We propose a di erential equation model for gene expression and provide two methods to construct the model from a set of temporal data. We model both transcription and translation by kinetic equations with feedback loops from translation products to transcription. Degradation of proteins and mRNAs is also incorporated. We study two methods to construct the model from experimental data: Minimum Weight Solutions to Linear Equations (MWSLE), which determines the regulation by solving under-determined linear equations, and Fourier Transform for Stable Systems (FTSS), which re nes the model with cell cycle constraints. The results suggest that a minor set of temporal data may be su cient t o c o nstruct the model at the genome level. We also give a comprehensive discussion of other extended models: the RNA Model, the Protein Model, and the Time Delay Model.

If G is a Lie group, H ⊂ G is a closed subgroup, and τ is a unitary representation of H, then the authors give a sufficient condition on ξ ∈ ig * to be in the wave front set of Ind G H τ . In the special case where τ is the trivial representation, this result was conjectured by Howe. If G is a real, reductive algebraic group and π is a unitary representation of G that is weakly contained in the regular representation, then the authors give a geometric description of WF(π) in terms of the direct integral decomposition of π into irreducibles. Special cases of this result were previously obtained by Kashiwara-Vergne, Howe, and Rossmann. The authors give applications to harmonic analysis problems and branching problems.

A systems model that describes vesicle trafficking during pollen tube growth in Arabidopsis (Arabidopsis thaliana) was constructed. The model is composed of ordinary differential equations that connect the molecular functions of genes expressed in pollen. The current model requires soluble N-ethylmaleimide-sensitive fusion protein attachment protein receptors (SNAREs) and small GTPases, Arf or Rab, to reasonably predict tube growth as a function of time. Tube growth depends on vesicle trafficking that transports phospholipid and pectin to the tube tip. The vesicle trafficking genes identified by analyzing publicly available transcriptome data comprised 328 genes. Fourteen of them are up-regulated by the gibberellin signaling pathway during pollen development, which includes the SNARE genes SYP124 and SYP125 and the Rab GTPase gene RABA4D. The model results adequately fit the pollen tube growth of both previously reported wild-type and raba4d knockout lines. Furthermore, the difference of pollen tube growth in syp124/syp125 single and double mutations was quantitatively predicted based on the model analysis. In general, a systems model approach to vesicle trafficking arguably demonstrated the importance of the functional connections in pollen tube growth and can help guide future research directions.

In this paper, we discuss the positivity of the Hermitian form (, ) π introduced by Jian-Shu Li in [11]. Let (G 1 , G 2 ) be a type I dual pair with G 1 the smaller group. Let π be an irreducible unitary representation in the semistable range of θ(M G 1 , M G 2 ) (see [5]). We prove that the invariant Hermitian form (, ) π is positive semidefinite under certain restrictions on the size of G 2 and a mild growth condition on the matrix coefficients of π. Therefore, if (, ) π does not vanish, θ(M G 1 , M G 2 )(π) is unitary.Theta correspondence over R was established by Howe in ( [7]). Li showed that theta correspondence preserves unitarity for dual pairs in stable range. Our results generalize the results of Li for type I classical groups ( [11]). The main result in this paper can be used to construct irreducible unitary representations of classical groups of type I. Thus the second condition can be converted into a growth condition on the matrix coefficients of π (see Corollary 5.

The main purpose of this paper is to study theta correspondence from representation theoretic point of view. There are two problems we have in mind. One is the construction of unipotent representations of semisimple Lie group. The other is the parametrization of unitary dual of semisimple Lie group. In the first paper of this series, we define semistable range in the domain of theta correspondence. Roughly speaking, semistable range is a range where one can define certain averaging operator analytically. In this paper, we prove that if the averaging operator is not vanishing, then it produces the theta correspondence.