Statistical element

Kuhn has shown how a real polymer chain may be approximated by an equivalent freely jointed chain. Instead of taking the individual bonds as statistical elements, one may for this purpose choose sequences of m bonds each. In Fig. 79, arbitrarily chosen statistical elements consisting of five bonds are indicated, the displacement vectors for these elements being shown by the dashed lines. The direction assumed by a statistical element will be nearly independent of the direction of the preceding element, provided the number m of bonds per... 

Fig. 27. The negative packing entropy correction to the Gaussian term in unidirectional deformation according to DiMarzio (38). The numbers refer to the number of statistical elements in the network chains...

Realistic predichons of study results based on simulations can be made only with realistic simulation models. Three types of models are necessary to mimic real study observations system (drug-disease) models, covariate distribution models, and study execution models. Often, these models can be developed from previous data sets or obtained from literature on compounds with similar indications or mechanisms of action. To closely mimic the case of intended studies for which simulations are performed, the values of the model parameters (both structural and statistical elements) and the design used in the simulation of a proposed trial may be different from those that were originally derived from an analysis of previous data or other literature. Therefore, before using models, their appropriateness as simulation tools must be evaluated to ensure that they capture observed data reasonably well [19-21]. However, in some circumstances, it is not feasible to develop simulation models from prior data or by extrapolation from similar dmgs. In these circumstances, what-if scenarios or sensitivity analyses can be performed to evaluate the impact of the model uncertainty and the study design on the trial outcome [22, 23]. 

From the theoretical point of view, as has already been said in Section II, difficulties encountered are much greater than for the interpretation of dipole moments. At the present time, there is no hope of elaborating a complete theory based on the correct values of the internal field and accounting, in a rigorous manner, for all types of interaction. Even if one considers the individual polymer chains as independent of one another and of the solvent, the task of describing the movement of the resultant dipole is still quite overwhelming. In fact, consideration must be given to the simultaneous movements of the molecule as a whole, of the statistical elements, and of each dipole unit. In order to move, each molecule, submolecule, or dipole must overcome the viscous resistance of the... 

The interesting idea in this theory is that the statistical elements, which may be considered as completely independent units when the equilibrium distribution is investigated, are no longer independent from the kinetic point of view. 

C = monomer concentration vc = excluded volume l = length of the statistical element. 

Compare this situation with the classical description in which at some time,, we would specify precisely a value of momentum, p. At any subsequent time the momentum can be calculated from the classical laws. In quantum mechanics this precision is replaced by a probability. At any randomly chosen time, if by some magic we could ask the particle in the box for its momentum, it could only reply Plus nnh/L, with 50 % probability, or Minus nnh/L, with 50 % probability. Thus a statistical element is present in quantum mechanics. Certainty in classical mechanics is replaced by probability, or uncertainty, in quantum mechanics. 

By letting I be the length of a statistical element, the length of the fiber at maximum extension,, is... 

The geometrical parameters, V/Lm, cr, and I, required for this computation were obtained as follows. jLm has already been shown to be 0.41 for the polypeptide chain in the a-helical and fully extended forms. Since we have assumed that there is one amino acid residue per statistical element for simplicity, the value of V is 3.6 X 10 cm. A reasonable value for square centimeter cross section of the fiber, is obtained from X-ray data on poly-7-benzyl-L-glutamate. Pauling and Corey (1951) interpreted the X-ray data for this synthetic polypeptide in terms of a parallel array of a-helices with two helices per unit cell of cross section 25.0 X 14.42 A. Dividing the number of chains per unit cell by the cross-sectional... 

Here C is a dimensionless quantity of the order of unity and Ao is the number of statistical elements in the network. A similar expression has been derived by Schellman for the effect of intramolecular cross-linkages in stabilizing ordered polypeptide chains.(12)... 

A statistical element of the netwo-k bears (he same relation to a chain unit as the statistical element of an equivalent statistical chain does to the repeating unit of a real chain. See Ref. (9), pp. 410 ff. 

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