Model
Overview
Regarding the LuxI-LuxR-pLux quorum sensing system, designing experiments to accurately characterize its behavior presents a significant consideration. In 2023, team UCAS-China adopted a stepwise validation approach. Specifically, they separately verified the expression of the upstream component LuxI and assessed the functionality and strength of the downstream LuxR-pLuxR module using exogenously supplied AHL.
In contrast, our team's current experimental strategy focuses on evaluating the operational efficiency of the entire gene circuit as a unified system. Building on this holistic assessment, we further aim to investigate the impact of the putative pLuxR-LuxI positive feedback loop on overall system performance.
This difference in experimental design stems from our project's intended application scenario: wastewater treatment ponds. In such environments, the initial density of engineered bacteria is anticipated to be very low. Consequently, the concentration of the small signaling molecule AHL, which facilitates communication, is also expected to be relatively low at the outset .
Therefore, validating system functionality using high concentrations of AHL in experiments is considered less appropriate or non-representative of the actual conditions. Correspondingly, the potential role of the pLuxR-LuxI positive feedback loop becomes increasingly crucial under these conditions of low initial AHL concentration, as it may be essential for signal amplification and robust system activation.
To quantitatively assess the dynamics of the quorum sensing system and gain a deeper understanding of the critical role of the positive feedback loop under low cell density, we developed a kinetic model based on ordinary differential equations. The model core describes the interplay between the concentrations of the signal molecule AHL, the transcriptional activation complex, and the synthase LuxI over time.
Key variables and parameters are defined as:
The model is defined by the following system of equations:
1. Dynamics of AHL:
This equation states that the net rate of change of AHL equals its production rate (catalyzed by LuxI) minus its degradation/dilution rate.
2. Formation of the active complex [C] (approximated by a Hill function):
This equation describes how the concentration of the transcriptional activation complex [C] is determined cooperatively by the AHL concentration.
3. Dynamics of LuxI (Core Equation):
This equation is the soul of the model. The rate of change of LuxI comes from three terms:
①Ki0: Basal expression rate (typically very small, representing leaky expression).
②Positive Feedback Term:
This term describes the auto-amplification of LuxI gene expression after the active complex [C] binds to its own promoter pLuxm. The parameter β directly quantifies the strength of this positive feedback.
③−δi[I]: Degradation/dilution term for LuxI.
Analysis of Dynamical Behavior at Low Initial Density
The key hypothesis of this study is that the initial inoculation density of the engineered bacteria in the intended wastewater treatment scenario is extremely low, leading to a near-zero starting concentration of endogenous AHL. The system's ability to activate reliably under this condition is a prerequisite for its successful application.
We performed theoretical analysis and numerical simulations of the model to compare the system behavior without the positive feedback loop (QS1, β=0) and with the positive feedback loop (QS2, β>0).
1. Steady-State Analysis: The system reaches a steady state when dtd[I]=0and dtd[AHL]=0. Under low initial conditions, the system exhibits a low-expression steady state (the "OFF" state), where [I], [AHL], and [C]are all close to zero. The stability of this steady state determines whether the system can self-activate.
2. Case WITHOUT Positive Feedback (QS1): When β=0, the expression of LuxI relies entirely on the tiny basal expression ki0. Starting from a very low initial AHL concentration, the system remains stably trapped near the low-expression steady state. Escaping this state to achieve activation would require significant stochastic fluctuations or external intervention, implying an unacceptably high risk of activation failure or severe delay in practical applications.
3. Case WITH Positive Feedback (QS2): When β>0, the dynamics of the system are fundamentally altered. Although the low-expression steady state still exists, the positive feedback significantly destabilizes this state (or effectively shrinks its basin of attraction). Even starting from near-zero AHL concentrations, the minute amounts of AHL and complex [C] produced by basal expression can be captured and non-linearly amplified by the positive feedback loop. This allows the system to overcome the initial energy barrier, achieve self-activation, and rapidly transition to a new high-expression steady state (the "ON" state).
Conclusion
Both theoretical analysis and numerical simulations confirm that under the harsh condition of low initial AHL concentration, the introduction of the pLuxm-LuxI positive feedback loop significantly lowers the activation threshold of the system and greatly accelerates its response kinetics. This theoretical result perfectly validates our core concept: for engineered bacteria operating in real wastewater environments, the positive feedback loop is not merely an improvement but is a critical and necessary design ensuring reliable functional initiation. This model provides a solid mathematical foundation for our experimental observations (the superior low-concentration performance of the QS2 system).