Mathematical Biology

Monitoring marked individuals is a common strategy in studies of wild animals (referred to as mark-recapture or capture-recapture experiments) and hard to track human populations (referred to as multi-list methods or multiple-systems estimation). A standard assumption of these techniques is that individuals can be identified uniquely and without error, but this can be violated in many ways. In some cases, it may not be possible to identify individuals uniquely because of the study design or the choice of marks. Other times, errors may occur so that individuals are incorrectly identified. I will discuss work with my collaborators over the past 10 years developing methods to account for problems that arise when are only individuals are only partially identified. I will present theoretical aspects of this research, including an introduction to the latent multinomial model and algebraic statistics, and also describe applications to studies of species ranging from the golden mantella (an endangered frog endemic to Madagascar measuring only 20 mm) to the whale shark (the largest know species of fish, measuring up to 19m).

Exposure of bacteria to cidal stresses typically select for the emergence of stress-tolerant cells refractory to killing. Stress tolerance has historically been attributed to the regulation of discrete molecular mechanisms, including though not limited to regulating pro-drug activation or pumps abrogating antibiotic accumulation. However, fractions of mycobacterial mutants lacking these molecular mechanisms still maintain the capacity to broadly tolerate stresses. We have sought to understand the nature of stress tolerance through a largely overlooked axis of mycobacterial-environmental interactions, namely microbial biomechanics. We developed Long-Term Time-Lapse Atomic Force Microscopy (LTTL-AFM) to dynamically characterize nanoscale surface mechanical properties that are otherwise unobservable using other established advanced imaging modalities. LTTL-AFM has allowed us to revisit and redefine fundamental biophysical principles underlying critical bacterial cell processes targeted by a variety of cidal stresses and for which no molecular mechanisms have previously been described. I aim to highlighting the disruptive power of LTTL-AFM to revisit dogmas of fundamental cell processes like cell growth, division, and death. Our studies aim to uncover new molecular paradigms for how mycobacteria physically adapt to stress and provide expanded avenues for the development of novel treatments of microbial infections.

How far inside a domain does a flux of Brownian particles perturb a background concentration when particles can escape through a neighboring window? What motivates this question is the dynamics of ions entering and exiting nanoregions of excitable cells through ionic membrane channels. Here this is explored using a simple diffusion model consisting of the Laplace's equation in a domain whose boundary is everywhere reflective except for a collection of narrow circular windows, where either flux or absorbing boundary conditions are prescribed. We derive asymptotic formulas revealing the role of the influx amplitude, the diffusion properties, and the geometry, on the concentration difference. Lastly, a length scale to estimate how deep inside a domain a local diffusion current can spread is introduced. This is joint work with David Holcman at ENS.

Molecular interactions lie at the core of biochemistry and biology, and their understanding is crucial to the advancement of biotechnology, therapeutics, and diagnostics. Most existing tools make “ensemble” measurements and report a single result, typically averaged over millions of molecules or more. These measurements can miss rare events, averaging out the natural variations or sub-populations within biological samples, and consequently obscure insights into multi-step and multi-state reactions. The ability to make and connect robust and quantitative measurements on multiple scales - single molecules, cellular complexes, cells, tissues - is a critical unmet need. In this talk, I will introduce a general method called “CLiC” imaging to image molecular interactions one molecule at time with precision and control, and under cell-like conditions. CLiC works by mechanically confining molecules to the field of view in an optical microscope, isolating them in nanofabricated features, and eliminates the complexity and potential biases inherent to tethering molecules. By imaging the trajectories of many single molecules simultaneously and in a dynamic manner, CLiC allows us to investigate and discover the design rules and mechanisms which govern how therapeutic molecules or molecular probes interact with target sites on nucleic acids; and how molecular cargo is released inside cells from lipid nanoparticles. In this talk, I will discuss applications of our imaging platform to better understand DNA, RNA, protein interactions, as well as emerging classes of genetic medicines, gene editing and drug delivery systems. I will highlight current and potential future applications to connect our observations from the level of single molecule to single cells, and opportunities for collaboration as we set up our labs at UBC.

Recursive distributional equations (RDEs) are ubiquitous in probability. For example, the standard Gaussian distribution can be characterized as the unique fixed point of the following RDE

$$
X = (X_1 + X_2) / \sqrt{2}
$$

among the class of centered random variables with standard deviation of 1. (The equality in the equation is in distribution; the random variables and must all be identically distributed; and and must be independent.)

Recently, it has been discovered that the dynamics of certain recursive distributional equations can be solved using by using tools from numerical analysis, on the convergence of approximation schemes for PDEs. In particular, the framework for studying stability and convergence for viscosity solutions of nonlinear second order equations, due to Crandall-Lions, Barles-Souganidis, and others, can be used to prove distributional convergence for certain families of RDEs, which can be interpreted as tree- valued stochastic processes. I will survey some of these results, as well as the (current) limitations of the method, and our hope for further interplay between these two research areas.