Quantitative analysis of a fully generalized four-state kinetic scheme

To describe the macroscopic behavior of many ion channels, at a minimum a four-state kinetic scheme is needed to provide for three processes: a delay in activation development, the activation process, and inactivation. I present here an analytical solution for a fully generalized four-state kinetic scheme in which every state can transit to every other state and any initial conditions can be specified. The solution describes the time courses of the probabilities of occupancy of each state during a step change in the rate constants of the scheme and includes closed-form expressions for the relaxation time constants and steady-state probabilities of occupancy as functions of the rate constants. Solutions for several relevant special cases are also included along with demonstrations that the general solution yields the correct behavior for several reduced or special cases where the result is independently known.     

On mutations that uncouple sodium channel activation from inactivation

Computations on sodium channel gating were conducted using a closed-open-inactivated coupled kinetic scheme. The time constant of inactivation (tauh) derives a voltage dependency from coupling to voltage-dependent activation even when rate constants between inactivated and other states are strictly voltage independent. The derived voltage dependency does not require any physical, molecular link between the structures responsible for inactivation and the charges producing voltage-dependent activation. The only requirement is that the closed to inactivated rate constant (kCI) differs from the open to inactivated (kOI), consistent with experimental results. A number of mutations and other treatments uncouple sodium channel activation and inactivation in that the voltage dependency of tauh is substantially reduced while voltage-dependent activation persists. However, a clear basis for uncoupling has not been described. A variety of experimental results are accounted for just by changes in the difference between kOI and kCI. In wild type channels, kOI > kCI and inactivation develops with a delay whose time constant is just that for channel opening. Mutations that reduce the kOI - kCI difference reduce the amplitude of the delay process and the derived voltage dependency of tauh. If kOI = kCI, inactivation develops as a single exponential (no matter what the number of closed states), activation and inactivation become independent, parallel processes, and any voltage dependency of tauh is then entirely intrinsic to inactivation. If kOI < kCI, inactivation develops as the sum of exponentials, tauh at negative potentials speeds and then slows with more positive potentials. These predicted kOI < kCI effects have all been seen experimentally (O'Leary, M.E., L.-Q. Chen, R.G. Kallen, and R. Horn. 1995. J. Gen. Physiol. 106: 641-658). An open to closed rate constant of zero also removes the derived voltage dependency of tauh, but activation and inactivation are still coupled and the inactivation delay remains.     

Single-Channel Kinetic Analysis for Activation and Desensitization of Homomeric 5-HT3A Receptors

The 5-HT₃A receptor is a member of the Cys-loop family of ligand-gated ion channels. To perform kinetic analysis, we mutated the 5-HT3A subunit to obtain a high-conductance form so that single-channel currents can be detected. At all 5-HT concentrations (>0.1 μM), channel activity appears as openings in quick succession that form bursts, which coalesce into clusters. By combining single-channel and macroscopic data, we generated a kinetic model that perfectly describes activation, deactivation, and desensitization. The model shows that full activation arises from receptors with three molecules of agonist bound. It reveals an earlier conformational change of the fully liganded receptor that occurs while the channel is still closed. From this pre-open closed state, the receptor enters into an open-closed cycle involving three open states, which form the cluster whose duration parallels the time constant of desensitization. A similar model lacking the pre-open closed state can describe the data only if the opening rates are fixed to account for the slow activation rate. The application of the model to M4 mutant receptors shows that position 10′ contributes to channel opening and closing rates. Thus, our kinetic model provides a foundation for understanding structural bases of activation and drug action.     

Determining RuBisCO activation kinetics and other rate and equilibrium constants by simultaneous multiple non-linear regression of a kinetic model

The forward and reverse rate constants involved in carbamylation, activation, carboxylation, and inhibition of D-ribulose-1,5-bisphosphate carboxylase/oxygenase (RuBisCO) have been estimated by a new technique of simultaneous non-linear regression of a differential equation kinetic model to multiple experimental data. Parameters predicted by the model fitted to data from purified spinach enzyme in vitro included binding affinity constants for non-substrate CO2 and Mg2+ of 200+/-80 microM and 700+/-200 microM, respectively, as well as a turnover number (k(cat)) of 3.3+/-0.5 s(-1), a Michaelis half-saturation constant for carboxylation (K(M,C)) of 10+/-4 microM and a Michaelis constant for RuBP binding (K(M,RuBP)) of 1.5+/-0.5 microM. These and other constants agree well with previously measured values where they exist. The model is then used to show that slow inactivation of RuBisCO (fallover) in oxygen-free conditions at low concentrations of CO2 and Mg2+ is due to decarbamylation and binding of RuBP to uncarbamylated enzyme. In spite of RuBP binding more tightly to uncarbamylated enzyme than to the activated form, RuBisCO is activated at high concentrations of CO2 and Mg2+. This apparent paradox is resolved by considering activation kinetics and the fact that while RuBP binds tightly but slowly to uncarbamylated enzyme, it binds fast and loosely to activated enzyme. This modelling technique is presented as a new method for determining multiple kinetic data simultaneously from a limited experimental data set. The method can be used to compare the properties of RuBisCO from different species quickly and easily.     

Depressing synapse as a detector of frequency change

In this article we discuss the short-term synaptic depression using a mathematical model. We derive the model of synaptic depression caused by the depletion of synaptic vesicles for the case of infinitely short stimulation time and show that the analytical formulas for the postsynaptic potential (PSP) and kinetic functions take simple closed form. A solution in this form allows an analysis of the characteristics of depression as a function of the models parameters and the derivation of analytic formulas for measures of short time synaptic depression commonly used in experimental studies. Those formulas are used to validate the model by fitting it to two types of synapses described in the literature. Given the fitted parameters we discuss the behavior of the synapse in situations involving frequency change. We also indicate a possible role of depressing synapses in information processing as not only a filter of high frequency input but as a detector of the return from high frequency stimulation to the stimulation within frequency band specific for a given synapse.     

Rate of opening of a population of Na channels in frog skeletal muscle fibers

Linear Systems convolution analysis of muscle sodium currents was used to predict the opening rate of sodium channels as a function of time during voltage clamp pulses. If open sodium channel lifetimes are exponentially distributed, the channel opening rate corresponding to a sodium current obtained at any particular voltage, can be analytically obtained using a simple equation, given single channel information about the mean open-channel lifetime and current.Predictions of channel opening rate during voltage clamp pulses show that sodium channel inactivation arises coincident with a decline in channel opening rate.Sodium currents pharmacologically modified with Chloramine-T treatment so that they do not inactivate, show a predicted sustained channel opening rate.Large depolarizing voltage clamp pulses produce channel opening rate functions that resemble gating currents.The predicted channel opening rate functions are best described by kinetic models for Na channels which confer most of the charge movement to transitions between closed states.Comparisons of channel opening rate functions with gating currents suggests that there may be subtypes of Na channel with some contributing more charge movement per channel opening than others.Na channels open on average, only once during the transient period of Na activation and inactivation.After transiently opening during the activation period and then closing by entering the inactivated state, Na channels reopen if the voltage pulse is long enough and contribute to steady-state currents.The convolution model overestimates the opening rate of channels contributing to the steady-state currents that remain after the transient early Na current has subsided.     

An Improved Parameter Estimation Method for Hodgkin-Huxley Models

We consider whole-cell voltage-clamp data of isolated currents characterized by the Hodgkin-Huxley paradigm. We examine the errors associated with the typical parameter estimation method for these data and show them to be unsatisfactorally large especially if the time constants of activation and inactivation are not sufficiently separated. The size of these errors is due to the fact that the steady-state and kinetic properties of the current are estimated disjointly. We present an improved parameter estimation method that utilizes all of the information in the voltage-clamp conductance data to estimate steady-state and kinetic properties simultaneously and illustrate its success compared to the standard method using simulated data and data from P. interruptus shal channels expressed in oocytes.     

Mechanism of inactivation gating of human T-type (low-voltage activated) calcium channels

Recovery from inactivation of T-type Ca channels is slow and saturates at moderate hyperpolarizing voltage steps compared with Na channels. To explore this unique kinetic pattern we measured gating and ionic currents in two closely related isoforms of T-type Ca channels. Gating current recovers from inactivation much faster than ionic current, and recovery from inactivation is much more voltage dependent for gating current than for ionic current. There is a lag in the onset of gating current recovery at -80 mV, but no lag is discernible at -120 mV. The delay in recovery from inactivation of ionic current is much more evident at all voltages. The time constant for the decay of off gating current is very similar to the time constant of deactivation of open channels (ionic tail current), and both are strongly voltage dependent over a wide voltage range. Apparently, the development of inactivation has little influence on the initial deactivation step. These results suggest that movement of gating charge occurs for inactivated states very quickly. In contrast, the transitions from inactivated to available states are orders of magnitude slower, not voltage dependent, and are rate limiting for ionic recovery. These findings support a deactivation-first path for T-type Ca channel recovery from inactivation. We have integrated these concepts into an eight-state kinetic model, which can account for the major characteristics of T-type Ca channel inactivation.     

Enhanced Closed-state Inactivation in a Mutant Shaker K+ Channel

Many mutations that shift the voltage dependence of activation in Shaker channels cause a parallel shift of inactivation. The I2 mutation (L382I in the Shaker B sequence) is an exception, causing a 45 mV activation shift with only a 9 mV shift of inactivation midpoint relative to the wildtype (WT) channel. We compare the behavior of WT and I2 Shaker 29-4 channels in macropatch recordings from Xenopus oocytes. The behavior of WT channels can be described by both simple and detailed kinetic models which assume that inactivation proceeds only from the open state. The behavior of I2 channels requires that they inactivate from closed states as well, a property characteristic of voltage-gated sodium channels. A detailed ``multiple-state inactivation' model is presented that describes both activation and inactivation of I2 channels. The results are consistent with the view that residue L382 is associated with the receptor for the inactivation particles in Shaker channels. Received: 16 December 1996/Revised: 5 February 1997     

A molecular link between activation and inactivation of sodium channels

A pair of tyrosine residues, located on the cytoplasmic linker between the third and fourth domains of human heart sodium channels, plays a critical role in the kinetics and voltage dependence of inactivation. Substitution of these residues by glutamine (Y1494Y1495/QQ), but not phenylalanine, nearly eliminates the voltage dependence of the inactivation time constant measured from the decay of macroscopic current after a depolarization. The voltage dependence of steady state inactivation and recovery from inactivation is also decreased in YY/QQ channels. A characteristic feature of the coupling between activation and inactivation in sodium channels is a delay in development of inactivation after a depolarization. Such a delay is seen in wild-type but is abbreviated in YY/QQ channels at -30 mV. The macroscopic kinetics of activation are faster and less voltage dependent in the mutant at voltages more negative than -20 mV. Deactivation kinetics, by contrast, are not significantly different between mutant and wild-type channels at voltages more negative than -70 mV. Single-channel measurements show that the latencies for a channel to open after a depolarization are shorter and less voltage dependent in YY/QQ than in wild-type channels; however the peak open probability is not significantly affected in YY/QQ channels. These data demonstrate that rate constants involved in both activation and inactivation are altered in YY/QQ channels. These tyrosines are required for a normal coupling between activation voltage sensors and the inactivation gate. This coupling insures that the macroscopic inactivation rate is slow at negative voltages and accelerated at more positive voltages. Disruption of the coupling in YY/QQ alters the microscopic rates of both activation and inactivation.     

Kinetics of thermal inactivation of catalase in the presence of additives

The kinetics of catalase inactivation in buffer-free aqueous solutions within the temperature range 30–60 °C in the absence or presence of hydrogen peroxide was investigated and discussed taking into account the effect of NaCl, ethane-1,2-diol, propane-1,2,3-triol and sucrose, additives expected to increase the enzyme thermostability. Using the kinetic extended curves obtained by measuring the substrate absorbance in time and an isoconversional method, several simple kinetic models most frequently encountered in literature were fitted to the experimental data. The best model for inactivation was chosen on the basis of several statistical criteria. The half-times of inactivation and the activation energies were also reported and discussed.     

A sodium channel gating model based on single channel, macroscopic ionic, and gating currents in the squid giant axon.

Sodium channel gating behavior was modeled with Markovian models fitted to currents from the cut-open squid giant axon in the absence of divalent cations. Optimum models were selected with maximum likelihood criteria using single-channel data, then models were refined and extended by simultaneous fitting of macroscopic ionic currents, ON and OFF gating currents, and single-channel first latency densities over a wide voltage range. Best models have five closed states before channel opening, with inactivation from at least one closed state as well as the open state. Forward activation rate constants increase with depolarization, and deactivation rate constants increase with hyperpolarization. Rates of inactivation from the open or closed states are generally slower than activation or deactivation rates and show little or no voltage dependence. Channels tend to reopen several times before inactivating. Macroscopic rates of activation and inactivation result from a combination of closed, open and inactivated state transitions. At negative potentials the time to first opening dominates the macroscopic current due to slow activation rates compared with deactivation rates: channels tend to reopen rarely, and often inactivate from closed states before they reopen. At more positive potentials, the time to first opening and burst duration together produce the macroscopic current.     

Cross talk between activation and slow inactivation gates of Shaker potassium channels

This study addresses the energetic coupling between the activation and slow inactivation gates of Shaker potassium channels. To track the status of the activation gate in inactivated channels that are nonconducting, we used two functional assays: the accessibility of a cysteine residue engineered into the protein lining the pore cavity (V474C) and the liberation by depolarization of a Cs(+) ion trapped behind the closed activation gate. We determined that the rate of activation gate movement depends on the state of the inactivation gate. A closed inactivation gate favors faster opening and slower closing of the activation gate. We also show that hyperpolarization closes the activation gate long before a channel recovers from inactivation. Because activation and slow inactivation are ubiquitous gating processes in potassium channels, the cross talk between them is likely to be a fundamental factor in controlling ion flux across membranes.     

A tyrosine substitution in the cavity wall of a k channel induces an inverted inactivation

Ion permeation and gating kinetics of voltage-gated K channels critically depend on the amino-acid composition of the cavity wall. Residue 470 in the Shaker K channel is an isoleucine, making the cavity volume in a closed channel insufficiently large for a hydrated K(+) ion. In the cardiac human ether-a-go-go-related gene channel, which exhibits slow activation and fast inactivation, the corresponding residue is tyrosine. To explore the role of a tyrosine at this position in the Shaker channel, we studied I470Y. The activation became slower, and the inactivation faster and more complex. At +60 mV the channel inactivated with two distinct rates (tau(1) = 20 ms, tau(2) = 400 ms). Experiments with tetraethylammonium and high K(+) concentrations suggest that the slower component was of the P/C-type. In addition, an inactivation component with inverted voltage dependence was introduced. A step to -40 mV inactivates the channel with a time constant of 500 ms. Negative voltage steps do not cause the channel to recover from this inactivated state (tau > 10 min), whereas positive voltage steps quickly do (tau = 2 ms at +60 mV). The experimental findings can be explained by a simple branched kinetic model with two inactivation pathways from the open state.     

Kinetics of combined pressure-temperature inactivation of avocado polyphenoloxidase

Irreversible combined pressure-temperature inactivation of the food quality related enzyme polyphenoloxidase was investigated. Inactivation rate constants (k) were obtained for about one hundred combinations of constant pressure (0.1-900 MPa) and temperature (25-77.5 degrees C). According to the Eyring and Arrhenius equation, activation volumes and activation energies, respectively, representing pressure and temperature dependence of the inactivation rate constant, were calculated for all temperatures and pressures studied. In this way, temperature and pressure dependence of activation volume and activation energy, respectively, could be considered. Moreover, for the first time, a mathematical model describing the inactivation rate constant of a food quality-related enzyme as a function of pressure and temperature is formulated. Such pressure-temperature inactivation models for food quality-related aspects (e.g., the spoilage enzyme polyphenoloxidase) form the engineering basis for design, evaluation, and optimization of new preservation processes based on the combined effect of temperature and pressure. Furthermore, the generated methodology can be used to develop analogous kinetic models for microbiological aspects, which are needed from a safety and legislative point of view, and other quality aspects, e.g., nutritional factors, with a view of optimal quality and consumer acceptance.     

BTX modification of Na channels in squid axons. I. State dependence of BTX action

The state dependence of Na channel modification by batrachotoxin (BTX) was investigated in voltage-clamped and internally perfused squid giant axons before (control axons) and after the pharmacological removal of the fast inactivation by pronase, chloramine-T, or NBA (pretreated axons). In control axons, in the presence of 2-5 microM BTX, a repetitive depolarization to open the channels was required to achieve a complete BTX modification, characterized by the suppression of the fast inactivation and a simultaneous 50-mV shift of the activation voltage dependence in the hyperpolarizing direction, whereas a single long-lasting (10 min) depolarization to +50 mV could promote the modification of only a small fraction of the channels, the noninactivating ones. In pretreated axons, such a single sustained depolarization as well as the repetitive depolarization could induce a complete modification, as evidenced by a similar shift of the activation voltage dependence. Therefore, the fast inactivated channels were not modified by BTX. We compared the rate of BTX modification of the open and slow inactivated channels in control and pretreated axons using different protocols: (a) During a repetitive depolarization with either 4- or 100-ms conditioning pulses to +80 mV, all the channels were modified in the open state in control axons as well as in pretreated axons, with a similar time constant of approximately 1.2 s. (b) In pronase-treated axons, when all the channels were in the slow inactivated state before BTX application, BTX could modify all the channels, but at a very slow rate, with a time constant of approximately 9.5 min. We conclude that at the macroscopic level BTX modification can occur through two different pathways: (a) via the open state, and (b) via the slow inactivated state of the channels that lack the fast inactivation, spontaneously or pharmacologically, but at a rate approximately 500-fold slower than through the main open channel pathway.     

牛肉中单增李斯特菌的热失活模型

【目的】建立牛肉中单增李斯特菌的热失活动力学模型。【方法】将接种了3种不同血清型的单增李斯特菌混合菌液的牛肉分别在55℃、57.5℃、60℃、63℃、66℃和70℃进行热处理,在不同温度条件下单增李斯特菌数从109CFU/g下降至103CFU/g,其热失活曲线用修正的Gompertz模型进行了拟合;利用线性模型对单增李斯特菌的相对失活率(μ)和所持续时间(M)的自然对数值与温度(55℃-70℃)进行拟合;通过在59℃和64℃对牛肉中单增李斯特菌热处理,对所建的模型进行了验证。【结果】建立了牛肉中单增李斯特菌热失活动力学的一级模型和二级模型,经验证其准确因子和偏差因子均在可接受范围内。【结论】本研究所建立的模型能较好的模拟不同温度(55℃-70℃)对牛肉中单增李斯特菌热失活的影响。     

Conditional probability analysis for a domain model of Ca(2+)-inactivation of Ca2+ channels.

The domain model of Ca2+ inactivation of Ca2+ channels, which has been used to explain rapid inactivation of whole cell Ca2+ currents in pancreatic beta cells, is applied to single-time and conditional open probability measurements on guinea pig ventricular myocyte Ca2+ channels. These two measurements greatly constrain the choice of kinetic constants in the model. Calculations with the model provide a simple quantitative explanation of recent experimental results, including a slow increase in the inactivation rate.     

A cyclic model for bimodal activation of calcium activated potassium channels in radish vacuoles.

This paper presents the mathematical framework of a cyclic model proposed for describing the transition between a fast and a slow mode (fast-slow effect) induced by the application of step membrane potentials to ion channels from radish vacuoles. A voltage stimulation pulse with frequency in the range of 2 Hz or higher increased the activation time (slow mode) of the recorded currents. When the frequency of the stimulation pattern was restored to 0.1 Hz the activation time decreased twofold (fast mode). This experimental result cannot be explained by classical kinetic theory. The model, based on a simple extension of the Hodgkin and Huxley chain, describes the whole current experimental data and provides hints on the structural conformation of ion channels.