Mathematical modeling of transmembrane calcium transport kinetics in smooth muscle cells

A mathematical model, which describes kinetics of transmembrane calcium transport in a smooth muscular cell, has been elaborated and investigated taking into account that the change of calcium cations concentration within a cell is determined by two mutually opposite processes: an increase of a carrying capacity of calcium channels of plasma membrane under signal substance action and calcium removal from the intracellular space by Mg2+, ATP-dependent calcium pump localized on the plasma membrane. The fundamental difference of the proposed model against the models analyzed in literature before is that the cellular system returns to the initial stationary state after enzyme-catalysed transformation of the signal substance. The results of calculations showed that this model really described the experimental kinetics of the transmembrane calcium transport. In this paper the influence of different parameters (Michaelis constant and ultimate rate of calcium pump, initial concentrations of signal substance and enzyme decomposing it, rate constants) on kinetics of calcium transport through the plasma membrane has been investigated in detail.     

Mathematical models of the kinetics of glucose and insulin in plasma

On the assumption that plasma glucose and IRI curves are essentially non-periodic in character, two simple, linear, lumped-parameter mathematical models have been investigated, one consisting of a single compartment and the other of two compartments between which glucose exchanges. The two-compartment model has the stronger appeal in that it admits more readily of a variable hepatic output of glucose. Estimation of many of the rate constants involved has been effected by conforming the theoretical functions to data from intravenous glucose tolerance tests. Numerical values obtained for the rate constants associated with the two-compartment model are satisfactory, but some confusion still remains concerning the turnover time of insulin in plasma.     

Brain-derived adrenomedullin controls blood volume through the regulation of arginine vasopressin production and release

Central nervous system-derived adrenomedullin (AM) has been shown to be a physiological regulator of thirst. Administration of AM into the lateral ventricle of the brain attenuated water intake, whereas a decrease in endogenous AM, induced by an AM-specific ribozyme, led to exaggerated water intake. We hypothesized that central AM may control fluid homeostasis, in part by regulating plasma arginine vasopressin (AVP) levels. To test this hypothesis, AM or a ribozyme specific to AM was administered intracerebroventricularly, and alterations in plasma AVP concentrations were examined under basal and stimulated (hypovolemic) conditions. Additionally, we examined changes in blood volume, kidney function, and plasma electrolyte and protein levels, as well as changes in plasma aldosterone concentrations. Intracerebroventricular administration of AM increased plasma AVP levels, whereas AM ribozyme treatment led to decreased plasma AVP levels under stimulated conditions. During hypovolemic challenges, AM ribozyme treatment led to an increased loss of plasma volume compared with control animals. Although overall plasma osmolality did not differ between treatment groups during hypovolemia, aldosterone levels were significantly higher and, consequently, plasma potassium concentrations were lower in AM ribozyme-treated rats than in controls. These data suggest that brain-derived AM is a physiological regulator of vasopressin secretion and, thereby, fluid homeostasis.     

Plant potassium content modifies the effects of arbuscular mycorrhizal symbiosis on root hydraulic properties in maize plants

It is well known that the arbuscular mycorrhizal (AM) symbiosis helps the host plant to overcome several abiotic stresses including drought. One of the mechanisms for this drought tolerance enhancement is the higher water uptake capacity of the mycorrhizal plants. However, the effects of the AM symbiosis on processes regulating root hydraulic properties of the host plant, such as root hydraulic conductivity and plasma membrane aquaporin gene expression, and protein abundance, are not well defined. Since it is known that K(+) status is modified by AM and that it regulates root hydraulic properties, it has been tested how plant K(+) status could modify the effects of the symbiosis on root hydraulic conductivity and plasma membrane aquaporin gene expression and protein abundance, using maize (Zea mays L.) plants and Glomus intraradices as a model. It was observed that the supply of extra K(+) increased root hydraulic conductivity only in AM plants. Also, the different pattern of plasma membrane aquaporin gene expression and protein abundance between AM and non-AM plants changed with the application of extra K(+). Thus, plant K(+) status could be one of the causes of the different observed effects of the AM symbiosis on root hydraulic properties. The present study also highlights the critical importance of AM fungal aquaporins in regulating root hydraulic properties of the host plant.     

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.     

Theoretical analysis of lipid transport in sciatic nerve.

We modify our previous mathematical model of axonal transport to analyze data on the fast transport of lipids in rat sciatic nerve given in Toews et al. (J. Neurochem. 40, 555-562 (1983)). The theoretical model accounts well for the shapes of the profiles of phosphatidylcholine, phosphatidylethanolamine, cholesterol and diphosphatidylglycerol. The parameters obtained support the qualitative conclusions of Toews et al. and provide quantitative estimates of the underlying processes, e.g., rates of vesicle and mitochondria translocation, rate constants for association and dissociation between vesicles, kinesin and microtubules, rates of deposition and rates of loss of each class of lipid from the nerve by leakage or via removal by the retrograde transport system. The analysis suggests that two classes of vesicles moving at different speeds may be involved in the transport of phosphatidylcholine and phosphatidylethanolamine.     

Flow characteristics of red cell containing fluids through pores. The effect of filter plugging, a mathematical model

A mathematical model was developed that describes the effects of filter plugging on flow through 3 micron pore polycarbonate filters as a function of time, pressure, and cell concentration, both under stirring and nonstirring conditions. The mathematical constants for the model were derived from experimental data generated with a filtration apparatus, and were tested by using various concentrations of cells that are able to plug filter pores. A computer simulation program was written to test the model over a wide range of nonfilterable cell concentrations.     

Structure of MFP2 and its function in enhancing MSP polymerization in Ascaris sperm amoeboid motility

The simplicity and specialization of the cell motility machinery of Ascaris sperm provides a powerful system in which to probe the basic molecular mechanism of amoeboid cell motility. Although Ascaris sperm locomotion closely resembles that seen in many other types of crawling cell, movement is generated by modulation of a cytoskeleton based on the major sperm protein (MSP) rather than the actin present in other cell types. The Ascaris motility machinery can be studied conveniently in a cell-free in vitro system based on the movement of plasma membrane vesicles by fibres constructed from bundles of MSP filaments. In addition to ATP, MSP and a plasma membrane protein, reconstitution of MSP motility in this cell-free extract requires cytosolic proteins to orchestrate the site-specific assembly and bundling of MSP filaments that generates locomotion. One of these proteins, MFP2, accelerates the rate of movement in this assay. Here, we describe crystal structures of two isoforms of MFP2 and show that both are constructed from two domains that have the same fold based on a novel, compact beta sheet arrangement. Patterns of conservation observed in a structure-based analysis of MFP2 sequences from different nematode species identified regions that may be putative functional interfaces involved both in interactions between MFP2 domains and also with other components of the sperm motility machinery. Analysis of the growth of fibres in vitro in the presence of added MFP2 indicated that MFP2 increases the rate of locomotion by enhancing the effective rate of MSP filament polymerization. This observation, together with the structural data, suggests that MFP2 may function in a manner analogous to formins in actin-based motility.     

Mathematical modeling of the microtubule detyrosination/tyrosination cycle for cell-based drug screening design

Microtubules and their post-translational modifications are involved in major cellular processes. In severe diseases such as neurodegenerative disorders, tyrosinated tubulin and tyrosinated microtubules are in lower concentration. We present here a mechanistic mathematical model of the microtubule tyrosination cycle combining computational modeling and high-content image analyses to understand the key kinetic parameters governing the tyrosination status in different cellular models. That mathematical model is parameterized, firstly, for neuronal cells using kinetic values taken from the literature, and, secondly, for proliferative cells, by a change of two parameter values obtained, and shown minimal, by a continuous optimization procedure based on temporal logic constraints to formalize experimental high-content imaging data. In both cases, the mathematical models explain the inability to increase the tyrosination status by activating the Tubulin Tyrosine Ligase enzyme. The tyrosinated tubulin is indeed the product of a chain of two reactions in the cycle: the detyrosinated microtubule depolymerization followed by its tyrosination. The tyrosination status at equilibrium is thus limited by both reaction rates and activating the tyrosination reaction alone is not effective. Our computational model also predicts the effect of inhibiting the Tubulin Carboxy Peptidase enzyme which we have experimentally validated in MEF cellular model. Furthermore, the model predicts that the activation of two particular kinetic parameters, the tyrosination and detyrosinated microtubule depolymerization rate constants, in synergy, should suffice to enable an increase of the tyrosination status in living cells.     

Synthesis and degradation of xanthine dehydrogenase in chick liver. In vivo and in vitro studies.

The present study describes the (xanthine:NAD+ oxidoreductase, EC 1.2.1.37) synthesis and degradation of chick liver xanthine dehydrogenase in vivo and in organ cultures. The results indicate that control of xanthine dehydrogenase activity is mediated by changes in the rate of enzyme synthesis, but that degradation rates are unaffected. The results also suggest that xanthine dehydrogenase synthesis occurs through a previously unreported intermediate. Detected in cultures of liver tissue, this intermediate apparently is not converted into an active enzyme. A model of synthesis and degradation for xanthine dehydrogenase proposes that the synthesis of the enzyme is proportional to messenger RNA and includes an inactive enzyme precursor and a second inactive intermediate prior to degradation. Integrated mathematical solutions describing the concentration of intermediates as a function of time can be found explicitly for simple models. The appendix to this paper extrapolates solutions for one-, two- and three-step models to generate a mathematical solution for an 'n'-step model containing 'n' intermediates. The rate constants in the solutions can be found experimentally.     

A theory of drug tolerance and dependence I: a conceptual analysis

A mathematical model of drug tolerance and its underlying theory is presented. The model extends a first approach, published previously. The model is essentially more complex than the generally used model of homeostasis, which is demonstrated to fail in describing tolerance development to repeated drug administrations. 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 only 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 behavior 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 mathematical model verifies the proposed theory and provides a basis for the implementation of mathematical models of specific physiological processes. In addition, it establishes a relation between the drug dose at any moment, and the resulting drug effect and relates the magnitude of the reactions following withdrawal to the rate of tolerance and other parameters involved in the tolerance process. The present paper analyses the concept behind the model. The next paper discusses the mathematical model.     

The role of mRNA and protein stability in gene expression

How important is the stability of gene products in the process of gene expression? We use a dual-compartment mathematical model to demonstrate the effects that changing the rates of synthesis and degradation of hypothetical mRNAs and proteins would have on the final concentration of protein. The model predicts that the concentration of protein at steady state equals the product of the rate constants for synthesis of mRNA and protein (ks1 and ks2) divided by the product of the rate constants for degradation (kd1 and kd2) and that the rate at which protein concentration changes depends on the rate constants for degradation of both the mRNA and the protein. This permits great flexibility in controlling induction kinetics for particular gene products, since their synthesis, translation, and degradation may be regulated coordinately to permit induction to be stable or transient or to amplify the final yield of protein. We suggest single exons may encode structural features that cause both mRNAs and proteins to be labile, thereby ensuring that modal stabilities of highly regulated macromolecules are similar.     

Determination the rate constants of some biexponential reactions

Reactions that are described by biexponential functions are typical for many biological processes. The kinetics of these reactions is described by transcendental irrational equations interconnecting the reagent concentrations, time and rate constants. Meantime, their graphical representation in the semi-logarithmic coordinates can be decomposed into two straight lines that intercept at some angle. New simple methods for asymptotic numerical solution of the equations describing these reactions are suggested. These methods permit determining the rate constants using the kinetic data of initial substance concentration, which transform into final product according to a two-component model, a sequential model or a competitive model.     

Mathematical models of the balance between apoptosis and proliferation

As our understanding of cellular behaviour grows, and we identify more and more genes involved in the control of such basic processes as cell division and programmed cell death, it becomes increasingly difficult to integrate such detailed knowledge into a meaningful whole. This is an area where mathematical modelling can complement experimental approaches, and even simple mathematical models can yield useful biological insights. This review presents examples of this in the context of understanding the combined effects of different levels of cell death and cell division in a number of biological systems including tumour growth, the homeostasis of immune memory and pre-implantation embryo development. The models we describe, although simplistic, yield insight into several phenomena that are difficult to understand using a purely experimental approach. This includes the different roles played by the apoptosis of stem cells and differentiated cells in determining whether or not a tumour can grow; the way in which a density dependent rate of apoptosis (for instance mediated by cell-cell contact or cytokine signalling) can lead to homeostasis; and the effect of stochastic fluctuations when the number of cells involved is small. We also highlight how models can maximize the amount of information that can be extracted from limited experimental data. The review concludes by summarizing the various mathematical frameworks that can be used to develop new models and the type of biological information that is required to do this.     

Bench and mathematical modeling of the effects of breathing a helium/oxygen mixture on expiratory time constants in the presence of heterogeneous airway obstructions

ABSTRACT: BACKGROUND: Expiratory time constants are used to quantify emptying of the lung as a whole, and emptying of individual lung compartments. Breathing low-density helium/oxygen mixtures may modify regional time constants so as to redistribute ventilation, potentially reducing gas trapping and hyperinflation for patients with obstructive lung disease. In the present work, bench and mathematical models of the lung were used to study the influence of heterogeneous patterns of obstruction on compartmental and whole-lung time constants. METHODS: A two-compartment mechanical test lung was used with the resistance in one compartment held constant, and a series of increasing resistances placed in the opposite compartment. Measurements were made over a range of lung compliances during ventilation with air or with a 8/22% mixture of helium/oxygen. The resistance imposed by the breathing circuit was assessed for both gases. Experimental results were compared with predictions of a mathematical model applied to the test lung and breathing circuit. In addition, compartmental and whole-lung time constants were compared with those reported by the ventilator. RESULTS: Time constants were greater for larger minute ventilation, and were reduced by substituting helium/oxygen in place of air. Notably, where time constants were long due to high lung compliance (i.e. low elasticity), helium/oxygen improved expiratory flow even for a low level of resistance representative of healthy, adult airways. In such circumstances, the resistance imposed by the external breathing circuit was significant. Mathematical predictions were in agreement with experimental results. Time constants reported by the ventilator were wellcorrelated with those determined for the whole-lung and for the low-resistance compartment, but poorly correlated with time constants determined for the high-resistance compartment. CONCLUSIONS: It was concluded that breathing a low-density gas mixture, such as helium/oxygen, can improve expiratory flow from an obstructed lung compartment, but that such improvements will not necessarily affect time constants measured by the ventilator. Further research is required to determine if alternative measurements made at the ventilator level are predictive of regional changes in ventilation. It is anticipated that such efforts will be aided by continued development of mathematical models to include pertinent physiological and pathophysiological phenomena that are difficult to reproduce in mechanical test systems.     

Effect of exercise duration on lactate kinetics after short muscular exercise

Arterial blood lactate concentrations were measured in six normal males before, during and after 3- and 6-min bicycle exercises performed at three different work rates. The lactate recovery curves were fitted to a bi-exponential time function consisting of a rapidly increasing and a slowly decreasing component, which supplied an accurate representation of the changes in lactate concentration. Variations in the parameters of this mathematical model have been studied as a function of the duration of exercise and of the work rate, showing a clear dependence on exercise duration such that increasing exercise length decreases the velocity constants of the fitted curves. In terms of the functional meaning which can be given to these constants, this result indicates that extending exercise duration from 3 to 6 min reduces the ability of the whole body to exchange and remove lactate. This effect did not qualitatively modify the one already described, which is due to increased work rates, but it shifted the ability to exchange and remove lactate towards lower values. The main conclusion of the study is that lactate kinetic data vary as a function of time during exercise. This inference must be accounted for in the interpretation of lactate data obtained during muscular exercise.