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Mathematical model: boundary and initial conditions and steady-state solution in unperturbed condition
This protocol is extracted from research article:
Radiation-induced cell cycle perturbations: a computational tool validated with flow-cytometry data
Sci Rep, Jan 13, 2021;

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Model equations (3) govern the dynamics between cell-cycle phases. To solve the system and find the time t and DNA content x dependence of the four phases, $G1$(x,t), S(x,t), $G2$(x,t) and M(x,t), boundary and initial conditions are needed, that we now discuss. The following boundary condition (Eq. (4)):

ensures a positive DNA content in all cells at all times.

At first, as initial conditions, all cells are synchronised in $G1$-phase, while the other three compartments are empty. An approximation of the flow-cytometric profile of $G1$-phase at time t is given by Eq. (5):

where $m¯G1$ is the mean DNA content in $G1$-phase and $θG12$ is the corresponding variance. For simplicity, the mean parameter is normalized to a relative value $x=1$ for $G1$-phase, thus giving $x=2$ for $G2$-phase and M-phase, while the variance is chosen sufficiently small so that $G1x,0$ exists only for $x>0$, and can be adapted to simulate the experimental variance of the flow-cytometric profile.

Given the initial conditions as Eq. (5), the model is made evolve until $TSDD$ hours to reach a steady DNA distribution. After such time, the model is considered to give the cell-cycle distribution of cells in exponential growth. The variance $θ12$ for the Gaussian distribution of cells in $G1$-phase is fixed at 0.05, which is chosen based on the experimental variance of the $G1$-phase sham profiles (around 5% of the mean). The variance in $G2$-phase and M, $θG22$, is considered as two times $θG12$. In practice, starting from the initial condition, the full profile in x is superimposed to the solution of the problem in its matricial formulation, that gives the number of cells in each phase at each time (see next section).

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