Inverse Problems in Ion Channels (ctd.)

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

2. 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 rk of the form Inverse Problems in Ion Channels Lake Arrowhead, June 2006

3. PNP-DFT • Potentials are obtained as variations of an energy functional • Energy functional is of the form Inverse Problems in Ion Channels Lake Arrowhead, June 2006

4. 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 potentialcan be computed via integrals of the densities Inverse Problems in Ion Channels Lake Arrowhead, June 2006

5. 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 Inverse Problems in Ion Channels Lake Arrowhead, June 2006

6. Mobile and Confined Species • Mobile ions (Na, Ca, Cl, Ka, …) and mobile neutral species (H2O) can be controlled in the baths. No confining potential mk0 • 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“) Inverse Problems in Ion Channels Lake Arrowhead, June 2006

7. Mobile and Confined Species • Model case: L-type Ca Channel • M=5 species (Ca2+, Na+, Cl-, H2O, O-1/2) • Channel length 1nm + two surrounding baths of lenth 1.7 nm Inverse Problems in Ion Channels Lake Arrowhead, June 2006

8. Boundary Conditions • Dirichlet part left and right of baths, Neumann part above and below baths Inverse Problems in Ion Channels Lake Arrowhead, June 2006

9. Boundary Charge Neutrality • Only charge neutral combinations of the ions can be obtained in the bath, i.e. possible boundary values restricted by Inverse Problems in Ion Channels Lake Arrowhead, June 2006

10. Total Permanent Charge • In order to determine rM uniquely additional condition is needed • NM is the number of confined particles („total permanent charge“) Inverse Problems in Ion Channels Lake Arrowhead, June 2006

11. Simulation of PNP-DFT • L-type Ca Channel, U =50mV, N5 = 8 Inverse Problems in Ion Channels Lake Arrowhead, June 2006

12. 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 Inverse Problems in Ion Channels Lake Arrowhead, June 2006

13. 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 NM of confined particles and the constraining potential mM0 Inverse Problems in Ion Channels Lake Arrowhead, June 2006

14. 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 NM or / and mM0 from current measurements Inverse Problems in Ion Channels Lake Arrowhead, June 2006

15. 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 NM or / and mM0 to maximize (improve) selectivity measure Inverse Problems in Ion Channels Lake Arrowhead, June 2006

16. 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 Inverse Problems in Ion Channels Lake Arrowhead, June 2006

17. 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 zerobath concentration Inverse Problems in Ion Channels Lake Arrowhead, June 2006

18. Simple Case • 1 D PNP model in interval (-L,L) Inverse Problems in Ion Channels Lake Arrowhead, June 2006

19. Simple Case • Equations can be integrated to obtain fluxes Inverse Problems in Ion Channels Lake Arrowhead, June 2006

20. 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 Inverse Problems in Ion Channels Lake Arrowhead, June 2006

21. Simple Case • Note that where • There is a one-to-one relation between rM and V0,0. We can start by identifying V0,0 Inverse Problems in Ion Channels Lake Arrowhead, June 2006

22. 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 Inverse Problems in Ion Channels Lake Arrowhead, June 2006

23. 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 Inverse Problems in Ion Channels Lake Arrowhead, June 2006

24. Simple Case • With the above formula for V0,0 and we arrive at the linear integral equation Inverse Problems in Ion Channels Lake Arrowhead, June 2006

25. Simple Case • The equation ( ) is severely ill-posed (singular values decay exponentially) • Second step of computing permanent charge density is mildly ill-posed Inverse Problems in Ion Channels Lake Arrowhead, June 2006

26. 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 Inverse Problems in Ion Channels Lake Arrowhead, June 2006

27. Simple Case • Stability (instability) depends on Decay of singular values Inverse Problems in Ion Channels Lake Arrowhead, June 2006

28. 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 Inverse Problems in Ion Channels Lake Arrowhead, June 2006

29. 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 Inverse Problems in Ion Channels Lake Arrowhead, June 2006

30. Full Problem • Next step: identification of the constraining potential 8 4x2x2=16 data pts 6x3x3=54 data pts Inverse Problems in Ion Channels Lake Arrowhead, June 2006

31. Full Problem • Instability for 1% data noise Residual Error Inverse Problems in Ion Channels Lake Arrowhead, June 2006

32. 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) Inverse Problems in Ion Channels Lake Arrowhead, June 2006

33. 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 Inverse Problems in Ion Channels Lake Arrowhead, June 2006

34. 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 Inverse Problems in Ion Channels Lake Arrowhead, June 2006

35. Full Problem • Strong improvement in reconstruction quality, even in presence of noise Inverse Problems in Ion Channels Lake Arrowhead, June 2006

36. Full Problem • Further improvements needed to increase computational complexity • Multi-scale techniques for forward and inverse problem • Kaczmarz techniques to sweep over measurements Inverse Problems in Ion Channels Lake Arrowhead, June 2006

37. 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) Inverse Problems in Ion Channels Lake Arrowhead, June 2006

38. Design Problem • a = 200 Initial Value Optimal Potential Inverse Problems in Ion Channels Lake Arrowhead, June 2006

39. Design Problem • a = 0 Initial Value Optimal Potential Inverse Problems in Ion Channels Lake Arrowhead, June 2006

40. Design Problem • Objective functional for a = 200 (black) and a = 0 (red) Inverse Problems in Ion Channels Lake Arrowhead, June 2006

41. 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 Inverse Problems in Ion Channels Lake Arrowhead, June 2006

42. 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 Inverse Problems in Ion Channels Lake Arrowhead, June 2006