Empirical Quantification of Predictive Uncertainty Due to Model Discrepancy by Training with an Ensemble of Experimental Designs: An Application to Ion Channel Kinetics.

When using mathematical models to make quantitative predictions for clinical or industrial use, it is important that predictions come with a reliable estimate of their accuracy (uncertainty quantification). Because models of complex biological systems are always large simplifications, model discrepancy arises-models fail to perfectly recapitulate the true data generating process. This presents a particular challenge for making accurate predictions, and especially for accurately quantifying uncertainty in these predictions. Experimentalists and modellers must choose which experimental procedures (protocols) are used to produce data used to train models. We propose to characterise uncertainty owing to model discrepancy with an ensemble of parameter sets, each of which results from training to data from a different protocol. The variability in predictions from this ensemble provides an empirical estimate of predictive uncertainty owing to model discrepancy, even for unseen protocols. We use the example of electrophysiology experiments that investigate the properties of hERG potassium channels. Here, 'information-rich' protocols allow mathematical models to be trained using numerous short experiments performed on the same cell. In this case, we simulate data with one model and fit it with a different (discrepant) one. For any individual experimental protocol, parameter estimates vary little under repeated samples from the assumed additive independent Gaussian noise model. Yet parameter sets arising from the same model applied to different experiments conflict-highlighting model discrepancy. Our methods will help select more suitable ion channel models for future studies, and will be widely applicable to a range of biological modelling problems.

Venetoclax, alone and in combination with the BH3 mimetic S63845, depletes HIV-1 latently infected cells and delays rebound in humanized mice.

HIV-1 persists indefinitely in people living with HIV (PLWH) on antiretroviral therapy (ART). If ART is stopped, the virus rapidly rebounds from long-lived latently infected cells. Using a humanized mouse model of HIV-1 infection and CD4+ T cells from PLWH on ART, we investigate whether antagonizing host pro-survival proteins can prime latent cells to die and facilitate HIV-1 clearance. Venetoclax, a pro-apoptotic inhibitor of Bcl-2, depletes total and intact HIV-1 DNA in CD4+ T cells from PLWH ex vivo. This venetoclax-sensitive population is enriched for cells with transcriptionally higher levels of pro-apoptotic BH3-only proteins. Furthermore, venetoclax delays viral rebound in a mouse model of persistent HIV-1 infection, and the combination of venetoclax with the Mcl-1 inhibitor S63845 achieves a longer delay in rebound compared with either intervention alone. Thus, selective inhibition of pro-survival proteins can induce death of HIV-1-infected cells that persist on ART, extending time to viral rebound.

A necroptosis-independent function of RIPK3 promotes immune dysfunction and prevents control of chronic LCMV infection.

Necroptosis is a lytic and inflammatory form of cell death that is highly constrained to mitigate detrimental collateral tissue damage and impaired immunity. These constraints make it difficult to define the relevance of necroptosis in diseases such as chronic and persistent viral infections and within individual organ systems. The role of necroptotic signalling is further complicated because proteins essential to this pathway, such as receptor interacting protein kinase 3 (RIPK3) and mixed lineage kinase domain-like (MLKL), have been implicated in roles outside of necroptotic signalling. We sought to address this issue by individually defining the role of RIPK3 and MLKL in chronic lymphocytic choriomeningitis virus (LCMV) infection. We investigated if necroptosis contributes to the death of LCMV-specific CD8+ T cells or virally infected target cells during infection. We provide evidence showing that necroptosis was redundant in the pathogenesis of acute forms of LCMV (Armstrong strain) and the early stages of chronic (Docile strain) LCMV infection in vivo. The number of immune cells, their specificity and reactivity towards viral antigens and viral loads are not altered in the absence of either MLKL or RIPK3 during acute and during the early stages of chronic LCMV infection. However, we identified that RIPK3 promotes immune dysfunction and prevents control of infection at later stages of chronic LCMV disease. This was not phenocopied by the loss of MLKL indicating that the phenotype was driven by a necroptosis-independent function of RIPK3. We provide evidence that RIPK3 signaling evoked a dysregulated type 1 interferone response which we linked to an impaired antiviral immune response and abrogated clearance of chronic LCMV infection.

Endothelial Caspase-8 prevents fatal necroptotic hemorrhage caused by commensal bacteria.

Caspase-8 transduces signals from death receptor ligands, such as tumor necrosis factor, to drive potent responses including inflammation, cell proliferation or cell death. This is a developmentally essential function because in utero deletion of endothelial Caspase-8 causes systemic circulatory collapse during embryogenesis. Whether endothelial Caspase-8 is also required for cardiovascular patency during adulthood was unknown. To address this question, we used an inducible Cre recombinase system to delete endothelial Casp8 in 6-week-old conditionally gene-targeted mice. Extensive whole body vascular gene targeting was confirmed, yet the dominant phenotype was fatal hemorrhagic lesions exclusively within the small intestine. The emergence of these intestinal lesions was not a maladaptive immune response to endothelial Caspase-8-deficiency, but instead relied upon aberrant Toll-like receptor sensing of microbial commensals and tumor necrosis factor receptor signaling. This lethal phenotype was prevented in compound mutant mice that lacked the necroptotic cell death effector, MLKL. Thus, distinct from its systemic role during embryogenesis, our data show that dysregulated microbial- and death receptor-signaling uniquely culminate in the adult mouse small intestine to unleash MLKL-dependent necroptotic hemorrhage after loss of endothelial Caspase-8. These data support a critical role for Caspase-8 in preserving gut vascular integrity in the face of microbial commensals.

Uncertainty and error in SARS-CoV-2 epidemiological parameters inferred from population-level epidemic models.

During the SARS-CoV-2 pandemic, epidemic models have been central to policy-making. Public health responses have been shaped by model-based projections and inferences, especially related to the impact of various non-pharmaceutical interventions. Accompanying this has been increased scrutiny over model performance, model assumptions, and the way that uncertainty is incorporated and presented. Here we consider a population-level model, focusing on how distributions representing host infectiousness and the infection-to-death times are modelled, and particularly on the impact of inferred epidemic characteristics if these distributions are mis-specified. We introduce an SIR-type model with the infected population structured by 'infected age', i.e. the number of days since first being infected, a formulation that enables distributions to be incorporated that are consistent with clinical data. We show that inference based on simpler models without infected age, which implicitly mis-specify these distributions, leads to substantial errors in inferred quantities relevant to policy-making, such as the reproduction number and the impact of interventions. We consider uncertainty quantification via a Bayesian approach, implementing this for both synthetic and real data focusing on UK data in the period 15 Feb-14 Jul 2020, and emphasising circumstances where it is misleading to neglect uncertainty. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".

Epigenetic Silencing of RIPK3 in Hepatocytes Prevents MLKL-mediated Necroptosis From Contributing to Liver Pathologies.

BACKGROUND & AIMS: Necroptosis is a highly inflammatory mode of cell death that has been implicated in causing hepatic injury including steatohepatitis/ nonalcoholic steatohepatitis (NASH); however, the evidence supporting these claims has been controversial. A comprehensive, fundamental understanding of cell death pathways involved in liver disease critically underpins rational strategies for therapeutic intervention. We sought to define the role and relevance of necroptosis in liver pathology. METHODS: Several animal models of human liver pathology, including diet-induced steatohepatitis in male mice and diverse infections in both male and female mice, were used to dissect the relevance of necroptosis in liver pathobiology. We applied necroptotic stimuli to primary mouse and human hepatocytes to measure their susceptibility to necroptosis. Paired liver biospecimens from patients with NASH, before and after intervention, were analyzed. DNA methylation sequencing was also performed to investigate the epigenetic regulation of RIPK3 expression in primary human and mouse hepatocytes. RESULTS: Identical infection kinetics and pathologic outcomes were observed in mice deficient in an essential necroptotic effector protein, MLKL, compared with control animals. Mice lacking MLKL were indistinguishable from wild-type mice when fed a high-fat diet to induce NASH. Under all conditions tested, we were unable to induce necroptosis in hepatocytes. We confirmed that a critical activator of necroptosis, RIPK3, was epigenetically silenced in mouse and human primary hepatocytes and rendered them unable to undergo necroptosis. CONCLUSIONS: We have provided compelling evidence that necroptosis is disabled in hepatocytes during homeostasis and in the pathologic conditions tested in this study.

Modeling reveals posttranscriptional regulation of GA metabolism enzymes in response to drought and cold.

The hormone gibberellin (GA) controls plant growth and regulates growth responses to environmental stress. In monocotyledonous leaves, GA controls growth by regulating division-zone size. We used a systems approach to investigate the establishment of the GA distribution in the maize leaf growth zone to understand how drought and cold alter leaf growth. By developing and parameterizing a multiscale computational model that includes cell movement, growth-induced dilution, and metabolic activities, we revealed that the GA distribution is predominantly determined by variations in GA metabolism. Considering wild-type and UBI::GA20-OX-1 leaves, the model predicted the peak in GA concentration, which has been shown to determine division-zone size. Drought and cold modified enzyme transcript levels, although the model revealed that this did not explain the observed GA distributions. Instead, the model predicted that GA distributions are also mediated by posttranscriptional modifications increasing the activity of GA 20-oxidase in drought and of GA 2-oxidase in cold, which we confirmed by enzyme activity measurements. This work provides a mechanistic understanding of the role of GA metabolism in plant growth regulation.

Caspase-8 has dual roles in regulatory T cell homeostasis balancing immunity to infection and collateral inflammatory damage.

Targeting the potent immunosuppressive properties of FOXP3+ regulatory T cells (Tregs) has substantial therapeutic potential for treating autoimmune and inflammatory diseases. Yet, the molecular mechanisms controlling Treg homeostasis, particularly during inflammation, remain unclear. We report that caspase-8 is a central regulator of Treg homeostasis in a context-specific manner that is decisive during immune responses. In mouse genetic models, targeting caspase-8 in Tregs led to accumulation of effector Tregs resistant to apoptotic cell death. Conversely, inflammation induced the MLKL-dependent necroptosis of caspase-8-deficient lymphoid and tissue Tregs, which enhanced immunity to a variety of chronic infections to promote clearance of viral or parasitic pathogens. However, improved immunity came at the risk of lethal inflammation in overwhelming infections. Caspase-8 inhibition using a clinical-stage compound revealed that human Tregs have heightened sensitivity to necroptosis compared with conventional T cells. These findings reveal a fundamental mechanism in Tregs that could be targeted to manipulate the balance between immune tolerance versus response for therapeutic benefit.

The role of MKK4 in T-cell development and immunity to viral infections.

The stress-activated protein kinases (SAPKs)/c-Jun-N-terminal-kinases (JNK) are members of the mitogen-activated protein kinase family. These kinases are responsible for transducing cellular signals through a phosphorylation-dependent signaling cascade. JNK activation in immune cells can lead to a range of critical cellular responses that include proliferation, differentiation and apoptosis. MKK4 is a SAPK that can activate both JNK1 and JNK2; however, its role in T-cell development and function has been controversial. Additionally, loss of either JNK1 or JNK2 has opposing effects in the generation of T-cell immunity to viral infection and cancer. We used mice with a conditional loss of MKK4 in T cells to investigate the in vivo role of MKK4 in T-cell development and function during lymphocytic choriomeningitis virus (LCMV) infection. We found no physiologically relevant differences in T-cell responses or immunity to either acute or chronic LCMV in the absence of MKK4.

Parameter inference to motivate asymptotic model reduction: An analysis of the gibberellin biosynthesis pathway.

Developing effective strategies to use models in conjunction with experimental data is essential to understand the dynamics of biological regulatory networks. In this study, we demonstrate how combining parameter estimation with asymptotic analysis can reveal the key features of a network and lead to simplified models that capture the observed network dynamics. Our approach involves fitting the model to experimental data and using the profile likelihood to identify small parameters and cases where model dynamics are insensitive to changing particular individual parameters. Such parameter diagnostics provide understanding of the dominant features of the model and motivate asymptotic model reductions to derive simpler models in terms of identifiable parameter groupings. We focus on the particular example of biosynthesis of the plant hormone gibberellin (GA), which controls plant growth and has been mutated in many current crop varieties. This pathway comprises two parallel series of enzyme-substrate reactions, which have previously been modelled using the law of mass action (Middleton et al., 2012). Considering the GA20ox-mediated steps, we analyse the identifiability of the model parameters using published experimental data; the analysis reveals the ratio between enzyme and GA levels to be small and motivates us to perform a quasi-steady state analysis to derive a reduced model. Fitting the parameters in the reduced model reveals additional features of the pathway and motivates further asymptotic analysis which produces a hierarchy of reduced models. Calculating the Akaike information criterion and parameter confidence intervals enables us to select a parsimonious model with identifiable parameters. As well as demonstrating the benefits of combining parameter estimation and asymptotic analysis, the analysis shows how GA biosynthesis is limited by the final GA20ox-mediated steps in the pathway and generates a simple mathematical description of this part of the GA biosynthesis pathway.

Three-dimensional plant architecture and sunlit-shaded patterns: a stochastic model of light dynamics in canopies.

Background and Aims: Diurnal changes in solar position and intensity combined with the structural complexity of plant architecture result in highly variable and dynamic light patterns within the plant canopy. This affects productivity through the complex ways that photosynthesis responds to changes in light intensity. Current methods to characterize light dynamics, such as ray-tracing, are able to produce data with excellent spatio-temporal resolution but are computationally intensive and the resulting data are complex and high-dimensional. This necessitates development of more economical models for summarizing the data and for simulating realistic light patterns over the course of a day. Methods: High-resolution reconstructions of field-grown plants are assembled in various configurations to form canopies, and a forward ray-tracing algorithm is applied to the canopies to compute light dynamics at high (1 min) temporal resolution. From the ray-tracer output, the sunlit or shaded state for each patch on the plants is determined, and these data are used to develop a novel stochastic model for the sunlit-shaded patterns. The model is designed to be straightforward to fit to data using maximum likelihood estimation, and fast to simulate from. Key Results: For a wide range of contrasting 3-D canopies, the stochastic model is able to summarize, and replicate in simulations, key features of the light dynamics. When light patterns simulated from the stochastic model are used as input to a model of photoinhibition, the predicted reduction in carbon gain is similar to that from calculations based on the (extremely costly) ray-tracer data. Conclusions: The model provides a way to summarize highly complex data in a small number of parameters, and a cost-effective way to simulate realistic light patterns. Simulations from the model will be particularly useful for feeding into larger-scale photosynthesis models for calculating how light dynamics affects the photosynthetic productivity of canopies.

SIDT2 Transports Extracellular dsRNA into the Cytoplasm for Innate Immune Recognition.

Double-stranded RNA (dsRNA) is a common by-product of viral infections and acts as a potent trigger of antiviral immunity. In the nematode C. elegans, sid-1 encodes a dsRNA transporter that is highly conserved throughout animal evolution, but the physiological role of SID-1 and its orthologs remains unclear. Here, we show that the mammalian SID-1 ortholog, SIDT2, is required to transport internalized extracellular dsRNA from endocytic compartments into the cytoplasm for immune activation. Sidt2-deficient mice exposed to extracellular dsRNA, encephalomyocarditis virus (EMCV), and herpes simplex virus 1 (HSV-1) show impaired production of antiviral cytokines and-in the case of EMCV and HSV-1-reduced survival. Thus, SIDT2 has retained the dsRNA transport activity of its C. elegans ortholog, and this transport is important for antiviral immunity.

The BCL-2 pro-survival protein A1 is dispensable for T cell homeostasis on viral infection.

The physiological role of the pro-survival BCL-2 family member A1 has been debated for a long time. Strong mRNA induction in T cells on T cell receptor (TCR)-engagement suggested a major role of A1 in the survival of activated T cells. However, the investigation of the physiological roles of A1 was complicated by the quadruplication of the A1 gene locus in mice, making A1 gene targeting very difficult. Here, we used the recently generated A1-/- mouse model to examine the role of A1 in T cell immunity. We confirmed rapid and strong induction of A1 protein in response to TCR/CD3 stimulation in CD4+ as well as CD8+ T cells. Surprisingly, on infection with the acute influenza HKx31 or the lymphocytic choriomeningitis virus docile strains mice lacking A1 did not show any impairment in the expansion, survival, or effector function of cytotoxic T cells. Furthermore, the ability of A1-/- mice to generate antigen-specific memory T cells or to provide adequate CD4-dependent help to B cells was not impaired. These results suggest functional redundancy of A1 with other pro-survival BCL-2 family members in the control of T cell-dependent immune responses.

The 4-Dimensional Plant: Effects of Wind-Induced Canopy Movement on Light Fluctuations and Photosynthesis.

Physical perturbation of a plant canopy brought about by wind is a ubiquitous phenomenon and yet its biological importance has often been overlooked. This is partly due to the complexity of the issue at hand: wind-induced movement (or mechanical excitation) is a stochastic process which is difficult to measure and quantify; plant motion is dependent upon canopy architectural features which, until recently, were difficult to accurately represent and model in 3-dimensions; light patterning throughout a canopy is difficult to compute at high-resolutions, especially when confounded by other environmental variables. Recent studies have reinforced the expectation that canopy architecture is a strong determinant of productivity and yield; however, links between the architectural properties of the plant and its mechanical properties, particularly its response to wind, are relatively unknown. As a result, biologically relevant data relating canopy architecture, light- dynamics, and short-scale photosynthetic responses in the canopy setting are scarce. Here, we hypothesize that wind-induced movement will have large consequences for the photosynthetic productivity of our crops due to its influence on light patterning. To address this issue, in this study we combined high resolution 3D reconstructions of a plant canopy with a simple representation of canopy perturbation as a result of wind using solid body rotation in order to explore the potential effects on light patterning, interception, and photosynthetic productivity. We looked at two different scenarios: firstly a constant distortion where a rice canopy was subject to a permanent distortion throughout the whole day; and secondly, a dynamic distortion, where the canopy was distorted in incremental steps between two extremes at set time points in the day. We find that mechanical canopy excitation substantially alters light dynamics; light distribution and modeled canopy carbon gain. We then discuss methods required for accurate modeling of mechanical canopy excitation (here coined the 4-dimensional plant) and some associated biological and applied implications of such techniques. We hypothesize that biomechanical plant properties are a specific adaptation to achieve wind-induced photosynthetic enhancement and we outline how traits facilitating canopy excitation could be used as a route for improving crop yield.

RUNX2 Mediates Plasmacytoid Dendritic Cell Egress from the Bone Marrow and Controls Viral Immunity.

Plasmacytoid dendritic cells (pDCs) represent a unique immune cell type that responds to viral nucleic acids through the rapid production of type I interferons. Within the hematopoietic system, the transcription factor RUNX2 is exclusively expressed in pDCs and is required for their peripheral homeostasis. Here, we show that RUNX2 plays an essential role in promoting pDC localization and function. RUNX2 is required for the appropriate expression of the integrin-mediated adhesion machinery, as well as for the down-modulation of the chemokine receptor CXCR4, which allows pDC egress into the circulation. RUNX2 also facilitates the robust response to viral infection through the control of IRF7, the major regulator of type I interferon production. Mice lacking one copy of Runx2 have reduced numbers of peripheral pDCs and IFN-α expression, which might contribute to the reported difficulties of individuals with cleidocranial dysplasia, who are haploinsufficient for RUNX2, to clear viral infections.

Hydrodynamic Injection as a Method of Gene Delivery in Mice: A Model of Chronic Hepatitis B Virus Infection.

Gene delivery methods are important for both therapeutic intervention and as tools in research to address specific questions. Hydrodynamic injection (HDI) is a method that facilitates the delivery and expression of genetic material in target cells, namely hepatocytes, through an intravenous injection. HDI has great utility for research involving cell death and signaling pathways essential in the processes of cancer, inflammation, and transplant therapy, as well as representing a valuable technique to establish hepatitis B virus (HBV) expression in hepatocytes. This chapter describes in detail how to generate a model of chronic HBV infection in immunocompetent mice using HDI as a delivery method.

High-Resolution Three-Dimensional Structural Data Quantify the Impact of Photoinhibition on Long-Term Carbon Gain in Wheat Canopies in the Field.

Photoinhibition reduces photosynthetic productivity; however, it is difficult to quantify accurately in complex canopies partly because of a lack of high-resolution structural data on plant canopy architecture, which determines complex fluctuations of light in space and time. Here, we evaluate the effects of photoinhibition on long-term carbon gain (over 1 d) in three different wheat (Triticum aestivum) lines, which are architecturally diverse. We use a unique method for accurate digital three-dimensional reconstruction of canopies growing in the field. The reconstruction method captures unique architectural differences between lines, such as leaf angle, curvature, and leaf density, thus providing a sensitive method of evaluating the productivity of actual canopy structures that previously were difficult or impossible to obtain. We show that complex data on light distribution can be automatically obtained without conventional manual measurements. We use a mathematical model of photosynthesis parameterized by field data consisting of chlorophyll fluorescence, light response curves of carbon dioxide assimilation, and manual confirmation of canopy architecture and light attenuation. Model simulations show that photoinhibition alone can result in substantial reduction in carbon gain, but this is highly dependent on exact canopy architecture and the diurnal dynamics of photoinhibition. The use of such highly realistic canopy reconstructions also allows us to conclude that even a moderate change in leaf angle in upper layers of the wheat canopy led to a large increase in the number of leaves in a severely light-limited state.

Exploiting heterogeneous environments: does photosynthetic acclimation optimize carbon gain in fluctuating light?

Plants have evolved complex mechanisms to balance the efficient use of absorbed light energy in photosynthesis with the capacity to use that energy in assimilation, so avoiding potential damage from excess light. This is particularly important under natural light, which can vary according to weather, solar movement and canopy movement. Photosynthetic acclimation is the means by which plants alter their leaf composition and structure over time to enhance photosynthetic efficiency and productivity. However there is no empirical or theoretical basis for understanding how leaves track historic light levels to determine acclimation status, or whether they do this accurately. We hypothesized that in fluctuating light (varying in both intensity and frequency), the light-response characteristics of a leaf should adjust (dynamically acclimate) to maximize daily carbon gain. Using a framework of mathematical modelling based on light-response curves, we have analysed carbon-gain dynamics under various light patterns. The objective was to develop new tools to quantify the precision with which photosynthesis acclimates according to the environment in which plants exist and to test this tool on existing data. We found an inverse relationship between the optimal maximum photosynthetic capacity and the frequency of low to high light transitions. Using experimental data from the literature we were able to show that the observed patterns for acclimation were consistent with a strategy towards maximizing daily carbon gain. Refinement of the model will further determine the precision of acclimation.

Platelet production proceeds independently of the intrinsic and extrinsic apoptosis pathways.

BH3 mimetic drugs that target BCL-2 family pro-survival proteins to induce tumour cell apoptosis represent a new era in cancer therapy. Clinical trials of navitoclax (ABT-263, which targets BCL-2, BCL-XL and BCL-W) have shown great promise, but encountered dose-limiting thrombocytopenia. Recent work has demonstrated that this is due to the inhibition of BCL-XL, which is essential for platelet survival. These findings raise new questions about the established model of platelet shedding by megakaryocytes, which is thought to be an apoptotic process. Here we generate mice with megakaryocyte-specific deletions of the essential mediators of extrinsic (Caspase-8) and intrinsic (BAK/BAX) apoptosis. We show that megakaryocytes possess a Fas ligand-inducible extrinsic apoptosis pathway. However, Fas activation does not stimulate platelet production, rather, it triggers Caspase-8-mediated killing. Combined loss of Caspase-8/BAK/BAX does not impair thrombopoiesis, but can protect megakaryocytes from death in mice infected with lymphocytic choriomeningitis virus. Thus, apoptosis is dispensable for platelet biogenesis.

Modelling the regulation of telomere length: the effects of telomerase and G-quadruplex stabilising drugs.

Telomeres are guanine-rich sequences at the end of chromosomes which shorten during each replication event and trigger cell cycle arrest and/or controlled death (apoptosis) when reaching a threshold length. The enzyme telomerase replenishes the ends of telomeres and thus prolongs the life span of cells, but also causes cellular immortalisation in human cancer. G-quadruplex (G4) stabilising drugs are a potential anticancer treatment which work by changing the molecular structure of telomeres to inhibit the activity of telomerase. We investigate the dynamics of telomere length in different conformational states, namely t-loops, G-quadruplex structures and those being elongated by telomerase. By formulating deterministic differential equation models we study the effects of various levels of both telomerase and concentrations of a G4-stabilising drug on the distribution of telomere lengths, and analyse how these effects evolve over large numbers of cell generations. As well as calculating numerical solutions, we use quasicontinuum methods to approximate the behaviour of the system over time, and predict the shape of the telomere length distribution. We find those telomerase and G4-concentrations where telomere length maintenance is successfully regulated. Excessively high levels of telomerase lead to continuous telomere lengthening, whereas large concentrations of the drug lead to progressive telomere erosion. Furthermore, our models predict a positively skewed distribution of telomere lengths, that is, telomeres accumulate over lengths shorter than the mean telomere length at equilibrium. Our model results for telomere length distributions of telomerase-positive cells in drug-free assays are in good agreement with the limited amount of experimental data available.