Tissue engineering and regenerative medicine aim to develop materials that mimic some of the characteristics of the tissue they are replacing and control the growth and proliferation of cells. Despite exceptional advances in the range and quality of materials used, much remains to discover about the processes regulating interfaces between cells and their surroundings, or at cell-material interfaces. In order to study and control such interactions, scientists have produced engineered matrices aiming to mimic some of the feature of natural extra-cellular matrix (biochemistry, geometry/topography and mechanical properties). In order to pattern 2D-nanofibers on relatively large areas and throughput, allowing comprehensive biological studies, we developed a nano-fabrication technique based on the deposition of sparse mats of electrospun fibres with different diameters. These mats are used as masks to grow cell resistant polymer brushes from exposed areas. After removal of the fibres, the remaining brushes define a quasi-2D fibrous pattern onto which ECM molecules such as fibronectin can be adsorbed. Chapter 2 includes details of the techniques used to produce and characterize the fibrous nanopattern. Chapter 3 is focused on cell phenotype observed on the different nanofibres sizes. Adhesion assays showed that cell spreading, shape and polarity are regulated by the size of fibres but also the density of the nanofibres, similarly to previous observations made on circular nanopatterns. We then focused on the study of focal adhesion formation and maturations on these nanofibres and the role of key proteins involved in the regulation of the adhesion plaque: integrins and vinculin. Cells expressing different integrins were found to sense the nanoscale geometry differently. Vinculin sensing is the topic of Chapter 4. Although vinculin recruitment dynamics was affected by the nanofibrous patterns and focal adhesions arrange differently on the nanofibres, this protein does not seem to mediate nanoscale sensing. In Chapter 5, we finally focused on the role of the actin cytoskeleton as a direct sensor of nanoscale geometry. A gradual decrease in stress fibre formation was observed as the nanofibres dimensions decrease. Live imaging also demonstrated that the geometry of the extracellular environment strongly affects cytoskeleton rearrangement, stress fibres formation and disassembly. We identify the role of cytoskeleton contractility as an important sensor of the nanoscale geometry. Our study provides a deeper insight in understanding cell adhesion to the extracellular environment and the role of the matrix geometry and topography on such phenomena, but also raises questions regarding the more detailed molecular sensory elements enabling the direct sensing of nanoscale geometry through the actin cytoskeleton.

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The principle of coordinate invariance states that all physical laws must be formulated in a mathematical form that is independent of the geometrical properties of any particular coordinate system. Embracing this principle is the key to understand how to systematically incorporate curved material interfaces into a numerical solution of Maxwell’s equations. This dissertation describes how to generate a coordinate invariant representation of Maxwell’s equations in differential form, and it demonstrates why employing such representation is crucial to the development of robust finite-difference discretisations with consistent global error properties. As part of this process, two original contributions are presented that address the issue of constructing finite-difference approximations at the locations of material interfaces. The first contribution is a domain-decomposition procedure to enforce the tangential field continuity conditions with a second-order local truncation error that can be applied in 2-D or 3-D. The second contribution is a similar domain-decomposition procedure that enforces the tangential field continuity conditions with a local truncation of order 2L—where L is an integer greater or equal to one—but that can only be applied in 1-D. To conclude, the dissertation also describes the interesting connection that exists between the use of a coordinate invariant representation of Maxwell’s equations to design artificial materials and the use of the same representation to model curved material interfaces in a finite-difference discretisation.

The principle of coordinate invariance states that all physical laws must be formulated in a mathematical form that is independent of the geometrical properties of any particular coordinate system. Embracing this principle is the key to understand how to systematically incorporate curved material interfaces into a numerical solution of Maxwell’s equations. This dissertation describes how to generate a coordinate invariant representation of Maxwell’s equations in differential form, and it demonstrates why employing such representation is crucial to the development of robust finite-difference discretisations with consistent global error properties. As part of this process, two original contributions are presented that address the issue of constructing finite-difference approximations at the locations of material interfaces. The first contribution is a domain-decomposition procedure to enforce the tangential field continuity conditions with a second-order local truncation error that can be applied in 2-D or 3-D. The second contribution is a similar domain-decomposition procedure that enforces the tangential field continuity conditions with a local truncation of order 2L—where L is an integer greater or equal to one—but that can only be applied in 1-D. To conclude, the dissertation also describes the interesting connection that exists between the use of a coordinate invariant representation of Maxwell’s equations to design artificial materials and the use of the same representation to model curved material interfaces in a finite-difference discretisation.