shows simulated response curves of complexes of 19 chemoreceptor dimers to step increases in concentration of alpha-methyl aspartate (MeAsp), an attractant. The results shown include the dynamics of CheR and CheB (see Model) and are similar to those obtained with static ANs [18
]. Precise adaptation occurs over four orders of magnitude of MeAsp concentration, with methylation levels increasing to compensate for drops in activity due to increases in attractant concentration.
Averaged Response Curves for Step Increases in Attractant Concentration (0–100 mM)
In , the Tar-only complexes exemplify two different limits of precise adaptation at high attractant concentrations, as in the static AN model [18
]. For the Tar-only complex with higher
(dot-dashed curve), the activity continues to adapt precisely, but the activity stops responding to increases of MeAsp. In this case, the receptors become saturated, and further increases in MeAsp do not produce changes in the free-energy difference between the on and off states of the complex. The average methylation of the complex reaches a constant value, below full methylation (B). In contrast, for the complex with lower
(dashed curve), the activity approaches zero at high concentrations. In this case, full methylation occurs before saturation of the receptors with MeAsp. When MeAsp concentrations increase further, the resulting increase in the free-energy difference between the on and off state of the complex cannot be compensated by additional methylation, so the activity drops without recovering.
Compared to the Tar-only complexes, the heterogeneous receptor complex with six Tar and 13 Tsr receptors (solid curve) continues to respond to MeAsp increases and to adapt precisely over an extended range. The Tar receptors in this complex have
, as in the second case considered above (dashed curve); these six Tar receptors allow for a sensitive response at low concentrations of MeAsp. In contrast, the 13 Tsr receptors in the complex have low affinity for MeAsp. Therefore, at low MeAsp concentrations the Tsr receptors act as extra methylation sites, increasing the range of precise adaptation. As MeAsp concentrations increase, the Tar receptors become fully saturated, but the Tsr receptors begin to bind MeAsp. This increases the upper limit of response to well over 100 mM MeAsp.
For homogeneous complexes, the limit of adaptation at high attractant concentration depends on which occurs first, saturation of receptors by attractant or full methylation of receptors. Which of these occurs first depends on the ratio
. The crossover ratio between the two limits of adaptation can be estimated in mean field theory (Equation 2
). At high ligand concentrations:
is the number of receptors in the complex. Loss of activity occurs if the offset energy at full methylation r(8)
cannot compensate for the free-energy difference per receptor due to saturating attractant,
. Therefore, for F*
denoting the precisely adapted free-energy difference, if
or equivalently, if
, loss of activity will occur at high concentrations of attractant (dashed curve in A). In contrast, if
, loss of response will occur at high concentrations of attractant (dot-dashed curve in A). For fixed r(m)
, and complex size n
, the limit of adaptation depends only on the ratio
, not on the individual magnitudes of
. For our simulation, r(8)
= −30, n
= 19, and A*
= 1/3, so F*
= log2 = 0.693. Therefore, the expected crossover ratio is exp(F*
) = 20.8. For Tar-only complexes with
, so adaptation fails through loss of activity as observed in A (dashed curve). For Tar-only complexes with
, and adaptation fails through loss of response, as also observed in A (dot-dashed curve).
Experiments indicate that the adapted tumbling rate, and therefore, also the adapted receptor activity, increases with the concentration of CheR [26
]. At low levels of CheR, the binding rate
is proportional to the concentration of CheR. In , we show the adapted activity as a function of the CheR binding rate
. Adapted activity is the average activity calculated according to Equation 3
, after allowing the complex to reach equilibrium (see Methods
). As the binding rate of CheR increases, the proportion of CheR-bound receptors
also increases (, inset). The increase in
causes the rate of methylation for the whole complex to rise, therefore increasing activity. The complex with full ANs (including all nearest neighbors) closely follows the expected mean-field-theory result (Equation 7
), whereas the complex with ANs of size one deviates to higher activity over a wide range of CheR binding rates. In these simulations, no attractant is present, and therefore, the average methylation level of the receptors is low. Consequently, complete demethylation of individual receptors is likely to occur in the AN = 1 model (
), leading to missed demethylation attempts, and therefore, to an increase in adapted activity according to Equation 6
. In effect, for the AN = 1 model, the demethylation rate is lower than it “should be” because by chance, some individual receptors are already fully demethylated, and therefore, CheB fails to act sometimes when it “should.”
Figure 3 Adapted Activity as a Function of CheR Binding Rate
In , we explore precision of adaptation over a broad range of MeAsp concentrations for several variants of our model. In general, deviations from precise adaptation occur if and only if the rates of methylation or demethylation cease to depend exclusively on complex activity (Equation 6
). We find that large AN sizes, saturated kinetics of CheR/CheB, and short CheR/CheB dwell times favor precise adaptation. In all cases, we consider the same complex of 19 receptors (A) composed of six Tar receptors and 13 Tsr receptors. For comparison, we also show the mean field theory result (see Model).
In A, we show the effect of AN size on precise adaptation. Within each AN, there is a “ladder” of possible methylation levels. Fluctuations cause the methylation level to move up and down the ladder, deviating from the average. For small ANs, the ladder is shorter, and fluctuations are more likely to produce fully methylated or fully demethylated neighborhoods. At low levels of MeAsp and low average methylation, fluctuations are likely to produce fully demethylated neighborhoods, lowering the rate of demethylation and increasing activity according to Equation 6
. Similarly, at high levels of MeAsp and high average methylation, neighborhoods may become fully methylated, lowering the rate of methylation by CheR and decreasing activity.
As shown in A, complexes with ANs of size one have a drastically reduced precision of adaptation, but half neighborhood complexes have a precision of adaptation close to that of full AN complexes. Beyond a certain AN size, the methylation ladder is already long enough to effectively prevent fluctuations from causing full methylation or full demethylation of neighborhoods. Therefore, increasing AN size improves precision of adaptation only up to a point, beyond which AN size only affects activity near the concentration at which all receptors become fully methylated. For our parameters, receptors do not become fully demethylated even at zero attractant concentration, but full demethylation could be induced by addition of repellent.
We also performed simulations with varying degrees of saturation of CheR and CheB (B). Specifically, we introduced a factor of N/(N + Msat) into the rates of CheR and CheB action, where N is the total number of available sites for methylation/demethylation and Msat is a constant (see Model). For all other simulations, CheR and CheB were assumed to work at saturation, independent of methylation level (Msat = 0). Increasing Msat makes the rate of action of CheR and CheB more dependent on the number of available modification sites. For finite Msat, in low concentrations of MeAsp, and therefore, low average methylation levels, the rate of demethylation by CheB is significantly lower than the saturated (maximal) rate. Conversely, there are many available sites for methylation, so the rate of methylation is near maximal. Therefore NB/(NB + Msat) < NR/(NR + Msat) ≈ 1. A relative decrease in the rate of demethylation by CheB compared to the rate of methylation by CheR causes an increase in the activity of the complex as seen below 0.1 mM MeAsp in B. As the average methylation level increases with increasing MeAsp concentration, the inequality is reversed so that NR/(NR + Msat) < NB/(NB + Msat) ≈ 1 results in a relative decrease in the rate of methylation by CheR compared to the rate of demethylation by CheB. Therefore, at high MeAsp concentration, above 0.1 mM, the adapted activity decreases below the expected value for precise adaptation.
Within mean field theory for Msat
> 0, we can approximate the crossover concentration, i.e., the concentration of attractant at which the activity of complexes is equal to the expected precisely adapted activity. The crossover occurs when the saturation factors of CheR and CheB are equal, NB
) = NR
). This occurs when the average methylation level
is 4, which occurs at 0.36 mM MeAsp. This is close to the crossover concentration observed in our simulations (B).
Simulations were also performed in which the average dwell time of CheR and CheB was varied (C). The average dwell time is equal to
, whereas the average number of enzymes bound to the complex depends on the ratios
). Therefore, in order to change the average dwell time while conserving the average number of CheR and CheB enzymes bound to the complex, we altered both
by the same factor. In the model, when a CheR or CheB is bound for a long time, the enzyme catalyzes the same reaction numerous times before unbinding. The methylation level in the neighborhood will therefore move along the ladder in one direction, possibly reaching the end, i.e., full methylation or full demethylation. As for the AN = 1 model in A, the result in C for long CheR and CheB dwell times is higher activity at low MeAsp concentrations (where full demethylation is more likely) and lower activity at high MeAsp (where full methylation is more likely).
Deviations from mean field theory occur if the average dwell time of CheR or CheB is long enough to allow full methylation or demethylation of neighborhoods. Below 0.001 mM MeAsp, the average adapted methylation level per receptor homodimer is ≈ 2.2. Since there are on average 5.4 receptors per AN, the average distance to the bottom of the methylation ladder is 2.2 × 5.4 ≈ 12. Therefore, precise adaptation is expected to fail when the demethylation rate is ≈12 times the CheB unbinding rate (
). For our parameters, A*
= 1/3 and kB
= 0.2 s−1
, we expect precise adaptation to fail for
. Consistent with this calculation, our simulations show that deviations from precise adaptation begin to occur at low MeAsp concentrations for
around 0.01 s−1
The fact that most receptors are either fully methylated or fully demethylated for long dwell times of CheR and CheB is clearly shown by the distribution of methylation levels for different average dwell times (C, inset). As dwell time increases, the single-peaked methylation distribution flattens out and becomes bimodal, i.e., most receptors become fully methylated or fully demethylated. Addition of ligand causes a shift in the amplitudes of the two peaks, but the peak positions, at m
= 0 and m
= 8, do not change. We can exploit this fact along with mean field theory to estimate the crossover attractant concentration where the activity of the complex crosses A*
. The average methylation (demethylation) rate has a correction factor equal to the proportion of not fully methylated (not fully demethylated) ANs (Equation 6
). The crossover attractant concentration will occur where these two correction factors are equal, namely where
, which implies
. For our mean field–adapted activity of A*
= 1/3, and requiring
, we obtain a crossover concentration of 30 mM MeAsp, consistent with the simulation results shown in C.
In all our simulations, the methylation levels of receptors fluctuate, translating into fluctuations in complex activity. shows the distribution of activities due to fluctuating methylation levels at 0 mM, 1 mM, and 100 mM of MeAsp. Within the MWC model, complex activity is strictly either zero or one. However, we assume that switching between these two states is rapid, so we consider the distribution of thermally averaged complex activities given by Equation 3
. Even for the full AN model, for which adaptation is precise, there is a broad range of complex activities. Note though that for the observed variation in activity of ≈50% for a single complex and assuming ≈500 independent receptor complexes per cell, the resulting variation in total activity would be only ≈2.5%. As shown in , for size-one ANs at 0 mM and 100 mM MeAsp, the activity distributions are shifted relative to the activity distributions for full ANs because adaptation is not precise when CheR and CheB act only on single bound receptors (cf. A). Also shown in , long dwell times of CheR and CheB cause a bimodal distribution of complex activities, corresponding to the bimodal distribution of receptor methylation levels (cf. C, inset).
Distribution of Adapted Complex Activities (Reflecting Distribution of Complex Methylation Levels) at Different MeAsp Concentrations
Within our model, noise is caused by fluctuations in both binding/unbinding of CheR and CheB and methylation/demethylation by CheR/CheB. For short average dwell times, fluctuations in the number of bound CheR and CheB enzymes are rapidly averaged out, and the dominant source of noise is the discrete methylation/demethylation events by receptor-bound CheR/CheB. We have estimated the resulting variance in complex methylation
with the linear noise approximation (see Model and ). In this limit, the only factor that affects the variance
is the free-energy difference δ
per methyl group, with
. In the opposite limit of long average dwell times, fluctuations in the number of CheR and CheB enzymes bound to the complex add to the variance in methylation levels and thus activity. As seen in A and C, low binding and unbinding rates
cause an increase in noise over the calculated theoretical noise limit due to the discreteness of methylation and demethylation events. Increasing complex size can decrease the noise due to CheR and CheB binding/unbinding, but not the noise due to CheR/CheB methylation/demethylation. Therefore, increasing complex size only decreases noise for long average dwell times of CheR and CheB, but has no effect in the case of short dwell times, where noise is near the theoretical limit (B and D).
Variance in Complex Methylation and Activity Levels
It was observed experimentally by Chalah and Weis [27
] that CheR and CheB methylate/demethylate the four different methyl-attachment sites on each receptor monomer at different rates. These observations suggest two possible scenarios: either CheR and CheB have different rates of action on different modification sites, or CheR and CheB divide their time unequally among the sites (or some combination of these two). To test the first scenario, we extended our model to include variation in the rates of action of CheR and CheB, with the results shown in . Specifically, we assumed that when a CheR or CheB is tethered to a receptor, it divides its time equally among all available modification sites in the AN. The total rate of action by a bound CheR or CheB is therefore the average over the rates for all available modification sites in the AN. The catalytic rates for a methylation/demethylation reaction were assumed to vary in the ratio 1:2:4:8 for the four different sites [27
]. We studied two cases. In the first case, the ratios of methylation and demethylation matched for each site (i.e., for sites 1–4, the ratios for both kB
were 1:2:4:8). In the second case, the ratios for methylation and demethylation were inverted relative to each other (i.e., for sites 1–4, the ratios for kB
are 1:2:4:8 and for kR
Precision of Adaptation for Receptors with Site-Dependent and Site-Independent Methylation/Demethylation Rates
As shown in , when the ratios of methylation and demethylation match for each site, precise adaptation is preserved. In this case, since every site has the same ratio of kB/kR as every other site, the average methylation levels of all sites remain the same, as shown in the inset. The average methylation and demethylation rates over sites is therefore constant, independent of ligand concentration, preserving precise adaptation. In contrast, inverted ratios of methylation and demethylation rates among the sites fail to produce precise adaptation. In this case, the ratio kB/kR varies among the four methylation sites, causing varying equilibrium methylation levels (, inset). The sites with a low kB/kR ratio are the first to become methylated at low concentrations of MeAsp, leading to low average rates of demethylation compared to methylation, and therefore to high adapted activity. As average methylation levels rise with increasing MeAsp, these low kB/kR sites “fill up,” leading to high average rates of demethylation compared to methylation, and therefore to low adapted activity.
The second scenario suggested by the Chalah and Weis data [27
], namely different dwell times for CheR and CheB among the modification sites, leads more robustly to precise adaptation. As long as CheR and CheB work near saturation, differences in dwell times between sites will not affect total rates of methylation and demethylation, and precise adaptation will be preserved, according to Equation 6
. Indeed, as shown in , even if the relative dwell times for each site are inverted for CheR and CheB, precise adaptation is preserved.
Experiments by Berg and Brown [9
] on wild-type E. coli
indicate that whereas adaptation to aspartate is precise over a large concentration range, precise adaptation to serine fails at relatively low concentrations. In , we compare our model to these experiments. In both cases, adaptation to aspartate (or MeAsp) is precise over four orders of magnitude. However, adaptation to serine fails at approximately 0.1 mM. Within our model, this difference with respect to attractants reflects the presence of more Tsr receptors (13) in the complex than Tar receptors (six). More Tsr receptors amplify the change in complex free energy due to serine, which results in an increased sensitivity at low concentrations, but also results in full methylation of the complex and loss of activity beginning at 0.1 mM serine.
Adapted Complex Activity versus Concentration of MeAsp (Filled Circles) or Serine (Open Circles)
We tested robustness of our theoretical model by randomly varying parameters as described in the Model section. The results shown in demonstrate that precise adaptation is a robust property of our model. Almost ideal adaptation occurs for all parameter sets up to a total parameter variation of K
. For larger parameter variations, in the range of K
, 77% of the altered models still have a precision of adaptation within 10%. These results are similar to those obtained from the simple single-receptor model of Barkai and Leibler [16
]. However, in one regard, our MWC model with ANs is more robust than the single-receptor model. In the single-receptor model, precise adaptation requires that the activity of the receptor is zero at full demethylation and one at full methylation. Our model has the property of precise adaptation without this assumption.
Robustness of Assistance-Neighborhood Model for Adaptation