scholarly journals The Joy of Markov Models—Channel Gating and Transport Cycling Made Easy

2021 ◽  
Author(s):  
G. Zifarelli ◽  
P. Zuccolini ◽  
S. Bertelli ◽  
M. Pusch
Keyword(s):  
Single Molecule ◽  
Protein Function ◽  
Markov Models ◽  
Single Channel ◽  
Real Life ◽  
Power Spectra ◽  
K Channel ◽  
Modular Architecture ◽  
Gating Currents ◽  
Discrete State

ABSTRACT The behavior of ion channels and transporters is often modeled using discrete state continuous-time Markov models. Such models are helpful for the interpretation of experimental data and can guide the design of experiments by testing specific predictions. Here, we describe a computational tool that allows us to create Markov models of chosen complexity and to calculate the predictions on a macroscopic scale, as well on a single-molecule scale. The program calculates steady-state properties (current, state probabilities, and cycle frequencies), deterministic macroscopic and stochastic time courses, gating currents, dwell-time histograms, and power spectra of channels and transporters. In addition, a visual simulation mode allows us to follow the time-dependent stochastic behavior of a single channel or transporter. After a basic introduction into the concept of Markov models, real-life examples are discussed, including a model of a simple K+ channel, a voltage-gated sodium channel, a 3-state ligand-gated channel, and an electrogenic uniporter. In this manner, the article has a modular architecture, progressing from basic to more advanced topics. This illustrates how the MarkovEditor program can serve students to explore Markov models at a basic level but is also suited for research scientists to test and develop models on the mechanisms of protein function.

scholarly journals On the Mechanism by which 4-Aminopyridine Occludes Quinidine Block of the Cardiac K+ Channel, hKv1.5

1998 ◽  
Vol 111 (4) ◽  
pp. 539-554 ◽  
Author(s):  
Fred S.P. Chen ◽  
David Fedida
Keyword(s):  
Open Channel ◽  
Channel Blocker ◽  
Single Channel ◽  
K Channel ◽  
Gating Currents ◽  
Channel Activation ◽  
Gating Charge ◽  
Channel Opening ◽  
Conformational Rearrangements ◽  
A Site

4-Aminopyridine (4-AP) binds to potassium channels at a site or sites in the inner mouth of the pore and is thought to prevent channel opening. The return of hKv1.5 off-gating charge upon repolarization is accelerated by 4-AP and it has been suggested that 4-AP blocks slow conformational rearrangements during late closed states that are necessary for channel opening. On the other hand, quinidine, an open channel blocker, slows the return or immobilizes off-gating charge only at opening potentials (>−25 mV). The aim of this study was to use quini-dine as a probe of open channels to test the kinetic state of 4-AP-blocked channels. In the presence of 0.2–1 mM 4-AP, quinidine slowed charge return and caused partial charge immobilization, corresponding to an increase in the Kd of ∼20-fold. Peak off-gating currents were reduced and decay was slowed ∼2- to 2.5-fold at potentials negative to the threshold of channel activation and during depolarizations shorter than normally required for channel activation. This demonstrated access of quinidine to 4-AP-blocked channels, a lack of competition between the two drugs, and implied allosteric modulation of the quinidine binding site by 4-AP resident within the channel. Single channel recordings also showed that quinidine could modulate the 4-AP-induced closure of the channels, with the result that frequent channel reopenings were observed when both drugs were present. We propose that 4-AP-blocked channels exist in a partially open, nonconducting state that allows access to quinidine, even at more negative potentials and during shorter depolarizations than those required for channel activation.

scholarly journals A Gastropod Toxin Selectively Slows Early Transitions in the Shaker K Channel's Activation Pathway

2004 ◽  
Vol 123 (6) ◽  
pp. 685-696 ◽  
Author(s):  
Jon T. Sack ◽  
Richard W. Aldrich ◽  
William F. Gilly
Keyword(s):  
Single Channel ◽  
K Channel ◽  
Channel Activity ◽  
Gating Currents ◽  
K Channels ◽  
Voltage Sensors ◽  
Essentially Normal ◽  
Channel Opening ◽  
Single Channel Activity ◽  
Kinetics Of

A toxin from a marine gastropod's defensive mucus, a disulfide-linked dimer of 6-bromo-2-mercaptotryptamine (BrMT), was found to inhibit voltage-gated potassium channels by a novel mechanism. Voltage-clamp experiments with Shaker K channels reveal that externally applied BrMT slows channel opening but not closing. BrMT slows K channel activation in a graded fashion: channels activate progressively slower as the concentration of BrMT is increased. Analysis of single-channel activity indicates that once a channel opens, the unitary conductance and bursting behavior are essentially normal in BrMT. Paralleling its effects against channel opening, BrMT greatly slows the kinetics of ON, but not OFF, gating currents. BrMT was found to slow early activation transitions but not the final opening transition of the Shaker ILT mutant, and can be used to pharmacologically distinguish early from late gating steps. This novel toxin thus inhibits activation of Shaker K channels by specifically slowing early movement of their voltage sensors, thereby hindering channel opening. A model of BrMT action is developed that suggests BrMT rapidly binds to and stabilizes resting channel conformations.

NOISE ANALYSIS IN STUDIES OF PROTEIN DYNAMICS AND MOLECULAR TRANSPORT

2004 ◽  
Vol 04 (01) ◽  
pp. L23-L31 ◽  
Author(s):  
SERGEY M. BEZRUKOV
Keyword(s):  
Protein Dynamics ◽  
Single Molecule ◽  
Protein Function ◽  
Molecular Mechanisms ◽  
Single Channel ◽  
Noise Analysis ◽  
Ionic Current ◽  
Low Frequency ◽  
Molecular Transport ◽  
Protein Molecule

Understanding the role of noise at cellular and higher hierarchical levels depends on our knowledge of the physical mechanisms of its generation. Conversely, noise is a rich source of information about these mechanisms. Using channel-forming protein molecules reconstituted into artificial 5-nm-thick insulating lipid films, it is possible to investigate noise in single-molecule experiments and to relate its origins to protein function. Recent progress in this field is reviewed with an emphasis on how this experimental technique can be used to study low-frequency protein dynamics, including not only reversible ionization of sites on the channel-forming protein molecule, but also molecular mechanisms of 1/f-noise generation. Several new applications of the single-molecule noise analysis to membrane transport problem are also addressed. Among those is a study on antibiotic translocation across bacterial walls. High-resolution recording of ionic current through the single channel, formed by the general bacterial porin, OmpF, enables us to resolve single-molecule events of antibiotic translocation.

Markov models for ion channels: versatility versus identifiability and speed

2009 ◽  
Vol 367 (1896) ◽  
pp. 2161-2179 ◽  
Author(s):  
Martin Fink ◽  
Denis Noble
Keyword(s):  
Parameter Estimation ◽  
Cardiac Electrophysiology ◽  
Markov Models ◽  
Single Channel ◽  
Model Development ◽  
Computational Cost ◽  
Computation Time ◽  
State Variables ◽  
Gating Currents ◽  
State Dependent

Markov models (MMs) represent a generalization of Hodgkin–Huxley models. They provide a versatile structure for modelling single channel data, gating currents, state-dependent drug interaction data, exchanger and pump dynamics, etc. This paper uses examples from cardiac electrophysiology to discuss aspects related to parameter estimation. (i) Parameter unidentifiability (found in 9 out of 13 of the considered models) results in an inability to determine the correct layout of a model, contradicting the idea that model structure and parameters provide insights into underlying molecular processes. (ii) The information content of experimental voltage step clamp data is discussed, and a short but sufficient protocol for parameter estimation is presented. (iii) MMs have been associated with high computational cost (owing to their large number of state variables), presenting an obstacle for multicellular whole organ simulations as well as parameter estimation. It is shown that the stiffness of models increases computation time more than the number of states. (iv) Algorithms and software programs are provided for steady-state analysis, analytical solutions for voltage steps and numerical derivation of parameter identifiability. The results provide a new standard for ion channel modelling to further the automation of model development, the validation process and the predictive power of these models.

scholarly journals Kinetic time constants independent of previous single-channel activity suggest Markov gating for a large conductance Ca-activated K channel.

1989 ◽  
Vol 94 (6) ◽  
pp. 1037-1070 ◽  
Author(s):  
O B McManus ◽  
K L Magleby
Keyword(s):  
Rate Constants ◽  
Inverse Relationship ◽  
Markov Models ◽  
Single Channel ◽  
K Channel ◽  
Channel Kinetics ◽  
Channel Activity ◽  
Time Constants ◽  
Single Channel Activity ◽  
Markov Assumption

Models for the gating of ion channels usually assume that the rate constants for leaving any given kinetic state are independent of previous channel activity. Although such discrete Markov models have been successful in describing channel gating, there is little direct evidence for the Markov assumption of time-invariant rate constants for constant conditions. This paper tests the Markov assumption by determining whether the single-channel kinetics of the large conductance Ca-activated K channel in cultured rat skeletal muscle are independent of previous single-channel activity. The experimental approach is to examine dwell-time distributions conditional on adjacent interval durations. The time constants of the exponential components describing the distributions are found to be independent of adjacent interval duration, and hence, previous channel activity. In contrast, the areas of the different components can change. Since the observed time constants are a function of the underlying rate constants for transitions among the kinetic states, the observation of time constants independent of previous channel activity suggests that the rate constants are also independent of previous channel activity. Thus, the channel kinetics are consistent with Markov gating. An observed dependent (inverse) relationship between durations of adjacent open and shut intervals together with Markov gating indicates that there are two or more independent transition pathways connecting open and shut states. Finally, no evidence is found to suggest that gating is not at thermodynamic equilibrium: the inverse relationship was independent of the time direction of analysis.

scholarly journals Linking Exponential Components to Kinetic States in Markov Models for Single-Channel Gating

2008 ◽  
Vol 132 (2) ◽  
pp. 295-312 ◽  
Author(s):  
Christopher Shelley ◽  
Karl L. Magleby
Keyword(s):  
Rate Constants ◽  
Time Distribution ◽  
Dwell Time ◽  
Markov Models ◽  
Single Channel ◽  
Closed Interval ◽  
Discrete State ◽  
Closed Intervals ◽  
Gating Mechanism ◽  
The Relationship

Discrete state Markov models have proven useful for describing the gating of single ion channels. Such models predict that the dwell-time distributions of open and closed interval durations are described by mixtures of exponential components, with the number of exponential components equal to the number of states in the kinetic gating mechanism. Although the exponential components are readily calculated (Colquhoun and Hawkes, 1982, Phil. Trans. R. Soc. Lond. B. 300:1–59), there is little practical understanding of the relationship between components and states, as every rate constant in the gating mechanism contributes to each exponential component. We now resolve this problem for simple models. As a tutorial we first illustrate how the dwell-time distribution of all closed intervals arises from the sum of constituent distributions, each arising from a specific gating sequence. The contribution of constituent distributions to the exponential components is then determined, giving the relationship between components and states. Finally, the relationship between components and states is quantified by defining and calculating the linkage of components to states. The relationship between components and states is found to be both intuitive and paradoxical, depending on the ratios of the state lifetimes. Nevertheless, both the intuitive and paradoxical observations can be described within a consistent framework. The approach used here allows the exponential components to be interpreted in terms of underlying states for all possible values of the rate constants, something not previously possible.

Coaction of Electrostatic and Hydrophobic Interactions: Dynamic Constraints on Disordered TrkA Juxtamembrane Domain

2019 ◽  
Author(s):  
Zichen Wang ◽  
Huaxun Fan ◽  
Xiao Hu ◽  
John Khamo ◽  
Jiajie Diao ◽  
...  
Keyword(s):  
Single Molecule ◽  
Protein Function ◽  
Hydrophobic Interactions ◽  
Transmembrane Helix ◽  
Point Mutations ◽  
Salt Bridge ◽  
Receptor Activation ◽  
Membrane Dynamics ◽  
Juxtamembrane Domain ◽  
Joint Work

<p>The receptor tyrosine kinase family transmits signals into cell via a single transmembrane helix and a flexible juxtamembrane domain (JMD). Membrane dynamics makes it challenging to study the structural mechanism of receptor activation experimentally. In this study, we employ all-atom molecular dynamics with Highly Mobile Membrane-Mimetic to capture membrane interactions with the JMD of tropomyosin receptor kinase A (TrkA). We find that PIP<sub>2 </sub>lipids engage in lasting binding to multiple basic residues and compete with salt bridge within the peptide. We discover three residues insertion into the membrane, and perturb it through computationally designed point mutations. Single-molecule experiments indicate the contribution from hydrophobic insertion is comparable to electrostatic binding, and in-cell experiments show that enhanced TrkA-JMD insertion promotes receptor ubiquitination. Our joint work points to a scenario where basic and hydrophobic residues on disordered domains interact with lipid headgroups and tails, respectively, to restrain flexibility and potentially modulate protein function.</p>

scholarly journals Membrane-delimited Inhibition of Maxi-K Channel Activity by the Intermediate Conductance Ca2+-activated K Channel

2006 ◽  
Vol 127 (2) ◽  
pp. 159-169 ◽  
Author(s):  
Jill Thompson ◽  
Ted Begenisich
Keyword(s):  
Single Channel ◽  
Expression System ◽  
Chemical Activation ◽  
K Channel ◽  
Channel Activity ◽  
Single Family ◽  
Open Probability ◽  
K Channels ◽  
Channel Proteins ◽  
K Activity

The complexity of mammalian physiology requires a diverse array of ion channel proteins. This diversity extends even to a single family of channels. For example, the family of Ca2+-activated K channels contains three structural subfamilies characterized by small, intermediate, and large single channel conductances. Many cells and tissues, including neurons, vascular smooth muscle, endothelial cells, macrophages, and salivary glands express more than a single class of these channels, raising questions about their specific physiological roles. We demonstrate here a novel interaction between two types of Ca2+-activated K channels: maxi-K channels, encoded by the KCa1.1 gene, and IK1 channels (KCa3.1). In both native parotid acinar cells and in a heterologous expression system, activation of IK1 channels inhibits maxi-K activity. This interaction was independent of the mode of activation of the IK1 channels: direct application of Ca2+, muscarinic receptor stimulation, or by direct chemical activation of the IK1 channels. The IK1-induced inhibition of maxi-K activity occurred in small, cell-free membrane patches and was due to a reduction in the maxi-K channel open probability and not to a change in the single channel current level. These data suggest that IK1 channels inhibit maxi-K channel activity via a direct, membrane-delimited interaction between the channel proteins. A quantitative analysis indicates that each maxi-K channel may be surrounded by four IK1 channels and will be inhibited if any one of these IK1 channels opens. This novel, regulated inhibition of maxi-K channels by activation of IK1 adds to the complexity of the properties of these Ca2+-activated K channels and likely contributes to the diversity of their functional roles.

scholarly journals Correlation between Charge Movement and Ionic Current during Slow Inactivation in Shaker K+ Channels

1997 ◽  
Vol 110 (5) ◽  
pp. 579-589 ◽  
Author(s):  
Riccardo Olcese ◽  
Ramón Latorre ◽  
Ligia Toro ◽  
Francisco Bezanilla ◽  
Enrico Stefani
Keyword(s):  
Time Course ◽  
Voltage Dependence ◽  
Ionic Current ◽  
K Channel ◽  
Charge Movement ◽  
Slow Inactivation ◽  
Gating Currents ◽  
Similar Time ◽  
K Channels ◽  
Recovery From Inactivation

Prolonged depolarization induces a slow inactivation process in some K+ channels. We have studied ionic and gating currents during long depolarizations in the mutant Shaker H4-Δ(6–46) K+ channel and in the nonconducting mutant (Shaker H4-Δ(6–46)-W434F). These channels lack the amino terminus that confers the fast (N-type) inactivation (Hoshi, T., W.N. Zagotta, and R.W. Aldrich. 1991. Neuron. 7:547–556). Channels were expressed in oocytes and currents were measured with the cut-open-oocyte and patch-clamp techniques. In both clones, the curves describing the voltage dependence of the charge movement were shifted toward more negative potentials when the holding potential was maintained at depolarized potentials. The evidences that this new voltage dependence of the charge movement in the depolarized condition is associated with the process of slow inactivation are the following: (a) the installation of both the slow inactivation of the ionic current and the inactivation of the charge in response to a sustained 1-min depolarization to 0 mV followed the same time course; and (b) the recovery from inactivation of both ionic and gating currents (induced by repolarizations to −90 mV after a 1-min inactivating pulse at 0 mV) also followed a similar time course. Although prolonged depolarizations induce inactivation of the majority of the channels, a small fraction remains non–slow inactivated. The voltage dependence of this fraction of channels remained unaltered, suggesting that their activation pathway was unmodified by prolonged depolarization. The data could be fitted to a sequential model for Shaker K+ channels (Bezanilla, F., E. Perozo, and E. Stefani. 1994. Biophys. J. 66:1011–1021), with the addition of a series of parallel nonconducting (inactivated) states that become populated during prolonged depolarization. The data suggest that prolonged depolarization modifies the conformation of the voltage sensor and that this change can be associated with the process of slow inactivation.

scholarly journals Computing Optimal Properties of Drugs Using Mathematical Models of Single Channel Dynamics

2018 ◽  
Vol 6 (1) ◽  
pp. 41-64 ◽  
Author(s):  
Aslak Tveito ◽  
Mary M. Maleckar ◽  
Glenn T. Lines
Keyword(s):  
Differential Equations ◽  
Markov Model ◽  
Stochastic Models ◽  
Markov Models ◽  
Single Channel ◽  
Probability Density Functions ◽  
Density Functions ◽  
Mathematical Framework ◽  
Wild Type ◽  
Channel Dynamics

AbstractSingle channel dynamics can be modeled using stochastic differential equations, and the dynamics of the state of the channel (e.g. open, closed, inactivated) can be represented using Markov models. Such models can also be used to represent the effect of mutations as well as the effect of drugs used to alleviate deleterious effects of mutations. Based on the Markov model and the stochastic models of the single channel, it is possible to derive deterministic partial differential equations (PDEs) giving the probability density functions (PDFs) of the states of the Markov model. In this study, we have analyzed PDEs modeling wild type (WT) channels, mutant channels (MT) and mutant channels for which a drug has been applied (MTD). Our aim is to show that it is possible to optimize the parameters of a given drug such that the solution of theMTD model is very close to that of the WT: the mutation’s effect is, theoretically, reduced significantly.We will present the mathematical framework underpinning this methodology and apply it to several examples. In particular, we will show that it is possible to use the method to, theoretically, improve the properties of some well-known existing drugs.

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