Endpoint correlations

Mathematical appendices introduce (5-functious, Fourier transform, or d-dimensional integration over Gaussian functions, respectively. 

Ignoring the excluded volume interaction we deal with independent chains. It thus is sufficient to consider a single chain system. The spatial dimension is found to play an important role in the theory, and we therefore work with general dimension d. Wo write the configurational probability for a noninteracting Gaussian chain as 

For this model only the spatial correlations among segments are of interest. Let us first consider the distribution function of the end-to-end distance, which is defined as 

With our model the evaluation of PtJ(r n) is an exercise in doing Gaussian integrals. (See Appendix A 3.3.) Wo only note here the results, leaving the explicit calculation to Sect. 3.3. We find 

We often will be concerned with the Fourier transform (see Appendix A 3.2) 

4 his again amounts to a Gaussian integral which yields 

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All the other correlation functions are to be defined according to Eq. (5.11). For instance, being interested in endpoint correlations, we introduce the local endpoint density of the m-th chain... 

The corresponding results are easily established for endpoint correlations. 

The relevance of a measurement endpoint is the degree to which the measurement can be associated to the assessment endpoint under consideration. Perhaps the most direct measurement endpoints are those that reflect the mechanism of action, such as inhibition of a protein or mortality of members of the species under protection. Although correlated functions can and are used as measurement endpoints, correlations do not necessarily imply cause and effect. 

Rouguet, R., Cotovia, J. Gaetani, Q., Dossou K. G. Rougier, A. 1993. Cytotoxicity of 28 MEIC chemicals to rat hepatocytes using two viabihty endpoints correlation with acute toxicity data in rat and man. ATLA 1 216-224. Ruelius, H. W. 1987. Extrapolation from animals to man predictions, pitfalls and perspectives. Xenobiotica 17 255-265. 

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