Human iodine requirements determined by the saturation kinetics model

Iodine plays a decisive role in metabolism and the process of early growth and development of most organs, especially of the brain. Effects of iodine deficiency include goiter, stillbirth and miscarriage, neonatal and juvenile thyroid deficiency, dwarfism, mental defects, deaf mutism, spastic weakness and paralysis. In this study, the application of a mathematical model (derived from Machaelis-Menten enzyme kinetics) to iodine measured in urine samples from a randomly selected group derived from the Egyptian village of West El-Mawhoub in the Dakhlah Oasis resulted in the conclusion that iodine excretion parameters can be used to characterize iodine utilization and accurately predict the level of salt iodination required to maintain proper physiological functions. The four parameter saturation kinetics model analysis indicated that a salt iodination level of 63 mg/kg reduced the severity of IDD, with 83% of the studied subjects having urinary excretion levels of 1.18 micromol/L. This gives a convenient mechanism for providing adequate dietary iodine with a non-invasive index for the avoidance of IDD. Commercially available salt was analyzed using standard iodiometric titration methods to determine iodination levels. Analysis revealed that only 20% of the commercially available salt complied with the manufacturer's label and revealed the presence of large individual variability between batches amounting to -95 to +150% of the claimed iodine level. Therefore, salt iodination requires careful supervision to ensure that promised iodine levels are being delivered and consumed.     

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 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 describing kinetics of conversion of violaxanthin to zeaxanthin via intermediate antheraxanthin by the xanthophyll cycle enzyme violaxanthin de-epoxidase

The xanthophyll cycle is one of the mechanisms protecting the photosynthetic apparatus against the light energy excess. Its action is still not well understood on the molecular level.Our model makes it possible to follow independently the kinetics of the two de-epoxidation steps occurring in the xanthophyll cycle: the conversion of violaxanthin into antheraxanthin and the conversion of antheraxanthin into zeaxanthin. Using a simple form of the transition rates of these two conversions, we model the time evolution of the concentration pattern of violaxanthin, antheraxanthin and zeaxanthin during the de-epoxidation process. The model has been applied to describe the reactions of de-epoxidation in a system of liposome membranes composed of phosphatidylcholine and monogalactosyldiacylglycerol. Results obtained within the model fit very well with the experimental data. Values of the transition probabilities of the violaxanthin conversion into antheraxanthin and the antheraxanthin conversion into zeaxanthin calculated by means of the model indicate that the first stage of the de-epoxidation process is much slower than the second one.     

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 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 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 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.     

Development of predictive mathematical model for the growth kinetics of Staphylococcus aureus by response surface model

A response surface model was developed for predicting the growth rates of Staphylococcus aureus in tryptic soy broth (TSB) medium as a function of combined effects of temperature, pH, and NaCl. The TSB containing six different concentrations of NaCl (0, 2, 4, 6, 8, and 10%) was adjusted to an initial of six different pH levels (pH 4, 5, 6, 7, 8, 9, and 10) and incubated at 10, 20, 30, and 40 degrees C. In all experimental variables, the primary growth curves were well (r2=0.9000 to 0.9975) fitted to a Gompertz equation to obtain growth rates. The secondary response surface model for natural logarithm transformations of growth rates as a function of combined effects of temperature, pH, and NaCl was obtained by SAS's general linear analysis. The predicted growth rates of the S. aureus were generally decreased by basic (pH 9-10) or acidic (pH 5-6) conditions and higher NaCl concentrations. The response surface model was identified as an appropriate secondary model for growth rates on the basis of correlation coefficient (r=0.9703), determination coefficient (r2=0.9415), mean square error (MSE=0.0185), bias factor (B(f)=1.0216), and accuracy factor (A(f)=1.2583). Therefore, the developed secondary model proved reliable for predictions of the combined effect of temperature, NaCl, and pH on growth rates for S. aureus in TSB medium.     

A mathematical model of airway and pulmonary arteriole smooth muscle

Airway hyperresponsiveness is a major characteristic of asthma and is believed to result from the excessive contraction of airway smooth muscle cells (SMCs). However, the identification of the mechanisms responsible for airway hyperresponsiveness is hindered by our limited understanding of how calcium (Ca²⁺), myosin light chain kinase (MLCK), and myosin light chain phosphatase (MLCP) interact to regulate airway SMC contraction. In this work, we present a modified Hai-Murphy cross-bridge model of SMC contraction that incorporates Ca²⁺ regulation of MLCK and MLCP. A comparative fit of the model simulations to experimental data predicts 1), that airway and arteriole SMC contraction is initiated by fast activation by Ca²⁺ of MLCK; 2), that airway SMC, but not arteriole SMC, is inhibited by a slower activation by Ca²⁺ of MLCP; and 3), that the presence of a contractile agonist inhibits MLCP to enhance the Ca²⁺ sensitivity of airway and arteriole SMCs. The implication of these findings is that murine airway SMCs exploit a Ca²⁺-dependent mechanism to favor a default state of relaxation. The rate of SMC relaxation is determined principally by the rate of release of the latch-bridge state, which is predicted to be faster in airway than in arteriole. In addition, the model also predicts that oscillations in calcium concentration, commonly observed during agonist-induced smooth muscle contraction, cause a significantly greater contraction than an elevated steady calcium concentration.     

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.     

A mathematical model of the proliferative kinetics of cells labelled with brominated DNA precursors

A mathematical model of cell division is presented. It permits analyzing cell proliferation obtained by using BrdU. Numerical experiments show that this model adequately describes experimental data. Application of this model permits decreasing experimental data bulk.     

A mathematical model of the unidirectional block caused by the pulmonary veins for anatomically induced atrial reentry

It is widely believed that the pulmonary veins (PVs) of the left atrium play the central role in the generation of anatomically induced atrial reentry but its mechanism has not been analytically explained. To understand this mechanism, a new analytic approach is proposed by adapting the geometric relative acceleration analysis from spacetime physics based on the hypothesis that a large relative acceleration can translate to a dramatic increase in the curvature of a wavefront and subsequently to conduction failure. By verifying the strong dependency of the propagational direction and the magnitude of anisotropy for conduction failure, this analytic method reveals that a unidirectional block can be generated by asymmetric propagation toward the PVs. This model is validated by computational tests in a T-shaped domain, computational simulations for three-dimensional atrial reentry and previous in-silico reports for anatomically induced atrial reentry.     

A mathematical model of protein degradation by the proteasome

The proteasome is the major protease for intracellular protein degradation. The influx rate of protein substrates and the exit rate of the fragments/products are regulated by the size of the axial channels. Opening the channels is known to increase the overall degradation rate and to change the length distribution of fragments. We develop a mathematical model with a flux that depends on the gate size and a phenomenological cleavage mechanism. The model has Michaelis-Menten kinetics with a V(max) that is inversely related to the length of the substrate, as observed in the in vitro experiments. We study the distribution of fragment lengths assuming that proteasomal cleavage takes place at a preferred distance from the ends of a protein fragment, and find multipeaked fragment length distributions similar to those found experimentally. Opening the gates in the model increases the degradation rate, increases the average length of the fragments, and increases the peak in the distribution around a length of 8-10 amino acids. This behavior is also observed in immunoproteasomes equipped with PA28. Finally, we study the effect of re-entry of processed fragments in the degradation kinetics and conclude that re-entry is only expected to affect the cleavage dynamics when short fragments enter the proteasome much faster than the original substrate. In summary, the model proposed in this study captures the known characteristics of proteasomal degradation, and can therefore help to quantify MHC class I antigen processing and presentation.     

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 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.     

A mathematical model of cancer treatment by immunotherapy

In this paper, a detailed mathematical study of cancer immunotherapy will be presented. General principles of cancer immunotherapy and the model equations and hypotheses will be discussed. Mathematical analyses of the model equations with regard to dissipativity, boundedness of solutions, invariance of non-negativity, nature of equilibria, persistence, extinction and global stability will be analyzed. It will also be shown that bifurcations can occur, and criteria for total cure will also be derived.     

A mathematical model of the urethra

Models having the form of surfaces of revolution may be used to represent the urethra under pre-voiding pressure. From such models are derived formulas for calculating muscle tension from the shape of a urethragram.