We develop and describe a Bayesian statistical analysis to solve the surface brightness equations for Cepheid distances and stellar properties. Our analysis provides a mathematically rigorous and objective solution to the problem, including immunity from Lutz-Kelker bias. We discuss the choice of priors, show the construction of the likelihood distribution, and give sampling algorithms in a Markov chain Monte Carlo approach for efficiently and completely sampling the posterior probability distribution. Our analysis averages over the probabilities associated with several models rather than attempting to pick the '' best model '' from several possible models. Using a sample of 13 Cepheids we demonstrate the method. We discuss diagnostics of the analysis and the effects of the astrophysical choices going into the model. We show that we can objectively model the order of Fourier polynomial fits to the light and velocity data. By comparison with theoretical models of Bono et al. we find that EU Tau and SZ Tau are overtone pulsators, most likely without convective overshoot. The period-radius and period-luminosity relations we obtain are shown to be compatible with those in the recent literature. Specifically, we find logð R h iÞ ¼ ð0:693 AE 0:037Þ logðPÞ À 1:2 ½ þ ð2:042 AE 0:047Þ and M v h i ¼ Àð2:690 AE 0:169Þ logðPÞ À 1:2 ½ À ð 4:699 AE 0:216Þ.

Computing protein-protein association affinities is one of the fundamental challenges in computational biophysics/biochemistry. The overwhelming amount of statistics in the phase space of very high dimensions cannot be sufficiently sampled even with today's high-performance computing power. In this paper, we extend a potential of mean force (PMF)-based approach, the hybrid steered molecular dynamics (hSMD) approach we developed for ligand-protein binding, to protein-protein association problems. For a protein complex consisting of two protomers, P1 and P2, we choose m (≥3) segments of P1 whose m centers of mass are to be steered in a chosen direction and n (≥3) segments of P2 whose n centers of mass steered in the opposite direction. The coordinates of these m+n centers constitute a phase space of 3(m+n) dimensions (3(m+n)-D). All the other degrees of freedom of the proteins, ligands, solvents, and solutes are freely subject to the stochastic dynamics of the all-atom model system. Conducting SMD along a line in this phase space, we obtain the 3(m+n)-D PMF difference between two chosen states, one single state in the associated state ensemble and one single state in the dissociated state ensemble. This PMF difference is the first of four contributors to the protein-protein association energy. The second contributor is the 3(m+n−1)-D partial partition in the associated state counting for the rotations and fluctuations of the (m+n−1) centers while fixing one of the m+n centers of the P1-P2 complex. The two other contributors are the 3(m−1)-D partial partition of P1 and the 3(n−1)-D partial partition of P2 counting for the rotations and fluctuations of their m−1 or n−1 centers while fixing one of the m/n centers of P1/P2 in the dissociated state. Each of these three partial partitions can be factored exactly into a 6-D partial partition multiplying a remaining factor counting for the small fluctuations while fixing three of the centers of P1, P2, or the P1-P2 complex, respectively. These small fluctuations can be well approximated as Gaussian. And every 6-D partition can be reduced in an exact manner to three problems of “1-D sampling”, counting the rotations and fluctuations around one of the centers being fixed. We implement this hSMD approach to the Ras-RalGDS complex, choosing three centers on RalGDS and three on Ras (m=n=3). At a computing cost of about 71.6 wall-clock hours using 400 computing cores in parallel, we obtained the association energy, - 9.2±1.9 kcal/mol on the basis of CHARMM 36 parameters, which well agrees with the experimental data, - 8.4±0.2 kcal/mol.

Amyloid-β (Aβ) fibrils and plaques are one of the hallmarks of Alzheimer’s disease. While the kinetics of fibrillar growth of Aβ have been extensively studied, several vital questions remain. In particular, the atomistic origins of the Arrhenius barrier observed in experiments have not been elucidated. Employing the familiar thermodynamic integration method, we have directly simulated the dissociation of an Aβ(15–40) (D23N mutant) peptide from the surface of a filament along its most probable path (MPP) using all-atom molecular dynamics. This allows for a direct calculation of the free energy profile along the MPP, revealing a multipeak energetic barrier between the free peptide state and the aggregated state. By definition of the MPP, this simulated unbinding process represents the reverse of the physical elongation pathway, allowing us to draw biophysically relevant conclusions from the simulation data. Analyzing the detailed atomistic interactions along the MPP, we identify the atomistic origins of these peaks as resulting from the dock-lock mechanism of filament elongation. Careful analysis of the dynamics of filament elongation could prove key to the development of novel therapeutic strategies for amyloid-related diseases.

Computing the ligand-protein binding affinity (or the Gibbs free energy) with chemical accuracy has long been a challenge for which many methods/approaches have been developed and refined with various successful applications. False positives and, even more harmful, false negatives have been and still are a common occurrence in practical applications. Inevitable in all approaches are the errors in the force field parameters we obtain from quantum mechanical computation and/or empirical fittings for the intra- and inter-molecular interactions. These errors propagate to the final results of the computed binding affinities even if we were able to perfectly implement the statistical mechanics of all the processes relevant to a given problem. And they are actually amplified to various degrees even in the mature, sophisticated computational approaches. In particular, the free energy perturbation (alchemical) approaches amplify the errors in the force field parameters because they rely on extracting the small differences between similarly large numbers. In this paper, we develop a hybrid steered molecular dynamics (hSMD) approach to the difficult binding problems of a ligand buried deep inside a protein. Sampling the transition along a physical (not alchemical) dissociation path of opening up the binding cavity---pulling out the ligand---closing back the cavity, we can avoid the problem of error amplifications by not relying on small differences between similar numbers. We tested this new form of hSMD on retinol inside cellular retinol-binding protein 1 and three cases of a ligand (a benzylacetate, a 2-nitrothiophene, and a benzene) inside a T4 lysozyme L99A/M102Q(H) double mutant. In all cases, we obtained binding free energies in close agreement with the experimentally measured values. This indicates that the force field parameters we employed are accurate and that hSMD (a brute force, unsophisticated approach) is free from the problem of error amplification suffered by many sophisticated approaches in the literature.