An interactive program is described for calculating the
second virial coefficient contribution to the thermodynamic nonideality of solutions of rigid macromolecules based on their triaxial
dimensions. The FORTRAN-77 program, available in precompiled form for
the PC, is based on theory for the covolume of triaxial ellipsoid particles [Rallison, J. M., and S. E. Harding. (1985). J.
Colloid Interface Sci. 103:284-289
]. This covolume has the
potential to provide a magnitude for the second virial coefficient of
macromolecules bearing no net charge. Allowance for a charge-charge
contribution is made via an expression based on Debye-Hückel
theory and uniform distribution of the net charge over the surface of a
sphere with dimensions governed by the Stokes radius of the
macromolecule. Ovalbumin, ribonuclease A, and hemoglobin are used as
model systems to illustrate application of the COVOL routine.
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INTRODUCTION |
The interpretation of thermodynamic equilibrium
data such as those derived from sedimentation equilibrium distributions
in the analytical ultracentrifuge, as well as those from classical (static) light scattering and osmotic pressure measurements for biological and other macromolecules in terms of the molecular weight or
the stoichiometry and strength of interactions between macrolecules, is
often influenced by contributions from the thermodynamic nonideality of
the system. This nonideality, which exists at all finite
concentrations, derives from two sources (see, e.g., Tanford, 1961): an
excluded volume (covolume) contribution emanating from the large size
of macromolecules relative to that of solvent molecules; and, in
aqueous systems, a polyelectrolyte contribution deriving from the net
charge (valence) of many macromolecular species
particularly those of
biological origin (proteins, nucleic acids, polysaccharides, and
glycoconjugates). For some macromolecules at high dilution, such
contributions are sufficiently small to warrant their neglect. Alternatively, measurements at a series of concentrations may be
extrapolated to zero concentration to eliminate effects of nonideality
(Tanford, 1961). Unfortunately, both of those procedures tend to
compromise the analysis of properties that are concentration dependent;
in particular, the study of interactions between macromoleculesan area that underpins the whole of biological science (Schachman, 1989).
From the quantitative expression for the polyelectrolyte contribution
to thermodynamic nonideality for spherical macromolecular solutes
(Wills et al., 1980; Winzor and Wills, 1995), it is evident that the
extent of nonideality stemming from this source may be decreased either
by increasing the ionic strength of the solvent or, in the case of
proteins, by selecting a pH in the vicinity of the isoelectric point.
In contrast, the covolume contribution is independent of solvent
conditions. Covolume formulations are available for certain types of
centrosymmetric rigid structures. Of these, the simplest is the sphere,
but expressions have also been derived for ellipsoids of revolution
(Isihara, 1950; Ogston and Winzor, 1975) and for the triaxial ellipsoid
in which all three semiaxes differ in magnitude (Rallison and Harding,
1985). This triaxial ellipsoid with semiaxial dimensions a 
b c (Fig. 1) clearly
provides the most general example of a rigid centrosymmetric particle.
Although complicated structures, such as an immunoglobulin or
complement system, are not accurately described, rod-shaped (a
b 
c), disc-shaped (a b c),
globular shaped (a b c), and even
tape-shaped (a b c) macromolecules can all be represented adequately by such means. Indeed, because the required covolume is a time-averaged parameter for macromolecules under dominant
Brownian motion, the representation of even an immunoglobulin in terms
of one of the above shapes will almost certainly suffice for
description of thermodynamic nonideality effects on the magnitude of an
equilibrium thermodynamic property. However, for such irregular shaped
macromolecules, a recent development has been to extend, to the case of
covolume calculations, multiple-sphere or bead-modeling approaches for
a structure that, although approximate in terms of its hydrodynamics
and thermodynamics, can give better representations of
structure. For covolume, a Monte Carlo procedure has been
incorporated into the most recent version of the general bead-modeling
algorithm SOLPRO (Garcia de la Torre et al., 1999), by
sampling or "trialing" all possible orientations of two-particle
interactions and checking for overlap of any bead in one molecule with
any in its neighbor. The precision of this method obviously increases
with increase in the number of trials. With a moderate number of trials
taking, for example, 50 min in a Pentium 200 computer, estimates for
the covolume can be returned to a precision of better than ~10%.
However, for more regular structures, the exact covolume relations for triaxial ellipsoids are more useful. Both the bead and the general ellipsoidal deliberations are based on the premise that solutions are
sufficiently dilute to allow the consideration of thermodynamic nonideality solely in terms of two-particle interactions, whereupon effects of thermodynamic nonideality become manifested in the magnitude
of the second virial coefficient.
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FIGURE 1
Schematic representation of a rigid macromolecule as a
triaxial ellipsoid in which all three semiaxes (a,
b, c) can differ in length. Its shape is
characterized by the two axial ratios
(a/b, b/c)
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At present, considerable interest is centered on the use of scaled
particle theory to analyze the thermodynamic activity of concentrated
protein solutions in terms of a single parameterthe effective radius
of the hard particle (Ross and Minton, 1979; Berg, 1990; Guttman et
al., 1995). An obvious attribute of this approach is its ability to
extend the analysis of experimental data beyond the concentration range
for which description of nonideality in terms of a second virial
coefficient ceases to be a valid approximation. However, the lack of
any specific allowance for the charge-charge contribution to
thermodynamic nonideality means that the quantitative description in
such terms only applies to the system under the conditions (pH, ionic
strength) of the experiment subjected to analysis. Conclusions about
nonideality effects in solutions of the same protein at (say) a
different ionic strength are precluded because the change in the
charge-charge contribution necessitates redetermination of the
empirical-scaled particle parameter (effective radius) that is required
to describe the nonideality under the new condition. We therefore
retain the classical statistical-mechanical approach.
The triaxial ellipsoid expressions (Rallison and Harding, 1985) were
devised with the intention of combining hydrodynamic parameters with
measurements of the second virial coefficient to estimate
macromolecular shape in solution (Harding, 1989). However, a greater
potential now seems to be the use of macromolecular structure details
to predict the magnitude of thermodynamic activity coefficients that
are required to make allowance for nonideality effects in the
evaluation of equilibrium constants for macromolecular interactions
(Winzor and Wills, 1995; Wills et al., 1996). COVOL has been
developed with this objective in mind.
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THE SECOND THERMODYNAMIC VIRIAL COEFFICIENT |
The simplest way to represent thermodynamic nonideality of
macromolecular solutions, correct to first order in concentration, is
in terms of the second virial coefficient B (sometimes
designated as A2). It may be envisaged as a
measure of the extent to which a determined value of the apparent molar
mass Mapp, at finite concentration c,
underestimates the true parameter M. For molecular weight
measurement by osmotic pressure, the relationship is (Tanford, 1961)
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(1)
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where the additional subscript (n) signifies that the
number-average molecular weight is measured by osmometry. For
measurements of molecular weight from either absorption or Rayleigh
interference records of sedimentation equilibrium distributions in the
analytical ultracentrifuge, and also from classical (static) light
scattering data, the corresponding expression is
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(2)
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where the w subscript denotes the weight-average nature of the
molecular weight determined by these methods. Use of the qualifying coefficient in the concentration term of Equ. 2 allows retention of the
osmotic virial coefficient B for the description of
nonideality in the various experimental methods of molecular weight measurement.
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EXCLUDED VOLUME CONTRIBUTION, Bex |
The excluded volume (or covolume) of a macromolecule,
u, is the volume of solution (frequently expressed in mL)
from which the centers of two molecules are mutually excluded. For the
simple situation of an impenetrable spherical particle with radius
r, the distance of closest approach is 2r, in
which case u = 
(2r)3 = 8V, where
V is the volume of the particle. To obtain a normalized parameter related solely to shape, Rallison and Harding (1985) introduced the concept of a reduced covolume,
ured, defined as the excluded volume per unit
particle volume. The excluded volume then becomes the product of the
shape parameter, ured (with a minimal value of
8), and the particle volume, which takes into account the degree of
swelling of the macromolecule through solvation (u = Vured). By expressing V in terms of
the specific solvated volume vs, i.e., the
volume of the solvated particle per unit unsolvated mass, the
relationship between excluded volume and reduced excluded volume may be
written as
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(3a)
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where the ratio of molar mass (M) to Avogadro's number
(NA) has been substituted for the molecular
mass. This is equivalent to the approach (Tanford, 1961) in which
vs is regarded as the sum of the unsolvated
partial specific volume 
and a term for particle
solvation. So, vs = + 
/
o, where o is the solvent density and the extent of solute solvation (g solvent per g solute). Because of the greater popularity of this approach, Eq. 3a is
usually written in the form
![u={[<A><AC>v</AC><AC>&cjs1171;</AC></A>+(&dgr;/&rgr;<SUB><UP>o</UP></SUB>)]M/N<SUB><UP>A</UP></SUB>}u<SUB><UP>red</UP></SUB>.](/content/vol76/issue5/fulltext/2432/img006.gif)
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(3b)
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In the fields of colloid and polymer chemistry, the virial
expansion is traditionally defined with c expressed in g/mL,
whereupon the dimensions of the second virial coefficient B
become mL mol g
2. In these terms, the excluded (or
covolume) contribution to the second virial coefficient,
Bex, is given by the relationship
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(4)
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POLYELECTROLYTE CONTRIBUTION, BZ |
In studies of charged macromolecules such as proteins and
polyelectrolytes in aqueous solution, the effective distance of closest
approach is greater than that based on geometrical considerations because of the repulsive force opposing the approach of two particles bearing net charge (valence) Z. This additional contribution
to the second virial coefficient, BZ, has only
been evaluated explicitly for impenetrable spheres. For such systems,
the expression for the second virial coefficient, B, is
given by (Wills et al., 1980; Winzor and Wills, 1995)
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(5)
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where the factor of 1000 is introduced to accommodate the
conventional definition of ionic strength I (mol/L),
rs is the product of the inverse screening
length (Debye and Hückel, 1923) and the solvated radius,
rs, of the particle. The Stokes radius provides
an acceptable estimate of rs (cm), irrespective
of macromolecular shape, and the magnitude of (cm1)
may be evaluated from the expression = 3.27 × 107
at 20°C.
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CALCULATION OF Bex FROM THE TRIAXIAL
DIMENSIONS OF A RIGID IMPENETRABLE MACROMOLECULE |
As noted above, the simplest situation for which
ured is known is a sphere, where
ured = 8. This is the minimal value for ured of a triaxial ellipsoid, for which the
general expression is
![u<SUB><UP>red</UP></SUB>=2+[3/(2&pgr;abc)]SR,](/content/vol76/issue5/fulltext/2432/img010.gif)
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(6)
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where S and R are the double-integral
functions defined in Eqs. 3 and 4 of Rallison and Harding (1985).
Although it is possible to solve analytically one of the double
integrals in each of the expressions for S and R,
the results are sufficiently complicated that it is easier to perform
all of the integrals by numerical integration.
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SECOND VIRIAL COEFFICIENTS DEFINED ON A MOLAR BASIS |
From the viewpoint of allowing for effects of thermodynamic
nonideality in the characterization of macromolecular equilibria, there
is merit in defining the second virial coefficient on a molar rather
than a molecular basis. In terms of molar covolume, U = uNA, Eq. 3 becomes
![U=(v<SUB><UP>s</UP></SUB>M)u<SUB><UP>red</UP></SUB>=[(<A><AC>v</AC><AC>&cjs1171;</AC></A>+&dgr;/&rgr;<SUB><UP>o</UP></SUB>)]Mu<SUB><UP>red</UP></SUB>.](/content/vol76/issue5/fulltext/2432/img011.gif)
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(7)
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U, in turn, is related to the second virial coefficient
defined in molecular terms, Bex, by the relation
(Tanford, 1961; Ogston and Winzor, 1975; Jeffrey et al., 1977)
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(8)
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COVOL |
COVOL, an interactive FORTRAN 77 algorithm written for
PC, evaluates Bex by enumerating S
and R, and hence ured, from
user-specified values of the three semiaxes a, b,
and c (or, alternatively, a/b and
b/c because of the sole dependence of
ured upon shape), through Eq. 6. The double integrals
S and R of Eq. 6 are evaluated using the
Numerical Algorithms Group (1992) numerical integration routine D01DAF.
The next stage is the evaluation of the molar covolume U
from ured and user-specified values for the
molecular weight (M) and either the solvated specific volume
(vs) (Eq. 3a) or the unsolvated partial specific
volume (v), the solvation (), and the solvent density
(o), through Eq. 3b. The routine prints out the molar excluded volume, U, the molecular excluded volume,
u (= U/NA), and
Bex (Eq. 4). At that stage the program asks
whether there is an additional contribution to B from
polyelectrolyte behavior. If yes, the user enters the ionic strength
(mol/L) and net charge (valence) of the macroion, Z. After
evaluation of B according to Eq. 5, the routine concludes by
printing out the charge-charge contribution
(Bz) and the magnitude of the second virial
coefficient, B = Bex + Bz.
A flow chart for the program is given in Fig.
2. The FORTRAN 77 compiler, Salford
FTN77/486 system (Salford, 1991) and the Numerical Algorithms Group
(1991) numerical integration routine D01DAF are built into the program;
no separate FORTRAN or NAG compilers are required. COVOL is
available in either precompiled or source-code form from
Steve.Harding{at}nottingham.ac.uk or from the web page
http://www.nottingham.ac.uk/ncmh/.
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FIGURE 2
Flow chart of the COVOL routine for
calculating second virial coefficients from the triaxial ellipsoid
shape or dimensions and net charge (valence) of a rigid
macromolecule.
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USER INPUT OF AXIAL RATIOS BASED ON CRYSTALLOGRAPHIC COORDINATES
OF A MACROMOLECULE
An objective method for defining the triaxial shape of a protein
molecule from its atomic structural coordinates has been provided by
Taylor et al. (1983). This method, which is insensitive to small
deviations from ideal ellipsoidal form, is based on the inertial,
momental, or Cauchy ellipsoid dilated so that it forms a close
approximation to the protein surface. The procedure is in the form of a
FORTRAN 77 algorithm called ELLIPSE. A recent version of the
algorithm, implemented by Hubbard (1994), was used to calculate the
ratios of the principal axes of the equimomental ellipsoid for the
three-dimensional coordinates of a protein. These ratios can be used in
conjunction with a second algorithm, SURFNET, (Laskowski,
1995) to generate a three-dimensional surface representation of the
ellipsoid (Fig. 3). SURFNET
can be downloaded from page
http://www.biochem.ucl.ac.uk/~roman/surfnet/surfnet.html.
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FIGURE 3
Inertial ellipsoid fitted to the crystal structure for
ovalbumin (Stein et al. 1991). The axial ratios for the ellipsoid were
calulated using ELLIPSE and the surface diagram was
generated using SURFNET (Laskowski, 1995).
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APPLICATION OF COVOL |
The use of COVOL for prediction of the magnitude of
second virial coefficients is explored first by consideration of
ovalbumin, a protein whose high resolution crystal structure was
recently published (Stein et al., 1991). Fitting the crystal
coordinates to the inertial ellipsoid using ELLIPSE yielded
axial ratios (a/b, b/c) of
(1.87, 1.08). The resulting fit is shown in Fig. 3. Input of these
respective values for a/b, and
b/c into COVOL yields a reduced
covolume, ured, of 8.996. Conversion of this
reduced covolume to a covolume, Bex, depends
upon the magnitude assigned to the solvation parameter () for this
protein with a partial specific volume () of 0.748 mL/g (Dayhoff et al., 1952) and a molecular weight of 45,000 (Jeffrey
et al., 1977). The effect of the extent of solvation upon the magnitude
of Bex calculated by Eqs. 3b and 4 is summarized
by the solid line in Fig. 4, where the
intersecting horizontal dashed lines denote the estimates of
B deduced experimentally from sedimentation equilibrium
(Jeffrey et al., 1977) and size exclusion chromatography studies
(Shearwin and Winzor, 1990) of isoelectric ovalbumin (upper
and lower lines, respectively). It is noted that the
consequent estimates of 0.49 (± 0.05) and 0.39 (± 0.18) for the
extent of ovalbumin solvation are at the upper end of, or greater
than, the usually accepted range (0.3-0.4) for globular proteins
(Oncley, 1941; Tanford, 1961; Zhou, 1995). Experimental support for a
higher value is provided by concordance of estimates (2.92 nm) for the
Stokes radius and the effective radius deduced from the molar covolume, U = 
NAr3. A
similar conclusion about the extent of solvation stems from size-exclusion chromatography studies (Shearwin and Winzor, 1990) in
phosphate-chloride buffer, pH 7.4, I 0.156, conditions under which a
net charge (Z) of 
16 results in a polyelectrolyte
contribution to B (Eq. 5). The upper dependence
(dash-dot line of Fig. 4) summarizes the calculated
variation of B with , whereas the intersecting horizontal
line denotes the experimental value of B obtained by exclusion chromatography. On this basis, ovalbumin is hydrated to the
extent of 0.42 (± 0.09).
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FIGURE 4
Effect of the extent of solvation () upon the
magnitude of the second virial coefficient (B)
calculated by COVOL on the basis of ratios of triaxial
ellipsoid semiaxes of 1.87 (a/b) and 1.08 (b/c) for isoelectric ovalbumin ( ),
and for the same protein under conditions (pH 7.4, I
0.156) where it bears a net charge (valence) of 16
(- · - · -). · · · · · , corresponding dependence for
isoelectric ovalbumin modeled as a sphere. Horizontal
lines denote experimental estimates of B from
sedimentation equilibrium and exclusion chromatography studies of
ovalbumin.
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Analysis of the dependences of the second virial coefficient upon
extent of solvation for isoelectric ribonuclease A and hemoglobin are
presented in Fig. 5. For ribonuclease
A, the atom coordinates stored in the x-ray crystallographic database
(Borkakoti et al., 1984) signify semiaxial ratios of 1.53 and 1.23, which yield a reduced covolume, ured, of 8.692 for this enzyme, with M = 13,700 and
= 0.703 mL/g. The atom coordinates of human
deoxyhemoglobin (Fermi et al., 1984) give rise to semiaxial ratios of
1.27 and 1.07, and hence to a reduced covolume of 8.170 for this
protein, with M = 64,500 and = 0.746 mL/g. On the basis of the horizontal broken lines, which
correspond to experimentally determined values of B for
ribonuclease A (Shearwin and Winzor, 1990) and hemoglobin (Baghurst et
al., 1974), the respective extents of hydration are 0.25 and 0.43 g/g.
In the latter regard, we note that values of 0.35-0.54 g/g have been
reported by Guttman et al. (1995) by analysis of the thermodynamic
nonideality of concentrated hemoglobin solutions in terms of the Berg
(1990) adaptation of scaled particle theory (Ross and Minton, 1979).
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FIGURE 5
Effect of the extent of solvation () upon the
magnitude of the second virial coefficient (B) on the
basis of the ratios of triaxial ellipsoid semiaxes for isoelectric
ribonuclease and hemoglobin. Horizontal broken lines
denote experimental estimates of B obtained from
exclusion chromatography studies of the enzyme and from osmotic
pressure measurements for hemoglobin.
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An obvious difficulty with calculation of the second virial coefficient
by this means is the pronounced dependence of
Bex upon the magnitude assigned to , a
parameter for which the value is often very subjective because it has
rarely been determined. Indeed, this sensitivity of
Bex to the value of relegates to secondary
importance the relative magnitudes of the triaxial ellipsoid semiaxesa factor evident from the dotted dependence in Fig. 4, which
refers to isoelectric ovalbumin modeled as a sphere
(ured = 8.000). Values of 0.47-0.57 for the
extent of ovalbumin hydration are obtained from this model and the
exclusion chromatographic (Shearwin and Winzor, 1990) and sedimentation
equilibrium (Jeffrey et al., 1977) estimates of B.
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ALLOWANCE FOR NONIDEALITY IN SOLUTE SELF-ASSOCIATION |
Thus far, the investigation has been dominated by considerations
of the procedure for predicting magnitudes of the second virial
coefficienton the grounds that values need to be assigned to these
parameters to account for the effects of thermodynamic nonideality in
the quantitative characterization of macromolecular interactions. We
illustrate such use of second virial coefficients (defined initially on
the molar basis) by considering the situation for a reversibly
dimerizing solute.
For a monomer 
dimer system the molar thermodynamic activity of
monomer, z1, which differs from its molal
counterpart (a1) because of the different
constraints entailed in the definitions of the chemical potential of
solute (Winzor and Wills, 1995), is related to the base-molar solute
concentration C (weight-concentration divided by monomer
molecular weight M1) by the expression
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(9)
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where K2 is the dimerization constant
(L/mol) and B11 is the molar second virial
coefficient reflecting monomer-monomer excluded volume interactions
(Wills et al., 1996, 1997). Interpretation of the quadratic coefficient
of the dependence of total solute concentration upon monomer activity
in terms of the dimerization constant K2 is thus
predicated upon specification of a value for B11. Statistical-mechanical considerations
establish the relationship,
![B<SUB>11</SUB>=(U<SUB>11</SUB>/2)+(Z<SUP>2</SUP>/4I)[(1+2&kgr;r<SUB>1</SUB>)/(1+&kgr;r<SUB>1</SUB>)<SUP>2</SUP>],](/content/vol76/issue5/fulltext/2432/img015.gif)
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(10)
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where r1 is the effective monomer radius
and U11 is the molar monomer-monomer covolume.
On noting that K2 = X2M1/2 is the relationship between dimerization constants defined on molar
(K2) and weight (X2)
bases, Eq. 9 may also be written in terms of B for monomer
(Eq. 6) as
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(11)
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for the treatment of data analyzed in terms of total weight
concentration c.
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CONCLUDING REMARKS |
This investigation has demonstrated the use of COVOL to
calculate second virial coefficients for macromolecules that can be modeled as impenetrable triaxial ellipsoids; but has also identified the problem that realization of its full potential must await more
definitive means of assessing the magnitude of , the extent of
macromolecule solvation. In that regard, the extent of solvation has
usually been considered to be in the range 0.3 to 0.4 for globular
proteins (Oncley, 1941; Tanford, 1961), whereas experimental measurements of B for ovalbumin signify a higher value (0.4 to 0.6) for . The value of 0.25 for ribonuclease is marginally below the considered range. Measurements of B for a range of
proteins with known axial dimensions are clearly required to shed
further light on the likely magnitude of and, hence, on its
prediction on the geometrical basis of an assigned thickness to the
solvation layer extending over the surface of the protein molecules
(see, e.g., Jacobsen et al., 1996). It is, therefore, hoped that this investigation may stimulate renewed interest in accurate measurement of
osmotic virial coefficientsparameters for which the major use in the
past has merely been to guide the elimination of nonideality by
extrapolation of data to infinite dilution.
This investigation has been funded by United Kingdom Biotechnology
and Biomolecular Sciences Research Council and the Engineering and
Physical Sciences Research Council. Financial support from the
Australian Research Council is also gratefully acknowledged.
Address reprint requests to Professor Stephen E. Harding, National
Centre for Macromolecular Hydrodynamics, University of Nottingham,
Sutton Bonington, Leicestershire LE12 5RD, U.K. Tel.: +44-1159-516148;
Fax: +44-1159-516142; Email: Steve.Harding{at}nottingham.ac.uk.
TOP
ABSTRACT
INTRODUCTION
