Determining Molecular Weight Distributions of AntigenAntibody Complexes by Quasi-Elastic Light Scattering REGINA M. MURPHY, MARTIN 1. YARMUSH, and CLARK K. COLTON Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts
SYNOPSIS
Physiological properties of soluble antigen-antibody ( Ag-Ab) complexes depend in part on the size of the complexes. In previous work, the size distribution and structure of model Ag-Ab complexes were determined by electron microscopy. In this study,we used constrained regularization analysis of quasi-elasticlight scattering data to estimate molecular weight distributions of model Ag-Ab complexes. A conformational model was necessary to determine appropriate correlationsbetween molecular weight and diffision coefficient, and to estimate particle structure factors. Porod-Kratky theory proved to be an adequate conformational model for these purposes. The molecular weight distributions determined by constrained regularization compared favorably with distributions obtained either by electron microscopy or by thermodynamic modeling.
INTRODUCTION Many physiological properties of soluble antigenantibody ( Ag-Ab ) complexes, including clearance from the circulation, complement fixation, adherence to phagocytes, and differential tissue deposition, depend on the size and composition of these complexes.'-4 In our previous work, 5-7 we measured average sizes of model Ag-Ab complexes under a variety of conditions. Specifically, we obtained the weight-average molecular weight (M ) wdirectly from classical light scattering (CLS) data and the z-average hydrodynamic radius (&,)z with quasi-elastic light scattering (QLS ) using the method of cumulants for data analysis. Electron microscopy ( E M ) by rotary shadow casting was used to obtain size distributions at a single concentration.6 A thermodynamic model was developed that predicts the size distribution of the complexes at any concentration; parameters of the model were obtained by fitting to EM, CLS, and radioimmunoassay data.7 Determining size distribution by EM is tedious and subject to potential artifacts. More rapid and convenient techniques to determine molecular weight distributions in solution are desirable. In this Biopolymers, Val. 31, 1289-1295 (1991) CCC 0006-3525/91/111289-07$04.00 0 1991 ,John Wiley & Sons, Inc.
work, we examined the utility of QLS for estimating molecular weight distributions of model Ag-Ab complexes by analyzing autocorrelation data using the method of constrained regularization?-" In order to do this, some knowledge or assumption regarding the structure of the complexes was required. The conformational model developed previously6 was used in this analysis to estimate the particle structure factor, and to determine the relationship between diffusion coefficient and molecular weight.
MATERIALS A N D METHODS Ag-Ab complexes were prepared as described in detail previously.6 To briefly describe the procedure: bovine serum albumin (BSA: Pentex sulfhydryl modified, Miles, Elkhart, I N ) was dissolved in phosphate-buffered saline ( PBSA: 0.15M NaCI, 0.01M KH2PO4/KzHPO4,0.02% ( w / v ) sodium azide, pH 7.4), purified by gel permeation chromatography to remove oligomers, and concentrated by ultrafiltration. Monoclonal antibodies ( MAb ) were prepared from hybridoma cell lines designated as 5.1, 6.1, and 9.1. Each MAb bound noncompetitively to a different single epitope on BSA." MAb were isolated from ascites by affinity adsorption to BSA covalently linked to CNBr-Sepharose 4B (Phar1289
1290
MURPHY, YARMUSH, AND COLTON
macia, Upsalla, Sweden). MAb were eluted with 0.1M glycine/HCl buffer, pH 2.5, dialyzed against PBSA, concentrated by ultrafiltration, purified by gel permeation chromatography to remove oligomers, and concentrated again by ultrafiltration. AgAb complexes, designated as 5.1 6.1, 5.1 9.1, and 6.1 9.1, were prepared by dilution of the stock solutions of two of the three MAb and BSA in PBSA. Final concentrations were [BSA] = 8.25 X 10p7M, and [MAb,] = [MAb2] = 3.5 X lOP7M. QLS experiments were conducted as described in detail previously.6 Autocorrelation data taken a t a 60.2" scattering angle were analyzed by the method of cumulants to obtain (Rh)z. Results were reported previously6; (Rh)= was 99 4 2 A for 5.1 9.1, 114 f 1 A for 5.1 6.1, and 146 k 3 A for 6.1 9.1. In this work, the same autocorrelation data were analyzed further by the method of constrained regularization.
+
+
+
+ +
+
where G ( ri)is the fraction of the total intensity contributed by species i, and the summation is taken over all species i. This fraction can be expressed as
DATA ANALYSIS For a monodisperse solution, the normalized firstorder autocorrelation function is
G(I'i)
=
NiM?P(q)i
(5)
c NiMSP(q)i i
where Ni = the number of moles of species i, Mi is the molecular weight, and P(q)i is the particle structure factor. The value of P ( q ) depends on the size and shape of the particle as well as the scattering angle. Particle structure factors calculated for a variety of shapes and sizes from equations available in the literature 13,14are plotted in Figure 1.For the conditions of our experiments, with an argon ion laser operated at 488 nm wavelength, an aqueous solvent with a refractive index of 1.33, and a scattering angle of 60.2", q = 0.0017 k'.Thus, a particle with qRh = 1 has a hydrodynamic radius Rhof 580 A and a particle structure factor P(q) that ranges from 0.54 for a Gaussian coil to 0.82 for a solid sphere. For our AgAb complexes, the largest complexes seen in electron micrographs had an estimated Rh around 250 A, where qRh 0.43. Under these conditions, for complexes modeled as wormlike chains, P ( q ) 0.9.
-
-
where T is the autocorrelation delay time, D is the translational diffusion coefficient, q = (47rn/ Xo)sin(B/2) is the scattering vector, n is the refractive index of the solvent, A, is the wavelength of light in vacuo, and 0 is the scattering angle. For a polydisperse solution, the autocorrelation data includes contributions from all scattering particles, and must therefore be written as
L
where G( I') is the distribution of the intensity of scattered light and
J, G(I')dI'=
1
(3)
For a finite number of species, Eq. ( 2 ) can be written as
soft sphere-...
.
4
qRh Figure 1. Particle structure factor for several shapes. R h is the hydrodynamic radius and q is the scattering vector. For star-shaped molecules, f is the number of rays. For soft spheres (also called three-functional regularly branched chains with Gaussian behavior of the subchains), m is the number of units between branching points and n is the number of branching shells. It was assumed for these calculations that the branching unit has a length of 300 A. The particle structure factors for wormlike chains shown in the figure were calculated assuming that the particle had a length of 1025 A, a persistence length of 205 A, and a chain thickness of 110 A.
MOLECULAR WEIGHT DISTRIBUTIONS OF ANTIGEN-ANTIBODY
Constrained Regularization
The expression in Eq. ( 2 ) is an example of a general class of equations known as Fredholm integrals of the first kind.g These occur frequently in a variety of experimental techniques and can be described in general as
1291
tomatically by CONTIN such that the increase in V ( a ) over V ( a = 0 ) is due to chance alone (i.e., noisy data) about 50% of the time.’-’’ Determination of Molecular Weight Distributions
The function to be determined from QLS data is the weight-average molecular weight distribution f ( M ) , where f ( M ) d M is the weight fraction of all complexes with molecular weights between M and M d M and
+
where y ( t )are the (noisy) experimental data with error t , F ( K , t ) is a known function, and s ( K ) is the unknown function. The terms involving LiBi in the general case account for any constant terms. The problem of finding the desired function s ( K ) is in general an ill-posed one, since there may be many solutions that are equally good within the experimental error E . The program CONTIN9-1’ uses the method of constrained regularization to solve these kinds of equations. First, Eq. ( 6 ) is converted into a system of linear equations:
i= 1
m=l
where c, are the weights of the quadrature (e.g., Simpson’s rule) formula and N g are the grid points at which the solution is determined. This equation can be written more compactly as
and (M)w is the weight-average molecular weight. In this work, the minimum and maximum molecular weights are set equal to 50,000 and 5,000,000, respectively. The minimum weight was chosen to be slightly below that of BSA (66,000) and the maximum was chosen such that the size distribution decayed to zero before reaching this limit. From Eqs. ( 3 ) and ( l o b ) ,
Using this relation, and including the particle structure factor P ( q ) to properly weight the intensity of scattered light from different-sized complexes, Eq. ( 4 ) becomes
+
where N, = N g N L , A is the matrix containing the known c,F( K,, t ) and Li, and x is a vector containing the unknowns s ( K,) and B,. Two additional constraints are imposed on the solution. First, s ( K ) is required to be nonnegative. This eliminates any solutions that are physically impossible. Second, the requirement of parsimony (smoothness) is imposed. Parsimony is forced on the solution by modifying the weighted least-squares minimization problem by adding a regularizing term:
V ( a )= IM;”2(y - AX)l 2
+ a2
d2s(K) [ r dK= minimum ]
(9)
where M, is the covariance matrix of the errors e, y is the vector containing the data points y ( t ) , and a is the regularization parameter. The a is chosen au-
X exp(-h)f(M)dM
(12)
The correspondence between the right-hand side of Eq. ( 1 2 ) and the first term on the right-hand side of Eq. ( 6 ) is clear. A correlation between molecular weight and diffusion coefficient is required to solve Eq. ( 12) for the molecular weight distribution. This correlation was assumed to take the form14
D
=
aMB
(13)
To calculate the parameters a and p for our Ag-Ab complexes, the molecular weights Mi of complexes containing one to ten Ag-Ab units were converted to contour lengths based on previously obtained EM
1292
MURPHY, YARMUSH, AND COLTON
data.6 Di was calculated for each linear Ag-Ab complex of molecular weight Mi (Figure 2 ) using correlations that relate the diffusion coefficient of a Porod-Kratky wormlike chain to its contour length, persistence length, and chain thicknes~.'~.'~ The values of a and ,6 for the three Ag-Ab complexes are listed in Table I. This was incorporated into Eq. (12) by making the substitution
r
=
q 2 D = q2aMb
(14)
There is a small error in these parameters because cyclic complexes were not included in Figure 2. This error should be insignificant because only 1.7 and 6% of the complexes for 5.1 6.1 and 6.1 9.1, respectively, were cyclic as determined by EM.6 For 5.1 9.1,23% of the complexes were cyclic, but 86% of the cyclic complexes contain two Ab and the calculated diffusion coefficient for two-Ab cycles (16) is only 3% greater than that for two-Ab linear complexes. CONTIN was used to find the molecular weight distribution f ( M ) . The particle structure factor P ( q ) was calculated using the previously determined persistence length a and chain thickness d (5.1 6.1 : a = 430 A, d = 83 A, 5.1 9.1: a = 91 A, d = 85 A, 6.1 9.1: a = 205 A, d = 110 A ) and available correlations for Porod-Kratky wormlike chains.I3 A single constant baseline B attributable to "dust" terms was fit to the data, with N L and L, set equal to 1. For our work, M, was assumed to contain only diagonal elements, with variances estimated from a preliminary unweighted analysis of the data assuming Poisson statistics as described in more detail elsewhere."
+
+
+
+
+
+
.s! .Y s
8
0
70
10
:
0 5.1+6.1
-
A
5.1+9.1
A
6.1+9.1
100
7000
10000
Molecular weight Figure 2. Relationship between molecular weight Mi (kdaltons) of Ag-Ab complexes and diffusion coefficient
D, (cm2/s).Diwas calculated using correlations for PorodKratky wormlike chains, as described in the text.
Table I Conversion of Molecular Weights to Diffusion Coefficients" Sample 5.1 5.1 6.1
+ 6.1 + 9.1 + 9.1
01
P
7.39 x 10-6 3.68 X 3.84 X
-0.577 -0.443 -0.485
Diffusion coefficients are related to molecular weight according to D = a M B ,where D is the translational diffusion coefficient in cmZ/s at 293 K in a solvent with 1.01 cp viscosity and M is the molecular weight in thousands.
RESULTS AND DISCUSSION Size distributions of Ag-Ab complexes made from three pairs of MAb and BSA at identical concentrations and molar ratios ([BSA] = 8.25 X 10-7M, [ MAb,] = [ MAbz] = 3.5 X 10p7M)were estimated using four different methods. In method A, EM data described previously6 were analyzed by determining the number of Ab visible in the complexes and counting the frequency distribution, then converting this to a weight distribution. In method B, weight distributions were calculated using the thermodynamic model and the equilibrium constants derived earlier.7 In method C, the distribution generated by the thermodynamic model (method B ) was converted to a first-order autocorrelation function I g " ' ( T ) I by calculating the hydrodynamic radius of each complex using the estimated Porod-Kratky parameters and available correlations, l 5 , I 6 and converting this into a diffusion coefficient through the Stokes-Einstein equation. Then, the concentration calculated from the model, the molecular weight, the calculated diffusion coefficient, and the calculated particle structure factor for each complex were employed with Eqs. ( 4 ) and ( 5 ) . The resulting autocorrelation functions were analyzed by CONTIN. In method D, the autocorrelation functions obtained from QLS measurements were analyzed by CONTIN. In Figure 3, the smoothed distributions obtained by CONTIN analysis of the autocorrelation functions predicted using the thermodynamic model (method C ) and measured by QLS (method D ) were compared. The agreement between the distributions estimated by CONTIN analysis of simulated data from the model (method C ) and of QLS data (method D ) is good, particularly for 6.1 9.1. The molecular weight at the maximum and the shape of the two curves are very close. The skewing of the distribution toward smaller molecular weights is
+
1293
MOLECULAR WEIGHT DISTRIBUTIONS OF ANTIGEN-ANTIBODY
0.20
0.15
11
w
I
4 -
Method C (model)
0.10
0.05
0.00
0-
400
800
1200
1600
2000
linear complexes were lumped together. The contribution of unbound Ag to the distribution was not included, because free BSA was difficult to positively identify on the micrographs. The smoothed distribution from CONTIN analysis of QLS data (method D ) was used to estimate a discrete distribution, in order to make a more direct comparison with the distributions from the EM data and the thermodynamic model. The area ai, under each section of the curve was estimated by using the trapezoidal rule. so that
0.20
s
*r
$Q
E"
-
0.15
w Method C (model)
0.4
5.1+6.1 W Method A (EM) 0 Method B (model)
0.10 0.3
.9
5
1
Method D (QLS)
0.2
0.05
0.1
0.00
0
400
0.20
800
1200
--
0.15
- _m - -
1600
2000
0.0 1
6.1+9.1
2
3
.e t;
0.10
m k
0.05
P .9
g
0.00 400
800
5
6
7
8
9
Method C (model) Method D (QLSl
W Method A (EM) 0 Method B (model)
E
0
4
1200
1600
Method D (OLS)
0.3
1 0.2 0.1
2000 0.0
Molecular weight
1
Figure 3. Smoothed molecular weight distributions obtained by CONTIN analysis of the predicted autocorrelation data from the thermodynamic model (method C ) and of the measured autocorrelation data from QLS experiments (method D ).
2
3
4
5
6
7
8
9
0.5
6.1+9.1 W Method A (EM) 0 Method B (model)
0.3
Method D (QLS)
0.2
correctly ascertained by CONTIN analysis of QLS data. Also, CONTIN analysis detected the presence of larger complexes in 6.1 9.1 than in 5.1 6.1 or 5.1 9.1, in agreement with the thermodynamic model. In Figure 4, discrete distributions were plotted. The distributions from electron microscopy (method A) and from the thermodynamic model (method B ) were calculated assuming that each Ab in a complex is, on average, bound to one Ag and that each AbAg unit has a molecular weight of 220 kd. Cyclic and
+
+
+
0.1 0.0
1
2
3
4
5
6
7
8
9
Number of Ab in Complex Figure 4. Discrete molecular weight distributions obtained from electron microscopy data (method A ) , thermodynamic model (method B ) , or discretization of the smoothed CONTIN analysis of QLS data (method D ) , as described in the text.
1294
MURPHY, YARMUSH, AND COLTON
1 (hi + hj)(Mj - MI) 2
There are differences in the details of the distributions when any two methods are compared. E M results show a greater weight fraction of larger comwhere h, and hJ are the weight fraction, and M I and plexes than the model; as was noted p r e v i ~ u s l y , ~ MJ are the molecular weights a t grid points i and j , this was attributed to possible artifacts in analysis respectively. In order t o discretize the distribution, of the E M data. Discrepancies between the distria, attributable to complexes with nAb was calculated bution derived from QLS data and the other distria s follows. First, the molecular weight range of all butions probably arise from two sources: first, the complexes with n Ab was calculated, where Mm,,,,is resolving power of QLS is limited because of the the molecular weight of complexes containing n - 1 mathematical uncertainties inherent in solving Eq. Ag and n Ab, and M,,,,,,, is the molecular weight of ( 6 ) and second, the discretization of the smooth complexes containing n 1 Ag and n Ab. For exdistribution is approximate. ample, for all complexes containing 2 Ab, Mm,n,2 T h e results of the four methods of analyses are = molecular weight of a complex with 2 Ab and 1 shown in Table I1 in terms of number- ( ( M ) , ) , Ag, or in our case, 2 X 154,000 66,000 = 374,000. weight- (( M ) , ) , and z-average (( M ) , ) molecular Likewise, MmaX,? is the molecular weight of a complex weights. For comparison, ( M ) , as measured by CLS with 2 Ab and 3 Ag, or 506,000. For each section ij on the identical sample as used for method D is included in Table I1 (method E ) . Comparing average where M,,,,, IMIand MJ IM,,,,,, a, was attributed t o complexes with n Ab. Because M , , , , is less than molecular weights of method B and method C is useful for seeing the effect of CONTIN analysis on Mmln,n+lthe section of the smoothed distribution where MI 2 M,,,,, and MI IMmln,n+l must be disa known distribution (in this case, from the thertributed between complexes with n or n 1Ab. For modynamic model). CONTIN tends to spread out these sections, a midpoint value M m l d = 0.5 (M,,,,, the distribution slightly, by lowering ( M ) , and in two cases by increasing ( M ) * .Ag-Ab complexes ochZmrn,n+l) was calculated. T h e value of hmldwas determined by linear interpolation, and the areas cur only in discrete sizes, whereas CONTIN assumes aL,mld and amid,, were calculated and attributed t o the a smooth continuous distribution. This effect may appropriate complex. T h e areas for complexes with be most important a t the low end of the scale, ben Ab were summed, then divided by the total area cause the relative increase in size is greater. Agreefor all complexes t o determine the fraction of comment between (M ) , from CLS (method E ) and that plexes with n Ab. determined by CONTIN analysis of QLS data The comparison between E M data (method A ) , (method D ) on the identical sample is excellent with the model (method B ) , and discretization of CON5.1 6.1 and 6.1 9.1. However, for 5.1 9.1, the T I N analysis of QLS data (method D ) , shown in CONTIN-determined (M ) , is closer t o (M ) , from Figure 4, is generally satisfactory in terms of shape E M (method A ) or from the thermodynamic model and range of the molecular weight distribution. (method B ) than from CLS (method E ) .This could
a,
=-
(15)
)
+
+
)
+
+
+
+
+
Table I1 Molecular Weight Distributions for Ag-Ab Complexes" Method Sample 5.1
+ 6.1
5.1
+ 9.1
6.1
+ 9.1
M
A
B
C
D
(M>n (M>W (M>Z (M>n (M), (M>z (M>n (M>W (M>2
399 573 783 427 568 765 482 748 1015
394 554 721 424 532 635 433 657 897
360 520 678 373 584 806 374 631 959
422 552 670 354 527 697 384 674 1197
E 544
414
721
* All molecular weights are in thousands. Methods: (A) EM. (B) Thermodynamic model. ( C ) Size distributions from method B analyzed by CONTIN. (D) QLS data analyzed by CONTIN. (E) CLS.
MOLECULAR WEIGHT DISTRIBUTIONS OF ANTIGEN-ANTIBODY
indicate errors in the correlation between diffusion coefficient and molecular weight resulting from inaccurate determinations of persistence length and chain thickness. Alternatively, it may suggest some error in this single measurement of (M ) , by CLS. Our result, that CONTIN is effective in determining size distributions, is consistent with results from several other studies. Provencher et al? showed that measurements of (M),,, ( M ) w , and ( M ) * of polystyrene spheres by CONTIN analysis of QLS data were in excellent agreement with National Bureau of Standards determinations made by osmotic pressure, light scattering, sedimentation, and fractionation. ( M ) , values for poly ( 1,4-benzamide) from CONTIN analysis were within 1.07.5% of values obtained by CLS." In a comparison of several methods for analyzing simulated QLS data," CONTIN proved best in fitting unimodal distributions. For bimodal distributions, accurate estimates of the average hydrodynamic radius and weight fraction of each peak were obtained. However, the distributions tended to be oversmoothed. Similar conclusions were drawn when experimental data were analyzed. In another study, size distributions of vesicles were examined by QLS and EM." Both techniques gave similar values for hydrodynamic radius at the maximum and similar size ranges. The distribution from QLS was slightly bimodal, whereas the EM distribution was unimodal but asymmetric. The authors attributed the differences to slight changes that may have occurred during preparation of the EM samples. This is the first reported study of the use of CONTIN to determine size distributions of Ag-Ab complexes. In general, determination of molecular weight distributions from CONTIN analysis of QLS was satisfactory. The analysis required a correlation between diffusion coefficients and molecular weight, as well as some estimate of the particle structure factor. Once these were available, CONTIN analysis became a quicker method to determine ( M ) ,than CLS, and a much simpler and easier way to estimate molecular weight distributions than electron microscopy.
1295
REFERENCES 1. Brennan, F. M., Grace, S. A. & Elson, C. J. ( 1983) J. Immunol. Methods 56,149-158. 2. Segal, D. M., Dower, S. K. & Titus, J. A. ( 1983) J . Immunol. 130, 130-137. 3. Plotz, P. H., Kimberly, R. P., Guyer, R. L. & Segal, D. M. (1979) Mol. Immunol. 19,99-110. 4. Doekes, G., van Es, L. A. & Daha, M. R. (1984) Scand. J . Immunol. 19,99-110. 5. Yarmush, D. M., Murphy, R. M., Colton, C. K.,Fisch, M. & Yarmush, M. L. (1988) Mol. Immunol. 25,1732. 6. Murphy, R. M., Slayter, H., Schurtenberger, P., Chamberlin, R. A., Colton, C. K. & Yarmush, M. L. (1988) Biophys. J. 54,45-56. 7. Murphy, R. M., Chamberlin, R. A., Schurtenberger, P., Yarmush, M. L. & Colton, C. K. (1990) Biochemistry 29,10889-10899. 8. Koppel, D. E. (1972) J . Chem. Phys. 57,4814-4820. 9. Provencher, S. W. (1982) Comp. Phys. Commun. 27, 213-227, 10. Provencher, S. W. (1982) Comp. Phys. Commun. 27, 229-242. 11. Provencher, S. W. (1984) C O N T I N User's Manual, EMBL Technical Report DA07, European Molecular
Biology Laboratory. 12. Morel, G. A., Yarmush, D. M., Colton, C. K., Benjamin, D. C. & Yarmush, M. L. (1988) Mol. Immunol. 25,17-25. 13. Benoit, H. & Doty, P. (1953) J . Chem. Phys. 57, 958-963. 14. Burchard, W. (1983) Adv. Polym. Sci. 48,4-124. 15. Yamakawa, H. & Fujii, M. (1973) Macromolecules 6, 407-415. 16. Fujii, M. & Yamakawa, H. (1975) Macromolecules 8 , 792-799. 17. Provencher, S. W., Hendrix, J., De Maeyer, L. & Paulussen, N. (1978) J. Chem. Phys. 6 9 , 4273-4276. 18. Stock, R. S. & Ray, W. H. (1985) J. Polym. Sci. 23, 1393-1447. 19. Ying, Q. & Chu, B. (1987) Macromolecules 20, 871877. 20. Bayerl, T. A., Schmidt, C. F. & Sackmann, E. (1988) Biochemistry 27, 6078-6085.
Received June 26, 1990 Accepted June 28, 1991
SYNOPSIS
Physiological properties of soluble antigen-antibody ( Ag-Ab) complexes depend in part on the size of the complexes. In previous work, the size distribution and structure of model Ag-Ab complexes were determined by electron microscopy. In this study,we used constrained regularization analysis of quasi-elasticlight scattering data to estimate molecular weight distributions of model Ag-Ab complexes. A conformational model was necessary to determine appropriate correlationsbetween molecular weight and diffision coefficient, and to estimate particle structure factors. Porod-Kratky theory proved to be an adequate conformational model for these purposes. The molecular weight distributions determined by constrained regularization compared favorably with distributions obtained either by electron microscopy or by thermodynamic modeling.
INTRODUCTION Many physiological properties of soluble antigenantibody ( Ag-Ab ) complexes, including clearance from the circulation, complement fixation, adherence to phagocytes, and differential tissue deposition, depend on the size and composition of these complexes.'-4 In our previous work, 5-7 we measured average sizes of model Ag-Ab complexes under a variety of conditions. Specifically, we obtained the weight-average molecular weight (M ) wdirectly from classical light scattering (CLS) data and the z-average hydrodynamic radius (&,)z with quasi-elastic light scattering (QLS ) using the method of cumulants for data analysis. Electron microscopy ( E M ) by rotary shadow casting was used to obtain size distributions at a single concentration.6 A thermodynamic model was developed that predicts the size distribution of the complexes at any concentration; parameters of the model were obtained by fitting to EM, CLS, and radioimmunoassay data.7 Determining size distribution by EM is tedious and subject to potential artifacts. More rapid and convenient techniques to determine molecular weight distributions in solution are desirable. In this Biopolymers, Val. 31, 1289-1295 (1991) CCC 0006-3525/91/111289-07$04.00 0 1991 ,John Wiley & Sons, Inc.
work, we examined the utility of QLS for estimating molecular weight distributions of model Ag-Ab complexes by analyzing autocorrelation data using the method of constrained regularization?-" In order to do this, some knowledge or assumption regarding the structure of the complexes was required. The conformational model developed previously6 was used in this analysis to estimate the particle structure factor, and to determine the relationship between diffusion coefficient and molecular weight.
MATERIALS A N D METHODS Ag-Ab complexes were prepared as described in detail previously.6 To briefly describe the procedure: bovine serum albumin (BSA: Pentex sulfhydryl modified, Miles, Elkhart, I N ) was dissolved in phosphate-buffered saline ( PBSA: 0.15M NaCI, 0.01M KH2PO4/KzHPO4,0.02% ( w / v ) sodium azide, pH 7.4), purified by gel permeation chromatography to remove oligomers, and concentrated by ultrafiltration. Monoclonal antibodies ( MAb ) were prepared from hybridoma cell lines designated as 5.1, 6.1, and 9.1. Each MAb bound noncompetitively to a different single epitope on BSA." MAb were isolated from ascites by affinity adsorption to BSA covalently linked to CNBr-Sepharose 4B (Phar1289
1290
MURPHY, YARMUSH, AND COLTON
macia, Upsalla, Sweden). MAb were eluted with 0.1M glycine/HCl buffer, pH 2.5, dialyzed against PBSA, concentrated by ultrafiltration, purified by gel permeation chromatography to remove oligomers, and concentrated again by ultrafiltration. AgAb complexes, designated as 5.1 6.1, 5.1 9.1, and 6.1 9.1, were prepared by dilution of the stock solutions of two of the three MAb and BSA in PBSA. Final concentrations were [BSA] = 8.25 X 10p7M, and [MAb,] = [MAb2] = 3.5 X lOP7M. QLS experiments were conducted as described in detail previously.6 Autocorrelation data taken a t a 60.2" scattering angle were analyzed by the method of cumulants to obtain (Rh)z. Results were reported previously6; (Rh)= was 99 4 2 A for 5.1 9.1, 114 f 1 A for 5.1 6.1, and 146 k 3 A for 6.1 9.1. In this work, the same autocorrelation data were analyzed further by the method of constrained regularization.
+
+
+
+ +
+
where G ( ri)is the fraction of the total intensity contributed by species i, and the summation is taken over all species i. This fraction can be expressed as
DATA ANALYSIS For a monodisperse solution, the normalized firstorder autocorrelation function is
G(I'i)
=
NiM?P(q)i
(5)
c NiMSP(q)i i
where Ni = the number of moles of species i, Mi is the molecular weight, and P(q)i is the particle structure factor. The value of P ( q ) depends on the size and shape of the particle as well as the scattering angle. Particle structure factors calculated for a variety of shapes and sizes from equations available in the literature 13,14are plotted in Figure 1.For the conditions of our experiments, with an argon ion laser operated at 488 nm wavelength, an aqueous solvent with a refractive index of 1.33, and a scattering angle of 60.2", q = 0.0017 k'.Thus, a particle with qRh = 1 has a hydrodynamic radius Rhof 580 A and a particle structure factor P(q) that ranges from 0.54 for a Gaussian coil to 0.82 for a solid sphere. For our AgAb complexes, the largest complexes seen in electron micrographs had an estimated Rh around 250 A, where qRh 0.43. Under these conditions, for complexes modeled as wormlike chains, P ( q ) 0.9.
-
-
where T is the autocorrelation delay time, D is the translational diffusion coefficient, q = (47rn/ Xo)sin(B/2) is the scattering vector, n is the refractive index of the solvent, A, is the wavelength of light in vacuo, and 0 is the scattering angle. For a polydisperse solution, the autocorrelation data includes contributions from all scattering particles, and must therefore be written as
L
where G( I') is the distribution of the intensity of scattered light and
J, G(I')dI'=
1
(3)
For a finite number of species, Eq. ( 2 ) can be written as
soft sphere-...
.
4
qRh Figure 1. Particle structure factor for several shapes. R h is the hydrodynamic radius and q is the scattering vector. For star-shaped molecules, f is the number of rays. For soft spheres (also called three-functional regularly branched chains with Gaussian behavior of the subchains), m is the number of units between branching points and n is the number of branching shells. It was assumed for these calculations that the branching unit has a length of 300 A. The particle structure factors for wormlike chains shown in the figure were calculated assuming that the particle had a length of 1025 A, a persistence length of 205 A, and a chain thickness of 110 A.
MOLECULAR WEIGHT DISTRIBUTIONS OF ANTIGEN-ANTIBODY
Constrained Regularization
The expression in Eq. ( 2 ) is an example of a general class of equations known as Fredholm integrals of the first kind.g These occur frequently in a variety of experimental techniques and can be described in general as
1291
tomatically by CONTIN such that the increase in V ( a ) over V ( a = 0 ) is due to chance alone (i.e., noisy data) about 50% of the time.’-’’ Determination of Molecular Weight Distributions
The function to be determined from QLS data is the weight-average molecular weight distribution f ( M ) , where f ( M ) d M is the weight fraction of all complexes with molecular weights between M and M d M and
+
where y ( t )are the (noisy) experimental data with error t , F ( K , t ) is a known function, and s ( K ) is the unknown function. The terms involving LiBi in the general case account for any constant terms. The problem of finding the desired function s ( K ) is in general an ill-posed one, since there may be many solutions that are equally good within the experimental error E . The program CONTIN9-1’ uses the method of constrained regularization to solve these kinds of equations. First, Eq. ( 6 ) is converted into a system of linear equations:
i= 1
m=l
where c, are the weights of the quadrature (e.g., Simpson’s rule) formula and N g are the grid points at which the solution is determined. This equation can be written more compactly as
and (M)w is the weight-average molecular weight. In this work, the minimum and maximum molecular weights are set equal to 50,000 and 5,000,000, respectively. The minimum weight was chosen to be slightly below that of BSA (66,000) and the maximum was chosen such that the size distribution decayed to zero before reaching this limit. From Eqs. ( 3 ) and ( l o b ) ,
Using this relation, and including the particle structure factor P ( q ) to properly weight the intensity of scattered light from different-sized complexes, Eq. ( 4 ) becomes
+
where N, = N g N L , A is the matrix containing the known c,F( K,, t ) and Li, and x is a vector containing the unknowns s ( K,) and B,. Two additional constraints are imposed on the solution. First, s ( K ) is required to be nonnegative. This eliminates any solutions that are physically impossible. Second, the requirement of parsimony (smoothness) is imposed. Parsimony is forced on the solution by modifying the weighted least-squares minimization problem by adding a regularizing term:
V ( a )= IM;”2(y - AX)l 2
+ a2
d2s(K) [ r dK= minimum ]
(9)
where M, is the covariance matrix of the errors e, y is the vector containing the data points y ( t ) , and a is the regularization parameter. The a is chosen au-
X exp(-h)f(M)dM
(12)
The correspondence between the right-hand side of Eq. ( 1 2 ) and the first term on the right-hand side of Eq. ( 6 ) is clear. A correlation between molecular weight and diffusion coefficient is required to solve Eq. ( 12) for the molecular weight distribution. This correlation was assumed to take the form14
D
=
aMB
(13)
To calculate the parameters a and p for our Ag-Ab complexes, the molecular weights Mi of complexes containing one to ten Ag-Ab units were converted to contour lengths based on previously obtained EM
1292
MURPHY, YARMUSH, AND COLTON
data.6 Di was calculated for each linear Ag-Ab complex of molecular weight Mi (Figure 2 ) using correlations that relate the diffusion coefficient of a Porod-Kratky wormlike chain to its contour length, persistence length, and chain thicknes~.'~.'~ The values of a and ,6 for the three Ag-Ab complexes are listed in Table I. This was incorporated into Eq. (12) by making the substitution
r
=
q 2 D = q2aMb
(14)
There is a small error in these parameters because cyclic complexes were not included in Figure 2. This error should be insignificant because only 1.7 and 6% of the complexes for 5.1 6.1 and 6.1 9.1, respectively, were cyclic as determined by EM.6 For 5.1 9.1,23% of the complexes were cyclic, but 86% of the cyclic complexes contain two Ab and the calculated diffusion coefficient for two-Ab cycles (16) is only 3% greater than that for two-Ab linear complexes. CONTIN was used to find the molecular weight distribution f ( M ) . The particle structure factor P ( q ) was calculated using the previously determined persistence length a and chain thickness d (5.1 6.1 : a = 430 A, d = 83 A, 5.1 9.1: a = 91 A, d = 85 A, 6.1 9.1: a = 205 A, d = 110 A ) and available correlations for Porod-Kratky wormlike chains.I3 A single constant baseline B attributable to "dust" terms was fit to the data, with N L and L, set equal to 1. For our work, M, was assumed to contain only diagonal elements, with variances estimated from a preliminary unweighted analysis of the data assuming Poisson statistics as described in more detail elsewhere."
+
+
+
+
+
+
.s! .Y s
8
0
70
10
:
0 5.1+6.1
-
A
5.1+9.1
A
6.1+9.1
100
7000
10000
Molecular weight Figure 2. Relationship between molecular weight Mi (kdaltons) of Ag-Ab complexes and diffusion coefficient
D, (cm2/s).Diwas calculated using correlations for PorodKratky wormlike chains, as described in the text.
Table I Conversion of Molecular Weights to Diffusion Coefficients" Sample 5.1 5.1 6.1
+ 6.1 + 9.1 + 9.1
01
P
7.39 x 10-6 3.68 X 3.84 X
-0.577 -0.443 -0.485
Diffusion coefficients are related to molecular weight according to D = a M B ,where D is the translational diffusion coefficient in cmZ/s at 293 K in a solvent with 1.01 cp viscosity and M is the molecular weight in thousands.
RESULTS AND DISCUSSION Size distributions of Ag-Ab complexes made from three pairs of MAb and BSA at identical concentrations and molar ratios ([BSA] = 8.25 X 10-7M, [ MAb,] = [ MAbz] = 3.5 X 10p7M)were estimated using four different methods. In method A, EM data described previously6 were analyzed by determining the number of Ab visible in the complexes and counting the frequency distribution, then converting this to a weight distribution. In method B, weight distributions were calculated using the thermodynamic model and the equilibrium constants derived earlier.7 In method C, the distribution generated by the thermodynamic model (method B ) was converted to a first-order autocorrelation function I g " ' ( T ) I by calculating the hydrodynamic radius of each complex using the estimated Porod-Kratky parameters and available correlations, l 5 , I 6 and converting this into a diffusion coefficient through the Stokes-Einstein equation. Then, the concentration calculated from the model, the molecular weight, the calculated diffusion coefficient, and the calculated particle structure factor for each complex were employed with Eqs. ( 4 ) and ( 5 ) . The resulting autocorrelation functions were analyzed by CONTIN. In method D, the autocorrelation functions obtained from QLS measurements were analyzed by CONTIN. In Figure 3, the smoothed distributions obtained by CONTIN analysis of the autocorrelation functions predicted using the thermodynamic model (method C ) and measured by QLS (method D ) were compared. The agreement between the distributions estimated by CONTIN analysis of simulated data from the model (method C ) and of QLS data (method D ) is good, particularly for 6.1 9.1. The molecular weight at the maximum and the shape of the two curves are very close. The skewing of the distribution toward smaller molecular weights is
+
1293
MOLECULAR WEIGHT DISTRIBUTIONS OF ANTIGEN-ANTIBODY
0.20
0.15
11
w
I
4 -
Method C (model)
0.10
0.05
0.00
0-
400
800
1200
1600
2000
linear complexes were lumped together. The contribution of unbound Ag to the distribution was not included, because free BSA was difficult to positively identify on the micrographs. The smoothed distribution from CONTIN analysis of QLS data (method D ) was used to estimate a discrete distribution, in order to make a more direct comparison with the distributions from the EM data and the thermodynamic model. The area ai, under each section of the curve was estimated by using the trapezoidal rule. so that
0.20
s
*r
$Q
E"
-
0.15
w Method C (model)
0.4
5.1+6.1 W Method A (EM) 0 Method B (model)
0.10 0.3
.9
5
1
Method D (QLS)
0.2
0.05
0.1
0.00
0
400
0.20
800
1200
--
0.15
- _m - -
1600
2000
0.0 1
6.1+9.1
2
3
.e t;
0.10
m k
0.05
P .9
g
0.00 400
800
5
6
7
8
9
Method C (model) Method D (QLSl
W Method A (EM) 0 Method B (model)
E
0
4
1200
1600
Method D (OLS)
0.3
1 0.2 0.1
2000 0.0
Molecular weight
1
Figure 3. Smoothed molecular weight distributions obtained by CONTIN analysis of the predicted autocorrelation data from the thermodynamic model (method C ) and of the measured autocorrelation data from QLS experiments (method D ).
2
3
4
5
6
7
8
9
0.5
6.1+9.1 W Method A (EM) 0 Method B (model)
0.3
Method D (QLS)
0.2
correctly ascertained by CONTIN analysis of QLS data. Also, CONTIN analysis detected the presence of larger complexes in 6.1 9.1 than in 5.1 6.1 or 5.1 9.1, in agreement with the thermodynamic model. In Figure 4, discrete distributions were plotted. The distributions from electron microscopy (method A) and from the thermodynamic model (method B ) were calculated assuming that each Ab in a complex is, on average, bound to one Ag and that each AbAg unit has a molecular weight of 220 kd. Cyclic and
+
+
+
0.1 0.0
1
2
3
4
5
6
7
8
9
Number of Ab in Complex Figure 4. Discrete molecular weight distributions obtained from electron microscopy data (method A ) , thermodynamic model (method B ) , or discretization of the smoothed CONTIN analysis of QLS data (method D ) , as described in the text.
1294
MURPHY, YARMUSH, AND COLTON
1 (hi + hj)(Mj - MI) 2
There are differences in the details of the distributions when any two methods are compared. E M results show a greater weight fraction of larger comwhere h, and hJ are the weight fraction, and M I and plexes than the model; as was noted p r e v i ~ u s l y , ~ MJ are the molecular weights a t grid points i and j , this was attributed to possible artifacts in analysis respectively. In order t o discretize the distribution, of the E M data. Discrepancies between the distria, attributable to complexes with nAb was calculated bution derived from QLS data and the other distria s follows. First, the molecular weight range of all butions probably arise from two sources: first, the complexes with n Ab was calculated, where Mm,,,,is resolving power of QLS is limited because of the the molecular weight of complexes containing n - 1 mathematical uncertainties inherent in solving Eq. Ag and n Ab, and M,,,,,,, is the molecular weight of ( 6 ) and second, the discretization of the smooth complexes containing n 1 Ag and n Ab. For exdistribution is approximate. ample, for all complexes containing 2 Ab, Mm,n,2 T h e results of the four methods of analyses are = molecular weight of a complex with 2 Ab and 1 shown in Table I1 in terms of number- ( ( M ) , ) , Ag, or in our case, 2 X 154,000 66,000 = 374,000. weight- (( M ) , ) , and z-average (( M ) , ) molecular Likewise, MmaX,? is the molecular weight of a complex weights. For comparison, ( M ) , as measured by CLS with 2 Ab and 3 Ag, or 506,000. For each section ij on the identical sample as used for method D is included in Table I1 (method E ) . Comparing average where M,,,,, IMIand MJ IM,,,,,, a, was attributed t o complexes with n Ab. Because M , , , , is less than molecular weights of method B and method C is useful for seeing the effect of CONTIN analysis on Mmln,n+lthe section of the smoothed distribution where MI 2 M,,,,, and MI IMmln,n+l must be disa known distribution (in this case, from the thertributed between complexes with n or n 1Ab. For modynamic model). CONTIN tends to spread out these sections, a midpoint value M m l d = 0.5 (M,,,,, the distribution slightly, by lowering ( M ) , and in two cases by increasing ( M ) * .Ag-Ab complexes ochZmrn,n+l) was calculated. T h e value of hmldwas determined by linear interpolation, and the areas cur only in discrete sizes, whereas CONTIN assumes aL,mld and amid,, were calculated and attributed t o the a smooth continuous distribution. This effect may appropriate complex. T h e areas for complexes with be most important a t the low end of the scale, ben Ab were summed, then divided by the total area cause the relative increase in size is greater. Agreefor all complexes t o determine the fraction of comment between (M ) , from CLS (method E ) and that plexes with n Ab. determined by CONTIN analysis of QLS data The comparison between E M data (method A ) , (method D ) on the identical sample is excellent with the model (method B ) , and discretization of CON5.1 6.1 and 6.1 9.1. However, for 5.1 9.1, the T I N analysis of QLS data (method D ) , shown in CONTIN-determined (M ) , is closer t o (M ) , from Figure 4, is generally satisfactory in terms of shape E M (method A ) or from the thermodynamic model and range of the molecular weight distribution. (method B ) than from CLS (method E ) .This could
a,
=-
(15)
)
+
+
)
+
+
+
+
+
Table I1 Molecular Weight Distributions for Ag-Ab Complexes" Method Sample 5.1
+ 6.1
5.1
+ 9.1
6.1
+ 9.1
M
A
B
C
D
(M>n (M>W (M>Z (M>n (M), (M>z (M>n (M>W (M>2
399 573 783 427 568 765 482 748 1015
394 554 721 424 532 635 433 657 897
360 520 678 373 584 806 374 631 959
422 552 670 354 527 697 384 674 1197
E 544
414
721
* All molecular weights are in thousands. Methods: (A) EM. (B) Thermodynamic model. ( C ) Size distributions from method B analyzed by CONTIN. (D) QLS data analyzed by CONTIN. (E) CLS.
MOLECULAR WEIGHT DISTRIBUTIONS OF ANTIGEN-ANTIBODY
indicate errors in the correlation between diffusion coefficient and molecular weight resulting from inaccurate determinations of persistence length and chain thickness. Alternatively, it may suggest some error in this single measurement of (M ) , by CLS. Our result, that CONTIN is effective in determining size distributions, is consistent with results from several other studies. Provencher et al? showed that measurements of (M),,, ( M ) w , and ( M ) * of polystyrene spheres by CONTIN analysis of QLS data were in excellent agreement with National Bureau of Standards determinations made by osmotic pressure, light scattering, sedimentation, and fractionation. ( M ) , values for poly ( 1,4-benzamide) from CONTIN analysis were within 1.07.5% of values obtained by CLS." In a comparison of several methods for analyzing simulated QLS data," CONTIN proved best in fitting unimodal distributions. For bimodal distributions, accurate estimates of the average hydrodynamic radius and weight fraction of each peak were obtained. However, the distributions tended to be oversmoothed. Similar conclusions were drawn when experimental data were analyzed. In another study, size distributions of vesicles were examined by QLS and EM." Both techniques gave similar values for hydrodynamic radius at the maximum and similar size ranges. The distribution from QLS was slightly bimodal, whereas the EM distribution was unimodal but asymmetric. The authors attributed the differences to slight changes that may have occurred during preparation of the EM samples. This is the first reported study of the use of CONTIN to determine size distributions of Ag-Ab complexes. In general, determination of molecular weight distributions from CONTIN analysis of QLS was satisfactory. The analysis required a correlation between diffusion coefficients and molecular weight, as well as some estimate of the particle structure factor. Once these were available, CONTIN analysis became a quicker method to determine ( M ) ,than CLS, and a much simpler and easier way to estimate molecular weight distributions than electron microscopy.
1295
REFERENCES 1. Brennan, F. M., Grace, S. A. & Elson, C. J. ( 1983) J. Immunol. Methods 56,149-158. 2. Segal, D. M., Dower, S. K. & Titus, J. A. ( 1983) J . Immunol. 130, 130-137. 3. Plotz, P. H., Kimberly, R. P., Guyer, R. L. & Segal, D. M. (1979) Mol. Immunol. 19,99-110. 4. Doekes, G., van Es, L. A. & Daha, M. R. (1984) Scand. J . Immunol. 19,99-110. 5. Yarmush, D. M., Murphy, R. M., Colton, C. K.,Fisch, M. & Yarmush, M. L. (1988) Mol. Immunol. 25,1732. 6. Murphy, R. M., Slayter, H., Schurtenberger, P., Chamberlin, R. A., Colton, C. K. & Yarmush, M. L. (1988) Biophys. J. 54,45-56. 7. Murphy, R. M., Chamberlin, R. A., Schurtenberger, P., Yarmush, M. L. & Colton, C. K. (1990) Biochemistry 29,10889-10899. 8. Koppel, D. E. (1972) J . Chem. Phys. 57,4814-4820. 9. Provencher, S. W. (1982) Comp. Phys. Commun. 27, 213-227, 10. Provencher, S. W. (1982) Comp. Phys. Commun. 27, 229-242. 11. Provencher, S. W. (1984) C O N T I N User's Manual, EMBL Technical Report DA07, European Molecular
Biology Laboratory. 12. Morel, G. A., Yarmush, D. M., Colton, C. K., Benjamin, D. C. & Yarmush, M. L. (1988) Mol. Immunol. 25,17-25. 13. Benoit, H. & Doty, P. (1953) J . Chem. Phys. 57, 958-963. 14. Burchard, W. (1983) Adv. Polym. Sci. 48,4-124. 15. Yamakawa, H. & Fujii, M. (1973) Macromolecules 6, 407-415. 16. Fujii, M. & Yamakawa, H. (1975) Macromolecules 8 , 792-799. 17. Provencher, S. W., Hendrix, J., De Maeyer, L. & Paulussen, N. (1978) J. Chem. Phys. 6 9 , 4273-4276. 18. Stock, R. S. & Ray, W. H. (1985) J. Polym. Sci. 23, 1393-1447. 19. Ying, Q. & Chu, B. (1987) Macromolecules 20, 871877. 20. Bayerl, T. A., Schmidt, C. F. & Sackmann, E. (1988) Biochemistry 27, 6078-6085.
Received June 26, 1990 Accepted June 28, 1991
