TITLE Calcium ion accumulation with longitudinal and radial diffusion
COMMENT
PROCEDURE factors_cadifus() sets up the scale factors
needed to model radial diffusion.
These scale factors do not have to be recomputed
when diam or DFree is changed.
The amount of calcium in an annulus is ca[i]*diam^2*vol[i]
with ca[0] being the 2nd order correct concentration at the exact edge
and ca[NANN-1] being the concentration at the exact center.
Buffer concentration and rates are based on Yamada et al. 1989
model of bullfrog sympathetic ganglion cell.
ENDCOMMENT
NEURON {
SUFFIX cadp
USEION ca READ cao, cai, ica WRITE cai, ica
RANGE ica_pump
GLOBAL vol, TotalBuffer, TotalPump
RANGE cai0
THREADSAFE
}
DEFINE NANN 4
UNITS {
(molar) = (1/liter)
(mol) = (1)
(mM) = (millimolar)
(um) = (micron)
(mA) = (milliamp)
FARADAY = (faraday) (10000 coulomb)
PI = (pi) (1)
}
PARAMETER {
DCa = 0.6 (um2/ms)
: to change rate of buffering without disturbing equilibrium
: multiply the following two by the same factor
k1buf = 100 (/mM-ms)
k2buf = 0.1 (/ms)
TotalBuffer = 0.003 (mM)
cai0 = 50e-6 (mM) : Requires explicit use in INITIAL block
k1 = 1 (/mM-ms)
k2 = 0.005(/ms)
k3 = 1 (/ms)
k4 = 0.005 (/mM-ms)
: to eliminate pump, set Totalpump to 0 in hoc
TotalPump = 1e-11 (mol/cm2)
}
ASSIGNED {
diam (um)
ica (mA/cm2)
cai (mM)
vol[NANN] (1) : gets extra um2 when multiplied by diam^2
Kd (/mM)
B0 (mM)
cao (mM)
ica_pmp (mA/cm2)
parea (um)
}
CONSTANT{ volo = 1e10 (um2) }
STATE {
ca[NANN] (mM) <1e-6> : ca[0] is equivalent to cai
CaBuffer[NANN] (mM)
Buffer[NANN] (mM)
pump (mol/cm2)
pumpca (mol/cm2)
}
BREAKPOINT {
SOLVE state METHOD sparse
ica = ica_pmp
}
LOCAL factors_done
INITIAL {
MUTEXLOCK
if (factors_done == 0) {
factors_done = 1
factors()
}
MUTEXUNLOCK
cai = cai0
Kd = k1buf/k2buf
B0 = TotalBuffer/(1 + Kd*cai)
FROM i=0 TO NANN-1 {
ca[i] = cai
Buffer[i] = B0
CaBuffer[i] = TotalBuffer - B0
}
parea = PI*diam
pump = TotalPump/(1+(cai*k1/k2))
pumpca = TotalPump - pump
}
COMMENT
factors() sets up factors needed for radial diffusion
modeled by NANN concentric compartments.
The outermost shell is half as thick as the other shells
so the concentration is spatially second order correct
at the surface of the cell.
The radius of the cylindrical core
equals the thickness of the outermost shell.
The intervening NANN-2 shells each have thickness = r/(NANN-1)
(NANN must be >= 2).
ca[0] is at the edge of the cell,
ca[NANN-1] is at the center of the cell,
and ca[i] for 0 < i < NANN-1 is
midway through the thickness of each annulus.
ENDCOMMENT
LOCAL frat[NANN]
PROCEDURE factors() {
LOCAL r, dr2
r = 1/2 :starts at edge (half diam)
dr2 = r/(NANN-1)/2 :half thickness of annulus
vol[0] = 0
frat[0] = 2*r
FROM i=0 TO NANN-2 {
vol[i] = vol[i] + PI*(r-dr2/2)*2*dr2 :interior half
r = r - dr2
frat[i+1] = 2*PI*r/(2*dr2) :exterior edge of annulus
: divided by distance between centers
r = r - dr2
vol[i+1] = PI*(r+dr2/2)*2*dr2 :outer half of annulus
}
}
LOCAL dsq, dsqvol : can't define local variable in KINETIC block
: or use in COMPARTMENT
KINETIC state {
COMPARTMENT i, diam*diam*vol[i] {ca CaBuffer Buffer}
COMPARTMENT (1e10)*parea {pump pumpca}
COMPARTMENT volo {cao}
LONGITUDINAL_DIFFUSION i, DCa*diam*diam*vol[i] {ca}
:pump
~ ca[0] + pump <-> pumpca (k1*parea*(1e10), k2*parea*(1e10))
~ pumpca <-> pump + cao (k3*parea*(1e10), k4*parea*(1e10))
CONSERVE pump + pumpca = TotalPump *parea * (1e10)
ica_pmp = 2*FARADAY*(f_flux - b_flux)/parea
:all currents except pump
~ ca[0] << (-(ica - ica_pmp)*PI*diam/(2*FARADAY))
FROM i=0 TO NANN-2 {
~ ca[i] <-> ca[i+1] (DCa*frat[i+1], DCa*frat[i+1])
}
dsq = diam*diam
FROM i=0 TO NANN-1 {
dsqvol = dsq*vol[i]
~ ca[i] + Buffer[i] <-> CaBuffer[i] (k1buf*dsqvol, k2buf*dsqvol)
}
cai = ca[0]
}