ANALYTICAL

BIOCHEMISTRY

66, 1-l 1 (1975)

The Theory of the Difference

Sedimentation

Method

R. J. SKERRETT Department

of Chemistry, University of Birmingham, Birmingham B15 2TT. England

Received January 8, 1974; revised July 29, 1974

The difference sedimentation method (1) has previously been applied on the assumption that solutes are ideal and monodisperse. The present paper shows that the basic equation can be simplified for these systems, and that the method can still be applied when solutes are nonideal, or polydisperse, or where reaction is occurring, although the basic equation then needs to be modified. Several different values of AS/S can be obtained, depending on the treatment of the data, and it is important to distinguish these since they can be very different numerically. It is found that measuring the moment of the peak only, instead of that of the total difference curve, has several advantages, chiefly the wider applicability of the equation, and the removal of factors due to differences in initial concentration or cell length. The difference sedimentation method was originated and developed by Schachman and his co-workers (l-3). This method measures directly the difference in concentration of similar solutions in the sectors of a double-sector cell, either by interference, or (4), by split-beam absorption, optics. The graph of the moment of the difference curve plotted against a time function gives, from the slope, the relative difference in sedimentation coefficients. It has proved to be very successful in the detection and measurement of small differences between the sedimentation coefficients of similar macromolecules, and as such should be useful in a variety of biochemical and biophysical problems. Because of this usefulness, it is worthwhile to examine the theory of the method in order to see what errors may arise, how data may best be treated, and how generally it may be applied. The equation used in the work of Schachman and his co-workers is derived by the consideration of the conservation of mass in the case of a monodisperse solute with a nonconcentration-dependent sedimentation coefficient. In many systems of interest, however, the sedimentation coefficient may be concentration-dependent, or there may be polydispersity or chemical reaction, and Schachman’s equation may no longer apply. The present paper extends this treatment to these other systems, and also Copyright All rights

0 1975 by Academic Press, Inc. of reproduction in any form reserved

2

R. J.

SKERRETT

derives formulae from Faxen’s equation (5,6a). Difference sedimentation is normally performed in double-sector cells, with the solute initially present throughout the liquid column, a condition to which the Fujita and MacCosham (7) equation applies. However, it was shown by the latter authors that the simpler Fax& equation applies well to this system, for most solutes, from an early stage in the ultracentrifugation. THEORY The following symbols are used throughout: t = time (s), r = distance the center of the rotor (cm), rM = value of r at the meniscus, Y, = value of Y at an arbitrary but constant plane in the plateau, c,r+? = the second moment of the concentration gradient, c = concentration (g/100 ml), c,, = concentration at t =.O, c, = plateau concentration, s = sedimentation coefficient(s), o = angular velocity of rotor (x-ad s-l), D = diffusion coefficient (cm* s-l). Corresponding parameters in the two sectors are subscripted 1 and 2. For any parameter p, p = I/Z (pl + p2), Ap = p1 - p2, from

Sp =

E = 2Dlso2rM2,

‘/2Ap.

7 = 2&Pt,

z = Xh

- r),

x=

l/rM [e(@ - I)] 1’2, a(z) = 2W2

’ exp (-x2)&. I To estimate the magnitude of sime results, a hypothetical “standard” system is used, with S = 5 X lo- 13, D= 5 x 1W7, pM= 6, c$= 107. Difference

Equations

for the Ideal Solute

The Faxen equation for an ideal solute is c=F

(1 -(a(z)).

The equation of the difference curve can be obtained by taking logarithms of both sides of the Fax&n equation, differentiating, and approximating to the differentials by difference values, thus: lnc=In(llzc,)

--7+ln(l dc = dco _ C

co

-a(z)) .&,,2tds

- - 2 exp(--z*) d’2

dz (1 - Q(z)> .

(Notice that dt, dr, and dw = 0, since the difference is obtained at a given time, radius and speed.) Logarithmic differentiation will also be used in most of the subsequent calculations. Now, putting dc = AC, etc., AC=

+

AZ exp(-Z*). - EoAso2t ) ( 1 _ @((z))e-7 _ coe-T +I2

(I)

This is the equation of the difference curve obtained optically by the ultracentrifuge. It consists of the superposition of a gaussian (exp(-z2))

DIFFERENCE

3

SEDIMENTATION

curve, and an error-function (Q(Z)) curve, which will generally cause a change in the level of the base line on which the gaussian is superposed. By constructing this sloping base line beneath the gaussian, the total difference curve can be separated into the gaussian, or “peak” part, and the error function, or “step” part. Since 2 = X(r+

-r),y=

AZ 2

Ar

+ --As 2s

_ rc -r

A& teT e’-1’

AD 2D

therefore, Az=XAr,+

((

$-

Also, since rr = rMeSWPt,Ar, = (F,Aso”t

+ ArM)e7’2.

By substituting the expression for AZ into (1), we can calculate the first moment of the difference curve: Acrdr =

(1 - @(.Z))rdr --

AZ exp(-z*)rdr.

Using f

(1-B(i))rdr=[(l-O(i))fl:tf[pexp(-i2)r2di M

FM

and -m

exp(-z2)dr

= I

m

exp(-z2)

dy dz, dz

we obtain, finally, 1 c&2

r‘p Acrdr=~((~)?--I)-Asozt($~--~, I 7,

(2)

neglecting minor terms which change the result by less than l%, and which presumably arise from the approximations in the simple form of the Faxen equation. This is the same equation as that derived by Schachman and co-workers, when allowance is made for the different signs of As and Ar, in the present paper. 02t is normally replaced by l/s In@+ IjiM), and

4

R. J. SKERRETT

-AS/S can then be obtained

as the slope of 1/c^,pM2.

against (r,/L )” ln(P, /FM). An expression can be derived for the first moment from

rP

-7 i-p

I FM

I f-t.4

Acrdr = $$-

rpD J6l Acrdr

plotted

of the peak alone

AZ exp(-P)rdr.

This gives

This equation is preferable to (2), in that it is simpler, and eliminates AC,,. Kirschner and Schachman (1) measured the moments of the peak only, although they used (2) to find AS/S. Their correction factor, however, converts their value to that which would be obtained, as they point out, by use of an equation of the type (2a). An approximate minimum value for (r,/?+ )” can be obtained from an inequality (2.116) of Fujita’s (6a):

(3 ( 2

1 + $61)

li2)z.

(3)

Thus, if Eq. (2) is used in the treatment of measurements from the peak alone, the presence of the factor (r,/F* )’ will cause an error in the value of AS/S which is obtained. This error in AS/S is likely to be only about 1% for viruses, but may be 50% or more for small molecules such as ribonuclease. In addition, the graph will be slightly curved in the case of small molecules. The area of the difference curve can be calculated in the same way as the first moment. We find that (4)

This can also be used to find As/f. The Value of i;,

An important variable in all formulae used in the difference method is F%. By analogy with the normal concentration-gradient case, we would expect rx to be very close to the position, rmax, of the maximum of the curve. Whether this is so can be tested by differentiating (1) with respect to r and equating the derivative to zero. This gives, approximately,

AC05Asw 2t 2 - (

r max = r* -

__---_ CO

AD 20

D(eT-- 1) ~~2Ar+ ’

DIFFERENCE

5

SEDIMENTATION

Hence, Y,,, is generally not identical with T;*. To estimate the size of the difference, we use the standard system, and assume that Aplp = 0.01, t = 5000 s. rmax can be used to calculate So”t as ln(n /Pi). If r is close to F~, then ln(r/PM) = (r - P,)/?,, so that the relative error in In@, /P& given by using rmax, is (i;* - ?,,,)/(f* - fM). This will give an error of 0.24 or 0.86% if Ac,, is negative. It appears, therefore, that the use of rmax in place of P, will introduce an error which will generally be small, but may not be entirely negligible. An accurate value for Y, can be derived from (2b) and (4):

Difference

Equations

for More

Complicated

Systems

The previous equations have been derived assuming that s is independent of concentration. Schachman’s derivation can still be applied when this is not so. For a single sector, Schachman’s conservation of mass equation is 1/2co(r,2 - rM2 ) -I” -

crdr + rP2 I’ sw2cPdt. FM

0

In the plateau region, the Lamm equation simplifies to dc,/dt = -2sw2cp, which applies whatever the dependence of s upon c. crdr = =

l/2

co ( rp2 - rM2 ) - rP2

Y2co(rp2

-

Y&f2)

+l/2rp2(cp

Hence, in the difference case, by Schachman’s

t I0

- l/2dc, dt dt

-

co).

derivation,

(5) For a concentration by

dependent system the plateau concentration c, = coeeTf,

is given (6)

where f is some function of the system parameters and variables, T = 2soo2t, and so = s at zero concentration. Deriving AC, from Eq. (6), and substituting into Eq. (5), we obtain

6

R. J.

SKERRETT

2 eerAf ---. 2

Provided

that the square dilution

rule applies,

Ar,

(7)

PM

this can be written

as

Acrdr=%((?r-1)

- (?r

02t (A$,-&$)

-2.

(7a)

For systems in which the Johnston-Ogston effect occurs, and for polydisperse systems, if rmax is used for r+ , this equation will not be applicable, although the error may frequently be small. In interacting systems, the deviation from the square dilution rule may be considerable if rmax is used for t, , and the difference method, using this equation, may be inapplicable. Equation (7a) shows that, in systems where s is concentration-dependent, graphs plotted by Schachman’s method will not be strictly linear, and we need to discover where As can be defined clearly. From Eq. (6), dc,/dt

Comparing

= -2sow2cp

this to the Lamm equation,

s=so---

1 df and

202f dt

+ coe-rdf/dt.

we see that

As=As,-&A

In Eq. (7a), the slope of the curve is As, - 1/202 . Af/ft, so this slope will give the value of As only if Af/ft = A[( l/f)(df/dr) J. This will not be generally true, except at t = 0. If the equation of the concentration distribution is c = cOevTFz where F is some function of the system parameters and variables, then AC =

2

- 2As,,oPt

>

c + C@e-‘AF.

This is the general equation of the difference curve, of which Eq. (1) is a particular case. It consists, like Eq. (I), of “step” and “peak” expressions. In this case, however, part of the step function may be included in AF, so that we should write AC=

(

‘C’I-2Asoo2tta CO

>

C-~-F’,

@a>

where a and b are unknown, and bF’ is the peak function. The height of the step is now given by (Ac~/F~ - 2Asoo2t + a)E,, and

DIFFERENCE

this is equal to AC, = A(coe-‘f), peak alone is now given by

TPcrdr

I+=M

is given

7

SEDIMENTATION

which gives a = Af/f The moment of the

by the basic

conservation

of mass equation

as

1/2C0(rp2 - FM?) - l/2rp2(Fo - 15~). Hence, using Eq. (7),

The use of the peak moment only, then. removes the Ac,JF,~ expression. and gives (AS&, from the moment vs w’t slope at t = 0. In addition. Eq. (9) is applicable whether or not the square dilution rule applies. The Use of Diferent

Functions

oft

In these cases, In(F+ /fM) can no longer s = s,,( 1 - kc), where k is a constant, o*t=

(l/25,,)

be used for fo*t.

. In(((~~/IP,)“--Zc,,)/(l

-liF,,)).

When ( lOa)

When s = s,,/( 1 + kc), w*t=

(I/S,,)

(In(F+/lP,)

+1/21iCg(l

-

(FM/P,)‘))

(lob)

(see Ref. (6b)). There are several possible methods of treating data, using various time functions. To compare the various methods, we use the standard system. with s = s&l + kc), k =k=0.5, co = 1, As,/s, and AC, = 0.01. In this case, the appropriate time-function is In@%/P,) + l/z kc,( 1 - (F~/?* )*) (see Eq. (lob)). When Ak = 0, the reduced moment plotted against this expression gives a slope at t = 0 of 0.00444, which is the value of (As)~=,/s~,. When the reduced moment minus Aflf is plotted against this expression, the slope is -0.01 = -AsO/sO. If the reduced moment is plotted against In@, IF,), the slope at t = 0 is (-As/s)+~ = -0.00667. The mean slope of this graph, over a period of 10000 s, is -0.00683. This is close to the mean value of -As/s, which is -0.00674. Similar results are obtained when Ak = 20.005. It appears, therefore, that by the use of different time variables, one can obtain the values of As,/s,, (As),=,/s,, (AS/S),=,,, and (approximately) (Asl~he,. However, one needs to be clear which is being obtained, as their values can be very different. Diferences

Kirschner

in Optical

Path Lengths

and Schachman

(I) have implicated

a difference in optical

8

R. J.

SKERRETT

path lengths as a source of error in the difference method. The expression which they derived for the difference in optical path length is Al = RCAh, where I= optical path length, h = depth of cell, and R = specific refractive index increment = (dn/dc),,,. Al can be converted to an apparent change in concentration, AC,,, by the equation Al = R&AC,,. Hence

The same expression can be shown to be applicable optics are used. Adding this to Eq. (8a), we obtain AC apparent= AC +

when adsorption

c+bF’. (8b) f > sector lengths will modify the apparent total

$2As0~lf+~+f

This shows that different moment to

- ($r but will leave the peak moment

dt (ASK-&]

-2

(7b)

unchanged. RESULTS

Use of Di$erence Equations Some of the formulae derived here were used with the data of Kirschner and Schachman (1). In one experiment, these authors obtained a value for AS/S of 0.0114, using Eq. (2), although the value obtained from the reported data is 0.01187. The true value was 0.0119. The reported data were used with Eq. (2b), and with Eqs. (9) and (1 Oa). The values obtained for AS/S are given in Table 1, with the standard errors of the points from the best straight line. The value of 0.035 for EC, was obtained from the data of Hersh and Schachman (8). The TABLE

1

Time variable

A slS

(rfi.)Z hl(F*/Qa In(?*/i-MM)

0.01187 0.01215 0.01213

f(r)*

a Kirschner and Schachman’s reported data. bf(r) = ) ln(((F,FM)* - 0.035)/(1-0.035)) . (l-0.035 (F&*)3.

SE x

lo5

5.97 4.93 4.89

DIFFERENCE

3.4

-

3.2

_

3.0

-

2.8

r

2.6

-

*

’ 1

I

I

2

I 4

3 (r,/F*Y

9

SEDIMENTATION

h(r*/r,)-

-

I

I

I

5

6

7

x 100 *

FOG. 1. Graph of the reduced moment vs (r,/?* )” In(F,/PD1l for the “standard” system. using the Fujita and MacCosham equation. -, theoretical curve: .... theoretical points with superimposed random errors.

value of Asls given by the use of Eq. (2b) or (lOa) is about 2.5% higher than that given by the use of Eq. (2). If this were true for the total data, the value of As/s obtained would be about 0.0117. The standard errors are smaller when either Eq. (2b) or (10a) are used, but the latter is not significantly better than the former, presumably because kc, is so small. The Early Stages of Sedimentation

The behaviour of a system in the early stages of sedimentation was calculated from the Fujita and MacCosham equation (7), using a computer program based on that of McCallum and Spragg (9). The “standard” system was used, with AC, = 0, ArM = 0.01, and A.sjs = 0.01, and the calculations were done, firstly, for an error free system, and then

10

R. J. SKERRETT

with AC having random errors superimposed, the standard deviation of these being 0.01, of the order of magnitude of AC itself. The plots of the moments vs (r,lf% )“ln(c/&) are shown in Fig. 1. It is seen that the plot is curved in the early stages, until AC at r&$2is a small fraction (4% in the present case) of its maximum value. The value of AS/S obtained from the slope of the straight portion is 0.00915. This low value presumably arises from the approximations in the derivation of the Fujita and MacCosham equation. The value of As/b obtained from the second set of data is 0.00777, 15.1% lower than the value from the first set. Although this would be an unacceptable error for most purposes, it demonstrates that, as would be expected, the summation tends to cancel random error, so that the method can be used even with relatively large errors in the measurement of AC. Computation

Computer programs used to obtain results in this paper were written in BASIC and run on a Data General Corporation NOVA 1220 computer. DISCUSSION

Kirschner and Schachman (1) have shown that difference sedimentation is a very sensitive method of measuring small differences in sedimentation coefficient. The present work suggests that the method is applicable to most systems, although modifications need to be made to the basic equations when the solute is nonideal. When the solute is polydisperse, or when the Johnston-Ogston effect occurs, the method is not strictly applicable using the total moment, although it is unlikely that its use would introduce a large error in AS/S. It is only when chemical reactions are occurring that the method (using the total moment) may give seriously inaccurate results if rmax is used for r, . It will often be more convenient, in this method, to measure the moment of the peak alone, as Schachman and his co-workers have done. It is apparent, from the present work, that this procedure has additional advantages. First, the abscissa variable, (t-,/F* )21n(Pe /P& is replaced by the simpler In(F+ /P&, and r, need no longer be measured. Secondly, the value of the moment is no longer dependent on AC,,, except in so far as the latter affects As. Thirdly, the equation is applicable even when the square dilution rule does not apply. Fourthly, the value of the moment is no longer dependent on Ah. For most experiments, therefore, equation (9) should be used, with either Eq. (10a) or (lob) as appropriate, to give As,/s, (As)&bO, or with ln(r, /T&, to give (AS/S),-,. If the sedimentation coefficient can be regarded as independent of concentration, Eq. (2b) or (4) can be used,

DIFFERENCE

SEDIMENTATION

11

and, since the peak area is likely to be less tedious to measure than the moment, Eq. (4) will be the preferred alternative. In some cases, for instance where large random errors in the data make the peak hard to delineate, it will still be preferable to use the total, rather than the “peak only” moment, and Eq. (7a), or, if the sectors have different path lengths, Eq. (7b), will need to be used, together with Eq. (1Oa) or (lob). If the sedimentation coefficient is independent of concentration, then the simpler version, Eq. (2) (Schachman’s equation), can be used. Smaller solutes are less easy to treat by this method, firstly because diffusion will more quickly eliminate the solvent and plateau regions, and secondly because rmax and rP are harder to measure accurately. That the presence of a solvent region is necessary is shown by the results obtained by using Fujita and MacCosham’s equation. The difference method would be still more useful if it could be modified so that a solvent region is unnecessary and, since it has such potential, this may be a useful line of future research. ACKNOWLEDGMENTS Much of the work for this paper was done at Strangeways Research Laboratory, Cambridge. and I am grateful to the staff there, and to the Medical Research Council for finance. 1 am also grateful to Dr. S. P. Spragg of this department for useful discussions. and to the Science Research Council for present finance.

REFERENCES 1. Kirschner, M. W., and Schachman, H. K. (1971) Biochemistry 10, 1900-1919. 2. Schachman. H. K. (1959) in Ultracentrifugation in Biochemistry. pp. 170- 174. Academic Press, New York. 3. Richards, E. G.. and Schachman, H. K. (1959)J. Phys. Chem. 63, 1578-1591. 4. Lamers, K., Putney, F.. Steinberg, I. Z., and Schachman. H. K. (1963) Arch. Biochem. Biophys. 103, 379-400. 5. Fax&, H. (1929) Arkiv Mat. Astron. Fysik 21B (3); Faxen. H. (1936) A&iv Mar. Astron. Fysik 25B (13). 6. Fujita. H. (1962) in Mathematical Theory of Sedimentation Analysis, (a) pp. 64-72: (b) pp. 57-62; (c) p. 145, Academic Press, New York. 7. Fujita, H., and MacCosham, V. J. (1959) J. Chem. Phys. 30, 291-298. 8. Hersh. R. T., and Schachman, H. K. (1958) J. Phys. Chem. 62, 170-178. 9. McCallum, M. A., and Spragg, S. P. (1972) Biochem. J. 128, 389-402.

The theory of the difference sedimentation method.

ANALYTICAL BIOCHEMISTRY 66, 1-l 1 (1975) The Theory of the Difference Sedimentation Method R. J. SKERRETT Department of Chemistry, University o...
575KB Sizes 0 Downloads 0 Views