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
Motivation Modelling Analysis for stationary Model
Credits
this talk is based on joint work with
• martin burger (WWU Munster)
• marie-therese wolfram (Vienna)
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
Motivation
Modelling
Analysis for stationary Model
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
Motivation
aim: modelling of transport and diffusion with size exclusion
application:
• ion channels/nanopores
• human crowds
• swarming
• chemotaxis
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
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
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
Modelling
two possible approaches in microscopic modelling;
1. force-basedNewton equation of motion(macroscopic limit difficult)
2. jump exclusion processcellular automata
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
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
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
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
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
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
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
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
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
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
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
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
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
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
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
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.
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
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
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
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)
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
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
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
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
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
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.
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
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
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
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
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
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
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
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
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
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
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
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 ⇒
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
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
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
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
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
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
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
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.
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
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.
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
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)
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
Reduction to One Dimension
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)
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
Reduction to One Dimension
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
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
Reduction to One Dimension
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
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
Conductance for L-type Calcium selective Channel
PPPPPP
PPPP
PP
Na+Ca2+
Cl−
Na+Ca2+
Cl−O12−O
12−
0mV 100mV
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
Numerical Results for L-type Calcium selective Channel
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
Open Questions
• at the moment: same size for different speciesin reality: often different sizes for different spezies
• explanation of biological phenomena such as gating
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster
Motivation Modelling Analysis for stationary Model
thank you for your attention!
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster