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On logical bifurcation diagrams.

Wassim Abou-Jaoudé1, Pedro T Monteiro2

  • 1IBENS, Département de Biologie, Ecole Normale Supérieure, CNRS, Inserm, PSL Research University, F-75005 Paris, France.

Journal of Theoretical Biology
|January 19, 2019
PubMed
Summary
This summary is machine-generated.

This study formalizes logical bifurcation diagrams for discrete dynamical systems. It links continuous parameter changes to discrete model behaviors, enabling analysis of complex biological networks like p53-Mdm2.

Keywords:
Bifurcation diagramsHasse diagramsLogical modelingLogical parametersPiecewise-linear differential equationsp53-Mdm2 network

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Area of Science:

  • Systems Biology
  • Theoretical Biology
  • Computational Biology

Background:

  • Bifurcation theory analyzes differential systems dynamics with parameters.
  • Logical models offer a discrete framework for biological networks.
  • Logical bifurcation diagrams extend bifurcation analysis to discrete models.

Purpose of the Study:

  • To formally define logical bifurcation diagrams.
  • To bridge continuous parameter changes with discrete logical model dynamics.
  • To provide methods for analyzing these diagrams in Boolean networks.

Main Methods:

  • Utilizing the piecewise linear differential (PWLD) framework to introduce bifurcation parameters.
  • Mapping regulatory graphs to sets of PWLD and logical models.
  • Defining logical bifurcation diagrams as sequences of logical parameter valuations and attractors.
  • Developing conditions and characterizations for logical bifurcation diagrams, especially in the Boolean case.
  • Proposing a procedure for determining maximum-length logical bifurcation diagrams in the Boolean case.

Main Results:

  • A formal definition of logical bifurcation diagrams is proposed.
  • The PWLD framework successfully links continuous bifurcation parameters to discrete logical model transitions.
  • Necessary conditions and a characterization for logical bifurcation diagrams in the Boolean case are established.
  • A procedure for finding maximal logical bifurcation diagrams in Boolean networks is presented.

Conclusions:

  • The proposed framework provides a rigorous method for analyzing discrete dynamical systems using bifurcation theory.
  • The methodology allows for the study of parameter-dependent behaviors in biological networks.
  • Application to the p53-Mdm2 network demonstrates the practical utility of logical bifurcation diagrams.