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.

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. Since 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 potentially 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 observation 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.

Continuous Effector CD8(+) T Cell Production in a Controlled Persistent Infection Is Sustained by a Proliferative Intermediate Population.

Highly functional CD8(+) effector T (Teff) cells can persist in large numbers during controlled persistent infections, as exemplified by rare HIV-infected individuals who control the virus. Here we examined the cellular mechanisms that maintain ongoing T effector responses using a mouse model for persistent Toxoplasma gondii infection. In mice expressing the protective MHC-I molecule, H-2L(d), a dominant T effector response against a single parasite antigen was maintained without a contraction phase, correlating with ongoing presentation of the dominant antigen. Large numbers of short-lived Teff cells were continuously produced via a proliferative, antigen-dependent intermediate (Tint) population with a memory-effector hybrid phenotype. During an acute, resolved infection, decreasing antigen load correlated with a sharp drop in the Tint cell population and subsequent loss of the ongoing effector response. Vaccination approaches aimed at the development of Tint populations might prove effective against pathogens that lead to chronic infection.

Some deterministic and stochastic mathematical models of naïve T-cell homeostasis.

Humans live for decades, whereas mice live for months. Over these long timescales, naïve T cells die or divide infrequently enough that it makes sense to approximate death and division as instantaneous events. The population of T cells in the body is naturally divided into clonotypes; a clonotype is the set of cells that have identical T-cell receptors. While total numbers of cells, such as naïve CD4+ T cells, are large enough that ordinary differential equations are an appropriate starting point for mathematical models, the numbers of cells per clonotype are not. Here, we review a number of basic mathematical models of the maintenance of clonal diversity. As well as deterministic models, we discuss stochastic models that explicitly track the integer number of naïve T cells in many competing clonotypes over the lifetime of a mouse or human, including the effect of waning thymic production. Experimental evaluation of clonal diversity by bulk high-throughput sequencing has many difficulties, but the use of single-cell sequencing is restricted to numbers of cells many orders of magnitude smaller than the total number of T cells in the body. Mathematical questions associated with extrapolating from small samples are therefore key to advances in understanding the diversity of the repertoire of T cells. We conclude with some mathematical models on how to advance in this area.

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.

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.

Role of genetic heterogeneity in determining the epidemiological severity of H1N1 influenza.

Genetic differences contribute to variations in the immune response mounted by different individuals to a pathogen. Such differential response can influence the spread of infectious disease, indicating why such diseases impact some populations more than others. Here, we study the impact of population-level genetic heterogeneity on the epidemic spread of different strains of H1N1 influenza. For a population with known HLA class-I allele frequency and for a given H1N1 viral strain, we classify individuals into sub-populations according to their level of susceptibility to infection. Our core hypothesis is that the susceptibility of a given individual to a disease such as H1N1 influenza is inversely proportional to the number of high affinity viral epitopes the individual can present. This number can be extracted from the HLA genetic profile of the individual. We use ethnicity-specific HLA class-I allele frequency data, together with genome sequences of various H1N1 viral strains, to obtain susceptibility sub-populations for 61 ethnicities and 81 viral strains isolated in 2009, as well as 85 strains isolated in other years. We incorporate these data into a multi-compartment SIR model to analyse the epidemic dynamics for these (ethnicity, viral strain) epidemic pairs. Our results show that HLA allele profiles which lead to a large spread in individual susceptibility values can act as a protective barrier against the spread of influenza. We predict that populations skewed such that a small number of highly susceptible individuals coexist with a large number of less susceptible ones, should exhibit smaller outbreaks than populations with the same average susceptibility but distributed more uniformly across individuals. Our model tracks some well-known qualitative trends of influenza spread worldwide, suggesting that HLA genetic diversity plays a crucial role in determining the spreading potential of different influenza viral strains across populations.

How many TCR clonotypes does a body maintain?

We consider the lifetime of a T cell clonotype, the set of T cells with the same T cell receptor, from its thymic origin to its extinction in a multiclonal repertoire. Using published estimates of total cell numbers and thymic production rates, we calculate the mean number of cells per TCR clonotype, and the total number of clonotypes, in mice and humans. When there is little peripheral division, as in a mouse, the number of cells per clonotype is small and governed by the number of cells with identical TCR that exit the thymus. In humans, peripheral division is important and a clonotype may survive for decades, during which it expands to comprise many cells. We therefore devise and analyse a computational model of homeostasis of a multiclonal population. Each T cell in the model competes for self pMHC stimuli, cells of any one clonotype only recognising a small fraction of the many subsets of stimuli. A constant mean total number of cells is maintained by a balance between cell division and death, and a stable number of clonotypes by a balance between thymic production of new clonotypes and extinction of existing ones. The number of distinct clonotypes in a human body may be smaller than the total number of naive T cells by only one order of magnitude.

How many dendritic cells are required to initiate a T-cell response?

T-cell activation in lymph nodes relies on encounters with antigen (Ag)-bearing dendritic cells (DCs) but the number of DCs required to initiate an immune response is unknown. Here we have used a combination of flow cytometry, 2-photon imaging, and computational modeling to quantify the probability of T cell-DC encounters. We calculated that the chance for a T cell residing 24 hours in a murine popliteal lymph nodes to interact with a DC was 8%, 58%, and 99% in the presence of 10, 100, and 1000 Ag-bearing DCs, respectively. Our results reveal the existence of a threshold in DC numbers below which T-cell responses fail to be elicited for probabilistic reasons. In mice and probably humans, we estimate that a minimum of 85 DCs are required to initiate a T-cell response when starting from precursor frequency of 10(-6). Our results have implications for the rational design of DC-based vaccines.

A mathematical perspective on CD4(+) T cell quorum-sensing.

We analyse a mathematical model of the peripheral CD4(+) T cell population, based on a quorum-sensing mechanism, by which an optimum number of regulatory T cells can be established and maintained. We divide the population of a single T cell receptor specificity into four pools: naive, IL-2 producing, IL-2 non-producing, and regulatory CD4(+) T cells. Proliferation, death and differentiation of cells are introduced as transition probabilities of a stochastic Markov model, with the assumption that the amount of IL-2 available to CD4(+) T cells is proportional to the size of the population of IL-2 producing CD4(+) T cells. We explore the population dynamics both in the absence and in the presence of specific antigen. We study the establishment of the peripheral CD4(+) T cell pool from thymic output in the absence of antigen, and its return to homeostasis after an immune challenge, by steady state analysis of the deterministic approximation. The number of regulatory T cells at steady state is greater in the presence of antigen than in its absence. We also consider the stochastic dynamics of the model after an immune challenge, in particular the behaviour leading to ultimate extinction of the IL-2 producing and regulatory T cell populations.

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.

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.

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.

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.

Proofreading of misincorporated nucleotides in DNA transcription.

The accuracy of DNA transcription is crucial for the proper functioning of the cell. Although RNA polymerases demonstrate selectivity for correct nucleotides, additional active mechanisms of transcriptional error correction are required to achieve observed levels of fidelity. Recent experimental findings have shed light on a particular mechanism of transcriptional error correction involving: (i) diffusive translocation of the RNA polymerase along the DNA (backtracking) and (ii) irreversible RNA cleavage. This mechanism achieves preferential cleavage of misincorporated nucleotides by biasing the local rates of translocation. Here, we study how misincorporated nucleotides affect backtracking dynamics and how this effect determines the level of transcriptional fidelity. We consider backtracking as a diffusive process in a periodic, one-dimensional energy landscape, which at a coarse-grained level gives rise to a hopping process between neighboring local minima. We propose a model for how misincorporated nucleotides deform this energy landscape and hence affect the hopping rates. In particular, we show that this model can be used to derive both the theoretical limit on the fidelity (i.e. the minimum fraction of misincorporated nucleotides) and the actual fidelity relative to this optimum, achieved for specific combinations of the cleavage and polymerization rates. Finally, we study how external factors influencing backtracking dynamics affect transcriptional fidelity. We show that biologically relevant loads, similar to those exerted by nucleosomes or other transcriptional barriers, increase error correction.

Proofreading of misincorporated nucleotides in DNA transcription.

The accuracy of DNA transcription is crucial for the proper functioning of the cell. Although RNA polymerases demonstrate selectivity for correct nucleotides, additional active mechanisms of transcriptional error correction are required to achieve observed levels of fidelity. Recent experimental findings have shed light on a particular mechanism of transcriptional error correction involving: (i) diffusive translocation of the RNA polymerase along the DNA (backtracking) and (ii) irreversible RNA cleavage. This mechanism achieves preferential cleavage of misincorporated nucleotides by biasing the local rates of translocation. Here, we study how misincorporated nucleotides affect backtracking dynamics and how this effect determines the level of transcriptional fidelity. We consider backtracking as a diffusive process in a periodic, one-dimensional energy landscape, which at a coarse-grained level gives rise to a hopping process between neighbouring local minima. We propose a model for how misincorporated nucleotides deform this energy landscape and hence affect the hopping rates. In particular, we show that this model can be used to derive both the theoretical limit on the fidelity (i.e. the minimum fraction of misincorporated nucleotides) and the actual fidelity relative to this optimum, achieved for specific combinations of the cleavage and polymerization rates. Finally, we study how external factors influencing backtracking dynamics affect transcriptional fidelity. We show that biologically relevant loads, similar to those exerted by nucleosomes or other transcriptional barriers, increase error correction.

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.

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.