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.

Thermodynamic and kinetic models for acid chloride formation: A computational and theoretical mechanistic study.

The structural evolution pathways leading to the conversion of some α-chlorinated carboxylic acids (ClnH3-nCCOOH, n = 0, 1, 2, 3) to their respective acid chlorides by thionyl chloride are investigated via density functional theoretical (DFT) modelling. For all compounds where n = 0-3, acid chloride formation occurs via two competing pathways, consisting of three activation barriers in both cases, all of which are enthalpy-controlled and moderate (ΔGǂ < 190 kJ mol-1). Though both pathways are not limited by the same step, they are both composed of only cyclic activated complexes. Rapid intra-molecular small molecule transfer (HCl) allows one pathway to be slight more productive than the other. Whereas all acids evolve via both competing pathways, the evolution of formic acid occurs exclusively via that which involves intramolecular HCl transfer where all the constituent transition states are formed quasi-synchronously. Results for both pathways are summarized in a detail kinetic model which, of-course, is based on the thermodynamic profiles.

On the distribution of interspecies correlation for Markov models of character evolution on Yule trees.

Efforts to reconstruct phylogenetic trees and understand evolutionary processes depend fundamentally on stochastic models of speciation and mutation. The simplest continuous-time model for speciation in phylogenetic trees is the Yule process, in which new species are "born" from existing lineages at a constant rate. Recent work has illuminated some of the structural properties of Yule trees, but it remains mostly unknown how these properties affect sequence and trait patterns observed at the tips of the phylogenetic tree. Understanding the interplay between speciation and mutation under simple models of evolution is essential for deriving valid phylogenetic inference methods and gives insight into the optimal design of phylogenetic studies. In this work, we derive the probability distribution of interspecies covariance under Brownian motion and Ornstein-Uhlenbeck models of phenotypic change on a Yule tree. We compute the probability distribution of the number of mutations shared between two randomly chosen taxa in a Yule tree under discrete Markov mutation models. Our results suggest summary measures of phylogenetic information content, illuminate the correlation between site patterns in sequences or traits of related organisms, and provide heuristics for experimental design and reconstruction of phylogenetic trees.

Analytical expressions for proton transfer voltammetry: analogy to surface redox voltammetry with Frumkin interactions.

Theory for interfacial proton transfer voltammetry of a molecular film containing any acid/base loading has been developed under equilibrium conditions. Diagnostic criteria to disentangle the interplay between diffuse layer and ionization effects are outlined. Easy-to-use analytical expressions for the voltammetric features are derived for the particular case of an invariant diffuse layer effect, which turn out to be entirely analogous to those for a surface redox conversion with Frumkin interactions. It is demonstrated that, regardless of the electrolyte concentration, significant ionization of the external acid groups located nearby the diffuse layer is sufficient for the fulfillment of this relevant particular case. A strategy is outlined to determine the amount, the intrinsic pKa, and the burial depth of the voltammetrically active groups from the surface concentration dependence of the main voltammetric features. Self-assembled monolayers of 11-mercaptoundecanoic acid deposited on Au(111), containing higher amounts of buried carboxylic groups than previously reported, have been studied to assess more critically the influence of electrostatic effects on the ionization process. Preliminary evidence suggests that the protonation/deprotonation voltammetric wave involves physisorbed rather than chemisorbed thiol molecules. Application of the present theoretical approach to this system reveals that the voltammetrically active carboxylic groups are located close to the electrode surface and become more acidic upon increasing their surface concentration.

Proton transfer voltammetry at electrodes modified with acid thiol monolayers.

By combining a description of the potential profile at electrodes coated with acid thiol monolayers with a quadratic relationship between activation energy and electrode potential, a rather simple expression for proton transfer voltammograms is derived. Our electrostatic analysis shows that proton transfer can only produce narrow voltammetric peaks when the immobilized acid groups lie close to the metal substrate. Quantitative fits of experimental voltammograms obtained with an Au(111) electrode modified with a 11-mercaptoundecanoic monolayer at pH 8.5 reveal that less than 1% of the carboxylic groups in the monolayer participate in the potential induced proton transfer process and that these groups lay close to the metal surface. A preliminary analysis of the kinetic parameters suggests that the interfacial electric field facilitates an intrinsically slow proton exchange between a proton donor and acceptor pair that are not in close contact with each other at the interface.

Probability distributions of ancestries and genealogical distances on stochastically generated rooted binary trees.

The stationary birth-only, or Yule-Furry, process for rooted binary trees has been analysed with a view to developing explicit expressions for two fundamental statistical distributions: the probability that a randomly selected leaf is preceded by N nodes, or "ancestors", and the probability that two randomly selected leaves are separated by N nodes. For continuous-time Yule processes, the first of these distributions is presented in closed analytical form as a function of time, with time being measured with respect to the moment of "birth" of the common ancestor (which is essentially inaccessible to phylogenetic analysis), or with respect to the instant at which the first bifurcation occurred. The second distribution is shown to follow in an iterative manner from a hierarchy of second-order ordinary differential equations. For Yule trees of a given number n of tips, expressions have been derived for the mean and variance for each of these distributions as functions of n, as well as for the distributions themselves. In addition, it is shown how the methods developed to obtain these distributions can be employed to find, with minor effort, expressions for the expectation values of two statistics on Yule trees, the Sackin index (sum over all root-to-leaf distances), and the sum over all leaf-to-leaf distances.

Experimental study of the interplay between long-range electron transfer and redox probe permeation at self-assembled monolayers: evidence for potential-induced ion gating.

Evidence for the competition between long-range electron transfer across self-assembled monolayers (SAMs) and incorporation of the redox probe into the film is reported for the electroreduction of Ru(NH(3)) at hydroxyl- and carboxylic-acid-terminated SAMs on a mercury electrode, by using electrochemical techniques that operate at distinct time scales. Two limiting voltammetric behaviors are observed, consistent with a diffusion control of the redox process at mercaptophenol-coated electrodes and a kinetically controlled electron transfer reaction in the presence of neutral HS-(CH(2))(10)-COOH and HS-(CH(2))(n)()-CH(2)OH (n = 3, 5, and 10) SAMs. The monolayer thickness dependence of the standard heterogeneous electron transfer rate constant shows that the electron transfer plane for the reduction of Ru(NH(3)) at hydroxyl-terminated SAMs is located outside the film | solution interface at short times. However, long time scale experiments provide evidence for the occurrence of potential-induced gating of the adsorbed structure in some of the monolayers studied, which takes the form of a chronoamperometric spike. Redox probe permeation is shown to be a kinetically slow process, whose activation strongly depends on redox probe concentration, applied potential, and chemical composition of the intervening medium. The obtained results reveal that self-assembled monolayers made of mercaptobutanol and mercaptophenol preserve their electronic barrier properties up to the reductive desorption potential of a fully grown SAM, whereas those of mercaptohexanol, mercaptoundecanol, and mercaptoundecanoic acid undergo an order/disorder transition below a critical potential, which facilitates the approach of the redox probe toward the electrode surface.

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.