k-resonant toroidal polyhexes

A toroidal polyhex H(p, q, t) is a cubic bipartite graph embedded on the torus such that each face is a hexagon, which can be described by a string (p, q, t) of three integers (p≥ 1, q≥ 1, 0≤ t≤ p−1). A set of mutually disjoint hexagons of H(p, q, t) is called a resonant pattern if H(p, q, t) has a prefect matching M such that all haxgons in are M-alternating. A toroidal polyhex H(p, q, t) is k-resonant if any i (1 ≤ i ≤ k) mutually disjoint hexagons form a resonant pattern. In [16], Shiu, Lam and Zhang characterized 1, 2 and 3-resonant toroidal polyhexes H(p, q, t) for min(p, q)≥ 2. In this paper, we characterize k-resonant toroidal polyhexes H(p, 1, t). Furthermore, we show that a toroidal polyhex H(p, q, t) is k-resonant (k≥ 3) if and only if it is 3-resonant.      

The forcing number of toroidal polyhexes

The forcing number, denoted by f(G), of a graph G with a perfect matching is the minimum number of independent edges that completely determine the perfect matching of G. In this paper, we consider the forcing number of a toroidal polyhex H(p,q,t) with a torsion t, a cubic graph embedded on torus with every face being a hexagon. We obtain that f(H(p,q,t)) ≥ min{p,q}, and equality holds for p ≤ q or p > q and t∈{ 0,p−q,p−q + 1,..., p−1}. In general, we show that f(H(p,q,t)) is equal to the side length of a maximum triangle on H(p,q,t). Based on this result, we design a linear algorithm to compute the forcing number of H(p,q,t).     

A complete characterization for <Emphasis Type="Italic">k</Emphasis>-resonant Klein-bottle polyhexes

A hexagonal tessellation K(p, q, t) on Klein bottle, a non-orientable surface with cross-cap number 2, is a finite-sized elemental benzenoid which can be produced from a p × q-parallelogram of hexagonal lattice with usual identifications of sides and with torsion t. Unlike torus, Klein bottle polyhex K(p, q, t) is not transitive except for some degenerated cases. We shall show, however, that K(p, q, t) does not depend on t. Accordingly, criteria for K(p, q, t) to be k-resonant for every positive integer k will be given. Moreover, we shall show that K(3, q, t) of 3-resonance are fully-benzenoid.      

Maximal resonance of cubic bipartite polyhedral graphs

Let H{\mathcal{H}} be a set of disjoint faces of a cubic bipartite polyhedral graph G. If G has a perfect matching M such that the boundary of each face of H{\mathcal{H}} is an M-alternating cycle (or in other words, G-H{G-\mathcal{H}} has a perfect matching), then H{\mathcal{H}} is called a resonant pattern of G. Furthermore, G is k-resonant if every i (1 \leqslant i \leqslant k){i\,(1\,\leqslant\, i\, \leqslant\, k)} disjoint faces of G form a resonant pattern. In particular, G is called maximally resonant if G is k-resonant for all integers k \geqslant 1{k\,\geqslant\, 1}. In this paper, all the cubic bipartite polyhedral graphs, which are maximally resonant, are characterized. As a corollary, it is shown that if a cubic bipartite polyhedral graph is 3-resonant then it must be maximally resonant. However, 2-resonant ones need not to be maximally resonant.     

Graphical properties of polyhexes: Perfect matching vector and forcing

From the viewpoint of graph theory and its applications, subgraphs of the tiling of the plane with unit squares have long been studied in statistical mechanics, In organic chemistry, a much more relevant case concerns subgraphs of the tiling with unit hexagons. Our purpose here is to take a mathematical view of such polyhex graphsG and study two novel concepts concerning perfect matchingsM. First, the forcing number ofM is the smallest number of edges ofM which are not contained in any other perfect matching ofG. Second, the perfect matching vector ofM is written (n ₃,n ₂,n ₁,n ₀), wheren _k is the number of hexagons with exactlyk edges inM. We establish some initial results involving these two concepts and pose some questions.     

k-Resonance of Open-Ended Carbon Nanotubes

An open-ended carbon nanotube or a tubule is a part of some regular hexagonal tessellation of a cylinder. A tubule T is said to be k-resonant if for every k (or fewer) pairwise disjoint hexagons, the subgraph obtained from T by deleting all the vertices of these hexagons must have a Kekule structure (perfect matching) or must be empty. The 1-resonant tubules can be constructed by an approach provided in H. Zhang and F. Zhang, Discrete Appl. Math. 36 (1992) 291. In this paper, we give the construction method of k(k3)-resonant tubules. The lower bound of its Clar number of k(k3)-resonant tubules is also given. Note that the present paper does not consider the capped species.     

Embedding on alphabet overlap digraphs

Alphabet overlap digraphs can be viewed as a generalization of directed de Bruijn graphs. Given three integers α ≥ 1, k ≥ 2 and 1 ≤ i < k, the alphabet overlap digraph O(α, k ; i) is a digraph: the set of all words of length k over a certain alphabet with cardinality α is vertex set, and there is an arc from a vertex u to a vertex v if and only if the word of last k−i letters of u coincides with the word of first k−i letters of v. In this paper, we consider whether O(α, k ; i) can be embedded in O(α, k ; j) for given integers 1 ≤ i < j < k. In order to resolve this problem, we give an O(1)-time algorithm to decide whether there exists a permutation on {1, . . . ,k} from O(α, k ; i) to O(α, k ; j). If such a permutation exists, for any vertex of O(α, k ; i), we apply the permutation to change its label’s position and map it to a vertex of O(α, k ; j). Furthermore, we obtain an embedding from O(α, k ; i) to O(α, k ; j). Hence, we solve partly the problem. As a consequence, we show that every directed de Bruijn graph can be embedded in all alphabet overlap digraphs with the same parameters α and k.     

On the existence of multiply connected monolayered cyclofusenes with given parameters

For a given bipartite graph G its skewness is defined as the difference between the sizes of its classes of bipartition. We show that a multiply connected monolayered cyclofusene with m ≥ 2 internal holes and skewness k exists if and only if 0 ≤ k ≤ 2m − 2, thus settling in affirmative two conjectures raised in a recent paper by Karimi et al.     

Effects of glucose,glutamine, and malate on the metabolism of spodoptera frugiperda clone 9 (sf9) cells

Optimal design and operation of bioreactors for insect cell culture is facilitated by functional relations providing quantitative information on cellular metabolite consumption kinetics, as well as on the specific cell growth rates (μ_G). Initial specific consumption rates of glucose, malate, and oxygen, and associated changes in μ_G, were measured forSpodoptera frugiperda clone 9 (Sf9) cells grown in batch suspension culture in medium containing 7–35 mM glucose, 0–16 mM malate, and 4–16 mM glutamine. The initial specific glucose consumption rate (q _G ) could be described by a modified Michaelis-Menten equation treating malate as a “competitive” inhibitorK ₁ = 6.5 mM) and glutamine as a “noncompetitive” inhibitorK _I = 14 mM) ofq _G , with aK _m of 7.1 mM for glucose. All three carbon sources were found to increase μ_G in a saturable manner, and a modified Monod equation was employed to describe this relationship (μ_Gmax = 0.047 h^-1). The initial specific oxygen consumption rate (q_O2) in Sf9 cells could be related to μ_G by the maintenance energy model, and it was calculated that, under typical culture conditions, about 15–20% of the cellular energy demand comes from functions not related to growth. Fitted parameters in mathematical expression for μg: K₄, Monod constant for glucose (mM); K₅, modified Monod constant for malate (mM); K₆, Monod constant for glutamine (mM); m_o2, specific consumption rate of oxygen by the cells under zero-growth conditions (nmol/cell/h); q_F, initial specific fumarate production rate (nmol/cell/ h);q _G , initial specific glucose consumption rate (nmol/cell/h); q_Gmax, maximum initial specific glucose consumption rate (nmol/cell/h);q _M , initial specific malate consumption rate (nmol/cell/h); q_o2, initial specific oxygen consumption rate (nmol/cell/h); Y_o2, cell yield on oxygen (cells/nmol); μ, initial specific cell growth rate (h^-1); μ_g, initial specific cell growth rate (h^-1); μG_max, maximum initial specific cell growth rate (h^-1).     

Hexagonal resonance of (3,6)-fullerenes

A (3,6)-fullerene G is a plane cubic graph whose faces are only triangles and hexagons. It follows from Euler’s formula that the number of triangles is four. A face of G is called resonant if its boundary is an alternating cycle with respect to some perfect matching of G. In this paper, we show that every hexagon of a (3,6)-fullerene G with connectivity 3 is resonant except for one graph, and there exist a pair of disjoint hexagons in G that are not mutually resonant except for two trivial graphs without disjoint hexagons. For any (3,6)-fullerene with connectivity 2, we show that it is composed of n(n ≥ 1) concentric layers of hexagons, capped on each end by a cap formed by two adjacent triangles, and none of its hexagons is resonant.     

On maximal resonance of polyomino graphs

A polyomino graph is a finite plane 2-connected bipartite graph every interior face of which is bounded by a regular square of side length one. Let k be a positive integer, a polyomino graph G is k-resonant if the deletion of any i ≤ k vertex-disjoint squares from G results in a graph either having perfect matchings or being empty. If graph G is k-resonant for any integer k ≥ 1, then it is called maximally resonant. All maximally resonant polyomino graphs are characterized in this work. As a result, the least integer k such that a k-resonant polyomino graph is maximally resonant is determined.     

Theoretical Study of the Interactions between Isolated <Emphasis Type="Italic">DNA</Emphasis> Bases and Various Groups IA and IIA Metal Ions by <Emphasis Type="Italic">Ab Initio</Emphasis> Calculations

Summary. Interactions of the DNA bases adenine (A), guanine (G), cytosine (C), and thymine (T) with various metal ions (M) of groups IA and IIA of the periodic table of the elements were studied at the HF, MP2, and DFT levels of theory. The structures and thermodynamic stabilities of these species were studied at the gas phase. The calculations uphold that there exist two active sites in G and one in A, C, and T. The calculations also show that the O² atom in T is a more active site for metal ion bindings than that in C. The stability energies for G … M complexes are larger than those for A … M complexes and the stability energies for T … M complexes are larger than those for C … M complexes. As z/r ratio for the metal ion increases, the interaction energy for the complex increases systematically. Thermodynamic quantities such as ΔH, ΔG, ΔS, and ln K were determined for each complexation reaction, [Base+M ⁿ⁺ →(Base … M) ⁿ⁺]. A, G, and C complexation reactions except for C … Rb⁺ are exothermic. The situation is quite different for T complexation reactions and all except for T … Be²⁺ and T … Mg²⁺ are endothermic.     

Hole-phonon Scattering and Thermal Conductivity of p-type InSb from 2 to 100 K

The hole-phonon interaction contributing to the thermal conductivity of three p-type samples of InSb is analyzed between temperatures of 2 and 100 K. In addition to the phonon scattering by bound and free holes, other phonon scattering such as boundary, point defect and phonons are considered. Both relaxation rates for q≤2k _F and for q>2k _F are used for free hole-phonon scattering. The role of screening due to plasma on hole-phonon scattering is also included. The Callaway model for thermal conductivity is utilized from which an excellent fit to the experimental data is obtained over the whole temperature range. This revised version was published online in August 2006 with corrections to the Cover Date.     

Steric effects on the singlet-triplet energy gaps of five-membered ring R<Subscript>2</Subscript>C<Subscript>4</Subscript>H<Subscript>2</Subscript>M

Thermal internal energy gaps, ΔE _s−t; enthalpy gaps, ΔH _s−t; Gibbs free energy gaps, ΔD _s−t, between singlet (s) and triplet (t) states of R₂C₄H₂M (M = C, Si, and Ge) were calculated at B3LYP/6-311++G** level of theory. The ΔG _s−t of R₂C₄H₂C was increased in the order (in kcal/mol): R = −CH₃ (−10.51) > −H (−9.59) > i-Pr (−9.51) > t-Bu (−8.98). While, the ΔG _s−t of R₂C₄H₂Si and R₂C₄H₂Ge were increased in the order (in kcal/mol): −CH₃ (17.01) > i-Pr (15.30) > −H (15.26) > t-Bu (14.35) and -H (22.79) > −CH₃ (22.69) > i-Pr (21.66) > t-Bu (21.01), respectively.     

Purification and Characterization of Chitinase from <Emphasis Type="BoldItalic">Paenibacillus</Emphasis> sp. D1

A 56.56-kDa extracellular chitinase from Paenibacillus sp. D1 was purified to 52.3-fold by ion exchange chromatography using SP Sepharose. Maximum enzyme activity was recorded at pH 5.0 and 50 °C. MALDI-LC-MS/MS analysis identified the purified enzyme as chitinase with 60% similarity to chitinase Chi55 of Paenibacillus ehimensis. The activation energy (E _a) for chitin hydrolysis and temperature quotient (Q ₁₀) at optimum temperature was found to be 19.14 kJ/mol and 1.25, respectively. Determination of kinetic constants k _m, V _max, k _cat, and k _cat/k _m and thermodynamic parameters ΔH*, ΔS*, ΔG*, ΔG*_E–S, and ΔG*_E–T revealed high affinity of the enzyme for chitin. The enzyme exhibited higher stability in presence of commonly used protectant fungicides Captan, Carbendazim, and Mancozeb compared to control as reflected from the t _1/2 values suggesting its applicability in integrated pest management for control of soil-borne fungal phytopathogens. The order of stability of chitinase in presence of fungicides at 80 °C as revealed from t _1/2 values and thermodynamic parameters E _a(d) (activation energy for irreversible deactivation), ΔH*, ΔG*, and ΔS* was: Captan > Carbendazim > Mancozeb > control. The present study is the first report on thermodynamic and kinetic characterization of chitinase from Paenibacillus sp. D1.     

Electron-pair radial density functions

We introduce and discuss a generalized electron-pair radial density function G(q; a) that represents the probability density for the electron-pair radius |r ₁+ar ₂| to be q, where a is a real-valued parameter. The density function G(q; a) is a projection of the two-electron radial density D ₂(r ₁, r ₂) along lines r ₁ + ar ₂ ± q = 0 in the r ₁ r ₂ plane onto a point in the qa plane, and connects three densities S(s), D(r), and T(t), defined independently in the literature, as a smooth function of a: For an N-electron (N ≥ 2) system, S(s) = G(s; + 1), D(r) = 2G(r; 0)/(N − 1), and T(t) = G(|t|;−1)/2, where S(s) and T(t) are the electron-pair radial sum and difference densities, respectively, and D(r) is the single-electron radial density. Simple illustrations are given for the helium atom in the ground 1s² and the first excited 1s2s ³S states.     

Calculation of spectral characteristics of water clusters upon interaction with oxygen molecules and bromine ions

Interactions of (Br⁻) _i (H₂O)_50−i clusters (0 ≤ i ≤ 6) with molecular oxygen is studied by the molecular dynamics method using flexible molecule model. Values of real and imaginary parts of permittivity decrease in the 0 ≤ ω ≤ 3500 cm⁻¹ frequency range with increasing number of bromine ions in a cluster. The ability of cluster to absorb IR radiation decreases, whereas the reflectance and Raman light scattering remains nearly unchanged. An increase in the content of Br⁻ ions in the cluster lowers the power of emitted IR radiation and decreases the amount of active electrons participating in the interaction with IR radiation. However, when the concentration of Brions becomes substantially higher (at i = 5 and 6), the values of emitted power and the number of active electrons are restored to the values that are typical for water cluster in the absence of Br⁻ ions. At i ≥ 3, repelling Br⁻ ions acquire kinetic energy, which is sufficient to remove molecular oxygen from the system.     

A quartz crystal microbalance study of β-cyclodextrin self assembly on gold and complexation of immobilized β-cyclodextrin with adamantane derivatives

Quartz crystal microbalance (QCM) was used to study the self-assembly of per-6-thio-β-cyclodextrin (t₇-βCD) on gold surfaces, and the subsequent inclusion interactions of immobilized βCD with adamantane-poly(ethylene glycol) (5,000 MW, AD-PEG), 1-adamantanecarboxylic acid (AD-C) and 1-adamantylamine (AD-A). From a 50 μM solution of t₇-βCD in 60:40 DMSO:H₂O, a t₇-βCD layer was formed on gold with surface density of 71.7 ± 2.7 pmol/cm², corresponding to 80 ± 3% of close-packed monolayer coverage. Gold sensors with immobilized t₇-βCD were then exposed alternately to six different concentrations of AD-PEG, 500 μM AD-C or 500 μM AD-A aqueous solutions for association, and water for dissociation. Association of AD-PEG conformed to a Langmuir isotherm, with a best fit equilibrium constant K = 125,000 ± 18,000 M⁻¹. For AD-C and AD-A, association (k _a ) and dissociation (k _d ) rate constants were extracted from kinetic profiles by fitting to the Langmuir model, and equilibrium constants were calculated. The parameters for AD-C were found to be: k _a = 100 ± 5 M⁻¹ s⁻¹, k _d = 110 (±18) × 10⁻⁴ s⁻¹, and K = 9,400 ± 1,700 M⁻¹. For AD-A, k _a = 58 ± 6 M⁻¹ s⁻¹, k _d = 154 (±7) × 10⁻⁴ s⁻¹, and K = 3,800 ± 400 M⁻¹. The results demonstrate the utility of QCM as a tool for studying small molecule surface adsorption and guest–host interactions on surfaces. More specifically, the kinetic and thermodynamic data of AD-C, AD-A, and AD-PEG inclusion with immobilized t₇-βCD form a basis for further surface association studies of AD-X conjugates to advance surface sensory and coupling applications.     

Coordination formulation of tetrahedrally close packed structures: An addendum to the observations of Yarmolyuk and Kripyakevich

Ya. P. Yarmolyuk and P. I. Kripyakevich (Kristallographiya 19, 539 (1974)) showed that all tetrahedrally close packed (t.c.p.) structures have coordination formulae P_pQ_qR_rX_x → (PX₂)_i(Q₂R₂X₃)_j (R₃X)_k, where P, Q, R, and X represent coordination numbers (CN) 16, 15, 14, and 12 polyhedra respectively: p, q, r, and x indicate the numbers of such polyhedra in the unit cells of t.c.p. structures and i, j, and k are positive integers. We propose and demonstrate a limitation to the above formulation: if i ≥ 1 and k ≥ 1, then j ≥ 1 (or if both p> 0 and r> 0, then q> 0). We give reasons for this and discuss the Aufbauprinzip of t.c.p. structures and the results of C. B. Shoemaker and D. P. Shoemaker (Acta Crystallogr. B 42, 3 (1986)).     

Cyclical Edge-Connectivity of Fullerene Graphs and (k, 6)-Cages

It is shown that every fullerene graph G is cyclically 5-edge-connected, i.e., that G cannot be separated into two components, each containing a cycle, by deletion of fewer than five edges. The result is then generalized to the case of (k,6)-cages, i.e., polyhedral cubic graphs whose faces are only k-gons and hexagons. Certain linear and exponential lower bounds on the number of perfect matchings in such graphs are also established.