: Calcium ion accumulation with endogenous buffers, DCM and pump

COMMENT

The basic code of Example 9.8 and Example 9.9 from NEURON book was adapted as:

1) Extended using parameters from Schmidt et al. 2003.
2) Pump rate was tuned according to data from Maeda et al. 1999
3) DCM was introduced and tuned to approximate the effect of radial diffusion

Reference: Anwar H, Hong S, De Schutter E (2010) Controlling Ca2+-activated K+ channels with models of Ca2+ buffering in Purkinje cell. Cerebellum*

*Article available as Open Access

PubMed link: http://www.ncbi.nlm.nih.gov/pubmed/20981513

Written by Haroon Anwar, Computational Neuroscience Unit, Okinawa Institute of Science and Technology, 2010.
Contact: Haroon Anwar (anwar@oist.jp)

ENDCOMMENT


NEURON {
  SUFFIX cdp5StCmod
  USEION ca READ cao, cai, ica WRITE cai
  USEION nrvc READ nrvci VALENCE 1
  RANGE ica_pmp
  RANGE Nannuli, Buffnull2, rf3, rf4, vrat
    RANGE CAM0, CAM1C, CAM2C, CAM1N2C, CAM1N, CAM2N, CAM2N1C, CAM1C1N, CAM4, icazz
  RANGE TotalPump

}


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

PARAMETER {
	Nannuli = 10.9495 (1)
	celsius (degC)
        
	cainull = 45e-6 (mM)
        mginull =.59    (mM)

:	values for a buffer compensating the diffusion

	Buffnull1 = 0	(mM)
	rf1 = 0.0134329	(/ms mM)
	rf2 = 0.0397469	(/ms)

	Buffnull2 = 60.9091	(mM)
	rf3 = 0.1435	(/ms mM)
	rf4 = 0.0014	(/ms)

:	values for benzothiazole coumarin (BTC)
	BTCnull = 0	(mM)
	b1 = 5.33	(/ms mM)
	b2 = 0.08	(/ms)

:	values for caged compound DMNPE-4
	DMNPEnull = 0	(mM)
	c1 = 5.63	(/ms mM)
	c2 = 0.107e-3	(/ms)

:       values for Calbindin (2 high and 2 low affinity binding sites)

        :CBnull=	.16             (mM)
        :nf1   =43.5           (/ms mM)
        :nf2   =3.58e-2        (/ms)
        :ns1   =5.5            (/ms mM)
        :ns2   =0.26e-2        (/ms)

:       values for Parvalbumin

        PVnull  = .08           (mM)
        m1    = 1.07e2        (/ms mM)
        m2    = 9.5e-4                (/ms)
        p1    = 0.8           (/ms mM)
        p2    = 2.5e-2                (/ms)

:	Calmodulin concentration
	CAM_start 	= 0.03		(mM) :Pepke 2010

: 	Calmodulin Kinetic parameters. The values are the mean between max and min.
	:C-lobe
	Kd1C = 		0.00965	(mM)						: Kd - Equilibrium binding of 1st Ca2+ to CaM C-terminus 
	K1Coff = 	0.04	(/ms)						: From 0C to 1C with X ions on N-lobe
	K1Con = 	5.4	(/mM ms)					: From 1C to 0C with X ions on N-lobe
	Kd2C = 		0.00105	(mM)						: Kd - Equilibrium binding of 2nd Ca2+ to CaM C-terminus
	K2Coff = 	0.00925	(/ms)						: From 1C to 2C with X ions on N-lobe
	K2Con = 	15	(/mM ms)					: From 2C to 1C with X ions on N-lobe

	:N-lobe
	Kd1N = 		0.0275	(uM)						: Kd - Equilibrium binding of 1st Ca2+ to CaM N-terminus 
	K1Noff = 	2.5	(/ms)						: From 0N to 1N with X ions on C-lobe
	K1Non = 	142.5	(/mM ms)					: From 1N to 0N with X ions on C-lobe
	Kd2N = 		0.00615	(mM)						: Kd - Equilibrium binding of 2nd Ca2+ to CaM N-terminus
	K2Noff = 	0.75	(/ms)						: From 1N to 2N with X ions on C-lobe
	K2Non = 	175	(/mM ms)					: From 2N to 1N with X ions on C-lobe        


  	kpmp1    = 3e-3       (/mM-ms)
  	kpmp2    = 1.75e-5   (/ms)
  	kpmp3    = 7.255e-5  (/ms)
	TotalPump = 1e-9	(mol/cm2)	
	nrvci (nA)
}

ASSIGNED {
	diam      (um)
	ica       (mA/cm2)
	ica_pmp   (mA/cm2)
	parea     (um)     : pump area per unit length
	parea2	  (um)
	cai       (mM)
	mgi	(mM)
	vrat	(1)	
        icazz (nA)
}

CONSTANT { cao = 2	(mM) }

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		(mM)    <1e-3>
	mg		(mM)	<1e-6>
	
	Buff1		(mM)	
	Buff1_ca	(mM)

	Buff2		(mM)
	Buff2_ca	(mM)

	BTC		(mM)
	BTC_ca		(mM)

	DMNPE		(mM)
	DMNPE_ca	(mM)	

        :CB		(mM)
        :CB_f_ca		(mM)
        :CB_ca_s		(mM)
        :CB_ca_ca	(mM)

        PV		(mM)
        PV_ca		(mM)
        PV_mg		(mM)

:State for the Calmodulin      

	CAM0		(mM)

	:C-lobe mainly
	CAM1C		(mM)
	CAM2C		(mM)
	CAM1N2C		(mM)
	
	:N-Lobe Mainly
	CAM1N		(mM)
	CAM2N		(mM)
	CAM2N1C		(mM)

	:One ion on C-lobe and one on N-lobe
	CAM1C1N		(mM)

	:CaM complete
	CAM4		(mM)	

	
	pump		(mol/cm2) <1e-15>
	pumpca		(mol/cm2) <1e-15>

}

BREAKPOINT {
	SOLVE state METHOD sparse
}

LOCAL factors_done

INITIAL {
		factors()

		ca = cainull
		mg = mginull
		
		Buff1 = ssBuff1()
		Buff1_ca = ssBuff1ca()

		Buff2 = ssBuff2()
		Buff2_ca = ssBuff2ca()

		BTC = ssBTC()
		BTC_ca = ssBTCca()		

		DMNPE = ssDMNPE()
		DMNPE_ca = ssDMNPEca()

		:CB = ssCB( kdf(), kds())   
	        :CB_f_ca = ssCBfast( kdf(), kds())
       	 	:CB_ca_s = ssCBslow( kdf(), kds())
        	:CB_ca_ca = ssCBca( kdf(), kds())

        	PV = ssPV( kdc(), kdm())
        	PV_ca = ssPVca(kdc(), kdm())
        	PV_mg = ssPVmg(kdc(), kdm())

	:Calmodulin
	CAM0	= CAM_start		
	CAM1C	= 0
	CAM2C	= 0
	CAM1N2C = 0
	CAM1N   = 0
	CAM2N	= 0
	CAM2N1C = 0
	CAM1C1N = 0
	CAM4	= 0
		
  	parea = PI*diam
	parea2 = PI*(diam-0.2)
	ica = 0
	ica_pmp = 0
:	ica_pmp_last = 0
	pump = TotalPump
	pumpca = 0
	
	cai = ca
}

PROCEDURE factors() {
        LOCAL r, dr2
        r = 1/2                : starts at edge (half diam)
        dr2 = r/(Nannuli-1)/2  : full thickness of outermost annulus,
        vrat = PI*(r-dr2/2)*2*dr2  : interior half
        r = r - dr2
}


LOCAL dsq, dsqvol  : can't define local variable in KINETIC block
                   :   or use in COMPARTMENT statement

KINETIC state {
  COMPARTMENT diam*diam*vrat {ca mg Buff1 Buff1_ca Buff2 Buff2_ca BTC BTC_ca DMNPE DMNPE_ca PV PV_ca PV_mg}
  COMPARTMENT (1e10)*parea {pump pumpca}


	:pump
	~ ca + pump <-> pumpca  (kpmp1*parea*(1e10), kpmp2*parea*(1e10))
	~ pumpca <-> pump   (kpmp3*parea*(1e10), 0)
  	CONSERVE pump + pumpca = TotalPump * parea * (1e10)
	
	ica_pmp = 2*FARADAY*(f_flux - b_flux)/parea	
	: all currents except pump
	: ica is Ca efflux
	~ ca << (-ica*PI*diam/(2*FARADAY))

	:RADIAL DIFFUSION OF ca, mg and mobile buffers

	dsq = diam*diam
		dsqvol = dsq*vrat
		~ ca + Buff1 <-> Buff1_ca (rf1*dsqvol, rf2*dsqvol)
		~ ca + Buff2 <-> Buff2_ca (rf3*dsqvol, rf4*dsqvol)
		~ ca + BTC <-> BTC_ca (b1*dsqvol, b2*dsqvol)
		~ ca + DMNPE <-> DMNPE_ca (c1*dsqvol, c2*dsqvol)
		:Calbindin	
		:~ ca + CB <-> CB_ca_s (nf1*dsqvol, nf2*dsqvol)
	       	:~ ca + CB <-> CB_f_ca (ns1*dsqvol, ns2*dsqvol)
        	:~ ca + CB_f_ca <-> CB_ca_ca (nf1*dsqvol, nf2*dsqvol)
        	:~ ca + CB_ca_s <-> CB_ca_ca (ns1*dsqvol, ns2*dsqvol)

		:Paravalbumin
        	~ ca + PV <-> PV_ca (m1*dsqvol, m2*dsqvol)
        	~ mg + PV <-> PV_mg (p1*dsqvol, p2*dsqvol)

		:Calmodulin
		  :C-lobe
		~ ca + CAM0 <-> CAM1C (K1Con*dsqvol, K1Coff*dsqvol)
		~ ca + CAM1C <-> CAM2C (K2Con*dsqvol, K2Coff*dsqvol)
		~ ca + CAM2C <-> CAM1N2C (K1Non*dsqvol, K1Noff*dsqvol)
		~ ca + CAM1N2C <-> CAM4 (K2Non*dsqvol, K2Noff*dsqvol) 

		  :N-lobe
		~ ca + CAM0 <-> CAM1N (K1Non*dsqvol, K1Noff*dsqvol)
		~ ca + CAM1N <-> CAM2N (K2Non*dsqvol, K2Noff*dsqvol) 
		~ ca + CAM2N <-> CAM2N1C (K1Con*dsqvol, K1Coff*dsqvol)
		~ ca + CAM2N1C <-> CAM4 (K2Con*dsqvol, K2Coff*dsqvol)

		  :Mixed C and N lobes
		~ ca + CAM1C <-> CAM1C1N (K1Non*dsqvol, K1Noff*dsqvol) 
		~ ca + CAM1N <-> CAM1C1N (K1Con*dsqvol, K1Coff*dsqvol)
		~ ca + CAM1C1N <-> CAM1N2C (K2Con*dsqvol, K2Coff*dsqvol)
		~ ca + CAM1C1N <-> CAM2N1C (K2Non*dsqvol, K2Noff*dsqvol) 


  	cai = ca
	mgi = mg
        icazz = nrvci
}

FUNCTION ssBuff1() (mM) {
	ssBuff1 = Buffnull1/(1+((rf1/rf2)*cainull))
}
FUNCTION ssBuff1ca() (mM) {
	ssBuff1ca = Buffnull1/(1+(rf2/(rf1*cainull)))
}
FUNCTION ssBuff2() (mM) {
        ssBuff2 = Buffnull2/(1+((rf3/rf4)*cainull))
}
FUNCTION ssBuff2ca() (mM) {
        ssBuff2ca = Buffnull2/(1+(rf4/(rf3*cainull)))
}

FUNCTION ssBTC() (mM) {
	ssBTC = BTCnull/(1+((b1/b2)*cainull))
}

FUNCTION ssBTCca() (mM) {
	ssBTCca = BTCnull/(1+(b2/(b1*cainull)))
}

FUNCTION ssDMNPE() (mM) {
	ssDMNPE = DMNPEnull/(1+((c1/c2)*cainull))
}

FUNCTION ssDMNPEca() (mM) {
	ssDMNPEca = DMNPEnull/(1+(c2/(c1*cainull)))
}

:FUNCTION ssCB( kdf(), kds()) (mM) {
:	ssCB = CBnull/(1+kdf()+kds()+(kdf()*kds()))
:}
:FUNCTION ssCBfast( kdf(), kds()) (mM) {
:	ssCBfast = (CBnull*kds())/(1+kdf()+kds()+(kdf()*kds()))
:}
:FUNCTION ssCBslow( kdf(), kds()) (mM) {
:	ssCBslow = (CBnull*kdf())/(1+kdf()+kds()+(kdf()*kds()))
:}
:FUNCTION ssCBca(kdf(), kds()) (mM) {
:	ssCBca = (CBnull*kdf()*kds())/(1+kdf()+kds()+(kdf()*kds()))
:}
:FUNCTION kdf() (1) {
:	kdf = (cainull*nf1)/nf2
:}
:FUNCTION kds() (1) {
:	kds = (cainull*ns1)/ns2
:}
FUNCTION kdc() (1) {
	kdc = (cainull*m1)/m2
}
FUNCTION kdm() (1) {
	kdm = (mginull*p1)/p2
}
FUNCTION ssPV( kdc(), kdm()) (mM) {
	ssPV = PVnull/(1+kdc()+kdm())
}
FUNCTION ssPVca( kdc(), kdm()) (mM) {
	ssPVca = (PVnull*kdc())/(1+kdc()+kdm())
}
FUNCTION ssPVmg( kdc(), kdm()) (mM) {
	ssPVmg = (PVnull*kdm())/(1+kdc()+kdm())
}