Xist nucleates local protein gradients to propagate silencing across the X chromosome.

The lncRNA Xist forms ∼50 diffraction-limited foci to transcriptionally silence one X chromosome. How this small number of RNA foci and interacting proteins regulate a much larger number of X-linked genes is unknown. We show that Xist foci are locally confined, contain ∼2 RNA molecules, and nucleate supramolecular complexes (SMACs) that include many copies of the critical silencing protein SPEN. Aggregation and exchange of SMAC proteins generate local protein gradients that regulate broad, proximal chromatin regions. Partitioning of numerous SPEN molecules into SMACs is mediated by their intrinsically disordered regions and essential for transcriptional repression. Polycomb deposition via SMACs induces chromatin compaction and the increase in SMACs density around genes, which propagates silencing across the X chromosome. Our findings introduce a mechanism for functional nuclear compartmentalization whereby crowding of transcriptional and architectural regulators enables the silencing of many target genes by few RNA molecules.

Reliable ligand discrimination in stochastic multistep kinetic proofreading: First passage time vs. product counting strategies.

Cellular signaling, crucial for biological processes like immune response and homeostasis, relies on specificity and fidelity in signal transduction to accurately respond to stimuli amidst biological noise. Kinetic proofreading (KPR) is a key mechanism enhancing signaling specificity through time-delayed steps, although its effectiveness is debated due to intrinsic noise potentially reducing signal fidelity. In this study, we reformulate the theory of kinetic proofreading (KPR) by convolving multiple intermediate states into a single state and then define an overall "processing" time required to traverse these states. This simplification allows us to succinctly describe kinetic proofreading in terms of a single waiting time parameter, facilitating a more direct evaluation and comparison of KPR performance across different biological contexts such as DNA replication and T cell receptor (TCR) signaling. We find that loss of fidelity for longer proofreading steps relies on the specific strategy of information extraction and show that in the first-passage time (FPT) discrimination strategy, longer proofreading steps can exponentially improve the accuracy of KPR at the cost of speed. Thus, KPR can still be an effective discrimination mechanism in the high noise regime. However, in a product concentration-based discrimination strategy, longer proofreading steps do not necessarily lead to an increase in performance. However, by introducing activation thresholds on product concentrations, can we decompose the product-based strategy into a series of FPT-based strategies to better resolve the subtleties of KPR-mediated product discrimination. Our findings underscore the importance of understanding KPR in the context of how information is extracted and processed in the cell.

Order-of-Mutation Effects on Cancer Progression: Models for Myeloproliferative Neoplasm.

In some patients with myeloproliferative neoplasms (MPN), two genetic mutations are often found: JAK2 V617F and one in the TET2 gene. Whether one mutation is present influences how the other subsequent mutation will affect the regulation of gene expression. In other words, when a patient carries both mutations, the order of when they first arose has been shown to influence disease progression and prognosis. We propose a nonlinear ordinary differential equation, the Moran process, and Markov chain models to explain the non-additive and non-commutative mutation effects on recent clinical observations of gene expression patterns, proportions of cells with different mutations, and ages at diagnosis of MPN. Combined, these observations are used to shape our modeling framework. Our key proposal is that bistability in gene expression provides a natural explanation for many observed order-of-mutation effects. We also propose potential experimental measurements that can be used to confirm or refute predictions of our models.

Stochastic dynamics and ribosome-RNAP interactions in transcription-translation coupling.

Under certain cellular conditions, transcription and mRNA translation in prokaryotes appear to be "coupled," in which the formation of mRNA transcript and production of its associated protein are temporally correlated. Such transcription-translation coupling (TTC) has been evoked as a mechanism that speeds up the overall process, provides protection against premature termination, and/or regulates the timing of transcript and protein formation. What molecular mechanisms underlie ribosome-RNAP coupling and how they can perform these functions have not been explicitly modeled. We develop and analyze a continuous-time stochastic model that incorporates ribosome and RNAP elongation rates, initiation and termination rates, RNAP pausing, and direct ribosome and RNAP interactions (exclusion and binding). Our model predicts how distributions of delay times depend on these molecular features of transcription and translation. We also propose additional measures for TTC: a direct ribosome-RNAP binding probability and the fraction of time the translation-transcription process is "protected" from attack by transcription-terminating proteins. These metrics quantify different aspects of TTC and differentially depend on parameters of known molecular processes. We use our metrics to reveal how and when our model can exhibit either acceleration or deceleration of transcription, as well as protection from termination. Our detailed mechanistic model provides a basis for designing new experimental assays that can better elucidate the mechanisms of TTC.

Mathematical Characterization of Private and Public Immune Receptor Sequences.

Diverse T and B cell repertoires play an important role in mounting effective immune responses against a wide range of pathogens and malignant cells. The number of unique T and B cell clones is characterized by T and B cell receptors (TCRs and BCRs), respectively. Although receptor sequences are generated probabilistically by recombination processes, clinical studies found a high degree of sharing of TCRs and BCRs among different individuals. In this work, we use a general probabilistic model for T/B cell receptor clone abundances to define "publicness" or "privateness" and information-theoretic measures for comparing the frequency of sampled sequences observed across different individuals. We derive mathematical formulae to quantify the mean and the variances of clone richness and overlap. Our results can be used to evaluate the effect of different sampling protocols on abundances of clones within an individual as well as the commonality of clones across individuals. Using synthetic and empirical TCR amino acid sequence data, we perform simulations to study expected clonal commonalities across multiple individuals. Based on our formulae, we compare these simulated results with the analytically predicted mean and variances of the repertoire overlap. Complementing the results on simulated repertoires, we derive explicit expressions for the richness and its uncertainty for specific, single-parameter truncated power-law probability distributions. Finally, the information loss associated with grouping together certain receptor sequences, as is done in spectratyping, is also evaluated. Our approach can be, in principle, applied under more general and mechanistically realistic clone generation models.

Stochastic nucleosome disassembly mediated by remodelers and histone fragmentation.

We construct and analyze monomeric and multimeric models of the stochastic disassembly of a single nucleosome. Our monomeric model predicts the time needed for a number of histone-DNA contacts to spontaneously break, leading to dissociation of a non-fragmented histone from DNA. The dissociation process can be facilitated by DNA binding proteins or processing molecular motors that compete with histones for histone-DNA contact sites. Eigenvalue analysis of the corresponding master equation allows us to evaluate histone detachment times under both spontaneous detachment and protein-facilitated processes. We find that competitive DNA binding of remodeling proteins can significantly reduce the typical detachment time but only if these remodelers have DNA-binding affinities comparable to those of histone-DNA contact sites. In the presence of processive motors, the histone detachment rate is shown to be proportional to the product of the histone single-bond dissociation constant and the speed of motor protein procession. Our simple intact-histone model is then extended to allow for multimeric nucleosome kinetics that reveal additional pathways of disassembly. In addition to a dependence of complete disassembly times on subunit-DNA contact energies, we show how histone subunit concentrations in bulk solutions can mediate the disassembly process by rescuing partially disassembled nucleosomes. Moreover, our kinetic model predicts that remodeler binding can also bias certain pathways of nucleosome disassembly, with higher remodeler binding rates favoring intact-histone detachment.

Impact of random and targeted disruptions on information diffusion during outbreaks.

Outbreaks are complex multi-scale processes that are impacted not only by cellular dynamics and the ability of pathogens to effectively reproduce and spread, but also by population-level dynamics and the effectiveness of mitigation measures. A timely exchange of information related to the spread of novel pathogens, stay-at-home orders, and other measures can be effective at containing an infectious disease, particularly during the early stages when testing infrastructure, vaccines, and other medical interventions may not be available at scale. Using a multiplex epidemic model that consists of an information layer (modeling information exchange between individuals) and a spatially embedded epidemic layer (representing a human contact network), we study how random and targeted disruptions in the information layer (e.g., errors and intentional attacks on communication infrastructure) impact the total proportion of infections, peak prevalence (i.e., the maximum proportion of infections), and the time to reach peak prevalence. We calibrate our model to the early outbreak stages of the SARS-CoV-2 pandemic in 2020. Mitigation campaigns can still be effective under random disruptions, such as failure of information channels between a few individuals. However, targeted disruptions or sabotage of hub nodes that exchange information with a large number of individuals can abruptly change outbreak characteristics, such as the time to reach the peak of infection. Our results emphasize the importance of the availability of a robust communication infrastructure during an outbreak that can withstand both random and targeted disruptions.

Density- and elongation speed-dependent error correction in RNA polymerization.

Backtracking of RNA polymerase (RNAP) is an important pausing mechanism during DNA transcription that is part of the error correction process that enhances transcription fidelity. We model the backtracking mechanism of RNAP, which usually happens when the polymerase tries to incorporate a noncognate or 'mismatched' nucleotide triphosphate. Previous models have made simplifying assumptions such as neglecting the trailing polymerase behind the backtracking polymerase or assuming that the trailing polymerase is stationary. We derive exact analytic solutions of a stochastic model that includes locally interacting RNAPs by explicitly showing how a trailing RNAP influences the probability that an error is corrected or incorporated by the leading backtracking RNAP. We also provide two related methods for computing the mean times for error correction and incorporation given an initial local RNAP configuration. Using these results, we propose an effective interacting-RNAP lattice that can be readily simulated.

A statistical model of COVID-19 testing in populations: effects of sampling bias andtesting 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. This article is part of the theme issue 'Data science approach to infectious disease surveillance'.

Discrete stochastic models of SELEX: Aptamer capture probabilities and protocol optimization.

Antibodies are important biomolecules that are often designed to recognize target antigens. However, they are expensive to produce and their relatively large size prevents their transport across lipid membranes. An alternative to antibodies is aptamers, short (∼15-60 bp) oligonucleotides (and amino acid sequences) with specific secondary and tertiary structures that govern their affinity to specific target molecules. Aptamers are typically generated via solid phase oligonucleotide synthesis before selection and amplification through Systematic Evolution of Ligands by EXponential enrichment (SELEX), a process based on competitive binding that enriches the population of certain strands while removing unwanted sequences, yielding aptamers with high specificity and affinity to a target molecule. Mathematical analyses of SELEX have been formulated in the mass action limit, which assumes large system sizes and/or high aptamer and target molecule concentrations. In this paper, we develop a fully discrete stochastic model of SELEX. While converging to a mass-action model in the large system-size limit, our stochastic model allows us to study statistical quantities when the system size is small, such as the probability of losing the best-binding aptamer during each round of selection. Specifically, we find that optimal SELEX protocols in the stochastic model differ from those predicted by a deterministic model.

A mathematical model of reward-mediated learning in drug addiction.

Substances of abuse are known to activate and disrupt neuronal circuits in the brain reward system. We propose a simple and easily interpretable dynamical systems model to describe the neurobiology of drug addiction that incorporates the psychiatric concepts of reward prediction error, drug-induced incentive salience, and opponent process theory. Drug-induced dopamine releases activate a biphasic reward response with pleasurable, positive "a-processes" (euphoria, rush) followed by unpleasant, negative "b-processes" (cravings, withdrawal). Neuroadaptive processes triggered by successive intakes enhance the negative component of the reward response, which the user compensates for by increasing drug dose and/or intake frequency. This positive feedback between physiological changes and drug self-administration leads to habituation, tolerance, and, eventually, to full addiction. Our model gives rise to qualitatively different pathways to addiction that can represent a diverse set of user profiles (genetics, age) and drug potencies. We find that users who have, or neuroadaptively develop, a strong b-process response to drug consumption are most at risk for addiction. Finally, we include possible mechanisms to mitigate withdrawal symptoms, such as through the use of methadone or other auxiliary drugs used in detoxification.

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.

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.

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.

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.

A Mathematical Model of the Effects of Aging on Naive T Cell Populations and Diversity.

The human adaptive immune response is known to weaken in advanced age, resulting in increased severity of pathogen-born illness, poor vaccine efficacy, and a higher prevalence of cancer in the elderly. Age-related erosion of the T cell compartment has been implicated as a likely cause, but the underlying mechanisms driving this immunosenescence have not been quantitatively modeled and systematically analyzed. T cell receptor diversity, or the extent of pathogen-derived antigen responsiveness of the T cell pool, is known to diminish with age, but inherent experimental difficulties preclude accurate analysis on the full organismal level. In this paper, we formulate a mechanistic mathematical model of T cell population dynamics on the immunoclonal subpopulation level, which provides quantitative estimates of diversity. We define different estimates for diversity that depend on the individual number of cells in a specific immunoclone. We show that diversity decreases with age primarily due to diminished thymic output of new T cells and the resulting overall loss of small immunoclones.

The Effects of Statistical Multiplicity of Infection on Virus Quantification and Infectivity Assays.

Many biological assays are employed in virology to quantify parameters of interest. Two such classes of assays, virus quantification assays (VQAs) and infectivity assays (IAs), aim to estimate the number of viruses present in a solution and the ability of a viral strain to successfully infect a host cell, respectively. VQAs operate at extremely dilute concentrations, and results can be subject to stochastic variability in virus-cell interactions. At the other extreme, high viral-particle concentrations are used in IAs, resulting in large numbers of viruses infecting each cell, enough for measurable change in total transcription activity. Furthermore, host cells can be infected at any concentration regime by multiple particles, resulting in a statistical multiplicity of infection and yielding potentially significant variability in the assay signal and parameter estimates. We develop probabilistic models for statistical multiplicity of infection at low and high viral-particle-concentration limits and apply them to the plaque (VQA), endpoint dilution (VQA), and luciferase reporter (IA) assays. A web-based tool implementing our models and analysis is also developed and presented. We test our proposed new methods for inferring experimental parameters from data using numerical simulations and show improvement on existing procedures in all limits.

Reconstruction of Cell Focal Adhesions using Physical Constraints and Compressive Regularization.

We develop a method to reconstruct, from measured displacements of an underlying elastic substrate, the spatially dependent forces that cells or tissues impart on it. Given newly available high-resolution images of substrate displacements, it is desirable to be able to reconstruct small-scale, compactly supported focal adhesions that are often localized and exist only within the footprint of a cell. In addition to the standard quadratic data mismatch terms that define least-squares fitting, we motivate a regularization term in the objective function that penalizes vectorial invariants of the reconstructed surface stress while preserving boundaries. We solve this inverse problem by providing a numerical method for setting up a discretized inverse problem that is solvable by standard convex optimization techniques. By minimizing the objective function subject to a number of important physically motivated constraints, we are able to efficiently reconstruct stress fields with localized structure from simulated and experimental substrate displacements. Our method incorporates the exact solution for a stress tensor accurate to first-order finite differences and motivates the use of distance-based cutoffs for data inclusion and problem sparsification.

Mechanisms of blood homeostasis: lineage tracking and a neutral model of cell populations in rhesus macaques.

BACKGROUND: How a potentially diverse population of hematopoietic stem cells (HSCs) differentiates and proliferates to supply more than 10(11) mature blood cells every day in humans remains a key biological question. We investigated this process by quantitatively analyzing the clonal structure of peripheral blood that is generated by a population of transplanted lentivirus-marked HSCs in myeloablated rhesus macaques. Each transplanted HSC generates a clonal lineage of cells in the peripheral blood that is then detected and quantified through deep sequencing of the viral vector integration sites (VIS) common within each lineage. This approach allowed us to observe, over a period of 4-12 years, hundreds of distinct clonal lineages. RESULTS: While the distinct clone sizes varied by three orders of magnitude, we found that collectively, they form a steady-state clone size-distribution with a distinctive shape. Steady-state solutions of our model show that the predicted clone size-distribution is sensitive to only two combinations of parameters. By fitting the measured clone size-distributions to our mechanistic model, we estimate both the effective HSC differentiation rate and the number of active HSCs. CONCLUSIONS: Our concise mathematical model shows how slow HSC differentiation followed by fast progenitor growth can be responsible for the observed broad clone size-distribution. Although all cells are assumed to be statistically identical, analogous to a neutral theory for the different clone lineages, our mathematical approach captures the intrinsic variability in the times to HSC differentiation after transplantation.