Factorisation of Littlewood-Richardson coefficients

The hive model is used to show that the saturation of any essential Horn inequality leads to the factorisation of Littlewood-Richardson coefficients. The proof is based on the use of combinatorial objects known as puzzles. These are shown not only to account for the origin of Horn inequalities, but also to determine the constraints on hives that lead to factorisation. Defining a primitive Littlewood-Richardson coefficient to be one for which all essential Horn inequalities are strict, it is shown that every Littlewood-Richardson coefficient can be expressed as a product of primitive coefficients. Precisely the same result is shown to apply to the polynomials defined by stretched Littlewood-Richardson coefficients.     

Instance-optimality in probability with an -minimization decoder

Let Φ(ω), ωΩ, be a family of n×N random matrices whose entries _i,j are independent realizations of a symmetric, real random variable η with expectation and variance . Such matrices are used in compressed sensing to encode a vector by y=Φx. The information y holds about x is extracted by using a decoder . The most prominent decoder is the ℓ₁-minimization decoder Δ which gives for a given the element which has minimal ℓ₁-norm among all with Φz=y. This paper is interested in properties of the random family Φ(ω) which guarantee that the vector will with high probability approximate x in to an accuracy comparable with the best k-term error of approximation in for the range kan/log₂(N/n). This means that for the above range of k, for each signal , the vector satisfies

with high probability on the draw of Φ. Here, Σ_k consists of all vectors with at most k nonzero coordinates. The first result of this type was proved by Wojtaszczyk [P. Wojtaszczyk, Stability and instance optimality for Gaussian measurements in compressed sensing, Found. Comput. Math., in press] who showed this property when η is a normalized Gaussian random variable. We extend this property to more general random variables, including the particular case where η is the Bernoulli random variable which takes the values with equal probability. The proofs of our results use geometric mapping properties of such random matrices some of which were recently obtained in [A. Litvak, A. Pajor, M. Rudelson, N. Tomczak-Jaegermann, Smallest singular value of random matrices and geometry of random polytopes, Adv. Math. 195 (2005) 491–523].     

Mechanism of the chiral SHG activity of bacteriorhodopsin films

The nonlinear optical activity of bacteriorhodopsin (bR) Langmuir-Blodgett (LB) films were investigated computationally and experimentally. The second harmonic generation optical rotary dispersion (SHG-ORD) response was calculated directly from the known structure and orientation of the PSB retinal chromophore within bR with no adjustable parameters. The predicted results agree remarkably well in sign, magnitude, and trend with the experimental SHG-ORD measurements. The calculations indicated negligible chirality with the tensor for the PSB retinal chromophore, but significant chiral-specific activity for the thin film through a relatively simple orientational mechanism.     

Cation and anion transport through hydrophilic pores in lipid bilayers

To understand the origin of transmembrane potentials, formation of transient pores, and the movement of anions and cations across lipid membranes, we have performed systematic atomistic molecular dynamics simulations of palmitoyl-oleoyl-phosphatidylcholine (POPC) lipids. A double bilayer setup was employed and different transmembrane potentials were generated by varying the anion (Cl-) and cation (Na+) concentrations in the two water compartments. A transmembrane potential of approximately 350 mV was thereby generated per bilayer for a unit charge imbalance. For transmembrane potential differences of up to approximately 1.4 V, the bilayers were stable, over the time scale of the simulations (10-50 ns). At larger imposed potential differences, one of the two bilayers breaks down through formation of a water pore, leading to both anion and cation translocations through the pore. The anions typically have a short residence time inside the pore, while the cations show a wider range of residence times depending on whether they bind to a lipid molecule or not. Over the time scale of the simulations, we do not observe the discharge of the entire potential difference, nor do we observe pore closing, although we observe that the size of the pore decreases as more ions translocate. We also observed a rare lipid flip-flop, in which a lipid molecule translocated from one bilayer leaflet to the opposite leaflet, assisted by the water pore.     

New developments in the theory of the diffuse double layer

The role of ion size effects in determining diffuse layer properties is considered. Monte Carlo data relevant to this question are reviewed. Then the integral equation approach to the problem is considered with an emphasis on attempts to derive an analytical equation for the potential drop across the diffuse layer. An empirical model for ion size effects is described that allows one to estimate not only the diffuse layer potential drop but also the diffuse layer capacity, the ionic surface excesses, and the potential distribution in the diffuse layer. It is demonstrated that the resulting model can easily be applied by experimentalists to the analysis of experimental data for double layer phenomena in electrochemistry and colloid science.     

Encapsulation of cyanometalates by a tris-macrocyclic ligand tricopper(II) complex: syntheses, structural variation, and magnetic exchange coupling pathways

The reaction of [M(CN)(6)](3-) (M = Cr(3+), Mn(3+), Fe(3+), Co(3+)) and [M(CN)(8)](4-/3-) (M = Mo(4+/5+), W(4+/5+)) with the trinuclear copper(II) complex of 1,3,5-triazine-2,4,6-triyltris[3-(1,3,5,8,12-pentaazacyclotetradecane)] ([Cu(3)(L)](6+)) leads to partially encapsulated cyanometalates. With hexacyanometalate(III) complexes, [Cu(3)(L)](6+) forms the isostructural host-guest complexes [[[Cu(3)(L)(OH(2))(2)][M(CN)(6)](2)][M(CN)(6)]][M(CN)(6)]30 H(2)O with one bridging, two partially encapsulated, and one isolated [M(CN)(6)](3-) unit. The octacyanometalates of Mo(4+/5+) and W(4+/5+) are encapsulated by two tris-macrocyclic host units. Due to the stability of the +IV oxidation state of Mo and W, only assemblies with [M(CN)(8)](4-) were obtained. The Mo(4+) and W(4+) complexes were crystallized in two different structural forms: [[Cu(3)(L)(OH(2))](2)[Mo(CN)(8)]](NO(3))(8)15 H(2)O with a structural motif that involves isolated spherical [[Cu(3)(L)(OH(2))](2)[M(CN)(8)]](8+) ions and a "string-of-pearls" type of structure [[[Cu(3)(L)](2)[M(CN)(8)]][M(CN)(8)]](NO(3))(4) 20 H(2)O, with [M(CN)(8)](4-) ions that bridge the encapsulated octacyanometalates in a two-dimensional network. The magnetic exchange coupling between the various paramagnetic centers is characterized by temperature-dependent magnetic susceptibility and field-dependent magnetization data. Exchange between the CuCu pairs in the [Cu(3)(L)](6+) "ligand" is weakly antiferromagnetic. Ferromagnetic interactions are observed in the cyanometalate assemblies with Cr(3+), exchange coupling of Mn(3+) and Fe(3+) is very small, and the octacoordinate Mo(4+) and W(4+) systems have a closed-shell ground state.     

Chemically controlled self-assembly of protein nanorings

The exploitation of biological macromolecules, such as nucleic acids, for the fabrication of advanced materials is a promising area of research. Although a greater variety of structural and functional uses can be envisioned for protein-based materials, systematic approaches for their construction have yet to emerge. Consistent with theoretical models of polymer macrocyclization, we have demonstrated that, in the presence of dimeric methotrexate (bisMTX), wild-type Escherichia coli dihydrofolate reductase (DHFR) molecules tethered together by a flexible peptide linker (ecDHFR(2)) are capable of spontaneously forming highly stable cyclic structures with diameters ranging from 8 to 20 nm. The nanoring size is dependent on the length and composition of the peptide linker, on the affinity and conformational state of the dimerizer, and on induced protein-protein interactions. Delineation of these and other rules for the control of protein oligomer assembly by chemical induction provides an avenue to the future design of protein-based materials and nanostructures.     

Cold denaturation of encapsulated ubiquitin

Theoretical considerations suggest that protein cold denaturation can potentially provide a means to explore the cooperative substructure of proteins. Protein cold denaturation is generally predicted to occur well below the freezing point of water. Here NMR spectroscopy of ubiquitin encapsulated in reverse micelles dissolved in low viscosity alkanes is used to follow cold-induced unfolding to temperatures below -25 degrees C. Comparison of cold-induced structural transitions in a variety of reverse micelle-buffer systems indicate that protein-surfactant interactions are negligible and allow the direct observation of the multistate cold-induced unfolding of the protein.     

Reaction of cyanoacetylene HCCCN(X 1Sigma+) with ground-state carbon atoms C(3P) in cold molecular clouds

The reaction of the simplest cyanopolyyne, cyanoacetylene [HCCCN(X (1)Sigma(+))], with ground-state atomic carbon C((3)P) is investigated theoretically to explore the probable routes for the depletion of the famed interstellar molecule HCCCN, and the formation of carbon-nitrogen-bearing species in extraterrestrial environments particularly of ultralow temperature. Six collision complexes (c1-c6) without entrance barrier as a result of the carbon atom addition to the pi systems of HCCCN are located. The optimized geometries and harmonic frequencies of the intermediates, transition states, and products along the isomerization and dissociation pathways of each collision complex are obtained by utilizing the unrestricted B3YLP6-311G(d,p) level of theory, and the corresponding CCSD(T)/cc-pVTZ energies are calculated. Subsequently, with the facilitation of Rice-Ramsperger-Kassel-Marcus (RRKM) and variational RRKM rate constants at collision energy of 0-10 kcal/mol, the most probable paths for the titled reaction are determined, and the product yields are estimated. Five collision complexes (c1-c3, c5, and c6) are predicted to give the same products, a chained CCCCN (p2)+H, via the linear and most stable intermediate, HCCCCN (i2), while collision complex c4 is likely to dissociate back to C+HCCCN. The study suggests that this class of reaction is an important route to the destruction of cyanoacetylene and cyanopolyynes in general, and to the synthesis of linear carbon-chained nitriles at the temperature as low as 10 K to be incorporated in future chemical models of interstellar clouds.