The action potential signal of nerve and muscle is produced by voltage sensitive channels that include a specialized device to sense voltage. Gating currents of the voltage sensor are now known to depend on the back-and-forth movements of positively charged arginines through the hydrophobic plug of a voltage sensor domain. Transient movements of these permanently charged arginines, caused by the change of transmembrane potential, further drag the S4 segment and induce opening/closing of ion conduction pore by moving the S4-S5 linker. The ion conduction pore is a separate device from the voltage sensor, linked (in an unknown way) by the mechanical motion and electric field changes of the S4-S5 linker. This moving permanent charge induces capacitive current flow everywhere. Everything interacts with everything else in the voltage sensor so everything must interact with everything else in its mathematical model, as everything does in the whole protein. A PNP-steric model of arginines and a mechanical model for the S4 segment are combined using energy variational methods in which all movements of charge and mass satisfy conservation laws of current and mass. The resulting 1D continuum model is used to compute gating currents under a wide range of conditions, corresponding to experimental situations. Chemical-reaction-type models based on ordinary differential equations cannot capture such interactions with one set of parameters. Indeed, they may inadvertently violate conservation of current. Conservation of current is particularly important since small violations (<0.01%) quickly (<< 10-6 seconds) produce forces that destroy molecules. Our model reproduces signature properties of gating current: (1) equality of on and off charge in gating current (2) saturating voltage dependence in QV curve and (3) many (but not all) details of the shape of gating current as a function of voltage.
HALL effect - SemiConductors - and it's Applications - Engineering PhysicsTheerumalai Ga
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A 20 mins discussion on the "HALL EFFECT and it's applications" of Semiconductors and a brief explanation about Hall Sensors with a derivation and video attached. Engineering Physics - important area of discussion for Anna University examination- seminar
HALL effect - SemiConductors - and it's Applications - Engineering PhysicsTheerumalai Ga
Β
A 20 mins discussion on the "HALL EFFECT and it's applications" of Semiconductors and a brief explanation about Hall Sensors with a derivation and video attached. Engineering Physics - important area of discussion for Anna University examination- seminar
A brief walk through over the Electrical fundamentals with illustrations of basic Electrical terms, Laws, Equations, Symbols and a complete guidance for understanding Electricity
Case Study: Cyclic Voltametric MeasurementHasnain Ali
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The design of an ac Cyclic Voltammetric Measurement System for the in βsitu measurement of dissolved oxygen in sediment on the seabed. The measurement strategy should be based on linear ramp cyclic voltammetry
Current Electricity and Effects of CurrentOleepari
Β
Electric current, potential difference and electric current. Ohmβs law; Resistance, Resistivity,
Factors on which the resistance of a conductor depends. Series combination of resistors,
parallel combination of resistors and its applications in daily life. Heating effect of electric
current and its applications in daily life. Electric power, Interrelation between P, V, I and R
Computer simulations of realistic ion channel structures have always been challenging and
a subject of rigorous study. Simulations based on continuum electrostatics have proven to
be computationally cheap and reasonably accurate in predicting a channelβs behavior. In
this paper we discuss the use of a device simulator, SILVACO, to build a solid-state model for
KcsA channel and study its steady-state response. SILVACO is a well-established program,
typically used by electrical engineers to simulate the process flow and electrical characteristics
of solid-state devices. By employing this simulation program, we have presented an
alternative computing platform for performing ion channel simulations, besides the known
methods of writing codes in programming languages. With the ease of varying the different
parameters in the channelβs vestibule and the ability of incorporating surface charges,
we have shown the wide-ranging possibilities of using a device simulator for ion channel
simulations. Our simulated results closely agree with the experimental data, validating our
model.
A brief walk through over the Electrical fundamentals with illustrations of basic Electrical terms, Laws, Equations, Symbols and a complete guidance for understanding Electricity
Case Study: Cyclic Voltametric MeasurementHasnain Ali
Β
The design of an ac Cyclic Voltammetric Measurement System for the in βsitu measurement of dissolved oxygen in sediment on the seabed. The measurement strategy should be based on linear ramp cyclic voltammetry
Current Electricity and Effects of CurrentOleepari
Β
Electric current, potential difference and electric current. Ohmβs law; Resistance, Resistivity,
Factors on which the resistance of a conductor depends. Series combination of resistors,
parallel combination of resistors and its applications in daily life. Heating effect of electric
current and its applications in daily life. Electric power, Interrelation between P, V, I and R
Computer simulations of realistic ion channel structures have always been challenging and
a subject of rigorous study. Simulations based on continuum electrostatics have proven to
be computationally cheap and reasonably accurate in predicting a channelβs behavior. In
this paper we discuss the use of a device simulator, SILVACO, to build a solid-state model for
KcsA channel and study its steady-state response. SILVACO is a well-established program,
typically used by electrical engineers to simulate the process flow and electrical characteristics
of solid-state devices. By employing this simulation program, we have presented an
alternative computing platform for performing ion channel simulations, besides the known
methods of writing codes in programming languages. With the ease of varying the different
parameters in the channelβs vestibule and the ability of incorporating surface charges,
we have shown the wide-ranging possibilities of using a device simulator for ion channel
simulations. Our simulated results closely agree with the experimental data, validating our
model.
This paper was published by my former Supervisor and involves partly my calculations and the concepts used during my MSci Thesis at University College London.
Ion Channel Simulations for Potassium, Sodium, Calcium, and Chloride Channels...Iowa State University
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Computer simulations of realistic ion channel structures have always been challenging and a subject of rigorous study. Simulations based on continuum electrostatics have proven to be computationally cheap and reasonably accurate in predicting a channel's behavior. In this paper we discuss the use of a device simulator, SILVACO, to build a solid-state model for KcsA channel and study its steady-state response. SILVACO is a well-established program, typically used by electrical engineers to simulate the process flow and electrical characteristics of solid-state devices. By employing this simulation program, we have presented an alternative computing platform for performing ion channel simulations, besides the known methods of writing codes in programming languages. With the ease of varying the different parameters in the channel's vestibule and the ability of incorporating surface charges, we have shown the wide-ranging possibilities of using a device simulator for ion channel simulations. Our simulated results closely agree with the experimental data, validating our model.
https://www.sciencedirect.com/science/article/abs/pii/S0169260706002276
Design of sliding mode controller for chaotic Josephson-junction IJECEIAES
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It is known that a shunted nonlinear resistive-capacitive-inductance Josephson-junction (RCLSJ) model has a chaotic attractor. This attractor is created as a result of Hopf bifurcation that occurs when a certain direct current (DC) applied to one of the junction terminals. This chaotic attractor prevents the system from reaching the phase-locked state and hence degrade the performance of the junction. This paper aims at controlling and taming this chaotic attractor induced in this model and pulling the system to the phase-locked state. To achieve this task, a sliding mode controller is proposed. The design procedures involve two steps. In the first one, we construct a suitable sliding surface so that the dynamic of the system follows the sliding manifolds in order to meet design specifications. Secondly, a control law is created to force the chaotic attractor to slide on the sliding surface and hence stabilizes system trajectory. The RCLSJ model under consideration is simulated with and without the designed controller. Results demonstrate the validity of the designed controller in taming the induced chaos and stabilizing the system under investigation.
Cyclic voltammetry is the most widely used technique for acquiring qualitative information about electrochemical reactions. The power of cyclic voltammetry results from its ability to rapidly provide considerable information on the thermodynamics of redox processes and the kinetics of heterogeneous electron-transfer reactions and on coupled chemical reactions or adsorption processes. Cyclic voltammetry is often the first experiment performed in an electroanalytical study. In particular, it offers a rapid location of redox potentials of the electroactive species and convenient evaluation of the effect of media upon the redox process.
The Effect of High Zeta Potentials on the Flow Hydrodynamics in Parallel-Plat...CSCJournals
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This paper investigates the effect of the EDL at the solid-liquid interface on the liquid flow through a micro-channel formed by two parallel plates. The complete Poisson-Boltzmann equation (without the frequently used linear approximation) was solved analytically in order to determine the EDL field near the solid-liquid interface. The momentum equation was solved analytically taking into consideration the electrical body force resulting from the EDL field. Effects of the channel size and the strength of the zeta-potential on the electrostatic potential, the streaming potential, the velocity profile, the volume flow rate, and the apparent viscosity are presented and discussed. Results of the present analysis, which are based on the complete Poisson-Boltzmann equation, are compared with a simplified analysis that used a linear approximation of the Poisson-Boltzmann equation.
Maxwell's equations can be written without a dielectric constant. They are then universal and exact. They are also useful because they imply a universal and exact conservation of total current (including the displacement current). Kirchhoff's current law for circuits then also becomes universal and exact.
Maxwell's Equations can be written without a dielectric constant.
They then imply conservation of total current as a universal and exact physical law. In circuits they imply a generalization of Kirchhoff's current law that is universal and exact
Core Maxwell Equations are Scary April 11, 2021Bob Eisenberg
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When the Maxwell equations are written without a dielectric constant, they are universal and exact, for biological and technological applications, from inside atoms to between stars. Dielectric and polarization phenomena need then to be described by stress strain relations for charge, that show how charge redistributes when the electric field is changed, in each system of interest.
Conservation of total current (including the ethereal displacement current νΊ_ν ννβνν) is then as exact as the Maxwell equations themselves and independent of any property of matter. It is a consequence of the Lorentz invariance of the elementary charge, a property of all locally inertial systems, described by the theory of relativity.
Exact Conservation of Total Current allows a redefinition of Kirchhoffβs current law that is itself exact. In unbranched systems like circuit components or ion channels, conservation of total current becomes equality. Spatial dependence of total current disappears in that case. Hopping phenomena disappear. Spatial Brownian motion disappears. The infinite variation of a Brownian model of thermal noise becomes the zero spatial variation of total current. Maxwellβs Core Equations become a perfect (spatial) low pass filter.
An Exact and Universal theory of Electrodynamics is a scary challenge to scientists like me, trained to be skeptical of all sweeping claims to perfection.
Thermostatics vs. electrodynamics preprints 10.20944.v1Bob Eisenberg
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Thermodynamics has been the foundation of many models of biological and technological systems. But thermodynamics is static and is misnamed. A more suitable name is thermostatics.
Thermostatics does not include time as a variable and so has no velocity, flow or friction. Indeed, as usually formulated, thermostatics does not include boundary conditions. Devices require boundary conditions to define their input and output. They usually involve flow and friction. Thermostatics is an unsuitable foundation for understanding technological and biological devices. A time dependent generalization of thermostatics that might be called thermal dynamics is being
developed by Chun Liu and collaborators to avoid these limitations. Electrodynamics is not restricted like thermostatics, but in its classical formulation involves drastic assumptions about polarization and an over-approximated dielectric constant. Once the Maxwell equations are rewritten without a dielectric constant, they are universal and exact. Conservation of total current,including displacement current, is a restatement of the Maxwell equations that leads to dramatic simplifications in the understanding of one dimensional systems, particularly those without branches, like the ion channel proteins of biological membranes and the two terminal devices of electronic systems. The Brownian fluctuations of concentrations and fluxes of ions become the spatially independent total current, because the displacement current acts as an unavoidable low pass filter, a consequence of the Maxwell equations for any material polarization. Electrodynamics and thermal dynamics together form a suitable foundation for models of technological and biological systems.
Maxwell equations without a polarization field august 15 1 2020Bob Eisenberg
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Electrodynamics is almost always written using a polarization vector field to describe the response of matter to an electric field, or more specifically, to describe the change in distribution of charges as an electric field is applied or changed. This approach does not allow unique specification of a polarization field from measurements of the electric and magnetic fields and electrical current.
Many polarization fields will produce the same electric and magnetic fields, and current, because only the divergence of the polarization enters Maxwellβs first equation, relating charge and electric field. The curl of any function can be added to a polarization field without changing the electric field at all. The divergence of the curl is always zero. Models of structures that produce polarization cannot be uniquely determined from electrical measurements for the same reason. Models must describe charge distribution not just distribution of polarization to be unique.
I propose a different approach, using a different paradigm to describe field dependent charge, i.e., to describe the phenomena of polarization. I propose an operational definition of polarization that has worked well in biophysics where a field dependent, time dependent polarization provides the gating current that makes neuronal sodium and potassium channels respond to voltage. The operational definition has been applied successfully to experiments for nearly fifty years. Estimates of polarization have been computed from simulations, models, and theories using this definition and they fit experimental data quite well.
I propose that the same operational definition be used to define polarization charge in experiments, models, computations, theories, and simulations of other systems. Charge movement needs to be computed from a combination of electrodynamics and mechanics because βeverything interacts with everything elseβ.
The classical polarization field need not enter into that treatment at all.
Molecular Mean Field Theory of ions in Bulk and ChannelsBob Eisenberg
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Life and most of chemistry occurs in ionic solutions, but ionic solutions have only recently been recognized as the complex fluids that they are. The molecular view shows ions interacting with surrounding water and nearby ions. Everything is correlated in a complex way because ions and water have diameters comparable to their interaction length. The molecular scale shows only a small part of the correlation enforced by electrodynamics. Current defined as Maxwell did to include the ethereal current is exactly conserved, and therefore correlated, over all scales reaching to macroscopic boundary conditions some 10^9Γ larger than atoms crucial in batteries and nerve cells.
Jinn Liang Liu and I have built a molecular field theory PNPB Poisson Nernst Planck Bikerman that deals with water as molecules and describes local interactions with a steric potential that depends on the volume fraction of molecules and voids between them. The correlations of electrodynamics are described by a fourth-order differential operator that gives (as outputs) ion-ion and ion-water correlations; the dielectric response (permittivity) of ionic solutions; and the polarization of water molecules, all using a single correlation length parameter. The theory fits experimental data on activity and differential capacitance in ionic solutions of varying composition and content, including mixtures. Potassium channels, Gramicidin, L-type calcium channels, and the Na/Ca transporter are computed in three dimensions from structures in the Protein Data Bank.
Numerical analysis faces challenges
Geometric singularities of molecular surfaces
strong electric fields (100 mV/nm) and resulting exponential nonlinearities, and the
enormous concentrations (> 10 M) often found where ions are important, for example, near electrodes in batteries, in ion channels, and in active sites of proteins.
Wide ranging concentrations of Ca^(2+) in (> 10M) and near (10^(-2) to 10^(-8)M) almost every protein in biological cells make matters worse.
Challenges have been overcome using methods developed over many decades by the large community that works on the computational electronics of semiconductors.
Molecular Mean-Field Theory of Ionic Solutions: a Poisson-Nernst-Planck-Biker...Bob Eisenberg
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We have developed a molecular mean-field theory β fourth-order Poisson-
Nernst-Planck-Bikerman theory β for modeling ionic and water flows in biological ion channels
by treating ions and water molecules of any volume and shape with interstitial voids,
polarization of water, and ion-ion and ion-water correlations. The theory can also be used to
study thermodynamic and electrokinetic properties of electrolyte solutions in batteries, fuel
cells, nanopores, porous media including cement, geothermal brines, the oceanic system, etc.
The theory can compute electric and steric energies from all atoms in a protein and all ions
and water molecules in a channel pore while keeping electrolyte solutions in the extra- and
intracellular baths as a continuum dielectric medium with complex properties that mimic
experimental data. The theory has been verified with experiments and molecular dynamics
data from the gramicidin A channel, L-type calcium channel, potassium channel, and
sodium/calcium exchanger with real structures from the Protein Data Bank. It was also
verified with the experimental or Monte Carlo data of electric double-layer differential capacitance
and ion activities in aqueous electrolyte solutions. We give an in-depth review of
the literature about the most novel properties of the theory, namely, Fermi distributions of
water and ions as classical particles with excluded volumes and dynamic correlations that
depend on salt concentration, composition, temperature, pressure, far-field boundary conditions
etc. in a complex and complicated way as reported in a wide range of experiments.
The dynamic correlations are self-consistent output functions from a fourth-order differential
operator that describes ion-ion and ion-water correlations, the dielectric response (permit2
tivity) of ionic solutions, and the polarization of water molecules with a single correlation
length parameter.
1. INTRODUCTION
Water and ions give life. Their electrostatic and kinetic interactions play essential roles
in biological and chemical systems such as DNA, proteins, ion channels, cell membranes,
Hsinchu maxwell talk january 7 1 2020 for uploadBob Eisenberg
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Applying the Maxwell equations to mitochondria seems a hopeless task: there is so much complexity. But computers and their chips are nearly as complicated. Design of circuits is done every day by uncounted numbers of engineers and scientists, thanks to Kirchoff's Current Law, which is a conservation law in one dimensional (branched) systems of devices. Kirchoff's current law conserves flux, not current in its usual derivation. But Maxwell's equations do not conserve flux; they conserve total current. Total current J equals flux plus displacement current J+ eps_0 partial E/partial t . Maxwell's definition of current allows circuit laws to be applied to complex systems of devices, over a wide range of times and conditions. Channels and enzymes are devices because they localize current flow. Channels and enzymes can be analyzed by the methods of circuit theory, for that reason.
Lens as an Osmotic Pump, a Bidomain Model. DOI: 10.13140/RG.2.2.25046.80966Bob Eisenberg
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The lens of the eye has no blood vessels to interfere with vision. The lens is far too large for diffusion to provide food and clear wastes.
Experimental, theoretical and computational work has shown that the lens supports its own microcirculation. It is an osmotic pump that implements what physiologists have long believed βConvection provides what diffusion cannot.β
We introduce a general (non-electro-neutral) model that describes the steady-state relationships among ion fluxes, water flow and electric field inside cells, and in the narrow extracellular spaces within the lens.
Using asymptotic analysis, we derive a simplified model exploiting the numerical values of physiological parameters. The model reduces to first generation βcircuitβ models and shows the basis of computer simulations too large to easily understand. The full model helps resolve paradoxes that have perplexed molecular biologists: crucial physiological properties do not depend as expected on the permeability of the lens interior (to water flow).
Updating maxwell with electrons and charge version 6 aug 28 1Bob Eisenberg
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Maxwellβs equations describe the relation of charge and electric force almost perfectly even though electrons and permanent charge were not in his equations, as he wrote them. For Maxwell, all charge depended on the electric field. Charge was induced and polarization was described by a single dielectric constant.
Electrons, permanent charge, and polarization are important when matter is involved. Polarization of matter cannot be described by a single dielectric constant Ξ΅_(r )with reasonable realism today when applications involve 10^(-10) sec. Only vacuum is well described by a single dielectric constant Ξ΅_(0 ).
Here, Maxwell's equations are rewritten to include permanent charge and any type of polarization. Rewriting is in one sense petty, and in another sense profound, in either case presumptuous. Either petty or profound, rewriting confirms the legitimacy of electrodynamics that includes permanent charge and realistic polarization. One cannot be sure that a theory of electrodynamics without electrons or (permanent, field independent) charge (like Maxwellβs equations as he wrote them) would be legitimate or not. After all a theory cannot calculate the fields produced by charges (for example electrons) that are not in the theory at all!
After updating,
1) Maxwellβs equations seem universal and exact.
2) Polarization must be described explicitly to use Maxwellβs equations in applications.
3) Conservation of total current (including Ξ΅_0 βEββt) becomes exact, independent of matter, allowing precise definition of electromotive force EMF in circuits.
4) Kirchhoffβs current law becomes as exact as Maxwellβs equations themselves.
5) Conservation of total current needs to be satisfied in a wide variety of systems where it has not traditionally received much attention.
6) Classical chemical kinetics is seen to need revision to conserve current.
The name PNP was introduced by Eisenberg and Chen because it has important physical meaning beyond being the first letters of Poisson-Nernst-Planck. PNP also means Positive-Negative-Positive, the signs of majority current carriers in different regions of a PNP bipolar transistor. PNP transistors are two diodes in series PN + NP that rectify by changing the shape of the electric field. Transistors can function as quite different types of nonlinear devices by changing the shape of the electric field. Those realities motivated Eisenberg and Chen to introduce the name PNP.
The pun βPNP = Poisson-Nernst-Planck = Positive-Negative-Positiveβ has physical content. It suggests that Poisson-Nernst-Planck systems like open ionic channels should not be assumed to have constant electric fields. The electric field should be studied and computed because its change of shape is likely to be important in the function of biological systems, as it is in semiconductor systems.
Updating maxwell with electrons and charge Aug 2 2019Bob Eisenberg
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Maxwellβs equations describe the relation of charge and electric force almost perfectly even though electrons and permanent charge were not in his equations, as he wrote them. For Maxwell, all charge depended on the electric field. Charge was induced and polarization was described by a single dielectric constant.
Electrons, permanent charge, and polarization are important when matter is involved. Polarization of matter cannot be described by a single dielectric constant Ξ΅_(r )with reasonable realism today when applications involve 10^(-10) sec. Only vacuum is well described by a single dielectric constant Ξ΅_(0 ).
Here, Maxwell's equations are rewritten to include permanent charge and any type of polarization. Rewriting is in one sense petty, and in another sense profound, in either case presumptuous. Either petty or profound, rewriting confirms the legitimacy of electrodynamics that includes permanent charge and realistic polarization. One cannot be sure ahead of time that a theory of electrodynamics without electrons or (permanent, field independent) charge (like Maxwellβs equations as he wrote them) would be legitimate or not. After all a theory cannot calculate the fields produced by charges (for example electrons) that are not in the theory at all!
After updating,
Maxwellβs equations seem universal and exact.
Polarization must be described explicitly to use Maxwellβs equations in applications.
Conservation of total current (including Ξ΅_0 βEββt) becomes exact, independent of matter, allowing precise definition of electromotive force EMF in circuits.
Kirchhoffβs current law becomes as exact as Maxwellβs equations themselves.
Classical chemical kinetics is seen to need revision to conserve current.
Dielectric Dilemma 1901.10805 v2 feb 4 2019Bob Eisenberg
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A dielectric dilemma faces scientists because Maxwell's equations are poor approximations as usually written, with a single dielectric constant. Maxwell's equations are then not accurate enough to be useful in many applications. The dilemma can be partially resolved by a rederivation of conservation of current, where current is defined now to include the Βepolarization of the vacuumΒf ..0 .......... Conserveration of current becomes Kirchoff's current law with this definition, in the one dimensional circuits of our electronic technology. With this definition, Kirchoff's laws are valid whenever Maxwell's equations are valid, explaining why those laws reliably describe circuits that switch in nanoseconds.
Ions in solutions liquid plasma of chemistry & biology february 27 2019Bob Eisenberg
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Ions in water form a plasma of great importance in which much chemistry is done. All of life occurs in ionic solutions. We show how analysis based on a Poisson Fermi treatment of ionic solutions, based on the saturation which can occur when ions have finite volume, can account for the experimental properties of many solutions
What is different about life? it is inherited oberwolfach march 7 1 2018Bob Eisenberg
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What is different about life? Why do life sciences require different science and mathematics? I address these issues starting from the obvious: all of life is inherited from genes. Twenty thousand genes of say 30 atoms each control an animal of ~1e25 atoms. How is that possible? Answer: the structures of life form a hierarchy of devices that allow handfuls of atoms to control everything. A nerve signal involves meters of nerve but is controlled by a few atoms. Indeed, potassium and sodium differ only in the diameter of the atoms. Life depends on this difference in diameter. Sodium and potassium are otherwise identical. The task of the biological scientist is first to identify the hierarchy of devices and what they do. Then we want to know how the devices work. We want to understand life well enough to improve its devices, in disease and technology.
Electricity plays a special role in our lives and life. Equations of electron dynamics are nearly exact and apply from nuclear particles to stars. These Maxwell equations include a special term the displacement current (of vacuum). Displacement current allows electrical signals to propagate through space. Displacement current guarantees that current is exactly conserved from inside atoms to between stars, as long as current is defined as Maxwell did, as the entire source of the curl of the magnetic field. We show how the Bohm formulation of quantum mechanics allows easy definition of current. We show how conservation of current can be derived without mention of the polarization or dielectric properties of matter. Matter does not behave the way physicists of the 1800's thought it does with a single dielectric constant, a real positive number independent of everything. Charge moves in enormously complicated ways that cannot be described in that way, when studied on time scales important today for electronic technology and molecular biology. Life occurs in ionic solutions in which charge moves in response to forces not mentioned or described in the Maxwell equations, like convection and diffusion. Classical derivations of conservation of current involve classical treatments of dielectrics and polarization in nearly every textbook. Because real dielectrics do not behave that way, classical derivations of conservation of current are often distrusted or even ignored. We show that current is conserved exactly in any material no matter how complex the dielectric, polarization or conduction currents are. We believe models, simulations, and computations should conserve current on all scales, as accurately as possible, because physics conserves current that way. We believe models will be much more successful if they conserve current at every level of resolution, the way physics does.
Electricity is different august 16 1 2016 with doiBob Eisenberg
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Charges are everywhere because most atoms are charged. Chemical bonds are formed by electrons with their charge. Charges move and interact according to Maxwell's equations in space and in atoms where the equations of electrodynamics are embedded in SchrΓΆdinger's equation as the potential. Maxwell's equations are universal, valid inside atoms and between stars from times much shorter than those of atomic motion (0.1 femtoseconds) to years (32 mega-seconds). Maxwell's equations enforce the conservation of current. Analysis shows that the electric field can take on whatever value is needed to ensure conservation of current. The properties of matter rearrange themselves to satisfy Maxwell's equations and conservation of current. Conservation of current is as universal as Maxwell's equations themselves. Yet equations of electrodynamics find little place in the literature of material physics, chemistry, or biochemistry. Kinetic models of chemistry and Markov treatments of atomic motion are ordinary differential equations in time and do not satisfy conservation of current unless modified significantly. Systems at equilibrium, without macroscopic flow, have thermal fluctuating currents that are conserved according to the Maxwell equations although their macroscopic averages are zero. The macroscopic consequences of atomic scale fluctuating thermal currents are not known but are likely to be substantial because of the nonlinear interactions in systems like these, in which 'everything interacts with everything else'.
Device approach to biology and engineeringBob Eisenberg
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Device Approach to Biology (and Engineering)
The goal of biological research is often more to control than to understand. Devices in biology (like ion channels) control individual functions just as they do in our technology. Study of control requires a multiscale approach because a handful of atoms, moving in 10-15 sec, control biological functions extending meters and taking seconds. Structural biology and molecular dynamics are essential (and beautiful!) parts of this hierarchy, but so are the functions themselves, and the electric field equations that link structure and function on all scales from atoms to nerve cells. Analyzing biological systems as devices is usually successful, and almost always productive.
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...University of Maribor
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Slides from:
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), NiΕ‘, 3-6 June 2024
Track: Artificial Intelligence
https://www.etran.rs/2024/en/home-english/
Introduction:
RNA interference (RNAi) or Post-Transcriptional Gene Silencing (PTGS) is an important biological process for modulating eukaryotic gene expression.
It is highly conserved process of posttranscriptional gene silencing by which double stranded RNA (dsRNA) causes sequence-specific degradation of mRNA sequences.
dsRNA-induced gene silencing (RNAi) is reported in a wide range of eukaryotes ranging from worms, insects, mammals and plants.
This process mediates resistance to both endogenous parasitic and exogenous pathogenic nucleic acids, and regulates the expression of protein-coding genes.
What are small ncRNAs?
micro RNA (miRNA)
short interfering RNA (siRNA)
Properties of small non-coding RNA:
Involved in silencing mRNA transcripts.
Called βsmallβ because they are usually only about 21-24 nucleotides long.
Synthesized by first cutting up longer precursor sequences (like the 61nt one that Lee discovered).
Silence an mRNA by base pairing with some sequence on the mRNA.
Discovery of siRNA?
The first small RNA:
In 1993 Rosalind Lee (Victor Ambros lab) was studying a non- coding gene in C. elegans, lin-4, that was involved in silencing of another gene, lin-14, at the appropriate time in the
development of the worm C. elegans.
Two small transcripts of lin-4 (22nt and 61nt) were found to be complementary to a sequence in the 3' UTR of lin-14.
Because lin-4 encoded no protein, she deduced that it must be these transcripts that are causing the silencing by RNA-RNA interactions.
Types of RNAi ( non coding RNA)
MiRNA
Length (23-25 nt)
Trans acting
Binds with target MRNA in mismatch
Translation inhibition
Si RNA
Length 21 nt.
Cis acting
Bind with target Mrna in perfect complementary sequence
Piwi-RNA
Length ; 25 to 36 nt.
Expressed in Germ Cells
Regulates trnasposomes activity
MECHANISM OF RNAI:
First the double-stranded RNA teams up with a protein complex named Dicer, which cuts the long RNA into short pieces.
Then another protein complex called RISC (RNA-induced silencing complex) discards one of the two RNA strands.
The RISC-docked, single-stranded RNA then pairs with the homologous mRNA and destroys it.
THE RISC COMPLEX:
RISC is large(>500kD) RNA multi- protein Binding complex which triggers MRNA degradation in response to MRNA
Unwinding of double stranded Si RNA by ATP independent Helicase
Active component of RISC is Ago proteins( ENDONUCLEASE) which cleave target MRNA.
DICER: endonuclease (RNase Family III)
Argonaute: Central Component of the RNA-Induced Silencing Complex (RISC)
One strand of the dsRNA produced by Dicer is retained in the RISC complex in association with Argonaute
ARGONAUTE PROTEIN :
1.PAZ(PIWI/Argonaute/ Zwille)- Recognition of target MRNA
2.PIWI (p-element induced wimpy Testis)- breaks Phosphodiester bond of mRNA.)RNAse H activity.
MiRNA:
The Double-stranded RNAs are naturally produced in eukaryotic cells during development, and they have a key role in regulating gene expression .
This presentation explores a brief idea about the structural and functional attributes of nucleotides, the structure and function of genetic materials along with the impact of UV rays and pH upon them.
Cancer cell metabolism: special Reference to Lactate PathwayAADYARAJPANDEY1
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Normal Cell Metabolism:
Cellular respiration describes the series of steps that cells use to break down sugar and other Β chemicals to get the energy we need to function.
Energy is stored in the bonds of glucose and when glucose is broken down, much ofΒ that energy is released. Β
Cell utilize energy in the form of ATP.
The first step of respiration is called glycolysis. In a series of steps, glycolysis breaks glucose into two smaller molecules -Β a chemical called pyruvate. A small amount of ATP is formed during this process.Β
Most healthy cells continue the breakdownΒ inΒ a second process, called the Kreb's cycle. The Kreb's cycleΒ allows cells to βburnβ the pyruvates made in glycolysisΒ to get more ATP.
The last step in the breakdown of glucose is calledΒ oxidativeΒ phosphorylationΒ (Ox-Phos).
ItΒ takes place in specialized cell structuresΒ calledΒ mitochondria. This process produces a large amount of ATP. Β Importantly,Β cells need oxygen to complete oxidative phosphorylation.
If a cell completes only glycolysis, only 2 molecules of ATP are made per glucose. However, if the cell completes the entire respiration process (glycolysis -Β Kreb's -Β oxidative phosphorylation), about 36Β molecules of ATP are created, giving itΒ much moreΒ energy to use.
IN CANCER CELL:
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result,Β cancer cells need to use a lot more sugar molecules to get enoughΒ energy to survive.Β
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result,Β cancer cells need to use a lot more sugar molecules to get enoughΒ energy to survive.Β
introduction to WARBERG PHENOMENA:
WARBURG EFFECT Usually, cancer cells are highly glycolytic (glucose addiction) and take up more glucose than do normal cells from outside.
Otto Heinrich Warburg (; 8 October 1883 β 1 August 1970) In 1931 was awarded the Nobel Prize in Physiology for his "discovery of the nature and mode of action of the respiratory enzyme.
WARNBURG EFFECT : Β cancer cells under aerobic (well-oxygenated) conditions to metabolize glucose to lactate (aerobic glycolysis) is known as the Warburg effect. Warburg made the observation that tumor slices consume glucose and secrete lactate at a higher rate than normal tissues.
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
1. Available on arXiv and internet at https://arxiv.org/abs/1707.02566
1
https://arxiv.org/abs/1707.02566
CONTINUUM GATING CURRENT MODELS COMPUTED WITH CONSISTENT INTERACTIONS
Tzyy-Leng Horng1 , Robert S. Eisenberg2, Chun Liu3, Francisco Bezanilla4
1Department of Applied Mathematics, Feng Chia University, Taichung, Taiwan 40724
2Department of Molecular Biophysics and Physiology, Rush University, Chicago, Illinois 60612, United
States
3Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802,
United States
4Department of Biochemistry and Molecular Biology, University of Chicago, Chicago, Illinois 60637,
United States
Abstract
The action potential signal of nerve and muscle is produced by voltage sensitive channels that include a
specialized device to sense voltage. Not surprisingly the voltage sensor depends on the movement of
permanent charges of basic side chains in the changing electric field, as suggested long ago by Hodgkin
and Huxley and Bezanilla and Armstrong, for example. Gating currents of the voltage sensor are now
known to depend on the back-and-forth movements of positively charged arginines through the
hydrophobic plug of a voltage sensor domain. Transient movements of these permanently charged
arginines, caused by the change of transmembrane potential, further drag the S4 segment and induce
opening/closing of ion conduction pore by moving the S4-S5 linker. The ion conduction pore is a separate
device from the voltage sensor, linked (in an unknown way) by the mechanical motion and electric field
changes of the S4-S5 linker. This moving permanent charge induces capacitive current flow everywhere.
Everything interacts with everything else in the voltage sensor so everything must interact with everything else
in its mathematical model, as everything does in the whole protein. A PNP-steric model of arginines and a
mechanical model for the S4 segment are combined using energy variational methods in which all movements
of charge and mass satisfy conservation laws of current and mass. The resulting 1D continuum model is used
to compute gating currents under a wide range of conditions, corresponding to experimental situations.
Conservation laws are partial differential equations in space and time. Chemical-reaction-type models based on
ordinary differential equations cannot capture such interactions with one set of parameters. Indeed, they may
inadvertently violate conservation of current. Conservation of current is particularly important since small
violations (<0.01%) quickly (<< 10-6 seconds) produce forces that destroy molecules. Our model reproduces
signature properties of gating current: (1) equality of on and off charge in gating current (2) saturating voltage
dependence in QV curve and (3) many (but not all) details of the shape of gating current as a function of voltage.
The model computes gating current flowing in the baths produced by arginines moving in the voltage sensor.
The model also captures the capacitive pile up of ions from the bulk solution in the form of electric double
layers in vestibules adjacent to both ends of the hydrophobic plug. This pile up is coupled to the movement of
2. 2
arginines, and appears as a spike in the beginning of gating current that have been already observed as the early
transient gating current in experiments. Our results agree qualitatively well with experiments, and can obviously
be improved by adding more details of the structure and its correlated movements. The proposed
continuum model is a promising tool to explore the dynamics and mechanism of the voltage sensor that
avoids the sampling constraints that so limit the representation of biological concentrations (10-7- 5Γ10-1 M)
and currents (time scale > 10-4 sec) in molecular dynamics simulations that compute atomic motion on the 10-
15 sec time scale.
Keywords: ion channel; voltage sensor; gating current; arginine; S4; PNP-steric model
1. Introduction
Much of biology depends on the voltage across cell membranes. The voltage across the membrane
must be sensed before it can be used by proteins. Permanent charges1 move in the strong electric fields
within membranes, so carriers of gating and sensing charge were proposed as voltage sensors even
before membrane proteins were known to span lipid membranes [18-19, 21-23]. Movement of
permanent charges of the voltage sensor is gating current and movement is the voltage sensing
mechanism. Measurements of gating current were made much easier by biological preparations with
artificially increased densities of voltage sensors and blocked conduction pores.
Knowledge of membrane protein structure has allowed us to identify and look at the atoms that
make up the voltage sensor. Protein structures do not include the membrane potentials and macroscopic
concentrations that power gating currents, and therefore simulations are needed. Atomic level
simulations like molecular dynamics do not provide an easy extension from the atomic time scale 2Γ10-
16 sec to the biological time scale of gating currents that starts at 50Γ10-6 sec and reaches 50Γ10-3 sec.
Calculations of gating currents from simulations must average the trajectories (lasting 50Γ10-3 sec
sampled every 2Γ10-16 sec) of ~106 atoms all of which interact through the electric field to conserve
charge and current, while conserving mass. It is difficult to enforce continuity of current flow in
simulations of atomic dynamics because simulations compute only local behavior while continuity of
current is global, involving current flow far from the atoms that control the local behavior. It is impossible
to enforce continuity of current flow in calculations that assume equilibrium (zero flow) under all
conditions. Periodic boundary conditions are widely used in simulations. Such conditions take a box of
material and replicate it identically, so the potential at the corresponding edges of the box are identical.
If the potentials are identical, current will not flow. Periodic boundary conditions of this sort are
incompatible with current flow from one boundary to the other. Voltage clamp experiments, and natural
biological function involve current flow from one boundary to another.
A hybrid approach is needed, starting with the essential knowledge of structure, but computing
1
Permanent charge is our name for a charge or charge density independent of the local electric field, for example, the charge and charge distribution of Na+
but
not the charge in a highly polarizable anion like Brο
or the nonuniform charge distribution of H2O in the liquid state with its complex time dependent (and
perhaps nonlinear) polarization response to the local electric field.
3. 3
only those parts of the structure used by biology to sense voltage. In close packed (βcondensedβ) systems
like the voltage sensor, or ionic solutions, βeverything interacts with everything elseβ because electric fields
are long ranged as well as exceedingly strong [14]. In ionic solutions, ion channels, even in enzyme active
sites, steric interactions that prevent the overfilling of space in well-defined protein structures are also of
great importance as they produce long range correlations [17]. Closely packed charged systems are best
handled mathematically by energy variational methods. Energy variational methods guarantee that all
variables satisfy all equations (and boundary conditions) at all times and under all conditions and are
thus always consistent. We have used the energy variational approach developed in [12, 20] to derive a
consistent model of gating charge movement, based on the basic features of the structure of crystallized
channels and voltage sensors. The schematic of the model is shown below. The continuum model we use
simulates the mechanical dynamics in a single voltage sensor, while the experimental data from [7] is
from many independent voltage sensors. Ensemble averages of independent voltage sensors recording is
roughly equivalent to macroscopic continuum modeling in a single voltage sensor. It is unlikely one can
do better until experiments determine the structural source of the variance in the voltage sensor and its
time, voltage, and chemical dependence on ion type and concentration.
3.Mathematicalmodel
The reduced mechanical model for a voltage sensor is shown in Fig. 1 with four arginines Ri, i=1, 2,
3, 4, each attached to S4 helix by identical spring with spring constant K. The electric field will drag these
four arginines since each arginine carries +1 charge. In addition, the charged arginines move the group
itself. S4 connect to S3 and S5 at its two ends by identical springs with spring constant KS4/2. Once the
membrane is depolarized from say ο90mV inside negative, to +10mV inside positive, arginines together
with S4 will be driven towards extracellular side. When the membrane is repolarized from say +10mV
volts to ο90mV, the arginines move to the intracellular side. This movement is the basic voltage sensing
mechanism of ion channel. The movement of S4 triggers the opening or closing of the lower gate mainly
consisting of S6 of the permeant ion channel by a mechanism widely assumed to be mechanical, although
we hasten to say that electrical aspects of the linker motion are likely to be involved, as well, perhaps
acting as a deterministic trigger and control for the otherwise stochastic sudden discontinuous change of
current seen in single channel measurements of ion flow through the permeation channel.
When arginines are driven by electric field, they are forced to move through a hydrophobic plug,
composed by several nonpolar amino acids from S1, S2 and S3 [25]. Arginines moving from vestibule, a
hydrated lumen, on one side, though the hydrophobic plug, and entering into the vestibule on another
side would involve dehydration when passing through the hydrophobic plug. Therefore, arginines will
encounter a barrier in the potential of mean force, an energy barrier, mainly dominated by the solvation
energy [42]. Note that Na+ and Clο (used here as the only solution ions, for simplicity) can only exist in
vestibules and are not allowed into the hydrophobic plug. The bottoms of the two vestibules on each side
of the system act as impermeable walls for Na+ and Clο. When the voltage is turned on and off, these two
walls will form a pair of capacitors storing/releasing net ion charges in their electric double layers (EDL).
Molecular dynamics considers every particle individually and must deal with its enormous thermal
motion (more or less at the speed of sound 1,000 m/s or 1 nm per nanosecond), as it seeks to compute
gating currents on the biological time scale. We use a continuum treatment that deals with the thermal
4. 4
motion on the average, as temperature or diffusion. Such methods often include the correlations of the
average electric field, e.g., Poisson-Nernst-Planck (PNP), but not those that differ from atom to atom, as a
glance at the structure of any protein shows is important. Here we include a steric effect in the PNP model
[20], and so deal with many of the atomic scale correlations ignored in classical PNP. We hasten to add
that it is not at all clear how well molecular dynamics deals with correlations produced by the average
electric field, particularly when they include nonequilibrium components that drive current flow.
Classical molecular dynamics assumes equilibrium and uses periodic boundary conditions that have
certain difficulties dealing with the current flow and nonperiodic electric fields found in real devices and
channels, as we have previously mentioned. Continuum models can adequately describe the time
behavior of a voltage sensor in action, in which the time span is on the order of several hundreds of
milliseconds.
The four arginines Ri, i=1, 2, 3, 4, are described by their individual density distributions
(concentrations) ci, i=1, 2, 3, 4, allowing the arginines to interact with Na+ and Clο in vestibules. The
density (concentration) distributions represent probability density functions as shown explicitly in the
theory of stochastic processes, used to derive such equations in [29] using the general methods of [28].
This practice is used widely in quantum mechanics [3, 34]. The important issue here is how well the
correlations are captured in the continuum model. Some are more likely to be faithfully captured in
molecular dynamics simulations (e.g., more or less local hard sphere interactions), others in continuum
models (e.g., correlations induced by far field boundary conditions).
Here we treat the S4 itself as a rigid body so we can capture the basic mechanism of a voltage sensor
without considering the full details of structure, which might lead to a three dimensional model hard to
compute in the time available. We construct an axisymmetric 1D model with a three-zone geometric
configuration illustrated in Fig. 2 following Fig. 1. Zone 1 with π§ β [0, πΏ ] is the intracellular vestibule;
zone 2 with π§ β [πΏ , πΏ + πΏ] is the hydrophobic plug; zone 3 with π§ β [πΏ + πΏ, 2πΏ + πΏ] is the
extracellular vestibule. Arginines, Na+, and Clο can all reside in zone 1 and 3. Zone 2 only allows the
residence of arginines, albeit with a severe hydrophobic penalty because of their permanent charge, in a
region of low dielectric coefficient hence called hydrophobic. Spatial variation of dielectric coefficient
allows construction of many useful devices [15, 41]. Types of boundary conditions and relevant sizes of
this geometric configuration are also specified in Fig. 2. Note the no-flux boundary conditions specified
in Fig. 2. One prevents Na+ and Clο from entering the hydrophobic plug (zone 2) with low dielectric
coefficient. The other boundary condition constrains S4 motion and so prevents the arginines from
leaving the vestibules into intracellular/extracellular domains.
We first non-dimensionalize all physical quantities as follows,
πΜ = , π =
/
, π = , π Μ = , π‘Μ =
/
, π· = , π =
/
, π½ =
/
, πΌ = ,
where π is concentration of species i, with i=Na+, Clο, 1, 2, 3, and 4. Each is scaled by π which is the
bulk concentration of NaCl at the intracellular/extracellular domains. Here π is set to be 184mM, equal
on both sides, so that the Debye length π = is 1nm when the relative permittivity π = 80.
Note that π , i=1, 2, 3, and 4 needs to satisfy the following additional constraint β« π΄( π§) π ππ§ = 1 due
5. 5
to uniqueness of each arginine. π is the electric potential scaled by π π/π with π being the
Boltzmann constant; π the temperature; e the elementary charge. All relevant external potentials U are
scaled by π π. All sizes s are scaled by R, which is the radius of vestibule as shown in Fig. 2. R=1nm here.
The time t is scaled by π /π· , with π· being a diffusion coefficient that can be adjusted later to be
consistent with the time spans of on/off currents measured in experiments (caused by the movement of
arginines). The diffusion coefficient of species i is scaled by π· . The coupling constant π of PNP-steric
model based on combining rules of Lennard Jones, representing the strength of steric interaction between
species i and j, is scaled by π π/π [20]. Note that here we only consider steric interaction among
arginines, which is the most significant since arginines are generally crowded in hydrophobic plug and
vestibules. For simplicity, we assume π =
π, βπ β π
0, βπ = π
, π, π = 1,2,3,4. The flux density of species i, π½ , is
scaled by π π· /π , and therefore the electric current I is scaled by ππ π· π . For simplicity of notation, we
will drop ~ for all dimensionless quantities from here on.
Based on Fig. 2, the governing 1D dimensionless PNP-steric equations are expressed below. The first
one is Poisson equation:
β Ξπ΄ = β π§ π , π = Na, Cl, 1, 2, 3, 4, (1)
with π§ = 1, π§ = β1, π§ = π§ = 1, π = 1, 2, 3, 4, Ξ = and A(z) being the cross-sectional area. For
zones 1 and 3, Ξ = 1 since here the arginines are fully hydrated with π = 80. For zone 2, we assume a
hydrophobic environment π€ππ‘β π = 8, and therefore Ξ = 0.1.
The second equation is the species transport equation based on conservation law:
+ ( π΄π½ ) = 0, π = Na, Cl, 1 ,2 , 3, 4. (2)
with the content of Ji expressed below based on Nernst-Planck equation for Na+ and Cl-:
π½ = βπ· + π π§ , π = Na, Cl, π§ β zones 1 and 3, (3)
and based on Nernst-Planck equation with steric effect and some imposed potentials for 4 arginines ci,
i=1, 2, 3 and 4, based on Fig. 2,
π½ = βπ· + π§ πΆ + πΆ + + ππΆ + + , π§ β all zones, (4)
π½ = βπ· + π§ πΆ + πΆ + + ππΆ + + , π§ β all zones, (5)
π½ = βπ· + π§ πΆ + πΆ + + ππΆ + + , π§ β all zones, (6)
π½ = βπ· + π§ πΆ + πΆ + + ππΆ + + , π§ β all zones. (7)
The first and second terms in Eqs. (3-7) describe diffusion and electro-migration term respectively.
The third term in Eqs. (4-7) are external potential terms with Vi, i=1, 2, 3 and 4 being the constraint
potential for the 4 arginines ci to S4 represented here by a spring connecting each arginine ci to S4 as
shown in Fig. 1. It is expressed as
6. 6
π (π§, π‘) = πΎ(π§ β (π§ + π (π‘))) , (8)
where K is the spring constant, zi is the fixed anchoring position of the spring for each arginine ci on S4,
π (π‘) is the center-of-mass z position of S4 by treating S4 as a rigid body. Here we set z1=0.6, z2=0.2, z3=-
0.2, z4=-0.6 given from the structure that gives the arginine anchoring interval on S4 as 0.4nm. π (π‘)
follows the motion of equation based on spring-mass system:
π + π + πΎ (π β π , ) = β πΎ π§ , β ( π§ + π ) , (9)
where π , π and πΎ are mass, damping coefficient and restraining spring constant for S4. π , is
the natural position of π (π‘) when the net force on the right hand side of Eq. (9) vanishes. Here, π§ ,
is the center of mass for the set of arginines ci , which can be calculated by
π§ , =
β« ( )
β« ( )
, i=1, 2, 3, 4. (10)
We assume that the spring mass system for S4 is over-damped, which means the inertia term in Eq. (9)
can be neglected.
Another external potential in the third term of Eqs. (4-7) is the energy barrier π , which is nonzero
only in zone 2 representing mainly the solvation energy barrier from hydrophobicity. Generally, the
accurate shape of π in zone 2 requires a calculation of potential mean force (PMF) using molecular
dynamics simulations by the structure of protein and the possible change in the PMF with membrane
potential, ionic conditions, consistent with the distribution of charge and boundary conditions, and so on.
However, here we simply assume a hump shape for the PMF, while we await proper calibrated calculations
of the PMF and its dependence on membrane potential, etc.
π , tanh 5( π§ β πΏ ) β π‘ππβ 5( π§ β πΏ β πΏ ) β 1 , π§ β zone 2,
0, π§ β zones 1 and 3.
(11)
with π , set to be 5 in the current situation [42]. If we set π , too large, the gating current would
be very small since it would be very difficult for arginines to move across this barrier. Note that the tanh
function is designated to smooth the otherwise top-hat-shape barrier profile, with its awkward infinite
slopes not good for (numerical) differentiation. It is also generally believed that energy barrier in a
protein structure does not have a jump. The last term in Eqs. (4-7) is the steric term that accounts for
steric interaction among arginines [20]. Here we set g=0.5. Larger g implies larger steric effect, but g
cannot be arbitrarily large due to the limitation of stability.
Governing equations Eqs. (1-7) were derived by energy variational method based on the following
energy (in dimensional form):
πΈ = π π π ππππ β
π π
2
|βπ| + π§ π π π + ( π + π ) π
+
π
2
π π
,
ππ,
where the first term is entropy; second and third terms are electrostatic energy; fourth term is constraint
and barrier potential for arginines; last term is the steric energy term based on Lennard-Jones potential
[12, 20]. The Poisson equation Eq. (1) is derived by
πΏπΈ
πΏπ
= 0,
7. 7
and species flux densities in Eqs. (3-7) are derived by
π =
πΏπΈ
πΏπ
, π½ = β
π·
π π
π βπ ,
where π is the chemical potential of species i.
Also, here we assume quasi-steady state for Na+ and Cl-, which means = 0, π = Na, Cl. This is
justified by the fact that the diffusion coefficients of Na+ and Clο in vestibules are much larger than the
diffusion coefficient of arginine based on the very narrow time span of the leading spike of gating current
measured in experiments. The spike comes from the linear capacitive current of vestibule when the
command potential suddenly rises or drops. This quasi-steady state assumption is essential. Otherwise
using realistic diffusion coefficients for Na+ and Cl- would render Eqs. (1-7) too stiff to integrate in time.
The spike contaminating the gating current can be removed by a simple technique called P/n leak
subtraction in experiments (see Section 5.4). How to do this computationally will be discussed in section
5.3. Boundary and interface conditions for electric potential π are
π(0) = π, π(πΏ ) = π(πΏ ), Ξ(πΏ )π΄(πΏ )
ππ
ππ§
(πΏ ) = Ξ(πΏ )π΄(πΏ )
ππ
ππ§
(πΏ ),
π( πΏ + πΏ ) = π( πΏ + πΏ ), Ξ( πΏ + πΏ ) π΄( πΏ + πΏ ) ( πΏ + πΏ ) = Ξ( πΏ + πΏ ) π΄( πΏ + πΏ ) ( πΏ +
πΏ ), π(2πΏ + πΏ) = 0. (12)
These are Dirichlet boundary conditions at both ends and continuity of electric potential and
displacement at the interfaces between zones. Boundary and interface conditions for arginine are
π½ (0, π‘) = π½ (2πΏ + πΏ, π‘) = 0, π (πΏ , π‘) = π (πΏ , π‘), π΄(πΏ )π½ (πΏ , π‘) = π΄(πΏ )π½ (πΏ , π‘),
π (πΏ + πΏ , π‘) = π (πΏ + πΏ , π‘), π΄(πΏ + πΏ )π½ (πΏ + πΏ , π‘) = π΄(πΏ + πΏ )π½ (πΏ + πΏ , π‘), π = 1,2,3,4. (13)
No-flux boundary conditions are placed at both ends of the gating pore to prevent arginines and S4 from
entering intracellular/extracellular domains. The others are continuity of concentration and flux at
interfaces between zones. Boundary conditions for Na+ and Clο are
π (0, π‘) = π (0, π‘) = π (2πΏ + πΏ, π‘) = π (2πΏ + πΏ, π‘) = 1,
π½ (πΏ , π‘) = π½ (πΏ , π‘) = π½ (πΏ + πΏ, π‘) = π½ (πΏ + πΏ, π‘) = 0. (14)
Dirichlet boundary conditions are placed at both ends of the gating pore to describe the concentrations
for Na+ and Clο are bulk concentration over there and no-flux boundary conditions at both ends of
hydrophobic plug to describe the impermeability of Na+ and Clο into hydrophobic plug.
Besides the main input parameter π, which is the voltage bias (command potential) applied, other
parameters like Di, i=1, 2, 3, 4, K, KS4, bS4 are also required. Results are especially sensitive to the values of
K, KS4, bS4. We have tried and found Di=50, i=1,2,3,4, K=3, KS4=3, bS4=1.5 fit best with experiment [7]. We
compare computational outputs with important experiments: (1) gating current versus voltage curve (IV)
and (2) gating charge versus voltage curve (QV) as well as (3) gating current vs. time curves.
Usually the electric current in the ion channel is treated simply as the flux of charge and is uniform
in z when steady. This is not so in the present non-steady dynamic situation, since storing and releasing
of charge in vestibules and the channel are involved. Here the flux of charge at the middle of hydrophobic
plug, z= LR+L/2, was computed to estimate the experimentally observed gating current. However, it is
actually impossible to experimentally measure the current within the channel since it is impossible to put
8. 8
a probe in the middle of the hydrophobic plug and optical methods to record these variables are so far
unavailable. In experiments, the voltage clamp technique is used, and on/off gating current through the
membrane is measured in experiments, which should be equal to the flux of charge at z=0 in the present
frame work as shown in Fig. 2. The flux of charges at any z position πΌ(π§, π‘) can be related to the flux of
charges at z=0, πΌ(0, π‘), simply by charge conservation:
π (π§, π‘) = πΌ(0, π‘) β πΌ( π§, π‘), (15)
where
π ( π§, π‘) = β« π΄( π) β π§ π ππ,all (16)
and flux of charges at any z position πΌ(π§, π‘) is defined by
πΌ(π§, π‘) = π΄(π§) β π§ π½ (π§, π‘)all . (17)
We may as well identify π (π§, π‘) as the displacement current, and denote it as πΌ (π§, π‘), since it will
be shown later that Eq. (15) is equivalent to Ampereβs law in Maxwellβs equations, and π (π§, π‘) is
exactly the displacement current in Ampereβs law. A general discussion can be found in [10]. Hence, Eq.
(15) can be simply re-written as
πΌ ( π§, π‘) = πΌ( π§, π‘) + πΌ (π§, π‘) = πΌ(0, π‘), (18)
where we define the sum of displacement current and flux of charges as the total current πΌ (π§, π‘). The
z-distribution of the total current should be uniform by Kirchhoffβs law, which will be computationally
verified in section 5.3. We are also interested in observing the net charge at vestibules. Taking net charge
at intracellular vestibule, π (πΏ , π‘), as an example, the net charge consists of arginine charges and their
counter charges formed by the EDL of ionic solution over there. Generally, electro-neutrality is
approximate but will not be exact there. Note that the existence of EDLβs at vestibules, acting as a pair of
capacitors, is an unavoidable consequence of discontinuities in constitutive properties like permanent
charge density and dielectric coefficients. Realistic atomic scale simulations will have EDL and capacitive
currents. More about flux of charge, displacement current and net charge at vestibules will be discussed
in section 5.3.
Eq. (15) is consistent with Ampereβs law in Maxwellβs equations:
βΓ
β
= π π
β
+ π½β, (19)
or equivalently,
β β π π
β
+ π½β = 0, (20)
where πΈβ is the electric field and π½β is flux density of charge (current density). Eq. (20) tells us that the
current is conserved everywhere and it consists of flux of charges π½β and displacement current π π
β
.
Eq. (20) can be derived by Poisson equation and species transport equation like Eq. (1) and Eq. (2).
Starting from Poisson equation in dimensional form:
ββ β (π π βπ) = π + β π§ ππ , (21)
9. 9
or equivalently
β β π π πΈβ = π + β π§ ππ . (22)
Taking time derivative of Eq. (22),
β β π π
β
= β π§ π , (23)
and using species transport equation based on mass conservation,
+ β β π½β = 0, (24)
then
β β π π
β
= β π§ π = β β β β π§ ππ½β = ββ β π½β, (25)
which is exactly Eq. (20) by defining
π½β=β π§ ππ½β . (26)
Casting Eq. (20) into the present 1D framework and integrate in space from 0 to z, we have
π π π΄( π§)
( , )
+ πΌ( π§, π‘) = π π π΄(0)
( , )
+ πΌ(0, π‘). (27)
Comparing with Eq. (18),
π π π΄( π§)
( , )
β π π π΄(0)
( , )
= πΌ (π§, π‘), (28)
which justifies the naming of displacement current in Eq. (18).
To construct the QV curve, we calculate π = β« π΄(π§) β π ππ§, π = β« π΄(π§) β π ππ§, π =
β« π΄(π§) β π ππ§, which are the amounts of arginine found in zone 1, 2 and 3, respectively. Usually
π β 0 due to the energy barrier π in zone 2. Arginines tend to jump across zone 2 when driven from
zone 1 to zone 3 when the voltage V is turned on. The number of arginines that move and settles at zone
3 depends on the magnitude of π. Besides IV and QV curves, the time course of the movement of the
arginines and of gating charge (and π§ , (π‘) and π (π‘)), are important to report here. The movement
of arginines can (almost) be recorded in experiments nowadays by optical methods. Many qualitative
models accounting for the movement of S4 and conformation change of the voltage sensor have been
proposed. Readers are referred to the review paper [37] for more details.
4.Numericalmethod
High-order multi-block Chebyshev pseudospectral methods are used here to discretize Eqs. (1-2)
in space [38]. The resultant semi-discrete system is then a set of coupled ordinary differential equations
in time and algebraic equations (an ODAE system) [2]. The ordinary differential equations are chiefly
from Eq. (2), and algebraic equations are chiefly from Eq. (1) and boundary/interface conditions Eq. (12-
14). This system is further integrated in time by an ODAE solver (ODE15S in MATLAB [31-32]) together
with appropriate initial condition. ODE15S is a variable-order-variable-step (VSVO) solver, which is
highly efficient in time integration because it adjusts the time step and order of integration. High-order
10. 10
pseudospectral methods generally provide excellent spatial accuracy with economically practicable
resolutions. A combination of these two techniques makes the whole computation very efficient. This is
particularly important, since numerous computations have to be tried during the tuning of parameters.
5.Resultsanddiscussions
Here we explored several parameter values to obtain charge movement with kinetics and steady
state properties similar to the experimentally recorded gating currents. Numerical results based on the
mathematical model described above were calculated and compared with experimental measurements
[7]. Our one dimensional continuum model has advantages and disadvantages. The lack of three-
dimensional structural detail means of course that some details of the gating current and charge cannot
be reproduced. It should be noted however that to reproduce those, one needs more than just static
structural detail. One must also know how the structures (particularly their permanent and polarization
charge) change after a command potential is applied, in the ionic conditions of importance. Structural
detail that does not depend on time can be expected to give an incomplete, probably misleading image of
mechanism (imagining studying an internal combustion engine without seeing the pistons or valves
moving). The 1D model has the important advantages that it computes the actual experimental results on
the actual experimental time scale, in realistic ionic solutions and with far field boundary conditions
actually used in voltage clamp experiments. It also conserves current as we will discuss later
5.1 QV curve
When the membrane and voltage sensor is held at a large inside negative potential (e.g.,
hyperpolarized to inside negative say -90mV), S4 is in a down position and all arginines stay in the
intracellular vestibule. When the potential is made more positive (e.g., depolarized to +10mV, inside
positive), S4 is in the up position and all arginines are at the extracellular vestibule.
The voltage dependence of the charge (arginines) transferred from intracellular vestibule to
extracellular vestibule is characterized as a QV curve in experimental papers and it is sigmoidal in shape
[7]. Fig. 3(a) shows that our computed QV curveβ the dependence of π on Vβis in very good
agreement with the experiment [7]. Fig. 3(b) shows the steady-state distributions of Na+, Clο and arginines
in the inside negative, hyperpolarized situation (V=-90mV). As we can see, all arginines stay at
intracellular vestibule, and none of arginines transferred to extracellular vestibule (π β 0).
Fig. 3(c) shows the situation at V=-48mV, which is the midpoint of QV curve. As we can see, each
vestibule has half of the arginines staying in it (π = 2): R1 and R2 are expected in extracellular vestibule,
and R3 and R4 in intracellular vestibule. The concentration distributions of arginines shown in Fig. 3(c),
interpreted as individual probability density functions, show R1 and R2 residing more in extracellular
vestibule, and R3 and R4 on the contrary more in intracellular vestibule. Note that there are almost no
arginines in zone 2 (hydrophobic plug) due to the energy barrier in it. If the command potential at the
midpoint (π = 2) is V=0mV, the unforced natural position of S4 is at the middle of gating pore, i.e.,
π , = πΏ + 0.5πΏ. In experimental measurement [7], V is actually -48mV instead of 0mV for the midpoint.
This requires π , to be biased from πΏ + 0.5πΏ to π , = πΏ + 0.5πΏ + 1.591nm. Fig. 3(d) shows the
situation at full depolarization (V=-8mV). As we can see, all arginines move to extracellular vestibule
11. 11
(π β 4).
5.2 Gating current
Fig. 4 shows the time course of gating currents, observed as flux of charge at the middle of gating
pore πΌ(πΏ + πΏ/2, π‘), due to the movement of arginines when the membrane is largely depolarized, and
partially depolarized. In the case of large depolarization, V rises from -90mV to -8mV at t=10, holds on
till t=150, and drops back to -90mV as shown in Fig. 4(a). Time course of gating current and contributions
of individual arginines are shown in Fig. 4(b). We observe that the rising order of each current component
follows the moving order of R1, R2, R3 and R4 when depolarized, and that order is reversed when
repolarized later. The area under the gating current is the amount of charge moved. Since arginines move
forward and backward in this depolarization/repolarization scenario, the areas under the ON current
(arginines moving forward) and the OFF current (arginines moving backward) are same. The areas are
equal for each component of current as well. The equality of area is an important signature of gating
current that contrasts markedly with the properties of ionic current. Equality of area has in fact been used
as a signature that identifies gating current and separates it from ionic current [26, 5]. Since almost all
arginines move from intracellular to extracellular vestibule when the command potential is suddenly
made more positive in a large depolarization, the area under each current component is then very close
to 1. In the case of partial depolarization, V rises from -90mV to -50mV at t=10, holds on till t=150, and
drops back to -90mV as shown in Fig. 4(c). The time course of gating current and its four components
contributed by each arginine for this situation is shown in Fig. 4(d). Under this partial polarization, not
all arginines move past the middle of hydrophobic plug due to weaker driving force in partial
polarizations compared with large depolarization case. This can be seen in Fig. 4(d), where areas under
each component current are different, and based on these various areas R1 moves more towards
extracellular vestibule than R2; R2 more so than R3; and etc.
The gating currents shown above can be better understood by looking at a sequence of snapshots
showing the spatial distribution of electric potential, species concentration and electric current. The
distributions at several times are shown in Fig. 5 for the case of sudden change in command voltage to a
more positive value, a large depolarization, and Fig. 6 for the case of small positive going change in
potential, a partial depolarization. Fig. 5 shows that almost all arginines are driven from intracellular to
extracellular vestibules in case of large depolarization, but only part of the arginines move in case of
partial depolarization (Fig. 6). The electric potential profiles at t=13 and t=148 show that the profile of
electric potential as arginines move from left to right even though the voltage is maintained constant
across the sensor. This is not a constant field system at all! Note that slight bulges in electric potential
profile exist wherever arginines are dense. This can be easily understood by understanding the effect of
Eq. (1) on a concave spatial distribution of electric potential.
In Figs. 5 and 6, total current defined in Eq. (18), though changing with time, is always constant in
z at all times, which is no surprise because it must satsify Kirchhoffβs law. At t=13, a time that gating
current is not quiescent as shown in Fig. 4(b) and 4(d), we can particularly visualize the z-distributions
of flux of charges πΌ( π§, π‘), displacement of current πΌ ( π§, π‘) and total current πΌ ( π§, π‘) individually in
Figs. 5 and 6.
12. 12
5.3 Flux of charges at different locations
Flux of charges πΌ( π§, π‘), together with displacement of current πΌ ( π§, π‘) and total current
πΌ (π§, π‘), depicted in Figs. 5 and 6 deserves more discussions here. Flux of charges at the middle of gating
pore, πΌ(πΏ + πΏ/2, π‘), and both ends of gating pore, πΌ(0, π‘) and πΌ(2πΏ + πΏ, π‘), should be computed by
πΌ πΏ + , π‘ = π΄ πΏ + β π§ π½ πΏ + , π‘ , (29)
πΌ(0, π‘) = π΄(0) β π§ π½ (0, π‘), , (30)
πΌ(2πΏ + πΏ, π‘) = π΄(2πΏ + πΏ) β π§ π½ (2πΏ + πΏ, π‘), . (31)
Except πΌ πΏ + , π‘ , πΌ(0, π‘) and πΌ(2πΏ + πΏ, π‘) are trivially zero due to the implement of quasi-
steadiness = 0, π = Na, Cl, in vestibules, which causes π½ and π½ to be uniform in vestibules by Eq.
(2), and further become zero by the no-flux boundary conditions for ππ and πΆπ at the bottom of
vestibules as described in Eq. (14). We have to alternatively reconstruct πΌ(0, π‘) and πΌ(2πΏ + πΏ, π‘) by
charge conservation of ππ and πΆπ ,
πΌ(0, π‘) = β« π΄(π§) β π§ π ππ§Na,Cl , (32)
πΌ(2πΏ + πΏ, π‘) = β β« π΄(π§) β π§ π ππ§Na,Cl . (33)
After obtaining πΌ(0, π‘) and πΌ(2πΏ + πΏ, π‘), we can further reconstruct flux of charges πΌ(π§, π‘) at zone 1
(intracellular vestibule) and zone 3 (extracellular vestibule) by charge conservation of ππ and πΆπ
again,
πΌ( π§, π‘) = πΌ(0, π‘) β β« π΄( π§) β π§ π ππ§Na,Cl , π§ β [0, πΏ ], (34)
πΌ( π§, π‘) = πΌ(2πΏ + πΏ, π‘) + β« π΄( π§) β π§ π ππ§Na,Cl , π§ β [πΏ + πΏ, 2πΏ + πΏ]. (35)
Once flux of charges is analyzed, we can then compute the displacement current based on finding the time
derivative of Eq. (16). Finally we sum up flux of charges πΌ( π§, π‘) and displacement of current πΌ ( π§, π‘)
to obtain total current πΌ (π§, π‘) and verify that it satisfies Eq. (18) and is uniform in z. This verification
is shown in Figs. 5 and 6 at several various times, and is fact true at any time as well.
In the bottom rows of Figs. 5 and 6 at t=13, we observe that πΌ(π§, π‘) is generally non-uniform in z
and is accompanied by congestion/decongestion of arginines in between. However πΌ(π§, π‘) is almost
uniform at zone 2 (hydrophobic plug), which means almost no congestion/decongestion of arginines
occurs there, and therefore no contribution to the displacement current π (π§, π‘) from zone 2. This
is no surprise since arginines can hardly reside at zone 2 due to energy barrier in it. Because of the energy
barriers, once arginines leave zone 1 (intracellular vestibule), they immediately jump across zone 2 and
enter into zone 3 (extracellular vestibule).
Several things are worth noting in the time courses of πΌ πΏ + , π‘ , and πΌ(0, π‘) illustrated in Fig.
13. 13
7(a) under the case of large depolarization. First, πΌ πΏ + , π‘ is noticeably larger than πΌ(0, π‘) in ON
period. This is because their difference, exactly the displacement current πΌ , is always negative at zone
2 when depolarized, since arginines are leaving zone 1 and make π < 0 for zone 2. It is expected
the area under the time course of πΌ πΏ + , π‘ would be very close to 4e, as verified by the time courses
of Q3 in Fig. 7(b), while the counterpart area of experimentally measurable πΌ(0, π‘) would be less than 4e
due to its smaller magnitude compared with πΌ πΏ + , π‘ . This may account for some experimental
observations that at most 13e [4, 27, 30], instead of 16e, are moved during full depolarization in 4 voltage
sensors (for a single ion channel) based on calculating the area under voltage-clamp gating current.
Therefore, flux of charge at any location of zone 2, though impossible to measure in experiments so far,
will give us amount of arginines moved during depolarization more reliably than the measurable πΌ(0, π‘).
Second, we see in Fig. 7(a) with magnification in its inset plot that, as in experiments, πΌ(0, π‘), but
not πΌ πΏ + , π‘ , has contaminating leading spikes in ON and OFF parts of the current. These spikes are
linear capacitive currents from solution EDL of vestibules caused by sudden rising and dropping of
command potential, and are called the early transient gating current in experiments [33, 35-36]. In
voltage-clamp experiments, subtracting this linear capacitive component and removing the spike from
gating current is done by βleak subtractionβ, in various forms, e.g., P/4 (see Section 5.4) In reality, this
linear capacitive current that is subtracted in this procedure comes from both the lipid bilayer membrane
in parallel with the channel and vestibule solution in series with the channel. Here, we only considered
capacitive current from solution EDL of vestibule and ignored the capacitive current of the membrane in
parallel with the channel (which is actually much larger than vestibule capacitive current) because we
use Dirichlet boundary conditions for π at both ends of gating pore in Eq. (12). Following the idea of
experiment [7], we calculated πΌ(0, π‘) with V rising from -150mV to -140mV at t=10, holding on till t=150,
and dropping back to -150mV. The reason to choose the rising range of V from -150mV to -140mV is that
essentially none of the arginines move in this hyperpolarized situation. The voltage step only quickly
charges and discharges solution EDL in vestibules, and the computational time course of πΌ(0, π‘) is just
two spikes when the command potential rises suddenly and drops suddenly later. Subtracting this
hyperpolarized πΌ(0, π‘), multiplied by a proportion factor, from its original counterpart will then remove
the spikes, and the unspiked πΌ(0, π‘) is shown in Fig. 7(a).
Third, in Fig. 7(b), the time courses of Q1 and Q3 show that arginines move from the intracellular
vestibule to the extracellular one when depolarized and move back to intracellular vestibule again when
repolarized later. As arginines move from one vestibule to another, the concentrations of Na+ and Clο also
correspondingly change with time at the vestibules, form counter charges through EDL, and balance
arginine charges at vestibules. However, these movement only maintain approximate, not exact,
electroneutrality as shown in Fig. 7(b). The violation of electroneutrality is produced by the displacement
current, that is not negligible we note.
14. 14
As mentioned above, here we used flux of charges at the middle of hydrophobic plug, πΌ πΏ + , π‘ ,
instead of experimentally measurable πΌ(0, π‘) to represent βthe gating currentβ in discussions. This βgating
currentβ just leaves out the displacement current πΌ πΏ + , π‘ . We use this definition of βgating
currentβ for several reasons: (1) the area under time course of πΌ πΏ + , π‘ gives us the amount of
arginines moved during depolarization more faithfully than πΌ(0, π‘) as explained above. The fluxes of
charge for each arginine shown in Fig. 4 (b) and (d) carry important information about how each arginine
is moved by the electric field, that will be illustrated in Fig. 8. All these will not be easy to display and
comprehend if we use πΌ(0, π‘) instead. (2) Using πΌ(0, π‘) as a definition of gating current would require a
decontamination by removing the leading spikes in it, which is computationally costly by the procedure
described above. Especially, it would pose a heavy numerical burden when doing parameter fitting, where
numerous repeated computations need to be conducted; (3) The shape of πΌ πΏ + , π‘ is actually close
to the unspiked πΌ(0, π‘). Hence here we used the computationally handy πΌ πΏ + , π‘ to replace πΌ(0, π‘)
as the gating current.
5.4 Comparison with experimental records.
Our computations have limited fidelity at short times because of time step limitation in integrating
stiff systems. The spike artifacts are one example, described previously. Experimental measurements [16,
33] of the fast transient gating current are fascinating but our calculations must be extended to deal with
them.
A more general consideration is the subtraction procedure used in experiments to isolate gating
current from currents arising from other sources. Channels and their voltage sensors are always
accompanied by large amounts of lipid membranes. Currents through channels and in voltage sensors are
βin parallelβ with large capacitive currents through lipid membranes. The lipid membrane of cells
introduces a large capacitance ( πΆ β 6Γ10 farads/cm ) that has nothing to do with the capacitive
(i.e., displacement) currents produced by charge movement in the voltage sensor. Fortunately, this
capacitance πΆ is a nearly ideal circuit element. Current through the membrane capacitance is entirely
a displacement current accurately described by π = πΆ with a single constant πΆ . V is the
voltage across the lipid capacitor. Note that π does not include any current carried by the translocation
of ions across the lipid.
In experimental measurements, this displacement current π is always present, in large amounts,
because voltage sensors (and channels) are always embedded in lipid membranes. Experimental
measurements mix the displacement current of the lipid membrane and the displacement current of the
voltage sensor. Indeed, the lipid membrane current dominates the measurement of displacement current
in native preparations and remains large in systems mutated to have unnaturally large numbers of voltage
sensors.
15. 15
A procedure to remove the lipid membrane current is needed if the gating current of the voltage
sensor is to be measured. The procedure introduced by Schneider and Chandler [26] has been used ever
since in the improved P/4 version developed by Armstrong and Bezanilla [1] reviewed and discussed in
[8]. Also, see another approach in [6, 13]. Schneider and Chandlerβs procedure [26] estimates the so-
called linear current π = πΆ in conditions in which the voltage sensor and πΆ behaves as an ideal
circuit element. An ideal capacitor has a capacitance πΆ independent of voltage, time, current, or ionic
composition. The Schneider procedure then subtracts that linear current π from the total current
measured in conditions in which the voltage sensor does not behave as an ideal capacitor. The leftover
estimates the nonlinear properties of the charge movement in the voltage sensor. That is to say it
estimates the charge movement of the voltage sensor that is not proportional to the size of the voltage
step used in the measurement. The leftover is called gating current here and in experimental papers.
The gating current reported in experiments however is missing a component of the displacement
current of the voltage sensor, because of the Schneider and Chandlerβs procedure that estimates π . The
Schneider and Chandlerβs procedure does not measure π by itself. The linear current π estimated
by the Schneider and Chandlerβs procedure has more in it than the current through the lipid membrane
capacitor π . Rather, it contains the lipid membrane current π plus current through any structures
in the membrane (βin parallelβ) in which current follows the law π = πΆ .
Clearly, some of the current produced by movements of the arginines in the voltage sensor will be a
linear displacement current, a linear component of gating current. But that component is likely to be
included in π and so would not be present in the reported gating current. The linear component of
gating current is removed by the experimental procedure that is needed to remove the lipid membrane
displacement current π . Other systems may contribute to the linear displacement current as well, for
example, (1) all sorts of experimental and instrumentation artifacts and (2) displacement current in the
conduction channel itself. The conduction channel of field effect transistors produces a large capacitance
described by drift diffusion equations quite similar to the PNP equations of the open conduction channel.
Our procedure in the computations reported here subtracts a hyperpolarized current with arginines
not moving in this situation, thereby removing all the currents carried by arginine movement, linear and
nonlinear in voltage. Most systems have substantial motions that are linear in voltage (even if the system
is labelled βnonlinearβ). The response of most systems to a voltage step can be written as a sum of terms,
each with a different dependence on the size of the applied step of voltage V. The first term in that sum
changes linearly if the voltage step is changed. Higher order terms exhibit a nonlinear dependence on V.
The linear term is present in most systems, just as it is present in most Taylor expansions of nonlinear
functions.
The linear component missed in experiments, and removed in these calculations, may have
functional and structural significance. The voltage sensor works by sensing voltage, for example, by
producing a motion of arginines. That motionβthe response of the voltage sensor in this modelβ
includes a linear component. The signal passed to the conduction channel, to control gating, is likely to
include the linear component of sensor function, including the linear component of the motion of the
arginines. Confusion will result if the linear component is ignored when a model is created that links the
16. 16
voltage sensor to the gating process of the conduction channel.
Direct measurements of the movement of arginines (e.g., with optical methods) are likely to
include the linear component and so should not agree with experimental measurements of gating current
or with the currents reported here.
5.5 Time course of arginine and S4 translocation
Time course of Q (amount of arginines moved to extracellular vestibule, equal to Q3 here) and
center-of-mass trajectories of individual arginines (π§ , , i=1, 2, 3, 4) and S4 segment (π ) are shown in
Fig. 8 with (a) and (b) for the case of large depolarization and (c) and (d) for the case of partial
depolarization. In the case of large depolarization, from Fig. 8(b), we can see arginines and S4 are driven
towards extracellular vestibule once the electric field is turned on. Their z-positions quickly reach
individual steady states with almost all arginines transferred to extracellular vestibule as previously
shown in Fig. 5 and therefore Q is close to 4 as shown in Fig. 8(a). Arginines and S4 move back to
intracellular vestibule once the voltage drops back to -90mV. From Fig. 8(b), the forward moving order of
arginines is R1, R2, R3 and R4, and the backward moving order is on the opposite R4, R3, R2 and R1. This
obviously coincides with the structure. Note that S4 is initially farthest to the right but lags behind R1 and
R2 during movement in depolarization as shown in Fig. 8(b). This is certainly because S4 is relaxed to an
almost unforced situation close to its natural position π , . We can further calculate the displacements
of each arginine and S4 during this full depolarization, and find Ξπ§ , β Ξπ§ , β Ξπ§ , β 1.93nm,
Ξπ§ , =1.76nm, Ξπ =1.51nm. Besides almost the same displacements for R1, R2 and R3, their average
moving velocities are also very close to each other. This seems to suggest a synchronized movement
among R1, R2 and R3. Also, we can see the movements of arginines contribute significantly to the
movement of S4 segment. This can be seen by steady state z-position of S4 derived from Eq. (9),
π = β π§ , β π§ + π , = π , + β π§ , .
Experimental estimates of S4 displacement during full depolarization ranged from 2-20 A [24, 37].
This large deviation accommodates several models for voltage sensing: transporter model, helical screw
model and paddle model [37]. Our Ξπ =1.51nm here is large enough that seems to agree better with
experimental estimates requiring large displacements, such as the paddle model. However, our model is
more based on helical screw model, which is known to have shorter displacements. A plausible
explanation for our over-estimated Ξπ is that our 1D model has made the known helical path into a
linear path. It would be shorter for displacement perpendicular to the plane of the membrane. The
diameter of an alpha helix between alpha carbons is approximately 1 nm, therefore if the rotation were
90o, the displacement in 1D would be extended by 0.5Γ0.5ΓΟ=0.78 nm, making the linear translation
perpendicular to the membrane only 0.73 nm. This number is very close to the displacement
perpendicular to the membrane that is estimated when comparing the open-relaxed state crystal
structure of Kv1.2 [9] and the consensus closed structure that has been derived from experimental
measurements [40].
For partial depolarization, arginines and S4 are the same driven towards extracellular vestibule
once the electric field is turned on and return when voltage drops. However, since the driving force is
weaker than in a large depolarization, their z-positions do not reach steady states as they do during a full
depolarization before they return to intracellular vestibule once the voltage drops. This behavior can be
17. 17
seen from Fig. 8(c), that Q reaches 1.57 at most which should be 2 instead if equilibrium, which can be
reached if time is long enough, as shown in QV curve of Fig. 3(a). Fig. 8(d) shows that S4 initially farthest
to the right lags behind R1 during movement and is almost caught up by R2. The maximum displacements
of arginines and S4 calculated from Fig. 8(d) are Ξπ§ , = 1.36nm, Ξπ§ , = 0.966nm,
Ξπ§ , =0.459nm, Ξπ§ , =0.316nm, and Ξπ , =0.616nm.
5.6 Family of gating currents for a range of voltages
Fig. 9(a) shows the time courses of gating current for a range of voltages V, ranging from -62mV to
-8mV. We can see the area under gating current, for both on and off parts, increases with V since more
arginines are transferred to extracellular vestibule as V increases. However, this area will saturate with
further increasing V since all arginines would then be transferred to the extracellular vestibule. The
shapes of this family of gating currents agree well with experiment [7] in both magnitude and time span.
We can characterize the time span by fitting the time course (only the decay part) of a gating current by
ππ β
+ ππ β
, π < π , as generally conducted in experiment [7], where π is the fast time constant
and π is the slow time constant. Usually the movement of arginines is dominated by π . Here π was
calculated from simulation and compared with experiment [7] as shown in Fig. 9(b). Since in our
computation the time is in arbitrary unit, we have scaled the time to have the maximum π to fit with its
counterpart in experiment [7]. The trend of π versus V in our result agrees well with experiment [7].
For the left branches to the maximum point in Fig. 9(b), simulation result fits very well with experiment.
For the right branches to the maximum point, simulation result overestimates π compared with
experiment. This is consistent with the observation that the amount of transferred charges Q saturates
faster in experiment data than in present simulation as V increases as shown in the QV curve of Fig. 3(a).
This phenomenon is related to cooperativity of movement among arginines, that will be further discussed
below.
5.7 Effect of voltage pulse duration
Fig. 10 shows the effect of voltage pulse duration with Fig. 10(a) for the case of partial
depolarization and Fig. 10(b) for the case of full depolarization. Magnitude and time span of gating
current are influenced by pulse duration in both cases, but the shape will asymptotically approach the
same as pulse duration increases. This behavior occurs because driving arginines towards extracellular
vestibule by the command potential takes time. If the pulse duration is long enough, the time course of Q
will approach its steady state like in Fig. 8(a) for large depolarization. Partial depolarization takes longer
time to reach its steady state as demonstrated in Fig. 8(c). Again these shapes of gating currents in Fig. 10
compare favorably with experiment [7].
6. Conclusions
The present 1D mechanical model of the voltage sensor tries to capture the essential structural
details that are necessary to reproduce the basic features of experimentally recorded gating currents.
After finding appropriate parameters, we find that the general kinetic and steady-state properties are
18. 18
well represented by the simulations. The continuum approach seems to be a good model of voltage
sensors, provided that it takes into account all interactions, and satisfies conservation of current. The
continuum approach used here describes the mechanical behavior of a single voltage sensor, but
experimental measurements are ensemble averages of multiple ion channels (and hence multiple voltage
sensors). There is no conflict, however. Experiments average of measurements of large numbers of
voltage sensors as does our macroscopic model.
Here we simplify the profile of energy barrier in hydrophobic plug since the potential of mean
force (PMF) in that region, and its variation with potential and conditions, is unknown. Therefore the next
step is to model the details of interactions or the moving arginines with the wall of the hydrophobic plug.
There is plenty of detailed information on the amino acid side chains in the plug and how each one of
them has important effects in the kinetics and steady-state properties of gating charge movement [25].
The studied side chains reveal steric as well as dielectric interactions with the arginines that the present
model does not have. On the other hand, the power of the present mathematical modeling is precisely the
implementation of interactions, therefore we believe that when we add the dielectric details of the
channel a better prediction of the currents should be attained. Also, here the plug energy barrier Vb is
assumed to be independent of time. However, once the first arginine enters the hydrophobic plug by
carrying some water with it, this partial wetting of pore will lower Vb, chiefly consisting of solvation
energy, and enable the next arginine to enter the plug with less difficulty. This might explain the
cooperativity of movement among arginines when they jump through the plug. This remains to be
included in the future modeling. Besides, it has been reported that a very strong electric field might affect
the hydration equilibrium of the hydrophobic plug and would lower its hydration energy barrier as well
[39]. This cooperativity of movement may help explain the quick saturation in the upper right branch of
QV curve (and smaller Ο2). This also remains to be considered in the future modeling.
Also, further work must address the mechanism of coupling between the voltage sensor
movements and the conduction pore. It seems likely that the classical mechanical models will need to be
extended to include coupling through the electrical field. It is possible that the voltage sensor modifies
the stability of conduction current by triggering sudden transitions from closed to open state, in a
controlled process reminiscent of Coulomb blockade.
ACKNOWLEDGMENTS
This research was sponsored in part by the Ministry of Science and Technology of Taiwan under grant no.
MOST-102-2115-M-002-015-MY4 (T.L.H.) and MOST-104-2115-M-035 -002 -MY2 (T.L.H.). Dr. Horng also
thanks the support of National Center for Theoretical Sciences Mathematical Division of Taiwan
(NCTS/MATH), and Dr. Ren-Shiang Chen for the long-term benefiting discussions.
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Figure 1. Geometric configuration of the model including the attachments of arginines to the S4 segment.
Figure 2. Following Figure 1, an axisymmetric 3-zone domain shape is designated in r-z coordinate for
the current 1D model. Here the diameter of hydrophobic plug is 0.3nm (arginineβs diameter); L=0.7nm;
LR=1.5nm; radius of vestibule is R=1nm. BC means Boundary Condition
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Figure 3. (a) QV curve and comparison with [7]. Steady-state distributions for Na, Cl and arginines at (b)
V=-90mV, (c) V=-48mV, (d) V=-8mV.
Figure 4. (a) Time course of V rising from -90mV to -8mV at t=10, holds on till t=150, and drops back to -
90mV, (b) time course of gating current and its components corresponding to (a), (c) time course of V
rising from -90mV to -50mV at t=10, holds on till t=150, and drops back to -90mV, (d) time course of
gating current and its components corresponding to (c).
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Figure 5. The four panels on the top row are species concentration distributions at t=0, 13, 148, for the
case of large depolarization with V rising from -90mV to -8mV at t=10, holding on till t=150, and dropping
back to -90mV. The four panels on the middle row are concurrent electric potential profiles. The four
panels on the bottom row are concurrent electric current profiles with components of flux of charge,
displacement current and total current.
24. 24
Figure 6. The four panels on the top row are species concentration distributions at t=0, 13, 148, for the
case of large depolarization with V rising from -90mV to -50mV at t=10, holding on till t=150, and
dropping back to -90mV. The four panels on the middle row are concurrent electric potential profiles. The
four panels on the bottom row are concurrent electric current profiles with components of flux of charge,
displacement current and total current.
Figure 7. (a) Time courses of πΌ πΏ + , π‘ , πΌ(0, π‘) and unspiked πΌ(0, π‘) for the case of full
25. 25
depolarization with V rising from -90mV to -8mV at t=10, holding on till t=150, and dropping back to -
90mV. The inset plot is a blow-up of ON-current to visualize the difference of πΌ(0, π‘) and unspiked πΌ(0, π‘)
more clearly. (b) Time courses of π , π , β« (π β π ) ππ§, and β« (π β π ) ππ§ under the same
depolarization scenario as (a).
Figure 8. (a) and (c) are time courses of amount of arginines moved to extracellular vestibule. (b) and
(d) are center-of-mass trajectories of individual arginines and S4. (a) and (b) are the case of large
depolarization with V rising from -90mV to -8mV at t=10, holding on till t=150, and dropping back to -
90mV. (c) and (d) are the case of partial depolarization with V rising from -90mV to -50mV at t=10,
holding on till t=150, and dropping back to -90mV.
Figure 9. (a) Time courses of gating current with voltage rising from -90mV to VmV at t=10, holds on till
t=150, and drops back to -90mV, where V=-62, -50, β¦ -8 mV. (b) π versus V compared with experiment
26. 26
[7].
Figure 10. Effect of voltage pulse duration: (a) V increases from -90mV to -35mV at t=10 and drops back
to -90mV at various times, (b) V increases from -90mV to 0mV at t=10 and drops back to -90mV at
various times.