Statistics/statistical weight matrix

In the RIS model for PE described by Abe et al. [143], the conformational partition function, Z, of the unperturbed chain with n bonds is formulated using a well-known statistical weight matrix for bond i, Uj, that is usually written in a 3 x 3 form. 

A perfectly analogous procedure can be performed for the RIS model on the 2nnd lattice, using an expression for Z developed with the statistical weight matrix in Eq. (4.4) or (4.5) [153]. The replacement for Eq. (4.6) becomes ... 

Collect the statistical weights into statistical weight matrices (of dimension Vj.j X v , where v,- is the number of rotational isomeric states of bond i), one per skeletal bond subject to conformational change, indexing rows and columns of the matrices with the RIS of the bond. The generic statistical weight matrix for bond i is termed U,-. 

M Values for the interdiad statistical weight matrix, Up, in which all first-order interactions are included, are given in the original paper. Table VI. 

For CH2-CH2-CH2 (ethylene) bond pairs, the statistical weight matrix is designated Ue. 

The statistical weight matrix for bond i in branch j is denoted by jUj. Matrices are given as... 

For molecules limited to one helical sequence, the basic statistical weight matrix used is given by ... 

In order to calculate the partition function and molecular averages, a 4x4 statistical weight matrix for the fth residue is formulated to correlate the states of residues i-1, i, and / + 1 of the polymer chain ... 

The statistical weight matrix U used for an amino acid residue is given by U... 

Application of matrix methods to partially helical homopolypeptides is most easily achieved by using a 2 x 2 statistical weight matrix A. 

Matrix schemes which assign end effects to each end of a helical segment are also proposed. A 3 x3 statistical weight matrix B using this weighting scheme is used here. Row index states of amino acid residues i-1 and /, while columns index states of amino acid residues i and /+ 1. 

Helix-probability profiles are calculated using the method of Zlmm and Bragg. A 3 x3 statistical weight matrix is formulated ... 

Computing the partition function for an /V-statc chain requires enumerating all possible states. The clever trick associated with the helix-coil transition theory is to generalize this calculation using the statistical-weight matrix ... 

Flory has adopted the use of a matrix to represent the dependency between rotational states of neighboring bonds. For the typical case of a polymer with three rotational states such as trans (t), gauche plus (g+), and gauche minus (g ), the statistical weight matrix is... 

In the Monte Carlo method, the bonds are assigned to the different rotational states according to their conditional probabilities. These are calculated, using the eigenvalue method outlined by Flory (1969), from the corresponding statistical weight matrix ... 

Let the polymer chain be capable of assuming two different conformational states, coil and helix, the latter capable of binding iodine as I or I. Thus an amylose molecule may be described as a sequence of coil and helix sites, which for present purposes can each be taken to comprise approximately six glucose residues. Coil states can be denoted by c, unbound helix states by h, and helix states bound either to I or I by b. We define "complex" as any uninterrupted sequence of b states, regardless of length. Then in the usual fashion (3 ) we define the statistical weight matrix U indexed for sites i-1 and i as... 

Calculation of the Chains Unperturbed Dimensions. Inspection of the conformational energy maps (Figures 2 and 3) reveals that adoption of the familiar three-state (i.e., T, G", G") scheme is sufficient for application of the RIS theory to these chains.The corresponding statistical weight matrix U, inclusive of both first-order and second-order interactions (those interactions depending on, respectively, one and two skeletal-bond rotations), is represented by ... 

This equation was derived previously for the case of 1,2 dichloroethylene (equation 2.17). The possible permutations for neighboring bonds with a threefold potential barrier can be represented by a statistical weight matrix composed of elements based on individual statistical weights. The matrix methods involved are described in Appendix 2E. The reference bond angle Q, is placed in a right-handed coordinate system. Transformation matrices A[ (see for example equation 2C.25) are then used to transform the coordinates of the Ql bond into that of the (2, one. The process is repeated until the coordinates of all the bonds in a given conformation are generated. 

For linear polyethylene, the first order interaction can be given by a diagonal statistical weight matrix... 

Two matrices that describe conformations that alternate in sequence are used to specify conformation in tactic vinyl polymers. Tacticity introduces asymmetric elements into the chain. One convention used in specifying chain asymmetry consists of distinguishing between right handed chains termed d and left handed or 1 chains. This convention, admittedly arbitrary, is borrowed from the field of optical activity. For a tactic polymer with a threefold potential (Figure 2.13), the statistical weight matrix for a d placement between neighboring bonds is... 

As an example, the statistical weight matrix for polyethylene with its three-fold rotational states (trans, gauche plus, gauche minus) is... 

The statistical weight matrix is raised to the n-2 power thereby taking into account all the internal bonds... 

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