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Duality between decomposition and gluing: a theoretical biology via adjoint functors
Taichi Haruna1, Yukio-Pegio Gunji
1Graduate School of Science & Technology, Kobe University, Nada, Kobe 657-8501, Japan. cheetha@kcc.zaq.ne.jp
Bio Systems
|April 17, 2007
Summary
This study formalizes biological concepts like function decomposition and gluing using category theory on directed graphs. The findings reveal invariant structures implying biological conditions such as cycles and anticipatory diagrams.
Area of Science:
- Theoretical biology
- Category theory
- Graph theory
Background:
- Biological systems involve complex interactions that can be understood through functional decomposition and integration.
- Formalizing these biological concepts is crucial for developing predictive models.
Purpose of the Study:
- To mathematically formalize the biological concepts of 'decomposition into functions' and 'gluing functions' using category theory.
- To establish an adjunction between these formalized concepts within the category of directed graphs.
- To derive invariant structures from this adjunction and explore their biological implications.
Main Methods:
- Formalization of biological functions as endofunctors on the category of directed graphs.
- Proof of an adjunction between the decomposition and gluing endofunctors.
- Analysis of invariant structures arising from the established adjunction.
Main Results:
- The 'decomposition into functions' and 'gluing functions' are demonstrated to form an adjunction.
- Invariant structures of this adjunction are identified.
- These invariant structures lead to two significant biological conditions: the existence of cycles in finite graphs and the presence of anticipatory diagrams.
Conclusions:
- The category-theoretic formalization provides a rigorous framework for understanding biological system organization.
- The derived invariant structures offer novel insights into the fundamental properties of biological networks, including cyclicality and anticipation.
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