Global Free Energy Minimum Research Articles

AlphaFold2 Predicts Alternative Conformation Populations in Green Fluorescent Protein Variants.

Artificial intelligence-based protein structure prediction methods such as AlphaFold2 have emerged as powerful tools for characterizing proteins sequence-structure relationship offering unprecedented opportunities for the molecular interpretation of biological and biochemical phenomena. While initially confined to providing a static representation of proteins through their global free-energy minimum, AlphaFold2 has demonstrated the ability to partially sample conformational landscapes, providing insights into protein dynamics, which is fundamental for interpreting and potentially tuning the function of natural and artificial proteins. In this study, we show that targeted column masking of AlphaFold2's multiple sequence alignment enables the characterization and estimation of the population ratio of the two main conformations of engineered green fluorescent proteins with alternative β-strands. The possibility of quickly estimating relative populations through AlphaFold2 predictions is expected to speed-up the computational design of related systems for sensing applications.

Solvent Isotherms and Structural Transitions in Nanoparticle Superlattice Assembly.

We introduce a Molecular Theory for Compressible Fluids (MOLT-CF) that enables us to compute free energies and other thermodynamic functions for nanoparticle superlattices with any solvent content, including the dry limit. Quantitative agreement is observed between MOLT-CF and united-atom molecular dynamics simulations performed to assess the reliability and precision of the theory. Among other predictions, MOLT-CF shows that the amount of solvent within the superlattice decreases approximately linearly with its vapor pressure and that in the late stages of drying, solvent-filled voids form at lattice interstitials. Applied to single-component superlattices, MOLT-CF predicts fcc-to-bcc Bain transitions for decreasing vapor pressure and for increasing ligand length, both in agreement with experimental results. We explore the stability of other single-component phases and show that the C14 Frank-Kasper phase, which has been reported in experiments, is not a global free-energy minimum. Implications for precise assembly and prediction of multicomponent nanoparticle systems are discussed.

Impact of ethylene glycol on ions influencing corrosion in pores between iron oxide and calcium carbonate

ABSTRACT Injection wells for carbon capture and storage are made of iron casings supported by cement. Cement can transform into calcium carbonate and iron into iron oxide. Thus, these wells are subject to corrosion. Monoethylene glycol (MEG), used for hydrate prevention and natural gas dehydration, can adsorb both to calcium carbonate and iron oxide. As a first step towards an atomistic understanding of effects caused by adsorbed MEG, molecular dynamics simulations were applied to estimate free energy changes to hydronium and bicarbonate as MEG was adsorbed to the surface. We found that the global free energy minimum of both hydronium and bicarbonate was moved closer to the surfaces, due to adsorbed MEG, which may be caused by MEG replacing water molecules within the first water layers. This could have an effect on chemical reactions involving hydronium and bicarbonate. Minima in the free energy profiles other than the global one was found to originate from adsorbed water combined with interactions from the surfaces.

Protein folding problem: enigma, paradox, solution.

The ability of protein chains to spontaneously form their three-dimensional structures is a long-standing mystery in molecular biology. The most conceptual aspect of this mystery is how the protein chain can find its native, "working" spatial structure (which, for not too big protein chains, corresponds to the global free energy minimum) in a biologically reasonable time, without exhaustive enumeration of all possible conformations, which would take billions of years. This is the so-called "Levinthal's paradox." In this review, we discuss the key ideas and discoveries leading to the current understanding of protein folding kinetics, including folding landscapes and funnels, free energy barriers at the folding/unfolding pathways, and the solution of Levinthal's paradox. A special role here is played by the "all-or-none" phase transition occurring at protein folding and unfolding and by the point of thermodynamic (and kinetic) equilibrium between the "native" and the "unfolded" phases of the protein chain (where the theory obtains the simplest form). The modern theory provides an understanding of key features of protein folding and, in good agreement with experiments, it (i) outlines the chain length-dependent range of protein folding times, (ii) predicts the observed maximal size of "foldable" proteins and domains. Besides, it predicts the maximal size of proteins and domains that fold under solely thermodynamic (rather than kinetic) control. Complementarily, a theoretical analysis of the number of possible protein folding patterns, performed at the level of formation and assembly of secondary structures, correctly outlines the upper limit of protein folding times.

Free energy barrier in wetting parallel-structured surfaces

Parallel structures with various simple cross-section geometries, such as trapezoid, rectangle, and circle, can be patterned on surfaces to control wettability. Free energy barriers that hinder a system from reaching its global minimum of free energy can influence the wetting behaviors. Specifying the forming conditions of energy barriers help evaluate metastable contact angles and contact angle hysteresis. Therefore, a thermodynamic model relating free energy with the continuous movement of three-phase contact line is needed to evaluate the energy barrier. This paper reports an integrated thermodynamic model for various parallel structures with simple geometries. Forming conditions of energy barriers are derived from the thermodynamic model, and validated using experimental data by electrospun microfibers and existing theoretical works. For sharp-edge structures with an internal angle φ and an intrinsic contact angle θY, the initiating and ending apparent contact angles for energy barriers are θY+φ and θY–φ, respectively, in noncomposite state, but this range may be discontinuous when θY<φ/2 or θY>180˚–φ/2. For round structures, an energy barrier may form at every structure in noncomposite state. In our experiment, aligned electrospun microfibers (with a mean radius of 0.9 µm) can pin the droplet at the first advancing energy barrier and achieve contact angles greater than 150˚.

Phase separation in high-T c cuprates

We develop a minimal non-BCS model for the CuO2 planes with the on-site Hilbert space reduced to only three effective valence centers CuO4 with different charge, conventional spin, and orbital symmetry, combined in a charge triplet, to describe the low-energy electron structure and the phase states of HTSC cuprates. Using the S = 1 pseudospin algebra we introduce an effective spin-pseudospin Hamiltonian which takes into account local and nonlocal correlations, one- and two-particle transport, and spin exchange. The T-n phase diagrams of the complete spin-pseudospin model for the CuO2 planes were reproduced by means of a site-dependent variational approach within effective field approximation typical for spin-magnetic systems. Limiting ourselves to two-sublattice approximation and nn-couplings we arrived at several Néel-like phases in CuO2 planes for parent and doped systems with a single nonzero local order parameter: antiferromagnetic insulator, charge order, glueless d-wave Bose superfluid phase, and unusual metallic phase. However, the Maxwell’s construction shows the global minimum of free energy is realized for phase separated states which are bounded by the third-order phase transition line T*(n), which is believed to be responsible for the onset of the pseudogap phenomenon.

Effect of Ionic Strength on Ibuprofenate Adsorption on a Lipid Bilayer of Dipalmitoylphosphatidylcholine from Molecular Dynamics Simulations.

In this work, the free energy change in the process of transferring ibuprofenate from the bulk solution to the center of a model of the dipalmitoylphosphatidylcholine bilayer at different NaCl concentrations was calculated. Two minima were found in the free energy profile: a local minimum, located in the vicinity of the membrane, and the global free energy minimum, found near the headgroup region. The downward shift of free energy minima with increasing NaCl concentration is consistent with the results of previous works. Conversely, the upward shift of the free energy maximum with increasing ionic strength is due to the competition of sodium ions and lipids molecules to coordinate with ibuprofenate and neutralize its charge. In addition, normal molecular dynamics simulations were performed to study the effects of the ibuprofenate on the lipid bilayer and in the presence of a high ibuprofenate concentration. The effect of ionic strength on the properties of the lipid bilayer and on lipid-drug interactions was analyzed. The area per lipid shrinking with increasing ionic strength, volume of lipids, and thickness of the bilayer is consistent with the experimental results. At a very high ibuprofenate concentration, the lipid bilayer dehydrates, and it consequently transforms into the gel phase, thus blocking the permeation.

Is Protein Folding a Thermodynamically Unfavorable, Active, Energy-Dependent Process?

The prevailing current view of protein folding is the thermodynamic hypothesis, under which the native folded conformation of a protein corresponds to the global minimum of Gibbs free energy G. We question this concept and show that the empirical evidence behind the thermodynamic hypothesis of folding is far from strong. Furthermore, physical theory-based approaches to the prediction of protein folds and their folding pathways so far have invariably failed except for some very small proteins, despite decades of intensive theory development and the enormous increase of computer power. The recent spectacular successes in protein structure prediction owe to evolutionary modeling of amino acid sequence substitutions enhanced by deep learning methods, but even these breakthroughs provide no information on the protein folding mechanisms and pathways. We discuss an alternative view of protein folding, under which the native state of most proteins does not occupy the global free energy minimum, but rather, a local minimum on a fluctuating free energy landscape. We further argue that ΔG of folding is likely to be positive for the majority of proteins, which therefore fold into their native conformations only through interactions with the energy-dependent molecular machinery of living cells, in particular, the translation system and chaperones. Accordingly, protein folding should be modeled as it occurs in vivo, that is, as a non-equilibrium, active, energy-dependent process.

Accurate Binding Configuration Prediction of a G-Protein-Coupled Receptor to Its Antagonist Using Multicanonical Molecular Dynamics-Based Dynamic Docking.

We have performed dynamic docking between a prototypic G-protein-coupled receptor (GPCR) system, the β2-adrenergic receptor, and its antagonist, alprenolol, using one of the enhanced conformation sampling methods, multicanonical molecular dynamics (McMD), which does not rely on any prior knowledge for the definition of the reaction coordinate. Although we have previously applied our McMD-based dynamic docking protocol to various globular protein systems, its application to GPCR systems would be difficult because of their complicated design, which include a lipid bilayer, and because of the difficulty in sampling the configurational space of a binding site that exists deep inside the GPCR. Our simulations sampled a wide array of ligand-bound and ligand-unbound structures, and we measured 427 binding events during our 48 μs production run. Analysis of the ensemble revealed several stable and meta-stable structures, where the most stable structure at the global free energy minimum matches the experimental one. Additional canonical MD simulations were used for refinement and validation of the structures, revealing that most of the intermediates are sufficiently stable to trap the ligand in these intermediary states and furthermore validated our prediction results. Given the difficulty in reaching the orthosteric binding site, chemical optimization of the compound for the second ranking configuration, which binds near the pocket's entrance, might lead to a high-affinity allosteric inhibitor. Accordingly, we show that the application of our methodology can be used to provide crucial insights for the rational design of drugs that target GPCRs.

Ion dynamics and selectivity of Nav channels from molecular dynamics simulation

The voltage-gated sodium (Nav) channels are crucial in initiating and propagating action potentials in excitable cells. In this work, with atomistic molecular simulations employing conventional and advanced sampling techniques, we investigate the dynamics of a series of Nav channels. The thermodynamic picture along the ion permeation pathway is obtained and the ion dynamics in the channels are analyzed. The ion permeation pathway could be divided into several regions. The cation enters the channel through the selectivity filter, stays in the global free energy minimum at the central cavity, and waits for the open of the activation gate. Exchanges between neighboring ions are also observed. There is an obvious free energy barrier for K+ entering the channel, and the thermodynamic stability of K+ in the central cavity is similar to the extracellular side, while there is no explicit barrier for Na+ and the cation is more stable in the cavity.

Multi-replica biased sampling for photoswitchable π-conjugated polymers.

In recent years, π-conjugated polymers are attracting considerable interest in view of their light-dependent torsional reorganization around the π-conjugated backbone, which determines peculiar light-emitting properties. Motivated by the interest in designing conjugated polymers with tunable photoswitchable pathways, we devised a computational framework to enhance the sampling of the torsional conformational space and, at the same time, estimate ground- to excited-state free-energy differences. This scheme is based on a combination of Hamiltonian Replica Exchange Method (REM), parallel bias metadynamics, and free-energy perturbation theory. In our scheme, each REM samples an intermediate unphysical state between the ground and the first two excited states, which are characterized by time-dependent density functional theory simulations at the B3LYP/6-31G* level of theory. We applied the method to a 5-mer of 9,9-dioctylfluorene and found that upon irradiation, this system can undergo a dihedral inversion from -155° to 155°, crossing a barrier that decreases from 0.1 eV in the ground state (S0) to 0.05 eV and 0.04 eV in the first (S1) and second (S2) excited states. Furthermore, S1 and even more S2 were predicted to stabilize coplanar dihedrals, with a local free-energy minimum located at ±44°. The presence of a free-energy barrier of 0.08 eV for the S1 state and 0.12 eV for the S2 state can trap this conformation in a basin far from the global free-energy minimum located at 155°. The simulation results were compared with the experimental emission spectrum, showing a quantitative agreement with the predictions provided by our framework.

The C-terminal domain of transcription factor RfaH: Folding, fold switching and energy landscape.

We simulate the folding and fold switching of the C-terminal domain (CTD) of the transcription factor RfaH using an all-atom physics-based model augmented with a dual-basin structure-based potential energy term. We show that this hybrid model captures the essential thermodynamic behavior of this metamorphic domain, that is, a change in the global free energy minimum from an α-helical hairpin to a 5-stranded β-barrel upon the dissociation of the CTD from the rest of the protein. Using Monte Carlo sampling techniques, we then analyze the energy landscape of the CTD in terms of progress variables for folding toward the two folds. We find that, below the folding transition, the energy landscape is characterized by a single, dominant funnel to the native β-barrel structure. The absence of a deep funnel to the α-helical hairpin state reflects a negligible population of this fold for the isolated CTD. We observe, however, a higher α-helix structure content in the unfolded state compared to results from a similar but fold switch-incompetent version of our model. Moreover, in folding simulations started from an extended chain conformation we find transiently formed α-helical structure, occurring early in the process and disappearing as the chain progresses toward the thermally stable β-barrel state.

Genetic Algorithm with New Stochastic Greedy Crossover Operator for Protein Structure Folding Problem

Introduction. The spatial protein structure folding is an important and actual problem in biology. Considering the mathematical model of the task, we can conclude that it comes down to the combinatorial optimization problem. Therefore, genetic and mimetic algorithms can be used to find a solution. The article proposes a genetic algorithm with a new greedy stochastic crossover operator, which differs from classical approaches with paying attention to qualities of possible ancestors. The purpose of the article is to describe a genetic algorithm with a new greedy stochastic crossover operator, reveal its advantages and disadvantages, compare the proposed algorithm with the best-known implementations of genetic and memetic algorithms for the spatial protein structure prediction, and make conclusions with future steps suggestion afterward. Result. The work of the proposed algorithm is compared with others on the basis of 10 known chains with a length of 48 first proposed in [13]. For each of the chain, a global minimum of free energy was already precalculated. The algorithm found 9 out of 10 spatial structures on which a global minimum of free energy is achieved and also demonstrated a better average value of solutions than the comparing algorithms. Conclusion. The quality of the genetic algorithm with the greedy stochastic crossover operator has been experimentally confirmed. Consequently, its further research is promising. For example, research on the selection of optimal algorithm parameters, improving the speed and quality of solutions found through alternative coding or parallelization. Also, it is worth testing the proposed algorithm on datasets with proteins of other lengths for further checks of the algorithm’s validity. Keywords: spatial protein structure, combinatorial optimization, genetic algorithms, crossover operator, stochasticity.

Transient Trapping into Metastable States in Systems with Competing Orders

The quench dynamics of a system involving two competing orders is investigated using a Ginzburg-Landau theory with relaxational dynamics. We consider the scenario where a pump rapidly heats the system to a high temperature, after which the system cools down to its equilibrium temperature. We study the evolution of the order parameter amplitude and fluctuations in the resulting time dependent free energy landscape. Exponentially growing thermal fluctuations dominate the dynamics. The system typically evolves into the phase associated with the faster-relaxing order parameter, even if it is not the global free energy minimum. This theory offers a natural explanation for the widespread experimental observation that metastable states may be induced by laser induced collapse of a dominant equilibrium order parameter.

Solution of Levinthal's Paradox and a Physical Theory of Protein Folding Times.

“How do proteins fold?” Researchers have been studying different aspects of this question for more than 50 years. The most conceptual aspect of the problem is how protein can find the global free energy minimum in a biologically reasonable time, without exhaustive enumeration of all possible conformations, the so-called “Levinthal’s paradox.” Less conceptual but still critical are aspects about factors defining folding times of particular proteins and about perspectives of machine learning for their prediction. We will discuss in this review the key ideas and discoveries leading to the current understanding of folding kinetics, including the solution of Levinthal’s paradox, as well as the current state of the art in the prediction of protein folding times.

Programming patchy particles to form complex periodic structures.

We introduce a scheme to design patchy particles so that a given target crystal is the global free-energy minimum at sufficiently low temperature. A key feature is a torsional component to the potential that only allows binding when particles have the correct relative orientations. In all examples studied, the target crystal structures readily assembled on annealing from a low-density fluid phase, albeit with the simpler target structures assembling more rapidly. The most complex example was a clathrate with 46 particles in its primitive unit cell. We also explored whether the structural information encoded in the particle interactions could be further reduced. For example, removing the torsional restrictions led to the assembly of an alternative crystal structure for the BC8-forming design, but the more complex clathrate design was still able to assemble because of the greater remaining specificity.

Solvent-Mediated Affinity of Polymer-Wrapped Single-Walled Carbon Nanotubes for Chemically Modified Surfaces.

Semiconducting single-walled carbon nanotube (s-CNT) arrays are being explored for next-generation semiconductor electronics. Even with the multitude of alignment and spatially localized s-CNT deposition methods designed to control s-CNT deposition, fundamental understanding of the driving forces for s-CNT deposition is still lacking. The individual roles of the dispersant, solvent, target substrate composition, and the s-CNT itself are not completely understood because it is difficult to decouple deposition parameters. Here, we study poly[(9,9-dioctylfluorenyl-2,7-diyl)-alt-co-(6,6'-[2,2'-{bipyridine}])] (PFO-BPy)-wrapped s-CNT deposition from solution onto a chemically modified substrate. We fabricate various self-assembled monolayers (SAMs) to gain a greater understanding of substrate effects on PFO-BPy-wrapped s-CNT deposition. We observe that s-CNT deposition is dependent on both the target substrate and s-CNT dispersion solvent. To complement the experiments, molecular dynamics simulations of PFO-BPy-wrapped s-CNT deposition on two different SAMs are performed to obtain mechanistic insights into the effect of the substrate and solvent on s-CNT deposition. We find that the global free-energy minimum associated with favorable s-CNT adsorption occurs for a configuration in which the minimum of the solvent density around the s-CNT coincides with the minimum of the solvent density above a SAM-grafted surface, indicating that solvent structure near a SAM-grafted surface determines the adsorption free-energy landscape driving s-CNT deposition. Our results will help guide informative substrate design for s-CNT array fabrication in semiconductor devices.

General Purpose Water Model Can Improve Atomistic Simulations of Intrinsically Disordered Proteins.

Unconstrained atomistic simulations of intrinsically disordered proteins and peptides (IDP) remain a challenge: widely used, "general purpose" water models tend to favor overly compact structures relative to experiment. Here we have performed a total of 93 μs of unrestrained MD simulations to explore, in the context of IDPs, a recently developed "general-purpose" 4-point rigid water model OPC, which describes liquid state of water close to experiment. We demonstrate that OPC, together with a popular AMBER force field ff99SB, offers a noticeable improvement over TIP3P in producing more realistic structural ensembles of three common IDPs benchmarks: 55-residue apo N-terminal zinc-binding domain of HIV-1 integrase ("protein IN"), amyloid β-peptide (Aβ42) (residues 1-42), and 26-reside H4 histone tail. As a negative control, computed folding profile of a regular globular miniprotein (CLN025) in OPC water is in appreciably better agreement with experiment than that obtained in TIP3P, which tends to overstabilize the compact native state relative to the extended conformations. We employed Aβ42 peptide to investigate the possible influence of the solvent box size on simulation outcomes. We advocate a cautious approach for simulations of IDPs: we suggest that the solvent box size should be at least four times the radius of gyration of the random coil corresponding to the IDP. The computed free energy landscape of protein IN in OPC resembles a shallow "tub" - conformations with substantially different degrees of compactness that are within 2 kB T of each other. Conformations with very different secondary structure content coexist within 1 kB T of the global free energy minimum. States with higher free energy tend to have less secondary structure. Computed low helical content of the protein has virtually no correlation with its degree of compactness, which calls into question the possibility of using the helicity as a metric for assessing performance of water models for IDPs, when the helicity is low. Predicted radius of gyration ( R g) of H4 histone tail in OPC water falls in-between that of a typical globular protein and a fully denatured protein of the same size; the predicted R g is consistent with two independent predictions. In contrast, H4 tail in TIP3P water is as compact as the corresponding globular protein. The computed free energy landscape of H4 tail in OPC is relatively flat over a significant range of compactness, which, we argue, is consistent with its biological function as facilitator of internucleosome interactions.

Biologically inspired far-from-equilibrium materials

Abstract

Free-Energy Landscapes for Macromolecules that Form a Unique 3D Structure

Abstract—The entropic effects for the energy landscapes of macromolecules are considered. We use the ideas and methods of multidimensional geometry and topology, expansion into a multidimensional Fourier series for a potential-energy surface, and the Gaussian distribution function for the expansion coefficients to describe the interaction between conformational degrees of freedom to investigate the free energy surface in the space of dihedral (torsion) angles. We show that the free-energy surface for a macromolecule that forms a unique 3D structure in the framework of the proposed approach has the following features. The main properties of the free-energy surface can be represented in terms of a two-dimensional surface, which depends on two generalized variables. At low temperatures, the multidimensional space of torsion angles can be divided into region that belongs to the deepest central energy funnel and the region of satellite funnels. The funnels are separated by relatively high energy barriers, whose height decreases as the representative point moves up further away from the bottom of the funnel. The depths of satellite funnels decrease while the distances from the central funnel becomes larger. The funnels are surrounded by smooth entropic barriers. The height of entropic barrier is at a minimum for the central funnel, while the entropic barriers heights of satellite funnels increase as funnels move further away from the central funnel. As the temperature increases, the satellite funnels are destroyed, starting from the most distant central one. When the temperature is higher than the critical point (proportional to the specific gain in the potential energy during the formation of a unique 3D structure) the compact state becomes unstable and the global minimum of free energy disappears. As the number of interacting torsion angles (the length of the polymer chain) increases, the critical temperature becomes higher.