# Structural Properties of Dynamic Systems Biology Models: Identifiability, Reachability, and Initial Conditions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background: Nonlinear Observability, Reachability, and Identifiability

#### 2.1. Notation and Differential Geometry Concepts

- $\{{f}_{l}\left(x\right),\cdots ,{f}_{k}\left(x\right)\}$ is a linearly independent set for each $x\in U$.
- $\Delta \left(x\right)=\mathbf{span}\{{f}_{1}\left(x\right),\cdots ,{f}_{k}\left(x\right)\}$, $\forall x\in U$ [17].

#### 2.2. Observability

#### 2.2.1. Linear Observability

#### 2.2.2. Nonlinear Observability

**Theorem**

**1.**

**Observability Rank Condition (ORC).**

#### 2.3. Controllability and Reachability

#### 2.3.1. Linear Controllability and Reachability

#### 2.3.2. Nonlinear Controllability and Reachability

**Theorem**

**2.**

**Controllability Rank Condition (CRC).**

**Theorem**

**3.**

**Reachability Theorem.**

#### 2.4. Structural Identifiability as Observability

**Theorem**

**4.**

**Observability-Identifiability Condition (OIC).**

**Proof.**

## 3. Results

#### 3.1. Reachability Is Neither Necessary Nor Sufficient for Identifiability

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

#### 3.2. The Results of Identifiability Tests Can Be Misleading for Certain State Vectors

**Example**

**4.**

**Example**

**5.**

#### 3.3. Knowledge of Additional Initial Conditions Can Increase Identifiability

**Example**

**6.**

#### 3.4. Guaranteeing the Results of the Structural Identifiability Test

- Check the Observability-Identifiability condition (OIC, Theorem 4) for a generic symbolic augmented vector $\tilde{x}$. If the matrix is not full rank, i.e., $\mathrm{rank}\left({\mathcal{O}}_{I}\left(\tilde{x}\right)\right)<n+q$, then the model is structurally unidentifiable and no further tests are needed. If, however, it is full rank, i.e., $\mathrm{rank}\left({\mathcal{O}}_{I}\left(\tilde{x}\right)\right)=n+q$, the model is generically structurally identifiable. However, in this case there may be loss of identifiability for certain states; to assess this we proceed to the next step.
- Check the Observability-Identifiability condition (OIC, Theorem 4) for a particular vector of initial conditions of interest, ${\tilde{x}}_{0}$. If $\mathrm{rank}\left({\mathcal{O}}_{I}\left({\tilde{x}}_{0}\right)\right)<n+q$, there has been a loss of rank which indicates loss of local identifiability (as in Example 4), but that may be possible to overcome (as in Example 5). To assess this point, we go to step 3.
- Simulate the model starting from ${x}_{0}$ to see if there are any states that remain zero.

#### 3.5. Finding Specific Initial Conditions That Lead to Loss of Rank in the OIC

#### 3.5.1. Finding Solutions Analytically

#### 3.5.2. Finding Solutions via Optimization

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

ORC | Observability Rank Condition |

CRC | Controllability Rank Condition |

OIC | Observability-Identifiability Condition |

ODE | Ordinary Differential Equation |

EAR | Exact Arithmetic Rank |

SI | Structural Identifiability |

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**Figure 1.**Time courses of the ${M}_{3}$ model used in Example 3. Panel (

**A**) shows the evolution of the system starting from non-zero initial conditions, for two different parameter vectors. From this plot it would seem that the model is structurally identifiable, since different parameter vectors yield different model outputs; Panel (

**B**) shows the same time courses for initial condition ${x}_{2}(t=0)=0$; in this case the model output is identical for two different parameter vectors, which makes the parameters indistinguishable; Panel (

**C**) shows that it is actually impossible to distinguish between initial condition ${x}_{2}(t=0)=0$ and initial condition ${x}_{2}(t=0)=1$: the output of the model simulated with two wildly different parameter vectors can be the same. This illustrates that unidentifiability is inescapable if we only measure ${x}_{1}$; Panel (

**D**) shows the model output for two parameter vectors and initial condition ${x}_{1}(t=0)=0$; for this case the model output remains at zero, and there is a loss of reachability.

**Figure 2.**Time courses of the Robertson model used in Example 5. The vertical axis shows the concentration curves, which have been normalized to adopt values between 0 and 1 to facilitate the visualization.

**Table 1.**Main properties of the examples used in this paper. The term “generic” applied to SI (structural identifiability) or reachability means that the model has said property for almost all values of x. The last three columns refer to the possibility of losing said properties for certain values of x. N/A stands for Not Applicable.

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Villaverde, A.F.; Banga, J.R. Structural Properties of Dynamic Systems Biology Models: Identifiability, Reachability, and Initial Conditions. *Processes* **2017**, *5*, 29.
https://doi.org/10.3390/pr5020029

**AMA Style**

Villaverde AF, Banga JR. Structural Properties of Dynamic Systems Biology Models: Identifiability, Reachability, and Initial Conditions. *Processes*. 2017; 5(2):29.
https://doi.org/10.3390/pr5020029

**Chicago/Turabian Style**

Villaverde, Alejandro F, and Julio R Banga. 2017. "Structural Properties of Dynamic Systems Biology Models: Identifiability, Reachability, and Initial Conditions" *Processes* 5, no. 2: 29.
https://doi.org/10.3390/pr5020029