Organized by: James Holehouse (The Santa Fe Institute), Kaan Öcal (University of Melbourne) and Augustinas Sukys (University of Melbourne)

1. Daniel Muratore Santa Fe Institute
   "Cellular Macromolecular Dynamics Induce Emergent Viral Biogeography in the Pacific Ocean"
Viruses are the most numerically abundant biological entity in the ocean, and the success of viral infection is determined by the capacity of their microbial hosts to provide necessary macromolecular machinery to synthesize viral progeny. Stochastic processes governing the relative balance of the nucleic acid and protein production in the infected ‘virocell’ can disrupt viral replication and lead to the production of viral particles packaged with host, as opposed to viral, genomes. This talk will discuss a stochastic process model of viral infection informed by light availability (cellular energy input) that determine cell macromolecular production. We identify regimes under which different viral infection strategies prevail and compare them with known population distributions of marine bacteriophages. Latitudinal shifts in seasonality and average day length unveil a regime shift in viral infection efficacy that corresponds to a rapid restructuring of viral fitnesses, suggesting the sub cellular environment informs global-scale biogeographic trends in microbial pathogens in the ocean.
2. Anish Pandya UT Austin
   "Transcriptional noise tunes correlations between stages of the mRNA lifecycle"
Gene expression is a key process conserved in life. A central goal is to understand complex intracellular processes through construction of gene regulatory networks from biophysical mechanisms. Many models of Eukaryotic gene expression represent biophysical processes such as (multi-)promoter binding, post-transcriptional modifications, and product degradation as transitions between states in a Markov Chain. A key step is from correlations in co-expression data to inferring molecular mechanisms. We demonstrate the converse— deducing the expected Pearson correlation and squared coefficient of variation of mRNA waiting time distributions a priori from models—can pose indistinguishability problems. In particular, if the mRNA waiting time distribution contains combinations of reversible or (effectively) irreversible transitions and or the transcriptional reaction network contains cycles. We characterize the dependence of the mRNA Pearson correlation coefficients and the coefficient of variation on causal properties of transcriptional reaction graphs. With the linear noise approximation, we exactly calculate the expected properties of the covariance, Pearson correlation coefficient, and coefficient of variation. In addition, we investigate the degeneracy of transcriptional waiting time distributions to correlative measurements of post-transcriptional mRNA with few transcriptional gene states. In these models, we show causal relationships do not necessarily entail correlative relationships. To potentially mitigate spurious correlations, we discuss methods to potentially distinguish between causal generating mechanisms based on correlations between post-transcriptional products.
3. Ethan Levien Dartmouth College
   "Gene expression following abrupt antibiotic exposure"
Single-cell mother machine experiments have revealed that genetically homogeneous bacterial populations can exhibit divergent cell fates following abrupt antibiotic exposure. The mechanisms underlying this divergence remain unclear, particularly the respective roles of intrinsic and extrinsic factors. Here, we propose a simple model of single-cell gene expression and growth dynamics following sudden drug exposure, grounded in established scaling relations between proteome allocation and growth rate. In this model, resources allocated to the transcription of resistance genes behave analogously to generalized momenta, and their initial variation predicts eventual cell fate. Without parameter fitting, the model recapitulates key experimental observations, including the emergence of distinct phenotypic outcomes and the existence of a critical threshold in TetR production velocity that determines survival. We further derive a scaling law for the critical velocity as a function of external drug concentration, yielding a testable prediction for future experiments.
4. Lucy Ham University of Melbourne
   "Cell fate control in space and time: fundamental limits on spatial organisation in multicellular systems"
Genetically identical cells develop and maintain distinct identities over time, despite fluctuations in intracellular and extracellular conditions. This talk examines the mechanisms behind cell fate determination and spatial patterning in multicellular systems. Using spatial stochastic models, we investigate how gene regulatory networks interact with cell-to-cell communication to control cell fate decisions. Our results show that feedback loops and paracrine signalling act as biological switches that trigger transitions from temporary to stable cellular states. We provide mathematical expressions that predict the signalling thresholds needed for these transitions and identify a key physical constraint: the mean size of phenotypic regions scales with the cubic root of signalling strength. This relationship reveals why maintaining large, stable domains requires disproportionately high signalling costs. This work highlights the fundamental trade-offs between pattern stability and signalling efficiency that organisms must balance during development. Our findings contribute to a deeper understanding of the principles governing tissue organisation and multicellular patterning in biological systems.

Organized by: Eric Cytrynbaum (University of British Columbia), Tim Tian (University of British Columbia)

1. Hannah Scanlon Duke University
   "Mechanisms of Microtubule Polarity Regulation in Neuronal Regeneration"
Across many organisms, neurons in the peripheral nervous system (PNS) can regenerate injured axons while neurons in the central nervous system cannot. Experimentalists have identified responses in polarized, cytoskeletal filaments called microtubules which are key to facilitating axon regeneration in injured PNS neurons. In a healthy neuron, microtubules maintain a strict polarity distribution over the lifetime of the cell. In response to axon injury in the PNS, microtubules rearrange dramatically to facilitate axonal regeneration. While several mechanisms have been hypothesized to regulate microtubule polarity organization, they are difficult to verify experimentally. Motivated by experiments in fruit flies, we use multi-scale mathematical modeling to investigate mechanisms related to microtubule polarity regulation. This work seeks to assess the efficacy of hypothesized mechanisms at producing the microtubule polarity observed in healthy neurons and in response to axon injury.
2. Taeyoon Kim Purdue University
   "Reconstituting the Mechanical and Dynamic Behaviors of the Actin Cytoskeleton"
Actin cytoskeleton is a dynamic structural scaffold used by eukaryotic cells to provide mechanical integrity and resistance to deformation, while simultaneously remodeling itself and adapting to diverse extracellular stimuli. The actin cytoskeleton utilizes these properties to play crucial roles in essential cellular processes such as cell migration and division. However, despite its known mechanical role in cell behaviors, a clear understanding of the mechanical properties of actin cytoskeleton and the molecular origin of these properties still lacks, partly due to experimental limitations. Computer simulations can access time and length scales inaccessible by experiments, and thus aid in creating a descriptive model of the molecular interactions that evolve into the mechanical properties observed on cellular scales. To this end, we have developed a cutting-edge computational model which is designed to reproduce the mechanical and dynamic behaviors of actin cytoskeleton within cells. Guided by explicit experimental data, we systematically explored, via simulation, how the mechanics and dynamics of actins and actin-binding proteins determine the deformation, flow, and stiffness of the passive actin cytoskeleton. We also investigated how interactions between the passive cytoskeletal constituents and active molecular motors lead to force generation, contraction, and morphological changes in the active actin cytoskeleton. In this talk, we will briefly introduce our foundational works and discuss our recent studies designed to illuminate the mechanisms of various cellular phenomena, including the pulsed contraction of cell cortex, actin retrograde flow in the lamellipodia, and cell blebbing.
3. Calina Copos Northeastern University
   "Modeling insights into actin cytoskeleton regulation with external size changes"
Actin is one of the most abundant proteins in eukaryotic cells and a fundamental component of the cytoskeleton, playing a critical role in maintaining cell structure and enabling motility. A compelling preliminary experimental observation underpins our work: in micropatterned epithelial cells of increasing sizes, the mechanical energy does not scale linearly with size. Instead, an optimal force is generated at a critical cell size, suggesting a force response that combines both passive and active mechanical components. To explore this phenomenon, we present a mechanical model of the actin cytoskeleton in an adherent cell that captures the observed biphasic response in force production, arising from an underlying scaling law in cytoskeletal mechanical properties. Complementing this, we develop an agent-based model that simulates the microscopic dynamics of actin filament formation, incorporating crosslinkers and myosin motors. Within this framework, we test various hypotheses — such as the impact of limited resources — that could give rise to the scaling law identified in the macroscopic model. Together, these efforts constitute a multiscale approach aimed at uncovering the mechanisms by which cell size regulates cytoskeletal force generation.
4. Tim Y.Y. Tian University of British Columbia
   "Organization of Plant Cortical Microtubules"
The self-organization of cortical microtubule arrays within plant cells is an emergent phenomenon with important consequences for the synthesis of the cell wall, cell shape, and subsequently the structure of plants. Mathematical modelling and experiments have elucidated the underlying processes involved. However, the mechanical influence of membrane curvature on these elastic filaments has largely been ignored. We previously proposed a model to describe how the anchoring process may control the deflection of individual microtubules seeking to minimize bending on a cylindrical cell. We implement this process into a model of interacting microtubules and find the cell curvature influence should be significant: the array favours orientations parallel to the direction of elongation rather than the expected transverse direction. Even without elasticity, the geometry of large cells hinders robust microtubule organization. These results suggest the necessity of additional processes to overcome these factors. Alongside this, there has been growing interest in modelling the influence of various other processes such as nucleation and membrane tension. We present ongoing efforts in piecing together our results with others from increasingly complex models, with the goal of better understanding the bigger picture of microtubule organization.

Organized by: Tin Phan (Los Alamos National Laboratory), Yun Kang (Arizona State University); Tracy Stepien (University of Florida)

Note: this minisymposia has multiple sessions. The other sessions are: Part-1, Part-2, and Part-3.

1. Bruce Pell Lawrence Technological University
   "Stability Switching Induced by Cross-Immunity in a Two-Strain Virus Competition Model with Wastewater Data Validation"
We study a two-strain virus competition model incorporating temporary immunity through a discrete delay. After reducing and analyzing the system, we identify stability switching phenomena at the strain-1-only equilibrium, including the occurrence of Hopf bifurcations. A detailed characterization of the stability dynamics at this equilibrium is provided. We further validate the model using wastewater surveillance data and apply it to investigate the viral shedding behavior of recovered individuals.
2. Tianxu Wang University of Alberta
   "Derivations of Animal Movement Models with Explicit Memory"
Highly evolved animals continuously update their knowledge of social factors, refining movement decisions based on both historical and real-time observations. Despite its significance, research on the underlying mechanisms remains limited. In this study, we explore how the use of collective memory shapes different mathematical models across various ecological dispersal scenarios. Specifically, we investigate three memory-based dispersal scenarios: gradient-based movement, where individuals respond to environmental gradients; environment matching, which promotes uniform distribution within a population; and location-based movement, where decisions rely solely on local suitability. These scenarios correspond to diffusion advection, Fickian diffusion, and Fokker-Planck diffusion models, respectively. We focus on the derivation of these memory-based movement models using three approaches: spatial and temporal discretization, patch models in continuous time, and discrete-velocity jump process. These derivations highlight how different ways of using memory lead to distinct mathematical models. Numerical simulations reveal that the three dispersal scenarios exhibit distinct behaviors under memory-induced repulsive and attractive conditions. The diffusion advection and Fokker-Planck models display wiggle patterns and aggregation phenomena, while simulations of the Fickian diffusion model consistently stabilize to uniform constant states.
3. Lifeng Han Tulane University
   "A Simplified Model of Cancer Vaccine with Two Different Tumor-Immune Functional Responses"
This talk is dedicated to celebrating Dr. Yang Kuang’s profound influence on the field of mathematical biology and his pivotal role in shaping my own journey into mathematical oncology. In this work, I explore a simplified model of cancer vaccine incorporating two commonly used functional forms for immune-mediated tumor cell killing: the law of mass action (LMA) and the dePillis-Radunskaya Law (LPR). Through analytical techniques, we uncover how each functional response yields distinct biological insights. Notably, we find that under the LPR formulation, tumor elimination depends on the initial condition—offering mathematical support for the clinical practice of using cancer vaccines as an adjuvant therapy.
4. Tin Phan Los Alamos National Laboratory
   "The development and validation of a modeling framework for HIV treatment"
Most people living with HIV-1 experience rapid viral rebound once antiretroviral therapy is interrupted; however, a small fraction remain in viral remission for extended periods. Understanding the factors that determine whether viral rebound is likely after treatment interruption can inform the development of optimal treatment regimens and therapeutic interventions aimed at achieving a functional cure for HIV-1. Building upon the theoretical framework proposed by Conway and Perelson, we iteratively formulated and examined hundreds of dynamic models of virus–immune interactions to identify those that both recapitulate viral dynamics across all studies and generate predictions consistent with clinical observations. We evaluated these models using extensive longitudinal viral-load and immunological data from multiple clinical trials. The best-performing models accurately capture the heterogeneity of viral dynamics from the acute phase through rebound. Our results robustly demonstrate that the expansion capacity of effector cells is a key determinant of viral control.

Organized by: Diana White (Clarkson University)

Note: this minisymposia has been accepted, but the abstracts have not yet been finalized.

Organized by: Terry Easlick (Univeristé de Montréal/Centre de recherche Azrieli du CHU Sainte-Justine), Morgan Craig, Univeristé de Montréal/Centre de recherche Azrieli du CHU Sainte-Justine

Note: this minisymposia has multiple sessions. The other session is: and Part-1.

1. Mélanie Prague Université de Bordeaux/INRIA
   "Mechanistic Model of initial and persisting antibody response following Ebola vaccination: application to the PREVAC trial."
Antibody response dynamics following Ebola vaccination remain incompletely understood, particularly regarding the continuum of initial induction to long-term persistence. This study developed a mechanistic model of B-cell stimulation post-vaccination able to infer antigen presentation kinetics and propose an identifiable model based solely on anti-ZEBOV IgG levels. This study was based on the PREVAC randomized placebo-controlled trial (NCT02876328), which enrolled healthy adults and children, from Sierra Leone, Mali, guinea and Liberia, to evaluate the safety and immune responses of three vaccine strategies: Ad26.ZEBOV followed by MVA-BN-Filo 56 days later (the Ad26-MVA group, 799 participants), rVSV∆G-ZEBOV-GP followed by placebo 56 days later (the rVSV group, 802 participants), and rVSV∆G-ZEBOV-GP followed by rVSV∆G-ZEBOV- GP 56 days later (the rVSV–booster group, 399 participants). Our model was modified from Clairon et al. (2022, PLOS Comp. Biol.), which assumes that antigen stimulates the differentiation of naive B cells into long-lived and short-lived antibody-secreting cells. The two groups were modeled following similar pipelines. For both vaccine groups, a bell-shaped curve best described the dynamics of antigen presentation according to model information criterion indicating a long antigen presentation regardless of replication competence of the vaccine. Longer presentation times were found for rVSV (half-life t1/2=45 days at first dose and t1/2=1 days at second dose) than for Ad26–MVA (t1/2=35 days for Ad26.ZEBOV and t1/2=7 days for MVA-BN-Filo). We as well quantified effect of sex, age and geography on humoral dynamics. This work provides a foundation for in silico simulations of vaccination clinical trials, with the objective of optimizing booster strategies to ensure the long-term maintenance of immunogenicity in target populations according to patients characteristics, revaccination timing and vaccination strategy.
2. Elizabeth Amona Virginia Commonwealth University
   "Studying Disease Reinfection Rates, Vaccine Efficacy and the Timing of Vaccine Rollout in the context of Infectious Diseases"
The COVID-19 pandemic has highlighted the intricate nature of disease dynamics, extending beyond transmission patterns to the complex interplay of intervention strategies. In the post-COVID-19 era, reinfection has emerged as a critical factor, shaping how we model disease progression, evaluate immunity, and assess the effectiveness of public health interventions. This research uniquely explores the varied efficacy of existing vaccines and the pivotal role of vaccination timing in the context of COVID-19. Departing from conventional modeling, we introduce two models that account for the impact of vaccines on infections, reinfections, and deaths. We estimate model parameters under the Bayesian framework, specifically utilizing the Metropolis-Hastings Sampler. The study conducts data-driven scenario analyses for the State of Qatar, quantifying the potential duration during which the healthcare system could have been overwhelmed by an influx of new COVID-19 cases surpassing available hospital beds. Additionally, the research explores similarities in predictive probability distributions of cumulative infections, reinfections, and deaths, employing the Hellinger distance metric. Comparative analysis, utilizing the Bayes factor, underscores the plausibility of a model assuming a different susceptibility rate to reinfection, as opposed to assuming the same susceptibility rate for both infections and reinfections. Results highlight the adverse outcomes associated with delayed vaccination, emphasizing the efficacy of early vaccination in reducing infections, reinfections, and deaths. Our research advocates prioritizing early vaccination as a key strategy in effectively combating future pandemics, thereby providing vital insights for evidence-based public health interventions.
3. Cailan Jeynes-Smith University of Tennessee Health Science Centre
   "Dissecting Cytokine Production: Integrating Subset-Specific Data into Immunological Models"
Understanding cytokine regulation and its impact on the cellular response to infection is challenging. Immunological models often rely on implicit assumptions about cytokine production, as direct quantification of cytokine-producing cell subsets is uncommon. However, only a fraction of cells may be actively producing cytokines, and their dynamics frequently diverge from those of the broader population. To address this, we developed a mechanistic model of IFN-γ production during influenza A virus infection, integrating cell abundance data with integrated median fluorescence intensity (iMFI) measurements. This framework allowed us to quantify the relative contributions and nonlinear regulation in addition to demonstrating the necessity of using the iMFI to define the balance between production and uptake to explain observed IFN-γ levels. Our findings highlight the importance of incorporating both cell subset data and functional intensity (iMFI) into cytokine modeling, enabling more accurate inference of production mechanisms and improved model predictions.
4. Jonah Hall University of British Columbia/BC Children's Hospital Research Institute
   "Optimization of Pertussis Immunization Using Mathematical Modeling"
Pertussis disease (whooping cough), caused by the bacteria Bordetella pertussis, is most severe in young infants, with the majority of deaths occurring among unvaccinated children aged <3 months. Pertussis vaccination is a safe and effective approach for prevention of pertussis, with the DTaP (priming) and TdaP (booster) vaccine series. TdaP is given during pregnancy and DTaP is given in the first year of childhood. The phenomenon of immunomodulation, however, is known to dampen the IgG response of infants born following pertussis vaccination during pregnancy. We hypothesize that the timing of the vaccination series, while not the immunological cause, could be modified so as to decrease the effect of immunomodulation and thus increase the efficacy of childhood pertussis vaccination. While attempting to empirically test many different schedules would be ineffective, using mathematical modeling to evaluate several schedules simultaneously could be extremely useful in determining a more effective schedule. We will use a combination of a mathematical model of pertussis vaccination and an experimental mouse model of pertussis vaccination to identify the optimal immunization schedule for pregnancy and infancy. We will immunize pregnant and infant mice, according to a murine analog of the conventional vaccination schedule, with TdaP (pregnant) and DTaP (infant) pertussis vaccine doses. These data will form an input to our working mathematical model. Using the data, we will estimate experimentally inaccessible parameters that govern the mechanisms of the immune system. Once parametrized, we will use the mathematical model to propose immunization schedules that maximize the infant antibody response. We will test our proposed schedules via a second mouse experiment, comparing the immune responses between the two experiments to evaluate the efficacy of the mathematically-optimized schedule. Possible mechanisms of immunomodulation can be evaluated in the mathematical model, using the data collected in both experiments to ensure model accuracy.

Organized by: Folashade B. Agusto (University of Kansas), Chidozie Williams Chukwu

Note: this minisymposia has multiple sessions. The other session is: and Part-1.

1. Chidozie Williams Chukwu DePaul University, USA
   "Dynamic Multi-country Modeling for Forecasting and Controlling Tube"
In this talk, we present a multi-country analysis of Tuberculosis (TB) epidemic model. We develop a deterministic TB model incorporating optimal control strategies and analyze its dynamics using mathematical tools. The model was calibrated using the new TB incidence data from India, Lesotho, Angola, and Indonesia. Numerical simulations are conducted to assess the impact of effective mask usage and case detection as intervention strategies. Our results project future trends of TB in the four countries studied. These insights are crucial for mitigating the spread of TB and addressing future challenges associated with potential TB outbreaks, particularly in the context of global public health crises.
2. Hewan Shemtaga, Selim Sukhtaiev, and Dr. Wenxian Shen Auburn University, USA
   "Logistic Keller-Segel chemotaxis models on compact graphs"
Chemotaxis phenomena governs the directed movement of micro-organisms in response to chemical stimuli. We investigate a pair of logistic type Keller–Segel systems of reaction-advection-diffusion equations modeling chemotaxis on networks. The distinction between the two systems is driven by the rate of diffusion of chemo-attractant. We prove the global existence of classical solution subject to Neumann-Kirchhoff vertex conditions without any conditions on chemotaxis sensitivity. In addition, we show that solutions with a non-negative and non-zero initial function converge to a globally stable constant solution for relatively small chemotaxis sensitivity. However, as chemotaxis sensitivity increase, we prove the constant solution loses stability and there exist other non-constant steady states bifurcating from the constant solution.
3. Ousmane Seydi University Le Havre, France
   "Growth Bounds and Threshold Dynamics in Periodic Structured Population Models"
Understanding when a population will grow, decline, or persist over time is a central question in mathematical biology. In this talk, we present a general method for identifying the conditions under which population growth occurs, even in models that incorporate age-structure, nonlocal interactions, or delays. Our approach applies to a broad class of mathematical systems, and we provide tools to compute critical threshold values—such as reproduction numbers—and to explain how these values determine the long-term behavior of the population. This framework draws on ideas from operator theory and dynamical systems to gain insight into biological processes that are periodic in time.
4. Daniel Cooney University of Illinois Urbana-Champaign, USA
   "Modeling Cross-Scale Evolutionary Dynamics"
Natural selection often operates simultaneously across multiple levels of biological organization, with evolutionary tensions often arising between traits or behaviors that are favored at different levels of selection. One common example of this tension arises in the evolution of altruism in group-structured populations, in which actions that are costly and result in an individual-level disadvantage while providing a collective benefit to the individual’s group. In this talk, we will explore a variety of PDE models that use evolutionary game theory to describe the evolution of altruism under competition occur both within and among groups, and we will discuss how different formulations of individual-level and group-level birth and death events can impact the long-time support for cooperation under these PDE models.

Organized by: Abba Gumel (University of Maryland), Alex Safsten, Arnaja Mitra (both University of Maryland)

Note: this minisymposia has multiple sessions. The other sessions are: Part-1, and Part-2.

1. Casey O'Brien North Carolina State University
   "Modeling a Novel Gene Drive That Targets Immune Responses"
Gene drive technologies hold promise for controlling invasive pests, mitigating disease transmission, and protecting local ecosystems and agriculture. However, their deployment hinges on resolving safety concerns, particularly the risk of unintended spread into non-target populations. Current confinement strategies rely largely on invasion thresholds which take advantage of unstable equilibrium points in allele frequency, below which the drive will not spread. This maintains local confinement by preventing migrants from spreading the drive in surrounding populations. While this is an effective strategy for gene drives meant to introduce a trait to a population, its success has been more limited in suppression gene drives. We circumvent this issue by designing a novel suppression drive system that targets the immune response of an organism to a local stressor (i.e., endemic virus, fungus, or a specialized parasitoid). The drive system increases the target organism’s susceptibility to the stressor by increasing the likelihood of acquiring the infection or the impact of infection on the organism. This means that the drive system’s fitness cost is dependent on the abundance of the stressor. We model several drive systems to consider the efficacy of the system in different settings
2. Jackson Champer Peking University
   "Suppression gene drive for mosquito control: large scale spatial models and impact on disease transmission"
Gene drive alleles bias inheritance in the favor, allowing them to quickly spread throughout a population. They could combat disease by rapidly spreading a cargo gene that blocks pathogen transmission, or they could directly suppress vector populations. Progress has been made to reduce resistance allele formation, a main obstacle to successful gene drive in Anopheles stephensi mosquitoes, yielding efficient systems. However, computational analysis using individual-based models predicts that suppression drives may still not succeed in spatially structured natural populations due to the 'chasing' phenomenon that causes chaotic, long-term persistence of both drive and wild-type alleles. To assess this effect on malaria transmission, we developed a deep-learning model to allow assessment of many drive, ecology, and disease parameters without a large computational burden. We found that malaria could potentially be eliminated even if the mosquito population persists. Thus, despite unexpected complexity, gene drive remains a potentially powerful method to reduce malaria infections.

Organized by: Amy Hurford (Memorial University), Michael Li, Public Health Agency of Canada

Note: this minisymposia has multiple sessions. The other sessions are: Part-1, and Part-2.

1. Wade McDonald University of Saskatchewan
   "Use of Synthetic Data to Improve Wastewater-based Epidemiological Models in a Small Jurisdiction"
Previous studies have shown that applying methods such as Particle Filtering and Particle Markov-Chain Monte Carlo (pMCMC) to stochastic mechanistic epidemiological models can enhance model accuracy compared to simple parameter calibration. Addition of data streams to the filter can improve model fit even if those data are deemed to be of “low quality,” e.g., internet search volumes. In the present work, we employ a synthetic dataset, generated by an agent-based model, to explore the use of pMCMC with a compartmental epidemiological model, including wastewater-based epidemiology (WBE), in the context of a small jurisdiction facing an emerging pathogen. Predictive performance of the filtered model will be compared against synthetic ground truth using clinical cases alone versus clinical cases and WBE measures. Effects of structural mismatches between the synthetic ground truth and filtered model will be considered; for example, what if the synthetic ground truth admits waning of immunity (SIRS) but our filtered model assumes immunity is permanent (SIR)?
2. Matthew Betti Mount Allison University
   "Modeling healthcare demand during a disease outbreak"
One of the driving concerns during any epidemic is the strain on the healthcare system. During severe outbreaks, healthcare systems can become quickly overwhelmed. We develop a healthcare demand module that can take epidemiological data and healthcare parameters and can forecast number of doctor visits, hospital occupancy. Using real-world data we can estimate the length of stay of hospitalized individuals. The module can be extended to account for pharmaceutical and PPE usage at differing levels of conservation.
3. Sicheng Zhao McMaster University
   "Edge-based Modeling for Disease Transmission on Random Graphs – an Application to Mitigate a Syphilis Outbreak"
Edge-based random network models, especially those based on bond percolation methods, can be used to model disease transmission on complex networks and accommodate social heterogeneity while keeping tractability. Here we present an application of an edge-based network model to the spread of syphilis in the Kingston, Frontenac and Lennox & Addington (KFL&A) region of Southeastern Ontario, Canada. We compared the results of using a network-based susceptible-infectious-recovered (SIR) model to those generated from using a traditional mass action SIR model. We found that the network model yields very different predictions, including a much lower estimate of the final epidemic size. We also used the network model to estimate the potential impact of introducing a rapid syphilis point of care test (POCT) and treatment intervention strategy that has recently been implemented by the public health unit to mitigate syphilis transmission.
4. Caroline Mburu British Columbia Centre for Disease Control/Simon Fraser University
   "Wastewater-based modelling for Mpox surveillance among gbMSM in BC"
Background: The 2022 global outbreak of Mpox, caused by Clade IIb of the monkeypox virus (MPXV), primarily affected gay, bisexual, and other men who have sex with men (gbMSM). While clinical case surveillance has been central to the public health response, it faces limitations due to underreporting, social stigma, and asymptomatic infections. To complement case-based surveillance, wastewater-based surveillance (WBS), which had been valuable in monitoring other infections, including during the COVID-19 pandemic, was adopted to track MPXV circulation. Several studies have demonstrated correlations between MPXV viral loads in wastewater and reported Mpox cases, supporting the utility of WBS for population-level monitoring. In parallel, mechanistic models based solely on clinical case data have provided insights into Mpox transmission dynamics and the impact of interventions such as vaccination and behavioral change. However, to date, no modeling framework has integrated both data streams to jointly infer Mpox transmission dynamics. As a result, the mechanistic relationship between viral load in wastewater and underlying disease transmission remains poorly understood, particularly in the context of evolving behavioral patterns and vaccination uptake Methods: We developed a compartmental model to simulate Mpox transmission within the gbMSM population, incorporating heterogeneity through stratification by levels of sexual activity. The model integrates key data streams, including clinical case notifications, MPXV viral load signals from WBS, sexual network data and vaccination coverage. The framework explicitly incorporates viral shedding dynamics into wastewater, allowing for the exploration of the relationship between underlying infections and observed WBS signals. We use this model to evaluate the conditions under which wastewater viral load may act as leading or lagging indicators of reported cases, considering factors such as reporting delays, underreporting, asymptomatic infections, changes in sexual behavior, and the rollout of vaccination programs. Conclusions: This study bridges clinical and environmental surveillance through a mechanistic framework tailored to behaviorally structured populations. By jointly modeling case and WBS data, we aim to improve the interpretation of wastewater signals and support more accurate assessments of transmission in hard-to-reach or underreported populations. Findings will inform public health decision-making around Mpox surveillance and preparedness, particularly in contexts where traditional case-based reporting is limited.

Organized by: Jennifer Flegg (University of Melbourne), Prof Mat Simpson, Queensland University of Technology

Note: this minisymposia has multiple sessions. The other session is: and Part-1.

1. Zan Luthey-Schulten University of Illinois at Urbana-Champaign
   "Bringing a cell to life on a computer and in Minecraft"
I will describe our research into constructing 4D (x,y,z + time) models of a living minimal cell. The 4D simulations integrate data from -omics, cryo-electron tomograms, DNA contact maps, fluorescent imaging, and kinetic experiments to initialize a realistic cell state as well as validate the states as they progress in time. Fundamental behaviors emerge from these simulations that reveal how the cell balances the demands of its metabolism, genetic information processes, and growth, oQering insight into the principles of life. Validation by coarse-grained atomistic MD simulations and experiments are critical steps in building func2oning models for bacterial and eukaryo2c cells. As part of the education and knowledge transfer goals of the NSF STC for Quantitative Cell Biology, we are bringing these simulations to Minecraft, enabling players to explore a whole living cell in an immersive 3D environment.
2. Hilary Hunt Queensland University of Technology
   "Stress, stability, and systems biology: Modelling yeast’s mRNA panic rooms"
Messenger RNA (mRNA) is the biochemical link between genetic information and protein synthesis. Experimentally measuring the amount of mRNA present in the cell for each gene (transcriptomics) has become relatively cheap and reliable, especially compared to measurements of downstream processes like protein abundance or enzyme activity. However, the mapping from the amount of mRNA present in a cell to the amount of protein produced is inconsistent between mRNA species. Between mRNA transcription from DNA and its subsequent translation into protein, there are multiple regulatory processes that affect each molecule’s lifespan and rate of translation. We are particularly interested in how stress granules affect mRNA survival. Under specific environmental conditions, mRNA can be sequestered into phase-separated compartments known as stress granules and physically removed from other regulatory mechanisms. Using a minimal model of mRNA dynamics and post-transcriptional modifications, we explore the effect these granules have on mRNA distributions in yeast, factors that impact which molecules are protected when the cell is under pressure, and how this might improve our transcriptome to proteome mappings.
3. Abigail Kushnir University of Edinburgh
   "Effective Mesoscopic Rate Equations for Spatial Stochastic Systems"
Chemical master equations (CMEs) describe stochastic reaction kinetics at the mesoscopic level. Generally, their predictions for the mean molecule numbers do not agree with the predictions of the (macroscopic) deterministic rate equations. Effective mesoscopic rate equations (EMREs), derived from van Kampen's system size expansion of CMEs, correct the deterministic rate equations. Here I discuss work to extend EMREs to the spatial domain – resulting in reaction-diffusion – and discuss their implementation in the Julia programming language. I demonstrate that these spatial EMREs offer a rapid way to identify regions of parameter space where there are significant disagreements between deterministic and stochastic formulations of reaction-diffusion systems.
4. Mica Yang Stanford University
   "Whole-cell modeling of E. coli colonies enables quantification of single-cell heterogeneity in antibiotic responses"
Antibiotic response in bacterial colonies is often characterized by phenotypic heterogeneity. This heterogeneity may in turn be driven by stochastic expression of antibiotic resistance genes, linking variation in molecular-scale gene expression to population-scale phenotypes. To better understand heterogeneous antibiotic responses, we bridged the molecular and colony-level scales by embedding instances of an E. coli whole-cell model in a dynamic spatial environment model. The resulting simulations enabled us to study variations in colony-level response to two beta-lactam antibiotics with differing mechanisms of action, tetracycline and ampicillin.

Organized by: Wenjun Zhao (UBC/Wake Forest University), Khanh Dao Duc (UBC)

Note: this minisymposia has multiple sessions. The other session is: and Part-1.

1. Laurent Younes JHU
   "Aligning measures using large deformation diffeomorphic mapping for spatial transcriptomics"
Signed measures provide a powerful and flexible object representation for registration problems as they cover both continuous and singular objects and are naturally transformed by diffeomorphisms. They can, in particular, be used as tools describing functions taking values on arbitrary feature space, which makes them well adapted to the representation of spatial transcriptomic images. In this presentation, we will summarize the theoretical foundations of measure registration using large deformation diffeomorphic measure mapping, and provide applications to spatial transcriptomics, within and across modalities.
2. Luis F Pereira UCSB
   "Statistical shape analysis with Geomstats"
Geomstats is an open-source Python package for computations and statistics on Riemannian manifolds. It provides object-oriented and extensively unit-tested implementations. Manifolds can be equipped with Riemannian metrics with associated exponential and logarithmic maps, geodesics, and parallel transport. Building on this general framework, the shape module implements widely used shape spaces, such as the Kendall shape space and elastic spaces of discrete curves and surfaces, by leveraging the abstract mathematical structures of group actions, fiber bundles, and quotient spaces. The Riemannian geometry tools enable users to compare, average, and interpolate between shapes belonging to a given shape space. These essential operations can then be used to perform statistics on shape data. In this talk, we will present the object-oriented implementation of the shape module along with illustrative examples and demonstrate its use in performing statistics on shape spaces.
3. Qiyu Wang UBC
   "Studying SARS-CoV2 spike protein heterogeneity from large Cryo-EM dataset with linear subspace method and path analysis"
Recent advances in single particle cryogenic electron microscopy (cryo-EM) have allowed to capture biomolecules in various conformations through large image datasets. However, interpreting and quantifying such conformational heterogeneity remain computationally challenging, leading to a variety of recent methods. In the context of SARS-CoV-2, we developed and implemented a pipeline to process large datasets (~ millions) of 2D images of spike proteins, and apply REgularized COVARiance estimator (RECOVAR), to project the images into a latent linear subspace. Our pipeline also includes new methods for trajectory inference and transport-based segmentation that facilitate data analysis, revealing specific transitions between multiple conformations of the receptor binding domains (RBDs) in SARS-Cov2 spike protein. Our study notably led us to discover a state with three RBDs up, as well as finding a cooperativity mechanism from states with one RBD up, that goes towards the closed state before transiting to the state with two RBD’s up, offering valuable insights into the conformational landscape of SARS-CoV-2.
4. Willem Diepeveen UCLA
   "Curvature corrected tangent space-based approximation of manifold-valued data and applications in protein dynamics analysis"
When generalizing schemes for real-valued data approximation or decomposition to data living in Riemannian manifolds (widely used for modelling biological shapes), tangent space-based schemes are very attractive for the simple reason that these spaces are linear. An open challenge is to do this in such a way that the generalized scheme is applicable to general Riemannian manifolds, is global-geometry aware and is computationally feasible. Existing schemes have been unable to account for all three of these key factors at the same time. In this work, we take a systematic approach to developing a framework that is able to account for all three factors. In addition, we consider applications of our theory to analysis of protein dynamics data.

Organized by: Sebastian Schreiber (University of California, Davis)

Note: this minisymposia has multiple sessions. The other session is: and Part-1.

1. Junping Shi College of William and Mary
   "Effect of rotational grazing on plant and animal production"
It is a common understanding that rotational cattle grazing provides a better yield than continuous grazing, but a qualitative analysis is lacking in the agriculture literature. In rotational grazing, cattle periodically move from one paddock to another in contrast to continuous grazing, in which the cattle graze on a single plot for the entire grazing season. Here we quantitatively show how production yields and stockpiled forage are greater in rotational grazing in some harvesting models. We construct a vegetation grazing model on a fixed area, and by using parameters obtained from agricultural publications and keeping the minimum value of remaining forage constant, our result shows that both the number of cattle per acre and stockpiled forage increase for all tested rotational configurations than the continuous grazing. Some related spatial harvesting models are also discussed. This is a joint work with Mayee Chen.
2. Kate Meyers Carleton College
   "From deluges to drizzle: continuous limits of flow-kick models"
To incorporate ongoing disturbances into a differential equation (DE) model of biological processes, one might embed the disturbance continuously in the DE or resolve the disturbance discretely. In this talk we’ll explore the flow-kick approach to modeling repeated, discrete disturbances and examine the dynamic implications of this modeling choice. We’ll position continuous disturbances as limits of repeated, discrete ones and share recent results on how flow-kick systems both mimic and depart from their continuous analogs.
3. Rebecca Tyson University of British Columbia, Okanagan
   "Host-parasitoid systems are vulnerable to extinction via P-tipping: Forest Tent Caterpillar as an example"
Continuous-time predator-prey models admit limit cycle solutions that are vulnerable to the phenomenon of phase-sensitive tipping (P-tipping): The predator-prey system can tip to extinction following a rapid change in a key model parameter, even if the limit cycle remains a stable attractor. In this paper, we investigate the existence of P-tipping in an analogous discrete-time system: a host-parasitoid system, using the economically damaging forest tent caterpillar as our motivating example. We take the intrinsic growth rate of the consumer as our key parameter, allowing it to vary with environmental conditions in ways consistent with the predictions of global warming. We find that the discrete-time system does admit P-tipping, and that the discrete-time P-tipping phenomenon shares characteristics with the continuous-time one: Both require an Allee effect on the resource population, occur in small subsets of the phase plane, and exhibit stochastic resonance as a function of the autocorrelation in the environmental variability. In contrast, the discrete-time P-tipping phenomenon occurs when the environmental conditions switch from low to high productivity, can occur even if the magnitude of the switch is relatively small, and can occur from multiple disjoint regions in the phase plane. This is joint work with Bryce F. Dyck.
4. Sebastian Schreiber University of California, Davis
   "Coexistence and extinction in flow kick systems via Lyapunov exponents"
Natural populations experience a complex interplay of continuous and discrete processes: continuous growth and interactions are punctuated by discrete reproduction events, dispersal, and external disturbances. These dynamics can be modeled by impulsive or flow-kick systems, where continuous flows alternate with instantaneous discrete changes. To study species persistence in these systems, an invasion growth rate theory is developed for flow-kick models with state-dependent timing of kicks and auxiliary variables that can represent stage structure, trait evolution, and environmental forcing. The invasion growth rates correspond to Lyapunov exponents that characterize the average per-capita growth of species when rare. Two theorems are proven that use invasion growth rates to characterize permanence, a form of robust coexistence where populations remain bounded away from extinction. The first theorem uses Morse decompositions of the extinction set and requires that there exists a species with a positive invasion growth rate for every invariant measure supported on a component of the Morse decomposition. The second theorem uses invasion growth rates to define invasion graphs whose vertices correspond to communities and directed edges to potential invasions. Provided the invasion graph is acyclic, permanence and extinction are fully characterized by the signs of the invasion growth rates. Invasion growth rates are also used to identify the existence of extinction-bound trajectories and attractors that lie on the extinction set. To demonstrate the framework's utility, these results are applied to a microbial serial transfer model where state-dependent timing enables coexistence through a storage effect, a spatially structured consumer-resource model showing intermediate reproductive delays can maximize persistence, and an empirically parameterized Lotka-Volterra model demonstrating how disturbance can lead to extinction by disrupting facilitation. Mathematical challenges, particularly for systems with cyclic invasion graphs, and promising biological applications are discussed. These results reveal how the interplay between continuous and discrete dynamics creates ecological outcomes not found in purely continuous or discrete systems, providing a foundation for predicting population persistence and species coexistence in natural communities subject to gradual and sudden changes

Organized by: Dhananjay Bhaskar (Yale University), Bernadette Stolz-Pretzer

Note: this minisymposia has multiple sessions. The other session is: and Part-1.

1. Eunbi Park Georgia Institute of Technology
   "Topological data analysis of pattern formation of human induced pluripotent stem cell colonies"
Understanding the multicellular organization of stem cells is vital for determining the mechanisms that coordinate cell fate decision-making during differentiation; these mechanisms range from neighbor-to-neighbor communication to tissue-level biochemical gradients. Current methods for quantifying multicellular patterning tend to capture the spatial properties of cell colonies at a fixed scale and typically rely on human annotation. We present a computational pipeline that utilizes topological data analysis to generate quantitative, multiscale descriptors which capture the shape of data extracted from 2D multichannel microscopy images. By applying our pipeline to certain stem cell colonies, we detected subtle differences in patterning that reflect distinct spatial organization associated with loss of pluripotency. These results yield insight into putative directed cellular organization and morphogen-mediated, neighbor-to-neighbor signaling. Because of its broad applicability to immunofluorescence microscopy images, our pipeline is well-positioned to serve as a general-purpose tool for the quantitative study of multicellular pattern formation.

Organized by: Jana Gevertz (The College of New Jersey), Thomas Hillen (University of Alberta), Linh Huynh (Dartmouth College)

Note: this minisymposia has multiple sessions. The other session is: and Part-1.

1. John Metzcar University of Minnesota
   "Evaluation of mechanistic and machine learning modeling approaches for glioblastoma recurrence prediction using white blood cell dynamics"
Glioblastoma (GBM) is the most aggressive primary brain tumor, with median recurrence times of approximately 9–11 months following surgery, despite intensive standard-of-care interventions. Early detection of recurrence is crucial for timely enrollment in clinical trials, potentially improving patient outcomes. The significant impact of GBM and its associated therapies on the immune system suggests clinically obtained white blood cell (WBC) counts with differential as possible biomarkers for recurrence prediction. We explore how mechanistic ODE modeling, capturing tumor-immune interactions and treatment impacts, compares with data-driven techniques (GPR and CPH) in predicting GBM recurrence. We apply methods individually and in hybrid combinations to patient-specific WBC trajectories spanning the perioperative period through recurrence. This comparative analysis evaluates predictive accuracy, interpretability, and clinical relevance across methodologies. Our aim is to share preliminary insights from applying multiple modeling strategies to a common clinical problem. By evaluating how each technique performs in the context of GBM recurrence, we hope to better understand their respective advantages and limitations. This work serves as a step toward assessing whether integrating mechanistic with data-driven models enables improved recurrence prediction through a clinically determined, dynamic biomarker.
2. Lena Podina University of Waterloo
   "Universal Physics-Informed Neural Networks and Their Applications"
Differential equations are widely used to model systems such as predator-prey interactions, and the effect of chemotherapy on cancer cells. However, in order to construct these models, assumptions must be made about the behaviour of these systems, which may require significant manual distillation of the literature if the model is large. In this talk, I will discuss Universal Physics-Informed Neural Networks (UPINNs), and show how UPINNs can be used to learn unknown terms in ordinary and partial differential equations from sparse and noisy data. This approach allows one to use machine learning to identify the best way to model a system, rather than relying on prior assumptions. Physics Informed Neural Networks (PINNs) have been very successful in a sparse data regime (reconstructing entire ODE solutions using only a single point or entire PDE solutions with very few measurements of the initial condition). The Universal PINN approach (UPINN) adds a neural network that learns a representation of unknown hidden terms in the differential equation. These hidden term neural networks can then be converted into symbolic equations using symbolic regression techniques like AI Feynman. In our work, we demonstrate strong performance of UPINNs even when provided with very few measurements of noisy data in both the ODE and PDE regime. We apply UPINNs to learning predator-prey interaction in the Lotka-Volterra model, chemotherapy drug action terms in a model of cancer cell growth, and terms in Burgers’ PDE. UPINNs could be instrumental to paving the way to allow machine learning to help applied mathematicians model systems in a more automatic, data-driven way even when observations are sparse.
3. Kit Gallagher University of Oxford, Moffitt Cancer Center
   "Predicting Treatment Outcomes from Adaptive Therapy — A New Mathematical Biomarker"
Adaptive Therapy dynamically adjusts drug treatment to control, rather than minimize, the tumor burden of metastatic cancer, thus suppressing the growth of treatment-resistant cell populations and delaying patient relapse. Promising clinical results in prostate cancer indicate the potential of adaptive treatment protocols, but demonstrate broad heterogeneity in patient response. This naturally leads to the question: why does this heterogeneity occur, and is a ‘one-size-fits-all' protocol best for patients across this spectrum of responses? Using deep reinforcement learning, we obtain personalized and clinically-feasible treatment protocols based on individual patient dynamics, and present a framework to generate these treatment schedules based on the patient's response to the first treatment cycle. From a Lotka–Volterra tumor model, we also obtain a predictive expression for the expected benefit from Adaptive Therapy and propose new mathematical biomarkers that can identify the best responders from a clinical dataset after only the first treatment cycle. Overall, the proposed strategies offer personalized treatment schedules that consistently outperform clinical standard-of-care protocols.

Organized by: Mohit Kumar Jolly (Indian Institute of Science), Sarthak Sahoo (Indian Institute of Science)

Note: this minisymposia has multiple sessions. The other session is: and Part-1.

1. David P Cook Ottawa Hospital Research Institute
   "Phenotypic constraints in ovarian cancer - a new perspective on targeted therapy"
High-grade serous ovarian cancer (HGSC) remains the most lethal gynecological malignancy, with a five-year survival rate below 50%. Despite the adoption of PARP inhibitors for patients with BRCA1/2 mutations (20% of cases), clinical management has remained unchanged for decades. The complex genetic landscape of HGSC has not revealed opportunities for effective targeted therapies as seen in other cancer types. To address this challenge, we conducted a meta-analysis of single-cell RNA sequencing data from 471 tumor samples, coupled with spatial transcriptomics using the 10x Genomics Xenium platform. Our analysis revealed three recurrent malignant epithelial phenotypes ('epitypes') that mirror fallopian tube lineages: SecA, SecB, and Cil. These epitypes emerge through non-genetic plasticity and serve distinct functions in disease progression—SecA cells showing high proliferation while SecB cells predominate in metastatic sites and post-chemotherapy samples. Each exhibits unique regulatory patterns and microenvironment interactions. Our findings suggest that developmental regulatory networks constrain malignant phenotypes, creating opportunities for phenotype-targeted therapeutics independent of genetic alterations. Targeting cellular plasticity could restrict tumours' adaptive capabilities, potentially enhancing treatment response and immune recognition in this challenging cancer.
2. Jill Gallaher Moffitt Cancer Center
   "Dynamic evolvability during tumor growth and treatment"
Drug resistance is an ongoing problem for maintaining a treatment response in advanced cancers, which are often more heterogeneous and evolvable. There are benefits for cells to be evolvable, e.g. to easily respond to large shifts in the microenvironment with large heritable shifts in traits, like allowing metastases to survive a new environment and thrive even during treatment. However, evolvability may also be a detriment. With too much deviation from the parental phenotype, cells lose important functions necessary to survive. So, is there an optimal rate of evolvability for tumors to grow and survive treatment that can be exploited therapeutically? We use an off-lattice agent-based model to investigate how the rate of change through proliferation-resistance phenotype space affects tumor growth and response to treatment. During growth, proliferation is selected for, but more evolvability leads to more heterogeneity and faster recurrence under treatment. When evolvability can evolve without constraints, faster evolvability changes will lead to faster recurrence. When evolvability is costly, tumor survival depends on the rate and jump size of heritable changes to transiently lose proliferation fitness selected for during growth and gain resistance for survival. We consider how to design treatment strategies based on a tumor’s evolvability dynamics.
3. Cordelia McGehee Mayo Clinic
   " Chemotherapy dosing as a driver of population evolution in models of intra-tumoral cell-cell competition in cancer"
Despite ongoing therapeutic advances in the treatment of cancer, many advanced solid tumors recur after initial therapy. Minimizing the emergence of drug resistance is a central problem in cancer pharmacology. Dose and dose schedule of chemotherapy administration has traditionally followed the maximum tolerated dose principle which aims to quickly eradicate the tumor while minimizing drug toxicity for the patient. In a clonal drug-sensitive cell population, using the highest dose of drug and achieving maximum tumor killing is a logical strategy. However, when a pre-existing drug-resistant cell population resides within a cancer cell population, the rapid elimination of drug sensitive cells has been hypothesized to lead to proliferation of the resistant cell population. In such cases, an alternative dosing paradigm coined adaptive therapy has been proposed to maintain the sensitive cell population in a tumor and thus prevent unchecked proliferation of the drug-resistant cells. In this talk, we use a model of cellular competition to mathematically explore two distinct paradigms of adaptive therapy dosing: continuous dose modulation versus intermittent high dose therapy. We compare these regimens to standard dosing schemes to explore how dose and dose schedule can drive cellular population evolution.
4. Russell C Rockne Beckman Research Institute, City of Hope
   "State-transitions at the single cell and system levels in chronic and acute myeloid leukemia"
In this presentation, I will discuss experimental data and mathematical models used to study state transitions in chronic and acute myeloid leukemia (CML and AML). Our experimental approach involves inducible and constitutively activated mouse models of CML and AML, which are monitored longitudinally through blood sampling and RNA sequencing. The mathematical models employed are stochastic differential equations and their corresponding probability density functions. By integrating experimental data with these mathematical models, and iteratively validating the models while generating new hypotheses, we have demonstrated that state transitions can be detected at very early stages of disease initiation. Furthermore, these transitions can be used to predict responses to chemotherapy and tyrosine kinase inhibitor (TKI) therapies. We explore how state-transitions can be used to characterize and quantify resistance to therapy through analysis of gene programs within cell types over time.

Organized by: Kishore Hari and Pradyumna Harlapur (Postdoctoral Research Fellow, Center for Theoretical Biological Physics), Pradyumna Harlapur, PhD Candidate, Dept. of BioEngineering, Indian Institute of Science

Note: this minisymposia has been accepted, but the abstracts have not yet been finalized.