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 Optimal Design of Ion Channels and Nanopores

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Optimal Design of Ion Channels and Nanopores. Joint Work with. Kattrin Arning, Linz Mary Wolfram, Münster / Linz Bob Eisenberg, Chicago Heinz Engl, Linz Zuzanna Siwy, Irvine Rene Pinnau, Kaiserslautern. ~ 5 µ m. Ion Channels and Life. Most of human life occurs in cells - PowerPoint PPT Presentation

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Page 1: 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

Page 2: 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

Page 3: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 3

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

Page 4: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 4

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

Page 5: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 5

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

Page 6: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 6

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+

Page 7: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 7

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

Page 8: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 8

31.3.2008Martin Burger

Channel FunctionExperimental setup:

Bath of ions and water on both sidesof channel

Bath concentrations controlledVoltage applied across channel

Page 9: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 9

31.3.2008Martin Burger

GatingSingle channel current is a Random Signal

Page 10: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 10

31.3.2008Martin Burger

SelectivityObserved current-voltage curves as in microelectronicsCurves for different bath concentrationsindicate selectivity

OmpF KCl 1M 1M

||

OmpF CaCl2

1M 1M

||

Page 11: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 11

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¯

Page 12: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 12

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)

Page 13: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 13

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

Page 14: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 14

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

Page 15: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 15

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

Page 16: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 16

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

Page 17: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 17

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

Page 18: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 18

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

Page 19: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 19

31.3.2008Martin Burger

ModellingNumerical Simulation by stabilized mixed finite elements

L-type Ca channel with 8 half-charged oxygens

Applied Voltage 50mV

Page 20: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 20

31.3.2008Martin Burger

ModellingMulti-D Simulation (here 3D Ca2+

synthetic channel with rotational symmetry)

Simulations by Mary Wolfram

Na+

Cl-

Page 21: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 21

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

Page 22: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 22

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

||

Page 23: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 23

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

Page 24: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 24

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

Page 25: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 25

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)

Page 26: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 26

31.3.2008Martin Burger

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

Page 27: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 27

31.3.2008Martin Burger

Why optimal design ? Gating in nanopores

Page 28: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 28

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

Page 29: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 29

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

Page 30: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 30

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

Page 31: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 31

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

Page 32: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 32

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):

Page 33: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 33

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

Page 34: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 34

31.3.2008Martin Burger

Gummel

Page 35: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 35

31.3.2008Martin Burger

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

mb-Pinnau-Wolfram 08

Page 36: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 36

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

Page 37: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 37

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

Page 38: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 38

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

Page 39: Optimal Design of  Ion Channels and Nanopores

Ion Channels and Nanopores 39

31.3.2008Martin Burger

Download / Contact

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

[email protected]