In this paper, we discuss the local and global existence of week solutions for some hyperbolic-parabolic systems modelling chemotaxis.

Single-molecule localisation microscopy (SMLM), has allowed for optical microscopy to probe biological systems beyond the diffraction limit. The intrinsic 3D nature of biology has motivated the development of 3D-SMLM with novel techniques, including the double-helix point spread function (DHPSF). A bespoke microscope platform employing the DHPSF transformation was built, achieving ~10 nm lateral and ~20 nm axial localisation precision over a ~4 μm axial depth. Until recently, the DHPSF has been limited by spherical aberration present when imaging away from coverslip surfaces to the study of small volumes close to the coverslip. By matching the refractive index of the objective lens immersion liquid to that of the imaging media, this aberration can be minimised, facilitating large-volume imaging away from unphysiological flat surfaces. The work presented in this thesis illustrates the capabilities of the DHPSF for 3D-SMLM and single-particle tracking (SPT) in previously inaccessible areas of biological samples (e.g. in the nucleus and on the apical cell surface). Application of the DHPSF for SPT in eukaryotic cells are presented; tracking the motion of T-cell membrane proteins on the apical surface and components of the chromosome remodelling complex in the nucleus of embryonic stem cells. For these applications, meansquared displacement and jump distance diffusion analysis methodologies were extended into 3D and benchmarked against simulated datasets. A variety imaging applications that are facilitated by the extended depth of focus of the DHPSF are presented, focusing on quantification of T-cell membrane protein reorganisation upon immunological activation. Finally, the clustering distribution of the T-cell receptor is investigated by Ripley’s K analysis enabled by duel labelling of its position and the outer membrane in primary T cells.