Chain dimensions and fluctuations in elastomeric networks in which the junctions alternate regularly in their functionality

Abstract

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.

The importance of slow motions for protein functional loops

“…The Gaussian Network Model ( GNM ) is one of the simplest of elastic network models, originally applied to protein dynamics by Bahar et al (31) and Haliloglu et al (31,32) who applied the approach of Tirion et al (33) in a coarse-grained way to both bonded and non-bonded contacts in proteins and represent their interactions with a single universal spring parameter. This model has its deep origins in the rubber like elasticity theory of Flory, James and Guth, James and Guth, Kloczkowski et al , Skliros et al (34,35,36,37,38). Each normal mode corresponds to a different frequency of oscillation.…”

Small‐Angle Neutron Scattering from Elastomeric Networks in which the Junctions Alternate Regularly in their Functionality

“…In some earlier work, [17] we computed mean-square fluctuations of junctions and points along the chains, and correlations between instantaneous fluctuations of two points or junctions, even those separated by other intermediate junctions. In this study, we used these results to compute small-angle neutron scattering from a labeled (deuterated) path containing cross-links in such networks.…”

“…All these quantities were recently computed by us for phantom Gaussian network with alternating functionally. [17] Since the computations leading to these formulas are tedious and lengthy; here we use only the final results; details are provided elsewhere. [17] The most general solution for the mean-square fluctuations Dr ij À Á 2 D E 0 of the distance between two points i and j separated by several multifunctional junctions with alternating functionality depends on the functionality of the junctions that are the closest to the left for points i and j.…”

“…[17] Since the computations leading to these formulas are tedious and lengthy; here we use only the final results; details are provided elsewhere. [17] The most general solution for the mean-square fluctuations Dr ij À Á 2 D E 0 of the distance between two points i and j separated by several multifunctional junctions with alternating functionality depends on the functionality of the junctions that are the closest to the left for points i and j. There are four possible cases: i) both of these junctions have functionality f 1 , ii) the junction closest to the left for point i has functionality f 1 and closest to the left for point j has functionality f 2 , iii) both of them have functionality f 2 , and, iv) the junction closest to the left for point i has functionality f 2 and closest to the left for point j has functionality f 1 .…”

“…The calculations of Kloczkowski et al on chain dimensions and fluctuations in phantom Gaussian networks were later extended to deformed networks, [15] bimodal networks (networks composed of chains of two different lengths), [16] and, most recently, to networks with alternating functionally (a network in which one chain end has functionality f 1 , and another end has functionality f 2 ). [17] In the case of bimodal networks, SANS form factors were also computed. [16] An overview of most of these results is provided elsewhere.…”