A dynamic-signaling-team model for chemotaxis receptors in Escherichia coli

Abstract

The chemotaxis system of Escherichia coli is sensitive to small relative changes in ambient chemoattractant concentrations over a broad range. Interactions among receptors are crucial to this sensitivity, as is precise adaptation, the return of chemoreceptor activity to prestimulus levels in a constant chemoeffector environment through methylation and demethylation of receptors. Signal integration and cooperativity have been attributed to strongly coupled, mixed teams of receptors, but receptors become individually methylated according to their ligand occupancy states. Here, we present a model of dynamic signaling teams that reconciles strong coupling among receptors with receptor-specific methylation. Receptor trimers of dimers couple to form a honeycomb lattice, consistent with cryo-electron microscopy (cryoEM) tomography, within which the boundaries of signaling teams change rapidly. Our model helps explain the inferred increase in signaling team size with receptor modification, and indicates that active trimers couple more strongly than inactive trimers.

Adaptation to persistent stimuli is important in maintaining a broad range of response-sensitivity in numerous sensory systems, including visual (1), olfactory (2), and auditory (3). The best understood example of such adaptation is the Escherichia coli chemotaxis system (Fig. 1A), in which receptor signaling adapts precisely to persistent chemoeffector concentrations. Adaptation is implemented by the methylation and demethylation of transmembrane receptors through an integral-feedback control mechanism (4, 5). Different receptor types bind specific ligands, but signal cooperatively in mixed teams (6, 7). One feature of this system that has remained mysterious is that, at short times, adaptation leads to similar modification of all receptors, but at longer times, modification is receptor-type specific, affecting primarily the receptors stimulated by ligand (8). Here, we present a model of dynamic-signaling teams that reconciles receptor cooperativity with receptor-specific methylation.

Fig. 1.

Dynamic-signaling-team model of two-state chemotaxis receptors. (A) Schematic of the chemotaxis network (details in beginning of article) (B) Top view of a portion of the receptor array. Trimers of dimers are arranged in a fixed honeycomb lattice with Tar (dark-colored) and Tsr (light-colored) receptors randomly assorted in a 1∶2 ratio. Interactions between adjacent trimers of dimers (solid line segments) can form and break through thermal fluctuations, creating dynamic-signaling teams. Additionally, these teams can switch between two states: active (red) and inactive (blue). (C) Average activity response to addition and subsequent removal of 1 mM MeAsp. Inset: Precise adaptation occurs through the methylation of inactive receptors by CheR and demethylation of active receptors by CheB. CheR and CheB act on AN composed of adjacent trimers of dimers, whether or not these trimers are part of the same signaling team.

Receptors direct cell movement by biasing the switching between tumbling and straight-swimming states, producing a biased random walk up gradients of chemoattractants or down gradients of chemorepellents. Of the five receptor types in E. coli, the highest abundance are Tar (aspartate binding) and Tsr (serine binding). Receptors form homodimers that can each bind one molecule of ligand. Homodimers, which we will refer to as receptors, in turn form mixed trimers of dimers, and associate with the linker protein CheW and the histidine kinase CheA (9). Receptor signaling activates CheA autophosphorylation, and the phosphoryl group is transferred to the response regulator CheY (or to the methylesterase CheB). Phosphorylated CheY diffuses and binds to the flagellar motors, causing tumbling. CheY is dephosphorylated by the phosphatase CheZ. Receptor trimers of dimers form the basic signaling units, with single receptors unable to activate CheA (10). Adaptation occurs though methylation by CheR and demethylation by CheB of eight sites on each receptor (11). Methylation at each site neutralizes a charged glutamate and increases the activity of receptors, with demethylation doing the reverse. Each Tar or Tsr protein has a 35 amino-acid tether at its C terminus, with a pentapeptide site that can bind one CheR or CheB (12). When transiently bound to a receptor, each CheR or CheB can also act on 4–6 adjacent receptors, defining an “assistance neighborhood” (AN) (13).

Cooperativity between receptors enhances sensitivity at low ligand concentrations and provides gain at higher ligand concentrations. Both in vivo (7) and in vitro (14) measurements of the activity of homogeneously expressed receptors indicate signaling teams larger than trimers, with Hill coefficients > 3. Fits of in vivo dose-response measurements indicate that cooperativity increases with receptor modification (15). Fluorescent imaging (16–18) and cryoEM tomography (19–21) reveal large polar or lateral clusters of receptors, with cryoEM revealing a slightly disordered honeycomb lattice of trimers. Receptors within clusters are relatively stable, as demonstrated by fluorescence recovery after photobleaching (22) and by receptor cross-linking (9).

Monod-Wyman-Changeux (MWC) models have been used to explain a variety of experimental dose-response curves (7, 23–28). In these MWC models, two-state receptors form signaling teams, which are either active or inactive as a whole. However, fixed quasiindependent teams of trimers cannot account for the observed lattice or differential methylation of receptors. Instead, MWC models predict nearly uniform methylation of all receptor types. Here we introduce a model in which trimers form dynamic-signaling teams. Our dynamic equilibrium model reproduces experimental dose-response curves and the observed methylation of receptors, explains the inferred increase in signaling-team size with receptor modification (15), and reveals that active trimers couple more strongly than inactive trimers.

Model

We explore a dynamic signaling-team model, where trimers of chemoreceptors transiently interact in small teams within a fixed, large honeycomb array (see Fig. 1B). Each team of trimers is assumed to be either fully active or fully inactive, but the interactions defining the teams are dynamic.

We model each receptor as a two-state system, being either active (on) or inactive (off). The free-energy difference between the on and off states of a single receptor is ((25) and SI Text)

[1]

Here, [L] is the ligand concentration, and are the binding constants of the receptor-type r in the on and off states, m is the methylation level (m = 0,…8), and is the zero-ligand offset energy. All energies are in units of the thermal energy k_BT. The free-energy difference F_i of a trimer at lattice site i is the sum of the individual f_r(m) of the three receptors in the trimer. When multiple receptor types are present, we assume random mixing, consistent with cross-linking experiments (9).

Coupling between neighboring trimers to form signaling teams occurs through the formation of dynamic interactions, presumably representing tight protein-protein interfaces. These interactions can form and break, dictated by thermal equilibrium energetics. At any moment, a signaling team consists of all trimers joined through a path of connected interactions (Fig. 1B). We allow the nearest-neighbor coupling energy to depend on the activity state of the trimers. Letting σ_i = +1 (-1) for an active (inactive) trimer at lattice site i, the coupling energy is J(σ_i,σ_j). Adjacent on trimers couple with energy J(+1,+1) = J_on, adjacent off trimers couple with energy J(-1,-1) = J_off and adjacent trimers with different activity states couple with energy J(+1,-1) = J(-1,+1) = J_diff. To keep the signaling teams small, every pair of trimers within a signaling team also contributes a repulsive energy U, independent of activity state (see Discussion). Because the total repulsive energy grows with the square of the team size while the coupling energy grows only linearly with team size, the result is a finite optimal team size. The overall magnitudes of the Js and U control the thermal spread of the signaling-team-size distribution. In summary, letting 〈i,j〉 denote nearest-neighbor trimers and letting b_〈i,j〉 = 1 (0) denote the presence (absence) of an interaction, the total free-energy of the lattice is given by

[2]

where N_i is the size of the signaling team that contains trimer i. Both the interactions and the trimer activities are considered as free variables.

We assume J_on > 0 and J_off > 0 are favorable, while J_diff < 0 is strongly unfavorable (|J_diff|≫|J_on|, |J_off|). Therefore, each signaling team acts as an MWC cluster, being either fully on or off as a whole. For a fixed interaction conformation, the effective free-energy difference between the on and off states of a signaling-team k is

[3]

Eq. 3 is identical to the MWC model, but with the extra first term to account for the difference in interaction energies between on and off states. Therefore, for a fixed interaction conformation, the probability of signaling-team k being active (its “activity”) is given by A_k = 1/[1 + exp(F_k)], with the average activity of the whole lattice given by the sum over all signaling teams weighted by team size, .

Adaptation Mechanism.

We assume adaptation to be much slower than receptor conformation switching (6). We model adaptation using only a local feedback mechanism as in the Barkai and Leibler two-state model (4). We assume CheRs add methyl groups to off receptors at a rate k_R and CheBs remove methyl groups from on receptors at a rate k_B, both independent of the number of available sites. An increase in the methylation level m decreases the offset energy ϵ_r(m), favoring the active state. As in previous adaptation models, each CheR or CheB acts on an AN of receptors (26). Here we include static ANs of size six, consisting of all pairs of adjacent trimers of dimers, independent of the signaling-team boundaries (Fig. 1C inset). Large ANs extend the “methylation ladder,” decreasing the probability that a neighborhood will become fully methylated or fully demethylated by chance and allowing the activity to adapt precisely over a broad range of ligand concentrations (26). Fig. 1C demonstrates precise adaptation of our model, with the adapted activity independent of ligand concentration.

Simulation Algorithm.

For a fixed interaction configuration, total receptor activity A is computed directly. To simulate interaction dynamics, we used the Metropolis algorithm (29) on a honeycomb array of trimers with periodic boundary conditions. To simulate adaptation we used a Gillespie algorithm (30). We assume that the interaction addition/deletion and trimer on/off switching rates are much faster than the receptor modification rates. Simulation details, binding parameters and modification rates are given in SI Text.

Results

Activity Response of a Mixed-Receptor Array.

First, we show that our dynamic model agrees well with in vivo dose-response data from Sourjik and Berg (6). In Fig. 2A, we show the normalized experimental dose-response curves to methyl aspartate (MeAsp) for wild-type and mutant cells (either cheR or cheR cheB) with mixed Tar and Tsr receptors. For cheR cheB cells lacking the adaptation mechanism, Tsr receptors are QEQE and Tar receptors are EEEE, QEEE, QEQE, or QEQQ. Glutamine (Q) is neutral and behaves similarly to methylated glutamate (methylated E) (31). The in vivo response is measured through FRET, in which the proteins CheY and CheZ are fused to different fluorescent proteins, allowing measurement of the phosphorylation of CheY, and thereby of receptor signaling.

Fig. 2.

Receptor activity response to steps of the attractant MeAsp. (A) Normalized FRET response measured by Sourjik and Berg (6). The smooth curves are ad hoc fits to guide the eye. (B) Normalized activity response of the dynamic-signaling-team model with a Tar:Tsr ratio of 1∶2, trimer-interaction energy J = 5, and long-range repulsive energy U = 0.25. The experimental strains and our corresponding choices of offset energies for Tar and Tsr (ϵ_a, ϵ_s) are: wild type, 0.02, 0.02; cheR, 0.25, 0.1; cheR cheB mutants—Tar{EEEE}, 1.0, -1.3; Tar{QEEE}, -0.2, -1.3; Tar{QEQE}, -0.9, -1.3; Tar{QEQQ}, -1.4, -1.3. All energies are given in units of the thermal energy k_BT.

In Fig. 2B, we show normalized dose-reponse curves to MeAsp from the dynamic-signaling-team model with a Tar:Tsr ratio of 1∶2 for different values of offset energies ϵ_r, but no other changes of parameters. The average signaling team size is approximately 24 receptors. Like the static MWC model (25), the dynamic-signaling-team model accounts well for the experimental data shown in Fig. 2A. Specifically, the model reproduces the two regimes of receptor response for two-state-signaling teams ((25) and SI Text). In the absence of ligand, for average ϵ_r > 0 the off state is favored and the response is in the low-activity regime, while for average ϵ_r < 0 the on state is favored and the response is in the high-activity regime. An essential feature of the dynamic model, required to fit the data, is the strong coupling between all receptor types across all activity regimes, as in the MWC model.

The observed initial activity of the wild-type strain is ≳10× the cheR mutant activity, meaning that the average ϵ_r for wildtype must be lower than for the cheR mutant. Despite different initial activities, the shapes of the normalized dose-response curves are very similar for these two strains. The inhibition constant K_i ≈ 3–4 μM is nearly identical for the two cell types and at least 5 times smaller than the binding constant of a single Tar receptor (≈20–30 μM). Moreover, all receptors are fully inhibited by 100 μM MeAsp, well below the affinity of Tsr for MeAsp (≈100 mM). These properties indicate that strong coupling between different receptor types persists in the low-activity regime (27). Our model correctly predicts the similar dose-response curves for wild-type and cheR cells, because each signaling team acts as a strongly-coupled MWC cluster, even at low activity.

The model also correctly predicts the response of nonadapting mixed-receptor strains in the high activity regime, including the plateaus and shifts of K_i. In these strains, the initial activity decrease is due to Tar receptors binding MeAsp, followed by activity plateaus when the Tar receptors become saturated. The activities finally drop to zero when Tsr receptors bind MeAsp. The K_is of the first activity drop and the plateau levels increase with Tar modification because the average offset energy decreases, favoring the active state. A notable feature of the experimental data is that in the high activity regime, the Hill coefficients are not enhanced (i.e., are close to 1). A static MWC model with a single team size naively predicts that Hill coefficients in this regime should approximately equal the number of ligand-binding receptors in a team. Therefore within the MWC model, to match observation one must include signaling teams with a distribution of sizes. Teams of different sizes and different numbers of Tar and Tsr receptors have different K_i values, resulting in a low ensemble Hill coefficient. Compared to the static MWC model, with an arbitrary distribution of signaling-team sizes, the dynamic model has the advantage of providing a natural mechanism for a distribution of team sizes. Indeed, the dynamic model results in low Hill coefficients (< 2) for the heterogeneous teams in the high activity regime. Note that for homogeneous receptors, the dynamic-signaling-team model does correctly predict the observed high Hill coefficients in the high activity regime.

Methylation Response of a Mixed-Receptor Array.

In a classic experiment, Sanders and Koshland (8) measured the methylation response of cells to different attractants, and found different methylation responses of ligand-binding and non-ligand-binding receptors. These experiments measured the methylation levels of mixed Tar and Tsr receptors at time points after addition of the Tar-ligand aspartate or the Tsr-ligand serine. Although both receptor types initially increased in methylation following addition of attractant, Tar methylation increased to a higher level in response to aspartate than in response to serine, and Tsr methylation increased to a higher level in response to serine than in response to aspartate (Fig. 3A, inset).

Fig. 3.

Receptor methylation response to a step of attractant. (A) Static-signaling-team model results and (B) dynamic-signaling-team model results for the average methylation levels of Tar (dark-colored curves) and Tsr (light-colored curves) receptors in response to 1 mM MeAsp (red) or 1 mM serine (blue). All parameters are the same as the wild type in Fig. 2. Inset: Experimental measurement by Sanders and Koshland (8) of Tar and Tsr methylation-level changes in response to aspartate and serine, normalized by the maximum and minimum methylation levels. Experimental adaptation times are longer than in wild-type cells due to receptor overexpression.

The dynamic-signaling-team model naturally accounts for the observed higher methylation response of ligand-binding receptors. In contrast, a static-signaling-team model predicts an approximately uniform methylation response for all receptor types. In Fig. 3, we plot the methylation responses predicted by both the static- and dynamic-signaling-team models for step increases of 1 mM MeAsp or 1 mM serine. At these concentrations only Tar receptors appreciably bind MeAsp and only Tsr receptors appreciably bind serine. While the static model predicts almost identical methylation for both Tar and Tsr receptors, the dynamic model qualitatively reproduces the experimental data showing preferential methylation of ligand-binding receptors.

Both static and dynamic models consist of fixed, honeycomb lattices of trimers of dimers, with each trimer either active (on) or inactive (off). The number of Tar and Tsr receptors in each trimer is distributed according to a binomial distribution. Within the static-signaling-team model, different receptor types within a team are methylated/demethylated at the same rate, but different teams have varying numbers of Tar and Tsr receptors. The activities of teams with more Tar (Tsr) receptors will drop more strongly in response to MeAsp (serine) addition, resulting in higher methylation of those teams in order to restore activity. The result is a slight increase in the methylation level of the ligand-binding receptor type. When teams are large, as in Fig. 3A, the proportions of Tar and Tsr receptors vary little from team to team, resulting in similar methylation levels for both receptor types.

In the dynamic-signaling-team model, the boundaries of the signaling teams move. Therefore in order for the whole receptor array to adapt precisely, each trimer of dimers acts as an individual adapting unit and must adapt precisely. The result is more methylation of trimers that contain more ligand-binding receptors (Fig. 3B), in agreement with observation of higher methylation levels of ligand-binding receptors. Also, as seen in the experimental response of Tar receptors to serine, the model can produce an overshoot in the methylation level of the nonligand-binding receptor. For example, after addition of MeAsp, Tsr-dominated trimers initially become inactive due to their coupling to Tar-dominated trimers. Since all trimers are inactive, all trimers initially become methylated at approximately the same rate. As adapation proceeds, the Tsr-dominated trimers overshoot their target methylation level because the Tar-dominated trimers are still undermethylated and inactive, lowering the activity of nearby trimers. Finally, when the whole array approaches its adapted activity, the now hypermethylated, and consequently hyperactive, Tsr-dominated trimers lose their excess methylation. At steady state, the methylation of receptor types in the dynamic-signaling-team model is similar to what would be found for isolated trimers, given our experimentally motivated assumption that individual trimers are static (9).

Increasing Coupling Energy with Modification Explains the Inferred Increase in Team Size.

Endres et al. (15) used principal component analysis (PCA) to analyze FRET dose-response data from Tar-only cells to demonstrate that within the MWC model team size increases 2- to 3-fold with receptor modification. Here, we show that this increase in team size can be naturally understood within the dynamic-signaling-team model if the trimer-trimer-interaction energy J increases with receptor modification.

In Fig. 4, we plot the experimental dose-response data from (15) (symbols) for all strains at high (≈3.6× native) Tar expression levels, as well as the corresponding theoretical best fits to our model obtained using PCA (curves). The parameters inferred from the theoretical fits are the six trimer-trimer-interaction energies J (Fig. 4B), the six receptor offset energies ϵ (Fig. 4C), and the overall amplitude factor ϕ (see SI Text). We set the long-range repulsive energy to U = 0.25. We find that receptor offset energies decrease systematically with increasing modification (Fig. 4C), as expected because more highly modified (neutralized) receptors are more active. Additionally, the interaction energy increases with modification, with , J_QQQQ. The team sizes depend directly on receptor modification state, not on receptor activity, as the Tar{QEQE}, and the Tar{QEQQ} and Tar{QQQQ} strains are all fully active at low MeAsp concentrations, but have significantly different interaction energies. Larger interaction energies J imply larger average team sizes 〈N〉 as shown in Fig. 4D. In turn, larger team sizes mean increased cooperativity in the high activity regime and decreased inhibition constants in the low-activity regime. As J drops below 0, a lower limit of 〈N〉 ≈ 3 is reached as trimer-trimer interactions become unfavorable. Unlike the MWC model, the dynamic model provides a mechanism for the increase in team size with modification: neutral residues (either glutamines or methylated glutamates) not only favor the active state of receptors, but also increase the strength of allosteric trimer-trimer interactions.

Fig. 4.

Fits of the dynamic-signaling-team model to FRET data from cells expressing only Tar receptors. The high-expression Tar-only data and PCA fitting method are described in Endres et al. (15) and reviewed in SI Text. Cell types include adapting (CheRB⁺) and nonadapting, engineered cheR cheB mutants (QEEE, QEQE, QEQQ, and QQQQ). CheRB⁺ cells are adapted either to zero attractant (× symbols) or to 0.1 mM MeAsp (+ symbols). (A) Individual receptor-activity dose-response curves (symbols) and corresponding PCA fits (lines). The experimental data were normalized by the inverse amplitude ϕ^-1 (see SI Text). (B) Fitted trimer-interaction energies J. (C) Fitted receptor offset energies ϵ. The error bars for B and C are 95% confidence intervals. (D) Within the model, the average signaling-team-size 〈N〉 increases with the trimer-trimer-interaction energy J. The average signaling-team size is measured at A = 0.5 (f = 0). Inset: Neutral residues (glutamine or methylated glutamate) increase the interaction energy J.

Activity Response When J_on ≠ J_off.

In principle, there is no reason that the on-on and off-off trimer-trimer-interaction energies have to be the same in the dynamic-signaling-team-model. If J_on ≠ J_off, on and off trimers will tend to form differently sized signaling teams, implying that the mean signaling-team size will change with receptor activity. The change in team size with activity also means a change in cooperativity. For example if J_on≫J_off, large on teams and small off teams will dominate as shown schematically in the inset for Fig. 5A. In this case, for A ≈ 1 individual trimers can turn off, but for A ≈ 0 only large signaling teams can turn on. Therefore the cooperativity is low for A ≈ 1 but high for A ≈ 0: The opposite holds for J_on ≪ J_off, with high cooperativity near A ≈ 1 and low cooperativity near A ≈ 0.

Fig. 5.

Activity response of the dynamic-signaling-team model for J_on ≠ J_off. (A) Normalized activity response of a wild-type array with a Tar:Tsr receptor ratio of 1∶2 for J_on = J_off = 5, J_on≫J_off (J_on = 9, J_off = 0), and J_on ≪ J_off (J_on = 0, J_off = 9). Experimental wild-type FRET response data is taken from Sourjik and Berg (6). Inset: Top view of a portion of the receptor array with J_on≫J_off. Large on teams and small off teams dominate. (B) Normalized experimental activity responses of Tar receptors at 1×, 2×, and 6× normal expression levels from Sourjik and Berg (7) and theoretical results of our model for J_on > J_off (solid curves) and J_on = J_off (dashed curves). For the fits to the model we used the same coupling and offset energies for all three expression levels, but varied the long-range repulsive energies. Parameters sets are: J_on = 5.8, J_off = 3.8, U_1× = 2.5, U_2× = 1.0, U_6× = 0.25 and J_on = J_off = 1.2, U_1× = 0.30, U_2× = 0.25, U_6× = 0.25. (C) Plot of |A - 0.5| vs. |f| for both the experimental data (symbols) and theory (curves) for the 6× Tar expression level used in (B) for f < 0 (red crosses and curve) and f > 0 (blue pluses and curve). (D) Plot of |A - 0.5| vs. n(0)|f| for the high expression Tar data shown in Fig. 4. Here the free energy is scaled by n(0), the cooperativity of the response at f = 0.

In Fig. 5A, we show that while J_on ≈ J_off or J_on≫J_off could both be consistent with experiment, J_on ≪ J_off is not consistent with the data. Experimentally, total activity drops to near zero by ≈100 μM MeAsp, even though at this low concentration only Tar receptors bind MeAsp. In contrast, within our model, if J_on ≪ J_off, Tsr-only trimers would produce an observable activity plateau extending beyond 100 μM MeAsp. This plateau occurs because, for weak J_on, Tsr-only trimers can fluctuate on while the surrounding trimers remain off. Within the model, if J_on ≈ J_off or J_on≫J_off, then Tsr-only trimers can only turn on as part of a multitrimer team, which is sure to also contain Tar receptors. As a whole, the team will strongly favor the off state because of the Tar receptors, resulting in no activity plateau.

The data in Fig. 5A are ambigious, but our model also predicts that if J_on ≠ J_off, the effective offset energy will change with team size. This effect provides a practical way to measure the difference between J_on and J_off for a homogeneous array of receptors. Specifically, if we define the effective offset energy ϵ_eff as the free-energy difference between the on and off states of a receptor in the array in the absence of ligand, then ϵ_eff = ϵ + 0.5〈b〉(J_off - J_on), where 〈b〉 is the probability a trimer-trimer-interaction is present. So if J_on ≠ J_off, then an increase in 〈b〉, i.e., an increase in team size, will shift the effective offset energy. In practice for receptors in the high-activity regime, ϵ_eff ≲ -1, an increase in the effective offset energy will be directly observable as a decrease in the inhibition constant, K_i ∝ exp(-ϵ_eff).

Experimentally, increasing the receptor expression level increases both cooperativity and the inhibition constant (7, 15). Higher cooperativity implies larger signaling-team sizes (larger 〈b〉), so the concomitant increase in inhibition constant implies J_on > J_off within our model. In Fig. 5B, we plot experimental FRET dose-response data for cells expressing only Tar{QEQE} receptors at 1×, 2×, and 6× the normal expression level (7), as well as fits to our model. For the model curves, we set J_on = 5.8 and J_off = 3.8 for all three curves and increased team size 〈N〉 by decreasing the long-range repulsive energy U. We assume that U captures all long-range effects including receptor pressure from expression level. As expected, the inhibition constant increases in step with increasing cooperativity (team size), consistent with J_on > J_off.

For homogeneous receptors, our model predicts that when J_on > J_off, the activity response as a function of the free-energy difference f will be more cooperative near A ≈ 0 than near A ≈ 1. (Cooperativity can be quantified by n(f) where A = {1 + exp[fn(f)]}^-1). To test for this predicted asymmetric cooperativity, we plotted |A - 0.5| as a function of |f| in Fig. 5C using the same experimental data and theory curves for 6× expression as in Fig. 5B. The free-energy f is obtained from the effective free-energy offset and ligand concentration using Eq. 1. If J_on > J_off, the tail of |A - 0.5| will be larger for A > 0.5 than for A < 0.5. This asymmetry is visible in the expression data in Fig. 5C, supporting the conclusion that J_on > J_off.

In Fig. 5D, we plot |A - 0.5| vs. n(0)|f| for the high expression Tar data used in Fig. 4. Here, |f| is scaled by the observed cooperativity n(0) of the activity response at A = 0.5 (f = 0) so the data for different modification states with different cooperativities should collapse. Similarly, the data indicate a larger tail for A > 0.5 than for A < 0.5, consistent with J_on > J_off.

Discussion

The chemotaxis system in the bacterium E. coli is remarkably sensitive to small relative changes in ambient chemoattractant concentrations. Interactions among receptors are crucial to this sensitivity, as is precise adaptation, the return of chemoreceptor activity to prestimulus levels in a constant chemoeffector environment, achieved through methylation and demethylation of receptors. Here, we introduce a dynamic-signaling-team model which is consistent with: (i) FRET activity dose-response data, (ii) cryotomography visualizations of receptors, and (iii) the different methylation responses of different receptor types. Our model also provides a mechanism for the inferred increase in signaling-team-size with receptor modification. Finally, analysis of FRET data within our model suggests that the coupling between neighboring trimers is stronger for active than inactive receptors.

CryoEM tomography reveals that receptors form a slightly disordered honeycomb lattice of trimers (19–21). A honeycomb lattice implies strong directional interactions through specific protein-protein interface couplings, otherwise one would expect a triangular lattice for closest packing. Additionally, high Hill coefficients obtained from dose-response measurements of receptor activity indicate that multiple trimers of dimers act cooperatively (7, 14). We conclude that neighboring trimers are allosterically coupled, presumably via protein-protein interactions between proximal receptor dimers, and we model this coupling via a trimer-trimer-interaction energy J.

To reconcile allosteric coupling between neighboring trimers with the observation of finite signaling teams much smaller than the whole lattice, we included an ad hoc long-range repulsion energy U. This repulsion acts between all trimers within a signaling team, and together with J, sets an average signaling-team-size. One possible mechanism for this repulsion is a different intrinsic curvature of coupled trimers from the membrane curvature (32). Indeed, membrane curvature influences receptor localization (33). More generally, elastic membrane deformations (34) could increase the energy of large teams, as activity is known to be sensitive to receptor-membrane interactions (35), osmotic pressure (36), and membrane curvature as well (37).

The different methylation responses of ligand-binding vs. nonligand-binding receptors requires signaling teams to be dynamic, with trimer-trimer interactions forming and breaking rapidly. In this case, each trimer must adapt individually in order for the whole array to adapt. In our simulations, we treated interaction dynamics as slower than receptor on/off switching, but the results are independent of these rates, depending only on thermodynamics. However, interaction dynamics must be faster than receptor modification to yield receptor-specific methylation. If the time scales of interaction dynamics and receptor modification are comparable, then receptor methylation levels interpolate between the static and dynamic limits, with the resulting levels dependent on the ratio of time scales (Fig. S1). If interaction dynamics are too slow, and if J_on > J_off (as indicated by experiment), then there would be an additional slow drift in receptor activity following the initial response to a change in ligand concentration, as receptor teams equilibrated to a new signaling-team size: such a drift is not observed experimentally.

In addition to MWC models, Ising-type lattice models with relatively weak couplings have been used to explain receptor cooperativity (38, 39). Ising models with trimers as subunits can explain receptor-specific methylation and high sensitivity of the wild-type response, but cannot explain the high sensitivity of the cheR mutant (A ≪ 1), as cooperativity in these models is only high near A ≈ 0.5 (27). Instead this type of model predicts that cooperativity among trimers decreases to near zero as activity drops, which implies a higher K_i for cheR than for wildtype along with a noticeable plateau in activity for cheR at high MeAsp due to Tsr-only trimers (Fig. S2). In contrast, our dynamic model, like MWC models, predicts the high sensitivity of the cheR mutant, as signaling teams form strongly-coupled units even at low activities.

Our model provides a mechanism for the inferred increase in signaling-team size with modification (15). Specifically, the trimer-trimer-interaction energy J increases with modification. Presumably as charged residues are neutralized the electrostatic repulsion between neighboring trimers is reduced, which is consistent with cryoEM images that indicate trimer interactions are strongest near modification and CheA/CheW binding sites. Additionally, demethylation decreases the stability of polar receptor clusters (16). Other additional factors that could contribute to the increase in team size are changes in CheW/CheA receptor binding affinities or in receptor-membrane interactions with modification state (15). Our model does not depend on the actual mechanism of intertrimer coupling and is consistent with any one of these mechanisms.

Within our model, the increase in the inhibition constant K_i with cooperativity and the asymmetry of the FRET response in the high-activity regime indicate that the coupling energy between active trimers is larger than the coupling energy between inactive trimers (J_on > J_off). In a standard Ising lattice model, different on and off coupling energies have no effect, since J_on ≠ J_off is mathematically equivalent to a single J with a different offset energy. Within our dynamic-interaction model, there is no such equivalence because not all neighboring trimers are coupled. As for Ising models, for static MWC teams, different on and off coupling energies can be absorbed into the energy offset. However, in the case of MWC teams with a dispersion of sizes, different on and off coupling energies would cause a spread in K_i values and result in low cooperativity for receptors in the high-activity regime, inconsistent with experimental data. While we have assumed the same repulsion energy U for both active and inactive trimers, one could easily include U_on ≠ U_off. U_on ≠ U_off and J_on ≠ J_off act almost identically in allowing equilibrium teams sizes to differ for active and inactive signaling teams.

CryoEM studies (19, 21) indicate that the honeycomb lattice is disordered, which could be due to the dynamic trimer-trimer interactions or inherent in the underlying lattice structure. Additionally, CheW and CheA are important in interactions among receptors (7, 17) and could modulate interaction strengths between neighboring trimers. A spatially varying J would not significantly alter our model, as an average signaling-team size would still be set by average interaction energies. Indeed, cooperativity and K_i are much more robust for our dynamic-signaling-team model as compared to an Ising-type lattice model when possible trimer-trimer interactions are eliminated (Fig. S3).