Inverse Problems in
Ion Channels (ctd.)
Martin Burger
Bob Eisenberg
Heinz Engl
Johannes Kepler University Linz, SFB F 013, RICAM
1
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
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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. See talk of M. Wolfram for efficient
methods for PNP and sensitivity computations
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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 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“)
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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 rM 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 mM0
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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 mM0 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 desing problems: find NM or / and mM0 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 rM 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

a = 200
Initial Value
Optimal Potential
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Design Problem

a=0
Initial Value
Optimal Potential
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Design Problem
Objective functional for a = 200 (black) and a = 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]
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