A mathematical model for germinal centre kinetics and affinity maturation

We present a mathematical model which reproduces experimental data on the germinal centre (GC) kinetics of the primed primary immune response and on affinity maturation observed during the reaction. We show that antigen masking by antibodies which are produced by emerging plasma cells can drive affinity maturation and provide a feedback mechanism by which the reaction is stable against variations in the initial antigen amount over several orders of magnitude. This provides a possible answer to the long-standing question of the role of antigen reduction in driving affinity maturation. By comparing model predictions with experimental results, we propose that the selection probability of centrocytes and the recycling probability of selected centrocytes are not constant but vary during the GC reaction with respect to time. It is shown that the efficiency of affinity maturation is highest if clones with an affinity for the antigen well above the average affinity in the GC leave the GC for either the memory or plasma cell pool. It is further shown that termination of somatic hypermutation several days before the end of the germinal centre reaction is beneficial for affinity maturation. The impact on affinity maturation of simultaneous initiation of memory cell formation and somatic hypermutation vs. delayed initiation of memory cell formation is discussed.     

A mathematical model for the germinal center morphology and affinity maturation

During germinal center reactions, the appearance of two specific zones are observed: the dark and the light zone. Up to now, the origin and function of these zones are poorly understood. In the framework of a stochastic and discrete model, several possible pathways of zone development during germinal center reactions are investigated. The importance of the zones in the germinal center for affinity maturation, i.e. the process of antibody optimization is discussed.     

A mathematical model for insulin kinetics. III. Sensitivity analysis of the model

A non-linear mathematical model involving four variables and several constants incorporating beta-cell kinetics, a glucose-insulin feedback system and a gastrointestinal absorption term had been applied in earlier papers to various forms of diabetes mellitus. In this paper, we examine the response of the system to variations in the parameters and to initial conditions using sensitivity analysis. It is found that such a method leads to results that are consistent with clinical findings. Further, it is suggested that such an analysis could help in making some predictions regarding future directions in the therapy of diabetes mellitus.     

A discrete mathematical model of unlabelled granulocyte kinetics

The equations used in formulating the continuous model of granulocyte kinetics developed by O'Fallon et al. (1971) were analyzed to see if they could be altered to simulate a feedback mechanism operating on the production and development of granulocytes. After extensive study and modification of the continuous model, it was found that a discrete model based on a Leslie matrix procedure was more effective for simulating the feedback system. This discrete model was used to show experimentally, from a mathematical view point, that a feedback mechanism of some kind must be operating on the production and development of granulocytes. Further, the discrete model was subjected to preliminary tests (simultaneous and cascading feedback) to demonstrate that it has the capability of responding to feedback control.This work was completed while the first author was at North Carolina State University at Raleigh, Department of Statistics, Biomathematics Division, part of it under grant number 5T1 GM 678-15, National Institutes of Health, and part of it under grant number 5 F32 CA05964-02 from the National Cancer Institute     

Studies on the germinal center

Summary Ultrastructure of mitotic cells in human lymph node germinal centers was deliberately studied in contrast to that of plasma cells in mitosis which were rarely found in medullary cords or lymphatic sinuses of the same materials. It was demonstrated that the mitotic cells in germinal centers are evidently different from the latter in the absence of lamellary arranged endoplasmic reticulum with ribosomes and Golgi apparatus, and are quite similar to the ultrastructure of thymic lymphocytes in mitosis reported by Murray et al. It should be concluded from these findings that the cells produced locally within the germinal centers in human lymph nodes are lymphocytic as has been repeatedly suggested by the authors.Supported by Grant in Aid from the Ministry of Education of Japan (69-9254).     

Immunobiology of chicken germinal center: II. Accumulation of apoptotic cells within the germinal center

The germinal center (GC) develops after antigen stimulation and is thought to occur at the site of various immune responses. We observed apoptotic cells within the GC using in situ end labeling (TUNEL), small amount DNA ladder assay, and RT-PCR analysis of Bcl-2 mRNA expression. Apoptosis was detected within GCs at all phases of the GC reaction by both TUNEL and DNA ladder assays. The number of TUNEL⁺ nuclei within the GC did not increase over the course of the GC reaction. However, the density of DNA in the ladder assay was higher in later-phase GCs. Bcl-2 mRNA expression was detected within GCs during the early phases of the GC reaction. These results indicate that accumulation of apoptotic cells and rescue from apoptosis occur within chicken GCs. In the present paper, the reasons for the accumulation of apoptotic cells will be discussed.This work was supported by Grants-in-Aid for Scientific Research (Nos. 11670322 and 10306017) from the Ministry of Education, Science, Sport and Culture, and the Ministry of Agriculture, Forestry and Fisheries of Japan (Special Scientific Research and Pioneering Research Project in Biotechnology), as well as from the Bio-oriented Technology Research Advancement Institution (BRAIN)     

A mathematical model of kinetics of the biosynthesis of tetracyclines

Summary A mathematical model simulating the behaviour or Streptomyces aureofaciens in batch culture under conditions when tetracyclines are synthesized in excessive amounts has been formulated. The response of the mathematical model to the experimental conditions applied corresponds with data obtained in the experiments. The mathematical model demonstrated that the level of tetracycline production is determined during the period of culture growth beginning with exhaustion of inorganic phosphate from the medium and ending with inhibition of the synthesis of enzymes caused by the synthesized tetracyclines. Further tetracycline synthesis is then proportional to the amount of enzymes synthesized in this interval.List of symbols E Activity of ACT-oxygenase (10×nkat/g) - P Product concentration (mg/l) - k ₁-k ₆ Rate constants - K _S Saturation constant (g sugar/l) - K _I1 Inhibition constant (mg product/l) - K _I2 Inhibition constant (mM phosphate/l) - K _I3 Inhibition constant (mg product/l) - S ₁ Substrate sucrose (g sugar/l) - S ₂ Substrate concentration — phosphate (mM/l) - r _P Specific rate of product formation (mg product/g · h) - r _E Specific rate of enzyme synthesis (10×nkat/g² · h), Expressed by activity units - t Cultivation time (hour) - X Biomass dry weight (g/l) - Y _S/X Yield coefficient - Specific growth rate (h^-1)     

A mathematical model for the batch reactor kinetics of algae growth

A mathematical model based on the Einstein law of photochemical equivalence is proposed to describe the batch growth of unicellular algae. The model was applied in an integrated form to cell concentration versus growth time data taken over an extended range of cell concentrations which include both the regions of “exponential” and “linear” growth. It is shown that a certain function of cell concentration contained in the integrated form of the model is linearly dependent on the growth time over both the “exponential” and “linear” growth regions.     

A mathematical model for the kinetics of Methanobacterium bryantii M.o.H. considering hydrogen thresholds

We develop a kinetic model that builds on the foundation of classic Monod kinetics, but incorporates new phenomena such as substrate thresholds and survival mode observed in experiments with the H2-oxidizing methanogen Methanobacterium bryantii M.o.H. We apply our model to the experimental data presented in our companion paper on H2 thresholds. The model accurately describes H2 consumption, CH4 generation, biomass growth, substrate thresholds, and survival state during batch experiments. Methane formation stops when its Gibbs free energy is equal zero, although this does not interrupt H2 oxidation. The thermodynamic threshold for H2 oxidation occurs when the free energy for oxidizing H2 and transferring electrons to biomass is no longer negative, at approximately 0.4 nM. This threshold is not controlled by the Gibbs free energy equation of methanogenesis from H2 + HCO3- as we show in our companion paper. Beyond this threshold, the microorganisms shift to a low-maintenance metabolism called "the survival state" in response to extended H2 starvation; adding the starvation response as another new feature of the kinetic model. A kinetic threshold (or S (min)), a natural feature of the Monod kinetics, is also captured by the model at H2 concentration of around approximately 2,400 nM. S (min) is the minimum substrate concentration to maintain steady-state biomass concentration. Our model will be useful for interpreting threshold results and designing new studies to understand thresholds and their ecological implications.     

Functional properties of human germinal center B cells.

Germinal centers (GCs) are histologically defined areas where B cells undergo extensive proliferation and maturation, or die of apoptosis. GC B cells isolated from human tonsils can be phenotypically identified by expression of peanut agglutinin (PNA)-binding sites and can be further divided into subpopulations based on their expression of CD77. To assess the functional potential of GC B cells, we studied CD77+ PNA+ B cells isolated from tonsils by examining their differentiation status and their ability to proliferate in vitro to various cytokines and costimulants. We found that CD77+ GC B cells are less differentiated than CD77- GC B cells; GC B cells less frequently express cytoplasmic IgG and IgM, and spontaneously secrete less Ig compared to CD77- GC B cells. To identify conditions capable of inducing GC B cell proliferation, we examined IL-4, IL-2, IFN-gamma, low molecular weight BCGF (LMW-BCGF), and an MLR supernatant along with costimulants such as anti-IgM antibody, Staphylococcus aureus Cowan I (SAC), PMA, and pokeweed mitogen (PWM). While non-GC B cells proliferate strongly in response to these stimuli, GC B cells did not proliferate. However, CD77+ as well as CD77- GC B cells mounted a rapid and strong proliferative response upon stimulation with IL-4, but only in the presence of anti-CD40 antibody. Moreover, although nine additional cytokines were examined, only IL-4 was capable of supporting CD77+ GC B cell proliferation in the presence of anti-CD40 antibody. When cells were stimulated with IL-4 and anti-CD40 antibody, we also found that IFN-gamma consistently decreased the proliferative response of CD77+ GC B cells without affecting the response of non-GC B cells. Taken together, these data indicate that GC B cells have characteristic growth requirements and that IL-4 may be important for GC B cell growth in vivo.     

Shining a light on germinal center B cells

The mechanisms of B cell selection in lymphoid tissues are poorly understood. In this issue, Victora et?al. (2010) use imaging of photoactivatable green fluorescent protein to define the movements of B?cells in germinal centers and provide evidence that antibody affinity maturation is driven by competition for T?cell help.     

Ketone body kinetics in humans: a mathematical model

A model has been developed to account for ketone body kinetics in man based on data following bolus injections of [14C]acetoacetate (A) and [14C]beta-OH butyrate (B) into normal humans in the postabsorptive state. The model consists of separate compartments for blood A and B that are linked by a tissue compartment in which rapid interconversion of the ketone bodies occurs. The probability of movement from blood into this compartment was assumed to be the same for both ketone bodies. Two slowly equilibrating tissue compartments are required to account for the slow components in the tracer data, and thus a five-compartment model is proposed. By modeling the transient tracer data with the tracee in a steady state, ketone body kinetics were defined in terms of the rapid interconversions of A and B, and the slow exchanges of carbon within the tissues. The rates of release of new A and B into blood, (UA and UB) were calculated. These rates were less than the apparent production rates, PRA and PRB, as the PR's included carbon atoms first released as the other ketone body. The exchange constants between the compartments were determined in addition to the fractional catabolic rates (FCR) and metabolic clearance rates (MCR) of A and B. The initial space of distribution was 10 L and the mean values +/- SD (n = 11), normalized to this volume, were UA = 6.4 +/- 5.0, UB = 8.8 +/- 8.0 (mumol L-1 min-1), FCRA = 0.226 +/- 0.142, FCRB = 0.188 +/- 0.124 (min-1), MCRA = 2.26 +/- 1.42, MCRB = 1.87 +/- 1.23 (L min-1) and PRA = 11.1 +/- 7.6, PRB = 12.7 +/- 10.0 (mumol L-1 min-1).     

A mathematical model for the immunosenescence.

We propose a model for the dynamics of the immune system by considering the subpopulations of virgin and memory T lymphocytes on a time scale corresponding to the human life span. In the deterministic balance equation we introduce a fluctuating term in order to take into account the chronic antigenic stress. Starting from the hypothesis that the depletion of virgin cells with cytotoxic properties (CD8+) is a mortality marker, the model provides survival curves quite similar to the demographic curves.     

Lysosomal destabilization contributes to apoptosis of germinal center B-lymphocytes.

During germinal center (GC) reactions, B-lymphocytes with high-affinity B-cell receptors are selected. Regulation of apoptosis is a key process in selecting such wanted B-cells and in eliminating B-cells with unwanted specificities. In this paper, we show that apoptosis in human GC B-cells involves lysosomal destabilization, which is strictly controlled by caspase-8 activity, but not by caspase-9 activity. Ligation of CD40 provides resistance to lysosomal destabilization. Experimental lysosomal rupture by the lysosomotropic drug O-methyl-l-serine dodecylamide hydrochloride (MSDH) induces apoptosis in GC B-cells, including phosphatidyl serine exposure, mitochondrial inactivation, and DNA fragmentation. These apoptotic features occur in the absence of caspase-3 activity. Follicular dendritic cells (FDCs) protect binding B-lymphocytes from lysosomal destabilization, in both the absence and the presence of MSDH. Our study demonstrates that lysosomal leakage induces apoptosis of GC B-cells in a caspase-3-independent manner and that high-affinity binding to FDCsprevents lysosomal leakage and apoptosis in GC B-cells.     

A mathematical model of indicator extraction by the pulmonary endothelium via saturation kinetics

When a bolus containing a nonpermeating indicator and an indicator which permeates the endothelial cell membrane by a saturable process is injected into the blood flowing into the lung, the instantaneous extraction ratio curves measured in the pulmonary venous outflow are asymmetric with respect to the nonpermeating indicator curve. If the bolus contains a sufficient quantity of the permeating indicator that the capillary concentration begins to saturate the transfort mechanism, the extraction ratio curves are concave upward as well. The purpose of this study was to determine whether a mathematical model which represents endothelial extraction by Michaelis-Menten kinetics could explain the time variation in the instantaneous extraction ratio curves. The venous concentration curves were assumed to be the result of the endothelial transfort and distributed capillary input and transit times. In addition, we evaluated a method for estimating the kinetic parameters (K_m and V_max) of the saturable transfort process in such an organ. The results of simulations indicate that the important features of the data can be reproduced by the model, and that useful estimates of the kinetic parameters will be obtained from linear multiple regression analysis of the venous concentration curves if the standard deviation of the capillary input time distribution is not less than that of the capillary transit time distribution.     

A mathematical analysis of the generation and termination of calcium sparks

Calcium sparks are local regenerative releases of Ca(2+) from a cluster of ryanodine receptors on the sarcoplasmic reticulum. During excitation-contraction coupling in cardiac cells, Ca(2+) sparks are triggered by Ca(2+) entering the cell via the T-tubules (Ca(2+)-induced Ca(2+) release). However under conditions of calcium overload, Ca(2+) sparks can be triggered spontaneously. The exact process by which Ca(2+) sparks terminate is still an open question, although both deterministic and stochastic processes are likely to be important. In this article, asymptotic methods are used to analyze a single Ca(2+) spark model, which includes both deterministic and stochastic biophysical mechanisms. The analysis calculates both spark frequencies and spark duration distributions, and shows under what circumstances stochastic transitions are important. Additionally, a model of the coupling of the release channels via the FK-binding protein is analyzed.