Implicit solvent simulations of DPC micelle formation

The formation of micelles by dodecylphosphocholine (DPC) is modeled by treating the surfactants in atomic detail and the solvent implicitly, in the spirit of the EEF1 solvation model for proteins. The solvation parameters of the DPC atoms are carried over from those of similar atoms in proteins. A slight adjustment of the parameters for the headgroup was found necessary for obtaining an aggregation number consistent with experiment. Molecular dynamics simulations of 960 DPC molecules at different concentrations are used to obtain the aggregation number, the micelle size distribution, and the CMC. At 20 mM concentration we obtain an aggregation number of 53-56 and a CMC of 1.25 mM, values close to the experimental ones. At 100 mM the aggregation number increases to 90. Simulations of individual micelles of varying size show that the effective energy per surfactant molecule is initially a decreasing function of aggregation number but stabilizes at about 60 molecules. The van der Waals term and the desolvation of nonpolar groups contribute to micellization, whereas the desolvation of polar groups opposes it. From the difference between the effective energy and the free energy (calculated from the CMC), the translational and rotational entropy contributions to the free energy are estimated at about 7 kcal/mol per monomer. The micelles obtained here are more irregular than those obtained in explicit water simulations. This modeling approach allows the study of larger surfactant aggregates for longer times and the extraction of thermodynamic in addition to structural information.     

Membrane protein native state discrimination by implicit membrane models

Four implicit membrane models [IMM1, generalized Born (GB)‐surface area‐implicit membrane (GBSAIM), GB with a simple switching (GBSW), and heterogeneous dielectric GB (HDGB)] were tested for their ability to discriminate the native conformation of five membrane proteins from 450 decoys generated by the Rosetta‐Membrane program. The energy ranking of the native state and Z‐scores were used to assess the performance of the models. The effect of membrane thickness was examined and was found to be substantial. Quite satisfactory discrimination was achieved with the all‐atom IMM1 and GBSW models at 25.4 Å thickness and with the HDGB model at 28.5 Å thickness. The energy components by themselves were not discriminative. Both van der Waals and electrostatic interactions contributed to native state discrimination, to a different extent in each model. Computational efficiency of the models decreased in the order: extended‐atom IMM1 > all‐atom IMM1 > GBSAIM > GBSW > HDGB. These results encourage the further development and use of implicit membrane models for membrane protein structure prediction. © 2012 Wiley Periodicals, Inc.     

Locally symmetric and ricci-symmetric contact metric manifolds

We have characterized locally symmetric and Ricci-symmetric contact metric manifolds of dimension greater than 3, by assuming certain conditions on the curvature and Ricci curvature along the characteristic vector field of the contact structure.     

Menshov’s “adjustment theorem” with respect to general measures

We prove Menshov’s theorem in the setting of arbitrary Borel measures.     

Ruled surfaces with vanishing second Gaussian curvature

For a surface free of points of vanishing Gaussian curvature in Euclidean space the second Gaussian curvature is defined formally. It is first pointed out that a minimal surface has vanishing second Gaussian curvature but that a surface with vanishing second Gaussian curvature need not be minimal. Ruled surfaces for which a linear combination of the second Gaussian curvature and the mean curvature is constant along the rulings are then studied. In particular the only ruled surface in Euclidean space with vanishing second Gaussian curvature is a piece of a helicoid.     

A Characterization of Vanishing Mean Oscillation

We show that a function has vanishing mean oscillation with respect to a nonatomic measure if and only if it satisfies an asymptotic reverse Jensen inequality.     

Calculations of pH-dependent binding of proteins to biological membranes

Binding of proteins to membranes is often accompanied by titration of ionizable residues and is, therefore, dependent on pH. We present a theoretical treatment and computational approach for predicting absolute, pH-dependent membrane binding free energies. The standard free energy of binding, DeltaG, is defined as -RTln(P(b)/P(f)), where P(b) and P(f) are the amounts of bound and free protein. The apparent pK(a) of binding is the pH value at which P(b) and P(f) are equal. Proteins bind to the membrane in the pH range where DeltaG is negative. The components of the binding free energy are (a) the free energy cost of ionization state changes (DeltaG(ion)), (b) the effective energy of transfer from solvent to the membrane surface, (c) the translational/rotational entropy cost of binding, and (d) an ideal entropy term that depends on the relative volume of the bound and free state and therefore depends on lipid concentration. Calculation of the first term requires determination of pK(a) values in solvent and on the membrane surface. All energies required by the method are obtained from molecular dynamics trajectories on an implicit membrane (IMM1-GC). The method is tested on pentalysine and the helical peptide VEEKS, derived from the membrane-binding domain of phosphocholine cytidylyltransferase. The agreement between the measured and the calculated free energies of binding of pentalysine is good. The extent of membrane binding of VEEKS is, however, underestimated compared to experiment. Calculations of the interaction energy between two VEEKS helices on the membrane suggest that the discrepancy is mainly due to the neglect of protein-protein interactions on the membrane surface.     

Water at biomolecular binding interfaces

Water molecules are often found at the binding interface of biomolecular complexes mediating the interaction between polar groups via hydrogen bonds, or simply filling space providing van der Waals interactions. Recent studies have demonstrated the importance of taking such water molecules into account in docking and binding affinity prediction. Here, we review the recent experimental and theoretical work aimed at quantifying the influence of interfacial water on the thermodynamic properties of binding. We highlight especially our recent results obtained by inhomogeneous fluid solvation theory in several systems and the prediction of the thermodynamic consequences of displacement of the bound water molecule by ligand modification. Finally, we discuss possible directions for further progress in this field.