Representation of proteins as walks in 20-D space

A novel representation of proteins was introduced. It is independent of arbitrary decisions with respect to the choice of labels to be assigned to the 20 natural amino acids. The approach is based on an assignment of 20 unit vectors in 20-dimensional vector space to the 20 natural amino acids. Proteins are then represented by a walk, that is, a sequence of steps in the 20-dimensional space analogous to a walk in the (x, y) plane in the case of binary strings. A straightforward numerical characterization of proteins is obtained from the distance matrix associated with the walk representing the protein in 20-dimensional space combining the information on the Euclidean distance between various amino acids in protein sequence. The Line Distance matrix offers additional numerical characterization of proteins, while the lengths of steps of the walk in 20-D space allow construction of a "protein profile," which represents distribution of average lengths of the steps and their powers.     

Atomic walk counts of negative order

Atomic walk counts (awc's) of order k (k > or = 1) are the number of all possible walks of length k which start at a specified vertex (atom) i and end at any vertex j separated by m (0 < or = m < or = k) edges from vertex i. The sum of atomic walk counts of order k is the molecular walk count (mwc) of order k. The concept of atomic and molecular walk counts was extended to zero and negative orders by using a backward algorithm based on the usual procedure used to obtain the values of mwc's. The procedure can also be used in cases in which the adjacency matrix A related to the actual structure is singular and therefore A(-1) does not exist. awc's and mwc's of negative order may assume noninteger and even negative values. If matrix A is singular, atomic walk counts of zero order may not be equal to one.     

Segmental distribution properties of a polymer chain near an interacting barrier

The exact distribution of segments for self-avoiding walks of lengths N=4–14 bonds in the presence of an interacting barrier on the diamond lattice has been obtained by the method of direct enumeration. Behavior for the infinite chain was estimated and compared with Rubin's results for the normal random walk. It is shown that the onset of a well defined transition for the self-avoiding walk coincides with the location predicted for the normal random walk. It was found that for a self-avoiding walk a plot of θ₂ (the fraction of segments in level z) versus the interaction parameter φ is shifted to the right (higher φ), for all z, as compared with a similar plot for the normal random walk. Conditional probabilities for a self-avoiding walk having its t th segment in level z (when φ=0) are reported.     

Walk counts, labyrinthicity, and complexity of acyclic and cyclic graphs and molecules

It is demonstrated how the complexity of a (molecular) graph can be quantified in terms of the walk counts, extremely easily obtained graph invariants that depend on size, branching, cyclicity, and edge and vertex weights (unsaturation, heteroatoms). The influence of symmetry is easily accounted for. The term labyrinthicity is proposed for what is measured by walk counts alone, neglecting symmetry. The total walk count and recently advanced measures of labyrinthicity or complexity are compared with respect to the ordering of structures and to the computational effort required to obtain numerical values.     

A ring walk of methylene groups in toluene radical cations. An extension of the toluene-cycloheptatriene rearrangement of aromatic radical cations. Theory and experiment

The minimum energy reaction pathway (MERP) of the toluene-cycloheptatriene radical cation rearrangement (TOL/CHT-rearrangement) has been calculated by the UHF and DFT model at the level UHF/6-311+G(3df,2p)//UHF/6-31G(d) and B3LYP/6-311+G(3df,2p)//B3LYp/6-31G(d), respectively, including the ring walk of the substituent by a 1,2-shift around the aromatic ring. This ring walk corresponds to interconversion of distonic ions and norcaradiene radical cations (the two intermediates of the TOL/CHT-rearrangement) by making and breaking of the external C-C bonds of the cyclopropane moiety of the intermediate norcaradiene structure. For toluene radical cation 1, UHF calculations adequately reproduce earlier results(4) and show, that the ring walk of the CH(3)-substituents requires slightly more energy than formation of the cycloheptatriene radical cation. By the DFT model, the distonic ion, which is formed initially by a 1,2-H shift from CH(3) to the benzene ring, is not stable but the transition state of an interconversion of norcaradiene radical cations along a ring walk of the CH(3) substituent. The activation energy for this ring walk exceeds that for formation of the cycloheptatriene radical cation by c. 30 kJ mol(-1). Thus, isomerization of 1 by a ring walk of the CH(3)-substituent competes with the TOL/CHT-rearrangement likely only for excited 1. The calculation was repeated for the MERPs of a TOL/CHT-rearrangement of para-xylene radical cation 5 and ethylbenzene radical cation 2, yielding basically the same results as for 1. According to the calculation, polar substituents alter significantly the relative energies of the competing routes of isomerization. For benzylcyanide 3 (X = CN), the activation energy for a ring walk of the NC-CH(2)-substituent is distinctly below that of a ring enlargement. For benzyl methyl ether 4 (X = OCH(3)), the distonic intermediate along the UHF-MERP is unusually stable. Further, the 7-methoxy-norcaradiene radical ion is unstable and corresponds to a transition state between isomeric distonic intermediates differing by a 1,2-shift of the side chain. In contrast, the 7-methoxy-norcaradiene radical ion is the only intermediate of the DFT-MERP, and the distonic ion is the transition state for a 1,2-shift of the cyclopropane ring. A ring walk of the CH(3)OCH(2)-substituent is much more favorable than formation of a 7-methoxy-cycloheptatriene radical cation in both MERPs. The findings of the theoretical calculation are substantiated by the mass spectrometric fragmentations of meta- and para-methoxymethylated 1-phenylethanols 8 and 9 and of para-methoxymethyl substituted benzyl ethyl ether 10 and benzyl n-propyl ether 11. Important fragmentation routes of metastable molecular ions of these compounds correspond to elimination of alcohols. Use of deuterated derivatives shows that the elimination occurs by a "false" ortho-effect which requires migration of a ROCH(2)-substituent around the benzene ring. Results of particular interest are obtained for the asymmetric bis-ethers 10 and 11. Here, the MIKE spectra of the molecular ions of deuterated analogs reveal a selective ring walk of the C(2)H(5)OCH(2)- and n-C(3)H(7)OCH(2)-side chain, respectively.     

A growth walk model for estimating the canonical partition function of interacting self-avoiding walk

We have explained in detail why the canonical partition function of interacting self-avoiding walk (ISAW) is exactly equivalent to the configurational average of the weights associated with growth walks, such as the interacting growth walk (IGW), if the average is taken over the entire genealogical tree of the walk. In this context, we have shown that it is not always possible to factor the density of states out of the canonical partition function if the local growth rule is temperature dependent. We have presented Monte Carlo results for IGWs on a diamond lattice in order to demonstrate that the actual set of IGW configurations available for study is temperature dependent even though the weighted averages lead to the expected thermodynamic behavior of ISAW.     

On walks in molecular graphs.

Walks in molecular graphs and their counts for a long time have found applications in theoretical chemistry. These are based on the fact that the (i, j)-entry of the kth power of the adjacency matrix is equal to the number of walks starting at vertex i, ending at vertex j, and having length k. In recent papers (refs 13, 18, 19) the numbers of all walks of length k, called molecular walk counts, mwc(k), and their sum from k = 1 to k = n - 1, called total walk count, twc, were proposed as quantities suitable for QSPR studies and capable of measuring the complexity of organic molecules. We now establish a few general properties of mwc's and twc among which are the linear dependence between the mwc's and linear correlations between the mwc's and twc, the spectral decomposition of mwc's, and various connections between the walk counts and the eigenvalues and eigenvectors of the molecular graph. We also characterize the graphs possessing minimal and maximal walk counts.     

On describing the steady absorption of brownian particles by a restricted random walk

This paper is concerned with idealizing brownian motion as a random walk, using the diffusion equation, and finding the boundary condition at an absorbing surface - all with an eye towards chemical kinetics. Three models of random walk (due to Smoluchowski, Fermi, and Lorentz) are considered, and it is concluded that the lorentzian model is the most appropriate.     

Multicomponent effective medium-correlated random walk theory for the diffusion of fluid mixtures through porous media

Molecular transport in nanoconfined spaces plays a key role in many emerging technologies for gas separation and storage, as well as in nanofluidics. The infiltration of fluid mixtures into the voids of porous frameworks having complex topologies is common place to these technologies, and optimizing their performance entails developing a deeper understanding of how the flow of these mixtures is affected by the morphology of the pore space, particularly its pore size distribution and pore connectivity. Although several techniques have been developed for the estimation of the effective diffusivity characterizing the transport of single fluids through porous materials, this is not the case for fluid mixtures, where the only alternatives rely on a time-consuming solution of the pore network equations or adaptations of the single fluid theories which are useful for a limited type of systems. In this paper, a hybrid multicomponent effective medium-correlated random walk theory for the calculation of the effective transport coefficients matrix of fluid mixtures diffusing through porous materials is developed. The theory is suitable for those systems in which component fluxes at the single pore level can be related to the potential gradients of the different species through linear flux laws and corresponds to a generalization of the classical single fluid effective medium theory for the analysis of random resistor networks. Comparison with simulation of the diffusion of binary CO(2)/H(2)S and ternary CO(2)/H(2)S/C(3)H(8) gas mixtures in membranes modeled as large networks of randomly oriented pores with both continuous and discrete pore size distributions demonstrates the power of the theory, which was tested using the well-known generalized Maxwell-Stefan model for surface diffusion at the single pore level.     

用墨西哥帽小波研究DNA序列的分形特征

结合小波分析和分形理论，采用分数布朗运动(FBM)建立数学模型，研究脱氧 核糖核酸(DNA)序列的自相似性。用DNA walk的方式将DNA序列表达成为一个数字信 号，通过不同小波变换的尺度对应不同特征长度的碱基，选择墨西哥帽小波为母小 波，进行实验考察，结果发现小波系数的图形在许多尺度看上去很相似，大尺度对 应较多的碱基(小波变换尺度为2^7时，对应512个碱基)，能看到概貌；小尺度对应 较少的碱基(小波变换尺度为2^3时，对应32个碱基)，可看到细节。这表明其DNA序 列中存在分形结构，可以用分维数来作为定量描述。这种算法为进一步研究与基因 序列自相似结构有关的基因进化信息提供一种选择的途径。     

Walking backward: walk counts of negative order

A closed formula is derived for walk counts of negative order k in a graph or molecule, as defined recently by Lukovits and Trinajsti?. Some unexpected observations made by these authors easily follow from this formula. Gratifyingly, the formula is very similar to the one obtained earlier for usual walk counts. Moreover, while for walk counts of k --> + infinity the numerically largest eigenvalue of the adjacency matrix plays an important part, for walk counts of k --> - infinity the numerically smallest eigenvalue plays a corresponding part.     

Growing self avoiding walk trees

The Growing self avoiding walk model (GSAW) was proposed to explain statistical mechanics of the growth process in polymerization. Close examination of the model reveals that it is suited only for addition polymerization where only one monomer unit is added to the growing polymer chain. In this paper we propose a model for step growth or condensation polymerisation, where all of the monomer is converted first to dimer, all dimers are converted to tetramer etc. In the calculation of probabilities of the walker taking a step in a specified direction in the GSAW model the probability is the reciprocal of the number of steps available if the walker looks one step ahead. It is seen that our model for condensation polymerization GSAW can be generalized to the walker looking a finite number of steps ahead (instead of one only) for the purpose of calculation of probabilities. This is explained in detail in the first section of the paper. In the second section of the paper we explain the use of the Depth first search (DFS) algorithm in the calculation of probabilities and moments for the generalized growing self avoiding walk model. In the third section of the paper we report the exact values of the mean square end to end distances and relevant survival probabilities for the model. When GSAW was first proposed there was a controversy over whether it belonged to the same universality class as the True self avoiding walk $\left( \upsilon =\frac{2}{d+1}\right) $ or that of the self avoiding walk $\left( \upsilon =\frac{3}{d+2}\right) $ . At that time it was conclusively shown that the GSAW model cannot belong to the same universality class as the true self avoiding walk model. However it was never conclusively shown that $\upsilon =\frac{3}{d+2}$ for GSAW. In this paper we propose two methods of studying this problem. One method is rigorous analysis of the DFS algorithm. We explain how this algorithm can be used to study restricted random walks with finite memory and self avoiding walks. The second method proposed by us is a detailed analysis of the generalised GSAW model proposed by us. This paper is to be viewed as an introductory paper on a new model that should be of interest to both chemists and mathematicians.     

Formation,growth mechanism and packing sequences of binary alloy cluster anions from laser ablation of mixtures of lead and transition metals

By using laser ablation of the mixtures of a transition metal (M: Ti, Cr, Mn, Fe, Co, Ni, Cu, Zn, Pd, Ag) plus lead, M/Pb binary cluster anions were observed except for Zn, and the number of transition metal atoms contained in the binary clusters is at most 4. This behavior is different from that reported previously for M/Ge binary clusters. The experiments indicate that it is also very difficult to form Al/Pb clusters. The distribution patterns of M/Pb binary alloy cluster anions are remarkably similar to those of pure Pb clusters, consistent with a formation mechanism in which transition metal atoms are sequentially attached to [M(x-1)Pb(y)](-) clusters and thus form [M(x)Pb(y)](-) clusters by a simple condensation process. As the number of transition metal atoms increases, the intensities of binary clusters gradually decrease. It is proposed that [MPb(4)](-) and [MPb(5)](-) cluster anions might be the unit building blocks of M/Pb binary cluster anions, and the layer packing sequences for magic clusters are predicted on this basis. The [M(x)Pb(y)](-) binary clusters containing 13 atoms (x + y = 13; x not equal 0) are proposed to have an icosahedral structure.     

Exciton propagation via quantum walks based on non-Hermitian coin flip operations

We examine the coherent propagation of the one-dimensional Frenkel exciton (correlated electron-hole pair system) based on a model of a quantum walker in multi-dimensional Hilbert space. The walk is governed by a non-Hermitian coin flip operation coupled to a generalized shift process. The dissipative coin flip operation is associated with amplitude leakages at occupied sites, typical of processes which occur when an exciton is transferred along dimer sites in photosynthetic protein complexes. We analyze the characteristics probability distribution of the one-dimensional quantum walk for various system parameters, and examine the complex interplay between non-Markovian signatures and amplitude leakages within the Hilbert position subspace. The visibility of topological defects such as exceptional points, and non-Markovian signatures via quantum tomography based spectroscopic measurements is discussed.     

Excess around a central molecule with application to binary mixtures

It was shown by us (J. Phys. Chem. B, 2006, 110, 12707) that the excess (deficit) of any species i around a central molecule j in a binary mixture is not provided by c(i)G(ij) (where c(i) is the molar concentration of species i in the mixture and G(ij) are the Kirkwood-Buff integrals) as usually considered and that an additional term, involving a volume V(j) which is inaccessible to molecules of species i because of the presence of the central molecule j, must be included. In this paper, the new expression is applied to various binary mixtures and used to establish a simple criterion for preferential solvation in a binary system. First, it is applied to binary Lennard-Jones fluids. The conventional expression for the excess (deficit) in binary mixtures, c(i)G(ij), provides always deficits around any central molecule in such fluids. In contrast, the new expression provides excess for one species and deficit for the other one. In addition, two kinds of binary mixtures involving weak (argon/krypton) and strong (alcohols/water) intermolecular interactions were considered. Again, the conventional expression for the excess (deficit) in a binary mixture, c(i)G(ij), provides always deficits for any central molecule in the argon/krypton mixture, whereas the new expression provides excess for argon (a somewhat smaller molecule) and deficit for krypton. Three alcohol/water binary mixtures (1-propanol/water, tert-butanol/water and methanol/water) with strong intermolecular interactions were considered and compared with the available experimental information regarding the molecular clustering in solutions. We found (for 1-propanol/water and tert-butanol/water) a large excess of alcohols around a central alcohol molecule and a large excess of water around a central water molecule. For both mixtures the maximum of the calculated excess with respect to the concentration corresponds to the maximum in the cluster size found experimentally, and the range of alcohol concentrations in which the calculated excess becomes very small corresponds to the composition range in which no clusters could be identified experimentally.     

On the use of the dual-process Langmuir model for correlating unary equilibria and predicting mixed-gas adsorption equilibria

A new model has been developed for predicting mixed-gas adsorption equilibria from multicomponent gas mixtures based on the dual-process Langmuir (DPL) formulation. It predicts ideal, nonideal, and azeotropic adsorbed solution behavior from a knowledge of only single-component adsorption isotherms and the assertion that each binary pair in the gas mixture correlates in either a perfect positive (PP) or perfect negative (PN) fashion on each of the two Langmuir sites. The strictly PP and strictly PN formulations thus provide a simple means for determining distinct and absolute bounds of the behavior of each binary pair, and the PP or PN behavior can be confirmed by comparing predictions to binary experimental adsorption equilibria or from intuitive knowledge of binary pairwise adsorbate-adsorbent interactions. The extension to ternary and higher-order systems is straightforward on the basis of the pairwise additivity of the binary adsorbent-adsorbate interactions and two rules that logically restrict the combinations of PP and PN behaviors between binary pairs in a multicomponent system. Many ideal and nonideal binary systems and two ternary systems were tested against the DPL model. Each binary adsorbate-adsorbent pair exhibited either PP or PN behavior but nothing in between. This binary information was used successfully to predict ternary adsorption equilibria based on binary pairwise additivity. Overall, predictions from the DPL model were comparable to or significantly better than those from other models in the literature, revealing that its correlative and predictive powers are universally applicable. Because it is loading-explicit, simple to use, and also accurate, the DPL model may be one of the best equilibrium models to use in gas-phase adsorption process simulation.     

Thermodynamic study of the complexation of p-isopropylcalix[6]arene with Cs+ cation in dimethylsulfoxide-acetonitrile binary media

The complexation reactions between the macrocyclic ionophore, p-isopropylcalix[6]arene and Cs+ cation were studied in dimethylsulfoxide-acetonitrile (DMSO-AN) binary non-aqueous solvents at different temperatures using a conductometry method. The conductance data show that the stoichiometry of the (p-isopropylcalix[6]-arene·Cs)+ complex in all binary mixed solvents is 1:1. The stability of the complexes is affected by the composition of the binary solvent media and a non-linear behavior was observed for changes of log K(f) of the complex versus the composition of the binary mixed solvents. The thermodynamic parameters (DH°(c) and DS°(c)) for formation of (p-isopropyl-calix[6]arene·Cs)+ complex were obtained from temperature dependence of the stability constant and the obtained results show that the (p-isopropylcalix[6]arene·Cs)+ complex is enthalpy destabilized, but entropy stabilized, and the values of the mentioned parameters are affected strongly by the nature and composition of the binary mixed solvents.     

Essential energy space random walk via energy space metadynamics method to accelerate molecular dynamics simulations

To overcome the possible pseudoergodicity problem, molecular dynamic simulation can be accelerated via the realization of an energy space random walk. To achieve this, a biased free energy function (BFEF) needs to be priori obtained. Although the quality of BFEF is essential for sampling efficiency, its generation is usually tedious and nontrivial. In this work, we present an energy space metadynamics algorithm to efficiently and robustly obtain BFEFs. Moreover, in order to deal with the associated diffusion sampling problem caused by the random walk in the total energy space, the idea in the original umbrella sampling method is generalized to be the random walk in the essential energy space, which only includes the energy terms determining the conformation of a region of interest. This essential energy space generalization allows the realization of efficient localized enhanced sampling and also offers the possibility of further sampling efficiency improvement when high frequency energy terms irrelevant to the target events are free of activation. The energy space metadynamics method and its generalization in the essential energy space for the molecular dynamics acceleration are demonstrated in the simulation of a pentanelike system, the blocked alanine dipeptide model, and the leucine model.     

Arsenate uptake and arsenite simultaneous sorption and oxidation by Fe-Mn binary oxides: influence of Mn/Fe ratio, pH, Ca2+, and humic acid

Arsenate retention, arsenite sorption and oxidation on the surfaces of Fe-Mn binary oxides may play an important role in the mobilization and transformation of arsenic, due to the common occurrence of these oxides in the environment. However, no sufficient information on the sorption behaviors of arsenic on Fe-Mn binary oxides is available. This study investigated the influences of Mn/Fe molar ratio, solution pH, coexisting calcium ions, and humic acids have on arsenic sorption by Fe-Mn binary oxides. To create Fe-Mn binary oxides, simultaneous oxidation and co-precipitation methods were employed. The Fe-Mn binary oxides exhibited a porous crystalline structure similar to 2-line ferrihydrite at Mn/Fe ratios 1:3 and below, whereas exhibited similar structures to δ-MnO(2) at higher ratios. The As(V) sorption maximum was observed at a Mn/Fe ratio of 1:6, but As(III) uptake maximum was at Mn/Fe ratio 1:3. However, As(III) adsorption capacity was much higher than that of As(V) at each Mn/Fe ratio. As(V) sorption was found to decrease with increasing pH, while As(III) sorption edge was different, depending on the content of MnO(2) in the binary oxides. The presence of Ca(2+) enhanced the As(V) uptake under alkaline pH, but did not significantly influence the As(III) sorption by 1:9 Fe-Mn binary oxide; whereas the presence of humic acid slightly reduced both As(V) and As(III) uptake. These results indicate that As(III) is more easily immobilized than As(V) in the environment, where Fe-Mn binary oxides are available as sorbents and they represent attractive adsorbents for both As(V) and As(III) removal from water and groundwater.