electrostatic basis of selectivity in biological ion channels · 2016. 4. 15. · eisenberg,...

Outline

Electrostatic Basis of Selectivityin 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

2SGT Inc., Ames Research Center, Moffett Field, CA, USA

3Molecular Biophysics, Rush University, Chicago, USA

Lancaster – 12 April 2016

Gibby, Kaufman, Luchinsky, McClintock, Eisenberg Electrostatics of selectivity

Outline

Outline

1 IntroductionIon channelsPuzzles

2 Model channelElectrostaticsEnergy

3 ConclusionSummary

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)

Archetypal SR characteristic

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


IntroductionModel channel

Conclusion

Ion channelsPuzzles

Ion channels

Cell membrane with ion channels (+ pumps etc.).

Natural nanotubes through the membrane.

Allow ion exchange between inside and outside of cell.

Essential to cell physiology, from bacteria to humans.

Simple, passive device – but many longstanding puzzles.



Conclusion

Ion channelsPuzzles

Puzzles include...

1. Selectivity for cations?

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+.



Conclusion

Ion channelsPuzzles

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.



Conclusion

Ion channelsPuzzles

Atomic structure of KcsA Potassium ion channel

Structure of channel is very complicated.

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

To apply physics, need a simplified model.



Conclusion

ElectrostaticsEnergy

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.



Conclusion


Electrostatic potentials – no fixed charge

Electric field almost 1-D on account of dielectric mismatch.So if fixed charge Qf is zero, there is a huge potentialbarrier impeding permeation.So finite fixed Qf is essential for conduction.



Conclusion


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...4. Coulomb blockade!



Conclusion


Energy considerations

Consider “eigenstates” {n} for n ions inside SF.

{n} → {n + 1} is entry of new ion, and{n + 1} → {n} is escape of trapped ion.

Total energy Un for channel in state {n} is

Un = Un,s + Un,a + Un,int

where Un,s is self-energy, Un,a is energy of attraction, and Un,int is ions’mutual interaction energy.

Approximate Un as dielectric self-energy Un,s of the excess charge Qn, applyGauss’s Theorem, and use self-capacitance Cs of channel

Un =Q2

nL8πε0εw R2

=Q2

n

2Cs(Electrostatic energy)

Qn = zen + Qf (Excess charge).

Identical to electronic Coulomb blockade relations, except for z.



Conclusion


Ground state energy

Theory can be developed in close analogy to conventionalCoulomb blockade, e.g. in a quantum dot.

Calculate Un as a function of Qf for n = 0,1,2,3 and seekthe minimum, i.e. the ground state energy

UG(Qf ) = minn

(Un(Qf ,n))

To test the theory use Brownian dynamics simulations.

Theory and simulations both reveal conduction bands andstop bands as functions of Qf .



Conclusion


Brownian dynamics simulations v. CB theory

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)

BROWNIAN DYNAMICS

← Current

← Channel occupancy

COULOMB BLOCKADE THEORY

← Ground state energy

Zn = zen ± δZn, Coulomb blockade

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



Conclusion


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



Conclusion


Comparison with experiment

The conduction bands observed in the Brownian dynamics simulations canbe related to real ion channels –

Band ≈FixedCharge

Channel Locus Nominalcharge

L0 0.5e Sodium Nav DEKA 1eM0 1.0e Non-selective OmpF (K)RRRDE 1eZ1 2e Na-selective mutant EAEA 2eM1 3e Ca-selective mutant;

Ca L-typeEEEA;EEEE

3e;4e

Z2 4e Sodium NaChBac,NavAB

EEEE 4e

M2 5e Calcium RyR DDDD(ED) 6e

Results mostly agree with published data, and make sense of observedmutation phenomena.Amino acids aspartate (D) and glutamate (E) have negatively-charged side chains; lysine (K) and argenine (R) havepositively-charged side chains; others are neutral.



ConclusionSummary

What puzzles have we explained?

Well, arguably –

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

But more work is needed to develop a fuller picture...



ConclusionSummary

Where next?

Seeking to determine the fundamental origins of selectivity andconductivity in ion channels –

Biological experiments on mutants, testing predictions e.g.of effect of changing Qf (Steven Roberts and OlenaFedorenko, Lancaster).

Brownian dynamics simulations (Igor Kaufman, Lancaster).

Molecular dynamics simulations (Igor Khovanov & CarloGuardiani, Warwick).

Analytic theory (Dmitri Luchinsky & Will Gibby, Lancaster)especially for situation where hydration energy dominates,e.g. K+/Na+ selectivity.

Most results equally applicable to charged artificial nanopores.



ConclusionSummary

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, “Energetics of discrete selectivity bands andmutation-induced transitions in the calcium-sodium ion channelsfamily”, Phys. Rev. E 88, 052712 (2013).

2 I Kh Kaufman, P V E McClintock and R S Eisenberg, “Coulombblockade model of permeation and selectivity in biological ionchannels”, New J. Phys. 17, 083021 (2015).

3 I Kh Kaufman, D G Luchinsky, W A T Gibby, P V E McClintock and R SEisenberg, “Putative resolution of the EEEE selectivity paradox inL-type Ca2+ and bacterial Na+ biological ion channels”, J. Stat. Mech.,in press (2016).


electrostatic basis of selectivity in biological ion channels · 2016. 4. 15. · eisenberg,...

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