Local Free Energy Minima Research Articles

Micro-macro consistency in multiscale modeling: Score-based model assisted sampling of fast/slow dynamical systems.

A valuable step in the modeling of multiscale dynamical systems in fields such as computational chemistry, biology, and materials science is the representative sampling of the phase space over long time scales of interest; this task is not, however, without challenges. For example, the long term behavior of a system with many degrees of freedom often cannot be efficiently computationally explored by direct dynamical simulation; such systems can often become trapped in local free energy minima. In the study of physics-based multi-time-scale dynamical systems, techniques have been developed for enhancing sampling in order to accelerate exploration beyond free energy barriers. On the other hand, in the field of machine learning (ML), a generic goal of generative models is to sample from a target density, after training on empirical samples from this density. Score-based generative models (SGMs) have demonstrated state-of-the-art capabilities in generating plausible data from target training distributions. Conditional implementations of such generative models have been shown to exhibit significant parallels with long-established-and physics-based-solutions to enhanced sampling. These physics-based methods can then be enhanced through coupling with the ML generative models, complementing the strengths and mitigating the weaknesses of each technique. In this work, we show that SGMs can be used in such a coupling framework to improve sampling in multiscale dynamical systems.

Collective neural network behavior in a dynamically driven disordered system of superconducting loops

Collective properties of complex systems composed of many interacting components such as neurons in our brain can be modeled by artificial networks based on disordered systems. We show that a disordered neural network of superconducting loops with Josephson junctions can exhibit computational properties like categorization and associative memory in the time evolution of its state in response to information from external excitations. Superconducting loops can trap multiples of fluxons in many discrete memory configurations defined by the local free energy minima in the configuration space of all possible states. A memory state can be updated by exciting the Josephson junctions to fire or allow the movement of fluxons through the network as the current through them surpasses their critical current thresholds. Simulations performed with a lumped element circuit model of a 4-loop network show that information written through excitations is translated into stable states of trapped flux and their time evolution. Experimental implementation on a high-Tc superconductor YBCO-based 4-loop network shows dynamically stable flux flow in each pathway characterized by the correlations between junction firing statistics. Neural network behavior is observed as energy barriers separating state categories in simulations in response to multiple excitations, and experimentally as junction responses characterizing different flux flow patterns in the network. The state categories that produce these patterns have different temporal stabilities relative to each other and the excitations. This provides strong evidence for time-dependent (short-to-long-term) memories, that are dependent on the geometrical and junction parameters of the loops, as described with a network model.

Temperature Accelerated Sliced Sampling to Probe Ligand Dissociation from Protein.

Modeling ligand unbinding in proteins to estimate the free energy of binding and probing the mechanism presents several challenges. They primarily pertain to the entropic bottlenecks resulting from protein and solvent conformations. While exploring the unbinding processes using enhanced sampling techniques, very long simulations are required to sample all of the conformational states as the system gets trapped in local free energy minima along transverse coordinates. Here, we demonstrate that temperature accelerated sliced sampling (TASS) is an ideal approach to overcome some of the difficulties faced by conventional sampling methods in studying ligand unbinding. Using TASS, we study the unbinding of avibactam inhibitor molecules from the Class C β-lactamase (CBL) active site. Extracting CBL-avibactam unbinding free energetics, unbinding pathways, and identifying critical interactions from the TASS simulations are demonstrated.

Spiral Pitch Control in Cholesteric Liquid Crystal Layers with Hybrid Boundary Conditions

The optical spectra of the cholesteric liquid crystal (CLC) layers under conditions of hybrid anchoring show a short-wave shift under a pulsed electric field. This behavior is anomalous because it is associated with a decrease in the pitch of the cholesteric spiral, which is atypical at conditions when the electric field is perpendicular to the axis of the CLC spiral. An analytical model of the phenomenon is discussed, according to which the spiral pitch under hybrid boundary conditions can be greater than the natural pitch in an unlimited volume of CLC. An in-plane electric field, being localized near the homeotropic-alignment surface, can be treated as effectively influencing the azimuthal anchoring and leading to a variety of metastable states with both increased and decreased pitch. These metastable states with local minima of free energy prevent the spiral from unwinding, and corresponding bands of selective reflection can even be shifted to the short-wave region of the spectrum. The observed effect is also studied numerically. It is shown by numerical simulations that the localized electric field from short-pitch electrodes can also modify zenithal anchoring, which should allow for defect-free controlling of the spiral pitch and spectral stop-band location.

Response of Elementary Structural Transitions in Glassy Atactic Polystyrene to Temperature and Deformation.

The effects of temperature, pressure, and imposed strain on the structural transition pathways of glassy atactic polystyrene (aPS) are studied for a wide range of conditions. By employing an atomistic description of the system, we systematically explore its free energy landscape, emphasizing connections between local free energy minima. A triplet of two minima connected to each other via a first-order saddle point provides the full description of each elementary structural relaxation event. The basis of the analysis is the potential energy landscape (PEL), where efficient methods for finding saddle points and exploring transition pathways have been developed. We then translate the stationary points of the PEL to stationary points of the proper free energy landscape that obeys the macroscopically imposed constraints (either stress- or strain-controlled). By changing the temperature under isobaric conditions (i.e., Gibbs energy landscape), we probe the temperature dependence of the transition rates of the subglass relaxations of aPS, thus obtaining their activation energies by fitting to the Arrhenius equation. The imposition of different strain levels under isothermic conditions allows us to estimate the apparent activation volume of every elementary transition. Our findings are in good agreement with experimental observations for the same system, indicating that both length- and time-scales of the structural transitions of glassy aPS can be obtained by proper free energy minimization of atomistically detailed configurations.

Machine learning the metastable phase diagram of covalently bonded carbon

Conventional phase diagram generation involves experimentation to provide an initial estimate of the set of thermodynamically accessible phases and their boundaries, followed by use of phenomenological models to interpolate between the available experimental data points and extrapolate to experimentally inaccessible regions. Such an approach, combined with high throughput first-principles calculations and data-mining techniques, has led to exhaustive thermodynamic databases (e.g. compatible with the CALPHAD method), albeit focused on the reduced set of phases observed at distinct thermodynamic equilibria. In contrast, materials during their synthesis, operation, or processing, may not reach their thermodynamic equilibrium state but, instead, remain trapped in a local (metastable) free energy minimum, which may exhibit desirable properties. Here, we introduce an automated workflow that integrates first-principles physics and atomistic simulations with machine learning (ML), and high-performance computing to allow rapid exploration of the metastable phases to construct “metastable” phase diagrams for materials far-from-equilibrium. Using carbon as a prototypical system, we demonstrate automated metastable phase diagram construction to map hundreds of metastable states ranging from near equilibrium to far-from-equilibrium (400 meV/atom). We incorporate the free energy calculations into a neural-network-based learning of the equations of state that allows for efficient construction of metastable phase diagrams. We use the metastable phase diagram and identify domains of relative stability and synthesizability of metastable materials. High temperature high pressure experiments using a diamond anvil cell on graphite sample coupled with high-resolution transmission electron microscopy (HRTEM) confirm our metastable phase predictions. In particular, we identify the previously ambiguous structure of n-diamond as a cubic-analog of diaphite-like lonsdaelite phase.

Adsorption Free Energy of Cellulose Nanocrystal on Water-Oil Interface.

To investigate the amphiphilicity of cellulose, a series of molecular dynamics simulations were performed with a cellulose nanocrystal and a water–octane interfacial system. Assuming that the axis of cellulose is parallel to the water–octane interface, the freedoms of motion of the nanocrystal were restricted to two, the distance from the interface and the orientation around the axis. The mean force and the mean torque on the nanocrystal were evaluated with sufficiently long simulation at each crystal configuration, and their numerical integration gave a smooth free energy surface as the potential of mean force. The cellulose sample used here was found to be much more hydrophilic than oleophilic with the free energy difference kcal/mol. Three adsorption states with local minimum of adsorption free energy are distinguished in the free energy surface—the direct contact type which is similar to previously reported one, the hydrophilic-surface/water/octane type where a thin water layer is sandwiched between the surface and the octane phase, and the oleophilic/water/octane type where a thin water layer also exists. Water molecules in these water layers contribute to stabilize the adsorption states by taking a special orientational order and slow self-diffusion.

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.

Uncertainty and anharmonicity in thermally activated dynamics

The atomic trajectories of condensed phase systems often reduce to long periods of thermal vibration interspersed by transitions between local free energy minima. The resultant rare event dynamics are exponentially sensitive to the catalog of available transitions, meaning incomplete models can make catastrophically erroneous predictions. This review summarises some recent efforts towards quantifying this uncertainty. I show that Bayesian methods can rigorously measure sampling incompleteness, be propagated to yield a quantified prediction uncertainty and autonomously manage massively parallel simulations. These methods allow uncertainty-controlled investigation of complex atomistic processes with minimal end-user supervision, facilitating high-throughput workflows. For individual transitions rates, I also show how the activation free energy can be evaluated with full treatment of anharmonic thermal vibrations. The developed methods, all freely available, are demonstrated on a wide range of challenging materials science problems.

Synthesis of Metastable Inorganic Solids with Extended Structures.

The number of known inorganic compounds is dramatically less than predicted due to synthetic challenges, which often constrains products to only the thermodynamically most stable compounds. Consequently, a mechanism-based approach to inorganic solids with designed structures is the holy grail of solid state synthesis. This article discusses a number of synthetic approaches using the concept of an energy landscape, which describes the complex relationship between the energy of different atomic configurations as a function of a variety of parameters such as initial structure, temperature, pressure, and composition. Nucleation limited synthesis approaches with high diffusion rates are contrasted with diffusion limited synthesis approaches. One challenge to the synthesis of new compounds is the inability to accurately predict what structures might be local free energy minima in the free energy landscape. Approaches to this challenge include predicting potentially stable compounds thorough the use of structural homologies and/or theoretical calculations. A second challenge to the synthesis of metastable inorganic solids is developing approaches to move across the energy landscape to a desired local free energy minimum while avoiding deeper free energy minima, such as stable binary compounds, as reaction intermediates. An approach using amorphous intermediates is presented, where local composition can be used to prepare metastable compounds. Designed nanoarchitecture built into a precursor can be preserved at low reaction temperatures and used to direct the reaction to specific structural homologs.

Replica symmetry breaking in random matrix model: A toy model of wormhole networks

We study the replica symmetry breaking (RSB) in the Gaussian Unitary Ensemble (GUE) random matrix model. In the large N limit the RSB occurs at the transition temperature TRSB∼N−2/3. We argue that this transition originates from a landscape of almost degenerate local minima of free energy labeled by Young diagrams. We also discuss a possible implication of our findings for the multiply-coupled Sachdev-Ye-Kitaev models and their holographic dual Jackiw-Teitelboim gravity from the viewpoint of ER=EPR conjecture.

Transition path times in asymmetric barriers.

Biomolecular conformational transitions are usually modeled as barrier crossings in a free energy landscape. The transition paths connect two local free energy minima and transition path times (TPT) are the actual durations of the crossing events. The simplest model employed to analyze TPT and to fit empirical data is that of a stochastic particle crossing a parabolic barrier. Motivated by some disagreement between the value of the barrier height obtained from the TPT distributions as compared to the value obtained from kinetic and thermodynamic analyses, we investigate here TPT for barriers which deviate from the symmetric parabolic shape. We introduce a continuous set of potentials, that starting from a parabolic shape, can be made increasingly asymmetric by tuning a single parameter. The TPT distributions obtained in the asymmetric case are very well-fitted by distributions generated by parabolic barriers. The fits, however, provide values for the barrier heights and diffusion coefficients which deviate from the original input values. We show how these findings can be understood from the analysis of the eigenvalues spectrum of the Fokker-Planck equation and highlight connections with experimental results.

The Poincaré-Shannon Machine: Statistical Physics and Machine Learning Aspects of Information Cohomology

Previous works established that entropy is characterized uniquely as the first cohomology class in a topos and described some of its applications to the unsupervised classification of gene expression modules or cell types. These studies raised important questions regarding the statistical meaning of the resulting cohomology of information and its interpretation or consequences with respect to usual data analysis and statistical physics. This paper aims to present the computational methods of information cohomology and to propose its interpretations in terms of statistical physics and machine learning. In order to further underline the cohomological nature of information functions and chain rules, the computation of the cohomology in low degrees is detailed to show more directly that the k multivariate mutual information () are -coboundaries. The -cocycles condition corresponds to , which generalizes statistical independence to arbitrary degree k. Hence, the cohomology can be interpreted as quantifying the statistical dependences and the obstruction to factorization. I develop the computationally tractable subcase of simplicial information cohomology represented by entropy and information landscapes and their respective paths, allowing investigation of Shannon’s information in the multivariate case without the assumptions of independence or of identically distributed variables. I give an interpretation of this cohomology in terms of phase transitions in a model of k-body interactions, holding both for statistical physics without mean field approximations and for data points. The components define a self-internal energy functional and components define the contribution to a free energy functional (the total correlation) of the k-body interactions. A basic mean field model is developed and computed on genetic data reproducing usual free energy landscapes with phase transition, sustaining the analogy of clustering with condensation. The set of information paths in simplicial structures is in bijection with the symmetric group and random processes, providing a trivial topological expression of the second law of thermodynamics. The local minima of free energy, related to conditional information negativity and conditional independence, characterize a minimum free energy complex. This complex formalizes the minimum free-energy principle in topology, provides a definition of a complex system and characterizes a multiplicity of local minima that quantifies the diversity observed in biology. I give an interpretation of this complex in terms of unsupervised deep learning where the neural network architecture is given by the chain complex and conclude by discussing future supervised applications.

Stepwise Homogeneous Melting of Benzene Phase I at High Pressure

We investigate, using molecular dynamics simulations, the spontaneous homogeneous melting of benzene phase I under a high pressure of 1.0 GPa. We find an apparent stepwise transition via a metastable crystal phase, unlike the direct melting observed at ambient pressure. The transition to the metastable phase is achieved by rotational motions, without the diffusion of the center of mass of benzene. The metastable crystal completely occupies the whole space and maintains its structure for at least several picoseconds, so that the phase seems to have a local free energy minimum. The unit cell is found to be unique—no such crystalline structure has been reported so far. Furthermore, we discuss the influence of pressure control on the melting behavior.

A mathematical model of embodied consciousness

We introduce a mathematical model of embodied consciousness, the Projective Consciousness Model (PCM), which is based on the hypothesis that the spatial field of consciousness (FoC) is structured by a projective geometry and under the control of a process of active inference. The FoC in the PCM combines multisensory evidence with prior beliefs in memory and frames them by selecting points of view and perspectives according to preferences. The choice of projective frames governs how expectations are transformed by consciousness. Violations of expectation are encoded as free energy. Free energy minimization drives perspective taking, and controls the switch between perception, imagination and action. In the PCM, consciousness functions as an algorithm for the maximization of resilience, using projective perspective taking and imagination in order to escape local minima of free energy. The PCM can account for a variety of psychological phenomena: the characteristic spatial phenomenology of subjective experience, the distinctions and integral relationships between perception, imagination and action, the role of affective processes in intentionality, but also perceptual phenomena such as the dynamics of bistable figures and body swap illusions in virtual reality. It relates phenomenology to function, showing the computational advantages of consciousness. It suggests that changes of brain states from unconscious to conscious reflect the action of projective transformations and suggests specific neurophenomenological hypotheses about the brain, guidelines for designing artificial systems, and formal principles for psychology.

Uptake Dynamics in the LacY Membrane Protein Transporter

Membrane protein transporters govern important cellular processes and are therefore central to human health. To accomplish transport, these proteins rearrange their structures to alternatively expose an internal binding site to either side of the membrane. Recent advances in protein structural determination methods have resulted in a steadily increasing number of high-resolution structures of membrane transporters trapped in different intermediate states. However, to understand the underlying transport mechanism, the molecular details of uptake and release need to be determined. We have used specialized simulation hardware to simulate uptake of galactoside sugar into the Lactose permease (LacY) of Escherichia coli. The extended brute-force simulation revealed large-scale structural rearrangements, lipid and amino acid interactions, and hydration associated with sugar uptake. The free energy landscape of sugar entry was determined by parallel bias-exchange metadynamics simulations and identified a global free energy minimum that coincided with the crystallographic binding site and also a local free energy minimum in the larger periplasmic cavity of LacY. Together, our observations show a putative molecular mechanism for sugar uptake in this prototype membrane transporter.

Hampering Effect of Cholesterol on the Permeation of Reactive Oxygen Species through Phospholipids Bilayer: Possible Explanation for Plasma Cancer Selectivity

In recent years, the ability of cold atmospheric pressure plasmas (CAPS) to selectively induce cell death in cancer cells has been widely established. This selectivity has been assigned to the reactive oxygen and nitrogen species (RONS) created in CAPs. To provide new insights in the search for an explanation for the observed selectivity, we calculate the transfer free energy of multiple ROS across membranes containing a varying amount of cholesterol. The cholesterol fraction is investigated as a selectivity parameter because membranes of cancer cells are known to contain lower fractions of cholesterol compared to healthy cells. We find that cholesterol has a significant effect on the permeation of reactive species across a membrane. Indeed, depending on the specific reactive species, an increasing cholesterol fraction can lead to (i) an increase of the transfer free energy barrier height and width, (ii) the formation of a local free energy minimum in the center of the membrane and (iii) the creation of extra free energy barriers due to the bulky sterol rings. In the context of plasma oncology, these observations suggest that the increased ingress of RONS in cancer cells can be explained by the decreased cholesterol fraction of their cell membrane.

Nonequilibrium all-atom molecular dynamics simulation of the bubble cavitation and application to dissociate amyloid fibrils.

The cavitation of gas bubbles in liquids has been applied to different disciplines in life and natural sciences, and in technologies. To obtain an appropriate theoretical description of effects induced by the bubble cavitation, we develop an all-atom nonequilibrium molecular-dynamics simulation method to simulate bubbles undergoing harmonic oscillation in size. This allows us to understand the mechanism of the bubble cavitation-induced liquid shear stress on surrounding objects. The method is then employed to simulate an Aβ fibril model in the presence of bubbles, and the results show that the bubble expansion and contraction exert water pressure on the fibril. This yields to the deceleration and acceleration of the fibril kinetic energy, facilitating the conformational transition between local free energy minima, and leading to the dissociation of the fibril. Our work, which is a proof-of-concept, may open a new, efficient way to dissociate amyloid fibrils using the bubble cavitation technique, and new venues to investigate the complex phenomena associated with amyloidogenesis.

Exploring the Levinthal limit in protein folding.

According to the thermodynamic hypothesis, the native state of proteins is uniquely defined by their amino acid sequence. On the other hand, according to Levinthal, the native state is just a local minimum of the free energy and a given amino acid sequence, in the same thermodynamic conditions, can assume many, very different structures that are as thermodynamically stable as the native state. This is the Levinthal limit explored in this work. Using computer simulations, we compare the interactions that stabilize the native state of four different proteins with those that stabilize three non-native states of each protein and find that the nature of the interactions is very similar for all such 16 conformers. Furthermore, an enhancement of the degree of fluctuation of the non-native conformers can be explained by an insufficient relaxation to their local free energy minimum. These results favor Levinthal's hypothesis that protein folding is a kinetic non-equilibrium process.

Free Energies of Cavity and Noncavity Hydrated Electrons Near the Instantaneous Air/Water Interface.

The properties of the hydrated electron at the air/water interface are computed for both a cavity and a noncavity model using mixed quantum/classical molecular dynamics simulation. We take advantage of our recently developed formalism for umbrella sampling with a restrained quantum expectation value to calculate free-energy profiles of the hydrated electron's position relative to the water surface. We show that it is critical to use an instantaneous description of the air/water interface rather than the Gibbs' dividing surface to obtain accurate potentials of mean force. We find that noncavity electrons, which prefer to encompass several water molecules, avoid the interface where water molecules are scarce. In contrast, cavity models of the hydrated electron, which prefer to expel water, have a local free-energy minimum near the interface. When the cavity electron occupies this minimum, its absorption spectrum is quite red-shifted, its binding energy is significantly lowered, and its dynamics speed up quite a bit compared with the bulk, features that have not been found by experiment. The surface activity of the electron therefore serves as a useful test of cavity versus noncavity electron solvation.