Optimal Design of Ion Channels and Nanopores

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

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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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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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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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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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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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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Channel FunctionExperimental setup:

Bath of ions and water on both sidesof channel

Bath concentrations controlledVoltage applied across channel

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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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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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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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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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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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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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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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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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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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ModellingNumerical Simulation by stabilized mixed finite elements

L-type Ca channel with 8 half-charged oxygens

Applied Voltage 50mV

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ModellingMulti-D Simulation (here 3D Ca2+

synthetic channel with rotational symmetry)

Simulations by Mary Wolfram

Na+

Cl-

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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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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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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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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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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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Why optimal design ? Selectivity and I-Vcurves as for biological channels

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Why optimal design ? Gating in nanopores

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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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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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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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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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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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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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Gummel

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Optimal Design of Doping ProfilesOn-off state design of bipolar diode

mb-Pinnau-Wolfram 08

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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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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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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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www.math.uni-muenster.de/u/burger

martin.burger@uni-muenster.de