Narrow Results

We utilize the original Hodgkin–Huxley (HH) model to consider the effects of defective ion channels to the temporal response of neurons. Statistics of firing rate and inter-spike interval (ISI) reveal that production of action potentials (APs) in neurons is not sensitive to changes in membrane conductance for sodium and potassium ions, as well as to the reversal potential for sodium ions, as long as the relevant parameters do not exceed 13% from their normal levels. We also found that blockage of a critical fraction of either sodium or potassium channels (dependent on constant input current) respectively limits the firing activity or increases spontaneous spiking activity of neurons. Our model may be used to guide experiment designs related to ion channel control drug development.

Human heart is elegantly articulated to mechanically contract in response to electrical excitation. Cardiac electrical activity may be described as a multiscale process from sub-cellular to cellular to tissue level. Ion movement at the cellular level through ion channels results in an action potential that propagates as an electrical wave in tissue. A first-principles-based mathematical description of the cellular-level dynamics of cardiac electrophysiological behavior provides a better understanding of the functioning of the heart.

The mathematical models describing cellular dynamics often involve a coupled system of ordinary differential equations (ODEs) with variables including transmembrane voltage, ion concentrations and ion channel gating variables, whose evolution describes activation/inactivation of ion channels. In this study we discuss a mathematical model of the human ventricular myocyte (O’Hara–Rudy model), defined as a system of 41 ODEs, with variables involving membrane voltage, 29 gating variables describing activation of Na${}^{+}$, K${}^{+}$, Ca${}^{2+}$ channels, 11 variables describing ion concentrations and Ca${}^{2+}$ related flux. Runge–Kutta method with variable order and variable time step was adopted to solve the system numerically. We discuss the action potential (AP) profile of a healthy human ventricular myocyte and corresponding dominant ionic currents. We present a phase plot that describes the change in voltage and its rate as the system evolves over time. The phase plot seems to provide more details of the underlying events than the AP curve.

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.

The motion of an ion through a channel is described as a classical/quantum mechanical hopping process between the individual sites of a channel. The transition rates are due to the coupling of an ion to suitable reservoirs. If fluctuating forces are added to the rate equations for the occupation numbers, the equations become quantum mechanical operator equations. Using previous results, the fluctuating forces are uniquely determined by the requirement of quantum mechanical consistency. The resulting equations are solved for several cases and the occupation number fluctuations discussed. Particular emphasis is laid on a model of correlated transport.

We analyze the statistics of skipping events in a deterministic and a noisy thermoreceptor within the model developed by Braun et al (Int. J. Bif. Chaos 8 881 (1998)). The statistics of skips can be captured by an intermittent map, which, for sufficient noise, can be approximated by a one-step Markov process with transition amplitudes that depend on the noise intensity. The theoretical and model results are in reasonable agreement with experimental data on cat coldreceptors.

We present a nonequilibrium reaction rate model of the ionic transition through an open ion channel, taking account of the interaction between an ion at the entrance of the channel and an ion at the binding site in a self-consistent way. The electrostatic potential is calculated by solution of the Poisson equation for a channel modeled as a cylindrical tube. The transition rate, and the binding site occupancy as a function of the left bulk concentration are compared to 1D Brownian dynamics simulations. The analysis is performed for a single binding site of high-affinity, with the exit rate influenced by barrier fluctuations at the channel exit. The results are compared with experimental data for the permeation of the Na⁺ ion through the Gramicidin A channel, with which they are shown to be in good agreement.

Here, we have provided a qualitative theoretical description about how the action potential generation and its associated intrinsic properties such as ionic current, spiking frequency, action potential duration, gating dynamics, etc. are affected due to site selective ion channel blockers, by suitably adapting Gillespie’s stochastic simulation technique to an extended Hodgkin–Huxley Markov model, representing a very basic type of neuron. Considering different types and degrees of blocking potency of channel blockers to channel proteins, we have found that the nature of action potential termination process and corresponding ionic current profiles are very distinct from each other. With the increasing blocking affinity, the frequency of action potential spiking falls off exponentially in presence of sodium channel only blockers and dual type blockers having more sodium binding potency than potassium blockers, whereas in contrast, for potassium channel only blockers, dual type blockers having equal or higher potassium blocking affinity with respect to sodium blocking, the spiking frequency initially is enhanced followed by a gradual decrease due to the competition between channel number fluctuation and overall sodium and potassium conductances. Sodium channel blockers tend to shorten the action potential duration while the potassium channel blockers broaden it. The channel gating dynamics are also found to be changed drastically for different types of blockers. The final quiescent state arrival time and the quiescent state membrane voltage profiles show distinct features for different types of channel blockers with different applied external stimulus. Finally, we showed how consistent our results are with the existing literature of experimentally observed channel blocking effects in diverse systems and compared the similarities, dissimilarities and advantages of our model with an existing theoretical drug binding model with Langevin description. Our approach provides a qualitative pathway to investigate the effects of many other types of blocking mechanisms such as closed state, inactivated state blocking with desired level of structural and functional details.

The interaction of ladder polyethers of marine origin, like ciguatoxin 3C and brevenal, as well as hypothetic ent-brevenal, with the human voltage-gated Kv1.5 potassium ion channel is investigated in this work using homology modeling, automated docking, and energy scoring from molecular dynamics (MD) simulations. A 3D homology model of the pore region of the Kv1.5 channel, previously developed from the 2.9 Å resolution crystal structure of the mammalian Kv1.2 channel — which has a very similar pore sequence — is used here. While ciguatoxin 3C did not enter the pore, both brevenal and ent-brevenal were found into the pore, the latter one with the best score. Binding is attended by notable strain in the ligands, and the corresponding energy increase was evaluated for ent-brevenal by self consistent field (SCF) and density functional theory (DFT) procedures. Egress of ent-brevenal from the pore, as a microscopical reversal of the ingress, was investigated by a smart form of biased MD simulations. While this study indicates ample room and attractive interactions for both brevenal and its enantiomer into the pore, whether these molecules will be found to inhibit voltage-gated potassium ion currents depends upon the barriers in the real system to access the pore, with their thermodynamic and kinetic requirements.

The interaction of the K⁺-sparing agent amiloride — a synthetic chlorinated pyrimidine derivative — with the hASIC1a ion channel is investigated here along homology modeling of the pore region (using the crystal structure of the cASIC1 channel as a template and the known sequence of hASIC1a), automated docking (using the NMR solution structure of amiloride and its conjugated acid, refined by computations), and molecular dynamics simulations. This represents the first modeling and computational chemistry of the pore region of ASIC/DEG/ENaCs/FaNaCh channels. The results agree with the putative amiloride binding site for alphaENaC channel chimeras once the amiloride free base is considered, while its conjugated acid — in contrast with literature beliefs — is poorly scored on a nearby protein pocket. Different protonation conditions of the pore region are irrelevant because histidine residues are far from the binding sites. Mapping the amino acids of the homology model closest to amiloride can have heuristic value in stimulating in silico search of new pore-blocking agents, experimental studies of ASIC channels themselves, and development of code for constant-pH MD simulations.

How a mutation affects the binding free energy of a ligand is a fundamental problem in molecular biology/biochemistry with many applications in pharmacology and biotechnology, e.g. design of drugs and enzymes. Free energy change due to a mutation can be determined most accurately by performing alchemical free energy calculations in molecular dynamics (MD) simulations. Here we discuss the necessary conditions for success of free energy calculations using toxin peptides that bind to ion channels as examples. We show that preservation of the binding mode is an essential requirement but this condition is not always satisfied, especially when the mutation involves a charged residue. Otherwise problems with accuracy of results encountered in mutation of charged residues can be overcome by performing the mutation on the ligand in the binding site and bulk simultaneously and in the same system. The proposed method will be useful in improving the affinity and selectivity profiles of drug leads and enzymes via computational design and protein engineering.

Molecular dynamics simulations of wild type and two mutant (T248F and L251T) human $\alpha$7 nicotinic acetylcholine receptors (nAChR) have been performed. The channel transmembrane domains were modeled from the closed channel structure from torpedo ray (PDB ID 2BG9) and embedded in DPPC lipid bilayers, surrounded by physiological saline solution. An external electric field was used to obtain stable open channel structures. The adaptive biasing force (ABF) method was used to obtain potential of mean force (PMF) profiles for Na${}^{+}$ ion translocation through the wild type and mutant receptors. Based on the geometry and PMF profiles, the channel gate was found to be at one of the two hydrophobic conserved regions (V249-L251) near the lower end of the channel. The L251T mutation reduced the energetic barrier by 1.9kcal/mol, consistent with a slight increase in the channel radius in the bottleneck region. On the other hand, the T248F mutation caused a significant decrease in the channel radius (0.4 Å) and a substantial increase of 3.9kcal/mol in the energetic barrier. Ion permeation in all three structures was compared and found to be consistent with barrier height values. Using an external field in an incrementally increasing manner was found to be an effective way to obtain stable open, conducting channel structures.

It is well established that membrane receptors, transporters, and ion channels are organized into functional networks at the cell membrane by multiprotein complexes. The scaffolding proteins physically link these signaling membrane proteins to their intracellular effectors and actin skeleton. The last ten years of research in the field have revealed the nature, structure, and functions of some of these multiprotein complexes. Here, we will focus on those which are present at the excitatory glutamatergic synapse and describe some of their structural and functional aspects, as well as the main methods which are use to study them.

Voltage-gated channel proteins cooperate in the transmission of membrane potentials between nerve cells. With the recent progress in atomic-scaled biological chemistry, it has now become established that these channel proteins provide highly correlated atomic environments that may maintain electronic coherences even at warm temperatures. Here we demonstrate solutions of the Schrödinger equation that represent the interaction of a single potassium ion within the surrounding carbonyl dipoles in the Berneche–Roux model of the bacterial KcsA model channel. We show that, depending on the surrounding carbonyl-derived potentials, alkali ions can become highly delocalized in the filter region of proteins at warm temperatures. We provide estimations on the temporal evolution of the kinetic energy of ions depending on their interaction with other ions, their location within the oxygen cage of the proteins filter region, and depending on different oscillation frequencies of the surrounding carbonyl groups. Our results provide the first evidence that quantum mechanical properties are needed to explain a fundamental biological property such as ion selectivity in transmembrane ion currents and the effect on gating kinetics and shaping of classical conductances in electrically excitable cells.

The development of synthetic transmembrane anion transport systems is of considerable interest, not only for mimicking the functions of natural transmembrane proteins but also for practical applications. We have recently reported a porous organic cage, porphyrin box (PB(8)) having multiple windows surrounded by octyl chains as an iodide selective anion channel. Herein, we report the modulation of transmembrane transport of halides (${\mathrm{\text{Cl}}}^{-}$, ${\mathrm{\text{Br}}}^{-}$, and ${\mathrm{\text{I}}}^{-})$ by dynamic window size engineering of the cage with different alkyl chain lengths (hexyl PB(6), octyl PB(8) and decyl PB(10)). ‘Apparent’ transport rates were measured by the HPTS fluorescence assay, which shows a gradual decrease in the transport rate upon increasing the length of alkyl chains of PB. We calculate the transport rate per PB in order to make a fair comparison as the ‘apparent’ transport rate is proportional to the number of PBs embedded in the lipid membrane. The transport rate per PB reveals that increasing the length of the alkyl chains of PBs results in a substantial fall in the iodide transport rate while only marginally decreasing the transport rates of bromide and chloride, thereby decreasing the selectivity of iodide transport.

In this document, we present the articles of the Special Issue on measuring & solving single molecules. These include reviews and articles about the state-of-the-art experimental and mathematical methods and applications in life sciences, biophysics and materials science. Ways of solving pitfalls in this field are presented in various articles. This Special Issue can intrigue, inspire and help scientifically both young and established scientists working in this field.

In this article, we talk about the ways that scientists can solve single molecule trajectories. Solving single molecules, that is, finding the model from the data, is complicated at least as much as measuring single molecules. We must filter the noise and take care of every step in the analysis when constructing the most accurate model from the data. Here, we present valuable solutions. Ways that solve clean discrete data are first presented. We review here our reduced dimensions forms (RDFs): unique models that are canonical forms of discrete data, and the statistical and numerical toolbox that builds a RDF from finite, clean, two-state data. We then review our most recent filter that "tackles" the noise when measuring two state noisy photon trajectories. The filter is a numerical algorithm with various special statistical treatments that is based on a general likelihood function that we have developed recently. We show the strengths of the filter (also over other approaches) and talk about its various new variants. This filter (with minor adjustments) can solve the noise in any discrete state trajectories, yet, extensions are needed in "tackling" the noise from other data, e.g. continuous data. Only the combined procedures enable creating the most accurate model from noisy discrete trajectories from single molecules. These concepts and methods (with adjustments) are valuable also when solving continuous trajectories and fluorescence resonance energy transfer trajectories. We also present a set of simple methods that can help any scientist with treating the trajectory perhaps encouraging applying the involved methods. The involved methods will appear in software that we are developing now, helping therefore the experimentalists utilizing these methods on real data. Comparisons with other known methods in this field are made.

Special Issue Comment: This article about mathematical treatments when solving single molecules is related to the reviews in this Special Issue about measuring enzymes⁶⁷ and about FRET experiments² and about the software QUB.⁶

Cancer can be viewed as a "tissue", where neoplastic cells are immersed into a peculiar microenvironment (the "tumor microenvironment", TME) which modulates tumor cell behaviour during multistep tumorigenesis. Based on this concept, antineoplastic therapy should be tuned to target not only tumor cells but also the cellular constituents of the TME. Such necessity is well exemplified by considering tumor angiogenesis, a major aspect of cancer biology.

Ion channels and transporters are increasingly recognized as relevant players in the tumor cell-TME cross-talk. For example, during tumor neo-angiogenesis, soluble factors as well as fixed components of the extracellular matrix (ECM) and membrane proteins determine signal exchange between the TME and the implicated cell types. The signalling network is coordinated by functional "hubs", which may be constituted by integrin receptors associated with other proteins to form macromolecular signalling platforms at the adhesive sites. These complexes often include ion channels.

The K⁺ channels encoded by the human ether-à-go-go related gene (Kv11.1, or hERG1) are frequently overexpressed in human cancers and regulate intracellular signalling by physically associating with integrin subunits and growth factor/chemokine receptors. In colorectal cancer (CRC) we recently identified a novel signalling pathway centered on hERG1 channels and integrins. This pathway involves the p53 protein, which is encoded by a tumor suppressor gene often mutated in human cancers. p53 controls angiogenesis, through a mechanism regulated by hERG1 K⁺ channels. The central role played by hERG1 in CRC angiogenesis suggests that targeting hERG1 may be an effective therapeutic option in patients with advanced CRC.

To better understand the above process, it is necessary to study the interlaced dynamics of the key microscopic actors by using dedicated mathematical models. We here review a simple model, of reductionist inspiration, that explores the intimate connections between apoptosis and hypoxia, passing through angiogenesis. We show that a dynamical switch takes place between the normoxia and cellular death conditions. When oxygen lacks, cells can cross the transition line and so gain their way towards the normoxia regime, by implementing point mutations that affect the p53 production and activation rate, with the involvement of K⁺ ion homeostasis, in agreement with the experimental observations.

Increasing evidence shows that ion channels play a significant role in cell proliferation, migration, apoptosis and differentiation. Many research works in gynecological cancer suggest that ion channels are involved in aberrant tumor growth and upregulation or downregulation of ion channels results in tumor growth arrest. Channelopathies are a diverse set of inherited mutations of ion channels that result in altered biophysical properties. In normal cells, there is a switch between cell growth and cell death which apparently depend upon the temporal organization and magnitude of different ion channels. There is a need for specific ion channel blockers which can abrogate the cellular mechanism of the cancer cell. Typically the impact of ion channels on cancer depends upon the magnitude and temporal organization of the ion channel activation and the activity of other signaling mechanisms. Membrane proteins are responsible for ionic homeostasis in ion channels. As cancer can be linked with the altered biophysical properties of ion channels a border context of different ion channels in cancer seems absolutely appropriate. The field of onco-channelopathies is rapidly expanding and understanding the pathophysiological mechanisms underlying the development of cancer enables researchers to better diagnose and develop treatment options for cancer. This review focuses on the role of ion channels in breast cancer and other gynecological cancers including ovarian cancer and cervical cancer, and how they contribute to tumor development.

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 problems 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.

We present a nonequilibrium reaction rate model of the ionic transition through an open ion channel, taking account of the interaction between an ion at the entrance of the channel and an ion at the binding site in a self-consistent way. The electrostatic potential is calculated by solution of the Poisson equation for a channel modeled as a cylindrical tube. The transition rate, and the binding site occupancy as a function of the left bulk concentration are compared to 1D Brownian dynamics simulations. The analysis is performed for a single binding site of high-affinity, with the exit rate influenced by barrier fluctuations at the channel exit. The results are compared with experimental data for the permeation of the Na⁺ ion through the Gramicidin A channel, with which they are shown to be in good agreement.