1. Inverse Problems in Ion Channels
Designing Structure for electrophysiological function:
Inverse Problems in Ion Channels
Martin Burger
Johannes Kepler University Linz, SFB F 013, RICAM

2. Inverse Problems in Ion Channels
Joint work with
Heinz Engl, RICAM, IMCC, JKU Linz
Bob Eisenberg, Rush Medical University, Argonne National Lab, Chicago
Inverse Problems in Ion Channels
Linz, June 2006

3. Flow time scale is 0.1 msec to min
Ion channels
.. are proteins with a hole down the middle
.. control flow in and out of cells
~ 5 µm
Flow time scale is 0.1 msec to min
Inverse Problems in Ion Channels
Linz, June 2006

4. Flow time scale is 0.1 msec to min
~30 Å
Flow time scale is 0.1 msec to min K+
Ion channels
Ion channels are the main molecular controllers (valves) of biological function
Inverse Problems in Ion Channels
Linz, June 2006

5. Modelling Thermodynamics Device Equation Equilibrium Nonequilibrium
Configurations
Trajectories
Boltzmann Distribution
Fokker Planck Equation
Finite OPEN System
Modelling
Statistical Mechanics
Theory of Stochastic Processes
Thermodynamics
Device Equation
Inverse Problems in Ion Channels
Linz, June 2006
Schuss, Nadler, and Eisenberg

6. Inverse Problems in Ion Channels
Modelling
Variables for continuum modelling are electric potential and ion densities
Electrostatics: Poisson equation with right-hand side equal to the sum of charge densities
Transport: Nernst-Planck equation with diffusion term and additional potentials (electric and chemical)
Size Exclusion: incorporated into model for chemical potential
Inverse Problems in Ion Channels
Linz, June 2006

7. Inverse Problems in Ion Channels
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
Linz, June 2006

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

9. Inverse Problems in Ion Channels
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
Inverse Problems in Ion Channels
Linz, June 2006

10. Inverse Problems in Ion Channels
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.
Inverse Problems in Ion Channels
Linz, June 2006

11. 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
Linz, June 2006

12. 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
Linz, June 2006

13. Inverse Problems in Ion Channels
Boundary Conditions
Dirichlet part left and right of baths, Neumann part above and below baths
Inverse Problems in Ion Channels
Linz, June 2006

14. 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
Linz, June 2006

15. 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
Linz, June 2006

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

17. Inverse Problems in Ion Channels
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
Linz, June 2006

18. 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
Linz, June 2006

19. 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
Linz, June 2006

20. 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 design problems: find NM or / and mM0 to maximize (improve) selectivity measure
Inverse Problems in Ion Channels
Linz, June 2006

21. 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
Linz, June 2006

22. Inverse Problems in Ion Channels
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
Inverse Problems in Ion Channels
Linz, June 2006

23. Inverse Problems in Ion Channels
Simple Case
1 D PNP model in interval (-L,L)
Inverse Problems in Ion Channels
Linz, June 2006

24. Inverse Problems in Ion Channels
Simple Case
Equations can be integrated to obtain fluxes
Inverse Problems in Ion Channels
Linz, June 2006

25. Inverse Problems in Ion Channels
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
Linz, June 2006

26. Inverse Problems in Ion Channels
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
Linz, June 2006

27. Inverse Problems in Ion Channels
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
Linz, June 2006

28. Inverse Problems in Ion Channels
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
Linz, June 2006

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

30. Inverse Problems in Ion Channels
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
Linz, June 2006

31. Inverse Problems in Ion Channels
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
Linz, June 2006

32. Inverse Problems in Ion Channels
Simple Case
Stability (instability) depends on
Decay of singular values
Inverse Problems in Ion Channels
Linz, June 2006

33. Inverse Problems in Ion Channels
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
Linz, June 2006

34. Inverse Problems in Ion Channels
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
Linz, June 2006

35. Inverse Problems in Ion Channels
Full Problem
Next step: identification of the constraining potential x2x2=16 data pts 6x3x3=54 data pts
Inverse Problems in Ion Channels
Linz, June 2006

36. Inverse Problems in Ion Channels
Full Problem
Instability for 1% data noise
Residual Error
Inverse Problems in Ion Channels
Linz, June 2006

37. Inverse Problems in Ion Channels
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
Linz, June 2006

38. Inverse Problems in Ion Channels
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
Linz, June 2006

39. Inverse Problems in Ion Channels
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
Linz, June 2006

40. Inverse Problems in Ion Channels
Full Problem
Strong improvement in reconstruction quality, even in presence of noise
Inverse Problems in Ion Channels
Linz, June 2006

41. Inverse Problems in Ion Channels
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
Linz, June 2006

42. Inverse Problems in Ion Channels
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
Linz, June 2006

43. Inverse Problems in Ion Channels
Design Problem
a = 200
Initial Value Optimal Potential
Inverse Problems in Ion Channels
Linz, June 2006

44. Inverse Problems in Ion Channels
Design Problem
a = 0
Initial Value Optimal Potential
Inverse Problems in Ion Channels
Linz, June 2006

45. Inverse Problems in Ion Channels
Design Problem
Objective functional for a = 200 (black) and a = 0 (red)
Inverse Problems in Ion Channels
Linz, June 2006

46. Inverse Problems in Ion Channels
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
Linz, June 2006

47. Inverse Problems in Ion Channels
Download and Contact
Papers and Talks:
From October: wwwmath1.uni-muenster.de/num
Inverse Problems in Ion Channels
Linz, June 2006