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It is well-known that the resolution of traditional optical imaging system is limited by the so-called Rayleigh resolution or diffraction limit, which is of several hundreds of nanometers. By employing fluorescence techniques, modern microscopic methods can resolve point scatterers separated by a distance much lower than the Rayleigh resolution limit. Localization-based fluorescence subwavelength imaging techniques such as PALM and STORM can achieve spatial resolution of several tens of nanometers. However, these techniques have limited temporal resolution as they require tens of thousands of exposures. Employing sparsity-based models and recovery algorithms is a natural way to reduce the number of exposures, and hence obtain high temporal resolution. Nevertheless, to date fluorescence techniques suffer from the trade-off between spatial and temporal resolutions. In a recent work by the last four authors, a newly multi-illumination imaging technique called Brownian Excitation Amplitude Modulation microscopy (BEAM) is introduced. BEAM achieves a threefold resolution improvement by applying a compressive sensing recovery algorithm over only few frames. Motivated by BEAM, our aim in this paper is to pioneer the mathematical foundation for sparsity-based multi-illumination super-resolution. More precisely, we consider several diffraction-limited images from sample exposed to different illumination patterns and recover the source by considering the sparsest solution. We estimate the minimum separation distance between point scatterers so that they could be stably recovered. By this estimation of the resolution of the sparsity recovery, we reveal the dependence of the resolution on the cut-off frequency of the imaging system, the signal-to-noise ratio, the sparsity of point scatterers, and the incoherence of illumination patterns. Our theory particularly highlights the importance of the high incoherence of illumination patterns in enhancing the resolution. It also demonstrates that super-resolution can be achieved using sparsity-based multi-illumination imaging with very few frames, whereby the spatio-temporal super-resolution becomes possible. BEAM can be viewed as the first experimental realization of our theory, which is demonstrated to hold in both the one- and two-dimensional cases.

Eph receptors (Ephs) and their membrane bound ephrin ligands constitute one of the major guidance cues that control developmental and pathological cell positioning. As in other receptor tyrosine kinases (RTKs), ligand binding induces oligomerization of Ephs at the plasma membrane (PM), activation of their intrinsic autocatalytic activity and phosphorylation of key tyrosine residues in their intracellular domain. In this thesis, we asked how the autocatalytic activity of Eph is dynamically controlled inside cells to prevent spurious self-activation yet allow for robust activation upon exposure to ligand. The current view in Eph signaling is that ligand-driven oligomerization of Ephs converts inactive monomers to active oligomers. However, in this thesis we show that the mechanism of receptor clustering is not that simple. Using an FKBPbased dimerizing system, we investigated the effect of cluster size composition on EphB2 cellular output. We found that small-sized EphB2 clusters produce a functional response, whose strength is determined by the abundance of multimers over dimers within a cluster population. In addition, we identified a positive regulatory effect on EphB2 clustering by GRIP1 protein and a negative regulatory effect on ephrin-induced clustering by C-terminal domains of EphB2. Together these results showed how the cell’s pre-established molecular context could determine the response of Ephs to ephrins, which is then translated into clusters with different size, composition and/or stability. The autocatalytic activity of Eph receptors is regulated by reversible autophosphorylation and dephosphorylation cycles that are spatially and temporally partitioned within cells due to the interactions with protein tyrosine phosphatases (PTPs). Using quantitative cellular imaging approaches, we found that the endoplasmic reticulum (ER)-anchored PTP1B comes in close proximity to regions of cell-cell contact rich in Eph receptors and identified EphA2 as a new substrate for PTP1B specifically at those regions. These results highlighted ER-PM interactions as an emerging new paradigm in cellular signaling. To investigate how Eph conformational dynamics as well as its spatial organization within cells can affect its inherent autocatalytic activity, we designed a genetically encoded biosensor that monitors EphA2 conformation, termed Linker optimized Intramolecular-FRET based sensor for EphA2 (LIFEA2). By measurements of LIFEA2 conformational dynamics, we could describe EphA2 activation with precise spatial temporal resolution. In addition, by correlating the cell-to-cell variance in LIFEA2 expression level to its activity state, we demonstrated the propensity of this autocatalytic system to self-activate in the absence of ligand. Finally, we described two levels of regulation for EphA2 activation, a cis-inhibitory dynamic interaction between the kinase domain and the juxtamembrane segment, and a novel PM-recycling mediated mechanism. The continuous recycling of EphA2 to the PM acts as a safeguard mechanism by maintaining a low steady-state level of receptor at the PM and by trafficking the receptor through peri-nuclear areas with high PTP1B activity that dephosphorylates spuriously autoactivated receptors.