Model Mathematics

q_w_tot = q_w_gut + q_w_epi + q_w_cap

\frac{d}{d t} (q_Na_gut) = (- 2) v_SGLT1 \frac{d}{d t} (q_Na_epi) = (2 v_SGLT1 - (3 v_NKE)) + v_NKCC1 + v_Na

c_Na_gut = \frac{q_Na_gut K_mm}{q_w_gut} c_Na_epi = \frac{q_Na_epi K_mm}{q_w_epi} c_Na_cap = \frac{q_Na_cap K_mm}{q_w_cap}

q_Na_tot = q_Na_gut + q_Na_epi + q_Na_cap

\frac{d}{d t} (q_K_epi) = (2 v_NKE - v_K) + v_NKCC1 - v_KCC1

c_K_epi = \frac{q_K_epi K_mm}{q_w_epi} c_K_cap = \frac{q_K_cap K_mm}{q_w_cap}

\frac{d}{d t} (q_Cl_epi) = 2 v_NKCC1 - v_KCC1

c_Cl_epi = \frac{q_Cl_epi K_mm}{q_w_epi} c_Cl_cap = \frac{q_Cl_cap K_mm}{q_w_cap}

\frac{d}{d t} (q_e) = F ((2 v_SGLT1 - v_NKE - v_K) + v_Na) u_e = \frac{q_e}{C_m}

\frac{d}{d t} (q_Glc_epi) = v_SGLT1 - v_GLUT2 - v_AM

c_Glc_gut = \frac{q_Glc_gut K_mm}{q_w_gut} c_Glc_epi = \frac{q_Glc_epi K_mm}{q_w_epi} c_Glc_cap = \frac{q_Glc_cap K_mm}{q_w_cap}

u_w_epi = \frac{E_w_epi (q_w_epi - U_w_epi)}{{(L_w_epi - q_w_epi)}^{2}}

v_w_epiApex = k_w_m (u_w_gut - u_w_epi - (\frac{RT}{K_mm} (c_Na_gut - c_Na_epi))) v_w_epiBase = k_w_m (u_w_epi - u_w_cap - (\frac{RT}{K_mm} (c_Na_epi - c_Na_cap)))

\frac{d}{d t} (q_ATP) = (- v_NKE) + 32 v_AM \frac{d}{d t} (q_ADP) = v_NKE - (32 v_AM) \frac{d}{d t} (q_Pi) = v_NKE - (32 v_AM)

q_adenosine = q_ATP + q_ADP

c_ATP = \frac{q_ATP K_mm}{q_w_epi} c_ADP = \frac{q_ADP K_mm}{q_w_epi} c_Pi = \frac{q_Pi K_mm}{q_w_epi} c_H = \frac{q_H K_mm}{q_w_epi}

v_SGLT1 = \frac{kappa_SGLT1 (\frac{{(\frac{c_Na_gut}{c_Na_epi})}^{2} c_Glc_gut}{c_Glc_epi} - (ⅇ^{\frac{2 F u_e}{RT}}))}{(1 + {(\frac{c_Na_gut}{20})}^{2}) (1 + {(\frac{c_Na_epi}{100})}^{2})}

v_GLUT2 = \frac{kappa_GLUT2 (\frac{c_Glc_epi}{c_Glc_cap} - 1)}{1 + \frac{c_Glc_epi}{5} + \frac{c_Glc_cap}{10} + \frac{c_Glc_epi c_Glc_cap}{100}}

v_NKE = \frac{kappa_NKE (\frac{{(\frac{c_Na_epi}{c_Na_cap})}^{3} c_ATP k_ATP_eq}{c_ADP c_Pi c_H} - ({(\frac{c_K_epi}{c_K_cap})}^{2} ⅇ^{\frac{F u_e}{RT}}))}{(1 + {(\frac{c_Na_epi}{1})}^{3}) (1 + {(\frac{c_Na_cap}{100})}^{3})}

v_NKCC1 = \frac{kappa_NKCC1 (\frac{\frac{c_Na_cap}{c_Na_epi} c_K_cap}{c_K_epi} {(\frac{c_Cl_cap}{c_Cl_epi})}^{2} - 1)}{(1 + \frac{c_Na_cap}{70}) (1 + \frac{c_Na_epi}{5}) (1 + \frac{c_K_cap}{2}) (1 + \frac{c_K_epi}{60}) (1 + {(\frac{c_Cl_cap}{50})}^{2}) (1 + {(\frac{c_Cl_epi}{10})}^{2})}

v_KCC1 = \frac{kappa_KCC1 (\frac{\frac{c_K_epi}{c_K_cap} c_Cl_epi}{c_Cl_cap} - 1)}{(1 + \frac{c_Cl_cap}{50}) (1 + \frac{c_K_cap}{2}) (1 + \frac{c_Cl_epi}{10}) (1 + \frac{c_K_epi}{60})}

GHKterm = \frac{u_e}{ⅇ^{\frac{F u_e}{RT}} - 1} v_K = kappa_K (q_K_epi ⅇ^{\frac{F u_e}{RT}} - q_K_cap) GHKterm

v_Na = kappa_Na (q_Na_cap - (q_Na_epi ⅇ^{\frac{F u_e}{RT}})) GHKterm

v_AM = kappa_AM (\frac{k_AM_eq c_Glc_epi c_ADP}{c_ATP} - 1)

u_SGLT1_reversal = \frac{RT}{2 F} \ln (\frac{{(\frac{c_Na_gut}{c_Na_epi})}^{2} c_Glc_gut}{c_Glc_epi})

u_K_reversal = \frac{RT}{F} \ln (\frac{c_K_cap}{c_K_epi})

u_Na_reversal = \frac{RT}{F} \ln (\frac{c_Na_cap}{c_Na_epi})

c_Na_gutThreshold = c_Na_epi \sqrt{\frac{c_Glc_epi}{c_Glc_gut}} ⅇ^{\frac{F u_e}{RT}}

c_Na_epiEquilibrium = c_Na_cap {(\frac{\frac{c_ADP}{c_ATP} c_Pi c_H}{k_ATP_eq})}^{0.33334} {(\frac{c_K_epi}{c_K_cap})}^{0.66667} ⅇ^{\frac{F u_e}{3 RT}}

c_Cl_epiEquilibrium = c_Cl_cap \sqrt{\frac{\frac{c_Na_cap}{c_Na_epi} c_K_cap}{c_K_epi}}