A Quantitative Model of Gastric Smooth Muscle Cellular Activation

Abstract

A physiologically realistic quantitative description of the electrical behavior of a gastric smooth muscle (SM) cell is presented. The model describes the response of a SM cell when activated by an electrical stimulus coming from the network of interstitial cells of Cajal (ICC) and is mediated by the activation of different ion channels species in the plasma membrane. The conductances (predominantly Ca²⁺ and K⁺) that are believed to substantially contribute to the membrane potential fluctuations during slow wave activity have been included in the model. A phenomenological description of intracellular Ca²⁺ dynamics has also been included because of its primary importance in regulating a number of cellular processes. In terms of shape, duration, and amplitude, the resulting simulated smooth muscle depolarizations (SMDs) are in good agreement with experimentally recordings from mammalian gastric SM in control and altered conditions. This model has also been designed to be suitable for incorporation into large scale multicellular simulations.

Appendix

The main equations of the model are:

$$ C_{\rm m} \frac{dV_{\rm m}} {dt}=-(I_{\rm ion} +I_{\rm stim} ) $$

$$ I_{\rm ion} =I_{\rm CaL} +I_{\rm LVA} +I_{\rm Kr} +I_{\rm Ka} +I_{\rm BK} +I_{\rm Kb} +I_{\hbox{Na}} +I_{\rm NSCC} $$

$$ I_{\rm stim} = \left\{ {\begin{array}{ll} {G_{\rm couple} \ast \Delta V_{\rm ICC}} & {t < t_{\rm ICCpeak}} \\ {G_{\rm couple} \ast \Delta V_{\rm ICC} \ast \frac{1}{1+e^{\frac{t-8000}{1000}}}} & {t_{\rm ICCpeak} < t < t_{\rm ICCplateau}} \\ {I_{\rm stim} (t=t_{\rm ICCplateau} )\ast \frac{1.3}{1+e^\frac{t-8000}{150}}} & {t_{\rm ICCplateau} < t < t_{\rm ICC}} \\ \end{array}} \right. $$

$$ \frac{d[{\rm Ca}]_i} {dt}=-\frac{I_{\hbox{caL}} +I_{\hbox{caT}}} {2\ast F\ast V_c} -I_{\rm CaExt} $$

• Steady state parameters

Activation: $\frac{1}{1+e^{-\frac{V_{0.5} +V}{k}}}$ Inactivation: $\frac{1}{1+e^{\frac{V_{0.5} +V}{k}}}$

Activation			Inactivation
Variable name	V 0.5 (mV)	k	Variable name	V 0.5 (mV)	k
d	17	4.3	f	43	8.9
d LVA	27.5	10.9	f LVA	15.8	7
x r1	27	5	x r2*	58	10
x a1	26.5	7.9	x a2**	65	6.2
d Na	47	4.8	f Na	78	3
m NSCC	25	20

1. * For this variable the equation was used in the form: $\frac{0.8}{1+e^{\frac{V_{0.5} +V}{k}}}+0.2$
2. ** For this variable the equation was used in the form: $\frac{0.9}{1+e^{\frac{V_{0.5} +V}{k}}}+0.1$

• Time constants

Variable name	Time constants (ms)
d	0.47
f	86
d LVA	3
f LVA	$7.58\ast e^{0.00817\ast V_{\rm m}}$
x r1	80
x r2	$-707.0+1481\ast e^\frac{V_{\rm m} +36}{95}$
x a1	$31.8+175\ast e^{-0.5\ast (\frac{V_{\rm m} +44.4}{22.3})^{2}}$
x a2	90
d Na	−0.017* V m+0.44
f Na	−0.25* V m+5.5
m NSCC	$\frac{150}{1+e^{-\frac{V_{\rm m} +66}{26}}}$

Model parameters

Parameter name	Description	Value	Units
R	Ideal gas constant	8.314	J/(mol*K)
T	Temperature	310	K
F	Faraday constant	96486.7	C/mol
C m	Cell membrane capacitance	77	pF
A m	Cell surface	0.000041	cm2
V c	Total cytoplasmic volume	3500	μm3
Cao	Extracellular Ca2+ concentration	2.5	mM
Nao	Extracellular sodium concentration	137	mM
Na i	Intracellular sodium concentration	10	mM
K o	Extracellular potassium concentration	5.9	mM
K i	Intracellular potassium concentration	164	mM
Ach	Acetylcholine concentration in absence of muscarinic stimulation	10	nM
G CaL	Maximal conductance for L-type Ca2+ channels	65	nS
G LVA	Maximal conductance for low-voltage activated Ca2+ channels	0.18	nS
G Kr	Maximal conductance for delayed rectifier potassium channels	35	nS
G Ka	Maximal conductance for A-type potassium channels	9	nS
G BK	Maximal conductance for Ca2+-activated potassium channels	45.7	nS
G Kb	Maximal background potassium conductance	0.014	nS
G Na	Maximal conductance for sodium channels	3	nS
G NSCC	Maximal conductance for non-selective cationic channels	50	nS
G couple	Coupling conductance between ICC and SM	1.3	nS
h	Steepness for Ca2+ activation for BK channels	2	–
K bk	Steepness for voltage activation for BK channels	−17	mV
Caset	Ca2+ set point for BK channels	0.001	mM
h Ca	Half concentration for the fCa variable	201.4	nM
s Ca	Slope factor for the steady state fCa variable	13.1	nM
E NSCC	Reversal potential for NSCC channels	−28	mV
K m-NSCC	Half activation value for Ach activation of NSCC channels	10	μM
n Ach	Hill coefficient for Ach activation of NSCC channels	1	–
K Ca-NSCC	Half activation value for Ca2+ facilitation of NSCC channels	200	nM
n Ca	Hill coefficient for Ca2+ facilitation of NSCC channels	−4	–
Q10-Ca	Q10 value for Ca2+ channels	2.1	–
Q10-K	Q10 value for potassium channels	1.5	–
Q10-Na	Q10 value for sodium channels	2.45	–
ΔV ICC	Membrane potential fluctuation in an ICC	59	mV
t ICCpeak	Peak time of the slow wave in an ICC	98	ms
t ICCplateau	Plateau time of the slow wave in the ICC	7582	ms
t ICC	Total time of the slow wave in the ICC	10,000	ms