Mathematical models for CFSE labelled lymphocyte dynamics: asymmetry and time-lag in division.

Since their invention in 1994, fluorescent dyes such as carboxyfluorescein diacetate succinimidyl ester (CFSE) are used for cell proliferation analysis in flow cytometry. Importantly, the interpretation of such assays relies on the assumption that the label is divided equally between the daughter cells upon cell division. However, recent experimental studies indicate that division of cells is not perfectly symmetric and there is unequal distribution of protein between sister cell pairs. The uneven partition of protein or mass to daughter cells can lead to an overlap in the generations of CFSE-labelled cells with straightforward consequences for the resolution of individual generations. Numerous mathematical models developed so far for the analysis of CFSE proliferation assay incorporate the premise that the CFSE fluorescence intensity is halved in the two daughter cells. Here, we propose a novel modelling approach for the analysis of the CFSE cell proliferation assays which are characterized by poorly resolved peaks of cell generations in flow cytometric histograms. We formulate a mathematical model in the form of a system of delay hyperbolic partial differential equations which provides a good agreement with the CFSE histograms time-series data and allows an analytical treatment. The model is a further generalization of the recently proposed class of division- and label-structured models as it considers an asymmetric cell division. In addition, the basic structure of the cell cycle, i.e. the resting and cycling cell compartments, is taken into account. The model is used to estimate fundamental parameters such as activation rate, duration of the cell cycle, apoptosis rate, CFSE decay rate and asymmetry factor in cell division of monoclonal T cells during cognate interaction with dendritic cells.

Feedback regulation of proliferation vs. differentiation rates explains the dependence of CD4 T-cell expansion on precursor number.

The mechanisms regulating clonal expansion and contraction of T cells in response to immunization remain to be identified. A recent study established that there was a log-linear relation between CD4 T-cell precursor number (PN) and factor of expansion (FE), with a slope of ∼-0.5 over a range of 3-30,000 precursors per mouse. The results suggested inhibition of precursor expansion either by competition for specific antigen-presenting cells or by the action of other antigen-specific cells in the same microenvironment as the most likely explanation. Several molecular mechanisms potentially accounting for such inhibition were examined and rejected. Here we adopt a previously proposed concept, "feedback-regulated balance of growth and differentiation," and show that it can explain the observed findings. We assume that the most differentiated effectors (or memory cells) limit the growth of less differentiated effectors, locally, by increasing the rate of differentiation of the latter cells in a dose-dependent manner. Consequently, expansion is blocked and reversed after a delay that depends on initial PN, accounting for the dependence of the peak of the response on that number. We present a parsimonious mathematical model capable of reproducing immunization response kinetics. Model definition is achieved in part by requiring consistency with available BrdU-labeling and carboxyfluorescein diacetate succinimidyl ester (CFSE)-dilution data. The calibrated model correctly predicts FE as a function of PN. We conclude that feedback-regulated balance of growth and differentiation, although awaiting definite experimental characterization of the hypothetical cells and molecules involved in regulation, can explain the kinetics of CD4 T-cell responses to antigenic stimulation.

A systems immunology approach to plasmacytoid dendritic cell function in cytopathic virus infections.

Plasmacytoid dendritic cell (pDC)-mediated protection against cytopathic virus infection involves various molecular, cellular, tissue-scale, and organism-scale events. In order to better understand such multiscale interactions, we have implemented a systems immunology approach focusing on the analysis of the structure, dynamics and operating principles of virus-host interactions which constrain the initial spread of the pathogen. Using high-resolution experimental data sets coming from the well-described mouse hepatitis virus (MHV) model, we first calibrated basic modules including MHV infection of its primary target cells, i.e. pDCs and macrophages (Mphis). These basic building blocks were used to generate and validate an integrative mathematical model for in vivo infection dynamics. Parameter estimation for the system indicated that on a per capita basis, one infected pDC secretes sufficient type I IFN to protect 10(3) to 10(4) Mphis from cytopathic viral infection. This extremely high protective capacity of pDCs secures the spleen's capability to function as a 'sink' for the virus produced in peripheral organs such as the liver. Furthermore, our results suggest that the pDC population in spleen ensures a robust protection against virus variants which substantially down-modulate IFN secretion. However, the ability of pDCs to protect against severe disease caused by virus variants exhibiting an enhanced liver tropism and higher replication rates appears to be rather limited. Taken together, this systems immunology analysis suggests that antiviral therapy against cytopathic viruses should primarily limit viral replication within peripheral target organs.

Distributed parameter identification for a label-structured cell population dynamics model using CFSE histogram time-series data.

In this work we address the problem of the robust identification of unknown parameters of a cell population dynamics model from experimental data on the kinetics of cells labelled with a fluorescence marker defining the division age of the cell. The model is formulated by a first order hyperbolic PDE for the distribution of cells with respect to the structure variable x (or z) being the intensity level (or the log(10)-transformed intensity level) of the marker. The parameters of the model are the rate functions of cell division, death, label decay and the label dilution factor. We develop a computational approach to the identification of the model parameters with a particular focus on the cell birth rate alpha(z) as a function of the marker intensity, assuming the other model parameters are scalars to be estimated. To solve the inverse problem numerically, we parameterize alpha(z) and apply a maximum likelihood approach. The parametrization is based on cubic Hermite splines defined on a coarse mesh with either equally spaced a priori fixed nodes or nodes to be determined in the parameter estimation procedure. Ill-posedness of the inverse problem is indicated by multiple minima. To treat the ill-posed problem, we apply Tikhonov regularization with the regularization parameter determined by the discrepancy principle. We show that the solution of the regularized parameter estimation problem is consistent with the data set with an accuracy within the noise level in the measurements.

Numerical modelling of label-structured cell population growth using CFSE distribution data.

BACKGROUND: The flow cytometry analysis of CFSE-labelled cells is currently one of the most informative experimental techniques for studying cell proliferation in immunology. The quantitative interpretation and understanding of such heterogenous cell population data requires the development of distributed parameter mathematical models and computational techniques for data assimilation. METHODS AND RESULTS: The mathematical modelling of label-structured cell population dynamics leads to a hyperbolic partial differential equation in one space variable. The model contains fundamental parameters of cell turnover and label dilution that need to be estimated from the flow cytometry data on the kinetics of the CFSE label distribution. To this end a maximum likelihood approach is used. The Lax-Wendroff method is used to solve the corresponding initial-boundary value problem for the model equation. By fitting two original experimental data sets with the model we show its biological consistency and potential for quantitative characterization of the cell division and death rates, treated as continuous functions of the CFSE expression level. CONCLUSION: Once the initial distribution of the proliferating cell population with respect to the CFSE intensity is given, the distributed parameter modelling allows one to work directly with the histograms of the CFSE fluorescence without the need to specify the marker ranges. The label-structured model and the elaborated computational approach establish a quantitative basis for more informative interpretation of the flow cytometry CFSE systems.

Asymmetry of Cell Division in CFSE-Based Lymphocyte Proliferation Analysis.

Flow cytometry-based analysis of lymphocyte division using carboxyfluorescein succinimidyl ester (CFSE) dye dilution permits acquisition of data describing cellular proliferation and differentiation. For example, CFSE histogram data enable quantitative insight into cellular turnover rates by applying mathematical models and parameter estimation techniques. Several mathematical models have been developed using different types of deterministic or stochastic approaches. However, analysis of CFSE proliferation assays is based on the premise that the label is halved in the two daughter cells. Importantly, asymmetry of protein distribution in lymphocyte division is a basic biological feature of cell division with the degree of the asymmetry depending on various factors. Here, we review the recent literature on asymmetric lymphocyte division and CFSE-based lymphocyte proliferation analysis. We suggest that division- and label-structured mathematical models describing CFSE-based cell proliferation should take into account asymmetry and time-lag in cell proliferation. Utilization of improved modeling algorithms will permit straightforward quantification of essential parameters describing the performance of activated lymphocytes.

Computational analysis of CFSE proliferation assay.

CFSE based tracking of the lymphocyte proliferation using flow cytometry is a powerful experimental technique in immunology allowing for the tracing of labelled cell populations over time in terms of the number of divisions cells undergone. Interpretation and understanding of such population data can be greatly improved through the use of mathematical modelling. We apply a heterogenous linear compartmental model, described by a system of ordinary differential equations similar to those proposed by Kendall. This model allows division number-dependent rates of cell proliferation and death and describes the rate of changes in the numbers of cells having undergone j divisions. The experimental data set that we specifically analyze specifies the following characteristics of the kinetics of PHA-induced human T lymphocyte proliferation assay in vitro: (1) the total number of live cells, (2) the total number of dead but not disintegrated cells and (3) the number of cells divided j times. Following the maximum likelihood approach for data fitting, we estimate the model parameters which, in particular, present the CTL birth- and death rate "functions". It is the first study of CFSE labelling data which convincingly shows that the lymphocyte proliferation and death both in vitro and in vivo are division number dependent. For the first time, the confidence in the estimated parameter values is analyzed by comparing three major methods: the technique based on the variance-covariance matrix, the profile-likelihood-based approach and the bootstrap technique. We compare results and performance of these methods with respect to their robustness and computational cost. We show that for evaluating mathematical models of differing complexity the information-theoretic approach, based upon indicators measuring the information loss for a particular model (Kullback-Leibler information), provides a consistent basis. We specifically discuss methodological and computational difficulties in parameter identification with CFSE data, e.g. the loss of confidence in the parameter estimates starting around the sixth division. Overall, our study suggests that the heterogeneity inherent in cell kinetics should be explicitly incorporated into the structure of mathematical models.

Underwhelming the immune response: effect of slow virus growth on CD8+-T-lymphocyte responses.

The speed of virus replication has typically been seen as an advantage for a virus in overcoming the ability of the immune system to control its population growth. Under some circumstances, the converse may also be true: more slowly replicating viruses may evoke weaker cellular immune responses and therefore enhance their likelihood of persistence. Using the model of lymphocytic choriomeningitis virus (LCMV) infection in mice, we provide evidence that slowly replicating strains induce weaker cytotoxic-T-lymphocyte (CTL) responses than a more rapidly replicating strain. Conceptually, we show a "bell-shaped" relationship between the LCMV growth rate and the peak CTL response. Quantitative analysis of human hepatitis C virus infections suggests that a reduction in virus growth rate between patients during the incubation period is associated with a spectrum of disease outcomes, from fulminant hepatitis at the highest rate of viral replication through acute resolving to chronic persistence at the lowest rate. A mathematical model for virus-CTL population dynamics (analogous to predator [CTL]-prey [virus] interactions) is applied in the clinical data-driven analysis of acute hepatitis B virus infection. The speed of viral replication, through its stimulus of host CTL responses, represents an important factor influencing the pathogenesis and duration of virus persistence within the human host. Viruses with lower growth rates may persist in the host because they "sneak through" immune surveillance.