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### Inverse Problems in Ion Channels

Designing Structure for electrophysiological function:

Martin Burger

Johannes Kepler University Linz, SFB F 013, RICAM

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

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

Inverse Problems in Ion Channels

Linz, June 2006

Flow time scale is 0.1 msec to min

~30 Å

Ion channels- Ion channels are the main molecular controllers (valves) ofbiological function

Inverse Problems in Ion Channels

Linz, June 2006

Modelling

StatisticalMechanics

Theory of Stochastic Processes

Thermodynamics

Device Equation

Inverse Problems in Ion Channels

Linz, June 2006

Schuss, Nadler, and Eisenberg

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

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

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

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

Linz, June 2006

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

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

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

Boundary Conditions

- Dirichlet part left and right of baths, Neumann part above and below baths

Inverse Problems in Ion Channels

Linz, June 2006

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

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

Simulation of PNP-DFT

- L-type Ca Channel, U =50mV, N5 = 8

Inverse Problems in Ion Channels

Linz, June 2006

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

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

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

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

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

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

Linz, June 2006

Simple Case

- Equations can be integrated to obtain fluxes

Inverse Problems in Ion Channels

Linz, June 2006

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

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

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

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

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

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

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

Simple Case

- Stability (instability) depends on

Decay of singular values

Inverse Problems in Ion Channels

Linz, June 2006

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

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

Full Problem

- Next step: identification of the constraining potential 8 4x2x2=16 data pts 6x3x3=54 data pts

Inverse Problems in Ion Channels

Linz, June 2006

Full Problem

- Instability for 1% data noise

Residual Error

Inverse Problems in Ion Channels

Linz, June 2006

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

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

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

Full Problem

- Strong improvement in reconstruction quality, even in presence of noise

Inverse Problems in Ion Channels

Linz, June 2006

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

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

Design Problem

- a = 200

Initial Value Optimal Potential

Inverse Problems in Ion Channels

Linz, June 2006

Design Problem

- Objective functional for a = 200 (black) and a = 0 (red)

Inverse Problems in Ion Channels

Linz, June 2006

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

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

Linz, June 2006

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