Affine model

IR dichroism has also been particularly helpful in this regard. Of predominant interest is the orientation factor S=( 1/2)(3—1) (see Chapter 8), which can be obtained experimentally from the ratio of absorbances of a chosen peak parallel and perpendicular to the direction in which an elastomer is stretched [5,249]. One representation of such results is the effect of network chain length on the reduced orientation factor [S]=S/(72—2 1), where X is the elongation. A comparison is made among typical theoretical results in which the affine model assumes the chain dimensions to change linearly with the imposed macroscopic strain, and the phantom model allows for junction fluctuations that make the relationship nonlinear. The experimental results were found to be close to the phantom relationship. Combined techniques, such as Fourier-transform infrared (FTIR) spectroscopy combined with rheometry (see Chapter 8), are also of increasing interest [250]. 

Note that X =X if a=o, which represents no change from the affine models, and X =1 where network unfolding takes place upon deformation without chain extension. If a is non-zero, network unfolding can be introduced into the previous equations by subr-stitutlng X for X in Eqs. 18, 19 or 20. The most general result of the models discussed above is obtained by replacing Xz by Xz in Eq. 20. Appropriate choices of k and a lead to the results of Eq. 18 or 19 as well. 

In this review, we have given our attention to Gaussian network theories by which chain deformation and elastic forces can be related to macroscopic deformation directly. The results depend on crosslink junction fluctuations. In these models, chain deformation is greatest when crosslinks do not move and least in the phantom network model where junction fluctuations are largest. Much of the experimental data is consistent with these theories, but in some cases, (19,20) chain deformation is less than any of the above predictions. The recognition that a rearrangement of network junctions can take place in which chain extension is less than calculated from an affine model provides an explanation for some of these experiments, but leaves many questions unanswered. 

Summary Statistics for Predictive Estrogen Receptor Binding Affinity Models... 

We now consider some applications of the VSA descriptors to receptor affinity modeling. Figure 7 depicts a typical structure of a series of 72 compounds each of which has been assayed against each of thrombin, trypsin, and factor Xa (27). The PEOE VSA, SlogP VSA, and SMR VSA descriptors were calculated... 

Salzberg, A.C. and Huang, E.S. (2007) Structure-based maximal affinity model predicts small-molecule druggability. Nature Biotechnology, 25, 71-75. 

In the affine model of network deformation, the cross-links are viewed as firmly embedded in the elastomeric matrix, and thus as moving linearly with the imposed macroscopic strain [1-4, 20]. In the alternative phantom model, the chains are treated as having zero cross-sectional areas, with the ability to move through one another as phantoms [2-4]. The cross-links in this model undergo consider-... 

The simplest model is the local linear or affine model... 

Cheng, A. C., Coleman, R. G, Smyth, K. T., Cao, Q., Soulard, P., Caffrey, D. R., Salzberg, A. C., and Huang, E. S. (2007). Structure-based maximal affinity model predicts small-molecule druggability. Nature Biotechnology 25, 71-75. 

A treatment of the classic phantom network model is contained in Ref. [6], page 252-256. However, the statement on p. 256 that this model leads to the same stress-strain relation as the affine model is incorrect because it neglects the effect of junction fluctuations on the predicted shear modulus. See Graessley [17],... 

To illustrate the usefulness of birefringence measurements in orientation studies, we now briefly discuss two simple models of orientation leading to different expressions of the second moment of the orientation function the affine deformation model for rubbers and the pseudo-affine model more frequently used for semi-crystalline polymers. 

Pseudo-affine model, the deformation process of polymers in cold drawing is very different from that in the rubbery state. Elements of the structure, such as crystallites, may retain their identity during deformation. In this case, a rather simple deformation scheme [12] can be used to calculate the orientation distribution function. The material is assumed to consist of transversely isotropic units whose symmetry axes rotate on stretching in the same way as lines joining pairs of points in the bulk material. The model is similar to the affine model but ignores changes in length of the units that would be required. The second moment of the orientation function is simply shown to be ... 

Fig.2 compares the predictions of the pseudo-affine model and the affine model with different values of the nvunber n of links per chain. It is clearly seen that the pseudo-affine scheme gives a much more rapid initial orientation than the affine network model. 

Work has been done to computationally model transporter structure, and this work will gain in value as additional high-resolution three-dimensional membrane protein structures are solved. Similar to the QSAR studies for P-gp described in this chapter, protein structural and substrate affinity modeling approaches have also been applied to various other transporters (181). The resulting three-dimensional structure-function relationships should be useful to understanding how individual genetic differences in transporter function will affect drug transport. 

Comparison of Eqs. (3.36a) and (3.33) indicates that the value of modulus G obtained from the affine deformation model is two times the value corresponding to the phantom network. This would mean that the latter model is more applicable in the region of moderate deformations and the affine model is more suitable in the region of low deformations. 

Mekenyan O, Kamenska V, Schmieder PK, Ankley G, Bradbury S. A computationally based identification algorithm for estrogen receptor ligand Part 2. Evaluation of hERalpha binding affinity model. Toxicol Sci 2000 58 270-81. 

Consider an affine model of an incompressible network with polydisperse strands between crosslinks. Prove that the stress a in this network due to... 

A steel ball is attached to a rubber band of size 0.2cm x 0.2cm x 10cm and stretches it in the z direction to a 40 cm length, developing an engineering stress 2 x lO dyncm in the rubber at the temperature 20 °C. Assume that the rubber band obeys the affine model. 

Can one design a high-affinity model system What will this tell us about the largely ill-defined high-affinity systems that are found in Nature ... 

The simplest model is the statistical theory of rubber-like elasticity, also called the affine model or neo-Hookean in the solids mechanics community. It predicts the nonlinear behavior at high strains of a rubber in uniaxial extension with Fq. (1), where ctn is the nominal stress defined as F/Aq, with F the tensile force and Aq the initial cross-section of the adhesive layer, A is the extension ratio, and G is the shear modulus. 

Note that, with this model, the reduced stress defined by Eq. (3) depends on the deformation X, whereas it did not for the simple affine model. 

The implications of this new model class are in contrast to most term structure models discussed in the literature, which assume that the bond markets are complete and fixed income derivatives are redundant securities. Collin-Dufresne and Goldstein [ 18] and Heiddari and Wu [36] show in an empirical work, using data of swap rates and caps/floors that there is evidence for one additional state variable that drives the volatility of the forward rates. Following from that empirical findings, they conclude that the bond market do not span all risks driving the term structure. This framework is rather similar to the affine models of equity derivatives, where the volatility of the underlying asset price dynamics is driven by a subordinated stochastic volatility process (see e.g. Heston [38], Stein and Stein [71] and Schobel and Zhu [69]). 

A pseudo-affine model predicts the variation of 2 with the deformation of a semi-crystalline polymer. It assumes that the distribution of crystal c axes is the same as the distribution of network chain end-to-end vectors r, in a rubber that has undergone the same macroscopic strain. Figure 3.12 showed the affine deformation of an r vector with that of a rubber block. 

In a stretched rubber, the molecules elongate, and the r vectors move towards the tensile axis. Fience the variation of Pi with extension ratio will differ from the pseudo-affine model. For moderate strains the increase of Pi with extension ratio is linear, but at high extensions the approximation used in Eq. (3.12), that both q and q are large, breaks down. Treloar (1975) described models which consider the limited number of links in the network chains. Figure 3.33 shows that the orientation function abruptly approaches 1 as the extension ratio of the rubber exceeds v. Although the model is successful for rubbers, it fails for the amorphous phase in polypropylene (Fig. 3.32), presumably because the crystals deform and reduce the strain in the amorphous phase. 

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