coulomb blockade oscillations and amfe in calcium/sodium biological ion ... · coulomb blockade...

Coulomb Blockade Oscillations and AMFEin Calcium/Sodium Biological Ion Channels

Igor Kh. Kaufman1, Dmitry G. Luchinsky1,2, William A.T. Gibby1,Peter. V.E. McClintock1, Robert. S. Eisenberg3

1Department of Physics, Lancaster University, UK2SGT Inc., Ames Research Center, Moffett Field, CA, USA

3Molecular Biophysics, Rush University, Chicago, USA

ESGCO-2016

Kaufman, Luchinsky, Gibby, McClintock, EisenbergCoulomb Blockade in Ion Channels ESGCO-2016 1 / 23

Outline1 Background

Ion channels: overviewSelectivityResearch goals

2 Multi-ion conduction bandsGeneric model of Ca channelMulti-ion conduction bands

3 Ionic Coulomb blockadeIonic Coulomb blockadeHydration and concentration bands shiftsLine shapes

4 SelectivityResonance selectivityAMFEDivalent blockadeNaChBac: LESWAS, LEDWAS and LEEWAS

5 ConclusionsKaufman, Luchinsky, Gibby, McClintock, EisenbergCoulomb Blockade in Ion Channels ESGCO-2016 2 / 23

Ion channels

Biological ion channels arenatural nanopores in proteinhubs passing through thebilipid membranes of biologicalcells.Fast and selective transport ofphysiologically important ions(e.g. Na+, K+ and Ca2+).

Essential to physiology –bacteria to humans.Target for drugsHighly selective for particularions.More than four decades aftertheir discovery a great deal isnow known about ionchannels.Yet there remain many of theirbasic features are still notproperly understood.Physical origin of permeationand selectivity - is one of themain long-standing problems.


Calcium and sodium channels

Our study is focused on voltage-gated calcium and sodium ionchannels.

Voltage-gated Ca2+ and Na+ channels controlmuscle contraction, neurotransmittersecretion, and the transmission of actionpotentials.

They are opening on response to cellmembrane depolarization.

Sodium and calcium channels have verysimilar structures, but differing SF loci, andhence differing Qf .

The mammalian calcium channels supposedto have a EEEE locus (Qf =4e )

The mammalian sodium channel has a DEKAlocus (Qf = 1e)

Bacterial sodium channels ChNaBac, NavABposses a EEEE locus similar to calciumchannel.


Ionic selectivity

Selectivity between different ions is the main functional feature of ionchannels. Specialized ion channels allow selected species topermeate freely (close to the rate of free diffusion) while largelyprohibiting permeation by other ions.

Valence selectivity. Calcium channelfavours Ca2+ over Na+ by up to1000:1, even though they are thesame size.Alike charge selectivity, Potassiumchannel strongly disfavours Na+,even though Na+ is smaller K+.

Yet the physical origin of selectivity and its relation to channel structureremains a conundrum and it constitutes the central topic of our Project.


Mutation-induced transformation of selectivity

The charge Qf at the selectivity filter can be changed by mutatingthe genes coding for the protein side chains (i.e.DEKA→DEEA→DEEE)Mutations that influence Qf usually destroy the channel’sselectivity, and hence physiological functionality, leading to“channelopathies".A point mutation of the DEKA sodium channel can convert itinto a calcium-selective channel with a DEEA locus, and viceversa.: Heinemann, Nature, 1992.Generally, Calcium selectivity increases with growth of Qf .Similar transformations occur in bacterial sodium channels.Experiments on the mutants show that Qf is a crucial factor indetermining the Ca2+ vs. Na+ valence selectivity.These results are not yet properly explained.


Research goals

Despite of structure complexity of ion channels, there is strongexperimental evidence that a simple parameter - fixed charge Qfat the SF controls valence selectivity features. This fact should beconfirmed and explained.Recently introduced Ionic Coulomb blockade model has beenapplied to permeation and conductivity of biological ion channels.We are going to investigate ionic selectivity of bacterial sodiumchannels, wild types and mutant and explain them on the base ofCoulomb blockade model.We are going to verify CB model and develop multi-level(structural, electrostatic and stochastic) model of bacterialchannels.We are going to find direct experimental evidences ofCoulomb blockade model of permeation and selectivity ofcalcium/sodium ion channels.


Generic electrostatic model. Geometry

A water-filled, cylindrical hole,radius R = 3 Å and lengthL = 16 Å through the proteinhub in the cellular membrane.The selectivity filter (chargedresidues) represented by arigid ring of negative chargeQf = 0− 7e.Water and protein describedas continuous media.Dielectric constants εw = 80(water) and εp = 2 (protein)

Na+

Ca2+

Qf

Membrane protein

L

Left bath,

CL>0

Right bath,

CR=0

R X

Ra


Generic electrostatic model. Electrostatics

The electrostatic field is derived by self-consistent numericalsolution of Poisson’s equation:

−∇(εε0∇u) =∑

ezini + ρf

where ε0 and ε is the dielectric permittivity of vacuum and themedium (water or protein), u is the electric potential, e is theelementary charge, zi is the valence, ni is the number density ofions, and ρf is the density of fixed charge.This equation accounts for both ion-ion interaction and self-actionfor all ions in their current positions.Dedicated Finite-Volume Poisson Solver accounting for largepermittivity mismatch between water and protein.One result of the calculations is the axial potential energy profile =the self-consistent Potential of the Mean Force (PMF) used forBrownian Dynamics simulations.


Generic electrostatic model. Brownian dynamics

The BD simulations use numerical solution of the 1-Doverdamped, time-discretized, Langevin equation for the i-th ion:

dxi

dt= −Dizi

(∂u∂xi

)+√

2Diξ(t)

where xi stands for the ion’s position, Di is its diffusion coefficient,zi is the valence, u is the self-consistent potential in kBT/e units,and ξ(t) is normalized white noise.Numerical solution is implemented with the Euler forward scheme.We use an ion injection scheme that allows us to avoidsimulations in the bulk. The arrival rate jarr is connected to thebulk concentration C through the Smoluchowski diffusion rate:jarr = 2πDRC.


Calcium conduction and occupancy bands

0

1

J (a

.u.)

20mM40mM 80mM

0 1 2 3 4 5 6 70

1

2

3

Qf (e)

P

20mM40mM 80mM

0 1 2 3 4 5 60

50

0

1

Qf (e)[Ca] (mM)

J (a

.u.)

M2M1M0

Current J and occupation P as a functionof charge Qf at selectivity filter weresimulated for different Ca2+ concentrations[Ca].

Brownian Dynamics simulationrevealed an oscillatory patternof narrow conduction bandsand stop bands.Conduction bands occur attransitions in channeloccupancy P.Occupancy P represented aCoulomb staircase typical forCoulomb blockadeConduction bands correspondto single-ion (M0=1e) andmulti-ion (M1=3e, M2=5e)barrier-less conduction.


Ionic CB vs electronic CB in quantum dots

The results of simulations are analogous to Electronic Coulomb blockade oscillationsin e.g. quantum dots and may be interpreted as Ionic Coulomb blockade oscillations( Krems et al., J. Phys. Condens. Matter 25, 065101 (2013).)

Conduction bands in Ca-channel

0 1 2 3 4 5 60

10

20

|Qf|

U

0

2

P

0

1

J(ar

b.un

its)

20mM40mM80mM

Un

UG

EF

Z1

M2

M0

M2

M1

M0

n=2 n=3n=0 n=1

(c)

M1

Z2

Z3

(b)

(a)

CB oscillations in quantum dots

Ion trapped at SF.

Periodic conduction bands .

Classical mechanics of ion.

Stochastic motion.

Electron trapped in quantum dot.

Coulomb blockade oscillations.

Quantum mechanics for electron.

Quantum tunnelling for electron.


Ionic Coulomb blockade

Coulomb blockade appears in low-capacitance, discrete-statessystems , provided that the ground state {nG} is separated fromneighbouring {nG ± 1} states by large gap Us,z � kBT .We approximate Un as the dielectric self-energy Un,s of the excesscharge Qn:

Un,s =Q2

n2Cs

( Electrostatic energy )

Qn = zen + Qf ( Excess charge ).

We examine the minimum (i.e. ground state) energy

UG(Qf ) = minn

(Un(Qf ,n)) (Ground state energy)

for the ground state occupancy nG, and the excess charge QG, allas functions of Qf .


Hydration and concentration band shifts

The total potential energy Un for a {n}-channel can be expressedas:

Un = Un,s + Uh ( Potential energy sum )µ = µ0 + kBT ln Pb ( Chemical potential )

where Un is the electrostatic energy, Uh is dehydration energy, µ0is standard chemical potential (we assume µ0 = 0) and Pb is bulkoccupancy related to SF volume.Resonant value Mn of fixed charge Qf for {n} → {n + 1} transitionis defined from shift condition Un+1 − Un − µ = 0:

Mn = −ze(n + 1/2)−∆M∆M = Cs(Uh − kBT ln(Pb)

Dehydration and concentration terms lead to proper shifts ofelectrostatic conduction bands. These shifts define different kindof selectivity.


Stair-case transitions and resonant conductance

For Ca2+ ions Coulomb gap∆Un � kBT and SF statesform exclusive set:

m = {n,n + 1}; θn + θn+1 = 1;

For this set of states standardderivation via partition functionand GCE leads to Fermi-Diracstatistics for an excessoccupancy P∗

c = Pc mod 1:

P∗c = (1+P−1

b exp(Uc/kBT ))−1,

where Uc = Un+1 − Un, Pb is areference bulk occupancy.

−10 −5 0 5 100

0.5

1

−(U/kBT)

Pc, J

c/Jm

ax

J/J

c, "Classical"

P, Fermi−Dirac

A self-consistent calculation ofthe conductance via linearresponse theory leads to thestandard CB approximation:

Jc ∝ (Uc/kBT ) sinh−1(Uc/kBT )

The current Jc exhibits aresonant peak similar to thatof the tunneling current in aquantum dot.


Energetics of “barrier-less” conduction for M0 band

(a) Potential energy alongthe channel’s x−axisplotted vs. Qf .

(b) Energy difference ∆Eas a function of Qf .

(c) “Barrier-less” profile(red) comes from abalance betweenself-repulsion (blue)and attraction to Qf(dashed).

40200−20−40

01

2

−10

0

10

x(A)Qf (e)

E(k

BT)

−30−20−10 0 10 20 30−10

0

10

x(A)

E(k

BT)

0 1 20

5

10

15

Qf (e)

∆E(k

BT)

(a)

(c)(b)


Permeation processes for Ca2+


Divalent blockade

Coulomb blockade theory (Fermi-Dirac distribution) predictsexponential character for dependence of divalent (calcium) blockadethreshold IC50 on Qf .

Divalent blockade is anexample ofconcentration-related shift.Figure show clear exponentialdependence of thresholdcalcium concentration IC50 onnominal fixed charge at theselectivity filter.

0 1 2 3 4 5 610

0

101

102

103

104

105

|Qf/e|

IC50

, µ M

Schlief:96Heinemann:92Best fit

DEKADKEAEEKA

DEEADEDAEEEA

DEEEEEEE

DEKEDEKDDEQADQEANEEA

Dependence of IC50 on Qf for calcium andsodium channels and their mutants.


Anomalous mole fraction effect (AMFE)

AMFE is a special phenomenon of valence selectivity manifesting incalcium channels, combining divalent blockade and selective Ca2

current.

Calcium channels conductNa+ ions at [Ca]=0.Divalent Blockade of Na+

current by µ-molarquantities of Ca2+

Appearance of Ca2+

current at milli-molar Ca2+

concentrations.Blockade threshold IC50 isa significant analyticalparameter of channel.


Concentration dependence: Phase diagram of AMFE

AMFE is exlained as divalentblockade for 1-st Ca2+ ionfollowed by self-sustained2-ions current starting when2-nd Ca2+ ion arrives.The switching lines and AMFEtrajectory are shown at the“ion exchange phase diagram”for the calcium channel.In “zero ions” area puresodium conducts ion current.The first Ca2+ ion providesblockade of channel.The second Ca2+ ion providesknock-on fast Ca2+

conduction.

−2 0 21

2

3

4

5

6

lg([Ca])−lg([Ca]0)

Qf/e

2 ions

1 ion

0 ions

δQf=2e

Figure: AMFE in calcium channels(mixed bath). Phase diagram Qf vslog([Ca]), switching lines are: 0→1(green) and 1→2 (violet), AMFEtrajectory is shown as red dashed.


NaChBac: LESWAS, LEDWAS and LEEWAS

LESWAS - Selectivity

LEEWAS - selectivity

LEDWAS - AMFE

Selectivity and AMFE patch clamptracks for NaChBac channels, wildtype (LESWAS SF formula) andmutants (LEDWAS and LEEWAS)are reproduced with kind permissionof Olena Fedorenko and StephenRoberts (Division of Biomedical andLife Sciences, Lancaster University,UK)


Summary/conclusions

1 Ionic Coulomb blockade appear in low-capacitance ion channelsdue to charge/occupancy discreteness.

2 Coulomb staircase is fundamental electrostatic phenomenonrelated to charge discreteness.

3 Resonant conductivity is derived from Coulomb staircase viafluctuation-dissipation theorem.

4 Hydration energy and concentration-related entropy addappropriate shifts in positions of conduction resonance leadingalike and valence selectivity and AMFE.

5 Coulomb blockade model of permeation and selectivity ofcalcium/sodium ion channels provides general and transparentexplanation of divalent blockade and AMFE.

6 Coulomb blockade model is consistent with new mutation studiesof NaChBac channels.


Acknowledgement and selected publications

AcknowledgementWe are grateful to O. Fedorenko and S.K. Roberts for patch clamp data, and to A.Stefanovska, B. Shklovskii and M. Di Ventra for useful comments. We are grateful tothe Engineering and Physical Sciences Research Council (UK) for their support of thisresearch (Grant No. EP/M015831/1).

Recent publications1 I Kh Kaufman, P V E McClintock, R S Eisenberg, ”Coulomb blockade model of

permeation and selectivity in biological ion channels”, New Journal of Physics17, 083021 (2015).

2 I Kh Kaufman, D G Luchinsky, W Gibby, P V E McClintock, & R S Eisenberg,"Coulomb blockade oscillations in biological ion channels", in, 23nd Intern. Conf.on Noise and Fluctuations (ICNF), Xian, 2-5 June 2015 (IEEE Conf. Proc., 2015),DOI:10.1109/ICNF.2015.7288558.

3 I Kaufman, D G Luchinsky, R Tindjong, P V E McClintock, and R S Eisenberg,“Energetics of discrete selectivity bands and mutation-induced transitions in thecalcium-sodium ion channels family”, Phys. Rev. E 88, 052712 (2013).

4 I Kaufman, D G Luchinsky, R Tindjong, P V E McClintock, and R S Eisenberg,“Multi-ion conduction bands in a simple model of calcium ion channels”, Phys.Biol. 10, 026007 (2013).


coulomb blockade oscillations and amfe in calcium/sodium biological ion ... · coulomb blockade...

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