inverse problems in ion channels

1

Inverse Problems in Ion Channels

Martin Burger

Johannes Kepler University Linz, SFB F 013, RICAM

Designing Structure for electrophysiological function:


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Joint work with Heinz Engl, RICAM, IMCC, JKU Linz

Bob Eisenberg, Rush Medical University, Argonne National Lab, Chicago


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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


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Ion channels Ion channels are the main molecular controllers (valves) of biological function

~30 ÅFlow time scale is 0.1 msec to min

K+


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ModellingEquilibrium NonequilibriumConfigurations Trajectories

Boltzmann Distribution Fokker Planck Equation

Finite OPEN System

lim ,N V

Thermodynamics Device Equation

Schuss, Nadler, and Eisenberg

StatisticalMechanics

Theory of Stochastic Processes


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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


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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


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PNP-DFT Potentials are obtained as variations of an energy functional

Energy functional is of the form


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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


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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.


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Mobile and Confined Species Mobile ions (Na, Ca, Cl, Ka, …) and mobile neutral species (H2O) 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“)


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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


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Boundary Conditions Dirichlet part left and right of baths, Neumann part above and below baths


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Boundary Charge Neutrality Only charge neutral combinations of the ions can be obtained in the bath, i.e. possible boundary values restricted by


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Total Permanent Charge In order to determine M uniquely additional condition is needed

NM is the number of confined particles („total permanent charge“)


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Simulation of PNP-DFT L-type Ca Channel, U =50mV, N5 = 8


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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


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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 M

0


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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 M0 from

current measurements


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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 M0 to

maximize (improve) selectivity measure


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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


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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


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Simple Case 1 D PNP model in interval (-L,L)


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Simple Case Equations can be integrated to obtain fluxes


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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


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Simple Case Note that

where

There is a one-to-one relation between M and V0,0. We can start by identifying V0,0


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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


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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


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Simple Case With the above formula for V0,0 and

we arrive at the linear integral equation


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Simple Case The equation ( )

is severely ill-posed (singular values decay exponentially) Second step of computing permanent charge density is mildly ill-posed


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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


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Simple Case Stability (instability) depends on

Decay of singular values


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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


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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


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Full Problem Next step: identification of the constraining potential

8

4x2x2=16 data pts 6x3x3=54 data pts


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Full Problem Instability for 1% data noise

Residual Error


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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)


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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


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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


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Full Problem Strong improvement in reconstruction quality, even in presence of noise


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Full Problem Further improvements needed to increase computational complexity

Multi-scale techniques for forward and inverse problem

Kaczmarz techniques to sweep over measurements


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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)


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Design Problem = 200

Initial Value Optimal Potential


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Design Problem = 0

Initial Value Optimal Potential


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Design Problem Objective functional for = 200 (black) and = 0 (red)


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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


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Download and Contact

Papers and Talks:

www.indmath.uni-linz.ac.at/people/burger

From October: wwwmath1.uni-muenster.de/num

e-mail: [email protected]

inverse problems in ion channels

Documents

mobile species

incorporation of dft

formpnpdft potentials

chemical potentialpnpdft

species ca2

confined species model

mobile neutral species

electric potential