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The weighted histogram analysis method (WHAM) has become the standard technique for the analysis of umbrella sampling simulations. In this article, we address the challenges (1) of obtaining fast and accurate solutions of the coupled nonlinear WHAM equations, (2) of quantifying the statistical errors of the resulting free energies, (3) of diagnosing possible systematic errors, and (4) of optimally allocating of the computational resources. Traditionally, the WHAM equations are solved by a fixed-point direct iteration method, despite poor convergence and possible numerical inaccuracies in the solutions. Here, we instead solve the mathematically equivalent problem of maximizing a target likelihood function, by using superlinear numerical optimization algorithms with a significantly faster convergence rate. To estimate the statistical errors in one-dimensional free energy profiles obtained from WHAM, we note that for densely spaced umbrella windows with harmonic biasing potentials, the WHAM free energy profile can be approximated by a coarse-grained free energy obtained by integrating the mean restraining forces. The statistical errors of the coarse-grained free energies can be estimated straightforwardly and then used for the WHAM results. A generalization to multidimensional WHAM is described. We also propose two simple statistical criteria to test the consistency between the histograms of adjacent umbrella windows, which help identify inadequate sampling and hysteresis in the degrees of freedom orthogonal to the reaction coordinate. Together, the estimates of the statistical errors and the diagnostics of inconsistencies in the potentials of mean force provide a basis for the efficient allocation of computational resources in free energy simulations., (Copyright © 2011 Wiley Periodicals, Inc.)

Proton-coupled electron transfer (PCET) is an elementary chemical reaction crucial for biological oxidoreduction. We perform quantum chemical calculations to study the direct and water-mediated PCET between two stacked tyrosines, TyrO(•) + TyrOH → TyrOH + TyrO(•), to mimic a key step in the catalytic reaction of class Ia ribonucleotide reductase (RNR). The energy surfaces of electronic ground and excited states are separated by a large gap of ~20 kcal mol(-1), indicative of an electronically adiabatic transfer mechanism. In response to chemical substitutions of the proton donor, the energy of the transition state for direct PCET shifts by exactly half of the change in energetic driving force, resulting in a linear free energy relation with a Brønsted slope of ½. In contrast, for water-mediated PCET, we observe integer Brønsted slopes of 1 and 0 for proton acceptor and donor modifications, respectively. Our calculations suggest that the π-stacking of the tyrosine dimer in RNR results in strong electronic coupling and adiabatic PCET. Water participation in the PCET can be identified perturbatively in a Brønsted analysis.

Water in the protein interior serves important structural and functional roles and is also increasingly recognized as a relevant factor in drug binding. The nonpolar cavity in the protein interleukin-1β has been reported to be filled by water on the basis of some experiments and simulations and to be empty on the basis of others. Here we study the thermodynamics of filling the central nonpolar cavity and the four polar cavities of interleukin-1β by molecular dynamics simulation. We use different water models (TIP3P and SPC/E) and protein force fields (amber94 and amber03) to calculate the semigrand partition functions term by term that quantify the hydration equilibria. We consistently find that water in the central nonpolar cavity is thermodynamically unstable, independent of force field and water model. The apparent reason is the relatively small size of the cavity, with a volume less than ∼80 Å(3). Our results are consistent with the most recent X-ray crystallographic and simulation studies but disagree with an earlier interpretation of nuclear magnetic resonance (NMR) experiments probing protein-water interactions. We show that, at least semiquantitatively, the measured nuclear Overhauser effects indicating the proximity of water to the methyl groups lining the nonpolar cavity can, in all likelihood, be attributed to interactions with buried and surface water molecules near the cavity. The same methods applied to determine the occupancy of the polar cavities show that they are filled by the same number of water molecules observed in crystallography, thereby validating the theoretical and simulation methods used to study the water occupancy in the nonpolar protein cavity.

Nonequilibrium pulling experiments provide detailed information about the thermodynamic and kinetic properties of molecules. We show that unperturbed free energy profiles as a function of molecular extension can be obtained rigorously from such experiments without using work-weighted position histograms. An inverse Weierstrass transform is used to relate the system free energy obtained from the Jarzynski equality directly to the underlying molecular free energy surface. An accurate approximation for the free energy surface is obtained by using the method of steepest descent to evaluate the inverse transform. The formalism is applied to simulated data obtained from a kinetic model of RNA folding, in which the dynamics consists of jumping between linker-dominated folded and unfolded free energy surfaces.

We study the effects of nanoscopic roughness on the interfacial free energy of water confined between solid surfaces. SPC/E water is simulated in confinement between two infinite planar surfaces that differ in their physical topology: one is smooth and the other one is physically rough on a sub-nanometre length scale. The two thermodynamic ensembles considered, with constant pressure either normal or parallel to the walls, correspond to different experimental conditions. We find that molecular-scale surface roughness significantly increases the solid-liquid interfacial free energy compared to the smooth surface. For our surfaces with a water-wall interaction energy minimum of -1.2 kcal mol(-1), we observe a transition from a hydrophilic surface to a hydrophobic surface at a roughness amplitude of about 3 angstroms and a wavelength of 11.6 angstroms, with the interfacial free energy changing sign from negative to positive. In agreement with previous studies of water near hydrophobic surfaces, we find an increase in the isothermal compressibility of water with increasing surface roughness. Interestingly, average measures of the water density and hydrogen-bond number do not contain distinct signatures of increased hydrophobicity. In contrast, a local analysis indicates transient dewetting of water in the valleys of the rough surface, together with a significant loss of hydrogen bonds, and a change in the dipole orientation toward the surface. These microscopic changes in the density, hydrogen bonding, and water orientation contribute to the large increase in the interfacial free energy, and the change from a hydrophilic to a hydrophobic character of the surface.

The reactive flux between folded and unfolded states of a two-state protein, whose coarse-grained dynamics is described by a master equation, is expressed in terms of the commitment or splitting probabilities of the microstates in the bottleneck region. This allows one to determine how much each transition through a dividing surface contributes to the reactive flux. By repeating the analysis for a series of dividing surfaces or, alternatively, by partitioning the reactive flux into contributions of unidirectional pathways that connect reactants and products, insight can be gained into the mechanism of protein folding. Our results for the flux in a network with complex connectivity, obtained using the discrete counterpart of Kramers' theory of activated rate processes, show that the number of reactive transitions is typically much smaller than the total number of transitions that cross a dividing surface at equilibrium.

Dynamic force spectroscopy probes the kinetic and thermodynamic properties of single molecules and molecular assemblies. Here, we propose a simple procedure to extract kinetic information from such experiments. The cornerstone of our method is a transformation of the rupture-force histograms obtained at different force-loading rates into the force-dependent lifetimes measurable in constant-force experiments. To interpret the force-dependent lifetimes, we derive a generalization of Bell's formula that is formally exact within the framework of Kramers theory. This result complements the analytical expression for the lifetime that we derived previously for a class of model potentials. We illustrate our procedure by analyzing the nanopore unzipping of DNA hairpins and the unfolding of a protein attached by flexible linkers to an atomic force microscope. Our procedure to transform rupture-force histograms into the force-dependent lifetimes remains valid even when the molecular extension is a poor reaction coordinate and higher-dimensional free-energy surfaces must be considered. In this case the microscopic interpretation of the lifetimes becomes more challenging because the lifetimes can reveal richer, and even nonmonotonic, dependence on the force.

We present a unified framework for extracting kinetic information from single-molecule pulling experiments at constant force or constant pulling speed. Our procedure provides estimates of not only (i) the intrinsic rate coefficient and (ii) the location of the transition state but also (iii) the free energy of activation. By analyzing simulated data, we show that the resulting rates of force-induced rupture are significantly more reliable than those obtained by the widely used approach based on Bell's formula. We consider the uniqueness of the extracted kinetic information and suggest guidelines to avoid over-interpretation of experiments.

A class of simple expressions of increasing accuracy for the free-energy difference between two states is derived based on numerical thermodynamic integration. The implementation of these formulas requires simulations of the initial and final (and possibly a few intermediate) states. They involve higher free-energy derivatives at these states which are related to the moments of the probability distribution of the perturbation. Given a specified number of such derivatives, these integration formulas are optimal in the sense that they are exact to the highest possible order of free-energy perturbation theory. The utility of this approach is illustrated for the hydration free energy of water. This problem provides a quite stringent test because the free energy is a highly nonlinear function of the charge so that even fourth order perturbation theory gives a very poor estimate of the free-energy change. Our results should prove most useful for complex, computationally demanding problems where free-energy differences arise primarily from changes in the electrostatic interactions (e.g., electron transfer, charging of ions, protonation of amino acids in proteins). © 1996 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Examines the use of Ewald summation in calculating electrostatic interactions in computer simulations of polar and charged substances. Derivation of formula to be implemented in standard molecular dynamics; Difference between virial and thermodynamic pressures; Model used in investigating the difference between virial and thermodynamic pressures.

Water expulsion from the protein core is a key step in protein folding. Nevertheless, unusually large water clusters confined into the nonpolar cavities have been observed in the X-ray crystal structures of tetrabrachion, a bacterial protein that is thermostable up to at least 403 K (130 °C). Here, we use molecular dynamics (MD) simulations to investigate the structure and thermodynamics of water filling the largest cavity of the right-handed coiled-coil stalk of tetrabrachion at 365 K (92 °C), the temperature of optimal bacterial growth, and at room temperature (298 K). Hydrogen-bonded water clusters of seven to nine water molecules are found to be thermodynamically stable in this cavity at both temperatures, confirming the X-ray studies. Stability, as measured by the transfer free energy of the optimal size cluster, decreases with increasing temperature. Water filling is thus driven by the energy of transfer and opposed by the transfer entropy, both depending only weakly on temperature. Our calculations suggest that cluster formation becomes unfavorable at ∼384 K (110 °C), signaling the onset of drying just slightly above the temperature of optimal growth. ‘Drying’ thus precedes protein denaturation. At room temperature, the second largest cavity in tetrabrachion accommodates a five water molecule cluster, as reported in the X-ray studies. However, the simulations show that at 365 K the cluster is unstable and breaks up. We suggest that the large hydrophobic cavities may act as binding sites for two proteases, possibly explaining the unusual thermostability of the resulting protease-stalk complexes (up to ∼393 K, 120 °C). [ABSTRACT FROM AUTHOR]

We used molecular dynamics simulations to investigate the thermodynamics of filling of a (6,6) open carbon nanotube (diameter D = 0.806 nm) solvated in TIP3P water over a temperature range from 280 K to 320 K at atmospheric pressure. In simulations of tubes with slightly weakened carbon-water attractive interactions, we observed multiple filling and emptying events. From the water occupancy statistics, we directly obtained the free energy of filling, and from its temperature dependence the entropy of filling. We found a negative entropy of about -1.3 kB per molecule for filling the nanotube with a hydrogen-bonded single-file chain of water molecules. The entropy of filling is nearly independent of the strength of the attractive carbon-water interactions over the range studied. In contrast, the energy of transfer depends strongly on the carbon-water attraction strength. These results are in good agreement with entropies of about -0.5 kB per water molecule obtained from grand-canonical Monte Carlo calculations of water in quasi-infinite tubes in vacuum under periodic boundary conditions. Overall, for realistic carbon-water interactions we expect that at ambient conditions filling of a (6,6) carbon nanotube open to a water reservoir is driven by a favorable decrease in energy, and opposed by a small loss of water entropy. [ABSTRACT FROM AUTHOR]

Equilibrium free energy differences can be calculated accurately from repeated fast-growth thermodynamic integration (TI) based on Jarzynski's identity [Phys. Rev. Lett. 78, 2690 (1997)]. We derive expressions for the free energy differences. Error estimates allow us to quantify the relative efficiency of performing many fast-growth vs few slow-growth TIs for a given total computational cost. Fast-growth TI is illustrated through the calculation of the potential of mean force between two methane molecules and compared to umbrella sampling analyzed by using the weighted histogram analysis method. Fast-growth TI is well suited for parallel computer architectures, requiring only the simplest parallelism with repeated runs for different starting conditions. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]