# Finding and Breaking Lie Symmetries: Implications for Structural Identifiability and Observability in Biological Modelling

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Structural Identifiability and Observability

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Theorem**

**1.**

#### 2.2. Lie Symmetries and Structural Unidentifiability

**Definition**

**5**

**.**The infinitesimal generator of the one-parameter Lie group of transformations is the differential operator:

**Theorem**

**2.**

**Theorem**

**3**

**.**The one-parameter Lie group of transformations (9) is equivalent to:

#### 2.2.1. Computing Symmetries

- Univariate:$${\eta}_{i}\left(x\right)=\sum _{d=0}^{{d}_{max}}{r}_{i,d}{x}_{i}^{d},\phantom{\rule{5.69046pt}{0ex}}i=1,\dots ,{n}^{\ast}\phantom{\rule{3.33333pt}{0ex}}.$$
- Partially variate:$$\begin{array}{cc}\hfill {\eta}_{i}\left(x\right)& =\sum _{{d}_{i},{d}_{m+1},\dots ,{d}_{m+q}=0}^{|d|={d}_{max}}{r}_{i,d}{x}_{i}^{{d}_{i}}{x}_{m+1}^{{d}_{m+1}}\xb7\xb7\xb7{x}_{m+q}^{{d}_{m+q}},\phantom{\rule{5.69046pt}{0ex}}i=1,\dots m\phantom{\rule{3.33333pt}{0ex}},\hfill \\ \hfill {\eta}_{i}\left(x\right)& =\sum _{{d}_{m+1},\dots ,{d}_{m+q}=0}^{|d|={d}_{max}}{r}_{i,d}{x}_{m+1}^{{d}_{m+1}}\xb7\xb7\xb7{x}_{m+q}^{{d}_{m+q}},\phantom{\rule{5.69046pt}{0ex}}i=m+1,\dots m+q\phantom{\rule{3.33333pt}{0ex}},\hfill \\ \hfill {\eta}_{i}\left(x\right)& =\sum _{{d}_{i},{d}_{m+1},\dots ,{d}_{m+q}=0}^{|d|={d}_{max}}{r}_{i,d}{x}_{i}^{{d}_{i}}{x}_{m+1}^{{d}_{m+1}}\xb7\xb7\xb7{x}_{m+q}^{{d}_{m+q}},\phantom{\rule{5.69046pt}{0ex}}i=m+q+1,\dots {n}^{\ast}\phantom{\rule{3.33333pt}{0ex}}.\hfill \end{array}$$
- Multivariate:$$\begin{array}{cc}\hfill {\eta}_{i}\left(x\right)& =\sum _{{d}_{1},\dots ,{d}_{m+q}=0}^{|d|={d}_{max}}{r}_{i,d}{x}_{1}^{{d}_{1}}\xb7\xb7\xb7{x}_{m+q}^{{d}_{m+q}},\phantom{\rule{5.69046pt}{0ex}}i=1,\dots m\phantom{\rule{3.33333pt}{0ex}},\hfill \\ \hfill {\eta}_{i}\left(x\right)& =\sum _{{d}_{m+1},\dots ,{d}_{m+q}=0}^{|d|={d}_{max}}{r}_{i,d}{x}_{m+1}^{{d}_{m+1}}\xb7\xb7\xb7{x}_{m+q}^{{d}_{m+q}},\phantom{\rule{5.69046pt}{0ex}}i=m+1,\dots m+q\phantom{\rule{3.33333pt}{0ex}},\hfill \\ \hfill {\eta}_{i}\left(x\right)& =\sum _{{d}_{1},\dots ,{d}_{{n}^{\ast}}=0}^{|d|={d}_{max}}{r}_{i,d}{x}_{1}^{{d}_{1}}\xb7\xb7\xb7{x}_{{n}^{\ast}}^{{d}_{{n}^{\ast}}},\phantom{\rule{5.69046pt}{0ex}}i=1,\dots {n}^{\ast}\phantom{\rule{3.33333pt}{0ex}}.\hfill \end{array}$$

**Theorem**

**4**

**.**The system of ordinary differential equations admits a one-parameter Lie group of transformations defined by the infinitesimal generator (12) if and only if:

- Univariate and partial:$$\begin{array}{cc}\hfill {P}^{k}{Q}^{k}{\displaystyle \frac{\partial {\eta}_{k}}{\partial {x}_{k}}}-\sum _{i=1}^{{n}^{\ast}}{\eta}_{i}[{P}_{{x}_{i}}^{k}{Q}^{k}-{P}^{k}{Q}_{{x}_{i}}^{k}]=& 0,\phantom{\rule{5.69046pt}{0ex}}k=1,\dots m\phantom{\rule{3.33333pt}{0ex}},\hfill \\ \hfill \sum _{i=1}^{{n}^{\ast}}{\eta}_{i}[{R}_{{x}_{i}}^{l}{S}^{l}-{R}^{l}{S}_{{x}_{i}}^{l}]=& 0,\phantom{\rule{5.69046pt}{0ex}}l=1,\dots n\phantom{\rule{3.33333pt}{0ex}}.\hfill \end{array}$$
- Multivariate:$$\begin{array}{cc}\hfill \sum _{j=1}^{m}{P}^{j}{Q}^{k}\left(\prod _{b\ne j}{Q}^{b}\right){\displaystyle \frac{\partial {\eta}_{k}}{\partial {x}_{j}}}-\sum _{i=1}^{{n}^{\ast}}{\eta}_{i}\left(\prod _{b\ne k}{Q}^{b}\right)[{P}_{{x}_{i}}^{k}{Q}^{k}-{P}^{k}{Q}_{{x}_{i}}^{k}]=& 0,\phantom{\rule{5.69046pt}{0ex}}k=1,\dots m\phantom{\rule{3.33333pt}{0ex}},\hfill \\ \hfill \sum _{i=1}^{{n}^{\ast}}{\eta}_{i}[{R}_{{x}_{i}}^{l}{S}^{l}-{R}^{l}{S}_{{x}_{i}}^{l}]=& 0,\phantom{\rule{5.69046pt}{0ex}}l=1,\dots n\phantom{\rule{3.33333pt}{0ex}}.\hfill \end{array}$$

#### 2.2.2. Initial Conditions

#### 2.3. Implementation

## 3. Results

#### 3.1. Simple Chemical Reaction

**(Page 1)**, corresponding to the initial model, all states and parameters are unobservable; in the second one, Figure 3

**(Page 2)**, corresponding to the model with Lie transformations given by the second generator, states and parameters are observable.

#### 3.2. Pharmacokinetic Model

${\dot{{k}_{2}}}^{\ast}=-{k}_{1}^{\ast}\phantom{\rule{3.33333pt}{0ex}},$ | ${k}_{2}^{\ast}\left(0\right)={k}_{2}$ , |

${\dot{{k}_{3}}}^{\ast}=-{k}_{3}^{\ast}{\displaystyle \frac{({k}_{1}^{\ast}+{k}_{2}^{\ast})}{{k}_{2}^{\ast}}}\phantom{\rule{3.33333pt}{0ex}},$ | ${k}_{3}^{\ast}\left(0\right)={k}_{3}$ , |

${\dot{{k}_{7}}}^{\ast}={k}_{3}^{\ast}{\displaystyle \frac{({k}_{1}^{\ast}+{k}_{2}^{\ast})}{{k}_{2}^{\ast}}}\phantom{\rule{3.33333pt}{0ex}},$ | ${k}_{7}^{\ast}\left(0\right)={k}_{7}$ , |

${\dot{{s}_{3}}}^{\ast}={\displaystyle \frac{{k}_{1}^{\ast}{s}_{3}^{\ast}}{{k}_{2}^{\ast}}}\phantom{\rule{3.33333pt}{0ex}},$ | ${s}_{3}^{\ast}\left(0\right)={s}_{3}$ , |

${\dot{{x}_{3}}}^{\ast}=-{\displaystyle \frac{{k}_{1}^{\ast}{x}_{3}^{\ast}}{{k}_{2}^{\ast}}}\phantom{\rule{3.33333pt}{0ex}},$ | ${x}_{3}^{\ast}\left(0\right)={x}_{3}$ . |

#### 3.3. NF-$\kappa $B Signalling Pathway

- Scaling symmetry was found for the known input function and two parameters:$${u}^{\ast}=u{e}^{-\epsilon};\phantom{\rule{8.5359pt}{0ex}}{k}_{i}^{\ast}={k}_{i}{e}^{\epsilon}\phantom{\rule{5.69046pt}{0ex}}i=0,1\phantom{\rule{3.33333pt}{0ex}}.$$
- The second symmetry was also for the input function and one parameter, in this case, a Mobius and translation symmetry, respectively:$${u}^{\ast}=-{\displaystyle \frac{u}{\epsilon u-1}};\phantom{\rule{8.5359pt}{0ex}}{k}_{0}^{\ast}={k}_{0}+\epsilon \phantom{\rule{3.33333pt}{0ex}}.$$
- Another scaling symmetry involving one state and two parameters:$${x}_{7}^{\ast}={x}_{7}{e}^{-\epsilon};\phantom{\rule{8.5359pt}{0ex}}{k}_{6}^{\ast}={k}_{6}{e}^{-\epsilon};\phantom{\rule{8.5359pt}{0ex}}{k}_{8}^{\ast}={k}_{8}{e}^{\epsilon}\phantom{\rule{3.33333pt}{0ex}}.$$
- One scaling type symmetry is admitted using the parameter ${\rho}_{\mathrm{vol}}$. All the nucleus states, as well as four parameters, take part in the symmetry:$${x}_{i}^{\ast}={x}_{i}{e}^{\epsilon}\phantom{\rule{5.69046pt}{0ex}}i=5,6,9,10;\phantom{\rule{8.5359pt}{0ex}}{s}_{2}^{\ast}={s}_{2}{e}^{-\epsilon}\phantom{\rule{3.33333pt}{0ex}};$$$${k}_{i}^{\ast}={k}_{i}{e}^{-\epsilon}\phantom{\rule{5.69046pt}{0ex}}i=6,10,11;\phantom{\rule{8.5359pt}{0ex}}{\rho}_{\mathrm{vol}}^{\ast}={\rho}_{\mathrm{vol}}{e}^{\epsilon}\phantom{\rule{3.33333pt}{0ex}}.$$
- The last symmetry is the only one that involves the initial condition parameter, ${x}_{{1}_{0}}$. All of the states have a scaling type symmetry, compensated by the scaling factor of ${s}_{i}$ and ${k}_{10}$:$${x}_{i}^{\ast}={x}_{i}{e}^{\epsilon}\phantom{\rule{5.69046pt}{0ex}}i=1,\dots ,10;\phantom{\rule{8.5359pt}{0ex}}{s}_{i}^{\ast}={s}_{i}{e}^{-\epsilon}\phantom{\rule{5.69046pt}{0ex}}i=1,\dots ,4\phantom{\rule{3.33333pt}{0ex}};$$$${k}_{10}^{\ast}={k}_{10}{e}^{-\epsilon};\phantom{\rule{8.5359pt}{0ex}}{x}_{{1}_{0}}^{\ast}={x}_{{1}_{0}}{e}^{\epsilon}\phantom{\rule{3.33333pt}{0ex}}.$$

#### 3.4. Glucose-Insulin Regulation

${\dot{{q}_{1}}}^{\ast}=-{q}_{1}^{\ast}\phantom{\rule{3.33333pt}{0ex}},$ | ${q}_{1}^{\ast}\left(0\right)={q}_{1}$ , |

${\dot{{q}_{2}}}^{\ast}={\displaystyle \frac{u+{p}_{3}^{\ast}{q}_{1}^{\ast}}{{p}_{2}^{\ast}}}\phantom{\rule{3.33333pt}{0ex}},$ | ${q}_{2}^{\ast}\left(0\right)={q}_{2}$ , |

${\dot{{p}_{1}}}^{\ast}={p}_{3}^{\ast}\phantom{\rule{3.33333pt}{0ex}},$ | ${k}_{7}^{\ast}\left(0\right)={k}_{7}$ , |

${\dot{{p}_{2}}}^{\ast}=-{p}_{2}^{\ast}\phantom{\rule{3.33333pt}{0ex}},$ | ${p}_{2}^{\ast}\left(0\right)={p}_{2}$ , |

${\dot{{p}_{3}}}^{\ast}=-{p}_{3}^{\ast}\phantom{\rule{3.33333pt}{0ex}},$ | ${p}_{3}^{\ast}\left(0\right)={p}_{3}$ , |

${\dot{{p}_{4}}}^{\ast}={\displaystyle \frac{{p}_{1}^{\ast}{p}_{3}^{\ast}-{\left({p}_{3}^{\ast}\right)}^{2}+{p}_{2}^{\ast}{p}_{4}^{\ast}}{{p}_{2}^{\ast}}}\phantom{\rule{3.33333pt}{0ex}},$ | ${p}_{4}^{\ast}\left(0\right)={p}_{4}$ . |

${\dot{{V}_{p}}}^{\ast}=-{V}_{p}^{\ast}\phantom{\rule{3.33333pt}{0ex}},$ | ${V}_{p}^{\ast}\left(0\right)={V}_{p}$ . |

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

CR | Chemical reaction |

FISPO | Full input, state, and parameter observability |

ICS | Initial conditions |

IVP | Initial value problem |

ODE | Ordinary differential equation |

OIC | Observability-identifiability condition |

PDE | Partial differential equation |

PK | Pharmacokinetic |

SLI | Structurally locally identifiable |

SU | Structurally unidentifiable |

## References

- DiStefano, J., III. Dynamic Systems Biology Modeling and Simulation; Academic Press: Amsterdam, The Netherlands, 2015. [Google Scholar]
- Sontag, E.D. Some new directions in control theory inspired by systems biology. IET Syst. Biol.
**2004**, 1, 9–18. [Google Scholar] [CrossRef] [PubMed] - Åström, K.J.; Kumar, P.R. Control: A perspective. Automatica
**2014**, 50, 3–43. [Google Scholar] [CrossRef] - Bellman, R.; Åström, K.J. On structural identifiability. Math. Biosci.
**1970**, 7, 329–339. [Google Scholar] [CrossRef] - Miao, H.; Xia, X.; Perelson, A.S.; Wu, H. On identifiability of nonlinear ODE models and applications in viral dynamics. SIAM Rev.
**2011**, 53, 3–39. [Google Scholar] [CrossRef] - Chiş, O.T.; Banga, J.R.; Balsa-Canto, E. Structural identifiability of systems biology models: A critical comparison of methods. PLoS ONE
**2011**, 6, e27755. [Google Scholar] [CrossRef] [Green Version] - Raue, A.; Karlsson, J.; Saccomani, M.P.; Jirstrand, M.; Timmer, J. Comparison of approaches for parameter identifiability analysis of biological systems. Bioinformatics
**2014**, 30, 1440–1448. [Google Scholar] [CrossRef] [Green Version] - Villaverde, A.F. Observability and Structural Identifiability of Nonlinear Biological Systems. Complexity
**2019**, 2019, 8497093. [Google Scholar] [CrossRef] [Green Version] - Bluman, G.; Anco, S. Symmetry and Integration Methods for Differential Equations; Springer Science & Business Media: New York, NY, USA, 2008. [Google Scholar]
- Oliveri, F. Lie symmetries of differential equations: Classical results and recent contributions. Symmetry
**2010**, 2, 658–706. [Google Scholar] [CrossRef] [Green Version] - Arrigo, D.J. Symmetry Analysis of Differential Equations: An Introduction; John Wiley & Sons: New York, NY, USA, 2015. [Google Scholar]
- Yates, J.W.; Evans, N.D.; Chappell, M.J. Structural identifiability analysis via symmetries of differential equations. Automatica
**2009**, 45, 2585–2591. [Google Scholar] [CrossRef] [Green Version] - Vajda, S.; Godfrey, K.R.; Rabitz, H. Similarity transformation approach to identifiability analysis of nonlinear compartmental models. Math. Biosci.
**1989**, 93, 217–248. [Google Scholar] [CrossRef] - Evans, N.D.; Chapman, M.J.; Chappell, M.J.; Godfrey, K.R. Identifiability of uncontrolled nonlinear rational systems. Automatica
**2002**, 38, 1799–1805. [Google Scholar] [CrossRef] - Merkt, B.; Timmer, J.; Kaschek, D. Higher-order Lie symmetries in identifiability and predictability analysis of dynamic models. Phys. Rev. E
**2015**, 92, 012920. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Villaverde, A.F.; Barreiro, A.; Papachristodoulou, A. Structural identifiability of dynamic systems biology models. PLoS Comput. Biol.
**2016**, 12, e1005153. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Villaverde, A.F.; Evans, N.D.; Chappell, M.J.; Banga, J.R. Input-dependent structural identifiability of nonlinear systems. IEEE Control Syst. Lett.
**2019**, 3, 272–277. [Google Scholar] [CrossRef] [Green Version] - Villaverde, A.F.; Tsiantis, N.; Banga, J.R. Full observability and estimation of unknown inputs, states, and parameters of nonlinear biological models. J. R. Soc. Interface
**2019**. in review. [Google Scholar] [CrossRef] [Green Version] - Anguelova, M.; Karlsson, J.; Jirstrand, M. Minimal output sets for identifiability. Math. Biosci.
**2012**, 239, 139–153. [Google Scholar] [CrossRef] - Saccomani, M.P.; Audoly, S.; D’Angiò, L. Parameter identifiability of nonlinear systems: The role of initial conditions. Automatica
**2003**, 39, 619–632. [Google Scholar] [CrossRef] - Rocha Filho, T.M.; Figueiredo, A. [SADE] a Maple package for the symmetry analysis of differential equations. Comput. Phys. Commun.
**2011**, 182, 467–476. [Google Scholar] [CrossRef] [Green Version] - Raksanyi, A. Utilisation du calcul formel pour l’étude des systèmes d’équations polynomiales applications en modélisation. Ph.D. Thesis, Université de Paris-Dauphine, Paris, France, 1986. [Google Scholar]
- Lipniacki, T.; Paszek, P.; Brasier, A.R.; Luxon, B.; Kimmel, M. Mathematical model of NF-kB regulatory module. J. Theor. Biol.
**2004**, 228, 195–215. [Google Scholar] [CrossRef] - Bolie, V.W. Coefficients of normal blood glucose regulation. J. Appl. Physiol.
**1961**, 16, 783–788. [Google Scholar] [CrossRef] - Sedoglavic, A. A probabilistic algorithm to test local algebraic observability in polynomial time. J. Symb. Comput.
**2002**, 33, 735–755. [Google Scholar] [CrossRef] [Green Version] - Cheong, R.; Hoffmann, A.; Levchenko, A. Understanding NF-kB signaling via mathematical modeling. Mol. Syst. Biol.
**2008**, 4, 192. [Google Scholar] [CrossRef] [PubMed] - Cobelli, C.; DiStefano, J. Parameter and structural identifiability concepts and ambiguities: A critical review and analysis. Am. J. Physiol.-Regulat. Integr. Comp. Physiol.
**1980**, 239, R7–R24. [Google Scholar] [CrossRef] [PubMed]

**Figure 2.**Diagrams of the models analysed in this article. (

**A**) Simple chemical reaction. (

**B**) Pharmacokinetic model. (

**C**) NF-$\kappa $B signalling pathway. (

**D**) Glucose-insulin regulation system.

**Figure 3.**Output of STRIKE-GOLDD for the initial model

**(Page 1)**and the model with one-parameter Lie transformations

**(Page 2)**.

Model Name (and Acronym) | Reference | States | Parameters | Outputs |
---|---|---|---|---|

Simple chemical reaction (CR) | [15] | A | $k,{s}_{1},{s}_{2}$ | ${A}^{\mathrm{obs}}$ |

Pharmacokinetic model (PK) | [22] | ${x}_{1},{x}_{2},{x}_{3},{x}_{4}$ | ${k}_{1},{k}_{2},{k}_{3},{k}_{5},$ | ${x}_{2}^{\mathrm{obs}},{x}_{3}^{\mathrm{obs}}$ |

${k}_{6},{k}_{7},{s}_{2},{s}_{3}$ | ||||

NF-$\kappa $B signalling pathway (NFKB) | [23] | ${x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5},$ | ${k}_{0},{k}_{1},{k}_{1p},{k}_{2},{k}_{3},{k}_{4},$ | ${y}_{1},{y}_{2},{y}_{3},{y}_{4}$ |

${x}_{6},{x}_{7},{x}_{8},{x}_{9},{x}_{10}$ | ${k}_{5},{k}_{6},{k}_{7},{k}_{8},{k}_{9},{k}_{10},{k}_{11},$ | |||

${s}_{1},{s}_{2},{s}_{3},{s}_{4},{\rho}_{\mathrm{vol}},{I}_{{0}_{\mathrm{cyt}}},{I}_{{0}_{\mathrm{nuc}}}$ | ||||

Glucose-insulin regulation (Bolie) | [24] | ${q}_{1},{q}_{2}$ | ${p}_{1},{p}_{2},{p}_{3},{p}_{4},{V}_{p}$ | h |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Massonis, G.; Villaverde, A.F.
Finding and Breaking Lie Symmetries: Implications for Structural Identifiability and Observability in Biological Modelling. *Symmetry* **2020**, *12*, 469.
https://doi.org/10.3390/sym12030469

**AMA Style**

Massonis G, Villaverde AF.
Finding and Breaking Lie Symmetries: Implications for Structural Identifiability and Observability in Biological Modelling. *Symmetry*. 2020; 12(3):469.
https://doi.org/10.3390/sym12030469

**Chicago/Turabian Style**

Massonis, Gemma, and Alejandro F. Villaverde.
2020. "Finding and Breaking Lie Symmetries: Implications for Structural Identifiability and Observability in Biological Modelling" *Symmetry* 12, no. 3: 469.
https://doi.org/10.3390/sym12030469