MS07 - MFBM-09

Probability & stochastic processes in biology: models, methods, and community (Part 3)

Thursday, July 17 at 4:00pm

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Note: this minisymposia has multiple sessions. The other sessions are: Part-1, Part-2, and Part-4.

Jinsu Kim (POSTECH), Eric Foxall (The University of British Columbia - Okanagan Campus), and Linh Huynh (Dartmouth College)

Description:

Mathematical biology has a long and rich history of physics-based models and methods. These include both explicitly physics-based models, such as mechanical models of cell wall geometry and molecular models of protein translation and folding, and those inspired by physics, such as interacting particle system models in ecology and epidemiology, to name just a few. In some cases, existing approaches are not optimal for the study of biochemical systems due to various issues including the curse of dimensionality, infinite state spaces, and uncertainty in model structure. Methods from probability theory, especially new methods developed specifically for the study of biological phenomena, can simplify and strengthen our analysis and understanding of these phenomena. In this minisymposium, which has two parts due to significant interest in the subject, we gather researchers from diverse backgrounds and fields of study to exchange ideas, models and methods. The intention is to foster communication and collaboration in the study of stochastic processes across all of biology, and especially to avail researchers of recent developments that could be fruitful in unexpected areas of application. Special requirement: Some of our speakers are only available on July 14th. Therefore, we kindly request that our minisymposium be scheduled on July 14th.

Hwai-Ray Tung

University of Utah

"Extreme first passage times with fast immigration"

Many scientific questions can be framed as asking for a first passage time (FPT), which generically describes the time it takes a random 'searcher' to find a 'target.' The important timescale in a variety of biophysical systems is the time it takes the fastest searcher(s) to find a target out of many searchers. Previous work on such fastest FPTs assumes that all searchers are initially present in the domain, which makes the problem amenable to extreme value theory. In this paper, we consider an alternative model in which searchers progressively enter the domain at some 'immigration' rate, which may be constant, time inhomogeneous, or proportional to the population size. In the fast immigration rate limit, we determine the probability distribution and moments of the k-th fastest FPT. Our rigorous theory applies to many models of stochastic motion, including random walks on discrete networks and diffusion on continuous state spaces. Mathematically, our analysis involves studying the extrema of an infinite sequence of random variables which are both not independent and not identically distributed. Our results constitute a rare instance in which extreme value statistics can be determined exactly for strongly correlated random variables.

Minjun Kim

POSTECH

"A Path Method for Non-exponential Ergodicity of Stochastic Chemical Reaction Systems"

We present criteria for non-exponential ergodicity of continuous-time Markov chains on a countable state space in total variation norm. These criteria can be verified by examining the ratio of transition rates over certain paths. We applied this path method to explore the non-exponential convergence of microscopic biochemical interacting systems. Using reaction network descriptions, we identified special architectures of biochemical systems for non-exponential ergodicity. In essence, we found that reactions forming a cycle in the reaction network can induce non-exponential ergodicity when they significantly dominate other reactions across infinitely many regions of the state space. Interestingly, the special architectures allowed us to construct many detailed balanced and complex balanced biochemical systems that are nonexponentially ergodic. Some of these models are low-dimensional bimolecular systems with few reactions. Thus we suggest the possibility of discovering or synthesizing stochastic systems arising in biochemistry that possess either detailed balancing or complex balancing and slowly converge to their stationary distribution.

Daniel Coombs

University of British Columbia

"Resolving Single Particle Mobility with Hidden Markov Models"

Using single particle tracking approaches, we study the motion of individual molecules on cell surfaces. At larger space- and time-scales, we can also follow the motion of individual cells in vitro and in vivo. In this talk I will describe how we have applied hidden Markov models to provide a time-resolved view of particle behaviour. In this talk I will describe how our approach has developed from the simplest two-state model, to inferring multi-state models and incorporating noise using an infinite hidden Markov model framework.

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Annual Meeting for the Society for Mathematical Biology, 2025.