not then been developed. Thus we had recourse to the method of KEGELES AND NARASINGA RAO which proved fast, simple and reasonably accurate for this study. However, studies with a modification equilibrium method of LABAR AND BALDWIN 42 (see ref. 41) in recent years have indicated that it is capable of rather higher precision. Where measurements at very low concentrations are needed to determine whether dissociation is occurring the method of YPHANTIS e7 is of great help, but of only moderate precision. Complete stepwise formation constants are calculable in principle from 3~rw measurements, but high precision is needed. Studies to achieve this objective are in progress.

Optical rotatory dispersion The present studies show how optical rotatory dispersion measurements can be

of value in studying association-dissociation reactions. I t is surprising that there have been so few studies of the effect of association-dissociation on optical rotatory properties of proteins. When we made our original observations in 1958 we examined the literature and it appears that among the earliest studies are those of CARPENTER 4~'. He found the laevorotation of whole casein at pH 6.8 and 20 ° decreased from / E~lssg/----- lO5 ° to/[~]5s9/~- 99 ° as the concentration was increased from 1. 5 to 15 g/1. I t is of interest to note that the caseins undergo a temperature-dependent aggregation. I t is our present belief that these proteins have very little helical character and the aggregation involves hydrophobic binding. BOEDTKER AND DOTY 45 found that gelatin shows a decrease in laevorotation on gelling presumably as it goes from a relatively and associated random coil to a highly associated collagen like structure. SCHELLMAN 46 observed a change in laevorotation during the association of insulin and examined the effect of temperature on the laevorotation of several proteins. The present s tudy is the first one in which optical rotatory dispersion measurements have been used to calculate equilibrium constants for association-dissociation reactions.

Association appears to result in the cases we have studied of little change in bo but an appreciable change in ao. I t is of interest to consider how these changes arise. I t can be considered that the optical rotation of a protein arises from (a) the sum of the interactions between groups present in the backbone of the helix, (b) the sum of the interactions of side chains with one another, (c) the sum of the interactions of the side chains with the backbone of the helix, (d) the sum of the inherent rotations of the side chains. The magnitude of the changes in bo on formation of the octamer would seem to make it unlikely that major changes of Type (a) are the cause of the change in rotation on association. While the X-ray scattering studies have given a good picture of the geometry of the " te t ramer" it is not sufficiently detailed to predict even qualitatively the magnitude of the changes to be expected in (b) and (c), but such changes would seem to us to be the more likely causes of the change in ao on association. I t is of interest that the effect of temperature on the laevorotation of the fl-lactoglobulins at pH 4.7 is qualitatively in accord with the KAUZMANI~--]~YRING 4~ rule for a native molecule in which there is considerable constraint, as one of us has pointed out previously (see also in ref. 48).

In the following paper ARMSTRONG AND McKENzIE 49 present evidence that the interaction of carboxyl groups is important in the dimer-octamer reaction of fl- lactoglobulin A at pH 4.7. We have some evidence that the conformation of the protein may change slightly to a specific conformation in the region of pH 5 to enable the


9 ° H. A. MCKENZIE, W. H. SAWYER, M. B. SMITH

mutual orientation of carboxyl groups in volved in the association reaction. The optical rotation of the fl-lactoglobulins varies considerably over the pH range 2-1o. The pH dependence for the C variant at 20 ° for a concentration of IO g/1 is shown in Fig. 8. I t can be seen that the variation in [~l over the pH range 4-5 is small. The change for the A variant over this pH range is considerable at low temperature, as shown for IO ° in Fig. 8. The major part of this effect is due to the presence of octamer. However, the conformation of the dimer itself in this pH region would appear to be different from that in the adjacent pH region. This is shown by the extent of the difference at 3 °0 and 45 °, where the octamerization reaction is negligible.

40 • 20 ° /3 -C /~/t/~p/ o 45 ° ) A 3 0 ° o 20oi fl_zs ca lO°J /

I I I I I 3 5 g 7 8

pH Fig. 8. The effect of tempera ture on the p H dependence of the specific rotat ion of fl-lactoglobulin A. For comparison, the curve for /~-lactoglobulin C at 2o ° is also presented.

I t has been pointed out already that some departure from linearity of MOFFITT- YANG plots was experienced at lower wavelengths (< 330 m/~) in the region 313-58o m~. Such deviation could be avoided perhaps by a different choice of the value of Xto from 212 mk~. However, analysis of the dispersion data assuming values of ~lo between 206 and 218 mtz was made but it was not possible to effect improvement in linearity by this artifice. Since our measurements were carried out, A. BODANSZKY AND H. A. McKENZlE (private communication) made measurements of optical rotatory dispersion of fl-lactoglobulins down to 19o ink, at Princeton University. They found inter alia weak Cotton effects in the range 28o-3oo m~ near pH 5. The laevorotation showed a broad shallow minimum at 295-296 m/~, a maximum at 290-291 mtz and another (smaller) minimum at 287 re#t, rising again slightly near 283 m/~. These Cotton effects could be made to disappear at high pH values, when the iV[OFFITT--YANG equation was obeyed more closely in the region 300-600 m/~. Thus the slight departure from linearity observed by us at pH 4.7 in the region 313-58o m/~ may be due to the presence of these aromatic Cotton effects. Similar Cotton effects have been observed also recently by MYERS AND EDSALL 5° for carbonic anhydrase, and TIMASHEFF, TOWNEND AND MESCANT151 fo r fl-lactoglobulins, and by KRONMAN, BLUM AND HOLMES 52 fo r c¢-

lactalbumin.

Biochim. Biophys. Acta, I47 (1967) 73-92

fl-LACTOGLOBULIN ASSOCIATION-DISSOCIATION 9 1

An independent study of the effect of association on the optical rotatory properties of f~-lactoglobulin has been made by HERSKOVlTS, T0WNEND AND TIMA- SHEFF 20. They found, in agreement with our work, that ao varies with the extent of association. However, they did not calculate values of the association constant, Ka, from this variation, but used their values of Ka, obtained from light scattering, to calculate ao values and compare them with the experimental values, obtaining good agreement. They found a similar variation in ao with temperature to ours. However, contrary to our results, they found bo to be independent of temperature. This discrepancy may be explained partly by the fact that the American workers determined ao and bo graphically from their data, rather than using the method of least squares. They also stress the value of optical rotatory dispersion measurements in association studies. We do not agree with their general statement that in all cases of polymerization involving hydrophobic bonding there is an increase in laevorotation.

ACKNOWLEDGEMENTS

Grateful acknowledgement is due to Dr. L. W. NICHOL for helpful discussions and for permission to quote unpublished results. Thanks are due to officers of the Rural Veterinary Centre, University of Sydney for help in collection of the milk samples. One of us (W.H.S.) is grateful to the Australian National University for the award of a scholarship.

R E F E R E N C E S

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92 H . A . MCKENZIE, W. H. SAWYER, M. B. SMITH

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