For biotechnological drugs, it is desirable to formulate antibody solutions with low viscosities. We go beyond previous colloid theories in treating protein–protein self–association of molecules that are antibody–shaped and flexible and have spatially specific binding sites. We consider interactions either through fragment antigen (Fab–Fab) or fragment crystalizable (Fab–Fc) binding. Wertheim's theory is adapted to compute the cluster–size distributions, viscosities, second virial coefficients, and Huggins coefficients, as functions of antibody concentration. We find that the aggregation properties of concentrated solutions can be anticipated from simpler–to–measure dilute solutions. A principal finding is that aggregation is controllable, in principle, through modifying the antibody itself, and not just the solution it is dissolved in. In particular: (i) monospecific antibodies having two identical Fab arms can form linear chains with intermediate viscosities. (ii) Bispecific antibodies having different Fab arms can, in some cases, only dimerize, having low viscosities. (iii) Arm–to–Fc binding allows for three binding partners, leading to networks and high viscosities.

Protein aggregation is broadly important in diseases and in formulations of biological drugs. Here, we develop a theoretical model for reversible protein-protein aggregation in salt solutions. We treat proteins as hard spheres having square-well-energy binding sites, using Wertheim's thermodynamic perturbation theory. The necessary condition required for such modeling to be realistic is that proteins in solution during the experiment remain in their compact form. Within this limitation our model gives accurate liquid-liquid coexistence curves for lysozyme and γ IIIa-crystallin solutions in respective buffers. It provides good fits to the cloudpoint curves of lysozyme in buffer-salt mixtures as a function of the type and concentration of salt. It than predicts full coexistence curves, osmotic compressibilities, and second virial coefficients under such conditions. This treatment may also be relevant to protein crystallization.phase separation | protein aggregation | Hofmeister series P rotein molecules can aggregate with each other. This process is important in many ways (1). First, a key step in developing biotech drugs-which are mostly mABs-is to formulate proteins so that they do not aggregate. This is because good shelflife requires long-term solution stability, and because patient compliance requires liquids having low viscosities. The importance of such formulations comes from the fact that the world market for protein biologicals is about the same size as for smartphones. Second, protein aggregation in the cell plays a key role in protein condensation diseases, such as Alzheimer's, Parkinson's, Huntington's, and others. Third, much of structural biology derives from the 100,000 protein structures in the Protein Data Bank, a resource that would not have been possible without protein crystals, a particular state of protein aggregation. Also, it is not yet possible to rationally design the conditions for proteins to crystallize.However, protein aggregation is poorly understood. Atomistic-level molecular simulations are not practical for studying multiprotein interactions as a function of concentration, and in liquid solutions that are themselves fairly complicated-that account for salts of different types and concentrations as well as other ligands, excipients, stabilizers, or metabolites. So, a traditional approach is to adapt colloid theories, such as the DerjaguinLandau-Verwey-Overbeek (DLVO) (2) theory. In those treatments, proteins are represented as spheres that interact through spherically symmetric van der Waals and electrostatic interactions in salt water, using a continuum representation of solvent and a Debye-Hückel screening for salts. DLVO often gives correct trends for the pH and salt concentration dependencies. However, DLVO does not readily account for protein sequence-structure properties, salt bridges (which are commonly the "glue" holding protein crystals together), explicit waters in general, or Hofmeister effects, where different salts have widely different powers of protein precipitation...

This study presents the theory for liquid-liquid phase separation for systems of molecules modeling monoclonal antibodies. Individual molecule is depicted as an assembly of seven hard spheres, organized to mimic the Y-shaped antibody. We consider the antibody-antibody interactions either through Fab, Fab' (two Fab fragments may be different), or Fc domain. Interaction between these three domains of the molecule (hereafter denoted as A, B, and C, respectively) is modeled by a short-range square-well attraction. To obtain numerical results for the model under study, we adapt Wertheim's thermodynamic perturbation theory. We use this model to calculate the liquid-liquid phase separation curve and the second virial coefficient B. Various interaction scenarios are examined to see how the strength of the site-site interactions and their range shape the coexistence curve. In the asymmetric case, where an attraction between two sites is favored and the interaction energies for the other sites kept constant, critical temperature first increases and than strongly decreases. Some more microscopic information, for example, the probability for the particular two sites to be connected, has been calculated. Analysis of the experimental liquid-liquid phase diagrams, obtained from literature, is presented. In addition, we calculate the second virial coefficient under conditions leading to the liquid-liquid phase separation and present this quantity on the graph B versus protein concentration.

We analyze experimentally determined phase diagram of γD–βB1 crystallin mixture. Proteins are described as dumbbells decorated with attractive sites to allow inter–particle interaction. We use thermodynamic perturbation theory to calculate the free energy of such mixtures and, by applying equilibrium conditions, also the compositions and concentrations of the co–existing phases. Initially we fit the Tcloud versus packing fraction η measurements for pure (x2 = 0) γD solution in 0.1 M phosphate buffer at pH = 7.0. Another piece of experimental data, used to fix the model parameters, is the isotherm x2 vs η at T = 268.5 K, at the same pH and salt content. We use the conventional Lorentz–Berthelot mixing rules to describe cross interactions. This enables us to determine (i) model parameters for pure βB1 crystallin protein and to calculate (ii) complete equilibrium surface (Tcloud – x2 – η) for the crystallin mixtures. iii) We present the results for several isotherms, including the tie–lines, as also the temperature–packing fraction curves. Good agreement with the available experimental data is obtained. An interesting result of these calculations is an evidence of coexistence of three phases. This domain appears for the region of temperatures just out of the experimental range studied so far. The input parameters, leading good description of experimental data, revealed large difference between the numbers of the attractive sites for γD and βB1 proteins. This interesting result may be related to the fact that γD has more than nine times smaller quadrupole moment than its partner in the mixture.

The effects of additions of low-molecular-mass salts on the properties of aqueous lysozyme solutions are examined by using the cloud-point temperature, T cloud , measurements. Mixtures of protein, buffer, and simple salt in water are studied at pH = 6.8 (phosphate buffer) and pH = 4.6 (acetate buffer). We show that an addition of buffer in the amount above I buffer = 0.6 mol dm −3 does not affect the T cloud values. However, by replacing a certain amount of the buffer electrolyte by another salt, keeping the total ionic strength constant, we can significantly change the cloud-point temperature. All the salts de-stabilize the solution and the magnitude of the effect depends on the nature of the salt. Experimental results are analyzed within the framework of the onecomponent model, which treats the protein-protein interaction as highly directional and of short-range. We use this approach to predict the second virial coefficients, and liquid-liquid phase diagrams under conditions, where T cloud is determined experimentally.