**Time & Location**: All talks are on Wednesdays in Gibson 310 at 3:00 PM unless otherwise noted.

**Organizer**: Gustavo Didier

September 14

**Speaker**institution

**Abstract**: TBA

September 21

**Melanie Jensen**Tulane University

**Abstract**:

In recent years, particle tracking experiments have provided new insights into interactions between our immune system and foreign bodies like viruses and bacteria. Sam Lai (UNC-Chapel Hill, Pharmacy) and collaborators have demonstrated that certain types of antibodies have the capacity to essentially immobilize Herpes virus in mucus and it is believed that a similar effect will be true for HIV. Because antibodies are too small to be directly observed in these experiments, the physical mechanism underlying this effect remains unclear. Using particle tracking data for Herpes virions, we construct a multi-scale model for virion movement and virion-antibody-mucin reaction kinetics to investigate the impact of varying antibody concentrations on virion movement.

First, we develop a classification system for path data to distinguish among diffusive, subdiffusive, and stationary motions. We use a continuous-time Markov model to describe virion-antibody-mucin kinetics and introduce a multi-scale approximation to compute important properties of the system that help us predict what fraction of a virion population should be immobilized at a given time, and how long a virion's immobilization should last. To specify our model with the data, we use identifiability analysis to set mathematically optimal and biological feasible parameter values. Finally, we compare theoretical immobilization times with observed immobilization times to determine whether prominent qualitative features of the data are predicted by our linear stochastic model.

September 28

**Lukasz Sikora**Tulane University

**Abstract**:

An ongoing effort in the study of microparticle movement in biofluids is the proper characterization of subdiffusive processes i.e. processes whose mean-squared displacement scales as a sublinear power law. In order to describe phenomena that lead to subdiffusive behavior, a few models have been developed: fractional Brownian motion, the generalized Langevin equation, and random walks with dependent increments. We will present perhaps a simpler model that leads to subdiffusion and is designed to characterize systems where a regularly diffusive particle intermittently becomes trapped for long periods of time.

To start with, we introduce the rigorous model of switching diffusion. We present a stochastic differential equation perspective and, using heavy tail analysis methods, we will present a proof that switching diffusion with heavy-tailed immobilization times is asymptotically subdiffusive.

October 5

**John Fricks**Arizona State University

**Abstract**:

In living cells, Brownian forces play a dominant role in the movement of small and not so small particles, such as vesicles, organelles, etc. However, proteins and other macromolecules bind to one another, altering the underlying Brownian dynamics. In this talk, classical approaches in the biophysical literature to time series observations which switch between bound and unbound states will be presented, and an alternative approach using stochastic expectation-maximization algorithm (EM) combined with particle filters will be proposed along with extensions for non-quadratic potentials when the particle is bound.

October 12

**Speaker**institution

**Abstract**: TBA

October 19

**Veronica Ciocanel**Brown University

**Abstract**:

Messenger RNA (mRNA) localization is essential during the early development of many organisms, including during development of frog egg cells into embryos. This accumulation of RNA at the cell periphery is not well understood, but is thought to depend on diffusion, bidirectional movement and anchoring mechanisms. Our goal is to test these proposed mechanisms using dynamical systems and stochastic models and analysis, informed by parameter estimation. These methods allow us to extract asymptotic quantities such as effective velocity and diffusion, and to conclude that the PDEs considered have approximate traveling wave solutions. We confirm the hypothesis of bidirectional transport, and use the parameter estimates in numerical studies of localization.

October 26

**Speaker**institution

**Abstract**: TBA

November 2

**Speaker**institution

**Abstract**: TBA

November 9

**Sean Lawley**University of Utah

**Abstract**:

A number of diverse biological systems involve diffusion in a randomly switching environment. For example, such processes arise in brain biochemistry, insect respiration, intracellular trafficking, and biochemical reaction kinetics. These processes present interesting mathematical subtleties as they combine two levels of randomness: Brownian motion at the individual particle level and a randomly switching environment.

In this talk, we will describe the tools for analyzing these systems and highlight the interesting behavior that they can exhibit. Special attention will be given to establishing mathematical connections between these classes of stochastic processes. In particular, we will use these connections to study certain random PDEs by analyzing the local time of a Brownian particle in a random environment.

December 7

**Alexej Gossmann**Tulane University

**Abstract**:

The method of Sorted L-One Penalized Estimation, or SLOPE, is a relatively new sparse regression method introduced by Bogdan et. al. (2015). It can be used to identify significant predictor variables in a linear model that may have more unknown parameters than observations. When the correlations between predictor variables are small, the SLOPE method is shown to successfully control the false discovery rate (the expected proportion of the irrelevant among all selected predictors) at a user specified level. However, the requirement for nearly uncorrelated predictors is too restrictive for genomic data. A possible solution is to divide the predictor variables into nearly uncorrelated groups, and to modify the procedure to select entire groups with an overall significant group effect, rather than individual predictors. Following this motivation, we extend SLOPE in the spirit of Group LASSO to Group SLOPE, a method that can handle group structures between the predictor variables, which are ubiquitous in real genomic data. Our theoretical results show that Group SLOPE controls the group-wise false discovery rate (gFDR), when groups are orthogonal to each other. For use in general, non-orthogonal settings we propose several heuristics, which lead to gFDR control with Group SLOPE in simulations on real SNP (single-nucleotide polymorphism) data. As an illustration of the merits of this method, an application of Group SLOPE to a dataset from the Framingham Heart Study results in the identification of some known DNA sequence regions associated with bone health, as well as some new candidate regions. Additionally, if time permits, further SLOPE-based methods and topics for future research will be briefly introduced.

Date

**Speaker**institution

**Abstract**: TBA

Date

**Speaker**institution

**Abstract**: TBA

Mathematics Department, 424 Gibson Hall, New Orleans, LA 70118 504-865-5727 math@math.tulane.edu