MS06 - MFBM-09

Probability & stochastic processes in biology: models, methods, and community (Part 2)

Thursday, July 17 at 10:20am

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Note: this minisymposia has multiple sessions. The other sessions are: Part-1, Part-3, 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.

Ellen Baake

Bielefeld University

"Evolving genealogies in cultural evolution"

We consider a Moran-type model of cultural evolution, which describes how traits emerge, are transmitted, and get lost in populations. Our analysis focuses on the underlying cultural genealogies; they were first described by Aguilar and Ghirlanda (2015) and are closely related to the ancestral selection graph of population genetics, wherefore we call them emph{ancestral learning graphs}. We investigate their dynamical behaviour, that is, we are concerned with emph{evolving genealogies}. In particular, we consider the total length of the genealogy of a sample of individuals from a stationary population as a function of the (forward) time at which the sample is taken. This quantity shows a sawtooth-like dynamics with linear increase interrupted by collapses to near-zero at random times. We relate this to the metastable behaviour of the stochastic logistic model, which, in our context, describes the evolution of the number of ancestors, or equivalently, the number of descendants of a given sample. This is joint work with Joe Wakano (Tokyo), Hisashi Ohtsuki (Hayama), and Yutaka Kobayashi (Kochi).

Linh Huynh

Dartmouth College

"Spin glass model for Large Language Models and evolution"

In recent years, Large Language Models (LLMs) have revolutionized Natural Language Processing with their ability to generate human-like texts. However, a fundamental challenge remains in understanding the underlying mechanisms driving their emergent behaviors, particularly the randomness in their outputs. In this talk, I will discuss the application of spin glass theory as a mathematical framework to quantify the uncertainty of LLMs. By making connections between LLMs and spin glass models, which are traditionally used in statistical physics and probability to describe disordered networks with random interactions and frustrations (conflicting constraints), we can gain insights into the high-dimensional optimization landscapes of LLMs, the uncertainty in their outputs, and the role of noise in their learning process. I will conclude by making a connection to evolution.

Samuel Isaacson

Boston University

"Coarse-grained limits of particle-based stochastic reactive-transport models"

In many applications, both spatial transport and stochasticity in chemical reaction processes play critical roles in system dynamics. Particle-based stochastic reaction diffusion (PBSRD) models have been successfully use to study a variety of such reaction processes, particularly at the single-cell scale. However, as commonly used, they typically assume overdamped transport, ignoring inertial forces. In this talk we investigate how to construct more microscopic particle-based reactive Langevin Dynamics (PBRLD) models that include inertial forces, formulating models that are consistent with detailed balance of reaction fluxes at equilibrium. We show via asymptotic analysis that with appropriate scaling assumptions for the dependence of reaction kernels on friction/mass, PBRLD models converge to common PBSRD models in the overdamped limit. Finally, we identify and prove the large population mean-field limit of the new PBRLD models, obtaining systems of nonlocal kinetic reaction-diffusion equations.

Clément Soubrier

University of British Columbia

"Modeling the meiotic spindle using a spatial birth-death process."

In eukaryotes, during the second phase of meiosis, the two chromatids of each chromosomes are separated to form haploid gametes. This segregation is driven by a bi-polar mechanical and dynamical structure, the spindle, primarily composed of microtubules. Spindle defects, such as loss or split of a pole, lead to failure of the mitosis or to aneuploid gametes. In this talk, we model the spindle stability using a spatial birth-death process representing the position of micro-tubules attached to the spindle pole. In particular, we study the first transition of the process to a multipolar state. We define this state as having a large spatial gap between two consecutive micro-tubules. Our main result is an asymptotic estimate of the first passage time of the multipolar state, as a function of the spindle creation rate and spatial gap.

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Annual Meeting for the Society for Mathematical Biology, 2025.