Polynomial Vector Fields on the Clifford Torus

First, we characterize all the polynomial vector fields in [Formula: see text] which have the Clifford torus as an invariant surface. Then we study the number of invariant meridians and parallels that such polynomial vector fields can have on the Clifford torus as a function of the degree of these vector fields.

Second-Order Conditional Lie-Bäcklund Symmetry and Differential Constraint of Radially Symmetric Diffusion System

The classifications and reductions of radially symmetric diffusion system are studied due to the conditional Lie-Bäcklund symmetry method. We obtain the invariant condition, which is the so-called determining system and under which the radially symmetric diffusion system admits second-order conditional Lie-Bäcklund symmetries. The governing systems and the admitted second-order conditional Lie-Bäcklund symmetries are identified by solving the nonlinear determining system. Exact solutions of the resulting systems are constructed due to the compatibility of the original system and the admitted differential constraint corresponding to the invariant surface condition. For most of the cases, they are reduced to solving four-dimensional dynamical systems.

The endosomal TbTpr86/TbUsp7/SkpZ (TUS) complex controls surface protein abundance in trypanosomes

In trypanosomes the orthologs of human USP7 and VDU1 control abundance of a cohort of surface proteins, including invariant surface glycoproteins (ISGs) by functioning as deubiquitinases (DUBs) Silencing TbUsp7 partially inhibits endocytosis and invariant surface glycoprotein turnover. As a component of cullin E3 ubiquitin ligases, S-phase kinase-associated protein 1 (Skp1) has crucial roles in cell cycle progression, transcriptional regulation, signal transduction and other processes in animals and fungi. Unexpectedly, trypanosomes possess multiple Skp1 paralogs, including a divergent paralog designated SkpZ. SkpZ is implicated in suramin-sensitivity and endocytosis and decreases in abundance following TbUsp7 knockdown and physically interacts with TbUsp7 and TbTpr86. The latter is a tetratricopeptide-repeat protein also implicated in suramin sensitivity and located close to the flagellar pocket/endosomes, consistent with a role in endocytosis. Further, silencing SkpZ reduced abundance of TbUsp7 and TbTpr86 and many trans-membrane domain surface proteins. Our data indicate that TbTpr86, TbUsp7 and SkpZ form the ‘TUS’ complex that regulates abundance of a significant cohort of trypanosome surface proteins.

Conditional Lie-Bäcklund Symmetries and Differential Constraints of Radially Symmetric Nonlinear Convection-Diffusion Equations with Source

A conditional Lie-Bäcklund symmetry method and differential constraint method are developed to study the radially symmetric nonlinear convection-diffusion equations with source. The equations and the admitted conditional Lie-Bäcklund symmetries (differential constraints) are identified. As a consequence, symmetry reductions to two-dimensional dynamical systems of the resulting equations are derived due to the compatibility of the original equation and the additional differential constraint corresponding to the invariant surface equation of the admitted conditional Lie-Bäcklund symmetry.

Late ESCRT Machinery Mediates The Recycling And Rescue of Invariant Surface Glycoprotein 65 in Trypanosoma brucei

ABSTRACTThe Endosomal Sorting Complex Required for Transport machinery consists of four protein complexes (ESCRT 0-IV) and the post ESCRT ATPase Vps4. ESCRT mediates cargo delivery for lysosomal degradation via formation of multivesicular bodies. Trypanosoma brucei contains orthologues of ESCRT I-III and Vps4. Trypanosomes also have a ubiquitinylated invariant surface glycoprotein (ISG65) that is delivered to the lysosome by ESCRT, however, we previously implicated TbVps4 in rescue and recycling of ISG65. Here we use conditional silencing to investigate the role of TbVps24, a phosphoinositide-binding ESCRT III component, on protein trafficking. TbVps24 localizes to the TbRab7+ late endosome, and binds PI(3,5)P2, the product of the TbFab1 kinase, both of which also localize to late endosomes. TbVps24 silencing is lethal, and negatively affects biosynthetic trafficking of the lysosomal markers p67 and TbCathepsin L. However, the major phenotype of silencing is accelerated degradation and depletion of the surface pool of ISG65. Thus, TbVps24 silencing phenocopies that of TbVps4 in regard to ISG65 trafficking. This presents a paradox since we have previously found that depletion of TbFab1 completely blocks ISG65 turnover. We propose a model in which late ESCRT components operate at two sites, one PI(3,5)P2-dependent (degradation) and one PI(3,5)P2-independent (recycling), to regulate ISG65 homeostasis.

Suramin exposure alters cellular metabolism and mitochondrial energy production in African trypanosomes

Introduced about a century ago, suramin remains a frontline drug for the management of early-stage East African trypanosomiasis (sleeping sickness). Cellular entry into the causative agent, the protozoan parasite Trypanosoma brucei, occurs through receptor-mediated endocytosis involving the parasite's invariant surface glycoprotein 75 (ISG75), followed by transport into the cytosol via a lysosomal transporter. The molecular basis of the trypanocidal activity of suramin remains unclear, but some evidence suggests broad, but specific, impacts on trypanosome metabolism (i.e. polypharmacology). Here we observed that suramin is rapidly accumulated in trypanosome cells proportionally to ISG75 abundance. Although we found little evidence that suramin disrupts glycolytic or glycosomal pathways, we noted increased mitochondrial ATP production, but a net decrease in cellular ATP levels. Metabolomics highlighted additional impacts on mitochondrial metabolism, including partial Krebs' cycle activation and significant accumulation of pyruvate, corroborated by increased expression of mitochondrial enzymes and transporters. Significantly, the vast majority of suramin-induced proteins were normally more abundant in the insect forms compared with the blood stage of the parasite, including several proteins associated with differentiation. We conclude that suramin has multiple and complex effects on trypanosomes, but unexpectedly partially activates mitochondrial ATP-generating activity. We propose that despite apparent compensatory mechanisms in drug-challenged cells, the suramin-induced collapse of cellular ATP ultimately leads to trypanosome cell death.