Properties Of Energy Landscape Research Articles

Bayesian estimation of maximum entropy model for individualized energy landscape analysis of brain state dynamics.

The pairwise maximum entropy model (MEM) for resting state functional MRI (rsfMRI) has been used to generate energy landscape of brain states and to explore nonlinear brain state dynamics. Researches using MEM, however, has mostly been restricted to fixed‐effect group‐level analyses, using concatenated time series across individuals, due to the need for large samples in the parameter estimation of MEM. To mitigate the small sample problem in analyzing energy landscapes for individuals, we propose a Bayesian estimation of individual MEM using variational Bayes approximation (BMEM). We evaluated the performances of BMEM with respect to sample sizes and prior information using simulation. BMEM showed advantages over conventional maximum likelihood estimation in reliably estimating model parameters for individuals with small sample data, particularly utilizing the empirical priors derived from group data. We then analyzed individual rsfMRI of the Human Connectome Project to show the usefulness of MEM in differentiating individuals and in exploring neural correlates for human behavior. MEM and its energy landscape properties showed high subject specificity comparable to that of functional connectivity. Canonical correlation analysis identified canonical variables for MEM highly associated with cognitive scores. Inter‐individual variations of cognitive scores were also reflected in energy landscape properties such as energies, occupation times, and basin sizes at local minima. We conclude that BMEM provides an efficient method to characterize dynamic properties of individuals using energy landscape analysis of individual brain states.

Applications of atomic force microscopy in modern biology.

Single-molecule force spectroscopy (SMFS) is an emerging tool to investigate mechanical properties of biomolecules and their responses to mechanical forces, and one of the most-used techniques for mechanical manipulation is the atomic force microscope (AFM). AFM was invented as an imaging tool which can be used to image biomolecules in sub-molecular resolution in physiological conditions. It can also be used as a molecular force probe for applying mechanical forces on biomolecules. In this brief review, we will provide exciting examples from recent literature which show how the advances in AFM have enabled us to gain deep insights into mechanical properties and mechanobiology of biomolecules. AFM has been applied to study mechanical properties of cells, tissues, microorganisms, viruses as well as biological macromolecules such as proteins. It has found applications in biomedical fields like cancer biology, where it has been used both in the diagnostic phases as well as drug discovery. AFM has been able to answer questions pertaining to mechanosensing by neurons, and mechanical changes in viruses during infection by the viral particles as well as the fundamental processes such as cell division. Fundamental questions related to protein folding have also been answered by SMFS like determination of energy landscape properties of variety of proteins and their correlation with their biological functions. A multipronged approach is needed to diversify the research, as a combination with optical spectroscopy and computer-based steered molecular dynamic simulations along with SMFS can help us gain further insights into the field of biophysics and modern biology.

Reliable extraction of energy landscape properties from critical force distributions

The structural dynamics of a biopolymer is governed by a process of diffusion through its conformational energy landscape. In pulling experiments using optical tweezers, features of the energy landscape can be extracted from the probability distribution of the critical force at which the polymer unfolds. The analysis is often based on rate equations having Bell-Evans form, although it is understood that this modeling is inadequate and leads to unreliable landscape parameters in many common situations. Dudko and co-workers [Phys. Rev. Lett. 96, 108101 (2006)] have emphasized this critique and proposed an alternative form that includes an additional shape parameter (and that reduces to Bell-Evans as a special case). Their fitting function, however, is pathological in the tail end of the pulling force distribution, which presents problems of its own. We propose a modified closed-form expression for the distribution of critical forces that correctly incorporates the next-order correction in pulling force and is everywhere well behaved. Our claim is that this new expression provides superior parameter extraction and is valid even up to intermediate pulling rates. We present results based on simulated data that confirm its utility.

Potential energy landscape of the two-dimensional XY model: higher-index stationary points.

The application of numerical techniques to the study of energy landscapes of large systems relies on sufficient sampling of the stationary points. Since the number of stationary points is believed to grow exponentially with system size, we can only sample a small fraction. We investigate the interplay between this restricted sample size and the physical features of the potential energy landscape for the two-dimensional XY model in the absence of disorder with up to N = 100 spins. Using an eigenvector-following technique, we numerically compute stationary points with a given Hessian index I for all possible values of I. We investigate the number of stationary points, their energy and index distributions, and other related quantities, with particular focus on the scaling with N. The results are used to test a number of conjectures and approximate analytic results for the general properties of energy landscapes.

Spectral Hole Burning, Recovery, and Thermocycling in Chlorophyll–Protein Complexes: Distributions of Barriers on the Protein Energy Landscape

Chlorophyll-protein complexes are ideal model systems for protein energy landscape research. Here pigments, used in optical spectroscopy experiments as sensitive probes to local dynamics, are built into protein by Nature (in a large variety of local environments; without extraneous chemical manipulations or genetic engineering). Distributions of the tunneling parameter, λ, and/or protein energy landscape barrier heights, V, have been determined for (the lowest energy state of) the CP43 core antenna complex of photosystem II. We demonstrate that spectral hole burning (SHB) and hole recovery (HR) measurements are capable of delivering important information on protein energy landscape properties and spectral diffusion mechanism details. In particular, we show that tunneling rather than barrier hopping is responsible for both persistent SHB and subsequent HR at 5-12 K, which allows us to estimate the md(2) parameter of the tunneling entities as ~1.0 × 10(-46) kg·m(2). The subdistributions of λ actually contributing to the nonsaturated spectral holes (and affecting their recovery) differ from the respective full true distributions. In the case of the full λ-distribution being uniform (or the barrier height distribution ~1/√V, a model which has been widely employed in theories of amorphous solids at low temperatures and in HR analysis), the difference is qualitative, with λ subdistributions probed in the HR experiments being highly asymmetrical, and barrier V subdistributions deviating significantly from ~1/√V. Thus, the distribution of λ for the protein energy landscape tier directly probed by SHB is likely Gaussian and not uniform. Additionally, a Gaussian distribution of barriers, with parameters incompatible with those of the landscape tier directly probed by SHB, contributes to the thermocycling results.

Nature of interacting electron states of Coulomb glass in local energy minima

Non-equilibrium relaxation of Coulomb glass in disordered thin films is investigated by kinetic Monte Carlo simulation. We numerically confirm aging phenomena in the autocorrelation function C(t, tW ) in a quasi-two-dimensional system with finite thickness and clarify the effect of an external electric field on the elongated relaxation time due to aging. We also study the statistical properties of electron states belonging to local energy minima in random site models. Our results highlight the difference in the properties of energy landscape between two different models to describe Coulomb glass, called the random site model and the lattice model.

Energy landscapes and their relation to thermodynamic phase transitions

In order to better understand the occurrence of phase transitions, we adopt an approachbased on the study of energy landscapes: the relation between stationary points of thepotential energy landscape of a classical many-particle system and the analyticityproperties of its thermodynamic functions is studied for finite as well as infinite systems.For finite systems, each stationary point is found to cause a non-analyticity in themicrocanonical entropy, and the functional form of this non-analytic term can be derivedexplicitly. With increasing system size, the order of the non-analytic term growsunboundedly, leading to an increasing differentiability of the entropy. Therefore, in thethermodynamic limit, only asymptotically flat stationary points may cause a phasetransition to take place. For several spin models, these results are illustrated by predictingthe absence or presence of a phase transition from stationary points and their localcurvatures in microscopic configuration space. These results establish a relationshipbetween properties of energy landscapes and the occurrence of phase transitions. Such anapproach appears particularly promising for the simultaneous study of dynamical andthermodynamical properties, as is of interest for example for protein folding or the glasstransition.

Energy landscape analysis of protein folding in an off-lattice model

Protein folding is numerically analysed in an off-lattice model in terms of potential energy landscape properties. Relevant landscape points (inherent structures, saddles) sampled by the system during the folding process are investigated. A landscape signature of the folding mechanism emerges.

Model of Apollonian Networks with Aging Mechanism

Abstract In relation to the problem of Apollonian packing, the Apollonian network is introduced. It has interesting properties like being scale-free, small-world, space-filling, and may help to explain the properties of energy landscapes, and can be applied to porous media, road networks, electrical systems, and the layout of the planar network. For further research on the Apollonian network, we propose a model of Apollonian Networks with aging mechanism. Using mathematical analysis, we deduce that the degree distribution obeys power-law form, which indicates that the network is power-law. Also we obtain the expression of clustering coefficient and prove that the average path length increases in the form of logarithm of network size, which indicates that the network is small-world. And then, the simulation results are given out, which agrees with the theoretic calculations. At last, we calculate the distribution of age of nodes and prove that the age of a node negatively correlates to the probability of the node's getting new links. Therefore, the model proposed in this article performs better than the model of Random Apollonian Networks and is more effective to the practice of layout of planar networks, such as printed circuits.

Short range attractive colloids: dynamics and energy landscape properties

We discuss the static and dynamical properties of a model of short range attractivecolloidal systems. Our model is a binary mixture of particles interacting with a modifiedLennard-Jones potential that we simulated by molecular dynamics. As previously found, weconfirm the existence of a maximum in diffusivity on varying the temperature at constanthigh density. The possibility to characterize such a maximum by studying the properties ofthe potential energy landscape (PEL) is also discussed. This has been achieved bycalculating the configurational entropy following two independent methods. It is shownthat this quantity presents a maximum in correspondence with that of diffusivity.

Exploring the origins of the power-law properties of energy landscapes: An egg-box model

Multidimensional potential energy landscapes (PELs) have a Gaussian distribution for the energies of the minima, but at the same time the distribution of the hyperareas for the basins of attraction surrounding the minima follows a power-law. To explore how both these features can simultaneously be true, we introduce an “egg-box” model. In these model landscapes, the Gaussian energy distribution is used as a starting point and we examine whether a power-law basin area distribution can arise as a natural consequence through the swallowing up of higher-energy minima by larger low-energy basins when the variance of this Gaussian is increased sufficiently. Although the basin area distribution is substantially broadened by this process, it is insufficient to generate power-laws, highlighting the role played by the inhomogeneous distribution of basins in configuration space for actual PELs.

Gaussian excitations model for glass-former dynamics and thermodynamics

We describe a model for the thermodynamics and dynamics of glass-forming liquids in terms of excitations from an ideal glass state to a Gaussian manifold of configurationally excited states. The quantitative fit of this three parameter model to the experimental data on excess entropy and heat capacity shows that "fragile" behavior, indicated by a sharply rising excess heat capacity as the glass transition is approached from above, occurs in anticipation of a first-order transition--usually hidden below the glass transition--to a "strong" liquid state of low excess entropy. The distinction between fragile and strong behavior of glass formers is traced back to an order of magnitude difference in the Gaussian width of their excitation energies. Simple relations connect the excess heat capacity to the Gaussian width parameter, and the liquid-liquid transition temperature, and strong, testable, predictions concerning the distinct properties of energy landscape for fragile liquids are made. The dynamic model relates relaxation to a hierarchical sequence of excitation events each involving the probability of accumulating sufficient kinetic energy on a separate excitable unit. Super-Arrhenius behavior of the relaxation rates, and the known correlation of kinetic with thermodynamic fragility, both follow from the way the rugged landscape induces fluctuations in the partitioning of energy between vibrational and configurational manifolds. A relation is derived in which the configurational heat capacity, rather than the configurational entropy of the Adam-Gibbs equation, controls the temperature dependence of the relaxation times, and this gives a comparable account of the experimental observations without postulating a divergent length scale. The familiar coincidence of zero mobility and Kauzmann temperatures is obtained as an approximate extrapolation of the theoretical equations. The comparison of the fits to excess thermodynamic properties of laboratory glass formers, and to configurational thermodynamics from simulations, reveals that the major portion of the excitation entropy responsible for fragile behavior resides in the low-frequency vibrational density of states. The thermodynamic transition predicted for fragile liquids emerges from beneath the glass transition in case of laboratory water and the unusual heat capacity behavior observed for this much studied liquid can be closely reproduced by the model.

Energy landscape properties studied using symbolic sequences

We investigate a classical lattice system with N particles. The potential energy V of the scalar displacements is chosen as a ϕ 4 on-site potential plus interactions. Its stationary points are solutions of a coupled set of nonlinear equations. Starting with Aubry’s anti-continuum limit it is easy to establish a one-to-one correspondence between the stationary points of V and symbolic sequences σ = ( σ 1 , … , σ N ) with σ n = + , 0 , − . We prove that this correspondence remains valid for interactions with a coupling constant ϵ below a critical value ϵ c and that it allows the use of a “thermodynamic” formalism to calculate statistical properties of the so-called “energy landscape” of V . This offers an explanation for why topological quantities of V may become singular, like in phase transitions. In particular, we find that the saddle index distribution is maximum at a saddle index n s max = 1 / 3 for all ϵ < ϵ c . Furthermore there exists an interval ( v ∗ , v max ) in which the saddle index n s as a function of the average energy v ̄ is analytical in v ̄ and it vanishes at v ∗ , above the ground state energy v gs , whereas the average saddle index n ̄ s as a function of the energy v is highly nontrivial. It can exhibit a singularity at a critical energy v c and it vanishes at v gs , only. Close to v gs , n ̄ s ( v ) exhibits power law behavior which even holds for noninteracting particles.

Diffusivity, excess entropy, and the potential-energy landscape of monatomic liquids

The connection between thermodynamic, transport, and potential-energy landscape features is studied for liquids with Lennard-Jones-type pair interactions using both microcanonical molecular-dynamics and isothermal-isobaric ensemble Monte Carlo simulations. Instantaneous normal-mode and saddle-point analyses of two variants of the monatomic Lennard-Jones liquid have been performed. The diffusivity is shown to depend linearly on several key properties of instantaneous and saddle configurations-the energy, the fraction of negative curvature directions, and the mean, maximum, and minimum eigenvalues of the Hessian. Since the Dzugutov scaling relationship also holds for such systems [Nature (London) 381, 137 (1996)], the exponential of the excess entropy, within the two-particle approximation, displays the same linear dependence on energy landscape properties as the diffusivity.

Mean-field theory, mode-coupling theory, and the onset temperature in supercooled liquids.

We consider the relationship between the temperature at which averaged energy landscape properties change sharply (T(o)) and the breakdown of mean-field treatments of the dynamics of supercooled liquids. First, we show that the solution of the wave vector dependent mode-coupling equations undergoes an ergodic-nonergodic transition consistently close to T(o). Generalizing the landscape concept to include hard-sphere systems, we show that the property of inherent structures that changes near T(o) is governed more fundamentally by packing and free volume than potential energy. Lastly, we study the finite-size random orthogonal model (ROM), and show that the onset of noticeable corrections to mean-field behavior occurs at T(o). These results highlight connections between the energy landscape and mode-coupling approach to supercooled liquids, and identify which features of the relaxation of supercooled liquids are properly captured by mode-coupling theory.

Dynamical TAP approach to mean field glassy systems

The Thouless, Anderson, Palmer (TAP) approach to thermodynamics of mean field spin-glasses is generalised to dynamics. A method to compute the dynamical TAP equations is developed and applied to the p-spin spherical model. In this context we show to what extent the dynamics can be represented as an evolution in the free energy landscape. In particular the relationship between the long-time dynamics and the local properties of the free energy landscape shows up explicitly within this approach. Conversely, by an instantaneous normal modes analysis we show that the local properties of the energy landscape seen by the system during its dynamical evolution do not change qualitatively at the dynamical transition.