Polymer Molecular Weight Methods - ACS Publications

22 The Study of Mixed Associations by Sedimentation Equilibrium and by Light

Downloaded by EAST CAROLINA UNIV on January 25, 2016 | http://pubs.acs.org Publication Date: June 1, 1973 | doi: 10.1021/ba-1973-0125.ch022

Scattering Experiments A L L E N H . PEKAR,

1

P E T E R J. WAN, and Ε. T. A D A M S , Jr. 2

2

Illinois Institute of Technology, Chicago, Ill. 60616 and Texas A&M University, College Station, Tex. 77843

The analysis of mixed associations by light scattering and sedimentation equilibrium experiments has been restricted so far to ideal, dilute solutions. Also it has been necessary to assume that the refractive index increments as well as the partial specific volumes of the associating species are equal. These two restrictions are removed in this study. Using some simple assumptions, methods are reported for the analysis of ideal or nonideal mixed associations by either ex perimental technique. The advantages and disadvantages of these two techniques for studying mixed associations are discussed. The application of these methods to various types of mixed associations is presented.

.Associations between two macromolecules, A and B, of the type nA + mB

A B n

(n,m = 1,2, . . .), or

m

A + Β A B 2A A

(1) (2)

2

as well as other related associations are known as mixed associations. These associations can occur in a variety of ways, and as Equation 2 indicates both complex formation and self-association can occur simulPresent address: The Lilly Research Laboratories, Eli Lilly & Co., Indianapolis, Ind. 46206. 2 Present address: Texas A&M University, College Station, Tex. 77843. 1

260 In Polymer Molecular Weight Methods; Ezrin, Myer; Advances in Chemistry; American Chemical Society: Washington, DC, 1973.


22.

PEKAR E T

Mixed Associations

AL.

261

taneously. This behavior may occur in the formation of the t r y p s i n trypsin inhibitor complex (1, 2) or the insulin-protamine (3) complex. Perhaps the best studied example of a mixed association is the antigenantibody reaction (4, 5, 6). Although mixed associations have been studied by a variety of ways (7, 8, 9, 10, 11), relatively little has been reported on the study of mixed associations by light scattering experiments (1,6, 12), by sedimentation equilibrium experiments (13, 14, 15), or by the latter's relative, the Archibald experiment (14, 16, 17). W i t h ideal mixed associations one can utilize the redistribution of components in sedimentation equilibrium experiments to evaluate the equilibrium constant(s); this has been described extensively in previous publications (13, 14, 15). In nonideal systems, the analysis of mixed associations is more complicated so that one must resort to sedimentation equilibrium experiments at different speeds on the same solution. Here one calculates M (cell mass) or M (cell), the w e i g h t — ( M ) or ζ— ( M ) average molecular weights, over the cell at each speed. For nonideal solutions one calculates the apparent values of these quantities. These values are extrapolated to zero speed, in a manner similar to that used by Albright and Williams (18) with nonideal, nonassociating poly mer solutions. The required calculations are done with these values at zero speed. W i t h the Archibald experiment (14) one extrapolates the values of M , the apparent weight-average molecular weight, obtained at various times at the extremes (r or r ) of the solution column i n the ultracentrifuge, to zero time. In both cases the analysis becomes quite similar to that used with the light scattering experiment; hence the reason for discussing these methods together. w

z

w

z

w a

m

b

Ideal Solutions Preliminary Thermodynamic Relations. Here it w i l l be assumed that we are dealing with incompressible systems; this is a very good as sumption for aqueous solutions since the isothermal compressibility of water is so small. A t constant temperature the equilibrium condition for any mixed association (see Equations 1 and 2 for example) is ημ

Α

+ τημ

Β

=

MA B n

T O

(w,m

=

0,1,2, . . .)

(3)

Here (i = A , B, or A B ) is the molar chemical potential of reacting species i. Equation 3 is valid for self-associations as well since η or m is zero in that case. Under ideal (theta) solution conditions the activity coefficient tji of each of the associating species is one, so that n

m

CA B N

M

=

KC "CB A

In Polymer Molecular Weight Methods; Ezrin, Myer; Advances in Chemistry; American Chemical Society: Washington, DC, 1973.

(4)

262

POLYMER MOLECULAR WEIGHT METHODS

Here Κ = K . In the discussion that follows, we use the mixed asso ciation described by Equation 1 as an example since the methods we w i l l develop can, i n general, be applied to other mixed associations. The total concentration c (we w i l l use grams/liter) of all the associating species is given b y AnBm

+ c

c = c

A

+ Kc c n

A

B

= /(CA,C )

m

B

(5)

B

From Equations 1 and 5 we note that c is a function of c and c only. N o w note that we can differentiate c with respect to c or c ; thus A


A

B

B

{dc/dc )c

=

1 +

nKc - c ,

(dc/dc )c

=

1 +

mKcx c -

(7)

mM

(8)

A

B

B

A

n

A

l

and

m

B

n

m

B

1

(6)

Also, note that M

A

N

B

M

=

nM

A

+

B

For any mixed association the weight-average molecular weight ( M ) depends on the total concentration c of the associating species and also on the initial proportion of the reactants ( A and B ) . A solution for which c = 10 grams/liter could be made up of a variety of blends of A and B ; each blend would have the same value of c but they would each have a different value of M . This suggests that we use the follow ing procedure to simplify matters. L e t us assume concentrated stock solu tions of A and Β are prepared. In work with proteins ( or other polyelectrolytes ) this means that the concentrated stock solutions ( one for A and one for B ) are dialyzed against buffer, with several changes of buffer. N o w the dialyzed, concentrated stock solutions of A and Β are blended so that the initial concentration ratio of A to Β in the blend (assuming no chemical reaction has occurred ) is c /c ° = β. This blend is known as the working stock; dilutions are prepared from it, using the buffer that was i n dialysis equilibrium with the concentrated stock solution. Note that β is the same for a l l dilutions, yet c is different for each of the solutions (working stock plus dilutions). If this procedure is followed, M or its apparent value M w i l l vary with c since the chemical equi librium w i l l cause more A B to form as c increases. This effect is shown in Figure 1; here we have plotted 1 / M vs. c at constant β for four different situations: ideal, nonassociating mixture of A and Β (curve 1 ) ; nonideal, nonassociating mixture of A and Β (curve 2 ) ; ideal, mixed association of A and Β ( curve 3 ) ; nonideal, mixed association of A and Β (curve 4). Clearly such a plot gives much useful information. w

e q

w

e q

0

A

w

e q

w

n

B

a

m

w a


22.

Mixed Associations

PEKAR E T A L .

263

The quantity M can be defined by 1 / M = ( 1 / M ) + Be, where M is the true weight-average molecular weight and Β is the second virial coefficient. F o r ideal, nonassociating mixtures of macromolecules, Β is zero and 1 / M = 1 / M . Thus a plot of 1 / M vs. c w i l l have a slope of zero, which is illustrated i n curve 1 of Figure 1. For a nonideal, nonassociating mixture of macromolecules the second virial coefficient w i l l be greater than zero. Since M is a constant when there is no asso ciation, a plot of 1 / M vs. c w i l l have an intercept of 1 / M and a slope (or limiting slope) equal to B, the second virial coefficient. This is shown in curve 2 of Figure 1. w

a

w a

W

w

w a

W

w a

w


w a

W

c (grams/liter) Figure 1. Curve 1 could represent an ideal polymer solution containing A and Β but undergoing no asso ciation; the nonideal counterpart of this is shown in curve 2. An ideal mixed association between A and B, such as described by Equation 1 might be described by curve 3, whereas, curve 4 could represent a nonideal, mixed association. Solutions exhibiting greater deviations from ideal behavior would give a curve instead of the straight line shown in curve 2 of Figure 1. If a mixed association occurs, then M = M w i l l increase with c (at constant β) because of the chemical equilibrium; this means that 1 / M w i l l decrease with increasing c as is shown in curve 3 of Figure 1. Here M = M = M . F o r a nonideal, mixed association with the second virial coefficient ( Β ) greater than zero, one could encounter a minimum in the plot of 1 / M vs. c. This is because the term Be increases with increasing c while 1 / M decreases with increasing c; such a situation is illustrated in curve 4 of Figure 1. The quantity c M is defined by w

w

e q

w

w a

w

w

e q

w a

w

w

e q

e q


e q

264

POLYMER

cM

w

e q

+ cM

= cM A

A

B

B

=

C M (dc/dC ) ,CB

=

Σ CiMi(dc/dCi) ,Cj

A

A

Kc

+

+

A T

A

c

n

B

MOLECULAR WEIGHT METHODS

M

m

A n B m

CBM (dc/dCB)T,C B

(9)

A

=

T

A or B )

i

Equations 6-8 have been used to obtain Equation 9; Equation 9 is a general equation which can be applied to all mixed associations. The final form of Equation 9 indicates that there are only two solute com ponents ( independent variables ) involved in the mixed association. W h e n no association occurs, then (dc/dCi) , = 1. The superscript "eq" is used to indicate a mixed association is present. N o w that the quantity cM has been defined, how do we obtain M , and how do we use it (or its analogs) to obtain the equilibrium constant or constants and the nonideal terms if they are present?


T

w

Cj

e q

w

E v a l u a t i o n of M

w

e q

e q

. SEDIMENTATION EQUILIBRIUM EXPERIMENTS A N D

T H E A R C H I B A L D M E T H O D . A t constant temperature the condition for sedi mentation equilibrium is that the total potential ~JH for each associating species i (i = A , B, or A B ) be constant at every radial position r i n the solution column of the ultracentrifuge; thus W

W

Mi = M; — — 2 —

=

c o n s

tant

(10)

Here μι is the molar chemical potential of species i , ω = 2 ^ p m / 6 0 is the angular velocity (radians/second) of the rotor, and M is the molecular weight of associating species i . F r o m this relation it can be shown that (14) {

( dc\

_ (dcA\

\d(r*))

\d(r*))

T

/dc \

/dc\

\d(r*))

\dcA/

T

/dc\

B

T

TCB

W r . c

A

L E c i M ; (dc/dcdr.cj = LcM ^ w

(11)

i

Here, L = ( 1 — v )o> /2RT 9

2

(12)

In obtaining Equation 11 it has been assumed that the partial specific volumes ν of the associating species are equal; we have also assumed that the specific refractive index increments Ψ of the associating solutes are equal. I n Equation 12 R is the universal gas constant (8.314 Χ 10 ergs/deg-mole), ρ is the density of the solution ( g r a m / m l ) , and Τ is the absolute temperature. Equation 11 is also valid for the Archibald ex periment but only at r or r , the radial positions ( in the solution column of the ultracentrifuge cell) of the air-solution meniscus and of the cell 7

m

b


22.

265

Mixed Associations

PEKAR E T A L .

bottom, respectively. Archibald ( 19 ) was the first to point out that there is no flow at the extremes (r or r ) of the ultracentrifuge cell solution column since it is a closed system; this fact could be used to show that one can obtain M at various times t at r or r . Here one notes that m

6

wt

m

1 dc 2r~c~L dr

=

M

w

6

? i

at r orr m

(13)

6

The centrifugal field causes a redistribution of the reacting species i n such a way that chemical equilibrium is maintained; in order to get M at c it is necessary to extrapolate values of M to zero time, i.e.,


w

0

w

lim

M,

w t

= M

w

e q

e q

i

atc

(14)

0

To utilize Equation 13 it is necessary to know c or c ; here we use the following equations, provided a plateau ( a region where the concentration gradient dc/dr = 0) exists: Tn

rb

Here r is any radial position i n the plateau region. p

^-o + hS?,***

(16)

Equations 15 and 16 were first obtained by Kegeles and his co-workers (16, 20, 21) and they are valid only for cells with sector-shaped center pieces. Because of convective disturbances in the solution column of the ultracentfifuge cell during the transient state when nonsector-shaped cells are used, it is customary to do Archibald experiments i n cells with sectorshaped centerpieces which avoid this problem. For details on the A r c h i bald method—pitfalls, extrapolation to zero time—one should consult the papers of LaBar (17) and Fujita et al. (22). The quantity c M can also be evaluated from sedimentation equi librium experiments done at different speeds on the same solution. Here one gets to sedimentation equilibrium with a given solution at one speed; the necessary information (concentration and/or concentration gradient) is recorded. Then the speed is changed; once sedimentation equilibrium is attained at the second speed, c and/or dc/dr are recorded. This pro cedure is repeated at two or more additional speeds. Then one calculates M (cell mass) or M (cell vol) from the following equations: w

w

e q

w


266


M

(cell mass) = °I±

M

(cell vol) =

w

w

Here Λ = L(r»« - rj)

l n [ C f

=

( 1

/ C r

J

(18)

"

~

r

J

(19)

)

Because of the redistribution of reacting species by the centrifugal field, one notes (23) that M (cell mass) and M (cell vol) do not represent the value of M at c , the original concentration. However, if these values are extrapolated to zero speed, they become equal to M , i.e., w


»

(17)

w

e q

w

0

w

lim M Λ->0

(cell mass) =

w

lim

e q

(cell vol) = M

M

w

w

e q

at c

(20)

0

Λ^Ο

Thus one has two ways to obtain M M (cell) or M (cell vol) where z

e q

w

. In addition one can also calculate

z

_

(ldc\

\r drJ

rb

A

M

(

and M

z

n

η

(cell vol) =

1

Π

[(^

/ldc\

\r drJ

r

d i ) r j (r

J

^

(22)

Here one notes that lim M L->0

z

(cell) = lim M Λ->0

z

(cell vol) = M

B

at c

0

(23)

L I G H T SCATTERING E X P E R I M E N T S . This discussion is concerned only with elastic light scattering (24). In these experiments one measures the reduced scattering intensity (sometimes called the Rayleigh ratio) R(0), where ™

= Zo (1 +

cos' Θ)

>

(24

Here r is the distance of the photomultiplier tube from the center of the cell, ie is the intensity of the scattered light at an angle θ to the path of the incident light beam, and / is the intensity of the incident light. F o r molecules whose maximum dimension is less than λ/20, where λ is the 0


22.

267

Mixed Associations

PEKAR E T A L .

wavelength of the light being used, R(6) is usually independent of angle. For very large molecules a correction for the angular dependence of the light scattering must be made; this is usually done by means of a Zimm plot (24, 25, 26, 27). It w i l l be assumed that angular dependence, if present, has been overcome, and we w i l l use the symbol R ( 0 ) to indicate this. One can evaluate M from the equation w

e q

= cM^


^ g i ° >

(25)

Here Κ is the well known light scattering constant and is defined by Κ

=

2xW/XW

In this equation n is the refractive index of the solvent, λ is the wave length of light, and Ν is Avogadro's number. The quantity A R ( 0 ) i n Equation 25 is defined by 0

AR(0)

=

fi(0)solution

-

#(0)solvent

Analysis of Mixed Associations Using M TATION E Q U I L I B R I U M EXPERIMENTS.

w

e q

. CONVENTIONAL SEDIMEN

Usually sedimentation

equilibrium

experiments are run at one speed only for each concentration; these are known as conventional sedimentation equilibrium experiments. Although the choice of speed is arbitrary, practical considerations force one to choose a speed to evaluate the concentration (c) or the concentration gradient (dc/dr) near the cell bottom; otherwise blurring or defocusing of the optical images i n that region at high speeds results. F o r this situation the basic sedimentation equilibrium equation becomes

L

=

(i = A or B)

= LM

^!*L*

i9

(1 —

(26)

v )rf/2RT 9

for each component. This equation can be integrated between r and then converted to exponential form to give

m

where φ as

= LM (r

ίΓ

V = c C

A

r

τ/ι

/

r

LM

2)

r

(28)

+ Kc c 7 n A r

Br

A

irm

The total concentration c can be expressed

2

m

+ c

Ar

=

— r ).

2

{

(27)

c e i^-™

=

+

C e*Br 7n Br

and r

B

+

Kc ? A

τη

C ?fj%6 B

n < J ,

Ar +

«Φ Γ Β


268


for a mixed association described by Equation 1. One can take values of c at three different radial positions (not too close to each other) and obtain three simultaneous equations which can be solved by standard methods to give c , c , and Kc c (14, 15). From these quan tities one can calculate K. B y comparing results obtained with different radial positions and two or more experiments, one can test for ideal be havior since Κ would be the same i n all cases if the solution were ideal. One can also use the concentration gradient to evaluate Κ and test for ideality. First one notes that r


n

Brm

A r m

^ J=

cM^ = c M A

N o w divide Equation 29 by M

MM

a

aVr '

c

=

m

Arrn

Brm

B

+ Kc

B

A

u

C B M\ B m

N

M

(29)

and then subtract c to give

A

[Ml ~ V

C B

+ cM

A

+

K

C

A

C

B

\ MT ~ V

(30)

At this point apply Equation 27 to Equation 30 and then multiply both sides by e ' V ; this leads to Y

= {aw: % ~

(31)

A plot of Y vs. e ^ A r + ^ - ^ V w i l l have a slope equal to: Slope = Kc c rZ n Ar m

{^^

m

B

- *)

( > 32

The intercept at r = r is m

Intercept = c r B

-

m

+ (^f

m

~ l ) Kc c Z. n Ar m

Br

(33)

Since c = c + c r + * C A r c r one can calculate K. U p w a r d curvature i n the plot based on Equation 31 indicates the presence of higher aggregates whereas downward curvature may be caused by nonideal effects. Tm

Arm

B

m

c

n

w

B

m

m

ARCHIBALD EXPERIMENTS, SEDIMENTATION EQUILIBRIUM EXPERIMENTS A T D I F F E R E N T SPEEDS, A N D L I G H T SCATTERING E X P E R I M E N T S

T h e analysis

of a mixed association described by Equation 1 is similar for these three


22.

PEKAR E T A L .

269

Mixed Associations

methods. First of all let us introduce the conservation of mass equations. For n A + w B 0

~ C

A

C

S

H m

C °->0

B

= P

(46)

a

β = const

β = const

Thus by doing these experiments at two or more values of β, one can obtain two or more values of χ, which can be used to obtain K and K . In this situation one must use a limiting intercept; there is no simple plot available as there was from Equation 41. AB

Analysis of Mixed Associations When v

A

^

v

and Ψ

B

Α

A2

^Β· S O far

we have assumed that v = v = ν and Ψ = Ψ = Ψ; under these cir cumstances one could evaluate and use M . N o w suppose we wished to study the interaction of a bacterial hapten which was a polysaccharide with an antibody which was a protein. W h y should one expect the partial specific volumes or the specific refractive index increments to be the same A

B

Α

w

Β

e q


272


for these two reactants? Does this negate the analysis? The answer is no, and this is how we overcome these minor complications. Note that the mass balance equations become ΨΑΟΑ

,

nKc c n

A


nB° =

nB° — n

,

ΨΒ(·

ΨΒο

=

0

=

m

B

Β

N

M A

)

0

mKc CB

M

A

(47)

\

M

"MZBT

+

and

0

(47a)

\

M

for a mixed association described b y the first equation. Furthermore, note that η = n° +

nB° = η — n

A

= Ψο Α

+

Α

= Ψο Α

0

Α

Ψ ΟΒ

+

Β

+ Ψο Β

This means that Ψ by

K

+

Β

A

C

/

C

B

(48)

+ πιΨ Μ ]

[ΠΨΑΜ

M

Β

α

Β

W Kc "c ™ AB

A

B

, the refractive index increment of A B , is defined

ΑB

N

_ ηΜ Ψ

w

Α

W

+ ηιΜ Ψ

Α

Β

Β

,,

αΛ

nm

A

D

Similarly we w i l l define v

AnBm

nM v A

A

+ M

mM v B

B

(50)

A

ANALYSIS

O F MIXED

ASSOCIATIONS F R O M

C O N V E N T I O N A L SEDIMENTA

EQUILIBRIUM EXPERIMENTS. I n these experiments one measures a quantity M i (14, 28) instead of M . The basic sedimentation equi librium equation for each reactant is

TION

e q

w

e q

= n

(51)

= rii — n

{

= Wid

0

This can be integrated to give n

iTm

explLiM^r

- rj)]

= n

2

irm

βχρ[φ, ] Γ


(52)

22.

PEKAR E T A L .

Mixed

273

Associations

where φ = LMi(r — r ). The Rayleigh optical system gives us infor mation proportional to η = Xrii = η — the number of fringes / is re 2

ίτ

2

m

lated to η — no by

J = \ (n -

n)

λ

(53)

0

Here h is the thickness of the ultracentrifuge centerpiece (usually 12 m m ) , η — n is the refractive index difference between the solution (n) and the solvent ( n ) , and λ is the wavelength of light used (generally 546 μ). Thus with the Rayleigh optical system one can obtain n, which can be expressed as


0

0

n

r

=

VACAr^Ar

= n + n + W Kc c + W e*Br + ψ K A? C ^ e^r Ar

Br

n

AnBm

BCBrm

AnBm

A

m

C

(54)

m

B

B

+ m$Br

m

Equation 54 is analogous to Equation 28, and one can take three values of n at different radial positions to solve for c Br , and "c r c r , and from this evaluate K. W i t h the schlieren optics one obtains r

A

m

B

A r m >

c

m

m

dît d{r*) = L n M A

A

~ B

= nM^ 1

_

Ση Μί {

Here,

(1 -

= Ση,· (1 -

= i

+

+ LnM

A

B

B

( 1

L

~ ^

"

2RT

Vf?)

n

AnBm

p )

AnBm

( 5 5 )

(1 -

Ση Μ {

M

AnBm

2

{

ϋ,·ρ)

^ r - = ^ r j z ^ - r ι\·ρ) η (1 - y p)

(56)

w

= A , B , or A B n

m

η = Ση» (1 - v ç>) = Ση,·(1 - ν

Polymer Molecular Weight Methods - ACS Publications

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