# Topics in the mathematical modelling of nanotoxicology

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Over the last ten years questions related to the safety of nanoparticles and their possible toxic effects have become well-established. The government’s Health and Safety Laboratories (HSL) at Buxton are currently attempting to determine their possible toxicity in the workplace. It is their responsibility to establish what levels are exposure can be considered safe in the workplace. This project is a CASE studentship with HSL and aims to start developing mathematical models relating to nanotoxicology. After reviewing the available literature, three key mechanisms which are involved in the possible toxicity of nanoparticles emerge. One mechanism is the oxidative stress they cause once they enter individual cells. The second mechanism is the damage done to the surface of the lung if they are not successfully phagocytosed on inhalation. Finally, the third mechanism is their propensity to aggregate both when dispersed in the air or when they are found inside the body. These three topics are dealt with in Parts I, II and III respectively. There has been much concern over how carbon nanotubes (CNTs) may cause oxidative stress. Oxidative stress occurs when there is an overload negatively charged species in the cell. These are collectively known as Reactive Oxidative Species (ROS). ROS are always present in a cell as they are the natural product of the metabolic pathways. By their reactivity, they readily cause damage to other molecules in the cell, so every cell produces anti-oxidants in order to control the concentration of ROS. However, when the concentration of ROS becomes too high the concentration of anti-oxidants becomes depleted and the cell can become too damaged to function. In this case it dies by necrosis. When a cell dies by necrosis is can cause irritation and further damage to surrounding cells. Oxidative stress can also trigger the immune response so that the cell dies by self-programmed apoptotic cell death which limits this damage to surrounding cells. It is best to avoid unnecessary cell death, however, not undergoing apoptosis risks a more damaging necrotic death. Part I introduces develops models of Tumor Necrosis Factor-alpha (TNF-alpha) activated pathways. This model consists of three signalling cascades. One pathway triggers apoptosis while a second inhibits apoptosis. These two models are based on pre-existing models. This work introduces a third pathway which activates Activator Protein-1 (AP-1). This pathway includes two well-known ROS sensitive elements. These are the ROS-sensitive activation of the Mitogen-Activated Protein Kinase (MAPK) cascade and the ROS-sensitive deactivation of the MAPK phosphatases. These three pathways are regulated by three sets of inhibiting reactions and inhibitors to these inhibitors. The effect of these inhibitors is to introduce a time-lag between the initial TNF-alpha extracellular signal and the death of the cell by apoptosis. This time-lag is regulated by the concentration of intracellular ROS and the concentration of anti-oxidants. Different combinations of inhibitors can be switched on or off before running the model. The effectiveness of the oxidative stress sensitive elements in regulating apoptosis can therefore be optimised while different sets of inhibitors are active. Two qualitatively different types of solutions are found. The cell can be either only transiently active, over a shorter period of time, or persistently active, over a longer period of time. This could provide some guidance to biologists investigation TNF-alpha activation of the immune system. On inhalation, CNTs have been found to reach the alveoli, where air exchange occurs in the lung. The only mechanism available to remove debris in these delicate regions of the lung are lung macrophages. Macrophages work by enclosing unwanted matter in an organelle called a lysosome and then moving this debris away to where it can be cleared by cilia. Non-organic material does not trigger a macrophage response as strongly as organic material, which also triggers the immune system. The shape of fibrous material makes it more difficult for a macrophage to successfully form a lysosome and to move the material away once it has been engulfed. Frustrated phagocytosis releases harmful acids and enzymes which can damage the alveoli causing oxidative stress. If debris cannot be removed, then dead cells may form around the debris to protect the surrounding tissue, forming a granuloma. Both scarring from frustrated phagocytosis and granuloma formation will impair the function of the lung. In Part II, insight is gained on how a cell membrane can engulf an object with a high aspect ratio. The mechanisms of phagocytosis are complex in terms of both cell signalling cascades and the polymerisation and de-polymerisation of the actin network. In order to find a model which takes into account the geometry of a cell as a whole, this picture has been simplified. An energy minimisation approach is used where the surface of a cell is taken to be a surface of rotation around an axis, which is taken to be the axis of a fibre. In Chapter 4, the free energy is taken to be of a liquid drop, resting on a solid surface, in vapour where only the surface and volume energies are considered. The surface tension is taken to account for the tension in the lipid bilayer on the surface of the macrophage. In Chapter 5, the free energy is extended to also include a Helfrich or bending energy which specifically takes into account the energy taken to bend a lipid bilayer. It is assumed that, in order to conserve the limited resources of a macrophage, the shape of a lipid membrane which has successfully engulfed a particle will be energetically stable with regards to these surface, volume and bending energies as a macrophage reaches the final stages of phagocytosis. This does not take into account the energy required to remodel the cytoskeleton for the cell to reach this shape. However, the bending energy associated with cell membranes of increasing length can be used to suggest the amount of energy required in this dynamical process. It is found that in Chapter 4, when no Helfrich energy is included in the energy minimisation, the only limiting shape possible in the limit of increasing length to radius ratio of the fibre is a sphere. When the Helfrich energy is included, three different boundary conditions are imposed. The first boundary condition sets the forces associated with the bending energy to zero at the edge of the membrane. At the point of contact between the membrane and the fibre, the forces reduce to that of a classical solid/liquid/vapour interface. The second boundary condition is imposes the length of the droplet. This length can be incrementally increased to find solutions of increasing length. Finally, a third boundary condition is imposed which sets the contact angle of the membrane at the surface of the fibre to zero. By imposing these three boundary conditions, a variety of membrane shapes were obtained. These results are expected to be a useful guide to experimentalists observing different shapes of macrophages under different conditions. Part III in Section 7.1 pin-points frameworks of models which use concepts from polymer physics to possibly predict the volume of an aggregate of CNTs and also to understand how nanoparticles interact with chain-like protein. However, no new results are presented in Part III.