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HIF (Hypoxia Inducible Factor) is an oxygen-regulated transcription factor that mediates the intracellular response to hypoxia in human cells. There is increasing evidence that cell signaling pathways encode temporal information, and thus cell fate may be determined by the dynamics of protein levels. We have developed a mathematical model to describe the transient dynamics of the HIF-1�� protein measured in single cells subjected to hypoxic shock. The essential characteristics of these data are modeled with a system of differential equations describing the feedback inhibition between HIF-1�� and Prolyl Hydroxylases (PHD) oxygen sensors. Heterogeneity in the single-cell data is accounted for through parameter variation in the model. We previously identified the PHD2 isoform as the main PHD responsible for controlling the HIF-1�� transient response, and make here testable predictions regarding HIF-1�� dynamics subject to repetitive hypoxic pulses. The model is further developed to describe the dynamics of HIF-1�� in cells cultured as 3D spheroids, with oxygen dynamics parameterized using experimental measurements of oxygen within spheroids. We show that the dynamics of HIF-1�� and transcriptional targets of HIF-1�� display a non-monotone response to the oxygen dynamics. Specifically we demonstrate that the dynamic transient behavior of HIF-1�� results in differential dynamics in transcriptional targets.

HIF-1 (Hypoxia Inducible Factor-1) is an oxygen-regulated transcription factor that mediates the intracellular response to hypoxia in human cells, targeting specific genes that promote cell survival by inducing processes such as angiogenesis and glycolysis. The HIF-1 signalling pathway has been of considerable interest due to its role in mammalian development and in particular several pathologies such as ischemia and cancer. Low cellular oxygen levels are found in cancer due to the rapid proliferation of tumour cells distant to blood vessels (forming a hypoxic core) and the typically irregular vasculature is unable to properly perfuse the tumour. In the Centre for Cell Imaging at the University of Liverpool, time lapsed imaging in an oxygen-controlled environment has captured the transient dynamics of the oxygen regulated subunit of HIF-1, HIF-1α, and revealed heterogeneity between individual cells. The essential characteristics of this data are modelled with a system of differential equations describing the feedback inhibition between HIF-1α and the pathway’s effective oxygen sensors. This novel model was formulated by employing a minimalist approach initially, allowing us to use the rich variety of single-cell data to determine the structure of the feedback loop between two key pathway components. Once the central regulatory motif was identified, the model was expanded to include more complexity, including experimental measurements for model parameter values and additional system components. Oxygen plays an especially important role in the early stages of tumour formation. Measurements of oxygen within cells have been recorded in cultured spheres of neuroblastoma cells at different atmospheric conditions. This data is representative of the early development stages of an avascular tumour and a spatial oxygen-diffusion model was coupled with the single-cell HIF-1α model to describe the dynamics of HIF-1α across a developing tumour. This coupled model is used to study HIF-1α dynamics in the context of various oxygen-dependent cellular functions such as cell-cycle progression and apoptosis by integrating with modifications of published mathematical models.

Mathematical modelling details for microscale transport; parameterisation of passive diffusion; and translating the multiscale model to a simple continuum model.

Mathematical modelling details for microscale transport; parameterisation of passive diffusion; and translating the multiscale model to a simple continuum model.

Nanoparticles have the potential to increase the efficacy of anticancer drugs whilst reducing off-target side effects. However, there remain uncertainties regarding the cellular uptake kinetics of nanoparticles which could have implications for nanoparticle design and delivery. Polymersomes are nanoparticle candidates for cancer therapy which encapsulate chemotherapy drugs. Here we develop a mathematical model to simulate the uptake of polymersomes via endocytosis, a process by which polymersomes bind to the cell surface before becoming internalised by the cell where they then break down, releasing their contents which could include chemotherapy drugs. We focus on two in vitro configurations relevant to the testing and development of cancer therapies: a well-mixed culture model and a tumour spheroid setup. Our mathematical model of the well-mixed culture model comprises a set of coupled ordinary differential equations for the unbound and bound polymersomes and associated binding dynamics. Using a singular perturbation analysis we identify an optimal number of ligands on the polymersome surface which maximises internalised polymersomes and thus intracellular chemotherapy drug concentration. In our mathematical model of the spheroid, a multiphase system of partial differential equations is developed to describe the spatial and temporal distribution of bound and unbound polymersomes via advection and diffusion, alongside oxygen, tumour growth, cell proliferation and viability. Consistent with experimental observations, the model predicts the evolution of oxygen gradients leading to a necrotic core. We investigate the impact of two different internalisation functions on spheroid growth, a constant and a bond dependent function. It was found that the constant function yields faster uptake and therefore chemotherapy delivery. We also show how various parameters, such as spheroid permeability, lead to travelling wave or steady-state solutions.