cdp.mod

COMMENT
Calcium ion accumulation with radial and longitudinal diffusion, pump, 
and SERCA.

Diffusion geometry based on Ca accumulation models from chapter 9 
of The NEURON Book.

Mechanistic details of calcium pump and SERCA as described by Fink et al. 2000.

alpha = relative abundance of SERCA.

Current implementation assumes that ip3i is uniform across all compartments, 
i.e. that radial diffusion of IP3 is very fast compared to the diffusion of 
Ca and the mechanisms that drive Ca to change with time.
Indeed, simulations reveal this to be the case--IP3 concentration 
remains nearly identical across section diameters.
There are slight differences in the soma during the fast rising phase 
of the IP3 transient, but these resolve quickly.

Consequently, coupling between the shells of the ip3cum mechanism 
and the SERCA channels in the shells of this mechanism 
is a complexity that can be omitted--all shells of this mechanism
can use the concentration of IP3 in the outermost shell of the 
ip3cum mechanism, and discoverable by any mechanism that has a 
USEION ip3 READ ip3i VALENCE 1
statement in its NEURON block, and declares
ip3i (mM)
in its ASSIGNED block.

-------------
SERCA channel
-------------

jchnl = alpha * jmax * (1-(ca/caer)) * ( (ip3/(ip3+Kip3)) * (ca/(ca+Kact)) * h)^3
note:  jchnl is release from SER to cytoplasm
jmax = 3500 uM/s
caer = 400 uM
Kip3 = 0.8 uM
Kact = 0.3 uM

h' = kon * (Kinh - (ca + Kinh)*h)
kon = 2.7 /uM-s
Kinh = 0.2 uM

Recasting h in terms of kinetic scheme--
From RHS of ODE for h'
hinf = Kinh/(ca+Kinh) = alpha/(alpha+beta)
tauh = 1/(kon*(ca+Kinh)) = 1/(alpha+beta)
So alpha = kon*Kinh and beta = kon*ca

----------
SERCA pump
----------

jpump = alpha * vmax*ca^2 / (ca^2 + Kp^2)
note:  jpump is uptake from cytoplasm into SER

vmax = 3.75 uM/s
Kp = 0.27 uM

----------
SERCA leak
----------

jleak = alpha * L*(1 - (Ca/caer))
note:  jleak is leak from SER to cytoplasm

L = 0.1 uM/s nominally,
but adjusted so that
jchnl + jpump + jleak = 0
when
ca = 0.05 uM and h = Kinh/(ca + Kinh)

ENDCOMMENT

NEURON {
  SUFFIX cdp
  USEION ca READ cao, cai, ica WRITE cai, ica
  USEION ip3 READ ip3i VALENCE 1
  RANGE ica_pmp
  RANGE alpha : relative abundance of SERCA
  GLOBAL vrat, TBufs, TBufm
    : vrat must be GLOBAL--see INITIAL block
    : however TBufs and TBufm may be RANGE
}

DEFINE Nannuli 4

UNITS {
  (mol)   = (1)
  (molar) = (1/liter)
  (uM)    = (micromolar)
  (mM)    = (millimolar)
  (um)    = (micron)
  (mA)    = (milliamp)
  FARADAY = (faraday)  (10000 coulomb)
  PI      = (pi)       (1)
}

PARAMETER {
  cai0 = 50e-6 (mM)

  DCa   = 0.22 (um2/ms) : Fink et al. 2000

: Bufs--endogenous, stationary buffer
  TBufs = 0.450 (mM) : total Bufs
  : just make kfs fast, and calculate krs as kfs*KDs
  kfs = 1000 (/mM-ms) : try these for now
  KDs = 10 (uM)

: Bufm--fura2, for bradykinin experiments
  TBufm = 0.075 (mM) : total Bufm
  : just make kfm fast, and calculate krm as kfm*KDm
  kfm = 1000 (/mM-ms) : try these for now
  KDm = 0.24 (uM)
  DBufm = 0.050 (um2/ms)

: Bufm--calcium green, for uncaging experiments
:  TBufm = 0.075 (mM) : total Bufm
  : just make kfm fast, and calculate krm as kfm*KDm
:  kfm = 1000 (/mM-ms) : try these for now
:  KDm = 0.26 (/ms)
:  DBufm = 0.0184 (um2/ms)

  : to eliminate ca pump, set gamma to 0 in hoc
  cath = 0.2e-3 (mM) : threshold for ca pump activity
  gamma = 8 (um/s) : ca pump flux density

  : SERCA params
  alpha = 1 (1) : relative abundance of SERCA mechanism as per Fig. 3

  : SERCA pump
: jpump = alpha * vmax*ca^2 / (ca^2 + Kp^2)
: jpump is uptake from cytoplasm into SER
  vmax = 3.75e-6 (mM/ms)
  Kp = 0.27e-3 (mM)

  : SERCA channel
: jchnl is release from SER to cytoplasm
  jmax = 3.5e-3 (mM/ms)
  caer = 0.400 (mM)
  Kip3 = 0.8e-3 (mM)
  Kact = 0.3e-3 (mM)
  kon = 2.7 (/mM-ms)
  Kinh = 0.2e-3 (mM)

  : SERCA leak -- no fixed parameter other than caer
  : does have an adjustable parameter L
}

ASSIGNED {
  diam      (um)
  ica       (mA/cm2)
  ica_pmp   (mA/cm2)
  ica_pmp_last   (mA/cm2)
  parea     (um)     : pump area per unit length

  sump      (mM)

  cai       (mM)
  cao       (mM)
  vrat[Nannuli]  (1) : dimensionless
                     : numeric value of vrat[i] equals the volume 
                     : of annulus i of a 1um diameter cylinder
                     : multiply by diam^2 to get volume per um length

  bufs_0 (mM)
  bufm_0 (mM)

  ip3i (mM)

  L[Nannuli] (mM/ms) : 0.1e-6 mM/ms nominally, but adjusted so that
    : jchnl + jpump + jleak = 0  when  ca = 0.05 uM and h = Kinh/(ca + Kinh)
}

CONSTANT { volo = 1e10 (um2) }

STATE {
  : ca[0] is equivalent to cai
  : ca[] are very small, so specify absolute tolerance
  : let it be ~1.5 - 2 orders of magnitude smaller than baseline level
  ca[Nannuli]       (mM) <1e-7>
  bufs[Nannuli]    (mM) <1e-3>
  cabufs[Nannuli]  (mM) <1e-7>
  bufm[Nannuli]    (mM) <1e-4>
  cabufm[Nannuli]  (mM) <1e-8>
  hc[Nannuli]
  ho[Nannuli]
}

BREAKPOINT {
  SOLVE state METHOD sparse
  ica_pmp_last = ica_pmp
  ica = ica_pmp
}

LOCAL factors_done, jx

INITIAL {
   if (factors_done == 0) {  : flag becomes 1 in the first segment
      factors_done = 1       :   all subsequent segments will have
      factors()              :   vrat = 0 unless vrat is GLOBAL
   }

  cai = cai0

  bufs_0 = KDs*TBufs/(KDs + (1000)*cai0)
  bufm_0 = KDm*TBufm/(KDm + (1000)*cai0)

  FROM i=0 TO Nannuli-1 {
    ca[i] = cai
    bufs[i] = bufs_0
    cabufs[i] = TBufs - bufs_0
    bufm[i] = bufm_0
    cabufm[i] = TBufm - bufm_0
  }

  sump = cath
  parea = PI*diam

: reconsider and revise initialization comments
  ica=0
  ica_pmp = 0
  ica_pmp_last = 0
: If there is a voltage-gated calcium current, 
: this is almost certainly the wrong initialization. 
: In such a case, first do an initialization run, then use SaveState
: On subsequent runs, restore the initial condition from the saved states.

  FROM i=0 TO Nannuli-1 {
    ho[i] = Kinh/(ca[i]+Kinh)
    hc[i] = 1-ho[i]

  : jx = jp + jc
  : choose L so that jl = -jx
  : jl = L*(1 - (ca[i]/caer))
  : jp = (-vmax*ca[i]^2 / (ca[i]^2 + Kp^2))
  : jc = jmax*(1-(ca[i]/caer)) * ( (ip3i/(ip3i+Kip3)) * (ca[i]/(ca[i]+Kact)) * ho[i] )^3
    jx = (-vmax*ca[i]^2 / (ca[i]^2 + Kp^2))
    jx = jx + jmax*(1-(ca[i]/caer)) * ( (ip3i/(ip3i+Kip3)) * (ca[i]/(ca[i]+Kact)) * ho[i] )^3
    L[i] = -jx/(1 - (ca[i]/caer))
  }
}

LOCAL frat[Nannuli]  : scales the rate constants for model geometry

PROCEDURE factors() {
  LOCAL r, dr2
  r = 1/2                : starts at edge (half diam)
  dr2 = r/(Nannuli-1)/2  : full thickness of outermost annulus,
                         : half thickness of all other annuli
  vrat[0] = 0
  frat[0] = 2*r
  FROM i=0 TO Nannuli-2 {
    vrat[i] = vrat[i] + PI*(r-dr2/2)*2*dr2  : interior half
    r = r - dr2
    frat[i+1] = 2*PI*r/(2*dr2)  : outer radius of annulus
                                : div by distance between centers
    r = r - dr2
    vrat[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 statement

KINETIC state {
  COMPARTMENT i, diam*diam*vrat[i] {ca bufs cabufs bufm cabufm sump}
  COMPARTMENT volo {cao}
  LONGITUDINAL_DIFFUSION i, DCa*diam*diam*vrat[i] {ca}
  LONGITUDINAL_DIFFUSION i, DBufm*diam*diam*vrat[i] {bufm cabufm}

  : cell membrane ca pump
  ~ ca[0] <-> sump  ((0.001)*parea*gamma*u(ca[0]/(1 (mM)), cath/(1 (mM))), (0.001)*parea*gamma*u(ca[0]/(1 (mM)), cath/(1 (mM))))
  ica_pmp = 2*FARADAY*(f_flux - b_flux)/parea

  : all currents except cell membrane ca pump
  ~ ca[0] << (-(ica - ica_pmp_last)*PI*diam/(2*FARADAY))  : ica is Ca efflux
  : radial diffusion
  FROM i=0 TO Nannuli-2 {
    ~ ca[i] <-> ca[i+1]  (DCa*frat[i+1], DCa*frat[i+1])
    ~ bufm[i] <-> bufm[i+1]  (DBufm*frat[i+1], DBufm*frat[i+1])
  }
  : buffering
  dsq = diam*diam
  FROM i=0 TO Nannuli-1 {
    dsqvol = dsq*vrat[i]
    ~ ca[i] + bufs[i] <-> cabufs[i]  (kfs*dsqvol, (0.001)*KDs*kfs*dsqvol)
    ~ ca[i] + bufm[i] <-> cabufm[i]  (kfm*dsqvol, (0.001)*KDm*kfm*dsqvol)
  }
  : SERCA pump, channel, and leak
  FROM i=0 TO Nannuli-1 {
    dsqvol = dsq*vrat[i]
    : pump
    ~ ca[i] << (-dsqvol*alpha*vmax*ca[i]^2 / (ca[i]^2 + Kp^2))
    : channel
    ~ hc[i] <-> ho[i]  (kon*Kinh, kon*ca[i])
    ~ ca[i] << ( dsqvol*alpha*jmax*(1-(ca[i]/caer)) * ( (ip3i/(ip3i+Kip3)) * (ca[i]/(ca[i]+Kact)) * ho[i] )^3 )
    : leak
    ~ ca[i] << (dsqvol*alpha*L[i]*(1 - (ca[i]/caer)))
  }
  cai = ca[0]
}

FUNCTION u(x, th) {
  if (x>th) {
    u = 1
  } else {
    u = 0
  }
}