A mathematical model of fatigue in skeletal muscle force contraction

Abstract

The ability for muscle to repeatedly generate force is limited by fatigue. The cellular mechanisms behind muscle fatigue are complex and potentially include breakdown at many points along the excitation–contraction pathway. In this paper we construct a mathematical model of the skeletal muscle excitation–contraction pathway based on the cellular biochemical events that link excitation to contraction. The model includes descriptions of membrane voltage, calcium cycling and crossbridge dynamics and was parameterised and validated using the response characteristics of mouse skeletal muscle to a range of electrical stimuli. This model was used to uncover the complexities of skeletal muscle fatigue. We also parameterised our model to describe force kinetics in fast and slow twitch fibre types, which have a number of biochemical and biophysical differences. How these differences interact to generate different force/fatigue responses in fast- and slow- twitch fibres is not well understood and we used our modelling approach to bring new insights to this relationship.

Appendix 1 Model Equations

Sarcolemma and t-tubule action potential generation

Table 1 Model parameter values and definitions

Two compartment model

$$\begin{aligned} & {\text{I}}_{\text{T}} = ({\text{v}}_{\text{s}} - {\text{v}}_{\text{t}} )/{\text{R}}_{\text{a}} {\text{10}}^{\text{3}} \,\mu {\text{A}}/{\text{cm}}^{\text{2}} \,{\text{(access}}\,{\text{current)}} \\ & \frac{{{\text{dv}}_{\text{s}} }}{{{\text{dt}}}} = - \left( {{\text{I}}_{\text{s}}^{{\text{ionic}}} + {\text{I}}_{\text{T}} } \right)/{\text{C}}_{\text{m}} \,{\text{mV}}/{\text{ms}}\,{\text{(sarcolemma}}\,{\text{membrane}}\,{\text{voltage)}} \\ & \frac{{{\text{dv}}_{\text{t}} }}{{{\text{dt}}}} = - \left( {{\text{I}}_{\text{t}}^{{\text{ionic}}} - {\text{I}}_{\text{T}} {\text{/}}\gamma } \right)/{\text{C}}_{\text{m}} \,{\text{mV/ms}}\,{\text{(t - tubule}}\,{\text{membrane}}\,{\text{voltage)}} \\ & \left[ {\frac{{{\text{dh}}_{\text{s}} }}{{{\text{dt}}}},\frac{{{\text{dm}}_{\text{s}} }}{{{\text{dt}}}},\frac{{{\text{dn}}_{\text{s}} }}{{{\text{dt}}}},\frac{{{\text{dh}}_{\text{s}}^{\text{K}} }}{{{\text{dt}}}},\frac{{{\text{dS}}_{\text{s}} }}{{{\text{dt}}}},{\text{I}}^{{\text{C1}}} ,{\text{I}}^{{\text{IR}}} ,{\text{I}}^{{\text{DR}}} ,{\text{I}}^{{\text{Na}}} ,{\text{I}}^{{\text{NaK}}} } \right] = {\text{Z}}_{\text{s}} \left( {{\text{v}}_{\text{s}} ,{\text{h}}_{\text{s}} ,{\text{m}}_{\text{s}} ,{\text{n}}_{\text{s}} ,{\text{h}}_{\text{s}}^{\text{K}} ,{\text{S}}_{\text{s}} ,{\text{K}}_{\text{i}} ,{\text{K}}_{\text{e}} ,{\text{Na}}_{\text{i}} ,{\text{Na}}_{\text{e}} } \right) \\ & {\text{I}}_{\text{s}}^{{\text{ionic}}} = {\text{I}}^{{\text{C1}}} + {\text{I}}^{{\text{IR}}} + {\text{I}}^{{\text{DR}}} + {\text{I}}^{{\text{Na}}} + {\text{I}}^{{\text{NaK}}} - {\text{I}}_{{\text{app}}} {\text{(t}};{\text{t}}_{{\text{app}}} {\text{)}}\,\mu {\text{A}}/{\text{cm}}^{\text{2}} \,{\text{(sarcolemma}}\,{\text{current)}} \\ & \left[ {\frac{{{\text{dh}}_{\text{t}} }}{{{\text{dt}}}},\frac{{{\text{dm}}_{\text{t}} }}{{{\text{dt}}}},\frac{{{\text{dn}}_{\text{t}} }}{{{\text{dt}}}},\frac{{{\text{dh}}_{\text{t}}^{\text{K}} }}{{{\text{dt}}}},\frac{{{\text{dS}}_{\text{t}} }}{{{\text{dt}}}},{\text{I}}_{\text{t}}^{{\text{Cl}}} ,{\text{I}}_{\text{t}}^{{\text{IR}}} ,{\text{I}}_{\text{t}}^{{\text{DR}}} ,{\text{I}}_{\text{t}}^{{\text{Na}}} ,{\text{I}}_{\text{t}}^{{\text{NaK}}} } \right] = {\text{Z}}_{\text{t}} \left( {{\text{v}}_{\text{t}} ,{\text{h}}_{\text{t}} ,{\text{m}}_{\text{t}} ,{\text{n}}_{\text{t}} ,{\text{h}}_{\text{t}}^{\text{K}} ,{\text{S}}_{\text{t}} ,{\text{K}}_{\text{i}} ,{\text{K}}_{\text{t}} ,{\text{Na}}_{\text{i}} ,{\text{Na}}_{\text{t}} } \right) \\ & {\text{I}}_{\text{t}}^{{\text{ionic}}} = {\text{I}}_{\text{t}}^{{\text{Cl}}} + {\text{I}}_{\text{t}}^{{\text{IR}}} + {\text{I}}_{\text{t}}^{{\text{DR}}} + {\text{I}}_{\text{t}}^{{\text{Na}}} + {\text{I}}_{\text{t}}^{{\text{NaK}}} \mu {\text{A}}/{\text{cm}}^{\text{2}} \,({\text{t - tubule}}\,{\text{current}}) \\ & \frac{{{\text{dK}}_{\text{i}} }}{{{\text{dt}}}} = - {\text{f}}_{\text{T}} \left( {{\text{I}}_{\text{t}}^{{\text{IR}}} + {\text{I}}_{\text{t}}^{{\text{DR}}} - {\text{2I}}_{\text{t}}^{{\text{NaK}}} + {\text{I}}_{{\text{rest}}}^{\text{K}} } \right)/({\text{1000F}}\xi ) - \left( {{\text{I}}^{{\text{IR}}} + {\text{I}}^{{\text{DR}}} - {\text{2I}}^{{\text{NaK}}} + {\text{I}}_{{\text{rest}}}^{\text{K}} } \right)/({\text{1000F}}\xi _2 )\,{\text{mM}}/{\text{ms}}\,{\text{(intracellular}}\,{\text{[K}}^ + {\text{])}} \\ & \frac{{{\text{dK}}_{\text{t}} }}{{{\text{dt}}}} = \left( {{\text{I}}_{\text{t}}^{{\text{IR}}} + {\text{I}}_{\text{t}}^{{\text{DR}}} - {\text{2I}}_{\text{t}}^{{\text{NaK}}} + {\text{I}}_{{\text{rest}}}^{\text{K}} } \right)/({\text{1000F}}\xi ) - ({\text{K}}_{\text{t}} - {\text{K}}_{\text{e}} )/\tau _{\text{K}} \,{\text{mM}}/{\text{ms}}\,({\text{t - tubule}}\,{\text{[K}}^ + ]) \\ & \frac{{{\text{dK}}_{\text{e}} }}{{{\text{dt}}}} = \left( {{\text{I}}^{{\text{IR}}} {\text{ + I}}^{{\text{DR}}} - {\text{2I}}^{{\text{NaK}}} + {\text{I}}_{{\text{rest}}}^{\text{K}} } \right)/({\text{1000F}}\xi _3 ) + ({\text{K}}_{\text{t}} - {\text{K}}_{\text{e}} )/\tau _{{\text{K}}_{\text{2}} } \,{\text{mM}}/{\text{ms}}\,({\text{interstitial}}\,{\text{[K}}^ + {\text{]}}) \\ & {\text{dNa}}_{\text{i}} /{\text{dt}} = - {\text{f}}_{\text{T}} \left( {{\text{I}}_{\text{t}}^{{\text{Na}}} + {\text{3I}}_{\text{t}}^{{\text{NaK}}} + {\text{I}}_{{\text{rest}}}^{{\text{Na}}} } \right)/({\text{1000F}}\xi ) - \left( {{\text{I}}^{{\text{Na}}} {\text{ + 3I}}^{{\text{NaK}}} + {\text{I}}_{{\text{rest}}}^{{\text{Na}}} } \right)/({\text{1000F}}\xi _2 )\,{\text{mM/ms}}\,({\text{intracellular}}\,{\text{[Na}}^ + {\text{]}}) \\ & {\text{dNa}}_{\text{t}} /{\text{dt}} = \left( {{\text{I}}_{\text{t}}^{{\text{Na}}} {\text{ + 3I}}_{\text{t}}^{{\text{NaK}}} + {\text{I}}_{{\text{rest}}}^{{\text{Na}}} } \right)/({\text{1000F}}\xi ) - ({\text{Na}}_{\text{t}} - {\text{Na}}_{\text{e}} )/\tau _{{\text{Na}}} \,{\text{mM}}/{\text{ms}}\,({\text{t - tubule}}\,{\text{[Na}}^ + ]) \\ & {\text{dNa}}_{\text{e}} /{\text{dt}} = \left( {{\text{I}}^{{\text{Na}}} {\text{ + 3I}}^{{\text{NaK}}} + {\text{I}}_{{\text{rest}}}^{{\text{Na}}} } \right)/({\text{1000F}}\xi _3 ) + ({\text{Na}}_{\text{t}} - {\text{Na}}_{\text{e}} )/\tau _{{\text{Na}}_{\text{2}} } \,{\text{mM}}/{\text{ms}}\,({\text{interstitial}}\,{\text{[Na}}^ + {\text{]}}) \\ \end{aligned} $$

(1)

where the function Z_s is defined by

$$ \begin{aligned}&\left[\frac{{{\hbox{dh}}}}{{{\hbox{dt}}}}{\hbox{,}}\frac{{{\hbox{dm}}}}{{{\hbox{dt}}}}{\hbox{,}}\frac{{{\hbox{dn}}}}{{{\hbox{dt}}}}{\hbox{,}}\frac{{{\hbox{dh}}^{{\rm {K}}} }}{{{\hbox{dt}}}}{\hbox{,}}\frac{{{\hbox{dS}}}}{{{\hbox{dt}}}}{\hbox{,I}}^{{{\rm {Cl}}}} {\hbox{,I}}^{{{\rm {IR}}}} {\hbox{,I}}^{{{\rm {DR}}}} {\hbox{,I}}^{{{\rm {Na}}}} {\hbox{,I}}^{{{\rm {NaK}}}}\right] ={\hbox{Z}}^{{}}_{{\rm {s}}} {\hbox{(V,h,m,n,h}}^{{\rm {K}}} {\hbox{,S,K}}_{{\rm {i}}} {\hbox{,K}}_{{\rm {o}}} {\hbox{,Na}}_{{\rm {i}}} {\hbox{,Na}}_{{\rm {o}}} {\hbox{)}}\\ &\alpha _{{\rm {h}}}=\bar{\alpha}_{\rm h} \hbox{exp}(-(\hbox{V}-\hbox{V} _{\rm h})/\hbox{K}_{\alpha_{\rm h}}) \hbox{ ms}^{-1}\\ &\beta_{\rm h}=\bar{\beta}_{h}{/(1 + \exp(- (\hbox{V} - \hbox{V}}_{{\rm {h}}} {\hbox{)/K}}_{{\beta_{{\rm {h}}}}}{\hbox{)) ms}}^{{{\rm {-1}}}}\\ &\alpha _{{\rm {m}}}=\bar{\alpha}_{{\rm {m}}}{(\hbox{V} - \hbox{V}}_{{\rm {m}}} {)/(1 - \exp(- (\hbox{V} - {V}}_{{\rm {m}}} {\hbox{)/K}}_{{\alpha _{{\rm {m}}} }} {\hbox{)) ms}}^{{{\rm {-1}}}}\\ &\beta _{{\rm {m}}}=\bar{\beta}_{m} {\exp(- (\hbox{V} - \hbox{V}}_{{\rm {m}}} {\hbox{)/K}}_{{\beta _{{\rm {m}}} }} {\hbox{) ms}}^{{{\rm {-1}}}}\\ &\alpha _{{\rm {n}}} =\bar{\alpha}_{{\rm {n}}} {\hbox{(V} - \hbox{V}}_{{\rm {n}}} {)/(1 - \exp(- (\hbox{V} - {V}}_{{\rm {n}}} {\hbox{)/K}}_{{\alpha _{{\rm {n}}} }} {\hbox{)) ms}}^{{{\rm {-1}}}}\\ &\beta _{{\rm {n}}} =\bar{\beta}_{n} {\exp(- (\hbox{V} - \hbox{V}}_{{\rm {n}}} {\hbox{)/K}}_{{\beta _{{\rm {n}}} }} {\hbox{) ms}}^{{{\rm {-1}}}}\\ &{\hbox{h}}^{{\rm {K}}}_{\infty}{=1/(1 + \exp((\hbox{V} - \hbox{V}}_{{{\rm {h}}^{\infty }_{{\rm {K}}} }} {\hbox{)/A}}_{{{\rm {h}}^{\infty }_{{\rm {K}}} }} {\hbox{)) }}\\ &{\hbox{S}}^{\infty } { = 1/(1 + \exp((\hbox{V} - \hbox{V}}_{{{\rm {S}}^{\infty } }} {\hbox{)/A}}_{{{\rm {S}}^{\infty } }} {\hbox{)) }}\\ &\tau _{{{\rm {h}}^{{\rm {K}}} }} {\hbox{ = 1000}} \times {\exp(- (\hbox{V} + 40)/25}{\hbox{.75) ms}}\\ &\tau _{S} {\hbox{ = 8571/(0}}{\hbox{.2 + 5}}{\hbox{.65((V + V}}_{\tau } {\hbox{)/100)}}^{{\rm {2}}} {\hbox{) ms}}\\ &{\hbox{a} = 1/(1 + \exp((\hbox{V} - \hbox{V}}_{{\rm {a}}} {\hbox{)/A}}_{{\rm {a}}} {\hbox{))}}\\ \end{aligned} $$

(2)

$$\begin{aligned}&{\hbox{E}}_{\rm K}{\hbox{ = RT/(+ F)log(K}}_{{\rm {o}}}{\hbox{/K}}_{{\rm {i}}} {\hbox{) mV (K}}^{ + } {\hbox{ Nernst potential)}}\\ &{\hbox{E}}_{{{\rm {Cl}}}} {\hbox{ = E}}_{{\rm {K}}}{\hbox{(Cl}}^{{-}} {\hbox{ Nernst potential)}}\\&{\hbox{E}}_{{{\rm {Na}}}} {\hbox{ = RT/(+ F)log(Na}}_{{\rm {o}}}{\hbox{/Na}}_{{\rm {i}}} {\hbox{) mV (Na}}^{ + } {\hbox{ Nernst potential)}}\\ &{\hbox{Cl}}_{{\rm {i}}} = {\hbox{(128}} +{\hbox{5}} \times {\hbox{5}}{\hbox{.7)/(5}} + {{\exp((}} -{\hbox{FE}}_{{\rm {K}}} {\hbox{)/(RT)))}}\quad{\hbox{(Intracellular [Cl}}^{-} {\hbox{])}}\\ &{\hbox{Cl}}_{{\rm {o}}} = {\hbox{(128}}+ {\hbox{5}} \times {\hbox{5}}{.7) - \hbox{5Cl}}_{{\rm {i}}}\quad{\hbox{(Extracellular [Cl}}^{-} {\hbox{])}}\\ &{\hbox{J}}_{{{\rm{Cl}}}} = {\hbox{V(Cl}}_{{\rm {i}}} - {\hbox{Cl}}_{{\rm {o}}}{\exp\hbox{((FV)/(RT))})/(1 - \exp\hbox{((FV)/(RT))})}\\ &{\hbox{J}}_{{\rm{K}}} = {\hbox{V(K}}_{{\rm {i}}} - {\hbox{K}}_{{\rm {o}}}{\exp(} - {\hbox{(FV)/(RT)}))/(1 - \exp(} -{\hbox{(FV)/(RT)))}}\\ &{\hbox{J}}_{{{\rm {Na}}}} ={\hbox{V(Na}}_{{\rm {i}}} - {\hbox{Na}}_{{\rm {o}}} {\exp(} -{\hbox{(FV)/(RT)}))/(1 - \exp(} - {\hbox{(FV)/(RT)))}}\\&{\hbox{g}}_{{{\rm {Cl}}}} =\bar{\hbox{g}}_{{{\rm {Cl}}}} {\hbox{a}}^{4}\,{\hbox{mS/cm}}^{{\rm {2}}}\\ &{\hbox{K}}_{{\rm {R}}} {\hbox{ =K}}_{{\rm {o}}} {\exp(- }\delta {\hbox{E}}_{{\rm {K}}}{\hbox{F/(RT))\,mM}}\\ &\bar{\hbox{g}}_{{{\rm {IR}}}} {\hbox{ = G}}_{{\rm{K}}} {\hbox{K}}^{{\rm {2}}}_{{\rm {R}}} {\hbox{/(K}}_{{\rm {K}}}{\hbox{ + K}}^{{\rm {2}}}_{{\rm {R}}} {\hbox{)\,mS/cm}}^{{\rm {2}}}\\&{\hbox{y}=1 - (1 + (\hbox{K}}_{{\rm {S}}} {\hbox{/(S}}^{{\rm {2}}}_{{\rm{i}}} {\exp(2(1 - }\delta {\hbox{)VF/(RT))))}} \times{\hbox{(1 + K}}^{{\rm {2}}}_{{\rm {R}}} {\hbox{/K}}_{{\rm {K}}}{\hbox{))}}^{{{\rm {-1}}}}\\ &{\hbox{g}}_{{{\rm{IR}}}}=\bar{\hbox{g}}_{{{\rm {IR}}}} {\hbox{y\,mS/cm}}^{{\rm {2}}}\\&{\hbox{g}}_{{{\rm {DR}}}}=\bar{\hbox{g}}_{{\rm {K}}} {\hbox{n}}^{4}{\hbox{h}}^{{\rm {K}}} {\hbox{\,mS/cm}}^{{\rm {2}}}\\&{\hbox{g}}_{{{\rm {Na}}}}=\bar{\hbox{g}}_{{{\rm {Na}}}} {\hbox{m}}^{{\rm{3}}} {\hbox{hS\,mS/cm}}^{{\rm {2}}}\\ &{\hbox{I}}_{{{\rm {Cl}}}}=\hbox{g}_{{{\rm {Cl}}}} {\hbox{(J}}_{{{\rm{Cl}}}}/45)\,\hbox{ }\mu\hbox{A/cm}^{\rm 2}\,{\hbox{(Cl}}^{{\rm {-}}}\,{\hbox{current)}}\\ &{\hbox{I}}_{{{\rm {IR}}}}=\hbox{g}_{{{\rm{IR}}}} {\hbox{(J}}_{{\rm {K}}} {\hbox{/50)H(J}}_{{\rm {K}}}/50)\,\mu\hbox{A/cm}^{\rm 2}\hbox{ }(\hbox{K}^{+}{\hbox{\,IR current)}}\\&{\hbox{I}}_{{{\rm {DR}}}}=\hbox{g}_{{{\rm\,{DR}}}}{\hbox{(J}}_{{\rm {K}}}/50)\,\mu{\hbox{A/cm}}^{{\rm {2}}}\,{\hbox{(K}}^{ + } {\hbox{\,DR current)}}\\ &{\hbox{I}}_{{{\rm{Na}}}}=\hbox{g}_{{{\rm {Na}}}} {\hbox{(J}}_{{{\rm {Na}}}}/75)\,\mu{\hbox{A/cm}}^{{\rm {2}}}\,({\hbox{Na}}^{+}\,{\hbox{current)}}\\&\hbox{H(t)}={\left\{ \begin{array}{l} 1, t > 0\\ 0, t < 0\\\end{array} \right. }\\ \end{aligned} $$

(3)

$$ \begin{aligned} & \sigma=\frac{1} {7}{\hbox{(exp(Na}}_{{\rm {o}}}{\hbox{/67}}{.3) - 1)}\\ & {\hbox{f}}_{{\rm {1}}} {\hbox{ = (1 + 0}}{.12\exp(- 0} {\hbox{.1VF/(RT)) + 0}}{\hbox{.04}}\sigma {\exp(- \hbox{VF/(RT)))}}^{{{\rm {-1}}}}\\ & \bar{\hbox{I}}_{{{\rm {NaK}}}} ={\hbox{F}}\bar{\hbox{J}}_{{{\rm {NaK}}}} {\hbox{/((1 + K}}_{{{\rm {mK}}}} {\hbox{/K}}_{{\rm {o}}} {\hbox{)}}^{{\rm {2}}} \times {\hbox{(1 + K}}_{{{\rm {mNa}}}} {\hbox{/Na}}_{{\rm {i}}} {\hbox{)}}^{{\rm {3}}})\mu {\hbox{A/cm}}^{2}\\ & {\hbox{I}}_{{{\rm {NaK}}}}=\bar{\hbox{I}}_{{{\rm {NaK}}}}{\hbox{f}}_{1}\,\mu \hbox{A/cm}^{2}\hbox{ (Na-K exchanger)}\\ & \frac{{{\hbox{dh}}}} {{{\hbox{dt}}}}=\alpha _{{\rm {h}}} {(1-\hbox{h}) - } \beta _{{\rm {h}}} {\hbox{h (Na inactivation gating variable)}}\\ & \frac{{{\hbox{dm}}}} {{{\hbox{dt}}}}=\alpha _{{\rm {m}}} {(1 - \hbox{m}) - }\beta _{{\rm {m}}} {\hbox{m (Na activation gating variable)}}\\ & \frac{{{\hbox{dn}}}} {{{\hbox{dt}}}}=\alpha _{{\rm {n}}} {(1 - \hbox{n}) - }\beta _{{\rm {n}}} {\hbox{n (K-DR activation gating variable)}}\\ & \frac{{{\hbox{dh}}^{{\rm {K}}} }} {{{\hbox{dt}}}}{\hbox{ = (h}}^{{\rm {K}}}_{\infty } {-\hbox{h}}^{{\rm {K}}} {\hbox{)/}}\tau _{{{\rm {h}}^{{\rm {K}}} }} {\hbox{ (K-DR inactivation gating variable)}}\\ & \frac{{{\hbox{dS}}}} {{{\hbox{dt}}}}{\hbox{ = (S}}_{\infty } {-\hbox{S)/}}\tau _{{\rm {S}}} {\hbox{ (Na very slow inactivation gating variable)}}\\ \end{aligned} $$

(4)

and the function Z_t is defined by

$$ \begin{aligned} &\left[\frac{{{\hbox{dh}}_{{\rm{t}}} }} {{{\hbox{dt}}}}{\hbox{,}}\frac{{{\hbox{dm}}_{{\rm {t}}} }}{{{\hbox{dt}}}}{\hbox{,}}\frac{{{\hbox{dn}}_{{\rm {t}}} }}{{{\hbox{dt}}}}{\hbox{,}}\frac{{{\hbox{dh}}^{{\rm {K}}}_{{\rm {t}}}}} {{{\hbox{dt}}}}{\hbox{,}}\frac{{{\hbox{dS}}_{{\rm {t}}} }}{{{\hbox{dt}}}}{\hbox{,I}}^{{{\rm {Cl}}}}_{{\rm {t}}}{\hbox{,I}}^{{{\rm {IR}}}}_{{\rm {t}}} {\hbox{,I}}^{{{\rm{DR}}}}_{{\rm {t}}} {\hbox{,I}}^{{{\rm {Na}}}}_{{\rm {t}}}{\hbox{,I}}^{{{\rm {NaK}}}}_{{\rm {t}}} \right] ={\hbox{ Z}}^{{}}_{{\rm{t}}} {\hbox{(v}}_{{\rm {t}}} {\hbox{,h}}_{{\rm {t}}}{\hbox{,m}}_{{\rm {t}}} {\hbox{,n}}_{{\rm {t}}} {\hbox{,h}}^{{\rm{K}}}_{{\rm {t}}} {\hbox{,S}}_{{\rm {t}}} {\hbox{,K}}_{{\rm {i}}}{\hbox{,K}}_{{\rm {t}}} {\hbox{,Na}}_{{\rm {i}}} {\hbox{,Na}}_{{\rm{t}}} {\hbox{)}}\\ &\left[\frac{{{\hbox{dh}}_{{\rm {t}}} }}{{{\hbox{dt}}}}{\hbox{,}}\frac{{{\hbox{dm}}_{{\rm {t}}} }}{{{\hbox{dt}}}}{\hbox{,}}\frac{{{\hbox{dn}}_{{\rm {t}}} }}{{{\hbox{dt}}}}{\hbox{,}}\frac{{{\hbox{dh}}^{{\rm {K}}}_{{\rm {t}}}}} {{{\hbox{dt}}}}{\hbox{,}}\frac{{{\hbox{dS}}}}{{{\hbox{dt}}}}{\hbox{,I}}^{{{\rm {Cl}}}} {\hbox{,I}}^{{{\rm {IR}}}}{\hbox{,I}}^{{{\rm {DR}}}} {\hbox{,I}}^{{{\rm {Na}}}}{\hbox{,I}}^{{{\rm {NaK}}}} \right] = {\hbox{Z}}^{{}}_{{\rm {s}}}{\hbox{(v}}_{{\rm {t}}} {\hbox{,h}}_{{\rm {t}}} {\hbox{,m}}_{{\rm{t}}} {\hbox{,n}}_{{\rm {t}}} {\hbox{,h}}^{{\rm {K}}}_{{\rm {t}}}{\hbox{,S}}_{{\rm {t}}} {\hbox{,K}}_{{\rm {i}}} {\hbox{,K}}_{{\rm{t}}} {\hbox{,Na}}_{{\rm {i}}} {\hbox{,Na}}_{{\rm {t}}} {\hbox{)}}\\& \eta _{{{\rm {Cl}}}}=0{\hbox{.1 (Cl channel density in t-tubules relative to sarcolemma)}}\\ & \eta _{{{\rm {IR}}}}=1.0{\hbox{ (K{-}IR channel density in t-tubules relative to sarcolemma)}}\\ &\eta _{{{\rm {DR}}}}=0{\hbox{.45 (K{-}DR channel density in t-tubules relative to sarcolemma)}}\\ & \eta _{{{\rm{Na}}}}=0{\hbox{.1 (Na channel density in t-tubules relative to sarcolemma)}}\\ & \eta _{{{\rm {NaK}}}}=0{\hbox{.1 (Na/K exchanger density in t-tubules relative to sarcolemma)}}\\&{\hbox{I}}^{{{\rm {Cl}}}}_{{\rm {t}}} =\eta _{{{\rm {Cl}}}}{\hbox{I}}^{{{\rm {Cl}}}} \hbox{ }\mu\hbox{A/cm}^{2}\\ &{\hbox{I}}^{{{\rm{IR}}}}_{{\rm {t}}} =\eta _{{{\rm {IR}}}} {\hbox{I}}^{{{\rm{IR}}}}\hbox{ }\mu\hbox{A/cm}^{2}\\ &{\hbox{I}}^{{{\rm {DR}}}}_{{\rm{t}}} =\eta _{{{\rm {DR}}}} {\hbox{I}}^{{{\rm{DR}}}}\hbox{ }\mu\hbox{A/cm}^{2}\\ &{\hbox{I}}^{{{\rm {Na}}}}_{{\rm{t}}} =\eta _{{{\rm {Na}}}} {\hbox{I}}^{{{\rm{Na}}}}\hbox{ }\mu\hbox{A/cm}^{2}\\ &{\hbox{I}}^{{{\rm {NaK}}}}_{{\rm{t}}} =\eta _{{{\rm {NaK}}}} {\hbox{I}}^{{{\rm{NaK}}}}\hbox{ }\mu\hbox{A/cm}^{2}\\ \end{aligned} $$

(5)

PDE model

$$ \begin{aligned} & \bar{G}_{L} = G_{L} \tau f_{T} \quad \hbox{mS/cm}\\ &{\hbox{D}}_{{\rm {s}}}=1000 \times (R_{s}/(2(R_{i} + R_{e})))/C_{m}\quad {\hbox{cm}}^{{\rm {2}}} {\hbox{/ms}}\\ &D_{t}=\bar{G}_{L}/(C_{m} f_{T}/\xi)\quad{\hbox{cm}}^{{\rm {2}}} {\hbox{/ms}}\\ &\left[\frac{{{\hbox{dh}}_{{\rm {s}}} }} {{{\hbox{dt}}}}{\hbox{,}}\frac{{{\hbox{dm}}_{{\rm {s}}} }} {{{\hbox{dt}}}}{\hbox{,}}\frac{{{\hbox{dn}}_{{\rm {s}}} }} {{{\hbox{dt}}}}{\hbox{,}}\frac{{{\hbox{dh}}^{{\rm {K}}}_{{\rm {s}}} }} {{{\hbox{dt}}}}{\hbox{,}}\frac{{{\hbox{dS}}_{{\rm {s}}} }} {{{\hbox{dt}}}}{\hbox{,I}}^{{{\rm {Cl}}}}{\hbox{,I}}^{{{\rm {IR}}}} {\hbox{,I}}^{{{\rm {DR}}}} {\hbox{,I}}^{{{\rm {Na}}}} {\hbox{,I}}^{{{\rm {NaK}}}} \right] ={\hbox{ Z}}^{{}}_{{\rm {s}}} {\hbox{(v}}_{{\rm {s}}} {\hbox{,h}}_{{\rm {s}}} {\hbox{,m}}_{{\rm {s}}} {\hbox{,n}}_{{\rm {s}}} {\hbox{,h}}^{{\rm {K}}}_{{\rm {s}}} {\hbox{,S}}_{{\rm {s}}} {\hbox{,K}}_{{\rm {i}}} {\hbox{,K}}_{{\rm {e}}} {\hbox{,Na}}_{{\rm {i}}} {\hbox{,Na}}_{{\rm {e}}} {\hbox{)}}\\ &{\hbox{I}}^{{{\rm {ionic}}}}_{{\rm {s}}} {\hbox{ = I}}^{{{\rm {Cl}}}}{\hbox{ + I}}^{{{\rm {IR}}}} {\hbox{ + I}}^{{{\rm {DR}}}}{\hbox{ + I}}^{{{\rm {Na}}}} {\hbox{ + I}}^{{{\rm {NaK}}}} {-\hbox{I}}_{{{\rm {app}}}} {\hbox{(t;t}}_{{{\rm {app}}}}) \hbox{ }\mu\hbox{A/cm}^{2}{\hbox{ (sarcolemma current)}}\\ & \left[\frac{{{\hbox{dh}}_{{\rm {t}}} }} {{{\hbox{dt}}}}{\hbox{,}}\frac{{{\hbox{dm}}_{{\rm {t}}} }} {{{\hbox{dt}}}}{\hbox{,}}\frac{{{\hbox{dn}}_{{\rm {t}}} }} {{{\hbox{dt}}}}{\hbox{,}}\frac{{{\hbox{dh}}^{{\rm {K}}}_{{\rm {t}}} }} {{{\hbox{dt}}}}{\hbox{,}}\frac{{{\hbox{dS}}_{{\rm {t}}} }} {{{\hbox{dt}}}}{\hbox{,I}}^{{{\rm {Cl}}}}_{{\rm {t}}} {\hbox{,I}}^{{{\rm {IR}}}}_{{\rm {t}}} {\hbox{,I}}^{{{\rm {DR}}}}_{{\rm {t}}} {\hbox{,I}}^{{{\rm {Na}}}}_{{\rm {t}}} {\hbox{,I}}^{{{\rm {NaK}}}}_{{\rm {t}}} \right] = {\hbox{ Z}}^{{}}_{{\rm {t}}} {\hbox{(v}}_{{\rm {t}}} {\hbox{,h}}_{{\rm {t}}} {\hbox{,m}}_{{\rm {t}}} {\hbox{,n}}_{{\rm {t}}} {\hbox{,h}}^{{\rm {K}}}_{{\rm {t}}} {\hbox{,S}}_{{\rm {t}}} {\hbox{,K}}_{{\rm {i}}} {\hbox{,K}}_{{\rm {t}}} {\hbox{,Na}}_{{\rm {i}}} {\hbox{,Na}}_{{\rm {t}}} {\hbox{)}}\\ &{\hbox{I}}^{{{\rm {ionic}}}}_{{\rm {t}}} {\hbox{ = I}}^{{{\rm {Cl}}}}_{{\rm {t}}} {\hbox{ + I}}^{{{\rm {IR}}}}_{{\rm {t}}} {\hbox{ + I}}^{{{\rm {DR}}}}_{{\rm {t}}} {\hbox{ + I}}^{{{\rm {Na}}}}_{{\rm {t}}} {\hbox{ + I}}^{{{\rm {NaK}}}}_{{\rm {t}}} \hbox{ }\mu\hbox{A/cm}^{2}{\hbox{ (t-tubule current)}}\\ & \frac{{\partial v_{t} {\left({r,x} \right)}}} {{\partial t}} + \frac{{I^{{\rm ionic}}_{t} }} {{C_{m} }} = D_{t} {\left({\frac{{\partial ^{2} v^{{}}_{t} }} {{\partial r^{2} }} + \frac{1} {r}\frac{{\partial v^{{}}_{t} }} {{\partial r}}} \right)} {\hbox{ mV/ms (t-tubule membrane voltage)}}\\ & \left. {\frac{{\partial v_{t} }} {{\partial r}}} \right|_{{r = 0}} = 0\\ & {\hbox{I}}_{{\rm {T}}} = 1000 \times [v_{s} (x) - v_{t} (r = R_{s},x)]/R_{a} = \bar{G}_{L} \left. {\frac{{\partial v_{t} }} {{\partial r}}} \right|_{{r = R_{s} }}\hbox{ }\mu\hbox{A/cm}^{2} {\hbox{\,(access current)}}\\ & \frac{{\partial v_{s} (x)}} {{\partial t}} + \frac{{I^{{\rm ionic}}_{s} }}{{C_{m}}}=D_{s} \frac{{\partial ^{2} v^{{}}_{s} }} {{\partial x^{2} }},\,{\hbox{mV/ms (sarcolemma membrane voltage)}}\\ & \left.{\frac{{\partial v_{s} }} {{\partial x}}} \right|_{{x = 0}} = 0,\left. {\frac{{\partial v_{s} }} {{\partial x}}} \right|_{{x = l_{s} }} = 0\\ & \frac{{\partial K_{t} (r,x)}} {{\partial t}} = D_{K} {\left({\frac{{\partial ^{2} K^{{}}_{t} }} {{\partial r^{2} }} + \frac{1} {r}\frac{{\partial K^{{}}_{t} }} {{\partial r}}} \right)} + {\hbox{(I}}^{{{\rm {IR}}}}_{{\rm {t}}} {\hbox{ + I}}^{{{\rm {DR}}}}_{{\rm {t}}} {-\hbox{2I}}^{{{\rm {NaK}}}}_{{\rm {t}}} + {\hbox{I}}^{{\rm {K}}}_{{{\rm {rest}}}} {\hbox{)/(1000F}}\xi {\hbox{)}}\\ & \left. {\frac{{\partial K_{t} }} {{\partial r}}} \right|_{{r = 0}} = 0\\ & K_{t} (r = R_{s},x) = K_{e} (x)\quad{\hbox{mM/ms}}{\hbox{ (t-tubule [K}}^{ + } {\hbox{])}}\\ & \frac{{\partial Na_{t} (r,x)}} {{\partial t}} = D_{{Na}} {\left({\frac{{\partial ^{2} Na^{{}}_{t} }} {{\partial r^{2} }} + \frac{1} {r}\frac{{\partial Na^{{}}_{t} }} {{\partial r}}} \right)} + {\hbox{(I}}^{{{\rm {Na}}}}_{{\rm {t}}} {\hbox{ + 3I}}^{{{\rm {NaK}}}}_{{\rm {t}}} + {\hbox{I}}^{{{\rm {Na}}}}_{{{\rm {rest}}}} {\hbox{)/(1000F}}\xi {\hbox{)}}\\ & \left. {\frac{{\partial Na_{t} }} {{\partial r}}} \right|_{{r = 0}} = 0\\ & Na_{t} (r = R_{s},x) = Na_{e} (x)\;{\hbox{mM/ms }}{\hbox{(t-tubule [Na}}^{ + } {\hbox{])}}\\ & \frac{{\partial K_{e} (x)}} {{\partial t}} = {\hbox{(I}}^{{{\rm {IR}}}} {\hbox{ + I}}^{{{\rm {DR}}}}{-\hbox{2I}}^{{{\rm {NaK}}}} + {\hbox{I}}^{{\rm {K}}}_{{{\rm {rest}}}} {\hbox{)/(1000F}}\xi _{3} {\hbox{)}}\;{\hbox{mM/ms}}\;{\hbox{(extracellular [K}}^{ + } {\hbox{])}}\\ & \frac{{\partial Na_{e} (x)}}{{\partial t}} = {\hbox{(I}}^{{{\rm {Na}}}} {\hbox{ + 3I}}^{{{\rm {NaK}}}}+{\hbox{I}}^{{{\rm {Na}}}}_{{{\rm {rest}}}} {\hbox{)/(1000F}}\xi _{3}{\hbox{) }}{\hbox{mM/ms}}\;{\hbox{(extracellular [Na}}^{ + } {\hbox{])}}\\ & \frac{{\partial K_{i} (r,x)}} {{\partial t}} = {-\hbox{f}}_{{\rm {T}}} {\hbox{(I}}^{{{\rm {IR}}}}_{{\rm {t}}} {\hbox{ + I}}^{{{\rm {DR}}}}_{{\rm {t}}} {-\hbox{2I}}^{{{\rm {NaK}}}}_{{\rm {t}}} + {\hbox{I}}^{{\rm {K}}}_{{{\rm {rest}}}} {\hbox{)/(1000F}}\xi {\hbox{)}}{\hbox{ mM/ms}}\; {\hbox{(intracellular [K}}^{ + } {\hbox{])}}\\ & \frac{{\partial Na_{i} (r,x)}} {{\partial t}} = {-\hbox{f}}_{{\rm {T}}} {\hbox{(I}}^{{{\rm {Na}}}}_{{\rm {t}}} {\hbox{ + 3I}}^{{{\rm {NaK}}}}_{{\rm {t}}} + {\hbox{I}}^{{{\rm {Na}}}}_{{{\rm {rest}}}} {\hbox{)/(1000F}}\xi {\hbox{)}}{\hbox{ mM/ms}}\;{\hbox{(intracellular [Na}}^{ + } {\hbox{])}}\\ \end{aligned}$$

(6)

Ca²⁺ release from RyR’s

Table 2 Model parameter values and definitions

$$\begin{aligned} {\text{k}}_{{\text{C}}} &= 0.5 \alpha _{1}\exp(({\text{v}}_{{\text{t}}}-\overline{\text{V}} /(8\text{K}))\ ({\text{activation rate between closed states}})\\{\text{k}}_{{\text{Cm}}} &= 0.5 \alpha _{1} \exp(-({\text{v}}_{{\text{t}}}-\overline{\text{V}}/(8\text{K}))\ ({\text{inactivation rate between closed states}})\\\frac{{\text{dC}}_{0}}{{\text{dt}}} &= -{\text{k}}_{{\rm{L}}} {\text{C}}_{0} +{\text{k}}_{{{\text{Lm}}}}{\text{O}}_{0}- 4{\text{k}}_{{\text{C}}} {\text{C}}_{0}+{\text{k}}_{{{\text{Cm}}}} {\text{C}}_{1}\ ({\text{C}}_{0}-{\text{C}}_{4}\ {\text{are RyR closed states}})\\ \frac{{\text{dO}}_{0}}{{\text{dt}}} &=+{\text{k}}_{{\text{L}}}{\text{C}}_{0} - {\text{k}}_{{\text{Lm}}}{\text{O}}_{0}-{\text{4k}}_{{\text{C}}}/{\text{fO}}_{0}+{\text{fk}}_{{{\text{Cm}}}}{\text{O}}_{1}\ ({\text{O}}_{0} -{\text{O}}_{4} {\text{ are RyR open states}})\\ \frac{{{\text{dC}}_{1}}}{{{\text{dt}}}}&=+{\text{4k}}_{{\text{C}}}{\text{C}}_{0}-{\text{k}}_{{{\text{Cm}}}}{\text{C}}_{1}-{\text{k}}_{{\text{L}}}/{\text{fC}}_{1}+{\text{k}}_{{{\text{Lm}}}} {\text{fO}}_{1}-{\text{3k}}_{{\text{C}}}{\text{C}}_{1}+{\text{2k}}_{\text{Cm}}\text{C}_{2}\\\frac{{{\text{dO}}_{1}}}{\text{dt}}&=\text{k}_{\text{L}}/{\text{fC}}_{1}-{\text{k}}_{{{\text{Lm}}}}{\text{fO}}_{1}+{\text{4k}}_{{\text{C}}}/{\text{fO}}_{0}-{\text{fk}}_{{{\text{Cm}}}} {\text{O}}_{1}-{\text{3k}}_{{\rm{C}}}/\text{fO}_{1}+2\text{fk}_{\text{Cm}}\text{O}_{2}\\\frac{{\text{dC}}_{2}}{\text{dt}}&=3\text{k}_{\text{C}}\text{C}_{1}-2\text{k}_{\text{Cm}}\text{C}_{2}-\text{k}_{\text{L}}/\text{f}^{2}\text{C}_{2}+\text{k}_{\text{Lm}}\text{f}^{2}\text{O}_{2}-2\text{k}_{\text{C}}\text{C}_{2}+3\text{k}_{\text{Cm}}\text{C}_{\text{3}}\\\frac{\text{dO}_{2}}{\text{dt}}&=3{\text{k}}_{\text{C}}/\text{fO}_{1}-2\text{fk}_{\text{Cm}}\text{O}_{2}+\text{k}_{\text{L}}/\text{f}^{2}\text{C}_{2}-\text{k}_{\text{Lm}}{\text{f}}^{2}\text{O}_{2}-2\text{k}_{\text{C}}/\text{fO}_{2}+3\text{fk}_{\text{Cm}}\text{O}_{\text{3}}\\\frac{{\text{dC}}_{\text{3}}}{\text{dt}}&=2{\text{k}}_{\text{C}}{\text{C}}_{2}-3{\text{k}}_{\text{Cm}}{\text{C}}_{\text{3}}-{\text{k}}_{\text{L}}/{\text{f}}^{3}\text{C}_{3}+\text{k}_{\text{Lm}}\text{f}^{3}\text{O}_{3}-\text{k}_{\text{C}}\text{C}_{3}+4\text{k}_{\text{Cm}}\text{C}_{4}\\\frac{\text{dO}_{3}}{\text{dt}}&=\text{k}_{\text{L}}/\text{f}^{3}\text{C}_{\rm3}-\text{k}_{\text{Lm}}\text{f}^{\text{3}}\text{O}_{3}+2\text{k}_{\text{C}}/\text{fO}_{2}-3\text{fk}_{\text{Cm}}\text{O}_{3}-\text{k}_{\text{C}}/\text{fO}_{\text{3}}+4\text{fk}_{\text{Cm}}\text{O}_{\text{4}}\\\frac{\text{dC}_{\text{4}}}{\text{dt}}&=\text{k}_{\text{C}}\text{C}_{3}-4\text{k}_{\text{Cm}}\text{C}_{4}-\text{k}_{\text{L}}/\text{f}^{4}\text{C}_{4}+\text{k}_{\text{Lm}}\text{f}^{4}\text{O}_{4}\\\frac{\text{dO}_{4}}{\text{dt}}&=\text{k}_{\text{C}}/\text{fO}_{3}-4\text{fk}_{\text{Cm}}\text{O}_{4}+\text{k}_{\text{L}}/\text{f}^{4}\text{C}_{4}\text{-k}_{\text{Lm}}\text{f}^{4}\text{O}_{4}\end{aligned}$$

(7)

Ca²⁺ transport and XB dynamics

Table 3 Model parameter values and definitions

$$\begin{aligned} & {\text{T}}_0 = {\text{T}}_{{\text{tot}}} - {\text{Ca}}_2^{\text{T}} - {\text{Ca}}_2^{{\text{CaT}}} - {\text{D}}_0 - {\text{D}}_1 - {\text{D}}_2 - {\text{A}}_1 - {\text{A}}_2 \,\mu {\text{M}}\,([{\text{free}}\,{\text{troponin}}\,{\text{binding}}\,{\text{sites}}]) \\ & \frac{{{\text{dCa}}_1 }}{{{\text{dt}}}} = ({\text{i}}_2 ({\text{O}}_0 + {\text{O}}_1 + {\text{O}}_2 + {\text{O}}_3 + {\text{O}}_4 )) \times ({\text{Ca}}_1^{{\text{SR}}} - {\text{Ca}}_1 )/{\text{V}}_1 - \nu _{{\text{SR}}} {\text{Ca}}_1 /({\text{Ca}}_1 + {\text{K}}_{{\text{SR}}} )/{\text{V}}_1 {\text{ }} + {\text{ L}}_e ({\text{Ca}}_1^{{\text{SR}}} - {\text{Ca}}_1 )/{\text{V}}_1 \\ & - \tau _{\text{R}} ({\text{Ca}}_1 - {\text{Ca}}_2 )/{\text{V}}_1 - ({\text{k}}_{\text{P}}^{{\text{on}}} {\text{Ca}}_1 ({\text{P}}_{{\text{tot}}} - {\text{Ca}}_{\text{1}}^{\text{P}} - {\text{Mg}}_1^{\text{P}} ) - {\text{k}}_{\text{P}}^{{\text{off}}} {\text{Ca}}_1^{\text{P}} ) - ({\text{k}}_{{\text{CATP}}}^{{\text{on}}} {\text{Ca}}_1 \left[ {{\text{ATP}}} \right]_1 - {\text{k}}_{{\text{CATP}}}^{{\text{off}}} {\text{Ca}}_{\text{1}}^{{\text{ATP}}} )\hbox{ }\mu {\text{M}}\,({\text{TSR}}\,{\text{myoplasm}}\,[{\text{Ca}}]) \\ & \frac{{{\text{dCa}}_{\text{1}}^{{\text{SR}}} }}{{{\text{dt}}}} = - ({\text{i}}_2 ({\text{O}}_0 + {\text{O}}_1 + {\text{O}}_2 + {\text{O}}_3 + {\text{O}}_4 )) \times ({\text{Ca}}_{\text{1}}^{{\text{SR}}} - {\text{Ca}}_1 )/{\text{V}}_1^{{\text{SR}}} + \nu _{{\text{SR}}} {\text{Ca}}_1 /({\text{Ca}}_1 + {\text{K}}_{{\text{SR}}} )/{\text{V}}_1^{{\text{SR}}} {\text{ }} \\ & - {\text{L}}_{\text{e}} ({\text{Ca}}_{\text{1}}^{{\text{SR}}} - {\text{Ca}}_1 )/{\text{V}}_1^{{\text{SR}}} - \tau _{\text{R}}^{{\text{SR}}} ({\text{Ca}}_1^{{\text{SR}}} - {\text{Ca}}_2^{{\text{SR}}} )/{\text{V}}_1^{{\text{SR}}} - ({\text{k}}_{{\text{Cs}}}^{{\text{on}}} {\text{Ca}}_1^{{\text{SR}}} ({\text{Cs}}_{{\text{tot}}} - {\text{Ca}}_1^{{\text{Cs}}} ) - {\text{k}}_{{\text{Cs}}}^{{\text{off}}} {\text{Ca}}_{\text{1}}^{{\text{Cs}}} )\hbox{ }\mu {\text{M}}\hbox{ }({\text{TSR}}\,[{\text{Ca}}]) \\ & \frac{{{\text{dCa}}_2 }}{{{\text{dt}}}} = - \nu _{{\text{SR}}} {\text{Ca}}_2 /({\text{Ca}}_2 + {\text{K}}_{{\text{SR}}} )/{\text{V}}_2 + {\text{L}}_e ({\text{Ca}}_2^{{\text{SR}}} - {\text{Ca}}_2 )/{\text{V}}_2 + \tau _{\text{R}} ({\text{Ca}}_1 - {\text{Ca}}_2 )/{\text{V}}_2 - ({\text{k}}_{\text{T}}^{{\text{on}}} {\text{Ca}}_2 {\text{T}}_0 - {\text{k}}_{\text{T}}^{{\text{off}}} {\text{Ca}}_{\text{2}}^{\text{T}} \\ & \quad \quad \quad + {\text{k}}_{\text{T}}^{{\text{on}}} {\text{Ca}}_2 {\text{Ca}}_2^{\text{T}} - {\text{k}}_{\text{T}}^{{\text{off}}} {\text{Ca}}_{\text{2}}^{{\text{CaT}}} + {\text{k}}_{\text{T}}^{{\text{on}}} {\text{Ca}}_2 {\text{D}}_0 - {\text{k}}_{\text{T}}^{{\text{off}}} {\text{D}}_1 + {\text{k}}_{\text{T}}^{{\text{on}}} {\text{Ca}}_2 {\text{D}}_1 - {\text{k}}_{\text{T}}^{{\text{off}}} {\text{D}}_2 ) \\ & \quad \quad \quad - ({\text{k}}_{\text{P}}^{{\text{on}}} {\text{Ca}}_2 ({\text{P}}_{{\text{tot}}} - {\text{Ca}}_2^{\text{P}} - {\text{Mg}}_2^{\text{P}} ) - {\text{k}}_{\text{P}}^{{\text{off}}} {\text{Ca}}_2^{\text{P}} ) - ({\text{k}}_{{\text{CATP}}}^{{\text{on}}} {\text{Ca}}_2 \left[ {{\text{ATP}}} \right]_2 - {\text{k}}_{{\text{CATP}}}^{{\text{off}}} {\text{Ca}}_2^{{\text{ATP}}} )\hbox{ }\mu {\text{M}}\hbox{ }({\text{myoplasm}}\hbox{ }[{\text{Ca}}]) \\ & \frac{{{\text{dCa}}_2^{{\text{SR}}} }}{{{\text{dt}}}} = + \nu _{{\text{SR}}} {\text{Ca}}_2 /({\text{Ca}}_2 + {\text{K}}_{{\text{SR}}} )/{\text{V}}_2^{{\text{SR}}} - {\text{L}}_{\text{e}} ({\text{Ca}}_2^{{\text{SR}}} - {\text{Ca}}_2 )/{\text{V}}_2^{{\text{SR}}} + \tau _{\text{R}}^{{\text{SR}}} ({\text{Ca}}_1^{{\text{SR}}} - {\text{Ca}}_2^{{\text{SR}}} )/{\text{V}}_2^{{\text{SR}}} \\ & \quad \quad \quad - ({\text{k}}_{{\text{Cs}}}^{{\text{on}}} {\text{Ca}}_2^{{\text{SR}}} ({\text{Cs}}_{{\text{tot}}} - {\text{Ca}}_2^{{\text{Cs}}} ) - {\text{k}}_{{\text{Cs}}}^{{\text{off}}} {\text{Ca}}_2^{{\text{Cs}}} ) \\ & \quad \quad \quad - 0.001 \times [{\text{A}}_{\text{P}} ({\text{P}}_{{\text{SR}}} \times 0.001 \times {\text{Ca}}_{\text{2}}^{{\text{SR}}} - {\text{PP}}){\text{H}}({\text{P}}_{{\text{SR}}} \times 0.001 \times {\text{Ca}}_2^{{\text{SR}}} - {\text{PP}}) \times 0.001 \times {\text{P}}_{{\text{SR}}} {\text{Ca}}_2^{{\text{SR}}} \\ & \quad \quad \quad - {\text{B}}_{\text{P}} {\text{P}}_{{\text{SR}}}^{\text{C}} ({\text{PP}} - {\text{P}}_{{\text{SR}}} \times 0.001 \times {\text{Ca}}_2^{{\text{SR}}} ){\text{H}}({\text{PP}} - {\text{P}}_{{\text{SR}}} \times 0.001 \times {\text{Ca}}_2^{{\text{SR}}} )]\hbox{ }\mu {\text{M}}\hbox{ }({\text{SR}}\hbox{ }[{\text{Ca}}]){\text{ }} \\ & \frac{{{\text{dCa}}_2^{\text{T}} }}{{{\text{dt}}}} = {\text{k}}_{\text{T}}^{{\text{on}}} {\text{Ca}}_2 {\text{T}}_0 - {\text{k}}_{\text{T}}^{{\text{off}}} {\text{Ca}}_2^{\text{T}} - {\text{k}}_{\text{T}}^{{\text{on}}} {\text{Ca}}_2 {\text{Ca}}_2^{\text{T}} + {\text{k}}_{\text{T}}^{{\text{off}}} {\text{Ca}}_2^{{\text{CaT}}} - {\text{k}}_0^{{\text{on}}} {\text{Ca}}_2^{\text{T}} + {\text{k}}_0^{{\text{off}}} {\text{D}}_1 \,\mu {\text{M}}\,({\text{myoplasm}}\,[{\text{Ca - Troponin}}]) \\ & \frac{{{\text{dCa}}_1^{\text{P}} }}{{{\text{dt}}}} = {\text{k}}_{\text{P}}^{{\text{on}}} {\text{Ca}}_1 ({\text{P}}_{{\text{tot}}} - {\text{Ca}}_1^{\text{P}} - {\text{Mg}}_1^{\text{P}} ) - {\text{k}}_{\text{P}}^{{\text{off}}} {\text{Ca}}_1^P \,{\text{ }}\mu {\text{M}}\,({\text{TSR}}\,{\text{myoplasm}}\,[{\text{Ca}} - {\text{Parvalbumin}}]) \\ & \frac{{{\text{dCa}}_2^{\text{P}} }}{{{\text{dt}}}} = {\text{k}}_{\text{P}}^{{\text{on}}} {\text{Ca}}_2 ({\text{P}}_{{\text{tot}}} - {\text{Ca}}_2^{\text{P}} - {\text{Mg}}_2^{\text{P}} ) - {\text{k}}_{\text{P}}^{{\text{off}}} {\text{Ca}}_2^{\text{P}} \,\mu {\text{M}}\,({\text{myoplasm}}\,[{\text{Ca - Parvalbumin}}]) \\ & {\text{dMg}}_1^{\text{P}} /{\text{dt}} = {\text{k}}_{{\text{Mg}}}^{{\text{on}}} ({\text{P}}_{{\text{tot}}} - {\text{Ca}}_1^{\text{P}} - {\text{Mg}}_1^{\text{P}} ){\text{Mg}}_1 - {\text{k}}_{{\text{Mg}}}^{{\text{off}}} {\text{Mg}}_1^{\text{P}} \,\mu {\text{M}}\,({\text{TSR}}\,{\text{myoplasm}}\,[{\text{Mg - Parvalbumin}}]) \\ & {\text{dMg}}_2^{\text{P}} /{\text{dt}} = {\text{k}}_{{\text{Mg}}}^{{\text{on}}} ({\text{P}}_{{\text{tot}}} - {\text{Ca}}_2^{\text{P}} - {\text{Mg}}_2^{\text{P}} ){\text{Mg}}_2 - {\text{k}}_{{\text{Mg}}}^{{\text{off}}} {\text{Mg}}_2^{\text{P}} \,\mu {\text{M}}\,({\text{myoplasm}}\,[{\text{Mg - Parvalbumin}}]) \\ & {\text{dCa}}_1^{{\text{Cs}}} /{\text{dt}} = {\text{k}}_{{\text{Cs}}}^{{\text{on}}} {\text{Ca}}_1^{{\text{SR}}} ({\text{Cs}}_{{\text{tot}}} - {\text{Ca}}_1^{{\text{Cs}}} ) - {\text{k}}_{{\text{Cs}}}^{{\text{off}}} {\text{Ca}}_1^{{\text{Cs}}} \,\mu {\text{M}}\,({\text{TSR}}\,[{\text{Ca - Calsequestrin}}]) \\ & {\text{dCa}}_2^{{\text{Cs}}} /{\text{dt}} = {\text{k}}_{{\text{Cs}}}^{{\text{on}}} {\text{Ca}}_{\text{2}}^{{\text{SR}}} ({\text{Cs}}_{{\text{tot}}} - {\text{Ca}}_2^{{\text{Cs}}} ) - {\text{k}}_{{\text{Cs}}}^{{\text{off}}} {\text{Ca}}_2^{{\text{Cs}}} \,\mu {\text{M}}\,({\text{SR}}\,[{\text{Ca - Calsequestrin}}]) \\ & {\text{dCa}}_1^{{\text{ATP}}} /{\text{dt}} = {\text{k}}_{{\text{CATP}}}^{{\text{on}}} {\text{Ca}}_1 \left[ {{\text{ATP}}} \right]_1 - {\text{k}}_{{\text{CATP}}}^{{\text{off}}} {\text{Ca}}_1^{{\text{ATP}}} - \tau _{{\text{ATP}}} ({\text{Ca}}_1^{{\text{ATP}}} - {\text{Ca}}_2^{{\text{ATP}}} )/{\text{V}}_1 \,\mu {\text{M}}\,({\text{TSR}}\,{\text{myoplasm}}\,[{\text{Ca - ATP}}]) \\ & {\text{dCa}}_2^{{\text{ATP}}} /{\text{dt}} = {\text{k}}_{{\text{CATP}}}^{{\text{on}}} {\text{Ca}}_2 \left[ {{\text{ATP}}} \right]_2 - {\text{k}}_{{\text{CATP}}}^{{\text{off}}} {\text{Ca}}_2^{{\text{ATP}}} + \tau _{{\text{ATP}}} ({\text{Ca}}_1^{{\text{ATP}}} - {\text{Ca}}_2^{{\text{ATP}}} )/{\text{V}}_2 \,\mu {\text{M}}\,({\text{myoplasm}}\,[{\text{Ca - ATP}}]) \\ & {\text{dMg}}_1^{{\text{ATP}}} /{\text{dt}} = {\text{k}}_{{\text{MATP}}}^{{\text{on}}} {\text{Mg}}_1 \left[ {{\text{ATP}}} \right]_1 - {\text{k}}_{{\text{MATP}}}^{{\text{off}}} {\text{Mg}}_1^{{\text{ATP}}} - \tau _{{\text{ATP}}} ({\text{Mg}}_1^{{\text{ATP}}} - {\text{Mg}}_2^{{\text{ATP}}} )/{\text{V}}_1 \,\mu {\text{M}}\,({\text{TSR}}\,{\text{myoplasm}}\,[{\text{Mg - ATP}}]) \\ & {\text{dMg}}_2^{{\text{ATP}}} /{\text{dt}} = {\text{k}}_{{\text{MATP}}}^{{\text{on}}} {\text{Mg}}_2 \left[ {{\text{ATP}}} \right]_2 - {\text{k}}_{{\text{MATP}}}^{{\text{off}}} {\text{Mg}}_2^{{\text{ATP}}} + \tau _{{\text{ATP}}} ({\text{Mg}}_1^{{\text{ATP}}} - {\text{Mg}}_2^{{\text{ATP}}} )/{\text{V}}_2 \,\mu {\text{M}}\,({\text{myoplasm}}\,[{\text{Mg - ATP}}]) \\ \end{aligned}$$

(8)

$$ \begin{aligned} & \hbox{d}{\left[ \hbox{ATP} \right]}_{1}/ {{\hbox{dt}}}=-(\hbox{k}^{\rm on}_{\rm CATP}\hbox{Ca}_{1}\left[\hbox{ATP}\right]_{1}-\hbox{k}^{\rm off}_{\rm CATP}\hbox{Ca}^{\rm ATP}_{1})-(\hbox{k}^{\rm on}_{\rm MATP}\hbox{Mg}_{1}\left[\hbox{ATP}\right]_{1}-\hbox{k}^{\rm off}_{\rm MATP} \hbox{Mg}^{\rm ATP}_{1})\\ & -\tau _{\rm ATP}(\left[\hbox{ATP}\right]_{1} -\left[ \hbox{ATP}\right]_{2})/\hbox{V}_{1} \;\mu\hbox{M}\;(\hbox{TSR myoplasm [ATP]})\\ &\hbox{d}\left[\hbox{ATP}\right]_{2} /{{\hbox{dt}}}=-(\hbox{k}^{\rm on}_{\rm CATP} \hbox{Ca}_{2}\left[\hbox{ATP}\right]_{2}-\hbox{k}^{\rm off}_{\rm CATP} \hbox{Ca}^{\rm ATP}_{2})-(\hbox{k}^{\rm on}_{\rm MATP} \hbox{Mg}_{2}\left[\hbox{ATP}\right]_{2}-\hbox{k}^{\rm off}_{\rm MATP}\hbox{Mg}^{\rm ATP}_{2})\\ & +\tau_{\rm ATP}(\left[\hbox{ATP}\right]_{1}-\left[\hbox{ATP}\right]_{2})/\hbox{V}_{2}\;\mu\hbox{M}\;(\hbox{myoplasm [ATP]})\\ & \hbox{dMg}_{1}/\hbox{dt}=-(\hbox{k}^{\rm on}_{\rm Mg}(\hbox{P}_{\rm tot} -\hbox{Ca}^{\rm P}_{1}-\hbox{Mg}^{\rm P}_{1})\hbox{Mg}_{1} -\hbox{k}^{\rm off}_{\rm Mg}\hbox{Mg}^{\rm P}_{1})-(\hbox{k}^{\rm on}_{\rm MATP}\hbox{Mg}_{1}\left[\hbox{ATP}\right]_{1} -\hbox{k}^{\rm off}_{\rm MATP}\hbox{Mg}^{\rm ATP}_{1})\\ & - \tau _{\rm Mg}(\hbox{Mg}_{1}-\hbox{Mg}_{2})/\hbox{V}_{1} \;\mu\hbox{M}\;(\hbox{TSR myoplasm [Mg]})\\ & \hbox{dMg}_{2}/\hbox{dt}=-(\hbox{k}^{\rm on}_{\rm Mg} (\hbox{P}_{\rm tot}-\hbox{Ca}^{\rm P}_{2}-\hbox{Mg}^{\rm P}_{2})\hbox{Mg}_{2}-\hbox{k}^{\rm off}_{\rm Mg}\hbox{Mg}^{\rm P}_{2})-(\hbox{k}^{\rm on}_{\rm MATP}\hbox{Mg}_{2}\left[ \hbox{ATP}\right]_{2}-\hbox{k}^{\rm off}_{\rm MATP} \hbox{Mg}^{\rm ATP}_{2})\\ & + \tau _{\rm Mg}(\hbox{Mg}_{1}-\hbox{Mg}_{2})/\hbox{V}_{2}\;\mu\hbox{M}\;(\hbox{myoplasm [Mg]})\\& \frac{\hbox{dCa}^{\rm CaT}_{2}} {\hbox{dt}}=\hbox{k}^{\rm on}_{\rm T} \hbox{Ca}_{2}\hbox{Ca}^{\rm T}_{2}-\hbox{k}^{\rm off}_{\rm T} \hbox{Ca}^{\rm CaT}_{2}-\hbox{k}^{\rm on}_{\rm Ca}\hbox{Ca}^{\rm CaT}_{2} + \hbox{k}^{\rm off}_{\rm Ca}\hbox{D}_{2}\;\mu\hbox{M}\;(\hbox{myoplasm [Ca{-}Ca-Troponin]})\\ & \frac{\hbox{dD}_{0}} {\hbox{dt}} =-\hbox{k}^{\rm on}_{\rm T}\hbox{Ca}_{2}\hbox{D}_{0} + \hbox{k}^{\rm off}_{\rm T}\hbox{D}_{1} + \hbox{k}^{\rm on}_{0}\hbox{T}_{0} -\hbox{k}^{\rm off}_{0}\hbox{D}_{0}\;\mu\hbox{M}\;(\hbox{myoplasm [detached activated RU]})\\ & \frac{\hbox{dD}_{1}} {\hbox{dt}}=+\hbox{k}^{\rm on}_{\rm T} \hbox{Ca}_{2}\hbox{D}_{0}-\hbox{k}^{\rm off}_{\rm T} \hbox{D}_{1}+\hbox{k}^{\rm on}_{0}\hbox{Ca}^{\rm T}_{2}-\hbox{k}^{\rm off}_{0} \hbox{D}_{1}-\hbox{k}^{\rm on}_{\rm T}\hbox{Ca}_{2}\hbox{D}_{1} + \hbox{k}^{\rm off}_{\rm T} \hbox{D}_{2}\;\mu\hbox{M}\;\\ & \hbox{(myoplasm [detached activated RU-Ca])}\\ & \frac{\hbox{dD}_{2}}{\hbox{dt}}=+\hbox{k}^{\rm on}_{\rm T} \hbox{Ca}_{2}\hbox{D}_{1}-\hbox{k}^{\rm off}_{\rm T}\hbox{D}_{2} + \hbox{k}^{\rm on}_{\rm Ca}\hbox{Ca}^{\rm CaT}_{2}-\hbox{k}^{\rm off}_{\rm Ca}\hbox{D}_{2}-\hbox{f}_{\rm o}\hbox{D}_{2} + \hbox{f}_{\rm p}\hbox{A}_{1} + \hbox{g}_{\rm o}\hbox{A}_{2} \;\mu\hbox{M}\;\\ & \hbox{(myoplasm [detached activated RU{-}Ca{-}Ca])}\\ & \frac{\hbox{dA}_{1}} {\hbox{dt}} = + \hbox{f}_{\rm o}\hbox{D}_{2} -\hbox{f}_{\rm p}\hbox{A}_{1} + \hbox{h}_{\rm p}\hbox{A}_{2}-\hbox{h}_{\rm o} \hbox{A}_{1}\;\mu\hbox{M}\;(\hbox{myoplasm [attached pre-power stroke XB]})\\ & \frac{\hbox{dA}_{2}} {\hbox{dt}}=-\hbox{h}_{\rm p}\hbox{A}_{2} + \hbox{h}_{\rm o} \hbox{A}_{1} -\hbox{g}_{\rm o}\hbox{A}_{2}\;\mu\hbox{M (myoplasm [attached post-power stroke XB])}\\ & \frac{\hbox{dP}}{\hbox{dt}}=0.001\times\left(\hbox{h}_{\rm o}\hbox{A}_{1}-\hbox{h}_{\rm p}\hbox{A}_{2}\right)-\hbox{b}_{\rm P}\hbox{P}-\hbox{k}_{\rm P}(\hbox{P}-\hbox{P}_{\rm SR})/ \hbox{V}_{2},\hbox{ mM}\;\hbox{(myoplasmic phosphate)}\\& \frac{\hbox{dP}_{\rm SR}}{\hbox{dt}}=\hbox{k}_{\rm P}(\hbox{P}-\hbox{P}_{\rm SR})/\hbox{V}^{\rm SR}_{2}-[\hbox{A}_{\rm P} (\hbox{P}_{\rm SR}\times 0.001 \times \hbox{Ca}^{\rm SR}_{2} -\hbox{PP})\hbox{H}(\hbox{P}_{\rm SR} \times 0.001 \times \hbox{Ca}^{\rm SR}_{2}-\hbox{PP})\times 0.001 \times \hbox{P}_{\rm SR}\hbox{Ca}^{\rm SR}_{2}\\ & -\hbox{B}_{\rm P}\hbox{P}^{\rm C}_{\rm SR}(\hbox{PP} - \hbox{P}_{\rm SR}\times 0.001 \times \hbox{Ca}^{\rm SR}_{2})\hbox{H}(\hbox{PP}-\hbox{P}_{\rm SR}\times0.001\times \hbox{Ca}^{\rm SR}_{2})],\;\hbox{mM}\,\hbox{(SR phosphate)}\\ & \frac{\hbox{dP}^{\rm C}_{\rm SR}} {\hbox{dt}}=\hbox{A}_{\rm P}(\hbox{P}_{\rm SR}\times0.001\times \hbox{Ca}^{\rm SR}_{2}-\hbox{PP})\hbox{H}(\hbox{P}_{\rm SR} \times 0.001 \times \hbox{Ca}^{\rm SR}_{2}-\hbox{PP})\times 0.001 \times \hbox{P}_{\rm SR}\hbox{Ca}^{\rm SR}_{2}\\ & -\hbox{B}_{\rm P}\hbox{P}^{\rm C}_{\rm SR}(\hbox{PP}-\hbox{P}_{\rm SR} \times 0.001 \times \hbox{Ca}^{\rm SR}_{2})\hbox{H}(\hbox{PP}-\hbox{P}_{\rm SR}\times 0.001 \times \hbox{Ca}^{\rm SR}_{2}),\;\hbox{mM}\;(\hbox{SR phosphate-Ca}^{2+}\hbox{precipitate})\end{aligned}$$

(9)