Motivation
Model for Ion Channels
Analysis
Nonlinear Cross Diffusion Models for Ion
Transport through Channels
Bärbel Schlake
Westfälische Wilhelms-Universität Münster
Institute für Computational und Applied Mathematics
22 September 2009
Nonlinear Cross Diffusion Models for Ion Transport through Channels
Universität Münster
Motivation
Model for Ion Channels
Analysis
Motivation
Model for Ion Channels
Analysis
Nonlinear Cross Diffusion Models for Ion Transport through Channels
Universität Münster
Motivation
Model for Ion Channels
Analysis
What are Ion Channels?
I
ion channels are proteins with a
hole down their middle
I
regulate movement of inorganic
ions (Na+ , Ca+ , Cl− , Ca2+ )
through impermeable cell
membranes
I
found in biological membranes
(humans, animals, plants)
Nonlinear Cross Diffusion Models for Ion Transport through Channels
Universität Münster
Motivation
Model for Ion Channels
Analysis
Why are Ion Channels interesting?
I
ion channels play a vital role in
life
I
defects are reason for serious
diseases
I
structural properties of certain
channels not known so far
I
existing models have poor
agreement with experimental
data in some cases
Nonlinear Cross Diffusion Models for Ion Transport through Channels
Universität Münster
Motivation
Model for Ion Channels
Analysis
Motivation
aim: effective model for ion channels with cross diffusion
application of cross diffusion models:
I
swarming
I
human crowds
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chemotaxis
Nonlinear Cross Diffusion Models for Ion Transport through Channels
Universität Münster
Motivation
Model for Ion Channels
Analysis
Derivation of Model
self-consistent one-dimensional hopping model:
~
~
-
~
Probability
ci (x, t) = P(particle of species i is at position x at time t)
Nonlinear Cross Diffusion Models for Ion Transport through Channels
Universität Münster
Motivation
Model for Ion Channels
Analysis
Derivation of Model
I
transition rate:
i
Πc+/−
(x, t) =
P(jump of particle of species i from position x to x ±
1
h in (t, t + ∆t))· ∆t
· P(x ± h is empty)
Nonlinear Cross Diffusion Models for Ion Transport through Channels
Universität Münster
Motivation
Model for Ion Channels
Analysis
Derivation of Model
I
transition rate:
i
Πc+/−
(x, t) =
P(jump of particle of species i from position x to x ±
1
h in (t, t + ∆t))· ∆t
· P(x ± h is empty)
I
closure assumption:
P
P(position x is empty at time t)=1 − j cj (x, t)
Nonlinear Cross Diffusion Models for Ion Transport through Channels
Universität Münster
Motivation
Model for Ion Channels
Analysis
Derivation of Model
I
transition rate:
i
Πc+/−
(x, t) =
P(jump of particle of species i from position x to x ±
1
h in (t, t + ∆t))· ∆t
· P(x ± h is empty)
I
closure assumption:
P
P(position x is empty at time t)=1 − j cj (x, t)
I
probability that a particle of species i is located at position x
at time t + ∆t:
ci (x, t + ∆t) = ci (x, t)(1 − Π+ (x, t) − Π− (x, t))
+ci (x + h, t)Π− (x + h, t) + ci (x − h, t)Π+ (x − h, t)
Nonlinear Cross Diffusion Models for Ion Transport through Channels
Universität Münster
Motivation
Model for Ion Channels
Analysis
Derivation of Model
mass density m(x, t) =
P
i ci (x, t)
diffusion coefficient Di
scaled mobility constant µi
Asymptotic Limit h → 0 ∆t → 0
∂ci
∂t
=
∂
∂x (Di ((1
∂Wi
∂m
i
− m) ∂c
∂x + ci ∂x + µi ci (1 − m) ∂x ))
Nonlinear Cross Diffusion Models for Ion Transport through Channels
Universität Münster
Motivation
Model for Ion Channels
Analysis
Derivation of Model
Wi (x, t) = νi (x, t) + zi V (x, t),
νi external potential, zi charge
V (x, t) computed self consistently from Poisson equation:
P
λ2 ∆V (x, t) = − i zi ci (x, t) + f (x, t)
f (x, t) protein charge patterns
I
Multidimensional Model
∂t ci = ∇ · (Di ((1 − m)∇ci + ci ∇m + µi ci (1 − m)∇Wi ))
|
{z
}
ion flux Ji
Nonlinear Cross Diffusion Models for Ion Transport through Channels
Universität Münster
Motivation
Model for Ion Channels
Analysis
Equilibrium Solution
ionic fluxes equal zero in equilibrium case:
(1 − meq )∇cieq + cieq ∇meq + µi cieq (1 − meq )∇Wieq = 0
Generalized Boltzmann Distributions
cieq =
ki exp(µi Wieq )
P
1+ j kj exp(µj Wjeq ) ,
ki ≥ 0
Modified Poisson-Boltzmann
Equation
P
−λ2 ∆Veq (x) =
− P
i zi ki exp(µi (νi (x)+zi Veq (x)))
1+ i ki exp(µi (νi (x)+zi Veq (x)))
Nonlinear Cross Diffusion Models for Ion Transport through Channels
+ f (x)
Universität Münster
Motivation
Model for Ion Channels
Analysis
Entropy
Entropy for this Process:
E=
RP
i (ci
log ci + (1 − m) log(1 − m) + µi ci Wi )dx
Entropy Dissipation:
RP
2
Et = −
i ci (1 − m)|∇ηi | dx, with
ηi = log ci (x, t) − log(1 − m(x, t)) + µi Wi (x, t)
⇒ Entropy is decreasing during the process. In equilibrium state,
the entropy is minimal at fixed total mass.
Nonlinear Cross Diffusion Models for Ion Transport through Channels
Universität Münster
Motivation
Model for Ion Channels
Analysis
Boundary Conditions in Experiments
ΓB
ΓE
v
v
ΓB
v
v
v
ΓB
v
ΓE
ΓB
concentration: ci (x, t) = γi (x)
remaining part of system: Ji (x, t) · n = 0
X
charge neutrality:
zi γi (x) = 0,
x ∈ ΓB ⊂ ∂Ω,
x ∈ ∂Ω \ ΓB ,
i
electrical Potential: V (x, t) = VD0 (x) + UVD1 (x)
remaining part of system: ∇V (x, t) · n = 0
Nonlinear Cross Diffusion Models for Ion Transport through Channels
x ∈ ΓE ⊂ ∂Ω,
x ∈ ∂Ω \ ΓE .
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Motivation
Model for Ion Channels
Analysis
Numerical Results for Equilibrium
The solid line corresponds √
to λ = 0.1 in the Poisson equation, the
dashed-dotted line to λ = 0.1 and the dotted line to λ = 0.5.
Nonlinear Cross Diffusion Models for Ion Transport through Channels
Universität Münster
Motivation
Model for Ion Channels
Analysis
Transformation to Slotboom Variables
Stationary Model
0 = ∇ · (Di ((1 − m)∇ci + ci ∇m + µi ci (1 − m)∇Wi ))
The system is not easy to solve. Hence we try to transform the
variables to obtain a simplified system. In classical semiconductor
theory, the variables after the transformation are called
’Slotboom-Variables’.
I
transformation function Fi (c1 , ..., cn ) = log ci − log(1 − m)
I
transformation variable ui = Fi−1 (Fi (c1 , ..., cn ) + µi Wi )
I
⇒ ci = Fi−1 (Fi (u1 , ..., un ) − µi Wi ) =
Nonlinear Cross Diffusion Models for Ion Transport through Channels
P ui exp(−µi Wi )
1+ j uj (exp(−µj Wj )−1)
Universität Münster
Motivation
Model for Ion Channels
Analysis
Slotboom Transformation
System after Transformation
0=∇·
exp(−µi Wi )
(1+
P
j
uj (exp(−µj Wj )−1))
2
∇ui (1 −
P
j uj ) + ui
P
j ∇uj
⇒ no convective terms
linearization ui = uieq + ũi around uieq =
System after Linearization
P
exp(−µ W )(1+
k)
0=∇·
j j
P i ieq
(1+ j kj exp(−µi Wjeq ))2
∇ũi + ki
Nonlinear Cross Diffusion Models for Ion Transport through Channels
ki
P
1+ j kj :
P
j ∇ũj
Universität Münster
Motivation
Model for Ion Channels
Analysis
Problems in the Analysis
We consider a system of two equations from now on. We consider
red and blue particles. Density of red particles is labelled r ,
respectivley for blue particles.
Only difference: Response to potental V .
System of Equations
∂t r = ∇ · ((1 − m)∇r + r ∇m − µr r (1 − m)∇V )
∂t b = ∇ · (D((1 − m)∇b + b∇m + µb b(1 − m)∇V ))
nonlinear cross diffusion ⇒ few literature available
I
existence and uniqueness of a solution?
I
no a-priori estimates
I
no maximum principle (only 0 ≤ r , b, m ≤ 1)
Nonlinear Cross Diffusion Models for Ion Transport through Channels
Universität Münster
Motivation
Model for Ion Channels
Analysis
Entropy
Entropy
RE (r , b) =
(r log (r ) + blog (b) + (1 − m) log (1 − m) − µr rV + µb bV ) dx
E is decreasing during the process and minimal in equilibrium state
Equilibrium Solutions
k1 exp(µr V (x))
k1 exp(µr V (x))+k2 exp(−µb V (x))+1
2 exp(−µb V (x))
= k1 exp(µr Vk(x))+k
2 exp(−µb V (x))+1
req (x, t) =
beq (x, t)
special cases:
I
m ≡ 1 ⇒ req = k1 exp(µr V ),
I
b ≡ 0 ⇒ rt = ∇ · (∇r − r (1 − r )∇V ) ⇒ req =
Nonlinear Cross Diffusion Models for Ion Transport through Channels
beq = k2 exp(−µb V )
k1 exp(µr V )
k1 exp(µr V )+1
Universität Münster
Motivation
Model for Ion Channels
Analysis
Special Case V (x) = 0
∂t r = ∇ · ((1 − m)∇r + r ∇m)
∂t b = ∇ · (D((1 − m)∇b + b∇m))
Transformation u = F (r , b) = G
1−r −b
(r −b−α)α
α=
D−1
D+1
we can show that u satisfies a maximum-principle.
special cases:
D = 0:
u = G ((1 − m)(r − b − 1))
D = 1:
u = G (1 − m)
Nonlinear Cross Diffusion Models for Ion Transport through Channels
Universität Münster
Motivation
Model for Ion Channels
Analysis
Special Case V (x) = 0
System of Equations with V(x) = 0:
∂t r = ∇ · ((1 − m)∇r + r ∇m)
∂t b = ∇ · (D((1 − m)∇b + b∇m))
equilibrium solutions: constants r̄ , b̄
small perturbations u and v
linearization around r̄ + u and b̄ + v :
First Order Linearization
∂ut = (1 − b̄)∆u + r̄ ∆v
∂vt = D((1 − r̄ )∆v + b̄∆u)
Nonlinear Cross Diffusion Models for Ion Transport through Channels
Universität Münster
Motivation
Model for Ion Channels
Analysis
Stability around Equilibria
combine first order linearizations to equation for w = u + α1/2 v .
choose
p
D(1 − r̄ ) − (1 − b̄) ± ((1 − b̄) − D(1 − r̄ ))2 + 4Dr̄ b̄
α1/2 =
2D b̄
⇒ heat equation
∂t w = k1/2 ∆w
with
k1/2
1
=
2
q
2
D(1 − r̄ ) + (1 − b̄) ± ((1 − b̄) − D(1 − r̄ )) + 4Dr̄ b̄
⇒ k1 > 0 and k2 ≥ 0 (k2 = 0 if r̄ + b̄ = 1)
⇒ stability around equilibria
Nonlinear Cross Diffusion Models for Ion Transport through Channels
Universität Münster
Motivation
Model for Ion Channels
Analysis
Existence and Uniqueness near Equilibrium
Consider
∂t r = ∇ · ((1 − m)∇r + r ∇m)
(1)
∂t b = ∇ · (D((1 − m)∇b + b∇m))
(2)
with initial data r0 , b0 ∈ H 2 . Assume
kr0 − req kH 2 ≤ , kb0 − beq kH 2 ≤ .
Then, there exists a unique solution to (1), (2) in
BR = (r , b) : kr − req kX ≤ R, kb − beq kX ≤ R
with X := L∞ ((0, T ); H 2 ) ∩ L2 ((0, T ); H 3 ) ∩ H 1 ((0, T ); H 1 ).
where the constant R depends on only.
Nonlinear Cross Diffusion Models for Ion Transport through Channels
Universität Münster
Motivation
Model for Ion Channels
Analysis
Sketch of Proof
The proof is based on Banach’s fixed point theorem:
1. Step: Construction of fixed point operator
(n − neq )t − (1 − peq )∆n − neq ∆p =
(peq − p)∆(n − neq ) − (neq − n)∆(p − peq )
|
{z
}
:=F1 (n,p)
(p − peq )t − (1 − neq )∆p − peq ∆n =
(neq − n)∆(p − peq ) − (peq − p)∆(n − neq ) .
|
{z
}
:=F2 (n,p)
2. Step: Splitting of fixed point operator in heat operator and in a
part that is of order 2 in L2
Nonlinear Cross Diffusion Models for Ion Transport through Channels
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Motivation
Model for Ion Channels
Analysis
Sketch of Proof
3. Step: Define Operator S wich maps r − req , b − beq on linear
combinations w1,2 = (r − req ) + α1,2 (b − beq ). w1,2 fulfills heat
equation
w1,2t − k1,2 ∆w1,2 = F1 + α1,2 F2 =: F
4. Step: Apply well-known results for heat operator: w1 , w2 ∈ X
and kw1,2 kX ≤ R1,2 ()
5. Step: Reiterating we finally obtain R 0 ≤ C (R 2 + ) and thus
obtain an R with BR = (r , b) : kr − req kX ≤ R, kb − beq kX ≤ R
6. Step: The operator G = S 0 ◦ L ◦ S ◦ F is selfmapping (L is
”solution operator” of the heat equation)
7. Step: We estimate kF1 (r1 , b1 ) − F1 (r2 , b2 )kL2 and
kF2 (r1 , b1 ) − F2 (r2 , b2 )kL2 to show contractiveness of G
⇒ We have existence and uniqueness of solutions (r , b) in BR
Nonlinear Cross Diffusion Models for Ion Transport through Channels
Universität Münster
Motivation
Model for Ion Channels
Analysis
Open Questions
• existence for the general case?
• uniqueness?
• longtime behaviour for the general case?
Nonlinear Cross Diffusion Models for Ion Transport through Channels
Universität Münster
Motivation
Model for Ion Channels
Analysis
Acknowledgements
Joint Work/Discussions with:
Martin Burger, Münster
Jan Pietschmann, Cambridge
Marco Di Francesco, L’Aquila
Nonlinear Cross Diffusion Models for Ion Transport through Channels
Universität Münster
Motivation
Model for Ion Channels
Analysis
Thank you for your Attention!
Nonlinear Cross Diffusion Models for Ion Transport through Channels
Universität Münster