Techniques for stochastic spatial sensing in biology: from immunology to anthropology

There is a large branch of mathematics applied to biological problems, and many of these problems deal with the movement and behaviour of many bodies. Nevertheless, biology has many problems where the techniques used for mass movement is not accurate enough. In this thesis we use mathematical and computational tools applied to such biological problems where, furthermore, the relative position of each element is important to the outcome of the system. The first of the problems is in immunology, where we focus on the interaction between a protein hooked to the inside of a T cell and an enzyme floating in the cell's cytoplasm. The second biological problem we tackle on is in anthropology, where we ask ourselves whether nomadic groups of human can meet enough times in a year to maintain their friendships. In immunology we we propose models of integrate-partial differential equations with which we can compare experimental data from the laboratory or results from computational simulations, and gain information about the system’s reaction rates and particle lengths. These models use auto- and pair-correlation functions to take spatial effects into account. In Chapter 2 we deal with a dephosphorylating enzyme, SHP-1, and phosphorylated tails in the T cell membrane. In Chapter 3 we deal with a phosphorylating enzyme, ZAP-70, and both phosphorylated and dephosphorylated tails in the T cell membrane. In both chapters we find our models have a better fit to the experimentally obtained data, or the computational simulations in Chapter 3, than the traditional ODE models with mean-field approximation. These models also help as a proof of concept of what is thought to describe the interaction between the proteins involved.
Chapter 4 takes on the anthropological problem, where we seek to find what set of physical properties allow each human to meet 150 people (known as the Dunbar number) within a year. Although not many hunter-gatherer societies remain in the world, this question has more of a historical value, aiding in theories about the present-day hierarchical structure of society, the purpose of rituals such as gift-giving, and the importance of human interaction for mental health. Through the theoretical development we gain insight into stochastic movement in a grid. Agent-based simulations are used to provide the number of humans met for a range of possible parameters, such that, given a society's basic details, the particular number of meetings per person could be looked up. We find that, in the simplest case, hunter-gatherers would be able to meet with enough people during a year to maintain their community network in most scenarios. As is the nature of mathematics, the models here developed could in future be adapted for other biological or physical scenarios where elements engage similarly to the elements in each of our problems. After all, life is a mixture of disciplines to be described in the language of mathematics.

Keywords: stchastic movement
hunter-gatherer
protein interaction
pair-correlation
Dunbar number
SHP-1
ZAP-70
integro-differential
Advisor: Maini, P
Dunbar, R
Dushek, O
Woolley, T
Pearce, E