Nonlinear Poisson-Nernst Planck Equation for Ion Flux · 2. Step:operator Gis well de ned on a set...
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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
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Motivation Modelling Analysis for stationary Model
Credits
this talk is based on joint work with
• martin burger (WWU Munster)
• marie-therese wolfram (Vienna)
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Motivation Modelling Analysis for stationary Model
Motivation
Modelling
Analysis for stationary Model
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Motivation Modelling Analysis for stationary Model
Motivation
aim: modelling of transport and diffusion with size exclusion
application:
• ion channels/nanopores
• human crowds
• swarming
• chemotaxis
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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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)
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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
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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
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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.
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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
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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
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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
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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
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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
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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 ⇒
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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
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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
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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
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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.
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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.
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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)
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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)
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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
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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
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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
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Motivation Modelling Analysis for stationary Model
Numerical Results for L-type Calcium selective Channel
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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
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Motivation Modelling Analysis for stationary Model
thank you for your attention!
Nonlinear Poisson-Nernst Planck Equation for Ion Flux Universitat Munster