Nonlinear Poisson-Nernst Planck Equations for Ion
Flux through Confined Geometries
M Burger1 , B Schlake1 and M-T Wolfram2
1
Institute for Computational and Applied Mathematics, University of Münster,
Einsteinstr. 62, 48149 Münster, Germany
2 Department of Mathematics, University of Vienna, Nordbergstr. 15, 1090
Vienna, Austria
E-mail: martin.burger@wwu.de, baerbel.schlake@wwu.de,
marie-therese.wolfram@univie.ac.at
Abstract. The mathematical modelling and simulation of ion transport trough
biological and synthetic channels (nanopores) is a challenging problem, with direct
application in biophysics, physiology and chemistry. At least two major effects
have to be taken into account when creating such models: the electrostatic
interaction of ions and the effects due to size exclusion in narrow regions.
While mathematical models and methods for electrostatic interactions are welldeveloped and can be transfered from other flow problems with charged particles,
e.g. semiconductor devices, less is known about the appropriate macroscopic
modelling of size exclusion effects.
Recently several papers proposed simple or sophisticated approaches for
including size exclusion effects into entropies, in equilibrium as well as off
equilibrium. The aim of this paper is to investigate a second potentially important
modification due to size exclusion, which often seems to be ignored and is not
implemented in currently used models, namely the modification of mobilities due
to size exclusion effects. We discuss a simple model derived from a self-consisted
random walk and investigate the stationary solutions as well as the computation
of conductance. The need of incorporating nonlinear mobilities in high density
situations is demonstrated in an investigation of conductance as a function of bath
concentrations, which does not lead to obvious saturation effects in the case of
linear mobility.
AMS classification scheme numbers: 92C35,92C30,35J47,35J60,35Q84,35J70,65M06
Submitted to: Nonlinearity
Nonlinear PNP equatios for ion flux through confined geometries
2
1. Introduction
Mathematical models for crowded motion recently received strong attention due to
various application in ion transport (cf. [1], [2], [3] ), cell biology (cf. [4], [5], [6]) and
even human behaviour (cf. [7]). While various approaches for microscopic modelling,
either based on equations of motions with forces accounting for finite sizes (cf. [8]) or
on exclusion processes, have been investigated in detail, there are still open problems
in the transition to macroscopic models based on partial differential equations. In
equilibrium situations this transition and the resulting modification of entropies are
investigated by various approaches (cf. [9], [10]). Away from equilibrium usually
just standard equations with linear mobilities and the modified entropies are used,
whose appropriateness remains unclear. The main reason seems to be the strong local
interactions, which destroy propagation of chaos and thus the standard mean-field
limit. Recently, several authors have demonstrated that it is more appropriate to
use models with nonlinear saturating mobility instead of the classical linear mobility
models, for single (cf. [7], [11], [12], [13]) and in few papers also for multiple species
(cf. [7], [14]). Those results so far have been achieved for toy models rather than for
practical applications and are hence still widely neglected in applied sciences and the
simulation of real-life phenomena.
In this paper we want to make a first step in this direction by discussing a
modified macroscopic modelling of ion transport through confined regions, a problem
of particular importance in physiology (membrane ion channels, cf. [1], [2]) and
electrochemistry (polymer membranes and nanopores, cf. [15]). We present a
modification of the classical Poisson-Nernst-Planck equations (cf. [16]) accounting for
nonlinear mobility effects, provide a mathematical analysis of the arising equations,
and discuss some practical implications.
The classical macroscopic model for ion transport are the Nernst-Planck equations
for the ion concentrations ci (each i = 1, . . . , M denoting an ion species with charge
zi , diffusion coefficient Di , and mobility µi )
)
(
(1)
∂t ci = ∇ · Di (∇ci + µi ci (ezi ∇V + ∇Wi0 )) ,
in a domain Ω, where V is the electric and Wi0 an external potential. In order to
obtain a self-consistent model we supplement the Nernst-Planck equations with the
Poisson equation, given by
(∑
)
−∇ · (ϵ∇V ) = e
zj cj + f ,
(2)
where ϵ denotes the permittivity and f a permanent charge density. Using the Einstein
relation Di = µ−1
i ξi , this model can be written as a formal gradient flow
∂t ci = ∇ · (ξi ci ∇∂ci E(c1 , . . . , cM ))
for an entropy functional of the form
M ∫
∑
( −1
)
E(c1 , . . . , cM ) =
µi ci log ci + ezi ci V [c1 , . . . , cM ] + ci Wi0 dx,
i=1
(3)
(4)
Ω
where V [c1 , . . . , cM ] depends implicitely on the concentration vector via the solution
of the Poisson equation (2).
The gradient flow structure (3) with a linear mobility has been frequently
discussed in terms of optimal transport theory and Wasserstein metrics for probability
measures (cf. [17]). It can be derived in a mean-field limit from microscopic particle
Nonlinear PNP equatios for ion flux through confined geometries
3
models (cf. e.g. [18], [19]). The rigorous derivation however breaks down if volume
exclusion effects are included, an issue naturally arising in ion transport through
channels and pores since available volumes are not orders of magnitudes larger than
the ion sizes.
The remainder of the paper is organized as follows: In section 2 we review
the modified entropy and arising modified Poisson-Boltzmann equations, before we
sketch the derivation of a nonlinear mobility model for off-equilibrium computations
in section 3. The resulting generalized Poisson-Nernst-Planck (PNP) equations and
their mathematical properties are discussed in section 4, as well as some aspects
of dimension reduction in narrow regions. We turn our attention to a study of
conductance close to equilibrium as well as current in the stationary case, which
are computed numerically, in section 5. The latter will provide some insight into
current saturation at high concentrations, which can only be included via nonlinear
mobilities.
2. Equilibria and Modified Poisson-Boltzmann Equations
In the following we briefly recall the computation of equilibria (i.e. if all ion fluxes
vanish) by classical Poisson-Boltzmann equations [16] as well as the modified ones
recently introduced by Li et al. [20, 9, 10]. Standard equilibria are obtained from zero
flux in the Poisson-Nernst-Planck setup, i.e.
eq
eq
0 = Ji = Di (∇ceq
+ ∇Wi0 ))
i + µi ci (ezi ∇V
(
)
eq
−1
eq
= ξi ceq
+ Wi0 ) ,
i ∇ µi log ci + ezi V
(5)
(6)
which can be solved for nonzero concentration to
ceq
i
−
= ki e
ezi V eq +Wi0
−1
µ
i
,
(7)
with constants ki to be determined from the bath concentrations γi . The form of the
equilibria is called Boltzmann statistics, they can also be computed as minimizers of
the entropy (4).
Inserting the equilibrium concentrations into the Poisson equation (2) yields the
Poisson-Boltzmann equation
(
)
ez V eq +W 0
∑
− i −1 i
µ
i
zi ki e
−∇ · (ϵ∇V eq ) = e
+f
(8)
i
which is frequently studied in biochemistry. The Poisson-Boltzman equation is a
nonlinear elliptic equation with a unique solution. It can numerically be solved using
finite element or finite difference discretization and Newton iterations.
In the case of transport through narrow regions an implicit solvent model seems
more appropriate. Roughly speaking implicit solvent models assume that there is
a maximal possible volume density, which is indeed achieved everywhere, since the
remaining part is filled by the solvent. Taking the solvent as an electrically neutral
species c0 implies a relation of the form
1=
M
∑
αi ci ,
(9)
i=0
where αi is the maximal volume fraction of the i−th species (volume of a particle
times maximal density). For simplicity we set αi = 1 in the following (we refer to [10]
Nonlinear PNP equatios for ion flux through confined geometries
4
for the general case of species with nonuniform sizes). Then the solvent concentration
is computed as
c0 = 1 − ρ := 1 −
M
∑
ci .
(10)
i=1
Instead of an entropy for the m + 1 species including the solvent, one thus obtains a
reduced entropy functional
M ∫
∑
( −1
)
0
0
µi ci log ci + µ−1
E(c1 , . . . , cM ) =
0 (1 − ρ) log(1 − ρ) + ezi ci V + ci (Wi − W0 ) dx
i=1
Ω
The equilibria in the modified model can be computed from the first-order optimality
conditions (taking into account the constraint from mass conservation)
−1
0
0
λi = ∂ci E = µ−1
i log ci − µ0 log(1 − ρ) + zi eV + (Wi − W0 ),
i
with constant Lagrange multiplier λi ∈ R. Defining ki = exp( µλ−1
) and W̃i = Wi0 −W00
0
we find
−1
µ
i
−1
µ0
zi eV
ci
W̃i
= ki exp(− −1 − −1 ).
1−ρ
µ0
µ0
−1
If µ−1
0 = µi , then the equilibria can be computed explicitely in so-called FermiDirac statistics [8]
eq
ceq
i
− µW̃−1i )
ki exp(− ziµeV
−1
0
0
,
=
∑
zj eV eq
W̃
1 + j kj exp(− µ−1 − µ−1j )
0
(11)
0
with constants ki to be determined from the bath concentrations γi . Equation (11)
implies that 0 ≤ ci ≤ ρ ≤ 1 holds at each point. The associated modified PoissonBoltzmann equation (cf. [9]) reads


eq
ki exp(− ziµeV
− µW̃−1i )
∑
−1
0
0
+ f  , (12)
−∇ · (ϵ∇V eq ) = e 
zi
∑
W̃
zj eV eq
1 + j kj exp(− µ−1 − µ−1j )
i
0
0
for which one still can show that there exists a unique solution. Throughout the paper,
we assume in the following µ0 = µi = µ.
Several other approaches have been proposed to compute the resulting entropy
and consequently minimizers in the case of volume exclusion, e.g. mean spherical
approximation yielding similar local functionals (cf. [21]) or density functional theory
yielding nonlocal functionals with small support (cf. [2],[22]). Their qualitative
behaviour is very similar to the implicit solvent model above.
3. Derivation of a Modified Model
In order to compute flow through narrow pores an off-equilibrium description of ion
transport is needed. A simple and frequently used way to do so is the transport
gradient flow structure (3) with linear mobility (cf. [17]), which yields the standard
PNP model in the case of the logarithmic entropy (4). For implicit solvent models the
validity of a model with linear mobility is at least doubtfull for large volume densities
Nonlinear PNP equatios for ion flux through confined geometries
5
ρ, since one expects flow saturation due to volume filling. In order to illustrate the
need for nonlinear mobilities we sketch the derivation of a nonlinear mobility model
from a microscopic lattice-based model with volume exclusion, which serves as the
basis of our subsequent investigations.
3.1. Derivation from hopping Model
We derive a system of drift-diffusion equations from a self-consistent one-dimensional
hopping approach modelling local interactions. The problem-setup is as follows:
Let Th denote an equidistant grid of mesh size h, where every grid point can be
occupied by a particle with charge zi . The probability of finding a particle with charge
zi at time t at location x is given by:
ci (x, t) = P (particle of species i is at position x at time t),
where P denotes the probability. For charged particles the potential V (x, t) is
computed self-consistently from the Poisson equation
1 ∑
−ϵ∂xx V (x, t) = e
zi δ(x − xhi (t)) + ef (x),
(13)
N i
where N denotes the number of particles and xhi denotes its position. Equation (13)
converges to
∑
−ϵ∂xx V (x, t) = e
zi ci (x, t) + ef (x)
(14)
i
in the large N limit. In addition to the electrostatic potential, we model other forces
via an external potential Wi0 . With Wi (x, t) = zi V (x, t) + Wi0 (x, t) the transition
rates for each species are given by
Π̃+
ci (x, t) = k exp(−β(Wi (x + h, t) − Wi (x, t)))
(15)
Π̃−
ci (x, t)
(16)
= k exp(−β(Wi (x − h, t) − Wi (x, t))),
where k denotes the normalization constant and β denotes the mobility constant.
Taylor expansion up to order h2 and rescaling, gives
Π̃+
ci (x, t) = P (jump of ci from positiom x to x + h in (t, t + ∆t))
Π̃−
ci (x, t)
= αi − hβi ∂x Wi (x + h/2, t),
(17)
= P (jump of ci from position x to x − h in (t, t + ∆t))
= αi + hβi ∂x Wi (x − h/2, t),
(18)
where α denotes the diffusion constants.
We assume that the diameter of every ion equals h and take into account that
neighbouring sites might be occupied. We include these assumptions in the simple
model by
+
Π+
ci = Π̃ci · P (position x + h is at time t not occupied),
+
Π−
ci = Π̃ci · P (position x − h is at time t not occupied).
We make the closure assumption that the probability of a free site is
∑
P (position x is at time t not occupied) = 1 −
cj (x, t),
j
Nonlinear PNP equatios for ion flux through confined geometries
6
which corresponds to rigorous results for one species, cf. [12]. We mention here that
the derivation of this model is also justified in the usual stochastic setting. Here we
are able to show that the detailed balance condition is fullfilled up to order h2 .
The probability to find a particle of species ci at position x at time t + ∆t is
−
ci (x, t + ∆t) = ci (x, t)(1 − Π+
ci (x, t) − Πci (x, t))
−
+ ci (x + h, t)Π−
ci (x + h, t) + ci (x − h, t)Πci (x − h, t).
Therefore we have (supressing the index ci in Π)
ci (x, t + ∆t) − ci (x, t) =
ci (x, t)(Π+ (x − h, t) + Π− (x + h, t) − Π+ (x, t) − Π− (x, t))
+ (ci (x + h, t) − ci (x, t))Π− (x + h, t) + (ci (x − h, t) − ci (x, t))Π+ (x − h, t).
We obtain after Taylor expansion up to second order that
ci (x, t + ∆t) − ci (x, t) =
(
)
h2
ci (x, t) h(∂x Π− (x, t) − ∂x Π+ (x, t)) + (∂xx Π+ (x, t) + ∂xx Π− (x, t))
2
(
) h2
+ h∂x ci Π− (x + h, t) − Π+ (x − h, t) + ∂xx ci (Π− (x + h, t) + Π+ (x − h, t)).
2
In the following, all expressions are evaluated at (x, t) if not further specified. For the
probabilities we have
(
(
∑ ))
∑
+
Π−
cj
+ 2hαi
∂xx cj + O(h2 ),
x (x, t) − Πx (x, t) = 2hβi ∂x ∂x Wi 1 −
∑
∂xx Π+ (x, t) + ∂xx Π− (x, t) = − 2αi
∂xx cj + O(h),
(
∑ )
Π− (x + h, t) − Π+ (x − h, t) = 2hβ∂x Wi 1 −
cj + O(h2 ),
(
∑ )
Π− (x + h, t) + Π+ (x − h, t) = 2αi 1 −
cj + O(h),
which yields
ci (x, t + ∆t) − ci (x, t)
(
∑ )
∑ )
∂ (
=
2h2 βi ci
∂x Wi (1 −
cj ) + 2h2 βi ∂x ci ∂x Wi 1 −
cj
∂x
(
∑
∑ )
+ h2 ci αi
∂xx cj + h2 ∂xx ci αi 1 −
cj
(
)
)
∑
∑
∑
∂
∂ (
=
2h2 βi
ci (1 −
cj )∂x Wi + h2 αi
(1 −
cj )∂x ci + ci
∂x cj .
∂x
∂x
Thus, with an appropriate scaling ( α2i ≈ Di , Di being the diffusion coefficient for
species i) and time step ∆t = 2h2 , the resulting system of continuum equations reads
(
)
∑ )
∑
∑ )
∂ ((
∂t ci = Di
1−
cj ∂x ci + ci
∂x cj + µci 1 −
cj ∂x Wi ,
∂x
i
where µ is given by µ = 2β
αi .
An analogous derivation can
∑ be carried out in arbitrary dimension. Denoting
the volume density by ρ(x, t) =
cj (x, t) and the ionic current by Ji , we obtain the
system
∂t ci = ∇ · Ji ,
Ji = Di ((1 − ρ)∇ci + ci ∇ρ + µci (1 − ρ)∇Wi ).
(19)
Nonlinear PNP equatios for ion flux through confined geometries
7
3.2. Entropy
We want to investigate the behaviour in time of the entropy. The entropy for this
process is defined via
E(x, t) =
∫
∑(
)
ci (x, t) log ci (x, t) + (1 − ρ(x, t)) log(1 − ρ(x, t)) + κ−1
i ci (x, t)Wi (x, t) dx
i
We apply so-called entropy variables, cf. section 4.3:
ui (x, t) = ∂ci E(x, t) + const.
We consider the first derivative of the entropy in time under the assumption that
∂t Wi (x, t) = 0, a similar argument holds for Wi satisfying the Poisson equation. Then
we obatin
∫ ∑
(
)
(∂t ci log(ci ) − ∂t ρ log(1 − ρ)) + κ−1
∂t E =
i ∂t ci Wi dx
=
i
∫ ∑
∂t ci ui dx
i
=
∫ ∑
(∇ · (Di ci (1 − ρ)∇ui ) ui ) dx
i
=−
∫ ∑(
Di ci (1 − ρ) |∇ui |
2
)
dx.
i
Since 0 ≤ ci ≤ 1, 0 ≤ ρ ≤ 1 and Di > 1 we conclude that the entropy is decreasing in
time.
4. Modified Poisson-Nernst-Planck Equations
After the motivation of a modified PNP model we would like to discuss its scaling and
analysis. We recall that our modified PNP model is given by
(∑
)
−ϵ∆V = e
zj cj + f
(20)
(
)
∂t ci = ∇ · Di ((1 − ρ)∇ci + ci ∇ρ + ezi µi ci (1 − ρ)∇V + µi ci (1 − ρ)∇Wi0 ) , (21)
where ϵ = ϵ0 ϵr denotes the permittivity and e the elementary charge.
4.1. Scaling
First of all, we transform the above equations into an appropriate scaled and
dimensionless form, similar to standard scaling for PNP equations. Given a typical
length L̃, a typical voltage Ṽ and a typical ion concentration c̃, we define the new
variables
x = L̃xs ,
V = Ṽ Vs ,
ci = c̃cis , f = c̃fs and Di = D̃Dis .
The dimensionless formulation of system (20), (21) with an appropriatly scaled
external potential Wi0 is given by (omitting the subscript s)
∑
−λ2 ∆V =
zi ci + f
(22)
i
(
)
∂t ci = ∇ · Di ((1 − ρ)∇ci + ci ∇ρ + ηi zi ci (1 − ρ)∇V + ci (1 − ρ)∇Wi0 ) ,
(23)
Nonlinear PNP equatios for ion flux through confined geometries
8
with t = t̃ts , t̃ = L2 /D̃ and effective parameters
ϵ0 ϵr Ṽ
and ηi = eṼ µi .
eL̃2 c̃
The factor 1 − ρ is already given in a scaled form, thus no further scaling is necessary.
λ2 =
4.2. Boundary Conditions in Experiments
In the standard experimental setup used with patch-clamp techniques, there are
certain parts of the system where still no-flux conditions apply, but there are also
parts that need to be modeled via Dirichlet conditions since the system is not closed.
The concentrations are usually controlled in the left and right bath, which can be
modeled via
ci (x, t) = γi (x)
x ∈ ΓB ⊂ ∂Ω.
(24)
On the remaining part of the system no-flux boundary conditions apply, i.e.
Ji (x, t) · n = 0
x ∈ ∂Ω \ ΓB .
(25)
For the bath concentrations the restriction of charge neutrality applies, i.e.
∑
zi γi (x) = 0.
(26)
The electric potential is influenced via an applied potential between two electrodes.
This can be modeled by Dirichlet boundary conditions
V (x, t) = VD0 (x) + U VD1 (x)
x ∈ ΓE ⊂ ∂Ω,
(27)
where U is the applied voltage. On the remaining part of the system no-flux boundary
conditions apply, i.e.
∇V (x, t) · n = 0
x ∈ ∂Ω \ ΓE .
(28)
In a simple geometric setup one might choose ΓB = ΓE be the left and right end of
the domain, see figure 1.
ΓB
ΓE
ΓB
x
x
ΓB
x
ΓE
x
x
Figure 1. Experimental Set-Up
x
ΓB
Nonlinear PNP equatios for ion flux through confined geometries
9
4.3. Formulations of the Stationary Problem
In the following we shall focus on the stationary problem, which is of high importance
for computing flow characteristics such as current-voltage relations. The (scaled)
stationary problem is given by
∑
−λ2 ∆V =
zj cj + f,
(29)
j
(
)
0 = ∇ · Di ((1 − ρ)∇ci + ci ∇ρ + ηi zi ci (1 − ρ)∇V + ci (1 − ρ)∇Wi0 ) ,
(30)
∑
with ρ = cj and boundary conditions (24), (25), (27), (28).
The above formulation of the stationary problem in the natural physical density
variables is not necessarily the most suitable one for analysis and computation. As
in the standard PNP equations there are two possible transformations (often used in
semiconductor simulation), namely to entropy variables (called quasi-Fermi levels in
semiconductor theory) and so-called Slotboom-variables.
Fixing V , a natural entropy for the model is given by
∫ ∑
)
(
E(c1 , . . . , cM ) =
ci log ci + (1 − ρ) log(1 − ρ) + ηi zi ci V + ci Wi0 . dx
(31)
i
For the transient model with natural boundary conditions this entropy is decreasing
in time and a natural Lyapunov-functional for the analysis of existence and large-time
behaviour (cf. [23] ). Based on this convex entropy functional we can introduce a
standard duality transform to so-called entropy variables (cf. [24, 25])
ui = ∂ci E + const = log ci − log(1 − ρ) + ηi zi V + Wi0 .
(32)
The explicit inversion of this transform can be obtained from the exponential form
(
)
ci
= exp ui − ηi zi V − Wi0 ,
1−ρ
yielding after brief manipulations
ci =
exp(ui − ηi zi V − Wi0 )
.
∑M
1 + j=1 exp(uj − ηj zj V − Wi0 )
(33)
The stationary model (29), (30) in entropy variables can be written as
∑
zk exp(uk − ηk zk V − Wk0 )
= f,
−λ2 ∆V −
∑M
0
j=1 exp(uj − ηj zj V − Wj )
k 1+


(34)
exp(ui − ηi zi V − Wi0 )

)2 ∇ui  = 0.
∑M
1 + j=1 exp(uj − ηj zj V − Wj0 )
(35)

∇ · Di (
A particularly attractive feature of the transformation is the elimination of crossdiffusion, the coupling only occurs in the diffusion coefficients. Consequently, a
maximum principle holds for ui , it attains its maximum at ΓB , with the transformed
boundary conditions
∑
ui = log γi − log(1 −
γj ) + ηi zi (VD0 + U VD1 ) + Wi0 on ΓB ,
(36)
j
∇ui · n = 0
on ∂Ω \ ΓB .
(37)
A second transformation that is routinely used in semiconductors is the one to
so-called Slotboom-Variables, which we shall denote by vi in the following. In the
Nonlinear PNP equatios for ion flux through confined geometries
10
standard Nernst-Planck case those variables are simply obtained by multiplication
with exponentials of V , which is not directly useful in the modified case we consider.
However, this transformation can be related again to the entropy, namely by partly
reverting the transformation to entropy variables. For the sake of simple reading we
use the notation Fi for the functions
Fi (c1 , ..., cM ) = log ci − log(1 − ρ) = ui − ηi zi V − Wi0 ,
(38)
hence
Fi−1 (u1 , ..., uM ) =
1+
exp u
∑ i
.
j exp uj
(39)
Now we define Slotboom variables via
(
)
vi = Fi−1 (u1 , ..., uM ) = Fi−1 Fi (c1 , ..., cM ) + ηi zi V + Wi0 .
This transformation can be written explicitely as
ci =
1+
vi exp(−ηi zi V − Wi0 )
.
0
j vj (exp(−ηj zj V − Wj ) − 1)
∑
Hence, in the stationary case we obtain the transformed system in Slotboom variables
as
∑
v exp(−ηk zk V − Wk0 )
∑ k
=f
−λ2 ∆V −
zk
1 + j vj (exp(−ηj zj V − Wj0 ) − 1)
k



∑
∑
exp(−ηi zi V − Wi0 )


vj ) + vi
∇vj  = 0.
∇ · Di (
)2 ∇vi (1 −
∑
0
j
j
1 + j vj (exp(−ηj zj V − Wj ) − 1)
Due to the fact that equilibrium solutions are minimizers of the entropy we obtain that
both the entropy and Slotboom variables are constant in equilibrium. This property is
very favourable for linearization around equilibrium situations, in particular for small
applied voltages, since all the gradient terms drop out. As a consequence a certain
decoupling with the linearized Poisson equation appears, we shall discuss this issue in
detail below.
4.4. Existence
In the following we shall verify the existence of weak solutions ci ∈ H 1 (Ω) ∩ L∞ (Ω),
V ∈ H 1 (Ω) ∩ L∞ (Ω). For this sake we consider the transformed system in entropy
variables, since the maximum principle is of fundamental importance for obtaining
a-priori bounds. Throughout this section we shall make the following assumptions,
which appear reasonable in the kind of applications we investigate:
(A1) f ∈ L∞ (Ω), Wi0 ∈ L∞ (Ω) ∩ H 1 (Ω).
(A2) VD0 ∈ H 1/2 (ΓE ) ∩ L∞ (ΓE ), γi in H 1/2 (ΓB ) ∩ L∞ (ΓB ).
Note that assumptions (A1), (A2) imply that the transformed boundary values
for the entropy variables are elements of the same function spaces. In order to prove
existence we shall construct a fixed-point equation and apply Schauder’s theorem on
the set
M = {(u1 , . . . uM ) ∈ L2 (Ω)M | a ≤ ui ≤ b a.e. in Ω},
(40)
Nonlinear PNP equatios for ion flux through confined geometries
11
where
a = min inf uD
i (x),
i
x∈ΓB
b = max sup uD
i (x).
i
(41)
x∈ΓB
Here uD
i denotes the Dirichlet boundary values for the entropy variables. In order to
keep notation at a reasonable limit we set ηi = 1 throughout this section, the results
remain valid for arbitrary constant ηi .
We show existence by a fixed point argument. The corresponding operator is
constructed in the strong L2 -topology, and split into two parts. We set F = H ◦ G,
with operators G and H defined as follows:
G:
L2 (Ω)M
(u1 , . . . , uM )
→
7→
L2 (Ω)M × H 1 (Ω)
(u1 , . . . , uM , V ),
(42)
where V is the unique solution of the nonlinear Poisson equation
−λ2 ∆V =
∑
k
zk
exp(uk − zk V − Wk0 )
∑
+f
1 + j exp(uj − zj V − Wj0 )
(43)
with boundary conditions (27), (28). We define H by
H:
DH ⊂ L2 (Ω)M × H 1 (Ω)
(u1 , . . . , uM , V )
→
L2 (Ω)M
7
→
(v1 , . . . , vM ),
(44)
where the vi are the unique weak solutions of the linear elliptic equations (cf. (34),
(35))
)
(
exp(ui − zi V − Wi0 )
∑
∇vi = 0
(45)
∇·
(1 + j exp(uj − zj V − Wj0 ))2
subject to the boundary conditions
∂vi
= 0 on ∂Ω\ΓD
∂n
The domain of the operator H is set to
and
vi = uD
i on ΓD .
(46)
DH = G(L2 (Ω)M ).
Next we shall verify some favorable properties of G and H which are necessary in the
existence proof. We start with the well-definedness of G.
Lemma 4.1 Let M be given by (40) and K be a bounded subset of H 1 (Ω) × L∞ (Ω).
The operator G is well defined by (42), continuous on M, and it maps M into M × K.
Proof : Given (u1 , . . . , uM ) ∈ M, consider the functional


∫
∫
∑
λ2
2
J(V ) =
|∇V | dx +
log 1 +
γj exp(uj − zj V − Wj0 ) dx.
2 Ω
Ω
j
It is straight-forward to see that J is strictly convex and coercive on H 1 (Ω), thus
there exists a unique minimizer V ∈ H 1 (Ω) (respectively on the subspace representing
Dirichlet boundary condition), which is a weak solution of (43). Vice versa every
solution of (43) is a minimizer due to convexity. This implies existence and uniqueness
of a solution V ∈ H 1 (Ω).
Nonlinear PNP equatios for ion flux through confined geometries
12
From the structure of the right-hand side in the Poisson equation we see that
(
)
1 ∑
−∆V ≤ 2
|zi | + ∥f ∥∞
a.e. in Ω
λ
i
and
1
−∆V ≥ − 2
λ
(
∑
)
|zi | + ∥f ∥∞
a.e. in Ω
i
hold in a weak sense. Thus the maximum principle [26] provides a uniform bound for
V in L∞ (Ω). Moreover, by the Friedrichs inequality,
(
)
2
2
2
∥V ∥H 1 ≤ C ∥VD ∥H 1/2 (ΓE ) + ∥∇V ∥L2 (Ω)
2C
J(V )
λ2
2C
2
≤ C ∥VD ∥H 1/2 (ΓE ) + 2 J(ṼD ),
λ
2
≤ C ∥VD ∥H 1/2 (ΓE ) +
where ṼD is an arbitrary H 1 -extension of VD , we obtain a uniform bound for V in
H 1 (Ω).
Now let V and Ṽ be solutions of (43) for given (u1 , . . . , uM ) and (ũ1 , . . . , ũM ),
respectively. To simplify notation we introduce the new variable R given by
∑
zk exp(uk − zk V − Wk0 )
k∑
R(V, u) =
.
1 + j exp(uj − zj V − Wj0 )
Then, in a weak formulation we obtain
∫
∫
(
)
2
λ ∇(V − Ṽ )∇φ dx =
φ R(V, u) − R(Ṽ , ũ) dx
Ω
Ω
∫
∫
(
)
(
)
=
φ R(Ṽ , u) − R(Ṽ , ũ) dx +
φ R(V, u) − R(Ṽ , u) dx.
Ω
Ω
Choosing the test function φ = V − Ṽ and using the monotonicity of the second term
on the right-hand side we obtain
∫
2
2
2
2
λ ∇(V − Ṽ )
≤ λ ∇(V − Ṽ )
− (V − Ṽ )(R(V, u) − R(Ṽ , u)) dx
L2 (Ω)
L2 (Ω)
Ω
∫
= (V − Ṽ )(R(Ṽ , u) − R(Ṽ , ũ)) dx.
Ω
With the Friedrichs inequality (note that V − Ṽ vanishes on ΓE ) and the CauchySchwarz inequality we finally conclude
C ≤ 2 R(Ṽ , u) − R(Ṽ , ũ) 2 .
V − Ṽ 1
λ
H (Ω)
L (Ω)
Using the a-priori bounds for V in L∞ as well as those for ui defined by M, it is easy
to use the Lipschitz-continuity of the nonlinearity to further conclude that
√
C̃ ∑
2
∥uj − ũj ∥L2 (Ω) .
≤ 2
V − Ṽ 1
λ
H (Ω)
j
Hence, G is Lipschitz-continuous on M. The next step is to analyze the properties of the operator H on G(M).
Nonlinear PNP equatios for ion flux through confined geometries
13
Lemma 4.2 Let Q denote a compact subset of M. Then the operator H : G(M) → Q
is well defined by (44) and continuous on M × K.
Proof: It is straight-forward to see that
Ai = Di
exp(ui − zi V − Wi0 )
∑
∈ L∞ (Ω),
(1 + j exp(uj − zj V − Wj0 ))2
more precisely
0< (
Di exp(a − |zi |C − ∥Wi0 ∥∞ )
) 2 ≤ A i ≤ Di ,
∑
1 + j exp(b − |zj |C − ∥Wj0 ∥∞ )
where C is such that ∥V ∥L∞ ≤ C on G(M). Hence, standard theory [27] for elliptic
equations in divergence form implies the existence and uniqueness of a weak solution
vi of
∇ · (Ai ∇vi ) = 0 in Ω
with boundary conditions (46). Moreover, the maximum principle [26] for linear
elliptic equations implies a ≤ vi ≤ b with a, b defined in (41). Thus, H is well-defined
and maps into M. Due to the compactness of the embedding H 1 (Ω) ,→ L2 (Ω),
H(G(M)) is precompact.
To verify the continuity of H we consider the sequences V k → V in H 1 (Ω) and
k
ui → ui in L2 (Ω) . Then Aki → Ai in L2 (Ω) with the uniform bounds above. Let vik
be the weak solution of
∇ · (Aki ∇vi ) = 0,
then vi is uniformly bounded in H 1 (Ω) and hence there exists a weakly convergent
subsequence vikl ,→ v̂i for all i. Then
∫
∫
kl
kl
0=
Ai ∇vi ∇ϕ dx →
Ai ∇v̂i ∇ϕ dx.
Ω
Ω
for all test functions ϕ ∈ W 1,∞ (Ω). Since Ai ∈ L∞ (Ω) and W01,∞ (Ω) is dense in
H01 (Ω), we conclude that
∫
Ai ∇v̂i ∇ϕ dx
0=
Ω
also holds for ϕ ∈ H01 (Ω). With the trace theorem we can pass to the limit in (46)
and thus, v̂i is the weak solution of
∇ · (Ai ∇vi ) = 0 in Ω
with boundary condition (46).
By the uniqueness of the limit v̂i we conclude vik → vi weakly in H 1 (Ω) and thus
strongly in L2 (Ω) which implies the continuity of H. We can now employ Schauder’s Fixed Point Theorem [26], which assures the
existence of a fixed point of H(G(M)). This fixed point is a solution of (43), (45),
which we summarize in:
Theorem 4.3 (Global existence of stationary solutions) Let assumptions (A1),
(A2) be satisfied. Then, there exists a weak solution
(V, c1 , ..., cn ) ∈ H 1 (Ω)M +1 ∩ L∞ (Ω)M +1
Nonlinear PNP equatios for ion flux through confined geometries
of
−λ2 ∆V =
∑
zj cj + f
14
(47)
j
0 = ∇ · (Di ((1 − ρ)∇ci + ci ∇ρ + ηi ci (1 − ρ)∇Wi )) ,
(48)
with Wi = V +Wi0 and boundary conditions (27), (28) and (24),(25), such that further
0 ≤ ci ,
ρ≤1
a.e. in Ω.
Proof: We proved the global existence of a solution to (34), (35). To show the same
for (47),(48) the only thing left to do is to transform back to the original variables
ci =
exp(u − zi V − Wi0 )
∑ i
,
1 + j exp(uj − zj V − Wj0 )
(49)
and on the used function spaces we obtain the system in original variables c1 , ..., cM .
Thus, we obtain global existence for a stationary solution of (47), (48). 4.5. Regularity
Next we show higher regularity for the existence result presented in section 4.4. Of
course, improved regularity can only hold for smooth data. Thus, for the next two
sections we make in addition to (A1), (A2) the following assumptions:
(A3) Wi0 ∈ H 2 (Ω).
(A4) VD0 + U VD1 ∈ H 3/2 (ΓE ), γi ∈ H 3/2 (ΓB ).
With these assumptions, we obtain the following regularity for V, c1 , ..., cM : The righthand-side of (29) is obviously in L∞ (Ω) ,→ L2 (Ω), accordingly we have with (A4) that
∆V ∈ L2 (Ω) and this means V ∈ H 2 (Ω). For dimension n = 1, 2, 3 the Sobolev
embedding theorem ensures that H 2 (Ω) ⊂ L∞ (Ω), cf. [28]. From (30) we conclude
that
(1 − ρ)∆ci + ci ∆ρ = −∇(zi ci (1 − ρ)∇V + ci (1 − ρ)∇Wi0 ) ∈ L2 (Ω),
hence ∆ci ∈ L2 (Ω) and thus with (A4)
(V, c1 , ..., cm ) ∈ H 2 (Ω)M +1 .
4.6. Uniqueness in simpler Situations
In this section we take a closer look at the uniqueness of a solution of (29) and (30)
in the stationary case. We consider two special cases in which simplifications can be
made. In general, we cannot expect uniqueness and potential non-uniqueness may
even be related to interesting phenomena appearing in practice such as gating.
Unfortunately, the uniqueness proof cannot be performed for (V, c1 , ..., cM ) ∈
H 1 (Ω) ∩ L∞ (Ω). But the proof can be performed in H 2 (Ω), which is not a serious
restriction due to the regularity results of section 4.5. As above, we consider the
transformed system in entropy variables u1 , ..., uM with boundary condition
ui = ηi (γ1 , ..., γM )
x ∈ ΓB ⊂ ∂Ω.
Assumption (A4) leads to ηi ∈ H 3/2 (ΓB ) and thus, as ci ∈ H 2 (Ω),
ui = log ci − log(1 − ρ) + zi V + Wi0 ∈ H 2 (Ω).
Nonlinear PNP equatios for ion flux through confined geometries
15
Let u = (ui )i=1,...,M and η = (ηi )i=1,...,M and
F(U, η; V, u) : R × (H 3/2 )M × H 2 × (H 2 )M → H 3/2 × (H 3/2 )M × L2 × (L2 )M
denote the operator
V − VD0 − U VD1
ui − ηi
∑
exp(uk − zk V − Wk0 )
−λ2 ∆V −
−f
∑M
0
j=1 exp(uj − zj V − Wj )
k 1+
(
)
exp(ui − zi V − Wi0 )
∇ · Di
∇ui
∑M
(1 + j=1 exp[uj − zj V − Wj0 ])2
on ΓE
on ΓB
(50a)
(50b)
∈ L2
(50c)
∈ L2
(50d)
with boundary condition VD0 + U VD1 ∈ H 3/2 (ΓE ) and ηi ∈ H 3/2 (ΓB ).
The proof will be based on the implicit function theorem in Banach spaces [29].
Thus we have to show that F(U, η; V, u) is Frechet-differentiable with respect to V ,U ,η
and u. For the sake of brevity we only detail the existence of the Frechet-derivative
of the ith component of (50d) with respect to ui , which we denote with
Fi′ (ui ) : H 2 (Ω) → L2 (Ω).
We express this component as Fi (ui ) = ∇ · (Gi (u1 , ..., um )∇ui ) and conclude
(suppressing the dependence of Fi of uj for j ̸= i) for ui , v ∈ H 2 (Ω) ,→ L∞ (Ω)
∥Fi (ui + v) − Fi (ui ) − Fi′ (ui )v∥L2 (Ω)
∥v∥H 2 (Ω)
=
∥∇ · [G(ui + v)∇(ui + v) − G(ui )∇ui − G′ (ui )v∇ui − G(ui )∇v]∥L2 (Ω)
∥v∥H 2 (Ω)
.
(51)
The enumerator of (51) can be written as
∥∇ · [G(ui + v)∇(ui + v) − G(ui )∇ui − G′ (ui )v∇ui − G(ui )∇v]∥L2 (Ω) =
′
2
2
G (ui + v) |∇(ui + v)| + G(ui + v)∆(ui + v) − G′ (ui ) |∇ui | − G(ui )∆ui
2
−G′′ (ui )v |∇ui | − G′ (ui )∇v∇ui − G′ (ui )v∆ui − G′ (ui )∇u∇v − G(ui )∆v L2 (Ω)
.
Therefore we obtain
∥[G(ui + v) − G(ui ) − G′ (ui )v] ∆ui + [G(ui + v) − G(ui )] ∆v
+ [G′ (ui + v) − G′ (ui ) − G′′ (ui )v] |∇ui |2 + 2 [G′ (ui + v) − G′ (ui )] ∇ui ∇v
+G′ (ui + v)|∇v|2 L2 (Ω) .
(52)
Note that H 2 (Ω) ⊂ W 1,4 (Ω) for n = 1, 2, 3 cf. [28]. This ensures that all product
terms in (52) really are in L2 (Ω).
1
(52) ≤ G′′ (ξ2 )v 2 L∞ (Ω) ∥∆ui ∥L2 (Ω) + ∥G′ (ξ1 )v∥L∞ (Ω) ∥∆v∥L2 (Ω)
2
1
2
+ G′′′ (ξ3 )v 2 L∞ (Ω) ∥∇ui ∥L4 (Ω) + 2 ∥G′′ (ξ2 )v∥L∞ (Ω) ∥∇v∇u∥L2 (Ω)
2
2
+ ∥G′ (ξ1 )∥L∞ (Ω) ∥∇v∥L4 (Ω) .
Nonlinear PNP equatios for ion flux through confined geometries
16
Using the following L∞ -bounds
∥G∥L∞ (Ω) ≤ 1,
∥G′ ∥L∞ (Ω) ≤ 1,
as well as the fact that
∥∇v∥L4 (Ω) ≤ ∥∇v∥L2 (Ω) ,
∥G′′ ∥L∞ (Ω) ≤ 5 and ∥G′′′ ∥L∞ (Ω) ≤ 23,
{
}
∥v∥H 2 (Ω) ≥ ∥v∥L2 (Ω) , ∥∇v∥L2 (Ω) , ∥∆v∥L2 (Ω)
and
∥v∥H 2 (Ω) ≥ c1 ∥v∥L∞ (Ω) ,
we have
5
(51) ≤ c1 ∥v∥L∞ (Ω) ∥∆ui ∥L2 (Ω) + ∥v∥L∞ (Ω)
2
23
2
+
c2 ∥v∥L∞ (Ω) ∥∇ui ∥L4 (Ω) + 10 ∥v∥L∞ (Ω) ∥∇ui ∥L2 (Ω) + c3 ∥∇v∥L4 (Ω) .
2
As ∥∇v∥L2 (Ω) ≤ ∥v∥H 2 (Ω) and ∥∇v∥L∞ (Ω) ≤ c4 ∥v∥H 2 (Ω) , we conclude that
lim
∥v∥H 2 (Ω) →0
(53)
(53) → 0.
Therefore Fi′ (ui ) is a Frechet-derivative. All other derivatives can be estimated using
analoguous arguments. Next we prove uniqueness for small voltage and small bath
concentration.
4.6.1. Small Voltage In this case, we assume that a small voltage U is applied at
the right-hand side of the bath. In case U = 0, we obtain the equilibrium state. We
investigated this case in section 2, one can show the well-posedness of this problem
by standard techniques for elliptic equations. We regard the linearization around zero
voltage or in turn linearization around equilibrium. The linearized system in entropy
variables reads
∑ ∂cj
∑ ∂cj
−λ2 ∆Ṽ −
zj
Ṽ
= h1 ∈ L2 (Ω)
(54)
u˜k +
zj
∂uk
∂V
j
j,k
(
)
ki exp(−zi Veq + Wi0 )
∑
∇ · Di
∇ũi = hi+1 ∈ L2 (Ω).
(55)
(1 + j kj exp(−zj Veq + Wj0 ))2
The constants ki can be determined form the bath concentrations ηi via γi =
1+
ki
∑
j
kj .
It is again possible to show existence and uniqueness of a solution (Ṽ , ũ1 , ..., ũM ) via
standard theory for elliptic equations (note the partial decoupling of the equations in
the linearization). Furthermore, the left-hand side of (54), (55) is a Frechet-derivative
of (34), (35). We are now able to prove well-posedness of the problem for small voltage:
Theorem 4.4 (Well-posedness close to Equilibrium) Let assumptions (A1)(A4) be fulfilled and let ∥U ∥H 3/2 (ΓB ) be sufficiently small. Then, for each ηi ∈
(H 3/2 (ΓB ))M there exists a locally unique solution
(V, c1 , ..., cM ) ∈ H 2 (Ω)M +1
of problem (29), (30) and the transformed, linearized problem (54), (55) is well-posed.
Nonlinear PNP equatios for ion flux through confined geometries
17
Proof: We already showed that (V, c1 , ..., cM ) ∈ H 1 (Ω)M +1 ∩ L∞ (Ω)M +1 and
assumptions (A1)-(A4) imply that (V, c1 , ..., cM ) ∈ H 2 (Ω)M +1 , and thus u1 , ..., uM ∈
H 2 (Ω). The equation operator is Frechet-differentiable for ui ∈ H 2 . For U = 0,
the problem is well-posed and its Frechet-derivative exists with continuous inverse in
the respective function spaces. Thus, we can apply the implicit function theorem in
Banach spaces to conclude the existence of a locally unique solution of problem (29),
(30) around U = 0 and that the linearized, transformed problem is well-posed for
small U . After back transformation
exp(u − zi V − Wi0 )
∑ i
ci =
1 + j exp(uj − zj V − Wj0 )
we obtain the same result for (29), (30).
4.6.2. Small Bath Concentrations We now regard the stationary system around small
bath concentrations γ. Due to the transformation, γi = 0 implies ηi = 0. In case
γi = 0,
i = 1, ..., M
we can easily construct a solution:
Lemma 4.5 Let γi = 0, i = 1, ..., M . There exists a solution
(V, c1 , ..., cM ) ∈ H 2 (Ω)M +1
of problem (29), (30) which is given by
−λ2 ∆V0 = f
and
ci ≡ 0
for i = 1, ..., M .
Proof: The functions ci ≡ 0 satisfy (30) as well as the boundary conditions. Standard
results for the elliptic equation
−λ2 ∆V0 = f
with Neumann and Dirichlet boundary conditions on ΓE and ΓB imply existence and
uniqueness of the remaining problem.
The resulting system for the linearization around zero bath concentration is
∑ ∂cj
∑ ∂cj
zj
zj
Ṽ = gi ∈ L2 (Ω),
−λ2 ∆Ṽ −
u˜k +
∂uk
∂V
j
j,k
= gi+1 ∈ L2 (Ω).
(56)
∇ · (Di (∇c̃i + zi c̃i ∇(V0 + Wi0 )))
The equations are partially decoupled, thus the potential V0 is not computed via
the Poisson-Boltzmann equation anymore. Equation (56) is the stationary NernstPlanck equation. After a change of variables, the Slotboom transformation known
from semiconductor theory
vi = exp(zi (V0 + Wi0 ))ci ,
we obtain the system of linear elliptic equations


∑ ∂vj
∑ ∂vj
u˜k +
zj
Ṽ  = gi ∈ L2 (Ω),
(57)
−λ2 ∆Ṽ − exp(−zi (V0 + Wi0 )) 
zj
∂uk
∂V
j
j,k
∇ · (Di (exp(−zi (V0 +
= gi+1 ∈ L2 (Ω), (58)
whose well-posedness can be analyzed by standard techniques for elliptic equations
[30]. Existence and uniqueness of (58) is also found as a result in standard PNP
theory [8].
Wi0 ))∇vi )
Nonlinear PNP equatios for ion flux through confined geometries
18
Theorem 4.6 (Well-posedness for small bath concentrations) Let (A1)-(A4)
be fulfilled and let ∥γi ∥H 3/2 (ΓB ) be sufficiently small. Then, for each U ∈ H 3/2 (ΓB ),
there exists a locally unique solution
(V, c1 , .., cM ) ∈ H 2 (Ω)M +1
of problem (29), (30) and the linearized problem (57), (58) is well-posed.
Proof: Again, (V, c1 , ..., cM ) ∈ H 1 (Ω)M +1 ∩ L∞ (Ω)M +1 implies (V, c1 , ..., cM ) ∈
H 2 (Ω)M +1 and u1 , ..., uM ∈ H 2 (Ω). For η = 0, problem (29), (30) is well-posed.
The Frechet-derivative of (34), (35) exists with continuous inverse in the respective
function spaces. Furthermore, the equation operator is Frechet-differentiable, so that
we can apply the implicit function theorem in Banach Spaces to conclude the existence
of a locally unique solution of problem (29), (30) around η = 0 and that the linearized
problem (57), (58) is well-posed for small η. After back transformation
ci =
exp(u − zi V − Wi0 )
∑ i
1 + j exp(uj − zj V − Wj0 )
we obtain the same result for (29), (30).
As mentioned above, global uniqueness cannot be expected.
4.7. Reduction to One Dimension
The cross section of a filter inside an ion channel is much smaller than its longitudinal
extension, which is, e.g. in the example discussed in section 5, about 1nm. Therefore
transport through a channel is accordingly nearly a one-dimensional process. We try
to approximate the three-dimensional model by a one-dimensional one. Such a model
is faster and easier to handle computationally than the three-dimensional version. We
assume a domain of the form
Ωϵ = {x ∈ (−L, L), (y, z) ∈ Qϵ }
where
Qϵ = {(x, rϵ (x) cos θ, rϵ (x) sin θ) |0 ≤ rϵ (x) ≤ ϵr |θ ∈ [0, 2π)},
√
and
y 2 + z 2 ≤ r. The boundary conditions are Dirichlet at x = ±L, i.e.
ΓB = ΓE = {−L, +L} × Qϵ , and no-flux on the remaining part. We assume that the
boundary values for the potential and the densities are constant in the two segments
at x = −L and x = +L.
We rescale the variables describing the channel as x, y ϵ = ϵy, z ϵ = ϵz
with (y, z) ∈ Q1 . Starting with the Poisson equation, we rescale the potential
V ϵ (x, y ϵ , z ϵ ) = Ṽ ϵ (x, y, z). The same scaling is used for the densities cϵi (x, y ϵ , z ϵ )
as well as for the transformed densities uϵi (x, y ϵ , z ϵ ). From the existence proof we have
potential and densities in
V ϵ (x, y ϵ , z ϵ ),
cϵi (x, y ϵ , z ϵ )
uϵi (x, y ϵ , z ϵ )
∈ H 1 (Ωϵ ) ∩ L∞ (Ωϵ ),
with uniform bounds in ϵ in the supremum norm.
For the Poisson equation we obtain
)
(
1
1
ϵ
ϵ
2
ϵ
ϵ ϵ
2
ϵ
−λ ∆V (x, y , z ) = −λ ∂xx Ṽ (x, y, z) + 2 ∂yy Ṽ (x, y, z) + 2 ∂zz Ṽ (x, y, z)
ϵ
ϵ
∑
=
zj cϵj (x, y, z) + f (x).
(59)
j
Nonlinear PNP equatios for ion flux through confined geometries
19
The weak formulation of (59) is given by
)
∫ ∫ ∫ (
1
1
∂x Ṽ ϵ ∂x φ + 2 ∂y Ṽ ϵ ∂y Ṽ ϵ + 2 ∂z Ṽ ϵ ∂z Ṽ ϵ dx dy dz
λ2
ϵ
ϵ
Ω1


∫ ∫ ∫ ∑

=
zj cϵj + f  φ dx dy dz.
j
With the special test function
φ(x, y, z) = Ṽ ϵ (x, y, z) − g(x),
where g(x) denotes a linear function of x in Ωϵ such that Ṽ ϵ − g vanishes at x = ±L,
we obtain (
)
∫ ∫ ∫
1
1
2
ϵ
ϵ
ϵ
ϵ
ϵ
ϵ
λ
∂x Ṽ ∂x (Ṽ − g) + 2 ∂y Ṽ ∂y Ṽ + 2 ∂z Ṽ ∂z Ṽ
dx dy dz
ϵ
ϵ
Ω1


∫ ∫ ∫
(
)
∑

zj cϵj + f  Ṽ ϵ − g dx dy dz
=

≤
Ω1
∑
j

ϵ
|zj | cj L∞ (Ωϵ ) + ∥f ∥L∞ (Ωϵ )  Ṽ ϵ − g L∞ (Ω1 )
j
|Ω1 |.
(60)
The right-hand side is uniformly bounded and using the linearity of g we obtain
∫ ∫ ∫
∫ ∫
g(L) − g(−L)
λ2
∂x Ṽ ϵ ∂x g dx dy dz =
Ṽ ϵ (x, y, L) − Ṽ ϵ (x, y, −L) dy dz,
2L
1
Ω
which can be estimated uniformly in terms of the boundary values. We obtain an
estimate of the form
)
∫ ∫ ∫ (
1
1
λ2
|∂x Ṽ ϵ |2 + 2 |∂y Ṽ ϵ |2 + 2 |∂z Ṽ ϵ |2 dx dy dz ≤ k1 ,
ϵ
ϵ
Ω1
where k1 denotes a constant independent of ϵ. Thus, we conclude
∫ ∫ ∫ (
∫ ∫ ∫ (
)2
)2
∂y Ṽ ϵ
dx dy dz ≤ ϵ2 k1 and
∂z Ṽ ϵ
dx dy dz ≤ ϵ2 k1
Ω1
Ω1
as
∫ ∫well
∫ as(
)2
∂x Ṽ ϵ
dx dy dz ≤ k1 .
Ω1
Hence, for ϵ → 0 we have
∂y Ṽ ϵ 2
L (Ωϵ )
→0
and
∂z Ṽ ϵ L2 (Ωϵ )
→ 0,
and overall Ṽ ϵ is uniformly bounded in H 1 (Ω1 ). From that we conclude for ϵ → 0
along subsequences
V ϵ (x, ϵy, ϵz) = Ṽ ϵ (x, y, z) ⇀ V 0 (x)
in H 1 (Ωϵ ).
(61)
Next we consider the nernst-Planck equation in entropy variables with test
function φ(x, y, z) = ũϵi (x, y, z) − g(x), where g(x) is again a linear function as above.
We can use the uniform bounds in L∞ (Ω) to deduce that
0 < k2 ≤ (
exp(uϵi − ηi zi V − Wi0 )
)2 ≤ k3 ,
∑
ϵ
0
1 + j exp(uj − ηj zj V − Wj )
Nonlinear PNP equatios for ion flux through confined geometries
20
with constants k2 and k3 independent of ϵ, to derive analogous estimates for the
functions ũϵi as for Ṽ ϵ . As above, we conclude for ϵ → 0
∥∂y ũi ϵ ∥L2 (Ω1 ) → 0
and
∥∂z ũi ϵ ∥L2 (Ω1 ) → 0
and altogether uniform boundedness of ũϵi in H 1 (Ω). Thus, along subsequences for
ϵ → 0 we have
uϵi (x, ϵy, ϵz) = ũi ϵ (x, y, z) ⇀ u0i (x)
in H 1 (Ωϵ )
cϵi (x, ϵy, ϵz) = c˜i ϵ (x, y, z) ⇀ c0i (x)
in H 1 (Ωϵ ).
and
Choosing test functions φ(x, y, z) = φ(x), we have
∫ ∫ ∫
λ2 ϵ−2
∇V ϵ (x, y ϵ , z ϵ ) · ∇φ(x) dx dy dz =
ϵ
∫ ∫ ∫Ω
2 −2
∂x V ϵ (x, y ϵ , z ϵ )∂x φ(x) dx dy dz→ϵ→0
λ ϵ
Ωϵ
∫ ∫ ∫
2 −2
λ ϵ
∂x V 0 (x)∂x φ(x) dy dz dx =
Ωϵ
∫
∫ ∫
λ2 ϵ−2 ∂x V 0 (x)∂x φ(x)
dy dz dx =
|
{z
}
∫
− λ2
ϵ2 a(x)
(
)
∂x a(x)∂x V 0 (x) φ(x) dx,
with a(x) being the cross-sectional area of Ω1 at x. The right-hand-side of the Poisson
equation can be derived from


∫ ∫ ∫
∑

zj cϵj (x, y ϵ , z ϵ ) + f (x) φ(x, y, z) dx dy dz
ϵ−2
∫
→


Ωϵ
∑
j

c0j (x) + f (x) φ(x)
∫ ∫
ϵ−2 dy dz dx =
∫

a(x) 
j
∑

c0j (x) + f (x) φ(x) dx.
j
Accordingly, in the limit ϵ → 0 we obtain the one-dimensional Poisson equation


∑
(
)
c0j (x) + f (x) .
−λ2 ∂x a(x)∂x V 0 (x) = a(x) 
j
We proceed in a similar manner with the Nernst-Planck equations, using strong
Lp convergence to pass to the limit in the nonlinear mobilities. The resulting simplified
one-dimensional system is given by (suppressing the index 0 )


∑
− λ2 ∂x (a(x)∂x V )
= a(x) 
cj + f  ,


∂x a(x)Di (

exp(ui − ηi zi V − Wi )

)2 ∂x ui  = 0.
∑
1 + j exp(uj − ηj zj V − Wj )
j
Nonlinear PNP equatios for ion flux through confined geometries
PP
PP
PP
ΓL
ΓN
21
channel right bath ΓR
left bath
PP
PP
ΓN
PP
Figure 2. Sketch of the computational domain
5. Numerical simulations
In this section we shall illustrate the behaviour of the derived mathematical models
with numerical results. In particular we discuss the following three situations:
(i) Conductance close to equilibrium in a multi-dimensional model and its
dependence on concentration.
(ii) Concentration profiles for stationary solutions.
(iii) Current in the stationary case and its dependence on concentration.
(iv) Current vs voltage curves for the stationary system.
(v) Current for a changing charge profile.
We choose the following problem setup for all four problems if not mentioned
otherwise. We consider an L-type calcium selective ion channel. We assume that the
channel is modelled as cylinder with radius rc = 0.4nm and length lc = 1nm, which
is embedded into two bathes with length lb = 2nm and outer radius rb = 2.4nm. The
total length is accordingly L̃ = 5nm. We assume that the boundary is split into the
following parts: ΓB = ΓE = ΓL ∪ ΓR (see also figure 2).
We consider three species, Ca2+ , Na+ and Cl− inside the baths and channel, as
well as one confined species O−1/2 , which represents the permanent charge inside
the channel. The external potential Wi0 is set to zero, and, according to the
thermodynamic understanding, we assume µi = 1/kB T . We consider eight confined
O−1/2 particles in the channel, which represent the fixed charge. The physical
parameters are given in table 1, N denotes the number of particles.
We assume a particle radius of 0.15nm for all particles. According to that, the
typical or maximal concentration is corresponding to 61.5mol/l. The resulting effective
parameters after scaling and nondimensionalization are
λ2 =
ϵ0 ϵr Ṽ
eṼ
= 4.68 × 10−4 and η =
= 3.87.
2
k
eL̃ c̃
BT
5.1. Conductance close to equilibrium
Here we consider the linearized stationary case for nonlinear PNP in a two dimensional
rotationally symmetric domain in Slotboom variables given by:
∑
zj kj exp(−czj V eq )
∑
−λ2 ∆V eq =
+ cO
(62)
1 + kk exp(−czk V eq )
∑
(
)
)
∑
exp(−czi V eq )(1 + kj ) (
∑
0 =∇·
∇v˜i + ki
∇ṽj ) ,
(63)
eq
2
(1 + kj exp(−czj V ))
Nonlinear PNP equatios for ion flux through confined geometries
Meaning
Boltzmann constant kB
Temperature T
Avogadros constant NA
Vacuum permittivity ϵ0
Relative permittivity ϵr
Elementary charge e
Particle radius
Typical length L̃
Typical concentration c̃
Typical voltage Ṽ
Diffusion coefficient Ca2+
Diffusion coefficient Na+
Diffusion coefficient Cl−
Value
1.3806504 × 10−23
300
6.02214179 × 1023
8.854187817 × 10−12
78.4
1.602176 × 10−19
0.15
5
3.7037 × 1025
100
7.9 × 10−10
1.33 × 10−9
2.03 × 10−9
22
Unit
J/K
K
N/mol
F/m
C
nm
nm
N/l
mV
m2 /s
m2 /s
m2 /s
Table 1. Parameters for computation
where (63) holds for i = Na+ , Ca2+ and Cl− . The concentration cO denotes the fixed
density of O−1/2 ions inside the channel. The parameters ki can be calculated using
the initial condition V eq (ΓL ) = 0 on ceq
i , i.e.
cieq (ΓL ) =
1+
ki
∑
kj
= γiL ,
(64)
where γiL denotes the boundary condition on ΓL for ci . We want to compare the
30
nonlinear PNP
PNP
Conductance in 10
−20
S
25
20
15
10
5
0
0
2
4
6
8
10
12
Concentration added in mol/l
Figure 3. Linearized conductance for new model and PNP
behaviour of the conductance of (62), (63) with simulations of the classical, linearized
PNP model given by (after Slotboom transformation and linearization for PNP):
∑
−λ2 ∆V eq =
zj vj exp(−czj V eq ) + f,
0
= ∇ · (exp(−czi V eq )∇v˜i ) ,
i = Na+ , Ca2+ , Cl− .
Nonlinear PNP equatios for ion flux through confined geometries
23
In this case, only the qualitative behaviour is of interest. Therefore we neglect the
diffusion coefficients and assume the oxygens in the channel to be point charges in
both models. The conductance is therefore given by
∑∫
σ=e
zi J˜i · dn,
Γ0
i∈I
for I = {Ca2+ , Na+ , Cl− }. The function J˜i for the nonlinear PNP is given by
∑
(
)
)
∑
exp(−czi V eq )(1 + kj ) (
˜
∑
Ji =
∇
v
˜
+
k
∇ṽ
)
i∈I
i
i
j
(1 + kj exp(−czj V eq ))2
and for linear PNP by
J˜i = exp(−czi V eq )∇v˜i ,
i ∈ I.
The simulations were done using Comsol 3.5. figure 3 shows a concentration
versus conductance plot for several concentrations of species in the bathes. These
concentrations range from zero conzentration for all species to 12.3mol/l for NaCl
and CaCl2 . Due to the linearization, the boundary condition for ci are equal in both
bathes. Note that the maximum values chosen in this simulation correpsond to the
“full state” of the channel. The applied voltage is zero at the left-hand-side ΓL and
U = 100mV at the right-hand-side ΓR . We depicted the concentration-current plot
for the classical PNP system in figure 3 as well. Note that the current of the classical
PNP model increases no matter how “full” the channel is. On the contrary the current
of the nonlinear PNP model decreases to zero as the concentrations approach their
maximum values, which correspond to the “full state”.
5.1.1. Analytical computation of conductance Next we would like to give an
analytical explanation of the above detected phenomena. As we want to gain
insight into the qualitative behaviour, we consider only one species and calculate the
conductance and current analytically. The conductance for nonlinear PNP is given by
the linearized ion flux in entropy variables
k exp(−V eq )
σ=
∇ũ,
(65)
(1 + k exp(−V eq ))2
where k can be determined from (64), and we have set z = 1 and η = 1. We know
that the transformed boundary values are ũ(ΓL ) = 0 and ũ(ΓR ) = 1, according to
(32), which we have to derive with respect to the boundary value U for V to obtain
boundary values for ũ. After integrating (65) and taking into account that σ does not
depend on Ω we find
(∫
)−1
(1 + k exp(−V eq ))2
σ=
dω
.
k exp(−V eq )
Ω
Using (64) we deduce that
σ(γ) =
(1 −
γ)2
∫
Ω
exp(V
eq )
γ(1 − γ)
∫
.
dω + 2γ(1 − γ)Ω + γ 2 Ω exp(−V eq ) dω
For 0 ≤ γ ≤ 1, γ = 0 and γ = 1 are the only zero points and the denominator is
always larger than zero. For the Poisson Boltzmann model, it is not easy to determine
the dependence of
∫
S(γ) =
exp(±V eq (γ)) dω
Ω
Nonlinear PNP equatios for ion flux through confined geometries
24
of γ. In the following, we neglect the Poisson equation and assume a linear behaviour
of V , i.e.
V eq = U x.
This leads in the scaled domain to
∫
(exp(U ) − 1)
exp(U x) dω =
Γyz ,
U
Ω
∫ ∫
where Γyz is given by y z dy dz. Using Taylor expansion for U , i.e. exp(U ) =
1 + U + O(U 2 ), we finally arrive at
σ=
γ(1 − γ)
.
Γyz
The slope of σ is exactly as in figure 3. The current can be computed via
I = σU + O(U 2 ).
For the PNP model, we directly compute the current analytically. The current is
given by
I = c + c∇V = exp(−V )∇v,
where v denotes the transformation in Slotboom variables,which is given for PNP via
c = v exp(−V ). The boundary transforms according to v(ΓL ) = exp(V (ΓL ))γ. For
consistency, we assume equal boundary conditions for the concentration here as above.
Hence we obtain
(exp(U ) − 1)γ
I= ∫
.
exp(V ) dω
Ω
Neglecting the Poisson equation and assuming V = U x, the current for the resulting
Nernst Planck model is given by
γU
I=
+ O(U 2 ).
Γyz
As expected, the conductance, which is given by
γ
σ=
,
Γyz
shows a linear behaviour on γ, as can be seen in figure 3.
5.2. Concentration profiles for stationary solutions
Next we consider the stationary system in entropy variables in a one-dimensional
domain
(
)
∑ zk exp(uk − ηk zk V )(1 − cO )
2
∑
−λ ∇ · (a(x)∇V ) = a(x)
+f ,
(66)
1 + j exp(uj − ηj zj V )
k


0
exp(ui − ηi zi V )(1 − c0 )2


= ∇ · a(x)Di (
)2 ∇ui  ,
∑
1 + j exp(uj − ηj zj V )
(67)
for i ∈ I. Here, a(x) denotes the area function describing the cross section of the
cylindrical channel and bath. The variable cO denotes the concentration of oxygen
in the channel. This concentration is not transformed to the entropy variable uO
Nonlinear PNP equatios for ion flux through confined geometries
Potential V
25
Charge neutrality
20
−100
0
−200
−20
mV
0
−300
0
0.2
0.4
0.6
0.8
−40
1
0
0.2
20
6
15
4
2
0
0.8
1
0.8
1
0.8
1
10
5
0
0.2
0.4
0.6
0.8
0
1
0
0.2
Chlor
60
0.15
40
0.1
0
0
−20
0.2
0.4
0.6
0.6
20
0.05
0
0.4
Oxygen
0.2
mol
mol
0.6
Natrium
8
mol
mol
Calcium
0.4
0.8
1
0
0.2
0.4
0.6
Figure 4. Stationary profiles for nonlinear PNP
because it is zero in the bath. The contribution of cO to the transformed variables
ui , where i denotes Ca2+ , Na+ and Cl− , is given by (1 − cO ). We solve (66) and (67)
in an iterative manner: Let V 0 be the initial datum for the potential V and u0i the
corresponding initial entropy variable
(i) Given V j solve (67) for uji , i = Ca2+ , Na+ , Cl− .
(ii) Given uji solve the nonlinear Poisson equation (66) for V j+1 using Newton’s
method.
(iii) Go to i) until convergence.
We choose boundary conditions according to [2]: We assume 0.1mol/l for NaCl in
both bathes. We have 5 · 10−3 mol/l CaCl2 in the left bath and 10−1 mol/l CaCl2 in
the right bath. We assume that the external potential is set to zero and a potential
of −50mV is applied in the left bath. The physical parameters are given in table 4.
The area function gives the scaled cross section of the channel and bath, where the
radius of the bath evolves linearly from 0.4nm at the channel to 2.4nm at the end of
the bathes. The simulations were done in Matlab, we chose a mesh size h = 0.005.
The stationary solution is depicted in figure 4. Note that the channel is in the region
between 0.4 and 0.6 on the x-axis. The concentration of oxygen is high inside the
channel. As the nonlinear PNP model prevents overcrowding, the charge neutrality
condition is not fulfilled anymore inside of the channel due to the high concentration
of oxygens. As a comparison, the stationary profiles for PNP are shown in figure 5.
It can clearly be seen that the charge neutrality is fulfilled here, but the channel gets
overcrowded as the total mass inside the channel is above the maximal admissable
concentration. This overcrowding effect is a well-known problem for PNP.
Nonlinear PNP equatios for ion flux through confined geometries
Potential V
Charge neutrality
mV
0
5
−50
0
−100
−5
−150
−10
−200
0
0.2
0.4
0.6
0.8
−15
1
0
0.2
0.4
Calcium
1
0.8
1
0.8
1
20
mol
mol
0.8
30
0.6
0.4
10
0.2
0
0.2
0.4
0.6
0.8
0
1
0
0.2
0.4
Chlor
60
0.15
40
0.1
20
0.05
0
0
−20
0
0.2
0.4
0.6
0.6
Oxygen
0.2
mol
mol
0.6
Natrium
0.8
0
26
0.8
1
0
0.2
0.4
0.6
Figure 5. Stationary profiles for PNP
5.3. Current and its dependence on concentration
We take a closer look on the dependence of current in the stationary case for the
example of the L-type Calcium channel described above. The ion flux for the nonlinear
PNP model ist given by
Ji = −Di a((1 − ρ)∂x ci + ci ∂x ρ + ηi zi ci (1 − ρ)∂x V ),
(68)
or, in the transformed expression
exp(ui − ηi zi V )
Ji = −Di a (
)2 ∂x ui .
∑
1 + j exp(uj − ηj zj V )
(69)
The current I flowing through the channel from one bath to the other, which can be
experimentally measured, is given by
∑∫
I=e
zi Ji · dn,
i∈I
Γ0
where Γ0 denotes the cross section of the channel and I = {Ca2+ , Na+ , Cl− }. The
oxygen ions do not contribute to the flux as they are fixed inside the channel. The
model setup is chosen as in the previous section. For the boundary condition assume
an applied potential of 50mV inside the left bath, in order to obtain a positive current.
As in the previous section, a solution containing 0.1mol/l NaCl is in both baths. In
the right bath, we have 5 · 10−3 mol/l CaCl2 . During the simulation, we add CaCl2 to
the left bath, starting from 1 · 10−3 mol/l up to 3 · 10−1 mol/l. The resulting curves of
current versus concentration in the left bath are shown in figure 6. The ion flux for
the PNP model ist given by
Ji = −Di a(∂x ci + ηi zi ci ∂x V ) = −Di a exp(ui − ηi zi V )∂x ui .
(70)
Nonlinear PNP equatios for ion flux through confined geometries
27
15
nonlinear PNP
PNP
Current in pA
10
5
0
0
50
100
150
200
250
300
mMol Ca2+ added
Figure 6. Current for nonlinear PNP and PNP
Due to the overcroding that takes place in the nonlinear model, the current of nonlinear
PNP saturates. The current for PNP shows nearly a linear increase. Furthermore,
the nonlinear current is remarkably less than the current for PNP.
5.4. Current versus voltage curves for the stationary system
We are going to take a closer look at the current versus voltage relation, cf. [2]. We
use the same setup as in section 5.2. figure 7 shows this relation. The voltage applied
in the right bath is keep zero, and the voltage in the left bath is varied. As expected,
the current of the nonlinear model lies again significantly below the current for PNP.
140
mod. PNP
PNP
120
100
Current in pA
80
60
40
20
0
−20
−40
−60
−80
−60
−40
−20
0
20
Voltage in mVolt
Figure 7. Current vs. voltage relation
40
60
80
Nonlinear PNP equatios for ion flux through confined geometries
28
5.5. Current for a changing charge profile
Finally we would like to take a closer look on how current is depending on the charge
profile by varying the positions of fixed charges. We do not change the number of
fixed oxygens, we simply contract them to the center of the channel. We affect the
function describing the density distribution of oxygens in the following way:

x < 0.5 − 0.1ϵ

0
0.5 − 0.1ϵ ≤ x ≤ 0.5 + 0.1ϵ
f (x) = c0 /ϵ
(71)


0
x > 0.5 − 0.1ϵ
In case that ϵ = 1, the density of oxygens is constant inside the channel, which lies
between 0.4 and 0.6 on the x-axis. If we decrease ϵ, the total density remains the
same, but it extension in x direction gets smaller, therefore the top goes up. The
smallest ϵ we use is 0.4. This value leads to concentrations inside the channel which
is above the assumed maximal concentration. In figure 8 we show current versus 1 − ϵ
for nonlinear PNP and PNP. It shows that if the concentration in the channel is large
enough, nonlinear PNP detects crowding. The current breaks and gets zero in the
crowded state. PNP is not able to detect these crowding effects.
12
nonlinear PNP
PNP
10
Current in pA
8
6
4
2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
1−epsilon
Figure 8. Current vs. charge distribution
6. Conclusion
In this paper we analyzed a nonlinear Poisson Nernst-Planck model which takes the
size of ions into account. We presented the macroscopic derivation of this model, here
the size exclusion effects resulted in nonlinear mobilities. First numerical simulations
of ion channels with the modified PNP model showed interesting and promising
features. As expected, one can observe several effects that arise due to crowding:
The conductance, which acts like the inverse of the resistance, is linearly increasing in
the PNP model but shows a decreasing behavior in the nonlinear case. This leads to
current saturation which is experimentally measured, but which can not be detected
using PNP. To detect the volume exclusion effects with PNP, several approaches like
Nonlinear PNP equatios for ion flux through confined geometries
29
mean spherical approximation or density functional theory have proposed. Their
qualitative behavior is similar to the nonlinear PNP model analyzed in this paper.
Nonetheless, the derivation of this model is very intuitive and elementary and the
computational cost significantly smaller.
Several issues on this model remain open. The proposed model is based on the
assumption that all ions have the same radius, this should be generalized to different
radii. This point of high interest because the dimension of ions has an effect on the
volume selectivity of the channels. However the calcium channel studied in this paper
is indeed calcium selective, which is a result of the charge selectivity.
Furthermore we would like to study whether the model is able to reproduce biological
phenomena such as gating. For this purpose it is necessary to include a more detailed
structure of the membrane into the model than performed in this paper. In particular
the ability of the model to reproduce blocked states by relatively small changes of
permanent charge is encouraging in this direction. This modeling approach is also
interesting for other applications where size exclusion effects should be taken into
account, like chemotaxis with different cell types, swarming or pedestrian dynamics
with heterogeneous agents.
Acknowledgments
MTW acknowledges financial support of the Austrian Science Foundation FWF via
the Hertha Firnberg Project T456-N23. MB and BS acknowledge financial support
from Volkswagen Stiftung via the grant Multi-scale simulation of ion transport through
biological and synthetic channels. The authors thank Z. Siwy (UCI) for stimulating
discussions.
References
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SIAM J. Appl. Math., 52(5):1405–1425, 1992.
[2] D Gillespie, W Nonner, and R S Eisenberg. Coupling Poisson-Nernst-Planck and density
functional theory to calculate ion flux. J. Phys: Condens. Matter, 14:12129–12145, 2002.
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