Cell motility is closely linked to many important physiological and pathological events such as the immune response, wound healing, tissue differentiation, embryogenesis, in ammation, tumour invasion and metastasis. Understanding the ability of cells to alter their shape, deform and migrate is of vital importance in many biological studies. The rapid development in microscopy and imaging techniques has generated a huge amount of discrete data on migrating cells in vivo and in vitro. A key challenge is the use of discrete experimental observations to develop novel methods and algorithms that track cells and construct continuous trajectories of their motion as well as characterising key geometric quantities associated with cell migration. Therefore, in this work using robust numerical tools we focus on proposing and implementing mathematical methodologies for cell movement and apply them to model neutrophil cell migration. We derive and implement a computational framework that encompasses modelling of cell motility and cell tracking based on phase field and optimal control theory. The cell membrane is represented by an evolving curve and approximated by a diffuse interface; while the motion of the cell is driven by a force balance acting normal on the cell membrane. This approach allows us to characterise the locus of the centroid cell-surface position. In addition, we describe a surface partial differential equation framework that can be coupled with the phase-field framework, thereby offering a wholistic approach for modelling biochemical processes and biomechanics properties associated with cell migration.

RNA interference (RNAi), or Post-Transcriptional Gene Silencing (PTGS), is a biological process which uses small RNAs to regulate gene expression on a cellular level, typically by causing the destruction of specfic mRNA molecules. This biological pathway is found in both plants and animals, and can be used as an effective strategy in defending cells against parasitic nucleotide sequences, viruses and transposons. In the case of plants, it also constitutes a major component of the adaptive immune system. RNAi is characterised by the ability to induce sequence-specific degradation of target messenger RNA (mRNAs) and methylation of target gene sequences. The small interfering RNA produced within the initiated cell is not only used locally but can also be transported into neighbouring cells, thus acting as a mobile warning signal. In the first part of the thesis I develop and analyse a new mathematical model of the plant immune response to a viral infection, with particular emphasis on the role of RNA interference. The model explicitly includes two different time delays, one to represent the maturation period of undifferentiated cells, and another to account for the time required for the RNAi propagating signal to reach other parts of the plant, resulting in either recovery or warning of susceptible cells. Analytical and numerical bifurcation theory is used to identify parameter regions associated with recovery and resistant plant phenotypes, as well as possible chronic infections. The analysis shows that the maturation time plays an important role in determining the dynamics, and that long-term host recovery does not depend on the speed of the warning signal but rather on the strength of local recovery. At best, the warning signal can amplify and hasten recovery, but by itself it is not competent at eradicating the infection. In the second part of the thesis I derive and analyse a new mathematical model of plant viral co-infection with particular account for RNA-mediated cross-protection in a single plant host. The model exhibits four non-trivial steady states, i.e. a disease-free steady state, two one-virus endemic equilibria, and a co-infected steady state. I obtained the basic reproduction number for each of the two viral strains and performed extensive numerical bifurcation analysis to investigate the stability of all steady states and identified parameter regions where the system exhibits synergistic or antagonistic interactions between viral strains, as well as different types of host recovery. The results indicate that the propagating component of RNA interference plays a significant role in determining whether both viruses can persist simultaneously, and as such, it controls whether the plant is able to support some constant level of both infections. If the two viruses are sufficiently immunologically related, the least harmful of the two viruses becomes dominant, and the plant experiences cross-protection. In the third part of the thesis I investigate the properties of intracellular dynamics of RNA interference and its capacity as a gene regulator by extending a well known model of RNA interference with time delays. For each of the two amplification pathways of the model, I consider the cumulative effects of delay in dsRNA-primed synthesis associated with the non-instantaneous nature of chemical signals and component transportation delay. An extensive bifurcation analysis is performed to demonstrate the significance of different parameters, and to investigate how time delays can affect the bi-stable regime in the model.

The focus of this thesis is to propose and implement a highly efficient numerical method to study cell dynamics. Three key phases are covered: mathematical modelling, linear stability analytical theory and numerical simulations using the moving grid finite element method. This aim is to study cell deformation and cell movement by considering both the mechanical and biochemical properties of the cortical network of actin filaments and its concentration. These deformations are assumed to be a result of the cortical actin dynamics through its interaction with a protein known as myosin II in the cell cytoskeleton. The mathematical model that we consider is a continuum model that couples the mechanics of the network of actin filaments with its bio-chemical dynamics. Numerical treatment of the model is carried out using the moving grid finite element method. By assuming slow deformations of the cell boundary, we verify the numerical simulation results using linear stability theory close to bifurcation points. Far from bifurcation points, we show that the model is able to describe the deformation of cells as a function of the contractile tonicity of the complex formed by the association of actin filaments with the myosin II motor proteins. Our results show complex cell deformations and cell movements such as cell expansion, contraction, translation and protrusions in accordance with experimental observations. The migratory behaviour of cells plays a crucial role in many biological events such as immune response, wound healing, development of tissues, embryogenesis, inflammation and the formation of tumours.