Mathematical Problems in the Inference of Source Location with Applications to Cellular Directional Sensing

We consider the inverse problem of determining the source location of unbiased random walkers to complex target configurations in two and three dimensions. This problem is motivated by cellular scale chemosensing where a variety of cell types must determine the strength and location of diffusive sources through noisy impacts to their exterior surface.

In this thesis we develop an efficient Kinetic Monte Carlo (KMC) method that describes the trajectories of diffusing molecules using solvable projection steps, regardless of the configuration of the absorbers in the surface of either one or multiple targets. We develop a new method to solve the diffusion problem analytically using a combination of asymptotic analysis of a Laplace transform problem, followed by numerical inversion of the transform. This hybrid approach is validated using the KMC Method.

We identify several homogenization limits in which complex target configurations can be replaced by simpler ones while retaining overall capture characteristics. Homogenization allows for faster and easier solution to diffusion problems, by calculating a logarithmic capacitance of the absorbers and substituting into a simplified Robin Boundary Condition.

Following previous literature, we investigate the use of splitting probabilities to infer the source location. We discuss the shortcomings of some existing methods and design a novel means to infer the location of sources from receptor inputs using a maximum likelihood approach. We observe that the earliest arrivals to the cell convey the strongest directional information and analyze the accuracy of adopting the direction of the first impact. We show that despite the simplicity of this mechanism, in many reasonable biological scenarios, it yields a more accurate angular estimate.