The multiscale coarse-graining method. VII. Free energy decomposition of coarse-grained effective potentials

The potential of mean force (PMF) with respect to coarse-grained (CG) coordinates is often calculated in order to study the molecular interactions in atomistic molecular dynamics (MD) simulations. The multiscale coarse-graining (MS-CG) approach enables the computation of the many-body PMF of an atomistic system in terms of the CG coordinates, which can be used to parameterize CG models based on all-atom configurations. We demonstrate here that the MS-CG method can also be used to analyze the CG interactions from atomistic MD trajectories via PMF calculations. In addition, MS-CG calculations at different temperatures are performed to decompose the PMF values into energetic and entropic contributions as a function of the CG coordinates, which provides more thermodynamic information regarding the atomistic system. Two numerical examples, liquid methanol and a dimyristoylphosphatidylcholine lipid bilayer, are presented. The results show that MS-CG can be used as an analysis tool, comparable to various free energy computation methods. The differences between the MS-CG approach and other PMF calculation methods, as well as the characteristics and advantages of MS-CG, are also discussed.     

Direct observation of uncoated spectrin with atomic force microscope

Spectrin molecules extracted from human blood ceil membrane have been examined by atomic force microscopy (AFM) without using shadowing or staining procedures. A drop of the solution containing spectrin molecules was deposited on the freshly deaved mica substrate. After about 1 min, the residual solution was removed with a piece of filter paper. Afterwards the sample was imaged with a home-made atomic force microscope (AFM) in air in a constant force mode. The obtained AFM images revealed that the spectrin molecules prepared from the above procedures exhibit several kinds of structures as follows: (i) the compact rod-like spectrin heterodimers with a length of around 100 nm; (ii) bent or curved linear tetramers with a length of around 200 nm; (iii) somewhat curved spectrin hexamers, octomers or decamers with lengths of about 300, 400, or 500 nm; and (iv) high oligomers with a length above 1 000 nm.     

Unbinding of the streptavidin-biotin complex by atomic force microscopy: a hybrid simulation study

A hybrid molecular simulation technique, which combines molecular dynamics and continuum mechanics, was used to study the single-molecule unbinding force of a streptavidin-biotin complex. The hybrid method enables atomistic simulations of unbinding events at the millisecond time scale of atomic force microscopy (AFM) experiments. The logarithmic relationship between the unbinding force of the streptavidin-biotin complex and the loading rate (the product of cantilever spring constant and pulling velocity) in AFM experiments was confirmed by hybrid simulations. The unbinding forces, cantilever and tip positions, locations of energy barriers, and unbinding pathway were analyzed. Hybrid simulation results from this work not only interpret unbinding AFM experiments but also provide detailed molecular information not available in AFM experiments.     

Pinpointing Mechanochemical Bond Rupture by Embedding the Mechanophore into a Macrocycle

Mechanophores contain a mechanically labile bond that can be broken by an external mechanical force. Quantitative measurement and control of the applied force is possible through atomic force microscopy (AFM). A macrocycle was synthesized that contains both the mechanophore and an aliphatic chain that acts as a “safety line” upon bond breaking. This ring‐opening mechanophore unit is linked to poly(ethylene glycol) spacers, which allow investigation by single molecule force spectroscopy. The length increase upon rupture of the mechanophore was measured and compared with quantum chemical calculations.     

The multiscale coarse-graining method. II. Numerical implementation for coarse-grained molecular models

The multiscale coarse-graining (MS-CG) method [S. Izvekov and G. A. Voth, J. Phys. Chem. B 109, 2469 (2005); J. Chem. Phys. 123, 134105 (2005)] employs a variational principle to determine an interaction potential for a CG model from simulations of an atomically detailed model of the same system. The companion paper proved that, if no restrictions regarding the form of the CG interaction potential are introduced and if the equilibrium distribution of the atomistic model has been adequately sampled, then the MS-CG variational principle determines the exact many-body potential of mean force (PMF) governing the equilibrium distribution of CG sites generated by the atomistic model. In practice, though, CG force fields are not completely flexible, but only include particular types of interactions between CG sites, e.g., nonbonded forces between pairs of sites. If the CG force field depends linearly on the force field parameters, then the vector valued functions that relate the CG forces to these parameters determine a set of basis vectors that span a vector subspace of CG force fields. The companion paper introduced a distance metric for the vector space of CG force fields and proved that the MS-CG variational principle determines the CG force force field that is within that vector subspace and that is closest to the force field determined by the many-body PMF. The present paper applies the MS-CG variational principle for parametrizing molecular CG force fields and derives a linear least squares problem for the parameter set determining the optimal approximation to this many-body PMF. Linear systems of equations for these CG force field parameters are derived and analyzed in terms of equilibrium structural correlation functions. Numerical calculations for a one-site CG model of methanol and a molecular CG model of the EMIM(+)NO(3) (-) ionic liquid are provided to illustrate the method.     

The quest for the best nonpolarizable water model from the adaptive force matching method

The recently introduced adaptive force matching (AFM) method is used to develop a significantly improved pair‐wise nonpolarizable potential for water. A rigid version of the potential is also presented to enable larger time steps for biological simulations. In this work, it is demonstrated that the AFM method can be used to systematically assess the importance of each functional term during the construction of a force field. For a water potential, it is established that a single off‐atom charge center (M) in the plane of water outperforms two out‐of‐plane charge sites for reproducing intermolecular forces. The four‐site pair‐wise nonpolarizable force field developed in this work rivals some of the most sophisticated polarizable models in terms of reproducing accurate ab initio forces. The force fields are parameterized to perform best in the temperature range from 0 to 40°C. Equilibrium and dynamical properties calculated with the flexible and rigid force fields are in good agreement with experimental results. For the flexible model, the agreement improves when path integral simulation is performed. These force fields provide high‐quality results at a very low computational cost and are thus well suited to atomistic scale biological simulations. The AFM method provides a mechanism for selecting important terms in force field expressions and is a very promising tool for producing accurate force fields in condensed phases. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2011     

Linking atomistic and mesoscale simulations of water-soluble polymers

A link between molecular and mesoscopic simulations for water-soluble polymers is made using the potential of mean force (PMF) method. Mesoscale parameters are adjusted to match selected thermodynamic quantities such as the monomer second virial coefficient or the limiting slope of the monomer adsorption isotherm, which are computed directly from atomistic PMFs. The method is illustrated by computing the bulk and surface interaction parameters (chi parameters in self-consistent field theory) for the adsorption of ethylene oxide (EO) and propylene oxide (PO) polymers from aqueous solution onto silica.     

Modeling of the transition temperature for the helical denaturation of α-keratin intermediate filaments

Simulations of the stability of the secondary and tertiary structure of the α-keratin intermediate filament (IF) monomeric unit of wool are reported. Based on the assumed secondary structure three segments of the primary structure were selected: 1A, L12, and a part of 2B. Starting with an ideal α-helical conformation for each IF-segment, molecular dynamics simulations were carried out on the atomistic level at various temperatures in vaccum using the CFF91 force field. In either simulation the expected destabilization of the helical structure with increasing simulation temperature was observed. By use of different procedures of analysis, transition temperatures for the α-helical denaturation were determined that are significantly higher for the supposedly α-helical segments 1A and 2B than for the linker segment L12. The different stabilities of segments 1A and L12 were further verified through simulations in water environment that show the linker segment to be non-helical at room temperature. The lower transition temperature of segment L12 confirms the expectation that its amino acid sequence leads to increased conformational flexibility. The mobility of the water molecules surrounding the IF-segment is found to be significantly decreased by protein/water interactions.     

The multiscale coarse-graining method. I. A rigorous bridge between atomistic and coarse-grained models

Coarse-grained (CG) models provide a computationally efficient method for rapidly investigating the long time- and length-scale processes that play a critical role in many important biological and soft matter processes. Recently, Izvekov and Voth introduced a new multiscale coarse-graining (MS-CG) method [J. Phys. Chem. B 109, 2469 (2005); J. Chem. Phys. 123, 134105 (2005)] for determining the effective interactions between CG sites using information from simulations of atomically detailed models. The present work develops a formal statistical mechanical framework for the MS-CG method and demonstrates that the variational principle underlying the method may, in principle, be employed to determine the many-body potential of mean force (PMF) that governs the equilibrium distribution of positions of the CG sites for the MS-CG models. A CG model that employs such a PMF as a "potential energy function" will generate an equilibrium probability distribution of CG sites that is consistent with the atomically detailed model from which the PMF is derived. Consequently, the MS-CG method provides a formal multiscale bridge rigorously connecting the equilibrium ensembles generated with atomistic and CG models. The variational principle also suggests a class of practical algorithms for calculating approximations to this many-body PMF that are optimal. These algorithms use computer simulation data from the atomically detailed model. Finally, important generalizations of the MS-CG method are introduced for treating systems with rigid intramolecular constraints and for developing CG models whose equilibrium momentum distribution is consistent with that of an atomically detailed model.     

Homophilic interactions between cadherin fragments at the single molecule level: an AFM study

We report measurements of the adhesion forces between single E-cadherin fragments anchored on solid surfaces. These fragments consist of the two outermost extracellular domains of the protein. The specificity of the measured rupture forces was demonstrated by Ca2+ exchange experiments. Two series of experiments were performed using two linkers of different rigidity and length. We find that the pull-off force is distributed with a maximum value independent of the linker and logarithmically dependent on the velocity of separation of the two surfaces. Our dynamical results are compatible with previous flow chamber experiments performed with the same fragments and can be compared from a different perspective with previously reported AFM experiments on the full-length extracellular domain of the VE-cadherin. Interestingly, using a rigid linker, we have been able for the first time to evidence the deformation of the cadherin molecule under mechanical stress, a piece of information not accessible with more classical grafting strategies.     

Adhesion Mechanisms of the Contact Interface of TiO(2) Nanoparticles in Films and Aggregates

Fundamental knowledge about the mechanisms of adhesion between oxide particles with diameters of few nanometers is impeded by the difficulties associated with direct measurements of contact forces at such a small size scale. Here we develop a strategy based on AFM force spectroscopy combined with all-atom molecular dynamics simulations to quantify and explain the nature of the contact forces between 10 nm small TiO(2) nanoparticles. The method is based on the statistical analysis of the force peaks measured in repeated approaching/retracting loops of an AFM cantilever into a film of nanoparticle agglomerates and relies on the in-situ imaging of the film stretching behavior in an AFM/TEM setup. Sliding and rolling events first lead to local rearrangements in the film structure when subjected to tensile load, prior to its final rupture caused by the reversible detaching of individual nanoparticles. The associated contact force of about 2.5 nN is in quantitative agreement with the results of molecular dynamics simulations of the particle-particle detachment. We reveal that the contact forces are dominated by the structure of water layers adsorbed on the particles' surfaces at ambient conditions. This leads to nonmonotonous force-displacement curves that can be explained only in part by classical capillary effects and highlights the importance of considering explicitly the molecular nature of the adsorbates.     

Single-molecule force spectroscopy measurements of "hydrophobic bond" between tethered hexadecane molecules

The hydrophobic effect is important for many biological and technological processes. Despite progress in theory, experimental data clarifying water structure and the interaction between hydrophobic solutes at the nanometer scale are scarce due to the very low solubility of hydrophobic species. This article describes an AFM single molecule force spectroscopy method to probe the interaction between molecules with low solubility and reports measurements of the strength and the length scale of the "hydrophobic bond" between hexadecane molecules. Hexadecane molecules are tethered by flexible poly(ethylene glycol) linkers to AFM probes and substrates, removing the aggregation state uncertainty of solution-based approaches as well as spurious surface effects. A shorter hydrophilic polymer layer is added to increase the accessibility of hydrophobic molecules for the force spectroscopy measurements. Statistical analysis of the rupture forces yields a barrier width of 0.24 nm, and a dissociation rate of 1.1 s(-1). The results of single molecule measurements are related to the theoretical predictions of the free energy of cavitation in water and to the empirical model of micellization of nonionic surfactants. It is estimated that approximately one-quarter of each molecule's surface is hydrated during forced dissociation, consistent with an extended (nonglobular) conformation of the hexadecane molecules in the dimer.     

Developing ab initio quality force fields from condensed phase quantum-mechanics/molecular-mechanics calculations through the adaptive force matching method

A new method called adaptive force matching (AFM) has been developed that is capable of producing high quality force fields for condensed phase simulations. This procedure involves the parametrization of force fields to reproduce ab initio forces obtained from condensed phase quantum-mechanics/molecular-mechanics (QM/MM) calculations. During the procedure, the MM part of the QM/MM is iteratively improved so as to approach ab initio quality. In this work, the AFM method has been tested to parametrize force fields for liquid water so that the resulting force fields reproduce forces calculated using the ab initio MP2 and the Kohn-Sham density functional theory with the Becke-Lee-Yang-Parr (BLYP) and Becke three-parameter LYP (B3LYP) exchange correlation functionals. The AFM force fields generated in this work are very simple to evaluate and are supported by most molecular dynamics (MD) codes. At the same time, the quality of the forces predicted by the AFM force fields rivals that of very expensive ab initio calculations and are found to successfully reproduce many experimental properties. The site-site radial distribution functions (RDFs) obtained from MD simulations using the force field generated from the BLYP functional through AFM compare favorably with the previously published RDFs from Car-Parrinello MD simulations with the same functional. Technical aspects of AFM such as the optimal QM cluster size, optimal basis set, and optimal QM method to be used with the AFM procedure are discussed in this paper.     

Analysis of the unfolding process of green fluorescent protein by molecular dynamics simulation

Molecular dynamics simulation of the enforced stretching of circularly permuted green fluorescent protein (cpGFP) was performed to observe the detailed process of unfolding of beta-sheets in cpGFP and to clarify the structural change arising from the force. The simulation using the generalized Born method with original force field parameters enabled us to observe the unfolding process of the entire region of the protein and to clarify atom motion of the individual domain during the stretching. The force required for the stretching of cpGFP was estimated from the differential of the computed potential energy. A prominent rise in force appeared three times during the stretching. The amplitude and the position of these three peaks were consistent with the observation in atomic force microscopy (AFM) experiments. Further, the movements of atoms involved in each peak were shown to be closely related to the dissociation of hydrogen bonds. Additional simulations for the unfolding process of titin and spectrin also gave satisfactory interpretation of the results of previous AFM experiments. The difference in the enforced stretching process between cpGFP and wild-type GFP was further discussed through the MD simulation.     

Direct Identification of Protein–Protein Interactions by Single‐Molecule Force Spectroscopy

Single‐molecule force spectroscopy based on atomic force microscopy (AFM‐SMFS) has allowed the measurement of the intermolecular forces involved in protein‐protein interactions at the molecular level. While intramolecular interactions are routinely identified directly by the use of polyprotein fingerprinting, there is a lack of a general method to directly identify single‐molecule intermolecular unbinding events. Here, we have developed an internally controlled strategy to measure protein–protein interactions by AFM‐SMFS that allows the direct identification of dissociation force peaks while ensuring single‐molecule conditions. Single‐molecule identification is assured by polyprotein fingerprinting while the intermolecular interaction is reported by a characteristic increase in contour length released after bond rupture. The latter is due to the exposure to force of a third protein that covalently connects the interacting pair. We demonstrate this strategy with a cohesin–dockerin interaction.     

Microscopic simulations of molecular cluster decay: Does the carrier gas affect evaporation?

We develop a kinetic theory of cluster decay by considering the stochastic motion of molecules within an effective potential of mean force (PMF) due to the cluster. We perform molecular dynamics simulations on a 50-atom argon cluster to determine the mean radial force on a component atom and hence the confining potential of mean force. Comparisons between isolated clusters and clusters thermostatted through the presence of a 100-atom helium carrier gas show that the heat bath has only a slight effect upon the PMF. This confirms the validity of calculations of cluster properties using isolated cluster simulations. The PMF is used to calculate the atomic evaporation rate from these clusters, and results are compared with the predictions of the capillarity approximation together with detailed balance, both components of the classical theory of aerosol nucleation.     

Steered molecular dynamics simulations of Na+ permeation across the gramicidin A channel

The potential of mean forces (PMF) governing Na+ permeation through gramicidin A (gA) channels with explicit water and membrane was characterized using steered molecular dynamics (SMD) simulations. Constant-force SMD with a steering force parallel to the channel axis revealed at least seven energy wells in each monomer of the channel dimer. Except at the channel dimer interface, each energy well is associated with at least three and at most four backbone carbonyl oxygens and two water oxygens in a pseudo-hexahedral or pseudo-octahedral coordination with the Na+ ion. Repeated constant-velocity SMD by dragging a Na+ ion from each energy well in opposite directions parallel to the channel axis allowed the computation of the PMF across the gA channel, revealing a global minimum corresponding to Na+ binding sites near the entrance of gA at +/-9.3 A from the geometric center of the channel. The effect of volatile anesthetics on the PMF was also analyzed in the presence of halothane molecules. Although the accuracy of the current PMF calculation from SMD simulations is not yet sufficient to quantify the PMF difference with and without anesthetics, the comparison of the overall PMF profiles nevertheless confirms that the anesthetics cause insignificant changes to the structural makeup of the free energy wells along the channel and the overall permeation barrier. On average, the PMF appears less rugged in the outer part of the channel in the presence of anesthetics, consistent with our earlier finding that halothane interaction with anchoring residues makes the gA channel more dynamic. A causal relationship was observed between the reorientation of the coordinating backbone carbonyl oxygen and Na+ transit from one energy well to another, suggesting the possibility that even minute changes in the conformation of pore-lining residues due to dynamic motion could be sufficient to trigger the ion permeation. Because some of the carbonyl oxygens contribute to Na+ coordination in two adjacent energy wells, our SMD results reveal that the atomic picture of ion "hopping" through a gA channel actually involves a Na+ ion being carried in a relay by the coordinating oxygens from one energy well to the next. Steered molecular dynamics complements other computational approaches as an attractive means for the atomistic interpretation of experimental permeation studies.     

Computer simulations of atomic force microscopy: crystalline polymers and the effects of surface contaminants

The atomistic dynamics of the interaction of an atomic force microscopic (AFM) probe with a crystalline polyethylene surface was examined by using the molecular dynamics method. The results show that the internal dynamics of the polymer crystal is such that rapid relaxation occurs, providing for a large amount of structural reversibility and making it possible to perform nondestructive AFM experiments. However, surface and/or AFM tip defects or contaminants (such as those which can be induced by polar molecules adsorbed on the surface), can result in significant perturbations in the AFM images produced, causing large and sharp structures to appear on the surface topology. A rationale of the mechanisms responsible for the image distortions is presented, and a relationship to defects observed in AFM and STM experiments is given.     

Molecular dynamics simulation of the association of nonpolar spheres in water

We apply molecular dynamics (MD) simulations to the study of the association of nonpolar spheres of effective radii between 1.6 and 6.1 A dissolved in water. The constrained MD method is used to calculate the potential of mean force (PMF) of the interaction between spheres. The depth of the potential of mean force increases with increasing radius of the nonpolar sphere. Our results suggest that the PMF is largely governed by size or entropic effects, and that energetic effects associated with the breaking or distortion of hydrogen bonds are of minor importance.