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tfg_fucho_rius_mariona.pdf (3.35 MB)

Document type

Bachelor thesis

Publication date

2023-06-13

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cc-by-nc-nd (c) Mariona Fucho Rius, 2023
Please use this identifier to cite or link to this item: https://hdl.handle.net/2445/202975

Dynamical analysis of mushroom bifurcations: deterministic and stochastic approaches

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Abstract

[en] Bifurcation theory has found contemporary applications in synthetic biology, particularly in the field of biosensors [43]. The aim of this thesis is to expand upon the framework presented in the referenced paper, which introduces a model depicting the behavior of mushroom bifurcations. The mushroom bifurcation diagram exhibits four saddle-node bifurcations and involves bistability. Our goal is to develop a comprehensive mathematical formalism that can effectively describe this behavior, both deterministically and stochastically. By doing so, we seek to uncover additional properties regarding the transients exhibited by these biosensors, specifically focusing on optimizing their timer-effect, memory properties, and signaling capabilities. We will introduce stochastic dynamics by considering intrinsic noise in the molecular processes, allowing us to investigate the slowing-down effects in the vicinity of the saddle-nodes and transcritical bifurcations. To conduct this study, we will use three fundamental mathematical tools, which can be regarded as the backbone of our analysis. These mathematical vertebrae include the Lemma of Morse, the Weierstrass Preparation Theorem and, most notably, the Implicit Function Theorem. Through this rigorous analysis, we aim to enhance our understanding of the underlying dynamics of these biosensors and facilitate their further improvement and utilization in various applications.

Description

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2023, Director: Àlex Haro i Josep Sardanyés i Cayuela

Subject

Teoria de la bifurcació, Treballs de fi de grau, Biosensors, Anàlisi estocàstica

Subject (English)

Bifurcation theory, Bachelor's theses, Biosensors, Stochastic analysis

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Treballs Finals de Grau (TFG) - Matemàtiques

Citation

FUCHO RIUS, Mariona. Dynamical analysis of mushroom bifurcations: deterministic and stochastic approaches. [consulted: 8 of June of 2026]. Available at: https://hdl.handle.net/2445/202975

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