We consider the problem of sampling from the Potts model on random regular graphs. It is conjectured that sampling is possible when the temperature of the model is in the so-called uniqueness regime of the regular tree, but positive algorithmic results have been for the most part elusive. In this paper, for all integers q ≥ 3 and ∆ ≥ 3, we develop algorithms that produce samples within error o(1) from the q-state Potts model on random ∆-regular graphs, whenever the temperature is in uniqueness, for both the ferromagnetic and antiferromagnetic cases.The algorithm for the antiferromagnetic Potts model is based on iteratively adding the edges of the graph and resampling a bichromatic class that contains the endpoints of the newly added edge. Key to the algorithm is how to perform the resampling step efficiently since bichromatic classes can potentially induce linear-sized components. To this end, we exploit the tree uniqueness to show that the average growth of bichromatic components is typically small, which allows us to use correlation decay algorithms for the resampling step. While the precise uniqueness threshold on the tree is not known for general values of q and ∆ in the antiferromagnetic case, our algorithm works throughout uniqueness regardless of its value.In the case of the ferromagnetic Potts model, we are able to simplify the algorithm significantly by utilising the random-cluster representation of the model. In particular, we demonstrate
RecQL4, one of the five human RecQ helicases, is crucial for genomic stability and RecQL4 when mutated leads to premature aging phenotypes in humans. Unlike other human RecQ helicases, RecQL4 is found both in the nucleus and the cytoplasm. While the nuclear localization signal (NLS) and the retention domain at the N-terminus are responsible for the nuclear localization of RecQL4, the signal for its cytoplasmic localization is essentially unknown. In this study, two functional nuclear exporting signals (NESs; pNES2 and pNES3) were identified at the C-terminus of RecQL4. Deletion of pNES2 drastically diminished the cytoplasmic localization of RecQL4. Strikingly, addition of ubiquitination tail at the C-terminus of RecQL4 substantially enriched the cytoplasmic fraction of RecQL4 only in the presence of functional pNES2. Immunofluorescence studies revealed that the cytoplasmic RecQL4 was localized in mitochondria. Consistent with its mitochondrial localization, a regulatory role for RecQL4 in the maintenance of mitochondrial DNA (mtDNA) copy number was demonstrated. Elevation of ectopic expression of RecQL4 increased the mtDNA copy number in HEK293 cells while RecQL4 knock down markedly decreased the mtDNA copy number in U2OS cells. Additionally, a substantially increased level of mitochondrial superoxide production, and a markedly decreased repair capacity for oxidative DNA damage were observed in the mitochondria of both RecQL4 deficient human fibroblasts and RecQL4-suppressed cancer cells. These data strongly suggest a regulatory role for RecQL4 in mitochondrial stability and function. Collectively, our study demonstrates that NES-mediated RecQL4 export to the cytoplasm is essential for the maintenance of mitochondrial genome stability.
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