Poisson Geometry in Constrained Systems

Constrained Hamiltonian systems fall into the realm of presymplectic geometry. We show, however, that also Poisson geometry is of use in this context. For the case that the constraints form a closed algebra, there are two natural Poisson manifolds associated to the system, forming a symplectic dual pair with respect to the original, unconstrained phase space. We provide sufficient conditions so that the reduced phase space of the constrained system may be identified with a symplectic leaf in one of those. In the second class case the original constrained system may be reformulated equivalently as an abelian first class system in an extended phase space by these methods. Inspired by the relation of the Dirac bracket of a general second class constrained system to the original unconstrained phase space, we address the question of whether a regular Poisson manifold permits a leafwise symplectic embedding into a symplectic manifold. Necessary and sufficient for this is the vanishing of the characteristic form-class of the Poisson tensor, a certain element of the third relative cohomology.Comment: 41 pages, more detailed abstract in paper; v2: minor corrections and an additional referenc

Transformation of Mexican lime with an intron-hairpin construct expressing untranslatable versions of the genes coding for the three silencing suppressors of Citrus tristeza virus confers complete resistance to the virus

[EN] Citrus tristeza virus (CTV), the causal agent of the most devastating viral disease of citrus, has evolved three silencing suppressor proteins acting at intra- (p23 and p20) and/or intercellular level (p20 and p25) to overcome host antiviral defence. Previously, we showed that Mexican lime transformed with an intron-hairpin construct including part of the gene p23 and the adjacent 3' untranslated region displays partial resistance to CTV, with a fraction of the propagations from some transgenic lines remaining uninfected. Here, we transformed Mexican lime with an intron-hairpin vector carrying full-length, untranslatable versions of the genes p25, p20 and p23 from CTV strain T36 to silence the expression of these critical genes in CTV-infected cells. Three transgenic lines presented complete resistance to viral infection, with all their propagations remaining symptomless and virus-free after graft inoculation with CTV-T36, either in the nontransgenic rootstock or in the transgenic scion. Accumulation of transgene-derived siRNAs was necessary but not sufficient for CTV resistance. Inoculation with a divergent CTV strain led to partially breaking the resistance, thus showing the role of sequence identity in the underlying mechanism. Our results are a step forward to developing transgenic resistance to CTV and also show that targeting simultaneously by RNA interference (RNAi) the three viral silencing suppressors appears critical for this purpose, although the involvement of concurrent RNAi mechanisms cannot be excluded.We thank J.E. Peris for his excellent technical assistance and Dr. W.O. Dawson (University of Florida, C.R.E.C., Lake Alfred, FL, USA) for providing the GFP-tagged CTV strain CTV947R-GFP. N.S. was supported by a PhD fellowship from the IVIA. C.F. is recipient of a postdoctoral Ramon y Cajal contract from the Ministerio de Ciencia e Innovacion (MICINN). This research was supported by grants AGL2009-08052, co-financed by Fondo Europeo de Desarrollo Regional-MICINN, and Prometeo/2008/121 from the Generalitat Valenciana.Soler, N.; Plomer Sáez, M.; Fagoaga García, CC.; Moreno, P.; Navarro, L.; Flores Pedauye, R.; Peña Garcia, L. (2012). Transformation of Mexican lime with an intron-hairpin construct expressing untranslatable versions of the genes coding for the three silencing suppressors of Citrus tristeza virus confers complete resistance to the virus. Plant Biotechnology Journal. 10(5):597-608. https://doi.org/10.1111/j.1467-7652.2012.00691.xS59760810