On the number of limit cycles of the equation \frac{𝑑𝑦}𝑑𝑥=\frac{𝑃(𝑥,𝑦)}𝑄(𝑥,𝑦)’ where 𝑃 and 𝑄 are polynomials of the second degree

1958 ◽  
pp. 177-221 ◽  
Author(s):  
I. G. and Landis Petrovskiĭ
Keyword(s):  
Limit Cycles ◽  
Second Degree

scholarly journals Lac Operon Boolean Models: Dynamical Robustness and Alternative Improvements

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 600 ◽  
Author(s):  
Marco Montalva-Medel ◽  
Thomas Ledger ◽  
Gonzalo A. Ruz ◽  
Eric Goles
Keyword(s):  
Escherichia Coli ◽  
Limit Cycles ◽  
The Other ◽  
Lac Operon ◽  
Boolean Models ◽  
Bistable Behavior

In Veliz-Cuba and Stigler 2011, Boolean models were proposed for the lac operon in Escherichia coli capable of reproducing the operon being OFF, ON and bistable for three (low, medium and high) and two (low and high) parameters, representing the concentration ranges of lactose and glucose, respectively. Of these 6 possible combinations of parameters, 5 produce results that match with the biological experiments of Ozbudak et al., 2004. In the remaining one, the models predict the operon being OFF while biological experiments show a bistable behavior. In this paper, we first explore the robustness of two such models in the sense of how much its attractors change against any deterministic update schedule. We prove mathematically that, in cases where there is no bistability, all the dynamics in both models lack limit cycles while, when bistability appears, one model presents 30% of its dynamics with limit cycles while the other only 23%. Secondly, we propose two alternative improvements consisting of biologically supported modifications; one in which both models match with Ozbudak et al., 2004 in all 6 combinations of parameters and, the other one, where we increase the number of parameters to 9, matching in all these cases with the biological experiments of Ozbudak et al., 2004.

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