The stereodynamics of 1,2‐dipropyldiaziridines

N‐alkylated trans‐diaziridines are an intriguing class of compounds with two stereogenic nitrogen atoms which easily interconvert. In the course of our investigations of the nature of the interconversion process via nitrogen inversion or electrocyclic ring opening ring closure, we synthesized and characterized the three constitutionally isomeric diaziridines 1,2‐di‐n‐propyldiaziridine 1 , 1‐isopropyl‐2‐n‐propyldiaziridine 2 , and 1,2‐diisopropyldiaziridine 3 to study the influence of the substituents on the interconversion barriers. Enantiomer separation was achieved by enantioselective gas chromatography on the chiral stationary phase Chirasil‐β‐Dex with high separation factors α (1‐isopropyl‐2‐n‐propyldiaziridine: 1.18; 1, 2‐diisopropyldiaziridine: 1.24; 100°C 50 kPa He) for the isopropyl substituted diaziridines. These compounds showed pronounced plateau formation between 100 and 150°C, and peak coalescence at elevated temperatures. The enantiomerization barriers ΔG? and activation parameters ΔH? and ΔS? were determined by enantioselective dynamic gas chromatography (DGC) and direct evaluation of the elution profiles using the unified equation implemented in the software DCXplorer. Interestingly, 1‐isopropyl‐2‐n‐propyldiaziridine and 1,2‐diisopropyldiaziridine exhibit similar high interconversion barriers ΔG? (100°C) of 128.3 ± 0.4 kJ mol^?1 and 129.8 ± 0.4 kJ mol^?1, respectively, which indicates that two sterically demanding substituents do not substantially increase the barrier as expected for a distinct nitrogen inversion process. Chirality, 2010. © 2009 Wiley‐Liss, Inc.     

Estimating Haplotype Effects for Survival Data

Summary Genetic association studies often investigate the effect of haplotypes on an outcome of interest. Haplotypes are not observed directly, and this complicates the inclusion of such effects in survival models. We describe a new estimating equations approach for Cox's regression model to assess haplotype effects for survival data. These estimating equations are simple to implement and avoid the use of the EM algorithm, which may be slow in the context of the semiparametric Cox model with incomplete covariate information. These estimating equations also lead to easily computable, direct estimators of standard errors, and thus overcome some of the difficulty in obtaining variance estimators based on the EM algorithm in this setting. We also develop an easily implemented goodness‐of‐fit procedure for Cox's regression model including haplotype effects. Finally, we apply the procedures presented in this article to investigate possible haplotype effects of the PAF‐receptor on cardiovascular events in patients with coronary artery disease, and compare our results to those based on the EM algorithm.     

Modeling relapse in infectious diseases

An integro-differential equation is proposed to model a general relapse phenomenon in infectious diseases including herpes. The basic reproduction number R(0) for the model is identified and the threshold property of R(0) established. For the case of a constant relapse period (giving a delay differential equation), this is achieved by conducting a linear stability analysis of the model, and employing the Lyapunov-Razumikhin technique and monotone dynamical systems theory for global results. Numerical simulations, with parameters relevant for herpes, are presented to complement the theoretical results, and no evidence of sustained oscillatory solutions is found.     

Asymptotic behaviour of solutions to abstract logistic equations

We analyze the asymptotic behaviour of solutions of the abstract differential equation u'(t)=Au(t)-F(u(t))u(t)+f. Our results are applicable to models of structured population dynamics in which the state space consists of population densities with respect to the structure variables. In the equation the linear term A corresponds to internal processes independent of crowding, the nonlinear logistic term F corresponds to the influence of crowding, and the source term f corresponds to external effects. We analyze three separate cases and show that for each case the solutions stabilize in a way governed by the linear term. We illustrate the results with examples of models of structured population dynamics -- a model for the proliferation of cell lines with telomere shortening, a model of proliferating and quiescent cell populations, and a model for the growth of tumour cord cell populations.     

Calculation of pKas in RNA: on the structural origins and functional roles of protonated nucleotides

pK(a) calculations based on the Poisson-Boltzmann equation have been widely used to study proteins and, more recently, DNA. However, much less attention has been paid to the calculation of pK(a) shifts in RNA. There is accumulating evidence that protonated nucleotides can stabilize RNA structure and participate in enzyme catalysis within ribozymes. Here, we calculate the pK(a) shifts of nucleotides in RNA structures using numerical solutions to the Poisson-Boltzmann equation. We find that significant shifts are predicted for several nucleotides in two catalytic RNAs, the hairpin ribozyme and the hepatitis delta virus ribozyme, and that the shifts are likely to be related to their functions. We explore how different structural environments shift the pK(a)s of nucleotides from their solution values. RNA structures appear to use two basic strategies to shift pK(a)s: (a) the formation of compact structural motifs with structurally-conserved, electrostatic interactions; and (b) the arrangement of the phosphodiester backbone to focus negative electrostatic potential in specific regions.     

Stabilization of telomeres in nonlinear models of proliferating cell lines

We analyse an age-structured model of telomere loss in a proliferating cell population. The cell population is divided into telomere classes, which shorten each round of division. The model consists of a nonlinear system of partial differential equations for the telomere classes. We prove that if the highest telomere class is exempted from mortality, then all the classes stabilize to a nontrivial equilibrium dependent on the initial state of cells in the highest telomere class.     

Evolutionary graph theory: breaking the symmetry between interaction and replacement

We study evolutionary dynamics in a population whose structure is given by two graphs: the interaction graph determines who plays with whom in an evolutionary game; the replacement graph specifies the geometry of evolutionary competition and updating. First, we calculate the fixation probabilities of frequency dependent selection between two strategies or phenotypes. We consider three different update mechanisms: birth-death, death-birth and imitation. Then, as a particular example, we explore the evolution of cooperation. Suppose the interaction graph is a regular graph of degree h, the replacement graph is a regular graph of degree g and the overlap between the two graphs is a regular graph of degree l. We show that cooperation is favored by natural selection if b/c>hg/l. Here, b and c denote the benefit and cost of the altruistic act. This result holds for death-birth updating, weak-selection and large population size. Note that the optimum population structure for cooperators is given by maximum overlap between the interaction and the replacement graph (g=h=l), which means that the two graphs are identical. We also prove that a modified replicator equation can describe how the expected values of the frequencies of an arbitrary number of strategies change on replacement and interaction graphs: the two graphs induce a transformation of the payoff matrix.