Calculation of Second Virial Coefficients of Atomistic Proteins Using Fast Fourier Fransform

Abstract

The second virial coefficient, B₂, measures a protein solution’s deviation from ideal behavior. It is widely used to predict or explain solubility, crystallization condition, aggregation propensity, and critical temperature for liquid-liquid phase separation. B₂ is determined by the interaction energy between two protein molecules, specifically, by the integration of the Mayer f-function in the relative configurational space (translation and rotation) of the two molecules. Simple theoretical models, such as one attributed to Derjaguin, Landau, Verwey, and Overbeek (DLVO), can fit the dependence of B₂ on salt concentrations. However, model parameters derived often are physically unrealistic and hardly transferable from protein to protein. Previous B₂ calculations incorporating atomistic details were done with limited sampling in the configurational space, due to enormous computational cost. Our FMAP method, based on fast Fourier transform, can considerably accelerate such calculations, and here we adapt it to calculate B₂ values for proteins represented at the atomic level in implicit solvent. After tuning of a single parameter in the energy function, FMAPB2 predicts well the B₂ values for lysozyme and other proteins over wide ranges of solvent conditions (salt concentration, pH, and temperature). The method is available as a web server at http://pipe.rcc.fsu.edu/fmapb2.

Graphical Abstract

INTRODUCTION

The second virial coefficient, B₂, measures a protein solution’s deviation from ideal behavior. It is widely used to predict or explain various thermodynamic properties, including protein solubility,¹ crystallization condition,² aggregation propensity,³ and critical temperature for liquid-liquid phase separation.⁴ While its value is obtained through bulk experiments, microscopically, according to statistical thermodynamics,⁵ B₂ is determined by the interaction energy between two protein molecules. Over the years, considerable efforts have been expended to obtain B₂ values experimentally and by theoretical modeling or computation, yet significant barriers remain on both fronts. Conceptually, there is much similarity between B₂ as a measure for weak, nonspecific binding and binding constant as the measure for strong, specific binding. Given the many nonspecific interactions that proteins inescapably encounter in cellular environments, there is renewed interest in better characterizing nonspecific interactions.⁶ This study aimed to develop a computational method for B₂ that achieves both accuracy in terms of comparison against cross-validated experimental data and physical insight at the atomic level.

A variety of experimental techniques have been developed to obtain B₂ values, each with its own limitations. The most often used technique nowadays is static light scattering (SLS), where one plots the reciprocal of the Raleigh scattering intensity at 90° angle against protein concentration, with the slope yielding B₂. This technique consumes a great amount of time and protein samples, and is prone to artifacts including protein aggregation and other impurities. These limitations motivated continued improvements, including flow SLS, which measures the scattering intensity and protein concentration simultaneously in flow-mode using size exclusion chromatography,⁷ and concentration gradient SLS,⁸ which achieves similar goals using a programmable dual-syringe infusion pump for dispensing solutions with varying concentrations. Another technique is small angle X-ray or neutron scattering (SAXS/SANS).^9–10 The scattering intensity can be decomposed to obtain the structure factor, which contains information about the weak interactions between solute molecules, over the entire range of the magnitude, q, of the scattering vector. This technique requires specialized resources and again is time-consuming. In addition, extracting B₂ from SAXS/SANS data is an elaborate process that can generate considerable uncertainty, usually involving extrapolation to q = 0 of the structure factors at several protein concentrations and a subsequent linear fit of the concentration dependence. Aside from the inherent limitations of the experimental techniques, an even more serious problem is that the same technique when used by different researchers and the same researchers using different techniques can produce large discrepancies in B₂ values.

Theoretically, B₂ can be calculated by integrating the Mayer f-function in the 6-dimensional relative configurational space (translation and rotation) of the two molecules. When the interaction energy is modeled as spherically symmetric, dependence on 5 of the 6 degrees of freedom disappears and B₂ becomes a 1-dimensional integral, over the interprotein distance. Such spherical models, in particular one attributed to Derjaguin, Landau, Verwey, and Overbeek (DLVO),^11–12 have long been used to help interpret experimental data, e.g., for fitting salt dependence of B₂. However, the parameters derived from such fits often lack physical significance¹³ and are hardly transferable from one solvent condition to another or from one protein to another.

Various energy models with higher levels of realism have been introduced for B₂ calculations, necessitating sampling in the 6-dimensional space.^14–23 The representation of the protein ranged from sphere plus embedded charges¹⁶ to coarse-grained^{17, 20–23} to all atom.^{14–15, 18–19} While most used implicit solvent, one used explicit solvent²² and another used explicit ions;¹⁷ one other study considered limited protein flexibility.²³ B₂ was calculated in one of two ways. In the first, the configurational space was sampled to implement the integration of the Mayer f-function.^{14–16, 20–21} The number of configurations ranged from as few as six²⁰ to thousands¹⁴ to 10⁶ to 10⁹,^{15–16, 21} generated by uniform¹⁵ or Monte Carlo^{14, 16, 21} sampling. In the second way, a dilute protein solution was simulated by Brownian dynamics,^18–19 molecular dynamics,²² or Monte Carlo^{17, 23} to obtain the radial distribution function, which after subtracting by 1 equals the Mayer f-function. Several findings from these studies are worth noting. The pioneering study of Neal et al.,¹⁴ though with too few configurations to have convergent B₂, generated insightful observations, such as the outsized contributions of a small number of highly favorable configurations. Elcock and McCammon¹⁵ commented that the integration of the Mayer f-function “might be dramatically accelerated using Fast Fourier Transform (FFT) methods.” McGuffee and Elcock¹⁸ presented success in modeling the effects of single mutations on B₂. Stark et al.²² ran molecular dynamics simulations in the popular Martini coarse-grained force field²⁴ to calculate B₂, and found that 70% weakening of the strength of van der Waals interactions between proteins was necessary to reach agreement with experiment. The more realistic and/or more rigorous of these studies are so computationally demanding as to prohibit their applications on a large scale.

Here we present a method that breaks this last barrier, based on using FFT. In previous studies, we have already introduced a method called FMAP, or FFT-based modeling of atomistic protein-protein interactions, and demonstrated its superb speed in calculating the interaction energy of a probe molecule when placed at uniformly distributed positions inside a box of target molecules, enabling fast determination of the chemical potential.^25–28 Here we adapt FMAP to calculate B₂ for proteins represented at the atomic level in implicit solvent, and test FMAPB2 on five proteins for which extensive experimental data are available: lysozyme,^{1, 7, 9–10, 23, 29–30} chymotrypsinogen A,^{7, 9, 31} bovine pancreas trypsin inhibitor (BPTI),³² γD crystallin;³³ and bovine serum albumin (BSA).³⁴ After limited tuning of a single parameter in the interaction energy function, the calculated B₂ values, each involving more than 10¹¹ configurations, agree well with experimental data over wide ranges of solvent conditions (salt concentration, pH, and temperature). We make FMAPB2 available as a web server at http://pipe.rcc.fsu.edu/fmapb2.

THEORETICAL BACKGROUND

Virial Coefficients.

The equation of state of any gas approaches the ideal gas law

$$\beta P=\rho$$ [1]

at very low number densities (i.e., ρ → 0). Here P denotes pressure, and β = 1/(k_BT) with k_B and T denoting the Boltzmann constant and absolute temperature, respectively. A similar relation is valid for the osmotic pressure, Π, of a dilute solution,

$$\beta \Pi =\rho$$ [2]

where ρ now represents the number density of the solute. Indeed, according to the McMillan-Mayer solution theory,³⁵ a solution is exactly like a gas, if the intermolecular interaction energy is replaced by a potential of mean force (PMF) for a pair of solute molecules, with the solvent degrees of freedom averaged out. This correspondence between gas and solution is the basis for implicit modeling of the solvent. Below we describe how deviations from ideal behaviors are related to intermolecular interactions. The presentation is in the language for gases, but it is also applicable to solutions according to the just noted correspondence.

At higher densities, the pressure can be expanded in a series,⁵

$$\beta P=\rho +\sum _{n\ge 2}{B}_n{\rho }^n$$ [3]

where B₂, B₃, … are known as the second, third, etc. virial coefficients. For a monatomic gas, B₂ is determined by

$$B_2=-\frac{1}{2}\int dR\left[e^{-\beta U(R)}-1\right]$$ [4]

where R is the relative position vector between two gas molecules, and U(R) is the intermolecular interaction energy. Higher order virial coefficients are given by interaction energies among three, four, etc. gas molecules. The integrand, e^−βU(R) − 1, in eq [4] is known as a Mayer f-function. If U(R) is centrosymmetric, then the integration over the direction of R, as specified by polar angle θ and azimuthal angle ϕ, can be trivially done,

$$\begin{gathered} B_2=-\frac{1}{2}\int dRd\theta d\varphi R^2\text{sin}\theta \left[e^{-\beta U(R)}-1\right] \\ =-\frac{1}{2}\int dR4\pi R^2\left[e^{-\beta U(R)}-1\right] \end{gathered}$$ [5]

Even when U(R) is not centrosymmetric, we can express B₂ in the form of eq [5], by first integrating over θ and ϕ,

$$e^{-\beta W(R)}\equiv \frac{1}{4\pi }\int d\theta d\varphi \text{sin}\theta e^{-\beta U(R)}\equiv <e^{-\beta U(R)}{>}_{\theta ,\varphi }$$ [6]

leading to

$$B_2=-\frac{1}{2}\int dR4\pi R^2\left[e^{-\beta W(R)}-1\right]$$ [7]

Eq [6] serves to define a PMF W(R) along the distance R between two gas molecules.

For polyatomic gas molecules that are rigid, the interaction energy depends on both the relative position and the relative orientation. We denote the latter by Ω, which can be specified by 3 rotation angles such as the Euler angles. Then the second virial coefficient is given by

$$B_2=-\frac{1}{2}\frac{1}{8{\pi }^2}\int dRd\Omega \left[e^{-\beta U(R,\Omega )}-1\right]$$ [8]

where an additional prefactor has been added to ensure proper reduction to eq [4] when the dependence of U(R, Ω) on Ω disappears. It is assumed that any Jacobian associated with the integration over Ω is absorbed into dΩ and that ∫ dΩ = 8π². We again can cast eq [8] into the form of eq [7], by integrating over all the degrees of freedom except R. The PMF W(R) is now given by

$$e^{-\beta W(R)}\equiv \frac{1}{4\pi }\frac{1}{8{\pi }^2}\int d\theta d\varphi d\Omega \text{sin}\theta e^{-\beta U(R,\Omega )}\equiv <e^{-\beta U(R,\Omega )}{>}_{\theta ,\varphi ,\Omega }$$ [9]

When the polyatomic molecules are flexible, i.e., they possess internal degrees of freedom, collectively denoted by X, we can further expand the definition of the PMF to

$$e^{-\beta W(R)}\equiv \frac{1}{4\pi }\frac{1}{8{\pi }^2}\frac{1}{{\mathcal{V}}_{\mathbf{\text{X}}}}\int d\theta d\varphi d\Omega dX\text{sin}\theta e^{-\beta U(R,\Omega ,X)}\equiv <e^{-\beta U(R,\Omega ,X)}{>}_{\theta ,\varphi ,\Omega ,X}$$ [10]

where V_X = ∫ dX.

As can be easily seen from eq [8], repulsive interactions [i.e., U(R, Ω) > 0] make positive contributions to B₂, whereas attractive interactions [i.e., U(R, Ω) < 0] make negative contributions.

Simple Models of Intermolecular Interaction Energy.

Proteins have long been modeled as spherical. Often the protein spheres are assumed to be impenetrable, meaning that the interaction energy is

$$W_{\text{st}}(R)=\infty \text{ if }R<d$$ [11]

where the subscript “st” indicates steric repulsion and d is the diameter of the protein spheres. If there is no other interactions, then according to eq [5] the second virial coefficient for proteins with only steric repulsion is

$$B_2^0=\frac{1}{2}\underset{0}{\overset{d}{\int }}dR4\pi R^2\equiv \frac{1}{2}{V}_{\text{co}}$$ [12]

Eq [12] defines the co-volume V_co, which is the volume inaccessible to the center of a second protein molecule by the presence of a first protein molecule, and is 8 times the volume of a single protein sphere.

The DLVO theory^11–12 includes additional contributions to W(R) from van der Waals attraction and electrostatic interactions, respectively given by

$$W_{\text{vdW}}(R)=-\frac{A_{\text{H}}}{12}\left(\frac{d^2}{R^2}+\frac{d^2}{R^2-d^2}+2\text{ln}\frac{R^2-d^2}{R^2}\right)$$ [13]

and

$$W_{\text{elec }}(R)=332{\left(\frac{Q}{1+\kappa d/2}\right)}^2\frac{e^{-\kappa (R-d)}}{\epsilon R}$$ [14]

In the above equations, A_H is the Hamaker constant representing the strength of van der Waals attraction, κ = 2.24(βI/ε)^1/2 is the inverse of the Debye screening length, ε is the dielectric constant of water, Q is the net charge of the protein, and I is the ionic strength of the solution. Throughout this paper, lengths will be in units of Å, charges in units of electronic charge, ionic strengths in units of M, and energies in units of kcal/mol. The three terms, given by eqs [11], [13], and [14] are assumed to be additive, leading to

$$W(R)=W_{\text{st}}(R)+W_{\text{vdW}}(R)+W_{\text{elec}}(R)$$ [15]

The steric and electrostatic terms make B₂ positive whereas the van der Waals term make B₂ negative. Note that eq [13] becomes singular at R = d. The usual fix is to increase the range of steric repulsion from d to d + δ, while maintaining the value of d in eqs [13] and [14].

Atomistic Model of Protein-Protein Interaction Energy.

As in our previous work,^25–26 we treat protein molecules as rigid and the solvent implicitly. Importantly, we represent the protein molecules at the atomic level, and calculate the interaction energy U(R, Ω) by accumulating contributions from individual pairs of atoms between two protein molecules, each depending on the interatomic distance r_ij. Again, there are three types of energy terms:

$$U(R,\Omega )=U_{\text{st}}(R,\Omega )+U_{\text{n}-\text{a}}(R,\Omega )+U_{\text{elec}}(R,\Omega )$$ [16]

The steric term is

$$U_{\text{st}}(R,\Omega )=\infty \text{ if any }{r}_{ij}<\left({\sigma }_{ii}+{\sigma }_{jj}\right)/2$$ [17]

where σ_ii/2 denotes the hard-core radius of atom i. The nonpolar attraction term, including van der Waals and hydrophobic contributions, has the form of a Lennard-Jones potential,

$$U_{\text{n}-\text{a}}(R,\Omega )=v_{\text{s}}\sum _{ij}4{ϵ}_{ij}\left[{\left(\frac{{\sigma }_{ij}}{r_{ij}}\right)}^{12}-{\left(\frac{{\sigma }_{ij}}{r_{ij}}\right)}^6\right]$$ [18]

where ϵ_ij denotes the magnitude of the attraction of the i − j atom pair, σ_ij is the distance at which the nonpolar interaction energy is zero, and v_s is a scaling constant, which is the only parameter that we tune here (see Results and Discussion for details). The electrostatic term has the form of a Debye-Hückel potential,

$$U_{\text{elec }}(R,\Omega )=332\sum _{ij}\frac{q_i{q}_j}{\epsilon r_{ij}}{e}^{-\kappa r_{ij}}$$ [19]

where q_i is the partial charge on atom i.

To calculate the second virial coefficient, one has to average the Mayer f-function of over R and Ω. If the evaluation of U(R, Ω) for a pair of atomistic proteins at a single relative configuration is already expensive, the averaging in the 6-dimensional configurational space (see eq [8]) can be computational prohibitive. Below we explain how FMAP^25–26 makes this calculation fast.

Basic Idea Behind FMAP.

The speedup by FMAP arises from expressing each unique term of the interaction energy as a correlation function in R space. Let us fix the first protein molecule at the origin, and position the second molecule at R. For now we consider a single relative orientation between the molecules and drop the explicit reference to Ω for notational simplicity. To illustrate how we can turn an interaction term into a correlation function, we note that the electrostatic interaction energy can be expressed as the integral of the product between the electrostatic potential of molecule 1 and the charge density of molecule 2:

$$U_{\text{elec }}(R)=\int ds{\psi }_1(s){\rho }_2(s|R)$$ [20]

where

$${\psi }_1(s)=\sum _i\frac{q_i}{\epsilon |s-r_{i|}}{e}^{-\kappa |s-r_{i|}}$$ [21]

$${\rho }_2(s|R)=\sum _j{q}_j\delta (s-r_{j|})$$ [22]

with δ(s) denoting a delta function. We can recognize the integral in eq [20] as a correlation function in R space, once we allow the position R of molecule 2 to move and realize that the charge density ρ₂(s|R) only depends on the position vector s − R relative to molecule 2.

The same reasoning applies to exp[−βU_st(R)] and to each of the two terms of U_n−a(R) in eq [18].^25–26 In the latter case, we must use the geometric mean, α_ij = (α_iiα_jj)^1/2, as the combination rule for both ϵ_ij and σ_ij. The fact that (σ_ii + σ_jj)/2 ≥ (σ_iiσ_jj)^1/2 ensures that, for any relative configuration between two protein molecules that is free of interatomic steric clashes, the nonpolar contribution from any atom pair cannot be positive.

The calculation of U_elec(R) by FFT involves the following steps. (1) Introduce a cubic box, centered on either protein molecule; discretize the box into a grid, and evaluate the values of the electrostatic potential ψ₁(s) and the charge density ρ₂(s|R) on all the grid points. (2) Perform FFT on both of these quantities, and multiply their Fourier transforms. (3) An inverse FFT of the product yields U_elec on all the grid points at once. By a similar procedure, U_st and the two terms of U_n−a can be obtained, yielding U(R, Ω) over the entire R space and allowing for the averaging over R for a given Ω. Lastly the averaging over Ω is done by repeating the above process at many choices of Ω.

COMPUTATIONAL DETAILS

Preparation of Protein Structures and Charges.

The second virial coefficients of five proteins at different solvent conditions were calculated (Fig. 1). Atomic coordinates of the proteins were obtained from crystal structures in the Protein Data Bank (PDB), including entries 1BPI for BPTI, 1AKI for hen egg white lysozyme, 1HK0 for human γD crystallin, 2CGA for bovine chymotrypsinogen A, and 3V03 for BSA. To investigate possible effects of protein structures on B₂ values, we also did calculations for lysozyme using six other entries: 1LKS, 1V7T, 2D4I, 2D4J, 2FBB, and 2Z18.

Figure 1.

Structures of the five proteins studied. Molecular weight and net charge at the indicated pH are shown.

The PDB files were converted to PQR files, which contain coordinates of atoms along with their charges and radii, by the PDB2PQR program (version 2.1.1).³⁶ We chose charges in the PARSE set,³⁷ and, by default, used pK_as predicted by PROPKA (version 3.0)³⁸ to set discrete protonation states (either protonated or unprotonated) of ionizable side chains and main-chain termini. For lysozyme, we instead used experimental pK_as determined at 25 °C and 0.1 M ion strength.³⁹ The net charge of lysozyme as a function of pH, determined based on discrete protonation states, compares favorably with that based on fractional protonation according to the Henderson–Hasselbalch equation (Fig. S1).

FMAPB2 Implementation.

FMAP was originally developed to calculate free energies for transferring proteins from dilute solutions to crowded solutions (containing concentrated macromolecules).^25–26 Subsequently it was adapted to calculate chemical potentials of proteins for determining liquid-liquid phase equilibria.²⁷ Here we tailor the method once again to calculate second virial coefficients. The following aspects are adjusted:

1. Number of protein molecules in the solution box. Previously FMAP dealt with the interaction of a protein molecule with a cubic box of other protein molecules representing the solution. Here in FMAPB2, the interest is the interaction between a probe protein molecule and a partner protein molecule. The solution box therefore contains only the latter single protein molecule. This difference is trivial, and the only impact is a slight reduction in computational cost when evaluating the electrostatic potential ψ₁(s) (see eq [21]) and its counterparts for the other terms of U(R, Ω) at all the grid points before FFT.

2. Interatomic cutoff distance and size of cubic box. Previously in calculating chemical potentials by FMAP we chose an interatomic cutoff distance of 12 Å when evaluating ψ₁(s) (and counterparts) at the grid points. For B₂, which is an integral over the interprotein distance (see eq [7]), we increased the cutoff distance to 36 Å (to be denoted by r_cut) to minimize the effect of the cutoff on W(R) and hence on B₂. Also, the use of FFT transforms the solution box into an infinite periodic system, which is actually appropriate when the purpose is to calculate chemical potentials for macroscopic (hence effectively infinite) solutions. However, B₂ involves the interaction between only two molecules, and therefore we must ensure that the probe molecule interacts with a single partner molecule in the solution box; i.e., all periodic images must be beyond the interatomic cutoff distance. To ensure satisfaction of the latter condition, we set the side length of the cubic box at 2(d_max + σ_max + r_cut), where d_max/2 is the largest distance of any atom from the center of the protein and σ_max/2 is the largest atomic radius. The side lengths for the five proteins were 163.2, 186.0, 195.6, 190.8, and 274.8 Å.

3. Number of rotations of the probe molecule. The average over the orientation of the probe molecule needs to be done by repeating FMAP calculations with the probe molecule in different orientations. Previously in calculating chemical potentials, a given probe molecule was placed in different regions of the solution box; the need for orientational averaging was modest since different regions of the solution box effectively sense different relative orientations of the same probe molecule. So at most a few hundred orientations of the probe molecule, randomly generated, were used.²⁸ Here the solution box contains a single partner molecule; in each FMAPB2 calculation it senses only one relative orientation of the probe molecule, and so the need for orientational average is greater. To meet this need, we uniformly sample Ω by “successive orthogonal images” (rotations.mitchell-lab.org),⁴⁰ which are rotation matrices separated by a fixed angle α. The number of rotations required to cover the entire Ω space is 8π⁴/α⁵ (with α in radians). For example, this number is 4392 for α = π/12 (or 15°). To assess the number of rotations necessary to provide sufficient coverage of the Ω space as to produce a converged B₂ value, we compare B₂ values calculated using increasingly larger subsets of randomly selected rotations from the 4392. As illustrated in Fig. S2 for lysozyme at various ionic strengths, even with one quarter of the 4392 rotations, B₂ values are within 99% of those calculated using all the 4392 rotations. Errors, estimated using a Python code (https://github.com/manoharan-lab/flyvbjerg-std-err/) that implements Flyvbjerg and Petersen’s block decorrelation technique,⁴¹ progressively decreases with increasing number of rotations, down to less than 2% of B₂ values at 4392 rotations. All these findings indicate that 4392 rotations are more than sufficient for orientational averaging and we used these many rotations for all subsequent B₂ calculations.

For determining B₂ by integrating the Mayer f-function of interaction energy U(R, Ω) (eq [8]), we sum over all the grid points in the cubic box and over all the probe molecule orientations. We also obtain the PMF W(R), by collecting Boltzmann factors of U(R, Ω), from all the probe molecule orientations, at the grid points within 0.6-Å bins in R (eq [9]), for R up to half of the box side length. Other details, including the choice of the grid spacing (0.6 Å) for discretizing the cubic box and the attendant modifications of atomic radii and charges to cancel any numerical errors from space discretization, can be found in the original presentation of FMAP.²⁶

The expressions for B₂ in the preceding section have units of volume. Below, we use volume, ų to be precise, as the unit only for reporting the steric component $B_2^0$ In all other cases, we use 10⁻⁴ mol · ml · g⁻² as the unit, for easy comparison with experimental measurements of B₂. The conversion factor from the former to the latter unit is 10⁻²⁶N_A/M², where N_A is Avogadro’s number and M is protein molecular weight in kDa. In some literature a new symbol, A₂, is used after the unit conversion, but here we stick with B₂.

RESULTS AND DISCUSSION

Below we present FMAPB2 results for five proteins: BPTI, lysozyme, γD crystallin, chymotrypsinogen A, and BSA (Fig. 1). These proteins span small, medium, and large sizes. Experimental B₂ data are available for them over wide ranges of solvent conditions (salt concentration, pH, and temperature), with lysozyme richest in such data.^{1, 7, 9–10, 23, 29–30} Our calculations, using an atomistic interaction energy function, on these wide spans of proteins and solvent conditions allow for unprecedent in-depth probe into the physical determinants of the second virial coefficient. The experimental data also afforded an opportunity to parameterize the interaction energy function.

Parameterization of the Interaction Energy Function.

The atomistic interaction energy U(R, Ω) is given by eqs [16]]–[19]. For the steric and nonpolar attraction terms, we used parameters for the van der Waals term in Autodock4.⁴² These were modified from the Amber force field,⁴³ with a reduced set of parameters, based on element types instead of bonding types (e.g., the same σ_ii and ϵ_ii parameters for carbon regardless of its bonding types). In addition, we applied a scaling constant, v_s, to convert an energy term (eq [18]) originally meant for interatomic van der Waals interactions in gas phase into one modeling nonpolar attraction in solution. Similar scaling was done in previous work.^{14, 42}

We tuned v_s for FMAPB2. In short, the steric, nonpolar (with v_s set to 1), and electrostatic terms of U(R, Ω) were calculated via FFT as described in the preceding section. The nonpolar term was then multiplied by various scaling values before adding to the other two terms, and the Mayer f-function of the resulting U(R, Ω) was finally summed over the grid points and the probe molecule rotations (eq [8]) to yield B₂ values. Including the latter v_s-tuning step incurred minimal computational cost. Such results for lysozyme at 25 °C, pH 4.5, and ionic strengths from 0.05 to 0.9 M, along with the experimental data,^{1, 7, 9–10, 29} are shown in Fig. S3. With v_s = 0.16, excellent agreement with experiment is achieved. Note that a decrease in v_s from 0.16 leads to a gradual increase in B₂ whereas a change in v_s in the opposite direction leads to a rapid decrease in B₂, and the effects on B₂ are amplified at higher ionic strengths. For BSA, best agreement with experimental data is obtained at v_s = 0.12; the results calculated with v_s = 0.16 are much too negative. Inspections of these and corresponding results for the other three proteins suggested that the optimal v_s depends on protein size. We found that the following dependence on protein molecular weight (M, in kD)

$$v_{\text{s}}=\frac{0.18(0.27M+80)}{M+80}$$ [23]

captures well the optimal v_s for the five proteins.

We can rationalize the decrease in v_s with increasing protein size as follows. The nonpolar attraction is empirically known to be largely determined by protein surface area, as opposed to protein volume. Specifically, the hydrophobic contribution to the interaction energy between two protein molecules is often modeled as proportional to the buried surface area when the molecules come into contact.⁴⁴ However, the nonpolar component of B₂, calculated with the electrostatic term in our energy function turned off, scales with a power of protein volume (as represented by $B_2^0$) greater than unity (Fig. S4, blue line), and therefore tends to grow too rapidly with protein size. Eq [23], by tempering v_s for larger proteins, rectifies this bias. The resulting nonpolar component now scales approximately with the two thirds power of protein volume, i.e., linearly with protein surface area (Fig. S4, orange line).

We now turn to the electrostatic term. For the dielectric constant, we used that of water without further modification. The dielectric constant of water depends on temperature, and the dependence can be approximated by the following empirical relation⁴⁵

$$\epsilon =87.74-0.40008t+9.398\times {10}^{-4}{t}^2-1.41\times {10}^{-6}{t}^3$$ [24]

where t is temperature in °C. In previous FMAP calculations^25–28 we used Amber partial charges, but because Amber did not have neutral main-chain termini (i.e., deprotonated N-terminus and protonated C-terminus) in its library, here we opted for PARSE charges³⁷ to enable FMAPB2 to work for arbitrary pH values. In the pH range where the main-chain termini of a protein are charged, the calculated B₂ values according to PARSE and Amber charges are very similar (Fig. S5A, for lysozyme at pH 4.5 and 25 °C).

Steric Component of B₂.

When the net effect of U_n−a and U_elec is absent (e.g., due to the exact cancelation of the two terms or by artificially turning them off), we are left with $B_2^0$, the steric component for B₂. This situation is effectively reached in the high-temperature limit. In FMAPB2, $B_2^0$ is calculated by counting the grid points at which the two protein molecules have steric clash. Expectedly, $B_2^0$ increases with increasing molecular weight (Fig. 2, x axis). The value of $B_2^0$ is half the covolume of the pair of protein molecules (eq [12]). The corresponding diameters of the equivalent spheres are 29.3, 36.8, 41.9, 45.5, and 69.4 Å, respectively, for the five proteins studied here.

Figure 2.

The steric component $B_2^0$ is accurately predicted by the covolume from GFMT.

We have developed a generalized fundamental measure theory (GFMT),⁴⁶ which calculates the volume (V_p), surface area (S_p), and linear size (L_p) of an atomistic protein when probed by a spherical particle, and uses these geometric properties to predicts the steric component of the excess chemical potential of the protein inside a concentrated solution of the spherical particle. This type of theory⁴⁷ also predicts the covolume between the protein and the spherical probe as

$$V_{\text{co}}=2\left(V_{\text{p}}+L_{\text{p}}{S}_{\text{p}}\right)$$ [25]

V_co predicted by eq [25] depends on the probe radius (r_p), and has a minimum for each protein. The minima come about because V_p and S_p change in opposite directions as r_p increases, and occur at r_p values that drift upward, from 3 Å for BPTI to 7 Å for BSA, as the protein size increases. The GFMT minimal V_co yields $B_2^0$ that is in excellent agreement with the value calculated by FMAPB2 (Fig. 2). This quantitative account of $B_2^0$ by GFMT V_co underscores the nonspherical shape of the proteins.

High Anisotropy of U(R, Ω).

In contrast to spherical models like DLVO, the interaction energy U(R, Ω) calculated by FMAPB2 for our atomistic model is highly anisotropic (Fig. 3). Figure 3A displays 300 poses (gray spheres) with the lowest interaction energies for a lysozyme pair. These poses fall into at least five clusters (with representatives shown as large gray spheres). The energy map on a slice through space (Fig. 3A inset) shows a highly localized minimum (indicated by a green circle, corresponding to one of the clusters) next to the region of steric clash (gray area). Notably, most of the crystal contacts in four different space groups (colored spheres) are close to the lowest-energy poses and fall into the clusters of the poses, as observed in previous studies.^{14, 18} To present a global view of the highly anisotropic interaction energy, we collected 15524 lowest-energy poses from the 4392 rotations of the probe molecule and calculated an energy map over different directions of the position vector R. The poses were divided into 30 × 30 bins in longitude and latitude, and in each bin the energies of the poses were Boltzmann-averaged (effectively yielding a PMF in ϕ and θ). The results are displayed as a Hammer projection in Fig. 3B, with the afore-mentioned five clusters showing as low-energy regions. The structures of the five representative poses show that they differ not only in the direction by which the probe molecule comes into contact with the partner molecule but also in the orientation of probe molecule.

Figure 3.

Anisotropy of the interaction energy U(R, Ω), calculated for lysozyme at pH 4.5, I = 0.2 M, and 25 °C. (A). 300 lowest-energy poses, collected from the 4392 rotations of the probe molecules and shown as small gray spheres at the centers of the probe molecule, around a target molecule (cartoon representation). Five larger grays sphere show cluster representatives. Color spheres show crystal contacts in four space groups: red, P4₃2₁2 ; green, , P2₁2₁2₁; yellow, P1; and magenta, P6₁22. Inset: energy map on an xy plane, calculated for a single rotation of the probe molecule. The spectrum of energies from negative to zero is spanned by colors from dark red to yellow to white. (B) Energy map on a Hammer projection. Locations of cluster representatives are indicated numbers denoting their rankings in a collection of 15524 lowest-energy poses. Structures of the representative poses are shown with the target molecule as surface and probe molecule as trace.

Salt Dependence.

Adding salts is a simple way to perturb B₂, by screening electrostatic interactions between two protein molecules. At pH values away from the isoelectric point, the net charge on a protein can be high. The resulting electrostatic repulsion is significant at low salt but become muted at high salt. This qualitative behavior is well illustrated by the extensive data^{1, 7, 9–10, 29} for the salt dependence of lysozyme at pH 4.5 (Fig. 4A, symbols), showing a substantial decrease in B₂ with increasing ionic strength, from 3.92 × 10⁻⁴ mol · ml · g⁻² at I = 0.078 M to approximately −5.0 × 10⁻⁴ mol · ml · g⁻² at I = 0.85 M.

Figure 4.

Salt dependence of lysozyme B₂ at pH 4.5 and 25 °C. (A) Comparison of experimental and FMAPB2 results. Here as well in subsequent figures, a horizontal dashed line indicates $B_2^0$. (B) PMF over R. Results are shown for the total PMF and the nonpolar and electrostatic PMFS at I = 0.1 and 0. 5 M.

As already noted, B₂ values calculated for lysozyme at pH 4.5 are in excellent agreement with the experiment data over the entire ionic strength range (Fig. 4A, solid curve). At this pH, lysozyme has a net charge of +10e (Fig. 1). At I = 0.05 M, the electrostatic repulsion has a dominant role, and B₂ reaches (4.868 ± 0.004) × 10⁻⁴ mol · ml · g⁻², which is nearly 60% larger than the steric component, (3.076 ± 0.002) × 10⁻⁴ mol · ml · g⁻². At approximately I = 0.08 M, the electrostatic repulsion has weakened to such an extent that its effect cancels that of the nonpolar attraction, making B₂ equal to $B_2^0$. At higher ionic strengths, the electrostatic repulsion is further weakened and the nonpolar repulsion becomes dominant, leading to a very negative B₂, of (−4.07 ± 0.02) × 10⁻⁴ mol · ml · g⁻² at I = 0.9 M.

To further dissect the electrostatic and nonpolar contributions of lysozyme, we compare the PMF over R with those calculated with the electrostatic and nonpolar terms in the energy function turned off one at a time (Fig. 4B). For convenience, we refer to the latter as the nonpolar and electrostatic PMFs. The steric core extends to approximately R = 25 Å. Starting at 28.5 Å and all the way to 50 Å, there is significant nonpolar attraction, as indicated by negative values of the nonpolar PMF (blue curve on either the left or right half). The fact that the nonpolar PMF comes into play at a much shorter interprotein distance of 28.5 Å than the 36.8-Å diameter of the equivalent sphere based on $B_2^0$ reflects the nonspherical shape of lysozyme. The nonpolar attraction is countered by electrostatic repulsion. At I = 0.1 M, the latter (red curve in the left half) is strong enough such that, when both of these interaction terms are present, their effects in the R range of 30 to 50 Å nearly cancel, with |W| ≤ 0.5k_BT (black curve in the left half). However, note that the contributions of the nonpolar attraction and electrostatic repulsion actually are not additive. In particular, at R = 33 Å, W has a minimum value of −0.3k_BT, even though the nonpolar and electrostatic PMFs are 2.5k_BT and −1.4k_BT, respectively, and would thus predict W = 1.1k_BT according to additivity. Such nonadditivity has been commented previously,²⁸ and is due to correlation between nonpolar and electrostatic interactions (among different configurations at a fixed R in the present case). Note also that both the nonpolar and electrostatic PMFs almost die out at R = 60 Å, where many interprotein atom pairs are still within the cutoff distance of 36 Å, indicating that this choice of cutoff distance is appropriate. At I = 0.5 M, heightened screening lowers the electrostatic PMF, by amounts ranging from 1.3k_BT to 0.1k_BT over the R range of 30 to 50 Å (red solid curve compared to red dashed curve in the right half). Although that still leaves the electrostatic PMF significant (e.g., with value 3.9k_BT at R = 30 Å), the total PMF is now very close to the nonpolar PMF (black and blue curves in the right half), once again exposing nonadditivity.

We also studied the salt dependences of B₂ for the other four proteins. As shown in Fig. 5, our calculated B₂ values show reasonable agreement with the experimental data overall.^{7, 9, 32–34} BPTI at pH 4.9 (net charge +6e), γD crystallin at pH 5.5 (net charge +5e), and chymotrypsinogen A at pH 3 (+18e) all show the expected decrease in B₂ with increasing ionic strength. FMAPB2 underestimated this dependence for BPTI, with a noticeable underestimation of B₂ at low ionic strength. Among these three proteins, γD crystallin has the weakest salt dependence, which can be mainly attributed to its smallest net charge (and the larger size relative to BPTI). BSA shows relatively weak salt dependence over the entire pH range from 3.5 to 8. At the ends of this pH range, BSA carries significant net charges (+95e and −19e, respectively), and B₂ exhibits the familiar decrease with increasing ionic strength. However, near the isoelectric point (pI = 5.25), the salt dependence reverses the trend. Here the electrostatic repulsion due to the net charge disappears, and attraction between electric dipoles comes into play.^{9, 31} As salt screens this electrostatic attraction, B₂ increases with increasing ionic strength. This “reverse” salt dependence cannot be predicted by a spherical model like DLVO, although FMAPB2 also overestimated the magnitude of this reversal. Note that the minimum B₂ of BSA occurs at pH 4.7, not precisely at the pI, further emphasizing the importance of detailed charge distributions in protein-protein electrostatic interactions.

Figure 5.

Salt dependences of B₂ for four proteins. (A) BPTI at pH 4.9 and 20 °C. (B) γD crystallin at pH 5.5 and 27 °C. (C) Chymotrypsinogen A at pH 3 and 25 °C. (D) BSA at 25 °C.

Taking advantage of the many crystal structures of lysozyme in the PDB, we tested how sensitive the calculated B₂ is to variation in protein structure. In Fig. S5B, we compare the B₂ values calculated on PDB entry 1AKI with those calculated on six other PDB entries, at pH 4.5 and 25 °C and in the ionic strength range of 0.05 to 0.9 M. The spread in B₂ among the seven structures increases modestly with increasing ionic strength, suggesting a higher sensitivity of nonpolar attraction than electrostatic repulsion to structural variation. Still, even at I = 0.9 M, the standard deviation of B₂ among the seven structures is only 7.7% of the mean value. This deviation is much less than discrepancies in B₂ values obtained using different techniques (e.g., SLS vs. SANS) or by different researchers. We thus conclude that crystal structures of globular proteins can be used for B₂ calculations. Note that, for proteins with flexible regions, we can account for the flexibility by representing the flexible regions by an ensemble of structures and taking an ensemble average of calculated B₂ values.

The success of FMAPB2 noted above also highlights the limitation of simple theoretical models such as DLVO. In Fig. S6A, we compare the DLVO predictions for lysozyme B₂ at pH 4.5 and 25 °C over a range of ionic strength. Although the chosen parameters (R = 36 Å; δ = 1.8 Å; A_H = 5.01 kcal/mol; and Q = +10e)⁴⁸ allow for good matching with experimental data at high ionic strength (I > 0.2 M), DLVO significantly overestimates B₂ at low ionic strength. The latter result arises from modeling the atomic charges as a uniform distribution of the net charge on the protein surface, leading to significant overestimation of electrostatic repulsion.³⁰ The latter is illustrated by a comparison between our electrostatic PMF and the DLVO counterpart (Fig. S6B, solid and dashed red curves). Our nonpolar PMF becomes discernible at R = 28.5 Å and peaks at 32.1 Å, but the DLVO counterpart has a much narrower range (starting and peaking at R = 37.8 Å, close to the diameter of the equivalent sphere based on our $B_2^0$) and, to compensate, a much higher peak (solid and dashed blue curves). A final difference is that, whereas the van der Waals and electrostatic terms of the PMF over R are additive in DLVO (eq [15]), their counterparts in FMAPB2 are not.

pH Dependence.

The salt effects at different pH values on the B₂ of BSA (Fig. 5D) have already revealed that pH, by modifying the charges on a protein, can significantly perturb B₂. In Fig. 6 we display the pH dependences of B₂ values calculated by FMAPB2 for lysozyme and chymotrypsinogen A. Overall the FMAPB2 results again show reasonable agreement with experimental data,^{7, 9, 23, 30–31} although for lysozyme the agreement is perhaps somewhat worse than that shown in Fig. 4A for salt dependence at pH 4.5. Our agreement with the data of Velev et al.⁹ displayed in Fig. 6 is better than that by Elcock and McCammon,¹⁵ which we attribute in part to the extensive sampling afforded by FMAPB2. Here again different experimental techniques and different researchers reported widely varying data for nominally the same solvent conditions. In particular, for lysozyme at I = 0.1 M, using SLS, Velev et al.⁹ found a sharp decline in B₂ around pH 7, but Liu et al.’s data³⁰ at pH 7 and Prytkova’s data²³ at pH 6.9 suggest a less sharp decline (Fig. 6A). Our results at I = 0.1 M fall in the middle of these three sets of experimental data. In explaining their discrepancy from experimental data on pH dependence at I = 0.1 M, Elcock and McCammon¹⁵ suggested change in protonation states upon protein weak binding as a possible reason.

Figure 6.

pH dependences of B₂ for two proteins. (A) Lysozyme at 25 °C. (B) Chymotrypsinogen A at 25 °C.

For both proteins, the calculated B₂ decreases with increasing pH, due to deceasing net charge and hence reduced electrostatic repulsion. The pH dependences become weaker at increasing ionic strength, due to salt screening of electrostatic repulsion. For chymotrypsinogen A, as pH approaches the pI (10.15), we again see the start of a reversal of salt dependence observed above on BSA, i.e., B₂ near pH 7 becomes higher at higher ionic strength (Fig. 6B), in agreement with the experimental data.^{9, 31}

Temperature Dependence.

The temperature dependence of B₂ was measured by Bonneté et al.¹⁰ for lysozyme at pH 4.5. This effect is reproduced quite well by FMAPB2 calculations (Fig. 7). To understand how temperature affects B₂, we first note that it is an integral of the Mayer f-function, and the latter contains k_BT as part of the Boltzmann factor of the interaction energy. For the moment, let us assume that the interaction energy itself does not dependent on temperature. At infinite temperature, the Boltzmann factor goes to 1 and the f-function goes to 0, except when a steric term is present (see eq [8]). Therefore B₂ approaches the steric component $B_2^0$ at high temperature. In contrast, at low temperatures, small regions in the configurational space where the interaction energy is especially negative (see Fig. 3), corresponding to a large positive f-function, make disproportionate contributions to B₂, pushing it toward negative values (reminiscent of observations by Neal et al.¹⁴). Note that the decrease in B₂ with decreasing temperature becomes steeper at higher ionic strength, because then the interaction energy and hence B₂ at low temperatures are more negative.

Figure 7.

Temperature dependence of lysozyme B₂ at pH 4.5. The intersection of the horizontal solid line and a B₂ curve predicts the critical temperature for liquid-liquid phase separation (vertical arrow). The numbers in the legend indicate ionic strengths (in M).

The temperature dependence of our B₂ can be elaborated from a different angle. The foregoing reasoning also leads to the conclusion that our PMF over R is temperature-dependent, i.e., more negative at low temperatures than at high temperatures (see Fig. S7). This is in contrast to the PMFs of spherical models like DLVO, which typically are assumed to be (essentially) temperature-independent. Although even a temperature-independent W(R) yields a modest temperature dependence for B₂, our temperature-dependent W(R) amplifies the temperature dependence of B₂. While the temperature dependence of our B₂ is in line with experimental data, that predicted by DLVO prediction is too weak.

The weak temperature dependence of B₂ predicted by another spherical model, with a square-well interaction energy, was noted by Platten et al.⁴⁹ To compensate, they made the well depth temperature-dependent (stronger attraction at low temperatures). This modification in turn was borrowed from Lomakin et al.,⁵⁰ who introduced the temperature-dependent well depth in order to broaden the binodal of liquid-liquid phase separation, which otherwise is too narrow compared to experimental observations. Our atomistic interaction energy function produces a temperature-dependent PMF over R, which enables the correct prediction of the slope of the increase in B₂ with respect to T (and possibly the broadness of binodals).

Now the electrostatic term in our interaction energy function has direct temperature dependence, through T and the temperature-dependent ε (eq [24]) that enter κ (inverse of screening length) and through ε that scales the magnitude of the electrostatic term (eq [19]). However, the temperature dependence of the electrostatic term contributes little to the temperature dependence of our B₂ shown in Fig. 7, because it is largely determined by negative values of the interaction energy, which occur in regions where the nonpolar attraction is strong while the electrostatic interactions are relatively weak.

There is one outlier to an otherwise excellent agreement between the FMAPB2 results and experimental data in Fig. 7: at 10 °C and I = 0.52 M, FMAPB2 predicts B₂ = (−5.05 ± 0.03) × 10⁻⁴ mol · ml · g⁻², but the experimental value is much more negative, at −7.06 × 10⁻⁴ mol · ml · g⁻⁴.¹⁰ The consistent performance of FMAPB2 and instances of inconsistencies among experimental measurements embolden us to suggest that perhaps this last experimental value is an underestimate. As support, the critical temperature (T_c) for lysozyme liquid-liquid phase separation was determined under similar solvent conditions to be approximately 9 °C,⁵¹ close to the 10 °C temperature of interest here. It has been proposed that $B_2\left(T_{\text{c}}\right)\approx -6.6B_2^0/4$.^{4, 52} The latter relation predicts a B₂ of −5.1 × 10⁻⁴ mol · ml · g⁻² at 9 °C. This estimate agrees much better with the FMAPB2 result than with the one reported by Bonneté et al. The foregoing comparison suggests that, by calculating the temperature dependence of B₂ and using the $B_2\left(T_{\text{c}}\right)\approx -6.6B_2^0/4$ relation, FMAPB2 can predict T_c (intersection of blue curve with solid horizontal line, Fig. 7).

FMAPB2 Web Server.

To make FMAPB2 widely accessible, we have implemented it as a web server, available at http://pipe.rcc.fsu.edu/fmapb2. The server outputs B₂ and its steric component $B_2^0$ (Fig. 8A). In addition, their convergence at increasing numbers of rotations used for averaging over Ω (Fig. 8B), the PMF over R, and the diameter of the equivalent sphere based on $B_2^0$ are reported (Fig. 8C).

Figure 8.

Output of the FMAPB2 web server, for lysozyme at pH 4.5. (A) Summary of predicted results. (B) Convergence of B₂ at increasing number of probe molecule rotations. (C) PMF over R. A vertical dashed line indicates the diameter of the equivalent sphere based on $B_2^0$.

The input to the server is a structure file in PQR format, which contains not only atomic coordinates but also atomic charges and radii. The atomic radii are reassigned according to the van der Waals parameters in AutoDock4.⁴² The user also enters the ionic strength and temperature in the B₂ calculation. For convenience a portal is also provided for the user to start from a PDB file, which can be directly uploaded or, by entering a PDB entry name, retrieved from the Protein Data Bank (http://www.rcsb.org/pdb/). In this case the user also enters the desired pH value. The server then assigns atomic charges and radii using the PROPKA³⁸ and PDB2PQR³⁶ programs.

The B₂ calculation is offloaded to compute nodes maintained by the Research Computing Center at Florida State University. On a single node with 16 Intel Xeon E5–2650 cores at 2.6 GHZ, the calculation takes approximately 4 hours for lysozyme and 17 hours for BSA, involving 310³ and 458³ grid points, respectively, for each of the 4392 rotations, or a total of 1.31 × 10¹¹ and 4.22 × 10¹¹ configurations uniformly distributed in the 6-dimensional space.

CONCLUSION AND OUTLOOK

We have developed the FMAPB2 method for calculating the second virial coefficients of protein solutions, with the interprotein interaction energy modeled at the all-atom level in implicit solvent. The method has been tested on lysozyme and other proteins ranging in molecular weight from 6.5 to 65 kD, over wide ranges of solvent conditions (salt concentration, pH, and temperature). Our calculations show that the value of B₂ is determined by the steric covolume and by the balance between nonpolar attraction and electrostatic interactions; this balance can be shifted by salt, pH, and temperature. However, the effects of nonpolar attraction and electrostatic interactions are not simply additive. FMAPB2 will hopefully replace DLVO for future interpretation of experimental data on B₂ and join experimental techniques for cross validation, given that the latter can produce large discrepancies. It can be accessed at http://pipe.rcc.fsu.edu/fmapb2 by simply uploading a PDB file or entering a PDB entry name.

Further developments can be pursued in many directions. For salt effects, we have only accounted for screening of electrostatic interactions. The B₂ data that we gathered from the literature all used NaCl as the salt, for which electrostatic screening is the dominant effect, up to approximately 0.5 M concentration. Salts can have two other effects.^53–54 First, at high concentrations, they can change the surface tension of water and thereby modulate the strength of hydrophobic interactions. Kosmotropic ions such as SO₄²⁻ strengthen hydrophobic interactions whereas chaotropic ions such as NO₃⁻ and SCN⁻ weaken them, lowering and raising, respectively, the value of B₂. An overestimation of the lysozyme B₂ by FMAPB2 at ionic strength close to 1 M (Fig. 4A) can be attributed to this effect, given that Cl⁻ is a mild kosmotrope. Second, starting at low concentrations, chaotropic anions can bind to the surface of a cationic protein like lysozyme, thereby neutralizing the protein and reducing electrostatic repulsion. This effect would lower B₂. Indeed, B₂ values obtained using NaNO₃ and NaSCN are lower than those using NaCl.¹⁰ We can model these two types of salt effects into our interaction energy function. Similarly, cosolvents such as alcohols can change B₂;^{30, 32, 55} their effects can also be modeled into our energy function.

The treatment of atomistic details in FMAPB2 not only has yielded physical insight into B₂ but also allows us to begin to investigate how amino-acid sequence variations⁷ or even single mutations^{18, 29} affect B₂. For such investigations to be meaningful, further refinement of our interaction energy function may be needed and the corresponding experimental data may also need to be cross-validated. Another straightforward application of FMAPB2 is to the second virial cross coefficient, which depends on the interaction between two different proteins and arises in a virial expansion of the osmotic pressure in a binary solution.^56–58 The only modification needed for FMAPB2 is to use two different proteins, one for the probe molecule and one for the target molecule; the computational cost remains the same. Also worth exploring is whether FMAPB2 can be used for calculating the binding constants for protein-protein specific binding (some effort on protein-ligand binding constant has already been made⁵⁹), although it can be anticipated that the development of an implicit-solvent interaction energy function appropriate for specific binding and the neglect of induced fit in our approach can pose challenges.

In addition to B₂ values, FMAPB2 can be used to calculate other experimental observables. In particular, the Boltzmann factor, exp[−βW(R)], of the PMF over R is the radial distribution function in the dilute limit, which can be used to calculate the structure factor and hence the scattering intensity over the entire q range, enabling direct comparison with raw SAXS/SANS data. For this reason, our web server returns the PMF over R (Fig. 8C). The difficulties in precise measurements of B₂ prompted Platten et al.⁴⁹ to propose an indirect method for obtaining B₂, through determining the binodal of liquid-liquid phase separation. The idea behind is the postulate that all molecular systems with only short-range attraction can be mapped to the square-well model,⁶⁰ for which one knows the relation between the binodal and B₂. We hope that FMAPB2, with further improvements, will become reliable as to alleviate the need for going to such great lengths to obtain B₂. Moreover, FMAPB2 may even be extended to predict the binodal. We have already illustrated that, based on the relation $B_2\left(T_{\text{c}}\right)\approx -6.6B_2^0/4$, FMAPB2 can be used to determine the critical temperature for liquid-liquid phase separation (Fig. 7). The entire binodal can be determined from B₂ and a few higher order virial coefficients, based on a truncated virial expansion of the excess chemical potential^{27–28, 50}

$$\beta {\mu }_{\text{ex}}\approx \beta {\mu }_{\text{ex}}^0+\sum _{l=2}^4\frac{l}{l-1}\left(B_l-B_l^0\right){\rho }^{l-1}$$ [26]

where the superscript “0” denotes the steric component. B₃ and B₄ are given by “cluster integrals” in the relative configurational space (translation and rotation) of three and four protein molecules; the integrands are the products of pairwise Mayer f-functions. Following the use of Fourier transforms in B₃ and B₄ calculations for spherical models of molecular fluids,^61–63 it is feasible to extend FMAPB2 to these calculations for atomistic proteins in implicit solvent. One can then use eq [26] to predict not just the binodal but many other thermodynamic properties of a protein solution.

ABBREVIATIONS

BPTI

bovine pancreas trypsin inhibitor

BSA

bovine serum albumin

DLVO

Derjaguin, Landau, Verwey, and Overbeek

FFT

fast Fourier transform

FMAP

FFT-based modeling of atomistic protein-protein interactions

GFMT

generalized fundamental measure theory

PDB

Protein Data Bank

PMF

potential of mean force

STL

static light scattering

SANS

small angle neutron scattering

SAXS

small angle X-ray scattering

SLS

static light scattering