The Computational Complexity of Knot and Link Problems

We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, {\sc unknotting problem} is in {\bf NP}. We also consider the problem, {\sc unknotting problem} of determining whether two or more such polygons can be split, or continuously deformed without self-intersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in {\bf PSPACE}. We also give exponential worst-case running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.Comment: 32 pages, 1 figur

Inactivation of MARCH5 prevents mitochondrial fragmentation and interferes with cell death in a neuronal cell model

PURPOSE: To study the impact of the mitochondrial ubiquitin ligase MARCH5 on mitochondrial morphology and induction of apoptosis using an in vitro model of neuronal precursor cells exposed to glaucoma-relevant stress conditions. METHODS: RGC5 cells transfected with expression constructs for MARCH5, MARCH5(H43W), Dpr1(K38A) or vector control were exposed to either elevated pressure of 30 mmHg, oxidative stress caused by mitochondrial electron transport chain (ETC) inhibition, or hypoxia-reoxygenation conditions. Mitochondrial morphology of RGC5 cells was analyzed following staining of the mitochondrial marker cytochrome c and photoactivatable GFP (PAGFP) diffusion assay. Induction of apoptotic cell death in these cells was determined by analyzing the release of cytochrome c from mitochondria into the cytosol and flow cytometry. RESULTS: Exposure of RGC5 cells to oxidative stress conditions as well as to elevated pressure resulted in the fragmentation of the mitochondrial network in control cells as well as in cells expressing MARCH5. In cells expressing inactive MARCH5(H43W) or inactive Drp(K38A), mitochondrial fragmentation was significantly blocked and mitochondrial morphology was comparable to that of control cells under normal conditions. Exposure of RGC5 cells to elevated pressure or oxidative stress conditions induced apoptotic cell death as assessed by cytochrome c release and DNA staining, while expression of dominant-negative MARCH5(H43W) or Drp1(K38A) did significantly delay cell death. CONCLUSION: Preventing mitochondrial fragmentation through interference with the mitochondrial fission machinery protects neuronal cells from programmed cell death following exposure to stressors physiologically relevant to the pathogenesis of glaucoma

Structural features within the nascent chain regulate alternative targeting of secretory proteins to mitochondria

Protein targeting to specified cellular compartments is essential to maintain cell function and homeostasis. In eukaryotic cells, two major pathways rely on N-terminal signal peptides to target proteins to either the endoplasmic reticulum (ER) or mitochondria. In this study, we show that the ER signal peptides of the prion protein-like protein shadoo, the neuropeptide hormone somatostatin and the amyloid precursor protein have the property to mediate alternative targeting to mitochondria. Remarkably, the targeting direction of these signal peptides is determined by structural elements within the nascent chain. Each of the identified signal peptides promotes efficient ER import of nascent chains containing alpha-helical domains, but targets unstructured polypeptides to mitochondria. Moreover, we observed that mitochondrial targeting by the ER signal peptides correlates inversely with ER import efficiency. When ER import is compromised, targeting to mitochondria is enhanced, whereas improving ER import efficiency decreases mitochondrial targeting. In conclusion, our study reveals a novel mechanism of dual targeting to either the ER or mitochondria that is mediated by structural features within the nascent chain