Electrostatics of the Voltage Sensor of Ion Channels: Simulation as a science

(1)

Member

of

the

Helmholtz-Association

Abstract

Computation in nervous systems depend on the initiation, propagation and

inhibition of electrical impulses along the membranes of neurons, leading to

synaptic activity connecting neurons. The underlying electrical excitability of

cells is possible because the movement of a few charges controls the flow of

many charges. This is often mediated by the voltage sensors of voltage

gated ion channels. In particular, the S4 voltage sensor of the Shaker K

+

ion channel is an experimentally well characterized example presented here

in terms of a boundary element method for calculating the electrostatic

potential energy and thus predicting expectations of measures.

The path from a theoretical description of the voltage sensor to a prediction

of a model’s macroscopic consequences requires simulation since such

models have not been solved analytically. However, given that this behavior

depends on the electrostatic properties of complex three-dimensional

dielectric regions that are inhomogeneous, the implementation in simulation

must be treated as a scientific instrument with robust controls and sensitivity

analysis. By taking this approach, it is possible to build a model that is

comparable with biological experiment, can be decomposed for analysis and

hypothesis, and can be used as an element in larger scale models of full ion

channels.

(2)

Member of the Helmholtz-Association

Electrostatics of the

Voltage Sensor of

Ion Channels:

Simulation as a science

(3)

Member

of

the

Computation

Computation in nervous systems depend on the initiation,

propagation and inhibition of electrical impulses along the

membranes of neurons, leading to synaptic activity connecting

neurons.

Electrical excitability of cells is possible because the movement

of a few charges controls the flow of many charges.

24. July, 2014 | Alexander Peyser | 3/41

Excitation Action Current Displacement Structure Dynamics

(4)

Excitation (computed)

I

=

C

M

V

˙

+ ¯

g

K

n

4 (

V

−

V

K

) + ¯

g

Na

m

3 h

(

V

−

V

Na

) + ¯

g

l

(

V

−

V

l

)

Hodgkin and Huxley (1952, Fig. 13, Eq. 26)

(5)

Member

of

the

The Action Potential

Hodgkin and Huxley (1952, Fig. 1)

(6)

Member

of

the

The Action Potential

(7)

Member

of

the

The Action Potential

(8)

Member

of

the

Voltage Gated K

+

Currents

g

_K

(

t

) =

g

_K∞

n

1 ₋

h

1 ₋

p

4

g

_K0

/

g

_K∞

i

exp

(

−

t

/τ

n

)

o

4

(9)

Member

of

the

Voltage-sensor displacement

Armstrong and Bezanilla (1973, Fig. 2)

(10)

Member

of

the

Topology

Gandhi and Isacoff (2002, Fig. 1a)

(11)

Member

of

the

Topology

Gandhi and Isacoff (2002, Fig. 1a)

(12)

Member

of

the

Structure

Tao and MacKinnon (2008, Fig. 1)

(13)

Member

of

the

Molecular Dynamics

Jensen et al. (2012, Fig. 1B)

(14)

Geometry: Radial

ϵw= 80 Lipid ϵm= 2

S4

S1–S3 Gating Canal ϵp Guard Electrode −6 R (nm) −7.5 0 Z (n m ) 7.5 0

z

r

7

6

4

3

2

5

1

Model 1 2 3 4 5 6 7 α r 6.0 2.532 1.966 1.566 1.466 1.266 1.0 z 1.5 1.5 1.5 0.5015 0.5015 3.7515 3.7515 310 r_z 6.0 2.492 1.946 1.546 1.446 1.246 0.98_{1.5 1.5} _1.5 _{0.602 0.602 4.602 4.602} 7.5 ϵp' 0 6 Bath Electrode

Peyser and Nonner (2012a, Fig. 1)

Geometry Geometry: 3d ICC Discretization Controls Energetics

(15)

Member

of

the

Geometry: 3d

(16)

Member

of

the

Basis for Induced Charge Calculation

Electric field in dielectric

Continuity of Displacement

1 E

⊥

1 =

2 E

⊥

2 Gauss’ Law

E

⊥

₁

+

σ

0 n

=

E

⊥

2 Surface Thickness

E

⊥

=

E

⊥

1 +

E

⊥

2

(17)

Member

of

the

Induced Charge Calculation

Charges in dielectric:

ρ

i

(

r

) =

1 − (

r

)

(

r

)

ρ

s

₍

_r

₎

₋

∇(

r

)

(

r

)

·

0 E

(

r

)

ρ

e

(

r

) =

ρ

s

₍

_r

₎

(

r

)

.

Matrix equations:

4 π

0 E

(

r

) =

X

k

q

_k

e

r

−

r

k

|

r

₋

r

k

|

3 +

Z

B

σ

i

(

r

0 )

r

−

r

0 |

r

₋

r

0 _|

3 da

0 +

Z

E

σ

e

(

r

0 )

r

−

r

0 |

r

₋

r

0 _|

3 da

0

4 π

0 V

(

r

) =

X

k

q

_k

e

1 |

r

₋

r

k

|

+

Z

B

σ

i

(

r

0 )

1 |

r

₋

r

0 _|

da

0 +

Z

E

σ

e

(

r

0 )

1 |

r

₋

r

0 _|

da

0 σ

i

(

r

) =

−

∆(

r

)

¯

(

r

)

0 n

(

r

)

· E

(

r

)

(18)

Discretization

Form

Ah

=

c

A

ij

for dielectric surfaces i

[

A

ii

] =

1 a

_i

Z

a

i

r

0

· n

|

r

0 _|

3 d

r

0 +

4 π

[

A

i∼j

] =

∆

i

1 a

j

Z

a

j

r

0 _ij

_·

_^

a

i

|

r

0 _ij

|

3 d

r

0 ij

[

A

ij

] =

∆

i

r

ij

· _^

a

i

|

r

ij

|

3

(19)

Discretization

Form

Ah

=

c

A

ij

for Dirichlet surfaces i

[

A

ii

] =

1 a

i

Z

a

i

1 r

d

r

[

A

i∼j

] =

1 a

j

Z

a

j

1 |

r

ij

|

d

r

ij

[

A

ij

] =

1 |

r

ij

|

(20)

Discretization

Form

Ah

=

c

h

j

for all surfaces

[

h

j

] = σ

j

a

j

(21)

Discretization

Form

Ah

=

c

i

for dielectric surfaces i

[

c

i

] =

∆

i

X

k

q

k

r

ik

· _^

a

i

|

r

ik

|

3

(22)

Discretization

Form

Ah

=

c

i

for dielectric surfaces i

[

c

i

] =

V

i

−

X

k

q

k

1 |

r

ik

|

(23)

Oil Droplet

ǫ

₁

ǫ

₂

r

o

R

₂

q

o

z

θ

_r

r

o

< R

, r ≤ R, R

2 → ∞

φ

11 (

r

) =

₄

q

o

π

0

1

1 |

r

₋

r

o

|

−

∞

X

`=0

(` +

1 )(

2 −

1 )

`

1 + (` +

1 )

2

· r

_o

`

r

`

R

2`+1

P

`

(

cos

θ

r

)

!

(24)

Oil Droplet

ǫ

₁

ǫ

₂

r

o

R

₂

q

o

z

θ

_r

r

o

< R

, r ≥ R, R

2 → ∞

φ

12 (

r

) =

₄

q

o

π

o

∞

X

`=0

2 ` +

1 `

1 + (` +

1 )

2

· r

_o

`

r

`+1

P

`

(

cos

θ

r

)

(25)

Oil Droplet

ǫ

₁

ǫ

₂

r

o

R

₂

q

o

z

θ

_r

r

o

> R

, r ≤ R, R

2 → ∞

φ

11 (

r

) =

₄

q

o

π

0

1

1 |

r

₋

r

o

|

−

∞

X

`=0

(` +

1 )(

2 −

1 )

`

1 + (` +

1 )

2

· r

_o

`

r

`

R

2`+1

P

`

(

cos

θ

r

)

!

(26)

Oil Droplet

ǫ

₁

ǫ

₂

r

o

R

₂

q

o

z

θ

_r

r

o

> R

, r ≥ R, R

2 → ∞

φ

12 (

r

) =

₄

q

o

π

o

∞

X

`=0

2 ` +

1 `

1 + (` +

1 )

2

· r

_o

`

r

`+1

P

`

(

cos

θ

r

)

(27)

Oil Droplet

-60.0 -40.0 -20.0 20.0 40.0 60.0 -30.0 -20.0 -10.0 10.0 r’ = 8.0, z vs. delta-potential (A; mV) 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 11 11 11 1 11 1 11 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 11 1 11 1 11 11 11 11 111 111 11 11 11 11 1 11 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 11 1 11 11 11 11 111 111 11 1111 11111 1111 111 111 111 11 11 11 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 11 11 11 111 111 111 1111 11111 1111 1111111 11111111111 1111111111111 111 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 111 1111111111111 11111111111 1111111 111111111111111 111111111111 11111 1111 111 1 1 1 11 111 1111111 111 11 1 1 1 111 1111 11111 111111111111 111111111111111 111111111111111111111 1111111111111 111111 1 1111111 11111 1111111 1 111111 1111111111111 111111111111111111111 111111111111111111111111111111111 1111111 1 1111111111111111111 1 1111111 111111111111111111111111111111111 -60.0 -40.0 -20.0 20.0 40.0 60.0 -400.0 -300.0 -200.0 -100.0 100.0 200.0 r’ = 12.0, z vs. delta-potential (A; µV) 1 1 1 1 11 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 11 1 1 1 1 1 11 111111111 11111111 1111 111 11 11 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 11 111 1111 11111111 111111111 11 1 1 111 111111111111 11111111 1111 11 11 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 11 11 1111 11111111 111111111111 111 1 1 111 111111111111111 111111111 111 111 1 11 1 1 1 1 1 1 1 1 1 1 1 11111 1 1 1 1 1 1 1 1 1 1 1 11 1 111 111 111111111 111111111111111 111 1 1 1 1 11 1 11 1 11 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 11 1 11 1 11 1 1 1 1 1111 1111111111111111111111111111111 1111 1 111 111 11 111 11 11 1111 1 11111 111111111111111111111111111111 1111 1 1 1111 111111111111111111111111111111 11111 1 1111111111111111111 1 11111 111111111111111111111111111111 1111 1 1 1111 11111111111111111111111111111111111 1 1111111111111111111 1 11111111111111111111111111111111111 1111 1

(28)

Geometry: Radial

ϵw= 80 Lipid ϵm= 2

S4

S1–S3 Gating Canal ϵp Guard Electrode −6 R (nm) −7.5 0 Z (n m ) 7.5 0

z

r

7

6

4

3

2

5

1

Model 1 2 3 4 5 6 7 α r 6.0 2.532 1.966 1.566 1.466 1.266 1.0 z 1.5 1.5 1.5 0.5015 0.5015 3.7515 3.7515 310 r_z 6.0 2.492 1.946 1.546 1.446 1.246 0.98_{1.5 1.5} _1.5 _{0.602 0.602 4.602 4.602} 7.5 ϵp' 0 6 Bath Electrode

(29)

Member

of

the

Geometry: 3d

(30)

Gauss’ Law

−2

−1

0

1

2 Translation (nm)

2

3

4

5

6

7

8

9 Cha

rge

E

rro

r

(10

− 3

e

0

)

Q Q Q Q Q R R R R R T T T T T T T T U U U U U U U U 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 99 9 6 66 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 66 6 6 6 6 6 6 6 66 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 66 6

I

S

(

r

)

0 E

(

r

)

· n

(

r

)

da

=

Z

V

ρ

src

(

r

)

d

τ

Q

calc

−

Q

_Gauss

=

−

X

j

p

j

−

p

σ

ind

a

j

−

X

k

q

src

_k

(31)

Energetics

−3 −2 −1 0 1 2 3 T ranslation (nm) Q Q Q Q Q R R R R R T T T T T T T U U U U U U U A α-Helix Vm= 0 mV −0.5 0.0 0.5 1.0 1.5 2.0 2.5 ∆ W (eV) Q Q Q Q Q R R R R R T T T T T T T U U U U U U U U U U U U U U B 310-Helix Vm= 0 mV −200 −100 0 100 200 Rotation (deg) −3 −2 −1 0 1 2 3 T ranslation (nm) Q Q Q Q Q R R R R R T T T T T T T U U U U U U U C Vm=−100 mV −200 −100 0 100 200 Rotation (deg) Q Q Q Q Q R R R R R T T T T T T T U U U U U U U D Vm=−100 mV −3 −2 −1 0 1 2 3 Translation (nm) −0.5 0.0 0.5 1.0 1.5 h∆ W i (eV) Q Q Q Q Q Q Q R R R R R R R T T T T T U U U U U E -100 mV 0 mV −3 −2 −1 0 1 2 3 Translation (nm) Q Q Q Q Q Q Q R R R R R R R T T T T T U U U U U F -100 mV 0 mV

Peyser and Nonner 2012b

Partition function

Q =

X

i

exp

(

−∆

W

_i

/

k

_B

T

)

P

i

=

exp

(−∆

W

i

/

k

B

T

)/Q

(32)

Member

of

the

Output: Energy & Displacement

Offset of Center of Charge, nm

Char ge Displece, e 0 3 0. 2 0 1 . 0 . 0 1. 2 0 0 3. . 0 5 1. 1.0 0 5. 0 0.5 .0 1 5. − − − − − − 1 −2 −1 0 1 2 Translation (nm) −200 −100 0 100 200 300 400 500 600 700 800 ∆ W (m eV ) 11 1 1 1 1 1 1 1 1 1 1 111 1 111 1 1 11 1 1 1 1 1 1 1 1 1 1 111111111 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 11 1 1 11 1 11 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 111111111 1 1 1 1 1 1 1 1 1 1 111 1 1 11 1 111 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 111 1 11 11 1 1 1 1 1 1 1 1 1 1 1 1 11 111111111 11 11 11 11 1 11 11111 11 11 111111 11 11 11 11 11 11 111111 11 1111111111111 11 11 111111111111111111111111 11 1111111 1111111111 111111 1 1 1 1 11 11111 1 1 1 1 1 1 1 1 1 1 11 1 1111 111 1 1 1 1 1 1 1 1 1 11 1 1 11 1 1 1 1 1 1 1 1 1 111 11111 11 1 1 1 1 1 1 1 1 1 1 11111 11 1 1 1 1 11111 1111111111 11111111 11 111111111111111111111111 11 11 1111111111111 11 11111 11 11 11 11 11 11 1111111 11 11 1111 11 11 11 11 11 11 111111111 11 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 111 1 1 1 1 1 1 1 1 1 1 1 1 1

(33)

Shockley-Ramo Displacement

+1/2 V

-1/2 V

Shockley-Ramo Theorem (Integrated)

Q

=

−

X

k

q

k

V

o

(

r

k

)/(

1V

)

Shockley-Ramo Engineering Cluster Shockley-Ramo Energetics

(34)

Member

of

the

Shockley-Ramo Displacement

Nonner et al. (2004, Fig. 2)

Shockley-Ramo Theorem (Integrated)

Q

=

−

X

k

q

k

V

o

(

r

k

)/(

1V

)

(35)

Engineering

Environment

Deuterostome (D) language: Postscript extended for numerics.

Late-binding RPN, with data, operand and hash-table stacks.

History: 32-bit single-process OS9

⇒

OSX

⇒

64-bit parallel Linux.

Extensions

Structures

Bi-endian and Bi-word size.

Comms

Transport of hash-summed D executable objects

transported by TCP, Unix sockets and MPI.

Parallelized

Numerical C operators were multithreaded, D

procedures were multiprocessed, and program units

were distributed across nodes with

dependency-tracking D job server.

Libraries

Atlas BLAS and PETSc were integrated.

Numerics

Hand optimized.

(36)

Cluster

PETSc Deuter ostome Objec ts Gigabit Ethernet VA LINUX VA LINUX VA LINUX VA LINUX −1.5x−1.0x−0.5x 0.5x1.0x1.5x −800.0y −600.0y −400.0y −200.0y 200.0y 400.0y 600.0y ctr_30:2 ctr_35:2 ctr_40:2 YAXIS XAXIS 0y 0x R o u ter Nodes BL AS Thr eads P rocesses TCP/UDP MPI NFS File system i5 i7 12 or 24 GB

Cumulative speedup

O(

nodes

)

× O(

cores

/

node

)

× ∼

100 (software & hardware)

(37)

Member

of

the

Shockley-Ramo Energetics

Instead of using force integration or change in potential for each

charged particle, SR can be extended to energetic calculations:

W

=

1

2 X

k

q

_k

V

E

=0

₍

_r

k

)

−

QV

m

He (2001)

Then all results can be calculated from one SR energy

calculation with V

m

=

0, and one SR displacement calculation

V

m

=

1 V for each conformation.

This results in a

_O(

voltage steps

)

speedup, after LU decomp.

(38)

Displacement:

_h

Q

_i

−150 −100 −50

0

50

100 Voltage (mV)

−2

−1

0

1

2 C

ha

rg

e

D

is

pl

ac

ed

(e

0

)

−150 −100 −50

0

50

100 Voltage (mV)

α

₃₁₀

Peyser and Nonner (2012b, Fig. 3) Seoh et al. (1996, Fig. 2)

Charge Displacement Sensitivity Load Gating Hamiltonians Predictions

(39)

Countercharge Geometry

−100 −50 0 50 100 Voltage / mV −3 −2 −1 0 1 2 3 h∆ Q i / e0 Q Q Q Q Q R R R R R T T T T T T T U U U U U U U α/h α/s 310/s 310/h a −3 −2 −1 0 1 2 3 Translation / nm −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 h∆ W iφ / eV Q Q Q Q Q Q Q R R R R R R R T T T T T T T T U U U U U U U U b −100 −50 0 50 100 Voltage / mV −2 −1 0 1 2 h∆ Q i / e0 Q Q Q Q Q R R R R R T T T T T U U U U U 1/2 2/3 1/1 4/3 c −2 −1 0 1 2 Translation / nm −300 −200 −100 0 100 200 300 400 h∆ W iφ / meV Q Q Q Q Q R R R R R T T T T T T T T U U U U U U U U d

(40)

Dielectric coefficient

−100 −50 0 50 100 Voltage / mV −2 −1 0 1 2 h∆ Q i / e0 Q Q Q Q Q R R R R R T T T T T U U U U U 2 4 8 16 a −100 −50 0 50 100 Voltage / mV −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 h∆ Q i / e0 Q Q Q Q Q R R R R R T T T T T T T U U U U U U U c −2 −1 0 1 2 Translation / nm −100 0 100 200 300 400 500 h∆ W iφ / meV Q Q Q Q Q R R R R R T T T T T T T U U U U U U U b α-Helix −3 −2 −1 0 1 2 3 Translation / nm 0.0 0.5 1.0 1.5 2.0 h∆ W iφ / eV Q Q Q Q Q Q Q R R R R R R R T T T T T U U U U U d 310-Helix

(41)

Member

of

the

Gating canal geometry

−100 −50 0 50 100 Voltage / mV −2 −1 0 1 2 h∆ Q i / e0 Q Q Q Q Q R R R R R T T T T T U U U U U a α-Helix −100 −50 0 50 100 Voltage / mV −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 h∆ Q i / e0 Q Q Q Q Q R R R R R T T T T T T T U U U U U U U c 310-Helix 1 2 3 4 −3 −2 −1 0 1 2 3 Translation / nm −300 −200 −100 0 100 200 300 400 500 600 h∆ W iφ / meV Q Q Q Q Q Q Q R R R R R R R T T T T T T T T T T U U U U U U U U U U b −3 −2 −1 0 1 2 3 Translation / nm −0.5 0.0 0.5 1.0 1.5 2.0 h∆ W iφ / eV Q Q Q Q Q Q Q R R R R R R R T T T T T T U U U U U U d

(42)

Bath screening

−100 −50 0 50 100 Voltage / mV −3 −2 −1 0 1 2 3 h∆ Q i / e0 Q Q Q Q Q R R R R R T T T T T T T U U U U U U U L//L L//H H//H a α-Helix −100 −50 0 50 100 Voltage / mV −3 −2 −1 0 1 2 3 h∆ Q i / e0 Q Q Q Q Q R R R R R T T T T T T T U U U U U U U b 310-Helix −2 −1 0 1 2 Translation / nm −200 −100 0 100 200 300 400 h∆ W iφ / meV Q Q Q Q Q R R R R R T T T T T T T U U U U U U U c −3 −2 −1 0 1 2 3 Translation / nm −0.2 0.0 0.2 0.4 0.6 0.8 1.0 h∆ W iφ / eV Q Q Q Q Q Q Q R R R R R R R T T T T T T T U U U U U U U d

(43)

Surface charge

−100 −50 0 50 100 Voltage / mV −2 −1 0 1 2 h∆ Q i / e0 Q Q Q Q Q R R R R R T T T T T U U U U U n//n c//c n//c f //f n//f a α-Helix −100 −50 0 50 100 Voltage / mV −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 h∆ Q i / e0 Q Q Q Q Q R R R R R T T T T T T T U U U U U U U b 310-Helix −2 −1 0 1 2 Translation / nm −100 0 100 200 300 400 h∆ W iφ / meV Q Q Q Q Q R R R R R T T T T T T U U U U U U c −3 −2 −1 0 1 2 3 Translation / nm −0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 h∆ W iφ / eV Q Q Q Q Q Q Q R R R R R R R T T T T T T T T U U U U U U U U d

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Load

−3 −2 −1 0 1 2 3 Translation (nm) −6 −4 −2 0 2 4 6 ∆ WL (k T ) −100 −50 0 50 100 Voltage (mV) −2 −1 0 1 2 C ha rg e D is pl ac em en t (e0 ) −100 −50 0 50 100 Voltage (mV) −7 −6 −5 −4 −3 −2 −1 0 1 2 ∆ W2 (k T ) −100 −50 0 50 100 Voltage (mV) −4 −3 −2 −1 0 1 2 3 ∆ WL (k T ) 〈〈〉〉

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Multiscale gating

1.1 nm 1.8 nm HVS(z, Vm) HG(r) HC(z, r)

B

(

z

,

r

,

V

_m

) =

exp

{−β[H

G

(

r

) +

4 X

i=1

(

H

C,i

(

z

i

,

r

)

+

H

B,i

(

z

i

)

+

H

VS,i

(

z

i

,

V

m

))]

}

H

VS,i

(

z

i

,

V

m

) =

−β

−1

ln

X

j

exp

[

−βH

VS,i

(

z

i

, φ

j

,

V

m

)]

(46)

Member

of

the

Hamiltonians & lower scale inputs

−2 −1 0 1 2 VS translation, z / nm −3 −2 −1 0 1 2 3 Gating Cha rge, Q / e 0 Q Q Q Q Q R R R R R T T T T T T T U U U U U U U A −100 −50 0 50 100 Voltage / mV −2 −1 0 1 2 VS T ranslation, z / nm −10 −5 0 5 10 15 20 HVS / kT Q Q Q Q Q R R R R R T T T T T U U U U U U U U U U U U D 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Gate radius, r / nm 0.0 0.2 0.4 0.6 0.8 1.0 Pl Q Q Q Q Q Q Q R R R R R R R T T T T T T U U U U U U B −2 −1 0 1 2 VS Translation, z / nm 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Gate radius, r / nm 0 2 4 6 8 10 HC / kT Q Q Q Q Q R R R R R T T T T T T T U U U U U U U U U U U U U E 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Gate radius, r / nm 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 HG / kT Q Q Q Q Q Q Q R R R R R R R T T T T T T T T U U U U U U U U C −2 −1 0 1 2 VS translation, z / nm 0 1 2 3 4 5 6 7 8 9 HB / kT Q Q Q Q Q R R R R R T T T T T T T T T T U U U U U U U U U U F

(47)

Member

of

the

Full-channel experimental predictions

−150 −100 −50

0

50

100 Voltage / mV

−10

−5

0

5

10 G

at

in

g

C

ha

rg

e

/

e

0

−100

−50

0

50

100 Voltage / mV

0

100

200

300

400

500

600

700

800 N

or

m

al

iz

ed

C

on

du

ct

an

ce

/

10

− 3

B

Q(

V

_m

) =

N

r

X

l=1

N

z

X

k

1

=1

N

z

X

k

2

=k

1

N

z

X

k

3

=k

2

N

z

X

k

4

=k

3

n

(

k

₁

,

k

₂

,

k

₃

,

k

₄

)

B

(

z

_k

₁

,

z

_k

₂

,

z

_k

₃

,

z

_k

₄

,

r

_l

,

V

_m

)

(48)

Acknowledgments

HPC in Neuroscience Boris Orth Andrew Adinetz Martin Stöckle Bastian Tweddell Anne Do Lam-Ruschewski German Research School for

Simulation Sciences Justin Finnerty SimLab Neuroscience Abigail Morrison Rajalekshmi Deepu Mikaël Naveau Yury Zaytsev Wolfram Schenck Anna Lührs Sven Strohmer Susanne Kunkel Markus Butz-Ostendorf University of Miami Wolfgang Nonner Karl Magleby Alice Holohean Rush University Bob Eisenberg Dirk Gillespie Universität Tübingen Roland Roth Funding Deutsche Forschungsgemeinschaft Helmholtz Association:

SMBH: Supercomputing and Modelling for Human Brain JARA: Jülich Aachen Research Alliance

University of Miami National Institutes of Health NSF Graduate Research Fellowship

German Research School for Simulation Sciences Any opinions, findings, conclusions or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of any funders.

(49)

Member

of

the

References I

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gating particles of the sodium channels.

Nature

, 242(5398):459–461,

13 Apr. 1973. doi: 10.1038/242459a0.

C. S. Gandhi and E. Y. Isacoff. Molecular models of voltage sensing.

J

Gen Physiol

, 120:455–463, Oct. 2002. doi: 10.1085/jgp.20028678.

Z. He. Review of the Shockley-Ramo theorem and its application in

semiconductor gamma-ray detectors.

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current and its application to conduction and excitation in nerve.

J

Physiol

, 117(4):500–544, 1952. ISSN 00223751. URL

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1392413

.

(50)

Member

of

the

References II

M. O. Jensen, V. Jogini, D. W. Borhani, A. E. Leffler, R. O. Dror, and

D. E. Shaw. Mechanism of voltage gating in potassium channels.

Science

, 336(6078):229–233, 2012.

doi: 10.1126/science.1216533

.

W. Nonner, A. Peyser, D. Gillespie, and B. Eisenberg. Relating

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Phys

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doi: 10.1007/s00249-012-0847-z.

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Member

of

the

References III

S.-A. Seoh, D. Sigg, D. M. Papazian, and F. Bezanilla. Voltage-sensing

residues in the S2 and S4 segments of the

Shaker

K

+

channel.

Neuron

, 16(6):1159–1167, 1 June 1996. ISSN 0896-6273.

doi: 10.1016/S0896-6273(00)80142-7.

X. Tao and R. MacKinnon. Functional analysis of Kv1.2 and paddle

chimera Kv channels in planar lipid bilayers.

J Mol Biol

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Sept. 2008. doi: 10.1016/j.jmb.2008.06.085.