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Model Mathematics

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Component: environment

Component: Gi

\frac{d}{d time} (Gi) = Jglut - Jgk

Component: Ge

Ge = glubasa1 + gluamp (1 - glustep) \cos (\frac{time 2 π}{60000 gluper}) + GeStep GeStep = \{\begin{matrix} glustep gluamp if \frac{time - steptime}{1000.0} > 1.0 \\ \frac{time - steptime}{1000.0} glustep gluamp if \frac{time - steptime}{1000.0} > 0.0 \land \frac{time - steptime}{1000.0} < 1.0 \\ 0.0 if \frac{time - steptime}{1000.0} < 0.0 \end{matrix}

Component: G6P

\frac{d}{d time} (G6P) = kappa (Jgk - JPFK)

Component: FBP

\frac{d}{d time} (FBP) = kappa (JPFK - (0.5 JGPDH))

Component: F6P

F6P = 0.3 G6P

Component: Jglut

Jglut = Vglut \frac{(Ge - Gi) Kglut}{(Kglut + Ge) (Kglut + Gi)}

Component: Jgk

Jgk = \frac{Vgk {Gi}^{ngk}}{{Kgk}^{ngk} + {Gi}^{ngk}}

Component: JGPDH

JGPDH = 0.2 \sqrt{\frac{FBP}{1}}

Component: JPFK

JPFK = \frac{pfkbas cat topa16 + cat topb}{bottom16}

Component: w

weight1 = 1 topa1 = 0 bottom1 = 1 weight9 = \frac{AMP}{K1} topa9 = topa8 bottom9 = bottom8 + weight9 weight5 = \frac{FBP}{K2} topa5 = topa4 bottom5 = bottom4 + weight5 weight3 = \frac{{F6P}^{2}}{1 K3} topa3 = topa2 + weight3 bottom3 = bottom2 + weight3 weight2 = \frac{{ATP}^{2}}{1 K4} topa2 = topa1 bottom2 = bottom1 + weight2 weight13 = \frac{AMP FBP}{K1 K2} topa13 = topa12 bottom13 = bottom12 + weight13 weight11 = \frac{AMP {F6P}^{2}}{1 K1 K3 famp} topa11 = topa10 + weight11 bottom11 = bottom10 + weight11 weight10 = \frac{AMP {ATP}^{2}}{1 K1 K4 fmt} topa10 = topa9 bottom10 = bottom9 + weight10 weight6 = \frac{FBP {ATP}^{2}}{1 K2 K4 fbt} topa6 = topa5 bottom6 = bottom5 + weight6 weight4 = \frac{{(F6P ATP)}^{2}}{1 fatp K3 K4} topa4 = topa3 + weight4 bottom4 = bottom3 + weight4 weight7 = \frac{FBP {F6P}^{2}}{1 K2 K3 ffbp} topa7 = topa6 + weight7 bottom7 = bottom6 + weight7 weight15 = \frac{AMP FBP {F6P}^{2}}{1 K1 K2 K3 ffbp famp} topa15 = topa14 topb = weight15 bottom15 = bottom14 + weight15 weight8 = \frac{FBP {F6P}^{2} {ATP}^{2}}{1 K2 K3 K4 ffbp fbt fatp} topa8 = topa7 + weight8 bottom8 = bottom7 + weight8 weight12 = \frac{AMP {F6P}^{2} {ATP}^{2}}{1 K1 K3 K4 famp fmt fatp} topa12 = topa11 + weight12 bottom12 = bottom11 + weight12 weight14 = \frac{AMP FBP {ATP}^{2}}{1 K1 K2 K4 fbt fmt} topa14 = topa13 bottom14 = bottom13 + weight14 weight16 = \frac{AMP FBP {F6P}^{2} {ATP}^{2}}{1 K1 K2 K3 K4 ffbp famp fbt fmt fatp} topa16 = topa15 + weight16 bottom16 = bottom15 + weight16

Component: ATP

ATP = 0.5 (Atot + rad - ADP) rad = \sqrt{{(ADP - Atot)}^{2} - (4 {ADP}^{2})}

Component: AMP

AMP = \frac{{ADP}^{2}}{ATP}

Component: ADP

\frac{d}{d time} (ADP) = autoadp (+ \frac{ATP - (ADP ⅇ^{fback \frac{1 - Ca}{r1}})}{tau_a}) + 1 (1 - autoadp) (adpknot - ADP) y = \frac{ky JGPDH}{kg + JGPDH} fback = r + y

Component: membrane

\frac{d}{d time} (v) = \frac{- (I_K + I_Ca + I_K_Ca + I_K_ATP)}{cm}

Component: I_K

I_K = gK n (v - vK)

Component: n

\frac{d}{d time} (n) = \frac{n_infinity - n}{tau_n} tau_n = \frac{1}{0.035 \cosh (\frac{v - (-16)}{22.4})} n_infinity = 0.5 (1 + \tanh (\frac{v - (-16)}{11.2}))

Component: I_Ca

I_Ca = gCa m_infinity (v - vCa)

Component: m

m_infinity = 0.5 (1 + \tanh (\frac{v - (-20)}{24}))

Component: I_K_Ca

I_K_Ca = \frac{gkCa}{1 + {(\frac{KD}{Ca})}^{nh}} (v - vK)

Component: I_K_ATP

I_K_ATP = gkATP_bar katpo (v - vK) katpo = 20.0 \frac{topo}{bottomo} topo = 0.08 (1 + \frac{2 MgADP}{17}) + 0.89 {(\frac{MgADP}{17})}^{2} bottomo = {(1 + \frac{MgADP}{17})}^{2} (1 + \frac{ADP3}{26} + \frac{ATP4}{1}) MgADP = 0.165 ADP ADP3 = 0.135 ADP ATP4 = 0.05 ATP

Component: Ca

\frac{d}{d time} (Ca) = fcyt (Jmem + Jer)

Component: Caer

\frac{d}{d time} (Caer) = (- fer) sigmav Jer

Component: Jmem

Jmem = - (alpha 1 I_Ca + kPMCA Ca)

Component: Jer

Jer = Jleak - JSERCA

Component: JSERCA

JSERCA = kSERCA Ca

Component: Jleak

Jleak = pleak (Caer - Ca)

Component: I

\frac{d}{d time} (I) = \frac{I_infinity - I}{tau_I} I_infinity = I_max \frac{{Ca}^{delta}}{{ki}^{delta} + {Ca}^{delta}}

Component: model_parameters

Source: Derived from workspace Pedersen, Bertram, Sherman, 2005 at changeset 54a5b1477c85.

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Intra- and inter-islet synchronization of metabolically driven insulin secretion