Pairwise Energy Research Articles

Real-Time Video Synopsis via Dynamic and Adaptive Online Tube Resizing

Nowadays, with the increased numbers of video cameras, the amount of recorded video is growing. Efficient video browsing and retrieval are critical issues when considering the amount of raw video data to be condensed. Activity-based video synopsis is a popular approach to solving the video condensation problem. However, conventional synopsis methods always consists of complicated and pairwise energy terms that involve a time-consuming optimization problem. In this paper, we propose a simple online video synopsis framework in which the number of collisions of objects is classified first. Different optimization strategies are applied according to different collision situations to maintain a balance among the computational cost, condensation ratio, and collision cost. Secondly, tube-resizing coefficients that are dynamic in different frames are adaptively assigned to a newly generated tube. Therefore, a suitable mapping result can be obtained in order to represent the proper size of the activity in each frame of the synopsis video. The maximum number of activities can be displayed in one frame with minimal collisions. Finally, in order to remove motion anti-facts and improve the visual quality of the condensed video, a smooth term is introduced to constrain the resizing coefficients. Experimental results on extensive videos validate the efficiency of the proposed method.

Analytical Formulation and Field-Theoretic Simulation of Sequence-Specific Phase Separation of Protein-Like Heteropolymers with Short- and Long-Spatial-Range Interactions.

A theory for sequence-dependent liquid-liquid phase separation (LLPS) of intrinsically disordered proteins (IDPs) in the study of biomolecular condensates is formulated by extending the random phase approximation (RPA) and field-theoretic simulation (FTS) of heteropolymers with spatially long-range Coulomb interactions to include the fundamental effects of short-range, hydrophobic-like interactions between amino acid residues. To this end, short-range effects are modeled by Yukawa interactions between multiple nonelectrostatic charges derived from an eigenvalue decomposition of pairwise residue-residue contact energies. Chain excluded volume is afforded by incompressibility constraints. A mean-field approximation leads to an effective Flory-Huggins χ parameter, which, in conjunction with RPA, accounts for the contact-interaction effects of amino acid composition and the sequence-pattern effects of long-range electrostatics in IDP LLPS, whereas FTS based on the formulation provides full sequence dependence for both short- and long-range interactions. This general approach is illustrated here by applications to variants of a natural IDP in the context of several different amino-acid interaction schemes as well as a set of different model hydrophobic-polar sequences sharing the same composition. Effectiveness of the methodology is verified by coarse-grained explicit-chain molecular dynamics simulations.

Learning physics-consistent particle interactions.

Interacting particle systems play a key role in science and engineering. Access to the governing particle interaction law is fundamental for a complete understanding of such systems. However, the inherent system complexity keeps the particle interaction hidden in many cases. Machine learning methods have the potential to learn the behavior of interacting particle systems by combining experiments with data analysis methods. However, most existing algorithms focus on learning the kinetics at the particle level. Learning pairwise interaction, e.g., pairwise force or pairwise potential energy, remains an open challenge. Here, we propose an algorithm that adapts the Graph Networks framework, which contains an edge part to learn the pairwise interaction and a node part to model the dynamics at particle level. Different from existing approaches that use neural networks in both parts, we design a deterministic operator in the node part that allows to precisely infer the pairwise interactions that are consistent with underlying physical laws by only being trained to predict the particle acceleration. We test the proposed methodology on multiple datasets and demonstrate that it achieves superior performance in inferring correctly the pairwise interactions while also being consistent with the underlying physics on all the datasets. While the previously proposed approaches are able to be applied as simulators, they fail to infer physically consistent particle interactions that satisfy Newton's laws. Moreover, the proposed physics-induced graph network for particle interaction also outperforms the other baseline models in terms of generalization ability to larger systems and robustness to significant levels of noise. The developed methodology can support a better understanding and discovery of the underlying particle interaction laws, and hence, guide the design of materials with targeted properties.

Accurate Description of Solvent-Exposed Salt Bridges with a Non-polarizable Force Field Incorporating Solvent Effects.

The strength of salt bridges resulting from the interaction of cations and anions is modulated by their environment. However, polarization of the solvent molecules by the charged moieties makes the accurate description of cation-anion interactions in an aqueous solution by means of a pairwise additive potential energy function and classical combination rules particularly challenging. In this contribution, aiming at improving the representation of solvent-exposed salt-bridge interactions with an all-atom non-polarizable force field, we put forth here a parametrization strategy. First, the interaction of a cation and an anion is characterized by hybrid quantum mechanical/molecular mechanics (QM/MM) potential of mean force (PMF) calculations, whereby constantly exchanging solvent molecules around the ions are treated at the quantum mechanical level. The Lennard-Jones (LJ) parameters describing the salt-bridge ion pairs are then optimized to match the reference QM/MM PMFs through the so-called nonbonded FIX, or NBFIX, feature of the CHARMM force field. We apply the new set of parameters, coined CHARMM36m-SBFIX, to the calculation of association constants for the ammonium-acetate and guanidinium-acetate complexes, the osmotic pressures for glycine zwitterions, guanidinium, and acetate ions, and to the simulation of both folded and intrinsically disordered proteins. Our findings indicate that CHARMM36m-SBFIX improves the description of solvent-exposed salt-bridge interactions, both structurally and thermodynamically. However, application of this force field to the standard binding free-energy calculation of a protein-ligand complex featuring solvent-excluded salt-bridge interactions leads to a poor reproduction of the experimental value, suggesting that the parameters optimized in an aqueous solution cannot be readily transferred to describe solvent-excluded salt-bridge interactions. Put together, owing to their sensitivity to the environment, modeling salt-bridge interactions by means of a single, universal set of LJ parameters remains a daunting theoretical challenge.

Classical Intermolecular Hydrogen Bonding Motifs of Heterocyclic rac-2-Amino-3-carbonitrile Derivatives: Linking Hirshfeld Surface Analysis, CT-DNA Binding Affinity, and Molecular Docking

Four 2-amino-4-(aryl)-7,7-dimethyl-5-oxo-5,6,7,8-tetrahydro-4H-chromene-3-carbonitrile (Series I) and three 2-amino-1-phenyl-7,7-dimethyl-5-oxo-4-(aryl)-1,4,5,6,7,8-hexahydroquinoline-3-carbonitrile (Series II) derivatives were formed via multicomponent microwave reactions as racemic mixtures. The solid-state IR spectra revealed bands in the region between 3190 and 3414 cm–1 attributed to the amino functional group’s asymmetric, symmetric, and bending overtone vibration modes. Furthermore, the IR spectra of some of the compounds in each series were similar and this served as an indication of similar structural features in the solid state. This was confirmed via single-crystal X-ray diffraction. A detailed survey of the related 2-amino-3-carbonitrile derivatives in the Cambridge Structural Database revealed 12 classical intermolecular hydrogen-bonding motifs involving the amino functional group. The vast variation in the hydrogen bonding motifs was found to be caused by the type of group bonded to the 4H-pyran (in Series I) or dihydropyridine (in Series II) moiety. In this work, an unprecedented hydrogen bonding motif comprising a non-classical hydrogen bond acceptor was found in one instance. For both Series I and II, the N–H···N and N–H···O hydrogen bonds were investigated using noncovalent interaction (NCI) plots and molecular pairwise energy calculations. The contributions of reciprocal N···H and O···H contacts (attributed to N–H···N and N–H···O, respectively) toward the Hirshfeld surface range between 9.9 and 15.3% in Series I and between 10.8 and 15.0% in Series II. In addition, the compounds showed moderate to good binding affinity to calf thymus DNA (CT-DNA) with intrinsic binding constants (Kb) between 8.00 × 104 and 5.00 × 105 M–1 for Series I compounds and between 6.60 × 104 and 1.45 × 105 M–1 for Series II. A relationship was established between the reciprocal N···H to O···H contributions ratio from the Hirshfeld surface analysis and Kb. Molecular docking studies of each compound’s respective enantiomers into DNA were done using canonical–DNA dodecamer (B-DNA) as a model for CT-DNA. The compounds were found to interact with DNA via a minor groove-binding mode. The S-enantiomer for most of the compounds bound favorably compared to their respective R-enantiomers. The ratio of calculated ΔG for the DNA binding of the R-enantiomer to that of the S-enantiomer (ΔGR/ΔGS) follows the same positive trend as the experimental Kb value for each compound in Series I while a negative trend was observed for Series II. Finally, a relationship between the Hirshfeld surface analysis and calculated ΔG values was also established in this work.

Integrating multiple sequence features for identifying anticancer peptides

As one of the most terrible diseases, cancer causes millions of deaths worldwide every year. The popular treatment approaches, such as radiotherapy and chemotherapy, have been used in against cancer cells. However, those traditional therapies have side effects on normal cells, time-consuming and expensive. Recent studies showed that anticancer peptides (ACP) may be a potential choice instead of traditional approaches for treating cancer. Therefore, it is desired to develop a computational method to identify anticancer peptides. In this study, a support vector machine (SVM) based computational model was proposed to discriminate anticancer peptides from non-anticancer peptides. In the model, peptide sequences were firstly encoded by amino acids physicochemical (PC) properties and residue pairwise energy content matrix (RECM). Then, Pearson’s correlation coefficient, high-order correlation information, and discrete wavelet transform were employed to extract useful information from PC and RECM matrix. The least absolute shrinkage and selection operator (LASSO) algorithm was applied to select discriminative features. Finally, these selected features were fed into SVM for distinguishing ACP from non-ACP. Experimental results demonstrated that the proposed method is powerful, it indicates that our proposed method may be a hopeful tool in discriminating anticancer peptides from non-anticancer peptides. The codes and datasets used in current work are available at https://figshare.com/articles/online_resource/iACP/16866232.

Prediction of Plant Resistance Proteins Based on Pairwise Energy Content and Stacking Framework.

Plant resistance proteins (R proteins) recognize effector proteins secreted by pathogenic microorganisms and trigger an immune response against pathogenic microbial infestation. Accurate identification of plant R proteins is an important research topic in plant pathology. Plant R protein prediction has achieved many research results. Recently, some machine learning-based methods have emerged to identify plant R proteins. Still, most of them only rely on protein sequence features, which ignore inter-amino acid features, thus limiting the further improvement of plant R protein prediction performance. In this manuscript, we propose a method called StackRPred to predict plant R proteins. Specifically, the StackRPred first obtains plant R protein feature information from the pairwise energy content of residues; then, the obtained feature information is fed into the stacking framework for training to construct a prediction model for plant R proteins. The results of both the five-fold cross-validation and independent test validation show that our proposed method outperforms other state-of-the-art methods, indicating that StackRPred is an effective tool for predicting plant R proteins. It is expected to bring some favorable contribution to the study of plant R proteins.

A comparison of three crystalline forms of miconazole: solvent-free, ethanol monosolvate and hemihydrate.

The crystal structures of miconazole {MIC, C18H14Cl4N2O, systematic name (RS)-1-[2-(2,4-dichlorobenzyloxy)-2-(2,4-dichlorophenyl)ethyl]-1H-imidazole}, its ethanol monosolvate (C18H14Cl4N2O·C2H5OH) and its hemihydrate (C18H14Cl4N2O·0.5H2O) were compared. A detailed comparison of the molecular conformation of the miconazole molecules showed a structural similarity of the solvate forms, whereas the unsolvated form is related to the gas-phase structure. This suggests that the molecular conformation of miconazole is influenced by solvent molecules. The crystal architectures of the considered solvatomorphs are differentiated by the intermolecular interactions formed by ethanol and water molecules. The structural studies are enriched by Hirshfeld surface and energy framework analysis. The pairwise model energies of the dominant contacts were estimated to be in the range 20-70 kJ mol-1. It is interesting that the contribution of dispersive forces predominates over the electrostatic forces.

Parameterization and Application of the General Amber Force Field to Model Fluoro Substituted Furanose Moieties and Nucleosides.

Molecular mechanics force field calculations have historically shown significant limitations in modeling the energetic and conformational interconversions of highly substituted furanose rings. This is primarily due to the gauche effect that is not easily captured using pairwise energy potentials. In this study, we present a refinement to the set of torsional parameters in the General Amber Force Field (gaff) used to calculate the potential energy of mono, di-, and gem-fluorinated nucleosides. The parameters were optimized to reproduce the pseudorotation phase angle and relative energies of a diverse set of mono- and difluoro substituted furanose ring systems using quantum mechanics umbrella sampling techniques available in the IpolQ engine in the Amber suite of programs. The parameters were developed to be internally consistent with the gaff force field and the TIP3P water model. The new set of angle and dihedral parameters and partial charges were validated by comparing the calculated phase angle probability to those obtained from experimental nuclear magnetic resonance experiments.

Spectroscopic and theoretical studies of some 2-(2′-ethylsulfanyl)acetyl-5-substituted furans and thiophenes

The conformational equilibrium of 2-(2′-ethylsulfanyl)acetyl-5-substituted furans and thiophenes [EtSCH2C(O)(C4H2X)Y] (1–5; Scheme 1) was determined through the infrared analysis of the carbonyl stretching band (νCO) supported by DFT (B3LYP/aug-cc-pVTZ) and molecular dynamics simulations (OPLS-AA). Six stable conformations, syn(1–3) and anti(1–3), were obtained through the optimisation of the isolated systems, where the syn/anti nomenclature refers to the dihedral angle between the carbonyl and the aromatic heteroatom (O=C–C–X; X = O, S) and 1, 2, and 3 refer to the ethyl rotation (Scheme 2) (torsion around the γ dihedral angle). The three ethyl conformers resemble six- (1 and 2) or seven-(3) member pseudo-rings with an electrostatic intramolecular hydrogen bond (Oδ-···Hδ+). The IR spectra in solution show νCO doublets for all compounds in all solvents (n-C6H14, CCl4, CHCl3, CH2Cl2 and CH3CN). The polarisable continuum model and molecular dynamics data allow the lowest wavenumber and more intense νCO IR component to be ascribed to the three anti conformers of the furan derivatives and to the three syn ones of the thiophene compounds. The conformational preferences of all compounds are governed by a balance between orbital and coulombic interactions, as estimated by means of natural bond orbital, natural coulomb electrostatic, pairwise steric exchange energy, quantum theory of atoms in molecules, non-covalent interaction and short-contact analyses. While the conformational isomerism of the ethylthioacetyl moiety is determined by both quantum and columbic effects, the equilibrium involving the carbonyl and heterocyclic moiety is controlled largely by electrostatic interactions.

ITTCA-MFF: identifying tumor T cell antigens based on multiple feature fusion.

Cancer isa terrible disease, recent studies reported that tumor T cell antigens (TTCAs) may play a promising role in cancer treatment. Since experimental methods are still expensive and time-consuming, it is highly desirable to develop automatic computational methods to identify tumor T cell antigens from the huge amount of natural and synthetic peptides. Hence, in this study, a novel computational model called iTTCA-MFF was proposed to identify TTCAs. In order to describe the sequence effectively, the physicochemical (PC) properties of amino acid and residue pairwise energy content matrix (RECM) were firstly employed to encode peptide sequences. Then, two different approaches including covariance and Pearson's correlation coefficient (PCC) were used to collect discriminative information from PC and RECM matrixes. Next, an effective feature selection approach called the least absolute shrinkage and selection operator (LAASO) was adopted to select the optimal features. These selected optimal features were fed into support vector machine (SVM) for identifying TTCAs. We performed experiments on two different datasets, experimental results indicated that the proposed method is promising and may play a complementary role to the existing methods for identifying TTCAs. The datasets and codes can be available at https://figshare.com/articles/online_resource/iTTCA-MFF/17636120 .

Identification of tumor homing peptides by utilizing hybrid feature representation

Cancer is one of the serious diseases, recent studies reported that tumor homing peptides (THPs) play a key role in treatment of cancer. Due to the experimental methods are time-consuming and expensive, it is urgent to develop automatic computational approaches to identify THPs. Hence, in this study, we proposed a novel machine learning methods to distinguish THPs from non-THPs, in which the peptide sequences firstly encoded by pseudo residue pairwise energy content matrix (PseRECM) and pseudo physicochemical property (PsePC). Moreover, the least absolute shrinkage and selection operator (LAASO) was employed to select optimal features from the extracted features. All of these selected features were fed into support vector machine (SVM) for identifying THPs. We achieved 89.02%, 88.49%, and 94.58% classification accuracy on the Main, Small, and Main90 dataset, respectively. Experimental results showed that our proposed method outperforms the existing predictors on the same benchmark datasets. It indicates that the proposed method may be a useful tool in identifying THPs. The datasets and codes used in current study are available at https://figshare.com/articles/online_resource/iTHPs/16778770. Communicated by Ramaswamy H. Sarma

Hidden electrostatic energy contributions define dynamic allosteric communications within p53 during molecular recognition

Molecular recognition is fundamental to transcription regulation. As a transcription factor, the tumor suppressor p53 has to recognize either specific DNA sequences or repressor protein partners. However, the molecular mechanism underlying the p53 conformational switch from the DNA-bound to repressor-bound states is not fully characterized. The highly charged nature of these interacting molecules prompted us to explore the nonbonded energy contributions behind molecular recognition of either a DNA or the repressor protein iASPP by p53 DNA binding domain (p53DBD), using molecular dynamics simulation followed by rigorous analyses of energy terms. Our results illuminate the allosteric pathway by which iASPP binding to p53 diminishes binding affinity between p53 and DNA. Even though the p53DBD uses a common framework of residues for recognizing both DNA and iASPP, a comparison of the electrostatics in the two p53DBD complexes revealed significant differences in residue-wise contributions to the electrostatic energy. We found that an electrostatic allosteric communication path exists in the presence of both substrates. It consists of evolutionarily conserved residues, from residue K120 of the binding loop L1 to a distal residue R213 of p53DBD. K120 is near the DNA in the p53DBD-DNA complex, whereas iASPP binding moves it away from its DNA binding position in the p53DBD-iASPP complex. The “energy hubs” (the residues show a higher degree of connectivity with other residues in the electrostatic networks) determined from the electrostatic network analysis established that this conformational change in K120 completely rewires the electrostatic network from K120 to R213, thereby impeding DNA binding. Furthermore, we found shifting populations of hydrogen bonds and salt bridges reduce pairwise electrostatic energies within p53DBD in its DNA-bound state.

Core–shell particles in rotating electric and magnetic fields: Designing tunable interactions via particle engineering

Tunable interactions between colloidal particles, governed by external rotating electric or magnetic fields, yield rich capabilities for prospective self-assembly technologies of materials and fundamental particle-resolved studies of phase transitions and transport phenomena in soft matter. However, the role of the internal structure of colloidal particles in the tunable interactions has never been systematically investigated. Here, we study the tunable interactions between composite particles with core-shell structure in a rotating electric field and show that the engineering of their internal structure provides an effective tool for designing the interactions. We generalized an integral theory and studied the tunable interactions between core-shell particles with homogeneous cores (layered particles) and cores with nano-inclusions to reveal the main trends in the interactions influenced by the structure. We found that depending on the materials of the core, shell, and solvent, the interactions with the attractive pairwise part and positive or negative three-body part can be obtained, as well as pairwise repulsion with attractive three-body interactions (for triangular triplets). The latter case is observed for the first time, being unattainable for homogeneous particles but feasible with core-shell particles: Qualitatively similar interactions are inherent to charged colloids (repulsive pairwise and attractive three-body energies), known as a model system of globular proteins. The methods and conclusions of our paper can be generalized for magnetic and 3D colloidal systems. The results make a significant advance in the analysis of tunable interactions in colloidal systems, which are of broad interest in condensed matter, chemical physics, physical chemistry, materials science, and soft matter.

Underlying Mechanisms of Allopurinol in Eliminating Renal Toxicity Induced by Melamine-Uric Acid Complex Formation: A Computational Study.

Using molecular dynamics, we address uric acid (UA) replacement by a model small-molecule inhibitor, allopurinol (AP), from its aggregated cluster in a columnar fashion. Experimentally it has been affirmed that AP is efficient in preventing UA-mediated renal stone formation. However, no study has presented the underlying mechanisms yet. Hence, a theoretical approach is presented for mapping the AP, which binds to melamine (MM) and UA clusters. In AP's presence, the higher-order cluster of UA molecules turns into a lower-order cluster, which "drags" fewer MM to them. Consequently, the MM-UA composite structure gets reduced. It is worth noting that UA-AP and AP-MM hydrogen-bonding interactions often play an essential role in reducing the UA-MM cluster size. Interestingly, an AP around UA makes a pillar-like structure, confirmed by defining the point-plane distribution function. The decomposition of the preferential interaction by Kirkwood-Buff integral into different angles like 0°-30°, 30°-60°, and 60°-90° firmly establishes the phenomenon mentioned above. However, the structural order for such π-stacking interactions between AP and UA molecules is not hierarchical but rather more spontaneous. The driving force behind UA-AP-MM composite formation is the favorable complexation energy that can be inferred by computing pairwise binding free energies for all possible combinations. Performing enhanced sampling and quantum calculations further confirms the evidence for UA degradation.

Impact of Ni alloying on Fe-C martensite ageing: an atomistic investigation

We propose a computational investigation of Ni-alloying impact on Fe-C martensite ageing, at the atomic scale. Using the Climbing Image Nudged Elastic Band technique, we showed the repulsive nature of Ni-C pairwise interactions in α-iron, and demonstrated that Ni alloying lowers the migration energies and force dipole tensor components associated to an interstitial carbon in α-iron. Computed values of pairwise interactions, migration energies and dipole components were used to implement a Kinetic Monte-Carlo approach. We found that Ni alloying: (i) has negligible impact on martensite thermodynamical stability, (ii) accelerates ageing kinetics, by increasing carbon diffusivity.

Accuracy of Alternate Nonpolarizable Force Fields for the Determination of Protein-Ligand Binding Affinities Dominated by Cation-π Interactions.

Modifying pair-specific Lennard-Jones parameters through the nonbonded FIX (NBFIX) feature of the CHARMM36 force field has proven cost-effective for improving the description of cation-π interactions in biological objects by means of pairwise additive potential energy functions. Here, two sets of newly optimized CHARMM36 force-field parameters including NBFIX corrections, coined CHARMM36m-NBF and CHARMM36-WYF, and the original force fields, namely CHARMM36m and Amber ff14SB, are used to determine the standard binding free energies of seven protein-ligand complexes containing cation-π interactions. Compared with precise experimental measurements, our results indicate that the uncorrected, original force fields significantly underestimate the binding free energies, with a mean error of 5.3 kcal/mol, while the mean errors of CHARMM36m-NBF and CHARMM36-WYF amount to 0.8 and 2.1 kcal/mol, respectively. The present study cogently demonstrates that the use of modified parameters jointly with NBFIX corrections dramatically increases the accuracy of the standard binding free energy of protein-ligand complexes dominated by cation-π interactions, most notably with CHARMM36m-NBF.

Rapid Assessment of T-Cell Receptor Specificity of the Immune Repertoire.

Accurate assessment of TCR-antigen specificity at the whole immune repertoire level lies at the heart of improved cancer immunotherapy, but predictive models capable of high-throughput assessment of TCR-peptide pairs are lacking. Recent advances in deep sequencing and crystallography have enriched the data available for studying TCR-p-MHC systems. Here, we introduce a pairwise energy model, RACER, for rapid assessment of TCR-peptide affinity at the immune repertoire level. RACER applies supervised machine learning to efficiently and accurately resolve strong TCR-peptide binding pairs from weak ones. The trained parameters further enable a physical interpretation of interacting patterns encoded in each specific TCR-p-MHC system. When applied to simulate thymic selection of an MHC-restricted T-cell repertoire, RACER accurately estimates recognition rates for tumor-associated neoantigens and foreign peptides, thus demonstrating its utility in helping address the large computational challenge of reliably identifying the properties of tumor antigen-specific T-cells at the level of an individual patient's immune repertoire.

Motion in the Restricted Three-Body Problem at the Nanoscale

This paper studies the classical restricted three-body problem of a carbon atom in the vicinity of two carbon 60 fullerenes ( fullerenes) at the nanoscale. The total molecular energy between the two fullerenes is determined analytically by approximating the pairwise potential energies between the carbon atoms on the fullerenes by a continuous approach. Using software MATHEMATICA, we compute the positions of the stationary points and their stability for a carbon atom at the nanosacle and it is observed that for each set of values, there exists at least one complex root with the positive real part and hence in the Lyapunov sense, the stationary points are unstable. Since only attractive Van der Waals forces contribute to the orbiting behavior, no orbiting phenomenon can be observed for , where the Van der Waals forces becomes repulsive. Although the orbital is speculative in nature and also presents exciting possibilities, there are still many practical challenges that would need to be overcome before the orbital might be realized. However, the present theoretical study is a necessary precursor to any of such developments.

Motion in the Restricted Three-Body Problem at the Nanoscale

This paper studies the classical restricted three-body problem of a carbon atom in the vicinity of two carbon 60 fullerenes ( fullerenes) at the nanoscale. The total molecular energy between the two fullerenes is determined analytically by approximating the pairwise potential energies between the carbon atoms on the fullerenes by a continuous approach. Using software MATHEMATICA, we compute the positions of the stationary points and their stability for a carbon atom at the nanosacle and it is observed that for each set of values, there exists at least one complex root with the positive real part and hence in the Lyapunov sense, the stationary points are unstable. Since only attractive Van der Waals forces contribute to the orbiting behavior, no orbiting phenomenon can be observed for , where the Van der Waals forces becomes repulsive. Although the orbital is speculative in nature and also presents exciting possibilities, there are still many practical challenges that would need to be overcome before the orbital might be realized. However, the present theoretical study is a necessary precursor to any of such developments.