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A multiscale computational model of spatially resolved calcium cycling in cardiac myocytes: from detailed cleft dynamics to the whole cell concentration profiles.

Vierheller J, Neubert W, Falcke M, Gilbert SH, Chamakuri N - Front Physiol (2015)

Bottom Line: Mathematical modeling of excitation-contraction coupling (ECC) in ventricular cardiac myocytes is a multiscale problem, and it is therefore difficult to develop spatially detailed simulation tools.Our concept for a multiscale mathematical model of Ca(2+) -induced Ca(2+) release (CICR) and whole cardiomyocyte electrophysiology incorporates stochastic simulation of individual LC- and RyR-channels, spatially detailed concentration dynamics in dyadic clefts, rabbit membrane potential dynamics, and a system of partial differential equations for myoplasmic and lumenal free Ca(2+) and Ca(2+)-binding molecules in the bulk of the cell.We developed a novel computational approach to resolve the concentration gradients from dyadic space to cell level by using a quasistatic approximation within the dyad and finite element methods for integrating the partial differential equations.

View Article: PubMed Central - PubMed

Affiliation: Mathematical Cell Physiology, Max Delbrück Center for Molecular Medicine Berlin, Germany.

ABSTRACT
Mathematical modeling of excitation-contraction coupling (ECC) in ventricular cardiac myocytes is a multiscale problem, and it is therefore difficult to develop spatially detailed simulation tools. ECC involves gradients on the length scale of 100 nm in dyadic spaces and concentration profiles along the 100 μm of the whole cell, as well as the sub-millisecond time scale of local concentration changes and the change of lumenal Ca(2+) content within tens of seconds. Our concept for a multiscale mathematical model of Ca(2+) -induced Ca(2+) release (CICR) and whole cardiomyocyte electrophysiology incorporates stochastic simulation of individual LC- and RyR-channels, spatially detailed concentration dynamics in dyadic clefts, rabbit membrane potential dynamics, and a system of partial differential equations for myoplasmic and lumenal free Ca(2+) and Ca(2+)-binding molecules in the bulk of the cell. We developed a novel computational approach to resolve the concentration gradients from dyadic space to cell level by using a quasistatic approximation within the dyad and finite element methods for integrating the partial differential equations. We show whole cell Ca(2+)-concentration profiles using three previously published RyR-channel Markov schemes.

No MeSH data available.


Related in: MedlinePlus

State scheme for the RyR and LCC Markov Models. A(i) 4-state RyR-channel model of Stern et al. (1999), O denotes the open, I the inactivated, and R the resting state. The activation rate ko is a fourth order Hill function of dyadic Ca2+, the inactivation rate ki is a first order Hill function of Ca2+. The Ca2+ dependent rates are influenced by the Ca2+ concentration in the jSR. The model was originally developed by Stern et al. (1999) and was modified by Schendel et al. (2012). A(ii) 2-state RyR-channel model of Cannell et al. (2013) where O denotes the open and C the closed state and kopen and kclose are polynomial functions of [Ca2+]. A(iii) 2-state RyR-channel model of Walker et al. (2014) where O denotes the open and C the closed state, kopen is a polynomial function of both [Ca2+] and [Ca2+] jsr (i.e., kopen is influenced by the Ca2+concentration in the jSR) and kclose is a constant. (B) 7-state scheme for the LCC according to Mahajan et al. (2008a) with the states O open, I1Ca, I2Ca Ca2+ dependent inactivated states, C1, C2 closed states, I1Ba, I2Ba Ca2+ independent inactivated states (for details see Mahajan et al., 2008a).
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Figure 3: State scheme for the RyR and LCC Markov Models. A(i) 4-state RyR-channel model of Stern et al. (1999), O denotes the open, I the inactivated, and R the resting state. The activation rate ko is a fourth order Hill function of dyadic Ca2+, the inactivation rate ki is a first order Hill function of Ca2+. The Ca2+ dependent rates are influenced by the Ca2+ concentration in the jSR. The model was originally developed by Stern et al. (1999) and was modified by Schendel et al. (2012). A(ii) 2-state RyR-channel model of Cannell et al. (2013) where O denotes the open and C the closed state and kopen and kclose are polynomial functions of [Ca2+]. A(iii) 2-state RyR-channel model of Walker et al. (2014) where O denotes the open and C the closed state, kopen is a polynomial function of both [Ca2+] and [Ca2+] jsr (i.e., kopen is influenced by the Ca2+concentration in the jSR) and kclose is a constant. (B) 7-state scheme for the LCC according to Mahajan et al. (2008a) with the states O open, I1Ca, I2Ca Ca2+ dependent inactivated states, C1, C2 closed states, I1Ba, I2Ba Ca2+ independent inactivated states (for details see Mahajan et al., 2008a).

Mentions: The state dynamics of the LC- and RyR-channels are simulated with Markov models. For the RyRs we explored three models, the first is a model originally developed by Stern et al. (1999) in it's modified version by Schendel et al. (2012). It is a four state model [see Figure 3A(i)], including one open, one resting and two inactivated states. It features RyR inhibition in case of high cytosolic or low jSR Ca2+concentrations. For an in-depth description of this model see (Schendel et al., 2012). The second model is the two-state model [see Figure 3A(ii)] developed by Cannell et al. (2013), in which termination of CICR is mediated through the steep Ca2+dependence of the RyR closed time. With a small decline in jSR [Ca2+] the Ca2+flux via open RyRs declines, causing a decline in local dyadic [Ca2+], which in turn causes a decrease in the open probability of neighboring RyRs, a process known as induction decay. This RyR model does not rely on experimentally un-substantiated biophysical mechanisms for CICR termination, such as dyadic/cytoplasmic RyR Ca2+-dependent inactivation, or RyR-lumenal Ca2+-dependent inactivation. The third model is the two-state model [see Figure 3A(iii)] developed by Walker et al. (2014) (adapted from Williams et al., 2011) which incorporates modulation of the RyR-opening rate by junctional RyR-lumenal [Ca2+] (). This model has a fixed closing rate, and, in accordance with the experimental data of Cannell et al. (2013), there is only weak regulation of the RyR opening-rate when () is < 1 mM. It is of note that the opening rate of this last RyR-Model is theoretically unbound, however in our simulations opening rates larger than 0.7 ms−1 were not encountered.


A multiscale computational model of spatially resolved calcium cycling in cardiac myocytes: from detailed cleft dynamics to the whole cell concentration profiles.

Vierheller J, Neubert W, Falcke M, Gilbert SH, Chamakuri N - Front Physiol (2015)

State scheme for the RyR and LCC Markov Models. A(i) 4-state RyR-channel model of Stern et al. (1999), O denotes the open, I the inactivated, and R the resting state. The activation rate ko is a fourth order Hill function of dyadic Ca2+, the inactivation rate ki is a first order Hill function of Ca2+. The Ca2+ dependent rates are influenced by the Ca2+ concentration in the jSR. The model was originally developed by Stern et al. (1999) and was modified by Schendel et al. (2012). A(ii) 2-state RyR-channel model of Cannell et al. (2013) where O denotes the open and C the closed state and kopen and kclose are polynomial functions of [Ca2+]. A(iii) 2-state RyR-channel model of Walker et al. (2014) where O denotes the open and C the closed state, kopen is a polynomial function of both [Ca2+] and [Ca2+] jsr (i.e., kopen is influenced by the Ca2+concentration in the jSR) and kclose is a constant. (B) 7-state scheme for the LCC according to Mahajan et al. (2008a) with the states O open, I1Ca, I2Ca Ca2+ dependent inactivated states, C1, C2 closed states, I1Ba, I2Ba Ca2+ independent inactivated states (for details see Mahajan et al., 2008a).
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Related In: Results  -  Collection

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Figure 3: State scheme for the RyR and LCC Markov Models. A(i) 4-state RyR-channel model of Stern et al. (1999), O denotes the open, I the inactivated, and R the resting state. The activation rate ko is a fourth order Hill function of dyadic Ca2+, the inactivation rate ki is a first order Hill function of Ca2+. The Ca2+ dependent rates are influenced by the Ca2+ concentration in the jSR. The model was originally developed by Stern et al. (1999) and was modified by Schendel et al. (2012). A(ii) 2-state RyR-channel model of Cannell et al. (2013) where O denotes the open and C the closed state and kopen and kclose are polynomial functions of [Ca2+]. A(iii) 2-state RyR-channel model of Walker et al. (2014) where O denotes the open and C the closed state, kopen is a polynomial function of both [Ca2+] and [Ca2+] jsr (i.e., kopen is influenced by the Ca2+concentration in the jSR) and kclose is a constant. (B) 7-state scheme for the LCC according to Mahajan et al. (2008a) with the states O open, I1Ca, I2Ca Ca2+ dependent inactivated states, C1, C2 closed states, I1Ba, I2Ba Ca2+ independent inactivated states (for details see Mahajan et al., 2008a).
Mentions: The state dynamics of the LC- and RyR-channels are simulated with Markov models. For the RyRs we explored three models, the first is a model originally developed by Stern et al. (1999) in it's modified version by Schendel et al. (2012). It is a four state model [see Figure 3A(i)], including one open, one resting and two inactivated states. It features RyR inhibition in case of high cytosolic or low jSR Ca2+concentrations. For an in-depth description of this model see (Schendel et al., 2012). The second model is the two-state model [see Figure 3A(ii)] developed by Cannell et al. (2013), in which termination of CICR is mediated through the steep Ca2+dependence of the RyR closed time. With a small decline in jSR [Ca2+] the Ca2+flux via open RyRs declines, causing a decline in local dyadic [Ca2+], which in turn causes a decrease in the open probability of neighboring RyRs, a process known as induction decay. This RyR model does not rely on experimentally un-substantiated biophysical mechanisms for CICR termination, such as dyadic/cytoplasmic RyR Ca2+-dependent inactivation, or RyR-lumenal Ca2+-dependent inactivation. The third model is the two-state model [see Figure 3A(iii)] developed by Walker et al. (2014) (adapted from Williams et al., 2011) which incorporates modulation of the RyR-opening rate by junctional RyR-lumenal [Ca2+] (). This model has a fixed closing rate, and, in accordance with the experimental data of Cannell et al. (2013), there is only weak regulation of the RyR opening-rate when () is < 1 mM. It is of note that the opening rate of this last RyR-Model is theoretically unbound, however in our simulations opening rates larger than 0.7 ms−1 were not encountered.

Bottom Line: Mathematical modeling of excitation-contraction coupling (ECC) in ventricular cardiac myocytes is a multiscale problem, and it is therefore difficult to develop spatially detailed simulation tools.Our concept for a multiscale mathematical model of Ca(2+) -induced Ca(2+) release (CICR) and whole cardiomyocyte electrophysiology incorporates stochastic simulation of individual LC- and RyR-channels, spatially detailed concentration dynamics in dyadic clefts, rabbit membrane potential dynamics, and a system of partial differential equations for myoplasmic and lumenal free Ca(2+) and Ca(2+)-binding molecules in the bulk of the cell.We developed a novel computational approach to resolve the concentration gradients from dyadic space to cell level by using a quasistatic approximation within the dyad and finite element methods for integrating the partial differential equations.

View Article: PubMed Central - PubMed

Affiliation: Mathematical Cell Physiology, Max Delbrück Center for Molecular Medicine Berlin, Germany.

ABSTRACT
Mathematical modeling of excitation-contraction coupling (ECC) in ventricular cardiac myocytes is a multiscale problem, and it is therefore difficult to develop spatially detailed simulation tools. ECC involves gradients on the length scale of 100 nm in dyadic spaces and concentration profiles along the 100 μm of the whole cell, as well as the sub-millisecond time scale of local concentration changes and the change of lumenal Ca(2+) content within tens of seconds. Our concept for a multiscale mathematical model of Ca(2+) -induced Ca(2+) release (CICR) and whole cardiomyocyte electrophysiology incorporates stochastic simulation of individual LC- and RyR-channels, spatially detailed concentration dynamics in dyadic clefts, rabbit membrane potential dynamics, and a system of partial differential equations for myoplasmic and lumenal free Ca(2+) and Ca(2+)-binding molecules in the bulk of the cell. We developed a novel computational approach to resolve the concentration gradients from dyadic space to cell level by using a quasistatic approximation within the dyad and finite element methods for integrating the partial differential equations. We show whole cell Ca(2+)-concentration profiles using three previously published RyR-channel Markov schemes.

No MeSH data available.


Related in: MedlinePlus