1. 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

2. Martin Burger Ion Channels and Nanopores 2 31.3.2008 Joint Work with Kattrin Arning, Linz Mary Wolfram, Münster / Linz Bob Eisenberg, Chicago Heinz Engl, Linz Zuzanna Siwy, Irvine Rene Pinnau, Kaiserslautern

3. Martin Burger Ion Channels and Nanopores 3 31.3.2008 Ion Channels and Life Most of human life occurs in cells Transport through cell membrane is essential for biological function The transport or blocking of ions is controlled by channels Ion channels = proteins with a hole in their middle ~ 5 µ m

4. Martin Burger Ion Channels and Nanopores 4 31.3.2008 Ion Channels and Life Flow of ions creates / modifies electric potential Electrical field determines flow direction of ions A substantial fraction of drugs are designed to influence channel behaviour

5. Martin Burger Ion Channels and Nanopores 5 31.3.2008 Ion Channels and Life Figures by Raimund Dutzler, courtesy Bob Eisenberg Chemical Bonds are lines Surface is Electrical Potential Red is positive Blue is negative Chemist’s Vie w All Atoms View

6. Martin Burger Ion Channels and Nanopores 6 31.3.2008 Channel Function Ion channel control flow like a micro- electronic charge Proteins in the channel walls create a permanent charge in the channel (like the doping of a semiconductor device) Additional effects due to size exclusion in narrow channels ~30 Å K+K+

7. Martin Burger Ion Channels and Nanopores 7 31.3.2008 Channel Function Channel 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

8. Martin Burger Ion Channels and Nanopores 8 31.3.2008 Channel Function Experimental setup: Bath of ions and water on both sides of channel Bath concentrations controlled Voltage applied across channel

9. Martin Burger Ion Channels and Nanopores 9 31.3.2008 Gating Single channel current is a Random Signal

10. Martin Burger Ion Channels and Nanopores 10 31.3.2008 Selectivity Observed current-voltage curves as in microelectronics Curves for different bath concentrations indicate selectivity OmpF KCl 1M 1M || OmpF CaCl 2 1M 1M |

11. Martin Burger Ion Channels and Nanopores 11 31.3.2008 Modelling Microscopic model based on equations of motions Forces include interaction between ions, and with protein Positive cat ion, e.g., p = Na + Negative an ion, e.g., n = Cl ¯

12. Martin Burger Ion Channels and Nanopores 12 31.3.2008 Modelling Force f k includes - Excess „chemical“ force - Electrical force: Electrical potential to be computed from Poisson equation with sources from all charges (ions, protein)

13. Martin Burger Ion Channels and Nanopores 13 31.3.2008 Macroscopic Model for Open State Standard Coarse-Graining leads to Poisson-Nernst- Planck (Poisson-drift-diffusion) system for potential and ion concentrations Similar issues as in Semiconductor Simulation

14. Martin Burger Ion Channels and Nanopores 14 31.3.2008 Modelling Additional issues due to finite size (chemical) effects Excess chemical potential includes - Chemical interaction between the ions - Chemical interaction between ions and proteins

15. Martin Burger Ion Channels and Nanopores 15 31.3.2008 Modelling Computation 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 computational effort

16. Martin Burger Ion Channels and Nanopores 16 31.3.2008 Modelling Due 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

17. Martin Burger Ion Channels and Nanopores 17 31.3.2008 Modelling In 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 unknown Binding forces of the protein on its charges are encoded in a confining potential

18. Martin Burger Ion Channels and Nanopores 18 31.3.2008 Modelling Structure 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

19. Martin Burger Ion Channels and Nanopores 19 31.3.2008 Modelling Numerical Simulation by stabilized mixed finite elements L-type Ca channel with 8 half-charged oxygens Applied Voltage 50mV

20. Martin Burger Ion Channels and Nanopores 20 31.3.2008 Modelling Multi-D Simulation (here 3D Ca 2+ synthetic channel with rotational symmetry) Simulations by Mary Wolfram Na + Cl -

21. Martin Burger Ion Channels and Nanopores 21 31.3.2008 Modelling Gating 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

22. Martin Burger Ion Channels and Nanopores 22 31.3.2008 Why optimal design ? Compare function of OmpF and G 119D: huge difference OmpF KCl 1M 1M || G119D KCl 1M 1M | ompF KCl 0.05 M 0.05M || G119D KCl 0.05 M 0.05M ||

23. Martin Burger Ion Channels and Nanopores 23 31.3.2008 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 OmpfG119D

24. Martin Burger Ion Channels and Nanopores 24 31.3.2008 Why optimal design ? Selective channels can be built by controlled mutation Many labs try, but rational design is still missing Calcium selective As charge density increases, channel becomes calcium selective E rev  E Ca Miedema et al, Biophys J 87: 3137–3147 (2004) Unselective Wild Type MUTANT ─ Compound

25. Martin Burger Ion Channels and Nanopores 25 31.3.2008 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)

26. Martin Burger Ion Channels and Nanopores 26 31.3.2008 Why optimal design ? Selectivity and I-V curves as for biological channels

27. Martin Burger Ion Channels and Nanopores 27 31.3.2008 Why optimal design ? Gating in nanopores

28. Martin Burger Ion Channels and Nanopores 28 31.3.2008 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

29. Martin Burger Ion Channels and Nanopores 29 31.3.2008 Optimal Design of Doping Profiles Typical 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

30. Martin Burger Ion Channels and Nanopores 30 31.3.2008 Optimal Design of Doping Profiles Fast 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 system Globally convergent Gummel method for design

31. Martin Burger Ion Channels and Nanopores 31 31.3.2008 Optimal Design of Doping Profiles Works 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

32. Martin Burger Ion Channels and Nanopores 32 31.3.2008 On-/Off-State Design of Doping Profiles Minimize 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):

33. Martin Burger Ion Channels and Nanopores 33 31.3.2008 On-/Off-State Design of Doping Profiles On-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

34. Martin Burger Ion Channels and Nanopores 34 31.3.2008 Gummel

35. Martin Burger Ion Channels and Nanopores 35 31.3.2008 Optimal Design of Doping Profiles On-off state design of bipolar diode mb-Pinnau-Wolfram 08

36. Martin Burger Ion Channels and Nanopores 36 31.3.2008 Optimal Design of Doping Profiles Optimization of a MOSFET: trying to increase on-state current by 50%, keeping off-state current as small as possible

37. Martin Burger Ion Channels and Nanopores 37 31.3.2008 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 07 US Patent Application 2006 - Calibration of reduced models - Control of transition rates through channels Bezrukov et al, Marinoschi 07

38. Martin Burger Ion Channels and Nanopores 38 31.3.2008 Optimization goals for channels II Subject 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

39. Martin Burger Ion Channels and Nanopores 39 31.3.2008 Download / Contact www.math.uni-muenster.de/u/burger martin.burger@uni-muenster.de