1. 1 Inverse Problems in Ion Channels (ctd.) Martin Burger Bob Eisenberg Heinz Engl Johannes Kepler University Linz, SFB F 013, RICAM

2. Inverse Problems in Ion Channels Lake Arrowhead, June 20062 PNP-DFT As seen above, the flow in ion channels can be computed by PNP equations coupled to models for direct interaction Resulting system of PDEs for electrical potential V and densities  k of the form

3. Inverse Problems in Ion Channels Lake Arrowhead, June 20063 PNP-DFT Potentials are obtained as variations of an energy functional Energy functional is of the form

4. Inverse Problems in Ion Channels Lake Arrowhead, June 20064 PNP-DFT Excess electro-chemical energy models direct interactions (hard spheres). Various models are available, we choose DFT (Density functional theory) for statistical physics (Gillespie-Nonner-Eisenberg 03) Same idea to DFT in quantum mechanics, reduction of high-dimensional Fokker-Planck instead of Schrödinger Associated excess potential can be computed via integrals of the densities

5. Inverse Problems in Ion Channels Lake Arrowhead, June 20065 PNP-DFT Leading order terms in the differential equations are just PNP, incorporation of DFT is compact perturbation Mapping properties of forward problem are roughly the same as for pure PNP High additional computational effort for computing integrals in DFT. See talk of M. Wolfram for efficient methods for PNP and sensitivity computations

6. Inverse Problems in Ion Channels Lake Arrowhead, June 20066 Mobile and Confined Species Mobile ions (Na, Ca, Cl, Ka, …) and mobile neutral species (H 2 O) can be controlled in the baths. No confining potential  k 0 Confined ions (half-charged oxygens) cannot leave the channel, are assumed to be in equilibrium (corresponding potential is constant) Notation: Index 1,2,..,M-1 for mobile species. Index M for confined species („permanent charge“)

7. Inverse Problems in Ion Channels Lake Arrowhead, June 20067 Mobile and Confined Species Model case: L-type Ca Channel M=5 species (Ca 2+, Na +, Cl -, H 2 O, O -1/2 ) Channel length 1nm + two surrounding baths of lenth 1.7 nm

8. Inverse Problems in Ion Channels Lake Arrowhead, June 20068 Boundary Conditions Dirichlet part left and right of baths, Neumann part above and below baths

9. Inverse Problems in Ion Channels Lake Arrowhead, June 20069 Boundary Charge Neutrality Only charge neutral combinations of the ions can be obtained in the bath, i.e. possible boundary values restricted by

10. Inverse Problems in Ion Channels Lake Arrowhead, June 200610 Total Permanent Charge In order to determine  M uniquely additional condition is needed N M is the number of confined particles („total permanent charge“)

11. Inverse Problems in Ion Channels Lake Arrowhead, June 200611 Simulation of PNP-DFT L-type Ca Channel, U =50mV, N 5 = 8

12. Inverse Problems in Ion Channels Lake Arrowhead, June 200612 Fluxes and Current Flux density of each species can be computed as One cannot observe single fluxes, but only the current on the outflow boundary

13. Inverse Problems in Ion Channels Lake Arrowhead, June 200613 Function from Structure With complete knowledge of system parameters and structure, we can (approximately) compute the (electrophysiological) function, i.e. the current for different voltages and different bath concentrations Structure enters via the permanent charge, namely the number N M of confined particles and the constraining potential  M 0

14. Inverse Problems in Ion Channels Lake Arrowhead, June 200614 Structure from Function Real life is different, since we observe (measure) the electrophysiological function, but do not know the structure Hence we arrive at an inverse problem: obtain information about structure from function Identification problems: find N M or / and  M 0 from current measurements

15. Inverse Problems in Ion Channels Lake Arrowhead, June 200615 Structure for Function For synthetic channels, one would like to achieve a certain function by design Usual goal is related to selectivity, designed channel should prefer one species (e.g. Ca) over another one with charge of same sign (e.g. Na) Optimal desing problems: find N M or / and  M 0 to maximize (improve) selectivity measure

16. Inverse Problems in Ion Channels Lake Arrowhead, June 200616 Differences to Semiconductors Multiple species with charge of same sign Additional chemical interaction in forward model Richer data set for identification (current as function of voltage and bath concentrations) No analogue to selectivity in semiconductors. Design problems completely new

17. Inverse Problems in Ion Channels Lake Arrowhead, June 200617 Simple Case Start 1D (realistic for many channels being extremely narrow in 2 directions), ignore DFT part as a first step. Identify fixed permanent charge density (instead of total charge and potential) Consider case of small bath concentrations Linearization of equations around zero bath concentration

18. Inverse Problems in Ion Channels Lake Arrowhead, June 200618 Simple Case 1 D PNP model in interval (-L,L)

19. Inverse Problems in Ion Channels Lake Arrowhead, June 200619 Simple Case Equations can be integrated to obtain fluxes

20. Inverse Problems in Ion Channels Lake Arrowhead, June 200620 Simple Case For bath concentrations zero, it is easy to show that all mobile ion densities vanish For each applied voltage U, we obtain a Poisson equation of the form

21. Inverse Problems in Ion Channels Lake Arrowhead, June 200621 Simple Case Note that where There is a one-to-one relation between  M and V 0,0. We can start by identifying V 0,0

22. Inverse Problems in Ion Channels Lake Arrowhead, June 200622 Simple Case The first-order expansion of the currents around zero bath concentration is given by If we measure for small concentrations, then this is the main content of information

23. Inverse Problems in Ion Channels Lake Arrowhead, June 200623 Simple Case Since we can vary the linearized bath concentrations we can achieve that only one of the numerators does not vanish in This means we may know in particular

24. Inverse Problems in Ion Channels Lake Arrowhead, June 200624 Simple Case With the above formula for V 0,0 and we arrive at the linear integral equation

25. Inverse Problems in Ion Channels Lake Arrowhead, June 200625 Simple Case The equation ( ) is severely ill-posed (singular values decay exponentially) Second step of computing permanent charge density is mildly ill-posed

26. Inverse Problems in Ion Channels Lake Arrowhead, June 200626 Simple Case Identifiability: Knowledge of implies knowledge of all derivatives at zero Hence, all moments of f are known, which implies uniqueness (even for arbitrarily small

27. Inverse Problems in Ion Channels Lake Arrowhead, June 200627 Simple Case Stability (instability) depends on Decay of singular values

28. Inverse Problems in Ion Channels Lake Arrowhead, June 200628 Simple Case Note: in this analysis we have only used values around zero and still obtained uniqueness. Using more measurements away from zero the problem may become overdetermined

29. Inverse Problems in Ion Channels Lake Arrowhead, June 200629 Full Problem We attack the full inverse problem by brute force numerically, implemented iterative regularization First step: computing total charge only (1D inverse problem, no instability). 95 % accuracy with eight measurements

30. Inverse Problems in Ion Channels Lake Arrowhead, June 200630 Full Problem Next step: identification of the constraining potential 8 4x2x2=16 data pts6x3x3=54 data pts

31. Inverse Problems in Ion Channels Lake Arrowhead, June 200631 Full Problem Instability for 1% data noise ResidualError

32. Inverse Problems in Ion Channels Lake Arrowhead, June 200632 Full Problem Results are a proof of principle For better reconstruction we need to increase discretization fineness for parameters and in particular number of measurements No problem to obtain high amount of data from experiments Computational complexity increases (higher number of forward problem)

33. Inverse Problems in Ion Channels Lake Arrowhead, June 200633 Full Problem Forward problem PNP-DFT is computationally demanding even in 1D (due to many integrals and self-consistency iterations in DFT part) So far gradient evaluations by finite differencing Each step of Landweber iteration needs (N+1)K solves of PNP-DFT (N = number of grid points for the potential, M = number of measurements) Even for coarse discretization of inverse problem, hundreds of PNP-DFT solves per iteration

34. Inverse Problems in Ion Channels Lake Arrowhead, June 200634 Full Problem Improvement: Adjoint method for gradient evaluation (higher accuracy, lower effort) Test again for reconstruction of permanent charge density in pure PNP problem Used 5 x 16 x 16 = 1280 data points

35. Inverse Problems in Ion Channels Lake Arrowhead, June 200635 Full Problem Strong improvement in reconstruction quality, even in presence of noise

36. Inverse Problems in Ion Channels Lake Arrowhead, June 200636 Full Problem Further improvements needed to increase computational complexity Multi-scale techniques for forward and inverse problem Kaczmarz techniques to sweep over measurements

37. Inverse Problems in Ion Channels Lake Arrowhead, June 200637 Design Problem Optimal design problem: maximize relative selectivity measure preferring Na over Ca P* is favoured initial design, penalty ensures to stay as close as possible to this design (manufacture constraint)

38. Inverse Problems in Ion Channels Lake Arrowhead, June 200638 Design Problem  = 200 Initial Value Optimal Potential

39. Inverse Problems in Ion Channels Lake Arrowhead, June 200639 Design Problem  = 0 Initial Value Optimal Potential

40. Inverse Problems in Ion Channels Lake Arrowhead, June 200640 Design Problem Objective functional for  = 200 (black) and  = 0 (red)

41. Inverse Problems in Ion Channels Lake Arrowhead, June 200641 Conclusions Great potential to improve identification and design tasks in channels by inverse problems techniques Results promising, show that the approach works Many challenging questions with respect to improvement of computational complexity

42. Inverse Problems in Ion Channels Lake Arrowhead, June 200642 Download and Contact Papers and Talks: www.indmath.uni-linz.ac.at/people/burger From October: wwwmath1.uni-muenster.de/num e-mail: martin.burger@jku.at