Cooperative Effects Due to Calcium Binding by Troponin and Their Consequences for Contraction and Relaxation of Cardiac Muscle Under Various Conditions of Mechanical Loading
Geoffrey
Nunns
Bioengineering Institute, University of Auckland
Model Status
This model runs in PCEnv and COR, but does not recreate the published results. The units are continuous throughout. The equations for F_PE and F_SE are wrong, but the F_CE equation and all related equations (the bulk of the model) look good.
Model Structure
Abstract: A mathematical model for the regulation of mechanical activity in cardiac muscle has been developed based on a three-element rheological model of this muscle. The contractile element has been modeled taking into account the results of extensive mechanical tests that involved the recording of length-force and force-velocity relations and muscle responses to short-time deformations during various phases of the contraction-relaxation cycle. The best agreement between the experimental and the mathematical modeling results was obtained when a postulate stating two types of cooperativity to regulate the calcium binding by troponin was introduced into the model. Cooperativity of the first type is due to the dependence of the affinity of troponin C for Ca2+ on the concentration of myosin crossbridges in the vicinity of a given troponin C. Cooperativity of the second type assumes an increase in the affinity of a given troponin C for Ca2+ when the latter is bound by molecules neighboring troponin.
model diagram
Schematic diagram of the Izakov et al model - a classic rheological scheme of the heart muscle including
contractile element CE and two passive elastic elements PE (parallel
one) and SE (series one). The effects of calcium and troponin (Tn) in facilitating actin-myosin binding is also highlighted.
The complete original paper reference is cited below:
Cooperative Effects Due to Calcium Binding by Troponin and Their Consequences for Contraction and Relaxation of Cardiac Muscle Under Various Conditions of Mechanical Loading, Valery Ya. Izakov, Leonid B. Katsnelson, Felix A. Blyakhman, Vladamir S. Markhasin, Tatyana F. Shklyar, 1991,
Circulation Research
, 69, 1171-1184. (PDF versions of the article are available on the American Heart Association website.) PubMed ID: 1934350
$\mathrm{V\_1}=0.1\mathrm{V\_max}\mathrm{Ca}=\begin{cases}\mathrm{Ca\_m}(1-e^{-\mathrm{a\_c}\mathrm{time}^{2}})^{2} & \text{if $\mathrm{time}\le \mathrm{t\_d}$}\\ \mathrm{Ca\_m}((1-e^{-\mathrm{a\_c}\mathrm{time}^{2}})e^{-\mathrm{b\_c}(\mathrm{time}-\mathrm{t\_d})^{2}})^{2} & \text{otherwise}\end{cases}\mathrm{C\_2}=\mathrm{C\_20}\mathrm{pi\_n}\mathrm{phi\_A\_1}\mathrm{pi\_n}=\begin{cases}\mathrm{pi\_min} & \text{if $(0.75\le n)\land (n\le 1)$}\\ \mathrm{pi\_min}^{(2n-0.5)} & \text{if $(0.25\le n)\land (n< 0.75)$}\\ 1 & \text{otherwise}\end{cases}\mathrm{phi\_A\_1}=e^{-\mathrm{q\_k}\mathrm{A\_1}}S=0.5\mathrm{l\_1}+\mathrm{S\_0}A=\frac{\mathrm{A\_1}S}{1}\frac{d \mathrm{A\_1}}{d \mathrm{time}}=\mathrm{C\_1}\mathrm{Ca}(\mathrm{TnC}-\mathrm{A\_1})-\mathrm{C\_20}\mathrm{pi\_n}\mathrm{phi\_A\_1}\mathrm{A\_1}n=\mathrm{n\_1}\mathrm{n\_2}\mathrm{n\_1}=\begin{cases}0 & \text{if $\mathrm{g\_1}\mathrm{l\_1}+\mathrm{g\_2}< 0$}\\ \mathrm{g\_1}\mathrm{l\_1}+\mathrm{g\_2} & \text{if $(0\le \mathrm{g\_1}\mathrm{l\_1}+\mathrm{g\_2})\land (\mathrm{g\_1}\mathrm{l\_1}+\mathrm{g\_2}\le 1)$}\\ 1 & \text{otherwise}\end{cases}\frac{d \mathrm{n\_2}}{d \mathrm{time}}=\mathrm{q\_V}(\mathrm{m\_0}\mathrm{G\_V}-\mathrm{n\_2})\mathrm{q\_V}=\begin{cases}\mathrm{q\_1}-\frac{\mathrm{q\_2}V}{\mathrm{V\_max}} & \text{if $V\le 0$}\\ \mathrm{q\_3} & \text{otherwise}\end{cases}\mathrm{G\_V}=1+\frac{0.6V}{\mathrm{V\_max}}V=-\mathrm{dl\_1\_dt}\mathrm{F\_V}=\frac{a(1+\frac{V}{\mathrm{V\_max}})}{a-\frac{V}{\mathrm{V\_max}}}\mathrm{p\_V}=\frac{\mathrm{F\_V}}{\mathrm{G\_V}}\mathrm{P\_CE}=\mathrm{lambda}S\mathrm{A\_1}n\mathrm{p\_V}\mathrm{P\_PE}=\mathrm{beta\_2}(e^{\mathrm{alpha\_2}\mathrm{l\_2}}-1)\mathrm{P\_SE}=\mathrm{beta\_1}(e^{\mathrm{alpha\_1}(\mathrm{l\_2}-\mathrm{l\_1})}-1)$
$\mathrm{dl\_1\_dt}=0\mathrm{l\_1}=\begin{cases}0 & \text{if $(200\le \mathrm{time})\land (\mathrm{time}\le 201)$}\\ 0 & \text{otherwise}\end{cases}\mathrm{l\_2}=\mathrm{l\_1}+1.87$
Geoff Nunns
Geoffrey
Nunns
Rogan
This model runs in PCEnv and COR, but does not recreate the published results. The units are continuous throughout. The equations for F_PE and F_SE are wrong, but the F_CE equation and all related equations (the bulk of the model) look good. Further curation is required. The current version of the model simulates an isometric contraction.
This model runs in PCEnv and COR, but does not recreate the published results. The units are continuous throughout. The equations for F_PE and F_SE are wrong, but the F_CE equation and all related equations (the bulk of the model) look good. Further curation is required. The current version of the model simulates an isometric contraction.
Auckland Bioengineering Institute
CellML Team
Added components to increase modularity.
Geoffrey
Nunns
Rogan
Leonid
Katsnelson
B.
2008-07-25T10:38:05+12:00
keyword
calcium dynamics
myofilament mechanics
excitation-contraction coupling
Cooperative Effects Due to Calcium Binding by Troponin and Their Consequences for Contraction and Relaxation of Cardiac Muscle Under Various Conditions of Mechanical Loading
69
1171
1184
Felix
Blyakhman
A.
Vladamir
Markhasin
S.
gnunns1@jhem.jhu.edu
1934350
Geoff Nunns
2008-07-16T00:00:00+00:00
Valery
Izakov
Ya.
1991-00-00 00:00
Circulation Research
Tatyana
Shkylar
F.