Exact Relation between Potential of Mean Force and Free-Energy Profile

Entropy and Energy Profiles of Chemical Reactions

The description of chemical processes at the molecular level is often facilitated by the use of reaction coordinates or collective variables (CVs). The CV measures the progress of the reaction and allows the construction of profiles that track how specific properties evolve as the reaction progresses. Whereas CVs are routinely used, especially alongside enhanced sampling techniques, the links among reaction profiles, thermodynamic state functions, and reaction rate constants are not rigorously exploited. Here, we report a unified treatment of such reaction profiles. Tractable expressions are derived for the free-energy, internal-energy, and entropy profiles as functions of only the CV. We demonstrate the ability of this treatment to extract quantitative insight from the entropy and internal-energy profiles of various real-world physicochemical processes, including intramolecular organic reactions, ionic transport in superionic electrolytes, and molecular transport in nanoporous materials.

The effect of solvent on determining highest occupied molecular orbital energies of semiconducting organic molecules: Insight from a combined computational approach

Although cyclic voltammetry (CV) measurements in solution have been widely used to determine the highest occupied molecular orbital energy (E_HOMO ) of semiconducting organic molecules, an understanding of the experimentally observed discrepancies due to the solvent used is lacking. To explain these differences, we investigate the solvent effects on E_HOMO by combining density functional theory and molecular dynamics calculations for four donor molecules with a common backbone moiety. We compare the experimental E_HOMO values to the calculated values obtained from either implicit or first solvation shell theories. We find that the first solvation shell method can capture the E_HOMO variation arising from the functional groups in solution, unlike the implicit method. We further applied the first solvation shell method to other semiconducting organic molecules measured in solutions for different solvents. We find that the E_HOMO obtained using an implicit method is insensitive to solvent choice. The first solvation shell, however, produces E_HOMO values that are sensitive to solvent choices and agrees with published experimental results. The solvent sensitivity arises from a hierarchy of three effects: (1) the solute electronic state within a surrounding dielectric continuum, (2) ambient temperature or solvent atoms changing the solute geometry, and (3) electronic interactions between the solute and solvents. The implicit method, on the other hand, only captures the effect of a dielectric environment. Our findings suggest that E_HOMO obtained by CV measurements should account for the influence of solvent when the results are reported, interpreted, or compared to other molecules.

Molecular dynamics investigation of the asphaltene-kaolinite interactions in water, toluene, and water-toluene mixtures

Understanding the interactions of petroleum asphaltenes with mineral surfaces is important for diluted bitumen spill response and modeling. In this study, molecular dynamics and umbrella sampling simulations are performed using interfacially active and non-interfacially active asphaltene model compounds individually positioned near each of the surfaces of kaolinite in the presence of explicit solvent environments containing water, toluene, and mixtures of toluene and water in varying proportions. The interfacially active asphaltene bonds the strongest to the silicon oxide surface of kaolinite in pure water and the bonding weakens to nearly zero in toluene-water mixtures. The non-interfacially active asphaltenes bond to kaolinites silicon oxide surface in water about half as strongly as the interfacially active one in water and the bonding weakens in the presence of toluene. The number of non-hydrogen bonded contacts between the interfacially active asphaltene and the aluminum hydroxide surface of kaolinite increases as the proportion of toluene is increased and the contacts with water are decreased. In these conditions, the non-interfacially active asphaltenes do not form non-hydrogen bonded contacts with kaolinite. On the silicon oxide surface, the number of non-hydrogen bonded contacts of all asphaltenes with kaolinite tends to decrease as the proportion of toluene is increased and the contacts with water are decreased. The number of hydrogen bonds of the interfacially active asphaltene with water decreases as the proportion of toluene is increased. The radii of gyration indicate that the interfacially active asphaltene is extended in water and when adsorbed on kaolinite, and becomes compact as the proportion of toluene is increased. The simulation results highlight the competitive interfacial interactions in the complex scenario of diluted bitumen spills in the presence of water and clay minerals.

Free Energy Profiles of Proton Transfer Reactions: Density Functional Benchmark from Biased Ab Initio Dynamics

By coupling an enhanced sampling algorithm with an orbital-localized variant of Car-Parrinello molecular dynamics, the so-called atomic centered density matrix propagation model, we reconstruct the free energy profiles along reaction pathways using different density functional approximations (DFAs) ranging from locals to hybrids. In particular, we compare the computed free energy barrier height of proton transfer (PT) reactions to those obtained by a more traditional static approach, based on the intrinsic reaction coordinate (IRC), for two case systems, namely malonaldehyde and formic acid dimer. The obtained results show that both the IRC profiles and the potentials of mean force, derived from biased dynamic trajectories, are very sensitive to the density functional approximation applied. More precisely, we observe that, with the notable exception of M06-L, local density functionals always strongly underestimate the reaction barrier heights. More generally, we find that also the shape of the free energy profile is very sensitive to the density functional choice, thus highlighting the effect, often neglected, that the choice of DFA has also in the case of dynamics simulations.

Fighting Against Bacterial Lipopolysaccharide-Caused Infections through Molecular Dynamics Simulations: A Review

Lipopolysaccharide (LPS) is the primary component of the outer leaflet of Gram-negative bacterial outer membranes. LPS elicits an overwhelming immune response during infection, which can lead to life-threatening sepsis or septic shock for which no suitable treatment is available so far. As a result of the worldwide expanding multidrug-resistant bacteria, the occurrence and frequency of sepsis are expected to increase; thus, there is an urge to develop novel strategies for treating bacterial infections. In this regard, gaining an in-depth understanding about the ability of LPS to both stimulate the host immune system and interact with several molecules is crucial for fighting against LPS-caused infections and allowing for the rational design of novel antisepsis drugs, vaccines and LPS sequestration and detection methods. Molecular dynamics (MD) simulations, which are understood as being a computational microscope, have proven to be of significant value to understand LPS-related phenomena, driving and optimizing experimental research studies. In this work, a comprehensive review on the methods that can be combined with MD simulations, recently applied in LPS research, is provided. We focus especially on both enhanced sampling methods, which enable the exploration of more complex systems and access to larger time scales, and free energy calculation approaches. Thereby, apart from outlining several strategies for surmounting LPS-caused infections, this work reports the current state-of-the-art of the methods applied with MD simulations for moving a step forward in the development of such strategies.

Molecular Dynamics Simulations in Drug Discovery and Pharmaceutical Development

Molecular dynamics (MD) simulations have become increasingly useful in the modern drug development process. In this review, we give a broad overview of the current application possibilities of MD in drug discovery and pharmaceutical development. Starting from the target validation step of the drug development process, we give several examples of how MD studies can give important insights into the dynamics and function of identified drug targets such as sirtuins, RAS proteins, or intrinsically disordered proteins. The role of MD in antibody design is also reviewed. In the lead discovery and lead optimization phases, MD facilitates the evaluation of the binding energetics and kinetics of the ligand-receptor interactions, therefore guiding the choice of the best candidate molecules for further development. The importance of considering the biological lipid bilayer environment in the MD simulations of membrane proteins is also discussed, using G-protein coupled receptors and ion channels as well as the drug-metabolizing cytochrome P450 enzymes as relevant examples. Lastly, we discuss the emerging role of MD simulations in facilitating the pharmaceutical formulation development of drugs and candidate drugs. Specifically, we look at how MD can be used in studying the crystalline and amorphous solids, the stability of amorphous drug or drug-polymer formulations, and drug solubility. Moreover, since nanoparticle drug formulations are of great interest in the field of drug delivery research, different applications of nano-particle simulations are also briefly summarized using multiple recent studies as examples. In the future, the role of MD simulations in facilitating the drug development process is likely to grow substantially with the increasing computer power and advancements in the development of force fields and enhanced MD methodologies.

Homology model of the human tRNA splicing ligase RtcB

RtcB is an essential human tRNA ligase required for ligating the 2',3'-cyclic phosphate and 5'-hydroxyl termini of cleaved tRNA halves during tRNA splicing and XBP1 fragments during endoplasmic reticulum stress. Activation of XBP1 has been implicated in various human tumors including breast cancer. Here we present, for the first time, a homology model of human RtcB (hRtcB) in complex with manganese and covalently bound GMP built from the Pyrococcus horikoshii RtcB (bRtcB) crystal structure, PDB ID 4DWQA. The structure is analyzed in terms of stereochemical quality, folding reliability, secondary structure similarity with bRtcB, druggability of the active site binding pocket and its metal-binding microenvironment. In comparison with bRtcB, loss of a manganese-coordinating water and movement of Asn226 (Asn202 in 4DWQA) to form metal-ligand coordination, demonstrates the uniqueness of the hRtcB model. Rotation of GMP leads to the formation of an additional metal-ligand coordination (Mn-O). Umbrella sampling simulations of Mn binding in wild type and the catalytically inactive C122A mutant reveal a clear reduction of Mn binding ability in the mutant, thus explaining the loss of activity therein. Our results furthermore clearly show that the GTP binding site of the enzyme is a well-defined pocket that can be utilized as target site for in silico drug discovery.

Review of computer simulations of isotope effects on biochemical reactions: From the Bigeleisen equation to Feynman's path integral

Enzymatic reactions are integral components in many biological functions and malfunctions. The iconic structure of each reaction path for elucidating the reaction mechanism in details is the molecular structure of the rate-limiting transition state (RLTS). But RLTS is very hard to get caught or to get visualized by experimentalists. In spite of the lack of explicit molecular structure of the RLTS in experiment, we still can trace out the RLTS unique "fingerprints" by measuring the isotope effects on the reaction rate. This set of "fingerprints" is considered as a most direct probe of RLTS. By contrast, for computer simulations, oftentimes molecular structures of a number of TS can be precisely visualized on computer screen, however, theoreticians are not sure which TS is the actual rate-limiting one. As a result, this is an excellent stage setting for a perfect "marriage" between experiment and theory for determining the structure of RLTS, along with the reaction mechanism, i.e., experimentalists are responsible for "fingerprinting", whereas theoreticians are responsible for providing candidates that match the "fingerprints". In this Review, the origin of isotope effects on a chemical reaction is discussed from the perspectives of classical and quantum worlds, respectively (e.g., the origins of the inverse kinetic isotope effects and all the equilibrium isotope effects are purely from quantum). The conventional Bigeleisen equation for isotope effect calculations, as well as its refined version in the framework of Feynman's path integral and Kleinert's variational perturbation (KP) theory for systematically incorporating anharmonicity and (non-parabolic) quantum tunneling, are also presented. In addition, the outstanding interplay between theory and experiment for successfully deducing the RLTS structures and the reaction mechanisms is demonstrated by applications on biochemical reactions, namely models of bacterial squalene-to-hopene polycyclization and RNA 2'-O-transphosphorylation. For all these applications, we used our recently-developed path-integral method based on the KP theory, called automated integration-free path-integral (AIF-PI) method, to perform ab initio path-integral calculations of isotope effects. As opposed to the conventional path-integral molecular dynamics (PIMD) and Monte Carlo (PIMC) simulations, values calculated from our AIF-PI path-integral method can be as precise as (not as accurate as) the numerical precision of the computing machine. Lastly, comments are made on the general challenges in theoretical modeling of candidates matching the experimental "fingerprints" of RLTS. This article is part of a Special Issue entitled: Enzyme Transition States from Theory and Experiment.

Multiscale methods for computational RNA enzymology

RNA catalysis is of fundamental importance to biology and yet remains ill-understood due to its complex nature. The multidimensional "problem space" of RNA catalysis includes both local and global conformational rearrangements, changes in the ion atmosphere around nucleic acids and metal ion binding, dependence on potentially correlated protonation states of key residues, and bond breaking/forming in the chemical steps of the reaction. The goal of this chapter is to summarize and apply multiscale modeling methods in an effort to target the different parts of the RNA catalysis problem space while also addressing the limitations and pitfalls of these methods. Classical molecular dynamics simulations, reference interaction site model calculations, constant pH molecular dynamics (CpHMD) simulations, Hamiltonian replica exchange molecular dynamics, and quantum mechanical/molecular mechanical simulations will be discussed in the context of the study of RNA backbone cleavage transesterification. This reaction is catalyzed by both RNA and protein enzymes, and here we examine the different mechanistic strategies taken by the hepatitis delta virus ribozyme and RNase A.

Ab initio path-integral calculations of kinetic and equilibrium isotope effects on base-catalyzed RNA transphosphorylation models

Detailed understandings of the reaction mechanisms of RNA catalysis in various environments can have profound importance for many applications, ranging from the design of new biotechnologies to the unraveling of the evolutionary origin of life. An integral step in the nucleolytic RNA catalysis is self-cleavage of RNA strands by 2'-O-transphosphorylation. Key to elucidating a reaction mechanism is determining the molecular structure and bonding characteristics of transition state. A direct and powerful probe of transition state is measuring isotope effects on biochemical reactions, particularly if we can reproduce isotope effect values from quantum calculations. This article significantly extends the scope of our previous joint experimental and theoretical work in examining isotope effects on enzymatic and nonenzymatic 2'-O-transphosphorylation reaction models that mimic reactions catalyzed by RNA enzymes (ribozymes), and protein enzymes such as ribonuclease A (RNase A). Native reactions are studied, as well as reactions with thio substitutions representing chemical modifications often used in experiments to probe mechanism. Here, we report and compare results from eight levels of electronic-structure calculations for constructing the potential energy surfaces in kinetic and equilibrium isotope effects (KIE and EIE) computations, including a "gold-standard" coupled-cluster level of theory [CCSD(T)]. In addition to the widely used Bigeleisen equation for estimating KIE and EIE values, internuclear anharmonicity and quantum tunneling effects were also computed using our recently developed ab initio path-integral method, that is, automated integration-free path-integral method. The results of this work establish an important set of benchmarks that serve to guide calculations of KIE and EIE for RNA catalysis.

Free energetics of rigid body association of ubiquitin binding domains: a biochemical model for binding mediated by hydrophobic interaction

Weak intermolecular interactions, such as hydrophobic associations, underlie numerous biomolecular recognition processes. Ubiquitin is a small protein that represents a biochemical model for exploring thermodynamic signatures of hydrophobic association as it is widely held that a major component of ubiquitin's binding to numerous partners is mediated by hydrophobic regions on both partners. Here, we use atomistic molecular dynamics simulations in conjunction with the Adaptive Biasing Force sampling method to compute potentials of mean force (the reversible work, or free energy, associated with the binding process) to investigate the thermodynamic signature of complexation in this well-studied biochemical model of hydrophobic association. We observe that much like in the case of a purely hydrophobic solute (i.e., graphene, carbon nanotubes), association is favored by entropic contributions from release of water from the interprotein regions. Moreover, association is disfavored by loss of enthalpic interactions, but unlike in the case of purely hydrophobic solutes, in this case protein-water interactions are lost and not compensated for by additional water-water interactions generated upon release of interprotein and moreso, hydration, water. We further find that relative orientations of the proteins that mutually present hydrophobic regions of each protein to its partner are favored over those that do not. In fact, the free energy minimum as predicted by a force field based method recapitulates the experimental NMR solution structure of the complex.

Liquid-vapor interfacial properties of aqueous solutions of guanidinium and methyl guanidinium chloride: influence of molecular orientation on interface fluctuations

The guanidinium cation (C(NH2)3(+)) is a highly stable cation in aqueous solution due to its efficient solvation by water molecules and resonance stabilization of the charge. Its salts increase the solubility of nonpolar molecules ("salting-in") and decrease the ordering of water. It is one of the strongest denaturants used in biophysical studies of protein folding. We investigate the behavior of guanidinium and its derivative, methyl guanidinium (an amino acid analogue) at the air-water surface, using atomistic molecular dynamics (MD) simulations and calculation of potentials of mean force. Methyl guanidinium cation is less excluded from the air-water surface than guanidinium cation, but both cations show orientational dependence of surface affinity. Parallel orientations of the guanidinium ring (relative to the Gibbs dividing surface) show pronounced free energy minima in the interfacial region, while ring orientations perpendicular to the GDS exhibit no discernible surface stability. Calculations of surface fluctuations demonstrate that, near the air-water surface, the parallel-oriented cations generate significantly greater interfacial fluctuations compared to other orientations, which induces more long-ranged perturbations and solvent density redistribution. Our results suggest a strong correlation with induced interfacial fluctuations and ion surface stability. These results have implications for interpreting molecular-level, mechanistic action of this osmolyte's interaction with hydrophobic interfaces as they impact protein denaturation (solubilization).

Free energetics of arginine permeation into model DMPC lipid bilayers: coupling of effective counterion concentration and lateral bilayer dimensions

Mechanisms and underlying thermodynamic determinants of translocation of charged cationic peptides such as cell-penetrating peptides across the cellular membrane continue to receive much attention. Two widely held views include endocytotic and non-endocytotic (diffusive) processes of permeant transfer across the bilayer. Considering a purely diffusive process, we consider the free energetics of translocation of a monoarginine peptide mimic across a model DMPC bilayer. We compute potentials of mean force for the transfer of a charged monoarginine peptide unit from water to the center of a 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC) model lipid bilayer. We use fully atomistic molecular dynamics simulations coupled with the adaptive biasing force (ABF) method for free energy estimation. The estimated potential of mean force difference from bulk to bilayer center is 6.94 ± 0.28 kcal/mol. The order of magnitude of this prediction is consistent with past experimental estimates of arginine partitioning into physiological bilayers in the context of translocon-based experiments, though the correlation between the bench and computer experiments is not unambiguous. Moreover, the present value is roughly one-half of previous estimates based on all-atom molecular dynamics free energy calculations. We trace the differences between the present and earlier calculations to system sizes used in the simulations and the dependence of the contributions to the free energy from various system components (water, lipids, ions, peptide) on overall system size. By varying the bilayer lateral dimensions in simulations using only sufficient numbers of counterions to maintain overall system charge neutrality, we find the possibility of an inherent convergent transfer free energy value.

Spherical monovalent ions at aqueous liquid-vapor interfaces: interfacial stability and induced interface fluctuations

Ion-specific interfacial behaviors of monovalent halides impact processes such as protein denaturation, interfacial stability, and surface tension modulation, and as such, their molecular and thermodynamic underpinnings garner much attention. We use molecular dynamics simulations of monovalent anions in water to explore effects on distant interfaces. We observe long-ranged ion-induced perturbations of the aqueous environment, as suggested by experiment and theory. Surface stable ions, characterized as such by minima in potentials of mean force computed using umbrella sampling MD simulations, induce larger interfacial fluctuations compared to nonsurface active species, conferring more entropy approaching the interface. Smaller anions and cations show no interfacial potential of mean force minima. The difference is traced to hydration shell properties of the anions, and the coupling of these shells with distant solvent. The effects correlate with the positions of the anions in the Hofmeister series (acknowledging variations in force field ability to recapitulate essential underlying physics), suggesting how differences in induced, nonlocal perturbations of interfaces may be related to different specific-ion effects in dilute biophysical and nanomaterial systems.

Temperature dependence and energetics of single ions at the aqueous liquid-vapor interface

We investigate temperature-dependence of free energetics with two single halide anions, I(-) and Cl(-), crossing the aqueous liquid-vapor interface through molecular dynamics simulations. The result shows that I(-) has a modest surface stability of 0.5 kcal/mol at 300 K and the stability decreases as the temperature increases, indicating the surface adsorption process for the anion is entropically disfavored. In contrast, Cl(-) shows no such surface state at all temperatures. Decomposition of free energetics reveals that water-water interactions provide a favorable enthalpic contribution, while the desolvation of ion induces an increase in free energy. Calculations of surface fluctuations demonstrate that I(-) generates significantly greater interfacial fluctuations compared to Cl(-). The fluctuation is attributed to the malleability of the solvation shells, which allows for more long-ranged perturbations and solvent density redistribution induced by I(-) as the anion approaches the liquid-vapor interface. The increase in temperature of the solvent enhances the inherent thermally excited fluctuations and consequently reduces the relative contribution from anion to surface fluctuations, which is consistent with the decrease in surface stability of I(-). Our results indicate a strong correlation with induced interfacial fluctuations and anion surface stability; moreover, resulting temperature dependent behavior of induced fluctuations suggests the possibility of a critical level of induced fluctuations associated with surface stability.

The Theory of Ultra-Coarse-Graining. 1. General Principles

Coarse-grained (CG) models provide a computationally efficient means to study biomolecular and other soft matter processes involving large numbers of atoms correlated over distance scales of many covalent bond lengths and long time scales. Variational methods based on information from simulations of finer-grained (e.g., all-atom) models, for example the multiscale coarse-graining (MS-CG) and relative entropy minimization methods, provide attractive tools for the systematic development of CG models. However, these methods have important drawbacks when used in the "ultra-coarse-grained" (UCG) regime, e.g., at a resolution level coarser or much coarser than one amino acid residue per effective CG particle in proteins. This is due to the possible existence of multiple metastable states "within" the CG sites for a given UCG model configuration. In this work, systematic variational UCG methods are presented that are specifically designed to CG entire protein domains and subdomains into single effective CG particles. This is accomplished by augmenting existing effective particle CG schemes to allow for discrete state transitions and configuration-dependent resolution. Additionally, certain conclusions of this work connect back to single-state force matching and open up new avenues for method development in that area. These results provide a formal statistical mechanical basis for UCG methods related to force matching and relative entropy CG methods and suggest practical algorithms for constructing optimal approximate UCG models from fine-grained simulation data.