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Theory of Ion Channels
Harvard-MIT-BU Theoretical Chemistry Lectures
5th March, 2003
Benoît Roux
Department of Biochemistry Weill Medical College of Cornell University
[email protected]://thallium.med.cornell.edu/RouxLab
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Ion channels: Basic concepts
• Selectivity ]/ln[)/( intexteq CCqTkV B=
• Pore blockers (Ba2+, QA, neurotoxins) • Gating
ExtracellularIntracellular
• Ion conduction )eqmp VVI −(Λ=
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Dielectric barrier for ion permeation
Parsegian (1969)
Jordan (1981) Finite-difference Poisson-Boltzman (PBEQ)Wonpil Im, Dmitrii Beglov
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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
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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
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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
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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).
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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
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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
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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)
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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
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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 !
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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
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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
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Different Ions Channels
• The Gramicidin A Channel
• KcsA potassium channel
• OmpF porin
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The Gramicidin Channel
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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
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The gA channel
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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 −∆+=∆
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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).
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• 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
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9.5 Å
0.15 Vmp
gA channel
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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).
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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
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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)
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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)
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Free Energy Profile of K+ along the GA channel
K+
Allen, Andersen and Roux (2003)
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Free Energy Profile around the GA channel
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Valence selectivity in the gramicidin A channelPermeable to cations, impermeable to anions
in water in GA
K+
Cl-
+KK
-ClK
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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
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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)
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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:
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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
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Hydrogen bonded single file of water molecules
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MD simulation of the proton wire
Pomes & Roux (Biophys J, 1997)
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Incorporate quantum mechanical effects arising from the light mass of the hydrogen nucleus using discretized Feynmann Path Integral Simulations
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The KcsA Potassium Channel
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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)
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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)
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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)
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PMF: Umbrella Sampling Procedure
z
Biasing harmonic potential
u(z)=1/2K(z-zi )2
Wpmf (z1,z2,z3)
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Bernèche & Roux (Nature, 2001)
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Zhou et al, (2001)
S0S1S2S3S4
External
Cavity
There are 7 favorable (binding) sites for K+
Bernèche & Roux, (2001)
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Diffusion Constant Profile of the K+ ions
→
−+
−= +→
222
2
22
0
)(ˆ
)(ˆlim
zzsz
zssC
zzsCD
s
&&
&
δδδ
δ
δδ
Crouzy, Woolf & Roux, (1994)
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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)
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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
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Brownian Dynamics Simulations of K+ in KcsA
Extracellular
Intracellular
1 microsecond takes about one hour on one Pentium 800 mHz…
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Brownian Dynamics Simulations of K+ in KcsA
S0
S1S2
S3S4
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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)
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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
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Why is KcsA slightly outward rectifying?
Outward current Inward current
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Ion-ion repulsion manifests itself only at short distances,…but is essential for rapid conduction…
Ion-ion distance ≅ 4 Å
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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+?
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Intracellular blockade of K+ flow by Na+ and “punchthrough” phenomenon
500 mM KCl symmetric100 mM NaCl intracellular
S0
S1S2
S3S4
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OmpF bacterial porin
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X-ray structure of OmpF
L2L3
Cowan et. al., Nature 358, 727-733 (1992)
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Cross sectional-area of the pore
cross-sectional area
Z
Constriction zone
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• 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
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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°
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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
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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)
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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
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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
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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
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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
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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
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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−∆−−→ +=
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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
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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
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Ion Fluxes calculated from BD
Symmetric 1 M [KCl]
Asymmetric 1.0/0.1 M [KCl]
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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
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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
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Permeation Models for OmpF Porin
MD PB & PNPBD
Im & Roux (2002) J. Mol. Biol. (2002
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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
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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-
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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