Estimating Lymphocyte Division and Death Rates from CFSE Data

The division tracking dye, carboxyfluorescin diacetate succinimidyl ester (CFSE) is currently the most informative labeling technique for characterizing the division history of cells in the immune system. Gett and Hodgkin [Nat. Immunol. 1:239–244, 2000] have pioneered the quantitative analysis of CFSE data. We confirm and extend their data analysis approach using simple mathematical models. We employ the extended Gett and Hodgkin [Nat. Immunol. 1:239–244, 2000] method to estimate the time to first division, the fraction of cells recruited into division, the cell cycle time, and the average death rate from CFSE data on T cells stimulated under different concentrations of IL-2. The same data is also fitted with a simple mathematical model that we derived by reformulating the numerical model of Deenick et al. [J. Immunol. 170:4963–4972, 2003]. By a non-linear fitting procedure we estimate parameter values and confidence intervals to identify the parameters that are influenced by the IL-2 concentration. We obtain a significantly better fit to the data when we assume that the T cell death rate depends on the number of divisions cells have completed. We provide an outlook on future work that involves extending the Deenick et al. [J. Immunol. 170:4963–4972, 2003] model into the classical smith-martin model, and into a model with arbitrary probability distributions for death and division through subsequent divisions.     

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 vitroL (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.      

Fractal analysis of the root architecture of Gliricidia sepium for the spatial prediction of root branching,size and mass: model development and evaluation in agroforestry

Based on fractal and pipe model assumptions, a static three-dimensional model of the Gliricidia sepium root system was developed, in order to provide a basis for the prediction of root branching, size and mass in an alley cropping system. The model was built from observations about the topology, branching rules, link length and diameter, and root orientation, provided by in situ and extracted root systems. Evaluation tests were carried out at the plant level and at the field level. These tests principally concerned coefficients α and q –- the proportionality factor α between total cross-sectional area of a root before and after branching, and allocation parameter q that defines the partitioning of biomass between the new links after a branching event –- that could be considered as key variables of this fractal approach. Although independent of root diameter, these coefficients showed a certain variability that may affect the precision of the predictions. When calibrated, however, the model provided suitable predictions of root dry matter, total root length and root diameter at the plant level. At the field level, the simulation of 2D root maps was accurate for root distribution patterns, but the number of simulated root dots was underestimated in the surface layers. Hence recommendations were made to improve the model with regard to α and q. This static approach appeared to be well suited to study the root system of adult trees. Compared with explicit models, the main advantage of the fractal approach is its plasticity and ease of use. This revised version was published online in June 2006 with corrections to the Cover Date.     

Multi-type branching models to describe cell differentiation programs

Cell proliferation and differentiation is described by a multi-type branching process, a probability model that defines the inheritance of cell type. Cell type is defined by (i) a repression index related to the time required for S-phase entry and (ii) phenotype as determined by cell markers and division history. The inheritance of cell type is expressed as the expected number and type of progeny cells produced by a mother cell given her type. Expressions for the expected number and type of cells produced by a multi-cellular (bulk culture) system are derived from the general model by making the simplifying assumption that cell generation times are independent. The multi-type Smith-Martin model (MSM) makes the further assumption that cell generation times are lag-exponentially distributed with phenotype transitions occurring just before entry into S-phase. The inheritance-modified MSM (IMSM) model includes the influence of generation time memory so that mother and daughter generation times are correlated. The expansion of human cord blood CD34⁺ cells by haematopoietic growth factors was division tracked in bulk culture using carboxyfluorescein diacetate, succinimidyl ester (CFDA-SE). The MSM model was fitted to division tracking data to indentify cell cycle length, and the rates of CD34 antigen down-regulation and apoptosis. The IMSM model was estimated for mouse granulocyte-macrophage progenitors using live cell imaging data. Multi-type branching models describe cell differentiation dynamics at both single- and multi-cell scales, providing a new paradigm for systematic analysis of stem and progenitor cell development.     

Quantitative analysis of T cell homeostatic proliferation

T cell homeostatic proliferation occurs on transfer of T cells into lymphopenic recipients; transferred cells undergo several rounds of division in the absence of specific antigen stimulation. For a quantitative analysis of this phenomenon, we applied a mathematical method to describe proliferating T cells to match peak distributions from actual CFSE dilution data. For in vitro stimulation of T cells with anti-CD3/anti-CD28, our simulation confirmed a high proportion of cells entering cell cycle with a low proportion undergoing apoptosis. When applied to homeostatic proliferation, it described striking differences in CD4 and CD8 T cell proliferation rates, and accurately predicted that successive divisions were accompanied by higher rates of apoptosis, limiting the accumulation of proliferating cells. Thus, the presence of multiple CFSE dilution peaks cannot be considered equivalent to lymphocyte expansion. Finally, genetic effects were identified that may help explain links between homeostatic proliferation and autoimmunity.     

Nonlinearities make a difference: comparison of two common Hill-type models with real muscle

Compared to complex structural Huxley-type models, Hill-type models phenomenologically describe muscle contraction using only few state variables. The Hill-type models dominate in the ever expanding field of musculoskeletal simulations for simplicity and low computational cost. Reasonable parameters are required to gain insight into mechanics of movement. The two most common Hill-type muscle models used contain three components. The series elastic component is connected in series to the contractile component. A parallel elastic component is either connected in parallel to both the contractile and the series elastic component (model [CC+SEC]), or is connected in parallel only with the contractile component (model [CC]). As soon as at least one of the components exhibits substantial nonlinearities, as, e.g., the contractile component by the ability to turn on and off, the two models are mechanically different. We tested which model ([CC+SEC] or [CC]) represents the cat soleus better. Ramp experiments consisting of an isometric and an isokinetic part were performed with an in situ cat soleus preparation using supramaximal nerve stimulation. Hill-type models containing force–length and force–velocity relationship, excitation–contraction coupling and series and parallel elastic force–elongation relations were fitted to the data. To test which model might represent the muscle better, the obtained parameters were compared with experimentally determined parameters. Determined in situations with negligible passive force, the force–velocity relation and the series elastic component relation are independent of the chosen model. In contrast to model [CC+SEC], these relations predicted by model [CC] were in accordance with experimental relations. In conclusion model [CC] seemed to better represent the cat soleus contraction dynamics and should be preferred in the nonlinear regression of muscle parameters and in musculoskeletal modeling.     

Glycan-binding profile of DC-like cells

Modification of vaccine carriers by decoration with glycans can enhance binding to and even targeting of dendritic cells (DCs), thus augmenting vaccine efficacy. To find a specific glycan-“vector” it is necessary to know glycan-binding profile of DCs. This task is not trivial; the small number of circulating blood DCs available for isolation hinders screening and therefore advancement of the profiling. It would be more convenient to employ long-term cell cultures or even primary DCs from murine blood. We therefore examined whether THP-1 (human monocyte cell line) and DC2.4 (immature murine DC-like cell line) could serve as a model for human DCs. These cells were probed with a set of glycans previously identified as binding to circulating human CD14low/-CD16+CD83+ DCs. In addition, we tested a subpopulation of murine CD14low/-CD80+СD11c+CD16+ cells reported as relating to the human CD14low/-CD16+CD83+ cells. Manα1–3(Manα1–6)Manβ1–4GlcNAcβ1–4GlcNAcβ bound to both the cell lines and the murine CD14low/-CD80+СD11c+CD16+ cells. Primary cells, but not the cell cultures, were capable of binding GalNAcα1–3Galβ (Adi), the most potent ligand for binding to human circulating DCs. In conclusion, not one of the studied cell lines proved an adequate model for DCs processes involving lectin binding. Although the glycan-binding profile of BYRB-Rb (8.17)1Iem mouse DCs could prove useful for assessing human DCs, important glycan interactions were missing, a situation which was aggravated when employing cells from the BALB/c strain. Accordingly, one must treat results from murine work with caution when seeking vaccine targeting of human DCs, and certainly should avoid cell lines such as THP-1 and DC2.4 cells.     

A comparative analysis of multi-conductance neuronal models <Emphasis Type="Italic">in silico</Emphasis>

We demonstrate that a previously presented flexible silicon–neuron architecture can implement three disparate conductance-based neuron models with both fast and slow dynamics. By exploiting the real-time nature of this physical implementation, we mapped the model dynamics across a large region of parameter space. We also found that two of these dynamically different models represent points in a contiguous bursting space that spans between the two models. By systematically varying the model parameters, we also found that multiple, diverse trajectories in parameter space connected the two canonical bursting points. In addition, we found that the combination of parameter values keeps the neuron in the bursting region. These findings demonstrate the usefulness of the silicon–neuron architecture as a neural-modeling tool and illustrate its versatility as a platform for a multi-behavioral neuron that resembles its living analog.     

Estimating the variation in S phase duration from flow cytometric histograms

A stochastic model for interpreting BrdUrd DNA FCM-derived data is proposed. The model is based on branching processes and describes the progression of the DNA distribution of BrdUrd-labelled cells through the cell cycle. With the main focus on estimating the S phase duration and its variation, the DNA replication rate is modelled by a piecewise linear function, while assuming a gamma distribution for the S phase duration. Estimation of model parameters was carried out using maximum likelihood for data from two different cell lines. The results provided quite a good fit to the data, suggesting that stochastic models may be a valuable tool for analysing this kind of data.     

Evolutionary relationships among photosynthetic bacteria

To understand the evolution of photosynthetic bacteria it is necessary to understand how the main groups within Bacteria have evolved from a common ancestor, a critical issue that has not been resolved in the past. Recent analysis of shared conserved inserts or deletions (indels) in protein sequences has provided a powerful means to resolve this long-standing problem in microbiology. Based on a set of 25 indels in highly conserved and widely distributed proteins, all main groups within bacteria can now be defined in clear molecular terms and their relative branching orders logically deduced. For the 82 presently completed bacterial genomes, the presence or absence of these signatures in various proteins was found to be almost exactly as predicted by the indel model, with only 11 exceptions observed in 1842 observations. The branching order of different bacterial groups as deduced using this approach is as follows: low G+C Gram-positive (Heliobacterium chlorum) ↔ high G+C Gram-positive ↔ Clostridium–Fusobacterium–Thermotoga ↔ Deinococcus–Thermus ↔ green nonsulfur bacteria (Chloroflexus aurantiacus) ↔ Cyanobacteria ↔ Spirochetes ↔ Chlamydia–Cytophaga–Flavobacteria–green sulfur bacteria (Chlorobium tepidum) ↔ Aquifex ↔ Proteobacteria (δ and ∈) ↔ Proteobacteria (α) ↔ Proteobacteria (β) and ↔ Proteobacteria (γ). The Heliobacterium species, which contain an Fe–S type of reaction center (RC 1) and represent the sole photosynthetic phylum from the Gram-positive or monoderm bacteria (i.e., bounded by only a single membrane), is indicated to be the most ancestral of the photosynthetic lineages. Among the Gram-negative or diderm bacteria (containing both inner and outer cell membranes) the green nonsulfur bacteria, which contain a pheophytin-quinone type of reaction center (RC 2), are indicated to have evolved first. The later emerging photosynthetic groups which contain either one or both of these reaction centers could have acquired such genes from the earlier branching lineages by either direct descent or by means of lateral gene transfer. This revised version was published online in August 2006 with corrections to the Cover Date.     

A multi-approach and multi-scale platform to model CD4+ T cells responding to infections

Immune responses rely on a complex adaptive system in which the body and infections interact at multiple scales and in different compartments. We developed a modular model of CD4+ T cells, which uses four modeling approaches to integrate processes at three spatial scales in different tissues. In each cell, signal transduction and gene regulation are described by a logical model, metabolism by constraint-based models. Cell population dynamics are described by an agent-based model and systemic cytokine concentrations by ordinary differential equations. A Monte Carlo simulation algorithm allows information to flow efficiently between the four modules by separating the time scales. Such modularity improves computational performance and versatility and facilitates data integration. We validated our technology by reproducing known experimental results, including differentiation patterns of CD4+ T cells triggered by different combinations of cytokines, metabolic regulation by IL2 in these cells, and their response to influenza infection. In doing so, we added multi-scale insights to single-scale studies and demonstrated its predictive power by discovering switch-like and oscillatory behaviors of CD4+ T cells that arise from nonlinear dynamics interwoven across three scales. We identified the inflamed lymph node’s ability to retain naive CD4+ T cells as a key mechanism in generating these emergent behaviors. We envision our model and the generic framework encompassing it to serve as a tool for understanding cellular and molecular immunological problems through the lens of systems immunology.     

A mathematical model of the sleep/wake cycle

We present a biologically-based mathematical model that accounts for several features of the human sleep/wake cycle. These features include the timing of sleep and wakefulness under normal and sleep-deprived conditions, ultradian rhythms, more frequent switching between sleep and wakefulness due to the loss of orexin and the circadian dependence of several sleep measures. The model demonstrates how these features depend on interactions between a circadian pacemaker and a sleep homeostat and provides a biological basis for the two-process model for sleep regulation. The model is based on previous “flip–flop” conceptual models for sleep/wake and REM/NREM and we explore whether the neuronal components in these flip–flop models, with the inclusion of a sleep-homeostatic process and the circadian pacemaker, are sufficient to account for the features of the sleep/wake cycle listed above. The model is minimal in the sense that, besides the sleep homeostat and constant cortical drives, the model includes only those nuclei described in the flip–flop models. Each of the cell groups is modeled by at most two differential equations for the evolution of the total population activity, and the synaptic connections are consistent with those described in the flip–flop models. A detailed analysis of the model leads to an understanding of the mathematical mechanisms, as well as insights into the biological mechanisms, underlying sleep/wake dynamics.     

A stochastic model of cell cycle desynchronization

A general branching process model is suggested to describe cell cycle desynchronization. Cell cycle phase times are modeled as random variables and a formula for the expected fraction of cells in S phase as a function of time is established. The model is compared to data from the literature and is also compared to previously suggested deterministic and stochastic models.     

Elevation of Survivin Levels by Hematopoietic Growth Factors Occurs in Quiescent CD34+ Hematopoietic Stem and Progenitor Cells Before Cell Cycle Entry

Survivin is a member of the inhibitor of apoptosis protein (IAP) family that is over-expressed during G2/M phase in most cancer cells. In contrast, we previously reported that Survivin is expressed throughout the cell cycle in normal CD34+ hematopoietic stem and progenitor cells stimulated by the combination of Thrombopoietin (Tpo), Stem Cell Factor (SCF) and Flt3 ligand (FL). In order to address whether Survivin expression is specifically up-regulated by hematopoietic growth factors before cell cycle entry, we isolated quiescent CD34+ cells and investigated Survivin expression in response to growth factor stimulation. Survivin is up-regulated in CD34+ cells with 2N DNA content following growth factor addition, suggesting it becomes elevated during G0/G1. Survivin is barely detectable in freshly isolated umbilical cord blood (UCB) Ki-67negative and Cyclin Dnegative CD34+ cells, however incubation with Tpo, SCF and FL for 20 hrs results in up-regulation without entry of cells into cell cycle. Culture of G0 CD34+ cells isolated based on Hoechst 33342/PyroninY staining with Tpo, SCF and FL for 48 hrs, results in significantly elevated Survivin mRNA and protein levels. Moreover, labeling of fresh G0 CD34+ cells with 5-(and 6-) carboxyfluorescein diacetate succinimidyl ester (CFSE) before culture with growth factors for up to 72 hrs, revealed that Survivin expression was elevated in CFSEbright G0 CD34+ cells, indicating that up-regulation occurred before entry into G1. These results suggest that up-regulation of Survivin expression in CD34+ cells is an early event in cell cycle entry that is regulated by hematopoietic growth factors and does not simply reflect cell cycle progression and cell division.

Key Words:

Survivin, Cord blood, CD34+ cells, Cell cycle     

Changing ecophysiological processes and carbon budget in East Asian ecosystems under near-future changes in climate: implications for long-term monitoring from a process-based model

Using a process-based model, I assessed how ecophysiological processes would respond to near-future global changes predicted by coupled atmosphere–ocean climate models. An ecosystem model, Vegetation Integrative SImulator for Trace gases (VISIT), was applied to four sites in East Asia (different types of forest in Takayama, Tomakomai, and Fujiyoshida, Japan, and an Alpine grassland in Qinghai, China) where observational flux data are available for model calibration. The climate models predicted +1–3°C warming and slight change in annual precipitation by 2050 as a result of an increase in atmospheric CO₂. Gross primary production (GPP) was estimated to increase substantially at each site because of improved efficiency in the use of water and radiation. Although increased respiration partly offset the GPP increase, the simulation showed that these ecosystems would act as net carbon sinks independent of disturbance-induced uptake for recovery. However, the carbon budget response relied strongly on nitrogen availability, such that photosynthetic down-regulation resulting from leaf nitrogen dilution largely decreased GPP. In relation to long-term monitoring, these results indicate that the impacts of global warming may be more evident in gross fluxes (e.g., photosynthesis and respiration) than in the net CO₂ budget, because changes in these fluxes offset each other.     

Stochastic models of tumor growth and the probability of elimination by cytotoxic cells

The probability of tumor extinction due to the action of cytotoxic cell populations is investigated by several one dimensional stochastic models of the population growth and elimination processes of a tumor. The several models are made necessary by the nonlinearity of the processes and the different parameter ranges explored. The deterministic form of the model is where γ₀, k′₆ and k ₁ are positive constants. The parameter of most import is which determines the stability of the T = 0 equilibrium. With an initial tumor size of one, a (linear) branching process is used to estimate the extinction probability. However, in the case λ = 0 when the linearization of the deterministic model gives no information (T = 0 is actually unstable) the branching model is unsatisfactory. This makes necessary the utilization of a density-dependent branching process to approximate the population. Through scaling a diffusion limit is reached which enables one to again compute the probability of extinction. For populations away from one a sequence of density-dependent jump Markov processes are approximated by a sequence of diffusion processes. In limiting cases, the estimates of extinction correspond to that computed from the original branching process. Table 1 summarizes the results.     

Estimation of Cell Proliferation Dynamics Using CFSE Data

Advances in fluorescent labeling of cells as measured by flow cytometry have allowed for quantitative studies of proliferating populations of cells. The investigations (Luzyanina et al. in J. Math. Biol. 54:57–89, 2007; J. Math. Biol., 2009; Theor. Biol. Med. Model. 4:1–26, 2007) contain a mathematical model with fluorescence intensity as a structure variable to describe the evolution in time of proliferating cells labeled by carboxyfluorescein succinimidyl ester (CFSE). Here, this model and several extensions/modifications are discussed. Suggestions for improvements are presented and analyzed with respect to statistical significance for better agreement between model solutions and experimental data. These investigations suggest that the new decay/label loss and time dependent effective proliferation and death rates do indeed provide improved fits of the model to data. Statistical models for the observed variability/noise in the data are discussed with implications for uncertainty quantification. The resulting new cell dynamics model should prove useful in proliferation assay tracking and modeling, with numerous applications in the biomedical sciences.     

Mass-Balance Analyses of Boreal Forest Population Cycles: Merging Demographic and Ecosystem Approaches

UsingEcopath, a trophic mass-balance modeling framework, we developed six models of a Canadian boreal forest food web centered around snowshoe hares, which have conspicuous 10-year population cycles. Detailed models of four phases of the cycle were parameterized with long-term population data for 12 vertebrate taxa. We also developed five other models that, instead of observed data, used parameter values derived from standard assumptions. Specifically, in the basic model, production was assumed to equal adult mortality, feeding rates were assumed to be allometric, and biomass was assumed to be constant. In the actual production, functional response, and biomass change models, each of these assumed values from the basic model was replaced individually by field data. Finally, constant biomass models included actual production by all species and functional responses of mammalian predators and revealed the proportion of herbivore production used by species at higher trophic levels. By comparing these models, we show that detailed information on densities and demographics was crucial to constructing models that captured dynamic aspects of the food web. These detailed models reinforced an emerging picture of the causes and consequences of the snowshoe hare cycle. The snowshoe hare decline and low phases were coincident with times when per capita production was relatively low and predation pressure high. At these times, ecotrophic efficiencies (EE) suggest there was little production that remained unconsumed by predators. The importance of both production and consumption implies that bottom–up and top–down factors interacted to cause the cycle. EEs of other herbivores (ground squirrels, red squirrels, small mammals, small birds, grouse) were generally low, suggesting weak top–down effects. Predation rates on these “alternative” prey, except ground squirrels, were highest when predators were abundant, not when hares were rare; consequently, any top–down effects reflected predator biomass and were not a function of diet composition or functional responses. Finally, several predators (lynx, coyotes, great-horned owls) showed clear bottom–up regulation, reproducing only when prey exceeded threshold densities. Taken altogether, these results demonstrate that ecosystem models parameterized by population data can describe the dynamics of nonequilibrial systems, but only when detailed information is available for the species modeled. Received 30 November 2000; Accepted 6 September 2001.     

Cutting edge: B7/CD28 interactions regulate cell cycle progression independent of the strength of TCR signaling

The role of B7/CD28 signals in Ag-induced cell cycle progression of CD4(+) T cells was examined using the technique of CFSE dye dilution and flow cytometry. In wild-type T cells, proliferation was directly related to the concentration of Ag available to the APC. Consistent with this, the rate of G(0)-->G(1) cell cycle progression varied with the concentration of Ag. However, cell division by T cell blasts occurred at a constant rate, independent of Ag concentration. G(0)-->G(1) phase progression by CD28-deficient CD4(+) T cells or wild-type T cells cultured in the presence of neutralizing anti-B7 mAbs was slowed, confirming that a synergy does exist between TCR and CD28 signaling in the initial activation of the T cells. However, unlike the TCR, the strength of CD28 stimulation was also shown to play a unique role in controlling the rate of cell division by T cell blasts.     

Selective identification of HLA-DP4 binding T cell epitopes encoded by the MAGE-A gene family

Because of the high frequency of HLA-DP4 in the Caucasian population, we have selectively delineated HLA-DP4 restricted T cell epitopes in the MAGE-A tumor antigens. We identified 12 good binders to HLA-DP4 and investigated the capacity of the seven best binders to induce in vitro specific CD4+ T cell lines from HLA-DP4 healthy donors. We found that the MAGE-A1 90–104 peptide exhibited a high and constant frequency of CD4+ T cell precursors in all the six tested donors. The MAGE-A1 268–282 peptide was found immunogenic in only two donors but with a high precursor frequency. The MAGE-A12 127–141 peptide was T cell stimulating in six different donors and induced fewer T cell lines. The peptide-specific T cell lines were stimulated by DC loaded with the lysates of cells transfected with MAGE-A1 or MAGE-A12, or loaded with the recombinant protein. We also show that the immunoreactivity of CD4+ T cell epitopes restricted to the same HLA II molecule may vary from one individual to another, as a result of inter-individual variations in the CD4+ T cell repertoire.