The number and pattern of viral genomic reassortments are not necessarily identifiable from segment trees.

Reassortment is an evolutionary process common in viruses with segmented genomes. These viruses can swap whole genomic segments during cellular co-infection, giving rise to novel progeny formed from the mixture of parental segments. Because large-scale genome rearrangements have the potential to generate new phenotypes, reassortment is important to both evolutionary biology and public health research. However, statistical inference of the pattern of reassortment events from phylogenetic data is exceptionally difficult, potentially involving inference of general graphs in which individual segment trees are embedded. In this paper, we argue that, in general, the number and pattern of reassortment events are not identifiable from segment trees alone, even with theoretically ideal data. We call this fact the fundamental problem of reassortment, which we illustrate using the concept of the `first-infection tree', a typically but not always counterfactual genealogy that would have been observed in the segment trees had no reassortment occurred. Further, we illustrate four additional problems that can arise logically in the inference of reassortment events and show, using simulated data, that these problems are not rare and can potentially distort our perception of reassortment even in small data sets. Finally, we discuss how existing methods can be augmented or adapted to account for not only the fundamental problem of reassortment but also the four additional situations that can complicate the inference of reassortment.

A story of viral co-infection, co-transmission and co-feeding in ticks: how to compute an invasion reproduction number.

With a single circulating vector-borne virus, the basic reproduction number incorporates contributions from tick-to-tick (co-feeding), tick-to-host and host-to-tick transmission routes. With two different circulating vector-borne viral strains, resident and invasive, and under the assumption that co-feeding is the only transmission route in a tick population, the invasion reproduction number depends on whether the model system of ordinary differential equations possesses the property of neutrality. We show that a simple model, with two populations of ticks infected with one strain, resident or invasive, and one population of co-infected ticks, does not have Alizon's neutrality property. We present model alternatives that are capable of representing the invasion potential of a novel strain by including populations of ticks dually infected with the same strain. The invasion reproduction number is analysed with the next-generation method and via numerical simulations.

The reproduction number and its probability distribution for stochastic viral dynamics.

We consider stochastic models of individual infected cells. The reproduction number, R, is understood as a random variable representing the number of new cells infected by one initial infected cell in an otherwise susceptible (target cell) population. Variability in R results partly from heterogeneity in the viral burst size (the number of viral progeny generated from an infected cell during its lifetime), which depends on the distribution of cellular lifetimes and on the mechanism of virion release. We analyse viral dynamics models with an eclipse phase: the period of time after a cell is infected but before it is capable of releasing virions. The duration of the eclipse, or the subsequent infectious, phase is non-exponential, but composed of stages. We derive the probability distribution of the reproduction number for these viral dynamics models, and show it is a negative binomial distribution in the case of constant viral release from infectious cells, and under the assumption of an excess of target cells. In a deterministic model, the ultimate in-host establishment or extinction of the viral infection depends entirely on whether the mean reproduction number is greater than, or less than, one, respectively. Here, the probability of extinction is determined by the probability distribution of R, not simply its mean value. In particular, we show that in some cases the probability of infection is not an increasing function of the mean reproduction number.

Multi-variate model of T cell clonotype competition and homeostasis.

Diversity of the naive T cell repertoire is maintained by competition for stimuli provided by self-peptides bound to major histocompatibility complexes (self-pMHCs). We extend an existing bi-variate competition model to a multi-variate model of the dynamics of multiple T cell clonotypes which share stimuli. In order to understand the late-time behaviour of the system, we analyse: (i) the dynamics until the extinction of the first clonotype, (ii) the time to the first extinction event, (iii) the probability of extinction of each clonotype, and (iv) the size of the surviving clonotypes when the first extinction event takes place. We also find the probability distribution of the number of cell divisions per clonotype before its extinction. The mean size of a new clonotype at quasi-steady state is an increasing function of the stimulus available to it, and a decreasing function of the fraction of stimuli it shares with other clonotypes. Thus, the probability of, and time to, extinction of a new clonotype entering the pool of T cell clonotypes is determined by the extent of competition for stimuli it experiences and by its initial number of cells.

Last in first out: SIV proviruses seeded later in infection are harbored in short-lived CD4+ T cells.

HIV can persist in a latent form as integrated DNA (provirus) in resting CD4+ T cells of infected individuals and as such is unaffected by antiretroviral therapy (ART). Despite being a major obstacle for eradication efforts, the genetic variation and timing of formation of this latent reservoir remains poorly understood. Previous studies on when virus is deposited in the latent reservoir have come to contradictory conclusions. To reexamine the genetic variation of HIV in CD4+ T cells during ART, we determined the divergence in envelope sequences collected from 10 SIV infected rhesus macaques. We found that the macaques displayed a biphasic decline of the viral divergence over time, where the first phase lasted for an average of 11.6 weeks (range 4-28 weeks). Motivated by recent observations that the HIV-infected CD4+ T cell population is composed of short- and long-lived subsets, we developed a model to study the divergence dynamics. We found that SIV in short-lived cells was on average more diverged, while long-lived cells harbored less diverged virus. This suggests that the long-lived cells harbor virus deposited starting earlier in infection and continuing throughout infection, while short-lived cells predominantly harbor more recent virus. As these cell populations decayed, the overall proviral divergence decline matched that observed in the empirical data. This model explains previous seemingly contradictory results on the timing of virus deposition into the latent reservoir, and should provide guidance for future eradication efforts.

Modeling the evolution of segment trees reveals deficiencies in current inferential methods for genomic reassortment.

Reassortment is an evolutionary process common in viruses with segmented genomes. These viruses can swap whole genomic segments during cellular co-infection, giving rise to new viral variants. Large-scale genome rearrangements, such as reassortment, have the potential to quickly generate new phenotypes, making the understanding of viral reassortment important to both evolutionary biology and public health research. In this paper, we argue that reassortment cannot be reliably inferred from incongruities between segment phylogenies using the established remove-and-rejoin or coalescent approaches. We instead show that reassortment must be considered in the context of a broader population process that includes the dynamics of the infected hosts. Using illustrative examples and simulation we identify four types of evolutionary events that are difficult or impossible to reconstruct with incongruence-based methods. Further, we show that these specific situations are very common and will likely occur even in small samples. Finally, we argue that existing methods can be augmented or modified to account for all the problematic situations that we identify in this paper. Robust assessment of the role of reassortment in viral evolution is difficult, and we hope to provide conceptual clarity on some important methodological issues that can arise in the development of the next generation of tools for studying reassortment.

Antiretroviral therapy reveals triphasic decay of intact SIV genomes and persistence of ancestral variants.

The decay kinetics of HIV-1-infected cells are critical to understand virus persistence. We evaluated the frequency of simian immunodeficiency virus (SIV)-infected cells for 4 years of antiretroviral therapy (ART). The intact proviral DNA assay (IPDA) and an assay for hypermutated proviruses revealed short- and long-term infected cell dynamics in macaques starting ART ∼1 year after infection. Intact SIV genomes in circulating CD4+T cells showed triphasic decay with an initial phase slower than the decay of the plasma virus, a second phase faster than the second phase decay of intact HIV-1, and a stable third phase reached after 1.6-2.9 years. Hypermutated proviruses showed bi- or mono-phasic decay, reflecting different selective pressures. Viruses replicating at ART initiation had mutations conferring antibody escape. With time on ART, viruses with fewer mutations became more prominent, reflecting decay of variants replicating at ART initiation. Collectively, these findings confirm ART efficacy and indicate that cells enter the reservoir throughout untreated infection.

Why are cell populations maintained via multiple compartments?

We consider the maintenance of 'product' cell populations from 'progenitor' cells via a sequence of one or more cell types, or compartments, where each cell's fate is chosen stochastically. If there is only one compartment then large amplification, that is, a large ratio of product cells to progenitors comes with disadvantages. The product cell population is dominated by large families (cells descended from the same progenitor) and many generations separate, on average, product cells from progenitors. These disadvantages are avoided using suitably constructed sequences of compartments: the amplification factor of a sequence is the product of the amplification factors of each compartment, while the average number of generations is a sum over contributions from each compartment. Passing through multiple compartments is, in fact, an efficient way to maintain a product cell population from a small flux of progenitors, avoiding excessive clonality and minimizing the number of rounds of division en route. We use division, exit and death rates, estimated from measurements of single-positive thymocytes, to choose illustrative parameter values in the single-compartment case. We also consider a five-compartment model of thymocyte differentiation, from double-negative precursors to single-positive product cells.

Counting generations in birth and death processes with competing Erlang and exponential waiting times.

Lymphocyte populations, stimulated in vitro or in vivo, grow as cells divide. Stochastic models are appropriate because some cells undergo multiple rounds of division, some die, and others of the same type in the same conditions do not divide at all. If individual cells behave independently, then each cell can be imagined as sampling from a probability density of times to division and death. The exponential density is the most mathematically and computationally convenient choice. It has the advantage of satisfying the memoryless property, consistent with a Markov process, but it overestimates the probability of short division times. With the aim of preserving the advantages of a Markovian framework while improving the representation of experimentally-observed division times, we consider a multi-stage model of cellular division and death. We use Erlang-distributed (or, more generally, phase-type distributed) times to division, and exponentially distributed times to death. We classify cells into generations, using the rule that the daughters of cells in generation n are in generation [Formula: see text]. In some circumstances, our representation is equivalent to established models of lymphocyte dynamics. We find the growth rate of the cell population by calculating the proportions of cells by stage and generation. The exponent describing the late-time cell population growth, and the criterion for extinction of the population, differs from what would be expected if N steps with rate [Formula: see text] were equivalent to a single step of rate [Formula: see text]. We link with a published experimental dataset, where cell counts were reported after T cells were transferred to lymphopenic mice, using Approximate Bayesian Computation. In the comparison, the death rate is assumed to be proportional to the generation and the Erlang time to division for generation 0 is allowed to differ from that of subsequent generations. The multi-stage representation is preferred to a simple exponential in posterior distributions, and the mean time to first division is estimated to be longer than the mean time to subsequent divisions.

Monitoring VEGF-Stimulated Calcium Ion Flux in Endothelial Cells.

The endothelial response to vascular endothelial growth factor A (VEGF-A) regulates many aspects of animal physiology in health and disease. Such VEGF-A-regulated phenomena include vasculogenesis, angiogenesis, tumor growth and progression. VEGF-A binding to receptor tyrosine kinases such as vascular endothelial growth factor receptor 2 (VEGFR2 ) activates multiple signal transduction pathways and changes in homeostasis, metabolism, gene expression, cell proliferation, migration, and survival. One such VEGF-A-regulated response is a rapid rise in cytosolic calcium ion levels which modulates different biochemical events and impacts on endothelial-specific responses. Here, we present a series of detailed and robust protocols for evaluating ligand-stimulated cytosolic calcium ion flux in endothelial cells. By monitoring an endogenous endothelial transcription factor (NFATc2 ) which displays calcium-sensitive redistribution, we can assess the relevance of cytosolic calcium to protein function. This protocol can be easily applied to both adherent and non-adherent cultured cells to evaluate calcium ion flux in response to exogenous stimuli such as VEGF-A.

Receptor tyrosine kinases regulate signal transduction through a liquid-liquid phase separated state.

The recruitment of signaling proteins into activated receptor tyrosine kinases (RTKs) to produce rapid, high-fidelity downstream response is exposed to the ambiguity of random diffusion to the target site. Liquid-liquid phase separation (LLPS) overcomes this by providing elevated, localized concentrations of the required proteins while impeding competitor ligands. Here, we show a subset of phosphorylation-dependent RTK-mediated LLPS states. We then investigate the formation of phase-separated droplets comprising a ternary complex including the RTK, (FGFR2); the phosphatase, SHP2; and the phospholipase, PLCγ1, which assembles in response to receptor phosphorylation. SHP2 and activated PLCγ1 interact through their tandem SH2 domains via a previously undescribed interface. The complex of FGFR2 and SHP2 combines kinase and phosphatase activities to control the phosphorylation state of the assembly while providing a scaffold for active PLCγ1 to facilitate access to its plasma membrane substrate. Thus, LLPS modulates RTK signaling, with potential consequences for therapeutic intervention.

Pre-existing humoral immunity to human common cold coronaviruses negatively impacts the protective SARS-CoV-2 antibody response.

SARS-CoV-2 infection causes diverse outcomes ranging from asymptomatic infection to respiratory distress and death. A major unresolved question is whether prior immunity to endemic, human common cold coronaviruses (hCCCoVs) impacts susceptibility to SARS-CoV-2 infection or immunity following infection and vaccination. Therefore, we analyzed samples from the same individuals before and after SARS-CoV-2 infection or vaccination. We found hCCCoV antibody levels increase after SARS-CoV-2 exposure, demonstrating cross-reactivity. However, a case-control study indicates that baseline hCCCoV antibody levels are not associated with protection against SARS-CoV-2 infection. Rather, higher magnitudes of pre-existing betacoronavirus antibodies correlate with more SARS-CoV-2 antibodies following infection, an indicator of greater disease severity. Additionally, immunization with hCCCoV spike proteins before SARS-CoV-2 immunization impedes the generation of SARS-CoV-2-neutralizing antibodies in mice. Together, these data suggest that pre-existing hCCCoV antibodies hinder SARS-CoV-2 antibody-based immunity following infection and provide insight on how pre-existing coronavirus immunity impacts SARS-CoV-2 infection, which is critical considering emerging variants.

Quantification of Type I Interferon Inhibition by Viral Proteins: Ebola Virus as a Case Study.

Type I interferons (IFNs) are cytokines with both antiviral properties and protective roles in innate immune responses to viral infection. They induce an antiviral cellular state and link innate and adaptive immune responses. Yet, viruses have evolved different strategies to inhibit such host responses. One of them is the existence of viral proteins which subvert type I IFN responses to allow quick and successful viral replication, thus, sustaining the infection within a host. We propose mathematical models to characterise the intra-cellular mechanisms involved in viral protein antagonism of type I IFN responses, and compare three different molecular inhibition strategies. We study the Ebola viral protein, VP35, with this mathematical approach. Approximate Bayesian computation sequential Monte Carlo, together with experimental data and the mathematical models proposed, are used to perform model calibration, as well as model selection of the different hypotheses considered. Finally, we assess if model parameters are identifiable and discuss how such identifiability can be improved with new experimental data.

A Stochastic Intracellular Model of Anthrax Infection With Spore Germination Heterogeneity.

We present a stochastic mathematical model of the intracellular infection dynamics of Bacillus anthracis in macrophages. Following inhalation of B. anthracis spores, these are ingested by alveolar phagocytes. Ingested spores then begin to germinate and divide intracellularly. This can lead to the eventual death of the host cell and the extracellular release of bacterial progeny. Some macrophages successfully eliminate the intracellular bacteria and will recover. Here, a stochastic birth-and-death process with catastrophe is proposed, which includes the mechanism of spore germination and maturation of B. anthracis. The resulting model is used to explore the potential for heterogeneity in the spore germination rate, with the consideration of two extreme cases for the rate distribution: continuous Gaussian and discrete Bernoulli. We make use of approximate Bayesian computation to calibrate our model using experimental measurements from in vitro infection of murine peritoneal macrophages with spores of the Sterne 34F2 strain of B. anthracis. The calibrated stochastic model allows us to compute the probability of rupture, mean time to rupture, and rupture size distribution, of a macrophage that has been infected with one spore. We also obtain the mean spore and bacterial loads over time for a population of cells, each assumed to be initially infected with a single spore. Our results support the existence of significant heterogeneity in the germination rate, with a subset of spores expected to germinate much later than the majority. Furthermore, in agreement with experimental evidence, our results suggest that most of the spores taken up by macrophages are likely to be eliminated by the host cell, but a few germinated spores may survive phagocytosis and lead to the death of the infected cell. Finally, we discuss how this stochastic modelling approach, together with dose-response data, allows us to quantify and predict individual infection risk following exposure.

Quantifying T Cell Cross-Reactivity: Influenza and Coronaviruses.

If viral strains are sufficiently similar in their immunodominant epitopes, then populations of cross-reactive T cells may be boosted by exposure to one strain and provide protection against infection by another at a later date. This type of pre-existing immunity may be important in the adaptive immune response to influenza and to coronaviruses. Patterns of recognition of epitopes by T cell clonotypes (a set of cells sharing the same T cell receptor) are represented as edges on a bipartite network. We describe different methods of constructing bipartite networks that exhibit cross-reactivity, and the dynamics of the T cell repertoire in conditions of homeostasis, infection and re-infection. Cross-reactivity may arise simply by chance, or because immunodominant epitopes of different strains are structurally similar. We introduce a circular space of epitopes, so that T cell cross-reactivity is a quantitative measure of the overlap between clonotypes that recognize similar (that is, close in epitope space) epitopes.

Competitive binding of STATs to receptor phospho-Tyr motifs accounts for altered cytokine responses.

Cytokines elicit pleiotropic and non-redundant activities despite strong overlap in their usage of receptors, JAKs and STATs molecules. We use IL-6 and IL-27 to ask how two cytokines activating the same signaling pathway have different biological roles. We found that IL-27 induces more sustained STAT1 phosphorylation than IL-6, with the two cytokines inducing comparable levels of STAT3 phosphorylation. Mathematical and statistical modeling of IL-6 and IL-27 signaling identified STAT3 binding to GP130, and STAT1 binding to IL-27Rα, as the main dynamical processes contributing to sustained pSTAT1 levels by IL-27. Mutation of Tyr613 on IL-27Rα decreased IL-27-induced STAT1 phosphorylation by 80% but had limited effect on STAT3 phosphorgylation. Strong receptor/STAT coupling by IL-27 initiated a unique gene expression program, which required sustained STAT1 phosphorylation and IRF1 expression and was enriched in classical Interferon Stimulated Genes. Interestingly, the STAT/receptor coupling exhibited by IL-6/IL-27 was altered in patients with systemic lupus erythematosus (SLE). IL-6/IL-27 induced a more potent STAT1 activation in SLE patients than in healthy controls, which correlated with higher STAT1 expression in these patients. Partial inhibition of JAK activation by sub-saturating doses of Tofacitinib specifically lowered the levels of STAT1 activation by IL-6. Our data show that receptor and STATs concentrations critically contribute to shape cytokine responses and generate functional pleiotropy in health and disease.

Fusion and fission events regulate endosome maturation and viral escape.

Endosomes are intracellular vesicles that mediate the communication of the cell with its extracellular environment. They are an essential part of the cell's machinery regulating intracellular trafficking via the endocytic pathway. Many viruses, which in order to replicate require a host cell, attach themselves to the cellular membrane; an event which usually initiates uptake of a viral particle through the endocytic pathway. In this way viruses hijack endosomes for their journey towards intracellular sites of replication and avoid degradation without host detection by escaping the endosomal compartment. Recent experimental techniques have defined the role of endosomal maturation in the ability of enveloped viruses to release their genetic material into the cytoplasm. Endosome maturation depends on a family of small hydrolase enzymes (or GTPases) called Rab proteins, arranged on the cytoplasmic surface of its membrane. Here, we model endosomes as intracellular compartments described by two variables (its levels of active Rab5 and Rab7 proteins) and which can undergo coagulation (or fusion) and fragmentation (or fission). The key element in our approach is the "per-cell endosomal distribution" and its dynamical (Boltzmann) equation. The Boltzmann equation allows us to derive the dynamics of the total number of endosomes in a cell, as well as the mean and the standard deviation of its active Rab5 and Rab7 levels. We compare our mathematical results with experiments of Dengue viral escape from endosomes. The relationship between endosomal active Rab levels and pH suggests a mechanism that can account for the observed variability in viral escape times, which in turn regulate the viability of a viral intracellular infection.

Quantification of Ebola virus replication kinetics in vitro.

Mathematical modelling has successfully been used to provide quantitative descriptions of many viral infections, but for the Ebola virus, which requires biosafety level 4 facilities for experimentation, modelling can play a crucial role. Ebola virus modelling efforts have primarily focused on in vivo virus kinetics, e.g., in animal models, to aid the development of antivirals and vaccines. But, thus far, these studies have not yielded a detailed specification of the infection cycle, which could provide a foundational description of the virus kinetics and thus a deeper understanding of their clinical manifestation. Here, we obtain a diverse experimental data set of the Ebola virus infection in vitro, and then make use of Bayesian inference methods to fully identify parameters in a mathematical model of the infection. Our results provide insights into the distribution of time an infected cell spends in the eclipse phase (the period between infection and the start of virus production), as well as the rate at which infectious virions lose infectivity. We suggest how these results can be used in future models to describe co-infection with defective interfering particles, which are an emerging alternative therapeutic.

The Cytokine Receptor IL-7Rα Impairs IL-2 Receptor Signaling and Constrains the In Vitro Differentiation of Foxp3+ Treg Cells.

IL-7 receptor signaling is essential for the generation and maintenance of conventional T cells. Immunosuppressive Foxp3+ Treg cells, however, express uniquely low amounts of the IL-7-proprietary IL-7Rα so that they are impaired in IL-7 signaling. Because Treg cells depend on IL-2, the loss of IL-7Rα has been considered irrelevant for Treg cells. In contrast, here, we report that IL-7Rα downregulation is necessary to maximize IL-2R signaling. Although IL-7Rα overexpression promoted IL-7 signaling, unexpectedly, IL-2 signaling was suppressed in the same cells. Mechanistically, we found that γc, which is a receptor subunit shared by IL-7R and IL-2R, directly binds and pre-associates with IL-7Rα, thus limiting its availability for IL-2R binding. Consequently, overexpression of signaling-deficient, tailless IL-7Rα proteins inhibited IL-2R signaling, demonstrating that IL-7Rα sequesters γc and suppresses IL-2R signaling by extracellular interactions. Collectively, these results reveal a previously unappreciated regulatory mechanism of IL-2 receptor signaling that is governed by IL-7Rα abundance.

Stochastic dynamics of Francisella tularensis infection and replication.

We study the pathogenesis of Francisella tularensis infection with an experimental mouse model, agent-based computation and mathematical analysis. Following inhalational exposure to Francisella tularensis SCHU S4, a small initial number of bacteria enter lung host cells and proliferate inside them, eventually destroying the host cell and releasing numerous copies that infect other cells. Our analysis of disease progression is based on a stochastic model of a population of infectious agents inside one host cell, extending the birth-and-death process by the occurrence of catastrophes: cell rupture events that affect all bacteria in a cell simultaneously. Closed expressions are obtained for the survival function of an infected cell, the number of bacteria released as a function of time after infection, and the total bacterial load. We compare our mathematical analysis with the results of agent-based computation and, making use of approximate Bayesian statistical inference, with experimental measurements carried out after murine aerosol infection with the virulent SCHU S4 strain of the bacterium Francisella tularensis, that infects alveolar macrophages. The posterior distribution of the rate of replication of intracellular bacteria is consistent with the estimate that the time between rounds of bacterial division is less than 6 hours in vivo.