### Fluorescence and luminescence response of bulk culture

*V.fischeri* mutant JB10 is a derivative of the ES114 strain in which a chromosomal *gfp* reporter is inserted into the *lux* operon by allele exchange, producing *luxI-gfp-luxCDABEG* [40]. We prepared JB10 from a glycerol stock and grew the cells to exponential phase in defined artificial seawater medium [41] to which was added 0.3% casamino acids. Cells were then diluted and regrown to OD ~ 0.1-0.3 in fresh medium, washed three times, and then rediluted 100× into a 96-well assay plate containing fresh medium. The individual wells were preloaded with an 11 × 8 array of concentrations of the two HSL autoinducers *N*-3-oxohexanoyl-*L*-homoserine lactone (3OC6HSL, Sigma #K3007) and *N-*octanoyl-*L*-homoserine lactone (C8HSL, Cayman Chemical Co. #10011199). The well plate was then incubated in a Biotek Synergy 2 plate reader at 25°C, giving a growth rate 1.1 ± 0.1 hr^{-1}. Optical density was measured at 600 nm, and GFP fluorescence was measured using a 485/20 nm excitation filter and a 528/20 nm emission filter. The optical density, luminescence and GFP fluorescence values for each well were recorded at regular intervals during exponential growth (Figure 2). Data collected early in growth (*t* < 12 hrs) showed a sensitive dependence on the exogenous levels of both HSLs, indicating that endogenous HSL did not accumulate significantly during this interval.

### Competitive inhibition model for bulk response

In order to generate a mathematical representation of the *lux* response, as a function of the 3OC6HSL and C8HSL signals, we fit the JB10 well-plate data (fluorescence *vs* HSL concentrations) to the competitive inhibition model of Figure 1[11, 13]. In this model *lux* is regulated primarily through competition between C8HSL and 3OC6HSL to form LuxR complexes that act as transcriptional activators for the *lux* genes. The action of C8HSL on LuxR synthesis through AinR and the phosphorelay is not considered. We assume that 3OC6HSL and C8HSL diffuse freely across the cell envelope and form multimeric complexes with LuxR. We allow an arbitrary degree of multimerization but we do not consider heterocomplexes (*i.e*. involving both C8HSL and 3OC6HSL). Although it is simple to include the weak activation of *lux* by C8HSL-LuxR, which is evident in the bioluminescence data at low 3OC6HSL concentrations, this activation is scarcely visible in the GFP fluorescence data that is the target of our modeling. Therefore we omitted this mechanism from our model and considered C8HSL only in its role as a competitor for LuxR. That is, we assume that the GFP fluorescence is proportional to the concentration of the 3OC6HSL-LuxR complex. More biochemical accuracy could be included by introducing extra parameters, but the simpler model appears sufficient to describe the JB10 data.

The model allows C8HSL and 3OC6HSL to form multimeric complexes (of degree *m* and *n* respectively) with LuxR,

\begin{array}{c}m\phantom{\rule{2.77695pt}{0ex}}C8HSL+m\phantom{\rule{2.77695pt}{0ex}}LuxR\iff {\left(C8HSL-LuxR\right)}_{m}\\ n\phantom{\rule{2.77695pt}{0ex}}3OC6HSL+n\phantom{\rule{2.77695pt}{0ex}}LuxR\iff {\left(3OC6HSL-LuxR\right)}_{n}\end{array}

where the Hill coefficients *n* and *m* are not assumed to be integers. These equilibria are characterized by two dissociation constants, *K*_{1} and *K*_{2}:

\begin{array}{c}{K}_{1}^{2m-1}=\frac{{\left[LuxR\right]}^{m}{\left[C8HSL\right]}^{m}}{\left[{\left(C8HSL-LuxR\right)}_{m}\right]}\\ {K}_{2}^{2n-1}=\frac{{\left[LuxR\right]}^{n}{\left[3OC6HSL\right]}^{n}}{\left[{\left(3OC6HSL-LuxR\right)}_{n}\right]}\end{array}

*K*_{
1
} and *K*_{
2
} are defined so as to have units of concentration, regardless of the values of *m* and *n*. If [LuxR_{0}] is the average total concentration of LuxR, including complexes, then

\left[\mathsf{\text{Lux}}{\mathsf{\text{R}}}_{0}\right]=\left[\mathsf{\text{LuxR}}\right]+n\left[{\left(3\mathsf{\text{OC}}6\mathsf{\text{HSL}}-\mathsf{\text{LuxR}}\right)}_{\mathsf{\text{n}}}\right]+m\left[{\left(\mathsf{\text{C}}8\mathsf{\text{HSL}}-\mathsf{\text{LuxR}}\right)}_{\mathsf{\text{m}}}\right]

(3)

As we do not measure the actual LuxR copy number (although see [42]), it is convenient to redefine the dissociation constants in terms of [LuxR_{0}] and a dimensionless concentration *r*:

\begin{array}{l}{k}_{1}^{m}=\frac{{r}^{m}{\left[C8HSL\right]}^{m}}{\left[{\left(C8HSL-LuxR\right)}_{m}\right]/\left[Lux{R}_{0}\right]}\\ {k}_{2}^{n}=\frac{{r}^{n}{\left[3OC6HSL\right]}^{n}}{\left[{\left(3OC6HSL-LuxR\right)}_{n}\right]/\left[Lux{R}_{0}\right]}\end{array}

(4)

Here *k*_{
1
} and *k*_{
2
} have dimensions of (autoinducer) concentration. Then Eqn. (3) becomes

1=r+m{r}^{m}\frac{{\left[C8HSL\right]}^{m}}{{k}_{1}^{m}}+n{r}^{n}\frac{{\left[3OC6HSL\right]}^{n}}{{k}_{2}^{n}}

(5)

Starting from the HSL concentrations and an initial guess for the parameters (*k*_{
1
}*, k*_{
2
}*, m, n*), we solve Eqn. (5) to find *r*. Then Eqn. (4) gives the concentrations (relative to LuxR_{0}) of the two multimer species. We compare the model to the well-plate data by assuming that the GFP fluorescence *F* is a linear, non-saturating function of the two multimer concentrations:

\begin{array}{c}F={F}_{0}+{a}_{1}\frac{\left[{\left(C8HSL-LuxR\right)}_{m}\right]}{\left[Lux{R}_{0}\right]}+\dots \\ \phantom{\rule{1em}{0ex}}{a}_{2}\frac{\left[{\left(3OC6HSL-LuxR\right)}_{n}\right]}{\left[Lux{R}_{0}\right]}\end{array}

(6)

Here *F*_{0}, *a*_{1} and *a*_{2} are positive constants (see *Results*). As explained above, *a*_{1} is evident in luminescence but is scarcely detectable in the fluorescence; setting *a*_{1} = 0 does not impair the fit. Then the shape of the *2D* surface *F*(3OC6HSL, C8HSL) is determined solely by the four parameters *k*_{
1
}*, k*_{
2
} , *n*, and *m*, while the parameters *F*_{
0
} and *a*_{
2
} provide an instrument-dependent offset and amplitude that scale the *2d* model *F* surface onto the measured values. We estimate the four model parameters through a nonlinear least squares fit of the fluorescence response surfaces *F*(3OC6HSL, C8HSL) measured at optical densities 0.05-0.15 cm^{-1} to Eqn. (6), with the scale parameters *a*_{2} and *F*_{0} determined by linear regression. This provides a parametrization of the average response *F* as a function of the two HSL inputs (Table 1).

The data do not require that *m* and *n* are different. As Table 1 indicates, the fit yields similar values for the two Hill coefficients (*m* = 1.1 ± 0.4 and *n* = 1.35 ± 0.05), and in fact we obtain a very similar fit if we assume that the same coefficient applies for both autoinducers (*m* = *n* = 1.2 ± 0.2). Table 1 also shows (as expected from Eqn. (1)) that we obtain similar parameters when we fit Eqn. (6) to the square root of the measured luminescence *L*^{1/2} rather than to the GFP fluorescence *F*.

### Microfluidic studies of individual cells

To measure the effect of exogenous HSL signals on *lux* expression in individual JB10 cells we loaded cells into microfluidic perfusion chambers that supplied a flow of medium containing exogenous 3OC6HSL and C8HSL. Each microfluidic device consisted of three parallel and unconnected channels (Figure 4), with each channel having width 400 μm (parallel to the observation window but perpendicular to the fluid flow), depth 10-15 μm (perpendicular to the observation window), and length (parallel to observation window and to fluid flow) 10 mm. The devices were fabricated from PDMS silicone elastomer (Sylgard 184, Dow-Corning Corporation) by a standard soft-lithographic method in which a PDMS replica is cast from a reactive ion-etched silicon master [43]. The device channels were sealed by a glass coverslip bonded to the PDMS. In order to promote cell adhesion to the interior of the glass window, we coated the interior of the device by filling it with a solution of poly-*L*-lysine (1 mg/ml, MW 300 000) and incubating it for 24 hours at 5°C, prior to cell injection. This provided stable adhesion of the *V.fischeri* to the glass window.

JB10 cells for microfluidic studies were prepared in exponential phase as for the 96-well assay above: We grew cells to exponential phase in defined artificial seawater medium with casamino acids [41], then washed (3×) and rediluted the cells, and then regrew them to OD (600 nm) = 0.015-0.03 cm^{-1} in fresh medium. Once the cells and the microfluidic device were prepared, we flushed the poly-*L*-lysine solution by pumping the JB10 culture into all three parallel channels at 1-2 ml/hr with a syringe pump. We then placed the device (with glass window facing downward) on the stage of a Nikon TE2000U microscope and reduced the flow rate to ~0.02 ml/hr. At this slow flowrate the cells gradually settle and adhere to the glass window. Once a sufficient number of cells had adhered to the window (requiring 15-30 minutes), we supplied autoinducer by connecting the device inputs to syringe pumps that delivered defined medium containing exogenous 3OC6 HSL and/or C8HSL. Each of the three channels was supplied with a different combination of HSLs, flowing at a rate ~0.02 ml/hr during fluorescence measurements.

The 0.02 ml/hr flow rate of medium corresponds to an average flow velocity of ~1 mm/s within each channel. Both the device design and experimental testing ensured that this flow was sufficiently uniform and rapid to wash away endogenous (natively produced) autoinducer that might otherwise affect activation of the *lux* genes. First of all, control experiments in our flow system showed that - in the absence of any exogenous autoinducer (HSL) - *gfp* expression from the *lux* reporter strain was at its baseline level (and luminescence was unobservable). Moreover, the physical parameters of the flow system make it highly implausible that spatial heterogeneity in the flow could develop or allow experimentally relevant concentrations of HSL to accumulate near any of the cells under observation: First, the dimensions of the device and the flow rate of growth medium lead to fluid flow at a very low Reynolds number (*Re* ~0.03). At this *Re* the flow velocity profile is highly uniform across the width and length of the flow chamber, up to within ~10 μm of the chamber edges [44]. Second, the only significant heterogeneity in this flow velocity profile occurs along the depth of the channel (*i.e*. perpendicular to the window), which is 10-15 μm. However HSL requires only ~0.1 s to diffuse this distance. This is so much faster than other relevant time scales in the experiment that a meaningful HSL gradient cannot be established in this direction. Third, the chamber volume and the 1 mm/s flow rate together indicate that the entire volume of the cell chamber region (10 mm length) is completely flushed every ~10 seconds. However the only cells occupying the chamber (and producing HSL) are those forming a sparse single layer (cells are typically spaced > 20 μm apart) on the chamber window. Literature estimates of HSL production rates in *V.fischeri* indicate that such a sparse layer of individual cells, within a chamber that is flushed at this rate, would not be able to generate an endogenous HSL concentration above ~100 pM [20]. This concentration is at least two orders of magnitude smaller than the exogenous HSL concentrations that we are providing.

Finally, if the cells did generate enough HSL to affect local concentrations, we would expect that cells downstream would in general express more GFP than cells upstream. More generally we would expect the correlation *C*_{
ij
} between the GFP fluorescence *F*_{
i
}*, F*_{
j
} of a pair of cells *i, j*

{C}_{ij}=\left({F}_{i}-\mu \right)\left({F}_{j}-\mu \right)\u2215{\sigma}^{2}

(7)

to depend on their spatial separations *x*_{
ij
} , *y*_{
ij
} , or *r*_{
ij
} . (Here μ is the mean cell fluorescence and σ^{2} is the variance in *F*.) We analyzed our data for such spatial correlations and found none. For example, Figure 4 shows no relationship between *C*_{
ij
} and *r*_{
ij
} : the *gfp* expression of two neighboring cells is no more similar than that of two distant cells. In short the data and the system design argue strongly against any autoactivation of (or local crosstalk between) the individual cells under observation.

### Characterizing heterogeneity in *lux*activation

The three-channel device allowed us to collect the fluorescence histogram of cells under three different HSL signal combinations, as it evolved over 4-5 hours. Once HSLs were introduced to the device at *t =* 0, we collected phase contrast and fluorescence image pairs for each channel (HSL combination) at intervals of 20 minutes, using a 20×/0.50 NA phase objective and a GFP filter cube. Images were recorded by a Coolsnap HQ2 camera (Photometrics) at -30°C and corrected in software for dark current and flat-field.

For each experimental condition we evaluated the fluorescent emission from (typically) ~200 individual cells by first determining the physical locations (pixel coordinates) of single cells in a phase contrast image. We then used a homemade Matlab code to evaluate the fluorescence per cell pixel in the associated fluorescence image by summing the fluorescence emission (relative to background) of the contiguous bright pixels associated with the cell's pixel coordinates. Normalizing the histogram of individual cell fluorescence values gives a distribution *P*(*F* | ([3OC6HSL],[C8HSL])), representing the probability of cell fluorescence *F* given the HSL input concentrations.

### Calculating the mutual information

By combining mathematical parametrizations of both the *lux* response, *F*([3OC6HSL],[C8HSL]) and the probability distribution *P*(*F* | ([3OC6HSL],[C8HSL])), we calculate the mutual information [23] between the HSL signal inputs and *lux* output. The signal concentrations are inconvenient parameters for this calculation because only an infinite concentration of autoinducer can saturate the response. In their analysis of *V.harveyi* QS, Mehta *et al*. [25] defined new coordinates that describe the state of saturation of the autoinducer receptors. In similar fashion we replace [3OC6HSL] and [C8HSL] with coordinates *X* and *Y* that describe the state of the association equilibria for the HSL-LuxR complexes:

\begin{array}{c}X=\frac{{\left[3OC6HSL\right]}^{n}}{{\left[30C6HSL\right]}^{n}+{k}_{2}^{n}}\\ Y=\frac{{\left[C8HSL\right]}^{m}}{{\left[C8HSL\right]}^{m}+{k}_{1}^{m}}\end{array}

(8)

*X* = 1 or *Y* = 1 corresponds to complete saturation of the 3OC6HSL-LuxR or C8HSL-LuxR binding equilibrium respectively. With these coordinates the response surface *F*(*X,Y*) for the competitive inhibition model has a simple shape (Figure 7) that is independent of the parameters *k*_{
1
} , *n*, *k*_{
2
} , *m*.

The mutual information between a combination of inputs (*X,Y*) and the output *F* is then calculated as:

\begin{array}{c}I\left(F;\left(X,Y\right)\right)=\int dFdXdY\phantom{\rule{0.3em}{0ex}}P\left(F,\left(X,Y\right)\right)\phantom{\rule{0.3em}{0ex}}{log}_{2}\phantom{\rule{0.3em}{0ex}}\frac{P\left(F,\left(X,Y\right)\right)}{P\left(F\right)P\left(X,Y\right)}\\ =\int dFdXdY\phantom{\rule{0.3em}{0ex}}P\left(F\mid \left(X,Y\right)\right)\phantom{\rule{0.3em}{0ex}}P\left(X,Y\right){log}_{2}\frac{P\left(F\mid \left(X,Y\right)\right)}{P\left(F\right)}\end{array}

(9)

Here *P*(*F*) is the probability of finding output *F*, in the absence of any knowledge of the input (*X, Y*). *P*(*F*|(*X, Y*)) is the probability of *F*, given the combination (*X, Y*). *P*(*F*,(*X, Y*)) is the probability of observing the particular combination *F*, (*X, Y*):

P\left(F,\left(X,Y\right)\right)=P\left(F\phantom{\rule{0.3em}{0ex}}\mid \left(X,Y\right)\right)P\left(X,Y\right)

These probability distributions are normalized as follows:

\begin{array}{c}P\left(F\right)=\int dXdY\phantom{\rule{0.3em}{0ex}}P\left(F,\left(X,Y\right)\right)=\cdots \\ =\int dXdY\phantom{\rule{0.3em}{0ex}}P\left(F\mid \left(X,Y\right)\right)P\left(X,Y\right)\\ \int dF\phantom{\rule{0.3em}{0ex}}P\left(F\right)=1\\ \int dF\phantom{\rule{0.3em}{0ex}}P\left(F\mid \left(X,Y\right)\right)=1\\ \int dFdXdY\phantom{\rule{0.3em}{0ex}}P\left(F,\left(X,Y\right)\right)=1\\ \int dXdY\phantom{\rule{0.3em}{0ex}}P\left(X,Y\right)=1\end{array}

To evaluate Eqn. (9) we model *P*(*F*,(*X, Y*)) as the gamma distribution that has the same mean and variance as observed in the bulk and single-cell measurements respectively. The calculation also requires an estimate of *P*(*X, Y*), the prior probability of a particular combination (*X, Y*). *P*(*X, Y*) is not so easily predicted. However, given that *X* and *Y* are both bounded by 0 and 1 we made the straightforward assumption that *P*(*X, Y*) = *constant*. The mutual information Eqn. (9) is then found to be *I ≈* 0.53 bits. However this result is not sensitive to our assumptions about the prior probability: various *P*(*X, Y*) functions that were strongly bimodal in both *X* and *Y*, and either symmetric or asymmetric in *X* vs. *Y* [25], all gave similar values of *I* ≈ 0.5 bits.