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

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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