Model Mathematics

sovr_ze = \{\begin{matrix} \frac{len_thick}{2} if \frac{len_thick}{2} < \frac{SL}{2} \\ \frac{SL}{2} otherwise \end{matrix} sovr_cle = \{\begin{matrix} \frac{SL}{2} - (SL - len_thin) if \frac{SL}{2} - (SL - len_thin) > \frac{len_hbare}{2} \\ \frac{len_hbare}{2} otherwise \end{matrix} len_sovr = sovr_ze - sovr_cle SOVFThick = \frac{len_sovr 2}{len_thick - len_hbare} SOVFThin = \frac{len_sovr}{len_thin}

konT = \frac{{kdHCa}^{m} + {H_cons}^{m}}{{kdHCa}^{m} + H^{m}} kon {Qkon}^{\frac{TmpC - 310}{10}} H = 10e6 10^{- pH} H_cons = 10e6 10^{- 7.15} koffLT = koffL koffmod {Qkoff}^{\frac{TmpC - 310}{10}} koffHT = koffH koffmod {Qkoff}^{\frac{TmpC - 310}{10}} kn_pT = kn_p permtot {Qkn_p}^{\frac{TmpC - 310}{10}} kp_nT = kp_n inprmt {Qkp_n}^{\frac{TmpC - 310}{10}} permtot = \sqrt{| \frac{1}{1 + {(\frac{perm50}{Tropreg})}^{nperm}} |} inprmt = \{\begin{matrix} \frac{1}{permtot} if \frac{1}{permtot} < 100 \\ 100 otherwise \end{matrix} \frac{d}{d time} (TRPNCaL) = dTRPNCaL dTRPNCaL = konT Cai (1 - TRPNCaL) - (koffLT TRPNCaL) \frac{d}{d time} (TRPNCaH) = dTRPNCaH dTRPNCaH = konT Cai (1 - TRPNCaH) - (koffHT TRPNCaH) Tropreg = (1 - SOVFThin) TRPNCaL + SOVFThin TRPNCaH

fappT = fapp xbmodsp {Qfapp}^{\frac{TmpC - 310}{10}} gapslmd = 1 + (1 - SOVFThick) gslmod gappT = gapp gapslmd xbmodsp {Qgapp}^{\frac{TmpC - 310}{10}} hfmd = ⅇ^{\frac{- xXBprer}{| xXBprer |} hfmdc {(\frac{xXBprer}{x_0})}^{2}} hbmd = 1 hfT = hf hfmd xbmodsp {Qhf}^{\frac{TmpC - 310}{10}} hbT = hb hbmd xbmodsp {Qhb}^{\frac{TmpC - 310}{10}} gxbmd = \{\begin{matrix} ⅇ^{sigmap {(\frac{x_0 - xXBpostr}{x_0})}^{2}} if xXBpostr < x_0 \\ ⅇ^{sigman {(\frac{xXBpostr - x_0}{x_0})}^{2}} otherwise \end{matrix} gxbT = gxb gxbmd xbmodsp {Qgxb}^{\frac{TmpC - 310}{10}}

\frac{d}{d time} (N_NoXB) = kp_nT P_NoXB - (kn_pT N_NoXB) \frac{d}{d time} (P_NoXB) = kn_pT N_NoXB - (kp_nT P_NoXB) N = 1 - (P + XBprer + XBpostr) \frac{d}{d time} (P) = kn_pT N - ((kp_nT + alpha1_plus) P) \frac{d}{d time} (XBprer) = dXBprer dXBprer = alpha1_plus P + alpha2_minus XBpostr - ((alpha1_minus + alpha2_plus) XBprer) \frac{d}{d time} (XBpostr) = dXBpostr dXBpostr = alpha3_minus P + alpha2_plus XBprer - ((alpha2_minus + alpha3_plus) XBpostr) alpha1_plus = fappT alpha2_plus = hfT alpha3_plus = \frac{\frac{MgATP gxbT}{MgATP_cons} (kdADP + MgADP_cons)}{kdADP + MgADP} alpha1_minus = \frac{xPi gappT}{xPi_cons} alpha2_minus = \frac{\frac{\frac{H hbT}{H_cons} (kdADP + MgADP_cons)}{MgADP_cons} MgADP}{kdADP + MgADP} alpha3_minus = fxbT fxbT = \frac{\frac{kdADP fappT hfT gxbT}{MgATP_cons}}{\frac{\frac{gappT}{xPi_cons} hbT}{H_cons} kMgATP}

\frac{d}{d time} (xXBprer) = dxXBprer dxXBprer = \frac{dSL}{2} + \frac{xPsi}{dutyprer} ((- alpha1_plus) xXBprer + alpha2_minus (xXBpostr - (x_0 + xXBprer))) \frac{d}{d time} (xXBpostr) = dxXBpostr dxXBpostr = \frac{dSL}{2} + \frac{xPsi}{dutypostr} alpha2_plus (xXBprer + x_0 - xXBpostr) dutyprer = \frac{alpha3_minus alpha2_minus + alpha3_plus alpha1_plus + alpha2_minus alpha1_plus}{alpha1_plus alpha2_plus + alpha3_minus alpha1_minus + alpha3_minus alpha2_plus + alpha3_minus alpha2_minus + alpha3_plus alpha1_plus + alpha2_minus alpha1_plus + alpha2_plus alpha3_plus + alpha3_minus alpha1_minus + alpha3_plus alpha1_minus} dutypostr = \frac{alpha1_plus alpha2_plus + alpha3_minus alpha1_minus + alpha3_minus alpha2_plus}{alpha1_plus alpha2_plus + alpha3_minus alpha1_minus + alpha3_minus alpha2_plus + alpha3_minus alpha2_minus + alpha3_plus alpha1_plus + alpha2_minus alpha1_plus + alpha2_plus alpha3_plus + alpha3_minus alpha1_minus + alpha3_plus alpha1_minus}

fxb = \frac{\frac{kdADP fapp hf gxb}{MgATP_cons}}{\frac{\frac{gapp}{xPi_cons} hb}{H_cons} kMgATP} SSXBprer = \frac{fxb hb + gxb fapp + hbT fapp}{hf gxb + hb gapp + gxb gapp + fxb hb + gxb fapp + hb fapp + fxb gapp + fapp hf + fxb hb} SSXBpostr = \frac{fapp hf + fxb gapp + fxb hb}{hf gxb + hb gapp + gxb gapp + fxb hb + gxb fapp + hb fapp + fxb gapp + fapp hf + fxb hb} F_active = \frac{SOVFThick (xXBpostr XBpostr + xXBprer XBprer)}{x_0 SSXBpostr} ppforce_t = \frac{SL - SLrest}{| SL - SLrest |} PCon_t (ⅇ^{PExp_t | SL - SLrest |} - 1) ppforce_c = \{\begin{matrix} PCon_c (ⅇ^{PExp_c | SL - SL_c |} - 1) if SL > SL_c \\ 0 otherwise \end{matrix} ppforce = ppforce_t + ppforce_c preload = \frac{| SLset - SLrest |}{SLset - SLrest} PCon_t (ⅇ^{PExp_t | SLset - SLrest |} - 1) afterload = \{\begin{matrix} KSE (SLset - SL) if SEon = 1 \\ 0 otherwise \end{matrix} \frac{d}{d time} (SL) = dSL dSL = 0 \frac{d}{d time} (intf) = preload + afterload - (ppforce + F_active)

FrSBXB = \frac{XBpostr + XBprer}{SSXBpostr + SSXBprer} dFrSBXB = \frac{dXBpostr + dXBprer}{SSXBpostr + SSXBprer} dsovr_ze = \{\begin{matrix} (- 0.5) dSL if SL < len_thick \\ 0 otherwise \end{matrix} dsovr_cle = \{\begin{matrix} (- 0.5) dSL if 2 len_thin - SL > len_hbare \\ 0 otherwise \end{matrix} dlen_sovr = dsovr_ze - dsovr_cle dSOVFThin = \frac{dlen_sovr}{len_thin} dSOVFThick = \frac{2 dlen_sovr}{len_thick - len_hbare} TropTot = Trop_conc ((1 - SOVFThin) TRPNCaL + SOVFThin (FrSBXB TRPNCaH + (1 - FrSBXB) TRPNCaL)) dTropTot = Trop_conc ((- dSOVFThin) TRPNCaL + (1 - SOVFThin) dTRPNCaL + dSOVFThin (FrSBXB TRPNCaH + (1 - FrSBXB) TRPNCaL) + SOVFThin (dFrSBXB TRPNCaH + FrSBXB dTRPNCaH + (1 - FrSBXB) dTRPNCaL - (dFrSBXB TRPNCaL)))