scholarly journals Comparing homologies: Čech's theory, singular chains, integral flat chains and integral currents

2007 ◽  
pp. 143-189 ◽  
Author(s):  
Thierry De Pauw
Keyword(s):  
Flat Chains ◽  
Integral Currents

scholarly journals On the structure of flat chains modulo p

2018 ◽  
Vol 11 (3) ◽  
pp. 309-323 ◽  
Author(s):  
Andrea Marchese ◽  
Salvatore Stuvard
Keyword(s):  
Integral Current ◽  
Flat Chains ◽  
Quantitative Estimates ◽  
The Family ◽  
Flat Chain ◽  
Modulo P ◽  
Integral Currents ◽  
Flat Norm ◽  
Closedness Property ◽  
Dimensional Integral

AbstractIn this paper, we prove that every equivalence class in the quotient group of integral 1-currents modulo p in Euclidean space contains an integral current, with quantitative estimates on its mass and the mass of its boundary. Moreover, we show that the validity of this statement for m-dimensional integral currents modulo p implies that the family of {(m-1)}-dimensional flat chains of the form pT, with T a flat chain, is closed with respect to the flat norm. In particular, we deduce that such closedness property holds for 0-dimensional flat chains, and, using a proposition from The structure of minimizing hypersurfaces mod 4 by Brian White, also for flat chains of codimension 1.

scholarly journals Flat chains of finite size in metric spaces

2013 ◽  
Vol 30 (1) ◽  
pp. 79-100 ◽  
Author(s):  
Luigi Ambrosio ◽  
Francesco Ghiraldin
Keyword(s):  
Metric Spaces ◽  
Finite Size ◽  
Flat Chains

Stokes' Theorem for Integral Currents Modulo ν

1977 ◽  
Vol 99 (2) ◽  
pp. 379
Author(s):  
Sandra O. Paur
Keyword(s):  
Integral Currents

scholarly journals Mammalian retromer is an adaptable scaffold for cargo sorting from endosomes

2019 ◽  
Author(s):  
Amy K. Kendall ◽  
Boyang Xie ◽  
Peng Xu ◽  
Jue Wang ◽  
Rodger Burcham ◽  
...  
Keyword(s):  
Molecular Mechanisms ◽  
Core Complex ◽  
Sorting Nexin ◽  
Flat Chains ◽  
Biophysical Studies ◽  
Cargo Proteins ◽  
Golgi Network ◽  
Multi Body ◽  
Cargo Sorting

AbstractIn metazoans, retromer (VPS26/VPS35/VPS29) associates with sorting nexin (SNX) proteins to form coats on endosomal tubules and sort cargo proteins to the trans-Golgi network (TGN) or plasma membrane. This core complex is highly conserved from yeast to humans, but molecular mechanisms of metazoan retromer assembly remain undefined. Here we combine single particle cryo-electron microscopy with biophysical methods to uncover multiple oligomer structures formed by mammalian retromer. Two-dimensional class averages in ice reveal the retromer heterotrimer; dimers of trimers; tetramers of trimers; and flat chains. These species are further supported by biophysical studies in solution. We provide cryo-EM reconstructions of all species, including pseudo-atomic resolution detail for key sub-structures. Multi-body refinement demonstrates how retromer heterotrimers and dimers adopt a range of conformations. Our structures identify a flexible yet highly conserved electrostatic interface in dimers formed by interactions between VPS35 subunits. We generate a structure-based mutant to disrupt this key interface in vitro and introduce equivalent mutations into S. cerevisiae to demonstrate the mutant exhibits a cargo sorting defect. Together, structures and complementary functional data in budding yeast imply a conserved assembly interface across eukaryotes. These data further suggest mammalian retromer acts as an adaptable and plastic scaffold that accommodates interactions with different SNXs to sort multiple cargoes from endosomes their final destinations.

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