Gaussian overlap distributions factorization

Assuming Gaussian density distributions, we obtain in the limit of a complete overlap, rc,i = Tc,2, an expression identical to the internal excluded volume interaction energy as given by Eq. (2.91), apart from a factor 1/2. Omitting the numerical prefactor of order unity we can write... 

Since the two Gaussians factorize into the three Cartesian directions, we may factorize the overlap distribution in the same way... 

Fig. 6.3. Cutoff fluorescence selection for screening. Instrumentation, labeling, and biological noise introduce spreading into a fluorescence measurement, such that the fluorescence probability distributions for wild-type and mutant cells overlap. The logarithm of single-cell fluorescence as measured by flow cytometry is generally well-approximated by a symmetrical Gaussian curve. A cutoff fluorescence value is selected for screening, with all cells above that value sorted out. The enrichment factor forthe mutants is the ratio of (dotted + striped areas)/(striped area), and the probability of retention of a given mutant clone at a single pass is the (striped + dotted area)/(all area under mutant curve).

Fig. 6.5. Peak spreading strongly affects enrichment ratio at fixed probability of retention. The coefficient of variance CV is equal to the ratio of the standard deviation to the mean, and is a measure of peak breadth. For example, in both curves shown in Fig. 6.3 the CV is 1.0. The enrichment ratio was calculated for a situation in which mutant fluorescence intensity was double wild-type fluorescence intensity, the mutant was initially present at 1 in 106 cells, and the probability of retention was fixed at 95 %. The logarithmic fluorescence intensity was assumed to follow a Gaussian distribution. Fixing the probability of retention defines the cutoff fluorescence value for screening at a given CV. Enrichment ratio drops precipitously with increasing CV, as the mutant and wild-type fluorescence distributions begin to overlap. At a CV of 0.2, the enrichment factor is 600. However, at a CV of 0.4, the enrichment factor has dropped to 3 Clearly, every effort should be expended to minimize peak spreading and subsequent overlap of the mutant and wild-type fluorescence distributions.

Figure 9.14 Simulated recombination rate R(f) of geminate e...h pairs. Computer printouts show (a) the coincidence of the data obtained at temperatures 77, 100, 120, 240, 375, and 600 K, (b) the virtual independence of the normalized rate on the wave function overlap factor 2ya, and (c) the influence of the initial pair separation in units of the lattice constant a. The computer system was a sample of cubic symmetry in which the energy of the hopping sites was distributed according to a Gaussian distribution of variance 0.1 eV. (From Ries, B. and Bassler, H., J. Mol. Electron., 3, 15, 1987. With permission.)...

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