Modeling immune reactivity in secondary lymphoid organs

Models of the dynamical interactions important in generating immune reactivity have generally assumed that the immune system is a single well-stirred compartment. Here we explicitly take into account the compartmentalized nature of the immune system and show that qualitative conclusions, such as the stability of the immune steady state, depend on architectural details. We examine a simple model idiotypic network involving only two types of B cells and antibody molecules. We show, for model parameters used by De Boeret al. (1990,Chem. Eng. Sci. 45, 2375–2382), that the immune steady state is unstable in a one compartmental model but stable in a two compartment model that contains both a lymphoid organ, such as the spleen, and the circulatory system. This work was performed under the auspices of the U.S. Department of Energy.     

On the number of steady states in a multiple futile cycle

The multisite phosphorylation-dephosphorylation cycle is a motif repeatedly used in cell signaling. This motif itself can generate a variety of dynamic behaviors like bistability and ultrasensitivity without direct positive feedbacks. In this paper, we study the number of positive steady states of a general multisite phosphorylation–dephosphorylation cycle, and how the number of positive steady states varies by changing the biological parameters. We show analytically that (1) for some parameter ranges, there are at least n + 1 (if n is even) or n (if n is odd) steady states; (2) there never are more than 2n − 1 steady states (in particular, this implies that for n = 2, including single levels of MAPK cascades, there are at most three steady states); (3) for parameters near the standard Michaelis–Menten quasi-steady state conditions, there are at most n + 1 steady states; and (4) for parameters far from the standard Michaelis–Menten quasi-steady state conditions, there is at most one steady state.      

Understanding bursting oscillations as periodic slow passages through bifurcation and limit points

We consider a biochemical system consisting of two allosteric enzyme reactions coupled in series. The system has been modeled by Decroly and Goldbeter (J. Theor. Biol. 124, 219 (1987)) and is described by three coupled, first-order, nonlinear, differential equations. Bursting oscillations correspond to a succession of alternating active and silent phases. The active phase is characterized by rapid oscillations while the silent phase is a period of quiescence. We propose an asymptotic analysis of the differential equations which is based on the limit of large allosteric constants. This analysis allows us to construct a time-periodic bursting solution. This solution is jumping periodically between a slowly varying steady state and a slowly varying oscillatory state. Each jump follows a slow passage through a bifurcation or limit point which we analyze in detail. Of particular interest is the slow passage through a supercritical Hopf bifurcation. The transition is from an oscillatory solution to a steady state solution. We show that the transition is delayed considerably and characterize this delay by estimating the amplitude of the oscillations at the Hopf bifurcation point.     

Zum zeitlichen Verlauf von Substratkonzentrationsprofilen in einer Enzymträgerkugel

In this paper the time evolution of substrate concentration profiles in spherical particles with homogeneous enzyme distribution is investigated. It turns out that the concentration S ? S(x, t) is monotone increasing in the time variable t against the uniquely defined steady state ? ? ?(x). Explizit formulas are derived for the estimation of the difference between the steady state ? and the presteady state S, that hold for all t ≧ 0, i.c., it is not only determined the order of magnitude of the convergence for sufficiently large t.     

Modeling the adaptive immune response in HBV infection

The aim of this work is to investigate a new mathematical model that describes the interactions between Hepatitis B virus (HBV), liver cells (hepatocytes), and the adaptive immune response. The qualitative analysis of this as cytotoxic T lymphocytes (CTL) cells and the antibodies. These outcomes are (1) a disease free steady state, which its local stability is characterized as usual by R ₀ < 1, (2) and the existence of four endemic steady states when R ₀ > 1. The local stability of these steady states depends on functions of R ₀. Our study shows that although we give conditions of stability of these steady states, not all conditions are feasible. This rules out the local stability of two steady states. The conditions of stability of the two other steady states (which represent the complete failure of the adaptive immunity and the persistence of the disease) are formulated based on the domination of CTL cells response or the antibody response.     

Genetic and physiological alterations occurring in a yeast population continuously propagated at increasing temperatures with cell recycling

This work investigated the effects of increasing temperature from 30°C to 47°C on the physiological and genetic characteristics of Saccharomyces cerevisiae strain 63M after continuous fermentation with cell recycling in a system of five reactors in series. Steady state was attained at 30°C, and then the temperature of the system was raised so it ranged from 35°C in the last reactor to 43°C in the first reactor or feeding reactor with a 2°C difference between reactors. After 15 days at steady state, the temperature was raised from 37°C to 45°C for 25 days at steady state, then from 39°C to 47°C for 20 days at steady state. Starter strain 63M was a hybrid strain constructed to have a MAT a/α, LYS/lys, URA/ura genotype. This hybrid yeast showed vigorous growth on plates at 40°C, weak growth at 41°C, positive assimilation of melibiose, positive fermentation of galactose, raffinose and sucrose. Of 156 isolates obtained from this system at the end of the fermentation process, only 17.3% showed the same characteristics as starter strain 63M. Alterations in mating type reaction and in utilization of raffinose, melibiose, and sucrose were identified. Only 1.9% of the isolates lost the ability to grow at 40°C. Isolates showing requirements for lysine and uracil were also obtained. In addition, cell survival was observed at 39–47°C, but no isolates showing growth above 41°C were obtained.     

Rich dynamics of a hepatitis B viral infection model with logistic hepatocyte growth

Chronic hepatitis B virus (HBV) infection is a major cause of human suffering, and a number of mathematical models have examined within-host dynamics of the disease. Most previous HBV infection models have assumed that: (a) hepatocytes regenerate at a constant rate from a source outside the liver; and/or (b) the infection takes place via a mass action process. Assumption (a) contradicts experimental data showing that healthy hepatocytes proliferate at a rate that depends on current liver size relative to some equilibrium mass, while assumption (b) produces a problematic basic reproduction number. Here we replace the constant infusion of healthy hepatocytes with a logistic growth term and the mass action infection term by a standard incidence function; these modifications enrich the dynamics of a well-studied model of HBV pathogenesis. In particular, in addition to disease free and endemic steady states, the system also allows a stable periodic orbit and a steady state at the origin. Since the system is not differentiable at the origin, we use a ratio-dependent transformation to show that there is a region in parameter space where the origin is globally stable. When the basic reproduction number, R ₀, is less than 1, the disease free steady state is stable. When R ₀ > 1 the system can either converge to the chronic steady state, experience sustained oscillations, or approach the origin. We characterize parameter regions for all three situations, identify a Hopf and a homoclinic bifurcation point, and show how they depend on the basic reproduction number and the intrinsic growth rate of hepatocytes.     

Logical identification of all steady states: The concept of feedback loop characteristic states

Biological regulatory systems can be described in terms of non-linear differential equations or in logical terms (using an “infinitely non-linear” approximation). Until recently, only part of the steady states of a system could be identified on logical grounds. The reason was that steady states frequently have one or more variable located on a threshold (see below); those steady states were not detected because so far no logical status was assigned to threshold values. This is why we introduced logical scales with values 0,¹θ, 1²θ, 2, ..., in which¹θ,²θ, ... are the logical values assigned to the successive thresholds of the scale. We thus have, in addition to the regular logical states,singular states in which one or more variables is located on a threshold. This permits identifyingall the steady states on logical grounds. It was noticed that each feedback loop (or reunion of disjointed loops) can be characterized by a logical state located at the thresholds at which the variables of the loop operate. This led to the concept ofloop-characteristic state, which, as we will see, enormously simplifies the analysis.The core of this paper is a formal demonstration that among the singular states of a system, only loop-characteristic states can be steady. Reciprocally, given a loop-characteristic state, there are parameter values for which this state is steady; in this case, the loop is effective (i.e. it generates multistationarity if it is a positive loop, homeostasis if it is a negative loop). This not only results in the above-mentioned radical simplification of the identification of the steady states, but in an entirely new view of the relation between feedback loops and steady states.     

The functioning of cytochrome b in the electron transport to fumarate in Propionibacterium freudenreichii and Propionibacterium pentosaceum

Fumarase-free electron particles from Propionibacterium freudenreichii and P. pentosaceum were prepared by discontinuous sucrose gradient centrifugation, and the influence of 2-n-heptyl-4-hydroxy-quinoline-N-oxide (HQNO) and ultraviolet irradiation on the reduction of menaquinone and cytochrome b with l-lactate or glycerol-3-phosphate and the reoxidation by fumarate was studied. In the presence of HQNO the steady state reduction level of menaquinone during fumarate reduction was increased whereas the steady state reduction level of cytochrome b was decreased as compared with the reduction levels measured in the absence of HQNO. The steady state reduction level of menaquinone during electron transport to fumarate was not influenced by ultraviolet irradiation and the steady state reduction level of cytochrome b was decreased at increasing irradiation times. The data indicate that cytochrome b is involved in the electron transport to fumarate.Abbreviations HQNO 2-n-heptyl-4-hydroxyquinoline-N-oxide - NQNO 2-n-nonyl-4-hydroxyquinoline-N-oxide Visiting Professor at the Biological Laboratory     

Properties of solutions for a chemotaxis system

 We investigate mathematically the system of equations proposed by Chaplain and Stuart [2], to describe the chemotactic response of endothelial cells under the angiogenesis stimulus. In particular, we characterize the steady state endothelial cell density function, and give conditions on the chemotactic parameter k and cell proliferation parameter b that ensure that migration/ proliferation either does or does not occur in steady state. The time dependent problem is also treated. Received 12 September 1995; received in revised form 6 August 1996     

Dissipative Irreversibility from Nosé's Reversible Mechanics

Abstract

Nose's Hamiltonian mechanics makes possible the efficient simulation of irreversible flows of mass, momentum and energy. Such flows illustrate the paradox that reversible microscopic equations of motion underlie the irreversible behavior described by the second law of thermodynamics. This generic behavior of molecular many-body systems is illustrated here for the simplest possible system, with only one degree of freedom: a one-body Frenkel-Kontorova model for isothermal electronic conduction. This model system, described by Nosé-Hoover Hamiltonian dynamics, exhibits several interesting features: (1) deterministic and reversible equations of motion; (2) Lyapunov instability, with phase-space offsets increasing exponentially with time; (3) limit cycles; (4) dissipative conversion of work (potential energy) into heat (kinetic energy): and (5) phase-space contraction, a characteristic feature of steady irreversible flows. The model is particularly instructive in illustrating and explaining a paradox associated with steady-state statistical mechanics: the Gibbs entropy of a nonequilibrium steady state decreases continuously to minus infinity.     

A model of cell firing patterns during epileptic seizures

A simplified model is presented of the dynamics of excitatory and inhibitory neurons in the cerebral cortex. A key feature of the model is that neurons may cease to fire when strongly depolarized (spike inactivation). Computer simulations for different parameters reveal five classes of solutons: a) steady states in which neither excitatory nor inhibitory cells are active, b) steady states in which one or both types of cells fire repetitively, c) states in which one type of cell fluctuates rapidly between bursts of action potentials and inactivity due to strong depolarization, d) rhythmic activity in which both types of cells fire in unison followed by a period of spike inactivation and e) states similar to d but in which the inhibitory cells never produce action potentials. Solutions b, c, d, and e qualitatively resemble the different firing patterns observed during experimental seizures. It is shown that changes in those parameters that are functions of potassium concentration can induce changes in the type of solution. It is therefore proposed that the increase in extracellular potassium concentration during seizures may be responsible for the progressive changes observed in firing patterns and particularly for the transition from tonic to clonic patterns. A method is also outlined for testing the predictions of the model.     

Interactions in enzymatic reactions

The paper deals with interactions of substances via an enzymatic reaction (Bull. Math. Biophysics,25, 141–154, 1963). The substances are the activators, inhibitors and/or substrates of the reaction. Due to the bimolecularity of the processes in the reaction, the quantitative relation between the steady state amount of complexes and the amounts of the substances assumes a typical form. In multiple enzymatic reactions this form is more complicated, though basically similar. Because the substances may influence the steady state amounts of the complexes in opposite directions, the compensation and blocking effects are the properties of enzymatic reactions. The substances with the same direction of influence may potentiate each other. In the enzymatic reaction here considered, the potentiation is always non-negative.     

Transient behavior of a continuous stirred tank biological reactor utilizing phenol as an inhibitory substrate

Transient experiments were conducted on a Pseudomomas utilizing phenol in a continuous culture by disturbing the influent substrate concentration and dilution rate. Two stable steady states existed for some ranges of the parameters. Highly damped oscillations were observed in approaching a new high conversion steady state or in returning to a new high conversion steady state following a small disturbance. When a large disturbance was applied there was a smooth (overdamped) approach to a new low conversion steady state. The observed oscillatory behavior for small disturbances was predicted by a modified Powell-Ierusalemskii bottleneck model, but could not be predicted by a Monod-Haldane model; neither model was accurate for predicting the effect of large disturbances. A constant wall growth factor was used to account for microbial film activity, and the existence of two stable states was directly due to the presence of the film.     

Effects of methanol on expression of an anticoagulant hirudin in recombinant Hansenula polymorpha

A series of batch, fed-batch, and continuous cultures was carried out to analyze the effects of methanol on the fermentation characteristics of recombinant Hansenula polymorpha for the production of hirudin, an anticoagulant. Hirudin expression efficiencies were greatly influenced by the methanol concentrations in continuous and fed-batch culture modes. At a steady state of continuous culture, an optimum methanol concentration of 1.7 g l⁻¹ was determined at a dilution rate of 0.18 h⁻¹ with 1.8 mg l⁻¹ h⁻¹ hirudin productivity. Journal of Industrial Microbiology & Biotechnology (2001) 27, 58–61. Received 21 September 2000/ Accepted in revised form 10 June 2001     

Photosynthesis and calcification in the calcifying algae Halimeda discoidea studied with microsensors

With microsensors, we measured the steady‐state microprofiles of O₂, pH and Ca²⁺ on the topside of young segments of Halimeda discoidea, as well as the surface dynamics upon light–dark shifts. The effect of several inhibitors was studied. The steady‐state measurements showed that under high light intensity, calcium and protons were taken up, while O₂ was produced. In the dark, O₂ was consumed, the pH decreased to below seawater level and Ca²⁺ uptake was reduced to 50%. At low light intensity (12 mmol photons m^‐2 s^‐1), Ca²⁺ efflux was observed. Upon light–dark shifts, a complicated pattern of both the pH and calcium surface dynamics was observed. Illumination caused an initial pH decrease, followed by a gradual pH increase: this indicated that the surface pH of H. discoidea is determined by more than one light‐induced process. When photosynthesis was inhibited by dichlorophenyl dimethyl urea (DCMU), a strong acidification was observed upon illumination. The nature and physiological function of this putative pump is not known. The calcium dynamics followed all pH dynamics closely, both in the presence and absence of DCMU. The Ca‐channel blockers verapamil and nifedipine had no effect on the Ca²⁺ dynamics and steady‐state profiles. Thus, in H. discoidea, calcification is not regulated by the alga, but is a consequence of pH increase during photosynthesis. Acetazolamide had no effect on photosynthesis, whereas ethoxyzolamide inhibited photosynthesis at higher light intensities. Therefore, all carbonic anhydrase activity is intracellular. Carbonic anhydrase is required to alleviate the CO₂ limitation. Calcification cannot supply sufficient protons and CO₂ to sustain photosynthesis.     

Antisense RNA based down-regulation of RNaseE in E.coli

Background  

Messenger RNA decay is an important mechanism for controlling gene expression in all organisms. The rate of the mRNA degradation directly affects the steady state concentration of mRNAs and therefore influences the protein synthesis. RNaseE has a key importance for the general mRNA decay in E.coli. While RNaseE initiates the degradation of most mRNAs in E.coli, it is likely that the enzyme is also responsible for the degradation of recombinant RNAs. As RNaseE is essential for cell viability and knockout mutants cannot be cultured, we investigated the possibility for a down-regulation of the intracellular level of RNaseE by antisense RNAs. During this study, an antisense RNA based approach could be established which revealed a strong reduction of the intracellular level of RNaseE in E.coli.     

Proliferation-differentiation pathways of murine haemopoiesis: correlation of lineage fluxes

Abstract. A quantitative analysis of literature data on muiine bone marrow haemopoiesis was performed in order to determine how the haemolpoietic system fulfils requirements of cell turnover in the normal steady state. In particular, the production rates (fluxes) of erythrocytes, megakaryocytes, monocytes, neutrophils, eosinophils, basophils and mast cells have been calculated and normalized. Subsequently, based on available data on relationships of various lineages, a working model of the developmental schema of haemopoiesis was derived. Since, according to the principle of conservation, the effluxes of all lineages from bone marrow in the steady state must be equal to effluxes from the compartments of respective precursors divided by the coefficient of multiplication, the correlation of lineage fluxes was used to determine the position of branching points. It was concluded that all the quantita.tive data on lineage proliferation in the normal steady state in vivo, available for analysis, are fully consistent with a simple binary model of the sequential type, similar to that suggested by Brown and coworkers (Brown G., Jones NA, Bunce CM, Owen PJ, IPatton WN. (1988) Haemopoiesis: a lottery or determinism?Differentiation, 39 , 83).