nonlinear poisson-nernst planck equation for ion flux · 2. step:operator gis well de ned on a set...

Motivation Modelling Analysis for stationary Model

Nonlinear Poisson-Nernst Planck Equation for IonFlux

Barbel Schlake

Westfalische Wilhelms-Universitat MunsterInstitute fur Computational und Applied Mathematics

01 December 2010

Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster


Credits

this talk is based on joint work with

• martin burger (WWU Munster)

• marie-therese wolfram (Vienna)



Motivation

Modelling

Analysis for stationary Model



Motivation

aim: modelling of transport and diffusion with size exclusion

application:

• ion channels/nanopores

• human crowds

• swarming

• chemotaxis



Motivation

so far, mainly transport and diffusion for one species investigated

Fokker-Planck equation

∂tρ = ∇ · (D∇ρ+ ρ(1− ρ)∇V )

• linear diffusion

• logistic mobility

• size exclusion



Modelling

two possible approaches in microscopic modelling;

1. force-basedNewton equation of motion(macroscopic limit difficult)

2. jump exclusion processcellular automata



Jump Exclusion Process

lattice based modelling:

~~ ~-

• jump probability to neighbouring lattice sites given bydiffusion, external and interaction fields

• modified by exclusion principle, only jumps to free sites

• closure relation: probability of finding empty site instead ofexact exclusion in the ensemble average



Model with Size-Exclusion

rescaling of lattice

limit of lattice site distance to zero

Taylor expansion of master equation

resulting model:

∂tci = ∂x(Di ((1− ρ)∂xci+ci∂xρ+ zici (1− ρ)∂xV ))

total volume density ρ(x , t) =∑

cj(x , t)

1D ⇒ single file movement



Model with Size-Exclusion

multidimensional model:

∂tci = ∇ · (Di ((1− ρ)∇ci + ci∇ρ+ zici (1− ρ)∇V )︸︷︷︸flux −Ji

)

• movement is mainly driven by diffusion and interactionsamong particles and externally applied field

• mean field approach

• model describes average densities of particles



Ion Channels so far

PNP system in three dimensions for ion densities ci :

− λ2∆V =∑

zici + f Poisson equation

∂tci = ∇ · (Di (∇ci + cizi∇V ))

• λ2 permittivity

• f (x) protein charge

problem: size effects in small channels



Ion Channels so farentropie:

E =

∫Ω

∑(ci log ci + ziciV ) dx

equilibria:

0 = Ji∞ = −Di (∇ci∞ + zici∞∇Vi∞)

Boltzmann statistics:

ci∞ = ki exp (−ziVi∞) ki ≥ 0

Poisson-Boltzmann equation:

−λ2∆V∞ =∑

zjkj exp(−zjV∞) + f



Model including Size Effects

−λ2∆V =∑

zjcj + f

∂tci = ∇ · (Di ((1− ρ)∇ci+ci∇ρ+ zici (1− ρ)∇V ))

boundary conditions differ with different model setup



Boundary Conditions for Ion Channels

PPPPPP

PPPP

PP

channelΓD ΓD

ΓN

ΓN

left bath right bath

concentration: ci (x , t) = γi (x) x ∈ ΓD

no flux: Ji (x , t) · n = 0 x ∈ ΓN

charge neutrality:∑

zjγj(x) = 0 in bathes

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

D(x) x ∈ ΓD

no flux: ∇V (x , t) · n = 0 x ∈ ΓN



Entropy

entropy for this process:

E =

∫Ω

∑(ci log ci + (1− ρ) log(1− ρ) + ziciV ) dx

entropy variables: ui = ∂ciE = log ci − log (1− ρ) + ziV ,

entropy dissipation:

d

dtE = −

∫Ω

∑cj(1− ρ) |∇uj |2 dx

⇒ decreasingin equilibrium, entropy is minimal at fixed total mass.



Equilibriastationary solutions are minimizers of the entropy

0 =Ji∞ = −Di ((1− ρ∞)∇ci∞ + ci∞∇ρ∞ + zici∞(1− ρ∞)∇Vi∞)

generalized Boltzmann distributions:

ci∞ =ki exp (−ziVi∞)

1 +∑

kj exp (−zjVj∞)ki ≥ 0

modified Poisson-Boltzmann equation:

−ε∆V∞ =

∑zjkj exp(−zjV∞)

1 +∑

kj exp(−zjV∞)+ f

entropy variables ui∞ are constant



Analysis

system of equations:

−λ2∆V =∑

zjcj + f ,

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

several challenges

• double degeneracy

• no maximum principle (only 0 ≤ ciρ ≤ 1)

• coupling in highest order terms (cross diffusion)



Formulations of the Problem

ui = log ci − log (1− ρ) + ziV ,

system in entropy variables

−λ2∆V −∑ zk exp(uk − zkV )

1 +∑

exp(uj − zjV )= f

∇·

(Di

exp(ui − ziV )

(1 +∑

exp(uj − zjV ))2∇ui

)= 0

• no cross diffusion

• maximum principle for ui



Formulations of the Problem

Fi (c1, ..., cM) = log(ci )− log(1− ρ)vi = F−1

i (Fi (c1, ..., cM) + ziV )

system in Slotboom variables:

−λ2∆V −∑ zkvk exp(−zkV )

1 +∑

vj [exp(−zjV )− 1]= f

∇ ·

(Di

exp(−ziV ) (∇vi (1−∑

vj) + vi∑∇vj)

(1 +∑

vj [exp(−zjV )− 1])2

)= 0

• Nernst-Planck case: mulitplication with exponentials of V



Global Existence

(A1) f ∈ L∞(Ω)

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

Let assumptions (A1), (A2) be satisfied. Then, there exists a weaksolution

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

of

−λ2∆V =∑

zjcj + f

0 = ∇ · (Di ((1− ρ)∇ci + ci∇ρ+ zici (1− ρ)∇V )) ,

and boundary conditions as above, such that further

0 ≤ ci ≤ 1, ρ ≤ 1 a.e. in Ω.



Global Existence

proof:

1. Step: construction of fixed point operator F = H G2. Step: operator G is well defined on a set M3. Step: G is continuous on M and maps M into M×K, where K is

a bounded subset of H1(Ω)× L∞(Ω)

4. Step: H is well defined, continuous and maps G(M) into compactsubset of M

5. Step: Schauders fixed point theorem: fixed point which is solution



Global Existence

G :L2(Ω)M → L2(Ω)M × H1(Ω)

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

V is unique solution of nonlinear Poisson equation

−λ2∆V =∑

zkexp(uk − zkV )

1 +∑

exp(uj − zjV )+ f

with boundary conditions as above



Global Existence

H :DH ⊂ L2(Ω)M × H1(Ω) → L2(Ω)M

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

vi are unique weak solutions of linear elliptic equations

∇ ·(

exp(ui − ziV )

(1 +∑

exp(uj − zjV ))2∇vi

)= 0

subject to boundary conditions above



Global Existence

J(V ) =λ2

2

∫Ω|∇V |2 dx +

∫Ω

log(

1 +∑

exp(uj − zjV ))

dx

• strictly convex and coercive on H1(Ω) ⇒ unique minimizer V

• maximum principle ⇒ uniform bound for V in L∞(Ω)

• weak formulation of Poisson equation with Friedrichsinequality ⇒ G is Lipschitz-continuous on M

⇒ G well defined and continuous



Global Existence

Ai = Diexp(ui − ziV )

(1 +∑

exp(uj − zjV ))2∈ L∞(Ω),

standard theory ⇒ existence and uniqueness of weak solution of

∇ · (Ai∇vi ) = 0

H1(Ω) → L2(Ω) compact ⇒ H(G(M)) precompact⇒ H well defined and maps G(M) into compact subset of M



Global Existence

Continuity of H:

• Aki → Ai in L2(Ω)

• vi uniformly bounded in H1(Ω) ⇒ weakly convergingsubsequence vkli → vi

• 0 =∫

Ω Ai∇vi∇φ dx for φ ∈W 1,∞0 (Ω) and also for φ ∈ H1

0 (Ω)

• trace theorem ⇒ boundary condition

• uniqueness of limits: vki → vi weakly in H1(Ω) and strongly inL2(Ω)

Schauders fixed point theorem ⇒



Regularity

existence: (V , c1, ..., cm) ∈ (H1(Ω) ∩ L∞(Ω))M+1

−λ2∆V =∑

zjcj + f

• rhs in L2(Ω)⇒ ∆V ∈ L2(Ω)⇒ V ∈ H2(Ω)

• Sobolev embedding theorem ⇒ H2(Ω) ⊂ L∞(Ω) forn = 1, 2, 3

(1− ρ)∆ci + ci∆ρ = −∇(zici (1− ρ)∇V ) ∈ L2(Ω)

• ⇒ ∆ci ∈ L2(Ω)

• (V , c1, ..., cm) ∈ H2(Ω)M+1



Uniqueness

uniqueness in general case cannot be expected!

but we can find uniqueness in simpler situations applying theimplicit function theorem. We can show:

• uniqueness around small potential

• uniqueness around small boundary values



Uniqueness

F(U, η;V , u) denotes operator

V − V 0D − UV 1

D on ΓE

ui − γi on ΓB

−λ2∆V −∑k

exp(uk − zkV )

1 +∑

exp(uj − zjV )− f ∈ L2

∇ ·(Di

exp(ui − ziV )

(1 +∑

exp[uj − zjV ])2∇ui

)∈ L2

F(U, η;V , u) Frechet-differentiable with respect to V ,U, γ and u



Uniqueness for small Voltage

• U = 0: equilibrium state, well posed

• linearized system in entropy variables is partially decoupledand Frechet derivative and has continuous inverse

implicit function theorem:

‖U‖H3/2(ΓB) sufficiently small. Then, for γ ∈ (H3/2(ΓB))M thereexists a locally unique solution

(V , c1, ..., cM) ∈ H2(Ω)M+1

of the above problem and the transformed, linearized problem iswell-posed.



Uniqueness for small Bath Conzentration

• γi = 0 ⇒ ci = 0, V0 solution

• linearized system is partially decoupled

• after Slotboom transformation: system of linear ellipticequations

• Frechet derivative and has continuous inverse

implicit function theorem:

‖γi‖H3/2(ΓB) sufficiently small. Then, for U ∈ H3/2(ΓB), thereexists a locally unique solution

(V , c1, .., cM) ∈ H2(Ω)M+1

of the above problem and the transformed, linearized problem iswell-posed.



Reduction to One Dimension

• cross section of filter much smaller than longitudinal extension⇒ nearly one dimensional process

• approximate three dimensional model by one dimensional one

rescale: x , y ε = εy , zε = εz , (y , z) ∈ Ω1

V ε(x , y ε, zε) = V ε(x , y , z)

weak formulation with test function:

ϕ(x , y , z) = V ε(x , y , z)− g(x)




weak formulation:

−λ2

∫ ∫ ∫Ω1

(∣∣∣∂x V ε∣∣∣2 +

1

ε2

∣∣∣∂y V ε∣∣∣2 +

1

ε2

∣∣∣∂z V ε∣∣∣2) dx dy dz ≤ k

for ε→ 0: ∥∥∥∂y V ε∥∥∥L2(Ωε)

→ 0∥∥∥∂z V ε

∥∥∥L2(Ωε)

→ 0

andV ε(x , εy , εz) = V ε(x , y , z) V 0(x) in H1(Ωε)




uniform bounds:

0 < k1 ≤exp(uεi − ziV )(

1 +∑

exp(uεj − zjV ))2≤ k2

analogous estimates

‖∂y uεi ‖L2(Ωε) → 0 ‖∂z uεi ‖L2(Ωε) → 0

uεi (x , εy , εz) = uεi (x , y , z) u0i (x) in H1(Ωε)

same for ci




λ2

∫ ∫ ∫Ωε∂xV

ε(x , y ε, zε)∂xϕ(x) dx dy dz →

λ2

∫∂xV

0(x)∂xϕ(x)

∫ ∫dy dz dx

Assume a(x) =∫ ∫

dy dz denotes shape/ area function of tunnel

reduced one dimensional system

−λ2∂x(a∂xV ) = a(∑

cj + f)

Di∂x

(a

exp(ui − ziV )

(1 +∑

exp(uj − zjV ))2∂xui

)= 0



Conductance for L-type Calcium selective Channel

PPPPPP

PPPP

PP

Na+Ca2+

Cl−

Na+Ca2+

Cl−O12−O

12−

0mV 100mV



Numerical Results for L-type Calcium selective Channel



Open Questions

• at the moment: same size for different speciesin reality: often different sizes for different spezies

• explanation of biological phenomena such as gating



thank you for your attention!


nonlinear poisson-nernst planck equation for ion flux · 2. step:operator gis well de ned on a set...

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