Estimating production and loss rates of cell populations is essential but difficult. The current state-of-the-art method to estimate these rates involves mathematical modelling of deuterium labelling experiments. Current models typically assume that the labelling in the precursors of the population of interest (POI) is proportional to the deuterium enrichment in body water/glucose. This assumption is not always true and it is known that this can have a significant effect on the rates estimated from labelling experiments. Here we quantify the effect that different turnover (replacement) rates of the precursors can have on the estimated proliferation and loss rates of a POI by explicitly modelling the dynamics of the precursors. We first confirm earlier results that the labelling curve of the POI only reflects its own turnover rate if either the turnover rate of the precursors is sufficiently fast, or the contribution from the precursors is sufficiently small. Next, we describe three realistic scenarios with a slowly turning over precursor population, and show how this changes the interpretation of the different parameter estimates. Our analyses underpin that uniquely identifying the turnover rate of a POI requires measurements (or knowledge) on the turnover of its immediate precursors.

Successful mathematical modeling of biological processes relies on the expertise of the modeler to capture the essential mechanisms in the process at hand and on the ability to extract useful information from empirical data. A model is said to be structurally unidentifiable, if different quantitative sets of parameters provide the same observable outcome. This is typical (but not exclusive) of partially observed problems in which only a few variables can be experimentally measured. Most of the available methods to test the structural identifiability of a model are either too complex mathematically for the general practitioner to be applied, or require involved calculations or numerical computation for complex non-linear models. In this work, we present a new analytical method to test structural identifiability of models based on ordinary differential equations, based on the invariance of the equations under the scaling transformation of its parameters. The method is based on rigorous mathematical results but it is easy and quick to apply, even to test the identifiability of sophisticated highly non-linear models. We illustrate our method by example and compare its performance with other existing methods in the literature.

The production of anticompetitor toxins is widespread among bacteria.Because production of such toxins is costly, it is typically regulated.In particular, many toxins are produced only when the local cell density is high.It is unclear which selection pressures shaped the evolution of density-dependent regulation of toxin production.Here, we study the evolution of toxin production, resistance and the response to a cell-density cue in a model of an evolving bacterial population with spatial structure.We present results for two growth regimes: (i) an undisturbed, fixed habitat in which only small fluctuations of cell density occur, and (ii) a serial-transfer regime with large fluctuations in cell density.We find that density-dependent toxin production can evolve under both regimes.However, the selection pressures driving the evolution of regulation differ.In the fixed habitat, regulation evolves because it allows cells to produce toxin only when opportunities for reproduction are highly limited (because of a high local cell density), and the effective fitness costs of toxin production are hence low.Under serial transfers, regulation evolves because it allows cells to switch from a fast-growing non-toxic phenotype when colonising a new habitat, to a slower-growing competitive toxic phenotype when the cell density increases.Colonies of such regulating cells rapidly expand into unoccupied space because their edges consist of fast-growing, non-toxin-producing cells, but are also combative because cells at the interfaces with competing colonies do produce toxin.Because under the two growth regimes different types of regulation evolve, our results underscore the importance of growth conditions in the evolution of social behaviour in bacteria. Author summaryBacteria live in microbial communities, in which they compete with many other microbes for nutrients and space.In this competitive environment, almost all known bacterial strains produce toxins that impair or kill other bacteria.This chemical warfare is thought to be one of the major factors shaping microbial diversity.Many toxins are produced only if the local density of bacteria is high.To achieve this, bacteria respond to cell-density cues: signalling molecules or other indicators of the presence of other cells.Here, we use a

MOTIVATION: High Throughput Sequencing (HTS) has enabled researchers to probe the human T cell receptor (TCR) repertoire, which consists of many rare sequences. Distinguishing between true but rare TCR sequences and variants generated by polymerase chain reaction (PCR) and sequencing errors remains a formidable challenge. The conventional approach to handle errors is to remove low quality reads, and/or rare TCR sequences. Such filtering discards a large number of true and often rare TCR sequences. However, accurate identification and quantification of rare TCR sequences is essential for repertoire diversity estimation. RESULTS: We devised a pipeline, called Recover TCR (RTCR), that accurately recovers TCR sequences, including rare TCR sequences, from HTS data (including barcoded data) even at low coverage. RTCR employs a data-driven statistical model to rectify PCR and sequencing errors in an adaptive manner. Using simulations, we demonstrate that RTCR can easily adapt to the error profiles of different types of sequencers and exhibits consistently high recall and high precision even at low coverages where other pipelines perform poorly. Using published real data, we show that RTCR accurately resolves sequencing errors and outperforms all other pipelines. AVAILABILITY AND IMPLEMENTATION: The RTCR pipeline is implemented in Python (v2.7) and C and is freely available at http://uubram.github.io/RTCR/along with documentation and examples of typical usage. CONTACT: [email protected].

Stable isotope labeling is the state of the art technique for in vivo quantification of lymphocyte kinetics in humans. It has been central to a number of seminal studies, particularly in the context of HIV-1 and leukemia. However, there is a significant discrepancy between lymphocyte proliferation rates estimated in different studies. Notably, deuterated (2)H2-glucose (D2-glucose) labeling studies consistently yield higher estimates of proliferation than deuterated water (D2O) labeling studies. This hampers our understanding of immune function and undermines our confidence in this important technique. Whether these differences are caused by fundamental biochemical differences between the two compounds and/or by methodological differences in the studies is unknown. D2-glucose and D2O labeling experiments have never been performed by the same group under the same experimental conditions; consequently a direct comparison of these two techniques has not been possible. We sought to address this problem. We performed both in vitro and murine in vivo labeling experiments using identical protocols with both D2-glucose and D2O. This showed that intrinsic differences between the two compounds do not cause differences in the proliferation rate estimates, but that estimates made using D2-glucose in vivo were susceptible to difficulties in normalization due to highly variable blood glucose enrichment. Analysis of three published human studies made using D2-glucose and D2O confirmed this problem, particularly in the case of short term D2-glucose labeling. Correcting for these inaccuracies in normalization decreased proliferation rate estimates made using D2-glucose and slightly increased estimates made using D2O; thus bringing the estimates from the two methods significantly closer and highlighting the importance of reliable normalization when using this technique.

Cell migration is a complex process involving many intracellular and extracellular factors, with different cell types adopting sometimes strikingly different morphologies. Modeling realistically behaving cells in tissues is computationally challenging because it implies dealing with multiple levels of complexity. We extend the Cellular Potts Model with an actin-inspired feedback mechanism that allows small stochastic cell rufflings to expand to cell protrusions. This simple phenomenological model produces realistically crawling and deforming amoeboid cells, and gliding half-moon shaped keratocyte-like cells. Both cell types can migrate randomly or follow directional cues. They can squeeze in between other cells in densely populated environments or migrate collectively. The model is computationally light, which allows the study of large, dense and heterogeneous tissues containing cells with realistic shapes and migratory properties.

It has been suggested that HIV-1 has evolved its set-point virus load to be optimized for transmission. Previous epidemiological models and studies into the heritability of set-point virus load confirm that this mode of adaptation within the human population is feasible. However, during the many cycles of replication between infection of a host and transmission to the next host, HIV-1 is under selection for escape from immune responses, and not transmission. Here we investigate with computational and mathematical models how these two levels of selection, within-host and between-host, are intertwined. We find that when the rate of immune escape is comparable to what has been observed in patients, immune selection within hosts is dominant over selection for transmission. Surprisingly, we do find high values for set-point virus load heritability, and argue that high heritability estimates can be caused by the 'footprints' left by differing hosts' immune systems on the virus.

To fight infections, rare T cells must quickly home to appropriate lymph nodes (LNs), and reliably localize the antigen (Ag) within them. The first challenge calls for rapid trafficking between LNs, whereas the second may require extensive search within each LN. Here we combine simulations and experimental data to investigate which features of random T cell migration within and between LNs allow meeting these two conflicting demands. Our model indicates that integrating signals from multiple random encounters with Ag-presenting cells permits reliable detection of even low-dose Ag, and predicts a kinetic feature of cognate T cell arrest in LNs that we confirm using intravital two-photon data. Furthermore, we obtain the most reliable retention if T cells transit through LNs stochastically, which may explain the long and widely distributed LN dwell times observed in vivo. Finally, we demonstrate that random migration, both between and within LNs, allows recruiting the majority of cognate precursors within a few days for various realistic infection scenarios. Thus, the combination of two-scale stochastic migration and signal integration is an efficient and robust strategy for T cell immune surveillance.

The Beauchemin model is a simple particle-based description of stochastic lymphocyte migration in tissue, which has been successfully applied to studying immunological questions. In addition to being easy to implement, the model is also to a large extent mathematically tractable. This article provides a comprehensive overview of both existing and new analytical results on the Beauchemin model within a common mathematical framework. Specifically, we derive the motility coefficient, the mean square displacement, and the confinement ratio, and discuss four different methods for simulating biased migration of pre-defined speed. The results provide new insight into published studies and a reference point for future research based on this simple and popular lymphocyte migration model.

Natural killer (NK) cells are circulating lymphocytes that play an important role in the control of viral infections and tumors. Their functions are regulated by several activating and inhibitory receptors. A subset of these receptors in human NK cells are the killer immunoglobulin-like receptors (KIRs), which interact with the highly polymorphic MHC class I molecules. One important function of NK cells is to detect cells that have down-regulated MHC expression (missing-self). Because MHC molecules have non polymorphic regions, their expression could have been monitored with a limited set of monomorphic receptors. Surprisingly, the KIR family has a remarkable genetic diversity, the function of which remains poorly understood. The mouse cytomegalovirus (MCMV) is able to evade NK cell responses by coding "decoy" molecules that mimic MHC class I. This interaction was suggested to have driven the evolution of novel NK cell receptors. Inspired by the MCMV system, we develop an agent-based model of a host population infected with viruses that are able to evolve MHC down-regulation and decoy molecules. Our simulations show that specific recognition of MHC class I molecules by inhibitory KIRs provides excellent protection against viruses evolving decoys, and that the diversity of inhibitory KIRs will subsequently evolve as a result of the required discrimination between host MHC molecules and decoy molecules.

In theoretical ecology it is well known that the steady state expressions of the variables in a food chain crucially depend on the parity of the length of the chain. This poses a major problem for modeling real food webs because it is difficult to establish their true number of trophic levels, with sometimes rare predators and often rampant pathogens. Similar problems arise in the modeling of chronic viral infections. We review examples where seemingly general interpretations strongly depend on the number of levels in a model, and on its specific equations. This Perspective aims to open the discussion on this problem.

The cellular immune system screens peptides presented by host cells on MHC molecules to assess if the cells are infected. In this study we examined whether the presented peptides contain enough information for a proper self/nonself assessment by comparing the presented human (self) and bacterial or viral (nonself) peptides on a large number of MHC molecules. For all MHC molecules tested, only a small fraction of the presented nonself peptides from 174 species of bacteria and 1000 viral proteomes ([Formula: see text]0.2%) is shown to be identical to a presented self peptide. Next, we use available data on T-cell receptor-peptide-MHC interactions to estimate how well T-cells distinguish between similar peptides. The recognition of a peptide-MHC by the T-cell receptor is flexible, and as a result, about one-third of the presented nonself peptides is expected to be indistinguishable (by T-cells) from presented self peptides. This suggests that T-cells are expected to remain tolerant for a large fraction of the presented nonself peptides, which provides an explanation for the "holes in the T-cell repertoire" that are found for a large fraction of foreign epitopes. Additionally, this overlap with self increases the need for efficient self tolerance, as many self-similar nonself peptides could initiate an autoimmune response. Degenerate recognition of peptide-MHC-I complexes by T-cells thus creates large and potentially dangerous overlaps between self and nonself.

In many immunological processes chemoattraction is thought to play a role in guiding cells to their sites of action. However, based on in vivo two-photon microscopy experiments in the absence of cognate antigen, T cell migration in lymph nodes (LNs) has been roughly described as a random walk. Although it has been shown that dendritic cells (DCs) carrying cognate antigen in some circumstances attract T cells chemotactically, it is currently still unclear whether chemoattraction of T cells towards DCs helps or hampers scanning. Chemoattraction towards DCs could on the one hand help T cells to rapidly find DCs. On the other hand, it could be deleterious if DCs become shielded by a multitude of attracted yet non-specific T cells. Results from a recent simulation study suggested that the deleterious effect dominates. We re-addressed the question whether T cell chemoattraction towards DCs is expected to promote or hamper the detection of rare antigens using the Cellular Potts Model, a formalism that allows for dynamic, flexible cellular shapes and cell migration. Our simulations show that chemoattraction of T cells enhances the DC scanning efficiency, leading to an increased probability that rare antigen-specific T cells find DCs carrying cognate antigen. Desensitization of T cells after contact with a DC further improves the scanning efficiency, yielding an almost threefold enhancement compared to random migration. Moreover, the chemotaxis-driven migration still roughly appears as a random walk, hence fine-tuned analysis of cell tracks will be required to detect chemotaxis within microscopy data.

It has recently been estimated that a single HIV-1 infected cell produces between and more than viral particles over its life span. Since body-wide estimates of the ratio of free virus to productively infected cells are smaller than and much smaller than , individual virions must be cleared rapidly. This seems difficult to reconcile with the fact that most of the total body virus is trapped on follicular dendritic cells where it can survive for many months. It has also been difficult to reconcile the vast difference in the rates at which the virus is cleared from the blood in rhesus macaques and in chronically infected patients. Here we attempt to reconcile these seemingly contradictory observations by considering the virion clearance rate in various organs and the virion exchange rates between them. The main results are that the per capita clearance rate of free virus in lymphoid tissue should be fast, the virion exchange rate between lymphoid tissue and the blood should be slow, and the comparatively slow previous estimates for the virion clearance rate from the blood correspond to the rate of virion efflux from the blood to other organs where the virus is ultimately cleared.

Estimation of division and death rates of lymphocytes in different conditions is vital for quantitative understanding of the immune system. Deuterium, in the form of deuterated glucose or heavy water, can be used to measure rates of proliferation and death of lymphocytes in vivo. Inferring these rates from labeling and delabeling curves has been subject to considerable debate with different groups suggesting different mathematical models for that purpose. We show that the three most common models, which are based on quite different biological assumptions, actually predict mathematically identical labeling curves with one parameter for the exponential up and down slope, and one parameter defining the maximum labeling level. By extending these previous models, we here propose a novel approach for the analysis of data from deuterium labeling experiments. We construct a model of "kinetic heterogeneity" in which the total cell population consists of many sub-populations with different rates of cell turnover. In this model, for a given distribution of the rates of turnover, the predicted fraction of labeled DNA accumulated and lost can be calculated. Our model reproduces several previously made experimental observations, such as a negative correlation between the length of the labeling period and the rate at which labeled DNA is lost after label cessation. We demonstrate the reliability of the new explicit kinetic heterogeneity model by applying it to artificially generated datasets, and illustrate its usefulness by fitting experimental data. In contrast to previous models, the explicit kinetic heterogeneity model 1) provides a novel way of interpreting labeling data; 2) allows for a non-exponential loss of labeled cells during delabeling, and 3) can be used to describe data with variable labeling length.

Several studies have shown that cytotoxic T lymphocytes (CTLs) play an important role in controlling HIV/SIV infection. Notably, the observation of escape mutants suggests a selective pressure induced by the CTL response. However, it remains difficult to assess the definite role of the cellular immune response. We devise a computational model of HIV/SIV infection having a broad cellular immune response targeting different viral epitopes. The CTL clones are stimulated by viral antigen and interact with the virus population through cytotoxic killing of infected cells. Consequently, the virus population reacts through the acquisition of CTL escape mutations. Our model provides realistic virus dynamics and describes several experimental observations. We postulate that inter-clonal competition and immunodominance may be critical factors determining the sequential emergence of escapes. We show that even though the total killing induced by the CTL response can be high, escape rates against a single CTL clone are often slow and difficult to estimate from infrequent sequence measurements. Finally, our simulations show that a higher degree of immunodominance leads to more frequent escape with a reduced control of viral replication but a substantially impaired replicative capacity of the virus. This result suggests two strategies for vaccine design: Vaccines inducing a broad CTL response should decrease the viral load, whereas vaccines stimulating a narrow but dominant CTL response are likely to induce escape but may dramatically reduce the replicative capacity of the virus.

Mutations that allow SIV/HIV to avoid the cytotoxic T lymphocyte (CTL) response are well documented. Recently, there have been a few attempts at estimating the costs of CTL escape mutations in terms of the reduction in viral fitness and the killing rate at which the CTL response specific to one viral epitope clears virus-infected cells. Using a mathematical model we show that estimation of both parameters depends critically on the underlying changes in the replication rate of the virus and the changes in the killing rate over time (which in previous studies were assumed to be constant). We provide a theoretical basis for estimation of these parameters using in vivo data. In particular, we show that 1) by assuming unlimited virus growth one can obtain a minimal estimate of the fitness cost of the escape mutation, and 2) by assuming no virus growth during the escape, one can obtain a minimal estimate of the average killing rate. We also discuss the conditions under which better estimates of the average killing rate can be obtained.