# Lac Operon Boolean Models: Dynamical Robustness and Alternative Improvements

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Background

- Fixed point (or steady state): configuration $x\in {\{0,1\}}^{n}$ such that $F\left(x\right)=x$.
- Limit cycle: sequence of configurations ${x}^{0},...,{x}^{l-1}\in {\{0,1\}}^{n}$ pairwise distinct such that $F\left({x}^{j}\right)={x}^{j+1}$, for all $j=0,...,l-2$, $F\left({x}^{l-1}\right)={x}^{0}$ and $l>1$ is a integer named the length of the limit cycle.

## 3. The lac Operon Boolean Models to Be Considered: Main Aspects and Choice Justification

- 1.
- All of them have qualitative behaviors that match very well with the experiments performed in [7].
- 2.
- The interaction digraph is composed by nodes that represent mRNA, proteins, and sugars. Its edges represent the type of interaction between the nodes (activation/inhibition).
- 3.
- The dynamic is obtained considering the parallel update schedule.

#### 3.1. The Chosen Models of Veliz-Cuba and Stigler 2011: Those without Catabolite Repression

#### 3.2. The Dynamics Produced by the Original and Reduced Models

**Case****1**:- ${G}_{e}=1$. Any configuration eventually converges to the unique fixed point OFF = $(0,0,0,1,1,1,0,0,0,0)$.
**Case****2**:- ${G}_{e}={L}_{e}={L}_{em}=0$. Any configuration eventually converges to the unique fixed point OFF $=(0,0,0,1,1,1,0,0,0,0)$.
**Case****3**:- ${G}_{e}=0\wedge {L}_{e}={L}_{em}=1$. Any configuration eventually converges to the unique fixed point ON $=(1,1,1,1,0,0,1,1,1,1)$.
**Case****4**:- ${G}_{e}={L}_{e}=0\wedge {L}_{em}=1$. Any configuration eventually converges to one of the two fixed points; OFF $=(0,0,0,1,1,1,0,0,0,0)$ or ON $=(1,1,1,1,0,0,0,1,0,1)$, i.e., bistability is obtained.

#### 3.3. Stochastic Simulations in the Original Model Which Largely Coincide with the Biological Experiments of Ozbudak et al., 2004

- (1)
- Starting with the normal distribution $N(0.6,0.3)$ for $\mathcal{L}$, to generate randomly a set of values for $\mathcal{L}$ and calculate for each of them the corresponding value for the pair $({L}_{e},{L}_{em})$.
- (2)
- Assuming ${G}_{e}=0$, to find which are the steady states of the dynamic obtained for each value $({L}_{e},{L}_{em})$ of (1), it is simply one of the three possibilities showed in the second column of Table 1; OFF, bistable (i.e., OFF and ON) or ON.
- (3)
- To repeat (1) and (2) but for $N(\mu ,0.3)$ with $\mu \in \{0.8,1.0,1.2,...,2.8\}$.

#### 3.4. Our Justification for the Choice of Models without Catabolite Repression

## 4. Results: Dynamical Robustness of the Original and Reduced Models

#### 4.1. Dynamical Robustness of the Original Model

**Lemma**

**1.**

**Proof.**

#### 4.1.1. Cases 1, 2 and 3 for the Original Model

**Proposition**

**1.**

**Proof.**

**Cases****1****and****2**.- It is easy to check that $C=1$ and $L={L}_{m}=0$, $\forall t\ge 1$. This implies that, $A={A}_{m}=0$, $\forall t\ge 2$. Therefore, we have the left digraph of Figure 6.
**Case****3**.- Notice that $C={L}_{m}=1$, $\forall t\ge 1$. Therefore, we have the following sequence of implications; ${A}_{m}=1$, $\forall t\ge 2$⇒$R=0$, $\forall t\ge 3$⇒${R}_{m}=0$, $\forall t\ge 4$. Therefore, we have the right digraph of Figure 6.

#### 4.1.2. Case 4 for the Original Model

#### 4.2. Dynamical Robustness of the Reduced Model

#### 4.2.1. Cases 1, 2 and 3 for the Reduced Model

**Proposition**

**2.**

#### 4.2.2. Case 4 for the Reduced Model

## 5. Alternative Improvements for the Studied Models

#### 5.1. Improvement 1: The Original and Reduced Models Match in All 6 Parameter Combinations with Ozbudak et al., 2004

- (1)
**Bistability:**when $({G}_{e},{L}_{e},{L}_{em})\in \{(0,0,1),(1,1,1)\}$.- (2)
**OFF:**when $({G}_{e},{L}_{e},{L}_{em})\in \{(0,0,0),(1,0,0),(1,0,1)\}$.- (3)
**ON:**when $({G}_{e},{L}_{e},{L}_{em})=(0,1,1)$.

#### 5.2. Improvement 2: (Improved) Original Model Extended to 9 Parameters

## 6. Conclusions

- For the first 3 cases described in Section 3.2 and that included 5 of the 6 combinations of parameters allowed in the original and reduced models, we establish two Propositions proving the non-existence of any limit cycle, whatever the update schedule used.
- For the case 4, where bistability appears, we made an exhaustive analysis of all its possible dynamics generated with any update schedule. Here we detail for both models the average sizes of their attraction basins, the number of dynamics without limit cycles (i.e., only with fixed points) and the number of dynamics with limit cycles (being less than 30% in both models).
- Again in the case 4, its predominant attractor (those that have the bigger attraction basin), changes dramatically; OFF attraction basin being, in average, 8 times bigger than that of ON for the original model while that in the reduced one, the ON basin is almost 2 times bigger than that of OFF.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) The lac operon OFF state is characterized by transcription of low mRNA levels, due to the strong binding of the LacI repressor to operator sequences. (

**b**) The lac operon ON state involves transcription of high mRNA levels and increased production of the different lac proteins.

**Figure 2.**The original model of [20] (

**left**) and its local Boolean functions (

**right**). A solid (dashed) edge represents an activation (inhibition). We use the red box to differentiate the parameters of the model (outside) from its variables (inside).

**Figure 3.**The reduced model of [20] (

**left**) and its local Boolean functions (

**right**). Dashed/solid edges represent inhibitions/activations.

**Figure 4.**State transition graphs of the original model under parallel update schedule for: (

**a**) Case 1, (

**b**) Case 2, (

**c**) Case 3 and (

**d**) Case 4 (bistability appears). The circles represent all the ${2}^{10}=1024$ configurations $(M,P,B,C,R,{R}_{m},A,{A}_{m},L,{L}_{m})\in {\{0,1\}}^{10}$, in particular, the red one correspond to the steady state OFF while the green one correspond to the steady state ON. The sizes of its attraction basins are 1024 for cases 1, 2 and 3 and 1006 and 18 for the OFF and ON steady states of case 4, respectively.

**Figure 5.**State transition graphs of the reduced model under parallel update schedule. Case 4 (

**left**): low glucose and medium lactose. Case 3 (

**center**): low glucose and high lactose. Cases 1 and 2 (

**right**): the other four combination of parameters.

**Figure 6.**Acyclic digraphs without loops obtained when considering the parameters of the cases 1 (

**left**), 2 (left also) and 3 (

**right**) of Section 3.2.

**Figure 7.**Digraph obtained when considering the parameters of case 4 (

**left**) of Section 3.2 and its local functions after considering ${G}_{e}={L}_{e}=L=A=0$ and ${L}_{em}=C=1$ (

**right**).

**Figure 8.**State transition graph of the original model under the parameters of case 4 and considering that the first nodes $M,P,B,C,R,A,L,{L}_{m}$ are updated (in parallel) and then the nodes ${R}_{m},{A}_{m}$ (also in parallel). The size of the attraction basins are 18 and 98 for the steady states ON and OFF, respectively (both in red), and 908 considering the four limit cycles (in blue).

**Figure 9.**The original model (

**left**) and the improved original model (

**right**), where changes are marked in red.

**Figure 10.**The reduced model (

**left**) and the improved reduced model (

**right**), where changes are marked in red.

**Figure 12.**Dynamic of the improved reduced model associated with; low glucose and low lactose (

**top left**), low glucose and medium lactose (

**top center**), low glucose and high lactose (

**top right**), high glucose and low lactose (

**bottom left**), high glucose and medium lactose (

**bottom center**) and high glucose and high lactose (

**bottom right**).

**Figure 13.**The improved model shown in Figure 9 (right) but with the additional parameter ${G}_{em}$. Modified local functions appear in red.

**Figure 14.**Interaction digraph of the improved original model with additional parameter ${G}_{em}$ and the local functions of Figure 13.

Glucose | $\begin{array}{c}\mathbf{Low}\\ {\mathit{G}}_{\mathit{e}}=0\end{array}$ | $\begin{array}{c}\mathbf{High}\\ {\mathit{G}}_{\mathit{e}}=1\end{array}$ | |
---|---|---|---|

Lactose | |||

Low: $({L}_{e},{L}_{em})=(0,0)$ | $\begin{array}{c}\mathrm{Case}\phantom{\rule{4.pt}{0ex}}2\\ \left(\mathrm{OFF}\right)\end{array}$ | $\begin{array}{c}\mathrm{Case}\phantom{\rule{4.pt}{0ex}}1\\ \left(\mathrm{OFF}\right)\end{array}$ | |

Medium: $({L}_{e},{L}_{em})=(0,1)$ | $\begin{array}{c}\mathrm{Case}\phantom{\rule{4.pt}{0ex}}4\\ \left(\mathrm{Bistability}\right)\end{array}$ | $\begin{array}{c}\mathrm{Case}\phantom{\rule{4.pt}{0ex}}1\\ \left(\mathrm{OFF}\right)\end{array}$ | |

High: $({L}_{e},{L}_{em})=(1,1)$ | $\begin{array}{c}\mathrm{Case}\phantom{\rule{4.pt}{0ex}}3\\ \left(\mathrm{ON}\right)\end{array}$ | $\begin{array}{c}\mathrm{Case}\phantom{\rule{4.pt}{0ex}}1\\ \left(\mathrm{OFF}\right)\end{array}$ |

**Table 2.**Size of the attraction basin associated with the dynamics of cases 1, 2 and 3 for the original model (Section 3.2), and considering any of the $|{S}_{10}|=102,247,563$ possible update schedules.

Attractor | OFF | ON | |
---|---|---|---|

Case | |||

1: ${G}_{e}=1$ | 1024 | 0 | |

2: ${G}_{e}={L}_{e}={L}_{em}=0$ | 1024 | 0 | |

3: ${G}_{e}=0\wedge {L}_{e}={L}_{em}=1$ | 0 | 1024 |

**Table 3.**Average size of the attraction basin for case 4 of the original model (Section 3.2) and calculated over the set ${S}_{10}$, $FP\subseteq S$ (set of update schedules whose dynamic have only steady states) and $LC\subseteq S$ (set of update schedules whose dynamic have limit cycles). Its respective sizes are $|{S}_{10}|$ = 102,247,563 (100%), $\left|FP\right|$ = 71,891,966 (70.3%) and $\left|LC\right|$ = 30,355,597 (29.7%).

Case 4: ${\mathit{G}}_{\mathit{e}}={\mathit{L}}_{\mathit{e}}=0\wedge {\mathit{L}}_{\mathit{em}}=1$ (Bistability) | |||
---|---|---|---|

Attractors | $\mathit{S}$ | $\mathit{FP}$ | $\mathit{LC}$ |

OFF | 684.7 | 908.9 | 153.7 |

ON | 124.4 | 115.1 | 146.5 |

Limit cycles | 214.9 | 0 | 723.8 |

Total | 1024 | 1024 | 1024 |

**Table 4.**Size of the attraction basin associated with the dynamics of cases 1, 2 and 3 for the reduced model (Section 3.2) and considering any of the $|{S}_{3}|=13$ possible updates schedules.

Attractor | OFF | ON | |
---|---|---|---|

Case | |||

1: ${G}_{e}=1$ | 8 | 0 | |

2: ${G}_{e}={L}_{e}={L}_{em}=0$ | 8 | 0 | |

3: ${G}_{e}=0\wedge {L}_{e}={L}_{em}=1$ | 0 | 8 |

**Table 5.**Average size of the attraction basin for case 4 of the reduced model (Section 3.2) and calculated over the set ${S}_{3}$, $FP$ and $LC$ (see definitions in Table 3). Its respective sizes are $|{S}_{3}|=13$ (100%), $\left|FP\right|=10$ (77%) and $\left|LC\right|=3$ (23%).

Case 4: ${\mathit{G}}_{\mathit{e}}={\mathit{L}}_{\mathit{e}}=0\wedge {\mathit{L}}_{\mathit{em}}=1$ (Bistability) | |||
---|---|---|---|

Attractors | $\mathit{S}$ | $\mathit{FP}$ | $\mathit{LC}$ |

OFF | 2.5 | 2.8 | 1.3 |

ON | 4.6 | 5.2 | 2.7 |

Limit cycles | 0.9 | 0 | 4 |

**Table 6.**The possible cases in the improved original and reduced models for each of the six combinations of parameters.

Glucose | $\begin{array}{c}\mathbf{Low}\\ {\mathit{G}}_{\mathit{e}}=0\end{array}$ | $\begin{array}{c}\mathbf{High}\\ {\mathit{G}}_{\mathit{e}}=1\end{array}$ | |
---|---|---|---|

Lactose | |||

Low: $({L}_{e},{L}_{em})=(0,0)$ | OFF | OFF | |

Medium: $({L}_{e},{L}_{em})=(0,1)$ | Bistability | OFF | |

High: $({L}_{e},{L}_{em})=(1,1)$ | ON | Bistability |

Glucose | $\begin{array}{c}\mathbf{Low}\\ ({\mathit{G}}_{\mathit{e}},{\mathit{G}}_{\mathit{em}})=(0,0)\end{array}$ | $\begin{array}{c}\mathbf{Medium}\\ ({\mathit{G}}_{\mathit{e}},{\mathit{G}}_{\mathit{em}})=(0,1)\end{array}$ | $\begin{array}{c}\mathbf{High}\\ ({\mathit{G}}_{\mathit{e}},{\mathit{G}}_{\mathit{em}})=(1,1)\end{array}$ | |
---|---|---|---|---|

Lactose | ||||

$\begin{array}{c}\mathrm{Low}\\ ({L}_{e},{L}_{em})=(0,0)\end{array}$ | OFF | OFF | OFF | |

$\begin{array}{c}\mathrm{Medium}\\ ({L}_{e},{L}_{em})=(0,0)\end{array}$ | Bistability | Bistability | OFF | |

$\begin{array}{c}\mathrm{High}\\ ({L}_{e},{L}_{em})=(0,0)\end{array}$ | ON | ON | Bistability |

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Montalva-Medel, M.; Ledger, T.; Ruz, G.A.; Goles, E. Lac Operon Boolean Models: Dynamical Robustness and Alternative Improvements. *Mathematics* **2021**, *9*, 600.
https://doi.org/10.3390/math9060600

**AMA Style**

Montalva-Medel M, Ledger T, Ruz GA, Goles E. Lac Operon Boolean Models: Dynamical Robustness and Alternative Improvements. *Mathematics*. 2021; 9(6):600.
https://doi.org/10.3390/math9060600

**Chicago/Turabian Style**

Montalva-Medel, Marco, Thomas Ledger, Gonzalo A. Ruz, and Eric Goles. 2021. "Lac Operon Boolean Models: Dynamical Robustness and Alternative Improvements" *Mathematics* 9, no. 6: 600.
https://doi.org/10.3390/math9060600