Bacterial quorum sensing (QS) is a form of intercellular communication that relies on the production and detection of diffusive signaling molecules called autoinducers. Such a mechanism allows the bacteria to track their cell density in order to regulate group behavior, such as biofilm formation and bioluminescence. In a number of bacterial QS systems, including V. harveyi, multiple signaling pathways are integrated into a single phosphorylation–dephosphorylation cycle. In this paper, we propose a weight control mechanism, in which QS uses feedback loops to ‘decode’ the integrated signals by actively changing the sensitivity in different pathways. We first use a slow/fast analysis to reduce a single-cell model to a planar dynamical system involving the concentrations of phosphorylated signaling protein LuxU and a small non-coding RNA. In addition to identifying the weight control mechanism, we show that adding a feedback loop can lead to a bistable QS response in certain parameter regimes. We then combine the slow/fast analysis with a contraction mapping theorem in order to reduce a population model to an effective single-cell model, and show how the weight control mechanism allows bacteria to have a finer discrimination of their social and physical environment.
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PCB was supported by the National Science Foundation (DMS-1613048). GF was supported by the National Science Foundation (DMS-RTG 1148230).
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In this appendix, we show that a sufficient condition for the equilibrium of the single-cell model to be unique is given by Eq. (40). We already showed the uniqueness for the cases of \(u_1>u_2\) and \(u_1=u_2\), in Sect. 2.3. Next, we will work on the case of \(u_1<u_2\). Since the equilibrium satisfies \(f_1(\phi )=f_2(\phi )\), it follows after some algebra that
Here, because the two sides of (A.1) have to be the same sign, we have the following cases,
Since we are studying the case \(u_1<u_2\), it follows that \(u_1<h(\phi )<u_2\), which implies \(f_3(\phi )>0\).
By some algebra on Eq. (A.1), we get
Recall from Eq. (33) that \(h(\phi )\ge 0\). Since the AI concentration \(u_2\ge 0\), we have \(1-f_3(\phi )>0\). Uniqueness is guaranteed by the monotonicity of the function \(f_4(\phi )\). By taking the derivative with respect to \(\phi \) of \(u_2=f_4(\phi )\), we find
Since \(0<f_3<1\) and \(1+h>0\), one sufficient condition for \(f_4^\prime <0\) to be true is having \(f_3^\prime <0\). Also,
Since \(0\le \phi \le 1\), we require \(h(\phi )-h^\prime (\phi )\phi ^2+\phi h^\prime (\phi )<0\). By direct substitution, we have
Therefore, Eq. (40) is a sufficient condition for the response to be unique.
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Fan, G., Bressloff, P.C. Modeling the Role of Feedback in the Adaptive Response of Bacterial Quorum Sensing. Bull Math Biol 81, 1479–1505 (2019). https://doi.org/10.1007/s11538-019-00570-8
- Quorum sensing
- Cell signaling
- Phosphorylation–dephosphorylation cycles
- Feedback pathways
- Intercellular communication