Figure by Raimund Dutzler

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Figure by Raimund Dutzler
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ION CHANNELS Proteins with a Hole
Channels form a class of Biological Systemsthat
can be analyzed with Physics as
Usual Physics-Mathematics-Engineering are the
proper language for Ion Channels in my opinion
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Ion Channels can be analyzed with
Physics as Usual along with Biology as Usual
Why think? . . . Exhaustively experiment. Then,
think Claude Bernard Appropriate when nothing
was known of Inverse Problems!
Cited in The Great Influenza, John M. Barry,
Viking Penguin Group 2004
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Ion Channels are the Main Molecular Controllers
Valves of Biological Function
Figure by Raimund Dutzler
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ION CHANNELS as Physical Devices

• Channels control flow of Charged Spheres
• Channels have Simple Invariant Structure on the
  biological time scale.
• Why cant we predict the movement of Charged
  Spheres through a Hole?

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Physical Characteristics of Ion ChannelsNatural
Nanodevices
Figure by Raimund Dutzler
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ION CHANNELS as Technological Objects
Channels Control Macroscopic Flow with Atomic
Resolution
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Ion Channelsare Important Enough to be Worth
the Effort
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Goal Predict Function From Structure given Fundam
ental Physical Laws
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Gating Opening of Porin Trimer
John TangRush Medical Center
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Single Channel Currents have little variance
John TangRush Medical Center
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Lipid Bilayer SetupRecordings from a Single
Molecule
Axopatch
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Patch clamp and Bilayer apparatus clamp ion
concentrations in the baths and the voltage
across membranes.
Patch Clamp Setup Recordings from One Molecule
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Current depends on
Bilayer Setup
Voltage in baths Concentration in
baths Fixed Charge on channel protein
John TangRush Medical Center
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Current depends on type of ionSelectivity
John TangRush Medical Center
Bilayer Setup
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Goal Predict Function From Structure given Fundam
ental Physical Laws
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Structures Location of charges are known with
atomic precision (0.1 Å) in favorable cases.
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Charge Mutation in Porin
Structure determined by x-ray crystallography in
Tilman Schirmers lab Figure by Raimund Dutzler
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Goal Predict Function From Structure given Fundam
ental Physical Laws
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But What are the Fundamental Physical Laws?
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Verbal Models Are Popular with Biologists but Inad
equate
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• 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. 188
• In Maxwell on Heat and Statistical Mechanics,
  Garber, Brush and Everitt, 1995

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• I fear
• Biologists indulge themselves
• with verbal models
• of molecules
• where
• Maxwell abstained

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Verbal Models areVagueandDifficult to Test
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Verbal Modelslead to Interminable Argument
and Interminable Investigation
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thus,to Interminable Funding
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and so Verbal Models Are Popular
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Can Molecular Simulationsserve as Fundamental
Physical Laws?
Only if they count correctly !
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It is very difficult for Molecular
Dynamics to count well enough to
reproduce Conservation Laws (e.g., of number,
energy) Concentration (i.e., number density) or
activity Energy of Electric Field Ohms law
(in simple situations) Ficks law (in simple
situations) Fluctuations in number density
(e.g., entropy)
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Can Molecular Simulations serve as Fundamental
Physical Laws?
Only if Calibrated!
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Calibrated Molecular Dynamics may be possible
Pair Correlation Function in Bulk Solution

• MD without Periodic Boundary Conditions - HNC
  HyperNetted Chain

Saraniti Lab, IIT Aboud, Marreiro, Saraniti
Eisenberg
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Calibrated Molecular Dynamics may be possible
Pair Correlation Function in Bulk Solution

• MD without Periodic Boundary Conditions BioMOCA
  - Equilibrium Monte Carlo (ala physical
  chemistry)

van der Straaten, Kathawala, Trellakis, Eisenberg
Ravaioli
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Until Mathematics of Simulations is available we
take an Engineering Approach
Essence of Engineering is knowing What
Variables to Ignore! WC Randels quoted in
Warner IEEE Trans CT 482457 (2001)
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What variables should we ignore when 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)
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Use the scientific method Guess and
Check! Intelligent Guesses are MUCH more
efficient When theory works, need few checks
guesses are almost as good as experiments.
in
Mechanical Engineering, Electricity, Computer
Science, Hydrodynamics,
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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
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Channels are only Holes Why cant 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.
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We start with Electrostaticsbecause of biology
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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
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Start with electrostatics because
1 charge gives
in a Sphere of diameter 14 Å
500 µvolt has significant effect on IV curve
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We start without quantum chemistry(only at
first)
i.e., delocalization of orbitals of outer
electrons
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Although we start with electrostatics We will
soon add Physical Models of Chemical Effects
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It is appropriate to be skepticalof analysis in
which the only chemistry is physical But give me
a chance, ask I will be in Linz for a
week! (thanks to Heinz!) Or email
beisenbe_at_rush.edu for the papers
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Very Different from Traditional Structural
Biology which, more or less, ignores
Electrostatics
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Lowest resolution theory that includes
Electrostatics and Flux is (probably)
Poisson-Nernst-Planck (PNP)
PNP, Gouy-Chapman, (nonlinear) Poisson-Boltzmann,
Debye-Hückel, are fraternal twins or siblings
with similar resolution
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ION CHANNEL MODELS
Poissons Equation
Continuity Equation Drift
- Diffusion
For Derivation Stay tuned Schuss, Singer, Nadler
et al
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One Dimensional PNP
Poissons Equation
Drift-Diffusion Continuity
Equation
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Boundary conditions are Crucial Flared Access to
Channel Required in 1D
Bath
Bath
Membrane
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PNP Forward Problem
EXPERIMENTAL CONDITIONS Bath concentrations
Bath Potential Difference
I
V
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Current Voltage Relation Gramicidin 3D PNP
Uwe Hollerbach
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PNP -1 Inverse Problem
I
V
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Charge Mutation in Porin
Ompf
G119D
Structure determined by x-ray crystallography in
Tilman Schirmers lab
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Fit of 1D PNP Current-Voltage 100 mM KCl
OmpF G119D
Duan ChenJohn TangRush Medical Center
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Net Charge Difference 0.13 ? 1.1 ? 0.97e

Duan ChenJohn Tang
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Main Qualitative Result
Shielding Dominates Electric Properties of
Channels, Proteins, as it does Ionic Solutions
Shielding is ignored in traditional treatments
of Ion Channels and of Active Sites of proteins
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Main Qualitative ResultShielding in Gramicidin
Uwe HollerbachRush Medical Center
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PNP misfits in some cases even with optimal
nonuniform D(x)
Duan ChenJohn TangRush Medical Center
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Shielding/PNP is not enough PNP includes
Correlations only in the Mean Field PNP ignores
ion- ion correlations anddiscrete particle
effects Single Filing, Crowded
ChargeDielectric Boundary Force
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Neither Field Theory nor Statistical Mechanics
easily accommodates Finite Size of Ions and
Protein Side-chains How can that be
changed? Learn from Mathematicians and/or
Physical Chemists
Learned from Doug Henderson, J.-P. Hansen, among
othersThanks!
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Learning with Mathematicians
Zeev Schuss, Boaz Nadler, Amit SingerDepts of
MathematicsTel Aviv University, Yale
UniversityMolecular 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
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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 Univ Boaz
Nadler Yale Univ,
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Conditional PNP
Schuss, Nadler, Eisenberg
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Boaz Nadler and Uwe Hollerbach Yale University
Dept of Mathematics Rush Medical Center
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by derivation, not assumption
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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
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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
othersThanks!
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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
othersThanks!
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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 SimulationDFT Density Functional
Theory of Solutions
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back to channels Selectivity in Channels
Wolfgang Nonner, Dirk GillespieUniversity of
Miami and Rush Medical Center
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Wolfgang Nonner
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Goal
Understand Selectivity well enough to Make a
Calcium Channel using techniques of molecular
genetics, site-directed Mutagenesis
George Robillard, Henk Mediema, Wim Meijberg
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Sensitivity to Parameters
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Trade-offs 1.5 adjustable parameters
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Binding CurvesSensitivity to Parameters
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At Large Volumes Electrical Potential can Reverse
Positive
Negative
Negative
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Competition of Metal Ions vs. Ca in L-type Ca
Channel
Nonner Eisenberg
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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
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Best Result to Date with Atomic Detail Monte
Carlo, including Dielectric Boundary Force
Na
Dezso Boda, Dirk Gillespie, Doug Henderson,
Wolfgang Nonner
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Other Properties of Ion Channels are likely to
involve more subtle physics including orbital
delocalization and chemical binding
Selectivity apparently does not!
Learned from Doug Henderson, J.-P. Hansen, among
othersThanks!
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Ionic Selectivity in Protein Channels Crowded
Charge Mechanism Simplest Version MSA
How doesCrowded Charge give Selectivity?
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Ionic Selectivity in Protein Channels Crowded
Charge Mechanism
4 Negative Charges of glutamates of
protein DEMAND 4 Positive Chargesnearby either
4 Na or 2 Ca
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Ionic Selectivity in Protein Channels Crowded
Charge Mechanism Simplest Version MSA
2 Ca are LESS CROWDED than 4 Na, Ca SHIELDS
BETTER than Na, so Protein Prefers Calcium
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2 Ca are LESS CROWDED than 4 Na
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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.
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What does the protein do?
Protein provides Mechanical StrengthVolume of
PoreDielectric Coefficient/BoundaryPermanent
Charge Precise Arrangement of Atoms is not
involved in the model, to first
order. butParticular properties (measures) of
the protein are crucial!
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Implications for Artificial Channels Design
Goals are
Mechanical StrengthVolume of PoreDielectric
Coefficient/BoundaryPermanent Charge But not
the precise arrangement of atoms
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Implications for Traditional Biochemistry
Traditional Biochemistry focuses on Particular
locations of atoms
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Traditional Biochemistry assumes Rate
Constants Independent of Concentration
Conditions
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Implications for Traditional Biochemistry Tradit
ional Biochemistry(more or less) Ignores the
Electric Field
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But Rate Constants depend steeply on
Concentration and Electrical Properties
because of shielding, a fundamental property of
matter, independent of model, in my
opinion. nearly always
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Electrostatic Contribution to Dissociation
Constant is large and is an Important
Determinant of Biological Properties
Change of Dissociation Constantwith
concentration is large and is an Important
Determinant of Biological Properties
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Traditional BiochemistryignoresShielding and
Crowded Charge although Shielding
Dominates Properties of Ionic Solutions and
cannot be ignored in Channels and Proteins in my
opinion
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How can we use these ideas?
Make a Calcium Channel using techniques of
molecular genetics, site-directed Mutagenesis
George Robillard, Henk Mediema, Wim
Meijberg BioMaDe Corporation, Groningen,
Netherlands
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More?
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Function can be predicted From
Structure given Fundamental Physical
Laws (sometimes, in some cases).
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Strategy Use
site-directed mutagenesis to put in extra
glutamates and create an EEEE locus in the
selectivity filter of OmpF
George Robillard, Henk Mediema, Wim
Meijberg BioMaDe Corporation, Groningen,
Netherlands
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Zero-current potential or reversal potential
measure of ion selectivity
Henk Mediema Wim Meijberg
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SUMMARY OF RESULTS (1)
Ca2 over Cl- selectivity (PCa/PCl) recorded in 1
0.1 M CaCl2
Henk Mediema Wim Meijberg
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.
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Henk Mediema Wim Meijberg
SUMMARY OF RESULTS (2)
Ca2 over Na selectivity (PCa/PNa) recorded in
0.1 M NaCl 0.1 M CaCl2
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.
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Other Types of Channels
Selectivity Differsin Different Types of
Channels
Wolfgang Nonner Dirk Gillespie
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Selectivity of Different Channel Types
Ca channel Na channel Cl channel K channel
prefers Small ions Ca2 gt Na prefers Small ions Na gt Ca2 Na over K prefers Large ions prefers K gt Na
Selectivity filter EEEE 4 - charges Selectivity filter DEKA 2 -, 1 charge Selectivity filter hydrophobic partial charges Selectivity filter single filing partial charges
PNP/DFT Monte Carlo Bulk Approx Not modeled yet
The same crowded charge mechanism can explain
all these different channel properties with
surprisingly little extra physics.
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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 gt10 mM CaCl2 gives substantial Ca current
• Monte Carlo method is limited (so far) to a
  uniform dielectric
• Stay tuned.

Wolfgang Nonner Dirk Gillespie
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Ca
Na
Ca in bath (M)
Wolfgang Nonner Dirk Gillespie
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Wolfgang Nonner Dirk Gillespie
Na/Alkali Metal Competition in Na Channel

• Model gives small-ion selectivity.
• Result also applies to the calcium channel.

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New result from PNP/SPM combined
analysisSpatial Nonuniformity in Na Channel
Wolfgang Nonner Dirk Gillespie
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Na vs K Selectivity
Wolfgang Nonner Dirk Gillespie
Na
Na Channel
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Summary of Na Channel
Na Channels Select Small Na over Big K
because(we predict) Protein side chains are
smallallowing Small Na to Pack into Niches
K is too big for the niches!
Wolfgang Nonner Dirk Gillespie
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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
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Wolfgang Nonner Dirk Gillespie
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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 concentration

Wolfgang NonnerDirk GillespieDoug
HendersonDezso Boda
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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
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Chloride Channel

• 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)

Wolfgang Nonner Dirk Gillespie
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Conclusion

• Each channel type is a variation on a theme of
  Crowded Charge
• and Electrostatics,
• Each channel types uses particular physics as a
  variation.

Wolfgang Nonner Dirk Gillespie
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Function can be predicted From
Structure given Fundamental Physical
Laws (sometimes, in some cases).
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More? DFT
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Density Functional and Poisson Nernst
Planck model of Ion Selectivity in Biological
Ion Channels Dirk Gillespie Wolfgang
Nonner Department of Physiology and
Biophysics University of Miami School of
Medicine Bob Eisenberg Department of Molecular
Biophysics and Physiology Rush Medical College,
Chicago
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Density Functional Theory
HS excess chemical potential is from free energy
functional
Energy density depends on non-local densities
Nonner, Gillespie, Eisenberg
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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.
We have extended the functional to Charged
Inhomogeneous Systems with a bootstrap
perturbation method that fits MC simulations
nearly perfectly.
Nonner, Gillespie, Eisenberg
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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
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The Problem
We 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
potential held constant far from channel
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Modeling Ion Flux
The flux of ion species i is given by the
constitutive relationship
where Di is the diffusion coefficient ?i is the
number density ?i is the total chemical
potential of species i
The flux follows the gradient of the total
chemical potential.
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The chemical potential has three components

• concentration-independent
• geometric restrictions
• solvation (Born) terms

excess chemical potential the rest the
difference between the real solution and the
ideal solution

• ideal term
• electrochemical potential of point particles in
  the electrostatic mean-field
• includes Poisson equation

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Density Functional Theory
HS excess chemical potential is derived from free
energy functional
Energy density depends on non-local densities
Nonner, Gillespie, Eisenberg
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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.
We have extended the functional to Charged
Inhomogeneous Systems with a bootstrap
perturbation method that fits MC simulations
nearly perfectly.
Nonner, Gillespie, Eisenberg
139
Density Functional Theory
Energy density depends on non-local densities
Nonner, Gillespie, Eisenberg
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Nonner, Gillespie, Eisenberg
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The ES Excess Chemical Potential Density
Functional Theory
We use Rosenfelds 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)
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The Reference Fluid
In 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 equilibrated edge
effects fully computed
The dielectric constant was 78.4 throughout the
system.
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New Mathematics is Needed Analysis of
Simulations
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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 Ohms law (in simple
situations) Ficks law (in simple
situations) Conservation Laws (e.g., of
energy) Fluctuations in number density
150
Simulations as fundamental physical laws (?)
First Principle of Numerical Integration
The larger the calculation, the more work done,
the greater the error
First Principle of Experimentation
The more work done, the less the error
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How do we include Macroscopic Variables in
Atomic Detail Calculations?
Another viable approachisHierarchy of
Symplectic Simulations
152
Analysis of Simulations e.g., How do we include
Macroscopic Variables Conservation laws in
Atomic Detail Calculations?
Because mathematical answer is unknown, I use an
Engineering Approach Hierarchy of Low Resolution
Models
153
Why not simulate?
Simulations produce too many numbers 106
trajectories each 10-6 sec long, with 109
samples in each trajectory, in background of
1022 atoms
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