Modeling Facilitative Sugar Transporters: Transitions Between Single and Double Ligand Occupancy of Multiconformational Channel Models Explain Anomalous Kinetics

The four-state simple carrier model (SCM) is employed to describe ligand translocation by diverse passive membrane transporters. However, its application to systems like facilitative sugar transporters (GLUTs) is controversial: unidirectional fluxes under zero-trans and equilibrium-exchange experimental conditions fit a SCM, but flux data from infinite-cis and infinite-trans experiments appear not to fit the same SCM. More complex kinetic models have been proposed to explain this ``anomalous' behavior of GLUTs, but none of them accounts for all the experimental findings. We propose an alternative model in which GLUTs are channels subject to conformational transitions, and further assume that the results from zero-trans and equilibrium-exchange experiments as well as trans-effects corresponds to a single-occupancy channel regime, whereas the results from the infinite-cis and infinite-trans experiments correspond to a regime including higher channel occupancies. We test the plausibility of this hypothesis by studying a kinetic model of a two-site channel with two conformational states. In each state, the channel can bind the ligand from only one of the compartments. Under single-occupancy, for conditions corresponding to zero-trans and equilibrium-exchange experiments, the model behaves as a SCM capable of exhibiting trans-stimulations. For a regime including higher degrees of occupancy and infinite-cis and infinite-trans conditions, the same channel model can exhibit a behavior qualitatively similar to a SCM, albeit with kinetic parameters different from those for the single-occupancy regime. Numerical results obtained with our model are consistent with available experimental data on facilitative glucose transport across erythrocyte membranes. Hence, if GLUTs are multiconformational channels, their particular kinetic properties can result from transitions between single and double channel occupancies. Received: 12 April 1995/Revised: 28 August 1995     

Electrostatic interactions in gramicidin channels

A model based on the solution of the electrostatic potential for a geometry of three dielectric regions associated with a gramicidin A channel (GA) is presented. The model includes a cylindrical dielectric layer to represent the peptide backbone and dipole rings to account for dipolar side chains. Image potential and dipolar contributions for different orientations and positions along the channel are analyzed. The conductance of GA and two analogues obtained by substituting the amino acid at position 1 are studied. The numerical simulation reproduces experimental results (Barrett et al. 1986, Biophys J 49, 673–686) and supports the idea that electrostatic dipole-ion interactions are of primary importance in gramicidin channel function. Correspondence to: G. Martinez     

The use of automated parameter searches to improve ion channel kinetics for neural modeling

The voltage and time dependence of ion channels can be regulated, notably by phosphorylation, interaction with phospholipids, and binding to auxiliary subunits. Many parameter variation studies have set conductance densities free while leaving kinetic channel properties fixed as the experimental constraints on the latter are usually better than on the former. Because individual cells can tightly regulate their ion channel properties, we suggest that kinetic parameters may be profitably set free during model optimization in order to both improve matches to data and refine kinetic parameters. To this end, we analyzed the parameter optimization of reduced models of three electrophysiologically characterized and morphologically reconstructed globus pallidus neurons. We performed two automated searches with different types of free parameters. First, conductance density parameters were set free. Even the best resulting models exhibited unavoidable problems which were due to limitations in our channel kinetics. We next set channel kinetics free for the optimized density matches and obtained significantly improved model performance. Some kinetic parameters consistently shifted to similar new values in multiple runs across three models, suggesting the possibility for tailored improvements to channel models. These results suggest that optimized channel kinetics can improve model matches to experimental voltage traces, particularly for channels characterized under different experimental conditions than recorded data to be matched by a model. The resulting shifts in channel kinetics from the original template provide valuable guidance for future experimental efforts to determine the detailed kinetics of channel isoforms and possible modulated states in particular types of neurons.     

Electrostatic calculations for an ion channel. II. Kinetic behavior of the gramicidin A channel.

A theoretical model of the gramicidin A channel is presented and the kinetic behavior of the model is derived and compared with previous experimental results. The major assumption of the model is that the only interaction between ions in a multiply-occupied channel is electrostatic. The electrostatic calculations indicate in a multiply-occupied channel is electrostatic. The electrostatic calculations indicate that there will be potential wells at each end of the channel and, at high concentrations, that both wells can be occupied. The kinetics are based on two reaction steps: movement of the ion from the bulk solution to the well and movement between the two wells. The kinetics for this reaction rate approach are identical to those based on the Nernst-Planck equation in the limit where the movement between the two wells is rate limiting. The experimental results for sodium and potassium are consistent with a maximum of two ions per channel. To explain the thallium results it is necessary to allow three ions per channel. It is shown that this case is compatible with the electrostatic calculations if the presence of an anion is included. The theoretical kinetics are in reasonable quantitative agreement with the following experimental measurements: single channel conductance of sodium, potassium, and thallium; bi-ionic potential and permeability ratio between sodium-potassium and potassium-thallium; the limiting conductance of potassium and thallium at high applied voltages; current-voltage curves for sodium and potassium at low (but not high) concentrations; and the inhibition of sodium conductance by thallium. The results suggest that the potential well is located close to the channel mouth and that the conductance is partially limited by the rate going from the bulk solution to the well. For thallium, this entrance rate is probably diffusion limited.     

Discrete-state theory in nerve impulse modelling.

Hodgkin-Huxley models have been the standard for describing ionic current kinetics. However, many single channel behaviors cannot be described using traditional Hodgkin-Huxley models; they can be described by expanding the Hodgkin-Huxley models to have multiple resting and inactivated states. The model, based on charge translocation between a finite number of discrete Markovian states, is a biophysical kinetic model, according to current generalizations of channel structure, capable of reproducing channel behavior. The elaboration of the model is based on the Markov process. This type of model assumes that each channel has a discrete number of states that are connected by a kinetic diagram that defines the allowable transitions between these states and the rates at which these transitions occur. The application of the model presented here leads to results in accordance with the experimental data regarding the shape and characteristics of the nerve impulse registered along the nerve fibre. Unlike the traditional Hodgkin-Huxley models, the model based on the Markov processes has the advantage of removing the empirical equations, simplifying the computation of the membrane potential and revealing the single-channel variables. The average behavior is obtained by the repetition in one channel of the same stimulus, a number of times equal to the number of channels, which means that the macroscopic variables are predictable by the repetition, a certain number of times, of the same observations in a single channel.     

Optimization of 3D Poisson–Nernst‐Planck model for fast evaluation of diverse protein channels

We show the accuracy and applicability of our fast algorithmic implementation of a three‐dimensional Poisson–Nernst–Planck (3D‐PNP) flow model for characterizing different protein channels. Due to its high computational efficiency, our model can predict the full current‐voltage characteristics of a channel within minutes, based on the experimental 3D structure of the channel or its computational model structure. Compared with other methods, such as Brownian dynamics, which currently needs a few weeks of the computational time, or even much more demanding molecular dynamics modeling, 3D‐PNP is the only available method for a function‐based evaluation of very numerous tentative structural channel models. Flow model tests of our algorithm and its optimal parametrization are provided for five native channels whose experimental structures are available in the protein data bank (PDB) in an open conductive state, and whose experimental current‐voltage characteristics have been published. The channels represent very different geometric and structural properties, which makes it the widest test to date of the accuracy of 3D‐PNP on real channels. We test whether the channel conductance, rectification, and charge selectivity obtained from the flow model, could be sufficiently sensitive to single‐point mutations, related to unsignificant changes in the channel structure. Our results show that the classical 3D‐PNP model, under proper parametrization, is able to achieve a qualitative agreement with experimental data for a majority of the tested characteristics and channels, including channels with narrow and irregular conductivity pores. We propose that although the standard PNP model cannot provide insight into complex physical phenomena due to its intrinsic limitations, its semiquantitative agreement is achievable for rectification and selectivity at a level sufficient for the bioinformatical purpose of selecting the best structural models with a great advantage of a very short computational time. Proteins 2013; 81:1802–1822. © 2013 Wiley Periodicals, Inc.     

Voltage-dependent modulation of Ca channel current in heart cells by Bay K8644

We have investigated the voltage-dependent effects of the dihydropyridine Bay K8644 on Ca channel currents in calf Purkinje fibers and enzymatically dispersed rat ventricular myocytes. Bay K8644 increases the apparent rate of inactivation of these currents, measured during depolarizing voltage pulses, and shifts both channel activation and inactivation in the hyperpolarizing direction. Consequently, currents measured after hyperpolarizing conditioning pulses are larger in the presence of drug compared with control conditions, but are smaller than control if they are measured after positive conditioning pulses. Most of our experimental observations on macroscopic currents can be explained by a single drug-induced change in one rate constant of a simple kinetic model. The rate constant change is consistent with results obtained by others with single channel recordings.     

Numerical analysis of a microwave torch with axial gas injection

The characteristics of a microwave discharge in an argon jet injected axially into a coaxial channel with a shortened inner electrode are numerically analyzed using a self-consistent equilibrium gas-dynamic model. The specific features of the excitation and maintenance of the microwave discharge are determined, and the dependences of the discharge characteristics on the supplied electromagnetic power and gas flow rate are obtained. The calculated results are compared with experimental data.     

A 3-D Propagation Model for Emerging Land Mobile Radio Cellular Environments

A tunable stochastic geometry based Three-Dimensional (3-D) scattering model for emerging land mobile radio cellular systems is proposed. Uniformly distributed scattering objects are assumed around the Mobile Station (MS) bounded within an ellipsoidal shaped Scattering Region (SR) hollowed with an elliptically-cylindric scattering free region in immediate vicinity of MS. To ensure the degree of expected accuracy, the proposed model is designed to be tunable (as required) with nine degrees of freedom, unlike its counterparts in the existing literature. The outer and inner boundaries of SR are designed as independently scalable along all the axes and rotatable in horizontal plane around their origin centered at MS. The elevated Base Station (BS) is considered outside the SR at a certain adjustable distance and height w.r.t. position of MS. Closed-form analytical expressions for joint and marginal Probability Density Functions (PDFs) of Angle-of-Arrival (AoA) and Time-of-Arrival (ToA) are derived for both up- and down-links. The obtained analytical results for angular and temporal statistics of the channel are presented along with a thorough analysis. The impact of various physical model parameters on angular and temporal characteristics of the channel is presented, which reveals the comprehensive insight on the proposed results. To evaluate the robustness of the proposed analytical model, a comparison with experimental datasets and simulation results is also presented. The obtained analytical results for PDF of AoA observed at BS are seen to fit a vast range of empirical datasets in the literature taken for various outdoor propagation environments. In order to establish the validity of the obtained analytical results for spatial and temporal characteristics of the channel, a comparison of the proposed analytical results with the simulation results is shown, which illustrates a good fit for 10⁷ scattering points. Moreover, the proposed model is shown to degenerate to various notable geometric models in the literature by an appropriate choice of a few parameters.     

A model of the closed form of the nicotinic acetylcholine receptor m2 channel pore

The nicotinic acetylcholine receptor is a neurotransmitter-gated ion channel in the postsynaptic membrane. It is composed of five homologous subunits, each of which contributes one transmembrane helix--the M2 helix--to create the channel pore. The M2 helix from the delta subunit is capable of forming a channel by itself. Although a model of the receptor was recently proposed based on a low-resolution, cryo-electron microscopy density map, we found that the model does not explain much of the other available experimental data. Here we propose a new model of the M2 channel derived solely from helix packing and symmetry constraints. This model agrees well with experimental results from solid-state NMR, chemical reactivity, and mutagenesis experiments. The model depicts the channel pore, the channel gate, and the residues responsible for cation specificity.     

Effect of protein kinase A-induced phosphorylation on the gating mechanism of the brain Na+ channel: model fitting to whole-cell current traces.

The activity of the voltage-gated Na+ channel is subjected to modulation through covalent modifications. It has been previously shown that brain Na+ currents are reduced following the activation of the protein kinase A (PKA) pathway, but the effect of the phosphorylation on the gating mechanism of the channel has not been demonstrated so far. In this study, we analyze the whole-cell Na+ current recorded in the absence or presence of forskolin, which stimulates the PKA pathway. A minimal molecular model of the gating mechanism of the Na+ channel is defined to fit the experimental data: it consists of three closed states, one open state, and two inactivated states. We experimentally demonstrate that the kinetics of inactivation from the closed states are not affected by phosphorylation. The results obtained by computer fitting indicate that, among all the kinetic parameters describing the transitions between states, only one parameter is significantly modified in the presence of forskolin, and corresponds to the acceleration of the inactivation from the open state. This conclusion is supported by the analysis of current traces obtained from cells in the presence of a phosphatase inhibitor or loaded with the PKA catalytic unit, and is in agreement with previously reported single channel records.     

Permeation of ions across the potassium channel: brownian dynamics studies

The physical mechanisms underlying the transport of ions across a model potassium channel are described. The shape of the model channel corresponds closely to that deduced from crystallography. From electrostatic calculations, we show that an ion permeating the channel, in the absence of any residual charges, encounters an insurmountable energy barrier arising from induced surface charges. Carbonyl groups along the selectivity filter, helix dipoles near the oval chamber, and mouth dipoles near the channel entrances together transform the energy barrier into a deep energy well. Two ions are attracted to this well, and their presence in the channel permits ions to diffuse across it under the influence of an electric field. Using Brownian dynamics simulations, we determine the magnitude of currents flowing across the channel under various conditions. The conductance increases with increasing dipole strength and reaches its maximum rapidly; a further increase in dipole strength causes a steady decrease in the channel conductance. The current also decreases systematically when the effective dielectric constant of the channel is lowered. The conductance with the optimal choice of dipoles reproduces the experimental value when the dielectric constant of the channel is assumed to be 60. The current-voltage relationship obtained with symmetrical solutions is linear when the applied potential is less than approximately 100 mV but deviates from Ohm's law at a higher applied potential. The reversal potentials obtained with asymmetrical solutions are in agreement with those predicted by the Nernst equation. The conductance exhibits the saturation property observed experimentally. We discuss the implications of these findings for the transport of ions across the potassium channels and membrane channels in general.     

New insights of membrane environment effects on MscL channel mechanics from theoretical approaches

The prokaryotic mechanosensitive channel of large conductance (MscL) is a remarkable integral membrane protein. During hypo-osmotic shock, it responses to membrane tension through large conformational changes, that lead to an open state of the pore. The structure of the channel from Mycobacterium tuberculosis has been resolved in the closed state. Numerous experiments have attempted to trap the channel in its open state but they did not succeed in obtaining a structure. A gating mechanism has been proposed based on different experimental data but there is no experimental technique available to follow this process in atomic details. In addition, it has been shown that a decrease of the lipid bilayer thickness lowered MscL activation energy and stabilized a structurally distinct closed channel intermediate. Here, we use atomistic molecular dynamics simulations to investigate the effect of the lipid bilayer thinning on our model of the structure of the Escherichia coli. We thoroughly analyze simulations of the channel embedded in two pre-equilibrated membranes differing by their hydrophobic tail length (DMPE and POPE). The MscL structure remains stable in POPE, whereas a distinct structural state is obtained in DMPE in response to hydrophobic mismatch. This latter is obtained by tilts and kinks of the transmembrane helices, leading to a widening and a diminution of the channel height. Part of these motions is guided by a competition between solvent and lipids for the interaction with the periplasmic loops. We finally conduct a principal component analysis of the simulation and compare anharmonic motions with harmonic ones, previously obtained from a coarse-grained normal mode analysis performed on the same structural model. Significant similarities exist between low-frequency harmonic motions and those observed with essential dynamics in DMPE. In summary, change in membrane thickness permits to accelerate the conformational changes involved in the mechanics of the E. coli channel, providing a closed structural intermediate en route to the open state. These results give clues for better understanding why the channel activation energy is lowered in a thinner membrane.     

The binding site of sodium in the gramicidin A channel: comparison of molecular dynamics with solid-state NMR data.

The location of the main binding site for sodium in the gramicidin A (GA) channel was investigated with molecular dynamics simulations, using an atomic model of the channel embedded in a fully hydrated dimyristoyl phosphatidycholine (DMPC) bilayer. Twenty-four separate simulations in which a sodium was restrained at different locations along the channel axis were generated. The results are compared with carbonyl 13C chemical shift anisotropy solid-state NMR experimental data previously obtained with oriented GA:DMPC samples. Predictions are made for other solid-state NMR properties that could be observed experimentally. The combined information from experiment and simulation strongly suggests that the main binding sites for sodium are near the channel's mouth, approximately 9.2 A from the center of the dimer channel. The 13C chemical shift anisotropy of Leu10 is the most affected by the presence of a sodium ion in the binding site. In the binding site, the sodium ion is lying off-axis, making contact with two carbonyl oxygens and two single-file water molecules. The main channel ligand is provided by the carbonyl group of the Leu10-Trp11 peptide linkage, which exhibits the largest deviation from the ion-free channel structure. Transient contacts with the carbonyl group of Val8 and Trp15 are also present. The influence of the tryptophan side chains on the channel conductance is examined based on the current information about the binding site.     

Conduction mechanisms of chloride ions in ClC-type channels

The conduction properties of ClC-0 and ClC-1 chloride channels are examined using electrostatic calculations and three-dimensional Brownian dynamics simulations. We create an open-state configuration of the prokaryotic ClC Cl(-) channel using its known crystallographic structure as a basis. Two residues that are occluding the channel are slowly pushed outward with molecular dynamics to create a continuous ion-conducting path with the minimum radius of 2.5 A. Then, retaining the same pore shape, the prokaryotic ClC channel is converted to either ClC-0 or ClC-1 by replacing all the nonconserved dipole-containing and charged amino acid residues. Employing open-state ClC-0 and ClC-1 channel models, current-voltage curves consistent with experimental measurements are obtained. We find that conduction in these pores involves three ions. We locate the binding sites, as well as pinpointing the rate-limiting steps in conduction, and make testable predictions about how the single channel current across ClC-0 and ClC-1 will vary as the ionic concentrations are increased. Finally, we demonstrate that a ClC-0 homology model created from an alternative sequence alignment fails to replicate any of the experimental observations.     

Novel splice variants of amphiphysin I are expressed in retina

The MscL channel is a mechanosensitive channel which is gated by membrane stress or tension. Here, we describe a series of simulations which apply simulated mechanical stress to a molecular model of the MscL channel using two methods - direct force application to the transmembrane segments, and anisotropic pressure coupling. In the latter simulations, pressures less than that equivalent to a bilayer tension of 12 dyn/cm did not cause the channel to open, while pressures in excess of this value resulted in the channel opening. These results are in approximate agreement with experimental findings.     

Point spread functions and detail detection

Point spread functions, intended to characterize local spatial transfer of the visual system, can be obtained psychophysically using a perturbation technique. Data of such point spread functions are shown for three experimental conditions: foveally at adaptation levels of 1200 and 10 Td, and in the parafovea at an eccentricity of 2 deg using a 1200 Td adaptation level. The results are consistent with earlier findings (Blommaert, F. J.J. and Roufs, J.A.J. 1981, Vision Res. 21, 1223-1233). On the basis of such point spread functions a simplified multiple unit model was constructed, the parameters of which were fitted to thresholds of discs with varying diameter. Threshold predictions from this model for annuli, thin lines and broad lines were found to be in fair quantitative agreement with experimental results. It is argued that for a certain class of slender stimuli, including alphanumeric characters, thresholds can be described with a single channel model containing only the experimentally determined point spread function as a basic component.     

Threshold fluctuations in an N sodium channel model of the node of Ranvier.

Computer simulations of stochastic single-channel open-close kinetics are applied to an N sodium channel model of a node of Ranvier. Up to 32,000 voltage-gated sodium channels have been simulated with modified amphibian sodium channel kinetics. Poststimulus time histograms are obtained with 1000 monophasic pulse stimuli, and measurements are made of changes in the relative spread of threshold (RS) with changes in the model parameters. RS is found to be invariant with pulse durations from 100 microseconds to 3 ms. RS is approximately of inverse proportion to square-root of N. It decreases with increasing temperature and is dependent on passive electrical properties of the membrane as well as the single-channel conductance. The simulated RS and its independence of pulse duration is consistent with experimental results from the literature. Thus, the microscopic fluctuations of single, voltage-sensitive sodium channels in the amphibian peripheral node of Ranvier are sufficient to account for the macroscopic fluctuation if threshold to electrical stimulation.     

Transient relative motion of two cells in a channel flow

The hydrodynamic interaction of a red blood cell and a white blood cell in microvessels is studied, by use of a two-dimensional numerical model. The red blood cell, modeled as a small rigid circular cylinder, and the white blood cell, modeled as a larger rigid circular cylinder, are immersed in an incompressible Newtonian fluid in a two-dimensional channel. It is assumed that no external force or moment acts on the model cells, and the effect of inertia forces on the motion of the fluid and the cells is neglected. The velocity field of the suspending fluid and the instantaneous velocities of the two model cells are computed by the finite element method. Using the translational velocities of the model cells obtained, the trajectories of their relative motion are determined, for various initial positions. It is shown that the cells may or may not pass each other or separate, depending on the initial positions. The present results compare well to the experimental results.     

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.