Multidimensional persistence in biomolecular data

Persistent homology has emerged as a popular technique for the topological simplification of big data, including biomolecular data. Multidimensional persistence bears considerable promise to bridge the gap between geometry and topology. However, its practical and robust construction has been a challenge. We introduce two families of multidimensional persistence, namely pseudomultidimensional persistence and multiscale multidimensional persistence. The former is generated via the repeated applications of persistent homology filtration to high‐dimensional data, such as results from molecular dynamics or partial differential equations. The latter is constructed via isotropic and anisotropic scales that create new simiplicial complexes and associated topological spaces. The utility, robustness, and efficiency of the proposed topological methods are demonstrated via protein folding, protein flexibility analysis, the topological denoising of cryoelectron microscopy data, and the scale dependence of nanoparticles. Topological transition between partial folded and unfolded proteins has been observed in multidimensional persistence. The separation between noise topological signatures and molecular topological fingerprints is achieved by the Laplace–Beltrami flow. The multiscale multidimensional persistent homology reveals relative local features in Betti‐0 invariants and the relatively global characteristics of Betti‐1 and Betti‐2 invariants. © 2015 Wiley Periodicals, Inc.     

The Effect of Doping Different Heteroatoms on the Interaction and Adsorption Abilities of Fullerene

Density functional method has been employed to compare the interaction and adsorption abilities of simple and doped fullerenes with various heteroatoms (Al, B, Si, N, P, and S). Three sulfur‐containing molecules (H₂S, SO₂, and thiophene) were selected to study of their interactions with fullerenes. These interactions will be important in the design of new sensor, adsorption, and elimination of pollutants and chemical reactions. The calculated adsorption energies (E_ad) in the gas phase and solvents (water, using the polarized continuum model) showed that all adsorbates have exothermic interaction with all fullerenes. The maximum E_ad values were calculated for aluminum‐doped fullerene (AF) and nitrogen‐doped fullerene (NF), and the adsorption energies in solvent are not so different with those in the gas phase. Natural Bond Orbitals (NBO) calculations showed the complexes of AF and then boron‐doped fullerene (BF) have the highest E2 interaction energies, whereas simple fullerene (F) and phosphorus‐doped fullerene (PF) have the least E2 energies. Population analyses showed that doping by heteroatoms bearing extra electrons reduces the energy gap and this decrease is more than the decrease observed from doping by heteroatoms with electron defect. Moreover, the change in the energy gaps of the complexes, obtained from the density of states (DOSs) plots, showed that these structures could be used in sensor devices. All calculated data confirmed the better adsorption of SO₂ by fullerenes versus H₂S and thiophene and among all fullerenes, AF and then BF and NF are the best adsorbent for these structures.     

Resonance energies of fullerenes with 4-membered rings

We report the resonance energies (REs ) of several fullerenes with 4-membered rings and their isomers with only 5- and 6-membered rings computed using the conjugated-circuit model [RE (CC )] and the topological resonance energy (TRE ) model. Both aromaticity indices were normalized by dividing by the size of the considered fullerene [RE (CC )/e and TRE /e]. The results parallel the predictions by Gao and Herndon using the much more advanced SCF –UHF π-electron approach. A good linear correlation is found between the topologically defined indices [RE (CC )/e and TRE /e] and normalized SCF –UHF π-electron energy. © 1995 John Wiley & Sons, Inc.     

Stereoanalysis of fullerenes with a chiral molecular framework

Stereoanalysis of three fullerene molecules with a chiral molecular framework C₃₂, C₇₆, and C₇₈ and achiral fullerene C₆₀ molecule was carried out. Comparative quantitative analysis of the degree of chirality showed topology to be the major factor governing the chirality of fullerenes. A procedure for determining the relative contribution of topological chirality to the total chirality of the molecule is proposed. Structural fragments responsible for chirality are found. The title fullerenes are assigned to the corresponding subclasses of homochirality. A classification system of isomeric fullerenes is proposed.     

Some aspects of the symmetry and topology of possible carbon allotrope structures

Elemental carbon has recently been shown to form molecular polyhedral allotropes known as fullerenes in addition to the familiar graphite and diamond known since antiquity. Such fullerenes contain polyhedral carbon cages in which all vertices have degree 3 and all faces are either pentagons or hexagons. All known fullerenes are found to satisfy the isolated pentagon rule (IPR) in which all pentagonal faces are completely surrounded by hexagons so that no two pentagonal faces share an edge. The smallest fullerene structures satisfying the IPR are the known truncated icosahedral C₆₀ of I _h symmetry and ellipsoidal C₇₀ of D _5h symmetry. The multiple IPR isomers of families of larger fullerenes such as C₇₆, C₇₈, C₈₂ and C₈₄ can be classified into families related by the so-called pyracylene transformation based on the motion of two carbon atoms in a pyracylene unit containing two linked pentagons separated by two hexagons. Larger fullerenes with 3ν vertices can be generated from smaller fullerenes with ν vertices through a so‐called leapfrog transformation consisting of omnicapping followed by dualization. The energy levels of the bonding molecular orbitals of fullerenes having icosahedral symmetry and 60n ² carbon atoms can be approximated by spherical harmonics. If fullerenes are regarded as constructed from carbon networks of positive curvature, the corresponding carbon allotropes constructed from carbon networks of negative curvature are the polymeric schwarzites. The negative curvature in schwarzites is introduced through heptagons or octagons of carbon atoms and the schwarzites are constructed by placing such carbon networks on minimal surfaces with negative Gaussian curvature, particularly the so-called P and D surfaces with local cubic symmetry. The smallest unit cell of a viable schwarzite structure having only hexagons and heptagons contains 168 carbon atoms and is constructed by applying a leapfrog transformation to a genus 3 figure containing 24 heptagons and 56 vertices described by the German mathematician Klein in the 19th century analogous to the construction of the C₆₀ fullerene truncated icosahedron by applying a leapfrog transformation to the regular dodecahedron. Although this C₁₆₈ schwarzite unit cell has local O _h point group symmetry based on the cubic lattice of the D or P surface, its larger permutational symmetry group is the PSL(2,7) group of order 168 analogous to the icosahedral pure rotation group, I, of order 60 of the C₆₀ fullerene considered as the isomorphous PSL(2,5) group. The schwarzites, which are still unknown experimentally, are predicted to be unusually low density forms of elemental carbon because of the pores generated by the infinite periodicity in three dimensions of the underlying minimal surfaces. This revised version was published online in July 2006 with corrections to the Cover Date.     

Laser desorption/ionization mass spectrometric analysis of small molecules using fullerene‐derivatized silica as energy‐absorbing material

In spite of the growing acceptance of matrix‐assisted laser desorption/ionization time‐of‐flight mass spectrometry for the analysis of a wide variety of compounds, including polymers and proteins, its use in analyzing low‐molecular‐weight molecules (<1000m/z) is still limited. This is mainly due to the interference of matrix molecules in the low‐mass range. Here the derivatized fullerenes covalently bound to silica particles with different pore sizes are applied as thin layer for laser desorption/ionization (LDI) mass spectrometric analysis. Thus, an interference of intrinsic matrix ions can be eliminated or minimized in comparison with the state‐of‐the‐art weak organic acid matrices. The desorption/ionization ability of the developed fullerene–silica materials depends on the applied laser power, sample preparation and pore size of the silica particles. Thus, fullerene–silica serves as an LDI support for mass spectrometric analysis of molecules (<1500 Da). The performance of the fullerene–silica is demonstrated by the mass analysis of variety of small molecules such as carbohydrates, amino acids, peptides, phospholipids and drugs. Copyright © 2010 John Wiley & Sons, Ltd.     

Addition Reaction of Fe（CO）_n （n = 3～5） on Fullerene C_（50）, C_（60）, and C_（70）： A Density Functional Theory Study

The electronic structure and reactivities of Fe(CO)n (n = 3~5) addition to different fullerenes have been investigated through the first-principles calculations, and the results indicate that Fe(CO)3 and Fe(CO)4 can be adsorbed to the outside network of fullerene via hollow and bridge sites, respectively. Both of them have larger binding energy, but when Fe(CO)5 is adsorbed via the top site, the binding energy is relatively smaller. According to the directional curvature theory, the reactivities of Fe(CO)3 addition to the fullerenes are determined by KM of the ring center, and those of Fe(CO)4 addition by KD of the C-C bond curvature; while for Fe(CO)5, it presents weak reactivities in the addition reaction because of the larger volume effect. No matter whether the addition reaction takes place on the hollow or bridge site, the binding energies show a linear relationship with KD. This work further enriched the directional curvature theory and applied the isolobel analogy theory in the fullerene addition reactions.     

Strain Release of Fused Pentagons in Fullerene Cages by Chemical Functionalization

According to the isolated pentagon rule (IPR), for stable fullerenes, the 12 pentagons should be isolated from one another by hexagons, otherwise the fused pentagons will result in an increase in the local steric strain of the fullerene cage. However, the successful isolation of more than 100 endohedral and exohedral fullerenes containing fused pentagons over the past 20 years has shown that strain release of fused pentagons in fullerene cages is feasible. Herein, we present a general overview on fused‐pentagon‐containing (i.e. non‐IPR) fullerenes through an exhaustive review of all the types of fused‐pentagon‐containing fullerenes reported to date. We clarify how the strain of fused pentagons can be released in different manners, and provide an in‐depth understanding of the role of fused pentagons in the stability, electronic properties, and chemical reactivity of fullerene cages.     

Organometallic derivatives of fullerenes: a DFT study of (eta2-Cx)[Pt(PH3)2]n (x = 60, 70, 84; n = 1-6)

To determine the relationship among curvature, patch type, and reactivity of the C-C site, a series of density functional calculations were performed on several substituted fullerenes. [6:6] pyracylene-type sites are the most reactive sites in all analyzed cages: C(60), C(70), and C(84). The binding energy between the Pt(PH(3))(2) unit and fullerene is almost independent of the size of the cage and of the number of metals coordinated on the fullerene surface. Contrarily, curvature and type of carbon-carbon bond are determinant for the coordination strength. The use of relatively large basis sets is necessary to have consistent energies.     

Complexes of C60 fullerene with simple donor molecules: Theoretical study

The complex formation between fullerene C60 and simple donor molecules such as dimethyl ether, dimethylamine, dimethylsulfide, furan, pyrrole, and thiophene has been studied applying the hybrid MP2/6‐31G(d′):PM3 ONIOM approach for geometry optimization. Local implementation of Møller–Plesset perturbation theory in combination with 6‐31G(d) and 6‐311G(d,p) basis sets was used for binding energies estimation of fullerene complexes. Two factors were found to contribute most to the complex stability: the polarizability and molecular volume of donor molecule. As follows from positive stabilization energies at the Hartree–Fock level, the stabilization of fullerene complexes is entirely due to dispersion interactions in accordance with available experimental data. The calculations show that for donors of similar molecular volume the binding energy of molecular complex increases with polarizability of donor molecules. Similarly, for such complexes the partial charges on molecules increase with decreasing of ionization potentials of donor molecules. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Dynamics of Photoinduced Electron‐Transfer Processes in Fullerene‐Based Dyads: Effects of Varying the Donor Strength

Two classes of fullerene‐based donor–bridge–acceptor (D–B–A) systems containing donors of varying oxidation potentials have been synthesized. These systems include fullerenes linked to heteroaromatic donor groups (phenothiazine/phenoxazine) as well as substituted anilines (p‐anisidine/p‐toluidine). In contrast to the model compound, an efficient intramolecular electron transfer is observed from the fullerene singlet excited state in polar solvents. An increase in the rate constant and quantum yield of charge separation (k_cs and Φ_cs) has been observed for both classes of dyads, with decrease in the oxidation potentials of the donor groups. This observation indicates that the rates of the forward electron transfer fall in the normal region of the Marcus curve. The long‐lived charge separation enabled the characterization of electron transfer products, namely, the radical cation of the donor and radical anion of the pyrrolidinofullerene, by using nanosecond transient absorption spectroscopy. The small reorganization energy (λ) of C₆₀ coupled with large negative free energy changes (‐ΔG°) for the back electron transfer places the back electron process in the inverted region of Marcus curve, thereby stabilizing the electron transfer products.     

The smallest stable fullerene, M@C28 (m = Ti, Zr, U): stabilization and growth from carbon vapor

The smallest fullerene to form in condensing carbon vapor has received considerable interest since the discovery of Buckminsterfullerene, C(60). Smaller fullerenes remain a largely unexplored class of all-carbon molecules that are predicted to exhibit fascinating properties due to the large degree of curvature and resulting highly pyramidalized carbon atoms in their structures. However, that curvature also renders the smallest fullerenes highly reactive, making them difficult to detect experimentally. Gas-phase attempts to investigate the smallest fullerene by stabilization through cage encapsulation of a metal have been hindered by the complexity of mass spectra that result from vaporization experiments which include non-fullerene clusters, empty cages, and metallofullerenes. We use high-resolution FT-ICR mass spectrometry to overcome that problem and investigate formation of the smallest fullerene by use of a pulsed laser vaporization cluster source. Here, we report that the C(28) fullerene stabilized by encapsulation with an appropriate metal forms directly from carbon vapor as the smallest fullerene under our conditions. Its stabilization is investigated, and we show that M@C(28) is formed by a bottom-up growth mechanism and is a precursor to larger metallofullerenes. In fact, it appears that the encapsulating metal species may catalyze or nucleate endohedral fullerene formation.     

Fullerenes linked to photosynthetic pigments

The synthesis and photochemical characterization of two porphyrin-fullerene dyads, two zinc porphyrin-fullerene dyads, and a carotenobuckminsterfullerene are reviewed. In these molecules, the fullerene first excited singlet state may be formed by direct excitation or by singlet-singlet energy transfer from the attached pigment. In polar solvents, the dominant singlet-state decay pathway is photoinduced electron transfer to yield the pigment radical cation and fullerene radical anion. This charge-separated state has a long lifetime relative to the time constant for charge separation. In toluene, in cases where photoinduced electron transfer is slow for thermodynamic reasons, the fullerene singlet state decays by intersystem crossing, and the resulting triplet energy is partitioned between the components of the dyad according to their triplet energies. The results suggest that fullerenes can be valuable components of photochemically active multicomponent molecular systems.     

Investigations in the field of higher fullerenes

Halogenation of higher fullerene mixtures or their perfluoroalkylation with R^FI followed by HPLC separation of R^F derivatives and subsequent synchrotron X-ray crystallographic study made it possible to confirm cage connectivities of higher fullerenes and, in addition, to receive information concerning their reactivity in radical addition reactions. The data obtained are compared with theoretical predictions for higher fullerenes. Addition patterns of higher fullerene derivatives are discussed. Skeletal rearrangement of the D ₂-C₇₆ cage during chlorination has been observed for the first time.     

Further test of the isolated pentagon rule: Thermodynamic and kinetic stabilities of C84 fullerene isomers

Thermodynamic and kinetic stabilities of 73 C₈₄ fullerene isomers were estimated from the MM3 heats of formation and the recently defined bond resonance energies (BREs), respectively. The BRE represents the contribution of a given π bond in a molecule to the topological resonance energy (TRE). All π bonds shared by two pentagons turned out to be highly reactive without exceptions. C₈₄ fullerene isomers with such π bonds must be incapable of survival during harsh synthetic processes. Thus, the isolated pentagon rule (IPR) proved to be applicable to such large fullerene cages. For sufficiently large fullerenes like C₈₄, some isolated-pentagon isomers are also predicted to be very unstable with highly antiaromatic π bonds. © 1996 by John Wiley & Sons, Inc.     

Structures,stabilities, and IR and 13C-NMR spectra of dihedral fullerenes: A density functional theory study

Dihedral fullerenes are thermodynamically stable molecules with D nd or D nh symmetry.Based on experimental findings,two series of dihedral fullerenes with five-fold(C5) and six-fold(C6) symmetry have been studied using density functional theory(DFT).The DFT calculations showed that for both series the stabilities increased with increasing fullerene size.Structural analyses indicated that the stabilities are related to specific local geometries.In the case of the more abundant C5 series,the presence of approximately planar pentagons and hexagons on the top bowl favors their formation.That is to say,those fullerenes with small dihedral angles within the polygons are readily formed,because planar hexagons lead to strengthened conjugation which lowers average bonding energies(ABE) and increases thermodynamic stabilities.Non-planar hexagons at equatorial positions in tube-shaped fullerenes have an adverse effect on the conjugation and inhibit their formation.Calculations also demonstrated that fullerenes in the two series,including C 50(D 5h),C 60(D 6h),C 80(D 5d),C 96(D 6d),C 110(D 5h),and C 120(D 5d),have thermodynamically stable triplet structures with strong conjugation.The calculated IR and 13 C NMR spectra of the fullerenes show some similarities and regular trends due to their homogenous structures.The electronic structures indicate that short double bonds in hexagons with high electron occupancies are readily attacked by electrophilic agents and can also be coordinated by transition metals.Mechanistic discussions suggested that C 2 additions and C 2 losses constitute reversible processes at high temperature and C 2 additions in pentagonal fusions are crucial to the kinetics of the curvature of structures.C 3 additions lead to the formation of large fullerenes of other types.     

Synthesis and In Vitro Biological Evaluation of Psoralen‐Linked Fullerenes

Photodynamic therapy (PDT) is a widely used medicinal treatment for the cancer therapy that utilizes the combination of a photosensitizer (PS) and light irradiation. In this study, we synthesized two novel C₆₀ fullerene derivatives, compounds 1 and 2 , with a psoralen moiety that can covalently bind to DNA molecules via cross‐linking to pyrimidine under photoirradiation. Along with several fullerene derivatives, the biological properties of several novel compounds have been evaluated. Compounds 1 and 2 , which have been shown to induce the production of hydroxyl radicals using several ROS detecting reagents, induced DNA strand breaks with relatively weak activities in the in vitro detection system using a supercoiled plasmid. However, the psoralen‐bound fullerene with carboxyl groups ( 2 ) only showed genotoxicity in the genotoxicity assay system of the umu test. Compound 2 was also seen to have cytotoxic activities in several cancer cell lines at higher doses compared to water‐soluble fullerenes.     

Highly efficient implementation of pseudospectral time‐dependent density‐functional theory for the calculation of excitation energies of large molecules

We have developed and implemented pseudospectral time‐dependent density‐functional theory (TDDFT) in the quantum mechanics package Jaguar to calculate restricted singlet and restricted triplet, as well as unrestricted excitation energies with either full linear response (FLR) or the Tamm–Dancoff approximation (TDA) with the pseudospectral length scales, pseudospectral atomic corrections, and pseudospectral multigrid strategy included in the implementations to improve the chemical accuracy and to speed the pseudospectral calculations. The calculations based on pseudospectral time‐dependent density‐functional theory with full linear response (PS‐FLR‐TDDFT) and within the Tamm–Dancoff approximation (PS‐TDA‐TDDFT) for G2 set molecules using B3LYP/6‐31G*^* show mean and maximum absolute deviations of 0.0015 eV and 0.0081 eV, 0.0007 eV and 0.0064 eV, 0.0004 eV and 0.0022 eV for restricted singlet excitation energies, restricted triplet excitation energies, and unrestricted excitation energies, respectively; compared with the results calculated from the conventional spectral method. The application of PS‐FLR‐TDDFT to OLED molecules and organic dyes, as well as the comparisons for results calculated from PS‐FLR‐TDDFT and best estimations demonstrate that the accuracy of both PS‐FLR‐TDDFT and PS‐TDA‐TDDFT. Calculations for a set of medium‐sized molecules, including C_n fullerenes and nanotubes, using the B3LYP functional and 6‐31G^** basis set show PS‐TDA‐TDDFT provides 19‐ to 34‐fold speedups for C_n fullerenes with 450–1470 basis functions, 11‐ to 32‐fold speedups for nanotubes with 660–3180 basis functions, and 9‐ to 16‐fold speedups for organic molecules with 540–1340 basis functions compared to fully analytic calculations without sacrificing chemical accuracy. The calculations on a set of larger molecules, including the antibiotic drug Ramoplanin, the 46‐residue crambin protein, fullerenes up to C₅₄₀ and nanotubes up to 14×(6,6), using the B3LYP functional and 6‐31G^** basis set with up to 8100 basis functions show that PS‐FLR‐TDDFT CPU time scales as N^2.05 with the number of basis functions. © 2016 Wiley Periodicals, Inc.     

Electronic structure of IPR and non-IPR endohedral metallofullerenes: Connecting orbital and topological rules

The electronic structure of endohedral metallofullerenes is rationalized by connecting the apparently independent orbital and topological rules that explain the stability of this family of fullerenes. The separation of the 12 pentagons of the fullerene, which is maximized in order to minimize the Coulomb repulsion, is found to be correlated with the orbital energies of the cage that accepts the electron transfer from the internal cluster. An explanation for the absence of non-IPR cages in large-size EMFs is also provided.