This report presents a comprehensive biosafety module designed to prevent engineered bacterial escape through density-dependent suicide mechanisms. We developed three interconnected mathematical models: an ODE-based threshold model establishing critical density limits, a PDE-based spatial distribution model analyzing positional effects, and a fluid cloak model enhancing robustness against environmental perturbations. Our simulations identify precise suicide thresholds and demonstrate effective containment strategies under various conditions, providing a quantitative framework for synthetic biology biosafety applications.

The deployment of engineered microorganisms in practical applications necessitates robust containment strategies to prevent environmental escape and potential ecological disruption. Traditional physical containment methods often prove insufficient for dynamic environments, prompting the development of genetic safeguards that activate autonomously under specific conditions.

Our approach leverages the natural quorum sensing mechanism of Staphylococcus aureus, engineering a synthetic circuit that triggers bacterial suicide when population density falls below a critical threshold. This ensures that bacteria remain confined to their intended habitat while allowing normal growth within designated boundaries.

The biosafety module comprises three complementary computational models, each addressing distinct aspects of containment assurance:

• Threshold Model (ODE): Establishes fundamental density thresholds through ordinary differential equations
• Spatial Distribution Model (PDE): Incorporates diffusion and spatial heterogeneity through partial differential equations
• Anti-Interference Model (Fluid Cloak): Enhances robustness against environmental perturbations using virtual density compensation

The suicide switch operates through a carefully balanced toxin-antitoxin system regulated by quorum sensing:

• Quorum Sensing: Autoinducing peptide (AIP) accumulates proportionally to bacterial density
• Regulatory Cascade: AIP activates AgrA transcription factor, modulating toxin expression
• Toxin-Antitoxin Balance: MazF toxin and MazE antitoxin compete, with suicide triggered when MazF dominates
• Threshold Behavior: System exhibits bistability with clear transition points

The models build upon established population dynamics and reaction-diffusion theory, incorporating novel elements for biosafety applications:

Where \(\mathbf{u}\) represents state variables (bacteria, AIP, regulatory proteins), \(\mathbf{D}\) contains diffusion coefficients, and \(\mathbf{f}\) describes reaction kinetics.

Where \(\mathbf{u}\) represents state variables (bacteria, AIP, regulatory proteins), \(\mathbf{D}\) contains diffusion coefficients, and \(\mathbf{f}\) describes reaction kinetics.

The core objective of this module is to simulate bacterial density dynamics over time and identify the critical suicide threshold. We constructed an ordinary differential equation (ODE) model describing the dynamics of bacterial density (N), autoinducing peptide (AIP), regulatory protein (AgrA), toxin (MazF), and antitoxin (MazE).

Where:

• \(r\): Growth rate, \(K\): Carrying capacity, describing basic bacterial growth dynamics
• \(d(N)\): Density-dependent death rate (sharply increases below threshold due to suicide activation)
• \(k_{N}\): AIP secretion rate (proportional to bacterial density)
• \(\delta_{AIP}\): AIP degradation rate

The complete regulatory network includes:

We systematically varied initial bacterial concentrations from \(10^{4}\) to \(10^{8}\) CFU/mL to identify the precise suicide threshold \(N_{0}\). The simulation tracks 24-hour survival probability and key molecular concentrations.

To understand the observed concentration patterns, we analyzed the spatial distribution of AIP and bacterial density. Without loss of generality, we selected an arbitrary central position (x=200) and assumed radial symmetry for the analysis.

(A) \(1.0\times 10^{7}\) CFU/mL (survival case)

(B) \(7.2\times 10^{6}\) CFU/mL (critical bistable case)

(C) \(1.0\times 10^{5}\) CFU/mL (suicide case).

Figure 1: Bacterial dynamics under different initial concentrations.

• AIP Dynamics: AIP exhibits a continuous sharp peak pattern, indicating that peptide secretion follows an accumulative process where concentration builds up progressively over time at the colony center.
• Density Limitation: Bacterial density shows an upper plateau with steep lateral declines, reflecting the fundamental constraint that density cannot exceed the physical carrying capacity \(K\). This spatial constraint creates the observed platform effect.
• Symmetry Analysis: Assuming radial symmetry around the central point, the model captures the continuous concentration gradients that drive the threshold behavior.

The observed spatial patterns can be mathematically described by:

Where the AIP profile shows Gaussian distribution (continuous sharp peak) while bacterial density exhibits hyperbolic tangent behavior (upper plateau with sharp boundaries).

The threshold model successfully identifies a critical bacterial density of \(7.2\times 10^{6}\) CFU/mL that triggers suicide mechanisms. This threshold:

• Provides clear operational guidelines for biosafe applications
• Ensures rapid containment when populations drop below safe levels
• Exhibits robustness against parameter variations
• Forms the foundation for spatial and anti-interference extensions

The concentration profile analysis reveals important patterns:

• AIP exhibits continuous sharp peak accumulation due to progressive secretion
• Bacterial density shows upper plateau limitation with sharp lateral declines due to physical constraints
• These patterns are consistent across the colony when analyzed with symmetry assumptions

This quantitative framework enables precise control of engineered bacteria, preventing environmental escape while maintaining functionality within designated boundaries.

The engineered bacteria in this project possess the ability to form biofilms, enhancing their resistance to environmental disturbances. To simulate this protective effect, we developed a fluid cloak model that introduces virtual density compensation.

Where:

• \(N_{real}\): Actual bacterial density (subject to fluid flushing-induced drops)
• \(N_{v}\): Virtual density providing compensation effect
• \(\gamma\): Compensation coefficient (virtual to actual density convergence rate)
• \(f(t)\): Perturbation function (negative during 1-hour flushing simulation)

Signal molecules (AIP) and regulatory proteins (AgrA) synthesis rates depend on \(N_{v}\) rather than \(N_{real}\) to maintain system stability.

We simulated a 1-hour fluid flushing event at t=12 hours, with perturbation function:

We tested three compensation coefficients (\(\gamma=0.1, 0.5, 1.0\)) to evaluate system robustness.

The fluid cloak mechanism effectively maintains system stability during environmental perturbations:

Figure 2: Virtual density compensation under fluid flushing perturbation with different compensation coefficients.

We systematically evaluated the impact of compensation coefficient \(\gamma\) on system performance:

• Low Compensation (\(\gamma=0.1\)): Slow response, prolonged recovery period, high false positive suicide risk
• Medium Compensation (\(\gamma=0.5\)): Balanced response, optimal stability, minimal false positives
• High Compensation (\(\gamma=1.0\)): Overcompensation, potential for false negatives, reduced safety margin

The anti-interference model demonstrates that virtual density compensation can effectively enhance biosafety system robustness:

• Maintains suicide system stability during environmental perturbations
• Prevents false positive activation due to transient density fluctuations
• Provides quantitative guidelines for compensation parameter selection
• Extends applicability to dynamic environments with fluid flow

Optimal performance was achieved with \(\gamma=0.5\), balancing response speed and stability. This approach significantly improves the practical viability of engineered bacterial containment systems.

The three computational models form a comprehensive framework for biosafety assurance:

• Threshold Model: Provides fundamental density limits and bistable behavior analysis
• Spatial Distribution Model: Addresses heterogeneity and diffusion effects in real environments
• Anti-Interference Model: Enhances robustness against environmental perturbations

This multi-scale approach enables precise prediction and control of engineered bacterial behavior across different environmental conditions.

The identified suicide threshold of \(7.2\times 10^{6}\) CFU/mL provides a clear operational guideline for field applications. This value:

• Ensures rapid containment when populations drop below safe levels
• Maintains functionality within designated operational boundaries
• Provides sufficient safety margin against environmental fluctuations

While our models provide valuable insights, several limitations warrant consideration:

• Parameter sensitivity to environmental conditions (temperature, pH, nutrients)
• Long-term evolutionary stability of the suicide circuit
• Interactions with native microbial communities

Future work should focus on experimental validation, multi-species interactions, and adaptive parameter tuning for different application scenarios.

We have developed a comprehensive computational framework for engineered bacterial containment through density-dependent suicide mechanisms. Our integrated approach combining threshold analysis, spatial modeling, and anti-interference strategies provides:

• Quantitative Safety Standards: Precise suicide threshold identification (\(7.2\times 10^{6}\) CFU/mL)
• Spatial Robustness: Effective containment across heterogeneous environments
• Environmental Resilience: Stability maintenance under fluid perturbations
• Practical Guidelines: Parameter optimization for real-world applications

This work establishes a solid foundation for safe deployment of engineered microorganisms, addressing critical biosafety concerns while maintaining functional efficacy. The modular design allows for adaptation to various synthetic biology applications beyond heavy metal remediation.