Theory of Ion Channels

Harvard-MIT-BU Theoretical Chemistry Lectures

5th March, 2003

Benoît Roux

Department of Biochemistry Weill Medical College of Cornell University

benoit.roux@med.cornell.eduhttp://thallium.med.cornell.edu/RouxLab

Ion channels: Basic concepts

• Selectivity ]/ln[)/( intexteq CCqTkV B=

• Pore blockers (Ba2+, QA, neurotoxins) • Gating

ExtracellularIntracellular

• Ion conduction )eqmp VVI −(Λ=

Dielectric barrier for ion permeation

Parsegian (1969)

Jordan (1981) Finite-difference Poisson-Boltzman (PBEQ)Wonpil Im, Dmitrii Beglov

The transmembrane potential

The Nernst potential is essentially an equilibrium phenomenon that arises spontaneously for a semi-permeable membrane

The Nernst potential arises from an exceedingly small charge imbalance across the membrane. It is almost impractical to try to simulate this explicitly with MD. We will treat the ions in the “pore region” explicitly, and all the ions in the bulk region implicitly with some Poisson-Boltzmann continuum electrostatic theory.

F

VF=qEmp

−=

II

IqTk

ρρlnBmp

LqVqE mpmp ==

L

Traditional phenomenologiesF=qEmp

Z

Traditional approaches, such as Eyring Rate Theory or Nernst-Planck continuum electrodiffusion theory, picture the movements of ions across membrane channels as chaotic random displacements in a free energy profile W(z) driven by the transmembrane electric field

Traditional phenomenologies

Eyring Rate Theory represents the movements of ions as a sequence of sudden stochastic “hopping events” across free energy barriers separating energetically favorable discrete wells (Eyring, 1934).

ν ∆W‡

kTWek /TST‡∆−=ν

TSTk

Nernst-Planck theory represents the movements of ions along the axis of the channel as a random continuous diffusion process in a potential W(z)

Cz

WkT

DzCDJ

∂∂−

∂∂−= 1

Theory of ion flow through channels: A Road Map

First, we need to formulate a statistical mechanical theory representing the equilibrium situation rigorously

Then, we will extend this formulation to non-equilibrium situations such that it the equilibrium theory is recovered under proper conditions

There is no guarantee that such non-equilibrium theory is exact, but it is a useful tool to develop all the important concepts.

At least, we will know that this theory is built on a correct representation of equilibrium, which is certainly a necessary condition (but perhaps not sufficient).

Equilibrium Theory of Ion Channels

side I

It is useful to define a “pore region”

Probability Pn for having exactely n ions in the pore region

∫ ∑ −==pore

N

iiN rrdrrrrrn

1321 )(),,,,( δΚ

mpVside II

Grand Canonical Ensemble( ) TknrrW

nN

n

nBnedrdr

n/]),,([

10

1

!µρ −−

=∫∫∑=Ξ

ΚΛ eq porepore

n-ion association constant (related to the PMF)

( )( ) ( ) ( ) Λ++++

= 33

22

111 ρρρ

ρKKK

KPn

nn

TknrrWnn

Bnedrdrn

K /]),([1 1!1 µ−−

∫∫=ΚΛ eq

porepore

For a 1-ion pore (first order saturation)

( )( )ρ

ρ

1

11 1 K

KP+

=TkrW BedrK /])([11 1

µ−−∫= eq

pore

TkUN

TkUN

TkrW

B

B

B

edrdr

edrdre

/2

/2

/)(*1

1

−

−

−

∫∫

∫∫=

bulkbulk

bulkbulk

eq

Λ

Λ

Theory of transmembrane potential

[ ]

[ ]

[ ] [ ] II side 0)()()()(

pore 0)()(

I side 0)()()()(

mpmp2

mp

mp

mp2

mp

=−−∇⋅∇

=∇⋅∇

=−∇⋅∇

Vrrrr

rr

rrrr

φκφε

φε

φκφε

mpVside I side II

Roux (Biophys J, 1999)

The total potential of mean force (PMF)

For an equilibrium system, the total PMF can be rigorously separated into a voltage-independent and voltage-dependent contributions.

Using a cumulant explansion of the configuration integral for the PMF, we get

Equilibrium Multi-ion PMF under symmetric conditions

TransmembranepotentialTotal Multi-ion PMF

ΛΚΚ +∑+=i

iinn rqrrWVrrW )(),,();,,( 11 mpeqmptot φ

1-Ion free energy profile along the channel axis

Z

Weq(z) is the reversible work to move the ion along z

∫−=1

0

)'(')()( 0z

zzFdzzWzW (eq)eqeq

Can be calculated from MD !

Molecular Dynamics Simulations“The molecular dynamics or MD approach consists in, having represented the microscopic forces between the atoms with some potential function, generating a step-by-step trajectory of the atoms by numerically integrating the classical equation of motion of Newton, F=MA.”

t

t+∆t

t+2∆t

t+3∆t

…….

From position and velocity at some time, we calculate the position and velocity at a short time step later

ttFM

tVttV

ttFM

ttVtRttR

∆+=∆+

∆+∆+=∆+

)(1)()(

)(2

1)()()( 2

Input: U

Simulating Ion Flow

→ ionic current

→ µs simulations

→ just K+ ions

→ all atoms, channel, ions, water, lipids

→ ns simulations

→ equilibrium properties

→ dynamical diffusion constant Di

Molecular Dynamics (MD)

UM

tr ii

i ∇−=1)(&&

WeqFree energy potential of mean force from

MD simulations

Inputs: Di, Wtot

Brownian Dynamics (BD)

)()( tot tWTk

Dtr iiB

ii ξ+∇−=&

VmpTransmembrane

potential profile from PB-V continumelectrostatics

+ =Wtot

Total free energy governing the movements of

permeation ions

Different Ions Channels

• The Gramicidin A Channel

• KcsA potassium channel

• OmpF porin

The Gramicidin Channel

The Gramicidin A Channel15 residues, alternating L and D amino acidsFormyl-Val-Gly-Ala-Leu-Ala-Val-Val-Val-Trp-Leu-Trp-Leu-Trp-Leu-Trp-Ethanolamine

Head-to-head β-helical dimer 26 Ålong (Urry, 1971; Arseniev, 1984; Cross 1991)

Individual channel recordings at 200 mV with 500 mM NaCl

The gA channel

Free Energy Simulation Technique

Z

B. ROUX and M. KARPLUS, Biophys. J. 59, 961-980 (1991).

W(z) can be calculated for short steps along Z using MD simulations

)(

11

ln

)()()(

z

kTU

nnnn

ekT

zWzWzzW∆−

++

−=

−=→∆

with

)()( nn zUzzUU −∆+=∆

Free Energy Profile of Na+ along the gA channel

Binding site at 9.3 Å

B. ROUX and M. KARPLUS, J. Am. Chem. Soc. 115, 3250-3262 (1993).

• 14% of voltage, 9 Å from the channel center based on a complete 3B2S model with difusion limitation and interfacial polarization (Becker et al, 1992)

• 15.4% of voltage, 8.6 Å from the channel center based on a 3B2S model (Busath and Szabo, 1988).

• 14% of voltage, 9.0 Å from the channel center based on a 3B2S model with diffusion (Andersen and Procopio, 1980)

• 6% of voltage, 11 Å from the channel center, Eisenman & Sandbloom, 1983).

The GA cation binding site from current-voltage measurements

9.5 Å

0.15 Vmp

gA channel

The binding site of Na+ in the gA Channel

• 9.3 Å from the channel center based on a free energy PMF molecular dynamics calculation (Roux & Karplus, 1991).

• 9.6 Å from low angle scattering with Th+(Olah & Huang, 1991).

• Close contact with Leu-10 according to 13C chemical shift anistropy (CSA) solid state NMR measurements (Smith et al, 1990).

The 13C and 15N chemical shift tensor of proteins can be used in solid state NMR mesurements to extract oriential constraints on the polypeptide backbone

Leu10

Trp15

The position of the binding site of Na+ deduced from the solid state CSA data is around 9.2 Å (Woolf & Roux, 1997)

The GA channel in a lipid bilayer

MD simulation of GA in explicit DMPC bilayer membrane

Samples used in Solid State NMR

8:1 DMPC:GA ratio

45% weight water

4,500 atoms

Woolf & Roux (PNAS, 1994; PROT, 1996)Allen, Andersen, Roux (2003)

Free Energy Profile of K+ along the GA channel

K+

Allen, Andersen and Roux (2003)

Free Energy Profile around the GA channel

Valence selectivity in the gramicidin A channelPermeable to cations, impermeable to anions

in water in GA

K+

Cl-

+KK

-ClK

Valence selectivity in the gramicidin A channel

In liquid water ∆Gwater

K+ -80 kcal/molCl- -80 kcal/mol

TkUTkG Bee // ∆−∆− =B

Free energy MD simulations based on atomic modelAlchemical transformation of K+ into a Cl-

Roux, Biophys. J. 71, 3177 (1996) Free energy difference in GA is similar to the difference in solvation between liquid water and liquid amides such as formamide, acetamide, and N-methylacetamide (NMA).

∆∆ Gwater= 0 kcal/mol

∆∆ GGA =+58 kcal/mol

Molecular basis of valence selectivity of the gA channel

K+ -27 kcal/mol

Cl- -17 kcal/molRadial charge distribution around the ion in bulk water and in the GA channel

Cations are better solvated than anions

Interaction with NMA(N-methylacetamide)

Multiple occupancy and ion-ion interactions

DionKSionK

Single occupancy Double occupancy

−=∆ S

ion

Dion

BSDion ln K

KTkGRelative free energy double/single occupancy:

SDCs

SDRb

SDK

SDNa

SDLi GGGGG ∆≈∆≈∆>>∆>>∆In the GA channel:

Double occupancy in the gramicidin A channelFree energy MD simulations based on atomic model

Alchemical transformation ofLi+, Na+, K+, Rb+ and Cs+

Roux et al, Biophys, J. 68, 876 (1995).

∆∆GSD (kcal/mol)

Li+ 0.0Na+ -2.4K+ -3.8Rb+ -3.3Cs+ -3.6

SDCs

SDRb

SDK

SDNa

SDLi GGGGG ∆≈∆≈∆>>∆>>∆

Free energy decomposition analysisshows that the trends comes from the 6 single-file water molecules

Hydrogen bonded single file of water molecules

MD simulation of the proton wire

Pomes & Roux (Biophys J, 1997)

Incorporate quantum mechanical effects arising from the light mass of the hydrogen nucleus using discretized Feynmann Path Integral Simulations

The KcsA Potassium Channel

The KcsA K+ Channel

inner helices

outer helices

gate

Channel opens at low intracellular pH (Cuello et al, 1998; Heginbotham et al, 1999)

pore helices

central cavity

selectivity filter

X-ray structure (Doyle et al, 1998)

General topology of K+ channels

•Intrinsic membrane protein•Tetramer

Topology of one monomer:

1 2 3 4 5 6

6 TM segments

Voltage-gated (Shaker, Kv)Ca-activated Ligand-activated

2 TM segments

Inward rectifiers (Kir)Bacterial channels (KcsA, MthK)

Molecular Dynamics simulations of the KcsA K+ Channel

F=MA

150 mM KCl112 DPPC, 6500 watersOver 40,000 atomsNo cutoff of electrostatics (PME)Nanosecond simulations

Bernèche & Roux (Biophys J, 2000)

PMF: Umbrella Sampling Procedure

z

Biasing harmonic potential

u(z)=1/2K(z-zi )2

Wpmf (z1,z2,z3)

Bernèche & Roux (Nature, 2001)

Zhou et al, (2001)

S0S1S2S3S4

External

Cavity

There are 7 favorable (binding) sites for K+

Bernèche & Roux, (2001)

Diffusion Constant Profile of the K+ ions

→

−+

−= +→

222

2

22

0

)(ˆ

)(ˆlim

zzsz

zssC

zzsCD

s

&&

&

δδδ

δ

δδ

Crouzy, Woolf & Roux, (1994)

Transmembrane potential across the channel in the open state calculated from the PB-V continuum electrostatic equation

S0

S4

S3

S2

S1

Open state of KcsA based on X-ray structure of MthK (Jiang et al, 2002)

Simulating Ion Flow

Inputs: Di, Wtot

Brownian Dynamics (BD)

)()( tot tWTk

Dtr iiB

ii ξ+∇−=&

Molecular Dynamics (MD)

WpmfFree energy potential of mean force from

MD umbrella sampling calculations

VmpTransmembrane

potential from PB-V continum

electrostatics

+

Poisson-Boltzmann Voltage

=Wtot

Total free energy governing the movements of

permeation ions

Brownian Dynamics Simulations of K+ in KcsA

Extracellular

Intracellular

1 microsecond takes about one hour on one Pentium 800 mHz…

Brownian Dynamics Simulations of K+ in KcsA

S0

S1S2

S3S4

Elementary Microscopic Step for Ion Conduction

K+

K+

K+

K+

K+

K+

K+Channel is a narrow multiion pore with 2-3 ions

The “knock-on” mechanism of Hodgkin & Keynes (1955)

Simulating K+ fluxes across the KcsA channel

OutwardInward

→ Outward rectification (as observed experimentally)

→ Maximum conductance is 580 pS (outward) and 390 pS (inward)

→ NO adjustable parameters

Why is KcsA slightly outward rectifying?

Outward current Inward current

Ion-ion repulsion manifests itself only at short distances,…but is essential for rapid conduction…

Ion-ion distance ≅ 4 Å

Selectivity for K+ against Na+

+2.8 kcal/mol

+6.6 kcal/mol

S1S2S3S4

TkUTkG Bee // ∆−∆− =B

What happends to K+ flux in the presence of Na+?

Intracellular blockade of K+ flow by Na+ and “punchthrough” phenomenon

500 mM KCl symmetric100 mM NaCl intracellular

S0

S1S2

S3S4

OmpF bacterial porin

X-ray structure of OmpF

L2L3

Cowan et. al., Nature 358, 727-733 (1992)

Cross sectional-area of the pore

cross-sectional area

Z

Constriction zone

• total of 70693 atoms OmpF trimer + 124 DMPC molecules 1M [KCl] = 13470 H2O, 231 K+ and 201 Cl-

• Hexagonal periodic boundary conditions in the XY plane

• Particle Mesh Ewald (PME) for electrostatic interactions

• CPTA dynamics• 350 ps for equilibration • 5 ns for production• 5 months on the super-computer centers• 45 Gbytes trajectories

MD simulation of OmpF

On average, K+ and Cl- follow two well-separated left-handed screw-like paths spanning nearly over 40 Å along the axis of the pore crossing at the constriction zone

Three views rotated by 120°

Ion Solvation

• the total solvation number is similar to that in bulk solution throughout the channel• the total the contributions from water and the protein are complementary • at least 4 water molecules around both ions even in the constriction zone • the contributions from the protein is asymmetric

Ion Pairing• Si(n1=0,n2=0) represents the frequency that an ion i has zero counterion

neighbor in the first shell and zero in the second shell

• Si(n1=1,n2=0) represents the frequency that an ion i has one counterionneighbor in the first shell and zero in the second shell

• Si(n1=1,n2=1) represents the frequency that an ion i has one counterionneighbor in the first shell and one in the second shell

Si(n1=1,n2=0)Si(n1=1,n2=1)

Si(n1=0,n2=0)

Ion Pairing

• ion pairing is reduced in the membrane-solvent interface and inside the pore • ion pairing is NOT reduced in the constriction zone• a K+ ion alone can pass the constriction zone• a Cl- ion cannot go through the constriction zone without the presence of K+ ions

Diffusion Constant from MD trajectory

D = limt →∞

x(t) − x (0)[ ]2

2 t

Mean-square displacement relation applies only in isotropic medium, not valid inside a channel when the mean force is not zero

W=const.

t

Diffusion ConstantX

0.2

D

D = limt →∞

x(t) − x (0)[ ]2

2 t

0.2 Å2/ps 10 fsX

x(t + ∆t) = x(t) − DkBT

dWdx

∆t + 2D∆t R(t)W=ksin(x)

t

< ∆x (t) >X

0.2

D

D =∆x(t) − ∆x(t)[ ]2

2τ,

∆x (t) = x(t + τ ) − x (t)

X

Diffusion Constant along the OmpF pore

D =∆ z( t) − ∆z(t )[ ]2

2τ, ∆z(t) = z(t + τ ) − z(t)D = lim

t → ∞

z(t) − z(0)[ ]2

2t

inside the channel ion mobility is reduced to about 50% of the bulk valuethe diffusion profile can be used for BD and PNP calculations

GCMC/BD Algorithm for simulating ion flow

side I side II

mpV

• Propagate with BD in the “inner” region

• Using the equation for the total PMF

• Impose constant chemical potential in “buffer” regions I & II with GCMC

• Influence of ions and solvent in “outer” region is treated implicitly

)(tFTk

Dr iB

i ξ+=&

Inner region

Buffer regionOuter region

i

ni r

VrrWF

∂∂

−=);,,( 1 mpΚ

Grand Canonical Monte Carlo (GCMC)

( ) TknrrWn

N

n

nBnedrdr

n/]),,([

10

1

!µρ −−

=∫∫∑=Ξ

ΚΛ

regioninner

regioninner

Creation/annihilation of ions

with systVn ρ=

( ) TkWTkW

nn ennenp

B

B

/][

/][

1 1 µµ

−∆−

−∆−

+→ ++=

TkWnn ennnp

B/][1 µ−∆−−→ +=

Test the GCMC/BD algorithm with Fick’s Law

)( III CCDJ −−=

W (R1,R 2 ,⋅ ⋅ ⋅) =

core repulsivepotential

αγ∑ uαγ

ij∑ (rαi − rγ j ) +

α ,i∑U core (rαi ) + ∆W sf (R1,R 2,⋅ ⋅ ⋅) + ∆W rf (R1,R 2 ,⋅ ⋅ ⋅)

Multi-ion Potential of Mean Force (PMF)

ion-ion interactions

staticexternal field Reaction field

3d-gridmap

PB equations

multipolar basis-setexpansion methoduαγ (r) = 4εαγ

σ αγr

12

−σ αγ

r

6

+qα qγεbulk r

+ w sr (r)Im et. al. (2000) Biophys. J. 79: 788-801Im et. al. (2001) J. Chem. Phys. 114: 2924-2937Im & Roux (2001) J. Chem. Phys. 115: 4850-4861

Microscopic Information (inputs) from MD1. Profile of ion diffusion constant 2. Short-range ion-ion interactions

w sr (r) = c 0 expc1 − r

c 2

cos c 3 (c1 − r)π[ ]+ c 4

c1r

6

MDBDPMfmin = 0.5

DK=0.196 Å2/ps DCl=0.203 Å2/ps

3. Ion-exclusion radius

0.75 Å for all nitrogen atoms0.93 Å for all oxygen atoms1.00 Å for all other heavy atoms

Ion Fluxes calculated from BD

Symmetric 1 M [KCl]

Asymmetric 1.0/0.1 M [KCl]

Continuum Approach to Ion Channels:PNP Electrodiffusion Theory

Jα r( ) = −Dα r( ) ∇Cα r( )+qαkBT

Cα r( )∇φ r( )

Fick’s law dragging force

Nernst-Planck (NP) equation :side I side II

mpV

∇ • ε r( )∇φ r( )[ ]= −4π ρp r( ) + qαCα r( )α∑

Poisson equation :

∇ ⋅ ε r( )∇φ r( )[ ]= −4π ρprot r( )+ qαCαbulk

α∑ exp −qαφ r( )/kBT( )

non-linear PB Equation :under equilibrium conditions Jα(r)=0

PNP Electrodiffusion Theory in 3D

Jα r( )= −Dα r( )exp −qαφ r( )/kBT( )∇ Cα r( )exp qαφ r( )/kBT( )[ ]

∇ • Dα r( )exp −qαφ r( )/kBT( )∇Cα r( )exp qαφ r( )/kBT( )[ ]= 0

[ ] 0)(*)(* =∇⋅∇ rr φεThis is just like the Poisson equationwhich can be routinely solved using finite-difference methods

∇ •Jα r( ) = 0( )under steady-state condition

∇ • ε r( )∇φ r( )[ ]= −4π ρp r( ) + qαCα r( )α∑

Iα z( ) = qα dx dy∫∫ ) z • Jα x,y,z( ) Ion current

Kurnikova et. al. (1999) Biophys. J. 76:642-656

Permeation Models for OmpF Porin

MD PB & PNPBD

Im & Roux (2002) J. Mol. Biol. (2002

Permeation Models for OmpF Porin

MD BD PB & PNP

Newton’s classical equations of motion F=MAPotential energy function

Output Potential of mean forceProfiles of iondiffusion constantsShort-range ion-ion interactionsIon exclusion radius

Brownian (or Langevin) equations of motionMulti-ion PMF (continuum electrostatics)Rigid channel proteinsIon-ion correlations

Output I-V curveChannel conductanceReversal potential

Mean-field theoriesPNP electrodiffusion equationsRigid channel proteinsContinuum electrostatics

Output I-V curveChannel conductanceReversal potential

more details less details

K+Cl-

Comparisons with Experimental Data:Channel Conductance KCl Solution

00.51

1.52

2.53

3.5

2 1 0.5 0.2

exp.BDPNP

[KCl] (in M)

Con

duct

ance

(in

nS)

Comparisons with Experimental Data:Reversal Potential in 0.1:1M KCl Solution

ItotIKICl

27.4 mV 22.1 mV

exp. 24.3 mV (Schirmer, 1999)

The reversal potential can be understood using the Goldman-Hodgkin-Katz (GHK) voltage equation

++

−=III

IIICPCPCPCP

qTkV

ClK

ClKBrev ln)()()( zqzWzW i mpeqtot φ+=

K+Cl-

Conclusions• Statistical Mechanical Equilibrium Theory

• Ion flow can be understood using concepts of equilibrium PMF andtransmembrane potential

• Hierarchy of approaches (MD, PB, PNP, BD)

• In the case of narrow selective channels such as gA or KcsA, there are strong ion-channel interactions and meaningful studies require MD umbrella sampling to calculate the PMF accurately

• In the case of the KcsA channel, in which ion-ion interactions are very strong, it is necessary to characterize the free energy landscape governing ion movements with a multi-ion PMF

• In the case of a wide aqueous channel such as OpmF, there are too many ion-ion and ion-counterion interactions to compute a PMF with MD and it is necessary to account for such effects with GCMC/BD

• The mean-field continuum 3d-PNP electrodiffusion theory is qualitatively correct for a wide aqueous pore but is not quantitatively accurate

The GA cation binding site from current-voltage measurementsThe binding site of Na+ in the gA ChannelX-ray structure of OmpFCross sectional-area of the poreConclusions