We solve the problem of Apollonius by applying the Lorentz transformation within the frame- work of special relativity to go to a frame of simultaneity, where the solutions are the circum- center. We transform back to get the solutions in a general frame. We show that the solutions in a general frame are the foci of the ellipse going through the centers of the given 3 circles.

The PII protein is an evolutionary highly conserved regulatory protein from bacteria to higher plants. In bacteria it modulates the activity of several enzymes, transporters and regulatory factors by interacting with them and thereby regulating important metabolic hubs like carbon/nitrogen homeostasis. More than two decades ago the PII protein was characterized for the first time in plants, but its physiological role is still not sufficiently resolved. To gain more insights into the function of this protein, we investigated the interaction behaviour of AtPII with candidate proteins by BiFC and FRET/FLIM in planta and with GFP/RFP traps in vitro. In the course of these studies we found that AtPII interacts in chloroplasts with itself as well as with known interactors like NAGK in dot-like aggregates, which we named PII foci. In these novel protein aggregates AtPII interacts also with yet unknown partners, which are known to be involved in plastidic protein degradation. Further studies revealed that the C-terminal part of AtPII is crucial for the formation of PII foci. Altogether, the presented results indicate a novel mode of interaction for PII proteins with other proteins in plants, which may be a new starting point for the elucidation of physiological functions of PII proteins in plants.

Classical models of gene expression were built using genetics and biochemistry. Although these approaches are powerful, they have very limited consideration of the spatial and temporal organization of gene expression. Although the spatial organization and dynamics of RNA polymerase II (RNAPII) transcription machinery has fundamental functional consequences for gene expression, its detailed studies have been for long time abrogated by the limits of classical light microscopy. The advent of super-resolution microscopy (SRM) techniques allowed for the visualization of the RNAPII transcription machinery with nanometer resolution and millisecond precision. In this review, we summarize the recent methodological advances in SRM, focus on its application for studies of the nanoscale organization in space and time of RNAPII transcription, and discuss its consequences for the mechanistic understanding of gene expression.