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. © 2014 The Authors Journal of Computational Chemistry Published by Wiley Periodicals, Inc.     

Effect of salt bridges on the energy landscape of a model protein

The effect of introducing salt bridges (gatekeepers) into an off-lattice three-color, 46-bead model protein is investigated in terms of the effect on global optimization statistics. The global minima for all the gatekeepers that exhibited faster folding in previous molecular dynamics studies are located more rapidly than for the original potential, although the global minimum itself may change. Visualization of the underlying potential energy surface using disconnectivity graphs reveals that the gatekeepers exhibit structure intermediate between the original potential and a Go model. Competition between low-lying minima and the global minimum is reduced in the gatekeepers compared to the original potential, and interconversion barriers are generally smaller.     

Protonated water clusters described by an empirical valence bond potential

The properties of low-lying stationary points on the potential energy surfaces of singly protonated water clusters (H(2)O)(n)H(+), are investigated using an empirical valence bond potential. Candidate global minima are reported for n=2-4, 8, and 20-22. For n=8, the variation in the energies and structures of low-lying minima with the number of valence bond states included in the model is studied. For n=4 and 8, disconnectivity graphs are also reported and are compared to results for the equivalent neutral water clusters as described by the rigid TIP3P potential. For the larger clusters, n=20-22, the structural properties of the low energy minima are compared with recently published spectroscopic data on these systems. The observed differences between the n=20 and n=21 systems are qualitatively reproduced by the model potential, but the similarities between the n=21 and n=22 systems are not.     

How many dimensions are required to approximate the potential energy landscape of a model protein?

A scheme to approximate the multidimensional potential energy landscape in terms of a minimal number of degrees of freedom is proposed using a linear transformation of the original atomic Cartesian coordinates. For one particular off-lattice model protein the inherent frustration can only be reproduced satisfactorily when a relatively large number of coordinates are employed. However, when this frustration is removed in a Go-type model, the number of coordinates required is significantly lower, especially around the global potential energy minimum. To aid our interpretation of the results we consider modified disconnectivity graphs where a measure of the structural diversity and a metric relation between the stationary points are incorporated.     

Archetypal energy landscapes: dynamical diagnosis

Recent studies have identified several motifs for potential energy surfaces corresponding to distinct dynamic and thermodynamic properties. The corresponding disconnectivity graphs were identified as "palm tree," "willow tree," and "banyan tree" patterns. In the present contribution we present a quantitative analysis of the relation between the topography and dynamics for each of these motifs. For the palm tree and willow tree forms we find that the arrangement of the stationary points in the monotonic sequences with respect to the global minimum is the most important factor in establishing the kinetic properties. However, the results are somewhat different for motifs involving a rough surface with several deep basins (banyan tree motif), with large barriers relative to the energy differences between minima. Here it is the size of the barrier for escape from the region relative to the barriers at the bottom that is most important. The present results may be helpful in distinguishing between the dynamics of "structure seeking" and "glass forming" systems.     

Tetrahelix conformations and transformation pathways in Pt1Pd12 clusters

The threshold method is used to explore the potential energy surface of the Pt(1)Pd(12) bimetallic cluster, defined by the Gupta semiempirical potential. A set of helical structures, which follow a Bernal tetrahelix pattern, correspond to local minima for the Pt(1)Pd(12) cluster, characterizing the region of the energy landscape where these structures are present. Both right-handed and left-handed chiral forms were discovered in our searches. Energetic and structural details of each of the tetrahelices are reported as well as the corresponding transition probabilities between these structures and with respect to the icosahedron-shaped global minimum structure via a disconnectivity graph analysis.     

On the encapsulation of nickel clusters by molecular nitrogen

The structures and energetic effects of molecular nitrogen adsorbates on nickel clusters are investigated using an extended Huckel model coupled with two models of the adsorbate-nickel interaction. The potential parameters for the adsorbates are chosen to mimic experimental information about the binding strength of nitrogen on both cluster and bulk surface phases of nickel. The first model potential is a simple Lennard-Jones interaction that leads to binding sites in holes defined by sets of near-neighbor nickel atoms. The second model potential has a simple three-body form that forces the model nitrogen adsorbates to bind directly to single nickel atoms. Significant rearrangement of the core nickel structures are found in both models. A disconnectivity graph analysis of the potential energy surfaces implies that the rearrangements arise from low transition state barriers and the small differences between available isomers in the nickel core.     

Resonance in catacondensed benzenoid hydrocarbons

We consider resonance in cata-condensed benzenoids having six and seven fused benzene rings. The resonance relationship between the Kekule valence structures of the molecules is represented by the resonance graphs in which the vertices represent the Kekule valence structures, and the edges, the presence of the quantum chemical resonance integral involving permutation of three CC double bonds (within a single six-membered ring). The construction of the resonance graphs for large benzenoids is outlined and the properties of the derived resonance graphs are discussed. It is shown that the leading eigenvalue of the resonance graphs correlates with the resonance energy of benzenoids. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 585–600, 1997     

Introduction to Formal Graphs, a new approach to the classical formalism

The creation of a purely graphic language called Formal Graphs for modelling many physical and physico-chemical systems is described. It represents an improvement over traditional equivalent circuits used for modelling systems made of individual components and over bond graphs used mainly in physico-chemistry. In contradistinction with these graphs, which represent graphically only mounting equations and maintain algebraic equations for describing components behaviour, a formal graph is an oriented graph incorporating all the information contained in a usual algebraic model. Combination of paths considerably extends use to domains that were not accessible to quantitative graphs, such as relaxation processes, chemical reactivity or mass-transfer. Moreover, inclusion of space derivation allows representing graphically every physical law describing a process involving energy conservation or dissipation, such as particle diffusion. Physical meaning can be deduced from paths in a graph that can be followed by processes, as illustrated by the exponent of fractional derivation, which appears as bearing the information on the proportion of conserved versus dissipated energy. The numerous examples given in this introduction address several domains, electrodynamics, mechanics, thermodynamics, and physico-chemistry. They show common graph structures that reveal a striking unity of our classical formalism, bringing transversal insight and opening a new route towards unification. Differences also appear that are subjects of interrogation.     

The characteristic polynomials of structures with pending bonds

It is well known [1] that the calculation of characteristic polynomials of graphs of interest in Chemistry which are of any size is usually extremely tedious except for graphs having a vertex of degree 1. This is primarily because of numerous combinations of contributions whether they were arrived at by non-imaginative expansion of the secular determinant or by the use of some of the available graph theoretical schemes based on the enumeration of partial coverings of a graph, etc. An efficient and quite general technique is outlined here for compounds that have pending bonds (i.e., bonds which have a terminal vertex). We have extended here the step-wise pruning of pending bonds developed for acyclic structures by one of the authors [2] for elegant evaluation of the characteristic polynomials of trees by accelerating this process, treating pending group as a unit. Further, it is demonstrated that this generalized pruning technique can be applied not only to trees but to cyclic and polycyclic graphs of any size. This technique reduces the secular determinant to a considerable extent. The present technique cannot handle only polycyclic structures that have no pending bonds. However, frequently such structures can be reduced to a combination of polycyclic graphs with pending bonds [3] so that the present scheme is applicable to these structures too. Thus we have arrived at an efficient and quite a simple technique for the construction of the characteristic polynomials of graphs of any size.     

De novo generation of molecular structures using optimization to select graphs on a given lattice

A recurrent problem in organic chemistry is the generation of new molecular structures that conform to some predetermined set of structural constraints that are imposed in an endeavor to build certain required properties into the newly generated structure. An example of this is the pharmacophore model, used in medicinal chemistry to guide de novo design or selection of suitable structures from compound databases. We propose here a method that efficiently links up a selected number of required atom positions while at the same time directing the emergent molecular skeleton to avoid forbidden positions. The linkage process takes place on a lattice whose unit step length and overall geometry is designed to match typical architectures of organic molecules. We use an optimization method to select from the many different graphs possible. The approach is demonstrated in an example where crystal structures of the same (in this case rigid) ligand complexed with different proteins are available.     

Quantum chemical reaction networks,reaction graphs and the structure of potential energy hypersurfaces

Global properties of the Born-Oppenheimer energy expectation value functional, defined over the nuclear configuration space R, are analyzed. Quantum chemical reaction graphs and reaction networks are defined in terms of intersection graphs of connected sets of nuclear geometries, representing various chemical structures. The set of all possible reaction mechanisms on the given energy hypersurface and the associated activation energy conditions are analyzed using reachability matrices defined over digraphs D ^s() and D ^s(, E).     

AADS--an automated active site identification, docking, and scoring protocol for protein targets based on physicochemical descriptors

We report here a robust automated active site detection, docking, and scoring (AADS) protocol for proteins with known structures. The active site finder identifies all cavities in a protein and scores them based on the physicochemical properties of functional groups lining the cavities in the protein. The accuracy realized on 620 proteins with sizes ranging from 100 to 600 amino acids with known drug active sites is 100% when the top ten cavity points are considered. These top ten cavity points identified are then submitted for an automated docking of an input ligand/candidate molecule. The docking protocol uses an all atom energy based Monte Carlo method. Eight low energy docked structures corresponding to different locations and orientations of the candidate molecule are stored at each cavity point giving 80 docked structures overall which are then ranked using an effective free energy function and top five structures are selected. The predicted structure and energetics of the complexes agree quite well with experiment when tested on a data set of 170 protein-ligand complexes with known structures and binding affinities. The AADS methodology is implemented on an 80 processor cluster and presented as a freely accessible, easy to use tool at http://www.scfbio-iitd.res.in/dock/ActiveSite_new.jsp .     

Unusual random walks

For most structures (molecules, graphs, lattices) a count of random walks for nonequivalent sites will give different numbers, particularly for walks of many steps. Occasionally one finds the same count of walks for nonequivalent sites. These have been termed “unusual walks” and have been closely examined in the case of trivalent graphs. While it remains to be understood what structural factors are critical, some regularities have been observed and are discussed. Unusual walks within a single structure signal “isospectural” points in a graph. A number of structures possessing unusual walks have been displayed, and a few constructive steps which do not alter the “unusual” characteristics of selected vertices have been indicated.     

Cospectral polymers: Differentiation via semiflexibility

We consider polymer structures which are known in the mathematical literature as "cospectral." Their graphs have (in spite of the different architectures) exactly the same Laplacian spectra. Now, these spectra determine in Gaussian (Rouse-type) approaches many static as well as dynamical polymer characteristics. Hence, in such approaches for cospectral graphs many mesoscopic quantities are predicted to be indistinguishable. Here we show that the introduction of semiflexibility into the generalized Gaussian structure scheme leads to different spectra and hence to distinct macroscopic patterns. Moreover, particular semiflexible situations allow us to distinguish well between cospectral structures. We confirm our theoretical results through Monte Carlo simulations.     

A family of metrics for biopolymers based on counting independent sets

We introduce a new family of metrics for graphs of fixed size, based on counting-independent sets. Our definition is simpler and easier to calculate than the edge ideal metric family defined by Llabrés and Rosselló without loosing any of its abstract properties. We contrast them on some examples with graphs that represent protein secondary and three-dimensional (3D) structures. We conclude that although the edge ideal metrics are faster to calculate on some sparse graphs, in general, the independent set metrics are more tractable.     

Variation of spectral, structural, and vibrational properties within the intrinsically fluorescent proteins family: a density functional study

We report an extensive (time-dependent) density functional study of the whole series of the chromophores within the Intrinsically Fluorescent Protein family, in the relevant conformations and protonation states. Over 30 structures are considered, including three newly discovered chromophores (zFP, Kaede, and the Orange variant of DsRed). Ab initio calculations on selected structures are also performed in order to assess the performances of TDDFT along the family. The use of a uniform scheme for all structures allows to establish relationships between the absorption energy and electrostatic, structural or vibrational properties of the chromophores.     

Similarity searching using reduced graphs

Reduced graphs provide summary representations of chemical structures. In this work, the effectiveness of reduced graphs for similarity searching is investigated. Different types of reduced graphs are introduced that aim to summarize features of structures that have the potential to form interactions with receptors while retaining the topology between the features. Similarity searches have been carried out across a variety of different activity classes. The effectiveness of the reduced graphs at retrieving compounds with the same activity as known target compounds is compared with searching using Daylight fingerprints. The reduced graphs are shown to be effective for similarity searching and to retrieve more diverse active compounds than those found using Daylight fingerprints; they thus represent a complementary similarity searching tool.     

Correlation between the Estrada index and π-electronic energies for benzenoid hydrocarbons with applications to boron nanotubes

In this article, we present an efficient computer-based computational technique to compute the energy and Estrada index of graphs. It is shown that our computational method is more efficient and bears less computational and algorithmic complexity. We use our method to show the main result of this article, which asserts that the Estrada index correlates with the π-electronic energies of lower benzenoid hydrocarbons with correlation coefficient 0.9993. This enhances the practical applicability of the Estrada index and warrants its further usage in quantitative structure activity relationships. We further apply our computational technique in computing the energy and Estrada index of two infinite families of boron triangular nanotubes. We perform simulation based on certain computer software packages to study the graph-theoretic behavior of the obtained results. Our results help to correlate certain physicochemical properties of underlying chemical structures of these nanotubes.