A mass reducing flow for real-valued flat chains with applications to transport networks

An oriented transportation network can be modeled by a 1-dimensional chain whose boundary is the difference between the demand and supply distributions, represented by weighted sums of point masses. To accommodate efficiencies of scale into the model, one uses a suitable $\mathbf{M}^{\alpha}$ norm for transportation cost for $\alpha \in (0,1]$. One then finds that the minimal cost network has a branching structure since the norm favors higher multiplicity edges, representing shared transport. In this paper, we construct a continuous flow that evolves some initial such network to reduce transport cost without altering its supply and demand distributions. Instead of limiting our scope to transport networks, we construct this $\mathbf{M}^{\alpha}$ mass reducing flow for real-valued flat chains by finding a higher dimensional real chain whose slices dictate the flow. Keeping the boundary fixed, this flow reduces the $\mathbf{M}^{\alpha}$ mass of the initial chain and is Lipschitz continuous under the flat-$\alpha$ norm. To complete the paper, we apply this flow to transportation networks, showing that the flow indeed evolves branching transport networks to be more cost efficient.

Mammalian retromer is an adaptable scaffold for cargo sorting from endosomes

In 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.

On the structure of flat chains modulo p

In 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.