Modeling cells in tissue by connecting
electrodiffusion and Hodgkin−Huxley-theory.
Brigham Young University
Department of Mathematics
Viktoria R.T. Hsu
University of Utah
Department of Mathematics
Outline
• Introduction: Motivation and Roadmap.
• Review: The Classic Hodgkin−Huxley Neuron Model.
• Toward Tissue Modeling: The QSSA of Electrodiffusion.
• Simpler Models by Further Approximation of the QSSA.
• Comparing Models of Electrodiffusion to Hodgkin−Huxley:
◦ approach to equilibrium of a cell with ion channels but no pumps,
◦ a measure of performance for comparison of the models,
◦ approach to rest after an action potential; channels and pumps.
• Summary and Conclusions.
• Future Work.
Schematic of a Neuron Cell
Different Shapes of Neuron Cells
Limbic System of the Human Brain
Neuron Cells in the Hippocampus
Roadmap
HH−GHK
(GHK currents)
HH−cls
(classic HH model)
model circuit
(leaky capacitor)
QSSA
CFA
(const. field)
HH−plk
(pump−leak)
electrodiffusion
PNP−system
HH−GHK
(GHK currents)
HH−cls
(classic HH model)
model circuit
(leaky capacitor)
electrodiffusion
PNP−system
QSSA
CFA
(const. field)
HH−plk
(pump−leak)
The Classic
Hodgkin−Huxley Model
The Classic Hodgkin−Huxley Neuron Model
Assumptions:
• Infinite buffering: ion concentrations and cell volume remain constant.
• Passive transport: ion channels pass an Ohmic current with voltage dependent
conductance.
• Active transport: ion pumps maintain homeostasis (living state); represented by
reversal potentials.
• Quantify: the membrane behaves like a leaky capacitor.
• Parameters are fit to data from giant squid axon.
The Classic Hodgkin−Huxley Neuron Model
The Model Equations:
4
3
Cm dV
dt = −ḡK n (V − VK ) − ḡN a m h (V − VN a ) ...
(1)
−ḡL (V − VL) + Iapp
dm
= αm (1 − m) + βmm
dt
dn
= αn (1 − n) + βnn
dt
dh
= αh (1 − h) + βhh,
dt
where αx and βx, for x ∈ {m, n, h}, are functions of v = V − V∞,
the difference of the cross-membrane potential from the resting potential.
(2)
(3)
(4)
The Classic Hodgkin−Huxley Neuron Model
The Fast-Slow Phase-Plane:
• m has fastest dynamics, so m ≈ m∞, its resting value.
• Eliminate h by using FitzHugh’s observation that n + h ≈ 0.8.
• V and n then satisfy:
4
3
Cm dV
dt = −ḡK n (V − VK ) − ḡN a m∞ (0.8 − n) (V − VN a ) ...
(5)
−ḡL (V − VL) + Iapp
dn
= αn (1 − n) + βnn.
dt
(6)
The Classic Hodgkin−Huxley Neuron Model
Flow Field in the Fast-Slow Phase-Plane:
The Classic Hodgkin−Huxley Neuron Model
Sub-Threshold Stimulus:
The Classic Hodgkin−Huxley Neuron Model
Signal Generation by Super-Threshold Stimulus:
HH−GHK
(GHK currents)
HH−cls
(classic HH model)
model circuit
(leaky capacitor)
electrodiffusion
PNP−system
QSSA
CFA
(const. field)
HH−plk
(pump−leak)
That’s Great!
What Improvements Are Needed?
A Need for Accurate Modeling of Cells in Tissue
• Cells in tissue experience variation of their external environment, and volume.
• Variations can be large under conditions like ischemia (lack of metabolic supply).
• Recall: present multi-compartment models assume isolated cell in buffer.
◦ phenomenological, so electroneutrality is neglected,
◦ external concentrations and cell volume are static.
• Propose: extend multi-compartment models to single-cell micro-environment.
◦ physically consistent model for signal generation with varying external ion
concentrations and cell volume.
• Challenge: step away from the model circuit assumption (for now).
◦ in finite volume one has to consider mass conservation and electroneutrality.
◦ Thus, model charge-carrier transport by electrodiffusion.
HH−GHK
(GHK currents)
HH−cls
(classic HH model)
model circuit
(leaky capacitor)
electrodiffusion
PNP−system
QSSA
CFA
(const. field)
HH−plk
(pump−leak)
From Electrodiffusion
To Its QSSA.
Charge-Carrier Transport - the Equations
Alike Semiconductor-Device Equations:
∂ci
= ∇ · [Di (∇ci + zi∇ϕci) + Si]
∂t
∇ · (ε∇ϕ) +
X
i
• c = species’ concentration
• ϕ = electro-static potential
• D = Diffusion coefficient
• ε = Dielectric coefficient
• z = species’ valency
• i = species’ index
zici = −N
(7)
(8)
The Single-Cell Micro-Environment
Dbk
Dm
Dbk
bk
m
bk
cell membrane
R
L
internal
compartment
external compartment
Simplifying Assumptions and Reduction to 1D
• No ionic species are involved in chemical reactions.
• All medium is homogeneous and neutral.
• The membrane is of uniform width and is impermeable to some ionic species.
Charge-Carrier Transport - 1D Equations
Electrodiffusion and Poisson’s Equations:
∂ci
∂
∂ci
∂ϕ
=
Di
+ zi ci
∂t
∂x
∂x
∂x
(9)
X
∂
∂ϕ
ε
+
zici = 0
∂x
∂x
i
(10)
• c = species’ concentration
• ϕ = electro-static potential
• D = Diffusion coefficient
• ε = Dielectric coefficient
• z = species’ valency
• i = species’ index
More Assumptions Reduce the System to Its QSSA
• Diffusion in membrane medium is much slower than diffusion in bulk.
• The space occupied by membrane medium is small compared to each bulk compartment.
The Domain in 1D - End of Membrane Impermeability
C in
i , =0
C out
i ,
mid−membrane
internal
region
= +
external
region
p
p
p
p
(internal bulk)
(external bulk)
p
p
p
p
p
x
−L
0
membrane
region
R
The QSSA in Membrane Region
PNP System After Assumptions and at Steady-State:
− J i = Di
∂
∂ϕ
ε
∂x
∂x
• c = species’ concentration
• ϕ = electro-static potential
• J = species’ flux density
• D = Diffusion coefficient
• ε = Dielectric coefficient
• z = species’ valency
∂ϕ
∂ci
+ zi ci
∂x
∂x
+
X
i
zici = 0
(11)
(12)
Dynamic Equations of the QSSA
dcR
Ac
i
=
Ji
dt
vout
(13)
dcLi
Ac
= − Ji
dt
vin
(14)
From Almost Newton (AN) numerical method:
Ji = −Di
ci (R) exp (ziϕ (R)) − ci (L) exp (ziϕ (L))
=?
RR
L exp (zi ϕ (s)) ds
ϕ (x) =?
(15)
(16)
HH−GHK
(GHK currents)
HH−cls
(classic HH model)
model circuit
(leaky capacitor)
electrodiffusion
PNP−system
QSSA
CFA
(const. field)
HH−plk
(pump−leak)
Approximating the QSSA to Obtain
Simpler Models of Electrodiffusion.
Potential Profile from QSSA
Dynamic Equations of the QSSA - with ion channels
dcR
Ac
i
=
Ji , for ci ∈ {N a; K; Cl}
dt
vout
(17)
dcLi
Ac
= − Ji , for ci ∈ {N a; K; Cl}
dt
vin
(18)
dX
= α (V ) (1 − X) − β (V ) X , for X ∈ {m; n; h}
dt
(19)
Ji = −Di·(gating)
ci (R) exp (ziϕ (R)) − ci (L) exp (ziϕ (L))
=? , from AN method
RR
L exp (zi ϕ (s)) ds
(20)
∆ϕ = −
FV
= ϕR − ϕL =? , from AN method
R0 T
(21)
Dynamic Equations of the CFA
dcout
Ac
i
=
Ji , for ci ∈ {N a; K; Cl}
dt
vout
(22)
dcin
Ac
i
= − Ji , for ci ∈ {N a; K; Cl}
dt
vin
(23)
dX
= α (V ) (1 − X) − β (V ) X , for X ∈ {m; n; h}
dt
(24)
zi ∆ϕ
zi∆ϕ cout
− cin
i e
i
·
Ji = −Di · (gating)
z
∆ϕ
δ
e i −1
(25)
FV
δ
= ϕR − ϕL =
∆ϕ = −
R0 T
εAc
!
vout
X
zi
zicout
i
(26)
Relating Parameters of QSSA and CFA to those of HH-type Models
With the following assumptions,
∂ϕ
1
FV
(0) ≈ (ϕR − ϕL) = −
∂x
δ
δR0T
∂ln (ci)
1
(0) ≈
ln ci
∂x
δ
out
in
− ln ci
1
= ln
δ
(27)
cout
i
cin
i
(28)
the PNP system yields HH-type currents and voltage dynamics with
gi
zi2Dici (0)
=
Cm
ε
Cm =
εAc
δ
(29)
(30)
Dynamic Equations of the Hodgkin−Huxley-Type Model (HH-plk)
HH













pump − leak
Cm dV
dt +
dX
dt
P
i Ii
= Iapp
= α (V ) (1 − X) − β (V ) X , for X ∈ {m; n; h}
Ii = gi · (gating) (V − Vi) , for i ∈ {N a; K; Cl}
















 Vi =
dcout
i
dt
dcin
i
dt
R0 T
zi F ln
cout
i
cin
i
=
Ac
vout
· zIiFi
= − vAinc · zIiFi
, for i ∈ {N a; K; Cl}
HH−GHK
(GHK currents)
HH−cls
(classic HH model)
model circuit
(leaky capacitor)
electrodiffusion
PNP−system
QSSA
CFA
(const. field)
HH−plk
(pump−leak)
Comparing the Dynamics to Eqlb.
of QSSA, CFA, and HH-plk
with Ion Channels but No Pumps.
Concentration Dynamics to Equilibrium:
Current Density Dynamics to Equilibrium:
Electro-Static Potential Dynamics to Equilibrium:
HH−GHK
(GHK currents)
HH−cls
(classic HH model)
model circuit
(leaky capacitor)
electrodiffusion
PNP−system
QSSA
CFA
(const. field)
HH−plk
(pump−leak)
How can one judge
which model works best?
A Measure of Maintaining Net-Electroneutrality:
X
∂
∂ϕ
ε
=−
zici
∂x
∂x
i
(31)
2
2
X
√ Ji
1 ∂ϕ
1 ∂ϕ
L
ε (R) −
ε (L) = ε
cR
−
c
+
2
ε
i
i
2
∂x
2
∂x
Di
i
(32)
X Ji
X cout − cin
i
√ i
≈
−
Di
2 ε
i
i
(33)
A Measure of Maintaining Net-Electroneutrality
Relative Measure of Maintaining Net-Electroneutrality
HH−GHK
(GHK currents)
HH−cls
(classic HH model)
model circuit
(leaky capacitor)
electrodiffusion
PNP−system
QSSA
CFA
(const. field)
HH−plk
(pump−leak)
Comparing the Dynamics to Rest
of CFA, HH-plk, and HH-cls
with Ion Channels and Pumps.
A Simple Model for Ion Pump Currents
Motivation:
Ii,cls = gi V −
ViN P
+
gi ViN P
−
NP
Vi,rest
Ii,plk = gi V − ViN P + Iipump
(34)
(35)
Specific concentration gradients are maintained at steady-state by:
pump
NP
Iipump = Ii,rest
− gipump ViN P − Vi,rest
,
pump
• Ii,rest
, pump current density at rest, equals channel current density at rest.
• gipump, conductance of the pump.
• ViN P , present Nernst potential of species i.
NP
• Vi,rest
, Nernst potential of species i at rest, pump is to maintain.
(36)
Updated Dynamic Equations of CFA and HH-plk
• Jipump =
dcout
vout i = Ac Jich + Jipump ,
dt
(37)
dcin
pump i
ch
vin
= −Ac Ji + Ji
,
dt
(38)
1 pump
zi F Ii
• Jich, the previously defined, passive channel flux density
of the CFA or HH-plk model.
Current Densities by CFA, HH-plk and HH-cls
Action Potential by CFA, HH-plk and HH-cls
Relative Measure of Maintaining Net-Electroneutrality
Summary
• Electrodiffusion (ED) was approximated to yield several models of ED: QSSA,
CFA and HH-plk.
• The ODE-based models, CFA and HH-plk, were extended to models of electric
cell signalling and compared to HH-cls.
• A measure of the maintenance of net-electroneutrality (net-EN) was used to
compare the methods’ performance.
• No electroneutrality conditions were explicitly enforced.
HH−GHK
(GHK currents)
HH−cls
(classic HH model)
model circuit
(leaky capacitor)
QSSA
CFA
(const. field)
HH−plk
(pump−leak)
electrodiffusion
PNP−system
Future Work
• Incorporate applied currents and volume dynamics (swelling).
• Perform a full analysis of the CFA model.
• Study the importance of cell-cell interactions through gap junctions versus other
ephaptic means when two cells share a small external environment.
• Study interactions between neuron and glia cells.
• Incorporate energetics into the CFA model by, for example, including ATPsensitive, and ATP-consuming, pumps and transporters.
Thank you...
...for Your attention.
Questions
... ? ? ?
EXTRA SLIDES
A Measure of Closeness to the Full PDE:
X
∂
∂ϕ
ε
=−
zici
∂x
∂x
i
X ∂ϕ
∂ϕ ∂
∂ϕ
ε
ε
= −ε
zi ci
∂x ∂x
∂x
∂x
i
2 !
X ∂ci Ji
∂ 1 ∂ϕ
ε
=ε
+
∂x 2
∂x
∂x
Di
i
2
2
X
√ Ji
1 ∂ϕ
1 ∂ϕ
in
cout
−
c
+
2
ε (out) −
ε (in) = ε
ε
i
i
2
∂x
2
∂x
Di
i
X
√ Ji
in
0=
cout
−
c
+
2
ε
i
i
Di
i
X Ji
X cout − cin
i
√ i
−
=
Di
2 ε
i
i
(39)
(40)
(41)
(42)
(43)
(44)
Hodgkin−Huxley Gating Dynamics.
Coefficients of the HH-type Gating Variables
IN a = gN am3h (V − VN a) ,
IK = gK n4 (V − VK ) ,
ICl = gL (V − VL)
αx and βx, for x ∈ {m, n, h}, are the following functions of v = V − V∞,
the difference of the cross-membrane potential from the resting potential:
αm = 0.1 exp
25−v
( 25−v
10 )−1
v
αn = 0.07exp − 20
βm = 4exp
v
− 18
1
βn = exp 30−v
( 10 )+1
v
αh = 0.01 exp 10−v
β
=
0.125exp
−
h
80 .
( 10−v
10 )−1
(45)
More Intuitive Gating Equations
Defining the new functions x∞ and τx for x ∈ {m, n, h} according to
x∞ =
αx
αx + βx
and τx =
1
αx + βx
(46)
allows us to write the original gating equations in a more intuitive form, namely
dm
= m∞ (v) − m
dt
dn
τn (v)
= n∞ (v) − n
dt
dh
= h∞ (v) − h.
τh (v)
dt
τm (v)
(47)
(48)
(49)
What is a Quasi Steady-State Approximation?
Quasi Steady-State Approximation, in general

 εxt = F (x, y)

(50)
yt = G(x, y)

 0 = F (x, y)
(51)
 y = G(x, y)
t


x = f (y)
 y = G(f (y), y)
t
(52)
Detailed Derivation of the QSSA.
The fully transient PDE model in 1D
Electro-Diffusion and Poisson’s Equations:
 i
m
i
i
i
i
c
=
D
c
+
z
ϕ
c
 t
x x for −L ≤ x < − 2
x
B
i
cit = DM
cix + z iϕxci x for − m2 ≤ x ≤ m2
 i
m
ct = DBi cix + z iϕxci x for
2 <x ≤R

P
m
 (εB ϕx)x = − Pi z ici for −L ≤ x < − 2
(εM ϕx)x = − i z ici for − m2 ≤ x ≤ m2
P

m
(εB ϕx)x = − i z ici for
2 < x ≤ R,
It is understood that the natural length scales of each compartment are
vin
m
R − m2 = vout
and
L
−
=
A
2
A.
(53)
(54)
Rescale time and space for different time scales
i
Assume that maxi DM
mini DBi .
2 2 min
min
and time by t̄ = m DM t, where DM
Rescale space by x̄ =
i
= mini DM , then
 i i
2L
i
i
i
 σB ct̄ = cx̄ + z ϕx̄c x̄ for − m ≤ x̄ < −1
i i
σM
ct̄ = cix̄ + z iϕx̄ci x̄ for
−1 ≤ x̄ ≤ 1
 i i
σB ct̄ = cix̄ + z iϕx̄ci x̄ for 1 < x̄ ≤ 2R
m,
in which
i
σM
=
min
DM
i
DM
= O (1) and
σBi
=
min
DM
i
DB
1.
2x
m
(55)
Relaxation of membrane region to steady-state
Neglect small terms σBi cit̄ and approximate the

i
i
i
 0 = cx̄ + z ϕx̄c x̄ for
i i
σM
ct̄ = cix̄ + z iϕx̄ci x̄ for

0 = cix̄ + z iϕx̄ci x̄
for
dynamics by
− 2L
m ≤ x̄ < −1
−1 ≤ x̄ ≤ 1
1 < x̄ ≤ 2R
m ,
which implies in turn that

ci (t̄, x̄) = ciin for − 2L

m ≤ x̄ < −1
i i
σM
ct̄ = cix̄ + z iϕx̄ci x̄ for
−1 ≤ x̄ ≤ 1

ci (t̄, x̄) = ciout
for 1 < x̄ ≤ 2R
m .
(56)
(57)
Membrane at steady-state and bulk changes slowly
2L
Assume that 1 2R
m ≤ m and track the total mass by integrating over each
bulk compartment,
i
Z
d −1 i
2L
dcin
1
i
i
i c (t̄, x̄) dx =
−1
= i cx̄ + z ϕx̄c x̄=−1 .
(58)
dt̄ − 2L
m
d
t̄
σ
M
m
t̄
max
i
Rescaling time once more by τ = σmax 2R −1 , where σM = maxi σM = 1, then
)
M (m

i
i dcin
i
i
i 
 γin dτ = cx̄ + z ϕx̄c x̄=−1
i i
(59)
γM
cτ = cix̄ + z iϕx̄ci x̄
for −1 ≤ x̄ ≤ 1

i
 i dcout
γout dτ = − cix̄ + z iϕx̄ci x̄=1 ,
where
i
γout
=
i
σM
max
σM
= O (1),
i
γin
=
i 2L −1
σM
(m )
max 2R −1
σM
(m )
= O (1), and
i
γM
=
i
σM
max 2R −1
σM
(m )
1.
Membrane at steady-state and bulk obeys ODEs
i i
Neglect the small terms γM
cτ and approximate the dynamics by

i
i dcin
i
i
i 
 γin dτ = cx̄ + z ϕx̄c x̄=−1
0 = cix̄ + z iϕx̄ci x̄
for −1 ≤ x̄ ≤ 1

i
 i dcout
γout dτ = − cix̄ + z iϕx̄ci x̄=1 .
(60)
QSSA for Relaxation to Donnan Equilibrium, I
Equations (62) and (64) are to be satisfied together with Poisson’s equation.
Concentration profiles of permeant species in the membrane region are
i
cix̄
i
cioutez ϕ(1) − ciinez ϕ(−1)
+ z ϕx̄c =
= const. for − 1 ≤ x̄ ≤ 1 or
R1 i
z
ϕ(x̄)
dx̄
−1 e
R
R
i z i ϕ(−1) 1 z i ϕ(x̄)
i
z i ϕ(1) x̄ z i ϕ(x̄)
c
e
e
dx̄
+
c
e
dx̄
i
out
in
−1 e
x̄
,
ci (x̄) = e−z ϕ(x̄)
R1 i
z
ϕ(x̄)
dx̄
−1 e
i
i
(61)
(62)
while species impermeant to the membrane have Boltzmann densities,
cix̄ + z iϕx̄ci = 0 for − 1 ≤ x̄ ≤ 1 or
(
i z i (ϕ(−1)−ϕ(x̄))
c
for − 1 ≤ x̄ < 0
in e
ci (x̄) =
i
cioutez (ϕ(1)−ϕ(x̄)) for 0 < x̄ ≤ 1 .
(63)
(64)
QSSA for Relaxation to Donnan Equilibrium, II
A set of ODEs in time governs the dynamics of the bulk concentrations,
2R
i
i
ci (x̄) = ciin for − 2L
m ≤ x̄ < −1 and c (x̄) = cout for 1 < x̄ ≤ m , namely
i
i
i
cioutez ϕ(1) − ciinez ϕ(−1)
i dcin
γin
=
R1 i
dτ
ez ϕ(x̄)dx̄
(65)
−1
i
i
i
cioutez ϕ(1) − ciinez ϕ(−1)
i dcout
,
=−
γout
R1 i
dτ
ez ϕ(x̄)dx̄
−1
where
i
γout
=
i
σM
max
σM
and
i
γin
=
i 2L −1
σM
(m )
max 2R −1
σM
(m )
.
(66)
Relaxation Time to Donnan Equilibrium
Reconnecting the time τ with the original time t delivers an estimate for the
relaxation time to Donnan equilibrium. In particular,
τ = αt with α =
m
2
min
DM
m
R− 2
and we approximate the dynamic approach to Donnan equilibrium of the
bulk concentrations by
−αt
i
i
i
i
cin (t) = cin (∞) − cin (∞) − cin (0) e
ciout (t) = ciout (∞) − ciout (∞) − ciout (0) e−αt ,
(67)
(68)
(69)
where ciin,out (0) are the initial bulk concentrations, and ciin,out (∞) are the final
bulk concentrations at Donnan equilibrium.
QSSA: PNP Equation and
Boundary Conditions.
Setup for Mid-Membrane Impermeability
C in
i , =0
C out
i ,
mid−membrane
internal
= +
external
region
region
p
p
p
(internal bulk)
(external bulk)
p
p
p
x
−L
0
boundary layer
and membrane
R
The Electro-Diffusion and Poisson System in 1D
Recall that:
∂ci
∂
∂ci
∂ϕ
=
Di
+ zi ci
∂t
∂x
∂x
∂x
(70)
X
∂
∂ϕ
ε
+
zici = 0
∂x
∂x
i
(71)
• c = species’ concentration
• ϕ = electro-static potential
• D = Diffusion coefficient
• ε = Dielectric coefficient
• z = species’ valency
• i = species’ index
Poisson-Nernst-Planck System in Membrane Region
After Assumptions and at Steady-State:
− J i = Di
∂
∂ϕ
ε
∂x
∂x
• c = species’ concentration
• ϕ = electro-static potential
• J = species’ flux density
• D = Diffusion coefficient
• ε = Dielectric coefficient
• z = species’ valency
∂ϕ
∂ci
+ zi ci
∂x
∂x
+
X
i
zici = 0
(72)
(73)
Solving Nernst-Planck’s Equation
Smoothness and continuity of ϕ at mid-membrane:
ci (R) eziϕ(R) − ci (L) eziϕ(L)
,
Ji = −Di
RR
L exp (zi ϕ (s)) ds
(74)
Species permeant to the membrane obey:
−zi ϕ(x)
ci (x) = e
ci (L) e
zi ϕ(L)
RR
x
e
zi ϕ(s)
ds + ci (R) e
RR
L
zi ϕ(R)
Rx
L
eziϕ(s)ds
eziϕ(s)ds
Species impermeant to the membrane have Boltzmann densities:

 ci (L) e−zi(ϕ(x)−ϕ(L)) for x < 0
ci (x) =
.

ci (R) e−zi(ϕ(x)−ϕ(R)) for 0 < x
(75)
(76)
The Poisson-Nernst-Planck (PNP) Equation
Notation:
α̃jx
=
X
permeant i
zi = j
ci (x)
τjx
=
X
ci (x)
(77)
trapped i
zi = j
Poisson-Nernst-Planck (PNP) Equation,
to be Solved with an Almost-Newton (AN) Method:
h
X
∂
∂ϕ
−jϕ(x)
ε
=−
je
τjLejϕ(L)H (−x) + τjR ejϕ(R)H (x) + ...
∂x
∂x
all j
#
R
R
L jϕ(L) R jϕ(s)
R jϕ(R) x jϕ(s)
α̃j e
ds + α̃j e
ds
x e
L e
... +
,
RR
jϕ(s)
ds
L e
(78)
Boundary Conditions?
Charge-Carrier Transport in Various Disciplines
Neumann and
Dirichlet BCs
on el. potential
Mathematical Device:
PNP Equations
Dirichlet BCs
on el. potential
Charge−Carrier
Transport
Natural Device:
ionic species
cell membranes
Physical Device:
holes and electrons
semiconductors
current and el. potential
caused by carrier
concentration gradient
current caused by
applied el. potential
Natural Boundary Conditions by Gauss’ Law
Integrating Poisson’s equation over the entire domain and
using that the system is net-electroneutral,
Z
R
ε
L
ε
X
∂ 2ϕ
dx = −
zi
∂x2
i
Z
R
cidx
(79)
L
∂ϕ
∂ϕ
(R) − ε
(L) = 0.
∂x
∂x
(80)
Since L represents the interior of the cell,
∂ϕ
∂ϕ
(L) = 0 =
(R) .
∂x
∂x
Two Neumann boundary conditions do not define a well-posed problem!
(81)
Almost-Newton Method for Solving the QSSA.
The PNP Equation - Linearize for a Newton-Type Iteration Scheme
given ϕ̃, solve for δ:
∂2
ε 2 (ϕ̃ + δ) =
∂x
−
X
all
je−j ϕ̃(x) [Aj (x) (1 − jδ (x)) + jBj (x) δ (L) + jCj (x) δ (R) + ...
j
Z
x
... +jDj (x)
δ (s) ej ϕ̃(s)ds + jEj (x)
L
where Aj through Ej are highly nonlinear.
Z
x
R
δ (s) ej ϕ̃(s)ds ,
(82)
Coefficients of the Linearized PNP Equation
Aj (x) = Bj (x) + Cj (x)
(83)
RR
Bj (x) = ej ϕ̃(L) τjLH (−x) + α̃jL RxR
L
Rx
Cj (x) = ej ϕ̃(R) τjR H (x) + α̃jR R LR
L
e
j ϕ̃(s)
ds
!
ej ϕ̃(s)ds
e
j ϕ̃(s)
ds
(84)
!
ej ϕ̃(s)ds
(85)
R R j ϕ̃(s)
α̃jR ej ϕ̃(R) − α̃jLej ϕ̃(L)
e
ds
Dj (x) =
∗ RxR
RR
j ϕ̃(s) ds
j ϕ̃(s) ds
L e
L e
(86)
R x j ϕ̃(s)
α̃jR ej ϕ̃(R) − α̃jLej ϕ̃(L)
e
ds
∗ R LR
Ej (x) = −
RR
j ϕ̃(s) ds
j ϕ̃(s) ds
L e
L e
(87)
Coefficients of the Almost-Newton (AN) Iteration Scheme
Dj (x) replaced by
α̃jR ej ϕ̃(R)
Dj∗
= RR
L
Ej (x) replaced by
Ej∗
ej ϕ̃(s)ds
α̃jLej ϕ̃(L)
= RR
L
ej ϕ̃(s)ds
.
(88)
(89)
Comparing the equations defining AN and FN implies:
Z
R
L
δ (s) ejϕ(s)ds = 0
for all j .
(90)
Comparison of the FN, MG and AN Methods
• FN faces problems with catastrophic cancellation, especially as flux densities
become large. (Dj and Ej are similar terms with opposite signs, and proportional
to flux densities.)
• MG uses Aj through Cj only, Dj and Ej are neglected. Thus, the system matrix
is sparse but terms proportional to flux densities are neglected. MG as well faces
problems of convergence when flux densities become large.
• AN uses modified, simpler Dj and Ej , which arise when the PNP equation
RR
is linearized under the assumption that L ejϕ(s) = const.. AN requires two
Neumann BCs and performs well even for large flux densities.
Results: Convergence of PNP-solvers.
Results: Steady-State Study of PNP-solvers.
Summary for Almost-Newton Method (AN)
• The solution by AN is at least as accurate as solutions by MG or FN.
• Physiological bulk concentrations result in large negative flux densities, at which
AN converges more efficiently than MG or FN.
• For any flux density, AN converges with roughly the same, low number of iterations.
Including Sources in the QSSA
The Generalized PNP Equation, Including Sources
X
Notation: σ̃j =
permeant i
zi = j
Si
Di
Si
Di
X
σj =
trapped i
zi = j
(91)
Generalized Poisson-Nernst-Planck (PNP) Equation:
Z x
X
∂
∂ϕ
ε
=−
je−jϕ(x) τjLejϕ(L) −
σj (s) ejϕ(s)ds H (−x) + ...
∂x
∂x
L
all j
R
Z
... + τjR ejϕ(R) +
σj (s) ejϕ(s)ds H (x) + ...
x
... +
α̃jLejϕ(L)
Z
−
σ̃j (s) e
L
... +
α̃jR ejϕ(R) +
RR
x
Z
x
R
jϕ(s)
ds
jϕ(s)
ds
x e
RR
jϕ(s) ds
L e
Rx
σ̃j (s) ejϕ(s)ds R LR
L
e
jϕ(s)
+ ...
ds
ejϕ(s)ds
#
The generalized PNP Equation - Linearized, Including Sources
given ϕ̃, σ̃, and σ, solve for δ:
−
X
all
∂2
ε 2 (ϕ̃ + δ) =
∂x
je−j ϕ̃(x) [Aj (x) (1 − jδ (x)) + jBj (x) δ (L) + jCj (x) δ (R) + ...
j
Z
x
... +jDj (x)
δ (s) e
j ϕ̃(s)
x
x
δ (s) σ̃ (s) e
... +jFj (x)
j ϕ̃(s)
Z
δ (s) σ̃ (s) ej ϕ̃(s)ds
x
x
... +jKj (x)
R
ds + jGj (x)
L
Z
δ (s) ej ϕ̃(s)ds
ds + jEj (x)
L
Z
R
Z
δ (s) σ (s) ej ϕ̃(s)ds + jMj (x)
L
where Aj through Mj are highly nonlinear.
Z
x
R
δ (s) σ (s) ej ϕ̃(s)ds
Validity of the QSSA.
1st Set:
“Sharp” Initial Conditions
Flux Density Dynamics to Donnan Eq. - ”sharp” ICs:
Electro-Static Potential Dynamics to Donnan Eq. - ”sharp” ICs:
2nd Set:
“Steady-State” Initial Conditions
Flux Density Dynamics to Donnan Eq. - St.State ICs:
Electro-Static Potential Dynamics to Donnan Eq. - St.State ICs:
3rd Set:
“Far From Eqlb.” Initial Conditions
Flux Density Dynamics to Donnan Eq. - Far From Eq. ICs:
Electro-Static Potential Dyn. to Donnan Eq. - Far From Eq. ICs:
Summary for Approach to Donnan Equilibrium
• Boundary layer is established within a few ms.
• Quasi steady-state assumption holds well for the kinetic approach to Donnan
equilibrium. (accuracy!)
• An ODE system based on AN takes a few seconds to solve, whereas the full PDE
takes 32 hours to solve. (efficiency!)
• Thus, the implementation of the QSSA using AN yields an accurate and efficient
means of modeling electrodiffusion and, in particular, the kinetic approach to
Donnan equilibrium.
The Constant Field Approximation, CFA.
Deriving the Constant Field Approximation (CFA)
X
dϕ
εAc (0) = −vin
zicin
i ,
dx
i
The corresponding CFA uses

0,
for x < − m2

dϕ
vin P
m
in
(x) = − εA
i zi ci , for − 2 ≤ x ≤
c

dx
0,
for m2 < x
(92)
m
2
(93)
and thus



for x < − m2
P
m vin
m
in
−
x
+
ϕ (x) − ϕ (L) =
i zi ci , for − 2 ≤ x ≤
2 P
εAc

vin
m
in

−m εA
z
c
,
for
i
i
i
2 <x .
c
0 ,
m
2
(94)
Deriving the Constant Field Approximation (CFA)
With ϕ (x) piecewise linear, the steady-state flux-density, Ji can be simplified:
ci (R) eziϕ(R) − ci (L) eziϕ(L)
Ji = −Di
RR
L exp (zi ϕ (s)) ds
(95)
zi ∆ϕ
cout
− cin
i e
i
Ji = −Diziϕx (0) ·
ezi∆ϕ − 1
(96)
zi ∆ϕ
zi∆ϕ cout
− cin
i e
i
Ji = −Di
·
z
∆ϕ
δ
e i −1
(97)
Verify that the CFA of the QSSA
Holds at Far-From-Eqlb. Steady-States.
Potential Profiles by AN and CFA, No Species Trapped
Error in CFA Potential Profile, No Species Trapped
Closeness of Bulk Profile to Eqlb. Profile, No Species Trapped
Difference of Membrane Profile to Eqlb. Profile, No Species Trapped
Summary: CFA for End-of-Membrane Impermeability
• Bulk regions are almost equilibrated, even at far-from-equilibrium steady-states
(relative error of order 10−10).
• The membrane region is far from equilibrated at far-from-equilibrium steadystates (relative error of order 10−1).
• The steady-state potential profile and cross-membrane potential difference of the
QSSA are approximated reasonably well by the CFA (relative error within 5%).
THE END