Moments and distribution functions for polymer chains of finite length. II. Polymethylene chains

Abstract

Statistical mechanical averages of vectors and tensors characterizing the configurations of polymethylene chains with n = 5-200 bonds have been calculated, these quantities being expressed in the internal reference frame attached to the first two bonds. Specifically, the persistence vector a == <r) and the averaged center-of-mass vector <g) have been obtained as functions of chain length.They are compared to the corresponding vectors for various model chains. Configurational averages of the Cartesian tensors formed from p = r -a have been computed up to and including the tensor of seventh rank. Density distribution functions Wa (p), evaluated from these moment tensors using the three-dimensional Hermite series elaborated in the preceding paper, are approximately cyclindrically symmetric about a principal axis that lies close to the direction of a. The second moment (and, hence, the dispersion) along this axis is smaller than the moments perpendicular thereto. Acentricity and skewness of the distribution Wa (p) is large for n < 40 bonds; they are appreciable even at n = 100 bonds. For n :;:: 50, moments of the normalized density distribution can be represented in good approximation by a model chain consisting of freely jointed segments originating at the terminus of a, with each segment taken to be equivalent to 20 bonds of the real chain. The traditional approximation of the unnormalized density distribution about r = 0 by a freely jointed equivalent chain is subject to much greater error. The second moment tens9r of gyration, averaged over all configurations, is approximately cylindrically symmetric, but its component (and its dispersion) along the symmetry axis is greater than in the transverse directions. This tensor is contrasted with that obtained by summing moments along the principal axes for individual configurations.the rotational states being indexed in the order t, g+, g-. Although the gauche minima doubtless occur at rotations 5366

Sequence-Dependent Effects in the Cyclization of Short DNA

“…In one approach the configurations of unconstrained linear molecules which meet preassigned geometric criteria, e.g., the distance and relative orientation of chain ends, are collected through exhaustive simulation studies. 1 This method has been used extensively in analyses of the kinetics of chain cyclization, i.e., ring closure, [2][3][4][5][6] and the formation of closed loops. [7][8][9] An alternative approach to the ring-closure problem is to derive an analytical expression for the configurational partition function of a simplified polymer, such as a freely jointed (Gaussian) chain or a freely rotating (wormlike) chain, that satisfies the requisite end conditions.…”

Relationships between unfolded configurations of proteins and dynamics of folding to the native state

“…The persistence length of the RG chains, being 6.1 Å , is approximately equal to that of polyethylene chains. 47 The persistence length of the CG chains, on the other hand, is 15.1 Å , which is significantly higher than that of the random chains. The value obtained for the CG chains is somewhat smaller than the persistence length for polyalanine calculated as 22.5 Å by Conrad and Flory.…”

“…The persistence vector reflects the anisotropy of chain dimensions when the first two bonds are held fixed in the x-y plane. 36,[45][46][47] In Figure 3, we compare the different components of the persistence vector for RG and CG. Calculations are performed as follows: the first bond of each chain (the bond between the first and second atoms, N and C a atoms) is fixed along the x-axis of a Cartesian coordinate system O xyz .…”

“…12 The hydrophobic core is formed by packing a-helix to the bstrand 2. In the native state, the residues in contact at the hydrophobic core are 5,8,16,19,20,27,29,47,49,51,57, and 61. The coordinates for the atoms in the first 18 residues are missing in the PDB file; therefore, we renumbered the residues in our work starting from 1 at the residue number 19 in the original PDB file.…”

Molecular structure, conformation and properties of macromolecules

“…Results for polymethylene (PM) chains of varying length n are shown by the filled circles in Fig. 3, taken from the work of Yoon in collaboration with the author (11). The x-component <x> of a is plotted on the abscissa and the y-component <y> on the ordinate for chains having the numbers n of bonds indicated with the points.…”