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PubMed1996

A mathematical model of the crustacean stretch receptor neuron. Biomechanics of the receptor muscle, mechanosensitive ion channels, and macrotransducer properties.

1. A mathematical model of the primary transduction process in a mechanoreceptor, the slowly adapting stretch receptor organ of the crayfish, has been developed taking into account the viscoelastic properties of the accessory structures of the receptor, i.e., the receptor muscle, the biophysical properties of the mechanosensitive channels (MSCs) and the passive electrical properties of the neuronal membrane (leak conductance and capacitative properties). The work is part of an effort to identify and characterize the mechanical and ionic mechanisms in a complex mechanoreceptor. The parameters of the model are based mainly on results of our own experiments and to some extent on results from other studies. The performance of the model has been compared with the performance of the slowly adapting receptor. 2. The model resulted in nonlinear differential equations that were solved by an iterative, fourth order Range-Kutta method. For the calculations of potential, the cell was treated as an idealized spherical body. The extension of the receptor muscle was 0-30%, which is within the physiological limits for this receptor. 3. The mechanical properties of the receptor muscle were modeled by a simple Voigt element (a spring in parallel with a dashpot) in series with a nonlinear spring. This element can describe resonably well the tension development in the receptor muscle at least for large extensions (> 12%). However, for small extensions (< 12%), the muscle seems to be more stiff than for large extensions. 4. The receptor current at different extensions of the receptor was computed using typical viscoelastic parameters for a receptor muscle together with a transformation of muscle tension to tension in the neuronal dendrites and finally the properties of the mechanosensitive channels. The model fit was satisfactory in the high extension range whereas in the low extension range the deviation from the experimental results could be explained partly by insufficient modeling of the nonlinear viscoelastic properties. The voltage dependence of the receptor current was also well predicted by the model. 5. If the parameters of the viscoelastic model were adjusted for each extension so that each tension response closely resembled the experimental values, the fit of the current responses was improved but still deviated from the experimental currents. One factor that might explain the difference is the possibility that the MSCs in the stretch receptor neuron might have intrinsic adaptive properties. Introducing an exponential adaptive behavior of individual MSCs increased the ability of the model to predict the receptor current. 6. The receptor potential was calculated by modeling the neuronal membrane by a lumped leak conductance and capacitance The calculated receptor potential was higher than the experimental receptor potential. However, the fit of the receptor potential was improved substantially by introducing an adaptation of the MSCs as outlined in the preceding paragraph. the remaining discrepancy might be explained by insufficient blocking of K+ channels in the experiment. 7. The model can predict a wide range of experimental data from the slowly adapting stretch receptor neuron including the mechanical response of the receptor muscle, the receptor current and its voltage dependence, and the receptor potential. It also describes accurately the passive electrical properties of the neuronal membrane.

Adaptation, Physiological↗
PubMed1991

A mathematical model of the volume, pH, and ion content regulation in reticulocytes. Application to the pathophysiology of sickle cell dehydration.

We developed a mathematical model of the reticulocyte, seeking to explain how a cell with similar volume but much higher ionic traffic than the mature red cell (RBC) regulates its volume, pH, and ion content in physiological and abnormal conditions. Analysis of the fluxbalance required by reticulocytes to conserve volume and composition predicted the existence of previously unsuspected Na(+)-dependent Cl- entry mechanisms. Unlike mature RBCs, reticulocytes did not tend to return to their original state after brief perturbations. The model predicted hysteresis and drift in cell pH, volume, and ion contents after transient alterations in membrane permeability or medium composition; irreversible cell dehydration could thus occur by brief K+ permeabilization, transient medium acidification, or the replacement of external Na+ with an impermeant cation. Both the hysteresis and drift after perturbations were shown to depend on the pHi dependence of the K:Cl cotransport, a major reticulocyte transporter. This behavior suggested a novel mechanism for the generation of irreversibly sickled cells directly from reticulocytes, rather than in a stepwise, progressive manner from discocytes. Experimental tests of the model's predictions and the hypothesis are described in the following paper.

Anemia, Sickle Cell↗
PubMed1979

Mathematical modelling of ATP, K+ and Na+ interactions with (Na+ + K+)-ATPase occurring under equilibrium conditions.

The controlling effect of ATP, K+ and Na+ on the rate of (Na+ + K+)-ATPase inactivation by 7-chloro-4-nitrobenzo-2-oxa-1,3-diazole (NBD-C1) is used for the mathematical modelling of the interaction of the effectors with the enzyme under equilibrium conditions. 1. Of a series of conceivable interaction models, designed without conceptual restrictions to describe the effector control of inactivation kinetics, only one fits the experimental data described in a preceding paper. 2. The model is characterized by the coexistence of two binding sites for ATP and the coexistence of two separate binding sites for K+ and Na+ on the enzyme-ATP complex. On the basis of this model, the effector parameters fitting the experimental data most closely are estimated by means of nonlinear least-squares fits. 3. The apparent dissociation constants for ATP fo the enzyme-ATP complex or of the enzyme-(ATP)2 complex are computed to lie near 0.0024 mM and 0.34 mM, respectively, irrespective of whether K+ and Na+ were absent or K+ and K+ plus Na+, respectively, were present in the experiments. 4. The origin of the high and the low affinity site for binding of ATP to the (Na+ + K+)-ATPase molecule is traced back to the coexistence of two catalytic centres which, although primarily equivalent as to the reactivity of their thiol groups with NBD-C1, are induced into anticooperative communication by ATP binding and thus show an induced geometric asymmetry. 5. On the basis of the interaction model outlined under item 2 the apparent dissociation constant for K+ or Na+ in the (K+ + Na+)-liganded enzyme-ATP complex are computed to be 1.7 mM and 3.5 mM, respectively. 6. The conclusions concerning the coexistence of two primarily equivalent but anticooperatively interacting catalytic centres and the coexistence of two separate ionophoric centres for Na+ and K+ correspond to the appropriate basic postulates of the flip-flop concept of (Na+ + K+)-ATPase mechanism.

4-Chloro-7-nitrobenzofurazan↗
PubMed2004

Using mathematical models to help understand biological pattern formation.

One of the characteristics of biological systems is their ability to produce and sustain spatial and spatio-temporal pattern. Elucidating the underlying mechanisms responsible for this phenomenon has been the goal of much experimental and theoretical research. This paper illustrates this area of research by presenting some of the mathematical models that have been proposed to account for pattern formation in biology and considering their implications.

Animals↗
PubMed1987

A mathematical model of the penetration of polycyclic aromatic hydrocarbons through the bronchial lining layer.

The enhanced carcinogenicity of particle-associated polycyclic aromatic hydrocarbons (PAHs) has previously been explained by their prolonged retention in the lung due to the slow release of these substances from their carrier particles. However, the mathematical model presented here indicates that the lipid-aqueous heterogenic structure of the bronchial lining layer (BLL) in itself may constitute the major rate-determining step in the overall transport of PAHs from carrier particles to the bronchial epithelium. Conditions within the BLL are then as important for the resulting cellular burdens of PAHs in the lung as are the properties of the carrier particles for these hydrocarbons. Thus, in addition to other vital functions in the respiratory tract, the lipids in lung surfactant can also be important in protecting the lung from the action of lipophilic carcinogens.

Animals↗
PubMed1990

Mathematical modelling of the purine metabolism of the rat liver.

The regulation of the purine metabolism of the rat liver is studied on the basis of a mathematical model which comprises rate laws and kinetic constants of all physiologically relevant reactions. The computed stationary and time-dependent concentrations are in good accordance with experimental data obtained in the ischaemic rat liver and in isolated hepatocytes. In particular, model-based simulations of the adenine nucleotide metabolism have been performed for situations where ATP-deficient states of the cell (hypoxia, anoxia or ischaemia) of various length are followed by onset of ATP production (reoxygenation). These simulations confirm the experimentally observed incomplete recovery of ATP and of the total pool of adenine nucleotides within a few hours of reoxygenation after long-term ATP depletion. Therefore, it can be concluded that this phenomenon is an intrinsic regulatory property of the purine metabolism and not necessarily due to some irreversible changes in the activity of the enzymes involved.

Animals↗
PubMed2006

Mathematical model comparisons of potential non-typeable Haemophilus influenzae vaccine effects.

Vaccines to prevent acute otitis media (AOM) caused by non-typeable Haemophilus influenzae (NTHi) are under development. Because NTHi is highly variable and colonization rates are high, special vaccine characteristics and trial designs might be needed. We examined in mathematical models the equilibrium NTHi-caused AOM rate given hypothetical vaccines that generated immunity identical to corresponding maximal naturally acquired immunity. Vaccines were examined with single effects and combinations of immunity affecting (1) AOM rates given colonization (pathogenicity), (2) susceptibility to colonization, and (3) contagiousness given colonization. Percent reductions in AOM across all preschool children were (1) 34%, (2) 31%, (3) 9%, (1 and 2) 57%, (2 and 3) 50%, and (1, 2, and 3) 75%. Effects on children in daycare vs. not in daycare were (1) 18 vs. 48%, (2) -1 vs. 57%, (3) 13 vs. 5%, (1 and 2) 30 vs. 79%, (2 and 3) 33 vs. 60%, and (1, 2, and 3) 64 vs. 85%. Pure pathogenicity effects (1 alone) will need to be supplemented by transmission effects. The effects of susceptibility (2 alone) are diminished or negative because children protected against colonization have lower levels of immunity to (1) and (3) than unvaccinated children. For trials to predict population effects, both colonization and AOM outcomes must be studied and all three effects must be evaluated. This need arises because, unlike H. influenzae type B, high NTHi exposure diminishes cumulative vaccine effects and high colonization rates generate rapid accumulation of natural immunity that alters the indirect effects of vaccine immunity on transmission differently by age and daycare status.

Acute Disease↗
PubMed1992

Mathematical models of complex dose-response relationships: implications for experimental design in psychopharmacologic research.

We develop a mathematical model to account for the complex relationship between drug dose and clinical response in psychopharmacologic research. The model specifies relationships among drug dose, drug bioavailability, pharmacokinetic factors, course moderators, clinical response and the heterogeneity of the disorder, and allows for the derivation of results that have implications for experimental design in psychopharmacologic research. These results form the basis for computer simulations which indicate that random assignment to two fixed doses is more powerful and less sensitive to heterogeneity than assignment to clinically determined doses. Fixed dose designs, however, tend to overestimate the magnitude of drug bioavailability-clinical response relationships. Clinically determined dose designs are useful in some experimental situations; their effectiveness is enhanced by systematically reducing the clinically determined dose. Larger dose reductions improve the ability to detect bioavailability-clinical response relationships.

Antipsychotic Agents↗
PubMed1993

A mathematical model for dynamics of cardiovascular drug action: application to intravenous dihydropyridines in healthy volunteers.

A physiologically based mathematical model was built to describe the pharmacodynamic effects in response to the administration of intravenous (iv) dihydropyridine drugs in healthy volunteers. This model incorporates a limited number of hemodynamic variables, namely, mean arterial blood pressure (MAP), cardiac output (CO) or heart rate (HR), stroke volume (SV), and total peripheral resistance (TPR), into a closed-loop system supposed to represent essential features of the cardiovascular regulation. We also defined an additional auxiliary control variable (U) which is thought to represent primarily the role of the baroreceptor reflex. It was assumed that the variable U was related to MAP changes through both deviation- and rate-sensitive mechanisms. Other model parameters are the baseline levels for MAP, CO (or HR), and TPR, as well as time constants to account for further temporal aspects of the regulation. Finally, TPR was assumed to be linked to the plasma concentrations of dihydropyridine drugs via a conventional pharmacokinetic/pharmacodynamic (PK/PD) model, relying upon an effect compartment and a linear, hyperbolic, or sigmoidal relationship between the reduction in TPR and the drug concentrations at the effect site. The model characteristics were explored by studying the influence of various parameters, including baseline levels and deviation- and rate-sensitive control parameters, on the hemodynamic responses to a fictive constant rate i.v. infusion of a vasodilator drug. Attempts were also made to mimic literature data with nifedipine, following i.v. administration under both constant and exponentially decreasing infusion rates. The applicability of the model was demonstrated by fitting hemodynamic data following i.v. infusion of nicardipine to healthy volunteers, under experimental conditions similar to those described above for nifedipine. The effect model for the action of nicardipine on TPR, combined with the physiological model including a feedback control loop, allowed an adequate quantitative description of time profiles for both cardiac output and mean arterial pressure. The suggested model is a useful tool for integrated data analysis of hemodynamic responses to vasodilator drugs in healthy volunteers. Computer simulations suggest that a graded variation of a few model parameters--including baseline levels of TPR and MAP and the deviation-sensitive parameter of the arterial pressure control--would also be able to account for the pattern of hemodynamic response observed in hypertensive patients, which is qualitatively different to that seen in normotensive subjects. Extrapolation of drug response from the healthy volunteer to the hypertensive patient is allowed by our model. Its usefulness for an early evaluation of drug efficacy during drug development is under current investigation.

Adult↗
PubMed1996

Mathematical models for coordinated interpersonal timing in mother-infant interactions in the first year of life.

Two mathematical models of social contingency are examined in terms of their development over the first year of life. The interactions of 53 mothers and their infants were recorded at 6 weeks, 4 months, and 12 months. The infants' gazes at 6 weeks, the mothers' vocal behavior at 6 weeks, and the vocal behavior of the mother and infant at 4 and 12 months were automatically coded in terms of four states. The conditional dependence model and the response effects model were computed for each interaction at each age, and the coefficients of the models were examined as a function of age. The relative success of the models as estimates of moment-to-moment contingency as well as their variations with age are discussed.

Female↗
PubMedDate not supplied

A non-linear mathematical model for computerized analysis of mood curves: construction of the model and its application to the mood curves of depressive and schizophrenic inpatients.

A non-linear mathematical model for the computerized description of mood curves is presented. This model reaches a high goodness of fit to the real data and seems superior to a linear model recently proposed. Using this model in a computer program for describing the mood data of a large sample of psychiatric inpatients, significant and clinically meaningful group differences between the mood curves of schizophrenic, endogenous depressive, and neurotic depressive inpatients could be demonstrated. The application of this methodology might be helpful among others in the field of evaluative research.

Antidepressive Agents↗
PubMed2004

A mathematical model of volatile release in mouth from the dispersion of gelled emulsion particles.

This paper presents a mathematical model of in-mouth volatile release from gelled emulsion particles dispersed in a continuous aqueous phase. Data based on APCI MS-Breath analysis is presented to demonstrate the effect of particle size, oil content and oil-water partition coefficients. It is shown that in-mouth release of aroma from the dispersion of gelled emulsion particles follows a two-component kinetic equation with fast and slow components. Both the fast and slow rate constants depend on the particle size, oil content and oil water partition coefficient of the aroma. The relative amount of aroma contributing to the fast and slow components also depends on the size of the particles. In order to understand this unexpected behaviour, an analytical model was developed that considers the interplay between the mass transfer of flavour across the interface of the particles and that across the air-liquid interface. Analytical expressions for the two rate constants and the relative ratio of aroma contributing to the fast component have been derived. From this model, three regimes of in-mouth release of aroma from the dispersion of gelled emulsion particles were identified including, the emulsion regime, the transition regime and the gel particle regime. In the emulsion regime, changes in the size of gelled emulsion particles had negligible impact on the overall release. In the transition regime, the release was controlled by the interaction of flavour transfer from the particles with that across the air-water interface. In the gel particle regime, aroma release at long times was governed by the particles and that at short times was governed by the air-water interface, and the two processes were fully decoupled. A simple relationship was derived for the critical size above which the release of aroma from the dispersion of gelled emulsion particles is affected by the size of the particles.

Emulsions↗
PubMed2006

Mathematical modelling of 4-chlorophenol inhibition on COD and 4-chlorophenol removals in an activated sludge unit.

A mathematical model was developed for an activated sludge unit treating 4-chlorophenol (4-CP) containing synthetic wastewater composed of diluted molasses, urea, KH(2)PO(4) and MgSO(4) with COD and 4-CP contents of 2500 and 500 mg l(-1), respectively. The model included 4-CP inhibition on COD and 4-CP removals. Experimental data obtained at different hydraulic residence times (HRT=5-30h) and sludge ages (SRT, 3-30 days) were used to estimate the kinetic and inhibition constants for COD and 4-CP removal rates. 4-CP inhibition on COD removal was negligible while the inhibition on 4-CP removal was significant. The specific rate constant (k), saturation constant (K(s)) for COD oxidation were found to be 2.64 day(-1) and 559 mg l(-1), respectively. A similar model was used for 4-CP oxidation in the activated sludge unit and the constants were found to be k'=1.44 day(-1), K'(s)=25.7 mgl(-1), K"(CP)=559 mg l(-1),and K(I,CP)=17 mg l(-1). Increases in death rate constant because of 4-CP inhibition was also quantified and the inhibition constants were determined for both COD and 4-CP removals. Model predictions with the estimated kinetic constants were in good agreement with the experimental data. Developed model can be used to estimate the performance of an activated sludge unit treating 4-CP containing wastewater under the specified experimental conditions.

Biodegradation, Environmental↗
PubMed2005

Mathematical modelling of prolactin-receptor interaction and the corollary for prolactin receptor gene expression in skin.

A mathematical model of prolactin regulating its own receptors was developed, and compared with experimental data on a qualitative level. The model incorporates the kinetics of prolactin-receptor interactions and subsequent signalling by prolactin-receptor dimers to regulate the production of receptor mRNA and hence the receptor population. The model relates changes in plasma prolactin concentration to prolactin receptor (PRLR) gene expression, and can be used for predictive purposes. The cell signalling that leads to the activation of target genes, and the mechanisms for regulation of transcription, were treated empirically in the model. The model's parameters were adjusted so that model simulations agreed with experimentally observed responses to administration of prolactin in sheep. In particular, the model correctly predicts insensitivity of receptor mRNA regulation to a series of subcutaneous injections of prolactin, versus sensitivity to prolonged infusion of prolactin. In the latter case, response was an acute down-regulation followed by a prolonged up-regulation of mRNA, with the magnitude of the up-regulation increasing with the duration of infusion period. The model demonstrates the feasibility of predicting the in vivo response of prolactin target genes to external manipulation of plasma prolactin, and could provide a useful tool for identifying optimal prolactin treatments for desirable outcomes.

Animals↗
PubMedDate not supplied

A simple mathematical model of second-messenger mediated slow excitatory postsynaptic potentials.

We have developed a novel and simple mathematical model of a slow excitatory postsynaptic potential (EPSP) based on an abstraction of the processes of activation, inactivation, and summation of a cAMP, protein kinase A (PKA)-dependent second-messenger cascade. The model describes the activation of receptors, G-proteins, and production of cAMP as the first stage and uses first-order, non-rate-limited kinetics. The second stage corresponds to the release of active, PKA catalytic subunit and can use first- or higher-order kinetics. The third stage represents simple phosphorylation of ion channels and is limited by the number of channels available. The decay of each stage is based on first-order, mass-action kinetics. These equations and some variations were solved numerically and values of the parameters were determined by fitting to a variety of experimental data from myenteric neurons of the guinea-pig ileum. The model produced a slow EPSP with a nonlinear stimulus-response relationship that resulted from the underlying kinetics of the signaling cascade. This system of equations is suitable for incorporation into a large-scale computer simulation, and the methodology should be generalizable to other pathways.

Animals↗
PubMed2004

Mathematical modeling of RDX and HMX metabolism in poplar (Populus deltoides x Populus nigra, DN34) tissue culture.

Three mathematical models were developed based on a fate study as an approach to define transformation pathways of hexahydro-1,3,5-trinitro-1,3,5-triazine (RDX) and octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine (HMX) within plant cells. [U-14C]RDX and [U-14C]HMX were added in Murashige and Skoog (MS) liquid media containing Populus deltoides x P. nigra (DN34) tissue cultures. Radioactivity of samples was analyzed using HPLC, a bio-oxidizer and liquid scintillation counter. Based on information collected, transformation pathways of nitramine compounds were fitted with the raw data obtained and using a modified "green liver" model. Ordinary differential equations were developed and simulations were performed with MicroMath Scientist version 2.0 (MicroMath Inc., St. Louis, MO, USA). The three models, with different sequential transformation processes, were tested in order to support the raw data (model I) and the assumptions of the modified "green liver" model (models II and III). The results showed a high correlation between the collected data and the simulated concentrations for all models. Thus, the simplest model developed (model I) is the best model description of these particular results. The results obtained suggest that the principle of parsimony should be applied. The "green liver"-based models also demonstrated a reliable approach for the investigation of degradation pathways of nitramines within plant cells.

Azocines↗
PubMed1996

Calcium dynamics and homeostasis in a mathematical model of the principal cell of the cortical collecting tubule.

Calcium (Ca) dynamics are incorporated into a mathematical model of the principal cell in the cortical collecting tubule developed earlier in Strieter et al. (1992a. Am. J Physiol. 263:F1063-1075). The Ca components are modeled after the Othmer-Tang model for IP(3)-sensitive calcium channels (1993, in Experimental and Theoretical Advances in Biological Pattern Formation, 295-319). There are IP(3)-sensitive Ca channels and ATP-driven pumps on the membrane of the endoplasmic reticulum. Calcium enters the cell passively down its electrochemical gradient. A Ca pump and Na/Ca exchange in the basolateral membrane are responsible for the extrusion of cytoplasmic calcium. Na/Ca exchange can also operate in reverse mode to transport Ca into the cell. Regulatory effects of cytoplasmic Ca on the apical Na channels are modeled after experimental data that indicate apical Na permeability varies inversely with cytoplasmic Ca concentration. Numerical results on changes in intracellular Ca caused by decreasing NaCl in the bath and the lumen are similar to those from experiments in Bourdeau and Lau (1990. Am. J Physiol. 258:F1497-1503). This match of simulation and experiment requires the synergistic action of the Na/Ca exchanger and the Ca regulated apical Na permeability. In a homogeneous medium, cytoplasmic Ca becomes oscillatory when extracellular Na is severely decreased, as observed in experiments of cultured principal cells (Koster, H., C. van Os and R. Bindels. 1993. Kidney Int.43:828-836). This essentially pathological situation arises because the hyperpolarization of membrane potential caused by Na-free medium increases Ca influx into the cell, while the Na/Ca exchanger is inactivated by the low extracellular Na and can no longer move Ca out of the cell effectively. The raising of the total amount of intracellular Ca induces oscillatory Ca movement between the cytoplasm and the endoplasmic reticulum. Ca homeostasis is investigated under the condition of severe extracellular Ca variations. As extracellular Ca is decreased, Ca regulation is greatly impaired if Ca does not regulate apical ionic transport. The simulations indicate that the Na/Ca exchanger alone has only limited regulatory capacity. The Ca regulated apical sodium or potassium permeability are essential for regulation of cytoplasmic Ca in the principal cell of the cortical collecting tubule.

Calcium↗