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Analysis of biochemical reactions models with delays. (English) Zbl 1205.92030
Summary: Deterministic descriptions of three biochemical reaction channels considered by D. Bratsun et al. [Proc. Natl. Acad. Sci. USA 102, No. 41, 14593–14598 (2005)] are studied. These descriptions are based on the mass action law and for the simple protein production with delayed degradation differ from that proposed by Bratsun et al. An explicit solution to this model is calculated. For the model of reaction with negative feedback and delayed production, global stability of a unique positive steady state is proved. According to the models of these two reaction channels there cannot appear delayed induced oscillations. For the model of reaction with negative feedback, dimerisation and delayed protein production, local stability for a unique positive steady state is shown for some range of parameters. It is also proved that for some range of parameters the destabilisation due to the increasing delay can occur and delayed induced oscillations may appear.

MSC:
92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
34K60 Qualitative investigation and simulation of models involving functional-differential equations
37N25 Dynamical systems in biology
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