Research Supported by the NITMB
Research supported by the NSF-Simons National Institute for Theory and Mathematics in Biology focuses on developing mathematical frameworks that illuminate emergent capabilities of biological systems. We are developing the theory and mathematics needed to highlight the fundamental roles of physical, chemical, and biological constraints as organizing principles for understanding biological mechanisms. NITMB will focus on fields of mathematics where the constraints of biological systems show promise for novel developments, including geometry, topology, optimization theory, dynamical systems, high-dimensional statistics, mathematical machine learning, inverse problems, statistical inference, and stochastic processes. Understanding constraints from mathematical and biological perspectives provides a unique opportunity for interdisciplinary work, with mathematical research that will advance our knowledge of biology and biology research that will catalyze new mathematics.
Explore detailed project highlights for NITMB supported research on the Research Highlights page
Emergence of Bistability in Circadian Clock Temperature Response
Kwanghoon Jeong
(University of Chicago)
Chris Chi
(University of Chicago)
Yujia Liu
(University of Chicago)
Faculty Mentors: Michael Rust (University of Chicago) and Aaron Dinner (University of Chicago)
Abstract: A defining feature of biological time-keeping is temperature compensation, in which the amplitude of an oscillation varies such that the frequency is nearly invariant to temperature. Temperature compensation enables circadian rhythms to anticipate the length of the day correctly even as temperature varies. Because individual enzymatic reactions are typically sensitive to temperature, temperature compensation is thought to be an emergent property of a reaction network. What features of a reaction network can enable temperature compensation is an open mathematical question. In cyanobacteria, interactions of the proteins KaiA, KaiB, and KaiC give rise to near-24-hour oscillations in phosphorylation of KaiC. KaiA binding to KaiC promotes phosphorylation of KaiC, while KaiB binding to KaiC promotes dephosphorylation. Preliminary work indicates that, as temperature decreases, KaiC phosphorylation can occur in a KaiA-independent manner, which can be viewed as modulating the feedback in the network. Furthermore, multiple steady states that depend on initial conditions appear at low temperatures. A computational approach based on algebraic geometry will be used to map the fixed points and bifurcations of models of the Kai system at various temperatures and, in turn, to elucidate general principles about the role of feedback in temperature compensation. The computational results will be tested experimentally using mutants of the Kai proteins that alter the KaiA-KaiC interaction. By revealing how feedback in a reaction network is related to the fixed-point and bifurcation structure elucidated by the approach based on algebraic geometry, the project will advance dynamical systems theory.
Quantifying Natural Movement Variation in the Brain and Behavior
Faculty Mentors: Stephanie Palmer (University of Chicago) & Jason MacLean (University of Chicago)
Abstract: The ultimate goal of neural processing is to drive reliable behaviors in an animal's natural environment, maximizing fitness in a complex environment. Our hypothesis posits a causal, evolved relationship between the complexity and structure of the brain and behavior and requires new mathematical approaches to both quantifying and recapitulating this matching between natural and neural state space. To deepen our understanding of motor encoding and control, we will integrate behavioral recording in freely moving mice executing a seed reach-to-grasp task with extensive, longitudinal tracking of neuronal activity across various layers in the motor cortex. This involves pairing detailed behavioral observations with comprehensive neuronal population recordings to characterize the mapping between the brain's control space and the resulting movement space exhibited by the arm and paw during the challenging task. To establish a direct connection between behavioral and neural data, we will utilize machine learning tools such as VAEs and U-nets to quantify the latent space of both datasets. Our primary objective with these advanced machine learning approaches is to identify interpretable features within the representations, and to develop new mathematics to define trajectories in this feature space. The goodness of fit will be assessed by training models on behavioral data and evaluating their ability to generate realistic limb and paw trajectories, with the constraints that these are differentiable and low-dimensional. Throughout our investigation, our specific focus will be on uncovering the features of the neural response that drive variable yet successful reach movements. By examining how the brain's code aligns or deviates from behavioral complexity, our goal is to reveal new principles of motor encoding and control that operate over both evolutionary and organismal timescales
Uncovering the Genetic Fitness Landscape Behind Bacterial Motility in Complex Environments
Haibei Zhang
(University of Chicago)
Leone Luzzatto
(Northwestern University)
Faculty Mentors: Jasmine Nirody (University of Chicago) & István Kovács (Northwestern University)
Abstract: Bacterial motility is a complex phenomenon that plays a fundamental role in widespread biological processes including pathogenesis and bioremediation. Motile bacteria perform chemotaxis – migration under the influence of a chemical gradient – to find conditions optimal for their fitness and survival. This movement can be either towards (positive chemotaxis) or away from (negative chemotaxis) a chemical stimulant. One important feature of this network is its ability to adapt to changes in the environment, allowing cells to maintain a high sensitivity to their environment over a wide range of chemical backgrounds. In natural environments, this sensory process also takes place in a cluttered and noisy mechanical background, as cells are constantly exposed to heterogeneous, variable physical cues. Despite this, the vast majority of studies into bacterial motility and chemotaxis have been performed in unconfined liquid media or along flat surfaces. In this project, we aim to develop both mechanistic and evolutionary insights into bacterial motility and adaptation under various environmental conditions. A key impact of the environment is posed by limiting the free path length for the bacteria. We will characterize the corresponding chord-length statistics and develop a modeling framework that takes into account geometric constraints. We will also revise the current theoretical models on chord-length statistics (Levitz&Tchoubar, 1992), as they rely on assumptions that are not valid in the planned experiments, leading to qualitative differences. We will also develop a hyper-network framework to infer fitness consequences of changes in the chemotactic gene regulatory network. Combining biophysical experiments using a novel microfluidic setup and modeling with predictive hyper-network analysis, we outline an investigation to characterize how the chemotactic sensing pathway adapts over multiple timescales to improve bacterial performance and fitness in a range of complex, naturalistic environments.
Learning Rules of Epithelial Tissue Dynamics
Anthea Weng
(Northwestern University)
Jesse Lin
(University of Chicago)
Matthew Schmitt
(University of Chicago)
Keanne Marie Quinn
(Northwestern University)
Faculty Mentors: Margaret Gardel (University of Chicago), Cara J. Gottardi (Northwestern University), & Vincenzo Vitelli (University of Chicago)
DNA as a Phosphate Reservoir: Spatiotemporal Modeling of Extracellular DNA (eDNA) Dynamics in Biofilms
Garth Fisher
(Northwestern University)
Blake Everett
(Northwestern University)
Mingjie Pei
(Northwestern University)
Faculty Mentors: Arthur Prindle (Northwestern University) & David Chopp (Northwestern University)
Abstract: DNA is the genetic code found inside all living cells and its molecular stability can also be utilized outside the cell. While extracellular DNA (eDNA) has been identified as a structural polymer in bacterial biofilms, whether it persists stably or can be reclaimed for further cellular activity remains unknown. Here, by imaging eDNA dynamics within undomesticated Bacillus subtilis biofilms, we propose to test the hypothesis that DNA acts as a temporary structural scaffold that is later metabolized for cell growth. Specifically, we found that eDNA is produced throughout biofilm development before being degraded in a spatiotemporally coordinated pulse. We identified YhcR, a secreted Ca2+-dependent nuclease, as responsible for DNA degradation. As predicted by a preliminary mathematical model, biofilms lacking this nuclease fail to reclaim DNA for its phosphate contents, thereby decreasing biofilm fitness. Our results identify a secreted nuclease that is crucial for reclaiming eDNA during biofilm development, expanding our knowledge of DNA and suggesting new targets for biofilm control.
Natural Motion and Optimal Prediction in a Complete Retinal Population
Le Trung Tran
(Northwestern University)
Faculty Mentors: Gregory W. Schwartz (Northwestern University) & Stephanie Palmer (University of Chicago)
Abstract: Signals in the natural world are often characterized by a mixture of amplitude scales, like the quiet and loud segments of a musical recording. This property is manifested in the form of non-Gaussian, heavy-tailed distributions and nonlinear dependencies, both over time and across signal components. This is in strong contrast to the Gaussian and linear features typically assumed when modeling input signals. In the context of neuroscience, natural signals pose a serious challenge for sensory systems, which must adapt on the fly in order to efficiently encode them. Our recent work has demonstrated that the motion of objects in natural scenes also contains a mixture of scales, with a locally averaged velocity amplitude that fluctuates significantly on sub-second timescales. We have shown that this behavior can be modeled using an autoregressive Gaussian scale-mixture (ARGSM) model, which captures the temporal correlation structure of both the velocity and the fluctuating scale. Retinal responses to object motion have been characterized previously using carefully controlled artificial stimuli with Gaussian and linear statistics, revealing an efficient predictive code through the information bottleneck method. Here, we will extend this analysis to more naturalistic stimuli by incorporating a fluctuating scale variable matched to the statistics of natural scenes. We will bring together new experimental access to full RGC populations and our new theory about predictive coding and natural motion statistics. We will quantify predictive information about 1D and 2D motion trajectories in complete populations and sub-populations of mouse RGCs. Theoretically, this will require new calculations of information bottleneck-optimal representations under the ARGSM model. These will allow us to assess the performance of the retinal code using state-of-the-art recordings of mouse retinal ganglion cells (RGCs). Of particular interest are the contributions of the great diversity of RGC subtypes to the neural coding of these dynamically rich, naturalistic stimuli.
Understanding Synaptic Wiring Rules in the C. Elegans Brain
Bingie Hao
(Northwestern University)
Guodong Xie
(University of Chicago)
Tingyu "Mark" Zhao
(Northwestern University)
Leone Luzzatto
(Northwestern University)
Indya Weathers
(University of Chicago)
Faculty Mentors: István Kovács (Northwestern University) & Engin Özkan (University of Chicago)
Abstract: Ongoing advances in brain imaging and single-cell RNA sequencing have produced a massive amount of data on the genetic identity of neurons and their synaptic connections. However, these advances set up a complexity bottleneck: We need guiding frameworks to integrate and conceptualize this data, distill the key emergent patterns and aid new biology discovery. Addressing this knowledge gap, we will focus on the following critical questions: i) What are the connection rules of neural networks, governing the emergent network structure and wiring mechanisms? ii) How can we integrate the existing connectomics, proteomics and transcriptomics datasets into a coherent and predictive theoretical framework? To start, we need to decode the genetic programs behind synapse formation and maintenance to make sense of the data and gain insight into the network organization and functional circuitry of the brain. As a solution, we propose a scalable modeling framework building upon our recently pioneered Spatial Connectome Model (SCM). The central hypothesis of the SCM is that synapses emerge due to an underlying wiring rule network that connects pre-synaptic and post-synaptic neuron features. First, we will develop scalable solutions to the SCM, using two alternative approaches, a Bayesian framework and an expectation maximization route. While the original model was linear, the underlying biological rules are highly non-linear and we aim to introduce and solve the non-linear SCM. In addition, we will introduce and solve a local SCM, allowing for the wiring rules to vary over different parts of the network. We will also develop a novel mathematical framework to debias experimental data for protein-protein interactions, extending the maximum entropy framework. Our research strategy focuses on the C. elegans as a model organism and combines tools from neuroscience, molecular biology, network science, and statistical physics to capture complex wiring mechanisms as well as key biological constraints. We will provide a series of falsifiable predictions, starting with i) neuron wiring rules, and ii) inferring missing synapses from the input data, as well as iii) inferring changes in the connectome upon genetic perturbations. We will then experimentally validate the key predictions of our computational framework using in vivo and in vitro approaches in the C. elegans.
Modeling and Analysis of Synchronous Behavior in Biological Systems
Aaron Scheiner
(Northwestern University)
Kumar Utkarsh
(Northwestern University)
Bennet Sakelaris
(Northwestern University)
Issac Brown
(Northwestern University)
Faculty Mentors: Daniel Abrams (Northwestern University) & Guy Amichay (Northwestern University)
Abstract: How does collective synchronous behavior emerge in living systems? We focus on two readily observable and malleable systems: firefly swarms (flashing in unison) and groups of fiddler crabs (waving their large claws in sync). In both examples, these groups are composed of males attempting to woo females through such collective displays. Our research will use four complementary approaches: (i) fieldwork to compile unparalleled new datasets (including the development and employment of novel experimental paradigms, perturbing animals with artificial conspecifics in the wild), (ii) development of new mathematical models for understanding the mechanism underlying biological sync (with potential implications for broader evolutionary theory), (iii) monitoring of a firefly population for conservational efforts, and (iiii) outreach and dissemination of our models and results to the public via podcast, visual arts, and print journalism. On the mathematical side, we are particularly interested in novel coupled oscillator models that can exhibit both phase synchrony and "breathing" chimera states that have been glimpsed in preliminary data. These models include oscillators coupled not just through phase but also through amplitude and / or frequency. We will explore such models on coupling networks of increasing realism, and we also plan to study the model selection problem in the context of oversampled dynamical data, where conventional approaches may need to be modified.
Keeping Growing Clocks in Sync
Sneha Kachhara
(Northwestern University)
.jpeg)
Jorge Luis Ocampo Espindola
(Northwestern University)
Mingjie Pei
(Northwestern University)
Connor Harrison Puritz
(Northwestern University)
Eliza Duvall
(Northwestern University)
Lily Burton
(University of Chicago)
Faculty Mentors: Rosemary Braun (Northwestern University) & Michael Rust (University of Chicago)
Abstract: The circadian clock, an endogenous near-24 h rhythm that can be entrained to environmental time cues, is a ubiquitous feature of life on Earth. An ancient mechanism for circadian oscillation is found in cyanobacteria, photosynthetic microbes found across the globe. Oscillations are based on cyclic phosphorylation of KaiC molecules, which are coupled together to create coherent bulk oscillations, a phenomenon which can be reconstituted using purified proteins (KaiA, KaiB, KaiC). However, in some cyanobacteria, the growth rate can be as much as 10x the oscillator frequency. In such a situation, the vast majority of protein at the end of a circadian cycle will be new protein that was not present at the beginning of the cycle. We will answer fundamental questions in this system using a combination of theoretical approaches and in vivo and in vitro measurements: How are new KaiC molecules “brought up to speed” without disturbing the frequency? Is this achievable by Kai proteins in isolation or does it require in vivo mechanisms? Is there a fundamental upper limit to the growth rate that is still compatible with oscillation? In addition to studying specific chemical reaction models of the Kai system, we will study the generic effects of growth on coupled oscillator systems by introducing a birth-death process into Kuramoto-like models of coupled phase oscillators to study the transition between phase-locked rhythms and desynchrony."
What’s the Place for Planning?
Angel German Espinosa
Christopher Angeloni
(Northwestern University)
Alexander Lai
(Northwestern University)
Felix Maldonado
(Northwestern University)
Faculty Mentors: Malcolm MacIver (Northwestern University), Daniel Dombeck (Northwestern University), Matthew Kaufman (University of Chicago), & Bradly Stadie (Northwestern University)
Abstract: Our research began with a fundamental question in computational neuroscience: how do biological systems learn effectively across multiple time scales? Our initial approach using transformer-based RL (Q-Zero) led us to develop D2AC, a diffusion-based policy optimization method. However, this work revealed a more fundamental issue: standard RL agents optimize for population-level success at the cost of individual survival, taking unnecessary risks that no biological entity would consider rational. Our latest work, 'Of Mice and Machines,' systematically analyzes this survival-value disparity between artificial and biological learning, where mice consistently prioritize individual survival while RL agents sacrifice it for marginally better aggregate performance. We've developed new mathematical frameworks that modify the core RL objective function with variance-based components, making artificial agents value their 'lives' more like biological entities do. Our results show that these biologically-inspired modifications not only produce more natural behavior but also improve performance on navigation tasks. The anticipated outcomes include both a deeper understanding of biological risk-assessment principles and more effective artificial learning algorithms that better mirror natural learning processes. This funding also allowed the completion of a publication on converting static plans into time-varying movement commands, and several upcoming preprints on movement representation in mouse sensorimotor areas and dynamical inference in noisy nonlinear systems. The Northwestern PIs leading the effort are: Assistant Professor Bradly Stadie, Dept of Statistics and Data Science, Professor Malcolm A. MacIver, Dept of Mechanical Engineering, Dept of Biomedical Engineering, Professor Daniel A. Dombeck, Dept of Neurobiology. NU Trainees being funded: German Espinosa (Postdoc); Chris Angeloni (Postdoc). The University of Chicago PI is Assistant Professor Matthew Kaufman, Department of Organismal Biology and Anatomy;Zulfar Ghulam-Jelani, PhD student.
A Theory of Models for Complex Ecology
Siqi Liu
(Northwestern University)
Sagnik Ghosh
(Northwestern University)
Kiseok Lee
(University of Chicago)
Faculty Mentors: Seppe Kuehn (University of Chicago) & Madhav Mani (Northwestern University)
Abstract: This proposal aims to develop new mathematical techniques to gain a deeper understanding of the "Theory of Models" in complex living systems. We will employ this formalism to explore how dynamic variations in the soil microbiome contribute to metabolic stability in the face of environmental changes. Effective models are essential for interpreting complex ecosystem functions in response to environmental disturbances, allowing us to quantify data and propose underlying mechanisms. Ecosystems offer an ideal setting to enhance our understanding of the mathematical constraints associated with the Physics of Models. We will develop new algorithms to facilitate data-driven model discovery and the identification of collective variables. Building upon groundbreaking research related to "Sloppy Models"~\cite{machta_parameter_2013}, our approach will harness the power of massively parallel experiments on soil microcosms, precise quantification of metabolite dynamics, controlled perturbations, and quantitative sequencing data. Our robust mathematical framework will enable us to develop ecological models that are easily interpretable.
Inferring Models for Microbial Dynamics
.jpg)
Cody Fitzgerald
(Northwestern University)
.jpeg)
Srilena Kundu
(University of Chicago)
Christina Catlett
(Northwestern University)
Yifan Zhang
(Northwestern University)
Pablo Lechon
(University of Chicago)
Faculty Mentors: Stefano Allesina (University of Chicago), Niall Mangan (Northwestern University), Mary Silber (University of Chicago), & Rebecca Willett (University of Chicago)
Abstract: Microbial communities are widespread from the human gut to the deep ocean and influence systems including animal development, host health, and biogeochemical cycles. Characterization of complex communities is challenging, as morphology, physiology, evolution, and sensitivity to the environment all influence microbial interactions-- richness not captured in commonly used Lotka-Volterra-style models originally developed for macro-scale ecological systems. High throughput sequencing has enabled high-resolution quantification of populations within natural and synthetic communities, which could aid in the development of novel mathematical models to explain complex interactions such as diauxic shifts, cross-feeding, biofilm formation, and pH modification. Data-driven model development presents several mathematical challenges: 1) usually measurements of relative but not absolute abundances are available, 2) unmeasured dynamic variables such as nutrient levels can strongly impact populations, and 3) evaluation of all possible models and interactions is costly due to the combinatorial complexity of possible interactions. Challenges 1 and 2 manifest mathematically as identifiability issues; multiple models and parameter sets can produce the same trends in the data. To identify ensembles of possible models, we will perform parameter estimation across tens of thousands of possible models capturing the range of interaction mechanisms. Informed by commonalities of structure and behavior in the ensemble and statistical analysis of fluctuations we will develop identifiability-informed model sampling techniques to accelerate future screens and infer absolute abundance dynamics from relative abundance data.
Topological Analysis of Biological Data
Joseph Sifakis
(University of Chicago)
Ryan Robinett
(University of Chicago)
Maalavika Pillai
(Northwestern University)
Faculty Mentors: Samantha Riesenfeld (University of Chicago), Shmuel Weinberger (University of Chicago), & Richard Carthew (Northwestern University)
Abstract: Understanding the underlying geometry and topology of biological data is a challenging problem that is key to improving inference. This new collaboration will adapt topological data analysis (TDA) tools to high-dimensional biological data, focusing on two datasets with distinct, complementary features: (i) images of morphological variation across closely related species, and (ii) transcriptomic samples of gene expression variation across related tissue samples. These very different data sets will give us a chance to study two different aspects of the usual TDA pipeline: one is the possibility of defining “testable” invariants (along the lines of property testability), i.e. ones that do not require full consideration of all the data to be approximated, and seeing whether they have utility for biological applications. The second is whether the space underlying a data set is actually Euclidean, or whether its embedding in Euclidean space (induced by the use of a number of measurements of each datum) induces metric distortion. We hope to approach this using the theory of metric distortion, and hope that this initiates a new large scale approach to understanding the geometry of data.
Inverse problem of inferring adaptive strategies from the statistics of rare events
Ben Kuznets-Speck
(Northwestern University)
Kalki Kukreji
(Northwestern University)
Maryn Carlson
(NITMB Fellow)
Madeline Melzer
(Northwestern University)
Faculty Mentors: Arvind Murugan (University of Chicago) & Yogesh Goyal (Northwestern University)
Abstract: Biology has a diverse range of adaptation strategies to deal with changing environments that range from Darwinian multi-generational processes which play out over millions of years to within-a-lifetime learning. The underlying mechanistic basis of these strategies is highly varied and context dependent. The traditional time-consuming approach has been to distinguish these strategies with mechanistic experimental approaches. Here we propose building a mathematical framework to guide high throughput experiments that will use rare event sampling to reveal learning and adaptation strategies. We will apply our mathematical framework to experiments on drug resistance in cancer cells and in microbes. The proposed work here will (a) solve the inverse problem of inferring a broad class of adaptation strategies with finite heritability from the shape of rare-event distributions; (b) tailor proposed mathematics to specific regimes accessible in current high-throughput experiments, (c) develop novel experimental workflows for studying drug resistance in cancer cells and microbes.
Developing predictive frameworks for the control of non-equilibrium cellular membrane dynamics
Simon Fabiunke
(Northwestern University)
Lissa Rennert
(University of Chicago)
Sen Diptendu
(Northwestern University)
Gonzalo Ferrandez Quinto
(Northwestern University)
Faculty Mentors: Suriyanarayanan Vaikutanathan (University of Chicago) & Petia Vlahovska (Northwestern University)
Abstract: The material properties of biological membranes control a vast array of molecular processes. Biological lipid membranes behave like fluids in plane and exhibit elastic fluctuations out of plane. While the basic driving forces for describing membrane biophysics are easy to formulate, their emergent properties and morphologies they can elicit remain important open questions. In this project, we seek to leverage modern advances in non-equilibrium statistical mechanics along with ideas from representation learning and AI to identify low dimensional physical laws for the non-equilibrium dynamics of biomimetic membranes. We will use a combination of experiments by Petia Vlahovska and co-workers, and theory from Petia Vlahovska, Suri Vaikuntanathan and co-workers. Briefly, data from experiments studying the fluctuations of model lipid membranes in electric fields mimicking polarized cellular membranes will, to the best of our knowledge for the first time, be analyzed using dimensional reduction techniques to infer physical laws and constraints in low dimensional spaces. These constraints will then be related to modern non-equilibrium thermodynamic bounds. If successful, this integration of theoretical approaches based on thermodynamics and AI and experiments with biomimetic membranes will compactly reveal how biological lipid membrane dynamics can be described, controlled, and leveraged. The project will establish a new collaboration of faculty from University of Chicago and Northwestern University with complementary expertise in statistical and continuum mechanics modeling, and experimental biomimetic membrane systems.
Towards the Molecular Basis of Graft Compatibility
Thomas Wytock
(Northwestern University)
Evan Eifler
(Northwestern University)
Jorin Graham
(Northwestern University)
Le Trung Tran
(Northwestern University)
Christopher Brian Volger
(Northwestern University)
Stephen Zhao
(Northwestern University)
Soren Moore
(Northwestern University)
Faculty Mentors: Adilson Motter (Northwestern University) & Nyree Zerega (Northwestern University)
Abstract: Plant grafting, the process of joining separate plants into a single organism, has been used for millennia in horticulture and food production. Despite this long-standing practice, gaps remain in our knowledge of how to predict if and why two different species will or will not be compatible for grafting. This collaborative project will develop and apply a novel approach that combines mathematical tools from network theory and data from plant biophysics to systematically identify key determinants of graft compatibility across plant species on a large scale. The mathematical effort will be centered on the development of inference methods for incomplete data. Experiments will be conducted to validate predictions by testing novel species pairings. The result will be a data-driven framework to identify biophysical, phylogenetic, and molecular mechanisms underlying graft compatibility. Success will be measured by the ability of the approach to predict compatibility between untested species. The research will be conducted at Northwestern University, with a graduate student dedicated to mathematical development and a postdoctoral researcher specializing in plant biophysics. Broader impacts include the potential of this research to contribute to food production, conservation efforts, and carbon fixation. The project will lead to innovative mathematical and biological development and is well aligned with several NITMB themes, especially Learning & Adaptation and Fitness & Optimization.
Linking gene expression profiles to firing properties in the Drosophila thermosensory circuit
Michael Harris Alpert
(Northwestern University)
Nan Ding
(Northwestern University)
Faculty Mentors: William Kath (Northwestern University) & Marco Gallio (Northwestern University)
Abstract: We will use neurons that are part of the Drosophila thermosensory circuit as models to explore the extent to which molecular profiling data obtained through single-cell patch-sequencing can be used to predict a neuron’s firing properties. We will develop new information theory-based data filtering and similarity methods to compare patch-seq and single-cell data and obtain improved estimates of ion channel expression in these neurons. We will then produce anatomically realistic models of thermosensory neurons by using the expression data to populate models of these neurons with candidate ion channels and regulators. Furthermore, we will extend current evolutionary algorithms to incorporate ion expression correlation data to improve fits to in-vivo electrophysiological measurements of each cell type. Our goal is two fold: to determine the extent to which a neuron’s firing properties can be extrapolated from its gene expression profile and to explore how underlying variability in a neuron’s repertoire of components can nevertheless lead to robust firing properties. Overall, we expect that our methods and models will bring clarity regarding the patterns of expression of ion channels, receptors, and signal transduction components that are key determinants of the functional properties of neurons in the thermosensory circuit, point to nodes of the network that may be particularly sensitive to perturbation, and make specific predictions on the effects of such perturbations, eventually directing new experiments that exploit cell-type specific RNAi and genetic mutants.
A new mathematical framework for classification in cell state dynamics
Aurelia Leone
(Northwestern University)
Emanuelle Grody
(Northwestern University)
Leon Schwartz
(Northwestern University)
Faculty Mentors: Yogesh Goyal (Northwestern University) & Suriyanarayanan Vaikuntanathan (University of Chicago)
Abstract: Our proposal emerges from a crucial challenge in cell biology: how do transcriptionally identical cells make different fate decisions when exposed to therapeutic drugs? We have observed that while cancer cells may appear homogeneous, they can develop remarkably diverse resistance trajectories when treated with drugs. To address this fundamental question, we propose developing new mathematical tools that unite concepts from statistical mechanics of deep neural networks, non-equilibrium statistical mechanics, and information theory to understand how biological networks function as classifiers. Our work extends our recent findings showing how biochemical networks' classification capacity can be systematically tuned through factors like input promiscuity. We anticipate that this ambitious undertaking will establish a mathematically consistent framework for defining cell states and their transitions, elucidate the minimal requirements for biological networks to classify perturbations, and create predictive tools for cellular responses to therapeutic interventions.
Form and Function of Drifting Olfactory Representations in the Piriform Cortex
Ethan Baxter
(Northwestern University)
Maanasa Natrajan
(Northwestern University)
Faculty Mentors: James Fitzgerald (Northwestern University) & Andrew Fink (Northwestern University)
Abstract: Cognition and behavior are generated by patterns of neural activity. This has led many to equate brain functions with specific neural activity patterns, also called representations. However, recent data suggest that the mapping between neural representations, cognition, and behavior is more complicated. For instance, the set of neurons representing an odor’s identity in the olfactory (piriform) cortex changes over time. Such shifts in neural representations are termed “representational drift,” because they are devoid of discernible learning, forgetting, or behavioral alterations. Recent work has modeled representational drift in neural networks as the random exploration of representations that correctly produce a memorized set of input-output associations. This revealed that representational drift can benefit memory by finding sparse representations that make the system more robust to noise and continual learning. However, current models do not produce realistic representational drift, and it is unclear if this theoretical benefit occurs for biological systems. Here we will assess whether this robustness benefit applies to realistic models of representational drift in the piriform cortex. First, we will use data from the Fink Lab to quantify the geometry of representational drift and its statistics of change. Previous findings suggest that there is no linear stable subspace, so we will specifically search for nonlinear representational features that are invariant to drift. Second, we will generalize the Fitzgerald Lab’s analyses of neural network solution spaces from linear readouts to nonlinear readouts that better match the empirically stable dimensions. Finally, we will combine these results to build and analyze a representational drift model that realistically mimics the piriform cortex. This work will advance both biology and mathematics, as representational drift is a fundamental biological mechanism, and new mathematics will be needed to quantify representational geometry and neural network solution spaces.
Developing a mathematics for evolved systems
Bipul Pandey
Postdoctoral Scholar, Physics
(University of Chicago)
Benjamin A. Doran
Graduate Student
(University of Chicago)
Faculty Mentor: Arjun Raman (University of Chicago)
Abstract: Systems that arise through the evolutionary process—iterative selection and variation—are qualitatively distinct from engineered systems in many ways from a functional and evolutionary standpoint. However, our capacity to understand how evolved systems manifest these differences is substantially limited because evolved systems are apparently extremely complex, comprised of many parts that interact together in unintuitive ways. Our laboratory has demonstrated that naturally evolved systems are hierarchically organized into layers of information that encode the complex whole. This result motivates creating a new mathematics that can use data collected on natural systems to generate hierarchically layered, functional emergent systems. We anticipate that our findings will lay the foundation for the mathematics of natural emergent simplicity, thereby creating generative models qualitatively distinct from those that exist today. The personnel involved in this effort are Dr. Bipul Pandey (Ph.D., Physics; Postdoctoral Scholar, University of Chicago) and Benjamin A. Doran (Graduate Student; Pritzker School of Molecular Engineering, University of Chicago).
Invasions in a Four-Species Cyclically Competing Ecological Community
Elisheva Siegfried
(Northwestern University)
Faculty Mentors: Alvin Bayliss (Northwestern University) & Vladimir Volpert (Northwestern University)
Abstract: We consider ecological communities consisting of 4 species engaging in cyclic competition. One can visualize the competition scheme by considering the four species, call them u, v, w, and z, as located on a clock, at 12:00 o’clock, 3 o’clock, 6 o’clock and 9 o’clock, respectively, with each species having a competitive advantage over its counterclockwise neighbor, i.e., species v (3 o’clock) wins over species u (12 o’clock) and similarly for the other competition pairs. When the competition is strong such communities are known to be dynamically stable, with two stable alliances formed by species which do not directly compete, i.e., u − w and v − z alliances , the only possible long-time outcomes of the competition. However, in nature the competitive interactions will not be the same for each species, furthermore there can be internal competition (and possibly predation) within each alliance. We consider how to determine which alliance is stronger depending on parameters of the competition. Said another way, which alliance will be able to displace the other alliance (in colloquial terms invade its territory). This problem is a version of a classical mathematical problem that transcends ecology - given two stable states of a system how to determine which state is dominant. This problem can be reduced to analysis of a system of four coupled boundary problems on an infinite interval. This system cannot be solved exactly. We will develop mathematical methods to address this problem. We will primarily employ asymptotic analysis - generally formulating the problem as a singular system, with one or more small parameters, and developing methods to approximate the solution in appropriate parameter regimes. Our analysis will enable us to determine the steady state outcome of the competition scheme, i.e., the surviving alliance.
Adaptation and Evolvability through Reinforcement Learning
Rathi Kannan
(University of Chicago)
Faculty Mentors: Vincenzo Vitelli (University of Chicago), Seppe Kuehn (University of Chicago), & Bradly Stadie (Northwestern University)
Abstract: The research proposal addresses the biological question of how the complexity of the genotype-to-phenotype mapimpacts adaptability and evolvability in dynamic environments, using reinforcement learning (RL) as a framework. The new mathematics being developed is a analytical and computational interrogation of how the complexity of the underlying network governing the behavior of an RL agent impacts its performance on learning tasks. Anticipated outcomes include theoretical insights into evolutionary dynamics, testable predictions about adaptability and evolvability, and experimental validation using microbial systems such as algae under temporally correlated light and temperature stresses
Modeling RNA sequence-structure-function relationships with multiscale higher-order graph neural networks
Edric Choi
(Northwestern University)
Tina Fu
(Northwestern University)
Xavier Bower
(Northwestern University)
Laura Hertz
(Northwestern University)
Faculty Mentors: Julius Lucks (Northwestern University) & Risi Kondor (University of Chicago)
Abstract: RNAs play central roles in regulating, maintaining, and defending the genomes of all organisms, with regulatory RNA sequences controlling almost all aspects of gene expression. Many of these RNA functions are linked to RNA structures that mediate interactions amongst cellular gene expression machinery, bind ligands, and perform catalysis. A central goal in biology then has been to solve the ‘RNA folding problem’ – to understand how RNA sequence determines RNA folding which governs RNA function. Once deciphered, the solution to the RNA folding problem would improve our understanding of living systems and our ability to program RNAs for biotechnologies. Graph neural networks (GNNs) are a promising new mathematical approach to modeling biomolecules, but currently do not have the mathematical properties needed to capture the features of large RNA molecules that can exist in multiple states. Here we propose to develop a new theory of graph modeling that encodes multiscale interactions within the graph architecture, preserving necessary properties of equivariance. To do so, we will derive new mathematical relationships of multiscale equivariant message passing and prove that the resulting model is the most general possible permutation equivariant multiscale neural architecture. By advancing the theory of higher order multiscale GNNs we will create a new, broadly applicable general class of neural architectures, which will apply to create a new approach to modeling RNA sequence-structure-function relationships.
Uncovering the link between chromatin organization and global transcriptional regulation
Karla Medina
(Northwestern University)
Maalavika Pillai
(Northwestern University)
Faculty Mentors: Luís A. Nunes Amaral (Northwestern University) & Vadim Backman (Northwestern University)
Abstract: The goal of the proposed research is to identify and develop mathematical formulations that will enhance our mechanistic and quantitative understanding of how cell types and their idiosyncratic states can be maintained or can evolve over time or in response to stimuli. We will use experimental data to construct the multiple interaction layers — from transcript factor regulation to chromatin accessibility — regulating gene expression and will use a multiplex network formalism to model different cellular states. Our modeling will enable us to understand the processes by which, for example, a cancer state deregulates gene expression.
Characterizing excitability and its applications to immunity
Nico Romeo
(University of Chicago)
Eric Leisten
(University of Chicago)
Chris Chi
(University of Chic)
Faculty Mentors: Elizabeth Jerison (University of Chicago), Aaron Dinner (University of Chicago), & Hermann Riecke (Northwestern University)
Abstract: The functions of many biological systems — including spiking neurons and activating macrophages — depend on excitable dynamics: a small perturbation triggers a rapid, nonlinear ramp followed by a return to equilibrium. While this behavior is common biologically, excitability lacks a precise mathematical definition, and the generic properties of these systems remain unclear. Prior work in theoretical neuroscience has explored in detail specific models that produce excitable behavior. This work describes the emergence of excitable dynamics near different types of bifurcations, and shows that distinct mathematical structures describe different types of neurons and their computational properties. Importantly, the same perturbation can have opposite effects depending on the mathematical origin of the excitability and the timing of the perturbation relative to the state of the system. Thus predicting and controlling the behavior of these systems depends on understanding the underlying dynamical structure, which demands enumeration of the types of excitable systems and their mathematical properties. We will take a two-pronged approach to define and classify excitable systems. First, we will extend transition path theory (TPT) to excitable systems and use it to define excitability precisely. Second, we will combine TPT with machine learning to map dynamical behavior over a broad class of models to enumerate the types of excitability. Both normal and pathological immune responses ‘flare’ — immune signaling molecules (cytokines) and immune cell populations amplify transiently before returning to baseline. This behavior suggests that systemic immune responses may be excitable. Observational data and modeling of multiple sclerosis and the existence of genetic disorders that cause spontaneous recurrent hyperinflammatory flares support this hypothesis. As in neuroscience, minimal dynamical models of these excitations could be powerful tools to understand the information processing capabilities of the immune system and develop strategies to intervene in systemic immune responses. However, we generally lack tractable experimental systems in which to make the controlled perturbations and quantitative observations necessary to test these hypotheses. To enable investigation of the excitable properties of systemic immune responses, we will study a ‘cytokine storm’ response triggered by a systemic pulse of the pathogen-associated molecular pattern lipopolysaccharide (LPS) in larval zebrafish. We will use this system to test properties such as the existence of an excitation threshold, and use our new classification of excitable systems to develop models capturing key nonlinear features of the response. Ultimately, we aim to understand which perturbations, at what stage of the response, would be necessary to return a systemic immune response to a homeostatic fixed point, allowing for the design of dynamical interventions.
Applications of free probability to the diversity of response and variability in neuronal networks
Faculty Mentors: Brent Doiron (University of Chicago) & David Freedman (University of Chicago)
Abstract: Cognition is supported by neuronal activity that spans several brain regions, and understanding how this activity is coordinated is essential for a coherent theory of neuronal function. We will study how the structure of a visual categorization task in non-human primates influences the coordinated activity of three brain areas known to be essential for proper task performance. While neuronal connectivity does influence how brains perform tasks on average, a key signature of connectivity is how it also determines the fluctuations (or variability) of brain activity during tasks. We will model the trial-to-trial variability of distributed population responses with simple, yet nonlinear, recurrent circuit models. A key advance of our proposal is to consider both the input to a brain region and the wiring within and between brain regions to be randomly structured. This choice will require the use of novel analysis techniques based in free probability theory as applied to random matrices, which can untangle how two sources of randomness contribute to the variability of circuit response. Our theory will make testable predictions which can be explored in our experimental framework. In total, our work will provide a deep understanding of how diverse population brain circuitry supports the underlying mechanics of neuronal codes.
Multimodal Data Analysis for uncovering host-microbiome responses to environmental stress
Maria Hernandez Limon
(University of Chicago)
Ruisi Liu
(University of Chicago)
Faculty Mentors: Claire Donnat (University of Chicago) & Catherine Pfister (University of Chicago)
Abstract: Algae are essential to aquatic ecosystems, but their survival depends on complex interactions with microbes like bacteria and viruses. These partnerships, known as holobionts, influence how algae respond to environmental stress. Our research merges biology and data science to better understand these interactions. We develop statistical methods to identify microbial communities, quantify uncertainty in data-driven discoveries, and extract meaningful patterns from complex, high-dimensional datasets. By applying these tools to real-world samples, we aim to uncover key microbes that support algal resilience. This work aims to enhance our understanding of how aquatic ecosystems adapt to environmental challenges.
