Concentration oscillations are a ubiquitous characteristic of intracellular dynamics. The period of these oscillations can vary from few seconds to many hours, well known examples being calcium oscillations (seconds to minutes), glycolytic oscillations (minutes) and circadian rhythms (1 day). Considerable advances into understanding the mechanisms and functionality of concentration oscillations have been made since glycolytic oscillations were observed in the late 1950s, and mathematical methods have been an essential part of this process. With increasing ability to experimentally measure oscillations in single cells as well as in cell ensembles, the gold standard of modelling is to provide tools that can elucidate how the system-wide dynamics in complex organisms emerge from a system of single cells. Both abstract and detailed mechanistic models are complementary in the insight they can bring, and for networks of coupled cells considerations such as intrinsic intracellular noise, cellular heterogeneity and coupling strength are all expected to play a part. Here, we investigate separately the potential roles played by intrinsic noise arising from finite numbers of interacting molecules and by coupling among cellular oscillators. Regarding the former, it is well known that internal or molecular noise induces concentration oscillations in chemical systems whose deterministic models exhibit damped oscillations. We show, using the linear-noise approximation of the chemical master equation, that noise can also induce oscillations in biochemical systems whose deterministic descriptions admit no damped oscillations, i.e., systems with a stable node. This non-intuitive phenomenon is remarkable since, unlike noise-induced oscillations in systems with damped deterministic oscillations, it cannot be explained by noise excitation of the deterministic resonant frequency of the system. We here prove the following general properties of stable-node noise-induced oscillations for systems with two species: (i) the upper bound of their frequency is given by the geometric mean of the real eigenvalues of the Jacobian of the system, (ii) the upper bound of the Q-factor of the oscillations is inversely proportional to the distance between the real eigenvalues of the Jacobian, and (iii) these oscillations are not necessarily exhibited by all interacting chemical species in the system. The existence and properties of stable-node oscillations are verified by stochastic simulations of the Brusselator, a cascade Brusselator reaction system, and two other simple chemical systems involving autocatalysis and trimerization. We also show that external noise induces stable node oscillations with different properties than those stimulated by internal noise. Having demonstrated and explored this non-intuitive effect of noise, we extend the work to investigate the phenomenon of noise induced oscillations in cellular reaction systems characterised by the ‘bursty’ production of one or more species. Experiments have shown that proteins are typically translated in sharp bursts and similar bursty phenomena have been observed for protein import into subcellular compartments. We investigate the effect of such burstiness on the stochastic properties of downstream pathways by considering two identical systems with equal mean input rates, except in one system molecules (e.g., proteins) are input one at a time and in the other molecules are input in bursts according to some probability distribution. We find that the stochastic behaviour falls in three categories: (i) both systems display or do not display noise-induced oscillations; (ii) the non-bursty input system displays noiseinduced oscillations whereas the bursty input system does not; (iii) the reverse of (ii). We derive necessary conditions for these three cases to classify pathways involving autocatalysis, trimerization and genetic feedback loops. Our results suggest that single cell rhythms can be controlled by regulation of burstiness in protein production. We go on to investigate roles played by intercellular coupling in whole organ-level oscillations with an experimental analysis of circadian rhythms in Arabidopsis thaliana †. Circadian clocks in animals are known to be tightly coupled among the cells of some tissues, and this coupling profoundly affects cellular rhythmicity. However, research on the clock in plant cells has largely ignored intercellular coupling. Our research group used luciferase reporter gene imaging to monitor circadian rhythms in leaves of Arabidopsis thaliana plants, with both a lower resolution, high throughput method and a high-resolution (cellular level), lower throughput method. Leaves were grown and imaged in a variety of light conditions to test the relative importance of intercellular coupling and cellular coupling to the environmental signal. We analysed the high throughput data and described the circadian phase by the timing of peak expression. Leaves grown for three weeks without entrainment reproducibly showed spatio-temporal waves of gene expression, consistent with intercellular coupling. A range of patterns was observed among the leaves, rather than a unique spatio-temporal pattern, although some patterns were more often observed. All of the measured leaves exposed to lightdark entrainment cycles had fully synchronised rhythms, which could desynchronise rather quickly when placed in a non-entraining environment (i.e., constant light conditions). After four days in constant light some of these leaves were as desynchronised as leaves grown without any rhythmic input, as described by the phase coherence across the leaf. The same fast transition was observed in the reverse experimental scenario, i.e., applying light-dark cycles to leaves grown in constant light resulted in full synchronisation within two to four days. From these results we conclude that single-cell circadian oscillators were coupled far more strongly to external light-dark cycles than to the other cellular oscillators. Leaves did not spontaneously completely desynchronise, which is consistent with a presence of intercellular coupling among heterogeneous clocks. We also developed a methodology, based on the notion of two functional spatial scales of expression across the leaf, to analyse the high-resolution microscope data and determine whether there is a difference in the phase of circadian expression between vein cells and mesophyll cells in the leaf. The result from a single leaf suggests that the global phase wave dominates the phase behaviour but that there are small delays in the veins compared to their nearby mesophyll cells. This result can be validated by applying the methodology developed here to new microscope leaf data which is currently being collected in the research group. † This work was performed as a collaboration between David Toner (DT) and Benedicte Wenden (BW). BW designed and carried out the experiments, DT performed the data analysis and led on data visualisation.

<p>Intracellular motions are important signatures of living tissues, and intracellular dynamics reflect overall cell function and health. Traditional microscopy methods can track 2D cellular motions but do not provide an ensemble evaluation of intracellular activity. Biodynamic imaging (BDI) is a unique 3D imaging technique based on the phase shifts of dynamic light scattering and is highly sensitive to intracellular dynamics in living cells and their changes. This makes BDI a versatile tool to evaluate many different types of samples under various scenarios, and BDI has the potential to improve patient diagnosis and to provide valuable information for health care research. This may include evaluating sample activity, profiling patient chemotherapy response, and studying drug mechanisms. This thesis discusses the theory and modeling of BDI, the construction of BDI systems, sample heterogeneity analysis (TDSI), and the use of BDI to study cytoskeletal drug mechanisms, improve embryo selection and select therapies in pre-clinical trials.</p>

In this thesis, I discuss two main subjects coming from biology and I propose two models that mimic the behaviours of the biological networks studied. The first part of the thesis deals with intracellular transport of molecules. Proteins, RNA and, generally, any kind of cargo molecules move freely in the cytoplasm: intracellular transport as a consequence of Brownian motion is classically modelled as a diffusion process. Some specific proteins, like the tumour suppressor p53, use microtubules to facilitate their way towards the nucleus. Microtubules are a dense network of filaments that point towards the cell centre. Motor proteins bind to these filaments and move along, bearing a cargo bound to them. I propose a simplified bi-dimensional model of nucleocytoplasmic transport taking into account the kinetic processes linked to microtubule transport. Unlike in other models we know, I represented the position of a single MT filament. This model is given by a system of partial differential equations which are cast in different dimensions and connected by suitable exchange rules. A numerical scheme is introduced and several scenarios are presented and discussed to answer to the question of which proteins benefit from microtubule transport, depending on their diffusion coefficients. In the second part of the thesis, I design and analyse a physiologically based model representing the accumulation of protein p53 in the nucleus after triggering of the sentinel protein ATM by DNA damage. The p53 protein plays an essential role in the physiological maintenance of healthy tissue integrity in multicellular organisms (regulation of cell cycle arrest, repair pathways and apoptosis). Firstly, I developed a compartmental ODE model to represent the temporal dynamics of the protein. Since the p53 protein is known for its oscillatory behaviour, I performed a numerical bifurcation study to verify the existence, in the model, of stable periodic solutions. Next, I have expanded the model by the addition of a spatial variable and analysed the spatio-temporal dynamics of p53. After checking the existence of oscillations in the spatial setting, I have analysed the robustness of the system under spatial variations (diffusion and permeability coefficients, cell shape and size).