Loops in proteins connect secondary structures such as alpha-helix and beta-sheet, are often on the surface, and may play a critical role in some functions of a protein. The mobility of loops is central for the motional freedom and flexibility requirements of active-site loops and may play a critical role for some functions. The structures and behaviors of loops have not been much studied in the context of the whole structure and its overall motions, and especially how these might be coupled. Here we investigate loop motions by using coarse-grained structures (Cα atoms only) to solve for the motions of the system by applying Lagrange equations with elastic network models to learn about which loops move in an independent fashion and which move in coordination with domain motions, faster and slower, respectively. The normal modes of the system are calculated using eigen-decomposition of the stiffness matrix. The contribution of individual modes and groups of modes are investigated for their effects on all residues in each loop by using Fourier analyses. Our results indicate overall that the motions of functional sets of loops behave in similar ways as the whole structure. But, overall only a relatively few loops move in coordination with the dominant slow modes of motion, and that these are often closely related to function.
A matrix method is used to determine fluctuations of junctions and points along the polymer chains making up a phantom Gaussian network that has the topology of an infinite, symmetrically grown tree. The functionalities of the junctions alternates between 1 and 2 , such that one end of each network chain has functionality 1 , while the opposite end has functionality 2 . Quantities calculated include fluctuations of 1 -functional and 2 -functional junctions, and fluctuations of points along network chains, as well as correlations of these fluctuations. This was done for points and junctions along any path in the network, where these points and junctions were separated by no junctions or several junctions, Fluctuations have also been calculated for the distances between points and junctions. The present results represent significant generalizations of earlier work in this area ͓Kloczkowski et al., Macromolecules 22, 1423 ͑1989͔͒. These generalizations and extensions should be very useful in a number of contexts, such as interpreting small-angle neutron scattering results on labeled paths in polymer networks, or fluctuations of loops in the Gaussian network model of proteins.
This paper presents a new algorithm for generating the conformational statistics of lattice polymer models. The inputs to the algorithm are the distributions of poses (positions and orientations) of reference frames attached to sequentially proximal bonds in the chain as it undergoes all possible torsional motions in the lattice. If z denotes the number of discrete torsional motions allowable around each of the n bonds, our method generates the probability distribution in end-to-end pose corresponding to all of the z n independent lattice conformations in O(n D+1 ) arithmetic operations for lattices in D-dimensional space. This is achieved by dividing the chain into short segments and performing multiple generalized convolutions of the pose distribution functions for each segment. The convolution is performed with respect to the crystallographic space group for the lattice on which the chain is defined. The formulation is modified to include the effects of obstacles (excluded volumes), and to calculate the frequency of the occurrence of each conformation when the effects of pairwise conformational energy are included. In the latter case (which is for 3 dimensional lattices only) the computational cost is O(z 4 n 4 ). This polynomial complexity is a vast improvement over the O(z n ) exponential complexity associated with the brute force enumeration of all conformations. The distribution of end-to-end distances and average radius of gyration are calculated easily once the pose distribution for the full chain is found. The method is demonstrated with square, hexagonal, cubic and tetrahedral lattices.
We compute scattering form factors for SANS from labeled paths in Gaussian phantom networks in which junctions alternate regularly in their functionality (the number of chains emanating from a junction). Our calculations are based on the James‐Guth model of rubber‐like elasticity, which assumes that fluctuations are strain independent, while mean vectors transform affinely with the applied strain. Kratky plots for scattering from isotropic and uniaxially stretched bifunctional networks are computed and compared with corresponding plots for the simpler unifunctional networks. The results show the effects of the length of the labeled path, extent of deformation, direction of scattering with respect to the principal axis of the deformation and the functionalities of the network junctions.magnified image
This paper presents a new approach to study the statistics of lattice random walks in the presence of obstacles and local self-avoidance constraints (excluded volume). By excluding sequentially local interactions within a window that slides along the chain, we obtain an upper bound on the number of self-avoiding walks (SAWs) that terminate at each possible position and orientation. Furthermore we develop a technique to include the effects of obstacles. Thus our model is a more realistic approximation of a polymer chain than that of a simple lattice random walk, and it is more computationally tractable than enumeration of obstacle-avoiding SAWs. Our approach is based on the method of the lattice-motion-group convolution. We develop these techniques theoretically and present numerical results for 2-D and 3-D lattices (square, hexagonal, cubic and tetrahedral/ diamond). We present numerical results that show how the connectivity constant μ changes with the length of each self-avoiding window and the total length of the chain. Quantities such as 〈R〉 and others such as the probability of ring closure are calculated and compared with results obtained in the literature for the simple random walk case.
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