Optimal Design of Ion Channels and Nanopores
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Martin Burger Institute for Computational and Applied Mathematics
European Institute for
Molecular Imaging
Center for
Nonlinear Science CeNoS
Optimal Design of Ion Channels and Nanopores
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Ion Channels and Nanopores 2
31.3.2008Martin Burger
Joint Work withKattrin Arning, LinzMary Wolfram, Münster / LinzBob Eisenberg, ChicagoHeinz Engl, LinzZuzanna Siwy, Irvine
Rene Pinnau, Kaiserslautern
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31.3.2008Martin Burger
Ion Channels and LifeMost of human life occurs in cells
Transport through cell membraneis essential for biological function
The transport or blocking of ions is controlled by channels
Ion channels = proteins with ahole in their middle
~ 5 µm
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31.3.2008Martin Burger
Ion Channels and LifeFlow of ions creates / modifies electric potential
Electrical field determinesflow direction of ions
A substantial fraction of drugsare designed to influence channel behaviour
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31.3.2008Martin Burger
Ion Channels and LifeFigures by Raimund Dutzler, courtesy Bob Eisenberg
Chemical Bonds are linesSurface is Electrical Potential
Red is positiveBlue is negative
Chemist’s View
All Atoms View
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31.3.2008Martin Burger
Channel FunctionIon channel control flow like a micro-electronic charge
Proteins in the channel walls create apermanent charge in the channel (likethe doping of a semiconductor device)
Additional effects due to size exclusionin narrow channels
~30 Å
K+
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31.3.2008Martin Burger
Channel FunctionChannel function creates two observable effects:
- Gating: (random) opening (flow, current) and closing (no flow) of channels
- Selectivity: in the open state flow of certain ions preferred over others, some (almost) completely blocked
Corresponding experimental measures always related to currents at different voltages and concentrations
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31.3.2008Martin Burger
Channel FunctionExperimental setup:
Bath of ions and water on both sidesof channel
Bath concentrations controlledVoltage applied across channel
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31.3.2008Martin Burger
GatingSingle channel current is a Random Signal
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SelectivityObserved current-voltage curves as in microelectronicsCurves for different bath concentrationsindicate selectivity
OmpF KCl 1M 1M
||
OmpF CaCl2
1M 1M
||
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31.3.2008Martin Burger
ModellingMicroscopic model based on equations of motions
Forces include interaction between ions, and with protein
; 2kp
k x q pp pkk k
f kTm mx x w Positive cat ion,
e.g., p = Na+
;
Newton'sLaw Friction & Noise
2kn
k x q nn nkk k
f kTm mx x w
Negative an ion,
e.g., n = Cl¯
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31.3.2008Martin Burger
ModellingForce fk includes
- Excess „chemical“ force- Electrical force: Electrical potential to be computed from Poisson equation with sources from all charges (ions, protein)
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31.3.2008Martin Burger
Macroscopic Model for Open StateStandard Coarse-Graining leads to Poisson-Nernst-Planck(Poisson-drift-diffusion) system for potential and ion concentrations
Similar issues as in Semiconductor Simulation
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31.3.2008Martin Burger
ModellingAdditional issues due to finite size (chemical) effects
Excess chemical potential includes - Chemical interaction between the ions- Chemical interaction between ions and proteins
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31.3.2008Martin Burger
ModellingComputation of the macroscopic excess chemical potential is a hard problem
Various models and schemes at different resolution
We currently use density functional theory (DFT) of statistical physics. Consequence are many nonlinear integrals to be computed with fine resolution and self-consistency iterations: lead to enormous computationaleffort
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31.3.2008Martin Burger
ModellingDue to narrow size of channels in two dimensions and predominant flow in one direction, use of effective spatially one-dimensional models becomes attractive
Model derivation still quite open, mainly due to chemical forces
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31.3.2008Martin Burger
ModellingIn some channels, like the L-type Ca Channel, it is reasonable that structure is not frozen at the working temperature.
Hence, the concentration of the protein charges (modelled as half-charged oxygens for L-type Ca) needs to be modelled as an additional unknownBinding forces of the protein on its charges are encoded in a confining potential
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31.3.2008Martin Burger
ModellingStructure can be represented via confining potentials in a unified way (almost infinite to include rigid structures)
Confining potential can become the actual design variable in the model, when designing structure
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31.3.2008Martin Burger
ModellingNumerical Simulation by stabilized mixed finite elements
L-type Ca channel with 8 half-charged oxygens
Applied Voltage 50mV
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31.3.2008Martin Burger
ModellingMulti-D Simulation (here 3D Ca2+
synthetic channel with rotational symmetry)
Simulations by Mary Wolfram
Na+
Cl-
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31.3.2008Martin Burger
ModellingGating models hardly available, physical basis of gating still unclear, various possibilities- Bubble formation- Conformation changes in the protein - Protonization- Precipitation- ..
Active research, will get to suitable models in a few years
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31.3.2008Martin Burger
Why optimal design ? Compare function of OmpF and G 119D: huge difference
OmpF KCl 1M 1M||
G119D KCl 1M 1M
||ompF KCl0.05 M
0.05M
||G119D KCl
0.05 M 0.05M
||
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31.3.2008Martin Burger
Why optimal design ? Compare structure of OmpF and G 119D: one mutation !
Structure determined by x-ray crystallography in Lab of T.Schirmer, Basel. Figures by R.Dutzler
Ompf G119D
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31.3.2008Martin Burger
Why optimal design ? Selective channelscan be built by controlled mutation
Many labs try, but rational designis still missing
30 60
-30
30
60
0
pA
mV
LECE (-7e)
LECE-MTSES- (-8e)
LECE-GLUT- (-8e)ECa
ECl
WT (-1e)
Calcium selective
As charge density increases, channel becomes calcium selective Erev ECa
Miedema et al, Biophys J 87: 3137–3147 (2004)
Unselective
Wild Type
MUTANT ─ Compound
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31.3.2008Martin Burger
Why optimal design ? Synthetic channels (nanopore) with gating and selectivity properties can be built by track etching from plastic (Siwy, UC Irvine / Trautmann, GSI Darmstadt)
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31.3.2008Martin Burger
Why optimal design ? Selectivity and I-Vcurves as for biological channels
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31.3.2008Martin Burger
Why optimal design ? Gating in nanopores
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31.3.2008Martin Burger
Optimal design as usual ? Previous work on optimal design of Semiconductor devices
Related issues except chemistry
Hinze-Pinnau 01-06, mb-Pinnau 03, Wolfram 07, mb-Pinnau-Wolfram 08, mb-Engl-Markowich et al 01-04
MOSFETs, from st.com
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31.3.2008Martin Burger
Optimal Design of Doping ProfilesTypical design-goal: maximize on-state current, keeping small off-state (leakage current)
Possible non-uniqueness from primary design goal
Secondary design goal: stay close to reference state (currently built design)
Sophisticated optimization tools possible for Poisson-Drift-Diffusion models Hinze-Pinnau 02/06, mb-Pinnau-Wolfram 08
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31.3.2008Martin Burger
Optimal Design of Doping ProfilesFast optimal design by simple trick
Instead of C, define new design variable as the total charge W = -q(n-p-C)Partial decoupling, simpler optimality systemGlobally convergent Gummel method for design
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31.3.2008Martin Burger
Optimal Design of Doping ProfilesWorks for single applied voltage, additional tricks are needed for „multi-load design“ (multiple applied voltages)Kaczmarz method: sweep over all voltages and solve single-voltage subproblems
On-off state design: one drive current (on-state), treated like before, in additon off-state current (fluctuations around zero) – modeled by linearized model around zero
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31.3.2008Martin Burger
On-/Off-State Design of Doping ProfilesMinimize combined functional Q of I (on-state current)K (linearized off-state current)
Alternative: constraints Regularized functional in the end ( W is relative charge to reference state):
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31.3.2008Martin Burger
On-/Off-State Design of Doping ProfilesOn-state equations as before (rewritten in Slotboom variables), W defined in on-state
Off-state problem
C needs to be eliminated in favour of W: leads to one-sided coupling with on-state
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31.3.2008Martin Burger
Gummel
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31.3.2008Martin Burger
Optimal Design of Doping ProfilesOn-off state design of bipolar diode
mb-Pinnau-Wolfram 08
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31.3.2008Martin Burger
Optimal Design of Doping ProfilesOptimization of a MOSFET: trying to increase on-state current by 50%, keeping off-state current as small as possible
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31.3.2008Martin Burger
Optimization goals for channels I
- Identification of channel structure from I-V Data
- Design of synthetic channels with improved selectivity (based on appropriate selectivity measures) mb-Eisenberg-Engl 07US Patent Application 2006
- Calibration of reduced models- Control of transition rates through channels Bezrukov et al, Marinoschi 07
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31.3.2008Martin Burger
Optimization goals for channels IISubject to a suitable dynamic gating model, the following will become of interest- Design of synthetic channels with optimal gating properties
- Design of synthetic channels with improved selectivity (based on appropriate selectivity measures)
- Calibration of reduced models
- Optimal control of gating
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31.3.2008Martin Burger
Download / Contact
www.math.uni-muenster.de/u/burger