MS03 - MFBM-09

Probability & stochastic processes in biology: models, methods, and community (Part 1)

Tuesday, July 15 at 10:20am

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

Jinsu Kim

POSTECH

"Stability of stochastic biochemical reaction networks"

A reaction network is a graphical representation of interactions between chemical species (molecules). When the species' abundances in the system are low, the inherent randomness of molecular interactions significantly influences the system dynamics. In such cases, the abundances are modeled stochastically using a continuous-time Markov chain that evolves in a jump-by-jump fashion. A major challenge in this area is establishing the stability of the Markov chain—that is, proving the existence of a stationary distribution. Another goal is to find a closed form of the stationary distribution, which is often extremely difficult. In this talk, I will present structural conditions on reaction networks that guarantee stability. I will also introduce novel techniques, inspired by reaction network theory, that aid in deriving closed-form expressions for stationary distributions.

Daniel Schultz

Dartmouth College

"Emergence of heterogeneity during bacterial antibiotic responses"

Heterogeneity is a fundamental aspect of microbial ecology, conferring resilience and adaptability to microbial populations. Remarkably, this heterogeneity does not necessarily depend on complex mechanisms of differentiation into specialized phenotypes but can instead be achieved through fundamental processes that are common to all microbes. Yet, despite significant advances in describing microbial physiology, we still lack a framework to bridge across scales to connect cellular processes to emergent behaviors in microbial communities. Here, we aim to understand how stochastic and spatial variations affect cellular metabolism and thereby provide a mechanism to generate phenotypic diversity, giving rise to complex collective behaviors in microbial populations. To overcome difficulties in studying cell responses across multiple scales, we use single-cell and biofilm microfluidics to develop mathematical models of antibiotic response dynamics. Our single-cell microfluidic experiments capture a remarkable variety of phenotypes caused by stochastic fluctuations during drug responses. Similarly, our model predicts the coexistence of stable phenotypes corresponding to growing and arrested cells. We describe the nature and stability of these different phenotypes and connect single-cell heterogeneity to population-level growth. In our biofilm microfluidic experiments, the formation of nutrient gradients due to spatial variations across the population results in a range of metabolic states with different antibiotic susceptibilities. Drug exposures result in a major reorganization of the biofilm, giving rise to collective behaviors such as increased resistance and “memory” from past exposures. A spatial version of our model describes the contribution of spatial structure to the collective mechanisms of resistance provided by organization into biofilms, showing how spatially structured populations can survive much higher drug doses than planktonic populations. Together, these results elucidate how heterogeneity emerges in microbial populations and how it gives rise to complex behaviors at the population level.

Anna Kraut

St. Olaf College

"Evolution across fitness valleys in a changing environment"

The biological theory of adaptive dynamics aims to study the interplay between ecology and evolution under the basic mechanisms of heredity, mutation, and competition. The typical evolutionary behavior of a population can be studied mathematically by looking at macroscopic limits of large populations and rare mutations derived from microscopic individual-based Markov processes. Previous work has been focused on a variety of scaling regimes and the resulting stochastic and deterministic limit processes under the assumption of constant environmental parameters. In our present work, we relax this assumption and study repeating changes in the environment, allowing for all of the model parameters to vary over time as periodic functions on an intermediate time scale between those of stabilization of the resident population (fast) and exponential growth of mutants (slow). Biologically, this can for example be interpreted as the influence of seasons or the fluctuation of drug concentration during medical treatment. Analyzing the influence of the changing environment carefully on each time scale, we are able to determine the effective growth rates of emergent mutants and their invasion of the resident population. In recent work, we study the crossing of so-called fitness valleys, where multiple disadvantageous mutations need to be accumulated to gain a fitness advantage. A changing environment has interesting implications for the crossing rates of such valleys, particularly if some of the intermediate traits occasionally have positive growth rates and can serve as pit stops.

Eric Foxall

UBC Okanagan

"Perturbation theory of reproductive value for branching Markov processes"

The reproductive value gives an individual’s relative contribution to the success of a population, as a function of its type, and for suitably recurrent models can be computed via a renewal argument. We show that the same argument can be used to compute its sensitivity in terms of an associated Markov process that describes the trajectory of a distinguished lineage, known to probabilists as the spinal particle. In the case of age-structured models this leads to nice formulas for the dependence of reproductive value on parameters.

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Annual Meeting for the Society for Mathematical Biology, 2025.