Structure and thermodynamics of colloidal protein cluster formation: comparison of square-well and simple dipolar models

Abstract

Reversible formation of weakly associated protein oligomers or clusters is a key early step in processes such as protein aggregation and colloidal phase separation. A previously developed cell-based, quasichemical model for lattice fluids [T. M. Young and C. J. Roberts, J. Chem. Phys. 127, 165101 (2007)] is extended here to treat continuous-space systems. It is illustrated using two simplified limiting cases for globular proteins at the isoelectric point: spherical square-well (SW) particles with an isotropic short-ranged attraction and screened dipolar particles with SW attractions and square-shoulder repulsions. Cluster free energies (DeltaA(i)) and structures are analyzed as a function of the reduced second virial coefficient b(2)(*). DeltaA(i) values and the average structures of clusters up to pentamers have distinct differences due to the anisotropic nature of the dipolar interactions. However, DeltaA(i) values can be mapped semiquantitatively between the two cases if compared at common values of b(2)(*). Free energy landscapes of oligomerization are constructed, illustrating significant differences in landscape ruggedness for small clusters of dipolar versus SW fluids, and suggesting a possible molecular interpretation for empirical models of nucleation-dependent aggregation of proteins.

Figure 1

Comparison between QC and MC for the probability of observing an oligomer composed of particles. Each symbol represents a different T^∗ and volumetric ρ(=Nπσ³∕V): fixed ρ=10⁻³ with T^∗ of 0.40 (●), 0.50 (◼), 0.60 (◆), and 0.65 (▲); fixed T^∗=1.0 with ρ=0.01 (○), 0.02 (◻), and 0.05 (◇). The line is a guide to the eyes for an ideal 1:1 correlation.

Figure 2

(a) Oligomer equilibrium constants based on a standard state concentration of 1 μM, with a HS or effective HS diameter of 4 nm, as a function of the reduced second virial coefficient. Black and gray lines represent the SW and the DP models, respectively. The label on each curve denotes the oligomer stoichiometry i. (b) Free energy of formation comparison between the DP and SW models. Each symbol represents a different cluster size: dimer (●), trimer (◼), tetramer (◆), and pentamer (▲). There are 12 points for each cluster size, representing a range of $b_2^{\ast }$ values from −20 to −0.01. The solid line is fit to the data with a slope of 0.895+0.014 and a y-intercept of −0.11+0.33 (95% confidence intervals). The dashed line is provided to illustrate an ideal 1:1 correlation.

Figure 3

(a) Internal energy change per monomer and (b) entropy change per monomer for cluster formation as a function of the reduced second virial coefficient. Black and gray lines represent the SW and DP models, respectively. The arrow on each plot marks the direction of increasing i.

Figure 4

Free energy landscape as a function of R_g∕σ for each cluster stoichiometry for a (a) $b_2^{\ast }=-10$ and (b) $b_2^{\ast }=-2.5$. Panel (c) shows the average ⟨R_g⟩∕σ vs $b_2^{\ast }$ for each cluster stoichiometry. In all panels, black and gray lines indicate SW and DP results, respectively.

Figure 5

Average orientational order parameter (χ), as a function of $b_2^{\ast }$. Cluster stoichiometry is denoted by the number on each curve.

Figure 6

Free energy landscapes vs χ at $b_2^{\ast }=-10$ (left column) and$b_2^{\ast }=-2.5$ (right column). Each row represents a different cluster size: dimer (first row), trimer (second row), tetramer (third row), and pentamer (fourth row).