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 on germinal center kinetics and termination.

We devise a mathematical model to study germinal center (GC) kinetics. Earlier models for GC kinetics are extended by explicitly modeling 1) the cell division history of centroblasts, 2) the Ag uptake by centrocytes, and 3) T cell dynamics. Allowing for T cell kinetics and T-B cell interactions, we study the role of GC T cells in GC kinetics, GC termination, and B cell selection. We find that GC T cells play a major role in GC formation, but that the maintenance of established GC reactions requires very few T cells only. The results therefore suggest that the termination of a GC reaction is largely caused by lack of Ag on the follicular dendritic cells and is hardly influenced by Th cells. Ag consumption by centrocytes is the major factor determining the decay rate of the antigenic stimulus during a GC reaction. Investigating the effect of the Ag dose on GC kinetics, we find that both the total size of the GC and its duration are hardly influenced by the initial amount of Ag. In the model this is due to a buffering effect by competition for limited T cell help and/or competition between proliferating centroblasts.     

A mathematical model for metal affinity protein partitioning

A mathematical model of metal affinity partitioning has been derived and used to describe protein partitioning in Cu (II)PEG/dextran systems. A working model has been extended to account for inhibition, which for metal affinity extraction is the inhibition of protein-metal binding by hydrogen ion. PEG/dextran partitioning experiments were performed on four proteins, tuna heart cytochrome c, Candida krusei cytochrome c, horse myoglobin, and sperm whale myoglobin. The partition coefficients for these proteins are increased by the addition of Cu (II)PEG-IDA, due to the affinity between the chelated copper atom and metal-coordinating histidine residues on the protein surface. The results of experiments to determine the effects of the number of binding sites on the protein, the copper concentration, and pH on partitioning are all well-described by the mathematical model. The pK(a) value of the metal binding site was determined to be 6.5, which is in the range of pK(a) values commonly observed for surface histidines. The average association constant for the binding of Cu (II)PEG-IDA to accessible histidines was found to be 4.5 x 10(3). This value is comparable to stability constants measured by conventional potentiometry techniques for analogous small complexes.     

A spatial model of germinal center reactions: cellular adhesion based sorting of B cells results in efficient affinity maturation

Affinity maturation of humoral responses to T-cell-dependent antigens occurs in germinal centers (GC). In GCs antigen-specific B cells undergo rounds of somatic mutations that alter their affinity. High-affinity mutants take over GCs very soon after they appear; the replacement rate is as high as 4 per day (Radmacher et al., Immunol. Cell Biol. 76 (1998) 373). To gain more insight into this selection process, we present a spatial model of GC reactions, where B cells compete for survival signals from follicular dendritic cells (FDC). Assuming that high-affinity B cells have increased cellular adhesion to FDCs, we obtain an affinity-based sorting of B cells on the FDC. This sorting imposes a very strong selection and therefore results in a winner-takes-all behavior. By comparing our sorting model with "affinity-proportional selection models", we show that this winner-takes-all selection is in fact required to account for the fast rates at which high affinity mutants take over GCs. Another important feature of in vivo GC reactions is that they are non-mixed, i.e. GCs contain either no high-affinity cells at all or they are dominated by high-affinity cells. We here show that this all-or-none behavior can be obtained if B cells are sorted based on their affinity on the FDC surface. Affinity-proportional selection models, in contrast, always produce mixed GCs.     

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     

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.     

Study of germinal vesicle requirement for the normal kinetics of maturation/M-phase-promoting factor activity during porcine oocyte maturation

Mammalian immature oocytes contain large nuclei referred to as germinal vesicles (GVs). The translocation of maturation/M-phase promoting factor (MPF) into GVs just before the activation of MPF has been reported in several species. To examine whether the GV is required for MPF activation in mammalian oocytes, porcine immature oocytes were enucleated and their MPF activity and CCNB (also known as cyclin B) levels were investigated. The activation of MPF at the start of maturation was detected at normal levels in enucleated oocytes, whereas reactivation to induce the second meiosis was not observed. Although protein synthesis was found to be normal both qualitatively and quantitatively, even in the absence of the nucleus, CCNB1 did not sufficiently accumulate in the enucleated oocytes. The defects in the enucleated oocytes were reversed by the injection of GV material into the enucleated oocytes. Furthermore, the inhibition of CCNB1 degradation revealed drastic accumulation of CCNB1, indicating active synthesis of CCNB1 in enucleated oocytes. The mitogen-activated protein kinase cascade remained unaffected by enucleation. These results indicate that GV is not required for the activation of MPF during the first meiosis, but that it is required for the second meiosis because of its promotion of CCNB1 accumulation.     

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 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)     

Turnover and maturation kinetics in the hairless mouse epidermis. Continuous [3H]TdR labelling and mathematical model analyses

Hairless mice were continuously labelled with 10 microCi of tritiated thymidine ([3H]TdR) every 4 h for 8 d, and the proportions of labelled basal and differentiating cells were recorded separately. The mitotic rate was measured by the stathmokinetic method and the cell cycle distributions were measured by flow cytometry of isolated basal cells at intervals during the labelling period. The mitotic rate of the [3H]TdR-injected animals did not deviate from control values during the first 5 d. Computer simulations of the data based on various mathematical models were made, and three main conclusions were obtained: (1) a large spread in transit times through the G1 phase was found, together with a very narrow distribution in maturation time of differentiating cells; (2) about 20% of the differentiating cells were estimated to leave the basal cell layer directly after mitosis. This is consistent with results obtained from different sets of data; and (3) during continuous labelling more than 90% of the cells are labelled during each passage through the S phase.     

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 photoreceptor interactions

The interactions between rods and cones in the retina have been the focus of innumerable experimental and theoretical biological studies in previous decades yet the understanding of these interactions is still incomplete primarily due to the lack of a unified concept of cone photoreceptor organization and its role in retinal diseases. The low abundance of cones in many of the non-primate mammalian models that have been studied make conclusions about the human retina difficult. A more complete knowledge of the human retina is crucial for counteracting the events that lead to certain degenerative diseases, in particular those associated with photoreceptor cell death (e.g., retinitis pigmentosa). In an attempt to gain important insight into the role and interactions of the rods and the cones we develop and analyze a set of mathematical equations that model a system of photoreceptors and incorporate a direct rod-cone interaction. Our results show that the system can exhibit stable oscillations, which correspond to the rhythmic renewal and shedding of the photoreceptors. In addition, our results show the mathematical necessity of this rod-cone direct interaction for survival of both and gives insight into this mechanism.     

A mathematical model for calcium homeostasis

In vivo control of calcium is analysed under the assumption that hormonal influences via plasma levels of parathormone and calcitonin are of prime (but not absolutely dominating) importance. A brief review concerning the physiological significance of body calcium and the mode of action of these two hormones is presented as an introduction to the basic philosophy of the study. A theoretical quasi-linear lumped-parameter model is developed to describe variations in ionic calcium, parathormone and calcitonin plasma concentrations to specific input stimuli. Formal evaluation of the system response requires the determination of ten constants, together with quantitation of ingested calcium entry into the plasma compartment which isindependent of hormonal influences. Values for various parameters are deduced from published data and experimental procedures are outlined to facilitate determination of the remaining unknowns. It is suggested that the proposed model should prove useful for investigations concerning general hormonal actions on calcium homeostatic mechanisms in both normal and diseased states, with particular reference to calcitonin.     

A mathematical model for dorsal closure

During embryogenesis, drosophila embryos undergo epithelial folding and unfolding, which leads to a hole in the dorsal epidermis, transiently covered by an extraembryonic tissue called the amnioserosa. Dorsal closure (DC) consists of the migration of lateral epidermis towards the midline, covering the amnioserosa. It has been extensively studied since numerous physical mechanisms and signaling pathways present in DC are conserved in other morphogenetic events and wound healing in many other species (including vertebrates).We present here a simple mathematical model for DC that involves a reduced number of parameters directly linked to the intensity of the forces in the presence and which is applicable to a wide range of geometries of the leading edge (LE). This model is a natural generalization of the very interesting model proposed in Hutson et al. (2003). Being based on an ordinary differential equation (ODE) approach, the previous model had the advantage of being even simpler, but this restricted significantly the variety of geometries that could be considered and thus the number of modified dorsal closures that could be studied.A partial differential equation (PDE) approach, as the one developed here, allows considering much more general situations that show up in genetically or physically perturbed embryos and whose study will be essential for a proper understanding of the different components of the DC process. Even for native embryos, our model has the advantage of being applicable since an early stages of DC when there is no antero-posterior symmetry (approximately verified only in the late phases of DC).We validate our model in a native setting and also test it further in embryos where the zipping force is perturbed through the expression of spastin (a microtubule severing protein). We obtain variations of the force coefficients that are consistent with what was previously described for this setting.     

A mathematical model for radiation-induced myelopoiesis

A model for damage, repair, killing, and repopulation of myelopoietic marrow is presented. Evaluation produces time and dose-rate profiles during and following any complex irradiation. Equations model variable dose rates, multiple exposures, different sources, and arbitrary intervals between treatments. If factors which dominate the control of biological processes can be demonstrated, an option is to set biological rate constants to experimentally determined values. Previously, knowledge did not permit identification of dominating biological processes and their temporal rates. But a unique feature of this study is that unspecified lesions for killing and injury of cells are evaluated from mortality data on the animal species of choice. "Unspecified" is used to indicate a condition of assumption-free modeling of molecular processes, whereby rate constants for cellular effects are simply computed directly from animal mortality data. Coefficients (estimated by maximum-likelihood methods for nonspecific processes) are compared with experimental values for specific processes. The model has many uses, including modeling of the myelopoietic potential as a function of time. Another option is to calculate the whole-body survival curve for cells that control myelopoiesis as a result of the treatment schedule. Also through simple extensions of the model, an extremely complex protocol can be identified with an equivalent prompt dose value--even for partial-body, fractionated exposures.     

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 generalized mathematical model for the growth kinetics of Saccharomyces cerevisiae with experimental determination of parameters

A general model of the kinetics of microbial growth has been developed involving the kinetics of incorporation of substrate into biomass and the maintenance energy requirements. Results obtained from batch cultures of the yeast Saccharomyces cerevisiae growing in synthetic media at pH 5.1 and 30°C permitted all biological parameters in the model to be calculated. Values obtained for these parameters were: maximum specific glucose uptake rate (μS_m), 2.08 g/g biomass/hr; apparent Michaelis constant for glucose (K_S), 0.1 g/liter (5.5 × 10^?4M) apparent Michaelis constant for oxygen (K_L), 1.4% O₂ (3.2 × 10^?6 M) quantitative index of the Pasteur effect (b), 4.9 × 10^?4%^?1 O₂ (207 M ^?1). Under conditions of strongly substrate-repressed respiration the values obtained for Y_ATP and P/O were constant over the course of the exponential phase of growth (Y_ATP = 10.4 g biomass/mole ATP; P/O = 3 moles ATP/atom 0). Mass balances for aerobic and anaerobic cultures confirmed the results obtained form the generalized model. Results presented suggested the operation of a mechanism for regulating energy-yielding metabolism which involved an equilibrium between the systems of oxidative phosphorylation and dephosphorylation and was dependent upon the level of catbolite repression.