A mathematical model of hematopoiesis: II. Cyclical neutropenia

Cyclical neutropenia is a dynamical disease of the hematopoietic system marked by an oscillation in circulating leukocyte (e.g. neutrophil) numbers to near zero levels and then back to normal. This oscillation is also mirrored in the platelets and reticulocytes which oscillate with the same period. Cyclical neutropenia has an animal counterpart in the grey collie. Using the mathematical model of the hematopoietic system of Colijn and Mackey [A mathematical model of hematopoiesis: I. Periodic chronic myelogenous leukemia. Companion paper to the present paper.] we have determined what parameters are necessary to mimic laboratory and clinical data on untreated grey collies and humans, and also what changes in these parameters are necessary to fit data during treatment with granulocyte colony stimulating factor (G-CSF). Compared to the normal steady-state values, we found that the major parameter changes that mimic untreated cyclical neutropenia correspond to a decreased amplification (increased apoptosis) within the proliferating neutrophil precursor compartment, and a decrease in the maximal rate of re-entry into the proliferative phase of the stem cell compartment. For the data obtained during G-CSF treatment, good fits were obtained only when parameters were altered that would imply that G-CSF led to higher amplification (lower rate of apoptosis) in the proliferating neutrophil precursors, and a elevated rate of differentiation into the neutrophil line.     

Oscillations in cyclical neutropenia: new evidence based on mathematical modeling

We present a dynamical model of the production and regulation of circulating blood neutrophil number. This model is derived from physiologically relevant features of the hematopoietic system, and is analysed using both analytic and numerical methods. Supercritical Hopf bifurcations and saddle-node bifurcations of limit cycles are shown to exist. We make the estimation of kinetic parameters for dogs and then apply the model to cyclical neutropenia (CN) in the grey collie, a rare disorder in which oscillations in all blood cell counts are found. We conclude that the major cause of the oscillations in CN is an increased rate of apoptosis of neutrophil precursors which leads to a destabilization of the hematopoietic stem cell compartment.     

High frequency spikes in long period blood cell oscillations

Several hematological diseases are characterised by oscillations of various blood cell populations. Two of these are a variant of chronic myelogenous leukemia (CML) and cyclical neutropenia (CN). These oscillations typically have long periods ranging from 20 to 60 days, despite the fact that the stem cell cycling time is thought to be of the order of 2–3 days. Clinical data from humans and laboratory data from the grey collie animal model of CN is suggestive of the idea that these long period oscillations may also contain higher frequency spiky oscillations. We show how such oscillations can be understood in the context of slow periodic stem cell oscillations, by analysing a two component differential-delay equation model of stem cell and neutrophil populations.For Karl Hadeler, on his 70th birthday, leader, teacher, colleague and friend.     

Periodic oscillations in leukopoiesis models with two delays

The term leukopoiesis describes processes leading to the production and regulation of white blood cells. It is based on stem cells differentiation and may exhibit abnormalities resulting in severe diseases, such as cyclical neutropenia and leukemias. We consider a nonlinear system of two equations, describing the evolution of a stem cell population and the resulting white blood cell population. Two delays appear in this model to describe the cell cycle duration of the stem cell population and the time required to produce white blood cells. We establish sufficient conditions for the asymptotic stability of the unique nontrivial positive steady state of the model by analysing roots of a second degree exponential polynomial characteristic equation with delay-dependent coefficients. We also prove the existence of a Hopf bifurcation which leads to periodic solutions. Numerical simulations of the model with parameter values reported in the literature demonstrate that periodic oscillations (with short and long periods) agree with observations of cyclical neutropenia in patients.     

Multistability in an age-structured model of hematopoiesis: Cyclical neutropenia

Cyclical neutropenia (CN) is a rare hematopoietic disorder in which the patient's neutrophil level drops to extremely low levels for a few days approximately every three weeks. CN is effectively treated with granulocyte colony stimulating factor (G-CSF), which is known to interfere with apoptosis in neutrophil precursors and to consequently increase the circulating neutrophil level. However, G-CSF treatment usually fails to eliminate the oscillation. In this study, we establish an age-structured model of hematopoiesis, which reduces to a set of four delay differential equations with specific forms of initial functions. We numerically investigate the possible stable solutions of the model equations with respect to changes in the parameters as well as the initial conditions. The results show that the hematopoietic system possesses multistability for parameters typical of the normal healthy state. From our numerical results, decreasing the proliferation rate of neutrophil precursors or increasing the stem cell death rate are two possible mechanisms to induce cyclical neutropenia, and the periods of the resulting oscillations are independent of the changing parameters. We also discuss the dependence of the model solution on the initial condition at normal parameter values corresponding to a healthy state. Using insight from our results we design a hybrid treatment method that is able to abolish the oscillations in CN.     

Cost-effective G-CSF therapy strategies for cyclical neutropenia: mathematical modelling based hypotheses

Using computer simulations of a mathematical model for the regulation of stem cell and neutrophil production in dogs, we have studied the efficacy of four different treatment protocols for cyclical neutropenia involving granulocyte colony stimulating factor (G-CSF). The first treatment scheme is based on the bifurcation analysis of the mathematical model and proposes a daily, phase-dependent, protocol. The second involves alternate day administration of G-CSF. The third triggers G-CSF administration whenever neutrophil levels fall below a predetermined level, and the fourth one follows a random administration protocol. The computer simulations predict that clinically desirable results can be achieved with the three last methods, using far less G-CSF than would be needed with the standard daily treatment. If the results of this modelling are borne out clinically, they will entail a considerable financial savings for patients.     

A Model of Oscillatory Blood Cell Counts in Chronic Myelogenous Leukaemia

In certain blood diseases, oscillations are found in blood cell counts. Particularly, such oscillations are sometimes found in chronic myelogenous leukaemia, and then occur in all the derived blood cell types: red blood cells, white blood cells, and platelets. It has been suggested that such oscillations arise because of an instability in the pluri-potential stem cell population, associated with its regulatory control system. In this paper, we consider how such oscillations can arise in a model of competition between normal (S) and genetically altered abnormal (A) stem cells, as the latter population grows at the expense of the former. We use an analytic model of long period oscillations to describe regions of oscillatory behaviour in the S–A phase plane, and give parametric criteria to describe when such oscillations will occur. We also describe a mechanism which can explain dynamically how the transformation from chronic phase to acute phase and blast crisis can occur.     

The rate of apoptosis in post mitotic neutrophil precursors of normal and neutropenic humans

Abstract. Using data on the fraction of post-mitotic neutrophil precursors (CD15⁺ cells) displaying positive markers for apoptosis in 12 normal humans, and a simple mathematical model, we have estimated the apoptotic rate to be about 0.28/day in this compartment. This implies that the influx of myelocytes into the post-mitotic compartment exceeds twice the granulocyte turnover rate (GTR), and that about 55% of the cells entering this compartment die before being released into the blood. The normal half life of apoptotic post-mitotic neutrophil precursors is calculated to be 10.4 h. Comparable calculations for patients indicate apoptosis rates in the post-mitotic compartment of about 17 times normal for one myelokathexis patient and rates of about 13 times normal for the one cyclical neutropenic patient and two severe congenital neutropenic patients. The estimated half life for apoptotic post-mitotic neutrophil precursors in the myelokathexis patient was about 0.4 h, 1.4 h in the cyclical neutropenia patient, and about 0.6 h in the severe congenital neutropenic patients.     

The Dynamics of Stress p53-Mdm2 Network Regulated by p300 and HDAC1

We construct a stress p53-Mdm2-p300-HDAC1 regulatory network that is activated and stabilised by two regulatory proteins, p300 and HDAC1. Different activation levels of observed due to these regulators during stress condition have been investigated using a deterministic as well as a stochastic approach to understand how the cell responds during stress conditions. We found that these regulators help in adjusting p53 to different conditions as identified by various oscillatory states, namely fixed point oscillations, damped oscillations and sustain oscillations. On assessing the impact of p300 on p53-Mdm2 network we identified three states: first stabilised or normal condition where the impact of p300 is negligible, second an interim region where p53 is activated due to interaction between p53 and p300, and finally the third regime where excess of p300 leads to cell stress condition. Similarly evaluation of HDAC1 on our model led to identification of the above three distinct states. Also we observe that noise in stochastic cellular system helps to reach each oscillatory state quicker than those in deterministic case. The constructed model validated different experimental findings qualitatively.     

Oscillatory protein expression dynamics endows stem cells with robust differentiation potential

The lack of understanding of stem cell differentiation and proliferation is a fundamental problem in developmental biology. Although gene regulatory networks (GRNs) for stem cell differentiation have been partially identified, the nature of differentiation dynamics and their regulation leading to robust development remain unclear. Herein, using a dynamical system modeling cell approach, we performed simulations of the developmental process using all possible GRNs with a few genes, and screened GRNs that could generate cell type diversity through cell-cell interactions. We found that model stem cells that both proliferated and differentiated always exhibited oscillatory expression dynamics, and the differentiation frequency of such stem cells was regulated, resulting in a robust number distribution. Moreover, we uncovered the common regulatory motifs for stem cell differentiation, in which a combination of regulatory motifs that generated oscillatory expression dynamics and stabilized distinct cellular states played an essential role. These findings may explain the recently observed heterogeneity and dynamic equilibrium in cellular states of stem cells, and can be used to predict regulatory networks responsible for differentiation in stem cell systems.     

Current insights into neutrophil homeostasis

Neutrophil granulocytes represent the first immunologic barrier against invading pathogens, and neutropenia predisposes to infection. However, neutrophils may also cause significant collateral inflammatory damage. Therefore, neutrophil numbers are tightly regulated by an incompletely understood homeostatic feedback loop adjusting the marrow's supply to peripheral needs. Granulocyte colony-stimulating factor (G-CSF) is accepted to be the major determinant of neutrophil production, and G-CSF levels have, soon after its discovery, been described to be inversely correlated with neutrophil counts. A neutrophil sensor, or "neutrostat," has, therefore, been postulated. The prevailing feedback hypothesis was established in adhesion molecule-deficient mice; it includes macrophages and Th17 cells, which determine G-CSF levels in response to the number of peripherally transmigrated, apoptosing neutrophils. Recent work has deepened our understanding of homeostatic regulation of neutrophil granulopoiesis, but there are still inconsistent findings and unresolved questions when it comes to a plausible hypothesis, similar to the feedback control models of red cell or platelet homeostasis.     

A model of the oscillatory metabolism of activated neutrophils

We present a two-compartment model to explain the oscillatory behavior observed experimentally in activated neutrophils. Our model is based mainly on the peroxidase-oxidase reaction catalyzed by myeloperoxidase with melatonin as a cofactor and NADPH oxidase, a major protein in the phagosome membrane of the leukocyte. The model predicts that after activation of a neutrophil, an increase in the activity of the hexose monophosphate shunt and the delivery of myeloperoxidase into the phagosome results in oscillations in oxygen and NAD(P)H concentration. The period of oscillation changes from >200 s to 10-30 s. The model is consistent with previously reported oscillations in cell metabolism and oxidant production. Key features and predictions of the model were confirmed experimentally. The requirement of the hexose monophosphate pathway for 10 s oscillations was verified using 6-aminonicotinamide and dexamethasone, which are inhibitors of glucose-6-phosphate dehydrogenase. The role of the NADPH oxidase in promoting oscillations was confirmed by dose-response studies of the effect of diphenylene iodonium, an inhibitor of the NADPH oxidase. Moreover, the model predicted an increase in the amplitude of NADPH oscillations in the presence of melatonin, which was confirmed experimentally. Successful computer modeling of complex chemical dynamics within cells and their chemical perturbation will enhance our ability to identify new antiinflammatory compounds.     

Model of oscillatory activity in thalamic neurons: Role of voltage- and calcium-dependent ionic conductances

This paper describes a computer modeling study of the generation of 10 Hz oscillations in the electrical activity of guinea pig thalamic neurons in vitro. The computer model was based on experimental evidence suggesting that single thalamic neurons in guinea pig have a set of voltage- and calcium-dependent ionic conductances that is capable of generating self-sustained rhythmic oscillations. Simulation results are consistent with this hypothesis, and indicate that a model that contains dendritic calcium and calcium-dependent potassium conductances, as well as a voltage-dependent, slow sodium conductance, can indeed generate self-sustained oscillations like those seen in thalamic neurons. Moreover, simulations indicate that the occurrence of such oscillatory activity is strongly dependent on the location of the slow sodium conductance. Results predict that this slow sodium conductance is located in the dendrites.The authors express their appreciation to R. J. MacGregor for providing equations and computer programs for simulating a two-point neuronal model with active calcium-related conductances     

Mechanical and biochemical modeling of cortical oscillations in spreading cells

Actomyosin-based cortical contractility is a common feature of eukaryotic cells and is involved in cell motility, cell division, and apoptosis. In nonmuscle cells, oscillations in contractility are induced by microtubule depolymerization during cell spreading. We developed an ordinary differential equation model to describe this behavior. The computational model includes 36 parameters. The values for all but two of the model parameters were taken from experimental measurements found in the literature. Using these values, we demonstrate that the model generates oscillatory behavior consistent with current experimental observations. The rhythmic behavior occurs because of the antagonistic effects of calcium-induced contractility and stretch-activated calcium channels. The model makes several experimentally testable predictions: 1), buffering intracellular calcium increases the period and decreases the amplitude of cortical oscillations; 2), increasing the number or activity of stretch activated channels leads to an increase in period and amplitude of cortical oscillations; 3), inhibiting Ca²⁺ pump activity increases the period and amplitude of oscillations; and 4), a threshold exists for the calcium concentration below which oscillations cease.     

Population balance model of in vivo neutrophil formation following bone marrow rescue therapy

In this paper, we develop a simple four parameter population balance model of in vivo neutrophil formation following bone marrow rescue therapy. The model is used to predict the number and type of neutrophil progenitors required to abrogate the period of severe neutropenia that normally follows a bone marrow transplant. The estimated total number of 5 billion neutrophil progenitors is consistent with the value extrapolated from a human trial. The model provides a basis for designing ex vivo expansion protocols. This revised version was published online in August 2006 with corrections to the Cover Date.     

Two redundant oscillatory mechanisms in the yeast cell cycle

The cell division cycle requires oscillations in activity of B-type cyclin (Clb)-Cdk1 kinases. Oscillations are due to periodic cyclin degradation by the anaphase-promoting complex (APC) activated by Cdc20 or Cdh1, and to cyclical accumulation of the Sic1 inhibitor. The results presented here suggest that the regulatory machinery controlling Clb kinase levels embeds two distinct oscillatory mechanisms. One, a "relaxation oscillator," involves alternation between two meta-stable states: Clb high/inhibitors (Sic1/APC-Cdh1) low, and Clb low/inhibitors high. The other, a "negative feedback oscillator," involves Clb kinase activation of APC-Cdc20, leading to Clb degradation. Genetic analysis suggests that these two mechanisms can function independently, and inactivation of both mechanisms is required to prevent mitosis. Computational modeling confirms that two such mechanisms can be linked to yield a robust cell cycle control system.     

Modulation of two oscillatory networks in the peripheral olfactory system by gamma-aminobutyric acid, glutamate, and acetylcholine in the terrestrial slug Limax marginatus

The digit-like extensions (the digits) of the tentacular ganglion of the terrestrial slug Limax marginatus are the cell body rich region in the primary olfactory system, and they contain primary olfactory neurons and projection neurons that send their axons to the olfactory center via the tentacular nerves. Two cell clusters (the cell masses) at the bases of the digits form the other cell body rich regions. Although the spontaneous slow oscillations and odor responses in the tentacular nerve have been studied, the origin of the oscillatory activity is unknown. In the present study, we examined the contribution of the neurons in the digits and cell masses to generation of the tentacular nerve oscillations by surgical removal from the whole tentacle preparations. Both structures contributed to the tentacular oscillations, and surgical isolation of the digits from the whole tentacle preparations still showed spontaneous oscillations. To analyze the dynamics of odor-processing circuits in the digits and tentacular ganglia, we studied the effects of gamma-aminobutyric acid, glutamate, and acetylcholine on the circuit dynamics of the oscillatory network(s) in the peripheral olfactory system. Bath or local puff application of gamma-aminobutyric acid to the cell masses decreased the tentacular nerve oscillations, whereas the bath or local puff application of glutamate and acetylcholine to the digits increased the digits' oscillations. Our results suggest the existence of two intrinsic oscillatory circuits that respond differentially to endogenous neurotransmitters in the primary olfactory system of slugs.