We present a palette of brightly fluorescent genetically encoded voltage indicators (GEVIs) with excitation and emission peaks spanning the visible spectrum, sensitivities from 6 - 10% Delta F/F per 100 mV, and half-maximal response times from 1 - 7
ms. A fluorescent protein is fused to an Archaerhodopsin-derived voltage sensor. Voltage-induced shifts in the absorption spectrum of the rhodopsin lead to voltage-dependent nonradiative quenching of the appended fluorescent protein. Through a library screen, we identified linkers and fluorescent protein combinations which reported neuronal action potentials in cultured rat hippocampal neurons with a single-trial signal-to-noise ratio from 6.6 to 11.6 in a 1 kHz imaging bandwidth at modest illumination intensity. The freedom to choose a voltage indicator from an array of colors facilitates multicolor voltage imaging, as well as combination with other optical reporters and optogenetic actuators.
We consider the problem of partitioning the set of vertices of a given unit disk graph (UDG) into a minimum number of cliques. The problem is NP-hard and various constant factor approximations are known, with the current best ratio of 3. Our main res
ult is a {em weakly robust} polynomial time approximation scheme (PTAS) for UDGs expressed with edge-lengths, it either (i) computes a clique partition or (ii) gives a certificate that the graph is not a UDG; for the case (i) that it computes a clique partition, we show that it is guaranteed to be within $(1+eps)$ ratio of the optimum if the input is UDG; however if the input is not a UDG it either computes a clique partition as in case (i) with no guarantee on the quality of the clique partition or detects that it is not a UDG. Noting that recognition of UDGs is NP-hard even if we are given edge lengths, our PTAS is a weakly-robust algorithm. Our algorithm can be transformed into an $O(frac{log^* n}{eps^{O(1)}})$ time distributed PTAS. We consider a weighted version of the clique partition problem on vertex weighted UDGs that generalizes the problem. We note some key distinctions with the unweighted version, where ideas useful in obtaining a PTAS breakdown. Yet, surprisingly, it admits a $(2+eps)$-approximation algorithm for the weighted case where the graph is expressed, say, as an adjacency matrix. This improves on the best known 8-approximation for the {em unweighted} case for UDGs expressed in standard form.
