Fig 1.
Signals, statistics, and dynamics in quorum sensing.
A. A growing bacterial colony tracks the concentration of endogenously produced autoinducers (AIs) via the internal abundance of a monitor protein (MP). B. The quorum-sensing system responds to changing cell density ρ by regulating the concentration of external autoinducers at and the internal concentration of MPs mt. A succession of identical growth cycles yields stationary distributions for ρ, a, and m within the operational range of quorum sensing (shaded frames). C. Dependency graph modeling quorum sensing with feedback from MP expression to AI signaling and to AI sensing. D. Functions parametrizing the stochastic dynamics of the quorum-sensing response. The AI output rate fext and the self-regulation level fint are emphasized as the external and internal feedback, respectively.
Fig 2.
Quorum-sensing information channel.
A. In this illustration, bacteria engage sequentially in five fictitious collective tasks, represented by five colors, that are homogeneously distributed in time. B. During the growth of a colony, the increasing cell density drives the quorum-sensing system. C. To perform the desired tasks, bacteria need to resolve the five cell-density stages, whose probability is shaped by bacterial growth. D. At fixed cell density, individual bacteria exhibit fluctuating levels of MPs, with mean and variance . E. The smallest difference in cell density that a bacterium can resolve by reading out its fluctuating MP abundance specifies the resolution of the channel, defined as δρ = Σm(ρ)/m′(ρ). Thus, the information available to individual bacteria via quorum sensing depends both on the cell density dynamics and the channel resolution.
Fig 3.
Models for the regulation of the monitor protein (MP) expression.
A. In the TF regulation model, AI molecules induce the production of MPs by allosterically regulating the transcription factor (TF), which only binds to its cognate DNA regulatory sequence when complexed with AI. B. In the sRNA regulation model, a TF positively regulates a small regulatory RNA (sRNA) that represses MP expression. In both models, the expression of MP is positively regulated by binding of AIs to the TF and MP proteins regulate their own expression. For strong sRNA-mRNA pairing, sRNA regulation reduces the stochasticity in MP expression compared with TF regulation. The internal feedback regulates the transcription of the MP mRNA for TF regulation, whereas it regulates the sRNA level for sRNA regulation.
Fig 4.
Optimal quorum-sensing response.
TF regulation is shown by dashed curves and sRNA regulation by solid curves. A. The bare MP expression rate , in the absence of self-regulation of the MP, obeys a Hill function with h = 2 (inset). Temporal dynamics of the optimal mean AI concentration and B. temporal dynamics of the optimal mean MP abundance. C. Optimal external feedback with f− = a−/(τa ρ−) and D. optimal internal feedback . E. Fano function with optimal feedback F⋆ and without feedback F as functions of MP abundance m. The ratio of effective correlation times τ⋆/τ = F⋆/F is shown inset. In all panels, red curves indicate the optimal quorum-sensing response, black curves indicate the absence of all feedbacks . Parameter values: ρ+/ρ− = 104, a− = 0.1nM, a+ = 1mM, m− = 100nM, m+ = 600nM, fint,− = 2, fint,+ = 1/2, K = 15nM, h = 2, b = 20 and v = 1μm3.
Table 1.
Number of discernible cell-density stages.
Fig 5.
Optimal MI and optimal MI increase.
Dependence of the optimal MI I⋆ and the optimal MI increase ΔI⋆ (in bits) on the burst size b and on the level of self-repression fint,+ (in logarithmic scale) for both TF and sRNA regulations. The Xs indicate the values fint,+ = 1/2 and b = 20 for which Table 1 was computed. In the top panels, the white curves are isoinformation curves separating regions where the optimal quorum-sensing channel can discriminate the indicated number of cell-density ranges. In the bottom panels, the white curves represent parameters for which feedbacks cannot improve information transfer (0 bits) or can double the number of distinguishable cell density ranges (1 bit). Note that, in both cases, a decrease in fint,+ has to be matched by a larger bare MP output rate to ensure the boundary condition . Parameter values: ρ+/ρ− = 104, a− = 0.1nM, a+ = 1mM, m− = 100nM, m+ = 600nM, fint,− = 2, K = 15nM, h = 2 and v = 1μm3.