Model cell cycle automaton

TablelO.l Parameter values and initial conditions considered in the various figures based on numerical simulations of the cell cycle automaton model. All figures were established for a uniform distribution of durations of cell cycle phases around a mean value, with variability V. Entrainment The cell...

Fig. 10.2 Waves through cell cycle phases in absence (a, b) or presence (c, d) of entrainment by the circadian clock. The variability of durations for all cell cycle phases is equal to 0% (left column) or 15% (right column). The curves, generated by numerical simulations of the cell cycle automaton model, show the proportions of cells in Cl, S, G2 or M phase as a function of time, for days 10-13. The time step used for simulations is equal to 1 min. The duration of the cell cycle before or in the absence of entrainment is 22 h. The successive phases of the cell cycle have the following mean durations G1 (9 h),...

The Cell Cycle Automaton Model Relation with Other Types of Cellular Automata... 

To clarify the reason why different circadian schedules of 5-FU delivery have distinct cytotoxic effects, we used the cell cycle automaton model to determine the time evolution of the fraction of cells in S phase in response to different patterns of circadian drug administration, for a cell cycle variability of 15%. The results, shown in Fig. 10.5, correspond to the case considered in Fig. 10.4, namely, entrainment of a 22-h cell cycle by the circadian clock. The data for Fig. 10.5a clearly indicate why the circadian schedule with a peak at 4 a.m. is the least toxic. The reason is that the fraction of cells in S phase is then precisely in antiphase with the circadian profile of 5-FU. Since 5-FU only affects cells in the S phase, the circadian delivery of the anticancer drug in this case kills but a negligible amount of cells. 

The cell cycle automaton model permits us to clarify the reason why circadian delivery of 5-FU is least or most toxic when it peaks at 4 a.m. or 4 p.m., respectively. Indeed, the model allows us to determine the position of the peak in S-phase cells relative to that of the peak in 5-FU. As shown in Fig. 10.5, 5-FU is least cytotoxic when the fraction of S-phase cells oscillates in antiphase with 5-FU (when 5-FU peaks at 4 a.m.) and most toxic when both oscillate in phase (when 5-FU peaks at 4 p.m). Intermediate cytotoxicity is observed for other circadian patterns of 5-FU (when the drug peaks at 10 a.m. or 10 p.m.), for which the peak of 5-FU partially overlaps with the peak of S-phase cells. For the continuous infusion of 5-FU, the peak in S-phase cells necessarily occurs in the presence of a constant amount of 5-FU. Hence, the constant delivery pattern is nearly as toxic as the circadian pattern peaking at 4 p.m. 

The results presented here point to the interest of measuring, both in normal and tumor cell populations, parameters such as the duration of the cell cycle phases and their variability, as well as the presence or absence of entrainment by the circadian clock. As shown by the results obtained with the cell cycle automaton model, these data are crucial for using the model to predict the differential outcome of various anticancer drug delivery schedules on normal and tumor cell populations. In a sub-... 

Here, as in a previous publication [33], we used the cell cycle automaton model to probe the cytotoxic effect of various patterns of circadian or continuous 5-FU delivery. The results provide a framework to account for experimental and clinical observations, and to help us predict optimal modes of drug delivery in cancer chronotherapy. By explaining the differential cytotoxicity of various circadian schedules of 5-FU delivery, the model clarifies the foundations of cancer chronothera-peutics. In view of its versatility and reduced number of parameters, the automaton model could readily be applied to probe the administration schedules of other types of anticancer medications active on other phases of the cell cycle. 

Altinok, A., Levi, F., Goldbeter, A. A cell cycle automaton model for probing circadian patterns of anticancer drug delivery. Adv. Drug Deliv. Rev. 2007,... 

The automaton model for the cell cycle (Fig. 10.1a) is based on the following assumptions ... 

Fig. 10.1 (a) Scheme of the automaton model for the cell cycle. The automaton switches sequentially between the phases Cl, S, G2, and M after which the automaton cell divides and two cells enter a new G1 phase. Switching from one phase to the next one occurs in a random manner as soon as the end of the preceding phase is reached, according to a transition probability related to a duration distribution centered for each phase around a mean value D and a variability V (see text). Exit from the cell cycle occurs with a given... 

To determine the effect of circadian rhythms on anticancer drug administration, it is important to incorporate the link between the circadian clock and the cell cycle. Entrainment by the circadian clock can be included in the automaton model by considering that the protein Weel undergoes circadian variation, because the circadian clock proteins CLOCK and BMAL1 induce the expression of the Weel gene (see Fig. 10.1b) [3-5]. Weel is a kinase that phosphorylates and thereby inactivates the protein kinase cdc2 (also known as the cyclin-dependent kinase Cdkl) that controls the transition G2/M and, consequently, the onset of mitosis. 

In the cell cycle model, we consider that the probability (P) of transition from G2 to M, at the end of G2, decreases as Weel rises, according to Eq. (1). Conversely, we assume that the probability of premature transition from G2 to M (i.e. before the end of G2, the duration of which was set when the automaton entered G2) increases with the activity of Cdkl according to Eq. (2). The probability is first determined with respect to Cdkl if the G2/M transition has not occurred, the cell progresses in G2. Only at the end of G2 is the probability of transition to M determined as a function of Weel. 

The automaton model for the cell cycle represents a cellular automaton. Because the latter term has been used in a partly different context, it is useful to distinguish the present model from those considered in previous studies. Cellular automata are often used to describe the spatiotemporal evolution of chemical or biological... 

The present modeling approach to circadian cancer chronotherapy is based on an automaton model for the cell cycle. Continuous approaches to cell cycle progression have also been used to study the link between cell proliferation and circadian rhythms [44] and to determine, in conjunction with optimal control theory, the most efficient circadian schedules of anticancer drug administration [45]. Including more molecular details of the cell cycle in continuous models for cell populations represents a promising line for future research. Hybrid models incorporating molecular details into the automaton approach presented here will also likely be developed. 

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