It has been shown that the nonideal behavior of several proteins in highly concentrated and/or crowded solutions may be well described by simple structural models in which globular proteins and other macromolecules are represented by equivalent convex hard particles (see for example ^{10–12}). In what follows we shall present an extension of the equivalent hard particle model to treat the case of significant attractive nonspecific chemical interactions between test molecules and background molecules that can, depending upon their magnitude, attenuate or even override excluded volume effects upon reaction equilibria.

To facilitate numerical computation we shall make the following simplifying assumptions. In reversible dimerization scheme [

1] we shall represent the monomer A

_{1} by a sphere of radius

*r*_{1}, and the dimer A

_{2} by a spherocylinder of cylindrical radius

*r*_{2} =

*r*_{1} and a cylinder length equal to

*L* times the cylinder diameter. In order for the protein volume to be conserved upon dimerization,

*L* = 2/3. A comparison between this equivalent hard particle model and a more detailed atomic model for the acid dimerization of α-chymotrypsin in shown in . It may be seen that for the purpose of calculating volume excluded sterically to molecules of comparable size, the representation of molecular shape by simple convex particles is a reasonably accurate approximation. In addition, we represent the background species as another spherical particle of radius

*r*_{B} =

*r*_{1}. In order to calculate concentrations in molar units, we shall assume that monomer and crowder Bhave molar masses equal to that of α-chymotrypsin, 25,000. The specific excluded volume of all species is taken to be

*v*_{exc} = 1 cm

^{3}/g

^{3}. It follows that

*r*_{B} =

*r*_{1} = 21.5 Å, and the surface areas of spherical monomer (s

_{1}) and spherocylindrical dimer (s

_{2}) are respectively equal to 5755 and 9591 Å

^{2}.

Using this structural model, the excluded volume contribution to the free energy of transfer of monomer and dimer from ideal to crowded solution may be estimated using the scaled particle theory of hard particle mixtures

^{13–15}. According to this theory, the negentropic work associated with the insertion of a single hard spherocylinder with cylindrical radius

*r*_{C} and cylindrical axial ratio

*L* into a suspension of hard spheres of radius

*r*_{B} that occupy a fraction

of total solution volume is given by

and

*R = r*_{C}/

*r*_{B}. Note that

equation [10] also applies to insertion of a sphere of radius r

_{C} in this fluid when

*L* = 0.

The contribution of attractive interactions between crowder and test molecule to the free energy of transfer may be estimated by treating such interactions as formally equivalent to weak, unsaturable binding

^{16}. Let i-mer contain n

_{site,i} sites for the binding of background molecule B, each of which can independently “bind”, or attract, B according to the following scheme:

As an illustrative example of weak attractive intermolecular interactions describable as binding, we consider those between urea and an unfolded protein. At a fixed temperature, the dependence of the heat of binding of urea to unfolded ribonuclease A ^{17} may be well described by a simple independent binding site isotherm

where

*K*_{A} denotes the equilibrium association constant, and

*c*_{U} the molar concentration of urea. The results obtained by Makhatadze and Privalov

^{17} at multiple temperatures could be accurately described by

equation [11] with a temperature-dependent equilibrium association constant given by

where

*R* denotes the molar gas constant,

*T* the absolute temperature, Δ

*H*^{o} = −2000

*R* and Δ

*S*^{o} = −9.77

*R*. The calculated value of

*K*_{A} is plotted as a function of temperature in .

In the example presented here, we shall assume that weak nonspecific attractive interactions between background species B and *i*-mer may be approximated by that between urea and unfolded ribonuclease, with the same temperature-dependent site binding constant: *K*_{B}_{,}_{i}(*T*)=*K*_{A}(*T*). The free energy change associated with the binding of a nonideal ligand L to 1 mole of an ideal substrate S containing *n* identical and independent binding sites is given by ^{18}

where

*γ*_{L} and

*c*_{L} respectively denote the activity coefficient and molar concentration of ligand L.

Equation [12] may be generalized to the case of a nonideal substrate

^{9}:

The free energy of binding B to nonideal i-mer is then given approximately by

where

. Since we have chosen B to have the same size and spherical shape as monomer,

*γ*_{exvol,B} =

*γ*_{exvol,}_{1} and

*γ*_{exvol,B}_{:1} =

*γ*_{exvol,}_{2}. We shall in addition approximate the shape of the complex of dimer and B by a spherocylinder with r = r

_{1} and a volume equal to three times that of monomer, such that L = 4/3. (Test calculations indicated that the final results are insensitive to the choice of shape). For the purpose of calculating the value of

as a function of T and

*ϕ*, the activity coefficients of monomer, dimer, monomer:B and dimer:B are estimated using

equation [10]. Since the number of virtual binding sites for B on i-mer is in general unknown, we make the reasonable approximation that

*n*_{sites,i} is proportional to the surface area of i-mer, so

equation [14] simplifies to

where

*α* denotes a temperature-independent constant of proportionality that is equal for monomer and homo-dimer.

Given

equations [10],

[15], and the simplified structural and thermodynamic models described above, we may use

equation [9] to estimate the values of Γ

_{exvol}, Γ

_{chem}, and Γ for dimer formation as a function of

*ϕ*_{B} and temperature. In , the calculated dependence of ln Γ upon

*ϕ*_{B} is plotted for a series of temperatures. For purposes of illustration, the value of

*α,* which scales the magnitude of ln

*γ*_{chem,i} was selected so that the temperature dependence of the crowding effect qualitatively resembles that reported for the hetero-association of superoxide dismutase and catalase

^{6}.