ASYMPTOTIC PROFILE OF QUENCHING IN A SYSTEM OF HEAT EQUATIONS COUPLED AT THE BOUNDARY

We study finite time quenching for the radial solutions of a system of heat equations coupled at the boundary condition. This system exhibits simultaneous and non-simultaneous quenching. In particular, three kinds of simultaneous quenching profiles are obtained for different nonlinear exponent regions and appropriate initial data

Four-wave mixing heterodyne interferometric imaging of gold nanoparticles in cells

Gold nanoparticles (NPs) are widely utilised for bio-imaging applications owing to their facile synthesis, ease of surface functionalisation and bio-conjugation, as well as bio-compatibility. However, when embedded in highly heterogeneous and fluo- rescing environments such as biological cells and tissues, these NPs have to be large (typically >50 nm diameter) to be distinguished optically against backgrounds via their linear absorption and scattering at the surface plasmon resonance (SPR). As a result, cell imaging protocols often adopt the use of fluorophore tags attached onto the NP, and assume that the fluorophore is a reliable reporter. These constructs are routinely used in correlative microscopy techniques such as Correlative Light Elec- tron microscopy (CLEM) and presume that the fluorophore colocalises well with the electron dense NP. In this thesis, I present the application of our recently developed four-wave-mixing (FWM) imaging modality to investigate the spatial correlation be- tween a gold NP and a fluorescently labeled biomolecule attached onto it on a variety of gold NPs-fluorophore conjugate constructs inside biological cells. Owing to the specific nature of the FWM process, which is a non-linear light matter interaction with the free-electron gas in the gold metal, I was able to detect single gold NPs down to 10 nm diameter background-free and with high 3D spatial resolution even in highly heterogeneous and fluorescing cellular environments. The FWM signal from gold was spatially correlated with the fluorescence signal of the fluorophore at- tached onto the biomolecule of the construct. Surprisingly, even covalently attached conjugate constructs exhibited low colocalization highlighting the limitations of flu- orescence tagging as a means to address fundamental questions about intracellular pathways of high biological interest. This study opens new prospects to the use of FWM microscopy for imaging small gold NPs potentially even in live cells and to the understanding of the cellular uptake of bio-conjugated gold NPs