On the repressilator stability with time-delayed gene expression
DOI:
https://doi.org/10.17072/1994-3598-2018-2-75-87Keywords:
репрессилятор, запаздывание, колебания, математическая биологияAbstract
The repressor is the first genetic regulatory network in synthetic biology, which was artificially constructed at Harvard University in 2000. It is a closed network of three genetic elements lacI, λcI and tetR, which have a natural origin, but are not found in nature in such a combination. The promoter of each of the three genes controls the next cistron via the negative feedback, suppressing the expression of the neighboring gene. In this paper, the nonlinear dynamics of a repressilator, which has time delays in all parts of the regulatory network, has been studied for the first time. Delay can be both natural, i.e. arises during the transcription / translation of genes due to the multistage nature of these processes, and artificial, i.e. specially to be introduced into the work of the regulatory network using synthetic biology technologies. It is assumed that the regulation is carried out by proteins being in a dimeric form. In the paper, the nonlinear dynamics has been considered within the framework of the deterministic description. By applying the method of many time scales, the set of nonlinear differential equations with delay on a slow manifold has been obtained. It is shown that there exists a single equilibrium state which loses its stability in an oscillatory manner at certain values of the control parameters. For a symmetric repressilator, in which all three genes are identical, an analytical solution for the neutral Andronov-Hopf bifurcation curve has been obtained. For the general case of an asymmetric repressilator, neutral curves are found numerically. The place of the model proposed in the present work among other theoretical models of the repressilator is discussed.References
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