April 18, 2025 Gayashan Jayavilal (UC Grad Student, Mathbio Journal Club)

Finite population size effects on optimal communication for social foragers

Foraging is crucial for animals to survive. Many species forage in groups, as individuals communicate to share information about the location of available resources. For example, eusocial foragers, such as honey bees and many ants, recruit members from their central hive or nest to a known foraging site. However, the optimal level of communication and recruitment depends on the overall group size, the distribution of available resources, and the extent of interference between multiple individuals attempting to forage from a site. In this paper, we develop a discrete-time Markov chain model of eusocial foragers, who communicate information with a certain probability. We compare the stochastic model and its corresponding infinite population limit. We find that foraging efficiency tapers off when recruitment probability is too high, a phenomenon that does not occur in the infinite population model, even though it occurs for any finite population size. The marginal inefficiency at high recruitment probability increases as the population increases, similar to a boundary layer. In particular, we prove there is a significant gap between the foraging efficiency of finite and infinite population models in the extreme case of complete communication. We also analyze this phenomenon by approximating the stationary distribution of foragers over sites in terms of mean escape times from multiple quasi-steady states. We conclude that, for any finite group of foragers, an individual who has found a resource should only sometimes recruit others to the same resource. We discuss the relationship between our analysis and multiagent multiarm bandit problems. [link]

April 11, 2025 Ryan Kamp (UC Undergrad Student, Mathbio Journal Club)

Normative solutions to the speed-accuracy trade-off in perceptual decision-making

These notes contain a brief introduction to finding normative solutions to the speed-accuracy trade-off in perceptual decision-making. [link]

April 4, 2025 Junghyung Lee (UC Grad Student, Mathbio Journal Club)

Wake-sleep cycles are severely disrupted by diseases affecting cytoplasmic homeostasis

The circadian clock is based on a transcriptional feedback loop with an essential time delay before feedback inhibition. Previous work has shown that PERIOD (PER) proteins generate circadian time cues through rhythmic nuclear accumulation of the inhibitor complex and subsequent interaction with the activator complex in the feedback loop. Although this temporal manifestation of the feedback inhibition is the direct consequence of PER’s cytoplasmic trafficking before nuclear entry, how this spatial regulation of the pacemaker affects circadian timing has been largely unexplored. Here we show that circadian rhythms, including wake-sleep cycles, are lengthened and severely unstable if the cytoplasmic trafficking of PER is disrupted by any disease condition that leads to increased congestion in the cytoplasm. Furthermore, we found that the time delay and robustness in the circadian clock are seamlessly generated by delayed and collective phosphorylation of PER molecules, followed by synchronous nuclear entry. These results provide clear mechanistic insight into why circadian and sleep disorders arise in such clinical conditions as metabolic and neurodegenerative diseases and aging, in which the cytoplasm is congested. [link]

March 28, 2025 Prof. Naveen Vaidya (San Diego State U)

HIV Infection in Drug Abusers: Mathematical Modeling Perspective

Drugs of abuse, such as opiates, have been widely associated with enhancing susceptibility to HIV infection, intensifying HIV replication, accelerating disease progression, diminishing host-immune responses, and expediting neuropathogenesis. In this talk, I will present a variety of mathematical models to study the effects of the drugs of abuse on several aspects of HIV infection and replication dynamics. The models are parameterized using data collected from simian immunodeficiency virus infection in morphine-addicted macaques. I will demonstrate how mathematical modeling can help answer critical questions related to the HIV infection altered due to the presence of drugs of abuse. Our models, related theories, and simulation results provide new insights into the HIV dynamics under drugs of abuse. These results help develop strategies to prevent and control HIV infections in drug abusers.

March 14, 2025 Prof. Seulip Lee (Tufts U)

Efficient and Robust Multiphysics Simulations: Structure-preserving discretizations for fluid flow and transport models

This talk presents efficient and robust numerical methods for solving coupled fluid flow and transport problems, emphasizing structure-preserving discretizations as the key to accurate and stable multiphysics simulations.

For fluid flow in porous media, we consider a model that combines the Stokes and Darcy equations, requiring different finite element conformities depending on viscous parameters. Standard methods often suffer from pressure-dependent errors, making it challenging to develop a solver that performs uniformly across all viscosity regimes. To overcome this, we introduce a pressure-robust method that employs a velocity reconstruction operator, replacing velocity functions with a reconstructed velocity. This approach ensures error estimates independent of pressure, achieving stability and accuracy while preserving minimal degrees of freedom.

For transport problems in convection-dominated regimes, we develop polygonal monotone schemes that maintain numerical stability while accurately capturing boundary and internal layers. A key property is the discrete maximum principle (DMP), which guarantees physically meaningful solutions free from non-physical oscillations. The DMP follows from a monotonicity property, ensured by satisfying the M-matrix condition. To construct these schemes, we employ edge-averaged finite element (EAFE) methods, which naturally preserve the DMP while minimizing numerical dissipation. The proposed framework extends the EAFE concept to the finite volume method, the virtual element framework, and polygonal triangulation.

Together, these structure-preserving discretizations provide a foundation for efficient and robust multiphysics simulations that accurately capture both flow and transport phenomena. Finally, we explore real-world applications, such as cerebrospinal fluid (CSF) dynamics with glymphatic clearance in the brain, a critical process in waste removal linked to neurodegenerative diseases like Alzheimer’s.

March 7, 2025 Prof. Charles Prestigiacomo (UC Medicine)

On the Measure of an Aneurysm:  Towards Novel Assessments of Aneurysm Growth and Behavior

Cerebral aneurysms are focal dilatations of the vessel wall that have a propensity for rupture.  Though this risk is low, the rupture of a cerebral aneurysm is associated with a 50% mortality and a 25% morbidity.  Hence 3 of every 4 patients sustaining a ruptured aneurysm will be disabled or will die.  Understanding the genetic, histological and physical factors that are associated with rupture (and perhaps serve as predictors of rupture) ensures that only those patients at risk will be exposed to the risks of treatment, which in and of themselves are significant.

Over the years, clinician scientists and investigators have studied several of the biophysical factors involved in aneurysm formation, growth and rupture, as well as recurrence.  Indeed, as the power of mathematical and biophysical analysis grows, the ability to enhance the predictability of aneurysms at risk for rupture seems to increase. This seminar will introduce the broad topics of aneurysm growth, formation and rupture and specifically address the use of biomathematical models, fractal analysis and AI as tools for a better understanding of aneurysm behavior.

By the conclusion of this Seminar, attendees should be able to:

1. Identify the major morphometric components that have been identified as characteristics that predict rupture.

2. Describe the general principles of fractal dimensions and how they apply in a novel way to the development and growth of aneurysms.

3. Discuss the principles by which AI may be used to enhance the algorithms being developed to develop and enhance a predictive model of aneurysm rupture risk.

February 28, 2025 Prof. Mbani Sayi (UC Math)

High Accuracy Fitted Operator Methods for Solving Interior Layer Problem

Fitted operator finite difference methods (FOFDMs) for singularly perturbed problems have been explored for the last three decades. The construction of these numerical schemes is based on introducing a fitting factor along with the diffusion coefficient or by using principles of the non-standard finite difference methods. The FOFDMs based on the latter idea, are easy to construct and they are extendible to solve partial differential equations (PDEs) and their systems. Our work deals with extension of these methods to solve interior layer problems, something that was still outstanding. The idea is then extended to solve singularly perturbed time-dependent PDEs whose solutions possess interior layers. The second aspect of our work is to improve accuracy of these approximation methods via methods like Richardson extrapolation. Having met these objectives, we then extended our approach to solve singularly perturbed two-point boundary value problems with variable diffusion coefficients and analogous time-dependent PDEs. Finally, we carefully analyse and present extensive numerical simulations to support theoretical findings.

February 21, 2025 Prof. Grzegorz Rempala (Ohio State U)

Epidemic Dynamics on Networks: Law of Large Numbers, Correlation Equation, and Exact Closure Conditions

In this talk, I will present some recent results on a widely used closure method for modeling contagion dynamics on random networks, specifically within the configuration model framework. This method approximates the dynamics of node triples by using only pairwise interactions and is often justified heuristically. Using  the asymptotic  theory,  I will rigorously derive the necessary and sufficient conditions under which the closure is asymptotically exact.  The results provide a robust criterion for determining when closure methods can effectively capture contagion dynamics on large-scale networks. This work is in collaboration with Eben Kenah (OSU) and Istvan Kiss (Northeastern University).

February 14, 2025 Lora Newman (UC Grad Student, Mathbio Journal Club)

The impact of vaccination on human papillomavirus infection with disassortative geographical mixing: a two-patch modeling study

Human papillomavirus (HPV) infection can spread between regions. What is the impact of disassortative geographical mixing on the dynamics of HPV transmission? Vaccination is effective in preventing HPV infection. How to allocate HPV vaccines between genders within each region and between regions to reduce the total infection? Here we develop a two-patch two-sex model to address these questions. The control reproduction number R_0 under vaccination is obtained and shown to provide a critical threshold for disease elimination. Both analytical and numerical results reveal that disassortative geographical mixing does not affect R_0 and only has a minor impact on the disease prevalence in the total population given the vaccine uptake proportional to the population size for each gender in the two patches. When the vaccine uptake is not proportional to the population size, sexual mixing between the two patches can reduce R_0 and mitigate the consequence of disproportionate vaccine coverage. Using parameters calibrated from the data of a case study, we find that if the two patches have the same or similar sex ratios, allocating vaccines proportionally according to the new recruits in two patches and giving priority to the gender with a smaller recruit rate within each patch will bring the maximum benefit in reducing the total prevalence. We also show that a time-variable vaccination strategy between the two patches can further reduce the disease prevalence. This study provides some quantitative information that may help to develop vaccine distribution strategies in multiple regions with disassortative mixing. [link]

February 7, 2025 Selena Laikos (UC Grad Student, Mathbio Journal Club)

Phase-Response-Curves: Elucidating the Dynamics of Coupled Oscillators

Phase response curves (PRCs) are widely used in circadian clocks, neuroscience, and heart physiology. They quantify the response of an oscillator to pulse-like perturbations. Phase response curves provide valuable information on the properties of oscillators and their synchronization. This chapter discusses biological self-sustained oscillators (circadian clock, physiological rhythms, etc.) in the context of nonlinear dynamics theory. Coupled oscillators can synchronize with different frequency ratios, can generate toroidal dynamics (superposition of independent frequencies), and may lead to deterministic chaos. These nonlinear phenomena can be analyzed with the aid of a phase transition curve, which is intimately related to the phase response curve. For illustration purposes, this chapter discusses a model of circadian oscillations based on a delayed negative feedback. In a second part, the chapter provides a step-by-step recipe to measure phase response curves. It discusses specifications of this recipe for circadian rhythms, heart rhythms, neuronal spikes, central pattern generators, and insect communication. Finally, it stresses the predictive power of measured phase response curves. PRCs can be used to quantify the coupling strength of oscillations, to classify oscillator types, and to predict the complex dynamics of periodically driven oscillations. [link]

November 22, 2024 Prof. Donald French (UC)

Beginning Explorations of Numerical Methods for Fractional Derivative Problems

Fractional derivative initial and boundary value problems have become attractive to researchers over the last 10-20 years. In this talk, we describe some of the basic manipulations commonly used to analyze problems that involve these fractional integral & derivative non-local operators. As a beginning on our work, we provide an error estimate for the convolution spectiral method of Amiri-Hezaveh et al (2021-2023) when applied to a simple initial value problem.

November 15, 2024 Prof. Hyunjoong Kim (UC)

Should I pay more now for better information? Normative decisions for individuals and groups in changing environments

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, 2024 Prof. Patrick Murphy (San Jose State University)

Data-Driven Modeling of Bacterial Aggregation

The soil bacterium Myxococcus xanthus is a model organism with a set of diverse behaviors. These behaviors include the starvation-induced multicellular development program, in which cells move collectively to assemble multicellular aggregates. After initial aggregates have formed, some will disperse, with smaller aggregates having a higher chance of dispersal. Initial aggregation is driven by two changes in cell behavior: cells slow down inside of aggregates and bias their motion by reversing direction less frequently when moving toward aggregates. However, the cell behaviors that drive dispersal are unknown. In this talk, we will first show how cell behaviors can systematically be extracted and analyzed from fluorescent microscopy data. Using agent-based modeling, we then show dispersal is predominantly generated by a change in bias, which is strong enough to overcome slowdown inside aggregates. Notably, the change in reversal bias is correlated with the nearest aggregate size, connecting cellular activity to previously observed correlations between aggregate size and fate.

November 1, 2024 Lora Newman (UC Graduate Student)

Transmission dynamics and optimal control of brucellosis in Inner Mongolia of China

A multigroup model is developed to characterize brucellosis transmission, to explore potential effects of key factors, and to prioritize control measures. The global threshold dynamics are completely characterized by theory of asymptotic autonomous systems and Lyapunov direct method. We then formulate a multi-objective optimization problem and, by the weighted sum method, transform it into a scalar optimization problem on minimizing the total cost for control. The existence of optimal control and its characterization are well established by Pontryagin's Maximum Principle. We further parameterize the model and compute optimal control strategy for Inner Mongolia in China. In particular, we expound the effects of sheep recruitment, vaccination of sheep, culling of infected sheep, and health education of human on the dynamics and control of brucellosis. This study indicates that current control measures in Inner Mongolia are not working well and Brucellosis will continue to increase. The main finding here supports opposing unregulated sheep breeding and suggests vaccination and health education as the preferred necessary emergency intervention control. The policymakers must take a new look at the current control strategy, and, in order to control brucellosis better in Inner Mongolia, the governments have to preemptively press ahead with more effective measures. [DOI]

October 25, 2024 Prof. Hyukpyo Hong (University of Wisconsin-Madison)

Koopman representation: Linear representation of nonlinear dynamics

A system of ordinary differential equations (ODEs) is one of the most widely used tools to describe a deterministic dynamical system. In general, ODEs involve nonlinear equations, which make analysis of dynamical systems difficult. In this talk, we introduce Koopman theory, which offers a (possibly infinite dimensional) linear representation of nonlinear dynamics. In particular, we demonstrate a data-driven algorithm to find such linear representations. We also provide some results on convergence of approximations obtained by the data-driven algorithm.

October 18, 2024 Selena Laikos (UC Graduate Student)

Mathematical analysis of robustness of oscillations in models of the mammalian circadian clock (Mathbio Journal Club)

Circadian rhythms in a wide range of organisms are mediated by molecular mechanisms based on transcription-translation feedback. In this paper, we use bifurcation theory to explore mathematical models of genetic oscillators, based on Kim & Forger’s interpretation of the circadian clock in mammals. At the core of their models is a negative feedback loop whereby PER proteins (PER1 and PER2) bind to and inhibit their transcriptional activator, BMAL1. For oscillations to occur, the dissociation constant of the PER:BMAL1 complex, $\hat{K}_d$, must be ≤ 0.04 nM, which is orders of magnitude smaller than a reasonable expectation of 1–10 nM for this protein complex. We relax this constraint by two modifications to Kim & Forger’s ‘single negative feedback’ (SNF) model: first, by introducing a multistep reaction chain for posttranscriptional modifications of Per mRNA and posttranslational phosphorylations of PER, and second, by replacing the first-order rate law for degradation of PER in the nucleus by a Michaelis-Menten rate law. These modifications increase the maximum allowable $\hat{K}_d$ to ~2 nM. In a third modification, we consider an alternative rate law for gene transcription to resolve an unrealistically large rate of Per2 transcription at very low concentrations of BMAL1. Additionally, we studied extensions of the SNF model to include a second negative feedback loop (involving REV-ERB) and a supplementary positive feedback loop (involving ROR). Contrary to Kim & Forger’s observations of these extended models, we find that, with our modifications, the supplementary positive feedback loop makes the oscillations more robust than observed in the models with one or two negative feedback loops. However, all three models are similarly robust when accounting for circadian rhythms (~24 h period) with $\hat{K}_d$ ≥ 1 nM. Our results provide testable predictions for future experimental studies. [DOI]

October 4, 2024 Prof. Ruxandra Dima (UC Chemistry)

Exploration of allosteric signals in molecular machines using graph networks and machine learning

Allostery has been recognized as being important for the proper action of molecular machines. The nanomachine from the ATPases associated with various cellular activities superfamily, called spastin severs microtubules during cellular processes. To characterize the functionally important allostery in this nanomachine, we employed state-of-the-art methods from evolutionary information, to graph-based networks, to machine learning applied to atomistic molecular dynamics simulations of machines in its monomeric and the functional hexameric forms, in the presence or absence of ligands. This approach enabled us to identify all the potential allosteric sites determined in experiments and to predict additional sites. Importantly, our approach allowed us to characterize the potential allosteric sites, the pathways for allosteric signal propagation, and the direction of structural and energetic changes of the regions that are found to best describe the allosteric transitions.

September 27, 2024 Junghyun Lee (UC Graduate Student)

Beyond homogeneity: Assessing the validity of the Michaelis–Menten rate law in spatially heterogeneous environments (Mathbio Journal Club)

The Michaelis–Menten (MM) rate law has been a fundamental tool in describing enzyme-catalyzed reactions for over a century. When substrates and enzymes are homogeneously distributed, the validity of the MM rate law can be easily assessed based on relative concentrations: the substrate is in large excess over the enzyme-substrate complex. However, the applicability of this conventional criterion remains unclear when species exhibit spatial heterogeneity, a prevailing scenario in biological systems. Here, we explore the MM rate law’s applicability under spatial heterogeneity by using partial differential equations. In this study, molecules diffuse very slowly, allowing them to locally reach quasi-steady states. We find that the conventional criterion for the validity of the MM rate law cannot be readily extended to heterogeneous environments solely through spatial averages of molecular concentrations. That is, even when the conventional criterion for the spatial averages is satisfied, the MM rate law fails to capture the enzyme catalytic rate under spatial heterogeneity. In contrast, a slightly modified form of the MM rate law, based on the total quasi-steady state approximation (tQSSA), is accurate. Specifically, the tQSSA-based modified form, but not the original MM rate law, accurately predicts the drug clearance via cytochrome P450 enzymes and the ultrasensitive phosphorylation in heterogeneous environments. Our findings shed light on how to simplify spatiotemporal models for enzyme-catalyzed reactions in the right context, ensuring accurate conclusions and avoiding misinterpretations in in silico simulations. [DOI]