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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
Summary
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.
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.
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