Post on 13-Jan-2016
description
1
Chemical Bonds are linesSurface is Electrical Potential
Red is positiveBlue is negative
Chemist’s View
Ion Channels Proteins with a Hole
All Atoms View
Figure by Raimund Dutzler
~30 Å
2
Channels form a class of Biological Systemsthat can be analyzed with
Physics as Usual
Physics-Mathematics-Engineering are the proper language
for Ion Channelsin my opinion
ION CHANNELS: Proteins with a Hole
3
Physics as Usual along with
Biology as Usual
Ion Channels can be analyzed with
“Why think? ...
Exhaustively experiment.
Then, think”
Claude BernardAppropriate when nothing was known of Inverse Problems!
Cited in The Great Influenza, John M. Barry, Viking Penguin Group 2004
4
Channels control flow in and out of cells
ION CHANNELS as Biological Objects
~5 µm
5
~30 Å
Ion Channels are the
Main Molecular Controllers “Valves”
of Biological Function
Figure by Raimund Dutzler
6
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?
ION CHANNELS as Physical Devices
7
Physical Characteristics of Ion ChannelsNatural Nanodevices
Ion channels have Selectivity. K channel selects K+ over Na+ by ~104.
Ion channels Gate/Switch in response to pH, voltage, chemical species and mechanical force
from conducting to nonconducting state.
Ion channels have VERY Large Charge Densities critical to I-V characteristics and selectivity (and gating?)
Ion channels allow Mutations that modify conductance, selectivity, and function.
Ion channels are Device Elements that self-assemble into perfectly reproducible arrays.
Ion channels form Templates for design of bio-devices and biosensors.
~30Å
Figure by Raimund Dutzler
8
Channels Control Macroscopic Flow
with Atomic Resolution
ION CHANNELS as
Technological Objects
9
Ion Channelsare
Important Enough to be
Worth the Effort
10
Goal:Predict FunctionFrom Structure
givenFundamental
Physical Laws
11
Current in One Channel Molecule is a Random Telegraph Signal
Voltage Step Applied Here (+ 80 mV; 1M KCl)
Gating: Opening of Porin Trimer
John TangRush Medical Center
13
Single Channel Currents have little variance
John TangRush Medical Center
14
PLANAR LIPID BILAYER SET UPrecordings on a single molecule!
OmpFtrimer
ions
Trans Cis
OA
Rf
Phospholipid bilayer
-
+
Vcom
Vout
IfIf
IK
IV-converter
Voltage clamp:- voltage is set- current is measured
Functional SurfacesSignalsActuation Lipid Bilayer Setup
Recordings from a Single Molecule
Conflict of Interest
15
Patch clamp and Bilayer apparatus clamp ion concentrations in the baths and the voltage across membranes.
Patch Clamp SetupRecordings from One Molecule
16
Voltage in baths
Concentration in baths
Fixed Charge on channel protein
Current depends on
John TangRush Medical Center
OmpF KCl 1M 1M||
G119D KCl 1M 1M||
ompF KCl0.05 M
0.05M||
G119D KCl0.05 M
0.05M||
Bilayer Setup
17
Current depends on type of ion
SelectivityJohn TangRush Medical Center
OmpF KCl 1M 1M||
OmpF CaCl2
1M 1M||
Bilayer Setup
18
Goal:Predict FunctionFrom Structure
givenFundamental
Physical Laws
19
Structures…
Location of charges are known with atomic precision (~0.1 Å)
in favorable cases.
20
Charge Mutation in PorinOmpf
Structure determined by x-ray crystallography in Tilman Schirmer’s lab
Figure by Raimund Dutzler
G119D
21
Goal:Predict FunctionFrom Structure
givenFundamental
Physical Laws
22
But …
What are theFundamental
Physical‘Laws’?
23
Verbal ModelsAre Popular with
Biologistsbut
Inadequate
24
James Clerk Maxwell
“I carefully abstain from asking molecules
where they start…
I only count them….,
avoiding all personal enquiries which would only get me into trouble.”
Royal Society of London, 1879, Archives no. 188In Maxwell on Heat and Statistical Mechanics, Garber, Brush and Everitt, 1995
25
I fear
Biologists indulge themselves
with verbal models of molecules
where
Maxwell abstained
26
Verbal Models are
Vagueand
Difficult to Test
27
Verbal Modelslead to
Interminable Argument and
Interminable Investigation
28
thus,to Interminable Funding
29
and so
Verbal ModelsAre Popular
30
Can Molecular Simulationsserve as
“Fundamental Physical Laws”?
Only if they count correctly !
31
It is very difficult forMolecular Dynamicsto count well enough to reproduce
Conservation Laws (e.g., of number, energy)
Concentration (i.e., number density) or activity
Energy of Electric FieldOhm’s ‘law’ (in simple situations)
Fick’s ‘law’ (in simple situations)
Fluctuations in number density (e.g., entropy)
32
Can Molecular Simulations serve as
“Fundamental Physical Laws”?
Only if Calibrated!
33
Calibrated Molecular Dynamics may be possible
MD without Periodic Boundary Conditions
─ HNC HyperNetted Chain
Pair Correlation Function in Bulk Solution
Saraniti Lab, IIT: Aboud, Marreiro, Saraniti & Eisenberg
34
Calibrated Molecular Dynamics may be possible
MD without Periodic Boundary Conditions BioMOCA
─ Equilibrium Monte Carlo (ala physical chemistry)
Pair Correlation Function in Bulk Solution
van der Straaten, Kathawala, Trellakis, Eisenberg & Ravaioli
0.0
0.5
1.0
g+
+(r
)
0.0
0.5
1.0
g--
(r)
EMC
BioMOCA
0
2
4
g+
-(r)
0.0 5.0 15.0 20.0
r ( Å )
10.0
35
Calibrated MD may be possible, even in aGramicidin channel
Molecular Dynamics without Periodic Boundary Conditions BioMOCA
van der Straaten, Kathawala, Trellakis, Eisenberg & Ravaioli
0 1 2 3 4 5 6 7 8
Channel current (pA)
0
2
4
6
8
Cou
nt
Simulations 235ns to 300ns, totaling 4.3 μs.Mean I = 3.85 pA, 24 Na+ crossings per 1 μs
16 Na+ single channel currents
36
Essence of Engineering is knowing
What Variables to Ignore!
WC Randels quoted in Warner IEEE Trans CT 48:2457 (2001)
Until Mathematics of Simulations is available we take an
Engineering Approach
37
What variables should we ignorewhen we make low resolution models?
How can we tell when a model is helpful?
Use the scientific method
Guess and Check!
Intelligent Guesses are MUCH more efficient
Sequence of unintelligent guesses may not converge! (e.g., Rate/State theory of channels/proteins)
38
Use the scientific methodGuess and Check!
Intelligent Guesses are MUCH more efficient
When theory works, need few checks ‘guesses’ are almost as good as experiments.
Mechanical Engineering,
Electricity,
Computer Science,
Hydrodynamics,
in
39
Use Theory of Inverse Problems to replace or optimize “Guess and Check”
1) Measure only what can be measured(e.g., not two resistors in parallel).
2) Measure what determines important parameters.
3) Use efficient estimators.
4) Use estimators with known bias
5) no matter what the theory,
Be clever in estimation
40
Channels are only Holes
Why can’t we have a fully successful theory?Must know physical basis to make a good theory
Physical Basis of Gating is not known
Physical Basis of Permeation is known,in my opinion.
41
Proteins Bristle with Charge Cohn (1920’s) & Edsall (1940’s)
Ion Channels are no exception
We start with
Electrostaticsbecause of biology
42
Many atoms in a protein have
Permanent Charge ~1e
Permanent charge is the (partial) charge on the atom when the local electric field is zero.
Active Sites in Proteins have
Many Charges in a Small Place
43−0.55O
0.35HH_22
0.60C
0.50C_z
0.35HH_21
−0.45NH_2
0.35HH_12
−0.45NH_1
0.35HH_11
0.32Average Magnitude
0.30H_e
−0.40N_e
0.10C_d
0.00C_g
0.00C_b
0.10C_a
0.25H
−0.40N
Charge (units /e)Atom
Atom in Arginine Charge
according to CHARMM
44
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
= 1.2×1022 cm-3
45
Small Charge in Small Places
Large Potentials*
Start with electrostatics because
*500 µvolt has significant effect on IV curve
in a Sphere of diameter 14 Å
1 charge gives
1
12.5
V at psec times; dielectric constant = 2
mV at µsec times; dielectric constant = 80
46
We start without quantum chemistry*(only at first)
* i.e., delocalization of orbitals of outer electrons
47
Although we start with electrostatics
We will soon add
Physical Models of
Chemical Effects
48
It is appropriate to be skepticalof analysis in which the only chemistry is physical
But give me a chance, askI will be in Linz for a week!
(thanks to Heinz!)
Or email beisenbe@rush.edu
for the papers
49
Very Different from Traditional Structural Biology
which, more or less,
ignores Electrostatics
50
Poisson-Nernst-Planck (PNP)
Lowest resolution theory that includes
Electrostatics and Flux is (probably)
PNP, Gouy-Chapman, (nonlinear) Poisson-Boltzmann, Debye-Hückel, are fraternal twins or siblings with similar resolution
51
ION CHANNEL MODELSPoisson’s Equation
Continuity Equation & Drift - Diffusion
0 j j
j
e z x x x
Mobile &Fixed
For Derivation …Stay tuned Schuss, Singer, Nadler et al
1 ;i i i iD
kT J x x x x 0; J x
Electrochemical Potential
ex*
ln ii i iz e kT
xx x x
Special Chemistry
includes Fixed Charge on Protein
includes only
Mobile Charge
52
One Dimensional PNP
Poisson’s Equation
Drift-Diffusion & Continuity Equation
0
&
j j
j
d dx A x e z x
A x dx dx
mobile
fixed
0idJ
dx
includes Fixed Charge on Protein
ii i i
dJ D x A x x
dx
includes only
Mobile Charge
Electrochemical Potential
ex*
ln ii i iz e kT
xx x x
Special Chemistry
53
The Problem
ionic concentrations, electrostatic potential
held constant far from channel
membrane with ion channel
ionic concentrations, electrostatic
potentialheld constant
far from channel
54BathBath Membrane
Boundary conditions are Crucial Flared Access to Channel Required in 1D
55
I
V
INPUT OUTPUTTHEORY
PNP
STRUCTURE
Length, diameter,
Permanent Charge
Dielectric Coefficient
EXPERIMENTAL CONDITIONS
Bath concentrations
Bath Potential Difference
PNP Forward Problem
56
Current Voltage Relation Gramicidin 3D PNP Uwe Hollerbach
57
V
I
PNP -1 Inverse Problem
Input OutputTheory
EXPERIMENTAL CONDITIONS
Bath Concentrations
Bath Potential Difference
EXPERIMENTS
ELECTRICAL STRUCTURE
Length, diameter,
Permanent Charge
Dielectric Coefficient
PNP -1
Inverse Problem for MathematiciansWhat measurements
best determine electrical structure?
58
Charge Mutation in PorinOmpf
Structure determined by x-ray crystallography in Tilman Schirmer’s lab
G119D
59
Fit of 1D PNP Current-Voltage 100 mM KCl
OmpFG119D
Duan ChenJohn TangRush Medical Center
60
Net Charge Difference= 0.13 1.1= 0.97e
Duan ChenJohn Tang
61
Shielding DominatesElectric Properties of Channels,
Proteins, as it does Ionic Solutions
Shielding is ignored in traditional treatments
of Ion Channels and of Active Sites of proteins
Main Qualitative Result
62
Main Qualitative ResultShielding in Gramicidin
Uwe HollerbachRush Medical Center
63
PNP misfits in some cases
even with ‘optimal’ nonuniform D(x)
Duan ChenJohn TangRush Medical Center
64
Shielding/PNP is not enoughPNP includes Correlations
only in the Mean Field
PNP ignores ion- ion correlations
and
discrete particle effects:
Single Filing, Crowded Charge
Dielectric Boundary Force
65
Neither Field Theory nor Statistical Mechanics
easily accommodates Finite Size
of Ions and Protein Side-chainsHow can that be changed?
Learn from Mathematicians
and/or
Physical Chemists
Learned from Doug Henderson, J.-P. Hansen, among others…Thanks!
66
Learning with MathematiciansZeev Schuss, Boaz Nadler, Amit Singer
Dept’s of MathematicsTel Aviv University, Yale University
Molecular Biophysics, Rush Medical College
We extend usual chemical treatment to include flux and spatially nonuniform boundary conditions
We have concrete results only in the uncorrelated case!
We have learned how to derive PNP (by mathematics alone).
Count trajectories not states
67
Counting Langevin Trajectories in a Channel
(between absorbing boundary conditions)
implies PNP (with some differences)
PNP measures the density of trajectories (nearly)
Zeev Schuss, Amit Singer: Tel Aviv UnivBoaz Nadler: Yale Univ,
68
Conditional PNP
0 |
| |
yy y perme
p n
y x y
c y x c y x
Electric Force depends on Conditional Density of Charge
Nernst-Planck gives UNconditional Density of Charge
1
| 0x p y y xc x e y x DBF
m x
massfriction
Schuss, Nadler, Eisenberg
69
Boaz Nadler and Uwe HollerbachYale University Dept of Mathematics & Rush Medical Center
Dielectric Boundary ForceDBF
0
04
0
0 r r
r
r rF r r
PNP ignores correlations induced by
Discovered by Coalson/Kurnikova, Chung/Corry, … not us
Dielectric Boundary Force for point charge 0F r q r
70
Counting gives PNP with corrections
1) Correction for Dielectric Boundary Force*
2) NP (diffusion) equation describes probability of location but
3) Poisson equation depends on a conditional probability of location of charge, i.e., probability of location of trajectories, given that a positive ion is in a definite location.
4) Relation of is not known but can be estimated by closure relations, as in equilibrium statistical mechanics.
5) Closure relations involve correlations from single filing, finite volume of ions, boundary conditions, not well understood, … yet. Stay Tuned!
0 |r r
P
r
P
0 |r r r
P Pand
*by derivation, not assumption
71
Until mathematics is available, we
Follow the Physical Chemists, even if
their approximations are ‘irrational’, i.e., do not have error bounds.
Bob Eisenberg blames only himself for this approach
72
Physical Chemistry has shown that
Chemically Specific Properties of ions
come from their
Diameter and Charge (much) more than anything else.
Physical Models are Enough
Learned from Doug Henderson, J.-P. Hansen, among others…Thanks!
73
Physical Theories of Plasma of Ions
Determine (±1-2%) Activity* of ionic solutions
from
Infinite dilution,to
Saturated solutions, even in
Ionic melts.
*Free Energy per Mole
Learned from Doug Henderson, J.-P. Hansen, among others…Thanks!
74
1 ( ) ( ) ( )i i i ikT D J x x x}
Concentration-independent• geometric restrictions• solvation (Born) terms
Ideal term• electrochemical potential of point particles in the
electrostatic mean-field• includes Poisson equation
Excess chemical potential•Finite size effects•Spatial correlations
Chemical potential has three components
0 id ex( ) ( ) ( ) i i ix x x
75
Simplest Physical Theory MSA
1 2
2 2 2 20
0
Electrochemical Potential of species
ln
24 1 1 3
i
HS ESi i i i
HS i i i ii i
i i
i
z F RT
e z z
Ideal Excess
Electrostatic SpheresHard Spheres
Ionic radii I are known
values in bulk
Learned from Lesser Blum & Doug Henderson … Thanks! We use Simonin/Turq formulation.
76
Properties of Highly Compressible Plasma of Ions
Similar Results are computed by many
Different theories and Simulations
MSA is only simplest.
We (and others) have used MSA, SPM, MC, and DFT
MSA: Mean Spherical ApproximationSPM: Solvent Primitive ModelMC: Monte Carlo Simulation
DFT: Density Functional Theory of Solutions
Most Accurate
Atomic Detail
Calibrated !
Inhomogeneous Systems
77
back to channels
Selectivity in Channels
Wolfgang Nonner, Dirk GillespieUniversity of Miami and Rush Medical Center
78
Binding Curve
Wolfgang Nonner
79
O½
Selectivity Filter
Selectivity Filter Crowded with Charge
Wolfgang Nonner
80
Understand Selectivity well enough to
Make a Calcium Channelusing techniques of molecular genetics,
site-directed Mutagenesis
Goal:
George Robillard, Henk Mediema, Wim Meijberg
81
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 from the filter by crowded charge effectsCa2+ enters the filter and displaces Na+
Biological Case
Wolfgang Nonner Dirk Gillespie
82
Sensitivity to Parameters
VolumeDielectric
Coefficient
83
Trade-offs ‘1.5’ adjustable parameters
Volume
Die
lect
ric
Co
effi
cien
t
84
Binding CurvesSensitivity to Parameters
0.75 nm3 volume 0.20 nm3 volume
85
At Large Volumes
Electrical Potential can Reverse0.75 nm3 volume 0.20 nm3 volume
- - 0 Potential- -
0 Potential
Positive
Negative
Negative
86
Competition of Metal Ions vs. Ca++ in L-type Ca Channel
Nonner & Eisenberg
87
Similar Results have been found by
Henderson, Boda, et al.Hansen, Melchiona, Allen, et al.,
Nonner, Gillespie, Eisenberg, et al.,Using MSA, SPM, MC and DFT
for the L-type Ca Channel
MSA: Mean Spherical ApproximationSPM: Solvent Primitive ModelMC: Monte Carlo Simulation
DFT: Density Functional Theory of Solutions
Atomic Detail
Calibrated!
Most Accurate
Inhomogeneous Systems
88
Best Result to Date with Atomic Detail Monte Carlo,
including Dielectric Boundary Force
Dezso Boda, Dirk Gillespie, Doug Henderson, Wolfgang Nonner
Na+
Na+
Ca++
Ca++
89
Other
Properties of Ion Channels are likely to involve
more subtle physics including
orbital delocalizationand
chemical binding Selectivity apparently does not!
Learned from Doug Henderson, J.-P. Hansen, among others…Thanks!
90
Ionic Selectivity in Protein ChannelsCrowded Charge Mechanism
Simplest Version: MSA
How doesCrowded Charge give Selectivity?
91
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 from the filter by crowded charge effectsCa2+ enters the filter and displaces Na+
Biological Case
Wolfgang Nonner Dirk Gillespie
92
Ionic Selectivity in Protein ChannelsCrowded Charge Mechanism
4 Negative Charges of glutamates of protein
DEMAND 4 Positive Charges
nearby
either 4 Na+ or 2 Ca++
93
Ionic Selectivity in Protein ChannelsCrowded Charge Mechanism
Simplest Version: MSA
2 Ca++ are LESS CROWDED than 4 Na+,
Ca++ SHIELDS BETTER than Na+, so
Protein Prefers Calcium
942 Ca++ are LESS CROWDED than 4 Na+
95
What does the protein do?
Selectivity arises from Electrostatics and Crowding of Charge
Certain MEASURES of structure are
Powerful DETERMINANTS of Functione.g., Volume, Dielectric Coefficient, etc.
Precise Arrangement of Atoms is not involved in the model, to first order.
96
Protein provides Mechanical Strength
Volume of PoreDielectric 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?
97
Mechanical StrengthVolume of Pore
Dielectric Coefficient/BoundaryPermanent Charge
But not the precise arrangement of atoms
Implications for Artificial Channels
Design Goals are
98
Implications for Traditional Biochemistry
Traditional Biochemistry focuses on
Particular locations of atoms
99
Traditional Biochemistry assumes
Rate Constants Independent
of Concentration & Conditions
100
Implications for Traditional Biochemistry
Traditional Biochemistry(more or less)
Ignores the Electric Field
101
ButRate Constants depend steeply
on
Concentration and
Electrical Properties*
because of shielding, a fundamental property of matter, independent of model, in my opinion.
*nearly always
102
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
Important Determinant of Biological Properties
103
Traditional Biochemistryignores
Shielding and Crowded Charge although
Shielding DominatesProperties of Ionic Solutions
and cannot be ignored in Channels and Proteins
in my opinion
104
Make a Calcium Channelusing techniques of molecular genetics,
site-directed Mutagenesis
How can we use these ideas?
George Robillard, Henk Mediema, Wim Meijberg
BioMaDe Corporation, Groningen, Netherlands
105
More?
106
Functioncan be predictedFrom Structure
givenFundamental
Physical Laws(sometimes, in some cases).
107
Strategy
Use site-directed mutagenesis to put in extra glutamates
and create an EEEE locus in the selectivity filter of OmpF
Site-directed
mutagenesis
R132
R82E42
E132
R42 A82
Wild type EAE mutant
E117 E117
D113D113
George Robillard, Henk Mediema, Wim MeijbergBioMaDe Corporation, Groningen, Netherlands
108
-100 -50 50 100
-150
-50
50
150
ECa
WT
EAE
Current (pA)
Voltage (mV)
Cis Trans
1 M CaCl2 0.1 M CaCl2
Ca2+
Ca2+
IV-PLOT
Cis Trans Cis Trans
IV-plot EAE: current reverses at equilibrium potential of Ca2+ (ECa),
indicating the channel can discriminate between Ca2+ and Cl-
Zero-current potentialor reversal potential = measure of ion selectivity
Henk MediemaWim Meijberg
109
PCa/PCl
WT 2.8AAA 25EAE >100
Ca2+ over Cl- selectivity (PCa/PCl)recorded in 1 : 0.1 M CaCl2
SUMMARY OF RESULTS (1)
Conclusions:
- Taking positive charge out of the constriction zone (= -3, see control mutant AAA) enhances the cation over anion permeability.
- Putting in extra negative charge (= -5, see EAE mutant) further increases the cation selectivity.
Henk MediemaWim Meijberg
110
PCa/PNa
WT 2.2AAA 3.7EAE 4.2
Ca2+ over Na+ selectivity (PCa/PNa)recorded in 0.1 M NaCl : 0.1 M CaCl2
SUMMARY OF RESULTS (2)
Conclusion:
- Compared to WT, EAE shows just a moderate increase of the Ca2+ over Na+ selectivity.
- To further enhance PCa/PNa may require additional negative charge and/or a change of the ‘dielectric volume’.
Work in Progress!
Henk MediemaWim Meijberg
111
Selectivity Differs
in Different Types of ChannelsWolfgang Nonner
Dirk Gillespie
Other Types of Channels
112
CaCa channelchannel Na channelNa channel Cl channelCl channel K channelK channel
prefers
Small ions Ca2+ > Na+
prefers
Small ions Na+ > Ca2+
Na+ over K+
prefers
Large ionsprefers
K+ > Na+
Selectivity filter
EEEE 4 − charges
Selectivity filter
DEKA2 −, 1+ charge
Selectivity filter
hydrophobicpartial charges
Selectivity filter
single filingpartial charges
PNP/DFT Monte Carlo Bulk Approx Not modeled yet
Selectivity of Different Channel Types
The same crowded charge mechanism can explain allthese different channel properties
with surprisingly little extra physics.
113
Sodium Channel
(with D. Boda, D. Busath, and D. Henderson)
Related to Ca++ channel• removing the positive lysine (K) from the DEKA locus makes calcium-selective channel
High Na+ selectivity• 1 mM CaCl2 in 0.1 M NaCl gives all Na+ current (compare to
calcium channel)• only >10 mM CaCl2 gives substantial Ca++ current
Monte Carlo method is limited (so far) to a uniform dielectric
Stay tuned….
Wolfgang Nonner Dirk Gillespie
114
Na+
Ca++
Na+/Ca2+ Competition in the Sodium Channel
Biological Region
Ca++ in bath (M)
Wolfgang Nonner Dirk Gillespie
115
•Model gives small-ion selectivity. •Result also applies to the calcium channel.
Na+/Alkali Metal Competition in Na+ Channel
Biological Region
Wolfgang Nonner Dirk Gillespie
116
‘New’ result from PNP/SPM combined analysis
Spatial Nonuniformity in Na+ Channel
Wolfgang Nonner Dirk Gillespie
117
Na+ vs K+ Selectivity
Na+
K+ Channel
Protein
ProteinProtein
Protein
Na+ Channel
Wolfgang Nonner Dirk Gillespie
118
Na+ Channels Select Small Na+ over Big K+
because(we predict)
Protein side chains are small
allowing
Small Na+ to Pack into Niches
K+ is too big for the niches!Wolfgang Nonner Dirk Gillespie
Summary of Na+ Channel
119
Sodium Channel Summary
Na+ channel is a Poorly Selective
Highly Conducting Calcium channel,
which is Roughened so it prefers
Small Na+ over big K+
Wolfgang Nonner Dirk Gillespie
120
Cl− Selective ChannelSelective for Larger Anions
The Dilute ChannelWolfgang Nonner Dirk Gillespie
121
Chloride Channel
Channel prefers large anions in experiments,
Low Density of Charge (several partial charges in 0.75
nm3)
Selectivity Filter contains hydrophobic groups
• these are modeled to (slightly) repel water• this results in large-ion selectivity
Conducts only anions at low concentrations
• Conducts both anions and cations at high concentration
Current depends on anion type and concentrationWolfgang NonnerDirk GillespieDoug HendersonDezso Boda
122
Biological Case
Chloride ChannelSelectivity depends Qualitatively on concentration
Wolfgang Nonner Dirk Gillespie
123
The Dilute Channel: Anion Selective
Channel protein creates a
Pressure difference between bath and channel
Large ions like Cl– are Pushed into the channel more than smaller ions
like F–
Wolfgang Nonner Dirk Gillespie
124
Key: Hydrophobic Residues Repel Water giving• Large-ion selectivity
(in both anion and cation channels). • Peculiar non-monotonic conductance properties
and IV curves observed in experiments
Hydrophobic repulsion can give gating. ‘Vacuum lock’ model of gating
(M. Green, D. Henderson; J.-P. Hansen; Mark Sansom; Sergei Sukarev)
Chloride Channel
Wolfgang Nonner Dirk Gillespie
125
Conclusion
Each channel type is a variation on a theme of
Crowded Chargeand Electrostatics,
Each channel types uses particular physics as a variation.
Wolfgang Nonner Dirk Gillespie
126
Functioncan be predictedFrom Structure
givenFundamental
Physical Laws(sometimes, in some cases).
127
More?
DFT
128
Density Functionaland
Poisson Nernst Planckmodel of
Ion Selectivity in
Biological Ion Channels
Dirk GillespieWolfgang Nonner
Department of Physiology and BiophysicsUniversity of Miami School of Medicine
Bob EisenbergDepartment of Molecular Biophysics and Physiology
Rush Medical College, Chicago
129
We (following many others) have used
Many Theories of Ionic Solutions
asHighly Compressible Plasma of Ions
with similar results
MSA, SPM, MC and DFT
MSA: Mean Spherical ApproximationSPM: Solvent Primitive ModelMC: Monte Carlo Simulation
DFT: Density Functional Theory of Solutions
Most Accurate
Atomic Detail
Inhomogeneous Systems
130
Density Functional Theory
HS excess chemical potential is from free energy functional
HS k HS n d x x x
Energy density depends on “non-local densities”
Nonner, Gillespie, Eisenberg
131
HS kHSi
i
HSikT n d
n
xx
x
x x x x
HS excess chemical potential is
Free energy functional is due to Yasha Rosenfeld and is considered more than adequate by most physical chemists.
The double convolution is hard to compute efficiently.
Nonner, Gillespie, Eisenberg
We have extended the functional to
Charged Inhomogeneous Systems with a bootstrap perturbation method that fits MC simulations
nearly perfectly.
132
Example of an Inhomogeneous Liquid
A two-component hard-sphere fluid near a wall in equilibrium (a small and a large species).
Near the wall there are excluded-volume effects that cause the particles to pack in layers.
These effects are very nonlinear and are amplified in channels because of the high densities.
small species
large species
133
The ProblemWe are interested in computing the flux of ions between two baths of fixed ionic concentrations. Across the system an electrostatic potential is applied.
Separating the two baths is a lipid membrane containing an ion channel.
ionic concentrations and electrostatic potentialheld constant far from
channel
membrane with ion channel
ionic concentrations and electrostatic potentialheld constant far from
channel
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Modeling Ion Flux
The flux of ion species i is given by the constitutive relationship
1i i i iD
kT J x x x
The flux follows the gradient of the total chemical potential.
where Di is the diffusion coefficienti is the number densityi is the total chemical potential of species i
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1 ( ) ( ) ( )i i i ikT D J x x x}
0 id ex( ) ( ) ( ) i i ix x x
concentration-independent• geometric restrictions• solvation (Born) terms
ideal term• electrochemical potential of point particles in the
electrostatic mean-field• includes Poisson equation
excess chemical potentialthe “rest”: the difference between the “real” solution and the ideal solution
The chemical potential has three components
136
When ions are charged, hard spheres the excess chemical potential is split into two parts
1 ( ) ( ) ( )i i i ikT D J x x x}
0 id ex( ) ( ) ( )i i i x x x}HS ES( ) ( )i i x x
Electrostatic Componentdescribes the electrostatic effects of charging up the ions
Hard-Sphere Componentdescribes the effects of excluded volume
the centers of two hard spheres of radius R cannot come closer than 2R
137
Density Functional Theory
HS excess chemical potential is derived from free energy functional
HS k HS n d x x x
Energy density depends on “non-local densities”
Nonner, Gillespie, Eisenberg
138
HS kHSi
i
HSikT n d
n
xx
x
x x x x
HS excess chemical potential is
Free energy functional is due to Yasha Rosenfeld and is considered more than adequate by most physical chemists.
The double convolution is hard to compute efficiently.
Nonner, Gillespie, Eisenberg
We have extended the functional to
Charged Inhomogeneous Systems with a bootstrap perturbation method that fits MC simulations
nearly perfectly.
139
Density Functional Theory
Energy density depends on “non-local densities”:
1 2 1 20 3
3
332 22
2 223
ln 11
124 1
V VHS
V V
n nn n n
n
n
nn
n nx
n n
Nonner, Gillespie, Eisenberg
140Nonner, Gillespie, Eisenberg
The non-local densities ( = 0, 1, 2, 3, V1, V2) are averages of the local densities:
2 3
0 1 22
2 1 2
4 4
4
i i
i
i i i i
i i i i i
V V Vi i i i i
n d
R R
R R
R R
x x x x x
r r r r
r r r
rr r r r
r
1
where is the Dirac delta function, is the Heaviside step function, and Ri is the radius of species i.
1
141
We use Rosenfeld’s perturbation approach to compute the electrostatic component.
Specifically, we assume that the local density i(x) is a perturbation of a reference density iref(x):
i x i
ref x i x
The ES Excess Chemical PotentialDensity Functional Theory
142
The Reference FluidIn previous implementations, the reference fluid was chosen to be a bulk fluid. This was both appropriate for the problem being solved and made computing its ES excess chemical potential straight-forward.
However, for channels a bulk reference fluid is not sufficient. The channel interior can be highly-charged and so 20+ molar ion concentrations can result. That is, the ion concentrations inside the channel can be several orders of magnitude larger than the bath concentrations.
For this reason we developed a formulation of the ES functional that could account for such
large concentration differences.
143
Test of ES Functional
To test our ES functional, we considered an equilibrium problem designed to mimic a calcium channel.
two compartments were equilibratededge effects fully computed
24 M O-1/2
CaCl2
NaClorKCl
0.1 M
The dielectric constant was 78.4 throughout the system.
144
145
146
‘New’ Mathematics is Needed:
Analysis of Simulations
147
Can Simulations serve as
“Fundamental Physical Laws”?
Direct Simulations are Problematic Even today
148
Can simulations serve as fundamental physical laws?
Direct Simulations are Problematic Even today
Simulations so far cannot reproduce macroscopic variables and phenomena known
to dominate biology
149
Simulations so far often do not reproduce
Concentration (i.e., number density)
(or activity coefficient) Energy of Electric Field Ohm’s ‘law’ (in simple situations)
Fick’s ‘law’ (in simple situations)
Conservation Laws (e.g., of energy)
Fluctuations in number density
150
The larger the calculation, the more work done, the greater the error
First Principle of Numerical Integration
First Principle of Experimentation
The more work done, the less the error
Simulations as fundamental physical laws (?)
151
How do we include Macroscopic Variables in
Atomic Detail Calculations?
Another viable approachis
Hierarchy of Symplectic Simulations
152
Analysis of Simulations
e.g.,How do we include
Macroscopic VariablesConservation laws
in
Atomic Detail Calculations?
Because mathematical answer is unknown, I use an Engineering Approach
Hierarchy of Low Resolution Models
153
Simulations produce too many numbers
106 trajectories each 10-6 sec long, with 109 samples in each trajectory,
in background of 1022 atoms
Why not simulate?
154
Simulations need a theory that
Estimates Parameters (e.g., averages)
or
Ignores Variables
Theories and Models are Unavoidable! (in my opinion)
155
Symplectic integrators are precise in‘one’ variable at a time!
It is not clear (at least to me) that symplectic integrators can be precise in all relevant variables at one time