Stochastic transport models in confined networked environments

Thesis

Diffusion of macromolecules within the intracellular environment, tissue generation in scaffolds, and the transport of blood cells through aortic media, are examples of processes which hinge upon the optimal transport of biological agents within complex crowded environments. For example, the organelles within a typical mam- malian cell create an environment with a range of heterogeneous microscopic spatial structure known to impede macromolecular motion. A significant challenge for bio- mathematicians is to develop mathematical transport models on macroscopic length scales that incorporate the key effects of the microscopic spatial structure and the spatial exclusion between biological agents. A useful abstractification is to approx- imate complex environments as networked topologies that are capable of describing a huge range of heterogeneous spatial structures. This thesis focusses broadly on the extension of stochastic transport models confined to networked environments and their analysis. We introduce a framework for macromolecular transport in a domain that is deconstructed into two region types; one where molecules move freely, and another where there are strong macromolecular crowding effects. Through analysis of the equilibration time, algorithms for optimal network construction are designed and tested. Additionally, we explore search processes for multiple interacting searchers within a networked environment. Quantifying the efficiency of the search process via the parallel cover time, we provide optimal strategies that minimise the cover time of a fixed population of particles. Finally, we propose a new measure of particle displace- ment that accounts for the total winding of the path of a particle as it moves through a network, finding that statistics of the particle displacement are a function of network topology. Together, this work represents a key step in developing generic algorithms for topological optimisation of stochastic networked systems. Such concepts are of growing importance within modern information theory, biophysics, biotechnology, network science, statistical physics, traffic flow and robotics.

Authors: Wilson, D

• DOI: 10.5287/ora-5r2v78r0e
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Networks; Stochastic Modelling; Transport on networks; Stochastic transport
• Subjects: Mathematics; Mathematical Biology