Probability Distribution of Tree Age for the Simple Birth-Death Process, with Applications to Distributions of Number of Ancestral Lineages and Divergence Times for Pairs of Taxa in a Yule Tree.

In this contribution, a general expression is derived for the probability density of the time to the most recent common ancestor (TMRCA) of a simple birth-death tree, a widely used stochastic null-model of biological speciation and extinction, conditioned on the constant birth and death rates and number of extant lineages. This density is contrasted with a previous result which was obtained using a uniform prior for the time of origin. The new distribution is applied to two problems of phylogenetic interest. First, that of the probability of the number of taxa existing at any time in the past in a tree of a known number of extant species, and given birth and death rates, and second, that of determining the TMRCA of two randomly selected taxa in an unobserved tree that is produced by a simple birth-only, or Yule, process. In the latter case, it is assumed that only the rate of bifurcation (speciation) and the size, or number of tips, are known. This is shown to lead to a closed-form analytical expression for the probability distribution of this parameter, which is arrived at based on the known mathematical form of the age distribution of Yule trees of a given size and branching rate, which is derived here de novo, and a similar distribution which additionally is conditioned on tree age. The new distribution is the exact Yule prior for divergence times of pairs of taxa under the stated conditions and is potentially useful in statistical (Bayesian) inference studies of phylogenies.

Mikanolide from Jamaican Mikania micrantha.

Mikanolide [systematic names: 1,10:2,3-diepoxy-6,8-dihydroxy-11-vinylgermacr-4-ene 12,14-di-gamma-lactone and 7,10a-dimethyl-1a,1b,2a,6a,7,9a,10,10a-octahydro-4H-6,3-methenofuro[3,2-c]bisoxireno[f,h]oxacycloundecin-4,8(6H)-dione], C(15)H(14)O(6), derived from a variety of Mikania micrantha growing in Portland, Jamaica, contains a methylcyclodecane ring fused to an unsaturated planar alpha,gamma-lactone, an envelope-type near-planar vinyl-beta,gamma-lactone and two epoxide moieties. The crystal packing shows stacks of mikanolide molecules interlocked via a network of non-classical C-H...O hydrogen bonds between the lactone units.

The electrocapillary effect at an electrode modified with an insoluble redox-active self-assembled monolayer.

The interface between an electrolyte solution and a metal electrode coated with an oxidatively adsorbed, redox-active monolayer of long-chain thiols has been examined from a thermodynamic point of view. The electrode potential is assumed to vary within the region where no reductive desorption of the thiol occurs, so that the interface may formally be regarded as ideally polarizable. The analysis leads to an expression describing the potential dependence of interfacial tension in terms of the charge density on the metal, salt concentration, dielectric properties of the organic film, and the redox properties of the active terminal groups, which vary with the (average) distance from the electrode surface. This result generalizes the classical Lippmann equation to modified electrodes of the type considered.

The use of size distributions of circular domains in Langmuir monolayers for determining physical parameters of surfactants.

An analysis of equilibrium size distributions of circular liquid-condensed (LC) domains in amphiphilic monolayers at the air/water interface, gleaned from microscopic images, is presented in terms of Gaussians fitted to a theoretical expression derived previously. It is demonstrated how, in principle, important properties of the monolayer, such as molecular dipole moments of surfactant head groups and line tension of the liquid-condensed/liquid-expanded phase boundary, can be obtained by combining the results of this simplified analysis with information from X-ray diffraction or surface pressure measurements.