Junction point vector

Figure 1 Representation of an element in terms of its junction point vector length and orientation.

In the strained state, the junction point vector deforms to a length and new orientation angles 0 and cp, Choosing the strained state orientation angles as the independent variables, the elemental strain is characterized by three quantities, the extensional deformation function... 

Noting from Figure 1 that d is a tangential displacement of the junction point vector, a tangential force is defined by... 

The force given by eq. 24 acts tangent to the junction point vector, and differs from the classical value 3kTt/i by a factor This arises from the definition... 

When a macromolecular chain is considered as the constituent element of a network, there is some reason to doubt the coincidence of the junction point vector and the axis of time averaged conformations. The deformation of the network by any deviatoric stress field always results in a net reorientation of elements toward the principal axes and the environment of an element must reflect, on the average, this orientation gradient perpendicular to the junction point vector. Those elements that are oriented along the principal axes of deformation are exceptions to this effect because of the absence of reorientation as discussed in the previous section. 

Figure 4 Illustration of asyinmetric conformations In the rest state the symmetry axis and junction point vector coincide, whereas in the strained state the conformations are biased in the direction of elemental orientation.

One of the earlier assumptions regarding microscopic deformation in networks is that the junction points in the networks move affinely (linearly) with macroscopic deformation. It follows that chain end-to-end vectors deform affinely also, and... 

Consider a linear chain consisting of m + jV monomers. The ends of this chain are fixed in space. The x coordinates of the ends are Xf and X2, while the junction point fluctuates with x coordinate R. What is the mean-square x coordinate of the end-to-end vector of the section containing N monomers ((R - Z j) ) ... 

Assume that each network strand has A monomers. One network strand, shown in Fig. 7.2, has end-to-end vector Rq with projections along the x, y, and z directions of R o, RyQ, and R in the undeformed state. In the affine network model, the positions of the junction points (the ends of the strands) are always fixed at particular points in space by the deformation and not allowed to fluctuate. For affine deformation, the end-to-end vector of the same chain in the deformed state is R (see Fig. 7.2) with projections along the x, y, and z directions of... 

In the affine model of the network it is assumed all junction points are imbedded in the network, and each Cartesian component of the chain end-to-end vector transforms linearly with macroscopic deformation... 

The state of the ideal rubber can be specified by the locations of all the junction points, ij, and by fce end-to-end vectors for all tire chains connecting the junction points,. The first postulate of the statistical theory of rubber elasticity is that, in the rest state with no external constraints, the distribution fimction for the set of chain end-to-end vectors is a Gaussian distribution witii a mean-squared end-to-end distance that is proportional to the molecular weight of the chains between jimcnons ... 

A key assumption of the single molecular theory is that the junction points in the network move affinely with the macroscopic deformation that is, they remain fixed in the macroscopic body. It was soon proposed by James and Guth [9] that this assumption is unnecessarily restrictive. It was considered adequate to assume that the network junction points fluctuate around their most probable positions [9,10] and the chains are portrayed as being able to transect each other. This has been termed the phantom network model. The vector r joining the two junction points is considered as the sum of a time average mean r and the instantaneous fluctuation Ar from the mean so that... 

Equivalent equations hold for the distribution functions of the internal distance vectors rij between two junction points in the chain, which follow as... 

The extension ratios Ai,A2 and A3 also determine the shifts in the locations of all junction points and therefore the changes of the end-to-end distance vectors of all chains. More specifically, the end-to-end distance vector of chain i in the unstrained state... 

The system to be considered consists of a three-dimensional array of elements (the macromolecules) that are joined together by discrete crosslinks, or junction points, to form a network. The deformation of an element is described conveniently in terms of the vector that connects its two ends. Referring to Figure 1, the junction... 

It is noteworthy that there is another limiting factor in the choice of amino acid types at the junction sites which affect the enzymatic process of the intein. For example, in the case of SceVMA (also called PI-Seel) from the IMPACT system, proline, cysteine, asparagine, aspartic acid, and arginine cannot be at the C-terminus of the N-terminal target protein just before the intein sequence. The presence of these residues at this position would either slow down the N-S acyl shift dramatically or lead to immediate hydrolysis of the product from the N-S acyl shift [66]. The compatibility of amino acid types at the proximal sites depends on the specific inteins and needs to be carefully considered during the design of the required expression vectors. The specific amino acid requirements at a particular splicing site depends on the specific intein used and is thus a crucial point in this approach. 

Breaking of a single diagram line into two or more vectors at points away from a junction... 

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