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TITLE Calcium ion accumulation and diffusion with pump
: The internal coordinate system is set up in PROCEDURE coord_cadifus()
: and must be executed before computing the concentrations.
: The scale factors set up in this procedure do not have to be recomputed
: when diam or DFree are changed.
: The amount of calcium in an annulus is ca[i]*diam^2*vol[i] with
: ca[0] being the second order correct concentration at the exact edge
: and ca[NANN-1] being the concentration at the exact center
: Free internal calcium is in fast equilibrium with a non-saturable buffer.
: The ratio of bound to free is beta.
NEURON {
SUFFIX cadifpmp
USEION ca READ cao, ica, cai WRITE cai, ica
RANGE ica_pmp, last_ica_pmp
GLOBAL vol, pump0
}
DEFINE NANN 4
UNITS {
(mV) = (millivolt)
(molar) = (1/liter)
(mM) = (millimolar)
(um) = (micron)
(mA) = (milliamp)
(mol) = (1)
FARADAY = (faraday) (coulomb)
PI = (pi) (1)
R = (k-mole) (joule/degC)
}
PARAMETER {
celsius (degC)
DFree = .6 (um2/ms)
diam (um)
cao (mM)
beta = 50
k1 = 5e8 (/mM-s) :optional mm formulation
k2 = .25e6 (/s)
k3 = .5e3 (/s)
k4 = 5e0 (/mM-s)
pump0 = 3e-14 (mol/cm2) : set to 0 in hoc if this pump not wanted
}
ASSIGNED {
ica (mA/cm2)
cai (mM)
vol[NANN] (1) : gets extra cm2 when multiplied by diam^2
ica_pmp (mA/cm2)
last_ica_pmp (mA/cm2)
area1 (um2)
c1 (1+8 um5/ms)
c2 (1-10 um2/ms)
c3 (1-10 um2/ms)
c4 (1+8 um5/ms)
}
CONSTANT {
volo = 1 (liter)
}
STATE {
ca[NANN] (mM) : ca[0] is equivalent to cai
pump (mol/cm2)
pumpca (mol/cm2)
}
INITIAL {LOCAL total
parms()
FROM i=0 TO NANN-1 {
ca[i] = cai
}
pumpca = cai*pump*c1/c2
total = pumpca + pump
if (total > 1e-9) {
pump = pump*(pump/total)
pumpca = pumpca*(pump/total)
}
ica_pmp = 0
last_ica_pmp = 0
}
BREAKPOINT {
SOLVE state METHOD sparse
last_ica_pmp = ica_pmp
ica = ica_pmp
}
LOCAL frat[NANN] : gets extra cm when multiplied by diam
PROCEDURE coord() {
LOCAL r, dr2
: cylindrical coordinate system with constant annuli thickness to
: center of cell. Note however that the first annulus is half thickness
: so that the concentration is second order correct spatially at
: the membrane or exact edge of the cell.
: note ca[0] is at edge of cell
: ca[NANN-1] is at center of cell
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
}
}
KINETIC state {
COMPARTMENT i, (1+beta)*diam*diam*vol[i]*1(um) {ca}
COMPARTMENT (1e10)*area1 {pump pumpca}
COMPARTMENT volo*(1e15) {cao}
: all currents except pump
~ ca[0] << (-(ica-last_ica_pmp)*PI*diam*1(um)*(1e4)*frat[0]/(2*FARADAY))
:diffusion
FROM i=0 TO NANN-2 {
~ ca[i] <-> ca[i+1] (DFree*frat[i+1]*1(um), DFree*frat[i+1]*1(um))
}
:pump
~ ca[0] + pump <-> pumpca (c1, c2)
~ pumpca <-> pump + cao (c3, c4)
ica_pmp = (1e-4)*2*FARADAY*(f_flux - b_flux)/area1
cai = ca[0]
}
PROCEDURE parms() {
coord()
area1 = 2*PI*(diam/2) * 1(um)
c1 = (1e7)*area1 * k1
c2 = (1e7)*area1 * k2
c3 = (1e7)*area1 * k3
c4 = (1e7)*area1 * k4
}
FUNCTION ss() (mM) {
SOLVE state STEADYSTATE sparse
ss = cai
}
COMMENT
The combination of voltage independent current and calcium
accumulation is more difficult because care must be taken not to count
the pump current twice in the computation of the change ca[0]. Hence
the usage of last_ica_pmp to subtract the pump portion of the total
calcium current in ica so that its effect can be calculated implicitly
via the reaction "pumpca <-> pump + cao". This artifice makes the
pumping much more stable than the assumption of constant pump current
during the step. Otherwise, ca[0] is prone to become negative and that
crashes the simulation (especially the automatic computation of eca).
Calcium currents that are inward are generally safe to compute in
separate models.
ENDCOMMENT
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