Visualizing energy landscapes with metric disconnectivity graphs

Membrane lateral organization from potential energy disconnectivity graph

Understanding the thermodynamic and kinetic properties of biomolecules requires elucidation of their complex energy landscape. A disconnectivity graph analysis of the energy landscape provides a framework for mapping the multi-dimensional landscape onto a two-dimensional representation while preserving the key features of the energy landscape. Several studies show that the structure or shape of the disconnectity graph is directly associated with the function of protein and nucleic acid molecules. In this review, we discuss how disconnectivity analysis of the potential energy surface can be extended to lipid molecules to glean important information about membrane organization. The shape of the disconnectivity graphs can be used to predict the lateral organization of multi-component lipid bilayer. We hope that this review encourages the use of disconnectivity graphs routinely by membrane biophysicists to predict the lateral organization of lipids.

Revealing Free Energy Landscape From MD Data via Conditional Angle Partition Tree

Deciphering the free energy landscape of biomolecular structure space is crucial for understanding many complex molecular processes, such as protein-protein interaction, RNA folding, and protein folding. A major source of current dynamic structure data is Molecular Dynamics (MD) simulations. Several methods have been proposed to investigate the free energy landscape from MD data, but all of them rely on the assumption that kinetic similarity is associated with global geometric similarity, which may lead to unsatisfactory results. In this paper, we proposed a new method called Conditional Angle Partition Tree to reveal the hierarchical free energy landscape by correlating local geometric similarity with kinetic similarity. Its application on the benchmark alanine dipeptide MD data showed a much better performance than existing methods in exploring and understanding the free energy landscape. We also applied it to the MD data of Villin HP35. Our results are more reasonable on various aspects than those from other methods and very informative on the hierarchical structure of its energy landscape.

Representations of energy landscapes by sublevelset persistent homology: An example with n-alkanes

Encoding the complex features of an energy landscape is a challenging task, and often, chemists pursue the most salient features (minima and barriers) along a highly reduced space, i.e., two- or three-dimensions. Even though disconnectivity graphs or merge trees summarize the connectivity of the local minima of an energy landscape via the lowest-barrier pathways, there is much information to be gained by also considering the topology of each connected component at different energy thresholds (or sublevelsets). We propose sublevelset persistent homology as an appropriate tool for this purpose. Our computations on the configuration phase space of n-alkanes from butane to octane allow us to conjecture, and then prove, a complete characterization of the sublevelset persistent homology of the alkane C_mH_2m+2 Potential Energy Landscapes (PELs), for all m, in all homological dimensions. We further compare both the analytical configurational PELs and sampled data from molecular dynamics simulation using the united and all-atom descriptions of the intramolecular interactions. In turn, this supports the application of distance metrics to quantify sampling fidelity and lays the foundation for future work regarding new metrics that quantify differences between the topological features of high-dimensional energy landscapes.

Application of Optimization Algorithms in Clusters

The structural characterization of clusters or nanoparticles is essential to rationalize their size and composition-dependent properties. As experiments alone could not provide complete picture of cluster structures, so independent theoretical investigations are needed to find out a detail description of the geometric arrangement and corresponding properties of the clusters. The potential energy surfaces (PES) are explored to find several minima with an ultimate goal of locating the global minima (GM) for the clusters. Optimization algorithms, such as genetic algorithm (GA), basin hopping method and its variants, self-consistent basin-to-deformed-basin mapping, heuristic algorithm combined with the surface and interior operators (HA-SIO), fast annealing evolutionary algorithm (FAEA), random tunneling algorithm (RTA), and dynamic lattice searching (DLS) have been developed to solve the geometrical isomers in pure elemental clusters. Various model or empirical potentials (EPs) as Lennard-Jones (LJ), Born-Mayer, Gupta, Sutton-Chen, and Murrell-Mottram potentials are used to describe the bonding in different type of clusters. Due to existence of a large number of homotops in nanoalloys, genetic algorithm, basin-hopping algorithm, modified adaptive immune optimization algorithm (AIOA), evolutionary algorithm (EA), kick method and Knowledge Led Master Code (KLMC) are also used. In this review the optimization algorithms, computational techniques and accuracy of results obtained by using these mechanisms for different types of clusters will be discussed.

Effects of lattice constraints in coarse-grained protein models

We compare and contrast folding behavior in several coarse-grained protein models, both on- and off-lattice, in an attempt to uncover the effect of lattice constraints in these kinds of models. Using modern, extended ensemble Monte Carlo methods-Wang-Landau sampling, multicanonical sampling, replica-exchange Wang-Landau sampling, and replica-exchange multicanonical sampling, we investigate the thermodynamic and structural behavior of the protein Crambin within the context of the hydrophobic-polar, hydrophobic-"neutral"-polar (H0P), and semi-flexible H0P model frameworks. We uncover the folding process in all cases; all models undergo, at least, the two major structural transitions observed in nature-the coil-globule collapse and the folding transition. As the complexity of the model increases, these two major transitions begin to split into multi-step processes, wherein the lattice coarse-graining has a significant impact on the details of these processes. The results show that the level of structural coarse-graining is coupled to the level of interaction coarse-graining.

The entropic bond in colloidal crystals

A vast array of natural phenomena can be understood through the long-established schema of chemical bonding. Conventional chemical bonds arise through local gradients resulting from the rearrangement of electrons; however, it is possible that the hallmark features of chemical bonding could arise through local gradients resulting from nonelectronic forms of mediation. If other forms of mediation give rise to "bonds" that act like conventional ones, recognizing them as bonds could open new forms of supramolecular descriptions of phenomena at the nano- and microscales. Here, we show via a minimal model that crowded hard-particle systems governed solely by entropy exhibit the hallmark features of bonding despite the absence of chemical interactions. We quantitatively characterize these features and compare them to those exhibited by chemical bonds to argue for the existence of entropic bonds. As an example of the utility of the entropic bond classification, we demonstrate the nearly equivalent tradeoff between chemical bonds and entropic bonds in the colloidal crystallization of hard hexagonal nanoplates.

Go-Kit: A Tool To Enable Energy Landscape Exploration of Proteins

Coarse-grained Go̅-like models, based on the principle of minimal frustration, provide valuable insight into fundamental questions in the field of protein folding and dynamics. In conjunction with commonly used molecular dynamics (MD) simulations, energy landscape exploration methods like discrete path sampling (DPS) with Go̅-like models can provide quantitative details of the thermodynamics and kinetics of proteins. Here we present Go-kit, a software that facilitates the setup of MD and DPS simulations of several flavors of Go̅-like models. Go-kit is designed for use with MD (GROMACS) and DPS (PATHSAMPLE) simulation engines that are open source. The Go-kit code is written in python2.7 and is also open source. A case study for the ribosomal protein S6 is discussed to illustrate the utility of the software, which is available at https://github.com/gokit1/gokit .

Isomers and energy landscapes of micro-hydrated sulfite and chlorate clusters

We present putative global minima for the micro-hydrated sulfite SO₃^2-(H₂O) _N and chlorate ClO₃^-(H₂O) _N systems in the range 3≤N≤15 found using basin-hopping global structure optimization with an empirical potential. We present a structural analysis of the hydration of a large number of minimized structures for hydrated sulfite and chlorate clusters in the range 3≤N≤50. We show that sulfite is a significantly stronger net acceptor of hydrogen bonding within water clusters than chlorate, completely suppressing the appearance of hydroxyl groups pointing out from the cluster surface (dangling OH bonds), in low-energy clusters. We also present a qualitative analysis of a highly explored energy landscape in the region of the global minimum of the eight water hydrated sulfite and chlorate systems.This article is part of the theme issue 'Modern theoretical chemistry'.

A Statistical Physics Characterization of the Complex Systems Dynamics: Quantifying Complexity from Spatio-Temporal Interactions

Biological systems are frequently categorized as complex systems due to their capabilities of generating spatio-temporal structures from apparent random decisions. In spite of research on analyzing biological systems, we lack a quantifiable framework for measuring their complexity. To fill this gap, in this paper, we develop a new paradigm to study a collective group of N agents moving and interacting in a three-dimensional space. Our paradigm helps to identify the spatio-temporal states of the motion of the group and their associated transition probabilities. This framework enables the estimation of the free energy landscape corresponding to the identified states. Based on the energy landscape, we quantify missing information, emergence, self-organization and complexity for a collective motion. We show that the collective motion of the group of agents evolves to reach the most probable state with relatively lowest energy level and lowest missing information compared to other possible states. Our analysis demonstrates that the natural group of animals exhibit a higher degree of emergence, self-organization and complexity over time. Consequently, this algorithm can be integrated into new frameworks to engineer collective motions to achieve certain degrees of emergence, self-organization and complexity.

Isomers and Energy Landscapes of Perchlorate-Water Clusters and a Comparison to Pure Water and Sulfate-Water Clusters

Hydrated ions are crucially important in a wide array of environments, from biology to the atmosphere, and the presence and concentration of ions in a system can drastically alter its behavior. One way in which ions can affect systems is in their interactions with proteins. The Hofmeister series ranks ions by their ability to salt-out proteins, with kosmotropes, such as sulfate, increasing their stability and chaotropes, such as perchlorate, decreasing their stability. We study hydrated perchlorate clusters as they are strongly chaotropic and thus exhibit different properties than sulfate. In this study we simulate small hydrated perchlorate clusters using a basin-hopping geometry optimization search with empirical potentials. We compare topological features of these clusters to data from both computational and experimental studies of hydrated sulfate ions and draw some conclusions about ion effects in the Hofmeister series. We observe a patterning conferred to the water molecules within the cluster by the presence of the perchlorate ion and compare the magnitude of this effect to that observed in previous studies involving sulfate. We also investigate the influence of the overall ionic charge on the low-energy structures adopted by these clusters.

Visualizing energy landscapes with metric disconnectivity graphs

The visualization of multidimensional energy landscapes is important, providing insight into the kinetics and thermodynamics of a system, as well the range of structures a system can adopt. It is, however, highly nontrivial, with the number of dimensions required for a faithful reproduction of the landscape far higher than can be represented in two or three dimensions. Metric disconnectivity graphs provide a possible solution, incorporating the landscape connectivity information present in disconnectivity graphs with structural information in the form of a metric. In this study, we present a new software package, PyConnect, which is capable of producing both disconnectivity graphs and metric disconnectivity graphs in two or three dimensions. We present as a test case the analysis of the 69-bead BLN coarse-grained model protein and show that, by choosing appropriate order parameters, metric disconnectivity graphs can resolve correlations between structural features on the energy landscape with the landscapes energetic and kinetic properties.