A reduced mathematical model of the acute inflammatory response II. Capturing scenarios of repeated endotoxin administration

Bacterial lipopolysaccharide (LPS; endotoxin) is a potent immunostimulant that can induce an acute inflammatory response comparable to a bacterial infection. Experimental observations demonstrate that this biological response can be either blunted (tolerance) or augmented (potentiation) with repeated administration of endotoxin. Both phenomena are of clinical relevance. We show that a four-dimensional differential equation model of this response reproduces many scenarios involving repeated endotoxin administration. In particular, the model can display both tolerance and potentiation from a single parameter set, under different administration scenarios. The key determinants of the outcome of our simulations are the relative time-scales of model components. These findings support the hypothesis that endotoxin tolerance and other related phenomena can be considered as dynamic manifestations of a unified acute inflammatory response, and offer specific predictions related to the dynamics of this response to endotoxin.     

A dynamical model of human immune response to influenza A virus infection

We present a simplified dynamical model of immune response to uncomplicated influenza A virus (IAV) infection, which focuses on the control of the infection by the innate and adaptive immunity. Innate immunity is represented by interferon-induced resistance to infection of respiratory epithelial cells and by removal of infected cells by effector cells (cytotoxic T-cells and natural killer cells). Adaptive immunity is represented by virus-specific antibodies. Similar in spirit to the recent model of Bocharov and Romanyukha [1994. Mathematical model of antiviral immune response. III. Influenza A virus infection. J. Theor. Biol. 167, 323-360], the model is constructed as a system of 10 ordinary differential equations with 27 parameters characterizing the rates of various processes contributing to the course of disease. The parameters are derived from published experimental data or estimated so as to reproduce available data about the time course of IAV infection in a na?ve host. We explore the effect of initial viral load on the severity and duration of the disease, construct a phase diagram that sheds insight into the dynamics of the disease, and perform sensitivity analysis on the model parameters to explore which ones influence the most the onset, duration and severity of infection. To account for the variability and speed of adaptation of the adaptive response to a particular virus strain, we introduce a variable that quantifies the antigenic compatibility between the virus and the antibodies currently produced by the organism. We find that for small initial viral load the disease progresses through an asymptomatic course, for intermediate value it takes a typical course with constant duration and severity of infection but variable onset, and for large initial viral load the disease becomes severe. This behavior is robust to a wide range of parameter values. The absence of antibody response leads to recurrence of disease and appearance of a chronic state with nontrivial constant viral load.     

A mathematical model of pulmonary gas exchange under inflammatory stress

During a severe local or systemic inflammatory response, immune mediators target lung tissue. This process may lead to acute lung injury and impaired diffusion of gas molecules. Although several mathematical models of gas exchange have been described, none simulate acute lung injury following inflammatory stress. In view of recent laboratory and clinical progress in the understanding of the pathophysiology of acute lung injury, such a mathematical model would be useful. We first derived a partial differential equations model of gas exchange on a small physiological unit of the lung (≈25 alveoli), which we refer to as a respiratory unit (RU). We next developed a simple model of the acute inflammatory response and implemented its effects within a RU, creating a single RU model. Linking multiple RUs with various ventilation/perfusion ratios and taking into account pulmonary venous blood remixing yielded our lung-scale model. Using the lung-scale model, we explored the predicted effects of inflammation on ventilation/perfusion distribution and the resulting pulmonary venous partial pressure oxygen level during systemic inflammatory stresses. This model represents a first step towards the development of anatomically faithful models of gas exchange and ventilation under a broad range of local and systemic inflammatory stimuli resulting in acute lung injury, such as infection and mechanical strain of lung tissue.     

Word organization in coding DNA: A mathematical model

This article deals with the relationship between vocabulary (total number of distinct oligomers or “words”) and text-length (total number of oligomers or “words”) for a coding DNA sequence (CDS). For natural human languages, Heaps established a mathematical formula known as Heaps' law, which relates vocabulary to text-length. Our analysis shows that Heaps' law fails to model this relationship for CDSs. Here we develop a mathematical model to establish the relationship between the number of type of words (vocabulary) and the number of words sampled (text-length) for CDSs, when non-overlapping nucleotide strings with the same length are treated as words. We use tangent-hyperbolic function, which captures the saturation property of vocabulary. Based on the parameters of the model, we formulate a mathematical equation, known as “equation of word organization”, whose parameters essentially indicate that nucleotide organization of coding sequences are different from one another. We also compare the word organization of CDSs with the random word distribution and conclude that a CDS is neither similar to a natural human language nor to a random one. Moreover, these sequences have their unique nucleotide organization and it is completely structured for specific biological functioning. IM and AS contributed equally to this work.     

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.     

A mathematical model of parathyroid hormone response to acute changes in plasma ionized calcium concentration in humans

A complex bio-mechanism, commonly referred to as calcium homeostasis, regulates plasma ionized calcium (Ca²⁺) concentration in the human body within a narrow range which is crucial for maintaining normal physiology and metabolism. Taking a step towards creating a complete mathematical model of calcium homeostasis, we focus on the short-term dynamics of calcium homeostasis and consider the response of the parathyroid glands to acute changes in plasma Ca²⁺ concentration. We review available models, discuss their limitations, then present a two-pool, linear, time-varying model to describe the dynamics of this calcium homeostasis subsystem, the Ca-PTH axis. We propose that plasma PTH concentration and plasma Ca²⁺ concentration bear an asymmetric reverse sigmoid relation. The parameters of our model are successfully estimated based on clinical data corresponding to three healthy subjects that have undergone induced hypocalcemic clamp tests. In the first validation of this kind, with parameters estimated separately for each subject we test the model’s ability to predict the same subject’s induced hypercalcemic clamp test responses. Our results demonstrate that a two-pool, linear, time-varying model with an asymmetric reverse sigmoid relation characterizes the short-term dynamics of the Ca-PTH axis.     

Perturbation analysis analyzed—mathematical modeling of intact and perturbed gene regulatory circuits for animal development

Gene regulatory networks for animal development are the underlying mechanisms controlling cell fate specification and differentiation. The architecture of gene regulatory circuits determines their information processing properties and their developmental function. It is a major task to derive realistic network models from exceedingly advanced high throughput experimental data. Here we use mathematical modeling to study the dynamics of gene regulatory circuits to advance the ability to infer regulatory connections and logic function from experimental data. This study is guided by experimental methodologies that are commonly used to study gene regulatory networks that control cell fate specification. We study the effect of a perturbation of an input on the level of its downstream genes and compare between the cis-regulatory execution of OR and AND logics. Circuits that initiate gene activation and circuits that lock on the expression of genes are analyzed. The model improves our ability to analyze experimental data and construct from it the network topology. The model also illuminates information processing properties of gene regulatory circuits for animal development.     

A mathematical model of hematopoiesis--I. Periodic chronic myelogenous leukemia

Periodic chronic myelogenous leukemia (PCML) is an interesting dynamical disease of the hematopoietic system in which oscillating levels of circulating leukocytes, platelets and/or reticulocytes are observed. Typically all of these three differentiated cell types have the same oscillation period, but the relation of the oscillation mean and amplitude to the normal levels is variable. Given the appearance of the abnormal Philadelphia chromosome in all of the nucleated progeny of the hematopoietic stem cells (HSCs), the most parsimonious conclusion is that chronic myelogenous leukemia, and its periodic variant, arise from derangements partially involving the dynamics of the stem cells. Here, we have synthesized several previous mathematical models of HSC dynamics, and models for the regulation of neutrophils, platelets and erythrocytes into a comprehensive model for the regulation of the hematopoietic system. Based on estimates of parameters for a typical normal human, we have systematically explored the changes in some of these parameters necessary to account for the quantitative data on leukocyte, platelet and reticulocyte cycling in 11 patients with PCML. Our results indicate that the critical model parameter changes required to simulate the PCML patient data are an increase in the amplification in the leukocyte line, an increase in the differentiation rate from the stem cell compartment into the leukocyte line, and the rate of apoptosis in the stem cell compartment. Our model system is particularly sensitive to changes in stem cell apoptosis rates, suggesting that changes in the numbers of proliferating stem cells may be important in generating PCML.     

A mathematical model of plants as ecosystem engineers

Understanding the structure and dynamics of plant communities in water-limited systems often calls for the identification of ecosystem engineers--key species that modify the landscape, redistribute resources and facilitate the growth of other species. Shrubs are excellent examples; they self-organize to form patterns of mesic patches which provide habitats for herbaceous species. In this paper we present a mathematical model for studying ecosystem engineering by woody plant species in drylands. The model captures various feedbacks between biomass and water including water uptake by plants' roots and increased water infiltration at vegetation patches. Both the uptake and the infiltration feedbacks act as mechanisms for vegetation pattern formation, but have opposite effects on the water resource; the former depletes the soil-water content under a vegetation patch, whereas the latter acts to increase it. Varying the relative strength of the two feedbacks we find a trade-off between the engineering capacity of a plant species and its resilience to disturbances. We further identify two basic soil-water distributions associated with engineering at the single patch level, hump-shaped and ring-shaped, and discuss the niches they form for herbaceous species. Finally, we study how pattern transitions at the landscape level feedback to the single patch level by affecting engineering strength.     

A mathematical model of blood,cerebrospinal fluid and brain dynamics

Using first principles of fluid and solid mechanics a comprehensive model of human intracranial dynamics is proposed. Blood, cerebrospinal fluid (CSF) and brain parenchyma as well as the spinal canal are included. The compartmental model predicts intracranial pressure gradients, blood and CSF flows and displacements in normal and pathological conditions like communicating hydrocephalus. The system of differential equations of first principles conservation balances is discretized and solved numerically. Fluid–solid interactions of the brain parenchyma with cerebral blood and CSF are calculated. The model provides the transitions from normal dynamics to the diseased state during the onset of communicating hydrocephalus. Predicted results were compared with physiological data from Cine phase-contrast magnetic resonance imaging to verify the dynamic model. Bolus injections into the CSF are simulated in the model and found to agree with clinical measurements.

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A mathematical model of human heart including the effects of heart contractility varying with heart rate changes

Incorporating the intrinsic variability of heart contractility varying with heart rate into the mathematical model of human heart would be useful for addressing the dynamical behaviors of human cardiovascular system, but models with such features were rarely reported. This study focused on the development and evaluation of a mathematical model of the whole heart, including the effects of heart contractility varying with heart rate changes. This model was developed based on a paradigm and model presented by Ottesen and Densielsen, which was used to model ventricular contraction. A piece-wise function together with expressions for time-related parameters were constructed for modeling atrial contraction. Atrial and ventricular parts of the whole heart model were evaluated by comparing with models from literature, and then the whole heart model were assessed through coupling with a simple model of the systemic circulation system and the pulmonary circulation system. The results indicated that both atrial and ventricular parts of the whole heart model could reasonably reflect their contractility varying with heart rate changes, and the whole heart model could exhibit major features of human heart. Results of the parameters variation studies revealed the correlations between the parameters in the whole heart model and performances (including the maximum pressure and the stroke volume) of every chamber. These results would be useful for helping users to adjust parameters in special applications.     

Investigating the link between epithelial-mesenchymal transition and the cancer stem cell phenotype: A mathematical approach

Under the cancer stem cell (CSC) hypothesis, sustained metastatic growth requires the dissemination of a CSC from the primary tumour followed by its re-establishment in a secondary site. The epithelial-mesenchymal transition (EMT), a differentiation process crucial to normal development, has been implicated in conferring metastatic ability on carcinomas. Balancing these two concepts has led researchers to investigate a possible link between EMT and the CSC phenotype—indeed, recent evidence indicates that, following induction of EMT in human breast cancer and related cell lines, stem cell activity increased, as judged by the presence of cells displaying the CD44^high/CD24^low phenotype and an increase in the ability of cells to form mammospheres. We mathematically investigate the nature of this increase in stem cell activity. A stochastic model is used when small number of cells are under consideration, namely in simulating the mammosphere assay, while a related continuous model is used to probe the dynamics of larger cell populations. Two scenarios of EMT-mediated CSC enrichment are considered. In the first, differentiated cells re-acquire a CSC phenotype—this model implicates fully mature cells as key subjects of de-differentiation and entails a delay period of several days before de-differentiation occurs. In the second, pre-existing CSCs experience accelerated division and increased proportion of self-renewing divisions; a lack of perfect CSC biomarkers and cell sorting techniques requires that this model be considered, further emphasizing the need for better characterization of the mammary (cancer) stem cell hierarchy. Additionally, we suggest the utility of comparing mammosphere data to computational mammosphere simulations in elucidating the growth characteristics of mammary (cancer) stem cells.     

Stability and phase transitions in a mathematical model of Duchenne muscular dystrophy

We present a mathematical model to investigate the role of the immune system in the Duchenne muscular dystrophy disease. It is based on the assumption that the immune system contributes to the tissue damage and indeed its interaction with the muscle tissue after an initial endogenous damage can be described as a predator-prey system showing typical oscillations. In this article we investigate the dynamical properties of the system. We find that, for a biologically relevant parameters range, it shows two phase transitions between qualitative different behaviors corresponding to complete recover or to a state where muscle regeneration and degeneration coexist.     

A theory of drug tolerance and dependence II: the mathematical model

The preceding paper presented a model of drug tolerance and dependence. The model assumes the development of tolerance to a repeatedly administered drug to be the result of a regulated adaptive process. The oral detection and analysis of exogenous substances is proposed to be the primary stimulus for the mechanism of drug tolerance. Anticipation and environmental cues are in the model considered secondary stimuli, becoming primary in dependence and addiction or when the drug administration bypasses the natural-oral-route, as is the case when drugs are administered intravenously. The model considers adaptation to the effect of a drug and adaptation to the interval between drug taking autonomous tolerance processes. Simulations with the mathematical model demonstrate the model's behaviour to be consistent with important characteristics of the development of tolerance to repeatedly administered drugs: the gradual decrease in drug effect when tolerance develops, the high sensitivity to small changes in drug dose, the rebound phenomenon and the large reactions following withdrawal in dependence. The present paper discusses the mathematical model in terms of its design. The model is a nonlinear, learning feedback system, fully satisfying control theoretical principles. It accepts any form of the stimulus-the drug intake-and describes how the physiological processes involved affect the distribution of the drug through the body and the stability of the regulation loop. The mathematical model verifies the proposed theory and provides a basis for the implementation of mathematical models of specific physiological processes.     

A mathematical and experimental study of ant foraging trail dynamics

In this article, we present a mathematical model coupled to an experimental study of ant foraging trails. Our laboratory experiments on Tetramorium caespitum do not find a strong relationship between ant densities and velocities, a common assumption in traffic modeling. Rather, we find that higher order effects play a major role in observed behavior, and our model reflects this by including inertial terms in the evolution equation. A linearization of the resulting system yields left- and right-moving waves, in agreement with laboratory measurements. The linearized system depends upon Froude numbers reflecting a ratio of the energy stored in the foraging trail to the kinetic energy of the ants. The model predicts and the measurements support the existence of two distinct phase velocities.     

Synchronization of oscillations of proliferation of keratinocytes in psoriatic skin by external periodic force: a mathematical model

A mathematical model of mitotic activity of epidermis in normal skin and skin afflicted with psoriasis is presented as a system of two autonomous nonlinear differential equations. Its qualitative analysis was carried out and numerical solutions were obtained at the parameter values corresponding to these states. It was shown that in the norm, a single stable equilibrium of a "focus" type exists in the system; whereas in psoriasis, owing to an increase in the growing fraction, hyperproliferation, and enhanced migration of interacting keratinocytes, a stable limit cycle arises from the state of unstable focus. In this paper we also report on the results of computer modeling of synchronization of self-excited oscillations of keratinocyte population density in psoriatic lesions by an external periodic force. This synchronization is viewed as a possible mechanism of the clinically observed dependence of psoriasis course on some natural factors of cyclic nature.     

A logical analysis of the process of T cell activation: different consequences depending on the state of CD28 engagement

The adaptive immune system is a complex organized action of several immune cell types like, T cells, B cells, dendritic cells, mast cells, and their ability to recognize self and foreign molecular information. Based on logical analysis, a model has been developed that describes TCR-ligand association coupled to intracellular signaling events that result in a proliferation signal. The model demonstrates that after TCR-ligand binding, the activation of tyrosine kinases in one of the paths leads to oscillations between the subsequent states of activation and deactivation of Ca(2+) initiation. In our studies the effect of costimulation on the primary signal has also been explored. Analysis reveals that costimulation increases by more than 2.5 fold the number of paths rendering a cell proliferation signal compared to the outcome when costimulation is blocked. Traversal of 97% of these paths attains a costimulation threshold of activation. We also examined a hypothesis that couples the primary signal and costimulation by modeling costimulation to act as an inhibitor on the Inhibitor proteins. Using this hypothesis our analysis showed a 25% increase in the number of paths leading to cell proliferation in comparison to when costimulation is blocked. Our model also reveals that this hypothesis actually decrease by approximately 50% the number of paths attaining cell proliferation compared to the number of available paths leading to cell proliferation when costimulation does not act as an inhibitor on Inhibitor proteins. This suggests that costimulation influences cell proliferation by providing a greater diversity of paths that converge to this state. However, costimulation should be thought independent of its regulatory interaction with the inhibitor proteins.     

A mathematical model of the methionine cycle

Building on the work of Martinov et al. (2000), a mathematical model is developed for the methionine cycle. A large amount of information is available about the enzymes that catalyse individual reaction steps in the cycle, from methionine to S-adenosylmethionine to S-adenosylhomocysteine to homocysteine, and the removal of mass from the cycle by the conversion of homocysteine to cystathionine. Nevertheless, the behavior of the cycle is very complicated since many substrates alter the activities of the enzymes in the reactions that produce them, and some can also alter the activities of other enzymes in the cycle. The model consists of four differential equations, based on known reaction kinetics, that can be solved to give the time course of the concentrations of the four main substrates in the cycle under various circumstances. We show that the behavior of the model in response to genetic abnormalities and dietary deficiencies is similar to the changes seen in a wide variety of experimental studies. We conduct computational "experiments" that give understanding of the regulatory behavior of the methionine cycle under normal conditions and the behavior in the presence of genetic variation and dietary deficiencies.