The last twenty years have seen a revolution in tracking data of biological agents across an impressive range of spatial and temporal scales. An important and ubiquitous observation from these studies is that path trajectories of living organisms are often poorly described by the classical model for random movement in continuous time, Brownian motion. To abuse a sentiment from Tolstoy: Brownian diffusions are all alike, but every anomalous diffusion is anomalous in its own way. In the details of the non-Brownian statistics lie vital clues to fundamental physical and physiological mechanisms of transport and interaction in living systems. The primary focus of my research in Applied Stochastic Processes has been to work directly with bioengineers and ecologists to determine the implications of anomalous diffusive data for both small scale physiology and for large scale animal population dynamics.

Examples of biological non-Brownian movement are numerous. At the microscopic scale, it has been observed in the behavior of the Adeno-associated virus in cytoplasm, lipid granules in living yeast cells , tracer particles in F-actin networksRNA in E.~coli and telomeres in the nucleus of mammalian cells. In recent personal research, I was part of a team that characterized non-Brownian behavior in the passive motion of micron-diameter beads in human bronchial epithelial cell culture mucus [Hill et al, 2014]. At a larger scale, animal tracking researchers have repeatedly reported non-Brownian statistics in the distribution of inter-observation relocation distances. A few examples include the movement patterns of fruit flies, honey bees, a variety of marine predators, and even humans. Along with a graduate student in UF’s Wildlife Ecology and Conservation Department, we studied non-Brownian behavior among a population of Florida panthers [Van de Kirk et al, 2014].

While these biological systems of interest are diverse, there is a common mathematical framework underlying investigations of organismal movement. Through modern stochastic process theory and innovative statistical analysis, we can develop the iterative cycle of predictive modeling and rigorous experimentation that is crucial for the bioengineering applications of the present and future.

These investigations fall into two categories: Description and Extrapolation. In the Descriptive phase of the microscale research, investigators have used the non-Brownian character of trajectories to learn details about the microstructure of biological fluids, and to assess the effectiveness of treatments meant to aid the diffusive transport of drug carrier particles. Meanwhile, in the emerging field known as Movement Ecology, researchers use animal tracking data to characterize home territories, locating nesting regions and favored resource locations; and to discover daily and annual activity cycles. The controversy arises when researchers engage in Extrapolation. Each field is prone to rely on idiomatic summary statistics that provide evidence that trajectories are non-Brownian, but are not sufficiently detailed to inform a mechanistic stochastic process model. The consequences for engineering applications can be profound. A misunderstanding of the behavior of outliers in a viral population can lead to wildly inaccurate predictions for the likelihood of infection [Chen et al, 2014]. A naive extrapolation of individual animal search behavior can lead to misidentification of the scaling of encounter rates at the population scale [Hein & McKinley, 2013].

What fascinates me about these problems is that paying attention to the details of the statistics of movement data forces us to forgo some of the basic assumptions we as mathematicians are prone to make. The data suggests that these paths contain significant memory effects, are not independent path-by-path, and are strongly shaped by the surrounding environment. As a result, off-the-shelf limit theorems and multi-scale analysis techniques have to be reconsidered. In my own work, I employ a combination of mathematical analysis and computation to answer the questions that drive me. At times I put significant energy into statistical analysis because the intrinsic noisiness and sparseness of biological data requires a creative and rigorous touch.