Mathematical Models of Blood-Brain Barrier Transport of Monoclonal Antibodies Targeting the Transferrin Receptor and the Insulin Receptor

We develop and analyze mathematical models for receptor-mediated transcytosis of monoclonal antibodies (MAb) targeting the transferrin receptor (TfR) or the insulin receptor (IR), which are expressed at the blood-brain barrier (BBB). The mass-action kinetic model for both the TfR and IR antibodies were solved numerically to generate predictions for the concentrations of all species in all compartments considered. Using these models, we estimated the rates of MAb endocytosis into brain capillary endothelium, which forms the BBB in vivo, the rates of MAb exocytosis from the intra-endothelial compartment into brain extracellular space, and the rates of receptor recycling from the endothelial space back to the luminal endothelial plasma membrane. Our analysis highlights the optimal rates of MAb association with the targeted receptor. An important role of the endogenous ligand, transferrin (Tf) or insulin, in receptor-mediated-transport (RMT) of the associated MAb was found and was attributed to the five order magnitude difference between plasma concentrations of Tf (25,000 nM) and insulin (0.3 nM). Our modeling shows that the very high plasma concentration of Tf leads to only 5% of the endothelial TfR expressed on the luminal endothelial membrane.

A statistical model of COVID-19 testing in populations: effects of sampling bias and testing errors

We develop a statistical model for the testing of disease prevalence in a population. The model assumes a binary test result, positive or negative, but allows for biases in sample selection and both type I (false positive) and type II (false negative) testing errors. Our model also incorporates multiple test types and is able to distinguish between retesting and exclusion after testing. Our quantitative framework allows us to directly interpret testing results as a function of errors and biases. By applying our testing model to COVID-19 testing data and actual case data from specific jurisdictions, we are able to estimate and provide uncertainty quantification of indices that are crucial in a pandemic, such as disease prevalence and fatality ratios.

Using excess deaths and testing statistics to determine COVID-19 mortalities

Factors such as varied definitions of mortality, uncertainty in disease prevalence, and biased sampling complicate the quantification of fatality during an epidemic. Regardless of the employed fatality measure, the infected population and the number of infection-caused deaths need to be consistently estimated for comparing mortality across regions. We combine historical and current mortality data, a statistical testing model, and an SIR epidemic model, to improve estimation of mortality. We find that the average excess death across the entire US from January 2020 until February 2021 is 9[Formula: see text] higher than the number of reported COVID-19 deaths. In some areas, such as New York City, the number of weekly deaths is about eight times higher than in previous years. Other countries such as Peru, Ecuador, Mexico, and Spain exhibit excess deaths significantly higher than their reported COVID-19 deaths. Conversely, we find statistically insignificant or even negative excess deaths for at least most of 2020 in places such as Germany, Denmark, and Norway.

Uniformly accurate nonlinear transmission rate models arising from disease spread through pair contacts

We derive and asymptotically analyze mass-action models for disease spread that include transient pair formation and dissociation. Populations of unpaired susceptible individuals and infected individuals are distinguished from the population of three types of pairs of individuals: both susceptible, one susceptible and one infected, and both infected. Disease transmission can occur only within a pair consisting of one susceptible individual and one infected individual. We use perturbation expansion to formally derive uniformly valid approximations for the dynamics of the total infected and susceptible populations under different conditions including combinations of fast association, fast transmission, and fast dissociation limits. The effective equations are derived from the fundamental mass-action system without implicitly imposing transmission mechanisms, such as those used in frequency-dependent models. Our results represent submodels that show how effective nonlinear transmission can arise from pairing dynamics and are juxtaposed with density-based mass-action and frequency-based models.

Using excess deaths and testing statistics to improve estimates of COVID-19 mortalities

Factors such as non-uniform definitions of mortality, uncertainty in disease prevalence, and biased sampling complicate the quantification of fatality during an epidemic. Regardless of the employed fatality measure, the infected population and the number of infection-caused deaths need to be consistently estimated for comparing mortality across regions. We combine historical and current mortality data, a statistical testing model, and an SIR epidemic model, to improve estimation of mortality. We find that the average excess death across the entire US is 13% higher than the number of reported COVID-19 deaths. In some areas, such as New York City, the number of weekly deaths is about eight times higher than in previous years. Other countries such as Peru, Ecuador, Mexico, and Spain exhibit excess deaths significantly higher than their reported COVID-19 deaths. Conversely, we find negligible or negative excess deaths for part and all of 2020 for Denmark, Germany, and Norway.

Using excess deaths and testing statistics to improve estimates of COVID-19 mortalities

Factors such as non-uniform definitions of mortality, uncertainty in disease prevalence, and biased sampling complicate the quantification of fatality during an epidemic. Regardless of the employed fatality measure, the infected population and the number of infection-caused deaths need to be consistently estimated for comparing mortality across regions. We combine historical and current mortality data, a statistical testing model, and an SIR epidemic model, to improve estimation of mortality. We find that the average excess death across the entire US is 13$\%$ higher than the number of reported COVID-19 deaths. In some areas, such as New York City, the number of weekly deaths is about eight times higher than in previous years. Other countries such as Peru, Ecuador, Mexico, and Spain exhibit excess deaths significantly higher than their reported COVID-19 deaths. Conversely, we find negligible or negative excess deaths for part and all of 2020 for Denmark, Germany, and Norway.

Machine learning and statistical prediction of patient quality-of-life after prostate radiation therapy

Thanks to advancements in diagnosis and treatment, prostate cancer patients have high long-term survival rates. Currently, an important goal is to preserve quality of life during and after treatment. The relationship between the radiation a patient receives and the subsequent side effects he experiences is complex and difficult to model or predict. Here, we use machine learning algorithms and statistical models to explore the connection between radiation treatment and post-treatment gastro-urinary function. Since only a limited number of patient datasets are currently available, we used image flipping and curvature-based interpolation methods to generate more data to leverage transfer learning. Using interpolated and augmented data, we trained a convolutional autoencoder network to obtain near-optimal starting points for the weights. A convolutional neural network then analyzed the relationship between patient-reported quality-of-life and radiation doses to the bladder and rectum. We also used analysis of variance and logistic regression to explore organ sensitivity to radiation and to develop dosage thresholds for each organ region. Our findings show no statistically significant association between the bladder and quality-of-life scores. However, we found a statistically significant association between the radiation applied to posterior and anterior rectal regions and changes in quality of life. Finally, we estimated radiation therapy dose thresholds for each organ. Our analysis connects machine learning methods with organ sensitivity, thus providing a framework for informing cancer patient care using patient reported quality-of-life metrics.

Mathematical modeling of depressive disorders: Circadian driving, bistability and dynamical transitions

The hypothalamus-pituitary-adrenal (HPA) axis is a key neuroendocrine system implicated in stress response, major depression disorder, and post-traumatic stress disorder. We present a new, compact dynamical systems model for the response of the HPA axis to external stimuli, representing stressors or therapeutic intervention, in the presence of a circadian input. Our work builds upon previous HPA axis models where hormonal dynamics are separated into slow and fast components. Several simplifications allow us to derive an effective model of two equations, similar to a multiplicative-input FitzHugh-Nagumo system, where two stable states, a healthy and a diseased one, arise. We analyze the effective model in the context of state transitions driven by external shocks to the hypothalamus, but also modulated by circadian rhythms. Our analyses provide mechanistic insight into the effects of the circadian cycle on input driven transitions of the HPA axis and suggest a circadian influence on exposure or cognitive behavioral therapy in depression, or post-traumatic stress disorder treatment.

Why case fatality ratios can be misleading: individual- and population-based mortality estimates and factors influencing them

Different ways of calculating mortality during epidemics have yielded very different results, particularly during the current COVID-19 pandemic. For example, the “CFR” has been interchangeably called the case fatality ratio, case fatality rate, and case fatality risk, often without standard mathematical definitions. The most commonly used CFR is the case fatality ratio, typically constructed using the estimated number of deaths to date divided by the estimated total number of confirmed infected cases to date. How does this CFR relate to an infected individual’s probability of death? To explore such issues, we formulate both a survival probability model and an associated infection duration-dependent SIR model to define individual- and population-based estimates of dynamic mortality measures to show that neither of these are directly represented by the case fatality ratio. The key parameters that affect the dynamics of different mortality estimates are the incubation period and the time individuals were infected before confirmation of infection. Using data on the recent SARS-CoV-2 outbreaks, we estimate and compare the different dynamic mortality estimates and highlight their differences. Informed by our modeling, we propose more systematic methods to determine mortality during epidemic outbreaks and discuss sensitivity to confounding effects and uncertainties in the data arising from, e.g., undertesting and heterogeneous populations.

Why estimating population-based case fatality rates during epidemics may be misleading

Different ways of calculating mortality ratios during epidemics can yield widely different results, particularly during the COVID-19 pandemic. We formulate both a survival probability model and an associated infection duration-dependent SIR model to define individual- and population-based estimates of dynamic mortality ratios. The key parameters that affect the dynamics of the different mortality estimates are the incubation period and the length of time individuals were infected before confirmation of infection. We stress that none of these ratios are accurately represented by the often misinterpreted case fatality ratio (CFR), the number of deaths to date divided by the total number of infected cases to date. Using available data on the recent SARS-CoV-2 outbreaks and simple assumptions, we estimate and compare the different dynamic mortality ratios and highlight their differences. Informed by our modeling, we propose a more systematic method to determine mortality ratios during epidemic outbreaks and discuss sensitivity to confounding effects and errors in the data.

Diversity in biology: definitions, quantification and models

Diversity indices are useful single-number metrics for characterizing a complex distribution of a set of attributes across a population of interest. The utility of these different metrics or sets of metrics depends on the context and application, and whether a predictive mechanistic model exists. In this topical review, we first summarize the relevant mathematical principles underlying heterogeneity in a large population, before outlining the various definitions of 'diversity' and providing examples of scientific topics in which its quantification plays an important role. We then review how diversity has been a ubiquitous concept across multiple fields, including ecology, immunology, cellular barcoding experiments, and socioeconomic studies. Since many of these applications involve sampling of populations, we also review how diversity in small samples is related to the diversity in the entire population. Features that arise in each of these applications are highlighted.

PDE MODELS OF ADDER MECHANISMS IN CELLULAR PROLIFERATION

Cell division is a process that involves many biochemical steps and complex biophysical mechanisms. To simplify the understanding of what triggers cell division, three basic models that subsume more microscopic cellular processes associated with cell division have been proposed. Cells can divide based on the time elapsed since their birth, their size, and/or the volume added since their birth-the timer, sizer, and adder models, respectively. Here, we propose unified adder-sizer models and investigate some of the properties of different adder processes arising in cellular proliferation. Although the adder-sizer model provides a direct way to model cell population structure, we illustrate how it is mathematically related to the well-known model in which cell division depends on age and size. Existence and uniqueness of weak solutions to our 2+1-dimensional PDE model are proved, leading to the convergence of the discretized numerical solutions and allowing us to numerically compute the dynamics of cell population densities. We then generalize our PDE model to incorporate recent experimental findings of a system exhibiting mother-daughter correlations in cellular growth rates. Numerical experiments illustrating possible average cell volume blowup and the dynamical behavior of cell populations with mother-daughter correlated growth rates are carried out. Finally, motivated by new experimental findings, we extend our adder model cases where the controlling variable is the added size between DNA replication initiation points in the cell cycle.

Dynamics of T cell receptor distributions following acute thymic atrophy and resumption

Naive human T cells are produced and developed in the thymus, which atrophies abruptly and severely in response to physical or psychological stress. To understand how an instance of stress affects the size and "diversity" of the peripheral naive T cell pool, we derive a mean-field autonomous ODE model of T cell replenishment that allows us to track the clone abundance distribution (the mean number of different TCRs each represented by a specific number of cells). We identify equilibrium solutions that arise at different rates of T cell production, and derive analytic approximations to the dominant eigenvalues and eigenvectors of the mathematical model linearized about these equilibria. From the forms of the eigenvalues and eigenvectors, we estimate rates at which counts of clones of different sizes converge to and depart from equilibrium values-that is, how the number of clones of different sizes "adjusts" to the changing rate of T cell production. Under most physiological realizations of our model, the dominant eigenvalue (representing the slowest dynamics of the clone abundance distribution) scales as a power law in the thymic output for low output levels, but saturates at higher T cell production rates. Our analysis provides a framework for quantitatively understanding how the clone abundance distribution evolves under small changes in the overall T cell production rate.

Nonspecific probe binding and automatic gating in flow cytometry and fluorescence activated cell sorting (FACS)

Flow cytometry is extensively used in cell biology to differentiate cells of interest (mutants) from control cells (wild-types). For mutant cells characterized by expression of a distinct membrane surface structure, fluorescent marker probes can be designed to bind specifically to these structures while the cells are in suspension, resulting in a sufficiently high fluorescence intensity measurement by the cytometer to identify a mutant cell. However, cell membranes may have relatively weak, nonspecific binding affinity to the probes, resulting in false positive results. Furthermore, the same effect would be present on mutant cells, allowing both specific and nonspecific binding to a single cell. We derive and analyze a kinetic model of fluorescent probe binding dynamics by tracking populations of mutant and wild-type cells with differing numbers of probes bound specifically and nonspecifically. By assuming the suspension is in chemical equilibrium prior to cytometry, we use a two-species Langmuir adsorption model to analyze the confounding effects of non-specific binding on the assay. Furthermore, we analytically derive an expectation maximization method to infer an appropriate estimate of the total number of mutant cells as an alternative to existing, heuristic methods. Lastly, using our model, we propose a new method to infer physical and experimental parameters from existing protocols. Our results provide improved ways to quantitatively analyze flow cytometry data.

Evolution of Coagulation-Fragmentation Stochastic Processes Using Accurate Chemical Master Equation Approach

Coagulation and fragmentation (CF) is a fundamental process in which smaller particles attach to each other to form larger clusters while existing clusters break up into smaller particles . It is a ubiquitous process that plays important roles in many physical and biological phenomena. CF is typically a stochastic process that often occurs in confined spaces with a limited number of available particles . Here, we study the CF process formulated with the discrete Chemical Master Equation (dCME). Using the newly developed Accurate Chemical Master Equation (ACME) method, we examine the time-dependent behavior of the CF system. We investigate the effects of a number of important factors that influence the overall behavior of the system, including the dimensionality, the ratio of attachment to detachment rates among clusters, and the initial conditions. By comparing CF in one and three dimensions, we conclude that systems in three dimensions are more likely to form large clusters. We also demonstrate how the ratio of the attachment to detachment rates affects the dynamics and the steady-state of the system. Finally, we demonstrate the relationship between the formation of large clusters and the initial condition.

A Dynamical Bifurcation Model of Bipolar Disorder Based on Learned Expectation and Asymmetry in Mood Sensitivity

Bipolar disorder is a common psychiatric dysfunction characterized by recurring episodes of mania and depression. Despite its prevalence, the causes and mechanisms of bipolar disorder remain largely unknown. Recently, theories focusing on the interaction between emotion and behavior, including those based on dysregulation of the so-called behavioral approach system (BAS), have gained popularity. Mathematical models built on this principle predict bistability in mood and do not invoke intrinsic biological rhythms that may arise from interactions between mood and expectation. Here we develop and analyze a model with clinically meaningful and modifiable parameters that incorporates the interaction between mood and expectation. Our nonlinear model exhibits a transition to limit cycle behavior when a mood-sensitivity parameter exceeds a threshold value, signaling a transition to a bipolar state. The model also predicts that asymmetry in response to positive and negative events can induce unipolar depression/mania, consistent with clinical observations. We analyze the model with asymmetric mood sensitivities and show that large unidirectional mood sensitivity can lead to bipolar disorder. Finally, we show how observed effects of lithium- and antidepressant-induced mania can be explained within the framework of our proposed model.

Modeling large fluctuations of thousands of clones during hematopoiesis: The role of stem cell self-renewal and bursty progenitor dynamics in rhesus macaque

In a recent clone-tracking experiment, millions of uniquely tagged hematopoietic stem cells (HSCs) and progenitor cells were autologously transplanted into rhesus macaques and peripheral blood containing thousands of tags were sampled and sequenced over 14 years to quantify the abundance of hundreds to thousands of tags or “clones.” Two major puzzles of the data have been observed: consistent differences and massive temporal fluctuations of clone populations. The large sample-to-sample variability can lead clones to occasionally go “extinct” but “resurrect” themselves in subsequent samples. Although heterogeneity in HSC differentiation rates, potentially due to tagging, and random sampling of the animals’ blood and cellular demographic stochasticity might be invoked to explain these features, we show that random sampling cannot explain the magnitude of the temporal fluctuations. Moreover, we show through simpler neutral mechanistic and statistical models of hematopoiesis of tagged cells that a broad distribution in clone sizes can arise from stochastic HSC self-renewal instead of tag-induced heterogeneity. The very large clone population fluctuations that often lead to extinctions and resurrections can be naturally explained by a generation-limited proliferation constraint on the progenitor cells. This constraint leads to bursty cell population dynamics underlying the large temporal fluctuations. We analyzed experimental clone abundance data using a new statistic that counts clonal disappearances and provided least-squares estimates of two key model parameters in our model, the total HSC differentiation rate and the maximum number of progenitor-cell divisions.

Hematopoiesis of virally tagged cells in rhesus macaques is analyzed in the context of a mechanistic and statistical model. We find that the clone size distribution and the temporal variability in the abundance of each clone (viral tag) in peripheral blood are consistent with (i) stochastic HSC self-renewal during bone marrow repair, (ii) clonal aging that restricts the number of generations of progenitor cells, and (iii) infrequent and small-size samples. By fitting data, we infer two key parameters that control the level of fluctuations of clone sizes in our model: the total HSC differentiation rate and the maximum proliferation capacity of progenitor cells. Our analysis provides insight into the mechanisms of hematopoiesis and a framework to guide future multiclone barcoding/lineage tracking measurements.

Perturbing the Hypothalamic-Pituitary-Adrenal Axis: A Mathematical Model for Interpreting PTSD Assessment Tests

We use a dynamical systems model of the hypothalamic-pituitary-adrenal (HPA) axis to understand the mechanisms underlying clinical protocols used to probe patient stress response. Specifically, we address dexamethasone (DEX) and ACTH challenge tests, which probe pituitary and adrenal gland responses, respectively. We show that some previously observed features and experimental responses can arise from a bistable mathematical model containing two steady-states, rather than relying on specific and permanent parameter changes due to physiological disruption. Moreover, we show that the timing of a perturbation relative to the intrinsic oscillation of the HPA axis can affect challenge test responses. Conventional mechanistic hypotheses supported and refuted by the challenge tests are reexamined by varying parameters in our mathematical model associated with these hypotheses. We show that (a) adrenal hyposensitivity can give rise to the responses seen in ACTH challenge tests and (b) enhanced cortisol-mediated suppression of the pituitary in subjects with PTSD is not necessary to explain the responses observed in DEX stress tests. We propose a new two-stage DEX/external stressor protocol to more clearly distinguish between the conventional hypothesis of enhanced suppression of the pituitary and bistable dynamics hypothesized in our model.