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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 Munster,Einsteinstr. 62, 48149 Munster, Germany2 Department of Mathematics, University of Vienna, Nordbergstr. 15, 1090Vienna, Austria

E-mail: [email protected], [email protected],[email protected]

Abstract. The mathematical modelling and simulation of ion transport troughbiological and synthetic channels (nanopores) is a challenging problem, with directapplication in biophysics, physiology and chemistry. At least two major effectshave to be taken into account when creating such models: the electrostaticinteraction of ions and the effects due to size exclusion in narrow regions.While mathematical models and methods for electrostatic interactions are well-developed and can be transferred from other flow problems with charged particles,e.g. semiconductor devices, less is known about the appropriate macroscopicmodelling of size exclusion effects.

Recently several papers proposed simple or sophisticated approaches forincluding size exclusion effects into entropies, in equilibrium as well as offequilibrium. The aim of this paper is to investigate a second potentially importantmodification due to size exclusion, which often seems to be ignored and is notimplemented in currently used models, namely the modification of mobilities dueto size exclusion effects. We discuss a simple model derived from a self-consistedrandom walk and investigate the stationary solutions as well as the computationof conductance. The need of incorporating nonlinear mobilities in high densitysituations is demonstrated in an investigation of conductance as a function of bathconcentrations, which does not lead to obvious saturation effects in the case oflinear mobility.

AMS classification scheme numbers: 92C35,92C30,35J47,35J60,35Q84,35J70,65M06

Submitted to: Nonlinearity

Nonlinear PNP equations for ion flux through confined geometries 2

1. Introduction

Mathematical models for crowded motion recently received strong attention due tovarious application in ion transport (cf. [1], [2], [3] ), cell biology (cf. [4], [5], [6]) andeven human behaviour (cf. [7]). While various approaches for microscopic modelling,either based on equations of motions with forces accounting for finite sizes (cf. [8]) oron exclusion processes, have been investigated in detail, there are still open problemsin the transition to macroscopic models based on partial differential equations. Inthermodynamic equilibrium this transition and the resulting modification of entropiesare investigated in various approaches (cf. [9], [10]). Away from equilibrium usuallyjust standard equations with linear mobilities and the modified entropies are used,whose appropriateness remains unclear. The main reason seems to be the strong localinteractions, which destroy propagation of chaos and thus the standard mean-fieldlimit. Recently, several authors have demonstrated that it is more appropriate touse models with nonlinear saturating mobility instead of the classical linear mobilitymodels, for a single mobility (cf. [7], [11], [12], [13]) and in few papers also for multiplespecies (cf. [7], [14]). Those results so far have been achieved for toy models ratherthan for practical applications and are hence still widely neglected in applied sciencesand the simulation of real-life phenomena.

The classical macroscopic model for ion transport are the Nernst-Planck equationsfor the ion concentrations ci (each i = 1, . . . ,M denoting an ion species with chargezi, diffusion coefficient Di, and mobility µi)

∂tci = ∇ ·(Di(∇ci + µici(ezi∇V +∇W 0

i ))), (1)

in a domain Ω, where e denotes the elementary charge, V is the electric and W 0i

an external potential. In order to obtain a self-consistent model we supplement theNernst-Planck equations with the Poisson equation, given by

−∇ · (ǫ∇V ) = e(∑

zjcj + f)

, (2)

where ǫ denotes the permittivity and f a permanent charge density. Using the rescaling(resampling) Di = µ−1

i ξi, this model can be written as a formal gradient flow

∂tci = ∇ · (ξici∇∂ciE(c1, . . . , cM )) (3)

for an entropy functional of the form

E(c1, . . . , cM ) =M∑

i=1

∫

Ω

(µ−1i ci log ci + eziciV [c1, . . . , cM ] + ciW

0i

)dx, (4)

where V [c1, . . . , cM ] depends implicitly on the concentration vector via the solution ofthe Poisson equation (2).

The gradient flow structure (3) with a linear mobility has been frequentlydiscussed in terms of optimal transport theory and Wasserstein metrics for probabilitymeasures (cf. [15]). It can be derived in a mean-field limit from microscopic particlemodels (cf. e.g. [16], [17]). The rigorous derivation however breaks down if volumeexclusion effects are included, an issue naturally arising in ion transport throughchannels and pores since available volumes are not orders of magnitudes larger thanthe ion sizes.

The standard ansatz to overcome this problem is to derive entropies based onthermodynamic principles, like density functional theory (DFT) (see [2],[18]) or meanspherical approximation (MSA), see [19]. Here two problems arise, first that the


derivation is done in equilibrium. We see that the modification of the entropy isnot enough to describe saturation effects in confined geometries. Therefore we derivea mathematical model with a modified entropy and mobility, which replicates theseeffect. A second problem is that DFT is computationally intractable. Thereforeonly 1D implementations can be found in the literature (see e.g. [20]). A differentapproach in this direction is the local density approximation (LDA) which is asolid mathematical approximation of the DFT functional, although it violates thethermodynamic formalism, see cf [21].

In this paper we want to discuss a modified macroscopic modelling of ion transportthrough confined regions, a problem of particular importance in physiology (membraneion channels, cf. [1], [2]) and electrochemistry (polymer membranes and nanopores,cf. [22]). We present a modification of the classical Poisson-Nernst-Planck equations(cf. [23]) accounting for nonlinear mobility effects, provide a mathematical analysis ofthe arising equations, and discuss some practical implications.

The remainder of the paper is organised as follows: In section 2 we reviewthe modified entropy and arising modified Poisson-Boltzmann equations, before wesketch the derivation of a nonlinear mobility model for off-equilibrium computationsin section 3. The resulting generalised Poisson-Nernst-Planck (PNP) equations andtheir mathematical properties are discussed in section 4, as well as some aspectsof dimension reduction in narrow regions. We turn our attention to a study ofconductance close to equilibrium as well as current in the stationary case, whichare computed numerically, in section 5. The latter will provide some insight intocurrent saturation at high concentrations, which can only be included via nonlinearmobilities.

2. Equilibria and Modified Poisson-Boltzmann Equations

In the following we briefly recall the computation of equilibria (i.e. if all ion fluxesvanish) by classical Poisson-Boltzmann equations [23] as well as the modified onesrecently introduced by Li et al. [24, 9, 10]. Standard equilibria are obtained from zeroflux in the Poisson-Nernst-Planck setup, i.e.

0 = Ji = Di(∇ceqi + µiceqi (ezi∇V eq +∇W 0

i ))

= ξiceqi ∇

(µ−1i log ceqi + eziV

eq +W 0i )), (5)

which results in the nonzero concentration

ceqi = kie−

eziVeq+W0

i

µ−1i . (6)

Note that varying the constants ki also defines the possible equilibrium boundaryconditions. The form of the equilibria is called Boltzmann statistics, they can also becomputed as minimiser of the entropy (4).

Inserting the equilibrium concentrations into the Poisson equation (2) yields thePoisson-Boltzmann equation

−∇ · (ǫ∇V eq) = e

(∑

i

zikie−

eziVeq+W0

i

µ−1i + f

)

, (7)

which is frequently studied in biochemistry, cf. [9]. The Poisson-Boltzman equationis a nonlinear elliptic equation with a unique solution. It can numerically be solvedusing finite element or finite difference discretization and Newton iterations.


In the case of transport through narrow regions an implicit solvent model seemsmore appropriate. Roughly speaking implicit solvent models assume that there isa maximal possible volume density, which is indeed achieved everywhere, since theremaining part is filled by the solvent. Taking the solvent as an electrically neutralspecies c0 implies a relation of the form

1 =M∑

i=0

αici, (8)

where αi is the maximal volume fraction of the i−th species (volume of a particletimes maximal density). For simplicity we set αi = 1 in the following (we refer to [10]for the general case of species with nonuniform sizes). Then the solvent concentrationis computed as

c0 = 1− ρ := 1−

M∑

i=1

ci. (9)

Instead of an entropy for the m + 1 species including the solvent, one thus obtains areduced entropy functional

E(c1, . . . , cM ) =M∑

i=1

∫

Ω

(µ−1i ci log ci + µ−1

0 (1− ρ) log(1− ρ) + eziciV + ci(W0i −W 0

0 ))dx

The equilibria in the modified model can be computed from the first-order optimalityconditions (taking into account the constraint from mass conservation)

λi = ∂ciE = µ−1i log ci − µ−1

0 log(1− ρ) + zieV + (W 0i −W 0

0 ),

with constant Lagrange multiplier λi ∈ R. Defining ki = exp( λi

µ−10

) and Wi = W 0i −W 0

0

we find

c

µ−1i

µ−10

i

1− ρ= ki exp(−

zieV

µ−10

−Wi

µ−10

).

If µ−10 = µ−1

i , then the equilibria can be computed explicitly in so-called Fermi-Dirac statistics [8]

ceqi =ki exp(−

zieVeq

µ−10

− Wi

µ−10

)

1 +∑

j kj exp(−zjeV eq

µ−10

−Wj

µ−10

), (10)

with constants ki to be determined from the bath concentrations γi. Equation (10)implies that 0 ≤ ceqi ≤ ρ ≤ 1 holds at each point. The associated modified Poisson-Boltzmann equation (cf. [9]) reads

−∇ · (ǫ∇V eq) = e

∑

i

ziki exp(−

zieVeq

µ−10

− Wi

µ−10

)

1 +∑

j kj exp(−zjeV eq

µ−10

−Wj

µ−10

)+ f

, (11)

for which we will show in Lemma 4.1 that there exists a unique solution. Throughoutthe paper, we assume in the following µ0 = µi = µ.


3. Derivation of a Modified Model

In order to compute flow through narrow pores an off-equilibrium description of iontransport is needed. A simple and frequently used way to do so is the transportgradient flow structure (3) with linear mobility (cf. [15]), which yields the standardPNP model in the case of the logarithmic entropy (4). For implicit solvent models thevalidity of a model with linear mobility is at least doubtfull for large volume densitiesρ, since one expects flow saturation due to volume filling. In order to illustrate theneed for nonlinear mobilities we sketch the derivation of a nonlinear mobility modelfrom a microscopic lattice-based model with volume exclusion, which serves as thebasis of our subsequent investigations.

3.1. Derivation from Hopping Model

We derive a system of drift-diffusion equations from a self-consistent one-dimensionalhopping 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 beoccupied by a particle with charge zi. The probability of finding a particle with chargezi 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) iscomputed self-consistently from the Poisson equation

−ǫ∂xxV (x, t) = e1

N

∑

i

ziδ(x− xhi (t)) + ef(x), (12)

where N denotes the number of particles and xhi their position. Equation (12)

converges to

−ǫ∂xxV (x, t) = e∑

i

zici(x, t) + ef(x) (13)

in the large N limit. In addition to the electrostatic potential, we model other forcesvia an external potential W 0

i . With Wi(x, t) = ziV (x, t) + W 0i (x, t) the transition

rates for each species are given by

Π+ci(x, t) = αi exp(−β(Wi(x+ h, t)−Wi(x, t))) (14)

Π−ci(x, t) = αi exp(−β(Wi(x− h, t)−Wi(x, t))), (15)

where αi denotes a normalisation constant and β denotes the mobility constant. Taylorexpansion up to order h2 and rescaling, gives

Π+ci(x, t) = P (jump of ci from position x to x+ h in (t, t+∆t))

= αi − hβi∂xWi(x+ h/2, t) +O(h2), (16)

Π−ci(x, t) = P (jump of ci from position x to x− h in (t, t+∆t))

= αi + hβi∂xWi(x− h/2, t) +O(h2), (17)

with βi = αiβ.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 simplemodel by

Π+ci = Π+

ci · P (position x+ h is at time t not occupied),

Π−ci = Π+

ci · 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−∑

j

cj(x, t),

which corresponds to rigorous results for one species, cf. [12]. We mention here thatthe derivation of this model is also justified in the usual stochastic setting. Here weare 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)−Π−

ci(x, t))

+ ci(x+ h, t)Π−ci(x+ h, t) + ci(x− h, t)Π−

ci(x− h, t).

Therefore we have (suppressing 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) =

ci(x, t)

(

h(∂xΠ−(x, t)− ∂xΠ

+(x, t)) +h2

2(∂xxΠ

+(x, t) + ∂xxΠ−(x, t))

)

+ h∂xci(Π−(x+ h, t)−Π+(x− h, t)

)+

h2

2∂xxci(Π

−(x+ h, t) + Π+(x− h, t)).

In the following, all expressions are evaluated at (x, t) if not further specified. For theprobabilities we have

Π−x (x, t)−Π+

x (x, t) = 2hβi∂x(

∂xWi

(

1−∑

cj

))

+ 2hαi

∑

∂xxcj +O(h2),

∂xxΠ+(x, t) + ∂xxΠ

−(x, t) = − 2αi

∑

∂xxcj +O(h),

Π−(x+ h, t)−Π+(x− h, t) = 2hβ∂xWi

(

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)

= 2h2βici∂

∂x

(

∂xWi(1−∑

cj))

+ 2h2βi∂xci∂xWi

(

1−∑

cj

)

+ h2ciαi

∑

∂xxcj + h2∂xxciαi

(

1−∑

cj

)

= 2h2βi∂

∂x

(

ci(1−∑

cj)∂xWi

)

+ h2αi∂

∂x

(

(1−∑

cj)∂xci + ci∑

∂xcj

)

.

Thus, with an appropriate scaling (αi

2 ≈ Di, Di being the diffusion coefficient forspecies i) and time step ∆t = 2h2, the resulting system of continuum equations reads

∂tci = Di∂

∂x

((

1−∑

cj

)

∂xci + ci∑

∂xcj + µci

(

1−∑

cj

)

∂xWi

)

,

where µ is given by µ = 2βi

αi.

An analogous derivation can be carried out in arbitrary dimension. Denotingthe volume density by ρ(x, t) =

∑cj(x, t) and the ionic current by Ji, we obtain the

system

∂tci = ∇ · Ji, Ji = Di((1− ρ)∇ci + ci∇ρ+ µci(1− ρ)∇Wi). (18)


3.2. Entropy

We want to investigate the behaviour in time of the entropy. The entropy for thisprocess is defined via

E(x, t) =M∑

i=1

∫

(ci(x, t) log ci(x, t) + (1− ρ(x, t)) log(1− ρ(x, t)) + µici(x, t)Wi(x, t)) dx.

We apply so-called entropy variables, cf. section 4.3:

ui(x, t) = ∂ciE(x, t) + const.

We consider the first derivative of the entropy in time under the assumption that∂tWi(x, t) = 0, a similar argument holds for Wi satisfying the Poisson equation. Thenwe obtain

∂tE =

∫ M∑

i=1

((∂tci log(ci)− ∂tρ log(1− ρ)) + µi∂tciWi) dx

=

∫ M∑

i=1

∂tciuidx

=

∫ M∑

i=1

(∇ · (Dici(1− ρ)∇ui)ui) dx

= −

∫ M∑

i=1

(

Dici(1− ρ) |∇ui|2)

dx.

Since 0 ≤ ci ≤ 1, 0 ≤ ρ ≤ 1 and Di > 1 we conclude that the entropy is decreasing intime.

4. Modified Poisson-Nernst-Planck Equations

After the motivation of a modified PNP model we would like to discuss its scaling andanalysis. We recall that our modified PNP model is given by

−ǫ∆V = e(∑

zjcj + f)

(19)

∂tci = ∇ ·(Di((1− ρ)∇ci + ci∇ρ+ eziµici(1− ρ)∇V + µici(1− ρ)∇W 0

i )), (20)

where ǫ = ǫ0ǫr denotes the permittivity and e the elementary charge.

4.1. Scaling

First of all, we transform the above equations into an appropriate scaled anddimensionless form, similar to standard scaling for PNP equations. Given a typicallength L, a typical voltage V and a typical ion concentration c, we define the newvariables

x = Lxs, V = V Vs, ci = ccis, f = cfs and Di = DDis.


The dimensionless formulation of system (19), (20) with an appropriately scaledexternal potential W 0

i is given by (omitting the subscript s)

−λ2∆V =∑

i

zici + f (21)

∂tci = ∇ ·(Di((1− ρ)∇ci + ci∇ρ+ ηizici(1− ρ)∇V + ci(1− ρ)∇W 0

i )), (22)

with t = tts, t = L2/D and effective parameters

λ2 =ǫ0ǫrV

eL2cand ηi = eV µi.

The factor 1− ρ is already given in a scaled form, thus no further scaling is necessary.

4.2. Boundary Conditions in Experiments

The patch clamp method (see, e.g. [25]) is a technique which allows measurements offlows through single ion channels. In the standard experimental setup used with patch-clamp techniques, there are certain parts of the system where still no-flux conditionsapply, but there are also parts that need to be modeled via Dirichlet conditions sincethe system is not closed. The concentrations are usually controlled in the left andright bath, which can be modeled via

ci(x, t) = γi(x) x ∈ ΓB ⊂ ∂Ω. (23)

On the remaining part of the system no-flux boundary conditions apply, i.e.

Ji(x, t) · n = 0 x ∈ ∂Ω \ ΓB . (24)

For the bath concentrations the restriction of charge neutrality applies, i.e.∑

ziγi(x) = 0. (25)

The electric potential is influenced via an applied potential between two electrodes.This can be modeled by Dirichlet boundary conditions

V (x, t) = V 0D(x) + UV 1

D(x) x ∈ ΓE ⊂ ∂Ω, (26)

where U is the applied voltage. On the remaining part of the system no-flux boundaryconditions apply, i.e.

∇V (x, t) · n = 0 x ∈ ∂Ω \ ΓE . (27)

In a simple geometric setup one might choose ΓB = ΓE be the left and right end ofthe domain, see figure 1. We assume that ∂Ω, ΓE and ΓB are piecewise C1 in thefollowing. Moreover, we shall use the definition

H10 (Ω) :=

u ∈ H1(Ω)|u|ΓB

= 0. (28)


ΓE ΓE

ΓB

ΓB

ΓB

ΓB

x

xx

xx

x

Figure 1. Experimental Set-Up

4.3. Formulations of the Stationary Problem

In the following we shall focus on the stationary problem, which is of high importancefor computing flow characteristics such as current-voltage relations. The (scaled)stationary problem is given by

−λ2∆V =∑

j

zjcj + f, (29)

0 = ∇ ·(Di((1− ρ)∇ci + ci∇ρ+ ηizici(1− ρ)∇V + ci(1− ρ)∇W 0

i )), (30)

with ρ =∑

cj and boundary conditions (23), (24), (26), (27).The above formulation of the stationary problem in the natural physical density

variables is not necessarily the most suitable one for analysis and computation. Asin the standard PNP equations there are two possible transformations (often used insemiconductor simulation), namely to entropy variables (called quasi-Fermi levels insemiconductor theory) and so-called Slotboom-variables.

Fixing V , a natural entropy for the model is given by

E(c1, . . . , cM ) =

∫∑

i

(ci log ci + (1− ρ) log(1− ρ) + ηiziciV + ciW

0i .)dx (31)

For the transient model with natural boundary conditions this entropy is decreasingin time and a natural Lyapunov-functional for the analysis of existence and large-timebehaviour (cf. [26] ). Based on this convex entropy functional we can introduce astandard duality transform to so-called entropy variables (cf. [27, 28])

ui = ∂ciE + const = log ci − log(1− ρ) + ηiziV +W 0i . (32)

The explicit inversion of this transform can be obtained from the exponential formci

1− ρ= exp

(ui − ηiziV −W 0

i

),

yielding after brief manipulations

ci =exp(ui − ηiziV −W 0

i )

1 +∑M

j=1 exp(uj − ηjzjV −W 0i )

. (33)

The stationary model (29), (30) in entropy variables can be written as

−λ2∆V −∑

k

zk exp(uk − ηkzkV −W 0k )

1 +∑M

j=1 exp(uj − ηjzjV −W 0j )

= f, (34)

∇ · (Diexp(ui − ηiziV −W 0

i )

(1 +∑M

j=1 exp(uj − ηjzjV −W 0j ))

2∇ui) = 0. (35)


A particularly attractive feature of the transformation is the elimination of cross-diffusion, the coupling only occurs in the diffusion coefficients. Consequently, amaximum principle holds for ui - it attains its maximum and minimum at ΓB - withthe transformed boundary conditions

ui = log γi − log(1−∑

j

γj) + ηizi(V0D + UV 1

D) +W 0i on ΓB , (36)

∇ui · n = 0 on ∂Ω \ ΓB . (37)

A second transformation that is routinely used in semiconductors is the one toso-called Slotboom-Variables, which we shall denote by vi in the following. In thestandard Nernst-Planck case those variables are simply obtained by multiplicationwith 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 partlyreverting the transformation to entropy variables. For the sake of simple reading weuse the notation Fi for the functions

Fi(c1, ..., cM ) = log ci − log(1− ρ) = ui − ηiziV −W 0i , (38)

hence

F−1i (u1, ..., uM ) =

expui

1 +∑

j expuj. (39)

Now we define Slotboom variables via

vi = F−1i (u1, ..., uM ) = F−1

i

(Fi(c1, ..., cM ) + ηiziV +W 0

i

).

This transformation can be written explicitly as

ci =vi exp(−ηiziV −W 0

i )

1 +∑

j vj(exp(−ηjzjV −W 0j )− 1)

.

Hence, in the stationary case we obtain the transformed system in Slotboom variablesas

−λ2∆V −∑

k

zkvk exp(−ηkzkV −W 0

k )

1 +∑

j vj(exp(−ηjzjV −W 0j )− 1)

= f

∇ · (Diexp(−ηiziV −W 0

i )

(1 +∑

j vj(exp(−ηjzjV −W 0j )− 1))2

(∇vi(1−∑

j

vj) + vi∑

j

∇vj)) = 0.

Due to the fact that equilibrium solutions are minimisers of the entropy we obtain thatboth the entropy and Slotboom variables are constant in equilibrium. This propertyis very favourable for linearisation around equilibrium situations, in particular forsmall applied voltages, since all the gradient terms vanish. As a consequence a certaindecoupling with the linearised Poisson equation appears, we shall discuss this issue indetail below.

4.4. Existence

In the following we shall verify the existence of weak solutions ci ∈ H1(Ω) ∩ L∞(Ω),V ∈ H1(Ω) ∩ L∞(Ω). For this sake we consider the transformed system in entropyvariables, since the maximum principle is of fundamental importance for obtaininga-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), W 0i ∈ L∞(Ω) ∩H1(Ω).

(A2) V 0D ∈ H1/2(ΓE) ∩ L∞(ΓE), γi in H1/2(ΓB) ∩ L∞(ΓB).

Note that assumptions (A1), (A2) imply that the transformed boundary valuesfor the entropy variables are elements of the same function spaces. In order to proveexistence we shall construct a fixed-point equation and apply Schauder’s theorem onthe set

M = (u1, . . . uM ) ∈ L2(Ω)M | a ≤ ui ≤ b a.e. in Ω, (40)

where

a = mini

infx∈ΓB

uDi (x), b = max

isupx∈ΓB

uDi (x). (41)

Here uDi 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 resultsremain valid for arbitrary constant ηi.

We show existence by a fixed point argument. The corresponding operator isconstructed 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 → L2(Ω)M ×H1(Ω)

(u1, . . . , uM ) 7→ (u1, . . . , uM , V ),(42)

where V is the unique solution of the nonlinear Poisson equation

−λ2∆V =∑

k

zkexp(uk − zkV −W 0

k )

1 +∑

j exp(uj − zjV −W 0j )

+ f (43)

with boundary conditions (26), (27). We define H by

H :DH ⊂ L2(Ω)M ×H1(Ω) → L2(Ω)M

(u1, . . . , uM , V ) 7→ (v1, . . . , vM ),(44)

where the vi are the unique weak solutions of the linear elliptic equations (cf. (34),(35))

∇ · (Diexp(ui − ziV −W 0

i )

(1 +∑

j exp(uj − zjV −W 0j ))

2∇vi) = 0 (45)

subject to the boundary conditions

∂vi∂n

= 0 on ∂Ω\ΓD and vi = uDi on ΓD. (46)

The domain of the operator H is set to

DH = G(L2(Ω)M ).

Next we shall verify some favourable properties of G and H which are necessary in theexistence 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 H1(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

J(V ) =λ2

2

∫

Ω

|∇V |2dx+

∫

Ω

log(1 +∑

j

γj exp(uj − zjV −W 0j )) dx.

It is straight-forward to see that J is strictly convex and coercive on H1(Ω), thusthere exists a unique minimiser V ∈ H1(Ω) (respectively on the subspace representingDirichlet boundary condition), which is a weak solution of (43). Vice versa everysolution of (43) is a minimiser due to convexity. This implies existence and uniquenessof a solution V ∈ H1(Ω).

From the structure of the right-hand side in the Poisson equation we see that

−∆V ≤1

λ2

(∑

i

|zi|+ ‖f‖∞

)

a.e. in Ω

and

−∆V ≥ −1

λ2

(∑

i

|zi|+ ‖f‖∞

)

a.e. in Ω

hold in a weak sense. Thus the maximum principle [29] provides a uniform bound forV in L∞(Ω). Moreover, by the Friedrich’s inequality,

‖V ‖2H1 ≤ C

(

‖VD‖2H1/2(ΓE) + ‖∇V ‖

2L2(Ω)

)

≤ C ‖VD‖2H1/2(ΓE) +

2C

λ2J(V )

≤ C ‖VD‖2H1/2(ΓE) +

2C

λ2J(VD),

where VD is an arbitrary H1-extension of VD, we obtain a uniform bound for V inH1(Ω).

Now let V and V be solutions of (43) for given (u1, . . . , uM ) and (u1, . . . , uM ),respectively. To simplify notation we introduce the new variable R given by

R(V, u) =

∑

k zk exp(uk − zkV −W 0k )

1 +∑


.

Then, in a weak formulation we obtain∫

Ω

λ2∇(V − V )∇ϕ dx =

∫

Ω

ϕ(

R(V, u)−R(V , u))

dx

=

∫

Ω

ϕ(

R(V , u)−R(V , u))

dx+

∫

Ω

ϕ(

R(V, u)−R(V , u))

dx.

We choose as test function ϕ = V − V and use the following monotonicity of thesecond term on the right-hand side in V : V > V ⇔ R(V ) < R(V ). Hence we obtain

λ2∥∥∥∇(V − V )

∥∥∥

2

L2(Ω)≤ λ2

∥∥∥∇(V − V )

∥∥∥

2

L2(Ω)−

∫

Ω

(V − V )(R(V, u)−R(V , u)) dx

=

∫

Ω

(V − V )(R(V , u)−R(V , u)) dx.

With the Friedrich’s inequality (note that V − V vanishes on ΓE) and the Cauchy-Schwarz inequality we finally conclude

∥∥∥V − V

∥∥∥H1(Ω)

≤C

λ2

∥∥∥R(V , u)−R(V , u)

∥∥∥L2(Ω)

.


Using the a-priori bounds for V in L∞ as well as those for ui defined by M, it is easyto use the Lipschitz-continuity of the nonlinearity to further conclude that

∥∥∥V − V

∥∥∥H1(Ω)

≤C

λ2

√∑

j

‖uj − uj‖2L2(Ω).

Hence, G is Lipschitz-continuous on M. The next step is to analyse the properties of the operator H on G(M).

Lemma 4.2 Let Q denote a compact subset of M. Then the operator H : G(M) → Qis well defined by (44) and continuous on M×K.

Proof: It is straight-forward to see that

Ai = Diexp(ui − ziV −W 0

i )

(1 +∑

j exp(uj − zjV −W 0j ))

2∈ L∞(Ω),

more precisely

0 <Di exp(a− |zi|C − ‖W 0

i ‖∞)(

1 +∑

j exp(b− |zj |C − ‖W 0j ‖∞)

)2 ≤ Ai ≤ Di,

where C is such that ‖V ‖L∞ ≤ C on G(M). Hence, standard theory [30] for ellipticequations in divergence form implies the existence and uniqueness of a weak solutionvi of

∇ · (Ai∇vi) = 0 in Ω

with boundary conditions (46). Moreover, the maximum principle [29] for linearelliptic equations implies a ≤ vi ≤ b with a, b defined in (41). Thus, H is well-definedand maps into M. Due to the compactness of the embedding H1(Ω) → L2(Ω),H(G(M)) is precompact.

To verify the continuity of H we consider the sequences V k → V in H1(Ω) anduki → ui in L2(Ω) . Then Ak

i → Ai in L2(Ω) with the uniform bounds above. Let vkibe the weak solution of

∇ · (Aki∇vi) = 0,

then vi is uniformly bounded in H1(Ω) and hence there exists a weakly convergentsubsequence vkl

i → vi for all i. Then

0 =

∫

Ω

Akli ∇vkl

i ∇φ dx →

∫

Ω

Ai∇vi∇φ dx.

for all test functions φ ∈ W 1,∞(Ω)∩H10 (Ω). Since Ai ∈ L∞(Ω) and W 1,∞(Ω) is dense

in H10 (Ω), we conclude that

0 =

∫

Ω

Ai∇vi∇φ dx

also holds for φ ∈ H10 (Ω). With the trace theorem we can pass to the limit in (46)

and thus, vi is the weak solution of

∇ · (Ai∇vi) = 0 in Ω with boundary condition (46).

By the uniqueness of the limit vi we conclude vki → vi weakly in H1(Ω) and thusstrongly in L2(Ω) which implies the continuity of H.

We can now employ Schauder’s Fixed Point Theorem [29], which assures theexistence of a fixed point of H(G(M)). This fixed point is a solution of (43), (45),which we summarise in:


Theorem 4.3 (Global existence of stationary solutions) Let assumptions (A1),(A2) be satisfied. Then, there exists a weak solution

(V, c1, ..., cn) ∈ H1(Ω)M+1 ∩ L∞(Ω)M+1

of

−λ2∆V =∑

j

zjcj + f (47)

0 = ∇ · (Di((1− ρ)∇ci + ci∇ρ+ ηici(1− ρ)∇Wi)) , (48)

with Wi = V +W 0i and boundary conditions (26), (27) and (23),(24), 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 samefor (47),(48) the only thing left to do is to transform back to the original variables

ci =exp(ui − ziV −W 0

i )

1 +∑


, (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. Ofcourse, improved regularity can only hold for smooth data. Thus, for the next twosections we make in addition to (A1), (A2) the following assumptions:

(A3) W 0i ∈ H2(Ω).

(A4) V 0D + UV 1

D ∈ H3/2(ΓE), γi ∈ H3/2(ΓB).

Furthermore, we assume Ω to be polygonal such that the angle between segmentsof Dirichlet and Neumann boundaries is less than π/2, and there are no adjacentDirichlet segments (as in figure 2). Since we only consider homogeneous Neumannboundaries, the angle between segments of adjacent Neumann boundaries is not equalto a multiple of π (otherwise they could be joined). We can apply Grisvard Theorem5.1.2.4 ([31]) to conclude that V ∈ H2(Ω) for ∆V ∈ L2(Ω), as the right-hand-side of(29) is obviously in L∞(Ω) → L2(Ω). For dimension n = 1, 2, 3 the Sobolev embeddingtheorem ensures that H2(Ω) ⊂ L∞(Ω), cf. [32]. From (30) we have that

(1− ρ)∆ci + ci∆ρ = −∇(zici(1− ρ)∇V + ci(1− ρ)∇W 0i ) ∈ L2(Ω),

hence ∆ci ∈ L2(Ω). With (A4) and boundary conditions

ci = γi on ΓB

∇ci · n = 0 on ∂Ω \ ΓB

we conclude in an analougus way

(V, c1, ..., cm) ∈ H2(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 canbe made. In general, we cannot expect uniqueness. Potential non-uniqueness mayeven be related to interesting phenomena appearing in practice such as gating, whichdenotes the (random) opening and closing of the channel ([21]). The frequency canbe driven by external factors.

Unfortunately, the uniqueness proof cannot be performed for (V, c1, ..., cM ) ∈H1(Ω) ∩ L∞(Ω). But the proof can be performed in H2(Ω), which is not a seriousrestriction due to the regularity results of section 4.5. As above, we consider thetransformed system in entropy variables u1, ..., uM with boundary condition

ui = ηi(γ1, ..., γM ) x ∈ ΓB ⊂ ∂Ω.

Assumption (A4) leads to ηi ∈ H3/2(ΓB) and thus, as ci ∈ H2(Ω),

ui = log ci − log(1− ρ) + ziV +W 0i ∈ H2(Ω).

Let u = (ui)i=1,...,M and η = (ηi)i=1,...,M and

F(U, η;V, u) : R× (H3/2)M ×H2 × (H2)M → H3/2 × (H3/2)M × L2 × (L2)M

denote the operator

V − V 0D − UV 1

D on ΓE (50a)

ui − ηi on ΓB (50b)

−λ2∆V −∑

k

exp(uk − zkV −W 0k )

1 +∑M

j=1 exp(uj − zjV −W 0j )

− f ∈ L2 (50c)

∇ ·

(

Diexp(ui − ziV −W 0

i )

(1 +∑M

j=1 exp[uj − zjV −W 0j ])

2∇ui

)

∈ L2 (50d)

with boundary condition V 0D + UV 1

D ∈ H3/2(ΓE) and ηi ∈ H3/2(ΓB).The proof will be based on the implicit function theorem in Banach spaces [33].

Thus we have to show that F(U, η;V, u) is Frechet-differentiable with respect to V ,U ,ηand u. For the sake of brevity we only detail the existence of the Frechet-derivativeof the ith component of (50d) with respect to ui, which we denote with

F ′i(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 6= i) for ui, v ∈ H2(Ω) → L∞(Ω)

‖Fi(ui + v)−Fi(ui)−F ′i(ui)v‖L2(Ω)

‖v‖H2(Ω)

=

‖∇ · [G(ui + v)∇(ui + v)−G(ui)∇ui −G′(ui)v∇ui −G(ui)∇v]‖L2(Ω)

‖v‖H2(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) =∥∥∥G′(ui + v) |∇(ui + v)|

2+G(ui + v)∆(ui + v)−G′(ui) |∇ui|

2−G(ui)∆ui

−G′′(ui)v |∇ui|2−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 (52)

+G′(ui + v)|∇v|2∥∥L2(Ω)

.

Note that H2(Ω) ⊂ W 1,4(Ω) for n = 1, 2, 3 cf. [32]. This ensures that all productterms in (52) really are in L2(Ω).

(52) ≤1

2

∥∥G′′(ξ2)v

2∥∥L∞(Ω)

‖∆ui‖L2(Ω) + ‖G′(ξ1)v‖L∞(Ω) ‖∆v‖L2(Ω)

+1

2

∥∥G′′′(ξ3)v

2∥∥L∞(Ω)

‖∇ui‖2L4(Ω) + 2 ‖G′′(ξ2)v‖L∞(Ω) ‖∇v∇u‖L2(Ω)

+ ‖G′(ξ1)‖L∞(Ω) ‖∇v‖2L4(Ω) .

Using the following L∞-bounds

‖G‖L∞(Ω) ≤ 1, ‖G′‖L∞(Ω) ≤ 1, ‖G′′‖L∞(Ω) ≤ 5 and ‖G′′′‖L∞(Ω) ≤ 23,

as well as the fact that

‖∇v‖L4(Ω) ≤ ‖∇v‖L2(Ω) , ‖v‖H2(Ω) ≥

‖v‖L2(Ω) , ‖∇v‖L2(Ω) , ‖∆v‖L2(Ω)

and

‖v‖H2(Ω) ≥ c1 ‖v‖L∞(Ω) ,

we have

(51) ≤5

2c1 ‖v‖L∞(Ω) ‖∆ui‖L2(Ω) + ‖v‖L∞(Ω)

+23

2c2 ‖v‖L∞(Ω) ‖∇ui‖

2L4(Ω) + 10 ‖v‖L∞(Ω) ‖∇ui‖L2(Ω) + c3 ‖∇v‖L4(Ω) . (53)

As ‖∇v‖L2(Ω) ≤ ‖v‖H2(Ω) and ‖∇v‖L∞(Ω) ≤ c4 ‖v‖H2(Ω), we conclude that

lim‖v‖H2(Ω)→0

(53) → 0.

Therefore F ′i(ui) is a Frechet-derivative. All other derivatives can be estimated using

analogous arguments. Next we prove uniqueness for small voltage and small bathconcentration.

4.6.1. Small Voltage In this case, we assume that a small voltage U is applied atthe right-hand side of the bath. In case U = 0, we obtain the equilibrium state. Weinvestigated this case in section 2, one can show the well-posedness of this problemby standard techniques for elliptic equations. We regard the linearisation around zerovoltage or in turn linearisation around equilibrium. The linearised system in entropyvariables reads

−λ2∆V −∑

j,k

zj∂cj∂uk

uk +∑

j

zj∂cj∂V

V = h1 ∈ L2(Ω) (54)

∇ ·

(

Diki exp(−ziVeq +W 0

i )

(1 +∑

j kj exp(−zjVeq +W 0j ))

2∇ui

)

= hi+1 ∈ L2(Ω). (55)

The constants ki can be determined form the bath concentrations ηi via γi =ki

1+∑

j kj.

It is again possible to show existence and uniqueness of a solution (V , u1, ..., uM ) via


standard theory for elliptic equations (note the partial decoupling of the equations inthe linearisation). Furthermore, the left-hand side of (54), (55) is a Frechet-derivativeof (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‖H3/2(ΓB) be sufficiently small. Then, for each ηi ∈

(H3/2(ΓB))M there exists a locally unique solution

(V, c1, ..., cM ) ∈ H2(Ω)M+1

of problem (29), (30) and the transformed, linearised problem (54), (55) is well-posed.

Proof: We already showed that (V, c1, ..., cM ) ∈ H1(Ω)M+1 ∩ L∞(Ω)M+1 andassumptions (A1)-(A4) imply that (V, c1, ..., cM ) ∈ H2(Ω)M+1, and thus u1, ..., uM ∈H2(Ω). The equation operator is Frechet-differentiable for ui ∈ H2. For U = 0,the problem is well-posed and its Frechet-derivative exists with continuous inverse inthe respective function spaces. Thus, we can apply the implicit function theorem inBanach spaces to conclude the existence of a locally unique solution of problem (29),(30) around U = 0 and that the linearised, transformed problem is well-posed forsmall U . After back transformation


i )

1 +∑


we obtain the same result for (29), (30).

4.6.2. Small Bath Concentrations We now regard the stationary system around smallbath 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 ) ∈ H2(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. Standardresults for the elliptic equation

−λ2∆V0 = f

with Neumann and Dirichlet boundary conditions on ΓE and ΓB imply existence anduniqueness of the remaining problem.

The resulting system for the linearisation around zero bath concentration is

−λ2∆V −∑

j,k

zj∂cj∂uk

uk +∑

j

zj∂cj∂V

V = gi ∈ L2(Ω),

∇ · (Di(∇ci + zici∇(V0 +W 0i ))) = gi+1 ∈ L2(Ω). (56)


The equations are partially decoupled, thus the potential V0 is not computed viathe Poisson-Boltzmann equation anymore. Equation (56) is the stationary Nernst-Planck equation. After a change of variables, the Slotboom transformation knownfrom semiconductor theory

vi = exp(zi(V0 +W 0i ))ci,

we obtain the system of linear elliptic equations

−λ2∆V − exp(−zi(V0 +W 0i ))

∑

j,k

zj∂vj∂uk

uk +∑

j

zj∂vj∂V

V

= gi ∈ L2(Ω), (57)

∇ · (Di(exp(−zi(V0 +W 0i ))∇vi) = gi+1 ∈ L2(Ω), (58)

whose well-posedness can be analysed by standard techniques for elliptic equations[34]. Existence and uniqueness of (58) is also found as a result in standard PNPtheory [8].

Theorem 4.6 (Well-posedness for small bath concentrations) Let (A1)-(A4)be fulfilled and let ‖γi‖H3/2(ΓB) be sufficiently small. Then, for each U ∈ H3/2(ΓB),there exists a locally unique solution

(V, c1, .., cM ) ∈ H2(Ω)M+1

of problem (29), (30) and the linearised problem (57), (58) is well-posed.

Proof: Again, (V, c1, ..., cM ) ∈ H1(Ω)M+1 ∩ L∞(Ω)M+1 implies (V, c1, ..., cM ) ∈H2(Ω)M+1 and u1, ..., uM ∈ H2(Ω). For η = 0, problem (29), (30) is well-posed.The Frechet-derivative of (34), (35) exists with continuous inverse in the respectivefunction spaces. Furthermore, the equation operator is Frechet-differentiable, so thatwe can apply the implicit function theorem in Banach Spaces to conclude the existenceof a locally unique solution of problem (29), (30) around η = 0 and that the linearisedproblem (57), (58) is well-posed for small η. After back transformation


i )

1 +∑


we obtain the same result for (29), (30).

As mentioned above, there are biological phenomena such as gating which suggestthat global uniqueness cannot be expected.

5. Numerical Simulations for an Ion-Channel Model

In this section we shall illustrate the behaviour of the derived mathematical modelswith numerical results. In particular we discuss the following three situations:

(i) Conductance close to equilibrium in a multi-dimensional model and itsdependence on concentration, cf. section 5.3.

(ii) Concentration profiles for stationary solutions, cf. section 5.4.

(iii) Current in the stationary case and its dependence on concentration, cf. section5.5.

(iv) Current vs voltage curves for the stationary system, cf. section 5.6.

(v) Current for a changing charge profile, cf. section 5.7.


PPP

PPP

PP

PP

PP

channelΓL ΓR

ΓN

ΓN

left bath right bath

Figure 2. Sketch of the computational domain

5.1. Model Presentation

We choose the following problem setup for all four problems if not mentionedotherwise. We consider an L-type calcium selective ion channel. We assume that thechannel is modelled as cylinder with radius rc = 0.4nm and length lc = 1nm, whichis embedded into two bathes with length lb = 2nm and outer radius rb = 2.4nm. Thetotal length is accordingly L = 5nm. We assume that the boundary is split into thefollowing parts: ΓB = ΓE = ΓL ∪ ΓR (see also figure 2).

We mention that due to the microscopic scale, the use of drift-diffusion typemodels may be debated, a final justification can merely be obtained from areproduction of experimental measurements such as currents, cf. [2], [35]. Notehowever that at least we do not expect quantum effects in such channels.

We consider three species, calcium (Ca2+), sodium (Na+) and chloride(Cl−)inside the baths and channel, as well as one confined species, oxygen (O−1/2), whichrepresents the permanent charge inside the channel. The external potential W 0

i is setto zero, and, according to the thermodynamic understanding, we assume µi = 1/kBT .We consider eight confined O−1/2 particles in the channel, which represent the fixedcharge. The physical parameters are given in table 1, N denotes the number ofparticles.

Meaning Value UnitBoltzmann constant kB 1.3806504× 10−23 J/KTemperature T 300 KAvogadros constant NA 6.02214179× 1023 N/molVacuum permittivity ǫ0 8.854187817× 10−12 F/mRelative permittivity ǫr 78.4Elementary charge e 1.602176× 10−19 CParticle radius 0.15 nm

Typical length L 5 nmTypical concentration c 3.7037× 1025 N/l

Typical voltage V 100 mVDiffusion coefficient Ca2+ 7.9× 10−10 m2/sDiffusion coefficient Na+ 1.33× 10−9 m2/sDiffusion coefficient Cl− 2.03× 10−9 m2/s

Table 1. Parameters for computation

We assume a particle radius of 0.15nm for all particles. According to that, thetypical or maximal concentration is corresponding to 61.5mol/l. The resulting effective


parameters after scaling and nondimensionalization are

λ2 =ǫ0ǫrV

eL2c= 4.68× 10−4 and η =

eV

kBT= 3.87.

5.2. Reduction to One Dimension

The cross section of a filter inside an ion channel is much smaller than the longitudinalextension of the system and the setup forces the transport through a channel to benearly a one-dimensional process. We hence try to approximate the three-dimensionalmodel by a one-dimensional one. Such a model is faster and easier to handlecomputationally than the three-dimensional version. We assume a domain of theform

Ωǫ = x ∈ (−L,L), (y, z) ∈ Qǫ

where

Qǫ = (x, rǫ(x) cos θ, rǫ(x) sin θ) |0 ≤ rǫ(x) ≤ ǫr |θ ∈ [0, 2π),

ǫr denotes the maximal radius of the channel and√

y2 + z2 ≤ r. The boundaryconditions are Dirichlet at x = ±L, i.e. ΓB = ΓE = −L,+L × Qǫ, and no-flux onthe remaining part. We assume that the boundary values for the potential and thedensities are constant in the two segments at x = −L and x = +L.

We rescale the variables describing the channel as x, yǫ = ǫy, zǫ = ǫz with (y, z) ∈Q1. Starting with the Poisson equation, we rescale the potential V ǫ(x, yǫ, zǫ) =V ǫ(x, y, z). The same scaling is used for the densities cǫi(x, y

ǫ, zǫ) = cǫi(x, y, z) aswell as for the transformed densities uǫ

i(x, yǫ, zǫ) = uǫ

i(x, y, z). From the existenceproof we have potential and densities in

V ǫ(x, yǫ, zǫ), cǫi(x, yǫ, zǫ) uǫ

i(x, yǫ, zǫ) ∈ H1(Ωǫ) ∩ L∞(Ωǫ),

with uniform bounds in ǫ in the supremum norm.For the Poisson equation we obtain

−λ2∆V ǫ(x, yǫ, zǫ) = −λ2

(

∂xxVǫ(x, y, z) +

1

ǫ2∂yyV

ǫ(x, y, z) +1

ǫ2∂zzV

ǫ(x, y, z)

)

=∑

j

zjcǫj(x, y, z) + f(x). (59)

The weak formulation of (59) is given by

λ2

∫ ∫ ∫

Ω1

(

∂xVǫ∂xϕ+

1

ǫ2∂yV

ǫ∂yϕ+1

ǫ2∂zV

ǫ∂zϕ

)

dx dy dz

=

∫ ∫ ∫

∑

j

zjcǫj + f

ϕ dx dy dz.

With the special test function

ϕ(x, y, z) = V ǫ(x, y, z)− g(x),

where g(x) denotes a linear function of x in Ωǫ such that V ǫ − g vanishes at x = ±L,we obtain

λ2

∫ ∫ ∫

Ω1

(

∂xVǫ∂x(V

ǫ − g) +1

ǫ2∂yV

ǫ∂yVǫ +

1

ǫ2∂zV

ǫ∂zVǫ

)

dx dy dz


=

∫ ∫ ∫

Ω1

∑

j

zjcǫj + f

(

V ǫ − g)

dx dy dz

≤

∑

j

|zj |∥∥cǫj∥∥L∞(Ωǫ)

+ ‖f‖L∞(Ωǫ)

∥∥∥V ǫ − g

∥∥∥L∞(Ω1)

|Ω1|. (60)

The right-hand side is uniformly bounded and using the linearity of g we obtain

λ2

∫ ∫ ∫

Ω1

∂xVǫ∂xg dx dy dz =

g(L)− g(−L)

2L

∫ ∫

V ǫ(x, y, L)− V ǫ(x, y,−L) dy dz,

which can be estimated uniformly in terms of the boundary values. We obtain anestimate of the form

λ2

∫ ∫ ∫

Ω1

(

|∂xVǫ|2 +

1

ǫ2|∂yV

ǫ|2 +1

ǫ2|∂zV

ǫ|2)

dx dy dz ≤ k1,

where k1 denotes a constant independent of ǫ. Thus, we conclude∫ ∫ ∫

Ω1

(

∂yVǫ)2

dx dy dz ≤ ǫ2k1 and

∫ ∫ ∫

Ω1

(

∂zVǫ)2

dx dy dz ≤ ǫ2k1

as well as∫ ∫ ∫

Ω1

(

∂xVǫ)2

dx dy dz ≤ k1.

Hence, for ǫ → 0 we have∥∥∥∂yV

ǫ∥∥∥L2(Ωǫ)

→ 0 and∥∥∥∂zV

ǫ∥∥∥L2(Ωǫ)

→ 0,

and overall V ǫ is uniformly bounded in H1(Ω1). From that we conclude for ǫ → 0along subsequences

V ǫ(x, ǫy, ǫz) = V ǫ(x, y, z) V 0(x) in H1(Ωǫ). (61)

Next we consider the Nernst-Planck equation in entropy variables with testfunction ϕ(x, y, z) = uǫ

i(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(uǫ

i − ηiziV −W 0i )

(

1 +∑

j exp(uǫj − ηjzjV −W 0

j ))2 ≤ k3,

with constants k2 and k3 independent of ǫ, to derive analogous estimates for thefunctions uǫ

i as for V ǫ. As above, we conclude for ǫ → 0

‖∂yuiǫ‖L2(Ω1) → 0 and ‖∂zui

ǫ‖L2(Ω1) → 0

and altogether uniform boundedness of uǫi in H1(Ω). Thus, along subsequences for

ǫ → 0 we have

uǫi(x, ǫy, ǫz) = ui

ǫ(x, y, z) u0i (x) in H1(Ωǫ)

and

cǫi(x, ǫy, ǫz) = ciǫ(x, y, z) c0i (x) in H1(Ωǫ).


Choosing test functions ϕ(x, y, z) = ϕ(x), we have

λ2ǫ−2

∫ ∫ ∫

Ωǫ

∇V ǫ(x, yǫ, zǫ) · ∇ϕ(x) dx dy dz =

λ2ǫ−2

∫ ∫ ∫

Ωǫ

∂xVǫ(x, yǫ, zǫ)∂xϕ(x) dx dy dz→ǫ→0

λ2ǫ−2

∫ ∫ ∫

Ωǫ

∂xV0(x)∂xϕ(x) dy dz dx =

λ2ǫ−2

∫

∂xV0(x)∂xϕ(x)

∫ ∫

dy dz

︸︷︷︸

ǫ2a(x)

dx =

− λ2

∫

∂x(a(x)∂xV

0(x))ϕ(x) dx,

with a(x) being the cross-sectional area of Ω1 at x. The right-hand-side of the Poissonequation can be derived from

ǫ−2

∫ ∫ ∫

Ωǫ

∑

j

zjcǫj(x, y

ǫ, zǫ) + f(x)

ϕ(x, y, z) dx dy dz

→

∫

∑

j

c0j (x) + f(x)

ϕ(x)

∫ ∫

ǫ−2 dy dz dx =

∫

a(x)

∑

j

c0j (x) + f(x)

ϕ(x) dx.

Accordingly, in the limit ǫ → 0 we obtain the one-dimensional Poisson equation

−λ2∂x(a(x)∂xV

0(x))= a(x)(

∑

j

c0j (x) + f(x)).

We proceed in a similar manner with the Nernst-Planck equations, using strongLp convergence to pass to the limit in the nonlinear mobilities. The resulting simplifiedone-dimensional system is given by (suppressing the index 0)

− λ2∂x (a(x)∂xV ) = a(x)(∑

j

cj + f),

∂x(a(x)Diexp(ui − ηiziV −Wi)

(1 +∑

j exp(uj − ηjzjV −Wj))2∂xui) = 0.

5.3. Conductance close to equilibrium

Here we consider the linearised stationary case for nonlinear PNP in a two dimensionalrotationally symmetric domain in Slotboom variables vi given by:

−λ2∆V eq =

∑zjkj exp(−ηzjV

eq)

1 +∑

kk exp(−ηzkV eq)+ cO (62)

0 = ∇ ·

(exp(−ηziV

eq)(1 +∑

kj)

(1 +∑

kj exp(−ηzjV eq))2

)(

∇vi + ki∑

∇vj))

, (63)

where (63) holds for i = Na+,Ca2+ and Cl−. The concentration cO denotes the fixeddensity of O−1/2 ions inside the channel. The parameters ki can be calculated usingthe initial condition V eq(ΓL) = 0 on ceqi , i.e.

cieq(ΓL) =ki

1 +∑

kj= γiL, (64)


0 2 4 6 8 10 120

5

10

15

20

25

30

Concentration added in mol/l

Con

duct

ance

in 1

0−20

S

nonlinear PNPPNP

Figure 3. Conductance for nonlinear PNP and PNP

where γiL denotes the boundary condition on ΓL for ci. We want to compare thebehaviour of the conductance of (62), (63) with simulations of the classical, linearisedPNP model given by (after Slotboom transformation and linearisation for PNP):

−λ2∆V eq =∑

zjkj exp(−ηzjVeq) + f,

0 = ∇ · (exp(−ηziVeq)∇vi) , i = Na+,Ca2+,Cl−.

In this case, only the qualitative behaviour is of interest. Therefore we neglect thediffusion coefficients and assume the oxygens in the channel to be point charges inboth models. The conductance is therefore given by

σ = e∑

i∈I

∫

Γ0

ziJi · dn,

for I = Ca2+,Na+,Cl−. The function Ji for the nonlinear PNP is given by

Ji =

(exp(−ηziV

eq)(1 +∑

kj)

(1 +∑

kj exp(−ηzjV eq))2

)(

∇vi + ki∑

∇vj))

i ∈ I

and for linear PNP by

Ji = exp(−µziVeq)∇vi, i ∈ I.

The simulations were done using Comsol 3.5. Figure 3 shows a concentrationversus conductance plot for several concentrations of species in the bathes. Theseconcentrations range from zero concentration for all species to 12.3mol/l for NaCland CaCl2. Due to the linearisation, the boundary condition for ci are equal in bothbathes. Note that the maximum values chosen in this simulation correspond to the“full state” of the channel. The applied voltage is zero at the left-hand-side ΓL andU = 100mV at the right-hand-side ΓR. We depicted the concentration-current plotfor the classical PNP system in figure 3 as well. Note that the current of the classicalPNP model increases no matter how “full” the channel is. On the contrary the currentof the nonlinear PNP model decreases to zero as the concentrations approach theirmaximum values, which correspond to the “full state”.


5.3.1. Analytical computation of conductanceNext we would like to give an analytical explanation of the above detected

phenomena. As we want to gain insight into the qualitative behaviour, we consideronly one species and calculate the conductance and current analytically from thereduced 1D model in the unit interval with a(x) ≡ 1. The conductance for nonlinearPNP is given by the linearised ion flux in entropy variables u

σ =k exp(−V eq)

(1 + k exp(−V eq))2∂xu, (65)

where k can be determined from (64), and we have set z = 1 and η = 1. We knowthat the transformed boundary values are u(ΓL) = u(0) = 0 and u(ΓR) = u(1) = 1,according to (32), which we have to derive with respect to the boundary value U forV to obtain boundary values for u. After integrating (65) and taking into accountthat σ does not depend on Ω we find

σ =

(∫ 1

0

(1 + k exp(−V eq))2

k exp(−V eq)dx

)−1

.

Using (64), esp. γ = k/(1 + k), we deduce that

σ(γ) =γ(1− γ)

(1− γ)2∫ 1

0exp(V eq) dx+ 2γ(1− γ) + γ2

∫ 1

0exp(−V eq) dx

.

For 0 ≤ γ ≤ 1, γ = 0 and γ = 1 are the only zero points and the denominator isalways larger than zero. For the Poisson Boltzmann model, it is not easy to determinethe dependence of

S(γ) =

∫ 1

0

exp(±V eq(γ)) dx

of γ. In the following, we neglect the Poisson equation and assume a linear behaviourof V , i.e.

V eq = Ux.

This leads in the scaled domain to∫ 1

0

exp(Ux) dx =(exp(U)− 1)

U.

Using Taylor expansion for U , i.e. exp(U) = 1 + U +O(U2), we finally arrive at

σ = γ(1− γ).

The slope of σ is exactly as in figure 3. The current can be computed via

I = σU +O(U2).

For the PNP model, we directly compute the current analytically. The current isgiven by

I = c+ c∂xV = exp(−V )∂xv,

where v denotes the transformation in Slotboom variables, which is given for PNPvia c = v exp(−V ). The boundary transforms according to v(0) = exp(V (0))γ. Forconsistency, we assume equal boundary conditions for the concentration here as above.Hence we obtain

I =(exp(U)− 1)γ∫ 1

0exp(V ) dx

.


Neglecting the Poisson equation and assuming V = Ux, the current for the resultingNernst Planck model is given by

I = γU +O(U2).

As expected, the conductance, which is given by

σ = γ,

shows a linear behaviour dependent on γ, as can be seen in figure 3.

5.4. Concentration profiles for stationary solutions

Next we consider the stationary system in entropy variables in a one-dimensionaldomain

−λ2∇ · (a(x)∇V ) = a(x)

(∑

k

zk exp(uk − ηkzkV )(1− cO)

1 +∑

j exp(uj − ηjzjV )+ f

)

, (66)

0 = ∇ ·

a(x)Di

exp(ui − ηiziV )(1− cO)2

(

1 +∑

j exp(uj − ηjzjV ))2∇ui

, (67)

for i ∈ I. Here, a(x) denotes the area function describing the cross section of thecylindrical channel and bath. The variable cO denotes the concentration of oxygen inthe channel. This concentration is not transformed to the entropy variable uO becauseit is zero in the bath. The contribution of cO to the transformed variables ui, wherei denotes Ca2+, Na+ and Cl−, is given by (1− cO).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−10

0

10

20

30

40

50

60

Oxygen

mol

Figure 4. Oxygen inside channel region

We solve (66) and (67) in an iterative manner: Let V 0 be the initial datum forthe potential V and u0

i 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 bothbathes. We have 5 ·10−3mol/l CaCl2 in the left bath and 10−1mol/l CaCl2 in the rightbath. We assume that the external potential is set to zero and a potential of −50mV


0 0.2 0.4 0.6 0.8 1−200

−150

−100

−50

0Potential V

mV

0 0.2 0.4 0.6 0.8 1−20

−10

0

10

20Charge neutrality

0 0.2 0.4 0.6 0.8 10

5

10

15Calcium

mol

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5Sodium

mol

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4Chloride

mol

0 0.2 0.4 0.6 0.8 10

0.5

1scaled total concentration

Figure 5. Stationary profiles for nonlinear PNP

is applied in the left bath. The physical parameters are given in table 5. The areafunction gives the scaled cross section of the channel and bath, where the radius ofthe 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 stationarysolution is depicted in figure 5. Note that the channel is in the region between 0.4 and0.6 on the x-axis. The concentration of oxygen is high inside the channel, see figure 4.As the nonlinear PNP model prevents overcrowding, the charge neutrality conditionis 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 6. It can clearlybe seen that the charge neutrality is fulfilled here, but the channel gets overcrowdedas the total mass inside the channel is above the maximal admissible concentration.This overcrowding effect is a well-known problem for PNP.

5.5. Current and its dependence on concentration

We take a closer look on the dependence of current in the stationary case for theexample of the L-type Calcium channel described above. The ion flux for the nonlinearPNP model is given by

Ji = −Dia((1− ρ)∂xci + ci∂xρ+ ηizici(1− ρ)∂xV ), (68)

or, in the transformed expression

Ji = −Diaexp(ui − ηiziV )

(

1 +∑

j exp(uj − ηjzjV ))2 ∂xui. (69)


0 0.2 0.4 0.6 0.8 1−100

−50

0Potential V

mV

0 0.2 0.4 0.6 0.8 1−10

−5

0

5Charge neutrality

0 0.2 0.4 0.6 0.8 10

5

10

15Calcium

mol

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2Sodium

mol

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4Chloride

mol

0 0.2 0.4 0.6 0.8 10

0.5

1

scaled total concentration

Figure 6. Stationary profiles for PNP

The current I flowing through the channel from one bath to the other, which can beexperimentally measured, is given by

I = e∑

i∈I

∫

Γ0

ziJi · dn,

where Γ0 denotes the cross section of the channel and I = Ca2+,Na+,Cl−. Theoxygen ions do not contribute to the flux as they are fixed inside the channel. The

0 50 100 150 200 250 3000

5

10

15

mMol Ca2+ added

Cur

rent

in p

A

nonlinear PNPPNP

Figure 7. Current for nonlinear PNP and PNP


model setup is chosen as in the previous section. For the boundary condition assumean 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. Inthe right bath, we have 5 · 10−3mol/l CaCl2. During the simulation, we add CaCl2 tothe left bath, starting from 1 · 10−3mol/l up to 3 · 10−1mol/l. The resulting curves ofcurrent versus concentration in the left bath are shown in figure 7. The ion flux forthe PNP model is given by

Ji = −Dia(∂xci + ηizici∂xV ) = −Dia exp(ui − ηiziV )∂xui. (70)

Due to the overcrowding that takes place in the nonlinear model, the current ofnonlinear PNP saturates. The current for PNP shows nearly a linear increase.Furthermore, the nonlinear current is remarkably less than the current for PNP.

5.6. Current versus voltage curves for the stationary system

We are going to take a closer look at the current versus voltage relation, cf. [2]. Weuse the same setup as in section 5.4. figure 8 shows this relation. The voltage appliedin 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.

−80 −60 −40 −20 0 20 40 60 80−60

−40

−20

0

20

40

60

80

100

120

140

Voltage in mVolt

Cur

rent

in p

A

mod. PNPPNP

Figure 8. Current vs. voltage relation

5.7. Current for a changing charge profile

Finally we would like to take a closer look on how current is depending on the chargeprofile by varying the positions of fixed charges. We do not change the number offixed oxygens, we simply contract them to the centre of the channel. We affect thefunction describing the density distribution of oxygens in the following way:

f(x) =

0 x < 0.5− 0.1ǫ

c0/ǫ 0.5− 0.1ǫ ≤ x ≤ 0.5 + 0.1ǫ

0 x > 0.5− 0.1ǫ

(71)


In case that ǫ = 1, the density of oxygens is constant inside the channel, which liesbetween 0.4 and 0.6 on the x-axis. If we decrease ǫ, the total density remains thesame, but it extension in x direction gets smaller, therefore the top goes up. Thesmallest ǫ we use is 0.4. This value leads to concentrations inside the channel whichis above the assumed maximal concentration. In figure 9 we show current versus 1− ǫfor nonlinear PNP and PNP. It shows that if the concentration in the channel is largeenough, nonlinear PNP detects crowding. The current breaks and gets zero in thecrowded state. PNP is not able to detect these crowding effects.

0 0.1 0.2 0.3 0.4 0.5 0.60

2

4

6

8

10

12

1−epsilon

Cur

rent

in p

A

nonlinear PNPPNP

Figure 9. Current vs. charge distribution

6. Conclusion

In this paper we analysed a nonlinear Poisson Nernst-Planck model which takes thesize of ions into account. We presented the macroscopic derivation of this model, herethe size exclusion effects resulted in nonlinear mobilities. First numerical simulationsof ion channels with the modified PNP model showed interesting and promisingfeatures. 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 inthe PNP model but shows a decreasing behaviour in the nonlinear case. This leads tocurrent saturation which is experimentally measured, but which can not be detectedusing PNP. To detect the volume exclusion effects with PNP, several approaches likemean spherical approximation or density functional theory have proposed. Theirqualitative behaviour is similar to the nonlinear PNP model analysed in this paper.Nonetheless, the derivation of this model is very intuitive and elementary and thecomputational cost significantly smaller.

Several issues on this model remain open. The proposed model is based on theassumption that all ions have the same radius, this should be generalised to differentradii. This point of high interest because the dimension of ions has an effect on thevolume selectivity of the channels. However the calcium channel studied in this paperis 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 detailedstructure of the membrane into the model than performed in this paper. In particularthe ability of the model to reproduce blocked states by relatively small changes ofpermanent charge is encouraging in this direction. This modeling approach is alsointeresting for other applications where size exclusion effects should be taken intoaccount, like chemotaxis with different cell types, swarming or pedestrian dynamicswith heterogeneous agents.

Acknowledgments

MTW acknowledges financial support of the Austrian Science Foundation FWF viathe Hertha Firnberg Project T456-N23. MB and BS acknowledge financial supportfrom Volkswagen Stiftung via the grant Multi-scale simulation of ion transport throughbiological and synthetic channels. The authors thank Z. Siwy (UCI) for stimulatingdiscussions.

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