Protein aggregation in salt solutions Miha Kastelica, Yurij V. Kalyuzhnyib, Barbara Hribar-Leea, Ken A. Dillc,1, and Vojko Vlachya aFaculty of Chemistry and Chemical Technology, University of Ljubljana, 1000 Ljubljana, Slovenia; bInstitute for Condensed Matter Physics, 79011 Lviv, Ukraine; and cLaufer Center for Physical and Quantitative Biology and Departments of Physics and Chemistry, Stony Brook University, Stony Brook, NY 11794 Contributed by Ken A. Dill, April 20, 2015 (sent for review January 16, 2015; reviewed by Shekhar Garde and John M. Prausnitz) Protein aggregation is broadly important in diseases and in address phenomena such as protein aggregation and self- formulations of biological drugs. Here, we develop a theoretical assembly” (9, 10), typically yielding coexistence curves that are model for reversible protein–protein aggregation in salt solutions. too narrow and/or overestimating the critical temperature of We treat proteins as hard spheres having square-well-energy bind- the phase diagram. Accounting for the discreteness of protein ing sites, using Wertheim’s thermodynamic perturbation theory. charges (11, 12) leads to better predictions, for example of small- The necessary condition required for such modeling to be realistic angle neutron scattering (13). Another approach for treating the is that proteins in solution during the experiment remain in their potential asymmetries is to assume heterogeneously charged compact form. Within this limitation our model gives accurate liq- surfaces, in the so-called patchy models (9, 14–17). Recent re- uid–liquid coexistence curves for lysozyme and γ IIIa-crystallin so- sults for the gas–liquid and fluid–solid coexistence equilibria for lutions in respective buffers. It provides good fits to the cloud- this type of model are in good agreement with experiments on point curves of lysozyme in buffer–salt mixtures as a function of lysozyme solutions, but it requires the assumption of a temper- the type and concentration of salt. It than predicts full coexis- ature-dependent pseudo-potential (16). However, another ap- tence curves, osmotic compressibilities, and second virial coeffic- proach is to assume the range and directionality of attractive ients under such conditions. This treatment may also be relevant interactions varies around the protein (9, 12, 18). A key con- to protein crystallization. clusion from these works is that to properly capture protein liquid-phase equilibria seems to require that the range of in- phase separation | protein aggregation | Hofmeister series teractions between proteins be short (18–22). Modeling proteins as rigid bodies has severe limitations, as rotein molecules can aggregate with each other. This process pointed out in the recent paper of Sarangapani et al. (23) and Pis important in many ways (1). First, a key step in developing highlighted by Prausnitz (24). When analyzing protein aggregation biotech drugs—which are mostly mABs—is to formulate pro- by such models, these studies indicate the importance of knowing teins so that they do not aggregate. This is because good shelf- that during the experiment the native structure is preserved. It is life requires long-term solution stability, and because patient recommended to use CD as a tool for verifying that no changes in compliance requires liquids having low viscosities. The impor- protein conformation take place during experiments. tance of such formulations comes from the fact that the world The cloud-point temperature measurements modeled here market for protein biologicals is about the same size as for were described in refs. 25 and 26. In experiments the tempera- smartphones. Second, protein aggregation in the cell plays a ture was first decreased until the transmitted beam would dis- key role in protein condensation diseases, such as Alzheimer’s, appear and then raised until the original scattering intensity was Parkinson’s, Huntington’s, and others. Third, much of structural restored. Reversible opacification was indicative that protein biology derives from the 100,000 protein structures in the Protein remained in the compact state. The transition between the sin- Data Bank, a resource that would not have been possible without gle-phase and the two-phase region is reversible and without protein crystals, a particular state of protein aggregation. Also, it hysteresis also for other proteins (see ref. 1, p. 165). Further is not yet possible to rationally design the conditions for proteins evidence on stability of the lysozyme, which is the central mol- to crystallize. ecule of our analysis, is provided in ref. 27, figure 6. The CD However, protein aggregation is poorly understood. Atomis- tic-level molecular simulations are not practical for studying Significance 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 Protein aggregation is a problem in amyloid and other diseases, other ligands, excipients, stabilizers, or metabolites. So, a traditional and it is a challenge when formulating solutions of biological approach is to adapt colloid theories, such as the Derjaguin– drugs, such as monoclonal antibodies. The physical processes of Landau–Verwey–Overbeek (DLVO) (2) theory. In those treat- aggregation, especially in salt solutions, are not well understood. ments, proteins are represented as spheres that interact through We model a protein as having multiple binding sites to other spherically symmetric van der Waals and electrostatic interactions proteins, leading to orientational variations, dependent on salt. in salt water, using a continuum representation of solvent and a With few parameters and with knowledge of the cloud-point – temperatures as a function of added salt, the model gives good Debye Hückel screening for salts. DLVO often gives correct – trends for the pH and salt concentration dependencies. However, predictions for properties including the liquid liquid coexistence DLVO does not readily account for protein sequence-structure curves, the second virial coefficients, and others for lysozyme properties, salt bridges (which are commonly the ‟glue” holding and gamma-crystallin. protein crystals together), explicit waters in general, or Hofmeister Author contributions: B.H.-L. and V.V. designed research; M.K. and Y.V.K. performed effects, where different salts have widely different powers of research; Y.V.K. helped create software; V.V. supervised the research; and K.A.D. and protein precipitation (3–5). A more subtle treatment is required V.V. wrote the paper. for these effects (6–8). Reviewers: S.G., Rensselaer Polytechnic Institute; and J.M.P., University of California, Coarse-grained statistical mechanics is essential for describing Berkeley. the properties of complex solutions. For proteins, such modeling The authors declare no conflict of interest. needs to go beyond central-force approximations because it is 1To whom correspondence should be addressed. Email: [email protected]. ‟ generally noted that the isotropic models fail to describe the This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. phase diagram of protein solutions quantitatively and cannot 1073/pnas.1507303112/-/DCSupplemental. 6766–6770 | PNAS | May 26, 2015 | vol. 112 | no. 21 www.pnas.org/cgi/doi/10.1073/pnas.1507303112 Downloaded by guest on September 26, 2021 ∞ for r < σ, u ðrÞ = [2] R 0for r ≥ σ, −«W for jxABj < aW, uABðxABÞ = [3] 0for jxABj ≥ aW. Here «W (>0) is the square-well depth of the binding potential and aW is its range. The interaction acts only when the site–site distance jxABj is smaller than aW, the square-well width. We avoid nonphysical multisite bonding by applying the following restriction (32, 34): pffiffiffi 0 < aW < σ − 3d. [4] Next, we assume that the free energy A of the solution is additive: Fig. 1. Proteins interact as two spheres. They interact at M × M pairs of binding sites on the surfaces, one pair of which (A and B) is indicated here. A = Aid + Ahs + Aass, [5] where Aid is the ideal free energy, Ahs is the hard sphere, and Aass the site– spectra of the lysozyme remain unchanged during the solubility site association contribution to the free energy. This latter term is adopted studies performed at protein concentrations similar to (or even from Wertheim’s thermodynamic perturbation theory (32, 33, 35): higher than) those modeled here. βAass X 1 The perturbed hard-sphere models, similar to the one pro- = M ln X − + , [6] N 2 2 posed below, have been successfully used before (see, e.g., refs. −1 19, 22, 25, 28, and 29). In these and other similar models the where β = ðkBTÞ and kB is Boltzmann’s constant. The association parameter perturbation part of potential was assumed to be isotropic. In X defines the average fraction of molecules not bonded to any site and is contrast to this we treat the protein–protein interaction as di- determined by the mass-action law (35): rectional. We model proteins as hard spheres, with a number of 1 X = . [7] square-well attractive sites located on the surface, which we call 1 + MXρΔAB the ‟binding sites.” In this aspect our approach resembles some The term Δ is defined by the expression (34) simple water models (30, 31). The modification seems to have AB + significant consequences for the shape of the liquid–liquid co- 2Zd aW hs 2 existence curve. We treat the solution physics through the ther- ΔAB = 4πg ðσÞ f assðrÞr dr, [8] modynamic perturbation theory that was developed by Wertheim σ for liquids that are strongly associating (32, 33). The model is used where f assðrÞ is the angular average of the Mayer function, obtained as to analyze various experimental data. described in ref. 34 (see also Eqs. S9 and S10). The radial distribution func- hsð Þ – Methods tion g r is obtained from the analytical solution of the Ornstein Zernike integral equation within the Percus–Yevick (PY) approximation (36). We model the protein solution as a one-component system of N protein Once the free energy A of the solution is known, we compute the pressure molecules with number density ρ = N=V at temperature T and volume V.