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Optimal Design of Ion Channels and Nanopores

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

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Optimal Design of Ion Channels and Nanopores

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  1. Optimal Design of Ion Channels and Nanopores

  2. 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

  3. ~5 µm Ion Channels and Nanopores Ion Channels and Life Most 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 a hole in their middle

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

  6. K+ ~30 Å Ion Channels and Nanopores Channel Function Ion 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

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

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

  10. OmpF KCl 1M 1M || OmpF CaCl2 1M 1M || Ion Channels and Nanopores Selectivity Observed current-voltage curves as in microelectronics Curves for different bath concentrationsindicate selectivity

  11. Positivecation, e.g., p= Na+ Negativeanion, e.g., n= Cl¯ Ion Channels and Nanopores Modelling Microscopic model based on equations of motions Forces include interaction between ions, and with protein

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

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

  20. Ion Channels and Nanopores Modelling Multi-D Simulation (here 3D Ca2+synthetic channel with rotational symmetry) Simulations by Mary Wolfram Na+ • Cl-

  21. Ion Channels and Nanopores 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. OmpF KCl 1M 1M || G119D KCl 1M 1M || G119D KCl0.05 M 0.05M || ompF KCl0.05 M 0.05M || Ion Channels and Nanopores Why optimal design ? Compare function of OmpF and G 119D: huge difference

  23. G119D Ompf Ion Channels and Nanopores 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

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

  25. Ion Channels and Nanopores 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. Ion Channels and Nanopores Why optimal design ? Selectivity and I-V curves as for biological channels

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

  28. Ion Channels and Nanopores 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. Ion Channels and Nanopores 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. Ion Channels and Nanopores 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. Ion Channels and Nanopores 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. Ion Channels and Nanopores 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. Ion Channels and Nanopores On-/Off-State Design of Doping Profiles On-state equations as before (rewritten in Slotboom variables), W defined in on-state Off-state problemC needs to be eliminated in favour of W: leads to one-sided coupling with on-state

  34. Ion Channels and Nanopores Gummel

  35. Ion Channels and Nanopores Optimal Design of Doping Profiles On-off state design of bipolar diodemb-Pinnau-Wolfram 08

  36. Ion Channels and Nanopores 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. Ion Channels and Nanopores 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

  38. Ion Channels and Nanopores 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. Ion Channels and Nanopores Download / Contact • www.math.uni-muenster.de/u/burger • martin.burger@uni-muenster.de

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