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Abstract The rank of a tiling’s return module depends on the geometry of its tiles and is not a topological invariant. However, the rank of the first Čech cohomology$$\check H^1(\Omega )$$gives upper and lower bounds for the rank of the return module. For all sufficiently large patches, the rank of the return module is at most the rank of$$\check H^1(\Omega )$$. For a generic choice of tile shapes and an arbitrary reference patch, the rank of the return module is at least the rank of$$\check H^1(\Omega )$$. Therefore, for generic tile shapes and all sufficiently large patches, the rank of the return module is equal to the rank of$$\check H^1(\Omega )$$. 

Abstract We prove moderate deviations bounds for the lower tail of the number of odd cycles in a random graph. We show that the probability of decreasing triangle density by , is whenever . These complement results of Goldschmidt, Griffiths, and Scott, who showed that for , the probability is . That is, deviations of order smaller than behave like small deviations, and deviations of order larger than behave like large deviations. We conjecture that a sharp change between the two regimes occurs for deviations of size , which we associate with a single large negative eigenvalue of the adjacency matrix becoming responsible for almost all of the cycle deficit. We give analogous results for the ‐cycle density, for all odd . Our results can be interpreted as finite size effects in phase transitions in constrained random graphs. 

How does a single amino acid mutation occurring in the blinding disease, Leber’s hereditary optic neuropathy (LHON), impair electron shuttling in mitochondria? We investigated changes induced by the m.3460 G>A mutation in mitochondrial protein ND1 using the tools of Molecular Dynamics and Free Energy Perturbation simulations, with the goal of determining the mechanism by which this mutation affects mitochondrial function. A recent analysis suggested that the mutation’s replacement of alanine A52 with a threonine perturbs the stability of a region where binding of the electron shuttling protein, Coenzyme Q10, occurs. We found two functionally opposing changes involving the role of Coenzyme Q10. The first showed that quantum electron transfer from the terminal Fe/S complex, N2, to the Coenzyme Q10 headgroup, docked in its binding pocket, is enhanced. However, this positive adjustment is overshadowed by our finding that the mobility of Coenzyme Q10 in its oxidized and reduced states, entering and exiting its binding pocket, is disrupted by the mutation in a manner that leads to conditions promoting the generation of reactive oxygen species. An increase in reactive oxygen species caused by the LHON mutation has been proposed to be responsible for this optic neuropathy.