Sodium is not required for chloride efflux via chloride/bicarbonate exchanger from rat thymic lymphocytes.

Sodium-dependent Cl(-)/HCO3 (-) exchanger acts as a chloride (Cl(-)) efflux in lymphocytes. Its functional characterization had been described when Cl(-) efflux was measured upon substituting extracellular sodium (Na(+)) by N-methyl-D-glucamine (NMDG). For Na(+) and Cl(-) substitution, we have used D-mannitol or NMDG. Thymocytes of male Wistar rats aged 7-9 weeks were used and intracellular Cl(-) was measured by spectrofluorimetry using MQAE dye in bicarbonate buffers. Chloride efflux was measured in a Cl(-)-free buffer (Cl(-) substituted with isethionate acid) and in Na(+) and Cl(-)-free buffer with D-mannitol or with NMDG. The data have shown that Cl(-) efflux is mediated in the absence of Na(+) in a solution containing D-mannitol and is inhibited by H2DIDS. Mathematical modelling has shown that Cl(-) efflux mathematical model parameters (relative membrane permeability, relative rate of exchanger transition, and exchanger efficacy) were the same in control and in the medium in which Na(+) had been substituted by D-mannitol. The net Cl(-) efflux was completely blocked in the NMDG buffer. The same blockage of Cl(-) efflux was caused by H2DIDS. The study results allow concluding that Na(+) is not required for Cl(-) efflux via Cl(-)/HCO3 (-) exchanger. NMDG in buffers cannot be used for substituting Na(+) because NMDG inhibits the exchanger.

Thrombin-stimulated discharge of calcium stores in human platelets: analysis of experimental data.

The purpose of this research was to analyze experimental data concerning thrombin-stimulated discharge of calcium stores in human platelets contained in calcium-free medium in view of better understanding the mechanisms involved in calcium fluxes. The model curves are reasonably close to experimental data; the parameters of the models are related to the properties of the entities responsible for control or maintenance of cytosolic calcium concentration. It has been shown that: (a) time-course of calcium concentration in cytosol of human platelets can be acceptably modeled on the basis of reasonable assumptions concerning agonist stimulated calcium redistribution in cellular compartments; (b) those assumptions are of fundamental importance for the model (c) some parameters of the model (taken arbitrarily) cannot be estimated independently of others from fitting the model to experimental data available; (d) special experiments are necessary to determine the unknown parameters; (e) agonist-stimulated change of the permeability of endomembrane of calcium stores can be regarded as a pulse of the permeability; it can be modeled as a sequence of transitions of the system from inactive to active and to inactive state again.

Calcium fluxes into and out of cytosol in human platelets: analysis of experimental data.

The purpose of this research was to analyse experimental data concerning cytosolic calcium concentration in view of the mechanisms involved in calcium fluxes in human platelets. The parameters of model curves are related to the properties of the entities responsible for control or maintenance of cytosolic calcium concentration. It has been shown that: (a) biphasicity of increase in cytosolic calcium concentration caused by inhibition of SERCAs either by TBHQ and TG or by TG alone is related to fast and slow discharge of acidic calcium stores and DTS; (b) biphasicity of decline in cytosolic calcium concentration after its rise caused by stimulation of platelets by the agonists is related to non-synchronous extrusion of calcium by PMCA and NCX; (c) NCX is active only in calcium containing medium: calcium ion(s) are necessary to be bound to the site(s) located on the medium-facing side of the (macro)molecule; (d) PMCA is likely to be activated either by binding calcium ion(s) to the site(s) located on its cytosol-facing side or by unbinding identical ion(s) from the site(s) on its medium-facing side.

Minimal models of growth and decline of microbial populations.

Dynamics of growth and decline of microbial populations were analysed and respective models were developed in this investigation. Analysis of the dynamics was based on general considerations concerning the main properties of microorganisms and their interactions with the environment which was supposed to be affected by the activity of the population. Those considerations were expressed mathematically by differential equations or systems of the equations containing minimal sets of parameters characterizing those properties. It has been found that: (1) the factors leading to the decline of the population have to be considered separately, namely, accumulation of metabolites (toxins) in the medium and the exhaustion of resources; the latter have to be separated again into renewable ('building materials') and non-renewable (sources of energy); (2) decline of the population is caused by the exhaustion of sources of energy but no decline is predicted by the model because of the exhaustion of renewable resources; (3) the model determined by the accumulation of metabolites (toxins) in the medium does not suggest the existence of a separate 'stationary phase'; (4) in the model determined by the exhaustion of energy resources the 'stationary' and 'decline' phases are quite discernible; and (5) there is no symmetry in microbial population dynamics, the decline being slower than the rise. Mathematical models are expected to be useful in getting insight into the process of control of the dynamics of microbial populations. The models are in agreement with the experimental data.

Plasma membrane calcium pump and sodium-calcium exchanger in maintenance and control of calcium concentrations in platelets.

The purpose of this research was to elucidate the activity of the mechanisms responsible for control of cytosolic calcium concentration in platelets by modeling the time-course of the concentration changing in response to discharge of the intracellular stores or store-operated calcium entry (SOCE). The parameters estimated as a result of model fitting to experimental data are related to physiological or pathological state of the cells. It has been shown that: (a) the time-course is determined by the passive calcium fluxes and activities of the corresponding mechanisms; (b) the decline in the concentration (after its rise) develops due to activity of plasma membrane calcium ATPase (PMCA) both in the case of discharge of the stores of platelets contained in calcium-free medium and in the case of SOCE; (c) impulsive extrusion of calcium in response to its sudden influx, presumably, is the main function of PMCA; (d) the function of sodium-calcium exchanger (NCX) is to extrude calcium excess by permanent counteracting its influx.

Minimal models of multi-site ligand-binding kinetics.

Mathematical models based on the current understanding of co-operativity in ligand binding to the (macro) molecule and relating the dose-response (saturation) curve of the (macro) molecule ligation to intrinsic dissociation constants characterizing the affinities of ligand for binding sites of both unliganded and partly liganded (macro) molecule have been developed. The simplified models disregarding the structural properties and considerations concerning conformational changes of the (macro) molecule retain the ability to yield sigmoid curves of ligand binding and reflect the co-operativity. Model 1 contains only three parameters, parameter kappa (a multiplier characterising the change in the affinity) reflects also the existence and type of co-operativity of ligand binding: kappa<1 corresponds to positive co-operativity, kappa>1 to the negative and kappa=1 to the absence of any co-operativity. Model 2 contains an extra parameter, omega, equilibrium constant for the T(0)<-->R(0) transition but fails to produce dose-response, which would suggest negative co-operativity. For any fixed n>1, the deviation of the dose-response (saturation) curve from the Henri hyperbola depends either solely on parameter kappa (Model 1) or also on parameter omega (Model 2). The (macro) molecule being a receptor, both models yield a diversity of dose-response curves due to possible variety of efficacies of the (macro) molecule. The models may be considered as extensions of the Henri model: in case the dissociation constants remain unchanged, the proposed models are reduced to the latter.

Physical properties of two types of calcium stores and SERCAs in human platelets.

The aim of this work was to obtain experimental data depending on the properties of calcium stores and SERCAs, to analyse these data in terms of the models based on simulation of the cellular compartments as communicating vessels, and to relate this way the data to the above properties. The main characteristics of the stores and corresponding SERCAs have been estimated. Calcium content in the DTS is approximately 1.5 x 10(6) ions per cell, that in the acidic stores, approximately 0.64 x 10(6) ions per cell. The rate constant of background calcium leakage from the DTS is approximately 0.0033 s(-1), that from the acidic stores, approximately 0.1 s(-1). The background activity of SERCA2b is approximately 0.22 x 10(6 )s(-1) ions per cell, that of SERCA3, approximately 2.5 x 10(6 )s(-1) ions per cell. The properties of both calcium stores and the SERCAs and the characteristics found might be related to physiological or pathological state of the cells.

Dynamics of calcium fluxes in nonexcitable cells: mathematical modeling.

Mathematical models simulating the dynamics of calcium redistribution (elicited by experimental interference with the pathways of calcium fluxes) in cellular compartments have been developed, based on a minimal scheme of the pathways of calcium fluxes in nonexcitable cells suspended in calcium-free medium. The models are consistent with available experimental data. All parameters are quantitatively related to the intrinsic properties of calcium adenosine triphosphatases (ATPases) and cellular membranes; there is no interdependence between the parameters. The models can be used as the basis for quantitative analysis and interpretation of experimental data. The activities of plasma membrane and sarcoendoplasmic reticulum calcium ATPases (PMCA and SERCAs) are governed by different mechanisms. PMCA is likely to undergo transitions from inactive to active to "dormant" (not identical to the initial) and back to inactive states, the mean duration of the cycle lasting for minutes or longer. The sequence of the transitions is initiated, presumably, by an increase in cytosolic calcium concentration. The transition of PMCA from inactive to active (at least at low rates of increase in cytosolic calcium concentration) is likely to be slower than that from active to dormant. SERCA, presumably, transits from inactive to active state in response to increases in calcium leakage from calcium stores. Whereas PMCA extrudes excess calcium (a definite quantity of it) in a short pulse, SERCA retakes calcium back into the stores permanently at a high rate. The models presented here may be the best means for the moment to quantitatively relate the dynamics of calcium fluxes in nonexcitable cells with known or putative properties of the mechanisms underlying activation of calcium ATPases.

Growth of microbial populations. Mathematical modeling, laboratory exercises, and model-based data analysis.

This paper has arisen as a result of teaching Models in Biology to undergraduates of Bioengineering at the Gediminas Technical University of Vilnius. The aim is to teach the students to use a fresh approach to the problems they are familiar with, to come up with an articulate verbal model after a mental effort, to express it in rigorous mathematical terms, to solve (with the aid of computers) corresponding equations, and finally, to analyze and interpret experimental data in terms of their (mathematical) models. Investigation of microbial growth provides excellent possibilities to combine laboratory exercises, mathematical modeling, and model-based data analysis. Application of mathematics in this field proved to be very fruitful in getting deeper insight into the processes of microbial growth. The step-by-step modeling resulted in an extended model of the growth covering conventional "lag," "exponential," and "stationary" phases. In contrast to known models (differential equations that can be solved only numerically), the present model is expressed symbolically as a finite combination of elementary functions. The approach can be applied in other areas of modern biology, such as dynamics of various cellular processes, enzyme and receptor kinetics, and others.

Dynamics of calcium fluxes in human platelets assessed in calcium-free medium.

Dynamics of changes in cytosolic calcium concentration resulting from facilitation of calcium leakage from the stores and (or) blocking the pathways of its reuptake back into the stores or extrusion out of the cell (or both) have been investigated experimentally. It has been found that: (a) no mechanisms other than the membrane leakage, PMCA or SERCA, are involved in the discharge of calcium stores and calcium extrusion or reuptake; (b) the discharge of calcium stores in the absence of both its extrusion and reuptake back into the stores depends only on membrane leakage, the asymptotic calcium concentration in cytosol depending only on the initial content of the stores and being independent of the leakage; (c) the dynamics of the activity of both PMCA and SERCA depend on the initial rate of calcium influx, the dynamics differing from each other at high initial rates of calcium influx; (d) whereas there is no observable background activity of PMCA, background activity of SERCA is observed.

A minimal model of non-hyperbolic enzyme and receptor kinetics.

A simplified version of P.W. Kühl's Recovery Model [Biochem. J. 298 (1994) 171-180] has been developed in which the duration of the recovery phase of receptor or enzyme (macro)molecule was assumed to be a random value distributed exponentially like other model parameters. The model has been shown to retain all the properties of the original Recovery Model except for its ability to yield asymmetric dose-response curves (if plotted in semi-logarithmic scale). Due to its simplicity, the present model is applicable for routine fitting to experimental data. In enzyme kinetics, the model yields a diversity of non-hyperbolic dose-response curves both with higher and lower steepness than that of Henri-type ones. In receptor kinetics, the diversity of dose-response curves is further increased due to virtually no restraints being imposed on the efficacies of any state of the macromolecule.