Statistical Weight Matrix Research Articles

Improving quantitative precipitation estimates by radar-rain gauge merging and an integration algorithm in the Yishu River catchment, China

High-precision areal rainfall is crucial for hydrometeorological coupled forecasts. The accuracy of quantitative precipitation estimates (QPE) is improved by merging radar-rain gauge data with an integration approach based on a statistical weight matrix in the Yishu River catchment, China. First, a local Z-R relationship (Z = 85R1.82) is reconstructed using a genetic optimization algorithm to minimize the error from different precipitation patterns and climate zones. Next, based on the local Z-R relationship, six methods of merging radar-rain gauge data are respectively adapted to improve the accuracy of QPE, as follows: mean field bias (MFB), Kalman filter (KLM), optimum interpolation (OPT), variation method (VAR), two-step calibration of KLM and OPT (KOP), and two-step calibration of KLM and VAR (KVR). The results indicate that QPE accuracy is clearly improved, and is in good agreement with rain gauge observations, after the six merging methods are applied. Among these methods, KOP performs the best, reducing the mean relative error from 55.2 to 15.1%. An innovative aspect of this work is the inclusion of an integrated ideology based on a statistical weight matrix, which further improves the accuracy of QPE by incorporating the advantages of each estimation mode. The results further show that the accuracy of QPE derived from the integration approach is higher than that obtained by any individual method; QPE values are similar to those obtained the automatic rain gauge network in both the spatial distribution and location of the intense precipitation centers, and better reflects the precipitation status over the ground surface. This approach could serve as a promising conventional method for QPE in the study region.

Effect of the Pendant Ester Group Orientations on the Dipole Moment of Poly(alkyl acrylate)s

The dipole moment analysis of vinyl polymers carrying polar side groups such as poly(methyl acrylate) (PMA), poly(methyl methacrylate) (PMMA), and poly(ethyl methyacrylate) (PEMA) has been performed by the expanded matrix algebra method with due consideration of the ester group orientations. The stereochemical characteristics of the pendant ester groups have been exactly treated by introducing a Bernoullian probability λ representing the occurrence of the cis conformation around the first articulated Cα−C* bond. Special care is required for the orientational correlation of the neighboring ester groups when the skeletal conformations around the two successive bonds are both in trans. Such a meso-tt effect has been accommodated by assigning probability λ‘ or (1 − λ‘) to the proper elements of the statistical weight matrix. Experimental observations were found to be satisfactorily reproduced with λ = 0.3−0.4, indicating that the trans conformation is more preferred for the ester group. The orientational corr...

Estimation of Long-Range Interaction in Coarse-Grained Rotational Isomeric State Polyethylene Chains on a High Coordination Lattice

A new Monte Carlo simulation on a coarse-grained tetrahedral lattice was recently developed for rotational isomeric state polyethylene chains. The short-range interaction was included by extending the classical statistical weight matrix. To describe the cohesive nature of realistic polyethylene systems, nonbonded long-range interaction was considered. The second virial coefficient B2 of two chains under an interparticle potential was suitably written in a discretized form to be analyzed for this new lattice. Utilizing the vanishing B2 of a ϑ chain was previously suggested for the estimation of interaction parameters representing several nearest neighbors. This approach was shown to be sufficient for treatment of dilute solutions, where long-range interactions are relatively infrequent. This study presents an alternative method of parameter estimation which is more appropriate for the bulk state, where long-range interactions are abundant. Interaction parameters were defined from an averaging procedure of ...

Introduction of Short and Long Range Energies To Simulate Real Chains on the 2nnd Lattice

Recently, a new high-coordination lattice model for rotational isomeric state (RIS) chains was introduced, with the motivation of eventually providing a method for the more efficient simulation of dense, multichain systems by using RIS chains. Here that new method has been extended to include short and long range interaction energies, using polyethylene as the example. The short range energies were included via an extended RIS model. The classical statistical weight matrix was expanded in size and coarse-grained. The conditional probabilities derived from this extended RIS model were the basis for the probabilities of the single-bead moves used in Monte Carlo simulations on the 2nnd lattice. The long range energies were introduced by using the lattice representation of the second virial coefficient. The coarse-grained linear chains have the proper mean-square dimensions, and the width of the distribution function for the squared dimensions has the proper value. Coarse-grained polyethylene macrocycles also have the proper mean-square dimensions.

Dimensional Changes of Polypeptides in the Helix-Sense Inversion Region

Dimensional changes of polypeptides in the helix-sense inversion region are analyzed by a theory based on the three-state, nearest-neighbor interaction model. Each peptide residue in a chain is assumed to take on right- and left-handed α-helical states as well as a random coil state, and the helix stabilization effect due to hydrogen-bond interactions is taken into account. Such interactions between four consecutive residues have explicitly been discussed in a 9×9 statistical-weight matrix in a previous theory [Macromolecules, 17, 1599 (1984)], while in the present model we simplify this matrix into the form of a 3×3 matrix by considering the interactions only between two consecutive residues. Calculations of the average helix fraction and the number of helix sequences are correspondingly simplified with maintaining quantitative agreement in numerical results. The new theory is also applied to the analyses of the length distribution and the higher moments of helix. Theoretical transition curves illustrating the general characteristics of the sense inversion phenomena suggest a method for evaluating the transition parameters.

Stabilization of short helices by intramolecular cluster formation

AbstractThe intramolecular formation of multiple clusters of interacting helices has been characterized in a homopolymer. The configuration partition function permits the formation of clusters in which the number of interacting helices may be as large as the greatest integer in n/2, where n denotes the number of amino acid residues in the chain. The theoretical formulation has its origin in a recent [Mattice, W. L. & Scheraga, H. A. (1984) Biopolymers 23, 1701–1724], tractable matrix expression for the configuration partition function for intramolecular antiparallel β‐sheet formation. Reassignment of the expression for one of the n(n+3)/2 elements in the sparse statistical weight matrix, along with a simple change in notation, converts that treatment into a matrix formulation of the configuration partition function for a chain containing multiple clusters of interacting antiparallel helices. The five statistical weights used are δ, fl, w, and the Zimm‐Bragg σ and s. Each tight bend that connects two interacting helices contributes a factor of δ, fl is used in the weight for larger loops between interacting helices, and w arises from helix–helix interaction. The influence of the helix–helix interaction is well illustrated by two helix–coil transitions in a chain with n = 156 and σ = 0.001. In the absence of helix–helix interaction, the transition occurs by the nucleation and subsequent elongation of a small number of helices. When helix–helix interaction is attractive, the transition can occur by a different mechanism. Formation of a single pair of interacting helices is followed by addition of new helices to the initial cluster. In the latter process, individual helices experience relatively little growth after they are formed.

The Interaction of a Neutral Polymer with Small Ions in Solution. I. A Method for the Analysis of Ion Binding to a Neutral Polymer

Abstract A method for the analysis of ion binding to a neutral polymer, such as poly(oxyethylene) (POE), based on the one-dimensional lattice model has been proposed. The grand partition function for the one-dimensional lattice has been given by the product of the statistical weight matrix. Interaction between the ions bound to the lattice points has been well approximated by the Debye-Hückel pair-by-pair potential, and the distance between the bound ions has been assumed to be the root mean square end-to-end distance of the partial chain. Based on the model, an equilibrium constant has been defined which is referred to the binding constant.

Electrostatic interactions in ionic homopolypeptides in solutions of moderate ionic strength

AbstractAt sufficiently high ionic strength, long‐range electrostatic interactions in a polyelectrolyte such as poly(L‐glutamic acid) might be adequately approximated in matrix calculations by use of statistical weights representing second‐order interactions. The validity of this assumption has been investigated making use of experimental observations (CD spectra and titration curves) for poly(L‐glutamic acid) as a function of temperature in 0.1–0.5M sodium chloride. Theoretical analysis, using a statistical weight matrix proposed by Warashina and Ikegami, is based on the Zimm‐Rice theory. Implementation differs from that of Warashina and Ikegami in one respect. Refinement of the initial estimates is achieved using a form of the configuration partition function which does not assume diagonalization of the statistical weight matrix. This difference is of no consequence for the values of σ and s, but it does produce somewhat different values for the statistical weights used to represent the electrostatic interactions. The method used to treat electrostatic interactions in poly(L‐glutamic acid) in 0.1M sodium chloride can be viewed as successful in that it properly reproduces the helix–coil transition and titration curves in this solvent and the molecular‐weight dependence of the titration curves yields values for s in harmony with those obtained using a treatment which is independent of model, and gives a reasonable ionic‐strength dependence for the electrostatic parameters. Furthermore, the model can account for measured helix–coil transitions and titration curves in homopolypeptides in which the side chain is —(CH2)xNHCO(CH2)yCOOH. The model, however, is not exact. It does not properly account for the molecular‐weight dependence of the helical content for polymers of low degree of polymerization.

Pressure dependence of polymer chain conformational parameters

Configurational statistics of chain molecules relates the average spatial configuration and the other configuration-dependent properties of a chain to the relative free energy AF of rotational isomers for each rotatable skeletal bond. The statistical weights uij = exp[ Mij/(RT)] are assigned to the pairs of conformations about adjoining bonds and are arranged as elements of statistical weight matrix’). The comparison of calculated and experimental values of configuration-dependent properties is the usual way for the assignment of statistical weights uij . Alternatively, uij can be determined from the conformational energy calculations for the compounds. Customarily, only the dependence of uij on temperature is assumed in conformational studies. For low molecular weight compounds there are many examples of the striking environmental (especially solvent) influence on conformational equilibria2). In configurational statistics this effect can be accounted for by introducing the solventdependent statistical weights uij . For example, the solvent effect on mean chain dimensions is simulated by a variation of some uij values’). It is common belief that the environmental effect on torsional potentials about single bonds and on the associated conformational free energies AF is important in polar systems only. However, there are also indications of the importance of the environmental effect on the conformational equilibria for nonpolar e.g. hydrocarbon molecule^^^^). One of the possible reasons for it is the modification of torsional potential in the condensed phase due to the different volumes of trans and gauche conformers’). In other words, the statistical weights for hydrocarbon chain molecules in the condensed phase are affected by changes of internal or external pressure or by changes of sample volume. In this paper the pressure variation of statistical weights of the polymethylene chain is estimated as a representative case of non-polar polymers.

Statistical mechanical treatment of protein conformation. 5. A multistate model for specific-sequence copolymers of amino acids.

One-dimensional short-range interaction models for specific-sequence copolymers of amino acids have been developed in this series of papers. In the present paper, a multistate model (involving right-handed helical (hR), extended (epsilon), chain-reversal (R and S), left-handed helical (hL), right-handed bridge-region (zota R), left-handed bridge-region (zota L), and coil (or other) (c) states) is developed for the prediction of protein backbone conformation. This model involves ten parameters (WhR, UPSILONHR, V epsilon, VR, VS, WhL, VhL, U zota R, U zota L, and Uc) and requires a 10X10 statistical weight matrix. Assuming that the left-handed helical sequence cannot occur in proteins, this 10X10 matrix can be reduced to a 9X9 matrix with nine parameters (WhR, VhR, V epsilon, VR, VS, VhL, U zota R, U zota L, and Uc). A nearest neighbor approximation of this multistate model is also formulated; with the omission of left-handed helical sequences, and the inclusion of the left-handed bridge region in the c state, this approximate model requires a 7X7 matrix with statistical weights WhR, VhR, VS, VhL, U zota R, and Uc, expressed as values relative to the statistical weight of the epsilon state. The statistical weights for the multistate model are evaluated from the atomic coordinates of the X-ray structures of 26 native proteins. These statistical weights and the multistate model are applied in the prediction of the backbone conformations of proteins. The conformational probabilities of finding a residue in hR, epsilon, R, S, hL, zota R, or c states, defined as relative values with respect to their average values over the whole molecule, are calculated for bovine pancreatic trypsin inhibitor and clostridial flavodoxin, in order to select the most probable conformation for each residue of these proteins. The predicted results are compared to experimental observations and are discussed together with the reliability of the statistical weights. In the Appendix, the property of asymmetric nucleation of helical sequences is introduced into the (nearest neighbor) multistate model.

Statistical mechanical treatment of protein conformation. 4. A four-state model for specific-sequence copolymers of amino acids.

One-dimensional short-range interaction models for specific-sequence copolymers of amino acids are being developed in this series of papers. In this paper, our earlier three-state model [involving helical (h), extended (epsilon), and coil (or other) (c) states] is extended to a four-state model by preserving the h and epsilon states, introducing the chain-reversal state (R and S), and redefining the c state. This model involves six parameters (wh, vh, vepsilon, vR, vS, and uc) and requires a 6 X 6 statistical weight matrix. A nearest-neighbor approximation of the four-state model is also formulated; it requires a 5 X 5 matrix, involving the same six parameters. By expressing the statistical weights relative to that of the epsilon state, only five parameters (wh, vh, vR, vS, and uc) are required in both the 6 X 6 and 5 X 5 matrices. The statistical weights for the four-state model are evaluated from the atomic coordinates of the x-ray structures of 26 native proteins. These statistical weights, and the four-state model, are used to develop a procedure to predict the backbone conformations of proteins. Since the prediction of helical and extended conformations is carried out by the procedure described in papers 1-3 of this series, we focus particular attention on chain-reversal conformations in this paper. The conformational-sequence probabilities of finding a residue in h, epsilon, R, S, or c states, and of finding two consecutive residues in a chain-reversal conformation, defined as relative values with respect to their average values over the whole molecule, are calculated for 23 proteins. By comparing these conformational-sequence probabilities to experimental X-ray observations, it was found that, in addition to the prediction of helical and extended conformations (reported in paper 3), 219 chain-reversal regions out of 372 observed by x-ray diffraction studies of 23 proteins were predicted correctly. These results suggest that the assumption of the dominance of short-range interactions in determining chain-reversal (as well as helical or extended) conformations in proteins, on which the predictive scheme is based, is a reasonable one. Finally, in the Appendix, the property of asymmetric nucleation of helical sequences is introduced into the (nearest-neighbor) four-state model.

On the reduction of certain matrix expressions in the statistical theory of chain molecule configurations

It is shown that certain quantities related to the configuration of a chain molecule of finite length and made up of repeating units of a single type can be expressed in terms of the proper values of the statistical weight matrix for individual bond conformations. Among these quantities are the configurational energy and the mean fraction of bonds in a specified conformations. Results for symmetric chains with threefold potentials are given.

A Model for the Helix-Coil Transition in Specific-Sequence Copolymers of Amino Acids

A Model for the Helix-Coil Transition in Specific-Sequence Copolymers of Amino Acids

Configurational Averages for Chain Molecules: Higher Moments and Related Quantitites

The higher even moments 〈r2p〉0 of the end-to-end vector r of a chain molecule are formulated according to a straightforward procedure which is free of complications for large values of p. It is generally applicable also to averages of other configuration-dependent quantities, and simplifies their treatment. The condensation of self-direct matrix products derived by Nagai for reducing the complexity of computation of higher moments is achieved directly through an appropriate transformation, which is defined for such a product of any degree. Major reductions of the orders of the statistical weight matrix and associated expressions permissible for symmetric chains are achieved succinctly by similar matrix transformations. It is pointed out that further condensations can be realized through deletion of superfluous rows and columns from the generator matrices (G, G⊗G, etc.) for the moments.