# Context-Dependent Stability and Robustness of Genetic Toggle Switches with Leaky Promoters

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Mathematical Model to Account for the Scarcity of Resources

#### 2.2. Stability Analysis

#### 2.3. Robustness Analysis

#### 2.4. Population-Level Analysis

#### 2.5. Context Effects

## 3. Results

#### 3.1. Stability Analysis

#### 3.2. Robustness Analysis

#### 3.3. Population-Level Analysis

#### 3.4. Context Effects

## 4. Discussion

## Supplementary Materials

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Stability and robustness of genetic switches depend on their context [40,41]. (

**A**) In the absence of loading from its context, trajectories of toggle switch #1 converge to one of two metastable states (red and green). Once loading from the context is present, realized via the addition of the repressilator [42], the two distinct subpopulations coalesce (purple). (

**B**) Although toggle switch #2 behaves identically to toggle switch #1 in the absence of the context (red and green), the same perturbation only causes a slight shift of the two subpopulations towards each other (purple). (

**C**) Toggle switch #2 displays dramatically reduced robustness to noise due to its context. For more details on the stochastic simulations [43,44,45], see Section S1 and Section S6 in the SI.

**Figure 2.**Resource competition and promoter leakiness both act against bistability. (

**A**) In the absence of promoter leakiness, the toggle switch is bistable if $q=2(1+\beta )/\alpha <1$, and monostable if otherwise (SI Section S2.2). (

**B**) In the absence of competition for shared cellular resources, the dynamics are bistable if $(\alpha ,\nu )$ lies below the curve $({\alpha}_{w}\left(w\right),{\nu}_{w}\left(w\right))$, where ${\alpha}_{w}\left(w\right)={(1+{w}^{2})}^{2}/\left(2w\right)$ and ${\nu}_{w}\left(w\right)=({w}^{2}-1)/{(1+{w}^{2})}^{2}$, parameterized by $w\ge 1$ (SI Section S2.3).

**Figure 3.**In the presence of promoter leakiness and resource competition, the constraints ${q}_{i}\left(\nu \right)=1$ from (3) partition the parameter space into monostable, bistable, and tristable regions (red, green, and purple, respectively). See SI Section S6 for simulation parameters.

**Figure 4.**Robustness to noise is characterized using a potential landscape-based approach. (

**A**) Trajectories flow downhill along the potential surface (grey) towards one of the metastable fixed points (red and green) depending on the initial conditions. (

**B**) The potential barrier in the bistable case can be approximated by $h\approx {\psi}_{1}{({q}^{-1}-1)}^{{\psi}_{2}}$ with $({\psi}_{1},{\psi}_{2})=(0.545,2.039)$, and 95% confidence intervals $(0.544,0.547)$ and $(2.035,2.042)$, respectively [67]. (

**C**) This approximation (solid line) is well aligned with the data obtained from the numerical calculations of the potential barrier (circles). (

**D**) The height of the potential barrier separating the two metastable fixed points in case of bistable dynamics for different values of $\alpha $ (normalized to the maximal value, represented as 100%), together with the mean transition time between these metastable fixed points (normalized to the maximal value, represented as 100%) in case of different noise power. See SI Section S6 for the simulation parameters.

**Figure 5.**Population-level properties of the toggle switch are governed by the distribution of the random variable Q from (6) in the absence of promoter leakiness. (

**A**) The distribution of Q shifts right as ${\mu}_{\beta}$ increases, pushing the population towards unimodality [73]. (

**B**) Greater correlation $\rho $ between $\alpha $ and $\beta $ yields increased population-level uniformity. See SI Section S6 for the simulation parameters.

**Figure 6.**In the presence of promoter leakiness, population-level properties depend on the random variables ${Q}_{i}$ from (6). (

**A**) Increasing leakiness decreases the bistable fraction of the population. (

**B**) Unwanted trimodality can be eliminated, for instance, by increasing resource usage of the toggle switch for green/purple, and by decreasing it for red (Figure S9 in the SI). (

**C**) Increasing the expected value ${\mu}_{\beta}$ of $\beta $ first increases both the bistable fraction of the population (displayed in percentages) and the robustness of the metastable states to noise (measured via the potential barrier separating them), then this effect reverses as further increasing ${\mu}_{\beta}$ pushes the population away from the optimal region. See SI Section S6 for the simulation parameters.

**Figure 7.**Loading from the context affects the stability and robustness properties of genetic switches. (

**A**) Increasing ${\beta}_{c}$ causes a shift in the $(\alpha ,\beta )$ plane towards the origin according to $\alpha \leftarrow \alpha /(1+{\beta}_{c})$ and $\beta \leftarrow \beta /(1+{\beta}_{c})$. Distributions show $y-z$ in the steady state (for more details, see Figures S10 and S11 in the SI). (

**B**) Higher $\beta $ protects against loss of robustness to noise due to loading from the context (the parameters are chosen so that q is the same in all cases when ${\beta}_{c}=0$, yielding identical potential barriers). Solid lines correspond to predictions considering the approximation $h\approx {\psi}_{1}{({q}^{-1}-1)}^{{\psi}_{2}}$ of the potential barrier with $({\psi}_{1},{\psi}_{2})=(0.545,2.039)$, whereas circles represent simulation data using the potential landscape directly. See SI Section S6 for the simulation parameters.

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Gyorgy, A. Context-Dependent Stability and Robustness of Genetic Toggle Switches with Leaky Promoters. *Life* **2021**, *11*, 1150.
https://doi.org/10.3390/life11111150

**AMA Style**

Gyorgy A. Context-Dependent Stability and Robustness of Genetic Toggle Switches with Leaky Promoters. *Life*. 2021; 11(11):1150.
https://doi.org/10.3390/life11111150

**Chicago/Turabian Style**

Gyorgy, Andras. 2021. "Context-Dependent Stability and Robustness of Genetic Toggle Switches with Leaky Promoters" *Life* 11, no. 11: 1150.
https://doi.org/10.3390/life11111150