coulomb blockade of stochastic permeation in biological...

Outline

Coulomb Blockade of Stochastic Permeationin Biological Ion Channels

W.A.T. Gibby1 I.Kh. Kaufman1 D.G. Luchinsky1,2

P.V.E. McClintock1 R.S. Eisenberg3

1Department of Physics, Lancaster University, UK

2Mission Critical Technologies Inc., El Segundo, CA USA

3Molecular Biophysics, Rush University, Chicago, USA

UPoN Barcelona – 13 July 2015

Gibby, Kaufman, Luchinsky, McClintock, Eisenberg Coulomb Blockade Dynamics

Outline

Outline

1 The unsolved problemBackgroundElectrostatic modelEffect of the fixed charge

2 Solving the problem?Modelling – Brownian dynamicsIonic Coulomb blockadeMechanisms of permeation

3 ConclusionsProspectsSummary

40200−20−40

01

2

−15

−10

−5

0

5

10

x(A)Qf (e)

E(k

BT

)

−30 −20 −10 0 10 20 30−10

0

10

x(A)

E(k

BT

)

(a)

(b)

How are ions transported selectivelythrough Ca2+ and Na+ channels?


The unsolved problemSolving the problem?

Conclusions

BackgroundElectrostatic modelEffect of the fixed charge

Ion channels

Cellular membrane has numerous ion channels (+ pumpsand transporters).Ion channel is a natural nanotube through the membrane.Allows ion exchange between inside and outside of cell.Essential to physiology – bacteria to humans.Highly selective for particular ions.



Conclusions


Gating

Channels spontaneously “gating” open/shut.A stochastic process influenced by e.g. –

VoltageChemicals

We are interested only in open channels.



Conclusions


Puzzles

1. Selectivity?

E.g. Calcium channel favours Ca2+

over Na+ by up to 1000:1, eventhough they are the same size –example of valence selectivity.Also alike charge selectivity, e.g.potassium channel stronglydisfavours sodium, even thoughNa+ is smaller K+.

2. Fast permeation? Almost at the rateof free diffusion (open hole).

3. AMFE? Na+ goes easily through acalcium channel but is blocked bytiny traces of Ca2+.



Conclusions


Puzzles

4. Function of the fixed charge at the SF

Ion channels have narrow “selectivity filters” with fixednegative charge... somehow associated with selectivity.What does the charge do, and how does it determineselectivity?

5. Mutations

Mutations that alter the fixed charge (alone) can –

(a) Destroy the channel (so it no longer conducts), or(b) Change the channel selectivity, e.g. Ca2+ to Na+ or vice

versa.

6. Gating

Why/how does the channel continually open and close?



Conclusions


Atomic structure of KcsA Potassium ion channel

So-called “crystal structure” of abacterial ion channel.

Very complicated.

Knowledge of the structure does notimmediately explain the function – thefamous “structure-function problem”.

Which features are important?

Selectivity filter?Generic features of structure?

Need to pick out aspects important formodelling the permeation process.



Conclusions


Minimal model of calcium/sodium ion channel

Na+

Ca2+

Qf

Membrane protein

L

Left bath,

CL>0

Right bath,

CR=0

R X

Ra

A water-filled, cylindrical hole, radius R = 3 Å and lengthL = 16 Å through the protein hub in the cellular membrane.

Water and protein described as continuous media with dielectricconstants εw = 80 (water) and εp = 2 (protein)

The selectivity filter (charged residues) represented by a rigid ring ofnegative charge Qf = 0 − 6.5e.



Conclusions


Calcium and sodium channels

We focus on voltage-gated calcium and sodium ion channels –which control muscle contraction, neurotransmitter secretion,transmission of action potentials) – because:

They are very similar in structure, but have differing SF lociand hence different Qf .

The many mutation experiments to change Qf (destroyingthe channel or changing its selectivity) are potentiallyrevealing but not yet properly understood.

But results and conclusions are more generally applicable.



Conclusions


Permeation through an ion channelTypically, the fixed charge Qf provides a strong binding siteat the selectivity filter.Dimensions & electrostatics enforce single-file motion.

Arriving ions captured at binding site, then fluctuationalescape occurs over the potential barrier Eb.



Conclusions

Modelling – Brownian dynamicsIonic Coulomb blockadeMechanisms of permeation

Electrostatics

The electrostatic field is derived by self-consistentnumerical solution of Poisson’s equation:

−∇(εε0∇u) =∑

ezini

where ε0 is the dielectric permittivity of vacuum, ε is thedielectric permittivity of the medium (water or protein), u isthe electric potential, e is the elementary charge, zi is thevalence, and ni is the number density of ions.

This equation accounts for both ion-ion interaction andself-action for all ions in their current positions.

One result of the calculations is the axial potential energyprofile = the Potential of the Mean Force (PMF).



Conclusions


Brownian dynamics

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

dxi

dt= −Dizi

(

∂u∂xi

)

+√

2Diξ(t)

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



Conclusions


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 andoccupation P as afunction of charge Qf atselectivity filter.

For different Ca2+

concentrations [Ca].

We find –

Pattern of narrowconduction bandsand stop bands.

Conduction bandsoccur at transitionsin channeloccupancy P.



Conclusions


Electrostatic exclusion principle

In absence of fixed chargeQf , self-energy barrier Us

prevents entry of ion to SF.

But Qf compensates Us

and allows cation to enter.

This effectively restoresthe impermeable Us for2nd ion at channel mouth.

So for this Qf only one ioncan occupy the SF.

Implications1. The SF’s forbidden multi-occupancy is an electrostatic exclusion principle.2. Like the Pauli exclusion principle in quantum mechanics, it implies a

Fermi-Dirac occupancy distribution.3. For larger Qf similar arguments apply for occupancies of 2,3...



Conclusions


Coulomb blockade: channels vs. quantum dots

Ca2+ channel

0 1 2 3 4 5 60

10

20

|Qf/e|

U/(

k BT

)

0

2

4

PC

a

0

5

10

15

J Ca(1

07 /s)

20mM40mM80mM

Un

UG

(b)Z

1

M2

M0

Z2

(a) M1

M2

M1

M0

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

Z3

(c)

Quantum dot

Ion(s) trapped at SF ⇔ Electron(s) trapped in quantum dot

Periodic conduction bands ⇔ Coulomb blockade oscillations

Steps in occupation number ⇔ Coulomb staircase

Classical mechanics for ion ⇔ Quantum mechanics for electron

Stochastic permeation by ion ⇔ Quantum tunnelling by electron



Conclusions


Singular points in ionic Coulomb blockade

0 1 2 3 4 5 60

10

20

|Qf/e|

U/(

k BT

)

0

2

4

PC

a

0

5

10

15J C

a(107 /s

)

20mM40mM80mM

Un

UG

(b)Z

1

M2

M0

Z2

(a) M1

M2

M1

M0

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

Z3

(c)

Zn = zen ± δZn, Coulomb blockade

Mn = ze(n + 1/2)± δMn Resonant conduction



Conclusions


Coulomb blockade: ion channels vs. quantum dots

Feature Ionic CB (channels) Electronic CB (q. dots)

System type Classical QuantumCharge carriers Ions – Ca2+, Na+... ElectronsField equation Langevin SchrödingerExclusion principle Electrostatic ElectrostaticOccupancy statistics Fermi-Dirac (in channel) Fermi-DiracCarrier valence z = 1, 2, ... z = 1Control parameter Fixed charge Qf Potential U = Q2

f /2COccupancy structure Coulomb staircase Coulomb staircaseConduction peaks at Qf = ze(n + 1

2 ) Qf = e(n + 12 )

Peak shape Landauer approximation Landauer approximation



Conclusions


Permeation processes for Ca2+



Conclusions

ProspectsSummary

Where next?

Develop model to include hydration effects: expectsignificant corrections for valence selectivity; and the basisof alike selectivity e.g. Na+/K+.

Mutation experiments (biologist collaborator StephenRoberts) to “tune” Qf .

Will J and S behave asexpected?

How about trivalent ions?

0

10

20

JNa(1/s),x107

0

5

10

JCa(1/s),x107

0 1 2 3 4 5 60

5

10

Qf/e

JLa(1/s),x107

L0 L1 L2

M1M0 M2

(a)

(c)

(b)

T1T0

Molecular dynamics simulations (collaborator IgorKhovanov in Warwick) – underpinning for our higher-scaleBrownian dynamics simulations.



Conclusions

ProspectsSummary

What puzzles have we explained?

1 Selectivity? (valence selectivity) X2 Fast permeation? X

3 AMFE? X

4 Role of fixed charge at the selectivity filter? X

5 Effect of mutations in the selectivity filter? X

6 Shed light on gating? No, not really, but...



Conclusions

ProspectsSummary

Acknowledgement and selected publications

AcknowledgementWe are grateful to the Engineering and Physical Sciences Research Council(UK) for their support of this research.

Recent publications1 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 ionchannels”, Phys. Biol. 10, 026007 (2013).

2 R S Eisenberg, I Kaufman, D Luchinsky, Tindjong, and P V EMcClintock, “Discrete conductance levels in calcium channel models:multiband calcium selective conduction”, Biophys. J. 104 358a (2013).

3 I Kaufman, D G Luchinsky, R Tindjong, P V E McClintock, and R SEisenberg, “Energetics of discrete selectivity bands andmutation-induced transitions in the calcium-sodium ion channelsfamily”, Phys. Rev. E 88, 052712 (2013).

4 I Kaufman, D G Luchinsky, R Tindjong, P V E McClintock, and R SEisenberg, “Coulomb blockade model of permeation and selectivity inbiological ion channels”, New J. Phys. (in press).


coulomb blockade of stochastic permeation in biological...

Documents