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Mol Syst Biol. 2008; 4: 211.
Published online Aug 5, 2008. doi:  10.1038/msb.2008.49
PMCID: PMC2538909

Variable sizes of Escherichia coli chemoreceptor signaling teams

Abstract

Like many sensory receptors, bacterial chemotaxis receptors form clusters. In bacteria, large-scale clusters are subdivided into signaling teams that act as ‘antennas' allowing detection of ligands with remarkable sensitivity. The range of sensitivity is greatly extended by adaptation of receptors to changes in concentrations through covalent modification. However, surprisingly little is known about the sizes of receptor signaling teams. Here, we combine measurements of the signaling response, obtained from in vivo fluorescence resonance energy transfer, with the statistical method of principal component analysis, to quantify the size of signaling teams within the framework of the previously successful Monod–Wyman–Changeux model. We find that size of signaling teams increases 2- to 3-fold with receptor modification, indicating an additional, previously unrecognized level of adaptation of the chemotaxis network. This variation of signaling-team size shows that receptor cooperativity is dynamic and likely optimized for sensing noisy ligand concentrations.

Keywords: adaptation, chemotaxis, FRET, principal component analysis, receptor clusters

Introduction

Transmembrane receptors of the chemotaxis network in Escherichia coli allow bacteria to sense chemicals in the environment, allowing cells to swim toward nutrients (attractant chemicals) and away from repellents (toxic chemicals). The chemotaxis network possesses remarkable signaling properties, including high sensitivity to small changes in chemical concentration over a wide range of ambient concentrations. These signaling properties rely on receptor clustering (Bray et al, 1998), which occurs at multiple length scales. At a small scale, the chemotaxis receptors form stable homodimers, which then assemble into larger complexes in which receptors of different chemical specificities are intermixed (Studdert and Parkinson, 2004; Lai et al, 2005). Trimers of dimers (Kim et al, 1999; Ames et al, 2002; Studdert and Parkinson, 2004) are believed to be the smallest signaling unit (Boldog et al, 2006). At a larger scale, ~10 000 receptors form large polar and lateral receptor clusters (Maddock and Shapiro, 1993; Ames et al, 2002; Zhang et al, 2007). Theoretical analysis of in vivo fluorescence resonance energy transfer (FRET) data suggests that large receptor clusters are composed of smaller signaling teams each consisting of about 10 strongly coupled receptor dimers (see Box 1 for details) (Sourjik and Berg, 2004; Mello and Tu, 2005; Keymer et al, 2006; Skoge et al, 2006), which is consistent with receptor cooperativity (Hill coefficients of dose–response curves) observed in vitro (Li and Weis, 2000).

Model for receptor signaling

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Box 1 Chemoreceptors cluster at multiple scales ranging from receptor dimers, to trimers of dimers, to large polar and lateral clusters. Signaling properties are believed to arise from groups of several trimers of dimers forming signaling teams. Within a signaling team, receptors are assumed to be coupled strongly enough that the receptors are either all in the on state or all in the off state (A). In the on state, the receptor-associated kinase CheA is assumed to be active, autophosphorylating itself and transferring the phospho-group to the response regulators CheY and CheB. In the off state, CheA is inactive and unable to autophosphorylate (B). Importantly, the two-state property of the receptor dimers and signaling teams leads to two characteristic regimens of activity as described below. Single two-state receptor: We assume that an individual chemoreceptor (homodimer) has two states, on or off (Asakura and Honda, 1984). Attractant binding favors the off state, in which the receptor-bound kinase CheA is inactive, whereas modification of the receptor favors the on state, in which CheA is active. At equilibrium, the probability that the receptor dimer is on is

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where all energies are expressed in units of the thermal energy kBT. The receptor dimer free-energy difference between on and off states is given by

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Object name is msb200849-i13.jpg

where ligand binding in the on (off) state with dissociation constant KDon (KDoff) is included. For attractant binding KDoff<KDon, and for repellent binding, KDoff>KDon. The modification state of the receptor dimer enters only through the offset energy Δepsilon(m). The probability of the on state is considered to be the receptor activity. Two regimens of activity are apparent. (C) A schematic energy-level diagram as a function of attractant (ligand) concentration [L] for a fully demethylated (unmodified) receptor dimer (left, regimen I) and a fully modified receptor dimer (right, regimen II) is shown (Keymer et al, 2006). The receptor dimer can either be on or off. For clarity in the figure, only the off state can bind attractant (red disc), with ligand dissociation constant KDoff. Methyl groups or glutamines (diamonds) lower the free energy of the on state. In regimen I, the on-state free energy is above the off-state free energy (Δepsilon>0) in the absence of attractant, leading to a low activity; the crossing of the lowest levels occurs at [L]=KDoff (black dot). In regimen II, the on-state free energy is below the off-state free energy (Δepsilon<0) in the absence of a ligand, leading to a high activity; the crossing of the lowest levels occurs at an increased ligand concentration [L]=KDoffexp([mid ]Δepsilon[mid ]) (black dot). In both regimens I and II, the crossing of the lowest levels corresponds to the inhibition constant Ki (ligand concentration at half-maximal activity), obtainable from dose–response curves. Receptor signaling team: Within the allosteric Monod–Wyman–Changeux (MWC) model (Monod et al, 1965), two-state receptor dimers form signaling teams with all receptors in a team either on or off together (Sourjik and Berg, 2004; Mello and Tu, 2005; Keymer et al, 2006). Assuming for simplicity a single receptor type, such as Tar, the equilibrium probability that a signaling team of N receptor dimers will be on is

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Importantly, signaling-team formation enhances the difference between the two regimens (Keymer et al, 2006). In regimen I, where Δepsilon>0 (e.g. for Tar{EEEE}), receptor dimers have an even lower activity, ~exp(−NΔepsilon), and an inhibition constant Ki~KDoff/N, indicating an N times higher sensitivity than that of a single receptor dimer. In regimen II, where Δepsilon<0 (e.g. for Tar{QQQQ}), receptor dimers are even more fully active, and turn off at attractant concentration Ki~KDoffexp([mid ]Δepsilon[mid ]) with enhanced cooperativity, specifically with a Hill coefficient nH~N. The strong effect of signaling-team size on the sensitivity of demethylated (unmodified) receptors and on the Hill coefficient of more highly modified receptors allows us to deduce signaling-team size from signaling dose–response curves (Supplementary Table I; Keymer et al, 2006; Endres et al, 2007).

Signal transduction by receptor signaling teams requires association with the kinase CheA, and the adapter protein CheW; both CheA and CheW affect receptor cooperativity (Sourjik and Berg, 2004) and large-scale clustering (Maddock and Shapiro, 1993; Liberman et al, 2004; Shiomi et al, 2005). CheA's kinase activity is inhibited by attractant binding to receptors. When active, CheA autophosphorylates using ATP and transfers the phosphoryl group to the response regulator CheY. Phosphorylated CheY (CheY-P) diffuses to the flagellar motor and induces clockwise motor rotation and cell tumbling. CheY-P is dephosphorylated by its phosphatase CheZ. In the absence of CheY-P, flagellar motors rotate counterclockwise and the cell runs straight. Adaptation in wild-type cells relies on methylation/demethylation of glutamates at 4–6 specific modification sites (depending on the receptor type) by the enzymes CheR/CheB, the latter being activated through phosphorylation by CheA.

Despite the excellent characterization of much of the bacterial chemotaxis network, very little is known about the sizes of receptor signaling teams, which are difficult to resolve by fluorescence or electron microscopy. However, receptor methylation has been observed to affect polar clustering slightly. Receptor clusters are relatively stable, as demonstrated for polar clusters using fluorescence recovery after photobleaching (Schulmeister et al, 2008) and for trimers of dimers (Studdert and Parkinson, 2005) using crosslinking experiments. However, there is some evidence that demethylation or attractant binding decreases the stability of polar receptor clusters (Shrout et al, 2003; Homma et al, 2004; Lamanna et al, 2005; Vaknin and Berg, 2006), although the effects on the size of the polar and lateral clusters appear to be minor or none (Lybarger and Maddock, 1999; Liberman et al, 2004; Lybarger et al, 2005; Shiomi et al, 2005). In vitro, receptor cooperativity has been observed to increase with receptor modification (Li and Weis, 2000), but such an increase in cooperativity is expected with increasing activity even for fixed-size signaling teams (see Box 1; Keymer et al, 2006).

Here, we report a method to reliably extract the size of receptor signaling teams from variable in vivo FRET signaling data (Figure 1), and apply the method to cells that express only the aspartate-specific Tar receptor. We examined both adapting receptors in the presence of the modification enzymes CheR and CheB, as well as genetically engineered receptors in particular modification states in non-adapting cheRcheB cells. To analyze dose–response curves (activity versus MeAsp concentration), we first applied the statistical method of principal component analysis (PCA) to properly treat correlated variability in the data. Next, we fitted our previously established two-state model of signaling to the PCA-treated data to extract parameters with error bars, including signaling-team sizes and modification-dependent receptor energies. As a result of this quantitative data analysis, we discovered a new level of adaptation in the E. coli chemotaxis network. Specifically, we found that signaling-team size increases about three-fold with receptor modification, such as that occurs during adaptation to an attractant. Furthermore, we present a theory that the observed variation in signaling-team size is a novel adaptive mechanism to optimally measure noisy ligand concentrations.

Figure 1
Determining sizes of receptor signaling teams from signaling dose–response curves. Signaling-team size N, i.e. the number of receptor dimers per signaling team, may differ depending on the receptor modification state (glutamate (E) or glutamine ...

Results

To measure signal processing by the receptor signaling teams in living bacterial cells under defined conditions, we expressed the high-abundance aspartate chemoreceptor Tar in an otherwise receptorless strain of E. coli and used in vivo FRET to measure the concentration of CheY-P/CheZ pairs, which is proportional to total CheA activity (Sourjik and Berg, 2002, 2004; Sourjik et al, 2007; see Supplementary information). For this purpose, CheY and CheZ were expressed as fusions to yellow (YFP) and cyan (CFP) fluorescent proteins, respectively, which allows energy transfer upon pair formation. From the high sensitivity and cooperativity (Hill coefficient) of dose–response curves previously measured by FRET (Sourjik and Berg, 2002, 2004), and quantitatively interpreted within the Monod–Wyman–Changeux (MWC) model (see Box 1; Monod et al, 1965; Sourjik and Berg, 2004; Mello and Tu, 2005; Keymer et al, 2006), receptor signaling teams are believed to consist of approximately N=10–20 receptor dimers (3–7 trimers of dimers). In the MWC model, a receptor signaling team is an effective two-state system, where all receptors are either on (active) or off (inactive) (Asakura and Honda, 1984), and bind ligand with the dissociation constants KDon and KDoff, respectively. Attractant binding is more favorable in the off state, and hence a high concentration of attractant [L] tends to turn receptors off, whereas receptor modification favors the on state. The probability that a receptor signaling team is active depends only on the free-energy difference between its on and off states, which is N times the free-energy difference of a single receptor, ΔF=NΔfepsilon, KDon, KDoff[L]), where Δepsilon is the free-energy difference in the absence of ligand. Δepsilon does depend on receptor modification: for unmodified receptors, Δepsilon>0, whereas for modified receptors, Δepsilon<0. Only in their on state do receptors induce the kinase activity of receptor-bound CheA. This model has been very successful in describing high sensitivity to ligand, precise adaptation, and signal integration by mixed receptor types (Endres and Wingreen, 2006; Keymer et al, 2006), and is believed to reflect an organization of receptors in small tightly coupled signaling teams that are then interlinked to form the large polar and lateral clusters.

Previous studies indicated that increasing receptor modification can enhance protein interactions in receptor clusters (Shrout et al, 2003; Shiomi et al, 2005; McAndrew et al, 2006). To test whether the size of receptor signaling teams depends on the modification state (cf. top of Figure 1, showing a schematic of different sized signaling teams of trimers of dimers), Tar receptors were genetically engineered to have either a glutamate (E) or a glutamine (Q) at the four modification sites in the cytoplasmic domain, and expressed in cells lacking all other receptors (see Supplementary information). In chemotaxis, a glutamine (Q) is functionally similar to a methylated glutamate. For instance, strains expressing only Tar{QQQQ} are highly active at zero attractant concentration, whereas strains expressing only Tar{EEEE} are generally inactive. Using α-methyl-DL-aspartate (MeAsp), a non-metabolizable analog of aspartate, we measured dose–response curves, i.e. the activity at various attractant concentrations, for adapting cells (CheRB+) and for non-adapting mutants (cheRcheB) that express Tar receptors in different modification states QEEE, QEQE, QEQQ, and QQQQ. The resulting data for each strain show large day-to-day variation (illustrated by the different symbols in Figure 1, middle), despite the fact that the FRET data are intrinsically independent of the number of measured cells, as only fluorescence ratios are considered. The variation presumably stems from a fluctuating expression of the Tar receptor even at a defined inducer concentration. Because of the large error bars on the data, a standard χ2 analysis provides only a weak constraint on signaling-team size. However, the variation of the data is highly correlated, e.g. the overall amplitudes of dose–response curves vary considerably, whereas the shapes remain more consistent. To exploit this consistency of the data, we used the statistical method of PCA (Figure 2, see Materials and methods, and Supplementary information). Given a set of dose–response curves, PCA identifies the independently varying collective modes of the data, specifically a highly variable amplitude mode and multiple, less variable shape modes. Within standard χ2 analysis, deviations between model and data are weighted by the inverse of the variance at each data point. In contrast, our fits of the MWC model to the data are weighted by the inverse variance of the PCA modes, exploiting our greater confidence in the least variable modes of the data. A sufficient number of principal components are included to account for most of the data variation while not overinterpreting the data due to small sample size (Supplementary Figure 6). As a result, we were able to obtain tight error bars on model parameters and to draw quantitative conclusions on the variation of signaling-team size with receptor modification. As a schematic example, Figure 1 (bottom) shows illustrative high-confidence fits, indicating that the number of receptor dimers per Tar{QEEE} and Tar{QQQQ} signaling team is N~6 and ~18 receptor dimers, respectively.

Figure 2
Illustration of principal component analysis (PCA) applied to dose–response curves of receptor activity to obtain principal modes of data variation. Receptor activity at various concentrations of attractant (MeAsp) was measured through in vivo ...

Figure 3 shows experimental dose–response curves (symbols) for all strains at high (about 3.6 × native) Tar expression level, as well as the corresponding theoretical best fits obtained using PCA (curves) (see Supplementary Figure 7 for the data and fits at low (about 1.4 × native) Tar expression level). The results for both expression levels are summarized in Figure 4, which shows the inferred signaling-team sizes N (top) and receptor offset energies Δepsilon (bottom) for low (left) and high (right) Tar receptor expression levels: An increase in signaling-team size with increasing receptor modification is apparent, ranging from 5 to 7 receptor dimers for Tar{QEEE} to 16–19 receptor dimers for Tar{QQQQ} (depending on receptor expression). On the basis of our error bars, at low expression level, signaling-team sizes for Tar{QEEE} and Tar{QEQE} are conclusively smaller than for Tar{QQQQ}, i.e. NQEEE, NQEQE<NQQQQ, and at high expression level NQEEE<NQEQE<NQEQQ, NQQQQ (see Supplementary Table I for numerical values of parameters). Additionally, receptor offset energies Δepsilon decrease systematically with modification state and are approximately independent of receptor expression level, as expected for an intrinsic single-receptor property, providing further evidence for the reliability of our data acquisition and analysis.

Figure 3
Individual receptor-activity dose–response curves (symbols) and corresponding PCA fits (solid lines) for E. coli cells expressing only Tar receptors at high (~3.6 × native) expression level (see Supplementary information). Cell ...
Figure 4
Sizes of receptor signaling teams N (number of receptor dimers per signaling team, panels A and B) and offset energies Δepsilon (panels C and D) obtained from PCA fits shown in Figure 3 in the main text and Supplementary Figure 7. The left panels ...

Our results indicate that the size of receptor signaling teams depends directly on receptor modification state, not on receptor activity. The Tar{QEQE}, Tar{QEQQ}, and Tar{QQQQ} strains are all fully active at low MeAsp concentrations (see Figure 3 and Supplementary Figure 7), but signaling-team size nevertheless increases with the number of glutamines (Figure 4). To further test the dependence of signaling-team size on receptor modification at high Tar expression level, we made use of the fact that CheRB+ cells adapted to two different attractant concentrations, 0 and 0.1 mM MeAsp, have the same activity (indicated by arrows in Figure 3), but, respectively, low and high receptor modification levels. Compared to cells adapted to zero ambient, the receptor methylation level (as evident from the offset energy) and signaling-team size have significantly increased for cells adapted to 0.1 mM MeAsp (plus symbols in Figure 3 and stripe pattern in Figure 4B and D).

To test whether our results for the sizes of receptor signaling teams depended on the MWC model, we also obtained fits of the FRET data using alternative models in which trimers of dimers are coupled to their nearest neighbors in a triangular or in a honeycomb lattice (see Supplementary information). Although these Ising-type lattice models do not account well for the data from mostly demethylated receptors (Skoge et al, 2006), adequate fits to dose–response curves were obtained for the more highly modified receptors. The results strongly confirm our conclusion that effective signaling-team size increases significantly with receptor modification.

Do the observed variable complex sizes reflect an ‘optimal' receptor signaling-team size that changes with ligand concentration? We address this question briefly here, with details in the Supplementary information. Complexes of N receptor dimers respond more strongly to changes in ligand concentration than do single receptor dimers, as the free-energy difference of a receptor signaling team is NΔf, where Δf is the free-energy difference of a single receptor dimer. Hence larger signaling teams are better for detecting weak signals. However, larger signaling teams are also noisier than smaller signaling teams, as ligand noise gets amplified as well. For instance, the standard deviation of the free-energy difference of a signaling team due to ligand concentration noise δ[L] is

equation image

where the derivative is

equation image

A large standard deviation of the free-energy difference of a signaling team, i.e. σF>1 in units of the thermal energy kBT, means that many signaling teams are either fully active or fully inactive, making them unresponsive to small changes in the mean ligand concentration, and therefore of no use for chemotaxis. A restriction to σF[less, similar]1 sets an upper limit on the size of signaling teams. The lowest possible uncertainty in ligand concentration is given by the Berg and Purcell limit (Berg and Purcell, 1977) and generalized by Bialek and Setayeshgar (2005) An external file that holds a picture, illustration, etc.
Object name is msb200849-i3.jpg, where a is the receptor dimension, D is the ligand diffusion constant, and τ is the averaging time. On the basis of this estimate, equation (1) can be solved for the upper limit on the signaling-team size

equation image

As a result, to optimize sensitivity the signaling-team size increases as N~[L]1/2 for intermediate ligand concentrations (KDoff[double less-than sign][L][double less-than sign]KDon). In a nutshell, larger signaling teams are more sensitive to small changes in ligand concentration, but as a result are also more sensitive to ligand concentration noise. Optimally, signaling teams will be as large as possible without being saturated by this noise (i.e. fully active or fully inactive and hence unresponsive to ligand concentration changes). As the relative ligand noise decreases with concentration, the optimal signaling-team size will grow with ligand concentration, as we observed.

Discussion

The chemotaxis network of E. coli exhibits remarkable sensing and signaling properties that rely on receptor signaling teams. Despite recent high-resolution electron microscopy (Weis et al, 2003; McAndrew et al, 2004, 2005, 2006; Briegel et al, 2008), very little is known about what determines receptor signaling-team formation and size (Kentner and Sourjik, 2006; Kentner et al, 2006). Notably, because signaling-team size and signaling sensitivity are closely related (Keymer et al, 2006; Endres et al, 2007; Mello and Tu, 2007), receptor kinase activity can be used to probe signaling-team size. Starting from dose–response data on the activity of Tar receptors from in vivo FRET experiments, we developed a robust analysis method to extract signaling-team sizes and receptor offset energies (Figure 1), interpreted within a well-established model for cooperative receptor signaling (see Box 1). To achieve this, we applied the unbiased statistical method of PCA (see Figure 2, Materials and methods, and Supplementary information) to the data to separate correlated variation in the data from noise due to small sample size. Conventional χ2 fitting to individually normalized data curves misses the comparison of amplitudes, and when applied to averaged data values and their standard deviations overestimates the fluctuations by missing the correlations among the data points within a dose–response curve measured on the same day. Using our PCA analysis, we then fitted the MWC model for receptor signaling teams (see Box 1) to the principal components of the data, allowing us to obtain receptor parameters with tight confidence intervals (Figures 3 and and4,4, as well as Supplementary Figure 7). Importantly for chemotaxis, we found that signaling-team size increases significantly with receptor modification level (Figure 4). Motivated by this discovery, we presented a model describing how variable signaling-team size may enable cells to optimally measure ligand concentrations. Whereas both small changes in ligand concentration (signal) and ligand concentration noise are increasingly amplified by increasing signaling-team size, the noise decreases for increasing ambient concentration. This allows the cell to utilize larger signaling teams at larger ambient concentrations and hence receptor modification levels without saturation of the signaling teams by noise.

Our results based on Tar-only mutants suggest a previously unrecognized level of adaptation relying on increasing signaling-team size with receptor modification. Interestingly, the opposite trend, i.e. a diminishing ability to cluster with increasing receptor modification, was suggested to possibly downregulate overall sensitivity with adaptation (Bray et al, 1998). In the MWC model, however, this downregulation simply emerges from the two-state property (Box 1 and Keymer et al, 2006). To show that variable signaling-team sizes also apply to mixed receptor types, a similar detailed analysis would need to be conducted on FRET data of wild-type cells. As Tsr and Tar are the most abundant receptor types, at least the numbers of Tsr and Tar receptor dimers per receptor signaling team would need to be considered while ensuring that the overall ratio of Tsr/Tar stays independent of modification state. Previous standard analysis of such FRET data showed that mixed Tar/Tsr signaling teams contain a total of around 10–20 receptor dimers, consistent with our Tar signaling teams. Additionally, as wild-type cells show high degree of polar clustering (Maddock and Shapiro, 1993; Sourjik and Berg, 2000; Liberman et al, 2004), the deduced clustering ability of Tar receptors is not an artifact of the homogeneous receptor expression.

What are the possible mechanisms responsible for receptor signaling-team formation and the increase in signaling-team size with increasing receptor modification? Receptor modification reduces the electrostatic repulsion between neighboring dimers by removing charged glutamate residues, potentially stabilizing larger signaling teams. However, the modification sites of neighboring receptor dimers are at least 3 nm apart in the partial crystal structure (Kim et al, 1999). For a Debye–Hückel screening length of 1 nm, the resulting screened electrostatic interaction is only 1% of the thermal energy for a pair of charged glutamates. Therefore, other mechanisms may contribute to the observed dependence of signaling-team size on receptor modification: (1) binding of CheA and CheW affects both large-scale clustering of receptors (Maddock and Shapiro, 1993; Sourjik and Berg, 2000; Liberman et al, 2004) and the sizes of signaling teams (Sourjik and Berg, 2004), and the relevant binding affinities of CheA and CheW appear to depend on the level of receptor modification (Shrout et al, 2003); (2) receptor activity has been shown to respond to receptor–membrane interactions (Draheim et al, 2006) and changes in osmotic pressure (Vaknin and Berg, 2006), so inter-dimer coupling could be mediated by elastic membrane deformations, as proposed for the approximate two-state osmolarity-sensing MscL pore (Ursell et al, 2007). Our data also indicate that signaling-team size increases moderately with expression level (Figure 4 and Supplementary Table I), perhaps because more available receptors push the distribution of signaling teams toward larger sizes (Endres et al, 2007).

Many other sensory systems cluster, including B cell (Schamel and Reth, 2000), T cell (Germain and Stefanova, 1999), synaptic (Griffith, 2004), and ryanodine (Yin et al, 2005), indicating that receptor clustering is an important regulatory mechanism for the cell, e.g. to adjust signaling properties or by recruiting auxiliary proteins. As eukaryotic G-protein-coupled receptors become covalently modified as well, e.g. by phosphorylation, receptor modification may prove to be a general mechanism for dynamic regulation of cluster size and optimization of signal response.

Materials and methods

Model for receptor signaling team

To extract parameters of receptor signaling teams from fitting to in vivo FRET data, we describe signaling teams by the allosteric MWC model (Monod et al, 1965), as described in Box 1. Briefly, in this model signaling teams are composed of two-state receptors with all receptors in a signaling team either on or off together (Sourjik and Berg, 2004; Keymer et al, 2006). The number of receptor dimers per signaling team is treated as a continuous parameter, reflecting the degree of cooperativity among receptors, independent of the suborganization of dimers into trimers of dimers. Assuming a single receptor type, such as Tar, the equilibrium probability that an MWC signaling team of N receptor dimers will be on is

equation image

where Δepsilon(m) is the offset energy, which depends on the receptor modification level m, [L] is the attractant concentration, and KDon(off) is the attractant dissociation constant in the on (off) state. All energies are expressed in units of the thermal energy kBT. We obtain the signaling-team size N and the modification-dependent off-set energies Δepsilon(m) from the fitting procedure described next. The attractant dissociation constants for the on and off states are taken from Keymer et al (2006).

PCA

To obtain χ2 fits, including confidence intervals for parameters, it requires a representation of the data where deviations from average values are uncorrelated. This is achieved by diagonalizing the covariance matrix of the data for each strain to obtain the principal (uncorrelated) components of the data (see Supplementary Figure 5). Specifically, from the M dose–response curves xi=1,…,M (L), each measured for D different ligand concentrations L, the D × D covariance matrix C is calculated through

equation image

where L1 and L2 are ligand concentrations, and xave(L) is the average of all M curves at concentration L. Each covariance matrix is diagonalized by V−1CV=U where V is a matrix with the eigenvectors of C as the columns and U is the diagonal matrix of the eigenvalues λm=1,…,D. These eigenvectors and eigenvalues are the principal components of the data. Standard fitting of a calculated curve c(ν; L), which depends on a set of fitting parameters ν, to xave would require minimizing

equation image

where σL is the experimental standard deviation of the data at ligand concentration L. But this neglects the observed correlations within each dose–response curve (cf. Figure 2A). Instead, we express χ2 in the PCA basis set by using c=V−1(cxave) and replacing the sum over D ligand concentrations by the sum over D principal components

equation image

For fits using equation (7), we included only the 3–5 largest principal components (see Supplementary Figure 6). We fitted the data for all five strains QQQQ, QEQQ, QEQE, QEEE, and CheRB+ simultaneously by minimizing the total χ2PCA, as one fitting parameter, an overall amplitude factor [var phi], is common to all five strains. The calculated dose–response curve for each strain also depends on an offset energy Δepsilon and signaling-team size N, leading to a total of 11 fitting parameters ν=([var phi], Δepsilon1, N1, …, Δepsilon5, N5) for the low expression level data (cf. Supplementary Figure 7A and Figure 4A), and 13 fitting parameters ν=([var phi], Δepsilon1, N1, …, Δepsilon6, N6) for the high expression level data (cf. Figures 3 and and4B).4B). The dissociation constants for MeAsp binding by Tar in the on and off states were taken to be KDon=0.5 and KDoff=0.02 mM, respectively (Keymer et al, 2006).

Confidence intervals

We followed the method of Press et al (2002), but expressed the curvature matrix

equation image

in the PCA basis by transforming c=V−1(cxave) and calculating

equation image

with A=(V−1)TU−1V−1 and Bqk=[partial differential]c(Lq)/([partial differential]νk). Confidence intervals were obtained from the total (αPCAtot) summed over the five strains. From the corresponding covariance matrix Cα=(αPCAtot)−1, the confidence interval of parameter νk is given by An external file that holds a picture, illustration, etc.
Object name is msb200849-i11.jpg where Δχ2=4 for 95.4% confidence.

Experimental protocol

Cell growth, FRET measurements, and immunoblots were performed essentially as before (Sourjik and Berg, 2002, 2004; Sourjik et al, 2007). Briefly, receptorless cells co-expressing a CheZ-CFP/CheY-YFP FRET pair and Tar receptors in one of the modification states were grown to mid-exponential phase in the presence of appropriate antibiotics. Expression of the FRET pair was induced with 50 μM isopropyl β-D-thiogalactoside. Expression of receptors was induced with either 1 or 2 μM sodium salicylate, corresponding to low and high levels, respectively. Cells were harvested, washed, and assayed for attractant-induced changes in FRET signal in a flow chamber mounted on a custom-modified Zeiss Axiovert 200 microscope. Fluorescence of a field of 300–500 cells was monitored in each experiment. FRET was defined as the fractional change in cyan fluorescence due to energy transfer, and was calculated from changes in the ratios of yellow and cyan fluorescence signals (Sourjik and Berg, 2004). See Supplementary information for measurement details and for the list of strains and plasmids used in this study.

Supplementary Material

Supplementary Table 1

Supplementary Information

Acknowledgments

We thank Fred Hughson, Tom Shimizu, Monica Skoge, and Chris Wiggins for helpful suggestions. All authors (except CHH) acknowledge funding from the Human Frontier Science Program (HFSP).

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