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

Inverse Problems in Ion Channels Lake Arrowhead, June 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

Inverse Problems in Ion Channels Lake Arrowhead, June 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 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 Lake Arrowhead, June 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 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“)

Inverse Problems in Ion Channels Lake Arrowhead, June 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

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

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

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

Inverse Problems in Ion Channels Lake Arrowhead, June 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 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

Inverse Problems in Ion Channels Lake Arrowhead, June 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

Inverse Problems in Ion Channels Lake Arrowhead, June 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

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

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

Inverse Problems in Ion Channels Lake Arrowhead, June 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 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

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

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

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

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

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

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

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

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

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