Modeling Ionic Flow in the Retina Carl Gardner, Jeremiah ...gardner/ICIAM.pdf · 2. Model arrays of...
Transcript of Modeling Ionic Flow in the Retina Carl Gardner, Jeremiah ...gardner/ICIAM.pdf · 2. Model arrays of...
Modeling Ionic Flow in the Retina
Carl Gardner, Jeremiah Jones, Steve Baer, & Shaojie ChangArizona State University
http://webvision.med.utah.edu/
Schematic (Kamermans & Fahrenfort) of horizontal cell dendritecontacting cone pedicle (0.5 micron× 0.5 micron): simulate 400 nmmembranes× 40 nm gap & 20/40 nm openings at side of HC
Drift-Diffusion (PNP) Model
cone pedicle
horizontal cell
gap Ca
Cl
VCP+
Φ- Σi-
VHC+
Φ- Σi-
∂ni
∂t+∇· fi = 0, i = Ca2+, Na+, K+, Cl−, . . .
fi = ziµiniE − Di∇ni, zi =qi
qe, ji = qifi, j =
∑
i
ji
∇· (ǫ∇φ) = eN −∑
i
qini, E = −∇φ
parabolic/elliptic system of PDEs
A Model of the Membrane(similar to Mori-Jerome-Peskin)
minside
outside
+Σ
-Σ
Φ
Φ+
Φ-
Φ
ni
ni
Σi+
Σi-
Poisson-Boltzmann Equation
ni = nbi exp
{
−qiφ
kT
}
∇· (ǫ∇φ) = −∑
i
qinbi exp
{
−qiφ
kT
}
≈(
∑
i
q2i nbi
)
φ
kT
Debye lengthlD =√
ǫkT/(∑
i q2i nbi)
≈ 1 nm
For z ⊥ & near membraneφzz ≈ φ/l2D
φ ≈ φ±e−|z|/lD , ni ≈ n±bi
(
1− qiφ±
kTe−|z|/lD
)
Setσ+
i =∫∞
0 qi(
ni − n+bi
)
dz = qilD(
n+i − n+bi
)
Jump conditions for Poisson’s equation
[φ] ≡ φ+ − φ− = V =σ
Cm
[n · ∇φ] = 0
BCs for drift-diffusion equation (Mori-Jerome-Peskin), but we useσ±
i = qilD(
n±i − n±bi
)
∂σ+
i
∂t= qilD
∂n+i∂t
= −n̂ · j+i + jmi
∂σ−i
∂t= qilD
∂n−i∂t
= n̂ · j−i − jmi
σ ≡∑
i
σ+
i = −∑
i
σ−i
Drift-Diffusion Model with Membrane Boundary Conditions
∂ni
∂t+∇· (ziµiniE) = Di∇2ni, i = Ca2+, Na+, K+, Cl−
∇· (ǫ∇φ) = −∑
qini, E = −∇φ
BCs : n−i = n−bi +σ−
i
qilD, φ−
CP,HC = V+
CP,HC − σ
Cm
∂σ−i
∂t= n̂ · j−i − jmi, σ = −
∑
σ−i
jm,Ca =gCa (∆VCP − VCa)
1+ exp{(θ± −∆VCP) /λm}, jm,Cl = 0 (CP)
jm,Ca = 0, jm,Cl = gCl (∆VHC − VCl) (HC)
cone pedicle
horizontal cell
gap Ca
Cl
VCP+
Φ- Σi-
VHC+
Φ- Σi-
BCs at openings areambient: ni = nbi, n · ∇φ = 0
1. Apply 2D TRBDF2 drift-diffusion code to cone pedicle /horizontal cell problem with model of membrane
2. Investigate relative importance of electrical (ephaptic) [vs.chemical (GABA)] effects
TRBDF2 Numerical Method
dudt
= f (u, t), γ = 2−√
2
un+γ − γ∆tn2
f n+γ = un + γ∆tn2
f n (TR)
un+1 − 1− γ
2− γ∆tnf n+1 =
1γ(2− γ)
un+γ − (1− γ)2
γ(2− γ)un (BDF2)
Use Newton’s method iff (u) is nonlinear
TR
BDF2
n+1n+Γn
Known Biological Parameters
Parameter Value Descriptionnb,Ca 2 mM bath density of Ca2+
nb,Na 140 mM bath density of Na+
nb,K 2.5 mM bath density of K+
nb,Cl 146.5 mM bath density of Cl−
ǫ 80 dielectric coefficient of waterNs 20 number of spine heads per cone pedicleAm 0.1µm2 spine head areaCm 1 µF/cm2 membrane capacitance per areaVCl −55 mV reversal potential for Cl−
GCl 5 nS leak conductance for Cl in HC
Known Biological Parameters
Parameter Value DescriptionDCa 0.8 nm2/ns diffusivity of Ca2+
DNa 1.3 nm2/ns diffusivity of Na+
DK 2 nm2/ns diffusivity of K+
DCl 2 nm2/ns diffusivity of Cl−
µCa 32 nm2/(V ns) mobility of Ca2+
µNa 52 nm2/(V ns) mobility of Na+
µK 80 nm2/(V ns) mobility of K+
µCl 80 nm2/(V ns) mobility of Cl−
Fitting Parameters in Model for CP Transmembrane ICa
jm,Ca =gCa (∆VCP − VCa)
1+ exp{(θ± −∆VCP) /λ}
Parameter Value DescriptionGCa 1.2 nS Ca conductanceVCa 50 mV reversal potential for Ca2+
θ− −33 mV kinetic parameter, bg offθ+ −40 mV kinetic parameter, bg onλ 3.5 mV kinetic parameter
Note thatgCa,Cl = GCa,Cl/(NsAm) & that
ICa = Ns
∫
Am
jm,Ca da
Drift-diffusion simulations
Drift-diffusion simulations
Experimental IV curves (Kamermans & Fahrenfort)
−70 −60 −50 −40 −30 −20 −10 0 10−90
−80
−70
−60
−50
−40
−30
−20
−10
0
10
Membrane Potential (mV)
Cur
rent
(pA
)
bkgd offbkgd onbkgd off (exp.)bkgd on (exp.)
ICa vs.∆VCP shift turning on background illumination
−70 −60 −50 −40 −30 −20 −10 0 100
5
10
15
20
25
30
35
40
Membrane Potential (mV)
Cur
rent
Shi
ft (p
A)
d = 10/20d = 20/40d = 40/80
Ephaptic effect: Shift in ICa vs.∆VCP for varying opening widths
Current & Future Work
1. Effects of more realistic geometry?
& with Sharon Crook & Christian Ringhofer:
2. Model arrays of cone pedicles & horizontal cells—perhaps“homogenize” over small spatial scales
3. Multiscale modeling: “integrate out” shortest time scales indrift-diffusion model to obtain intermediate model, so we cantreat time-dependent illuminations of retina