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Parallel Measurement of Circadian Clock Gene Expression and Hormone Secretion in Human Primary Cell Cultures
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Discrete and ultradiscrete models for biological rhythms comprising a simple negative feedback loop.

Shingo Gibo1, Hiroshi Ito2

  • 1Graduate School of Design, Kyushu University, 4-9-1, Shiobaru Minami-ku, Fukuoka 815-8540, Japan.

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|May 5, 2015
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Summary

Discrete and ultradiscrete models with two variables can generate self-sustained oscillations, challenging previous findings for negative feedback systems. These models exhibit oscillations when degradation rates are lower than synthesis rates.

Keywords:
Biological rhythmsBoolean systemNeimark–Sacker bifurcationSelf-sustained oscillationsTropical discretization

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Area of Science:

  • Systems Biology
  • Theoretical Biology
  • Mathematical Biology

Background:

  • Biological rhythms are often regulated by negative feedback mechanisms.
  • Continuous models with two variables and negative feedback, as proven by Griffith (1968) and Kurosawa et al. (2002), generally do not produce self-sustained oscillations.
  • This limitation highlights a gap in understanding how simple negative feedback systems can generate dynamic biological rhythms.

Purpose of the Study:

  • To propose novel discrete and ultradiscrete feedback models with two variables capable of generating self-sustained oscillations.
  • To investigate the bifurcation structures and parameter conditions required for oscillations in these new models.
  • To explore the relationship between continuous, discrete, and ultradiscrete models in the context of biological rhythms.

Main Methods:

  • Application of tropical discretization and ultradiscretization to a continuous two-variable negative feedback model.
  • Analysis of bifurcation structures, specifically identifying Neimark-Sacker bifurcation.
  • Investigation of parameter conditions, focusing on the ratio of degradation to synthesis rates.
  • Exploration of the reduction of the ultradiscrete model to a Boolean system.

Main Results:

  • The proposed discrete and ultradiscrete models successfully exhibit self-sustained oscillations.
  • Oscillations occur via Neimark-Sacker bifurcation when the degradation rate is lower than the synthesis rate.
  • The ultradiscrete model demonstrates reducibility to a Boolean system under specific conditions.
  • These findings extend the understanding of oscillation generation in simplified biological feedback systems.

Conclusions:

  • Discrete and ultradiscrete modeling approaches can overcome limitations of continuous models in generating biological oscillations.
  • The ratio of synthesis to degradation rates is a critical factor in achieving oscillations in these models.
  • The developed models provide a new framework for studying biological rhythms and offer insights into the transition to simpler, potentially Boolean, regulatory logic.