Protein Association in Solution: Statistical Mechanical Modeling

Abstract

Protein molecules associate in solution, often in clusters beyond pairwise, leading to liquid phase separations and high viscosities. It is often impractical to study these multi-protein systems by atomistic computer simulations, particularly in multi-component solvents. Instead, their forces and states can be studied by liquid state statistical mechanics. However, past such approaches, such as the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, were limited to modeling proteins as spheres, and contained no microscopic structure-property relations. Recently, this limitation has been partly overcome by bringing the powerful Wertheim theory of associating molecules to bear on protein association equilibria. Here, we review these developments.

Keywords: Wertheim’s theory; antibodies; association; phase transition; proteins.

Figure 1

Driving forces of association as deduced from experiments. From left to right: proteins cluster beyond pairwise; electrostatics can be important; clustering is enthalpic; and hydration and hydrophobicity are important.

Figure 2

Model of the interactions of two globular proteins. Protein spheres interact at M × M pairs of binding sites on the surfaces, one pair of which (A and B) is indicated here. Reprinted by permission from [103].

Figure 3

Liquid–liquid phase equilibria: theory and experiment. Top: $\gamma$ IIIa-crystallin. Bottom: lysozyme. Solid curves are calculated from the model; see [103] for details. Experimental data shown by symbols taken from Taratuta [117] (upper symbols) and Broide [118] (lower symbols). Reprinted by permission from [103].

Figure 4

Experimental data modulation of protein interactions by salts in lysozyme solutions. $T_{cloud}$ for lysozyme as a function of ionic strength of the added alkali-halide salts $I_{ion}$; the symbols denote experimental data [117]. The lines are the results of Equation (12) from [103] (from top to bottom: KBr, NaBr, KCl, and NaCl salts). Reprinted by permission from [103].

Figure 5

The ions that bind most weakly to water most strongly affect the protein-protein attraction. Correlation between slope a in Equation (12) [103] and the Gibbs free energy of hydration $\Delta G_{hydr}$ for the corresponding anions. Reprinted by permission from [103].

Figure 6

(a,b) Seven-bead model molecules. Each Y-shaped molecule first assembles from seven individual beads via strong forces, which act only between the sticky spots of the same color. Next, these molecules associate into non-covalent clusters. A and B denote the Fab fragments (the region of the antibody that binds to antigens), while C denotes the Fc arm (called the fragment crystallizable region, which interacts with the cell surface receptors). Figure reprinted by permission from [56]. Copyright Elsevier (2017).

Figure 7

Relative viscosity $\eta /{\eta }_0$ as a function of the protein concentration [56]. From bottom to top: (i) model of bispecific antibodies, green curve; (ii) symmetric Fab–Fab model of antibodies, red curve; and (iii) model of interacting Fab–Fc terminals, blue curve. For more details, see the original paper. Figure reprinted by permission from [56]. Copyright Elsevier (2017).

Figure 8

Three types of antibody clustering studied in this work. The types of antibody clustering studied in this work are: (a) monospecific two-arm binding; (b) bispecific single-arm binding; and (c) arms-to-Fc binding. Figure 8 reprinted by permission from [56]. Copyright Elsevier (2017).

Figure 9

The liquid-liquid phase equilibria of mAb solutions. Temperature T vs. mAb concentration: calculated (full line) and symbols (experimental data) [121,128]. The interaction between sites A, B, and C is modeled using the short-range square-well attraction. The two-phase region is indicated by the colored area; for more details, see [123]. Reprinted with permission from [123]. Copyright 2018 American Chemical Society.

Figure 10

Liquid-liquid phase diagrams of antibody solutions in the presence of attractive obstacles. Phase diagrams $T^{*}$ vs. $\eta =\pi {\rho }_1^{*}/6$ coordinate frame for model of monoclonal antibodies in Yukawa hard-sphere porous media at bonding distance $0.05{\sigma }_1$ and for ${ϵ}_{AA}^{\left(as\right)}={ϵ}_{BB}^{\left(as\right)}={ϵ}_{AB}^{\left(as\right)}=ϵ$, ${ϵ}_{CC}^{\left(as\right)}=0$, ${ϵ}_{AC}^{\left(as\right)}={ϵ}_{BC}^{\left(as\right)}=ϵ$, obstacles packing fraction ${\eta }_0=0.1$, and different strengths of the Yukawa interaction: ${ϵ}_Y=0$ (blue (2) line), ${ϵ}_Y=0.06ϵ$ (black (5) line), and ${ϵ}_Y=0.1ϵ$ (red (7) line). The green (1) line denotes the result for the neat fluid (no obstacles present, ${\eta }_0=0$). Reproduced in part from Ref. [135]. Reproduced with permission of the Royal Society of Chemistry.