Dynamics of immunological models.

We analyse the effect of the regulatory T cells (Tregs) in the local control of the immune responses by T cells. We obtain an explicit formula for the level of antigenic stimulation of T cells as a function of the concentration of T cells and the parameters of the model. The relation between the concentration of the T cells and the antigenic stimulation of T cells is an hysteresis, that is unfold for some parameter values. We study the appearance of autoimmunity from cross-reactivity between a pathogen and a self antigen or from bystander proliferation. We also study an asymmetry in the death rates. With this asymmetry we show that the antigenic stimulation of the Tregs is able to control locally the population size of Tregs. Other effects of this asymmetry are a faster immune response and an improvement in the simulations of the bystander proliferation. The rate of variation of the levels of antigenic stimulation determines if the outcome is an immune response or if Tregs are able to maintain control due to the presence of a transcritical bifurcation for some tuning between the antigenic stimuli of T cells and Tregs. This behavior is explained by the presence of a transcritical bifurcation.

On reverse engineering of gene interaction networks using time course data with repeated measurements.

MOTIVATION: Gene expression measurements are the most common data source for reverse engineering gene interaction networks. When dealing with destructive sampling in time course experiments, it is common to average any available measurements for each time point and to treat this as the actual time series data for fitting the network, neglecting the variability contained in the repeated measurements. Proceeding in such a way can affect the retrieved network topology. RESULTS: We propose a fully Bayesian method for reverse engineering a gene interaction network, based on time course data with repeated measurements. The observations are treated as surrogate measurements of the underlying gene expression. As these measurements often contain outliers, we use a non-Gaussian specification for dealing with measurement error. The network interactions are assumed linear and an autoregressive model is specified, augmented with indicator variables that allow inference on the topology of the network. We analyse two in silico and one in vivo experiments, the latter dealing with the circadian clock in Arabidopsis thaliana. A systematic attenuation of the estimated regulation strengths and a concomitant overestimation of their precision is demonstrated when measurement error is disregarded. Thus, a clear improvement in the inferred topology for the synthetic datasets is demonstrated when this is included. Also, the influence of outliers in the retrieved network is demonstrated when using the in vivo data. AVAILABILITY: Matlab code and data used in the article are available from http://go.warwick.ac.uk/majuarez/home/materials.

An improved strip FRAP method for estimating diffusion coefficients: correcting for the degree of photobleaching.

Fluorescence recovery after photobleaching is a widely established method for the estimation of diffusion coefficients, strip bleaching with an associated recovery curve analysis being one of the simplest techniques. However, its implementation requires near 100% bleaching in the region of interest with negligible fluorescence loss outside, both constraints being hard to achieve concomitantly for fast diffusing molecules. We demonstrate that when these requirements are not met there is an error in the estimation of the diffusion coefficient D, either an under- or overestimation depending on which assumption is violated the most. We propose a simple modification to the recovery curve analysis incorporating the concept of the relative bleached mass m giving a revised recovery time parametrization tau=m(2)w(2)/4piD for a strip of width w. This modified model removes the requirement of 100% bleaching in the region of interest and allows for limited diffusion of the fluorophore during bleaching. We validate our method by estimating the (volume) diffusion coefficient of FITC-labelled IgG in 60% glycerol solution, D= 4.09 +/- 0.21 microm(2) s(-1), and the (surface) diffusion coefficient of a green-fluorescent protein-tagged class I MHC protein expressed at the surface of a human B cell line, D= 0.32 +/- 0.03 microm(2) s(-1) for a population of cells.

Nonequilibrium-driven motion in actin networks: comet tails and moving beads.

We present 3D dynamic Monte-Carlo simulations of the growth of an actin network close to an obstacle coated with Wiskott-Aldrich syndrome protein (WASP), an inducer of actin branching. Our simulations incorporate both elasticity and relaxation of the actin tail, thus allowing for local network compression. Whilst steady state motility derives mainly from polymerization at the leading edge, nonthermal stored elastic energy and retrograde flow are observed in a thin slab of material close to the obstacle. We observe a crossover from steady to hopping bead motion as the branching rate is decreased.

A Markov chain Monte Carlo method for estimating population mixing using Y-chromosome markers: mixing of the Han people in China.

We present a new approach for estimating mixing between populations based on non-recombining markers, specifically Y-chromosome microsatellites. A Markov chain Monte Carlo (MCMC) Bayesian statistical approach is used to calculate the posterior probability distribution of population parameters of interest, including the effective population size and the time to most recent common ancestor (MRCA). To test whether two populations are homogeneously mixed we introduce a "mixing" statistic defined for each coalescent event that weights the contribution of that ancestor's descendants to the two subpopulations, and an associated population "purity" statistic. Using simulated data with low levels of migration between two populations, we demonstrate that our method is more sensitive than other commonly used distance-based methods such as R(ST) and D(SW). To illustrate our method, we analysed mixing between 11 pre-defined Chinese ethnic/regional populations, using 5 microsatellite markers from the non-recombining region of the Y-chromosome (NRY), demonstrating a significant clustering of a subset of subpopulations with a high mutual relative degree of mixing (homogeneous mixing with support >0.99). Our analysis suggests that there is a strong correlation between effective population size and mixing with other subpopulations. Thus, despite considerable mixing between these groups, the purity statistic still identifies significant heterogeneity, suggesting that periods of historical isolation continue to leave a recoverable signal despite modern introgression.

T-cell activation: A queuing theory analysis at low agonist density.

We analyze a simple linear triggering model of the T-cell receptor (TCR) within the framework of queuing theory, in which TCRs enter the queue upon full activation and exit by downregulation. We fit our model to four experimentally characterized threshold activation criteria and analyze their specificity and sensitivity: the initial calcium spike, cytotoxicity, immunological synapse formation, and cytokine secretion. Specificity characteristics improve as the time window for detection increases, saturating for time periods on the timescale of downregulation; thus, the calcium spike (30 s) has low specificity but a sensitivity to single-peptide MHC ligands, while the cytokine threshold (1 h) can distinguish ligands with a 30% variation in the complex lifetime. However, a robustness analysis shows that these properties are degraded when the queue parameters are subject to variation-for example, under stochasticity in the ligand number in the cell-cell interface and population variation in the cellular threshold. A time integration of the queue over a period of hours is shown to be able to control parameter noise efficiently for realistic parameter values when integrated over sufficiently long time periods (hours), the discrimination characteristics being determined by the TCR signal cascade kinetics (a kinetic proofreading scheme). Therefore, through a combination of thresholds and signal integration, a T cell can be responsive to low ligand density and specific to agonist quality. We suggest that multiple threshold mechanisms are employed to establish the conditions for efficient signal integration, i.e., coordinate the formation of a stable contact interface.

Regulatory T cell adjustment of quorum growth thresholds and the control of local immune responses.

The consequences of regulatory T cell (Treg) inhibition of interleukine 2 secretion is examined by mathematical modelling. We demonstrate that cytokine dependent growth exhibits a quorum T cell population threshold that determines if immune responses develop on activation. Secretion inhibition manipulates the growth dynamics and effectively increases the quorum threshold, i.e. to develop immune responses a higher number of T cells need to be activated. Thus Treg induced secretion inhibition can provide a mechanism for tissue specific regulation of the balance between suppression (control) and immune responses, a balance that can be varied at the local tissue level through the regulation of the local active Treg population size. However, nonspecific inhibition is prone to escape of initially controlled autoimmune T cells through cross reactivity to pathogens and bystander proliferation on unrelated immune responses.

Three-dimensional dynamic Monte Carlo simulations of elastic actin-like ratchets.

We present three-dimensional dynamic Monte Carlo simulations of the growth of a semiflexible fiber against a fluctuating obstacle. The natural reference for our numerical study are the elastic and Brownian ratchet models previously analyzed semianalytically. We find that the decay of the velocity versus applied load is exponential to a good degree of accuracy, provided we include in the load the drag force felt by the moving obstacle. If the fiber and obstacle only interact via excluded volume, there are small corrections to the Brownian ratchet predictions which suggest that tip fluctuations play a minor role. If on the other hand fiber and obstacle interact via a soft potential, the corrections are much larger when the obstacle diffuses slowly. This means that microscopic assumptions can profoundly affect the dynamics. We also identify and characterize a novel "pushing catastrophe"--which is distinct from the usual fiber buckling--in which the growth of the fiber decouples from the obstacle movement. The time distribution of catastrophes can be explained via an approximate analytical treatment, and our numerics suggest that the time taken to lose propulsive force is largely dependent on the fiber incidence angle. Our results are a first step in realizing numerical polymer models for the motion of sets or networks of semiflexible fibers close to a fluctuating membrane or obstacle.

Quantifying the strength of ligand antagonism in TCR triggering.

Triggering of the T cell receptor (TCR) may be antagonized by ligands that are slight variants of the immunogenic peptide. This paper proposes a mathematical model to quantify the strength of the antagonistic effect. The model is based on the kinetics of association and dissociation of TCR and peptide/major histocompatibility (pMHC) molecules, and incorporates TCR triggering according to a kinetic proofreading mechanism. Model analysis indicates that while the average lifetime of the TCR/pMHC complex is the basic determinant of the contribution to TCR triggering made by the ligand, the affinity of the ligand and its MHC presentation level are also important. However, these contributions depend on the kinetic limitation regime. There is a continuum of limitation regimes, at the extremes of which are found TCR limitation and MHC limitation. Both ligand affinity and TCR and pMHC densities determine whether TCR triggering is TCR limited or MHC limited. The changing importance of affinity and antigen presentation level under various kinetic limitation regimes may explain the respective roles of antagonistic and agonistic self peptides in thymic selection. Moreover, TCR down-regulation under TCR-limited conditions may allow the T cell to differentiate between the average lifetime of the TCR/pMHC complex and the presentation level of the ligand. A method for experimental differentiation between passive and active antagonistic effects is proposed which exploits the differences between TCR and MHC limitation.

TCR dynamics on the surface of living T cells.

T lymphocyte activation by specific antigen requires prolonged TCR occupancy and sustained signaling. This is accomplished by the formation of a specialized signaling domain, the immunological synapse, at the T cell-antigen-presenting cell contact site. Surface receptors and signaling components are progressively recruited into this domain where they are organized in defined three-dimensional structures. To better understand how TCR are supplied to the signaling domain during the activation process, we measured (using confocal microscopy and photo-bleaching recovery techniques) lateral mobility of GFP-tagged TCR on living Jurkat cell surface. We show that: (i) surface-expressed TCR exhibit an intrinsic, actin cytoskeleton-independent, lateral mobility which allows them to passively diffuse over the entire T cell surface within approximately 60 min and (ii) non-stimulated TCR rapidly enter the signaling domain. Our results indicate that TCR lateral mobility per se is sufficient to ensure TCR supply to the immunological synapse in the course of sustained T cell activation.

Stability of a diverse immunological memory is determined by T cell population dynamics.

The correlation between properties of the T cell memory pool and the two regulatory mechanisms of cell death (apoptosis) and memory entry (differentiation) is investigated mathematically. Apoptosis of T cells occurs at the end of an immune response, removing unwanted activated T cells. T cells escaping apoptosis enter the memory pool composed of T cells specific for previously encountered antigens. We find that the relative efficiencies of these two pathways determine the clonal distribution and the long-term stability of the memory pool by regulating the number of new entries. The main result presented in this paper is that immunological memory of previously encountered pathogens cannot be erased by either severe or repeat infections with a particular pathogen (the diversity of the memory pool is ensured) only if apoptosis and/or memory differentiation are regulated by population dependent processes. Furthermore, vaccination properties are improved significantly by population dependent mechanisms and our mathematical analysis reveals that the T cell population must communicate with other parts of the immune system to ensure optimal performance of immunological memory.

Long-term stability of diverse immunological memory.

Mathematical modelling is used to examine the effects of T cell apoptosis and memory differentiation dynamics on memory retention (memory stability). Apoptosis by cytokine deprivation induced death, and competition for survival signals are incorporated. Our models indicate that such population dependent processes are essential for the preservation of specific memory to previously encountered pathogens. Memory stability and vaccination properties improve significantly when population dependent mechanisms are present. These mechanisms work synergistically together to further improve memory characteristics. Our models emphasize the role of competitive cellular mechanisms in regulating repertoire structure and characteristics.

A reliable and safe T cell repertoire based on low-affinity T cell receptors.

Antigens are presented to T cells as short peptides bound to MHC molecules on the surface of body cells. The binding between MHC/peptides and T cell receptors (TCRs) has a low affinity and is highly degenerate. Nevertheless, TCR-MHC/peptide recognition results in T cell activation of high specificity. Moreover, the immune system is able to mount a cellular response when only a small fraction of the MHC molecules on an antigen-presenting cell is occupied by foreign peptides, while autoimmunity remains relatively rare. We consider how to reconcile these seemingly contradictory facts using a quantitative model of TCR signalling and T cell activation. Taking into account the statistics of TCR recognition and antigen presentation, we show that thymic selection can produce a working T cell repertoire which will produce safe and effective responses, that is, recognizes foreign antigen presented at physiological levels while tolerating self. We introduce "activation curves" as a useful tool to study the repertoire's statistical activation properties.

Mathematical analysis of growth and interaction dynamics of streptomycetes and a bacteriophage in soil.

We observed the infection cycle of the temperate actinophage KC301 in relation to the growth of its host Streptomyces lividans TK24 in sterile soil microcosms. Despite a large increase in phage population following germination of host spores, there was no observable impact on host population numbers as measured by direct plate counts. The only change in the host population following infection was the establishment of a small subpopulation of KC301 lysogens. The interaction of S. lividans and KC301 in soil was analyzed with a population-dynamic mathematical model to determine the underlying mechanisms of this low susceptibility to phage attack relative to aquatic environments. This analysis suggests that the soil environment is a highly significant component of the phage-host interaction, an idea consistent with earlier observations on the importance of the environment in determining host growth and phage-host dynamics. Our results demonstrate that the accepted phage-host interaction and host life cycle, as determined from agar plate studies and liquid culture, is sufficient for quantitative agreement with observations in soil, using soil-determined rates. There are four significant effects of the soil environment: (i) newly germinated spores are more susceptible to phage lysis than are hyphae of developed mycelia, (ii) substrate mycelia in mature colonies adsorb about 98% of the total phage protecting susceptible young hyphae from infection, (iii) the burst size of KC301 is large in soil (>150, 90% confidence) relative to that observed in liquid culture (120, standard error of the mean [SEM], 6), and (iv) there is no measurable impact on the host in terms of reduced growth by the phage. We hypothesize that spatial heterogeneity is the principal cause of these effects and is the primary determinant in bacterial escape of phage lysis in soil.

Markov chain Monte Carlo analysis of human Y-chromosome microsatellites provides evidence of biased mutation.

We describe a Markov Chain Monte Carlo analysis of five human Y- chromosome microsatellite polymorphisms based on samples from five diverse populations. Our analysis provides strong evidence for mutational bias favoring increase in length at all loci. Estimates of population coalescent times and population size from our two largest samples, one African and one European, suggest that the African population is older but smaller and that the English East Anglian population has undergone significant expansion, being larger but younger. We conclude that Markov Chain Monte Carlo analysis of microsatellite haplotypes can uncover information not apparent when the microsatellites are considered independently. Incorporation of population size as a variable should allow us to estimate the timing and magnitude of major historical population trends.

Significance testing of clinical data using virus dynamics models with a Markov chain Monte Carlo method: application to emergence of lamivudine-resistant hepatitis B virus.

Bayesian analysis using a virus dynamics model is demonstrated to facilitate hypothesis testing of patterns in clinical time-series. Our Markov chain Monte Carlo implementation demonstrates that the viraemia time-series observed in two sets of hepatitis B patients on antiviral (lamivudine) therapy, chronic carriers and liver transplant patients, are significantly different, overcoming clinical trial design differences that question the validity of non-parametric tests. We show that lamivudine-resistant mutants grow faster in transplant patients than in chronic carriers, which probably explains the differences in emergence times and failure rates between these two sets of patients. Incorporation of dynamic models into Bayesian parameter analysis is of general applicability in medical statistics.

Capping and methylation of mRNA by purified recombinant VP4 protein of bluetongue virus.

The core of bluetongue virus (BTV) is a multienzyme complex composed of two major proteins (VP7 and VP3) and three minor proteins (VP1, VP4, and VP6) in addition to the viral genome. The core is transcriptionally active and produces capped mRNA from which all BTV proteins are translated, but the relative role of each core component in the overall reaction process remains unclear. Previously we showed that the 76-kDa VP4 protein possesses guanylyltransferase activity, a necessary part of the RNA capping reaction. Here, through the use of highly purified (>95%) VP4 and synthetic core-like particles containing VP4, we have investigated the extent to which this protein is also responsible for other activities associated with cap formation. We show that VP4 catalyzes the conversion of unmethylated GpppG or in vitro-produced uncapped BTV RNA transcripts to m7GpppGm in the presence of S-adenosyl-L-methionine. Analysis of the methylated products of the reaction by HPLC identified both methyltransferase type 1 and type 2 activities associated with VP4, demonstrating that the complete BTV capping reaction is associated with this one protein.

Dynamics of T-cell antagonism: enhanced viral diversity and survival.

In rapidly evolving viruses the detection of virally infected cells can possibly be subverted by the production of altered peptides. There are peptides with single amino acid changes that can dramatically change T-cell responses, e.g. a loss of cytotoxic activity. They are still recognized by the T cell, but the signals required for effector function are only partially delivered. Thus, altered peptide presenting cells can act as decoy targets for specific immune responses. The existence of altered peptides in vivo has been demonstrated in hepatitis B and HIV. Using a mathematical model we address the question of how these altered peptides can affect the virus-immune system dynamics, and demonstrate that virus survival is enhanced. If the mutation rate of the virus is sufficient, one observes complex dynamics in which the antagonism acts so as to maintain the viral diversity, possibly leading to the development of a mutually antagonistic network or a continual turnover of escape mutants. In either case the pathogen is able to outrun the immune system. Indeed, sometimes the enhancement is so great that a virus that would normally be cleared by the immune system is able to outrun it.