Analysis of cell kinetics using a cell division marker: mathematical modeling of experimental data

Blood Lipoproteins Shape the Phenotype and Lipid Content of Early Atherosclerotic Lesion Macrophages: A Dual-Structured Mathematical Model

Macrophages in atherosclerotic lesions exhibit a spectrum of behaviours or phenotypes. The phenotypic distribution of monocyte-derived macrophages (MDMs), its correlation with MDM lipid content, and relation to blood lipoprotein densities are not well understood. Of particular interest is the balance between low density lipoproteins (LDL) and high density lipoproteins (HDL), which carry bad and good cholesterol respectively. To address these issues, we have developed a mathematical model for early atherosclerosis in which the MDM population is structured by phenotype and lipid content. The model admits a simpler, closed subsystem whose analysis shows how lesion composition becomes more pathological as the blood density of LDL increases relative to the HDL capacity. We use asymptotic analysis to derive a power-law relationship between MDM phenotype and lipid content at steady-state. This relationship enables us to understand why, for example, lipid-laden MDMs have a more inflammatory phenotype than lipid-poor MDMs when blood LDL lipid density greatly exceeds HDL capacity. We show further that the MDM phenotype distribution always attains a local maximum, while the lipid content distribution may be unimodal, adopt a quasi-uniform profile or decrease monotonically. Pathological lesions exhibit a local maximum in both the phenotype and lipid content MDM distributions, with the maximum at an inflammatory phenotype and near the lipid content capacity respectively. These results illustrate how macrophage heterogeneity arises in early atherosclerosis and provide a framework for future model validation through comparison with single-cell RNA sequencing data.

DISCRETE MATURITY AND DELAY DIFFERENTIAL-DIFFERENCE MODEL OF HEMATOPOIETIC CELL DYNAMICS WITH APPLICATIONS TO ACUTE MYELOGENOUS LEUKEMIA

In the last few years, many efforts were oriented towards describing the hematopoiesis phenomenon in normal and pathological situations. This complex biological process is organized as a hierarchical system that begins with primitive hematopoietic stem cells (HSCs) and ends with mature blood cells: red blood cells, white blood cells and platelets. Regarding acute myelogenous leukemia (AML), a cancer of the bone marrow and blood, characterized by a rapid proliferation of immature cells, which eventually invade the bloodstream, there is a consensus about the target cells during the HSCs development which are susceptible to leukemic transformation. We propose and analyze a mathematical model of HSC dynamics taking into account two phases in the cell cycle, a resting and a proliferating one, by allowing just after division a part of HSCs to enter the resting phase and the other part to come back to the proliferating phase to divide again. The resulting mathematical model is a system of nonlinear differential-difference equations. Due to the hierarchical organization of the hematopoiesis, we consider [Formula: see text] stages of HSCs characterized by their maturity levels. We obtain a system of [Formula: see text] nonlinear differential-difference equations. We study the existence, uniqueness, positivity, boundedness and unboundedness of the solutions. We then investigate the existence of positive and axial steady states for the system, and obtain conditions for their stability. Sufficient conditions for the global asymptotic stability of the trivial steady state as well as conditions for its instability are obtained. Using neutral differential equation associated to the differential-difference system, we also obtain results on the local asymptotic stability of the positive steady state. Numerical simulations are carried out to show the influence of variations of the differentiation rates and self-renewal coefficients of the HSCs on the behavior of the system. In particular, we show that a blocking of differentiation at an early stage of HSC development can lead in an overexpression of very immature cells. Such situation corresponds to the observation in the case of AML.

β-micrustoxin (Mlx-9), a PLA2 from Micrurus lemniscatus snake venom: biochemical characterization and anti-proliferative effect mediated by p53

Background

Endogenous phospholipases A₂ (PLA₂) play a fundamental role in inflammation, neurodegenerative diseases, apoptosis and cellular senescence. Neurotoxins with PLA₂ activity are found in snake venoms from the Elapidae and Viperidae families. The mechanism of action of these neurotoxins have been studied using hippocampal and cerebellar neuronal cultures showing [Ca²⁺]i increase, mitochondrial depolarization and cell death. Astrocytes are rarely used as a model, despite being modulators at the synapses and responsible for homeostasis and defense in the central nervous system. Preserving the cell division ability, they can be utilized to study the cell proliferation process. In the present work cultured astrocytes and glioblastoma cells were employed to characterize the action of β-micrustoxin (previously named Mlx-9), a PLA₂ isolated from Micrurus lemniscatus snake venom. The β-micrustoxin structure was determined and the cell proliferation, cell cycle phases and the regulatory proteins p53, p21 and p27 were investigated.

Methods

β-micrustoxin was characterized biochemically by a proteomic approach. Astrocytes were obtained by dissociation of pineal glands from Wistar rats; glioblastoma tumor cells were purchased from ATCC and Sigma and cultured in DMEM medium. Cell viability was evaluated by MTT assay; cell proliferation and cell cycle phases were analyzed by flow cytometry; p53, p21 and p27 proteins were studied by western blotting and immunocytochemistry.

Results

Proteomic analysis revealed fragments on β-micrustoxin that aligned with a PLA₂ from Micrurus lemniscatus lemniscatus previously identified as transcript ID DN112835_C3_g9_i1/m.9019. β-micrustoxin impaired the viability of astrocytes and glioblastoma tumor cells. There was a reduction in cell proliferation, an increase in G2/M phase and activation of p53, p21 and p27 proteins in astrocytes.

Conclusion

These findings indicate that β-micrustoxin from Micrurus lemniscatus venom could inhibit cell proliferation through p53, p21 and p27 activation thus imposing cell cycle arrest at the checkpoint G2/M.

Nongenetic origins of cell-to-cell variability in B lymphocyte proliferation

Rapid antibody production in response to invading pathogens requires the dramatic expansion of pathogen-derived antigen-specific B lymphocyte populations. Whether B cell population dynamics are based on stochastic competition between competing cell fates, as in the development of competence by the bacterium Bacillus subtilis, or on deterministic cell fate decisions that execute a predictable program, as during the development of the worm Caenorhabditis elegans, remains unclear. Here, we developed long-term live-cell microscopy of B cell population expansion and multiscale mechanistic computational modeling to characterize the role of molecular noise in determining phenotype heterogeneity. We show that the cell lineage trees underlying B cell population dynamics are mediated by a largely predictable decision-making process where the heterogeneity of cell proliferation and death decisions at any given timepoint largely derives from nongenetic heterogeneity in the founder cells. This means that contrary to previous models, only a minority of genetically identical founder cells contribute the majority to the population response. We computationally predict and experimentally confirm nongenetic molecular determinants that are predictive of founder cells' proliferative capacity. While founder cell heterogeneity may arise from different exposure histories, we show that it may also be due to the gradual accumulation of small amounts of intrinsic noise during the lineage differentiation process of hematopoietic stem cells to mature B cells. Our finding of the largely deterministic nature of B lymphocyte responses may provide opportunities for diagnostic and therapeutic development.

Estimates and impact of lymphocyte division parameters from CFSE data using mathematical modelling

Carboxyfluorescein diacetate succinimidyl ester (CFSE) labelling has been widely used to track and study cell proliferation. Here we use mathematical modelling to describe the kinetics of immune cell proliferation after an in vitro polyclonal stimulation tracked with CFSE. This approach allows us to estimate a set of key parameters, including ones related to cell death and proliferation. We develop a three-phase model that distinguishes a latency phase, accounting for non-divided cell behaviour, a resting phase and the active phase of the division process. Parameter estimates are derived from model results, and numerical simulations are then compared to the dynamics of in vitro experiments, with different biological assumptions tested. Our model allows us to compare the dynamics of CD4+ and CD8+ cells, and to highlight their kinetic differences. Finally we perform a sensitivity analysis to quantify the impact of each parameter on proliferation kinetics. Interestingly, we find that parameter sensitivity varies with time and with cell generation. Our approach can help biologists to understand cell proliferation mechanisms and to identify potential pathological division processes.

Structured models of cell migration incorporating molecular binding processes

The dynamic interplay between collective cell movement and the various molecules involved in the accompanying cell signalling mechanisms plays a crucial role in many biological processes including normal tissue development and pathological scenarios such as wound healing and cancer. Information about the various structures embedded within these processes allows a detailed exploration of the binding of molecular species to cell-surface receptors within the evolving cell population. In this paper we establish a general spatio-temporal-structural framework that enables the description of molecular binding to cell membranes coupled with the cell population dynamics. We first provide a general theoretical description for this approach and then illustrate it with three examples arising from cancer invasion.

Analysis of CFSE time-series data using division-, age- and label-structured population models

MOTIVATION

In vitro and in vivo cell proliferation is often studied using the dye carboxyfluorescein succinimidyl ester (CFSE). The CFSE time-series data provide information about the proliferation history of populations of cells. While the experimental procedures are well established and widely used, the analysis of CFSE time-series data is still challenging. Many available analysis tools do not account for cell age and employ optimization methods that are inefficient (or even unreliable).

RESULTS

We present a new model-based analysis method for CFSE time-series data. This method uses a flexible description of proliferating cell populations, namely, a division-, age- and label-structured population model. Efficient maximum likelihood and Bayesian estimation algorithms are introduced to infer the model parameters and their uncertainties. These methods exploit the forward sensitivity equations of the underlying partial differential equation model for efficient and accurate gradient calculation, thereby improving computational efficiency and reliability compared with alternative approaches and accelerating uncertainty analysis. The performance of the method is assessed by studying a dataset for immune cell proliferation. This revealed the importance of different factors on the proliferation rates of individual cells. Among others, the predominate effect of cell age on the division rate is found, which was not revealed by available computational methods.

AVAILABILITY AND IMPLEMENTATION

The MATLAB source code implementing the models and algorithms is available from http://janhasenauer.github.io/ShAPE-DALSP/Contact: jan.hasenauer@helmholtz-muenchen.de

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Parameters identification for a model of T cell homeostasis

In this study, we consider a model of T cell homeostasis based on the Smith-Martin model. This nonlinear model is structured by age and CD44 expression. First, we establish the mathematical well-posedness of the model system. Next, we prove the theoretical identifiability regarding the up-regulation of CD44, the proliferation time phase and the rate of entry into division, by using the experimental data. Finally, we compare two versions of the Smith-Martin model and we identify which model fits the experimental data best.

Predicting pathogen-specific CD8 T cell immune responses from a modeling approach

The primary CD8 T cell immune response constitutes a major mechanism to fight an infection by intra-cellular pathogens. We aim at assessing whether pathogen-specific dynamical parameters of the CD8 T cell response can be identified, based on measurements of CD8 T cell counts, using a modeling approach. We generated experimental data consisting in CD8 T cell counts kinetics during the response to three different live intra-cellular pathogens: two viruses (influenza, vaccinia) injected intranasally, and one bacteria (Listeria monocytogenes) injected intravenously. All pathogens harbor the same antigen (NP68), but differ in their interaction with the host. In parallel, we developed a mathematical model describing the evolution of CD8 T cell counts and pathogen amount during an immune response. This model is characterized by 9 parameters and includes relevant feedback controls. The model outputs were compared with the three data series and an exhaustive estimation of the parameter values was performed. By focusing on the ability of the model to fit experimental data and to produce a CD8 T cell population mainly composed of memory cells at the end of the response, critical parameters were identified. We show that a small number of parameters (2-4) define the main features of the CD8 T cell immune response and are characteristic of a given pathogen. Among these parameters, two are related to the effector CD8 T cell mediated control of cell and pathogen death. The parameter associated with memory cell death is shown to play no relevant role during the main phases of the CD8 T cell response, yet it becomes essential when looking at the predictions of the model several months after the infection.

Antiproliferative effect of the jararhagin toxin on B16F10 murine melanoma

BACKGROUND

Malignant melanoma is a less common but highly dangerous form of skin cancer; it starts in the melanocytes cells found in the outer layer of the skin. Jararhagin toxin, a metalloproteinase isolated from Bothrops jararaca snake venom acts upon several biological processes, as inflammation, pain, platelet aggregation, proliferation and apoptosis, though not yet approved for use, may one day be employed to treat tumors.

METHODS

B16F10 murine melanoma cells were treated with jararhagin (jara), a disintegrin-like metalloproteinase isolated from Bothrops jararaca snake venom, and jari (catalytic domain inactivated with 1,10-phenanthroline). Viability and adhesion cells were evaluated by MTT assay. The expression of caspase-3 active, phases of the cell cycle and apoptosis were assessed by flow cytometry. We analyze in vivo the effects of jararhagin on melanoma growth, apoptosis and metastasis.

RESULTS

The tumor cells acquired round shapes, lost cytoplasmic expansions, formed clusters in suspension and decreased viability. Jari was almost 20 times more potent toxin than jara based on IC50 values and on morphological changes of the cells, also observed by scanning electron microscopy. Flow cytometry analysis showed 48.3% decrease in the proliferation rate of cells and 47.2% increase in apoptosis (jara) and necrosis (jari), following 1.2 μM jara and 0.1 μM jari treatments. Caspase-3 activity was increased whereas G0/G1 cell cycle phase was on the decline. Proliferative rate was assessed by staining with 5,6-carboxyfluoresceindiacetate succinimidyl ester, showing a significant decrease in proliferation at all concentrations of both toxins.

CONCLUSIONS

In vivo treatment of the toxins was observed reduction in the incidence of nodules, and metastasis and antiproliferative inhibition capacity. This data strengthens the potential use jararhagin as an anti-neoplastic drug.

Parameter identification for model of T cell proliferation in lymphopenia conditions

The number of T Lymphocytes (T cells) in the body is under homeostatic control. At equilibrium, the majority of naive T cells are non-dividing and express low levels of the surface protein CD44. In conditions of T cell deficiency (lymphopenia), naive T cells enter into a proliferative phase, undergoing cell division accompanied by a subtle change in their surface expression of CD44. In this study, we use a mathematical modelling approach to analyse the proliferative response of transgenic T cells in lymphopenic conditions. Our nonlinear model is composed of ordinary differential equations and partial differential equations structured by age (maturity of cell) and CD44 expression. To better understand the evolution of CD44 expression on the surface of T cells during cell division, we present a numerical analysis to solve a parameter identification problem. Finally, we show the parameters and the simulations that we obtain from the model and compare them to experimental data.

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.

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.

Quantitative analysis of the effects of iododeoxyuridine and ionising radiation treatment on the cell cycle dynamics of DNA mismatch repair deficient human colorectal cancer cells

DNA mismatch repair (MMR) is involved in processing DNA damage following treatment with ionising radiation (IR) and various classes of chemotherapy drugs including iododeoxyuridine (IUdR), a known radiosensitiser. In this study, the authors have developed asynchronous probabilistic cell cycle models to assess the isolated effects of IUdR and IR and the combined effects of IUdR + IR treatments on MMR damage processing. The authors used both synchronous and asynchronous MMR-proficient/MMR-deficient cell populations and followed treated cells for up to two cell cycle times. They have observed and quantified differential cell cycle responses to MMR damage processing following IR and IUdR + IR treatments, principally in the duration of both G1 and G2/M cell cycle phases. The models presented in this work form the foundation for the development of an approach to maximise the therapeutic index for IR and IUdR + IR treatments in MMR-deficient (damage tolerant) cancers.

Modeling sequence evolution in HIV-1 infection with recombination

Previously we proposed two simplified models of early HIV-1 evolution. Both showed that under a model of neutral evolution and exponential growth, the mean Hamming distance (HD) between genetic sequences grows linearly with time. In this paper we describe a more realistic continuous-time, age-dependent mathematical model of infection and viral replication, and show through simulations that even in this more complex description, the mean Hamming distance grows linearly with time. This remains unchanged when we introduce recombination, though the confidence intervals of the mean HD obtained ignoring recombination are overly conservative.

Quantifying T lymphocyte turnover

Peripheral T cell populations are maintained by production of naive T cells in the thymus, clonal expansion of activated cells, cellular self-renewal (or homeostatic proliferation), and density dependent cell life spans. A variety of experimental techniques have been employed to quantify the relative contributions of these processes. In modern studies lymphocytes are typically labeled with 5-bromo-2'-deoxyuridine (BrdU), deuterium, or the fluorescent dye carboxy-fluorescein diacetate succinimidyl ester (CFSE), their division history has been studied by monitoring telomere shortening and the dilution of T cell receptor excision circles (TRECs) or the dye CFSE, and clonal expansion has been documented by recording changes in the population densities of antigen specific cells. Proper interpretation of such data in terms of the underlying rates of T cell production, division, and death has proven to be notoriously difficult and involves mathematical modeling. We review the various models that have been developed for each of these techniques, discuss which models seem most appropriate for what type of data, reveal open problems that require better models, and pinpoint how the assumptions underlying a mathematical model may influence the interpretation of data. Elaborating various successful cases where modeling has delivered new insights in T cell population dynamics, this review provides quantitative estimates of several processes involved in the maintenance of naive and memory, CD4(+) and CD8(+) T cell pools in mice and men.

Proliferation of fibroblasts and endothelial cells is enhanced by treatment with Phyllocaulis boraceiensis mucus

Previously, mucus of some molluscs has been studied as a potential source of new natural compounds capable of inducing cell proliferation and of remodelling tissue. Here, the focus of the study is possible use of mucus released by Phyllocaulis boraceiensis - a compound inducing cell proliferation and enhancing collagen synthesis in dermal fibroblasts and inducing proliferation human endothelial cell cultures. Fibroblasts treated with P. boraceiensis mucus at concentrations below 0.012 μg/μl developed high rates of proliferation, as evaluated using MTT assay; the proliferative effect was dose-dependent. Production and secretion of extracellular matrix components and collagen type I fibres were enhanced after 24 h of treatment, revealing a hormesis effect, biphasic dose response - low dose for proliferation yet toxic at high dose. No significant change in proliferation was observed in treated endothelial cells and production of lipid polyunsaturated free radicals was low in both cell types. Treatment with P. boraceiensis mucus produced pronounced changes in fibroblast cell number and morphology, and in quantities of well-ordered collagen deposition. These results support the premise that Phyllocaulis boraceiensis mucus demonstrates proliferative properties in cells involved in the healing process.

Neutrophil dynamics after chemotherapy and G-CSF: the role of pharmacokinetics in shaping the response

Chemotherapy has profound effects on the hematopoietic system, most notably leading to neutropenia. Granulocyte colony stimulating factor (G-CSF) is often used to deal with this neutropenia, but the response is highly variable. In this paper we examine the role of pharmacokinetics and delivery protocols in shaping the neutrophil responses to chemotherapy and G-CSF. Neutrophil responses to different protocols of chemotherapy administration with varying dosages, infusion times, and schedules are studied through a mathematical model. We find that a single dose of chemotherapy produces a damped oscillation in neutrophil levels, and short-term applications of chemotherapy can induce permanent oscillations in neutrophil level if there is a bistability in the system. In addition, we confirm previous findings [Zhuge et al., J. Theor. Biol., 293(2012), 111-120] that when periodic chemotherapy is given, there is a significant period of delivery that induces resonance in the system and exacerbates the corresponding neutropenia. The width of this resonant period peak increases with the recovery rate after a single chemotherapy, which is given by the real part of the dominant eigenvalue pair at the steady state, and both are determined by a single cooperativity coefficient in the feedback function for the neutrophils. Our numerical studies show that the neutropenia caused by chemotherapy can be overcome if G-CSF is given early after chemotherapy but can actually be worsened if G-CSF is given later, consistent with results reported in Zhuge et al. (2012). The nadir in neutrophil level is found to be more sensitive to the dosage of chemotherapy than that of the G-CSF. Furthermore, dependence of our results with changes in key pharmacokinetic parameters as well as initial functions are studied. Thus, this study illuminates the potential for destructive resonance leading to neutropenia in response to periodic chemotherapy, and explores and explains why the timing of G-CSF is so crucial for successful reversal of chemotherapy induced neutropenia.

The contribution of age structure to cell population responses to targeted therapeutics

Cells grown in culture act as a model system for analyzing the effects of anticancer compounds, which may affect cell behavior in a cell cycle position-dependent manner. Cell synchronization techniques have been generally employed to minimize the variation in cell cycle position. However, synchronization techniques are cumbersome and imprecise and the agents used to synchronize the cells potentially have other unknown effects on the cells. An alternative approach is to determine the age structure in the population and account for the cell cycle positional effects post hoc. Here we provide a formalism to use quantifiable lifespans from live cell microscopy experiments to parameterize an age-structured model of cell population response.

Antigen-specific responses assessment for the evaluation of Bordetella pertussis T cell immunity in humans

Measurement of antigen-specific T cell responses is an adjunctive parameter to evaluate protection induced by a previous Bordetella pertussis infection or vaccination. The assessment of T cell responses is technically complex and usually performed on fresh peripheral blood mononuclear cells (PBMC). The objective of this study was to identify simplified methods to assess pertussis specific T cell responses and verify if these assays could be performed using frozen/thawed (frozen) PBMC. Three read-outs to measure proliferation were compared: the fluorescent dye 5,6-carboxylfluorescein diacetate succinimidyl ester (CFSE) dilution test, the number of blast cells defined by physical parameters, and the incorporation of (3)H-thymidine. The results of pertussis-specific assays performed on fresh PBMC were compared to the results on frozen PBMC from the same donor. High concordance was obtained when the results of CFSE and blast read-outs were compared, an encouraging result since blast analysis allows the identification of proliferating cells and does not require any use of radioactive tracer as well as any staining. The results obtained using fresh and frozen PBMC from the same donor in the different T cell assays, including IFNγ and TNFα cytokine production, did not show significant differences, suggesting that a careful cryopreservation process of PBMC would not significantly influence T cell response evaluation. Adopting blast analysis and frozen PBMC, the possibility to test T cell responses is simplified and might be applied in population studies, providing for new instruments to better define correlates of protection still elusive in pertussis.

DYNAMICS OF THE DELAY HEMATOLOGICAL CELL MODEL

In this paper, complex dynamics of a two-compartment model of production and regulation of the circulating blood neutrophil number are investigated. It is shown that the proliferative disorders may be possible due to factors of the apoptosis rate rsof the haematopoietic stem cell and the cell cycle duration τs. Applying a recent geometrical criterion for the Hopf bifurcation and transient behaviors of delay systems to this model, we separate the stable regime from the unstable regime on the rs- τsplane. Numerically, regimes of patterned periodic oscillations with low periodicity in the number of circulating blood cells appear on the rs- τsplane. It is found that the dominated period-adding bifurcation mechanism leads transitions from period-n to period-(n + 1), eventually changes to the complex attractor with high-periodicity or chaos.

Systems biology of JAK-STAT signalling in human malignancies

Originally implicated in the regulation of survival, proliferation and differentiation of haematopoietic cells, the JAK-STAT pathway has also been linked to developmental processes, growth control and maintenance of homeostasis in a variety of other cells and tissues. Although it remains a complex system, its relative simplicity and the availability of molecular data makes it particularly attractive for modelling approaches. In this review, we will focus on JAK-STAT signalling in the context of cancer and present efforts to investigate signalling dynamics with the help of mathematical models. We describe the modelling workflow that realises a systems biology approach and give an example for interferon-γ signalling in pancreatic stellate cells.

Evaluation of multitype mathematical models for CFSE-labeling experiment data

Carboxy-fluorescein diacetate succinimidyl ester (CFSE) labeling is an important experimental tool for measuring cell responses to extracellular signals in biomedical research. However, changes of the cell cycle (e.g., time to division) corresponding to different stimulations cannot be directly characterized from data collected in CFSE-labeling experiments. A number of independent studies have developed mathematical models as well as parameter estimation methods to better understand cell cycle kinetics based on CFSE data. However, when applying different models to the same data set, notable discrepancies in parameter estimates based on different models has become an issue of great concern. It is therefore important to compare existing models and make recommendations for practical use. For this purpose, we derived the analytic form of an age-dependent multitype branching process model. We then compared the performance of different models, namely branching process, cyton, Smith-Martin, and a linear birth-death ordinary differential equation (ODE) model via simulation studies. For fairness of model comparison, simulated data sets were generated using an agent-based simulation tool which is independent of the four models that are compared. The simulation study results suggest that the branching process model significantly outperforms the other three models over a wide range of parameter values. This model was then employed to understand the proliferation pattern of CD4+ and CD8+ T cells under polyclonal stimulation.

Integration of Principles of Systems Biology and Radiation Biology: Toward Development of in silico Models to Optimize IUdR-Mediated Radiosensitization of DNA Mismatch Repair Deficient (Damage Tolerant) Human Cancers

Over the last 7 years, we have focused our experimental and computational research efforts on improving our understanding of the biochemical, molecular, and cellular processing of iododeoxyuridine (IUdR) and ionizing radiation (IR) induced DNA base damage by DNA mismatch repair (MMR). These coordinated research efforts, sponsored by the National Cancer Institute Integrative Cancer Biology Program (ICBP), brought together system scientists with expertise in engineering, mathematics, and complex systems theory and translational cancer researchers with expertise in radiation biology. Our overall goal was to begin to develop computational models of IUdR- and/or IR-induced base damage processing by MMR that may provide new clinical strategies to optimize IUdR-mediated radiosensitization in MMR deficient (MMR⁻) “damage tolerant” human cancers. Using multiple scales of experimental testing, ranging from purified protein systems to in vitro (cellular) and to in vivo (human tumor xenografts in athymic mice) models, we have begun to integrate and interpolate these experimental data with hybrid stochastic biochemical models of MMR damage processing and probabilistic cell cycle regulation models through a systems biology approach. In this article, we highlight the results and current status of our integration of radiation biology approaches and computational modeling to enhance IUdR-mediated radiosensitization in MMR⁻ damage tolerant cancers.

Stochastic models of lymphocyte proliferation and death

Quantitative understanding of the kinetics of lymphocyte proliferation and death upon activation with an antigen is crucial for elucidating factors determining the magnitude, duration and efficiency of the immune response. Recent advances in quantitative experimental techniques, in particular intracellular labeling and multi-channel flow cytometry, allow one to measure the population structure of proliferating and dying lymphocytes for several generations with high precision. These new experimental techniques require novel quantitative methods of analysis. We review several recent mathematical approaches used to describe and analyze cell proliferation data. Using a rigorous mathematical framework, we show that two commonly used models that are based on the theories of age-structured cell populations and of branching processes, are mathematically identical. We provide several simple analytical solutions for a model in which the distribution of inter-division times follows a gamma distribution and show that this model can fit both simulated and experimental data. We also show that the estimates of some critical kinetic parameters, such as the average inter-division time, obtained by fitting models to data may depend on the assumed distribution of inter-division times, highlighting the challenges in quantitative understanding of cell kinetics.

An age-dependent branching process model for the analysis of CFSE-labeling experiments

BACKGROUND

Over the past decade, flow cytometric CFSE-labeling experiments have gained considerable popularity among experimentalists, especially immunologists and hematologists, for studying the processes of cell proliferation and cell death. Several mathematical models have been presented in the literature to describe cell kinetics during these experiments.

RESULTS

We propose a multi-type age-dependent branching process to model the temporal development of populations of cells subject to division and death during CFSE-labeling experiments. We discuss practical implementation of the proposed model; we investigate a competing risk version of the process; and we identify the classes of cellular dependencies that may influence the expectation of the process and those that do not. An application is presented where we study the proliferation of human CD8+ T lymphocytes using our model and a competing risk branching process.

CONCLUSIONS

The proposed model offers a widely applicable approach to the analysis of CFSE-labeling experiments. The model fitted very well our experimental data. It provided reasonable estimates of cell kinetics parameters as well as meaningful insights into the processes of cell division and cell death. In contrast, the competing risk branching process could not describe the kinetics of CD8+ T cells. This suggested that the decision of cell division or cell death may be made early in the cell cycle if not in preceding generations. Also, we show that analyses based on the proposed model are robust with respect to cross-sectional dependencies and to dependencies between fates of linearly filiated cells.

REVIEWERS

This article was reviewed by Marek Kimmel, Wai-Yuan Tan and Peter Olofsson.

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.

Multi-scale Modelling for Threshold Dependent Differentiation

The maintenance of a stable stem cell population in the epidermis is important for robust regeneration of the stratified epithelium. The population size is usually regulated by cell secreted extracellular signalling molecules as well as intracellular molecules. In this paper, a simple model incorporating both levels of regulation is developed to examine the balance between growth and differentiation for the stem cell population. In particular, the dynamics of a known differentiation regulator c-Myc, its threshold dependent differentiation, and feedback regulation on maintaining a stable stem cell population are investigated.

Interpreting CFSE obtained division histories of B cells in vitro with Smith-Martin and cyton type models

The fluorescent dye carboxyfluorescin diacetate succinimidyl ester (CFSE) classifies proliferating cell populations into groups according to the number of divisions each cell has undergone (i.e., its division class). The pulse labeling of cells with radioactive thymidine provides a means to determine the distribution of times of entry into the first cell division. We derive in analytic form the number of cells in each division class as a function of time using the cyton approach that utilizes independent stochastic distributions for the time to divide and the time to die. We confirm that our analytic form for the number of cells in each division class is consistent with the numerical solution of a set of delay differential equations representing the generalized Smith-Martin model with cell death rates depending on the division class. Choosing the distribution of time to the first division to fit thymidine labeling data for B cells stimulated in vitro with lipopolysaccharide (LPS) and either with or without interleukin-4 (IL-4), we fit CFSE data to determine the dependence of B cell kinetic parameters on the presence of IL-4. We find when IL-4 is present, a greater proportion of cells are recruited into division with a longer average time to first division. The most profound effect of the presence of IL-4 was decreased death rates for smaller division classes, which supports a role of IL-4 in the protection of B cells from apoptosis.

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.

Quantitative proliferation dynamics and random chromosome segregation of hair follicle stem cells

Regulation of stem cell (SC) proliferation is central to tissue homoeostasis, injury repair, and cancer development. Accumulation of replication errors in SCs is limited by either infrequent division and/or by chromosome sorting to retain preferentially the oldest 'immortal' DNA strand. The frequency of SC divisions and the chromosome-sorting phenomenon are difficult to examine accurately with existing methods. To address this question, we developed a strategy to count divisions of hair follicle (HF) SCs over time, and provide the first quantitative proliferation history of a tissue SC during its normal homoeostasis. We uncovered an unexpectedly high cellular turnover in the SC compartment in one round of activation. Our study provides quantitative data in support of the long-standing infrequent SC division model, and shows that HF SCs do not retain the older DNA strands or sort their chromosome. This new ability to count divisions in vivo has relevance for obtaining basic knowledge of tissue kinetics.

Identification of age-structured models: cell cycle phase transitions

A methodology is developed that determines age-specific transition rates between cell cycle phases during balanced growth by utilizing age-structured population balance equations. Age-distributed models are the simplest way to account for varied behavior of individual cells. However, this simplicity is offset by difficulties in making observations of age distributions, so age-distributed models are difficult to fit to experimental data. Herein, the proposed methodology is implemented to identify an age-structured model for human leukemia cells (Jurkat) based only on measurements of the total number density after the addition of bromodeoxyuridine partitions the total cell population into two subpopulations. Each of the subpopulations will temporarily undergo a period of unbalanced growth, which provides sufficient information to extract age-dependent transition rates, while the total cell population remains in balanced growth. The stipulation of initial balanced growth permits the derivation of age densities based on only age-dependent transition rates. In fitting the experimental data, a flexible transition rate representation, utilizing a series of cubic spline nodes, finds a bimodal G(0)/G(1) transition age probability distribution best fits the experimental data. This resolution may be unnecessary as convex combinations of more restricted transition rates derived from normalized Gaussian, lognormal, or skewed lognormal transition-age probability distributions corroborate the spline predictions, but require fewer parameters. The fit of data with a single log normal distribution is somewhat inferior suggesting the bimodal result as more likely. Regardless of the choice of basis functions, this methodology can identify age distributions, age-specific transition rates, and transition-age distributions during balanced growth conditions.

Vascular adaptation and mechanical homeostasis at tissue, cellular, and sub-cellular levels

Blood vessels exhibit a remarkable ability to adapt throughout life that depends upon genetic programming and well-orchestrated biochemical processes. Findings over the past four decades demonstrate, however, that the mechanical environment experienced by these vessels similarly plays a critical role in governing their adaptive responses. This article briefly reviews, as illustrative examples, six cases of tissue level growth and remodeling, and then reviews general observations at cell-matrix, cellular, and sub-cellular levels, which collectively point to the existence of a "mechanical homeostasis" across multiple length and time scales that is mediated primarily by endothelial cells, vascular smooth muscle cells, and fibroblasts. In particular, responses to altered blood flow, blood pressure, and axial extension, disease processes such as cerebral aneurysms and vasospasm, and diverse experimental manipulations and clinical treatments suggest that arteries seek to maintain constant a preferred (homeostatic) mechanical state. Experiments on isolated microvessels, cell-seeded collagen gels, and adherent cells isolated in culture suggest that vascular cells and sub-cellular structures such as stress fibers and focal adhesions likewise seek to maintain constant a preferred mechanical state. Although much is known about mechanical homeostasis in the vasculature, there remains a pressing need for more quantitative data that will enable the formulation of an integrative mathematical theory that describes and eventually predicts vascular adaptations in response to diverse stimuli. Such a theory promises to deepen our understanding of vascular biology as well as to enable the design of improved clinical interventions and implantable medical devices.

Adding self-renewal in committed erythroid progenitors improves the biological relevance of a mathematical model of erythropoiesis

We propose a new mathematical model of erythropoiesis that takes a positive feedback of erythrocytes on progenitor apoptosis into account, and incorporates a negative feedback of erythrocytes on progenitor self-renewal. The resulting model is a system of age-structured equations that reduces to a system of delay differential equations where the delays account for progenitor compartment duration and cell cycle length. We compare this model with experimental data on an induced-anemia in mice that exhibit damped oscillations of the hematocrit before it returns to equilibrium. When we assume no self-renewal of progenitors, we obtain an inaccurate fitting of the model with experimental data. Adding self-renewal in the progenitor compartment gives better approximations, with the main features of experimental data correctly fitted. Our results indicate the importance of progenitor self-renewal in the modelling of erythropoiesis. Moreover, the model makes testable predictions on the lifespan of erythrocytes confronted to a severe anemia, and on the progenitors behavior.

IL-2 regulates expansion of CD4+ T cell populations by affecting cell death: insights from modeling CFSE data

It is generally accepted that IL-2 influences the dynamics of populations of T cells in vitro and in vivo. However, which parameters for cell division and/or death are affected by IL-2 is not well understood. To get better insights into the potential ways of how IL-2 may influence the population dynamics of T cells, we analyze data on the dynamics of CFSE-labeled polyclonal CD4(+) T lymphocytes in vitro after anti-CD3 stimulation at different concentrations of exogenous IL-2. Inferring cell division and death rates from CFSE-delabeling experiments is not straightforward and requires the use of mathematical models. We find that to adequately describe the dynamics of T cells at low concentrations of exogenous IL-2, the death rate of divided cells has to increase with the number of divisions cells have undergone. IL-2 hardly affects the average interdivision time. At low IL-2 concentrations 1) fewer cells are recruited into the response and successfully complete their first division; 2) the stochasticity of cell division is increased; and 3) the rate, at which the death rate increases with the division number, increases. Summarizing, our mathematical reinterpretation suggests that the main effect of IL-2 on the in vitro dynamics of naive CD4(+) T cells occurs by affecting the rate of cell death and not by changing the rate of cell division.

Modeling T cell proliferation and death in vitro based on labeling data: generalizations of the Smith-Martin cell cycle model

The fluorescent dye carboxyfluorescein diacetate succinimidyl ester (CFSE) classifies proliferating cell populations into groups according to the number of divisions each cell has undergone (i.e., its division class). The pulse labeling of cells with radioactive thymidine provides a means to determine the distribution of times of entry into the first cell division. We derive in analytic form the number of cells in each division class as a function of time based on the distribution of times to the first division. Choosing the distribution of time to the first division to fit thymidine labeling data for T cells stimulated in vitro under different concentrations of IL-2, we fit CFSE data to determine the dependence of T cell kinetic parameters on the concentration of IL-2. As the concentration of IL-2 increases, the average cell cycle time is shortened, the death rate of cells is decreased, and a higher fraction of cells is recruited into division. We also find that if the average cell cycle time increases with division class then the qualify of our fit to the data improves.

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.

Reconstruction of cell population dynamics using CFSE

Background

Quantifying cell division and death is central to many studies in the biological sciences. The fluorescent dye CFSE allows the tracking of cell division in vitro and in vivo and provides a rich source of information with which to test models of cell kinetics. Cell division and death have a stochastic component at the single-cell level, and the probabilities of these occurring in any given time interval may also undergo systematic variation at a population level. This gives rise to heterogeneity in proliferating cell populations. Branching processes provide a natural means of describing this behaviour.

Results

We present a likelihood-based method for estimating the parameters of branching process models of cell kinetics using CFSE-labeling experiments, and demonstrate its validity using synthetic and experimental datasets. Performing inference and model comparison with real CFSE data presents some statistical problems and we suggest methods of dealing with them.

Conclusion

The approach we describe here can be used to recover the (potentially variable) division and death rates of any cell population for which division tracking information is available.

Asymptotic behaviour of solutions to abstract logistic equations

We analyze the asymptotic behaviour of solutions of the abstract differential equation u'(t)=Au(t)-F(u(t))u(t)+f. Our results are applicable to models of structured population dynamics in which the state space consists of population densities with respect to the structure variables. In the equation the linear term A corresponds to internal processes independent of crowding, the nonlinear logistic term F corresponds to the influence of crowding, and the source term f corresponds to external effects. We analyze three separate cases and show that for each case the solutions stabilize in a way governed by the linear term. We illustrate the results with examples of models of structured population dynamics -- a model for the proliferation of cell lines with telomere shortening, a model of proliferating and quiescent cell populations, and a model for the growth of tumour cord cell populations.

Comparative analysis between CFSE flow cytometric and tritiated thymidine incorporation tests for beryllium sensitivity

BACKGROUND

In this study, we evaluated alternative possibility for CFSE beryllium flow cytometric test against beryllium blood lymphocyte proliferation test (BeLPT) as a standard radioactive clinical screening method to identify sensitization to beryllium.

METHODS

Delta PD (the ratio of divided cell population to the total number of cells with subtracted counts of unstimulated cells) of specific beryllium-induced pathogenic CD3+ CD4+ T-lymphocytes and stimulation index (SI) in CFSE proliferation test was compared with delta counts per minute (mean test CPM minus mean control CPM) and SI in radioactive blood BeLPT.

RESULTS

Comparison analysis of CFSE and BeLPT demonstrated excellent agreement between delta PD and delta CPM (kappa = 0.845, P << 0.0001). We determined 6.8% positive subjects in the beryllium-exposed, Be-LPT-negative group. The decreased mean difference of these indexes to percentage of average and the long tail in the plot reflects increased sensitivity. CFSE/CD4+ T-cell proliferation assay has 100% specificity, significantly higher sensitivity and efficiency than BeLPT.

CONCLUSIONS

Both delta PD, measured by the precursor frequencies method in CFSE assay and delta CPM, defined by tritiated thymidine in BeLPT, can be used for the enumeration of beryllium specific CD4+ T-cell proliferation and may substantially improve the quality of the early diagnosis of beryllium hypersensitivity.

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.

Modelling Hematopoiesis Mediated by Growth Factors With Applications to Periodic Hematological Diseases

Hematopoiesis is a complex biological process that leads to the production and regulation of blood cells. It is based upon differentiation of stem cells under the action of growth factors. A mathematical approach of this process is proposed to understand some blood diseases characterized by very long period oscillations in circulating blood cells. A system of three differential equations with delay, corresponding to the cell cycle duration, is proposed and analyzed. The existence of a Hopf bifurcation at a positive steady-state is obtained through the study of an exponential polynomial characteristic equation with delay-dependent coefficients. Numerical simulations show that long-period oscillations can be obtained in this model, corresponding to a destabilization of the feedback regulation between blood cells and growth factors, for reasonable cell cycle durations. These oscillations can be related to observations on some periodic hematological diseases (such as chronic myelogenous leukemia, for example).