We describe a model for protein crystallization equilibria. The model includes four terms, (1) protein translational entropy opposes crystallization, (2) proteins are attracted to each other by a nonelectrostatic contact free energy favoring crystallization, (3) proteins in the crystal repel each other but, to a greater extent, attract counterions sequestered in the crystal, which favors crystallization, and (4) the translational entropy of the counterions opposes their sequestration into the crystal, opposing crystallization. We treat the electrostatics using the nonlinear Poisson−Boltzmann equation, and we use unit cell information from native protein crystals to determine the boundary conditions. This model predicts the stabilities of protein crystals as functions of temperature, pH, and salt concentrations, in good agreement with the data of Pusey et al. on tetragonal and orthorhombic crystal forms of lysozyme. The experiments show a weak dependence of crystal solubility on pH. According to the model, this is because the entropic cost to neutralize the crystal is compensated by favorable protein−salt interactions. Experiments also show that adding salt stabilizes the crystal. Cohn’s empirical law predicts that the logarithm of solubility should be a linear function of salt. The present theory predicts nonlinearity, in better agreement with the experiments. The model shows that the salting out phenomena is not due to more counterion shielding but to lowered counterion translational entropy. Models of this type may help guide faster and better ways to crystallize proteins.

New amino acid sequences of proteins are being learned at a rapid rate, thanks to modern genomics. The native structures and functions of those proteins can often be inferred using bioinformatics methods. We show here that it is also possible to infer the stabilities and thermal folding properties of proteins, given only simple genomics information: the chain length and the numbers of charged side chains. In particular, our model predicts ΔH(T), ΔS(T), ΔCp, and ΔF(T) —the folding enthalpy, entropy, heat capacity, and free energy—as functions of temperature T; the denaturant m values in guanidine and urea; the pH-temperature-salt phase diagrams, and the energy of confinement F(s) of the protein inside a cavity of radius s. All combinations of these phase equilibria can also then be computed from that information. As one illustration, we compute the pH and salt conditions that would denature a protein inside a small confined cavity. Because the model is analytical, it is computationally efficient enough that it could be used to automatically annotate whole proteomes with protein stability information.

Although many properties of electrolyte solutions can be successfully described by theories at the McMillan Mayer level of approximation, there are other phenomena that cannot be explained without taking into account the explicit nature of solvent molecules. One of these that have received much attention is the Hofmeister effect that describes the influence of different types of ions on the solubility of hydrophobic molecules in water. In this work we use two simple water models, the ‘fused-spheres’ and the two-dimensional ‘Mercedes-Benz’ models to study ion solvation in water, and test suppositions about their effect on hydrophobicity. Both models give good qualitative agreement with experiment, such as Samoilov ion hydration activation energies, and Setchenow coefficients, which describe the salt concentration dependence of the solubilities of hydrophobic solutes. The results suggest that the interactions of ions with water are governed mostly by the ionic charge densities. Water structure is determined by the balance of electrostatic forces and the tendency for hydrogen bond formation. Ions with a high charge density bind water molecules very tightly and therefore exclude the hydrophobe from their first shell, leading to salting-out. The effect decreases with decreasing charge density of the ion.

Proteins are complex molecules, yet their folding kinetics is often fast (microseconds) and simple, involving only a single exponential function of time (called two-state kinetics). The main model for two-state kinetics has been transition-state theory, where an energy barrier defines a slow step to reach an improbable structure. But how can barriers explain fast processes, such as folding? We study a simple model with rigorous kinetics that explains the high speed instead as a result of the microscopic parallelization of folding trajectories. The single exponential results from a separation of timescales; the parallelization of routes is high at the start of folding and low thereafter. The ensemble of rate-limiting chain conformations is different from in transition-state theory; it is broad, overlaps with the denatured state, is not aligned along a single reaction coordinate, and involves well populated, rather than improbable, structures.

We present a method for the computer-based iterative assembly of native-like tertiary structures of helical proteins from α-helical fragments. For any pair of helices, our method, called MATCHSTIX, first generates an ensemble of possible relative orientations of the helices with various ways to form hydrophobic contacts between them. Those conformations having steric clashes, or a large radius of gyration of hydrophobic residues, or with helices too far separated to be connected by the intervening linking region, are discarded. Then, we attempt to connect the two helical fragments by using a robotics-based loop-closure algorithm. When loop closure is feasible, the algorithm generates an ensemble of viable interconnecting loops. After energy minimization and clustering, we use a representative set of conformations for further assembly with the remaining helices, adding one helix at a time. To efficiently sample the conformational space, the order of assembly generally proceeds from the pair of helices connected by the shortest loop, followed by joining one of its adjacent helices, always proceeding with the shorter connecting loop. We tested MATCHSTIX on 28 helical proteins each containing up to 5 helices and found it to heavily sample native-like conformations. The average rmsd of the best conformations for the 17 helix-bundle proteins that have 2 or 3 helices is less than 2 Å; errors increase somewhat for proteins containing more helices. Native-like states are even more densely sampled when disulfide bonds are known and imposed as restraints. We conclude that, at least for helical proteins, if the secondary structures are known, this rapid rigid-body maximization of hydrophobic interactions can lead to small ensembles of highly native-like structures. It may be useful for protein structure prediction.

A central challenge in structure-based ligand design is the accurate prediction of binding free energies. Here we apply alchemical free energy calculations in explicit solvent to predict ligand binding in a model cavity in T4 lysozyme. Even in this simple site, there are challenges. We made systematic improvements, beginning with single poses from docking, then including multiple poses, additional protein conformational changes, and using an improved charge model. Computed absolute binding free energies had an RMS error of 1.9 kcal/mol relative to previously determined experimental values. In blind prospective tests, the methods correctly discriminated between several true ligands and decoys in a set of putative binders identified by docking. In these prospective tests, the RMS error in predicted binding free energies relative to those subsequently determined experimentally was only 0.6 kcal/mol. X-ray crystal structures of the new ligands bound in the cavity corresponded closely to predictions from the free energy calculations, but sometimes differed from those predicted by docking. Finally, we examined the impact of holding the protein rigid, as in docking, with a view to learning how approximations made in docking affect accuracy and how they may be improved.