Subalgebras with Converging Star Products in Deformation Quantization: An Algebraic Construction for \complex \mbox{\LARGE P}^n

Based on a closed formula for a star product of Wick type on \CP^n, which has been discovered in an earlier article of the authors, we explicitly construct a subalgebra of the formal star-algebra (with coefficients contained in the uniformly dense subspace of representative functions with respect to the canonical action of the unitary group) that consists of {\em converging} power series in the formal parameter, thereby giving an elementary algebraic proof of a convergence result already obtained by Cahen, Gutt, and Rawnsley. In this subalgebra the formal parameter can be substituted by a real number $\alpha$: the resulting associative algebras are infinite-dimensional except for the case $\alpha=1/K$, $K$ a positive integer, where they turn out to be isomorphic to the finite-dimensional algebra of linear operators in the $K$th energy eigenspace of an isotropic harmonic oscillator with $n+1$ degrees of freedom. Other examples like the $2n$-torus and the Poincar\'e disk are discussed.Comment: 16 pages, LaTeX with AMS Font

A Boolean Gene Regulatory Model of heterosis and speciation

Modelling genetic phenomena affecting biological traits is important for the development of agriculture as it allows breeders to predict the potential of breeding for certain traits. One such phenomenon is heterosis or hybrid vigor: crossing individuals from genetically distinct populations often results in improvements in quantitative traits, such as growth rate, biomass production and stress resistance. Heterosis has become a very useful tool in global agriculture, but its genetic basis remains controversial and its effects hard to predict. We have taken a computational approach to studying heterosis, developing a simulation of evolution, independent reassortment of alleles and hybridization of Gene Regulatory Networks (GRNs) in a Boolean framework. Fitness is measured as the ability of a network to respond to external inputs in a pre-defined way. Our model reproduced common experimental observations on heterosis using only biologically justified parameters. Hybrid vigor was observed and its extent was seen to increase as parental populations diverged, up until a point of sudden collapse of hybrid fitness. We also reproduce, for the first time in a model, the fact that hybrid vigor cannot easily be fixed by within a breeding line, currently an important limitation of the use of hybrid crops. The simulation allowed us to study the effects of three standard models for the genetic basis of heterosis and the level of detail in our model allows us to suggest possible warning signs of the impending collapse of hybrid vigor in breeding. In addition, the simulation provides a framework that can be extended to study other aspects of heterosis and alternative evolutionary scenarios.Comment: See online version for supplementary materia

Purification and analytical characterization of an anti- CD4 monoclonal antibody for human therapy

A purification process for the monclonal anti-CD4 antibody MAX.16H5 was developed on an analytical scale using (NH&SO, precipitation, anion-exchange chromatography on MonoQ or Q-Sepharose, hydrophobic interaction chromatography on phenyl- Sepharose and gel filtration chromatography on Superdex 200. The purification schedule was scaled up and gram amounts of MAX.16H5 were produced on corresponding BioPilot columns. Studies of the identity, purity and possible contamination by a broad range of methods showed that the product was highly purified and free from contaminants such as mouse DNA, viruses, pyrogens and irritants. Overall, the analytical data confirm that the monoclonal antibody MAX.16H5 prepared by this protocol is suitable for human therapy

Phase Space Reduction for Star-Products: An Explicit Construction for CP^n

We derive a closed formula for a star-product on complex projective space and on the domain $SU(n+1)/S(U(1)\times U(n))$ using a completely elementary construction: Starting from the standard star-product of Wick type on $C^{n+1} \setminus \{ 0 \}$ and performing a quantum analogue of Marsden-Weinstein reduction, we can give an easy algebraic description of this star-product. Moreover, going over to a modified star-product on $C^{n+1} \setminus \{ 0 \}$, obtained by an equivalence transformation, this description can be even further simplified, allowing the explicit computation of a closed formula for the star-product on \CP^n which can easily transferred to the domain $SU(n+1)/S(U(1)\times U(n))$.Comment: LaTeX, 17 page