MO.nb

Mitotic Oscillator

Based on the two-variable model in
Tyson, J.J. (1991) Modeling the cell division cycle: cdc2 and cyclin interactions. Proc. Natl. Acad. Sci., USA, 88:7328-7332

Create a Model without the event with the Model Builder

newModel[id "MO", name "Mitotic Oscillator with Events [Tyson 1 ... 4; 0.018}] ; m = loadSimulator[] ; s = SBMLNDSolve[m, 500] ; SBMLPlot[s, PlotRange All] ;

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The model has two variables; we will treat the "spikey" variable u as a variable that must pass a threshold for the cell to divide

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Solve for fixed point of the system

${u^′[t]  -1. u[t] + 180. (0.0001 + u[t]^2) z[t], z^′[t] 0.015 - 180. (0.0001 + u[t]^2) z[t]}$

Find the Jacobian

${-1. u + 180. (0.0001 + u^2) z, 0.015 - 180. (0.0001 + u^2) z}$

Find the eigenvalues of the Jacobian

${λ (-0.018 - 180. u^2 + 0. u z + 0. u^3 z)/((0. + 1.036 ) + (0. + 360. ) u^2 - (0. + 720. ) u z)}$

The eigenvalues at the fixed point are purely imaginary

Examine the Fixed point as a function of the parameters

Create model without values for the parameters and then solve for fixed point as a function of the parameter names

newModel[id "MO", name "Mitotic Oscillator with Events [Tyson 1 ... 2754; k4 * z * (k4prime/k4 + u^2)] ; q = loadSimulator[context None] ; odes = SBMLODES/.q

${u^′[t]  -k6 u[t] + k4 (k4prime/k4 + u[t]^2) z[t], z^′[t] k - k4 (k4prime/k4 + u[t]^2) z[t]}$

${{z[t]  (k k6^2)/(k^2 k4 + k4prime k6^2), u[t] k/k6}}$

Fixed Point

${(k k6^2)/(k^2 k4 + k4prime k6^2), k/k6}$

${-k6 u + k4 (k4prime/k4 + u^2) z, k - k4 (k4prime/k4 + u^2) z}$

Calculate Jacobian

( k4prime 2 ) k4 ... me 2 -k4 (------- + u ) -2 k4 u z k4

Find the eigenvalues of the Jacobian

${λ ( (k4prime k6 + k4 k6 u^2))/(2 k4prime + k6 + 2 k4 u^2 - 4 k4 u z)}$

Find a formula for the eigenvalues at the fixed point as a function of the parameters

${λ ( k6 (k^2 k4 + k4prime k6^2)^2)/(2 k^4 k4^2 + k^2 k4 (4 k4prime - 3 k6) k6^2 + k4prime k6^4 (2 k4prime + k6))}$

Add cell division to the model - add mass that divides on a threshold

mass grows exponentially;

event: if mass > threshold AND protein u > threshold, divide cell in half: mass -> mass/2

newModel[id "MO", name "Mitotic Oscillator with Events [Tyson 1 ... 0.175)), eventAssignment {cellmass cellmass/2}] ; createModel["MO.xml"]

Load the model into the simulate and run a simulation

{InterpolationSet[0, 34.2473, {MO`u[t] InterpolatingFunction[{{0., 34.2588}}, <> ... 0.}}, <>][t], MO`cellmass[t] InterpolatingFunction[{{497.02, 500.}}, <>][t]}]}

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Created by Mathematica (February 27, 2004)