Information theoretic approach to signal integration in the Vibrio harveyi quorum-sensing circuit. (A) V. harveyi produces three distinct quorum-sensing signaling molecules, called (AIs), which are all detected by a single phosphorelay circuit that controls the expression of downstream target genes. Each signaling molecule, AI-1 (red hexagons), AI-2 (blue ovals), and CAI-1 (gray squares), is detected by a cognate receptor. The receptors phosphorylate a shared phosphorelay protein, LuxU, which in turn phosphorylates LuxO. In the absence of AIs, LuxO is phosphorylated and activates expression of genes encoding five small regulatory RNAs (sRNAs) that work in conjuction with Hfq to destabilize the mRNA of LuxR, the master regulator of quorum-sensing genes. In the presence of the AIs, LuxO is not phosphorylated, the sRNAs are not produced, and LuxR is expressed. (Inset) The receptors can exist in two states: a kinase ‘on' state and kinase ‘off' state with ligand binding favoring the ‘off' state. (B) Dose–response surface of V. harveyi to various combinations of AI-1 and AI-2 reproduced from Figure 3C in Long et al (2009). Each vertex of the grid is the averaged normalized GFP fluorescence intensity obtained from a population of 100 cells exposed to the specified AI-1 and AI-2 concentrations using a qrr4-gfp transcriptional reporter fusion that is activated by phosphorylated LuxO. (C) Dose–response surface re-plotted as a contour plot. The straight lines are constant output Z contours as function of the receptor on-state probabilities X ≈ 1 /( 1 + [AI- 1] / K_I^AI-1) and Y ≈ 1 /( 1 + [AI- 2] / K_I^AI-2). Output is normalized by the maximum of Z.

Constant output Z contours for different relative kinase strengths of the two signaling branches carrying the inputs X and Y: (A) k_Y/k_X=1/8, (B) k_Y/k_X=1, (C) k_Y/k_X=8. Output is normalized by the maximum of Z and the color bar applies to all graphs.

Graphical representation of input–output relations in the presence of a positive feedback on receptor number restricted to the domain X>Y. Equally spaced, constant output Z contours for a signaling circuit with positive feedback on receptor number (see inset). The output is normalized by the maximum of Z. Parameters are K=6, C=1/8, δC=8.

(A) The mutual informations I(Z, X), I(Z, Y), and I(Z,(X, Y)) measure how much one can learn about the inputs, for example, autoinducer levels, X, Y, and (X, Y), respectively, from the output Z, for example, LuxR level. Mutual information is a function of the prior, q(X, Y), that is, the a priori probability of a given input (X, Y), and of the probabilistic transfer function P(Z∣X, Y) of the signaling circuit. (B) (Top) For an idealized multi-input channel without noise, an input (X, Y) gives rise to a single output Z. (Middle) In the presence of noise, a single input can give rise to many outputs with a distribution described by the noisy-transfer function, P(Z∣X, Y). (Bottom) When viewed as single-input channel with input X and output Z, the second signal, Y, effectively acts as an additional source of noise.

Mutual information as a function of the ratio of kinase strengths, k_Y/k_X. (A–C) Mutual information I(Z, X) between Z and X (red curves), and mutual information I(Z, X) between Z and Y, I(Z, Y) (dashed blue curves) as functions of k_Y/k_X, for (A) a flat prior, (B) a symmetric prior, bimodal in each input, and (C) a non-symmetric prior, bimodal in each input. (Insets) Graphical representations of the corresponding priors with brighter colors representing higher probability. (D, E) Mutual information as shown in the figure, with the restriction that the inputs obey X⩾Y.