17th International Conference
ENGINEERING MECHANICS 2011
Svratka, Czech Republic, 9 – 12 May 2011
A NEW QUANTITATIVE DESCRIPTION OF INTRACELLULAR Ca2+
DYNAMICS IN THE MODEL OF RAT VENTRICULAR MYOCYTE
M. Pásek*
, J. Šimurda**
Abstract: In this paper, a new description of intracellular Ca2+
dynamics in the model of rat ventricular
cardiomyocyte is presented. The principal modifications based on the recently published data comprise:
formulation of the function of peripheral dyads, incorporation of peripheral and tubular intracellular
subspaces, reformulation of inactivation properties of surface of tubular ICa and description of the
function of exogenous Ca2+
buffer in intracellular space. The modified model will be used to explore the
activity induced ion-concentration changes in rat transverse-axial tubular system in a more detail and to
investigate their effects on excitation – contraction coupling in ventricular cardiomyocytes.
Keywords: cardiac cell, intracellular Ca2+
dynamics, quantitative modelling
1. Introduction
In our previous work we developed a mathematical model of rat ventricular cell electrical activity
(Pásek et al., 2006) that firstly included a quantitative description of membrane transverse-axial
tubular system (TATS). The model was used to explore the extent of activity induced ion
concentration changes in rat TATS and their role in electromechanical activity of rat ventricular
myocytes. The experimental data that have been published in the recent years show, however, that
some cellular events related to intracellular Ca2+
dynamics are more complex than formulated in the
model. In this work, we describe a modified model of rat ventricular cardiomyocyte that includes a
novel description of intracellular Ca2+
handling respecting the recent findings. The principal
modifications of the model include: (i) partition of originally single dyadic space into two
compartments, one adjacent to surface membrane and another one adjacent to tubular membrane; (ii)
incorporation of peripheral and tubular intracellular subspaces; (iii) reformulation of calcium current
(ICa) inactivation including its differentiation between surface and tubular membrane; (iv) incorporation
of quantitative description of the function of exogenous Ca2+
buffers in intracellular space.
2. Modification of the model
2.1 Model structure
The structure of the modified model is based on the previous model (Pásek et al., 2006) and is
illustrated in Fig. 1. The presence of peripheral dyads (Brette et al., 2004) and ion gradients under the
membrane (Shannon et al., 2004) is taken into account by incorporation of the following new
compartments: dyadic space adjacent to surface membrane (surface dyadic space); junctional
compartment of sarcoplasmic reticulum adjacent to surface dyadic space (JSRs); subsarcolemmal
spaces adjacent to surface and tubular membranes (surface subsarcolemmal space and tubular
subsarcolemmal space). The volumes of all intracellular compartments are specified in Tab. 1.
Readjustment of parameters related to cellular membrane was performed according to the results
published in Pásek et al., 2008; the fractional area of tubular membrane was lowered to 49% and
the specific capacitances of tubular and surface membrane were set to 1.275 µF/cm2
and to
*
Doc. Ing. Michal Pásek, Ph.D.: Institute of Thermomechanics, Czech Academy of Science - branch Brno; Technická 2; 616
69 Brno and Department of Physiology, Medical Faculty of Masaryk University, Kamenice 5; 62500 Brno; Czech Republic
Czech Republic; e-mail: pasek.avcr@centrum.cz
**
Doc. RNDr. Ing. Jiří Šimurda, CSc.: Department of Physiology, Medical Faculty of Masaryk University, Kamenice 5;
62500; Brno; Czech Republic; e-mail: simurda@med.muni.cz
0.714 µF/cm2
, respectively. This adjustment is consistent with commonly used value of total
membrane specific capacitance of 1 µF/cm2
.
Fig. 1: Schematic diagram of the modified rat ventricular cell model. The description of electrical
activity of surface (s, sd) and tubular (t, td) membrane comprises formulations of the following ion
currents: fast sodium current (INa), calcium currents through L-type channels (ICa), transient outward
potassium current (Ito), steady-state outward potassium current (IKss), hyperpolarization-activated
potassium current (If); inward rectifying potassium current (IK1), background currents (IKb, INab, ICab),
sodium-calcium exchange current (INaCa), sodium-potassium pump current (INaK) and calcium pump
current (IpCa). The intracellular space contains the cytosolic space, surface and tubular
subsarcolemmal subspaces, surface and tubular dyadic spaces, the network and junctional
compartments of sarcoplasmic reticulum (NSR, JSRs, JSRt), the endogenous Ca2+
buffers (calmodulin
(Bcm), troponin (Bhtrpn, Bltrpn), calsequestrin (Bcs)) and the exogenous Ca2+
buffer (e.g. BAPTA or EGTA
(Bext)). Btats denotes the non-specific Ca2+
buffer associated with luminal part of tubular membrane.
The small filled rectangles in JSR membrane represent ryanodine receptors. The small bi-directional
arrows denote Ca2+
diffusion. Ionic diffusion between the tubular and the extracellular space is
_____________________________represented by the dashed arrow.
Tab. 1: Volumes of intracellular compartments.
symbol specification value [pl]
Vc cytosolic space 11.137
Vss surface subsarcolemmal subspace 0.1440
Vst tubular subsarcolemmal subspace 0.0775
Vds surface dyadic space 0.0001671
Vdt tubular dyadic space 0.0006682
VJSRs surface junctional compartment of sarcoplasmic reticulum 0.0078
VJSRt tubular junctional compartment of sarcoplasmic reticulum 0.0312
VNSR network compartment of sarcoplasmic reticulum 0.3508
Bcm
extracellular space
INa,s IKto,s IKss,s If,s IK1,s IKb,s INab,s ICab,s
[Na+
]st [Ca2+
]st [K+
]st
tubular
subsarcolemmal space
Bltrpn
Bltrpn
Bcm
cytosolic space
[Na+
]c
[Ca2+
]c
[K+
]c
Bcs
τdtst
τstc
pipette
τpss
[Bext]p
[Ca2+
]p
[Bext]st
[Bext]c
Jup
INaCa,s
INaK,s
IpCa,s
[Na+
]e
[Ca2+
]e
[K+
]e
Bcm,t
Bcs,t
[Bext]dt
ICa,td INaCa,td INa,t IKto,t IKss,t If,t IK1,t IKb,t INab,t ICab,t INaCa,t INaK,t IpCa,t
[Ca2+
]dt
Bcm
[Na+
]t [Ca2+
]t [K+
]t
τssc
surface
subsarcolemmal space 1
τssst
[Na+
]ss [Ca2+
]ss [K+
]ss [Bext]ss
ICa,sd
NSR
sarcoplasmic
reticulum
Bcm,s
[Bext]ds
[Ca2+
]ds
INaCa,sd
JSRs
Bcs,s
JSRt
surface
dyadic
space
tubular dyadic space
τdsss
Btats
tubular space
2.2 Membrane transport system
Voltage dependent inactivation (VDI) and Ca2+
dependent inactivation (CDI) of ICa was newly
formulated on the basis of experimental results of Brette et al. (2004). All parameters of CDI are
regarded as dependent on the level of calmodulin saturation with Ca2+
(Bcm,Casat, Shannon et al., 2004).
The steady state levels and time constants of VDI (ssVDI, τVDI) and CDI (ssCDI, τCDI,s, τCDI,s) are
described by the following equations:
ssVDI,x =1/(1+exp((Vm,x+26.7)/5.4)),
τVDI,x =1.15⋅(0.041⋅exp(-((Vm,x+47)/12) 2
)+0.08/(1+exp(-(Vm,x+55)/5))+0.015/(1+exp((Vm,x+75)/25))),
ssCDI,x =1/(1+0.244⋅( Bcm,Casat,x
4
+0.3184
)/ Bcm,Casat,x
4
),
τCDI,s =1/(43.827⋅Bcm,Casat,s
4
/( Bcm,Casat,s
4
+0.9764
)+25.006),
τCDI,t =1/(1160⋅Bcm,Casat,t
8
/( Bcm,Casat,t
8
+1.148
)+16.66).
While the formulations of ssVDI, τVDI and ssCDI in the description of ICa,s and ICa,t are identical (the
suffix x in the equations stands for s (surface) or t (tubular)), the τCDI of these two currents is
formulated differently (see τCDI,s and τCDI,t). This takes into account the observed different modulation
of surface and tubular ICa by Ca2+
released from SR (Brette et al., 2004).
The conductivity of Ito was increased by 20% for action potentials of the model to exhibit
physiological duration.
The fractions of membrane currents in the tubular membrane were set to meet the results
published in Pásek et al. (2008) except for the values of fICa,t and fIpCa,t that were set to 80% (Brette et
al., 2004) and 95% (Chase & Orchard, 2011), respectively.
2.3 Intracellular Ca2+
- handling
The formulation of the function of ryanodine receptors in JSR (RyR) was adopted from Shannon et al.
(2004). The constants ks and koCa were increased from 25 ms-1
to 250 ms-1
and from 10 mM-2
ms-1
to
50 mM-2
ms-1
, respectively, for Ca2+
transients in the dyadic space to reach magnitude close to 100 µM
at the level of free [Ca2+
] in NSR of 0.5 mM (Shannon et al., 2004). Description of SR Ca2+
pump (Jup)
was modified to be consistent with data of Shannon and Bers (1997). The constants used are:
Vmax = 286 µM/s, Kmf = 168 nM, Kmr = 1.176 mM, hf = 1.2 and hr = 1.287. The dissociation constant
(Kd) of calsequestrin in JSR was decreased from original value 0.8 µM to 0.65 µM (Shannon et al.,
2004). Ca2+
buffering by calmodulin was described by differential equations with kon =100000 mM-1
s-1
and kof = 238 s-1
.
The model was supplemented by the description of exogenous Ca2+
buffer diffusion (BAPTA or
EGTA) among the pipette, subsarcolemmal spaces, dyadic spaces and cytosol. All time constants
controlling the rate of exogenous Ca2+
buffer and Ca2+
diffusion between individual cellular
compartments are specified in Tab. 2 (the time constants of intracellular Na+
and K+
diffusion were set
to the same values as in the case of Ca2+
diffusion).
2.4 Ion diffusion between tubular and extracellular space
The time constants of ion exchange between the TATS lumen and the extracellular solution (τCa,TATS,
τK,TATS, τNa,TATS) were readjusted for the model to better reproduce the changes in ICa and resting voltage
following rapid decrease or increase of external ion concentrations at 37 °
C (Yao et al., 1997). To
reconstruct the biphasic time course of ICa-decrease after rapid exposure of myocytes to Ca2+
-free
external solution, the model was supplemented by a formulation of Ca2+
buffer in TATS (Btats). Using
the same pulse and solution change protocols the reconstructions led to the following values of buffer
parameters and time constants: kon= 2.2 s-1
mM-1
, koff= 2.398 s-1
, Btats = 2.6 mM, τCa,TATS = 155 ms and
τNa,TATS = τK,TATS = 150 ms. Finally, the time constants were corrected for the lower temperature of the
model cell (22 °
C, Q10 = 1.3) and their final values were: τCa,TATS = τNa,TATS = τK,TATS = 220 ms.
Tab. 2: Time constants related to intracellular transport of Ca2+
and Ca2+
-buffers.
symbol specification value.[s]
τpss,buffer-free controls buffer diffusion from the pipette into the surface subspace 1.36
τpss,buffer-Ca controls buffer-Ca2+
diffusion from the pipette into the cytosolic space 1.36
τpss,Ca controls Ca2+
diffusion from the pipette into the cytosolic space 1.36
τdsss,buffer-free controls buffer diffusion from the surface dyadic space into surface subspace 0.34E-3
τdsss,buffer-Ca controls buffer-Ca2+
diffusion from surface dyadic space into surface subspace 0.34E-3
τdsss,Ca controls Ca2+
diffusion from the surface dyadic space into the surface subspace 0.34E-3
τdtst,buffer-free controls buffer diffusion from the tubular dyadic space into tubular subspace 0.34E-3
τdtst,buffer-Ca controls buffer-Ca2+
diffusion from the tubular dyadic space into tubular subspace 0.34E-3
τdtst,Ca controls Ca2+
diffusion from the tubular dyadic space into tubular subspace 0.34E-3
τssc,buffer-free controls buffer diffusion from the surface subspace into cytosolic space 0.004
τssc,buffer-Ca controls buffer-Ca2+
diffusion from the surface subspace into cytosolic space 0.004
τssc,Ca controls Ca2+
diffusion from the surface subspace into cytosolic space 0.004
τstc,buffer-free controls buffer diffusion from the tubular subspace into cytosolic space 0.001
τstc,buffer-Ca controls buffer-Ca2+
diffusion from the tubular subspace into cytosolic space 0.001
τstc,Ca controls Ca2+
diffusion from the tubular subspace into cytosolic space 0.001
τssst,buffer-free controls buffer diffusion from the surface subspace into tubular subspace 0.1
τssst,buffer-Ca controls buffer-Ca2+
diffusion from the surface subspace into tubular subspace 0.1
τssst,Ca controls Ca2+
diffusion from the surface subspace into tubular subspace 0.1
3. Conclusions
The present novel description of intracellular Ca2+
dynamics in the model of rat ventricular myocytes
is an important step toward understanding of specific details of excitation-contraction coupling in
cardiac ventricular myocytes. The principal modifications based on the recently published data
comprise: formulation of the function of peripheral dyads, incorporation of peripheral and tubular
intracellular subspaces, reformulation of inactivation properties of surface of tubular ICa and
description of the function of exogenous Ca2+
buffer in the intracellular space. The modified model
will be used to further investigate the effects of activity induced ion-concentration changes in TATS
on electrical and mechanical activity of ventricular cardiomyocytes.
Acknowledgement
This study was supported by the project AV0Z 20760514 from the Institute of Thermomechanics of
Czech Academy of Sciences and by the project MSM 0021622402 from the Ministry of Education,
Youth and Sports of the Czech Republic.
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