On Multiple Time-step Algorithms and the Ewald Sum

Abstract

We show how standard multiple time-step algorithms devised for systems with short-range potentials can be used successfully in simulations of periodic systems with long-range (Coulombic) potentials. Three strategies for incorporating the Ewald sum into a multiple time-step algorithm are considered. These are (i) evaluation of reciprocal space terms every time-step (ii) evaluating reciprocal space terms once every n time-steps and placing these terms in with the slowly varying forces and energies (iii) a modified form of the second strategy in which primary shell (close) electrostatic interactions are evaluated directly and the more distant interactions handled by the Ewald sum (once every n time-steps). Only the first and third approaches give satisfactory thermodynamic results. The third strategy is much more efficient than the first. With the third strategy substantial savings in cpu time are acheived in both the real space and, most importantly, the reciprocal space terms of the Ewald sum. This is achieved without significant loss of accuracy or stability. Overall execution time is decreased by a factor of between 2 and 3.     

DNA polymorphism: a comparison of force fields for nucleic acids

The improvements of the force fields and the more accurate treatment of long-range interactions are providing more reliable molecular dynamics simulations of nucleic acids. The abilities of certain nucleic acid force fields to represent the structural and conformational properties of nucleic acids in solution are compared. The force fields are AMBER 4.1, BMS, CHARMM22, and CHARMM27; the comparison of the latter two is the primary focus of this paper. The performance of each force field is evaluated first on its ability to reproduce the B-DNA decamer d(CGATTAATCG)(2) in solution with simulations in which the long-range electrostatics were treated by the particle mesh Ewald method; the crystal structure determined by Quintana et al. (1992) is used as the starting point for all simulations. A detailed analysis of the structural and solvation properties shows how well the different force fields can reproduce sequence-specific features. The results are compared with data from experimental and previous theoretical studies.     

Molecular Dynamics Simulations of Double-Stranded DNA in an Explicit Solvent Model with the Zero-Dipole Summation Method

Molecular dynamics (MD) simulations of a double-stranded DNA with explicit water and small ions were performed with the zero-dipole summation (ZD) method, which was recently developed as one of the non-Ewald methods. Double-stranded DNA is highly charged and polar, with phosphate groups in its backbone and their counterions, and thus precise treatment for the long-range electrostatic interactions is always required to maintain the stable and native double-stranded form. A simple truncation method deforms it profoundly. On the contrary, the ZD method, which considers the neutralities of charges and dipoles in a truncated subset, well reproduced the electrostatic energies of the DNA system calculated by the Ewald method. The MD simulations using the ZD method provided a stable DNA system, with similar structures and dynamic properties to those produced by the conventional Particle mesh Ewald method.     

Molecular dynamics simulations on HIV-1 Tat

Molecular dynamics simulations are used to investigate dynamics and intramolecular interactions of the HIV-1 transactivator (Tat) in aqueous solution. The calculations are based on the AMBER force field with particle mesh Ewald treatment for long-range electrostatics. The Tat structure exhibits a large flexibility, consistent with its absence of secondary structure elements. From an analysis of the correlation matrix and of electrostatic interactions we suggest that segments expressed by the two exons (amino acids 1-72 and 73-86, respectively) exhibit rather separated dynamic and energetic properties. We also identify intramolecular interactions of importance for structure stabilization. In particular, significant electrostatic interactions are recognized between the N-terminus and the basic domain of the protein, consistent with site-directed mutagenesis performed in this work.     

The range and shielding of dipole-dipole interactions in phospholipid bilayers

In molecular dynamics simulations of lipid bilayers, the structure is sensitive to the precise treatment of electrostatics. The dipole-dipole interactions between headgroup dipoles are not long-ranged, but the area per lipid and, through it, other properties of the bilayer are very sensitive to the detailed balance between the perpendicular and in-plane components of the headgroup dipoles. This is affected by the detailed properties of the cutoff scheme or if long-range interactions are included by Ewald or particle-mesh Ewald techniques. Interaction between the in-plane components of the headgroup dipoles is attractive and decays as the inverse sixth power of distance. The interaction is screened by the square of a dielectric permittivity close to the value for water. Interaction between the components perpendicular to the membrane plane is repulsive and decays as the inverse third power of distance. These interactions are screened by a dielectric permittivity of the order 10. Thus, despite the perpendicular components being much smaller in magnitude than the in-plane components, they will dominate the interaction energies at large distances.     

Long-range electrostatic interactions in molecular dynamics: an endothelin-1 case study

An extensive conformational search in explicit solvent was performed in order to compare the influence of different long-range electrostatic interaction treatments in molecular dynamics. The short peptide endothelin-1 was selected as the subject of molecular dynamics studies that started from both X-ray and NMR obtained structures. Electrostatic interactions were treated using two of the most common methods--residue-based cutoff and particle mesh Ewald (PME). Analyses of free energy calculations (MM-PBSA method used), secondary structure elements and hydrogen bonds were performed, and there suggested that there is no unambiguous conclusion about which of the two methods of long-range electrostatics treatment should be used in MD simulations in this case. The most reliable data was provided by a trajectory that started with the NMR structure and used the cutoff method to treat electrostatic interactions. This leads to a recommendation that the choice of electrostatics treatment should be made carefully and not automatically by choosing the PME method simply because it is the most widely used.     

Incorporating residual dipolar couplings into the NMR solution structure determination of nucleic acids

NMR solution structures of nucleic acids are generally less well defined than similar-sized proteins. Most NMR structures of nucleic acids are defined only by short-range interactions, such as intrabase-pair or sequential nuclear Overhauser effects (NOEs), and J-coupling constants, and there are no long-range structural data on the tertiary structure. Residual dipolar couplings represent an extremely valuable source of distance and angle information for macromolecules but they average to zero in isotropic solutions. With the recent advent of general methods for partial alignment of macromolecules in solution, residual dipolar couplings are rapidly becoming indispensable constraints for solution NMR structural studies. These residual dipolar couplings give long-range global structural information and thus complement the strictly local structural data obtained from standard NOE and torsion angle constraints. Such global structural data are especially important in nucleic acids due to the more elongated, less-globular structure of many DNAs and RNAs. Here we review recent progress in application of residual dipolar couplings to structural studies of nucleic acids. We also present results showing how refinement procedures affect the final solution structures of nucleic acids.Copyright 2001 John Wiley & Sons, Inc.     

Long-range Electrostatic Interactions in Molecular Dynamics: An Endothelin-1 Case Study

Abstract

An extensive conformational search in explicit solvent was performed in order to compare the influence of different long-range electrostatic interaction treatments in molecular dynamics. The short peptide endothelin-1 was selected as the subject of molecular dynamics studies that started from both X-ray and NMR obtained structures. Electrostatic interactions were treated using two of the most common methods—residue-based cutoff and particle mesh Ewald (PME). Analyses of free energy calculations (MM-PBSA method used), secondary structure elements and hydrogen bonds were performed, and there suggested that there is no unambiguous conclusion about which of the two methods of long-range electrostatics treatment should be used in MD simulations in this case. The most reliable data was provided by a trajectory that started with the NMR structure and used the cutoff method to treat electrostatic interactions. This leads to a recommendation that the choice of electrostatics treatment should be made carefully and not automatically by choosing the PME method simply because it is the most widely used.     

Molecular alignment of denatured states of staphylococcal nuclease with strained polyacrylamide gels and surfactant liquid crystalline phases

Residual dipolar couplings reflect the orientation of vectors between pairs of magnetic nuclei relative to a unique set of molecular axes. Thus, unlike NOEs and scalar couplings, dipolar couplings provide access to long-range structural information. A prerequisite for measurement of these NMR parameters is imposition of a weak net alignment, most simply by forcing the macromolecules to tumble in an asymmetric environment that restricts some orientations more than others. In this report, several denatured forms of staphylococcal nuclease are aligned by using compressed and stretched polyacrylamide gels, a nonionic type of lipid bilayer disk or bicelle, and a liquid crystalline phase formed by a cationic lipid. All three types of media can be used at high urea concentrations. While polyacrylamide gels and bicelles produce similar alignment tensors through steric interactions, a liquid crystalline phase of cetylpyridinium bromide aligns denatured nuclease along a different set of axes, presumably through electrostatic effects. The analysis of residual dipolar couplings collected with two different alignment tensors may permit the calculation of ensembles of conformations. The dipolar couplings observed for staphylococcal nuclease denatured with urea, by low pH or by deletion of residues from both termini, suggest that all denatured forms share a common "topology", one which has been shown previously to be native-like. Although SDS/nuclease complexes give sharp and disperse (1)H-(15)N correlation spectra, only small couplings are observed in strained polyacrylamide gels.     

Refined structure of a flexible heptasaccharide using 1H-13C and 1H-1H NMR residual dipolar couplings in concert with NOE and long range scalar coupling constants

The heptasaccharide isolated from the cell wall polysaccharide of Streptococcus mitis J22 serves as an important model for the dynamics and conformation of complex polysaccharides, illustrating the nature of flexibility with rigid epitopes joined by flexible hinges. One-bond C-H residual dipolar couplings (¹D_CH) and long-range H-H residual dipolar couplings (ⁿD_HH) were measured for the heptasaccharide in a cetylpyridinium chloride/hexanol/brine lamellar liquid crystal medium. A method is proposed to determine the ⁿD_HH in natural abundance based on a ¹³C resolved ¹H TOCSY pulse sequence previously published to determine the homonuclear scalar couplings. Different methods for interpretation of the ¹D_CH and the ⁿD_HH residual dipolar coupling data obtained were compared and combined with the NOE and long-range H,C and C,C scalar couplings available for this heptasaccharide. A flexible model of the heptasaccharide was determined in which two structurally well-defined regions involving four and two sugar residues, respectively are joined by a flexible hinge which involves two 16 glycosidic linkages.     

Short-range, long-range and transition state interactions in the denatured state of ACBP from residual dipolar couplings

Residual dipolar couplings in the denatured state of bovine acyl-coenzyme A binding protein (ACBP) oriented in strained polyacrylamide gels have been shown to be a sensitive, sequence-specific probe for residual secondary structure. Results supporting this were obtained by comparing residual dipolar couplings under different denaturing conditions. The data were analyzed using the program molecular fragment replacement (MFR), which demonstrated alpha-helix propensity in four isolated stretches along the protein backbone, and these coincide with the location of native helices. This is in full agreement with earlier findings based on secondary chemical shift values. Furthermore, N-H residual dipolar couplings provided direct evidence for the existence of native-like hydrophobic interactions in the acid-denatured state of ACBP at pH 2.3. It was shown that replacement of the hydrophobic side-chain of residue Ile27 with alanine in helix A2 leads to large decreases of residual dipolar couplings in residues that form helix A4 in the native state. It is suggested that the Ile to Ala mutation changes the probability for the formation of long-range interactions, which are present in the acid-denatured state of the wild-type protein. These long-range interactions are similar to those proposed to form in the transition state of folding of ACBP. Therefore, the application of residual dipolar couplings in combination with a comparative mutation study has demonstrated the presence of precursors to the folding transition state under acid-unfolding conditions.     

The use of NMR residual dipolar couplings in aqueous dilute liquid crystalline medium for conformational studies of complex oligosaccharides

C-H dipolar coupling values were measured for a natural-abundance sample of the pentasaccharide beta-D-Galp-(1-->3)-[alpha-L-Fucp-(1-->4)]-beta-D-GlcNAcp-(1 -->3)-beta-D- Galp-(1-->4)-beta-D-Glcp ('lacto-N-fucopentaose 2') (LNF-2), in a 7.5% solution of dimyristoyl phosphatidylcholine-dihexanoyl phosphatidylcholine bicelle liquid crystals oriented in the NMR magnetic field. Interpretation of the dipolar coupling data and NOE confirms the conformational model for the Lewis(a) trisaccharide epitope based on NOE, molecular dynamics simulations, and scalar coupling data and provided new structural information for the remaining residues of the pentasaccharide. Since residual dipolar coupling provides information on long-range order, it is a valuable complement to other types of NMR data such as NOE and scalar coupling for exploring conformations of complex oligosaccharides.     

Solution structure of calcium-saturated cardiac troponin C bound to cardiac troponin I

Cardiac troponin C (TnC) is composed of two globular domains connected by a flexible linker. In solution, linker flexibility results in an ill defined orientation of the two globular domains relative to one another. We have previously shown a decrease in linker flexibility in response to cardiac troponin I (cTnI) binding. To investigate the relative orientation of calcium-saturated TnC domains when bound to cTnI, (1)H-(15)N residual dipolar couplings were measured in two different alignment media. Similarity in alignment tensor orientation for the two TnC domains supports restriction of domain motion in the presence of cTnI. The relative spatial orientation of TnC domains bound to TnI was calculated from measured residual dipolar couplings and long-range distance restraints utilizing a rigid body molecular dynamics protocol. The relative domain orientation is such that hydrophobic pockets face each other, forming a latch to constrain separate helical segments of TnI. We have utilized this structure to successfully explain the observed functional consequences of linker region deletion mutants. Together, these studies suggest that, although linker plasticity is important, the ability of TnC to function in muscle contraction can be correlated with a preferred domain orientation and interdomain distance.     

Determination of solution structures of paramagnetic proteins by NMR

Standard procedures for using nuclear Overhauser enhancements (NOE) between protons to generate structures for diamagnetic proteins in solution from NMR data may be supplemented by using dipolar shifts if the protein is paramagnetic. This is advantageous since the electron-nuclear dipolar coupling provides relatively long-range geometric information with respect to the paramagnetic centre which complements the short-range distance constraints from NOEs. Several different strategies have been developed to date, but none of these attempts to combine data from NOEs and dipolar shifts in the initial stages of structure calculation or to determine three dimensional protein structures together with their magnetic properties. This work shows that the magnetic and atomic structures are highly correlated and that it is important to have additional constraints both to provide starting parameters for the magnetic properties and to improve the definition of the best fit. Useful parameters can be obtained for haem proteins from Fermi contact shifts; this approach is compared with a new method based on the analysis of dipolar shifts in haem methyl groups with respect to data from horse and tuna ferricytochromes c. The methods developed for using data from NOEs and dipolar shifts have been incorporated in a new computer program, PARADYANA, which is demonstrated in application to a model data set for the sequence of the haem octapeptide known as microperoxidase-8. Received: 13 October 1997 / Accepted: 19 December 1997     

Molecular dynamics study of Desulfovibrio africanus cytochrome c3 in oxidized and reduced forms

A 5-ns molecular dynamics study of a tetraheme cytochrome in fully oxidized and reduced forms was performed using the CHARMM molecular modeling program, with explicit water molecules, Langevin dynamics thermalization, Particle Mesh Ewald long-range electrostatics, and quantum mechanical determination of heme partial charges. The simulations used, as starting points, crystallographic structures of the oxidized and reduced forms of the acidic cytochrome c(3) from Desulfovibrio africanus obtained at pH 5.6. In this paper we also report structures for the two forms obtained at pH 8. In contrast to previous cytochrome c(3) dynamics simulations, our model is stable. The simulation structures agree reasonably well with the crystallographic ones, but our models show higher flexibility and the water molecules are more labile. We have compared in detail the differences between the simulated and experimental structures of the two redox states and observe that the hydration structure is highly dependent on the redox state. We have also analyzed the interaction energy terms between the hemes, the protein residues, and water. The direct electrostatic interaction between hemes is weak and nearly insensitive to the redox state, but the remaining terms are large and contribute in a complex way to the overall potential energy differences that we see between the redox states.     

Direct demonstration of structural similarity between native and denatured eglin C

To characterize the long-range structure that persists in the unfolded form of the 70-residue protein eglin C, residual dipolar couplings (RDCs) for HN-N and HA-CA bond vectors were measured by NMR spectroscopy for both its low pH, urea denatured state and its native state. When the data sets for the two different structural states were compared, a statistically significant correlation was found, with both sets of dipolar couplings yielding a correlation coefficient of r = 0.47 to 0.51. This finding directly demonstrates that the denatured state of eglin C has a nativelike global structure, a conclusion reached indirectly for staphylococcal nuclease by combining two different types of NMR data. A simple computer simulation showed that the degree of variation in phi and psi angles that yields the RDC correlation of r = 0.5 was inversely dependent on the statistical segment length, ranging from +/-6 to +/-30 degrees at the upper limit. Stable nativelike topologies that persist on unfolding would explain the rapid refolding kinetics displayed by many proteins and might provide a natural barrier against amyloid fibril formation.     

Dipolar assisted rotational resonance NMR of tryptophan and tyrosine in rhodopsin

Two dimensional (2D) solid-state (13)C.(13)C dipolar recoupling experiments are performed on a series of model compounds and on the visual pigment rhodopsin to establish the most effective method for long range distance measurements in reconstituted membrane proteins. The effects of uniform labeling, inhomogeneous B(1) fields, relaxation and dipolar truncation on cross peak intensity are investigated through NMR measurements of simple amino acid and peptide model compounds. We first show that dipolar assisted rotational resonance (DARR) is more effective than RFDR in recoupling long-range dipolar interactions in these model systems. We then use DARR to establish (13)C-(13)C correlations in rhodopsin. In rhodopsin containing 4'-(13)C-Tyr and 8,19-(13)C retinal, we observe two distinct tyrosine-to-retinal correlations in the DARR spectrum. The most intense cross peak arises from a correlation between Tyr268 and the retinal 19-(13)CH(3), which are 4.8 A apart in the rhodopsin crystal structure. A second cross peak arises from a correlation between Tyr191 and the retinal 19-(13)CH(3), which are 5.5 A apart in the crystal structure. These data demonstrate that long range (13)C em leader (13)C correlations can be obtained in non-crystalline integral membrane proteins reconstituted into lipid membranes containing less than 150 nmoles of protein. In rhodopsin containing 2-(13)C Gly121 and U-(13)C Trp265, we do not observe a Trp-Gly cross peak in the DARR spectrum despite their close proximity (3.6 A) in the crystal structure. Based on model compounds, the absence of a (13)C em leader (13)C cross peak is due to loss of intensity in the diagonal Trp resonances rather than to dipolar truncation.     

Domain orientation and dynamics in multidomain proteins from residual dipolar couplings.

The data most commonly available for the determination of macromolecular structures in solution are NOE based distance estimates and spin-spin coupling constant based dihedral angle estimates. This information is, unfortunately, inherently short-range in nature. Thus, for many multidomain proteins, little information is available to accurately position weakly interacting domains with respect to each other. Recent studies of proteins aligned in dilute liquid crystalline solvents have shown the utility of measuring anisotropic spin interactions, such as residual dipolar couplings, to obtain unique long-range structural information. In this work, the latter approach is taken to explore the relative domain orientation in a two-domain fragment from the protein barley lectin. An approach based on singular value decomposition as opposed to simulated annealing is used to directly determine order tensors for each domain from residual (15)N-(1)H dipolar couplings, and the limitations of the two approaches are discussed. Comparison of the order tensor principal axis frames as separately determined for each domain indicates that the two domains are not oriented as in the crystal structure of wheat germ agglutinin, a highly homologous protein ( approximately 95% sequence identical). Furthermore, differences in the order tensor values suggest that the two domains are not statically positioned but are experiencing different reorientational dynamics and, to a large degree, may be considered to reorient independently. Data are also presented that suggest that a specific association occurs between one domain and the lipid bicelles comprising the liquid crystal solvent.