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

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 and, 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 .

Figure 1.

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

Figure 2.

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

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.

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.

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.

Figure 6.

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

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).

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 $B_2^0$.