# Symmetries in Dynamic Models of Biological Systems: Mathematical Foundations and Implications

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Concepts and Analysis Tools

#### 2.1. Lie Symmetries: Definitions

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Types of Lie symmetries.**We can classify symmetries according to their expressions. Below we list some of the most common ones, along with their infinitesimal generators.

#### 2.2. Finding Lie Symmetries

**Theorem**

**1.**

#### 2.3. Structural Identifiability and Observability

**Definition**

**7.**

**Definition**

**8.**

## 3. Connections: Symmetries and Other Properties

#### 3.1. Symmetries and SIO

#### 3.2. Symmetries and Biological Robustness

## 4. Discussion and Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DC | Dynamical compensation |

FCD | Fold-Change Detection |

IFFL | Incoherent Feed-Forward Loop |

IVP | Initial value problem |

ODE | Ordinary differential equation |

PDE | Partial differential equation |

SIM | Scaling Invariance Method |

SIO | Structural Identifiability and Observability |

SLI | Structurally locally identifiable |

SU | Structurally unidentifiable |

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**Figure 1.**(

**A**) An incoherent feedforward loop and (

**B**) a nonlinear integral feedback loop (NLIFL). The controlled variable is ${x}_{2}$, and the input is u. The equations are: (

**A**): ${\dot{x}}_{1}=u-{x}_{1}$, ${\dot{x}}_{2}=\frac{u}{{x}_{1}}-{x}_{2}$; (

**B**): ${\dot{x}}_{1}={x}_{1}({x}_{2}-{x}_{2}\left(0\right))$, ${\dot{x}}_{2}=\frac{u}{{x}_{1}}-{x}_{2}$. Adapted from [29].

**Figure 2.**Interplay between SIO and adaptation in a system with symmetries. (

**A**) Diagram of a glucose-insulin regulation circuit that keeps plasma glucose concentration within admissible levels. The model equations are given by (24)–(28). (

**B**) Time-courses of glucose concentration over the course of three meals, before and after the system adapts to a change in parameter ${s}_{i}$. The plot illustrates the phenomenon of dynamical compensation: both curves are exactly the same. (

**C**) Lack of identifiability is linked to lack of observability: the two different values of ${s}_{i}$ yield two different insulin curves.

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Villaverde, A.F.
Symmetries in Dynamic Models of Biological Systems: Mathematical Foundations and Implications. *Symmetry* **2022**, *14*, 467.
https://doi.org/10.3390/sym14030467

**AMA Style**

Villaverde AF.
Symmetries in Dynamic Models of Biological Systems: Mathematical Foundations and Implications. *Symmetry*. 2022; 14(3):467.
https://doi.org/10.3390/sym14030467

**Chicago/Turabian Style**

Villaverde, Alejandro F.
2022. "Symmetries in Dynamic Models of Biological Systems: Mathematical Foundations and Implications" *Symmetry* 14, no. 3: 467.
https://doi.org/10.3390/sym14030467