The problem of the spreading of a liquid film along a solid surface: A new mathematical formulation

A new mathematical model is proposed for the spreading of a liquid film on a solid surface. The model is based on the standard lubrication approximation for gently sloping films (with the no-slip condition for the fluid at the solid surface) in the major part of the film where it is not too thin. In the remaining and relatively small regions near the contact lines it is assumed that the so-called autonomy principle holds—i.e., given the material components, the external conditions, and the velocity of the contact lines along the surface, the behavior of the fluid is identical for all films. The resulting mathematical model is formulated as a free boundary problem for the classical fourth-order equation for the film thickness. A class of self-similar solutions to this free boundary problem is considered.

Microsatellite variability and genetic distances.

We analyze the within- and between-population dynamics of the distribution of the number of repeats at multiple microsatellite DNA loci subject to stepwise mutation. Analytical expressions for moments up to the fourth order within a locus and the variance of between-locus variance at mutation-drift equilibrium have been obtained. These statistics may be used to test the appropriateness of the one-step mutation model and to detect between-locus variation in the mutation rate. Published data are compatible with the one-step mutation model, although they do not reject the two-step model. Using both multinomial sampling and diffusion approximations for the analysis of the genetic distance introduced by Goldstein et al. [Goldstein, D. B., Linares, A. R., Cavalli-Sforza, L. L. & Feldman, M. W. (1995) Proc. Natl. Acad. Sci. USA 92, 6723-6727], we show that this distance follows a chi 2 distribution with degrees of freedom equal to the number of loci when there is no variation in mutation rates among the loci. In the presence of such variation, the variance of the distance is obtained. We conclude that the number of microsatellite loci required for the construction of phylogenetic trees with reliable branch lengths may be several hundred. Also, mutations that change repeat scores by several units, even though extremely rare, may dramatically influence estimates of population parameters.