Analytic stability analysis of threecomponent selfregulatory genetic circuit
Abstract
A selfregulatory genetic circuit, where a protein acts as a positive regulator of its own production, is known to be a simplest form of biological network with a positive feedback loop. Although at least three components, DNA, RNA, and the protein, are required to form such a circuit, the stability analysis of fixed points of the selfregulatory circuit has been performed only after reducing the system into to a twocomponent system consisting of RNA and protein only, assuming a fast equilibration of the DNA component. Here, the stability of fixed points of the threecomponent positive feedback loop is analyzed by obtaining eigenvalues of full three dimensional Hessian matrix. In addition to rigorously identifying the stable fixed points and the saddle points, detailed information can be obtained, such as the number of positive eigenvalues near a saddle point. In particular, complex eigenvalues is shown to exist for sufficiently slow binding and unbinding of the autoregulatory transcription factor to DNA, leading to oscillatory convergence to the steady states, a novel feature unseen in the twodimensional analysis.
 Publication:

arXiv eprints
 Pub Date:
 August 2014
 arXiv:
 arXiv:1408.0502
 Bibcode:
 2014arXiv1408.0502L
 Keywords:

 Physics  Biological Physics;
 Condensed Matter  Statistical Mechanics;
 Nonlinear Sciences  Adaptation and SelfOrganizing Systems;
 Quantitative Biology  Molecular Networks
 EPrint:
 16 pages, 1 figure (6 small figures), Typos corrected