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Copyright © 1999 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 77, Issue 6, 2902-2910, 1 December 1999

doi:10.1016/S0006-3495(99)77123-7

Biophysical Theory and Modeling

Novel Size-Independent Modeling of the Dilute Solution Conformation of the Immunoglobulin IgG Fab′ Domain Using SOLPRO and ELLIPS

Beatriz Carrasco^*, Jose Garcia de la Torre^*, Olwyn Byron^#, 1, David King^§, Chris Walters^#, Susan Jones^¶ and Stephen E. Harding^#, ,  

^* Departamento de Quimica Fisica, Facultad de Quimica, Universidad de Murcia, 30071 Murcia, Spain
^# National Centre for Macromolecular Hydrodynamics, University of Nottingham, School of Biological Sciences, Sutton Bonington LE12 5RD, England
^§ Celltech Therapeutics, Bath Road, Slough, Berkshire, England
^¶ Biomolecular Structure and Modelling Unit, Department of Biochemistry and Molecular Biology, University of London, London WC1 6BT, England

Address reprint requests to Dr. Stephen E. Harding, NCMH Unit, University of Nottingham, Sutton Bonington LE12 5RD, England. Tel.: 44-115-951-6148; Fax: 44-115-951-6142.

¹ Dr. Byron's present address is Division of Infection and Immunity, Institute of Biomedical and Life Sciences, University of Glasgow, Glasgow G12 8QQ, Scotland.

There are two approaches to the modeling of quasirigid macromolecules in dilute solution, using hydrodynamic and solution scattering-based methodologies. One is the whole-body or ellipsoid approach (Perrin, 1934,Perrin, 1936,Harding, 1989); the other is the multiple-sphere array or bead approach (Bloomfield et al,Garcia de la Torre and Bloomfield, 1981,Garcia de la Torre, 1989). Although with the latter the hydrodynamic theory is approximate rather than exact, it does allow the prospect of representations of complex structures such as antibodies (Byron, 1992). With either type of approach one of the key difficulties to be confronted is that of molecular volume: experimental hydrodynamic coefficients such as the sedimentation coefficient and diffusion coefficient (manifestations of the translational frictional characteristics of a macromolecule), or the intrinsic viscosity (bulk flow characteristics), radius of gyration (mass distribution characteristics about the centre of mass), molecular covolume (thermodynamic nonideality), or rotational diffusion decay times (rotational friction) all depend on the volume properties of a molecule as well as shape; in fact in many cases the shape contribution is secondary to molecular volume.

To facilitate this move away from the ambiguities caused by size-dependent approaches, “size-independent” or “universal” hydrodynamic shape functions have now been developed for both modeling strategies. Indeed, this has been used for ellipsoid modeling since the 1930s; for example, Perrin, 1936 “translational frictional ratio due to shape” or P function (Harding and Rowe, 1982a,Harding and Rowe, 1982b,Harding and Rowe, 1983) or Simha's, 1940 “viscosity increment” or ν function. A suite of algorithms for the PC has recently been made available (Harding et al) for ellipsoids of revolution (ELLIPS1, direct prediction of axial ratios from a user-specified shape function) and the much more general triaxial ellipsoids (ELLIPS2, rigorous evaluation of the complete set of hydrodynamic shape functions for user-specified triaxial dimensions or (two) axial ratios; ELLIPS3,4, for combining hydrodynamic measurements to evaluate the triaxial shape of a molecule). These size-independent shape functions are known as “universal” shape functions, and Table 1 lists them and the experimental parameters from which they are derived. Some, such as P and ν, require an estimate for the degree of solvent association or “hydration”; some combined functions such as β, R, ∏, and ∧_h do not.

Table 1 Universal hydrodynamic parameters

	Universal hydrodynamic parameter	Name	Experimental parameters required for its measurement
	v	Viscosity increment	Intrinsic viscosity [η], partial specific volume v, molecular hydration δ
	P	Perrin translational frictional ratio	Translational diffusion coefficient D (or sedimentation coefficient s and molecular weight M), , δ
	R	Wales–van Holde R-function	Sedimentation concentration dependence (“Gralen”) parameter ks, [η]
	β	Scheraga-Mandelkern function	D (or s and M), , [η]
	ured	Reduced excluded volume	Thermodynamic 2nd virial coefficient B, M,, δ, charge (valency) Z, ionic strength I
	∏	Pi-function	B, M, [η], Z, I
	G	G-function	*Radius of gyration, M,
	θ+,−	Reduced electrooptic decay constant	Electrooptic decay constants times, θ+,−; M,,
	δ+,−	Electrooptic delta shape functions	θ+,−, M,, [η]
	γ+,−	Electrooptic gamma shape functions	s (or D), θ+,−, M,
	τh/τ0	Harmonic mean fluorescence anisotropydepolarization time ratio	Harmonic mean fluorescence anisotropy depolarization time; M,, δ
	Ψh	Psi-function	Harmonic mean fluorescence anisotropy depolarization time; M,, s (or D)
	∧h	Lambda-function	Harmonic mean fluorescence anisotropy depolarization time; M,, [η]
	τi/τ0 (i=1–5)	Time-resolved fluorescence anisotropy relaxation time ratio	Time-resolved fluorescence anisotropy relaxation time; M,, δ
	∧i (i=1–5)	Time-resolved lambda function	Time-resolved fluorescence anisotropy depolarization times; M,, [η]
	Ψi (i=1–5)	Time resolved psi function	Time-resolved fluorescence anisotropy depolarization times; M,, s (or D)

* If the density of bound water is the same as free, then v ≈ .

Bead modeling would appear to benefit strongly from such a size-independent approach, particularly with regard to the considerable uncertainties/ambiguities concerning the volume of a particle and the volume of the beads. So the algorithm SOLPRO (Garcia de la Torre et al) was constructed for precisely this reason, as well as to provide improved estimates for rotational and scattering-based parameters. A more recent version of the algorithm (Garcia de la Torre et al) also permits the prediction of NMR-based relaxation times as well as molecular covolumes for general particles.

In the conventional application of bead modeling the volume occupied by the particle is filled with spheres of various sizes; the number of beads required is minimized by making them as large as possible. Thus taking the case, for example, of an ellipsoid of revolution, this can be modeled as a straight string of beads whose radii decrease from the center, tapering toward the ends (Bloomfield et al,Garcia de la Torre and Bloomfield, 1977). An inconvenience of this procedure is that the (relative) size of some beads may be quite large and close to the size of the full particle. In this way, one or a few beads dominate the hydrodynamic behavior of the model, and this has the consequence that the rotational coefficients and the intrinsic viscosity (or viscosity increment) are more or less affected. The origin of the problem and its solution, in the form of the so-called volume corrections, has already been described (Garcia de la Torre and Rodes, 1983,Garcia de la Torre and Carrasco, 1998).

Very recently, an alternative procedure has been proposed to avoid such problems (Carrasco, 1998,Carrasco and Garcia de la Torre, 1999). The hydrodynamic model does not fill the particle's volume; instead it is just the particle's surface that is represented by a shell constructed with many small identical beads. The shell-model procedure was actually proposed in the early work of Bloomfield et al. The serious drawback of shell modeling is that the number of frictional elements in the model is very large, and the calculations require large amounts of CPU time. Methods for building such models and for rigorously computing their solution properties have been devised; for the details, see Carrasco, 1998 and Carrasco and Garcia de la Torre, 1999. Indeed, it has been shown by these authors that the hydrodynamic properties of ellipsoids of revolution can be accurately reproduced by such a shell approach.

In this paper we focus on five aspects of molecular modeling in solution, based on the IgG Fab′ fragment (Fig. 1):

Display large version of this figure

Figure 1

Schematic Fab′ (B72.3 IgG). (A) Intact antibody. (B) Fab′ fragment. (C) Fab fragment.

An objective method for defining the triaxial shape of a protein molecule from its atomic structural coordinates has been provided by Taylor et al. This method, which is insensitive to small deviations from an ideal ellipsoidal form, is based on the inertial, momental, or “Cauchy” ellipsoid (see MacMillan, 1936,Synge and Griffin, 1959), dilated so that it forms a close approximation to the protein surface. The original procedure, recently implemented by Hubbard, 1994 in a FORTRAN algorithm, is used to calculate the ratios of the principal axes of the equimomental ellipsoid for the 3D coordinates of a protein. These ratios can be used in conjunction with a second algorithm, SURFNET (Laskowski, 1995), to generate a 3D surface representation of the ellipsoid; the combined algorithm is referred to as ELLIPSE. Figure 2A shows the fit to the crystal structure for chimaeric B72.3c Fab′ with (a/b, b/c)=(1.595, 1.418). Table 2 compares the corresponding axial ratios (a/b, b/c)’s for this protein with the chimeric Fab hA5B7, and wild-type human and mouse Fab fragments. It is clear that there is little species variation in Fab′ or Fab (gross) shape, based on the published crystal coordinates.

Display large version of this figure

Figure 2

(A) Shape models for B72.3 Fab′ triaxial ellipsoid fit to the crystal structure using PROTRUDER and SURFNET and (B) bead-shell model for the equivalent prolate ellipsoid using SOLPRO.

Table 3 compares the exact hydrodynamic parameters for a triaxial ellipsoid of axial ratios (a/b, b/c)=(1.595, 1.418) with those of the equivalent prolate ellipsoid, whose minor axes are taken to be equal to the mean of b and c, giving a revised (a/b, b/c) of ∼(1.83, 1.0). Both sets of data—exact to four significant figures—were evaluated using the program ELLIPS2 (Harding et al). It can be seen that the ellipsoid of revolution approximation leads to errors of ∼1% in the frictional ratio-based P function and ∼4% in both the intrinsic viscosity-based ν function and the radius of gyration-based G function and only leads to serious error (∼10% or above) in the rotational diffusion-based γ₊ and γ₋ functions; because with the frictional ratio-based P function this can be experimentally measured to a precision no better than ±1%, such an approximation is therefore a reasonable one.

Figure 2B shows the bead-shell model approximation to the surface of this ellipsoid of revolution. The final column of Table 3 gives the set of universal parameters calculated from SOLPRO for this model.

The procedure for arranging the beads is as follows: a number of small beads of radius σ are placed in such a way that their centers lie on the surface of the ellipsoid. Each bead is touching its closest neighbors, or closely tangent to them, with small gaps. The number of beads, N, is large and increases with decreasing σ; for instance, for the smallest radius considered in our calculations, σ=3.2Å (which is a fraction 0.04a of the longest semiaxis, a=80Å, of the ellipsoid), we have N=874. The computing time for the HYDRO calculation of the shell model grows as N³. Actually, we have made the calculations for various bead sizes, ranging from 0.112a to 0.04a. The resulting properties show a slight dependence on σ; the final results are obtained by extrapolating to the shell-model limit of σ=0. This is done by fitting the data to a polynomial of degree 1 or 2, depending on the cases. With the HYDRO evaluations completed before SOLPRO, the universal parameters are then evaluated (Table 3).

To assess the usefulness of this approximation we have given in parentheses ( ) the percentage error compared with the ellipsoid of revolution model; those in square brackets [ ] represent the total percentage error compared with the triaxial ellipsoid. Let us now consider each in turn.

The Fab′ fragment of the chimeric antibody B72.3 was expressed directly in CHO cells as previously described (King et al). Fab′ was purified from the CHO cell supernatant, using affinity chromatography with mucin-sepharose. Submaxillary mucin acts as a mimic of the tumor-associated antigen recognized by B72.3 (Hanisch et al). Bovine submaxillary mucin was coupled to cyanogen bromide-activated Sepharose by standard techniques, packed into a column, and equilibrated with phosphate-buffered saline. CHO cell supernatant was applied directly to the column, which was then washed with phosphate-buffered saline before bound material was eluted with 0.1M citric acid. The pH of eluted material was immediately adjusted to pH 6–7, and the purity was tested by sodium dodecyl sulfate-polyacrylamide gel electrophoresis. Purified material was demonstrated to be >95% pure after this single-step purification.

For sedimentation analysis the protein was dissolved in a standard phosphate chloride buffer (Green, 1933) of pH 6.8, I=0.10.

An Optima XL-A analytical ultracentrifuge (Giebeler, 1992) from Beckman Instruments (Palo Alto, CA) was employed to perform both sedimentation velocity and sedimentation equilibrium measurements. Solute distributions at 20.0°C were recorded via their UV absorption at 278nm.

A rotor speed of 49,000 rev/min was employed. Sedimentation coefficients in the buffer at 20°C, s_T,b, were evaluated using SVEDBERG (Philo, 1997), LAMM (Behlke and Ristau, 1997), and DCDT (Stafford, 1992). These routines also yield estimates for the translational diffusion coefficient, D_T,b. s_T,b values were corrected to standard solvent conditions, the density, ρ, and viscosity, η, of water at 20°C, using the formula (Schachman, 1959)

(1a)

A similar correction was employed for the translational diffusion coefficient:

(1b)

(2a)

(2b)

The low/intermediate speed method was employed (Creeth and Harding, 1982). Equilibrium rotor speeds of 12,000rpm were employed in 12-mm Yphantis-type multichannel cells. Only the radially innermost two channels were used, each with 80μl of solution (dialysate in the solvent sector). Equilibrium was established within 36h. Solute distributions at equilibrium were analyzed by the model-independent routine MSTARA (Cölfen and Harding, 1997); the apparent whole-cell weight-averaged molecular weight M_w,app was extracted from the limiting value at the cell base of the M* function (Creeth and Harding, 1982). A low loading concentration was used (0.5mg/ml); at this concentration nonideality effects can be assumed to be negligible, and hence M_w,app ≈ M_w.

Analytical ultracentrifugation was performed on the Fab′ fragment of B72.3 to 1) assess the monodispersity, 2) confirm the monomeric state (solution molecular weight), 3) determine the sedimentation coefficient and the corresponding frictional ratio (f/f_o), 4) combine f/f_o with the Perrin, 1936 function P values of Table 3 calculated from the crystal structure of the IgG Fab′ to obtain an estimate for the molecular hydration δ of the Fab′ molecule.

This was supported by 1) the observation of only single boundaries from sedimentation velocity (Fig. 3), 2) no evidence of an increase in sedimentation coefficients with increase an in concentration (Figure 4A), 3) linear sedimentation equilibrium plots of log (absorbance) versus the radial displacement squared parameter ξ defined by

(3)

Display large version of this figure

Figure 3

g(s*) versus s_T,b plot from the program DCDT (Stafford, 1992) for B72.3 Fab′. Loading concentration of 1.10mg/ml. Rotor speed=49,000 rev/min, temperature=20.0°C. g(s*) is the apparent (i.e., not corrected for diffusion) distribution of sedimentation coefficients. The thin line is a single Gaussian fit with peak s_T,b=3.63S.

Display large version of this figure

Figure 4

(A) Gralen plots of s_20,w versus sedimenting concentration, c (corrected for radial dilution) for B72.3 Fab′. (B) Corresponding plot of D_20,w versus c.

Extrapolation of the M* function to the cell base (Fig. 5) yielded a whole-cell weight-averaged molecular weight M_w=(47,000±1000) g/mol, in virtually exact agreement with the sequence molecular weight of 47,499 g/mol calculated from the amino acid sequence. The result is also in agreement with the sedimentation-diffusion result; s_20,w^o and D_20,w^o values of (3.92±0.01)×10⁻¹³ s and (7.1±0.2)×10⁻⁷ cm² s⁻¹ were obtained (Fig. 4). Combination via the Svedberg equation (Svedberg and Pedersen, 1940)

(4)

Display large version of this figure

Figure 5

Sedimentation equilibrium M* plot for the extraction of the molecular weight (whole cell weight average) for B72.3 Fab′: M*(ω → 1)=M_w,app=(47,000±2000) g/mol, and because of the low loading concentration, M_w,app ≈ M_w. Rotor speed=12,000 rev/min. Temperature=20.0°C.

The sedimentation coefficient s_20,w^o is related to molecular shape via the frictional coefficient, f:

(5)

f_o is simply the Stokes’ law friction coefficient,

(6)

Thus from Eqs. (5), the frictional ratio f/f_o is given by

(7)

The frictional ratio is related to the Perrin shape function P and the extent of hydration δ by the formula (see, e.g., Squire and Himmel, 1979):

(8)

Table 4 Sedimentation and related parameters for Fab′

		s20,w0 (S)	M (g/mol)	(ml/g)	f/f0	δ (P=1.045)	δapp (P=1.023)
	B72.3 Fab′	3.92±0.01	47,500	0.727	1.22±0.01	0.43±0.07	0.51±0.07

Our value of (0.43±0.07) for δ can be compared with values obtained in a similar way for other proteins. For example, an inertial ellipsoid fit to the crystal structure for deoxyhemoglobin yields axial ratios (a/b, b/c) of (1.27, 1.07). From ELLIPS2 this yields a value for the Perrin shape function P of 1.0071. Experimentally, the s_20,w^o value is 4.6S (Behlke and Scheler, 1972). With M=64500, =0.746 ml/g, we obtain a frictional ratio f/f_o=1.174. Combining f/f_o (Eq. (8)) with P yields a value for the hydration δ of 0.44, almost identical to that for Fab′ and in exact agreement with a recently published value of 0.43, on the basis of thermodynamic nonideality and the algorithm COVOL (Harding et al).

For ribonuclease the inertial ellipsoid fit to the crystal structure yields (a/b, b/c) of (1.53, 1.23). From ELLIPS2 this yields a value for the Perrin shape function P of 1.0282. Experimentally, the s_20,w^o value is 2.0S (see, e.g., Creeth, 1958). With M=13700, =0.703 ml/g we obtain a frictional ratio f/f_o=1.146. Combining f/f_o (Eq. (8)) with P yields a value for the hydration δ of 0.27, again in exact agreement with the recently published value of 0.25 on the basis of thermodynamic nonideality and the algorithm COVOL (Harding et al). Kumoninski and Pessen, 1982 and Pessen et al report a very similar value on the basis of low-angle x-ray scattering.

We have shown that for a regular triaxial ellipsoidal structure based on the crystal dimensions of the Fab′ fragment of an antibody, only small errors are induced in the calculated set of hydrodynamic parameters if a prolate ellipsoid of revolution model is taken instead. A bead-shell model of this ellipsoid of revolution reproduces the hydrodynamic parameters well, with the Perrin frictional-ratio base universal function returned with an accuracy better than ∼2% within either the triaxial ellipsoid or the ellipsoid of revolution values, and all of the other parameters reproduced favorably, thus validating the bead-shell approximation to structures of this type. Furthermore, by combination of the value evaluated for the Perrin shape function P with the experimentally measured frictional ratio we have been able to obtain an estimate for both the hydration and the “apparent hydration” for the bead-shell model, a value that will subsequently prove useful for further modeling of intact, immunologically active antibody molecules, in which the bead-shell rather than ellipsoid-based modeling strategies are appropriate. It is hoped that such an approach will complement the advances that are now being made in structural determinations by x-ray methods of intact, immunologically active antibody molecules in the crystallized state (Harris et al,Harris et al).