Ion Channels are the Valves* of Biological Function Main Controllers of Cellular Function

1

Channels control flow in and out of cells

~5 µm

2 Figure of ompF porin by Raimund Dutzler

Ion Channels

are the

Valves* of Biological Function

Main Controllers of Cellular Function

*Pun:

Valve

=

Vacuum

Tube

≈

PNP

Transistor

≈

FET

Transistor

BritSpeak

~30 Å

Flow time scale is 0.1 msec to min

K+

Chemical Bonds are lines Surface is Electrical Potential

Red is negative (acid)

3

Natural nano-valves* for atomic control of biological function

Ion channels

coordinate contraction in the heart, allowing the heart to function as a pump

control secretion and absorption

are proteins whose genes (blueprints) can

be manipulated by molecular genetics have structures shown by x-ray

crystallography in favorable cases are involved in thousands of diseases and

many drugs act on channels studied by thousands of scientists

with >20,000 AxoPatch or other amplifiers

*nearly pico-valves: diameter is 400900 pico-meters

Ion Channels are Biological Devices

~30 Å

K+

Conflict of Interest

4

Natural nano-valves* for atomic control of macroscopic flow

Ion channels

Calcium channel selects Ca++ _{over Na}+ _{by ~10}4

Ion channels

Control macroscopic flow with Atomic structures.

Gate/Switch from conducting to nonconducting state

Ion channel

Ion channels

Devices

that self-assemble into perfectly reproducible arrays

Ion channels

Ion channels

*nearly pico-valves: diameter is 400900 pico-meters

Ion Channels are Engineering Devices

~30 Å

K+

5

Ion Channels are Physical Devices

Channels control flow of Charged Spheres

Channels have Simple Invariant Structure

on the biological time scale

Why can’t we predict the movement of

Charged Spheres through a Hole?

Physics of Ionic Movement under biological conditions is

MUCH SIMPLER

than the physics of Fluids, Plasmas, or Semiconductors,

6

Mathematics

describes only a tiny part of life,

But

Mathematics* Creates

our

Standard of Living

*e.g.,

electricity, computers, fluid dynamics, optics, structural mechanics, ….





u

7

How can we use mathematics to describe

biological systems?

I believe

Some biology can be described by

Physics and Mathematics ‘as usual’

Guess, Solve, Check

But you have to know which biology!





u

8

Channels are only Holes

Why can’t we have a fully successful theory?

Must know physical basis to make a good theory

Where do we start?

9

Proteins Bristle with Charge

Cohn (1920’s) & Edsall (1940’s)

We start with

Electrostatics

because of biology

We will soon add

10

We will soon add

Finite Size Effects

Chemically Specific Properties

of ions

come from their

Diameter and Charge

not vaguely defined hydration shells or ‘chemical’

bonds

11

Active Sites

of Proteins are

Very

Charged

e.g. 7 charges ~

20 M net charge

Selectivity Filters and Gates of Ion Channels

are

Active Sites

1 nM

=

10

Å

= 1.2×10

22

_cm

-3

-+ -+ -+

+

+

-

-Ions are

Crowded

12

We start with Langevin equations of

charged particles

Simplest stochastic trajectories

are

Brownian Motion of Charged Particles

Einstein, Smoluchowski, and Langevin ignored charge

and therefore

do not describe Brownian motion of ions in solutions

Once we learn to count Trajectories of Brownian Motion of Charge,

we can count trajectories of Molecular Dynamics

Opportunity

and Need

We use

Theory of Stochastic Processes

to go

from Trajectories to Probabilities

13

James Clerk Maxwell

“I only count molecules ….,

avoiding all personal enquiries

which would only get me into trouble.”

Slightly rephrased from Royal Society of London, 1879, Archives no. 188

14 Schuss, Nadler, Singer, Eisenberg

Langevin

Equations

Electric Force

from

all charges

including

Permanent charge of

Protein

,

Dielectric Boundary charges,

Boundary condition charge



;



2

p

k

x q

p

p

p

k

k

k

f

_kT

m

m

x





 



x





w







Positive

cat

ion

,

e.g.,

p

=

Na



;



Newton's Law

_{Friction & Noise}

2

k

n

k

x q

n

n

n

k

k

k

f

kT

m

m

x





 



x





w







 

15

 

 

 

 



 

_{ }





 

_{ }



1

protein j j

j j

f

eP

e

t

e

t







































x



= div

x grad

x

x



x x



x x

Trajectories of

Negative Ions

Trajectories of Positive

Ions

Electric Force

from all charges including

P

P

Boundary condition charge

Electric Force

in each Trajectory

from Poisson Equation

Schuss, Nadler, Singer, Eisenberg

16 Schuss, Nadler, Singer, Eisenberg

 

 

 

_{ }









0

0

p

p

p n

k

k

xs

k

i i

i

i

e

e

f

_

_f

_

_q



_





_

_

_

_

_

_z











x

div

E

x

x





_

Electric Force from

AVERAGED Poisson Equation

Excess

‘Chemical

’

Force

Implicit Solvent

‘Primitive’ Model

or

Primitive Solvent Model

Electric Force

from all charges including

P

P

Boundary condition charge

17

Conditional PNP

Schuss, Nadler, Singer, Eisenberg

 

_

_

_{ }









0

|

|

|

y

y

_e

y

y x

y

y x

y x

 

_







_{ }











P











Electric Force depends on Conditional Density of Charge





Nernst-Planck gives

UN

conditional Density of Charge

 

 









1

|

0

y

x e

y

y x

_{y x}

m



x









_

_

_



_



_

_









_





Other Forces

Mass

Friction

Closures or Approximations

Needed

18

Finite OPEN System

Fokker Planck Equation

Boltzmann Distribution

Trajectories

Configurations

Nonequilibrium

Equilibrium

Thermodynamics

Thermodynamics

Device Equation

lim ,

N V

 

Statistical Mechanics

Theory of Stochastic Processes

19

Poisson’s Equation

Drift-diffusion & Continuity Equation

 

0;

 

J x



_i

 

D

_i

   

_i

1

_i

 

kT







J x



x

x



x

 

 

 

 

0

i i

i

e

e

z

















_

_







x

x





x



x





Chemical Potential

 

 

_ln

 

 

i

z e

kT



















_

_







x

x

x

x

Chemical Correlations

Poisson-Nernst-Planck

(PNP)

Channel

P

20

Semiconductor Equations:

One Dimensional PNP

     

i

i

i

i

d

J

D x A x

x

dx





 

Poisson’s Equation

Drift-diffusion & Continuity Equation

 

0

   

 

i i

 

d

d

x A x

e

x

e

z

x

A x dx

dx



_

_



_

_



_

_













0

dJ

dx



Chemical Potential

 

 

 

 

*

x

x

x

ln

x

i

z e

kT



















_

_









Chemical Correlations ‘by hand’: finite size

Channel

P

21

Poisson-Nernst-Planck

(PNP)

Counting at low resolution gives

‘Semiconductor Equations’

Gouy-Chapman, (nonlinear) Poisson-Boltzmann,

Debye-Hückel,

are siblings with similar resolution

but at equilibrium, without current or flux of any species

Devices do not exist at equilibrium

22

Compare with Biological Function!

How do we check the theory?

Our task is to

Discover & Understand, Control & Improve

Biological Function

Inverse Problem

Heinz Engl (Vienna, Linz), Martin Burger (Muenster, Linz)

Must use realistic concentrations

because they are the

Power Supply

23

Selectivity

Differs

in Different Types of

Channels

24

Selectivity

depends on Correlations

I put in

Correlations ‘by hand’

as do the chemists

because

I do not know how to do better!

25

Chemically Specific Properties

of ions come from

Correlations produced by

Diameter & Charge

(much) more than anything else.

Established by Regensburg School, Josef Barthel leader,

Learned from Doug Henderson, J.-P. Hansen, Stuart Rice, among others…Thanks!

Physical Models are Enough

Vaguely defined hydration shells

are

26

PNP/DFT

Selectivity

filter

DDDD

4

−

charges

RyR

RyR

Channel

Channel

PNP/DFT

Monte Carlo

PNP/DFT

Monte Carlo

Selectivity

filter

Various

many - charges

Selectivity

filter

DEKA

2

−, 1+ charge

Selectivity

filter

EEEE

4

−

charges

Synthetic

Synthetic

Ca Channel

Ca Channel

Sodium

Sodium

Channel

Channel

Calcium

Calcium

Channel

Channel

Selectivity of Different Channel Types Studied in Many Solutions

K channel Model of Benoit Roux is very Similar

Quantum Water/K

Model of Susan Rempe is very Similar,

but

Neither K model has yet been computed in a range of solutions

27

Understand Selectivity

well enough to

Make a Calcium Channel

using techniques of

molecular genetics,

site-directed Mutagenesis

Goal:

28 30 60 -30 30 60 0

pA

mV

LECE (-7e)

LECE-MTSES- (-8e)

LECE-GLUT- (-8e)

E

E

WT (-1e)

Calcium selective

Experiment

Two Synthetic Calcium Channels

Designed by Theory

As charge density increases, channel becomes calcium selective

E_rev  E_Ca

Unselective Wild Type

built by Henk Miedema, Wim Meijberg of BioMade Corp.,Groningen, Netherlands Miedema et al, Biophys J 87: 3137–3147 (2004)

MUTANT ─ Compound

29 -6 -4 -2 0 2 4 6

-100 -50 0 50 100

5 nm diameter

1.0 mM

KCl 10 mM _KCl

-2 0 2 4 6

-150 -100 -50 0 50 100 150

pA

mV

-100 -50 0 50 100

pA

mV

pA

mV

V_r ~ 10 mV

Double Gradients

Zuzanna Siwy & Yan He, UC Irvine

Polyethylene terephthalate

200 nm

30

Understand Selectivity

Goal:

For L-type Ca channel

Selectivity = Binding Curves

31

Binding Curve

32

O

Selectivity

Filter

Selectivity Filter

Crowded with Charge

Wolfgang Nonner

+

33 Dielectric Protein

Dielectric Protein

6 Å









μ

mobile ions

=

μ

mobile ions

Ion ‘Binding’ in Crowded Channel

Classical Donnan Equilibrium of Ion Exchanger

Side chains move within channel to their equilibrium position of minimal free energy. We compute the Tertiary Structure as the structure of minimal free energy.

Boda, Nonner, Valisko, Henderson, Eisenberg & Gillespie

34

We* use

Monte Carlo Simulations

over a wide range of concentrations

based on

Amino Acid Sequence

of the

Selectivity Filter

Three Dimensional Structures not known,

so we use Sieving Data and Side Chain Size

and compute

Tertiary Structure as an Output

35

Ion Diameters

Pauling Diameters

Ca

_{1.98 Å}

Na

2.00 Å

K

_{2.66 Å}

‘

Side Chain’ Diameter

Lysine K

3.00 Å

D or E

2.80 Å

Channel Diameter 6 Å

Parameters are Fixed in all calculations in all solutions for all mutants

Boda, Nonner, Valisko, Henderson, Eisenberg & Gillespie

‘Side Chains’

Free

Snap Shots of Contents

Crowded Ions

6Å

36 10

‘Titration Curve’

Classical Definition of Selectivity

Boda, Nonner, Valisko, Henderson, Eisenberg & Gillespie

Calcium

Channel

6Å

3 electron charges

High Occupancy

Sodium

Channel

6Å

1 electron charge

Low Occupancy

Biology

Na

2 Å

Ca

1.98 Å

We believe curves like these must be reproduced by a useful theory

Mutation O cc up an cy

Low

Occupancy

High

Occupancy

37

Charge Selectivity

Na

_vs

Ca

Depends on Dielectric

Dielectric Boundary Force = DBF

Dielectric is a Charge Amplifier

Na

+

_{2.0 Å}

Small Size

Large

Dielectric

Boundary

Force

DEKA

Na Channel,

Zero

Dielectric

Boundary Force

Large

Dielectric

Boundary Force

DBF = f   Q

 

 

Boda et al













Diameter

/

S

el

ec

ti

vi

t

y

/

1.98

Ca

ε



10









6 8 10

38

We turn from Charge Selectivity

to

Size Selectivity

Na

+

_vs

K

+

in

the DEKA Na Channel

39 Selectivity Filter Selectivity Filter Selectivity Filter Selectivity Filter Selectivity Filter Selectivity Filter Selectivity Filter Selectivity Filter

log C/C_ref

Binding Sites*

*Binding Sites are

outputs of our model,

not

structural inputs

Boda, et al

Ion Diameter

Ca++ _{1.98 Å}

Na+ _{2.00 Å}

K+ _{2.66 Å}

‘Side Chain’ Diameter

3.00 Å

2.80 Å

Na Channel DEKA6Å

1 2

O

+ 4

NH

Lys or K

D or E

[NaCl] = [KCl] = 50 mM

BLACK

=

Depletion=0

Na vs K Size Selectivity is in Depletion Zone

Size

Selectivity

Size

40

Size Selectivity is in the Depletion Zone

K

Depletion Zone

Na

Boda, et al

0 K+

in Depletion Zone

[NaCl] = 50 mM [KCl] = 50 mM

Selectivity Filter Channel Protein

Na

_vs.

K

Occupancy

of the DEKA Na Channel, 6 Å

41

Size Selectivity

Depends on Channel Size,

not

Protein

Dielectric Coefficient

Selectivity

for small ion

Na

+

2.00 Å

K

+

2.66 Å

in DEKA 6Å

Na Channel

K+ Na+

Boda, et al

Small Channel Diameter Large

42

Size Selectivity does not depend on protein dielectric

Occupancy Depends on Protein Dielectric,

Protein

Dielectric Amplifies Electrostatics

80

solvent





= Dielectric Coefficient =

of Protein

protein





protein

O

cc

up

an

cy

Boda, et al

DEKA Na Channel, 6 Å

Occupancy

Na

2.0 Å

K

+

43

Binding Sites* are

OUTPUTS

of our Calculations

*Selectivity is in the Depletion Zone,

NOT IN THE BINDING SITE

of the DEKA Na Channel

Our model has no preformed

structural binding sites

but

44

Repulsion

Binding Sites

Electrostatic Attraction

Steric Competition for Space

Location and Strength of Binding Sites

Depend on Ionic Concentration and

Ryanodine Receptor RyR

Model has 5 charged amino acids

found by mutation experiments to

determine selectivity and permeation

The model reproduces >100 data

curves of RyR with 8 adjustable

parameters (geometry and K

_{) and}

only 1 for non-K

ions. Parameters

determined by fits to tiny data set.

Once found, parameters were

never changed.

Model predicted all AMFE data of

RyR with no adjustable parameters

Model predicts effect of drastic

mutation (removal of fixed charge)

Fixed charge 13M



0

Water (pure) is 55 M

RyR

Receptor

Gillespie, Meissner, Le Xu, et al,

not

Bob Eisenberg



More than 100 combinations of solutions

&

mutants



7 mutants with significant effects fit successfully

Model reproduces >120 data curves of RyR with

8 adjustable parameters (geometry, K

) and

1 adjustable parameter, for each other ion type.

Parameters determined by fits to tiny data set.

47

Nonner, Gillespie, Eisenberg

DFT/PNP

vs

Monte Carlo Simulations

Concentration Profiles

48

Divalents

KCl

CaCl

CsCl

CaCl

NaCl

CaCl

KCl

MgCl

Misfit

49

LiCl

D. Gillespie et al., J. Phys. Chem. 109, 15598 (2005).

Misfit in unequal concentration

50

KCl

Theory fits Mutation with Zero Charge

No parameters adjusted

Gillespie et al

J Phys Chem 109 15598 (2005)

Protein charge density

wild type*

13

M

Water is 55

*some wild type curves not shown, ‘off the graph’



0

M

in

D4899

Theory Fits Mutant

in K

The model predicted an AMFE for Na

/Cs

mixtures

before it had been measured

Gillespie, Meissner, Le Xu, et al

Bulk Solution

Channel

62 measurements Thanks to Le Xu!

The model predicted that AMFE disappears

Prediction made

without any adjustable

parameters.

The Na

/Cs

mole

fraction experiment is

repeated with varying

amounts of KCl and

LiCl present in addition

to the NaCl and CsCl.

The model predicted

that the AMFE

disappears when other

cations are present.

This was later

confirmed by

experiment.

Mixtures of THREE Ions

The model reproduced the competition of cations for the pore

without any adjustable parameters.

Li

_&

K

Cs

Li

Na

Cs

55

Traditional Biochemistry

(more or less)

Ignores the Electric Field

56

But

Rate Constants depend steeply

on

Electrical Properties

and

Concentration*

because of shielding, a fundamental property of matter,

independent of model,

in my opinion.

57

Electrostatic Contribution

to ‘Dissociation Constant’ is large and is an

Important Determinant of

Biological Properties

Change of

Dissociation ‘Constant’

with concentration is large and is an

58

What does the protein do?

Selectivity arises from

Electrostatics and Crowding of Charge

Certain

MEASURES

of structure are

Powerful

DETERMINANTS

of Function

e.g., Volume, Dielectric Coefficient, etc

.

Precise Arrangement of Atoms is not involved

in the model,

to first order

.

59

Ionic Selectivity in Protein Channels

Crowded Charge Mechanism

4 Negative Charges

of glutamates of protein

DEMAND

4 Positive Charges

nearby

either 4 Na

+

or 2 Ca

++

60

Ionic Selectivity in Protein Channels

Crowded Charge Mechanism

2 Ca

++

are

LESS CROWDED

than 4 Na

+

Ca

++

SHIELDS BETTER

than Na

+

, so

Protein Prefers Calcium

61

2 Ca

++

_are

_{LESS CROWDED}

_{than 4 Na}

+

62

Protein maintains

Mechanical Forces

*

Volume of Pore

Dielectric Coefficient/Boundary

Permanent Charge

Precise Arrangement of Atoms is not involved

in the model,

to first order

.

but

Particular properties

(‘measures’)

of the protein are crucial!

What does the protein do?

Nonner and Eisenberg

63

Other

Properties of Ion Channels

are likely to involve

more subtle physics

including

orbital delocalization

and

chemical binding

Selectivity apparently does not!

64

Remember

We can build them

(reasonably well)

65 30 60 -30 30 60 0

pA

mV

LECE (-7e)

LECE-MTSES- (-8e)

LECE-GLUT- (-8e)

E

E

WT (-1e)

Calcium selective

Experiment

Two Synthetic Calcium Channels

Designed by Theory

As charge density increases, channel becomes calcium selective

E_rev  E_Ca

Unselective Wild Type

built by Henk Miedema, Wim Meijberg of BioMade Corp.,Groningen, Netherlands Miedema et al, Biophys J 87: 3137–3147 (2004)

66

5 nm diameter

1.0 mM

KCl 10 mM _KCl

-2 0 2 4 6

-150 -100 -50 0 50 100 150

pA

mV

-100 -50 0 50 100

pA

mV

-100 -50 0 50 100

pA

mV

Ca2+

selectivity

V_r ~ 10 mV

Double Gradients

Zuzanna Siwy & Yan He, UC Irvine

Polyethylene terephthalate

200 nm

67

Selectivity can be understood

by

Reduced Models

K

channels

Benoit Roux

Susan Rempe

Na

&

Ca

channels

Nonner,

et al,

RyR

channels

Gillespie

&

Meissner

70

Binding Curve

71

We* use

Monte Carlo Simulations

over a wide range of concentrations

based on

Amino Acid Sequence

of the

Selectivity Filter

Three Dimensional Structures not known,

so we use Sieving Data and Side Chain Size

and compute

Tertiary Structure as an Output

72

What about

Traditional View of Selectivity

and

Binding Sites

?

73

Electrodiffusion of

charged, hard spheres

Correlations put in ‘by Hand’

using best available theory of liquids

74

Binding Sites Come From

Forces

in this model.

Forces

in the model are

Electrostatic Attraction

Steric Competition for Space

75

Ryanodine Receptor

one type of Calcium Channel

work of Gillespie, Meissner, Le Xu,

et al

RyR

Receptor

Gillespie, Meissner, Le Xu, et al, studied



More than 100 combinations of solutions

&

mutants



7 mutants with significant effects fit successfully

Model reproduces >100 data curves of RyR with

8 adjustable parameters (geometry, K

) and

1 adjustable parameter, for each other ion type.

Parameters determined by fits to tiny data set.

77

Theory fits Mutation with Zero Charge

No parameters adjusted

Gillespie et al

J Phys Chem 109 15598 (2005)

Protein charge density

wild type*

13

M

Water is 55

*some wild type curves not shown, ‘off the graph’



0

M

in

D4899

Theory Fits Mutant

in K

Theory Fits Mutant

The model predicted an AMFE for Na

/Cs

mixtures

before it had been measured

Gillespie, Meissner, Le Xu, et al

80

We* use

Monte Carlo Simulations

over a wide range of concentrations

based on

Amino Acid Sequence

of the

Selectivity Filter

Three Dimensional Structures not known,

so we use Sieving Data and Side Chain Size

and compute

Tertiary Structure as an Output

81

We turn from Charge Selectivity

to

Size Selectivity

Na

+

_vs

K

+

in

the DEKA Na Channel

82

What about

Traditional View of Selectivity

and

Binding Sites

?

83

Binding Sites Come From

Forces

in this model.

Forces

in the model are

Electrostatic Attraction

Steric Competition for Space

84

Binding Sites* are

OUTPUTS

of our Calculations

Location and Strength of Binding Sites Depend

on Ionic Concentration and Temperature (etc)

*Selectivity is in the Depletion Zone,

NOT IN THE BINDING SITE

of the DEKA Na Channel

Our model has no preformed

structural binding sites

but

Selectivity is very specific

85

Ionic Selectivity in Protein Channels

Crowded Charge Mechanism

How does

86

General Biological Theme

(‘adaptation’)

Selectivity Arises in a Crowded Space

Biological case:

0.1 M NaCl and 1 M CaCl₂ in the baths

As the volume is decreased,

water is excluded

by crowded charge effects.

As the volume is decreased,

Ca

_{enters and displaces Na}

_.

Biological Case