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.     

Binary interactions and salt-induced coalescence of spherical micelles of cationic surfactants from molecular dynamics simulations

A direct estimation of salt-mediated potential of mean force (PMF) between spherical micelles of cationic surfactants is obtained for the first time using molecular dynamics (MD) simulations. Coarse-grained (CG) potentials benchmarked in an earlier study [Langmuir, 2011, 27(11), 6628-6638] are used to model a binary system of cetyltrimethylammonium chloride (CTAC) surfactant micelles at varying concentrations of sodium chloride (NaCl) or sodium salicylate (NaSal). The shape and structure of micelles are not subject to external constraints. NaSal is significantly more efficient in screening the intermicelle repulsive interactions shown by the PMF compared to NaCl due to a stronger binding of salicylate counterions to the micelle corona. Upon contact with each other, the micelles coalesce in the presence of NaSal to form a cylindrical structure which is stabilized by the adsorbed salicylate anions. Comparison of the PMF with Derjaguin-Landau-Verwey-Overbeek (DLVO) potentials shows qualitative agreement, while the magnitude of PMF is significantly greater than that of the DLVO potentials. To understand this discrepancy, PMF is evaluated by turning off (a) long-ranged electrostatic interactions and (b) solvent polarizability. The above effects are shown to play an important role in determining the solvent-mediated and ion-correlated interactions between the two micelles, which are not explicitly captured by mean-field double layer theories such as DLVO.     

Potential of mean force between hydrophobic solutes in the Jagla model of water and implications for cold denaturation of proteins

Using the Jagla model potential we calculate the potential of mean force (PMF) between hard sphere solutes immersed in a liquid displaying water-like properties. Consistent estimates of the PMF are obtained by (a) umbrella sampling, (b) calculating the work done by the mean force acting on the hard spheres as a function of their separation, and (c) determining the position dependent chemical potential after calculating the void space in the liquid. We calculate the PMF for an isobar along which cold denaturation of a model protein has previously been reported. We find that the PMF at contact varies non-monotonically, which is consistent with the observed cold denaturation. The Henry constant also varies non-monotonically with temperature. We find, on the other hand, that a second (solvent separated) minimum of the PMF becomes deeper as temperature decreases. We calculate the solvent-solvent pair correlation functions for solvents near the solute and in the bulk, and show that, as temperature decreases, the two pair correlation functions become indistinguishable, suggesting that the perturbation of solvent structure by the solute diminishes as temperature decreases. The solvent-solute pair correlation function at contact grows as the temperature decreases. We calculate the cavity correlation function and show the development of a solvent-separated peak upon decrease of temperature. These observations together suggest that cold denaturation occurs when the solvent penetrates between hydrophobic solutes in configurations with favorable free energy. Our results thus suggest that cold denatured proteins are structured and that cold denaturation arises from strong solvent-solute interactions, rather than from entropic considerations as in heat denaturation.     

Effects of size constraint on water filling process in nanotube

Molecular dynamics (MD) simulation and the potential of mean force (PMF) analysis are used to investigate the structural properties of water molecules near the end of nanotube for the whole process from the initial water filling up to the configuration stabilization inside the carbon nanotubes (CNTs). Numerical simulations showed that when a small-sized nanotube is immersed into the water bath, the size constraint will induce a prevailing orientation for the water molecule to diffuse into the tube and this effect can persist approximately 3.3 angstroms from the end of CNT. As the structure within the CNTs stabilizes, the ambient structural properties can indirectly reflect their corresponding properties inside the nanotube. Our results also showed that there exists a close correlation between the PMF analysis and the results of MD simulations, and the properties at nanometer scale are closely related to the size-constraint effect.     

Molecular dynamics study of the inclusion of cholesterol into cyclodextrins

The interaction of three cyclodextrins (CDs), viz. beta-CD, heptakis (2,6-di-O-methyl)-beta-CD (DM-beta-CD), and 2-hydroxypropyl-beta-CD (HP-beta-CD), with cholesterol was investigated using molecular dynamics (MD) simulations. The free energy along the reaction pathway delineating the inclusion of cholesterol into each CD was computed using the adaptive biasing force method. The association constant and the corresponding association free energy were derived by integrating the potential of mean force (PMF) over a representative ordering parameter. The results show that the free energy profiles possess two local minima corresponding to roughly equally probable binding modes. Among the three CDs, DM-beta-CD exhibits the highest propensity to associate with cholesterol. Ranking for binding cholesterol, viz. DM-beta-CD > HP-beta-CD > beta-CD, agrees nicely with experiment. Partitioning of the PMF into free energy components illuminates that entering of cholesterol into the CD cavity is driven mainly by electrostatic interactions, whereas deeper inclusion results from van der Waals forces and solvation effects. Additional MD simulations were performed to investigate the structural stability of the host-guest complexes near the free energy minima. The present results demonstrate that association of cholesterol and CDs follows two possible binding modes. Although the latter are thermodynamically favorable for all CDs, one of the two inclusion complexes appears to be preferred kinetically in the case of DM-beta-CD.     

Molecular origin of anticooperativity in hydrophobic association

The structuring of water molecules in the vicinity of nonpolar solutes is responsible for hydrophobic hydration and association thermodynamics in aqueous solutions. Here, we studied the potential of mean force (PMF) for the formation of a dimer and trimers of methane molecules in three specific configurations in explicit water to explain multibody effects in hydrophobic association on a molecular level. We analyzed the packing and orientation of water molecules in the vicinity of the solute to explain the effect of ordering of the water around nonpolar solutes on many-body interactions. Consistent with previous theoretical studies, we observed cooperativity, manifested as a reduction of the height of the desolvation barrier for the trimer in an isosceles triangle geometry, but for linear trimers, we observed only anticooperativity. A simple mechanistic picture of hydrophobic association is drawn. The free energy of hydrophobic association depends primarily on the difference in the number of water molecules in the first solvation shell of a cluster and that in the monomers of a cluster; this can be approximated by the molecular surface area. However, there are unfavorable electrostatic interactions between the water molecules from different parts of the solvation shell of a trimer because of their increased orientation induced by the nonpolar solute. These electrostatic interactions make an anticooperative contribution to the PMF, which is clearly manifested for the linear trimer where the multibody contribution due to changes in the molecular surface area is equal to zero. The information theory model of hydrophobic interactions of Hummer et al. also explains the anticooperativity of hydrophobic association of the linear trimers; however, it predicts anticooperativity with a qualitatively identical distance dependence for nonlinear trimers, which disagrees with the results of simulations.     

Potential of mean force of hydrophobic association: dependence on solute size

The potentials of mean force (PMFs) were determined for systems involving formation of nonpolar dimers composed of methane, ethane, propane, isobutane, and neopentane, respectively, in water, using the TIP3P water model, and in vacuo. A series of umbrella-sampling molecular dynamics simulations with the AMBER force field was carried out for each pair in either water or in vacuo. The PMFs were calculated by using the weighted histogram analysis method (WHAM). The shape of the PMFs for dimers of all five nonpolar molecules is characteristic of hydrophobic interactions with contact and solvent-separated minima and desolvation maxima. The positions of all these minima and maxima change with the size of the nonpolar molecule, that is, for larger molecules they shift toward larger distances. The PMF of the neopentane dimer is similar to those of other small nonpolar molecules studied in this work, and hence the neopentane dimer is too small to be treated as a nanoscale hydrophobic object. The solvent contribution to the PMF was also computed by subtracting the PMF determined in vacuo from the PMF in explicit solvent. The molecular surface area model correctly describes the solvent contribution to the PMF together with the changes of the height and positions of the desolvation barrier for all dimers investigated. The water molecules in the first solvation sphere of the dimer are more ordered compared to bulk water, with their dipole moments pointing away from the surface of the dimer. The average number of hydrogen bonds per water molecule in this first hydration shell is smaller compared to that in bulk water, which can be explained by coordination of water molecules to the hydrocarbon surface. In the second hydration shell, the average number of hydrogen bonds is greater compared to bulk water, which can be explained by increased ordering of water from the first hydration shell; the net effect is more efficient hydrogen bonding between the water molecules in the first and second hydration shells.     

Self-assembling cyclic peptides: molecular dynamics studies of dimers in polar and nonpolar solvents

The self-assembly of cyclic D,L-alpha-peptides into hollow nanotubes is a crucial mechanistic step in their application as antibacterial and drug-delivery agents. To understand this process, molecular dynamics (MD) simulations were performed on dimers of cyclic peptides formed from cyclo [(-L-Trp-D-N-MeLeu-)4-]2 and cyclo [(-L-Trp-D-Leu-)4-]2 subunits in nonpolar (nonane) and polar (water) solvent. The dimers were observed to be stable only in nonpolar solvent over the full 10 ns length of the MD trajectory. The behavior of the dimers in different solvents is rationalized in terms of the intersubunit hydrogen bonding, hydrogen bonding with the solvent, and planarity of the rings. It is shown that the phi and psi dihedral angles of a single uncapped ring in nonane lie in the beta-sheet region of the Ramachandran plot, and the ring stays in a flat conformation. Steered MD (SMD) simulations based on Jarzynski's equality were performed to obtain the potential of mean force as a function of the distance between the two rings of the capped dimer in nonane. It is also shown that a single peptide subunit prefers to reside close to the nonane/water interface rather than in bulk solvent because of the amphiphilic character of the peptide ring. The present MD results build the foundation for using MD simulations to study the mechanism of the formation of cyclic peptide nanotubes in lipid bilayers.     

Interactions between spherical colloids mediated by a liquid crystal: a molecular simulation and mesoscale study

Monte Carlo simulations and dynamic field theory (DyFT) are used to study the interactions between dilute spherical particles, dispersed in nematic and isotropic phases of a liquid crystal. A recently developed simulation method (expanded ensemble density of states) was used to determine the potential of mean force (PMF) between the two spheres as a function of their separation and size. The PMF was also calculated by a dynamic field theory that describes the evolution of the local tensor order parameter. Both methods reveal an overall attraction between the colloids in the nematic phase; in the isotropic phase, the overall attraction between the colloids is much weaker, whereas the repulsion at short range is stronger. In addition, both methods predict a new topology of the disclination lines, which arises when the particles approach each other. The theory is found to describe the results of simulations remarkably well, down to length scales comparable to the size of the molecules. At separations corresponding to the width of individual molecular layers on the particles' surface, the two methods yield different defect structures. We attribute this difference to the neglect of density inhomogeneities in the DyFT. We also investigate the effects of the size of spherical colloids on their interactions.     

Hamiltonian and distance replica exchange method studies of Met-enkephalin

The conformational states of the zwitterionic form of the pentapeptide Met-enkephalin were explored with the use of explicit solvent molecular dynamics (MD). The N and C termini are ionized, as appropriate to polar solvent conditions, and consequently, there is a competition between open forms driven by polar solvation of the ammonium and carboxylate groups and closed forms driven by their salt-bridge formation. Normal MD started from an open state does not sample closed conformations. Sampling was enhanced with a distance replica exchange method (DREM) and with a Hamiltonian replica exchange method (HREM). The potential of mean force (PMF) along an end-to-end distance reaction coordinate was obtained with the DREM. The PMF shows a stable salt-bridge state and the presence of a large region of open states, as hypothesized for conformationally promiscuous small opiate peptides. The HREM systems differ by scaling the peptide-peptide and peptide-solvent electrostatic and Lennard-Jones potentials, with the goal of improving the sampling efficiency with a limited number of systems. A small number of systems were found to be sufficient to sample closed and open states. A principal component analysis (PCA) shows that the HREM-generated fluctuations are dominated by the first two principal modes. The first corresponds to the end-to-end reaction coordinate found in the DREM, and the first mode PMF is similar to the DREM PMF. The second mode describes the presence of two conformations, both of which correspond to the salt-bridge state distance. The conformers differ in the values of neighboring psi and phi dihedral angles, since such psi/phi compensation can still produce the same end-to-end distance. The two-dimensional PMF constructed from the first two PCA modes captures most of the significant backbone conformational space of Met-enkephalin.     

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.     

Alternative approaches to potential of mean force calculations: Free energy perturbation versus thermodynamic integration. Case study of some representative nonpolar interactions

We investigated the convergence behavior of potential of mean force (PMF) calculations using free energy perturbation (FEP), thermodynamic integration (TI), and “slow growth” (SG) techniques. The critical comparison of these alternative approaches is illustrated by the study of three different systems: two tagged argon atoms in a periodic box of argon, two methane molecules, and two benzene molecules maintained in a “T-shaped” conformation, both dimers embedded in a periodic box of water. The complete PMF simulations were carried out considering several protocols, in which the number of intermediate “λ” states, together with the amount of sampling per individual state, were varied. In most cases, as much as 1 ns of molecular dynamics (MD) sampling was used to derive each free energy profile. For the different systems examined, we find that FEP and TI unquestionably constitute robust computational methods leading to results of comparable accuracy. We also show that proper convergence of the free energy calculations, and further quantitative interpretation of the PMFs, requires total simulation times much higher than has been hitherto estimated. In some circumstances, the free energy profiles derived from FEP calculations tend to be slightly poorer than those obtained with TI, as a probable consequence of the greater sensitivity of FEP to the window spacing δλ. In the context of TI, and to a lesser extent FEP, simulations, it appears preferable to employ a limited number of “λ” points of the integrand involving extensive sampling, rather than numerous points with fewer samplings. Finally, we note that, at least in the case of nonpolar interactions, PMFs of reasonable quality can be generated using SG, and at a substantially lower cost than with either FEP or TI. © 1996 by John Wiley & Sons, Inc.     

Effective interactions between charged nanoparticles in water: What is left from the DLVO theory?

The keystone of the modelling of complex systems is the potential of mean force (PMF) between particles. This review focuses on recent numerical simulation studies that concern the computation of the PMF between charged nanoparticles in solution. Such simulations explicitly sample the configurations of the microions or water molecules over which the potential is averaged out. The studies rely on different levels of modelling and permit to quantify the relative amplitude of the different factors governing the interaction, such as the structure of the nanoparticle, the polarisability of microions, or hydrophobic interactions. We discuss the conditions in which the potential of mean force can safely be expressed as a DLVO potential, and why in some cases such a simple analytical expression cannot be used.     

Molecular dynamics simulation study of the water-mediated interaction between zwitterionic and charged surfaces

We calculated the potential of mean force (PMF) for the interaction between a model zwitterionic bilayer and a model charged bilayer. To understand the role of water, we separated the PMF into two components: one due to direct interaction and the other due to water-mediated interaction. In our calculations, we observed that water-mediated interaction is attractive at larger distances and repulsive at shorter. The calculation of the entropic and enthalpic contributions to the solvent-mediated components of the PMF showed that attraction is entropically dominant, while repulsion is dominated by the enthalpy.     

Defect structures and three-body potential of the mean force for nanoparticles in a nematic host

We have investigated the defect structures and potential of mean force (PMF) of three colloidal spheres immersed in a nematic liquid crystal (for linear and equilateral-triangular configurations), using a coarse-grained theory for the tensor order parameter. At large separations, each sphere is surrounded by an equatorial Saturn ring defect; at short separations, the theory predicts a drastic reorganization of the disclination lines: additional disclination rings are observed in planes perpendicular to the original ones. For both types of configurations, the PMF is attractive and always greater than a pairwise sum of binary PMFs. Comparing the PMFs of linear and equilateral-triangular configurations, we have found the triangular configuration to be more stable than the linear configuration. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 1033-1040, 2005     

苯磺酰胺从碳酸酐酶II中脱离过程的分子动力学模拟

综合运用分子动力学模拟和自由能计算方法研究了苯磺酰胺分子从碳酸酐酶II (CA II)的活性位点脱离过程中底物与酶之间的动态相互作用. 脱离过程的平均力势(PMF)显示, 底物脱离时存在一个特殊的结合状态. 其中, 静电相互作用占据了主导地位. 轨迹分析显示, 除了金属离子的配位作用之外, 底物脱离路径上的关键残基Leu198、Thr199和Thr200通过与底物磺胺基的氢键作用阻碍了底物从酶中的脱离. 当前的研究对于深入认识磺胺类药物与CA II的详细结合过程和相关的药物改良与设计具有重要的指导意义.     

Temperature dependence of three-body hydrophobic interactions: potential of mean force, enthalpy, entropy, heat capacity, and nonadditivity

Temperature-dependent three-body hydrophobic interactions are investigated by extensive constant-pressure simulations of methane-like nonpolar solutes in TIP4P model water at six temperatures. A multiple-body hydrophobic interaction is considered to be (i) additive, (ii) cooperative, or (iii) anti-cooperative if its potential of mean force (PMF) is (i) equal to, (ii) smaller than, or (iii) larger than the corresponding pairwise sum of two-methane PMFs. We found that three-methane hydrophobic interactions at the desolvation barrier are anti-cooperative at low to intermediate T, and vary from essentially additive to slightly cooperative at high T. Interactions at the contact minimum are slightly anti-cooperative over a wider temperature range. Enthalpy, entropy, and heat capacity are estimated from the computed PMFs. Contrary to the common expectation that burial of solvent-accessible nonpolar surface area always leads to a decrease in heat capacity, the present results show that the change in heat capacity upon three-methane association is significantly positive at the desolvation barrier and slightly positive at the contact minimum. This suggests that the heat capacity signature of a hydrophobic polymer need not vary uniformly nor monotonically with conformational compactness. Ramifications for protein folding are discussed.     

A comparison of methods to compute the potential of mean force.

Most processes occurring in a system are determined by the relative free energy between two or more states because the free energy is a measure of the probability of finding the system in a given state. When the two states of interest are connected by a pathway, usually called reaction coordinate, along which the free-energy profile is determined, this profile or potential of mean force (PMF) will also yield the relative free energy of the two states. Twelve different methods to compute a PMF are reviewed and compared, with regard to their precision, for a system consisting of a pair of methane molecules in aqueous solution. We analyze all combinations of the type of sampling (unbiased, umbrella-biased or constraint-biased), how to compute free energies (from density of states or force averaging) and the type of coordinate system (internal or Cartesian) used for the PMF degree of freedom. The method of choice is constraint-bias simulation combined with force averaging for either an internal or a Cartesian PMF degree of freedom.     

Calculation of the absolute thermodynamic properties of association of host-guest systems from the intermolecular potential of mean force

The authors report calculations of the intermolecular potential of mean force (PMF) in the case of the host-guest interaction. The host-guest system is defined by a water soluble calixarene and a cation. With an organic cation such as the tetramethylammonium cation, the calixarene forms an insertion complex, whereas with the Lanthane cation, the supramolecular assembly is an outer-sphere complex. The authors apply a modified free energy perturbation method and the force constraint technique to establish the PMF profiles as a function of the separation distance between the host and guest. They use the PMF profile for the calculation of the absolute thermodynamic properties of association that they compare to the experimental values previously determined. They finish by giving some structural features of the insertion and outer-sphere complexes at the Gibbs free energy minimum.     

Dynamic information for cardiotoxin protein desorption from a methyl-terminated self-assembled monolayer using steered molecular dynamics simulation

Dynamic information, such as force, structural change, interaction energy, and potential of mean force (PMF), about the desorption of a single cardiotoxin (CTX) protein from a methyl-terminated self-assembled monolayer (SAM) surface was investigated by means of steered molecular dynamics (SMD) simulations. The simulation results indicated that Loop I is the first loop to depart from the SAM surface, which is in good agreement with the results of the nuclear magnetic resonance spectroscopy experiment. The free energy landscape and the thermodynamic force of the CTX desorption process was represented by the PMF and by the derivative of PMF with respect to distance, respectively. By applying Jarzynski's equality, the PMF can be reconstructed from the SMD simulation. The PMFs, calculated by different estimators based upon Jarzynski's equality, were compared with the conventional umbrella sampling method. The best estimation was obtained by using the fluctuation-dissipation estimator with a pulling velocity of v = 0.25 nm/ns for the present study.