Gerhard Hummer

The Hydration Number of Li+ in Liquid Water

The hydration of ions in water is not only fundamental to physical chemistry, but is also relevant to the current issue of selectivity of biological ion channels. In the context of potassium channels for example, the free energies for replacement of inner shell water ligands with peptide carbonyls donated by proteins of the channel, specifically for the preference of K{sup +} over Na{sup +}. Studies to elucidate the thermodynamic features of such inner shell exchange reactions require prior knowledge of the ion hydration structures and energetics. Simulations have produced a range of results including both four and six inner shell water neighbors with considerable statistical dispersion. Simulations are typically not designed to provide sole determinations of such properties, although they do shed light on the issues determining the hydration number of ions in water. The theoretical scheme used here to address these problems for the Li{sup +}(aq) ion is based upon the quasi-chemical organization of solution theory, which is naturally suited to these problems.

Ion sizes and finite-size corrections for ionic-solvation free energies

Free energies of ionic solvation calculated from computer simulations exhibit a strong system size dependence. We perform a finite-size analysis based on a dielectric-continuum model with periodic boundary conditions. That analysis results in an estimate of the Born ion size. Remarkably, the finite-size correction applies to systems with only eight water molecules hydrating a sodium ion and results in an estimate of the Born radius of sodium that agrees with the experimental value.

Water penetration and escape in proteins.

The kinetics of water penetration and escape in cytochrome c (cyt c) is studied by molecular dynamics (MD) simulations at various temperatures. Water molecules that penetrate the protein interior during the course of an MD simulation are identified by monitoring the number of water molecules in the first coordination shell (within 3.5 Å) of each water molecule in the system. Water molecules in the interior of cyt c have 0–3 water molecules in their first hydration shell and this coordination number persists for extended periods of time. At T = 300 K we identify over 200 events in which water molecules penetrate the protein and reside inside for at least 5 picoseconds (ps) within a 1.5 nanoseconds (ns) time period. Twenty‐seven (27) water molecules reside for at least 300 ps, 17 water molecules reside in the protein interior for times longer than 500 ps, and two interior water molecules do not escape; at T = 360 K one water molecule does not escape; at 430 K all water molecules exchange. Some of the internal water molecules show mean square displacements (MSD) of 1 Å2 characteristic of structural waters. Others show MSD as large as 12 Å2, suggesting that some of these water molecules occupy transient cavities and diffuse extensively within the protein. Motions of protein‐bound water molecules are rotationally hindered, but show large librations. Analysis of the kinetics of water escape in terms of a survival time correlation function shows a power law behavior in time that can be interpreted in terms of a broad distribution of energy barriers, relative to κBT, for water exchange. At T = 300 K estimates of the roughness of the activation energy distribution is 4–10 kJ/mol (2–4 κBT). Activation enthalpies for water escape are 6–23 kJ/mol. The difference in activation entropies between fast exchanging (0.01 ns) and slow exchanging (0.1–1 ns) water molecules is −27 J/K/mol. Dunitz (Science 1997;264:670.) has estimated the maximum entropy loss of a water molecule due to binding to be 28 J/K/mol. Therefore, our results suggest that the entropy of interior water molecules is similar to entropy of bulk water. Proteins 2000;38:261–272. Published 2000 Wiley‐Liss, Inc.

Calculation of free‐energy differences from computer simulations of initial and final states

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).

Bound water in the proton translocation mechanism of the haem-copper oxidases

We address the molecular mechanism by which the haem‐copper oxidases translocate protons. Reduction of O2 to water takes place at a haem iron‐copper (CuB) centre, and protons enter from one side of the membrane through a ‘channel’ structure in the enzyme. Statistical‐mechanical calculations predict bound water molecules within this channel, and mutagenesis experiments show that breaking this water structure impedes proton translocation. Hydrogen‐bonded water molecules connect the channel further via a conserved glutamic acid residue to a histidine ligand of CuB. The glutamic acid side chain may have to move during proton transfer because proton translocation is abolished if it is forced to interact with a nearby lysine or arginine. Perturbing the CuB ligand structure shifts an infrared mode that may be ascribed to the OH stretch of bound water. This is sensitive to mutations of the glutamic acid, supporting its connectivity to the histidine. These results suggest key roles of bound water, the glutamic acid and the histidine copper ligand in the mechanism of proton translocation.

Ion pair potentials-of-mean-force in water

Motivated by the surprising dielectric model predictions of alkali-halide ion pair potentials-of-mean-force in water due to Rashin, we reanalyze the theoretical bases of that comparison. We discuss recent, pertinent molecular simulation and integral equation results that have appeared for these systems. We implement dielectric model calculations to check the basic features of Rashins calculations. We confirfn that the characteristic structure of contact and solvent-separated minima does appear in the dielectric model results for the pair potentials-of-mean-force for oppositely charged ions in water under physiological thermodynamic conditions. Comparison of the dielectric model results with the most current molecular level information indicates that the dielectric model does not, however, provide an accurate description of these potentials-of-mean-force. Since literature results indicate that dielectric models can be helpfully accurate on a coarse, or chemical energy scale, we consider how they might be based more firmly on molecular theory. The objective is a parameterization better controlled by molecular principles and thus better adapted to the prediction of quantities of physical interest. Such a result might be expected to describe better the thermal-level energy changes associated with simple molecular rearrangements, i.e. ion pair potentials-of-mean-force. We note that linear dielectric models correspond to modelistic implementations of second-order thermodynamic perturbation theory for the excess chemical potential of a distinguished solute molecule at infinite dilution. Therefore, the molecular theory corresponding to the dielectric models is second-order thermodynamic perturbation theory for that excess chemical potential. Examination of the required formulae indicate that this corresponding molecular theory should be quite amenable to computational implementation. The second-order, or fluctuation, term raises a technical computational issue of treatment of long-ranged interactions similar to the one which arises in calculation of the dielectric constant of the solvent. Satisfactory calculation of that term will require additional theoretical consideration of those issues. It is contended that the most important step for further development of dielectric models would be a separate assessment of the first-order perturbative term (equivalently thepotential at zero charge) which vanishes in the dielectric models but is generally nonzero. Parameterization of radii and molecular volumes should then be based on the second-order perturbative term alone. Illustrative initial calculations are presented and discussed.

Conformational dynamics of cytochrome c: correlation to hydrogen exchange.

We study the dynamical fluctuations of horse heart cytochrome c by molecular dynamics (MD) simulations in aqueous solution, at four temperatures: 300 K, 360 K, 430 K, and 550 K. Each simulation covers a production time of at least 1.5 nanoseconds (ns). The conformational dynamics of the system is analyzed in terms of collective motions that involve the whole protein, and local motions that involve the formation and breaking of intramolecular hydrogen bonds. The character of the MD trajectories can be described within the framework of rugged energy landscape dynamics. The MD trajectories sample multiple conformational minima, with basins in protein conformational space being sampled for a few hundred picoseconds. The trajectories of the system in configurational space can be described in terms of diffusion of a particle in real space with a waiting time distribution due to partial trapping in shallow minima. As a consequence of the hierarchical nature of the dynamics, the mean square displacement autocorrelation function, 〈|x(t) − x(0)|2〉, exhibits a power law dependence on time, with an exponent of around 0.5 for times shorter than 100 ps, and an exponent of 1.75 for longer times. This power law behavior indicates that the system exhibits suppressed diffusion (sub‐diffusion) in sampling of configurational space at time scales shorter than 100 ps, and enhanced (super‐diffusion) at longer time scales. The multi‐basin feature of the trajectories is present at all temperatures simulated. Structural changes associated with inter‐basin displacements correspond to collective motions of the Ω loops and coiled regions and relative motions of the α‐helices as rigid bodies. Similar motions may be involved in experimentally observed amide hydrogen exchange. However, some groups showing large correlated motions do not expose the amino hydrogens to the solvent. We show that large fluctuations are not necessarily correlated to hydrogen exchange. For example, regions of the proteins forming α helices and turns show significant fluctuations, but as rigid bodies, and the hydrogen bonds involved in the formation of these structures do not break in proportion to these fluctuations. Proteins 1999;36:175–191. Published 1999 Wiley‒Liss, Inc.

STRUCTURE AND DYNAMICS OF A PROTON SHUTTLE IN CYTOCHROME C OXIDASE

Abstract Protein-assisted transport of protons across the bioenergetic membrane is mediated by hydrogen-bonded networks. These networks involve titratable amino acid residues of membrane-spanning protein assemblies as well as internal water molecules. In cytochrome c oxidase, the so-called D-channel defines such a network for the uptake of protons from the cytoplasmic side of the membrane. It has been proposed that conformational changes of a Glu residue are required for the establishment of a proton linkage from the channel into the active site. The thermodynamic basis for the conformational isomerization of this residue is investigated using simulated annealing and free energy molecular dynamics simulations. The results support the existence of metastable conformations of the side chain, and their interchange through local structural fluctuations of neighboring residues and nearby internal chains of water molecules. The conformational isomerization of both protonated and unprotonated states of Glu, coupled with the reorganization of hydrogen bonds, suggests a kinetically competent mechanism for proton shuttling.

Multi-basin dynamics of a protein in a crystal environment

Abstract The dynamics of the small protein crambin is studied in the crystal environment by means of a 5.1 nanoseconds molecular dynamics (MD) simulation. The resulting trajectory is analyzed in terms of a small set of nonlinear dynamical modes that best describe the molecules fluctuations. These modes are nonlinear in the sence that they describe a trajectory exhibiting multiple transitions among local minima at various timescales. Nonlinear modes are responsible for most of the protein atomic fluctuations. An ultrametric hierarchy of sampled local minima describes the protein trajectory when structures are classified in terms of their interconfigurational mean squared distance. Transitions among minima involve small changes in the relative atomic positions of many atoms in the protein. The character of the MD trajectory fits within the framework of rugged energy landscape dynamics. This MD simulation clarifies the unique statistical features of the barriers between minima in the energy-like configurational landscape. Longer timescale dynamics seem to sample transitions between minima separated by relatively higher barriers. The MD trajectory of the system in configurational space can be described in terms of diffusion of a particle in real space with a waiting time distribution due to partial trapping in shallow minima. A description of the dynamics in terms of an open Newtonian system (the protein) coupled to a stochastic system (the solvent and fast quasiharmonic modes of the protein) reveals that the system loses memory of its configurational space within a few picoseconds. The diffusion of the protein in configurational space is anomalous in the sense that the mean square displacement increases sublinearly with time, i.e., as a power law with an exponent that is smaller than unity.

Pressure calculation in polar and charged systems using Ewald summation: Results for the extended simple point charge model of water

Ewald summation and physically equivalent methods such as particle-mesh Ewald, kubic-harmonic expansions, or Lekner sums are commonly used to calculate long-range electrostatic interactions in computer simulations of polar and charged substances. The calculation of pressures in such systems is investigated. We find that the virial and thermodynamic pressures differ because of the explicit volume dependence of the effective, resummed Ewald potential. The thermodynamic pressure, obtained from the volume derivative of the Helmholtz free energy, can be expressed easily for both ionic and rigid molecular systems. For a system of rigid molecules, the electrostatic energy and the forces at the atom positions are required, both of which are readily available in molecular dynamics codes. We then calculate the virial and thermodynamic pressures for the extended simple point charge (SPC/E) water model at standard conditions. We find that the thermodynamic pressure exhibits considerably less system size dependence th...