On the accurate estimation of free energies using the jarzynski equality

Coordination of inorganic disulfide species to ferric N-acetyl microperoxidase 11

The interest in chemical interactions between inorganic sulfur species and heme compounds has grown significantly in recent years due to their physiological relevance. The model system ferric N-acetyl microperoxidase 11 (NAcMP11FeIII) enables the exploration of the mechanistic aspects of the interaction between the ferric heme group and binding sulfur ligands, without the constraints imposed by a protein matrix and the stabilizing effects of distal amino acids. In this study, we investigated the coordination of disulfane (HSSH) and its conjugate base hydrodisulfide (HSS-) to NAcMP11FeIII. Kinetic estimations of the binding constant retrieved a pH-independent kon= (1.5 ± 0.7) x105 M-1s-1, for 6.4 ≤ pH ≤ 7.2, and a similar value for the intrinsic constant for HSS-, the predominant species. To obtain a molecular description of the binding process, we resorted to two complementary theoretical approaches. Firstly, using multiple steered molecular dynamics, we calculated the free energy profiles for the migration of the neutral species HSSH and the monoanionic HSS-, and also for the siblings hydrogen sulfide, H2S, and hydrosulfide, HS-. Our results reveal that both neutral and anionic species can achieve the distal cavity, as expected considering the highly solvent exposed heme group in NAcMP11FeIII. Secondly, we explored the ligand-exchange reaction using a combination of nudged elastic band (NEB) and hybrid quantum mechanics/molecular mechanics (QM/MM) calculations, which suggest that the monoanionic species can displace the water molecule coordinated to the heme iron more efficiently than the neutral ones. Altogether, our results provide a molecular description of the ligand binding process of these sulfur species to ferric heme proteins.

Binding mechanism of oligopeptides on solid surface: assessing the significance of single-molecule approach

This paper addresses the complementarity and potential disparities between single-molecule and ensemble-average approaches to probe the binding mechanism of oligopeptides on inorganic solids. Specifically, we explore the peptide/gold interface owing to its significance in various topics and its suitability to perform experiments both in model and real conditions. Experimental results show that the studied peptide adopts a lying configuration upon adsorption on the gold surface and interacts through its peptidic links and deprotonated thiolate extremities, in agreement with theoretical predictions. Single-molecule force spectroscopy (SMFS) measurements revealed the existence of a wide panel of adhesion forces, resulting from the interaction between individual peptide moieties and the abundant surface sites. We therefore propose methodological developments for sorting the events of interest to understand the peptide adsorption mechanism. Thermodynamic and kinetic aspects of the peptide adsorption are probed using both static and dynamic force spectroscopy measurements. Specifically, we show the possibility of providing a reasonable estimate of the peptide free energy of adsorption ΔadsG° by exploring the fluctuations of the adhesion work, based on the Jarzynski equality, and by using a parametric Gamma estimator. The proposed approach offers a relevant method for studying the different factors influencing the peptide adsorption and evaluating their impact on ΔadsG° as an alternative to exploring adhesion forces that may lead to misinterpretations. This is illustrated by the comparison of the adsorption of two peptides with specific amino acids substitution. Our method provides insights into the overall mechanism by which peptides interact with the surface and allows an integration of the single-molecule versus ensemble-average points of view.

Mechanism of Peroxynitrite Interaction with Ferric M. tuberculosis Nitrobindin: A Computational Study

Nitrobindins (Nbs) are all-β-barrel heme proteins present along the evolutionary ladder. They display a highly solvent-exposed ferric heme group with the iron atom being coordinated by the proximal His residue and a water molecule at the distal position. Ferric nitrobindins (Nb(III)) play a role in the conversion of toxic peroxynitrite (ONOO-) to harmless nitrate, with the value of the second-order rate constant being similar to those of most heme proteins. The value of the second-order rate constant of Nbs increases as the pH decreases; this suggests that Nb(III) preferentially reacts with peroxynitrous acid (ONOOH), although ONOO- is more nucleophilic. In this work, we shed light on the molecular basis of the ONOO- and ONOOH reactivity of ferric Mycobacterium tuberculosis Nb (Mt-Nb(III)) by dissecting the ligand migration toward the active site, the water molecule release, and the ligand binding process by computer simulations. Classical molecular dynamics simulations were performed by employing a steered molecular dynamics approach and the Jarzynski equality to obtain ligand migration free energy profiles for both ONOO- and ONOOH. Our results indicate that ONOO- and ONOOH migration is almost unhindered, consistent with the exposed metal center of Mt-Nb(III). To further analyze the ligand binding process, we computed potential energy profiles for the displacement of the Fe(III)-coordinated water molecule using a hybrid QM/MM scheme at the DFT level and a nudged elastic band approach. These results indicate that ONOO- exhibits a much larger barrier for ligand displacement than ONOOH, suggesting that water displacement is assisted by protonation of the leaving group by the incoming ONOOH.

Binding mechanism of disulfide species to ferric hemeproteins: The case of metmyoglobin

The interactions of the heme iron of hemeproteins with sulfide and disulfide compounds are of potential interest as physiological signaling processes. While the interaction with hydrogen sulfide has been described computationally and experimentally, the reaction with disulfide, and specifically the molecular mechanism for ligand binding has not been studied in detail. In this work, we study the association process for disulfane and its conjugate base disulfanide at different pH conditions. Additionally, by means of advanced sampling techniques based on multiple steered molecular dynamics, we provide free energy profiles for ligand migration for both acid/base species, showing a similar behavior to the previously reported for the related H2S/HS¯ pair. Finally, we studied the ligand interchange reaction (H2O/H2S, HS¯ and H2O/HSSH, HSS¯) by means of hybrid quantum mechanics-molecular mechanics calculations. We show that the anionic species are able to displace more efficiently the H2O bound to the iron, and that the H-bond network in the distal cavity can help the neutral species to perform the reaction. Altogether, we provide a molecular explanation for the experimental information and show that the global association process depends on a fine balance between the migration towards the active site and the ligand interchange reaction.

Dissociation Rate Calculation via Constant-Force Steered Molecular Dynamics Simulation

Steered molecular dynamics (SMD) simulations are used to study molecular dissociation events by applying a harmonic force to the molecules and pulling them at a constant velocity. Instead of constant-velocity pulling, we use a constant force: the constant-force SMD (CF-SMD) simulation. The CF-SMD simulation employs a constant force to reduce the activation barrier of molecular dissociation, thereby enhancing the dissociation event. Here, we present the capability of the CF-SMD simulation to estimate the dissociation time at equilibrium. We performed all-atom CF-SMD simulations for NaCl and protein-ligand systems, producing dissociation time at various forces. We extrapolated these values to the dissociation rate without a constant force using Bell's model or the Dudko-Hummer-Szabo model. We demonstrate that the CF-SMD simulations with the models predicted the dissociation time in equilibrium. A CF-SMD simulation is a powerful tool for estimating the dissociation rate in a direct and computationally efficient manner.

Free energy and kinetic rate calculation via non-equilibrium molecular simulation: application to biomolecules

Non-equilibrium molecular dynamics (NEMD) simulation has been recognized as a powerful tool for examining biomolecules and provides fruitful insights into not only non-equilibrium but also equilibrium processes. We review recent advances in NEMD simulation and relevant, fundamental results of non-equilibrium statistical mechanics. We first introduce Crooks fluctuation theorem and Jarzynski equality that relate free energy difference to work done on a physical system during a non-equilibrium process. The theorems are beneficial for the analysis of NEMD trajectories. We then describe rate theory, a framework to calculate molecular kinetics from a non-equilibrium process; this theoretical framework enables us to calculate a reaction time-mean-first passage time-from NEMD trajectories. We, in turn, present recent NEMD techniques that apply an external force to a system to enhance molecular dissociation and introduce their application to biomolecules. Lastly, we show the current status of an appropriate selection of reaction coordinates for NEMD simulation.

Benchmarking Adaptive Steered Molecular Dynamics (ASMD) on CHARMM Force Fields

The potentials of mean force (PMFs) along the end-to-end distance of two different helical peptides have been obtained and benchmarked using the adaptive steered molecular dynamics (ASMD) method. The results depend strongly on the choice of force field driving the underlying all-atom molecular dynamics, and are reported with respect to the three most popular CHARMM force field versions: c22, c27 and c36. Two small peptides, ALA 10 and 1PEF, serve as the particular case studies. The comparisons between the versions of the CHARMM force fields provides both a qualitative and quantitative look at their performance in forced unfolding simulations in which peptides undergo large changes in structural conformations. We find that ASMD with the underlying c36 force field provides the most robust results for the selected benchmark peptides.

Insights into the product release mechanism of dengue virus NS3 helicase

The non-structural protein 3 helicase (NS3h) is a multifunctional protein that is critical in RNA replication and other stages in the flavivirus life cycle. NS3h uses energy from ATP hydrolysis to translocate along single stranded nucleic acid and to unwind double stranded RNA. Here we present a detailed mechanistic analysis of the product release stage in the catalytic cycle of the dengue virus (DENV) NS3h. This study is based on a combined experimental and computational approach of product-inhibition studies and free energy calculations. Our results support a model in which the catalytic cycle of ATP hydrolysis proceeds through an ordered sequential mechanism that includes a ternary complex intermediate (NS3h-Pi-ADP), which evolves releasing the first product, phosphate (Pi), and subsequently ADP. Our results indicate that in the product release stage of the DENV NS3h a novel open-loop conformation plays an important role that may be conserved in NS3 proteins of other flaviviruses as well.

Mycobacterium tuberculosis DosS binds H2S through its Fe3+ heme iron to regulate the DosR dormancy regulon

Mycobacterium tuberculosis (Mtb) senses and responds to host-derived gasotransmitters NO and CO via heme-containing sensor kinases DosS and DosT and the response regulator DosR. Hydrogen sulfide (H2S) is an important signaling molecule in mammals, but its role in Mtb physiology is unclear. We have previously shown that exogenous H2S can modulate expression of genes in the Dos dormancy regulon via an unknown mechanism(s). Here, we test the hypothesis that Mtb senses and responds to H2S via the DosS/T/R system. Using UV-Vis and EPR spectroscopy, we show that H2S binds directly to the ferric (Fe3+) heme of DosS (KDapp = 5.30 μM) but not the ferrous (Fe2+) form. No interaction with DosT(Fe2+-O2) was detected. We found that the binding of sulfide can slowly reduce the DosS heme iron to the ferrous form. Steered Molecular Dynamics simulations show that H2S, and not the charged HS- species, can enter the DosS heme pocket. We also show that H2S increases DosS autokinase activity and subsequent phosphorylation of DosR, and H2S-mediated increases in Dos regulon gene expression is lost in Mtb lacking DosS. Finally, we demonstrate that physiological levels of H2S in macrophages can induce DosR regulon genes via DosS. Overall, these data reveal a novel mechanism whereby Mtb senses and responds to a third host gasotransmitter, H2S, via DosS(Fe3+). These findings highlight the remarkable plasticity of DosS and establish a new paradigm for how bacteria can sense multiple gasotransmitters through a single heme sensor kinase.

Accelerated Jarzynski estimator with deterministic virtual trajectories

The Jarzynski estimator is a powerful tool that uses nonequilibrium statistical physics to numerically obtain partition functions of probability distributions. The estimator reconstructs partition functions with trajectories of the simulated Langevin dynamics through the Jarzynski equality. However, the original estimator suffers from slow convergence because it depends on rare trajectories of stochastic dynamics. In this paper, we present a method to significantly accelerate the convergence by introducing deterministic virtual trajectories generated in augmented state space under the Hamiltonian dynamics. We theoretically show that our approach achieves second-order acceleration compared to a naive estimator with the Langevin dynamics and zero variance estimation on harmonic potentials. We also present numerical experiments on three multimodal distributions and a practical example in which the proposed method outperforms the conventional method, and we provide theoretical explanations.

A modified Jarzynski free-energy estimator to eliminate non-conservative forces and its application in nanoparticle-membrane interactions

Computational methods to understand interactions in bio-complex systems are however limited to time-scales typically much shorter than in Nature. For example, on the nanoscale level, interactions between nanoparticles (NPs)/molecules/peptides and membranes are central in complex biomolecular processes such as membrane-coated NPs or cellular uptake. This can be remedied by the application of e.g. Jarzynski's equality where thermodynamic properties are extracted from non-equilibrium simulations. Although, the out of equilibrium work leads to non-conservative forces. We here propose a correction Pair Forces method, that removes these forces. Our proposed method is based on the calculation of pulling forces in backward and forward directions for the Jarzynski free-energy estimator using steered molecular dynamics simulation. Our results show that this leads to much improvement for NP-membrane translocation free energies. Although here we have demonstrated the application of the method in molecular dynamics simulation, it could be applied for experimental approaches.

Reweighted Jarzynski Sampling: Acceleration of Rare Events and Free Energy Calculation with a Bias Potential Learned from Nonequilibrium Work

We introduce a simple enhanced sampling approach for the calculation of free energy differences and barriers along a one-dimensional reaction coordinate. First, a small number of short nonequilibrium simulations are carried out along the reaction coordinate, and the Jarzynski equality is used to learn an approximate free energy surface from the nonequilibrium work distribution. This free energy estimate is represented in a compact form as an artificial neural network and used as an external bias potential to accelerate rare events in a subsequent molecular dynamics simulation. The final free energy estimate is then obtained by reweighting the equilibrium probability distribution of the reaction coordinate sampled under the influence of the external bias. We apply our reweighted Jarzynski sampling recipe to four processes of varying scales and complexities─spanning chemical reaction in the gas phase, pair association in solution, and droplet nucleation in supersaturated vapor. In all cases, we find reweighted Jarzynski sampling to be a very efficient strategy, resulting in rapid convergence of the free energy to high precision.

Gamma estimator of Jarzynski equality for recovering binding energies from noisy dynamic data sets

A fundamental problem in thermodynamics is the recovery of macroscopic equilibrated interaction energies from experimentally measured single-molecular interactions. The Jarzynski equality forms a theoretical basis in recovering the free energy difference between two states from exponentially averaged work performed to switch the states. In practice, the exponentially averaged work value is estimated as the mean of finite samples. Numerical simulations have shown that samples having thousands of measurements are not large enough for the mean to converge when the fluctuation of external work is above 4 k_BT, which is easily observable in biomolecular interactions. We report the first example of a statistical gamma work distribution applied to single molecule pulling experiments. The Gibbs free energy of surface adsorption can be accurately evaluated even for a small sample size. The values obtained are comparable to those derived from multi-parametric surface plasmon resonance measurements and molecular dynamics simulations.

Measuring interaction energies from experimentally measured single-molecular interactions is challenging. Here, the authors report a gamma work distribution applied to single molecule pulling events for estimating peptide absorption free energy.

Single molecule force spectroscopy of a streptomycin-binding RNA aptamer: An out-of-equilibrium molecular dynamics study

Here, we investigate the unfolding behavior of a streptomycin-binding ribonucleic acid (RNA) aptamer under application of force in shear geometry. Using Langevin out-of-equilibrium simulations to emulate the single-molecule force spectroscopy (SMFS) experiment, we were able to understand the hierarchical unfolding process that occurs in the RNA molecule under application of stretching force and the influence of streptomycin modifying this unfolding. Subsequently, the application of the Jarzynski equality to the force profiles obtained in the pulling simulations shows that the free energies for individual systems and the difference of unfolding free energy upon streptomycin binding to the RNA free aptamer are in fair agreement with the experimental values, obtained through SMFS by Nick et al. [J. Phys. Chem. B 120, 6479 (2016)].

Study of the role of Mg2+ in dsRNA processing mechanism by bacterial RNase III through QM/MM simulations

The ribonuclease III (RNase III) cleaves dsRNA in specific positions generating mature RNAs. RNase III enzymes play important roles in RNA processing, post-transcriptional gene expression, and defense against viral infection. The enzyme's active site contains Mg²⁺ ions bound by a network of acidic residues and water molecules, but there is a lack of information about their specific roles. In this work, multiple steered molecular dynamics simulations at QM/MM level were performed to explore the hydrolysis reaction carried out by the enzyme. Free energy profiles modifying the features of the active site are obtained and the role of Mg²⁺ ions, the solvent molecules and the residues of the active site are discussed in detail. Our results show that Mg²⁺ ions carry out different roles in the hydrolysis process positioning the substrate for the attack from a coordinated nucleophile and activating it to perform hydrolysis reaction, cleaving the dsRNA backbone in a S_N2 substitution. In addition, water molecules present in the active site lower the energy barrier of the process. RNase III hydrolyzes dsRNA to generate mature RNAs. For this purpose, its active site contains Mg²⁺ which has an important role during the reaction. Results show that the Mg²⁺ activates the solvent molecule that produces the nucleophilic attack and the surrounding waters contribute significantly to the hydrolysis process.

Lessons learned about steered molecular dynamics simulations and free energy calculations

The calculation of free energy profiles is central in understanding differential enzymatic activity, for instance, involving chemical reactions that require QM-MM tools, ligand migration, and conformational rearrangements that can be modeled using classical potentials. The use of steered molecular dynamics (sMD) together with the Jarzynski equality is a popular approach in calculating free energy profiles. Here, we first briefly review the application of the Jarzynski equality to sMD simulations, then revisit the so-called stiff-spring approximation and the consequent expectation of Gaussian work distributions and, finally, reiterate the practical utility of the second-order cumulant expansion, as it coincides with the parametric maximum-likelihood estimator in this scenario. We illustrate this procedure using simulations of CO, both in aqueous solution and in a carbon nanotube as a model system for biologically relevant nanoheterogeneous environments. We conclude the use of the second-order cumulant expansion permits the use of faster pulling velocities in sMD simulations, without introducing bias due to large dispersion in the non-equilibrium work distribution.