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T R Chay

Publications and source records attributed to T R Chay.

At least 37 records · Page 2Linked to original sources
PubMed1988

Analyzing stochastic events in multi-channel patch clamp data.

In an effort to understand the gating properties of ionic channels in biological membranes, an efficient method was developed to estimate the kinetic constants from opening-closing events of the channels. Our method is suitable to single channel patch-clamp recordings that contain several ionic channels functioning simultaneously. It is different from the maximum likelihood method previous developed by Horn and Lange, in that our method is a continuum approach and makes uses of analytic expressions of the probability density functions of the event times. Combinatorial analysis was necessary to correctly include more typical multi-channel recordings. This yields computationally quicker results than the method of Horn and Lange, which uses a discretized time series. Model-dependent portions of the code are minimal and easily modified. To illustrate the goodness of our method, we have generated the open-close processes of the patch-clamp records on a digital computer using the exponential random number generators. For multi-channel patches, we have introduced a few plausible approximations to make our algorithm more efficient. The soundness of the our approximations were tested with such measures as the fraction of the open state at time t, Popen (t), and frequencies of the number of openings per run. A copy of the computer code implementing this algorithm can be obtained from the authors.

Computer Simulation↗
PubMed1988

Role of single-channel stochastic noise on bursting clusters of pancreatic beta-cells.

To study why pancreatic beta-cells prefer to burst as a multi-cellular complex, we have formulated a stochastic model for bursting clusters of excitable cells. Our model incorporated a delayed rectifier K+ channel, a fast voltage-gated Ca2+ channel, and a slow Cai-blockable Ca2+ channel. The fraction of ATP-sensitive K+ channels that may still be active in the bursting regime was included in the model as a leak current. We then developed an efficient method for simulating an ionic current component of an excitable cell that contains several thousands of channels opening simultaneously under unclamped voltage. Single channel open-close stochastic events were incorporated into the model by use of binomially distributed random numbers. Our simulations revealed that in an isolated beta-cell [Ca2+]i oscillates with a small amplitude about a low [Ca2+]i. However, in a large cluster of tightly coupled cells, stable bursts develop, and [Ca2+]i oscillates with a larger amplitude about a higher [Ca2+]i. This may explain why single beta-cells do not burst and also do not release insulin.

Animals↗
PubMed1987

The effect of inactivation of calcium channels by intracellular Ca2+ ions in the bursting pancreatic beta-cells.

Based on recently determined ionic channel properties, a simple theoretical model for the burst activity of the pancreatic beta-cell is formulated in this paper. The model contains an inward voltage-activated Ca2+ current which is inactivated by intracellular calcium ions and an outward K+ current that is activated by the membrane potential. The probability of opening of the channel gates is represented by Boltzmann equations. Our model is applicable in a regime where an ATP-blockable K+ channel is inhibited. In this regime, glucose is treated as an activator for the rate of efflux of intracellular Ca2+ ions, and hence its effect is equated to kca, the efflux rate constant. In addition, intracellular H+ ion, which is a byproduct of the glycolytic metabolic process, is treated as a competitive inhibitor for Ca2+ ion. Since H+ is a competitive inhibitor (according to our assumption), its effect is equated to the strength of the Cai dissociation constant Kh. In the model, a Ca2+ binding site is assumed to exist in the inner membrane of the voltage-gated Ca2+ channel. The model predicts that a spike and burst electrical pattern can be generated by varying kca and that a given pattern may produce different levels of intracellular Ca2+ depending on Kh. In other words, it predicts that levels of [Ca2+]i can be separated from the electrical activity by controlling the concentration of glucose and pH appropriately. This may account for the experimental observation of Lebrun et al. (1985) that insulin secretion is not correlated to the burst of electrical activity.

Action Potentials↗
PubMed1987

Theoretical studies on the electrical activity of pancreatic beta-cells as a function of glucose.

The electrical activity of pancreatic beta-cells, which has been closely correlated both with intracellular Ca2+ concentration and insulin release, is characterized by a biphasic response to glucose and bursts of spiking action potentials. Recent voltage clamp and single channel patch clamp experiments have identified several transmembrane ionic channels that may play key roles in the electrophysiological behavior of beta-cells. There is a hypothesis that Ca2+-activated K+ channels are responsible for both the resting potential during low glucose concentration and the silent phase during bursting. The discovery of the ATP-inactivated K+ channel raises the possibility that the current for this latter K+ channel may dominate the resting potential, while the Ca2+-activated K+ current dominates the silent phase potential between bursts. The recent discovery that Ca2+-activated K+ channels are pH sensitive raises an interesting possibility for the biphasic electrical response. In this paper, numerical methods are presented for evaluating these hypotheses against experimental evidence.

Animals↗
PubMed1986

On the effect of the intracellular calcium-sensitive K+ channel in the bursting pancreatic beta-cell.

Based on the observation that the calcium-activated K+ channel in the pancreatic islet cells can also be activated by the membrane potential, we have formulated a mathematical model for the electrical activity in the pancreatic beta-cell. Our model contains two types of ionic channels, which are active above the subthreshold glucose concentration in the limit-cycle region: a Ca2+-activated, voltage-gated K+ channel and voltage-gated Ca2+ channel. Numerical simulation of the model generates bursts of electrical activity in response to a variation of kCa, the rate constant for sequestration of intracellular calcium ions. The period and duration of the bursts in response to kCa are in good agreement with experiment. The model predicts that a combined spike and burst pattern can be created using only single species of inward and outward currents, the inactivation kinetics (i.e., h) in the inward current is not a necessary condition for the generation of the pattern, and a given pattern or intensity of electrical activity may produce different levels of intracellular Ca2+ depending on the set of certain electrical parameters.

Animals↗
PubMed1985

Glucose response to bursting-spiking pancreatic beta-cells by a barrier kinetic model.

A mathematical model of the pancreatic beta-cell electrical activity was developed using a barrier kinetic model. Our model incorporates the glucose sensitive channel which is known to conduct K+ in the absence of glucose. The model also incorporates Cai sensitive K+ channels which are inhibited by intracellular H+ ions. It is described by three non-linear simultaneous differential equations. Numerical integration of these equations allowed us to examine the effect of glucose and of external Ca2+ ions on the electrical and cellular activity of the beta-cell. Our results show that the contribution of glucose-sensitive channel activity, if not completely inhibited, plays a very important role in determining the bursting periodicity. Our results also shows that even a small decrease in pHi is sufficient to change a bursting beta-cell to a spiking one. The voltage dependence of calcium sensitive K+ channels, however, affects little to the bursting mode of the electrical activity. Our simulation supports an incomplete selectivity of the voltage dependent calcium channel for calcium ions with low external [Ca2+]. It also supports the role of [Ca]i as an inhibitor of this channel when [Ca]i becomes unusually high.

Animals↗
PubMed1985

Theory of the effect of extracellular potassium on oscillations in the pancreatic beta-cell.

Based on the observation that potassium ions are compartmentalized near the surface of pancreatic beta-cells in mouse islets (Perez-Armendariz, E.M., I. Atwater, and E. Rojas 1985, Biophys. J. 48:741-749), we present a theoretical treatment of the effect of external potassium on oscillations in the pancreatic beta-cell. Our model includes the effects of ionic diffusion, the Ca2+-activated K+ channel, voltage-gated K+ and Ca2+ channels, and some of the effects of glucose. It is described by four ordinary differential equations. Numerical integration of these equations allows us to examine the effect of glucose, external K+, quinine, and tetraethylammonium ion (TEA) on the oscillations in membrane potential, intracellular Ca2+, and compartmentalized K+. The results are in good agreement with experiment.

Animals↗
PubMed1985

Bursting, beating, and chaos in an excitable membrane model.

We have studied periodic as well as aperiodic behavior in the self-sustained oscillations exhibited by the Hodgkin-Huxley type model of Chay, T. R., and J. Keizer (Biophys. J., 1983, 42:181-190) for the pancreatic beta-cell. Numerical solutions reveal a variety of patterns as the glucose-dependent parameter kCa is varied. These include regimes of periodic beating (continuous spiking) and bursting modes and, in the transition between these modes, aperiodic responses. Such aperiodic behavior for a nonrandom system has been called deterministic chaos and is characterized by distinguishing features found in previous studies of chaos in nonbiophysical systems and here identified for an (endogenously active) excitable membrane model. To parallel the successful analysis of chaos in other physical/chemical contexts we introduce a simplified, but quantitative, one-variable, discrete-time representation of the dynamics. It describes the evolution of intracellular calcium (which activates a potassium conductance) from one spike upstroke to the next and exhibits the various modes of behavior.

Animals↗
PubMed1985

Phase resetting and bifurcation in the ventricular myocardium.

With the dynamic differential equations of Beeler, G. W., and H. Reuter (1977, J. Physiol. [Lond.]. 268:177-210), we have studied the oscillatory behavior of the ventricular muscle fiber stimulated by a depolarizing applied current I app. The dynamic solutions of BR equations revealed that as I app increases, a periodic repetitive spiking mode appears above the subthreshold I app, which transforms to a periodic spiking-bursting mode of oscillations, and finally to chaos near the suprathreshold I app (i.e., near the termination of the periodic state). Phase resetting and annihilation of repetitive firing in the ventricular myocardium were demonstrated by a brief current pulse of the proper magnitude applied at the proper phase. These phenomena were further examined by a bifurcation analysis. A bifurcation diagram constructed as a function of I app revealed the existence of a stable periodic solution for a certain range of current values. Two Hopf bifurcation points exist in the solution, one just above the lower periodic limit point and the other substantially below the upper periodic limit point. Between each periodic limit point and the Hopf bifurcation, the cell exhibited the coexistence of two different stable modes of operation; the oscillatory repetitive firing state and the time-independent steady state. As in the Hodgkin-Huxley case, there was a low amplitude unstable periodic state, which separates the domain of the stable periodic state from the stable steady state. Thus, in support of the dynamic perturbation methods, the bifurcation diagram of the BR equation predicts the region where instantaneous perturbations, such as brief current pulses, can send the stable repetitive rhythmic state into the stable steady state.

Action Potentials↗
PubMed1984

Abnormal discharges and chaos in a neuronal model system.

Using the mathematical model of the pacemaker neuron formulated by Chay, we have investigated the conditions in which a neuron can generate chaotic signals in response to variation in temperature, ionic compositions, chemicals, and the strength of applied depolarizing current.

Animals↗
PubMed1984

Impulse responses of automaticity in the Purkinje fiber.

We examined the effects of brief current pulses on the pacemaker oscillations of the Purkinje fiber using the model of McAllister , Noble, and Tsien (1975. J. Physiol. [Lond.]. 251:1-57). This model was used to construct phase-response curves for brief electric stimuli to find "black holes," where rhythmic activity of the Purkinje fiber ceases. In our computer simulation, a brief current stimulus of the right magnitude and timing annihilated oscillations in membrane potential. The model also revealed a sequence of alternating periodic and chaotic regimes as the strength of a steady bias current is varied. We compared the results of our computer simulations with experimental work on Purkinje fibers and pointed out the importance of modeling results of this kind for understanding cardiac arrhythmias.

Animals↗
PubMed1983

On the mechanism of spiking and bursting in excitable cells.

A mathematical model previously developed to explain beta-cell membrane potential oscillations has been modified to accommodate the external variation of K+, Na+ and Ca2+ concentrations. Our model, which is applicable to excitable cells, incorporates the barrier kinetics. Hodgkin-Huxley-type gating mechanism, and an electrogenic Na+-K+ pump. Numerical solutions of our model are in agreement with many of the experimental results reported in the literature on excitable cells.

Action Potentials↗
PubMed1983

Minimal model for membrane oscillations in the pancreatic beta-cell.

Following the experimental findings of Atwater et al. (In Biochemistry Biophysics of the Pancreatic-beta-Cell, George Thieme Verlag, New York, 100-107), we have formulated a mathematical model for ionic and electrical events that take place in pancreatic-beta-cells. Our formulation incorporates a Hodgkin-Huxley type gating mechanism for Ca2+ and K+ channels, in addition to Ca2+ gated K+-channels. Consistent with the experimental observations, our model generates spikes and bursts in beta-cell membrane potentials and gives the correct responses to additions of glucose, quinine, and tetraethylammonium ions. The response of the oscillations to ouabain and changing concentrations of external K+ can be incorporated into the present model, although a more complete treatment would require inclusion of the Na+/K+ pump.

Animals↗
PubMed1982

Model for cooperativity of biological membranes.

We present a mathematical model for the complex cooperativity observed in biological membranes. In our model, it is assumed that the proteins bound on the membrane are noncooperative and possess a Bohr proton. It is further assumed that the net charge of the unliganded state of the protein is different from that of the liganded state owing to the structural change upon binding the ligand. With this model, we show how an all-or-none response, a graded response, and a noncooperative response arise in the binding curve of such biological membranes. In addition, we show how an effector, which can alter the pKa involved in the binding site, induces a complex cooperativity.

Ligands↗
PubMed1982

On exploring the basis for slow and fast oscillations in cellular systems.

We show that interesting oscillatory patterns may arise from an immobilized enzyme system, when the enzyme has the properties that it is inhibited by a substrate, produces H+ and has a pH-activity curve of bell-shaped form. In agreement with experimental observations in cellular excitations, the system generates fast oscillations that are superimposed on a slow cycle, at certain salt and enzyme concentrations.

Animals↗
PubMed1981

Oscillations in vesicular compartments.

The oscillatory phenomena which occur in metabolic processes are of great interest to chemists and biologists for a better understanding of far-from-equilibrium behavior that exists in biological systems. In this paper, we present a model for metabolic oscillations in a vesicular compartment and show how oscillations of the components involved in an energy transducing system may arise from our model under certain external conditions.

Adenine Nucleotides↗
PubMed1981

A model for biological oscillations.

A model and computation for oscillatory phenomena observed in some biological processes that utilize ion gradients across a membrane is presented. The model contains two main features: (i) active H+ transport pathways in the membrane and (ii) key enzymes having a pH-dependent activity profile and either translocating H+ from outside or producing H+ as a product. With this model a very long period of oscillations, as observed in mitochondrial and circadian rhythms, could be quantitatively demonstrated.

Biological Transport, Active↗