September 6
Speaker: Al Holder (Rose-Hulman Institute of Technology
Title: An Introduction to Flux Balance Analysis and a Robust Extension
Abstract: Flux balance analysis (FBA) is a computational, whole-cell metabolic model with numerous applications. We introduce FBA and review a robust extension that permits stochastic interpretation and that heightens concomitant fidelity to the motivating biological intent. We conclude with possible continuations for research and application.
September 13
Speaker: Jichun Xie (Duke University)
Title: Disentangling Cellular Heterogeneities and Activities from the Topology Structures of Gene Co-expression Graphs
Abstract: Gene co-expression graphs are a rich source of information, revealing critical insights into cellular functions, states, and activities. Yet, extracting meaningful signals from these graphs presents a formidable challenge. This complexity arises due to the presence of multiple, overlapping sources of heterogenous and cellular activity signals, and large inherent technical bias and noise in single-cell data. In this talk, we explore our latest endeavors and progress in unraveling these intricate signals in gene co-expression graphs. We focus on the identification and interpretation of various topological structures and demonstrate how these structures help in pinpointing feature genes and assessing cellular pathway activeness by further leveraging domain knowledge in online knowledge databases.
September 20
Speaker: Samantha Linn (University of Utah)
Title: Exhaustive search with stochastic resetting
Abstract: Cover times quantify the speed of exhaustive search. In this talk we will review pre-existing results on cover time theory. We will then consider cover times of stochastic searchers undergoing stochastic resetting, a mechanism which can reduce mean search times, hence whose utility spans biology and other sciences. Computing these cover time statistics exactly is intractable in complicated domains. By considering the frequent resetting limit, we will approximate cover time moments on arbitrary discrete networks and in d-dimensional continuous space. These approximations apply to a large class of resetting time distributions and search processes including diffusion and Markov jump processes.
September 27
Speaker: Guowei Wei (Michigan State University)
Title: Mathematical AI Paradigm for Biosciences
Abstract: Mathematics underpins fundamental theories in physics such as quantum mechanics, general relativity, and quantum field theory. Nonetheless, its success in modern biology, namely cellular biology, molecular biology, chemical biology, genomics, and genetics, has been quite limited. Artificial intelligence (AI) has fundamentally changed the landscape of science, engineering, and technology in the past decade and holds a great future for discovering the rules of life. However, AI-based biological discovery encounters challenges arising from the intricate complexity, high dimensionality, nonlinearity, and multiscale biological systems. We tackle these challenges by a mathematical AI paradigm. We overcome AI’s limitations by new algorithms derived from algebraic topology, differential geometry, and geometric topology. Using our mathematical AI paradigm, my team has been the top winner in D3R Grand Challenges, a worldwide annual competition series in computer-aided drug design and discovery for years. By further integrating mathematical AI with millions of genomes isolated from patients, we discovered the mechanisms of SARS-CoV-2 evolution and accurately forecast emerging dominant SARS-CoV-2 variants months in advance.
October 4
Speaker: Polly Yu (University of Illinois Urbana-Champaign)
Title: A new framework for generalized Lotka-Volterra models
Abstract: Generalized Lotka-Volterra (GLV) systems, as models of ecological communities, can display dynamics ranging from globally stable equilibrium, to periodic orbits and even chaos. We propose a new framework of studying GLV systems, by borrowing ideas from reaction network theory. By associating a GLV system with a directed graph embedded in R^n, we can prove theorems of the form: "If the graph has property P (e.g., strongly connected), then for all possible parameter values, the associated GLV system has dynamics X (e.g., no extinction)." In this talk, we will introduce this framework, and present several results pertaining to globally stable equilibrium, persistence, and permanence. (Joint work with Gheorghe Craciun and Diego Rojas La Luz.)
October 11
Speaker: Laurel Ohm (University of Wisconsin - Madison)
Title: Free boundary dynamics of an elastic filament in 3D Stokes flow
Abstract: Motivated by biophysical applications, we consider a free boundary problem for a thin elastic filament immersed in 3D Stokes flow. The 3D fluid is coupled to the quasi-1D filament dynamics via a novel type of angle-averaged Neumann-to-Dirichlet operator. Much of the difficulty in the analysis lies in understanding this operator. We show that the principal part of this NtD map is the corresponding operator about a straight, periodic filament, for which we derive an explicit symbol. It is then possible to establish local well-posedness for an immersed filament evolving via a simple elasticity law. This establishes a mathematical foundation for the myriad computational results based on slender body approximations for thin immersed elastic structures.
October 18
Speaker: Hongsong Feng (Michigan State University)
Title: Mathematical AI models for drug design and discovery
Abstract: The effectiveness of machine learning in molecular bioscience and biophysics research is often hindered by the curse of dimensionality associated with biomolecular structures and the varying sizes of biomolecules, which limits the large-scale use of biological data. To overcome these challenges, I have employed advanced mathematical theories, such as topological data analysis, geometric data analysis, and multiscale differential geometry, to provide mathematical representations of high dimensional data. These approaches encode critical biological information from high-dimensional spaces into low-dimensional representations. Integrating these mathematical models with machine learning algorithms has led to the development of cutting-edge mathematical AI models that consistently outperform state-of-the-art methods in numerous benchmarks. Additionally, I apply these mathematical AI models to address real-world biological challenges, with a particular focus on AI-assisted drug design/discovery.
October 25
Speaker: Alexandria Volkening (Purdue University)
Title: Forecasting U.S. elections with compartmental models
Abstract: Election dynamics are a rich complex system, and forecasting U.S. elections is a high-stakes problem with many sources of subjectivity and uncertainty. In this talk, I take a dynamical-systems perspective on election forecasting, with the goal of helping to shed light on the forecast process and raising questions for future work. By adapting a Susceptible-Infected-Susceptible model to account for interactions between voters in different states, I will show how to combine a compartmental approach with polling data to produce forecasts of senatorial, gubernatorial, and presidential elections at the state level, including the 2024 cycle. Our results for the last two decades of U.S. elections are largely in agreement with those of popular analysts. We use our modeling framework to determine how weighting polling data by polling organization affects our forecasts, and explore how our forecast accuracy changes in time in the months leading up to each election. I will discuss both our historical accuracy and 2024 forecasts.
November 1
Speaker: Hyunjoong Kim (University of Cincinnati)
Title: Should I pay more now for better information? Normative decisions for individuals and groups in changing environments
Abstract: How and when do animals, including humans, focus their attention? They may concentrate their cognitive resources on a few options when they encounter many, or they may selectively allocate these resources dynamically depending on a changing environment. In this talk, we consider the latter case and determine the optimal foraging policy for ideal individuals and groups by using sequential sampling model and dynamic programming. Ideal agents make decisions to optimize both current and future value based on their beliefs about the current state of the resource environment. Through this research, we aim to understand mathematically how individuals or groups behave in dynamic environments. We introduce an experiment designed to understand how humans actually behave by comparing their behavior with that of ideal agent. For groups, allocating cognitive resources is akin to assigning tasks to multiple agents. We explore how the group’s optimal behavior changes depending on their beliefs about the resource environment and also analyze the asymptotics of optimal strategies for large groups.
November 8
Speaker: Kelsey Gasior (University of Notre Dame)
Title: Untangling the Molecular Interactions Underlying Intracellular Phase Separation
Abstract: An emerging mechanism for intracellular organization is liquid-liquid phase separation (LLPS). Found in both the nucleus and the cytoplasm, liquidlike droplets condense to create compartments that are thought to localize factors, such as RNAs and proteins, and promote biochemical interactions. Many RNA-binding proteins interact with different RNA species to create droplets necessary for cellular functions, such as polarity and nuclear division. Additionally, the proteins that promote phase separation are frequently coupled to multiple RNA binding domains and several RNAs can interact with a single protein, leading to a large number of potential multivalent interactions. This work focuses on a multiphase, Cahn-Hilliard diffuse interface model to examining the RNA-protein interactions driving LLPS. Using a ‘start simple, build up’ approach to model construction, these models explore how the molecular interactions underlying protein-RNA dynamics and RNA species competition control observable, droplet-scale phenomena. Numerical simulations reveal that RNA competition for free protein molecules contributes to intra-droplet patterning and the emergence of a heterogeneous droplet field. More in-depth analysis using combined sensitivity analysis techniques, such as Morris Method screening and Sobol’ method, highlights the complicated relationships underlying protein-RNA interactions and the results we can measure. Ultimately, this targeted approach to intracellular LLPS begins to peel back the layers of complex molecular dynamics controlling observable LLPS phenomena that contribute to droplet regulation and, ultimately, cellular function.
November 15
Speaker: Wai-Tong Louis Fan (Indiana University Bloomington)
Title: Stochastic partial differential equations as robust models in spatial population genetics
Abstract: Spatial-temporal data on population dynamics offer important evidence of how populations evolve over time and space, whether the populations consist of groups of living organisms, the cancer cells of a tumor, or the virus particles within a single host cell. To explain these data and make predictions, mechanistic models are essential for understanding population dynamics. These models are necessarily spatial and stochastic, due to the noisy nature of the data, making them challenging to study. Individual-based models can capture fine details, including the randomness and discreteness of individuals, that are not considered in continuum models such as partial differential equations (PDE). However, they are sensitive to minor changes and often intractable. The challenge lies in how to simultaneously retain key information from microscopic models while ensuring the efficiency and robustness of macroscopic models. In this talk, I will discuss how this challenge can be overcome by elucidating the probabilistic connections between individual-based models and PDEs. These connections will illuminate how certain stochastic partial differential equations (SPDEs) emerge as robust models—uniquely characterized by their resilience to changes in minute details. I will present an innovative class of SPDEs that offer insights about virus infection spread and about expanding populations in general. These robust models hold considerable promise for advancing the application of spatial-temporal data analysis in understanding dynamic populations.
November 22
Speaker: Zhe Su (Michigan State University)
Title: De Rham Hodge Theory in Topological Data Analysis
Abstract: Topological data analysis has become a trending topic in data science and engineering, with many techniques developed for point cloud data. However, these techniques do not work directly for one commonly occurring data format, data on manifolds or volumetric data. The limitation highlights the need to develop effective computational algorithms for such data in practical applications. De Rham Hodge theory for manifolds with boundary offers a promising solution by providing deep insights into both the topological and geometric information of manifolds through differential forms. Nevertheless, developing de Rham Hodge-enabled frameworks is significantly challenging due to the complicated boundary conditions and the geometric complexity associated with realistic shapes. In this talk, I will talk about our currently developed de Rham Hodge-enabled frameworks for the topological analysis of data on manifolds, which have shown to be effective and efficient, and also convenient for machine learning use.
December 6
Speaker: Gregory Handy (University of Minnesota)
Title: Ensemble Dynamics in Visual Cortex: Probing the Spatial and Feature Dependencies of Recurrent Activity
Abstract: Although most synapses within the neocortex are recurrent, the functional impact of this recurrence remains unclear. Prior experimental and theoretical work in the primary visual cortex (V1) suggests that recurrent excitation amplifies responses when signals are weak to optimize detection, while recurrent inhibition suppresses responses when signals are strong to optimize discrimination. Deepening our understanding of the logic for when recurrent activity facilitates cooperation among cortical ensembles and when it mediates competition beyond these initial results is fundamental to understanding cortical computation.
To isolate the impact of local recurrence, we used multiphoton holographic optogenetics to activate neuron ensembles in V1 without external stimuli. We found that both the spatial arrangement and stimulus preference of the ensemble shape the activity response, with the strongest effects within 30 µm of a stimulated cell. Compact, co-tuned ensembles recruit nearby iso-tuned neurons while suppressing others. Computational modeling reveals that local excitatory connections and selective inhibition explain these patterns, highlighting a trade-off between excitatory and suppressive pathways. This combination of in vivo and in silico circuit interrogation explains how recurrent cortical circuits selectively amplify or suppress cortical activity depending on the precise pattern of co-active neurons.
November 22
Speaker: Zhe Su (Michigan State University)
Title: De Rham Hodge Theory in Topological Data Analysis
Abstract: Topological data analysis has become a trending topic in data science and engineering, with many techniques developed for point cloud data. However, these techniques do not work directly for one commonly occurring data format, data on manifolds or volumetric data. The limitation highlights the need to develop effective computational algorithms for such data in practical applications. De Rham Hodge theory for manifolds with boundary offers a promising solution by providing deep insights into both the topological and geometric information of manifolds through differential forms. Nevertheless, developing de Rham Hodge-enabled frameworks is significantly challenging due to the complicated boundary conditions and the geometric complexity associated with realistic shapes. In this talk, I will talk about our currently developed de Rham Hodge-enabled frameworks for the topological analysis of data on manifolds, which have shown to be effective and efficient, and also convenient for machine learning use.
December 20
Speaker: Erik Amezquita Morataya (University of Missouri)
Title: Characterizing single-cell transcriptomic spatial patterns with Topological Data Analysis
Abstract: To gain their unique biological function, plant cells regulate protein biosynthesis through gene activation and repression along with multiple post-transcriptional, translational, and post-translational mechanisms. Additionally, the differential trafficking and subcellular localization of mRNAs have been reported as a complementary regulatory mechanism of the biology of fungi, yeast, and animal cells. However, studies comprehensively reporting the impact of mRNA localization in plant cells are lacking.
Here, we set to mathematically model the spatial distribution of sub-cellular cytosolic transcripts across multiple cell types and developmental stages. Through the use of high-resolution spatial transcriptomic technology, we first report the comprehensive and differential mapping of millions of plant transcripts between the nuclear and cytoplasmic compartments of various soybean nodule cell types. We then characterize key mathematical features of these transcriptomic spatial distributions using Topological Data Analysis (TDA). TDA offers a comprehensive pattern-quantifying framework that is robust to variations in cell shape, size, and orientation. TDA thus provides us with a common ground to mathematically compare and contrast intrinsic differences in sub-cellular transcript distributions and patterns across cell types and expressed genes.
Our analyses reveal distinct patterns and spatial distributions of plant transcripts between the nucleus and cytoplasm, varying both between and within genes, as well as across different cell types. We believe this differential distribution is an additional, less understood, regulatory mechanism controlling protein translation and localization, cell identity, and cell state and reveals the influence of the sub-compartmentalization of transcripts as another post-transcriptional regulatory mechanism.