Spatio-temporal models of synthetic genetic oscillators

Signal transduction pathways play a major role in many important aspects of cellular function e.g. cell division, apoptosis. One important class of signal transduction pathways is gene regulatory networks (GRNs). In many GRNs, proteins bind to gene sites in the nucleus thereby altering the transcription rate. Such proteins are known as transcription factors. If the binding reduces the transcription rate there is a negative feedback leading to oscillatory behaviour in mRNA and protein levels, both spatially (e.g. by observing fluorescently labelled molecules in single cells) and temporally (e.g. by observing protein/mRNA levels over time). Recent computational modelling has demonstrated that spatial movement of the molecules is a vital component of GRNs and may cause the oscillations. These numerical findings have subsequently been proved rigorously i.e. the diffusion coefficient of the protein/mRNA acts as a bifurcation parameter and gives rise to a Hopf bifurcation. In this paper we first present a model of the canonical GRN (the Hes1 protein) and show the effect of varying the spatial location of gene and protein production sites on the oscillations. We then extend the approach to examine spatio-temporal models of synthetic gene regulatory networks e.g. n-gene repressilators and activator-repressor systems.PostprintPeer reviewe

Mathematical modelling of p53 signalling during DNA damage response: a survey

No gene has garnered more interest than p53 since its discovery over 40 years ago. In the last two decades, thanks to seminal work from Uri Alon and Ghalit Lahav, p53 has defined a truly synergistic topic in the field of mathematical biology, with a rich body of research connecting mathematic endeavour with experimental design and data. In this review we survey and distill the extensive literature of mathematical models of p53. Specifically, we focus on models which seek to reproduce the oscillatory dynamics of p53 in response to DNA damage. We review the standard modelling approaches used in the field categorising them into three types: time delay models, spatial models and coupled negative-positive feedback models, providing sample model equations and simulation results which show clear oscillatory dynamics. We discuss the interplay between mathematics and biology and show how one informs the other; the deep connections between the two disciplines has helped to develop our understanding of this complex gene and paint a picture of its dynamical response. Although yet more is to be elucidated, we offer the current state-of-the-art understanding of p53 response to DNA damage

Spatial-stochastic modelling of synthetic gene regulatory networks

Funding: EPSRC Grant No. EP/N014642/1 (EPSRC Centre for Multiscale Soft Tissue Mechanics - With Application to Heart & Cancer) (MAJC,CKM).Transcription factors are important molecules which control the levels of mRNA and proteins within cells by modulating the process of transcription (the mechanism by which mRNA is produced within cells) and hence translation (the mechanism by which proteins are produced within cells). Transcription factors are part of a wider family of molecular interaction networks known as gene regulatory networks (GRNs) which play an important role in key cellular processes such as cell division and apoptosis (e.g. the p53-Mdm2, NFκB pathways). Transcription factors exert control over molecular levels through feedback mechanisms, with proteins binding to gene sites in the nucleus and either up-regulating or down-regulating production of mRNA. In many GRNs, there is a negative feedback in the network and the transcription rate is reduced. Typically, this leads to the mRNA and protein levels oscillating over time and also spatially between the nucleus and cytoplasm. When experimental data for such systems is analysed, it is observed to be noisy and in many cases the actual numbers of molecules involved are quite low. In order to model such systems accurately and connect with the data in a quantitative way, it is therefore necessary to adopt a stochastic approach as well as take into account the spatial aspect of the problem. In this paper, we extend previous work in the area by formulating and analysing stochastic spatio-temporal models of synthetic GRNs e.g. repressilators and activator-repressor systems.PostprintPeer reviewe

Computational modelling and simulation of cancer growth and migration within a 3D heterogeneous tissue: The effects of fibre and vascular structure

The term cancer covers a multitude of bodily diseases, broadly categorised by having cells which do not behave normally. Since cancer cells can arise from any type of cell in the body, cancers can grow in or around any tissue or organ making the disease highly complex. Our research is focused on understanding the specific mechanisms that occur in the tumour microenvironment via mathematical and computational modeling. We present a 3D individual-based model which allows one to simulate the behaviour of, and spatio-temporal interactions between, cells, extracellular matrix fibres and blood vessels. Each agent (a single cell, for example) is fully realised within the model and interactions are primarily governed by mechanical forces between elements. However, as well as the mechanical interactions we also consider chemical interactions, for example, by coupling the code to a finite element solver to model the diffusion of oxygen from blood vessels to cells. The current state of the art of the model allows us to simulate tumour growth around an arbitrary blood-vessel network or along the striations of fibrous tissue

The period ratio P₁/2P₂ in coronal waves

Increasing observational evidence of wave modes brings us to a closer understanding of the solar corona. Coronal seismology allows us to combine wave observations and theory to determine otherwise unknown parameters. The period ratio, P₁/2P₂, between the period P₁ of the fundamental mode and the period P₂ of its first overtone is one such tool of coronal seismology and its departure from unity provides information about the structure of the corona. In this thesis we consider the period ratio P₁/2P₂ of coronal loops from a theoretical standpoint. Previous theory and observations indicate that the period ratio is likely to be less than unity for oscillations of coronal loops. We consider the role of damping and density structuring on the period ratio. In Chapter 2 we consider analytically the one-dimensional wave equation with the inclusion of a generic damping term for both uniform and non-uniform media. Results suggest that the period ratio is dominated by longitudinal structuring rather than damping. In Chapter 3 we consider analytically the effects of thermal conduction and compressive viscosity on the period ratio for a longitudinally propagating sound wave. We find that damping by either thermal conduction or compressive viscosity typically has a small effect on the period ratio. For coronal values of thermal conduction the effect on the period ratio is negligible. For compressive viscosity the effect on the period ratio may become important for some short hot loops. In Chapter 4 we extend the analysis of Chapter 3 to include radiative cooling and find that it too has a negligible effect on the period ratio for typical coronal values. As an extension to the investigation, damping rates are considered for thermal conduction, compressive viscosity and radiative cooling. The damping time is found to be optimal for each mechanism in a different temperature range, namely below 1 MK for radiative cooling, 2 − 6 MK for thermal conduction and above 6 MK for compressive viscosity. In Chapter 5 we consider analytically the period ratio for the fast kink, sausage and n = N modes of a magnetic slab, discussing both an Epstein density profile and a simple step function profile. We find that transverse density structuring in the form of an Epstein profile or a step function profile may contribute to the shift of the period ratio for long thin slab-like structures. The similarity in the behaviour of the period ratio for both profiles means either can be used as a robust model. We consider also other profiles numerically for the kink mode, which are found to be either slab-like or Epstein-like suggesting again that it is not necessary to distinguish the nature of the density profile when considering the period ratio

Biomechanical modelling of cancer:agent‐based force‐based models of solid tumours within the context of the tumour microenvironment

Once cancer is initiated, with normal cells mutated into malignant ones, a solid tumour grows, develops and spreads within its microenvironment invading the local tissue; the disease progresses and the cancer cells migrate around the body leading to metastasis, the formation of distant secondary tumours. Interactions between the tumour and its microenvironment drive this cascade of events which have devastating, if not fatal, consequences for the human host/patient. Among these interactions, biomechanical interactions are a vital component. In this review paper, key biomechanical relationships are discussed through a presentation of modelling efforts by the mathematical and computational oncology community. The main focus is directed, naturally, towards lattice‐free agent‐based, force‐based models of solid tumour growth and development. In such models, interactions between pairs of cancer cells (as well as between cells and other structures of the tumour microenvironment) are governed by forces. These forces are ones of repulsion and adhesion, and are typically modelled via either an extended Hertz model of contact mechanics or using Johnson–Kendal–Roberts theory, both of which are discussed here. The role of the extracellular matrix in determining disease progression is outlined along with important cell‐vessel interactions which combined together account for a great proportion of Hanahan and Weinberg's Hallmarks of Cancer

Biomechanical modelling of cancer : agent‐based force‐based models of solid tumours within the context of the tumour microenvironment

CKM gratefully acknowledges the support of EPSRC Grant No. EP/N014642/1 (EPSRC Centre for Multiscale Soft Tissue Mechanics - With Application to Heart & Cancer).Once cancer is initiated, with normal cells mutated into malignant ones, a solid tumour grows, develops and spreads within its microenvironment invading the local tissue; the disease progresses and the cancer cells migrate around the body leading to metastasis, the formation of distant secondary tumours. Interactions between the tumour and its microenvironment drive this cascade of events which have devastating, if not fatal, consequences for the human host/patient. Among these interactions, biomechanical interactions are a vital component. In this review paper, key biomechanical relationships are discussed through a presentation of modelling efforts by the mathematical and computational oncology community. The main focus is directed, naturally, towards lattice‐free agent‐based, force‐based models of solid tumour growth and development. In such models, interactions between pairs of cancer cells (as well as between cells and other structures of the tumour microenvironment) are governed by forces. These forces are ones of repulsion and adhesion, and are typically modelled via either an extended Hertz model of contact mechanics or using Johnson–Kendal–Roberts theory, both of which are discussed here. The role of the extracellular matrix in determining disease progression is outlined along with important cell‐vessel interactions which combined together account for a great proportion of Hanahan and Weinberg's Hallmarks of Cancer.Publisher PDFPeer reviewe

The period ratio P₁/2P₂ in coronal waves

Increasing observational evidence of wave modes brings us to a closer understanding of the solar corona. Coronal seismology allows us to combine wave observations and theory to determine otherwise unknown parameters. The period ratio, P₁/2P₂, between the period P₁ of the fundamental mode and the period P₂ of its first overtone is one such tool of coronal seismology and its departure from unity provides information about the structure of the corona. In this thesis we consider the period ratio P₁/2P₂ of coronal loops from a theoretical standpoint. Previous theory and observations indicate that the period ratio is likely to be less than unity for oscillations of coronal loops. We consider the role of damping and density structuring on the period ratio. In Chapter 2 we consider analytically the one-dimensional wave equation with the inclusion of a generic damping term for both uniform and non-uniform media. Results suggest that the period ratio is dominated by longitudinal structuring rather than damping. In Chapter 3 we consider analytically the effects of thermal conduction and compressive viscosity on the period ratio for a longitudinally propagating sound wave. We find that damping by either thermal conduction or compressive viscosity typically has a small effect on the period ratio. For coronal values of thermal conduction the effect on the period ratio is negligible. For compressive viscosity the effect on the period ratio may become important for some short hot loops. In Chapter 4 we extend the analysis of Chapter 3 to include radiative cooling and find that it too has a negligible effect on the period ratio for typical coronal values. As an extension to the investigation, damping rates are considered for thermal conduction, compressive viscosity and radiative cooling. The damping time is found to be optimal for each mechanism in a different temperature range, namely below 1 MK for radiative cooling, 2 − 6 MK for thermal conduction and above 6 MK for compressive viscosity. In Chapter 5 we consider analytically the period ratio for the fast kink, sausage and n = N modes of a magnetic slab, discussing both an Epstein density profile and a simple step function profile. We find that transverse density structuring in the form of an Epstein profile or a step function profile may contribute to the shift of the period ratio for long thin slab-like structures. The similarity in the behaviour of the period ratio for both profiles means either can be used as a robust model. We consider also other profiles numerically for the kink mode, which are found to be either slab-like or Epstein-like suggesting again that it is not necessary to distinguish the nature of the density profile when considering the period ratio.EThOS - Electronic Theses Online ServiceCarnegieGBUnited Kingdo

Spatio-temporal models of synthetic genetic oscillators

Signal transduction pathways play a major role in many important aspects of cellular function e.g. cell division, apoptosis. One important class of signal transduction pathways is gene regulatory networks (GRNs). In many GRNs, proteins bind to gene sites in the nucleus thereby altering the transcription rate. Such proteins are known as transcription factors. If the binding reduces the transcription rate there is a negative feedback leading to oscillatory behaviour in mRNA and protein levels, both spatially (e.g. by observing fluorescently labelled molecules in single cells) and temporally (e.g. by observing protein/mRNA levels over time). Recent computational modelling has demonstrated that spatial movement of the molecules is a vital component of GRNs and may cause the oscillations. These numerical findings have subsequently been proved rigorously i.e. the diffusion coefficient of the protein/mRNA acts as a bifurcation parameter and gives rise to a Hopf bifurcation. In this paper we first present a model of the canonical GRN (the Hes1 protein) and show the effect of varying the spatial location of gene and protein production sites on the oscillations. We then extend the approach to examine spatio-temporal models of synthetic gene regulatory networks e.g. n-gene repressilators and activator-repressor systems

Computational modelling and simulation of cancer growth and migration within a 3D heterogeneous tissue : the effects of fibre and vascular structure

Funding: MAJC and CKM gratefully acknowledge the support of EPSRC Grant No. EP/N014642/1 (EPSRC Centre for Multiscale Soft Tissue Mechanics - With Application to Heart & Cancer).The term cancer covers a multitude of bodily diseases, broadly categorised by having cells which do not behave normally. Since cancer cells can arise from any type of cell in the body, cancers can grow in or around any tissue or organ making the disease highly complex. Our research is focused on understanding the specific mechanisms that occur in the tumour microenvironment via mathematical and computational modeling. We present a 3D individual-based model which allows one to simulate the behaviour of, and spatio-temporal interactions between, cells, extracellular matrix fibres and blood vessels. Each agent (a single cell, for example) is fully realised within the model and interactions are primarily governed by mechanical forces between elements. However, as well as the mechanical interactions we also consider chemical interactions, for example, by coupling the code to a finite element solver to model the diffusion of oxygen from blood vessels to cells. The current state of the art of the model allows us to simulate tumour growth around an arbitrary blood-vessel network or along the striations of fibrous tissue.PostprintPeer reviewe