# Stability and Robustness of Unbalanced Genetic Toggle Switches in the Presence of Scarce Resources

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Mathematical Model and Parameters

#### 2.2. Stability Analysis

#### 2.3. Robustness Analysis

## 3. Results

#### 3.1. Stability Properties

#### 3.2. Robustness Properties

#### 3.3. Balancing via Optimized Competition

#### 3.4. Context Effects

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A. Stability Profile of the Unbalanced Toggle Switch

**Theorem**

**A1**

## Appendix B. Computation of the Potential Landscape and the Potential Barriers

- locate the unique unstable and the two stable fixed points of (6);
- create a set of initial conditions in the range $(y,z)\in [{\alpha}_{y},{\alpha}_{z}]$;
- compute the potential decrease along system trajectories starting from the above initial points according to (9);
- using k-means clustering, partition the endpoints of these trajectories into two clusters;
- having identified the two regions of convergence (${\Omega}_{y}$ and ${\Omega}_{z}$), adjust the initial potentials in both of them so that trajectories converging to the same stable fixed point have the same end potential;
- adjust the initial potentials alongside the border separating ${\Omega}_{y}$ and ${\Omega}_{z}$ so that trajectories starting close but on different sides share the same potential; and

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**Figure 1.**Resource sequestration and unbalancedness both act against bistability. Fixed points and stability profile are determined by the intersection of the nullclines ${f}_{y}(y,z)=0$ and ${f}_{z}(y,z)=0$ depicted in blue; stable and unstable fixed points are denoted by full and empty circles, respectively. (

**a**) Stability profile in case of a balanced toggle switch ($a=1$). Middle panel: moderate resource sequestration ($\alpha >2{\beta}_{0}$) yields bistable dynamics (${\alpha}_{y}={\alpha}_{z}=10$, ${\beta}_{y}={\beta}_{z}=1$). Right panel: increasing resource sequestration above the critical threshold ($2{\beta}_{0}>\alpha $) eventually results in monostable dynamics (${\alpha}_{y}={\alpha}_{z}=10$, ${\beta}_{y}={\beta}_{z}=10$). (

**b**) In the general case when $a\ge 1$, fixed points and stability profile are determined by the intersection of the nullclines with the manifold given by the constraint in (7), depicted with solid gray lines. Middle panel: in case of a balanced toggle switch we have $a=1$, thus (7) simplifies to $y=z$ or $yz=1$. Right panel: increasing a pulls the two branches of (7) apart from each other, thus pushing the dynamics towards monostability.

**Figure 2.**Resource competition and unbalancedness both decrease robustness to noise. (

**a**) In case of balanced bistable toggle switches, the two potential barriers are identical (${h}_{y}={h}_{z}$,

**middle panel**), and resource sequestration lowers both these potential barriers (

**left panel**). Conversely, unbalancedness increases one of the potential barriers at the expense of the other (

**right panel**). Simulation parameters: ${\alpha}_{y}={\alpha}_{z}=10$, ${\beta}_{y}={\beta}_{z}=0.25$ in the left panel; ${\alpha}_{y}={\alpha}_{z}=10$, ${\beta}_{y}={\beta}_{z}=0$ in the middle panel; ${\alpha}_{y}=9.33$, ${\alpha}_{z}=10.72$, ${\beta}_{y}={\beta}_{z}=0$ in the right panel (thus $a=1.15$). In all panels ${\alpha}_{0}=10$. (

**b**) Based on the robustness of the metastable fixed points, cells switch states with probabilities ${p}_{y}$ and ${p}_{z}$ (STEP 1), followed by their doubling yielding two identical cells preserving the same state (STEP 2). Before the ith doubling, ${n}_{y}$ and ${n}_{z}$ cells preserve their y-dominated and z-dominated states, respectively, and the rest switch states (${m}_{y}$ and ${m}_{z}$ from the former to the latter and vice versa, respectively). The random variable ${Y}_{i}^{\prime}$ denotes the number of cells in the y-dominated state between STEP 1 and STEP 2, just before the ith doubling, so that ${Y}_{i}=2{Y}_{i}^{\prime}$.

**Figure 3.**Stability properties of the toggle switch are shaped by resource competition and parameter asymmetry. Gray regions denote parameter combinations yielding bistable dynamics. Bistability/monostability was determined numerically by simulating 100 trajectories with randomly chosen initial conditions for each parameter value and clustering the endpoints. (

**a**) Stability profile in case of balanced dynamics. Values of $({\beta}_{0},b)$ in the checkered regions are not possible with ${\beta}_{y},{\beta}_{z}\ge 0$. (

**b**) Stability profile in case of unbalanced dynamics. (

**c**) Stability profile in case of unbalanced dynamics with ${\beta}_{0}=0$ in the first panel and ${\beta}_{0}=6$ in the other three panels.

**Figure 4.**Robustness properties of the toggle switch are shaped by resource competition and parameter asymmetry. Solid lines denote the values with the indicated resource sequestration, whereas dotted lines correspond to the case when $({\beta}_{y},{\beta}_{z})=(0,0)$. Potential barriers are all normalized by the same factor so that the maximum across all plots is 1.

**Figure 5.**Evolution of population-level distribution of cells starting from a z-dominant and y-dominant initial cell (${Y}_{0}=0$ and ${Y}_{0}=1$, respectively). (

**a**) Population-level distribution of cells in the y-dominant state in generation 5. (

**b**) The average number of cells in the y-dominant state after successive doublings. (

**c**) The mean percentage of the population in the y-dominant state at equilibrium. (

**d**) The number of generations required to (approximately, within 0.1% range) reach the steady state distribution. Simulation parameters: $({p}_{y},{p}_{z})=(0.02,0.2)$, $({p}_{y},{p}_{z})=(0.1,0.05)$, $({p}_{y},{p}_{z})=(0.1,0.15)$ for red, green, and purple, respectively.

**Figure 6.**Resource competition can be leveraged to increase the balancedness of the toggle switch (uncolored regions correspond to parameter combinations that yield monostable dynamics). Contour values represent ${e}_{\Psi}$ in percentages. (

**a**) In the absence of resource competition (i.e., ${\beta}_{y}={\beta}_{z}=0$), differences between ${\alpha}_{y}$ and ${\alpha}_{z}$ lead to significant error ${e}_{\Psi}$. (

**b**) By the optimal choice of $({\beta}_{y},{\beta}_{z})$ the error ${e}_{\Psi}$ is greatly reduced.

**Figure 7.**Resource competition arising in the genetic context of toggle switches can fundamentally alter their behavior. (

**a**) Bistable toggle switches can render each other monostable due to increased resource sequestration. Simulation parameters are $({\alpha}_{y},{\alpha}_{z})=(25,20)$ and $({\beta}_{y},{\beta}_{z})=(1,1.5)$ with 100 randomly selected initial conditions from ${[0,{\alpha}_{0}]}^{2}$. (

**b**) The critical number ${N}_{crit}$ in case of unbalanced realizations (i.e., ${\alpha}_{y}\ne {\alpha}_{z}$). (

**c**) The critical number ${N}_{crit}$ in case of balanced realizations (i.e., ${\alpha}_{y}={\alpha}_{z}$). Values of $({\beta}_{0},b)$ in the checkered region are not possible with ${\beta}_{y},{\beta}_{z}\ge 0$.

Symbol | Meaning | Typical Value | Unit | Reference |
---|---|---|---|---|

$\kappa $ | RNAP dissociation constant | 1 | $\mathsf{\mu}$M | [22,26] |

k | ribosome dissociation constant | 10 | $\mathsf{\mu}$M | [26,81] |

K | repressor dissociation constant | 0.1 | nM | [82] |

D | DNA concentration | 100–1000 | nM | [26,83] |

${\lambda}^{\mathrm{TX}}$ | transcriptional rate constant | 100 | 1/h | [26,81] |

${\lambda}^{\mathrm{TL}}$ | translational rate constant | 1000 | 1/h | [26,81] |

$\delta $ | mRNA decay rate constant | 10 | 1/h | [84] |

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## Share and Cite

**MDPI and ACS Style**

Yong, C.; Gyorgy, A. Stability and Robustness of Unbalanced Genetic Toggle Switches in the Presence of Scarce Resources. *Life* **2021**, *11*, 271.
https://doi.org/10.3390/life11040271

**AMA Style**

Yong C, Gyorgy A. Stability and Robustness of Unbalanced Genetic Toggle Switches in the Presence of Scarce Resources. *Life*. 2021; 11(4):271.
https://doi.org/10.3390/life11040271

**Chicago/Turabian Style**

Yong, Chentao, and Andras Gyorgy. 2021. "Stability and Robustness of Unbalanced Genetic Toggle Switches in the Presence of Scarce Resources" *Life* 11, no. 4: 271.
https://doi.org/10.3390/life11040271