Calculating Structure Factors of Protein Solutions by Atomistic Modeling of Protein-Protein Interactions

Abstract

We present a method, FMAPS(q), for calculating the structure factor, $S\left(q\right)$ , of a protein solution, by extending our fast Fourier transform-based modeling of atomistic protein-protein interactions (FMAP) approach. The interaction energy consists of steric, nonpolar attractive, and electrostatic terms that are additive among all pairs of atoms between two protein molecules. In the present version, we invoke the free-rotation approximation, such that the structure factor is given by the Fourier transform of the protein center-center distribution function $g_C\left(R\right)$ . At low protein concentrations, $g_C\left(R\right)$ can be approximated as $e^{-\beta W\left(R\right)}$ , where $W\left(R\right)$ is the potential of mean force along the center-center distance $R$ . We calculate $W\left(R\right)$ using FMAPB2, a member of the FMAP class of methods that is specialized for the second virial coefficient [Qin and Zhou, J Phys Chem B 123 (2019) 8203-8215]. For higher protein concentrations, we obtain $S\left(q\right)$ by a modified random-phase approximation, which is a perturbation around the steric-only energy function. Without adjusting any parameters, the calculated structure factors for lysozyme and bovine serum albumin at various ionic strengths, temperatures, and protein concentrations are all in reasonable agreement with those measured by small-angle X-ray or neutron scattering. This initial success motivates further developments, including removing approximations and parameterizing the interaction energy function.

Keywords: second virial coefficient; small-angle X-ray scattering; small-angle neutron scattering; structure factor.

Fig. 1.

Structures of the two proteins studied here. Structures are from PDB entries 1AKI and 3V03; charged sidechains are displayed as sticks. Circles are drawn with the diameters determined by the steric part of the atomistic interaction energy function.

Fig. 2.

Pair potential of mean force, low-concentration limit of the structure factor, and mRPA prediction. Results are for lysozyme at 25 °C, pH 6.8 (charge +8e), and $I=0.02\mathrm{M}$. (a) Potentials of mean force presented in the form of Boltzmann factors, for the full interaction energy, the steric-only system, or the hard-sphere reference. (b) Fourier transforms of the Mayer functions for the steric-only system and the hard-sphere reference, or the difference in Mayer function between the full system and the hard-sphere reference. All the results are multiplied by a concentration of 1 mg/mL. (c) Differences in Mayer functions between the full system and the hard-sphere reference or the steric-only system. (d) mRPA predictions at 1, 10, and 100 mg/mL lysozyme. The prediction at 1 mg/mL is essentially identical to the low-concentration limit (LCL). The shaded top portion is also enlarged in a separate panel on top, where a red arrow indicates a low-$q$ peak.

Fig. 3.

Effects of ionic strength and temperature on the structure factor of lysozyme. (a) Boltzmann factors at $I=0.01,0.06$, and 0.15 M. The temperatures are at 25, 25, and 23 °C, respectively; the pH values (net charges) are 8 (+7e), 8 (+7e), and 7.4 (+8e), respectively. (b) Comparison of mRPA (solid curves) and experimental (dashed curves) structure factors; the latter are from SAXS measurements by Tanouye et al. [4]. A magenta curve is added to demonstrate that the experimental curve at $I=0.15\mathrm{M}$ can be reproduced well by scaling up $v_{\mathrm{s}}$ slightly, from 0,.16 to 0.174. (b) Corresponding results at 10 °C. (c) Corresponding results at higher temperatures (40, 40, and 37 °C). Here and in the remaining figures, experimental structure factors refer to those obtained by fitting the experimental scattering intensity to a model where the structure factor is an approximate analytical solution for a simple potential such as the two-Yukawa potential.

Fig. 4.

Dependence of lysozyme structure factors on concentration. (a) 50 mg/mL. (b) 100 mg/ml. (c) 175 mg/mL. (d) 225 mg/mL. The temperature is 25 °C, ionic strength is 0.02 M, and pH values (net charges) are 6.8 (+8e), 5.3 (+10e), 4.5 (+10e), and 4.5 (+10e). The mRPA predictions, Percus-Yevick structure factors, and SANS measurements [3] are shown as red, green, and black curves, respectively.

Fig. 5.

Effects of ionic strength on BSA pair interactions and structure factor. (a) Boltzmann factors at $I=0.005,0.01,0.015,0.05,0.1,0.3,0.5\mathrm{M}$. The local peak at $R=60\mathrm{\AA }$ is also shown in a zoomed view. Temperature = 20 °C; pH = 7 (charge −15e). (b) Comparison of mRPA (solid curves) and SAXS (dashed curves) [5] structure factors at a concentration of 100 mg/mL and ionic strengths from 0.015 to 0.5 M [in colors matching those in (a)]. Curves are shifted in the vertical direction to avoid overlap. (c) A preferential complex from the FMAPB2 sampling, stabilized by local electrostatic and nonpolar attraction. A zoomed view shows salt bridges between two BSA molecules. (d) The electrostatic potential of BSA shows a positive region in a mainly negatively charged surface. A zoomed view shows that the positive patch is lined by negatively charged residues of the second BSA molecule.

Fig. 6.

Dependence of BSA structure factors on concentration. (a) Comparison of mRPA (solid curves) and Brownian dynamics [19] (dotted curves) predictions and SAXS structure factors (dashed curves) [6] at concentrations from 0.9 to 90 mg/mL. (b) Comparison of mRPA (solid curves) and SAXS structure factors (dashed curves) [5] at concentrations from 20 to 200 mg/mL. Curves are shifted in the vertical direction to avoid overlap.