Mathematical model: boundary and initial conditions and steady-state solution in unperturbed condition

LL Leonardo Lonati SB Sofia Barbieri IG Isabella Guardamagna AO Andrea Ottolenghi GB Giorgio Baiocco

This protocol is extracted from research article:

Radiation-induced cell cycle perturbations: a computational tool validated with flow-cytometry data

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Sci Rep**,
Jan 13, 2021;
DOI:
10.1038/s41598-020-79934-3

Radiation-induced cell cycle perturbations: a computational tool validated with flow-cytometry data

Procedure

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, ${\text{G}}_{1}$(x,t), S(x,t), ${\text{G}}_{2}$(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 ${\text{G}}_{1}$-phase, while the other three compartments are empty. An approximation of the flow-cytometric profile of ${\text{G}}_{1}$-phase at time t is given by Eq. (^{5}):

where ${\overline{m}}_{{G}_{1}}$ is the mean DNA content in ${\text{G}}_{1}$-phase and ${\theta}_{{G}_{1}}^{2}$ is the corresponding variance. For simplicity, the mean parameter is normalized to a relative value $\mathit{x}=1$ for ${\text{G}}_{1}$-phase, thus giving $\mathit{x}=2$ for ${\text{G}}_{2}$-phase and M-phase, while the variance is chosen sufficiently small so that ${G}_{1}\left(x,,,0\right)$ 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 ${\text{T}}_{\mathrm{SDD}}$ 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 ${\theta}_{1}^{2}$ for the Gaussian distribution of cells in ${\text{G}}_{1}$-phase is fixed at 0.05, which is chosen based on the experimental variance of the ${\text{G}}_{1}$-phase sham profiles (around 5% of the mean). The variance in ${\text{G}}_{2}$-phase and M, ${\theta}_{{G}_{2}}^{2}$, is considered as two times ${\theta}_{{G}_{1}}^{2}$. 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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