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Molecular Dynamics Computations for Proteins: A Case Study in Membrane Ion Permeation
Abstract
Computer simulation can provide an atomic-level view of protein structure and function that is unattainable with experiments alone. In this chapter, we demonstrate how molecular dynamics (MD) simulation can reveal the microscopic mechanisms of biomolecular activity. In particular, we illustrate the quantitative value of MD simulation by exploring ion channel proteins that allow selective permeation of charged molecules across cell membranes to control our nervous systems and chemical activity in the body. We begin by introducing the basic statistical mechanical formulation and methodological aspects of modern day MD simulations. We then discuss how to analyze a simulation to obtain mechanistic insight through calculation of various molecular properties. Special attention is given to quantitative thermodynamic calculations, which are the driving forces of protein function. We explain how to calculate free energies and equilibrium constants using potential of mean force (PMF) and free energy perturbation (FEP) techniques as well as the use of more advanced sampling approaches. These methods are then used to study ion permeation through the gramicidin A (gA) channel as well as the energetics of arginine side chain translocation across a lipid bilayer to assess models of voltage-gated ion channel activation.
Keywords:
mechanical force fields;
MD methods;
MD simulation;
MD trajectories;
MM force fields
... In the simple case of water, correct gas-phase properties have been sacrificed to get accurate condensed-phase properties in most additive water models. 91 A water dipole changes dramatically between gas and water phases, and this cannot be included in a fixed charge model. A polarizable model, however, can simultaneously achieve accurate gas-phase (dipole moment, water structure, and energetics) and condensed-phase (density, heat of vaporization, and dielectric constant) properties. ...
... One may also group interactions into contributions that help explain the process, such as direct interactions and indirect strain energies, involving subset of terms. 91 Such analyses help us understand how ion−ion, ion− ligand, and ligand−ligand energy terms contribute to complex formation within an ion channel. A fine balance of these large opposing contributions, determined by system geometry and force field parameters, may control ion conduction, as explained below, as well as some channel activation and inactivation mechanisms. ...
Membrane ion channels are the fundamental electrical components in the nervous system. Recent developments in X-ray crystallography and cryo-EM microscopy have revealed what these proteins look like in atomic detail but do not tell us how they function. Molecular dynamics simulations have progressed to the point that we can now simulate realistic molecular assemblies to produce quantitative calculations of the thermodynamic and kinetic quantities that control function. In this review, we summarize the state of atomistic simulation methods for ion channels to understand their conduction, activation, and drug modulation mechanisms. We are at a crossroads in atomistic simulation, where long time scale observation can provide unbiased exploration of mechanisms, supplemented by biased free energy methodologies. We illustrate the use of these approaches to describe ion conduction and selectivity in voltage-gated sodium and acid-sensing ion channels. Studies of channel gating present a significant challenge, as activation occurs on longer time scales. Enhanced sampling approaches can ensure convergence on minimum free energy pathways for activation, as illustrated here for pentameric ligand-gated ion channels that are principal to nervous system function and the actions of general anesthetics. We also examine recent studies of local anesthetic and antiepileptic drug binding to a sodium channel, revealing sites and pathways that may offer new targets for drug development. Modern simulations thus offer a range of molecular-level insights into ion channel function and modulation as a learning platform for mechanistic discovery and drug development.
... In this regard, molecular dynamics (MD) simulations have become an indispensable method to explore the dynamics of biomolecular systems. Indeed, as a technique, MD has become a key tool in structural biology [42][43][44][45][46]. It enables an atomistic-level view of processes at a resolution, both spatially and temporally, that is currently inaccessible by experimental means. ...
... Electrostatic interactions, however, do not fall off as rapidly as Lennard-Jones interactions and the use of the minimum image convention can lead to artifacts when dealing with electrostatics. To deal with this, lattice methods such as the Ewald summation are employed which enables electrostatics interactions to be calculated in full without distance cutoffs [43,59,60]. ...
Computer simulations have become an indispensable tool in studying molecular biological systems. The unmatched spatial and temporal resolution that it offers enables for microscopic-level views into the dynamics and mechanics of biological systems. Recent advances in hardware resources have also opened up to computer simulations the investigation of longer timescale biological processes and larger systems. The study of membrane proteins or peptides especially benefits from simulations due to difficulties related to crystallization of such proteins in a membrane environment. In this chapter, we outline the method of molecular dynamics and how it is applied to simulations that involve a peptide and lipid bilayers. In particular, the simulation of a membrane-curvature sensing peptide is examined, and ways of employing computational simulations to design such peptides are discussed.
A systematic analysis is performed on the effectiveness of removing degrees of freedom from hydrogen atoms and/or increasing hydrogen masses to increase the efficiency of molecular dynamics simulations of hydrogen-rich systems such as proteins in water. In proteins, high-frequency bond-angle vibrations involving hydrogen atoms limit the time step to 3 fs, which is already a factor of 1.5 beyond the commonly used time step of 2 fs. Removing these degrees of freedom from the system by constructing hydrogen atoms as dummy atoms, allows the time step to be increased to 7 fs, a factor of 3.5 compared with 2 fs. Additionally, a gain in simulation stability can be achieved by increasing the masses of hydrogen atoms with remaining degrees of freedom from 1 to 4 u. Increasing hydrogen mass without removing the high-frequency degrees of freedom allows the time step to be increased only to 4 fs, a factor of two, compared with 2 fs. The net gain in efficiency of sampling configurational space may be up to 15% lower than expected from the increase in time step due to the increase in viscosity and decrease in diffusion constant. In principle, introducing dummy atoms and increasing hydrogen mass do not influence thermodynamical properties of the system and dynamical properties are shown to be influenced only to a moderate degree. Comparing the maximum time step attainable with these methods (7 fs) to the time step of 2 fs that is routinely used in simulation, and taking into account the increase in viscosity and decrease in diffusion constant, we can say that a net gain in simulation efficiency of a factor of 3 to 3.5 can be achieved. (C) 1999 John Wiley & Sons, Inc.
Nose has modified Newtonian dynamics so as to reproduce both the canonical and the isothermal-isobaric probability densities in the phase space of an N-body system. He did this by scaling time (with s) and distance (with V¹D/ in D dimensions) through Lagrangian equations of motion. The dynamical equations describe the evolution of these two scaling variables and their two conjugate momenta p/sub s/ and p/sub v/. Here we develop a slightly different set of equations, free of time scaling. We find the dynamical steady-state probability density in an extended phase space with variables x, p/sub x/, V, epsilon-dot, and zeta, where the x are reduced distances and the two variables epsilon-dot and zeta act as thermodynamic friction coefficients. We find that these friction coefficients have Gaussian distributions. From the distributions the extent of small-system non-Newtonian behavior can be estimated. We illustrate the dynamical equations by considering their application to the simplest possible case, a one-dimensional classical harmonic oscillator.
The molecular dynamics algorithm of Verlet is extended to study dense neutral assemblies of charged particles under specified V-T conditions. Thermodynamic quantities are obtained for liquid alkali chlorides at 1273°K which are in satisfactory agreement with experiment.
Explicit reversible integrators, suitable for use in large-scale
computer simulations, are derived for extended systems generating the
canonical and isothermal-isobaric ensembles. The new methods are
compared with the standard implicit (iterative) integrators on some
illustrative example problems. In addition, modification of the proposed
algorithms for multiple time step integration is outlined.
The extended Lagrangian method is applied to incorporate induced polarization effects in pure water, using molecular dynamics simulations, at a cost in computer time which is only twice that of modelling the corresponding pairwise additive system. Thermodynamic, structural and dynamical properties of the mean-field SPC and the polarizable PSPC models are compared. All simulations were performed in the canonical ensemble by use of the Nosé thermostat method. We confirm that explicitly including induced polarization effects considerably improves the quality of the SPC model for the transport properties. We show that within some difference in simulation conditions, the method of the extended Lagrangian gives quite satisfactory results when compared to the more classical self-consistent iterative/predictive procedure previously used to calculate the non-additive effects. The latter part of this study assesses the importance of many-body effects in determining the properties of liquid water. This question is addressed with simulations, using a water model, where the induced dipole moment of the PSPC is represented by a single fixed point dipole positioned at the oxygen. It is shown that, whereas the many-body effects appear to have little influence on static properties and on the self-diffusion coefficient, their accurate and explicit representation significantly affects the orientational times.
NAMD is a molecular dynamics program designed for high performance simulations of large biomolecular systems on parallel computers. An object-oriented design imple mented using C++ facilitates the incorporation of new algorithms into the program. NAMD uses spatial decom position coupled with a multithreaded, message-driven design, which is shown to scale efficiently to multiple processors. Also, NAMD incorporates the distributed par allel multipole tree algorithm for full electrostatic force evaluation in O(N) time. NAMD can be connected via a communication system to a molecular graphics program in order to provide an interactive modeling tool for viewing and modifying a running simulation. The application of NAMD to a protein-water system of 32,867 atoms illus trates the performance of NAMD.
Empirical force-field calculations on biological molecules represent an effective method to obtain atomic detail information on the relationship of their structure to their function. Results from those calculations depend on the quality of the force field. In this manuscript, optimization of the CHARMM27 all-atom empirical force field for nucleic acids is presented together with the resulting parameters. The optimization procedure is based on the reproduction of small molecule target data from both experimental and quantum mechanical studies and condensed phase structural properties of DNA and RNA. Via an iterative approach, the parameters were primarily optimized to reproduce macromolecular target data while maximizing agreement with small molecule target data. This approach is expected to ensure that the different contributions from the individual moieties in the nucleic acids are properly balanced to yield condensed phase properties of DNA and RNA, which are consistent with experiment. The quality of the presented force field in reproducing both crystal and solution properties are detailed in the present and an accompanying manuscript (MacKerell and Banavali, J Comput Chem, this issue). The resultant parameters represent the latest step in the continued development of the CHARMM all-atom biomolecular force field for proteins, lipids, and nucleic acids. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 86–104, 2000
A new force field has been developed for alkanes and cycloalkanes, excluding small rings, to improve the calculation of vibrational frequencies, rotational barriers, and numerous relatively small errors that were observed to result from the use of the MM3 force field. © 1996 by John Wiley & Sons, Inc.
Molecular modeling has undergone a remarkable transformation in the last 20 years, as biomolecular simulation moves from the realm of specialists to the wider academic community. Molecular Modeling of Proteins provides thorough introductions and a compilation of step-by-step methods applicable to problems faced by non-specialists – especially those new to the software packages used in molecular modeling. Tips on troubleshooting and avoiding common pitfalls are included in this book, along with chapters covering a wide range of subjects ranging from free energy calculation to applications for drug design. Written by an internationally distinguished panel of investigators and outlining the most striking developments in the field, Molecular Modeling of Proteins is an invaluable resource for those in the industry, as well as a cutting-edge reference for students and professionals in the fields of chemistry, biochemistry, biology, biophysics and bioinformatics.
The total dipole moment of a pair of isolated water molecules that employ a point polarizable dipole is compared with the total dipole moment obtained from MP4 calculations for the same intermolecular orientation and distance. At all reasonable intermolecular distances, the point dipole models give total dipole moments that are generally in very good agreement with MP4. This suggests that the point polarizable dipole approximation for water is reasonable and that the high dipole moments obtained frequently in molecular simulations of polarizable water must be attributed to something other than the inadequacy of the point dipole approximation.
Lipid membranes are an essential component of all living cells. A molecular description of the structure and dynamics of such membranes from either experimental or theoretical approaches is still lacking. This is due in part to the two-dimensional fluid character of membranes (Singer and Nicolson, 1972), which makes difficult a detailed analysis by X-ray diffraction, neutron diffraction, or nuclear magnetic resonance. Detailed structural data of lipid molecules based on X-ray crystallography are available only for the nearly anhydrous crystalline state (Pascher et al, 1992; Small, 1986).
Technical aspects of the constant pressure molecular dynamics (MD) method proposed by Andersen and extended by Parrinello and Rahman to allow changes in the shape of the MD cell are discussed. The new MD method is extended to treat molecular systems and to include long range charge-charge interactions.Results on the conservation laws, the frequency of oscillation of the MD cell, and the equations which constrain the shape of the MD cell are also given. An additional constraint is introduced to stop the superfluous MD cell rotation which would otherwise complicate the analysis of crystal structures. The method is illustrated by examining the behaviour of solid nitrogen at high pressure.
This chapter examines the titration behavior. The titration behavior of a protein is similar to that of a mixture of the constituent amino acids with their α-amino and α-carboxyl groups blocked, except for the terminal amino acids. The information obtained from titration curves of enzymes is based on an analytical nature. However, there is an important difference between the titration of a protein and the corresponding mixture of blocked amino acids. An electrometric titration curve represents the relation between pH and the number of moles of protons bound by, or removed from, 1 mole of protein in the reaction. The isoionic point depends on the protein concentration. Spectrophotometric titration of tyrosine groups, thermodynamic interpretation, and group counting is also discussed. The natural choice is the titration curve in a denaturing medium in which the protein is in the unfolded form, with virtually all titratable groups exposed to the solvent. The preferred denaturing agent is guanidine hydrochloride, because of its greater ability to unfold proteins than other reagents, such as urea, and its stability in solution. The result obtained from the titration curve of native proteins is also provided.
Explicit reversible integrators, suitable for use in large-scale computer simulations, are derived for extended systems generating the canonical and isothermal-isobaric ensembles. The new methods are compared with the standard implicit (iterative) integrators on some illustrative example problems. In addition, modification of the proposed algorithms for multiple time step integration is outlined.
1. Introduction; 2. Basic molecular dynamics; 3. Simulating simple
systems; 4. Equilibrium properties of simple fluids; 5. Dynamical
properties of simple fluids; 6. Alternative ensembles; 7. Nonequilibrium
dynamics; 8. Rigid molecules; 9. Flexible molecules; 10. Geometrically
constrained molecules; 11. Internal coordinates; 12. Many-body
interactions; 13. Long-range interactions; 14. Step potentials; 15.
Time-dependent phenomena; 16. Granular dynamics; 17. Algorithms for
supercomputers; 18. More about software; 19. The future.
Using a transformation of the rigid body equations of motion due to
Evans [4], a new algorithm is presented for the molecular dynamics
simulation of rigid polyatomic molecules. The algorithm consists of
solving the eulerian rigid body equations, using quaternions to
represent orientations, by a fifth-order predictor corrector method.
Compared to previous methods, it is shown that this algorithm leads to
an order of magnitude increase in computing speed.
This is a presentation of the main ideas and methods of modern nonequilibrium statistical mechanics. It is the perfect introduction for anyone in chemistry or physics who needs an update or background in this time-dependent field. Topics covered include fluctuation-dissipation theorem; linear response theory; time correlation functions, and projection operators. Theoretical models are illustrated by real-world examples and numerous applications such as chemical reaction rates and spectral line shapes are covered. The mathematical treatments are detailed and easily understandable and the appendices include useful mathematical methods like the Laplace transforms, Gaussian random variables and phenomenological transport equations.
An investigation into the effects of the anisotropic nature of the Ewald potential for the treatment of long range electrostatic interactions in liquid solutions has been performed. The rotational potential energy surface for two simple charge distributions, and a small protein, have been studied under conditions typically implemented in current biomolecular simulations. A transition between hindered and free rotation is observed which can be modeled quantitatively for simple charge distributions. For most systems in aqueous solution, the transition involves an energy change well below kBT. It is argued that, for solvents with a reasonably high relative permittivity, Ewald artifacts will be small and in many cases may be safely ignored.
The authors report the parameters for a new generic force field, DREIDING, that they find useful for predicting structures and dynamics of organic, biological, and main-group inorganic molecules. The philosophy in DREIDING is to use general force constants and geometry parameters based on simple hybridization considerations rather than individual force constants and geometric parameters that depend on the particular combination of atoms involved in the bond, angle, or torsion terms. Thus all bond distances are derived from atomic radii, and there is only one force constant each for bonds, angles, and inversions and only six different values for torsional barriers. Parameters are defined for all possible combinations of atoms and new atoms can be added to the force field rather simply. This paper reports the parameters for the nonmetallic main-group elements (B, C, N, O, F columns for the C, Si, Ge, and Sn rows) plus H and a few metals (Na, Ca, Zn, Fe). The accuracy of the DREIDING force field is tested by comparing with (i) 76 accurately determined crystal structures of organic compounds involving H, C, N, O, F, P, S, Cl, and Br, (ii) rotational barriers of a number of molecules, and (iii) relative conformational energies and barriers of a number of molecules. The authors find excellent results for these systems.
The mechanism for transporting a chloride ion or a cesium ion across a water-carbon tetrachloride liquid/liquid interface is characterized using molecular dynamics techniques. The results obtained in these studies provided new physical insight into both the free energies and solvent structures as the ion moved across the interface. The computed free-energy profiles of ion transfer for both ions increased monotonically from water to carbon tetrachloride. No free-energy minima were observed at the liquid/liquid interface. The first hydration shells of the ions were significantly reduced as the ion in the nonaqueous phase, which has a similar characteristic found in the aqueous ionic cluster studies.
are only difficultly accessible in the laboratory. One of the two basic problems in the field of molecular modeling and simulation is how to efficiently search the vast configuration space which is spanned by all possible molecular conformations for the global low (free) energy regions which will be populated by a molecular system in thermal equilibrium. The other basic problem is the derivation of a sufficiently accurate interaction energy function or force field for the molecular system of interest. An important part of the art of computer simulation is to choose the unavoidable assumptions, approximations and simplifications of the molecular model and computational procedure such that their contributions to the overall inaccuracy are of comparable size, without affecting significantly the property of interest. Methodology and some practical applications of computer simulation in the field of (bio)chemistry will be reviewed.
The different roles the attractive and repulsive forces play in forming the equilibrium structure of a Lennard-Jones liquid are discussed. It is found that the effects of these forces are most easily separated by considering the structure factor (or equivalently, the Fourier transform of the pair-correlation function) rather than the pair-correlation function itself. At intermediate and large wave vectors, the repulsive forces dominate the quantitative behavior of the liquid structure factor. The attractions are manifested primarily in the small wave vector part of the structure factor; but this effect decreases as the density increases and is almost negligible at reduced densities higher than 0.65. These conclusions are established by considering the structure factor of a hypothetical reference system in which the intermolecular forces are entirely repulsive and identical to the repulsive forces in a Lennard-Jones fluid. This reference system structure factor is calculated with the aid of a simple but accurate approximation described herein. The conclusions lead to a very simple prescription for calculating the radial distribution function of dense liquids which is more accurate than that obtained by any previously reported theory. The thermodynamic ramifications of the conclusions are presented in the form of calculations of the free energy, the internal energy (from the energy equation), and the pressure (from the virial equation). The implications of our conclusions to perturbation theories for liquids and to the interpretation of x-ray scattering experiments are discussed.
Andrews has suggested that the restoring forces in polyatomic molecules can best be chosen as harmonic restoring forces along the directions of the chemical bonds and prependicular to them. In order to test the suggestion, we have calculated the vibrational frequencies of tetrahedral molecules with this choice of forces. The agreement between calculated and observed values is unsatisfactory. It seems that there may be repulsive forces between the corner atoms of the type between ions of crystals or the inert gas atoms. The introduction of terms in the potential energy proportional to 1rjn, where rj is the distance between two corner atoms, makes it possible to secure very good agreement between calculated and observed frequencies, in the case of CCl4, SiCl4, SnCl4, CBr4, and SnBr4 but not such close agreement in that of TiCl4. The calculated frequencies are not very sensitive to the value of n which may be anywhere from 5 to 9. The repulsive forces necessary are of the same order of magnitude as those calculated from crystal properties and the viscosities of the inert gases. In the case of the SO4= and ClO4-, the inverse high power repulsive force is not sufficient, but the addition of terms, e2rj, e being the electronic charge, as well as an inverse higher power term does give very good agreement between calculated and observed frequencies of these molecules.
The use of Ewald and related methods to handle electrostatic interactions in explicit-solvent simulations of solutions imposes an artificial periodicity on systems which are inherently nonperiodic. The consequences of this approximation should be assessed, since they may crucially affect the reliability of those computer simulations. In the present study, we propose a general method based on continuum electrostatics to investigate the nature and magnitude of periodicity-induced artifacts. As a first example, this scheme is applied to the solvation free-energy of a spherical ion. It is found that artificial periodicity reduces the magnitude of the ionic solvation free-energy, because the solvent in the periodic copies of the central unit cell is perturbed by the periodic copies of the ion, thus less available to solvate the central ion. In the limit of zero ionic radius and infinite solvent permittivity, this undersolvation can be corrected by adding the Wigner self-energy term to the solvation free-energy. For ions of a finite size or a solvent of finite permittivity, a further correction is needed. An analytical expression for this correction is derived using continuum electrostatics. As a second example, the effect of artificial periodicity on the potential of mean force for the interaction between two spherical ions is investigated. It is found that artificial periodicity results in an attractive force between ions of like charges, and a repulsive force between ions of opposite charges. The analysis of these two simple test cases reveals that two individually large terms, the periodicity-induced perturbations of the Coulomb and solvation contributions, often cancel each other significantly, resulting in an overall small perturbation. Three factors may prevent this cancellation to occur and enhance the magnitude of periodicity-induced artifacts: (i) a solvent of low dielectric permittivity, (ii) a solute cavity of non-negligible size compared to the unit cell size, and (iii) a solute bearing a large overall charge.
Jarzynski’s equality provides a non-equilibrium method for calculation of free energies, which could have wide-ranging applications in complex molecules. Tests of Jarzynski’s equality have mostly focused on simple systems or near-equilibrium processes. Here we present a benchmark test that compares the results of free energy calculations for a potassium ion in carbon nanotube and gramicidin channel, which can be identified with simple and complex systems. A good agreement is found between the free energy results obtained using Jarzynski’s equality and established methods in the carbon nanotube, but discrepancies occur in gramicidin, indicating relaxation problems in application of Jarzynski’s equality to complex biomolecular processes.
All the possible methyl-substituted guanidines have been prepared and their dissociation constants measured. Contrary to a previous report, they are all very strong bases.
An N·log(N) method for evaluating electrostatic energies and forces of large periodic systems is presented. The method is based on interpolation of the reciprocal space Ewald sums and evaluation of the resulting convolutions using fast Fourier transforms. Timings and accuracies are presented for three large crystalline ionic systems. The Journal of Chemical Physics is copyrighted by The American Institute of Physics.
With use of a Lagrangian which allows for the variation of the shape and size of the periodically repeating molecular-dynamics cell, it is shown that different pair potentials lead to different crystal structures.
A method and parametrization scheme which allow fast and accurate calculations of hydration free energies are described. The solute is treated as a polarizable cavity of a shape defined by the molecular surface, containing point charges at the location of atomic nuclei. Electrostatic contributions to solvation are derived from:finite difference solutions of the Poisson equation (FDPB method). Nonpolar (cavity/van der Waals) energies are added as a surface area dependent term, with a single surface tension coefficient (gamma) derived from hydrocarbon solubility in water. Atomic charges and radii are obtained by modifying existing force-field or quantum-mechanically-derived values, by fitting to experimental solvation energies of small organic molecules. A new, simple parameter set (parameters for solvation energy, PARSE) is developed specifically for the FDPB/gamma method, by choosing atomic charges and radii which reproduce the estimated contributions to solvation of simple functional groups. The PARSE parameters reproduce hydration free energies for a test set of 67 molecules with an average error of 0.4 kcal/mol. For amino acid side chain and peptide backbone analogs the average error is only 0.1 kcal/mol.
A simple polarizable water model is developed and optimized for molecular dynamics simulations of the liquid phase under ambient conditions. The permanent charge distribution of the water molecule is represented by three point charges: two hydrogen sites and one additional M site positioned along the HOH bisector. Electronic induction is represented by introducing a classical charged Drude particle attached to the oxygen by a harmonic spring. The oxygen site carries an equal and opposite charge, and is the center of an intermolecular Lennard-Jones interaction. The HOH gas-phase experimental geometry is maintained rigidly and the dipole of the isolated molecule is 1.85 D, in accord with experiment. The model is simulated by considering the dynamics of an extended Lagrangian in which a small mass is attributed to the Drude particles. It is parametrized to reproduce the salient properties of liquid water under ambient conditions. The optimal model, refered to as SWM4-DP for ``simple water model with four sites and Drude polarizability,'' yields a vaporization enthalpy of 10.52 kcal/mol, a molecular volume of 29.93 Å3, a static dielectric constant of 79+/-5, a self-diffusion constant of (2.30+/-0.04)×10-5 cm2/s, and an air/water surface tension of 66.9+/-0.9 dyn/cm, all in excellent accord with experiments. The energy of the water dimer is -5.18 kcal/mol, in good accord with estimates from experiments and high level ab initio calculations. The polarizability of the optimal model is 1.04 Å3, which is smaller than the experimental value of 1.44 Å3 in the gas phase. It is likely that such a reduced molecular polarizability, which is essential to reproduce the properties of the liquid, arises from the energy cost of overlapping electronic clouds in the condensed phase due to Pauli's exclusion principle opposing induction.
The past decade has seen intense development of what are anticipated to be the next generation of classical force fields to be used in computational statistical mechanical approaches to studying a broad class of physical and biological systems. Among the several approaches being actively pursued currently is the fluctuating charge (or equivalently charge equilibration or electronegativity equalization) method. Within this formalism, dynamical electronic degrees of freedom are introduced and propagated in time in order to allow an electrostatic response to the local chemical environment. In this article, we present a review of our recent development and application efforts of a polarizable biomolecular force field based on the fluctuating charge formalism and founded on the CHARMM non-polarizable force field. We will discuss aspects of the parameterization, as well as recent applications to a spectrum of chemical and biological systems such as small-molecule liquid–vapor interfaces, solvated proteins/peptides, and physiological membrane systems. We will conclude with brief comments on aspects that require continued effort in terms of future development of such novel potentials.
A complete set of intermolecular potential functions has been developed for use in computer simulations of proteins in their native environment. Parameters are reported for 25 peptide residues as well as the common neutral and charged terminal groups. The potential functions have the simple Coulomb plus Lennard-Jones form and are compatible with the widely used models for water, TIP4P, TIP3P, and SPC. The parameters were obtained and tested primarily in conjunction with Monte Carlo statistical mechanics simulations of 36 pure organic liquids and numerous aqueous solutions of organic ions representative of subunits in the side chains and backbones of proteins. Bond stretch, angle bend, and torsional terms have been adopted from the AMBER united-atom force field. As reported here, further testing has involved studies of conformational energy surfaces and optimizations of the crystal structures for four cyclic hexapeptides and a cyclic pentapeptide. The average root-mean-square deviation from the X-ray structures of the crystals is only 0.17 Å for the atomic positions and 3% for the unit cell volumes. A more critical test was then provided by performing energy minimizations for the complete crystal of the protein crambin, including 182 water molecules that were initially placed via a Monte Carlo simulation. The resultant root-mean-square deviation for the non-hydrogen atoms is still ca. 0.2 Å and the variation in the errors for charged, polar, and nonpolar residues is small. Improvement is apparent over the AMBER united-atom force field which has previously been demonstrated to be superior to many alternatives.
This article concludes a series of papers concerned with the flow of electric current through the surface membrane of a giant nerve fibre (Hodgkinet al., 1952,J. Physiol.116, 424–448; Hodgkin and Huxley, 1952,J. Physiol.116, 449–566). Its general object is to discuss the results of the preceding papers (Section 1), to put them into mathematical form (Section 2) and to show that they will account for conduction and excitation in quantitative terms (Sections 3–6).
A theory of many-particle systems is developed to formulate transport, collective motion, and Brownian motion from a unified, statistical-mechanical point of view. This is done by, first, rewriting the equation of motion in a generalized form of the Langevin equation in the stochastic theory of Brownian motion and then, either studying the average evolution of a non-equilibrium system or calculating the linear response function to a mechanical perturbation. (1) An expression is obtained for the damping function φ(t), the real part of whose Laplace transform gives the damping constnat of collective motion. (2) A general equation of motion for a set of dynamical variables At) is derived, which takes the form
where is a frequency matrix determining the collective oscillation of A(t). The quantity f(t) consists of those terms which are either non-linear in A(s), t ≧s ≧0, or dependent on the other degrees-of-freedom explicitly, and its time-correlation function is connected with the damping function φ(t) by (f(t1), f(t2)*) = φ(t1 − t2)·(A, A*). (3) An expression is obtained for the linear after-effect function to thermal disturbances such as temperature gradient and strain tensor. Both the conjugate fluxes and the time dependence differ from those of the mechanical response function. The conjugate fluxes are random parts of the fluxes of the state variables, thus depending on temperature. (4) The difference in the time dependence arises from a special property of the time evolution of f(t) and ensures that the damping function and the thermal after-effect function are determined by the microscopic processes in strong contrast to the mechanical response function. The difficulty of the plateau value problem in the previous theories of Brownian motion and transport coefficients is thus removed. (5) The theory is illustrated by dealing with the motion of inhomogeneous magnetization in ferromagnets and the Brownian motion of the collective coordinates of fluids. (6) Explicit expressions are derived for the thermal after-effect functions and the transport coefficients of multi-component systems
We describe a computer program we have been developing to build models of molecules and calculate their interactions using empirical energy approaches. The program is sufficiently flexible and general to allow modeling of small molecules, as well as polymers. As an illustration, we present applications of the program to study the conformation of actinomycin D. In particular, we study the rotational isomerism about the D-Val-, L-Pro, and L-Pro-Sar amide bonds as well as comparing the energy and structure of the Sobell model and the x-ray structure of actinomycin D.
A new method for molecular dynamics computer simulations, called the multiple time-step (MTS) method, is described, in which two or more time steps of different lengths are used to integrate the equations of motion in systems governed by continuous potential functions. With this method computing speeds have been increased by factors of three to eight over conventional molecular dynamics methods in simulations of monatomic and polyatomic fluids, with only marginal increases in computer storage.
The structure of sperm whale metmyoglobin has been refined using new intensity data to 2·0 Å collected on a four-circle diffractometer. Starting with the original phase angles determined by isomorphous replacement with heavy atoms (Kendrew et al., 1960) an electron density map was calculated, and atomic parameters were subjected to real-space refinement. Phases derived from these improved atomic parameters were used to calculate a new electron density map, which served as a basis for the next real-space refinement. Several cycles of this procedure led to an R factor of 0·235. The C-terminal residues and many atoms in side chains which had not been clearly defined in the original map were located, as were 82 solvent molecules, including the two sulphate ions originally described.
The iron atom is displaced by 0·40 Å from the mean plane of the haem and lies at a distance of 2·13 Å from N (ε) of the proximal histidine. At least two of the four pyrrole rings are tilted out of the haem plane, while the exact inclination of the other two could not be determined. The haem is in van der Waals' contact with 83 atoms of the globin (excluding hydrogens). These include three large clusters of hydrophobic side chains, each with spare room for a foreign molecule. The tertiary structure of the globin is stabilized by many hydrogen bonds between the various helical and non-helical segments, many of which have been described before (Watson, 1969).
Metmyoglobin differs from the α-and β-chains of horse methaemoglobin in the orientation of the haem and in several details of the tertiary structure of the globin, which is not surprising seeing that the number of homologies between myoglobin and the α- and the β-chains are only 32 and 30, respectively (Dayhoff, 1972).
A novel method to calculate transition pathways between two known protein conformations is presented. It is based on a molecular dynamics simulation starting from one conformational state as initial structure and using the other for a directing constraint. The method is exemplified with the T ↔ R transition of insulin. The most striking difference between these conformational states is that in T the 8 N-terminal residues of the B chain are arranged as an extended strand whereas in R they are forming a helix. Both the transition from T to R and from R to T were simulated. The method proves capable of finding a continuous pathway for each direction which are moderately different. The refolding processes are illustrated by a series of transient structures and pairs of Ø, ψ angles selected from the time course of the simulations. In the T → R direction the helix is formed in the →last third of the transition, while in the R → T direction it is preserved during more than half of the simulation period. The results are discussed in comparison with those of an atternative method recently apptied to the T → R transition of insulin which is based on targeted energy minimisation.