Invariant manifold reductions for Markovian ion channel dynamics

We show that many Markov models of ion channel kinetics have globally attracting stable invariant manifolds, even when the Markov process is time dependent. The primary implication of this is that, since the dimension of the invariant manifold is often substantially smaller than the full master equation system, simulations of ion channel kinetics can be substantially simplified, with no approximation. We show that this applies to certain models of potassium channels, sodium channels, ryanodine receptors and IP₃ receptors. We also use this to show that the original Hodgkin–Huxley formulations of potassium channel conductance and sodium channel conductance are the exact solutions of full Markov models for these channels.      

Cooperative gating of chloride channel subunits in endothelial cells.

New methods are described to detect subconductance levels and to analyse ion channel gating. These methods are applied to simulated and experimental data. Single chloride channel records from inside-out membrane patches excised from human umbilical venous endothelial cells (HUVEC) exhibit, in addition to the full closed and full open configurations, intermediate subconductance levels which are multiple of an elementary conductance of 112.5 pS. Analysis of transitions from one state to another and the comparison of real data with simulated data leads to the proposal of a cooperative model of gating for the observed subunits of a chloride channel.     

Parameterization for In-Silico Modeling of Ion Channel Interactions with Drugs

Since the first Hodgkin and Huxley ion channel model was described in the 1950s, there has been an explosion in mathematical models to describe ion channel function. As experimental data has become richer, models have concomitantly been improved to better represent ion channel kinetic processes, although these improvements have generally resulted in more model complexity and an increase in the number of parameters necessary to populate the models. Models have also been developed to explicitly model drug interactions with ion channels. Recent models of drug-channel interactions account for the discrete kinetics of drug interaction with distinct ion channel state conformations, as it has become clear that such interactions underlie complex emergent kinetics such as use-dependent block. Here, we describe an approach for developing a model for ion channel drug interactions. The method describes the process of extracting rate constants from experimental electrophysiological function data to use as initial conditions for the model parameters. We then describe implementation of a parameter optimization method to refine the model rate constants describing ion channel drug kinetics. The algorithm takes advantage of readily available parallel computing tools to speed up the optimization. Finally, we describe some potential applications of the platform including the potential for gaining fundamental mechanistic insights into ion channel function and applications to in silico drug screening and development.     

Quasi-steady State Reduction of Molecular Motor-Based Models of Directed Intermittent Search

We present a quasi-steady state reduction of a linear reaction-hyperbolic master equation describing the directed intermittent search for a hidden target by a motor-driven particle moving on a one-dimensional filament track. The particle is injected at one end of the track and randomly switches between stationary search phases and mobile nonsearch phases that are biased in the anterograde direction. There is a finite possibility that the particle fails to find the target due to an absorbing boundary at the other end of the track. Such a scenario is exemplified by the motor-driven transport of vesicular cargo to synaptic targets located on the axon or dendrites of a neuron. The reduced model is described by a scalar Fokker–Planck (FP) equation, which has an additional inhomogeneous decay term that takes into account absorption by the target. The FP equation is used to compute the probability of finding the hidden target (hitting probability) and the corresponding conditional mean first passage time (MFPT) in terms of the effective drift velocity V, diffusivity D, and target absorption rate λ of the random search. The quasi-steady state reduction determines V, D, and λ in terms of the various biophysical parameters of the underlying motor transport model. We first apply our analysis to a simple 3-state model and show that our quasi-steady state reduction yields results that are in excellent agreement with Monte Carlo simulations of the full system under physiologically reasonable conditions. We then consider a more complex multiple motor model of bidirectional transport, in which opposing motors compete in a “tug-of-war”, and use this to explore how ATP concentration might regulate the delivery of cargo to synaptic targets.     

钾离子通道两态跳跃的通透机制

采用两态跳跃模型研究离子通道的通透机制,从两态动力学方程得到了平衡态下的能斯特方程、稳态条件下的米氏动力学关系。得出：若电压小于100mV,电导-电压关系是线性的;在电流-浓度关系中,电流具有饱和特性。这些与实验结果是一致的。此外,还讨论了钾离子通道到达稳态前的暂态过程,并用特征时间来描述这一过程。发现采用两态跳跃模型用较少的参数就可以说明离子通透的机制。     

Modelling the activation,opening, inactivation and reopening of the voltage-gated sodium channel.

A model of the voltage-gated sodium channel is put forward suggesting that the four S4 voltage-sensors behave as screw-helices making a series of discrete transitions that carry one elementary charge for each notch of the screw helix. After the channel has been activated by the first two steps R in equilibrium with P in equilibrium with A in all four domains, followed by a voltage-independent rearrangement, it is opened by a third cooperative step A in equilibrium with B in domains I, II and III in conjunction with hydration. Inactivation is a voltage-dependent process controlled by the third step A in equilibrium with I in sensor IVS4, and the closing of the channel is brought about its dehydration. From the inactivated steady state the channel may be reopened by a fourth step, I in equilibrium with C in sensor IVS4 and rehydration. The computed kinetics of the model are shown to conform closely with those observed experimentally.     

Permeable ions differentially affect gating kinetics and unitary conductance of L-type calcium channels

Although ion permeation and gating of L-type Ca(2+) channels are generally considered separate processes controlled by distinct components of the channel protein, ion selectivity can vary with the kinetic state. To test this possibility, we studied single-channel currents (cell-attached) of recombinant L-type channels (Ca(V)1.2, beta(2a), and alpha(2)delta) transiently expressed in tsA201 cells in the presence of the channel agonist BayK 8644 which promotes long channel openings (Mode 2 openings). We found that both the brief (Mode 1) and long (Mode 2) mean open times in the presence of Ca(2+) were relatively longer than those with Ba(2+). The unitary slope conductance with Ba(2+) was significantly larger (p<0.05) in Mode 2 openings than for brief Mode 1 openings, whereas the conductance with Ca(2+) did not vary with mode gating. Consequently, the gamma(Ba):gamma(Ca) ratio was greater for Mode 2 than Mode 1 openings. Our findings indicate that both ion permeation and gating kinetics of the L-type channel are differentially modulated by permeable ions. Ca(2+) binding to the L-type channel may stabilize the alteration of channel ion permeability mediated by gating kinetics, and thus, play a role in preventing excessive ion entry when the activation gating of the channel is promoted to the prolonged open state.     

Description of interacting channel gating using a stochastic Markovian model

Single-channel recordings from membrane patches frequently exhibit multiple conductance levels. In some preparations, the steady-state probabilities of observing these levels do not follow a binomial distribution. This behavior has been reported in sodium channels, potassium channels, acetylcholine receptor channels and gap junction channels. A non-binomial distribution suggests interaction of the channels or the presence of channels with different open probabilities. However, the current trace sometimes exhibits single transitions spanning several levels. Since the probability of simultaneous transitions of independent channels is infinitesimally small, such observations strongly suggest a cooperative gating behavior. We present a Markov model to describe the cooperative gating of channels using only the all-points current amplitude histograms for the probability of observing the various conductance levels. We investigate the steady-state (or equilibrium) properties of a system ofN channels and provide a scheme to express all the probabilities in terms of just two parameters. The main feature of our model is that lateral interaction of channels gives rise to cooperative gating. Another useful feature is the introduction of the language of graph theory which can potentially provide a different avenue to study ion channel kinetics. We write down explicit expressions for systems of two, three and four channels and provide a procedure to describe the system ofN channels.     

Synaptic integration of NMDA and non-NMDA receptors in large neuronal network models solved by means of differential equations

Alpha functions are commonly used to describe different receptor channel kinetics (non-NMDA, GABA_A and GABA_B). In this paper we show that they may be represented as solutions to simple ordinary differential equations. This alternative method is compared with the commonly used direct summation of the alpha function conductances in a high-level neuronal circuit model. A parametric study shows that the differential equation method greatly speeds up the previous summation method. The forward Euler method used to solve this differential equation is shown to be accurate for this type of simulation. The modelling of NMDA receptor channel kinetics is also discussed. Received: 18 December 1992/Accepted in revised form: 28 June 1993     

Kinetics of Genetic Switching into the State of Bacterial Competence

Nonlinear amplification of gene expression of master regulators is essential for cellular differentiation. Here we investigated determinants that control the kinetics of the genetic switching process from the vegetative state (B-state) to the competent state (K-state) of Bacillus subtilis, explicitly including the switching window which controls the probability for competence initiation in a cell population. For individual cells, we found that after initiation of switching, the levels of the master regulator [ComK](t) increased with sigmoid shape and saturation occurred at two distinct levels of [ComK]. We analyzed the switching kinetics into the state with highest [ComK] and found saturation after a switching period of length 1.4 ± 0.3 h. The duration of the switching period was robust against variations in the gene regulatory network of the master regulator, whereas the saturation levels showed large variations between individual isogenic cells. We developed a nonlinear dynamics model, taking into account low-number stochastic effects. The model quantitatively describes the probability and timescale of switching at the single cell level and explains why the ComK level in the K-state is highly sensitive to extrinsic parameter variations. Furthermore, the model predicts a transition from stochastic to deterministic switching at increased production rates of ComK in agreement with experimental data.     

Ion conduction in substates of the batrachotoxin-modified Na+ channel from toad skeletal muscle.

Batrachotoxin-modified Na+ channels from toad muscle were inserted into planar lipid bilayers composed of neutral phospholipids. Single-channel conductances were measured for [Na+] ranging between 0.4 mM and 3 M. When membrane preparations were made in the absence of protease inhibitors, two open conductance states were identified: a fully open state (16.6 pS in 200 mM symmetrical NaCl) and a substate that was 71% of the full conductance. The substate was predominant at [Na+] > 65 mM, whereas the presence of the fully open state was predominant at [Na+] < 15 mM. Addition of protease inhibitors during membrane preparation stabilized the fully open state over the full range of [Na+] studied. In symmetrical Na+ solutions and in biionic conditions, the ratio of amplitudes remained constant and the two open states exhibited the same permeability ratios of PLi/PNa and PCs/PNa. The current-voltage relations for both states showed inward rectification only at [Na+] < 10 mM, suggesting the presence of asymmetric negative charge densities at both channel entrances, with higher charge density in the external side. An energy barrier profile that includes double ion occupancy and asymmetric charge densities at the channel entrances was required to fit the conductance-[Na+] relations and to account for the rectification seen at low [Na+]. Energy barrier profiles differing only in the energy peaks can give account of the differences between both conductance states. Estimation of the surface charge density at the channel entrances is very dependent on the ion occupancy used and the range of [Na+] tested. Independent evidence for the existence of a charged external vestibule was obtained at low external [Na+] by identical reduction of the outward current induced by micromolar additions of Mg2+ and Ba2+.     

Permeation mechanism of a two-state potassium channel

A two-state hopping model was proposed to study the permeation of ion channel. The Nernst equation in equilibrium and the Michaelis-Menten relation in steady state were derived from the two-state kinetic model. The current-voltage relationship obtained in the symmetrical solutions case was linear when the applied potential was less than 100 mV, which met Ohm’s law. The conductance-concentration relationship exhibited the saturation property. Moreover, the characteristic time reaching the steady state of the KcsA channel was also discussed. Translated from Acta Biophysica Sinica, 2005, 21(4): 289–294 [译自: 生物物理学报]     

Physico-chemical model of a voltage-gated sodium channel

A permeant ion is known to create in the channel pore a local electrical field, the intensity of which exceeds the intensity of an electrical field produced by the membrane potential. In our study, we consider a sodium channel model, in which the effects of a permeant ion, an inactivating particle, and pharmacological agents on mobile charged groups of the channel are semi-phenomenologically taken into account by using motion equations for a generalized structural variable. Stationary solutions for the equation correspond to “open,” “closed,” and “inactivated” channel states. Because of this, the channel free energy profile, as a function of the structural variable, has three local minima. The three energy values of these states depend both on the electrical field applied externally and on the near-membrane concentrations of permeant ions and acting pharmacological agents. Sodium channel activation and inactivation kinetics are considered resulting from relative changes of the free energy typical of the above three states of the channel. The results we obtained in the course of channel activation and inactivation modeling and their voltage dependence are qualitatively consistent with the commonly known experimental data. The proposed model allows one to qualitatively predict the dependence of the sodium channel kinetic characteristics on the concentrations of permeant ions and pharmacological agents.     

Stochastic Markovian modeling of electrophysiology of ion channels: reconstruction of standard deviations in macroscopic currents

Markovian models of ion channels have proven useful in the reconstruction of experimental data and prediction of cellular electrophysiology. We present the stochastic Galerkin method as an alternative to Monte Carlo and other stochastic methods for assessing the impact of uncertain rate coefficients on the predictions of Markovian ion channel models. We extend and study two different ion channel models: a simple model with only a single open and a closed state and a detailed model of the cardiac rapidly activating delayed rectifier potassium current. We demonstrate the efficacy of stochastic Galerkin methods for computing solutions to systems with random model parameters. Our studies illustrate the characteristic changes in distributions of state transitions and electrical currents through ion channels due to random rate coefficients. Furthermore, the studies indicate the applicability of the stochastic Galerkin technique for uncertainty and sensitivity analysis of bio-mathematical models.     

Collective Equilibrium Behaviour of Ion Channel Gating in Cell Membranes: An Ising Model Formulation

A statistical mechanical model for voltage-gated ion channels in cell membranes is proposed using the transfer matrix method. Equilibrium behavior of the system is studied. Representing the distribution of channels over the cellular membrane on a one-dimensional array with each channel having two states (open and closed) and incorporating channel–channel cooperative interactions, we calculate the fraction of channels in the open state at equilibrium. Experimental data obtained from batrachotoxin-modified sodium channels in the squid giant axon, using the cut-open axon technique, is best fit by the model when there is no interaction between the channels.     

Can electromagnetic radiations induce changes in the kinetics of voltage-dependent ion channels?

Ion channels are protein molecules, which can assume distinct open and closed conformational states, a phenomenon termed ion channel kinetics. The transitions from one state to another depend on the potential energy barrier that separates those two states. Therefore, it is rational to suppose that electromagnetic waves could interact with this barrier and induce changes in the rate transitions of this kinetic process. Our aim is to answer the question: can electromagnetic radiations induce changes in the kinetics of voltage-dependent ion channels? We simulated the effects of the low and high frequency electromagnetic waves on the sodium and potassium channels of the giant axon of Loligo. The key parameter measured was the fractional open time (fv), because it reflects the voltage dependence of the kinetics of channels. The electromagnetic radiations induced the following changes in the kinetics of the potassium and sodium channels: i/ low frequency waves kept the potassium channel 50% of the time open independent on the mean voltage applied through the membrane; ii/ a gradual inhibition of the inactivation on the sodium channel, when the amplitudes of the low frequency waves were increased; iii/ high frequency waves on the potassium channel, decreased both Vo (voltage in which the channel stays 50% open) and the steepness of fv (d fv/dV) as the amplitudes of the waves increased, and iv/ high frequency and low amplitude radiations on the sodium channel decreased the maximum value of fv (in relation to control), while high amplitudes increased this value. In conclusion, high and low frequency electromagnetic radiations were able to change the kinetics of the potassium and sodium channels in a squid giant axon model.     

Cooperative block of the plant endomembrane ion channel by ruthenium red.

Effects of ruthenium red (RR) on the slow Ca(2+)-activated Ca(2+)-permeable vacuolar channel have been studied by patch-clamp technique. Applied to the cytosolic side of isolated membrane patches, RR at concentrations of 0.1-5 microM produced two distinct effects on single channel kinetics, long lasting closures and a flickering block of the open state. The first effect was largely irreversible, whereas the second one could be washed out. The extent of flickering block steeply increased (zdelta = approximately 1.35) with the increase of cytosol-positive voltage, dragging RR into the channel pore. At least two RR ions are involved in the block according to Hill coefficient n = approximately 1.30 for the dose response curves. The on-rate rate of the drug binding linearly depended on the RR concentration, implying that one RR ion already plugged the pore. The blocked state was further stabilized by binding of the second RR. This stabilization was in excess of that predicted by independent binding as the dependence of unblocking rate on RR concentration revealed. A cooperative model was therefore employed to describe the kinetic behavior of RR binding. At zero voltage the half-blocking RR concentration of 36 microM and the bimolecular on-rate constant of 1.8 x 10(8) M(-1) s(-1) were estimated.     

Noncooperative vs. cooperative reactivation of D-beta-hydroxybutyrate dehydrogenase: multiple equilibria for lecithin binding are determined by the physical state (soluble vs. bilayer) and composition of the phospholipids

D-beta-Hydroxybutyrate dehydrogenase (BDH) is a lecithin-requiring mitochondrial enzyme that catalyzes the interconversion of beta-hydroxybutyrate and acetoacetate. The purified soluble enzyme devoid of lipid (i.e., the apodehydrogenase) can be reactivated with soluble lecithin or by insertion into phospholipid vesicles containing lecithin. Lipid activation curves have a sigmoidal shape, and two models have been proposed to explain them. We have previously reported that the kinetics of reactivation with short-chain lecithins in the soluble state is consistent with a model in which the enzyme enzyme contains two identical, noninteracting lecithin binding sites, both of which must be occupied to activate the enzyme [noncooperative mechanism; Cortese, J.D., Vidal, J.C., Churchill, P., McIntyre, J.O., & Fleischer, S. (1982) Biochemistry 21, 3899-3908]. More recently a kinetic model involving cooperative interactions between lecithin binding sites was proposed for the reactivation of the membrane-bound enzyme [Sandermann, H., Jr., McIntyre, J.O., & Fleischer, S. (1986) J. Biol. Chem. 261, 6201-6208]. This study reinvestigates the basis for the different conclusions in these two studies. The previous study with soluble lecithins was limited to about 34% of maximal activation compared with mitochondrial phospholipid, due to inactivation of the enzyme at the critical micellar concentration. We could now extend this study to 91% activation by increasing the ethanol concentration. This experimental evidence confirms that the soluble system follows a noncooperative equation. We provide a new kinetic approach to test the cooperative model. A velocity equation is derived for a Hill-type cooperative ligand binding system interacting with a mixture of ligands. This equation predicts a proportionality between an overall weighted cooperative dissociation constant [Kcoop(w)] and a dissociation constant for a single lecithin (PC) species from interacting sites (KPC), regulated by the PC molar fraction (XPC): 1/Kcoop(w) = XPC/KPC. The equation was applied to the data of Sandermann et al. [Sandermann, H., Jr., McIntyre, J.O., & Fleischer, S. (1986) J. Biol. Chem. 261, 6201-6208] as well as to newly obtained data. The results obtained over a wide range of PC molar fractions and different mixtures of bilayer phospholipids fit this equation, confirming the cooperative behavior. We conclude that BDH has a different mode of reactivation depending on the nature of the lipid environment. With soluble lecithin, the activation is noncooperative, whereas in the bilayer, mixtures of phospholipids give cooperative behavior that fits a Hill equation.(ABSTRACT TRUNCATED AT 400 WORDS)     

Free energy of a potassium ion in a model of the channel formed by an amphipathic leucine-serine peptide

We use molecular dynamics simulations to investigate the position-dependent free energy of a potassium ion in a model of an ion channel formed by the synthetic amphipathic leucine-serine peptide, LS3. The channel model is a parallel bundle of six LS3 helices around which are packed 146 methane-like spheres in order to mimic a membrane. At either end of and within the channel are 1051 water molecules, plus four ions (two potassium and two chloride). The free energy of a potassium ion in the channel was estimated using the weighted histogram analysis (WHAM) method. This is the first time to our knowledge that such a calculation has been carried out as a function of the position of an ion in three dimensions within a channel. The results indicate that for this channel, which is lined by hydrophilic serine sidechains, there is a relatively weak dependence of the free energy on the axial/off-axial position of the ion. There are some off-axis local minima, especially in the C-terminal half of the channel. Using the free energy results, a single channel current-voltage curve was estimated using a one-dimensional Nernst-Planck equation. Although reasonable agreement with experiment is achieved for K(+) ions flowing from the N-terminal to the C-terminal mouth, in the opposite direction the current is underestimated. This underestimation may be a consequence of under-sampling of the conformational dynamics of the channel. We suggest that our simulations may have captured, for example, a sub-conductance level (i.e. an incompletely open state) of the LS3 channel.     

Mathematical model of electrogenic transport through biomembranes via oligomeric channels capable of conformational transitions

A mathematical model of electrogenic ion transport across biomembranes by oligomeric channels liable to conformational transformations has been derived. The model describes changes with time of the membrane potential and near--membrane ion concentrations. Different types of the channel conductance regulation such as activation or inhibition by the permeating ions and membrane potential have been considered. It appears that in the presence of such regulations 1) the channel voltage-current curves have negative resistance regions; 2) the dependence of the quasisteady state (or resting) potential on the ion concentrations in the solution is of hysteresis nature; 3) the model may have multiple steady-state and oscillating solutions.