Effect of receptor clustering on chemotactic performance of Escherichia coli: sensing versus adaptation
aa r X i v : . [ q - b i o . CB ] S e p Sensing versus adaptation in bacterial chemotaxis
Shobhan Dev Mandal and Sakuntala Chatterjee
Department of Theoretical Sciences, S. N. Bose National Centre for Basic Sciences,Block JD, Sector 3, Salt Lake, Kolkata 700106, India.
We show how the competition between sensing and adaptation can result in a performance peakin E.coli chemotaxis using extensive numerical simulations in a detailed theoretical model. Receptorclustering amplifies the input signal coming from ligand binding which enhances chemotactic effi-ciency. However, large clusters also induce large fluctuations in the activity which causes adaptationto take over. The activity, and hence the run-tumble motility now gets controlled by methylationlevels which are part of adaptation module, rather than ligand binding. This reduces chemotacticefficiency.
The behavior of a cell is controlled by the biochemical reaction network inside it. This reaction pathway is oftennoisy since various protein levels inside the cell are subject to fluctuations [1, 2]. With the advent of sophisticatedtechniques [3, 4] to measure single-cell response in experiments, an important question has emerged: how pathwaynoise affects the cell response [5, 6]. In this paper we address this question for E.coli chemotaxis, one of the bestcharacterized systems in biology [7].The chemotaxis describes the migration tendency of the E.coli cell towards the region of higher nutrient concen-tration. The underlying biochemical network has two main modules, sensing and adaptation, which are coupled toeach other through the activity of the transmembrane chemoreceptors. The receptor activity changes with binding ofthe receptor to the nutrient ligand molecules, and with methylation. There are few thousand receptors in a cell andthey show strong cooperativity where the neighboring receptors form clusters or ‘teams’ and switch between activeand inactive states in unison. This helps in amplification of the input signal coming from ligand binding and allowsthe cell to respond to even weak gradient of nutrient concentration [8–10]. In recent experiments involving singlecell FRET measurements it was observed that receptor clustering results in surprisingly large activity fluctuationsinside a cell [11, 12] even in absence of methylation noise. This observation was striking since methylation was longbelieved to be the most important source of noise in a chemotaxis network [13–17]. The experiments in [11, 12] showedthat receptor clustering is an independent and equally important noise source in the pathway. The immediate andimportant question here is, how this newly found noise source is related to the chemotactic performance of the cell.In this work, we address this question within a detailed theoretical model and find that there is an optimum size ofthe receptor cluster at which the chemotactic performance is at its best. Since receptor clustering amplifies the inputsignal coming from ligand binding, it is expected to enhance the cell performance [8–10]. However, when clustersbecome significantly large, the total number of clusters go down proportionately. The total activity of the cell, whichis the sum of activity of all the clusters, starts showing large fluctuations since the sum is now performed over asmall number of signaling teams. When the activity gets too high or too low, the adaptation comes into play andthe receptor methylation level undergoes large change to restore the activity to its mean value. Our data show thatthe total activity which controls the run-and-tumble motility of the cell is guided by methylation, rather than ligandbinding for large receptor clusters. This reduces the chemotactic efficiency of the cell and its performance goes down.Our study brings out a fundamentally important point: how competition between sensing and adaptation may resultin a performance peak. We demonstrate this by monitoring several different quantities as measures of performance. Inpresence of a spatially varying nutrient concentration profile we define a good chemotactic performance by measuringhow fast the cell is able to climb up the gradient, or how strongly it is able to localize itself in the nutrient-rich regions.A good performance implies a strong ability of the cell to distinguish between regions with high and low nutrientconcentration. We find that for an optimal size of the receptor cluster this ability is most pronounced. Interestingly,our conclusion remains valid even in the absence of a run-and-tumble motion, when the cell is tethered. In this casewe define the performance by the differential response of the cell when the nutrient level at its location is increased ordecreased. The rotational bias of the flagellar motors shows maximum difference between the ramped up and rampeddown inputs at a specific size of the receptor cluster.
Model:
In an E.coli cell the chemoreceptors pair up to form homodimers and three such homodimers form a trimerof dimers (TD) [18, 19]. In our description, a signaling team of size n contains n number of TDs. The free energydifference (in units of K B T ) between the active and inactive states of a dimer is calculated according to Monod-Wyman-Changeux model [20–22]: ǫ [ m, c ( x )] = 1 + log 1 + c ( x ) /K min c ( x ) /K max − m (1)where c ( x ) is the nutrient concentration at the cell location x and m is the methylation level of the dimer which can takeinteger values between 0 and 8. The constants K min and K max set the range within which a chemical concentrationcan be sensed by the cell. The total free energy of the cluster is the sum of free energy of the individual dimers. Alldimers in a cluster change their activity states simultaneously and the transition probability depends on the clusterfree energy [23]. The methylation level of a dimer is controlled by methylating enzyme CheR and demethylatingenzyme CheB-P. A dimer can bind to one enzyme molecule at a time. An inactive dimer gets methylated by CheRand probability to find it in active state increases. On the other hand, an active dimer gets demethylated by CheB-P and its activity decreases. Unphosphorylated CheB receives its phosphate group from autophosphorylation ofactive receptors. This constitutes a negative feedback in the reaction network and is responsible for adaptation.Autophosphorylation of active receptors also supplies phosphate group to another protein CheY and the resultingCheY-P binds to the flagellar motors and induces tumbling in the cell motion. A high value of total activity implieslarge tumbling probability. × -4 P ( x ) ( µ m - ) x ( µ m) n=10n=75n=300n=400 × -2 U ( µ m / s ) n (B) V ( µ m / s ) n (A) × -5 S l op e o f P ( x ) ( µ m - ) n FIG. 1. Peak in localization and drift velocity as a function of receptor cluster size. (A) The x -position distribution of the cellshows steepest variation at an optimum n . Inset shows form of P ( x ) for few representative n values. (B) Chemotactic driftvelocity measured from net displacement in a run and net displacement in a fixed time-interval T = 10 s (inset) both show peakfor a specific n . We have used a linearly varying nutrient concentration profile here. Each data point have been averaged overat least 10 histories. The simulation parameters are given in Table S1. However, the number of enzyme molecules is far too low compared to the number of dimers in a cell [24] and ittakes a long time for a dimer to bind to an unbound enzyme molecule in cell cytoplasm [25]. To reconcile the lowabundance of enzyme molecules with near-perfect adaptation of the cell [26, 27], few mechanisms like ‘brachiation’ or‘assistance neighborhood’ have been proposed [28–30] and also experimentally verified [31, 32] which allow a singlebound enzyme to modify the methylation level of more than one dimers before it unbinds and returns to the cellcytoplasm [21, 30, 33]. We include a flavor of this mechanism in our model. We have included a complete descriptionof our model and other simulation details in [23]. We perform Monte Carlo simulations on this model in one and twospatial dimensions. We present the data for two dimensions below and include those for one dimension in [23].
Performance peak at an optimal size of receptor cluster:
For a swimming cell with a linearly varying c ( x ), the steadystate position distribution P ( x ) of the cell also assumes an almost linear form (see inset of Fig. 1A). Clearly, a goodperformance implies a steep slope of P ( x ). In Fig. 1A (main plot) we plot this slope as a function of receptor clustersize and find a peak. A related quantity h C i = R P ( x ) c ( x ) dx which gives the average nutrient amount experienced bythe cell, is often used to characterize performance when c ( x ) or P ( x ) is not linear [15 ? , 16]. We find a similar peak in h C i also (data not shown here). Chemotactic drift velocity V measures how fast the cell climbs up the concentrationgradient and a large V implies a good performance. To extract V from the run-and-tumble trajectory of the cellwe measure net displacement of the cell in a run and divide it by the mean run duration. We present our data inFig. 1B main plot which shows a pronounced peak. Another possible way to measure the drift velocity is from thenet displacement in a fixed time interval T and divide that by T . In the inset of Fig. 1B we show the plot for thisquantity and find similar peak.At the core of chemotactic sensing lies the differential behavior of the cell when the nutrient level in its environmentgoes up or down. This difference should be large for a good performance. When a cell is running in the direction ofincreasing nutrient concentration, its tumbling rate decreases and the run is extended. Similarly, for a run towardslower nutrient level, the tumbling rate increases and the run is shortened. We measure the time till the first tumbleduring an uphill run and a downhill run and plot their difference in Fig. 2A. This difference shows a peak at a specificsize of the receptor cluster. Interestingly, we can use a similar measure to quantify performance for a tethered cell aswell, which is more commonly used in experiments. In this case we apply a nutrient concentration that is increasing(decreasing) linearly with time while the flagellar motors are rotating in the counter clockwise (CCW) direction. We (B) τ ↑ - τ ↓ ( s ) n (A) τ R - τ L ( s ) n FIG. 2. Motor response of the cell shows highest sensitivity at a specific size of receptor cluster. (A) For a swimming cell, themean first passage time to the tumble mode for uphill run ( τ R ) and downhill run ( τ L ) shows largest difference at a particular n . (B) For a tethered cell in CCW mode, the mean first passage time to CW mode when the nutrient level is ramped up ( τ ↑ )and ramped down ( τ ↓ ) at a rate 0 . µM/s shows largest difference at a specific n . All data have been averaged over at least10 histories. The simulation parameters are as in Fig. 1 measure the average time till the transition to clockwise (CW) rotation mode. In order to compare with the swimmingcell, we change the nutrient level at the same rate as that experienced by a swimming cell during a run. We plot thedifference between ramped up and ramped down cases in Fig 2B and find a peak at the optimal cluster size. Competition between sensing and adaptation:
The probability to find a receptor cluster in the active state is[1 + exp( F L − F m )] − , where F L is the sum of ligand binding energy of all dimers in the cluster and F m is thetotal methylation of all those dimers. Since the contribution due to ligand binding is the same for all dimers, F L isproportional to n . As the cell swims up (down) the ligand concentration gradient, F L increases (decreases) with time(see Eq. 1) and this change is proportionately larger with n . This means as n increases, the activity of a receptorcluster decreases (increases) quickly during an uphill (downhill) run, thereby elongating (shortening) the run. This iswhy the chemotactic performance gets better with n . For large n , however, activity fluctuations increase. Switchingthe activity state of one large cluster brings about large change in the total activity of the cell. For example, whenthe activity gets too low, all the inactive dimers in a cluster tend to get methylated. This increases F m significantlyand the change in F m overrides the change in F L . See Fig. 3 for a typical time-series of cluster activity, F m and F L for a large n value. In Fig. 4 we plot the average change ∆ F m in methylation free energy as a function of time duringan uphill run of the cell for various different n . The change in F L has been shown by a continuous line for reference.These plots clearly show for large n the change in F m overtakes the change in F L . The variation in cluster free energyis then controlled by F m . The cell is now less sensitive to ligand concentration profile. This reduces its chemotacticefficiency. Conclusions:
In this work we have investigated the role of receptor clustering on the chemotactic efficiency of acell. Although receptor cooperativity amplifies the cell sensitivity towards small variation in nutrient level, it alsoincreases the activity fluctuations inside the cell. Large deviation of activity from its mean value triggers large changein methylation levels to ensure adaptation in the biochemical network. The ligand binding energy cannot keep upwith such large change in methylation energy and the free energy difference between the active and inactive statesgets controlled by methylation now. Above interplay gives rise to a performance peak at an intermediate value of thereceptor cluster size.For a noisy nutrient environment, an optimal size of the receptor signaling team was reported in earlier studies[34, 35]. It was argued that receptor cooperativity does not only amplify the ligand signal, but also the noise present in (A)(B)(C) c l u s t e r ac ti v it y t(s) F L ( k B T ) t(s) F m ( k B T ) t(s) FIG. 3. Typical time-series of activity along with methylation component and ligand component of free energy of a receptorcluster of size n = 200. The time-series has been recorded in steady state over a time-window of 40 s . (A) shows few transitionsof activity state of the cluster. (B) records variation of methylation free energy of the cluster which is seen to roughly followthe activity transitions. (C) shows simultaneous variation of free energy due to ligand binding which directly captures therun-tumble trajectory of the cell. The scale of variation of ligand binding energy is negligible compared to that of methylationfor the present value of n . (A) n=10 ∆ F m ( k B T ) t (s) (B) n=20 ∆ F m ( k B T ) t (s) (C) n=30 ∆ F m ( k B T ) t (s) (D) n=100 ∆ F m ( k B T ) t (s) FIG. 4. Average change ∆ F m in methylation free energy (discrete points) of a cluster for first 0 . s during an uphill run for 4different n values. The continuous lines show the change in ligand free energy of the cluster. For small n the change in ligandfree energy dominates but as n increases, ∆ F m takes over. These data have been averaged over at least 2 × histories. it. For optimal performance, therefore, a trade-off is required where the signaling team size should be large enough forsensitive detection of small changes in ligand concentration, but small enough such that the amplified noise does notinsensitize the cell response [34]. Moreover, when both ligand noise and intracellular biochemical noise are considered,the receptor clustering is beneficial as long as the amplified ligand noise stays below the biochemical noise [35]. Onthe other hand, we find an optimal team size even when the ligand concentration profile does not fluctuate with time,and the origin of this optimality is completely new.It should be possible to test our results qualitatively in experiment. Both for swimming cell and tethered cell wehave observed the best chemotactic performance at a specific size of the receptor cluster. Experimentally varyingreceptor cluster size for a tethered cell has already been possible recently [11, 12]. Using a similar set up it may bepossible to verify the existence of a performance peak. In our model the best performance is observed for clusters whichcontain ∼
70 TDs. However, it may not be possible to find accurate quantitative agreement between our model andexperiments. To keep our model simple and tractable, we have not considered few aspects of the intracellular reactionnetwork, like hexagonal geometry of the spatial arrangement of the receptor array [18, 19], or more importantly, theenergy cost due to curvature of the cell membrane induced by the receptor clusters [36–38]. But our main conclusionsshould not get affected by these assumptions and the interplay between ligand free energy and methylation free energycan be experimentally investigated as the receptor cluster size is varied. Finally, our study opens up the importantquestion of competition between sensing and adaptation, which is relevant in a wide variety of biological systems[39–41]. It would be interesting to see if this competition gives rise to similar performance peaks in this broad classof systems as well.
ACKNOWLEDGEMENTS
S.D.M acknowledges a research fellowship [Grant No. 09/575(0122)/2019-EMR -I] from the Council of Scientific andIndustrial Research (CSIR), India. S.C. acknowledges financial support from the Science and Engineering ResearchBoard, India (Grant No: MTR/2019/000946). [1] Michael B Elowitz, Arnold J Levine, Eric D Siggia, and Peter S Swain. Stochastic gene expression in a single cell.
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There are three major steps in our simulation of the chemotactic reaction network and cell motion. These are • Activity switching of the receptor clusters • Binding/unbinding dynamics of CheR and CheB-P enzymes to the receptor dimers and modification of dimermethylation levels by these enzymes • Calculation of CheY-P level from total activity and the resulting run-and-tumble motion of the cell.Below we describe each of these steps in details.A receptor cluster containing n trimers of dimers can switch between active state ( a = 1) and inactive state ( a = 0).The free energy difference between the two states is given by the Monod-Wyman-Changeux model [1] F = 3 n (cid:18) c ( x ) /K min c ( x ) /K max (cid:19) − n X i =1 m i (1)where K min and K max set the range of sensitivity of the cell, i.e. the cell can sense a ligand concentration level c ( x )for K min < c ( x ) < K max . We have used K max = 3 mM and two values of K min which are 7 µM [2] and 18 µM [3].We do not find any significant difference between these two choices. In our model we do not explicitly consider thebinding events of the nutrient molecules to the receptors and assume all dimers experience a mean nutrient level c ( x )that depends on the current cell position. The methylation level of the i -th dimer in the cluster is m i which can takeany integer value between 0 and 8. A completely demethylated dimer has m i = 0 and a completely methylated dimerhas m i = 8.The probability to find a receptor cluster in active state is [1 + exp( F )] − . The transition rate between the twoactivity states is chosen based on local detailed balance [2]. From a = 0 state the receptor cluster switches to a = 1state with rate w a F ) and the reverse switch happens with rate w a exp( F )1 + exp( F ) . Here, w a is the characteristic time-scale of the transition. Here we have presented data for w a = 0 . /s which is a bit higher than the value w a = 0 . /s used in [2] to explain the experimental results. We have verified (data not shown here) that no significant changeresults from this difference.We consider a total of 140 CheR molecules and 240 CheB molecules in the cell [4]. An unbound CheR moleculewhich is in the cell cytoplasm can bind to a receptor dimer, provided no other enzyme is bound to it. The bindingcan take place at the receptor tether or modification site [5–7]. Although both these types of binding are slow, thebinding at the tether is relatively faster than that at the modification site [5, 8]. Because of this we assume in ourmodel that an unbound CheR only binds to the tether of the receptor with binding rate w r . Once bound, the CheRenzyme can raise the methylation level of the dimer with rate k r provided the dimer belongs to an inactive clusterand its methylation level is less than 8. With rate w u CheR can unbind from the dimer and can either reattach toanother unoccupied dimer within the same cluster, or return to the cytoplasmic bulk.Note that due to very slow binding of the enzyme to the receptors, if one binding results in only one methylationevent, then it becomes difficult for the system to maintain perfect adaptation [5]. To circumvent this, many differentmechanisms were proposed, including assistance neighborhood [9–11] and brachiation [12]. In assistance neighborhoodmodel, a bound enzyme can modify the methylation level of neighboring receptors and in brachiation mechanism abound enzyme can perform random walk on the receptor array before unbinding from it. Both these methods involvemethylation of multiple receptors from a single binding event. We incorporate this aspect in our model by allowingrebinding of CheR to other dimers within the same cluster. This step is simple and effective in our model, since wedo not explicitly include spatial geometry of the receptor cluster.A CheB molecule in the cytoplasmic bulk can undergo phosphorylation by an active receptor with the rate w p .In the phosphorylated state, an unbound CheB-P molecule can undergo dephosphorylation with the rate w dp . Thebinding-unbinding dynamics of a CheB-P molecule is very similar to what has been described above for CheR, withthe tether binding rate w b . In the bound state a CheB-P molecule can demethylate an active receptor dimer withrate k b provided its methylation level is non-zero. We implement the rebinding process in this case too in the sameway as mentioned above.The CheY-P level depends on the total activity of all the receptors, which is calculated as A = P k a k /N c where a k is the activity of the k − th cluster and N c is the total number of clusters in the cell. In our simple model we assumeall clusters are of equal size, and hence N c = N dim / n . Therefore, A measures the fraction of active receptor clustersin the cell. Define Y P = [CheY-P][CheY] and the rate equation that governs Y P is [3] dY P dt = K Y A (1 − Y P ) − K Z Y P (2)where K Y and K Z are rate constants for phosphorylation and dephosphorylation, respectively. In our simulation wedirectly use Y P = AA + K Z /K Y . The run-tumble motility of the cell is controlled by Y P . If the cell is in the runmode, it can switch to tumble mode with rate ω exp( − G ) where G = ∆ − ∆ Y /Y P and the opposite switch fromtumble to run happens with the rate ω exp( G ).In our simulations we consider motion of the cell in one and two spatial dimensions. In the one dimensional case,the cell can either move to the left or to the right with speed v during a run and stays put in the same spot during atumble. The smallest time-step in our simulation is dt = 0 . s during which a running cell can move by an amount vdt which which sets the lattice spacing in our one dimensional model. We set up a nutrient concentration profile c ( x ) = c (1 + x/x ) in a region of length L and when the cell reaches the boundaries of this region at x = 0 , L , itgets reflected back. For a two dimensional motion of the cell we consider an L x × L y box with reflecting boundariesat the four walls. The nutrient concentration gradient is present along x -direction, as in one dimension, and along y direction the concentration is homogeneous. Due to rotational diffusion the trajectory of the cell does not remain aperfect straight line during a run. It shows gradual bending. After each tumble the cell chooses a random directionto start a new run. We measure all response functions in the long time limit when the system has reached a steadystate. MODEL PARAMETERS
TABLE S1.
Symbol Description Value References N dim Total number of receptor dimers 7200 [4, 5] N R Total number of CheR protein molecules 140 [4, 5] N B Total number of CheB protein molecules 240 [4, 5] K min Minimum concentration receptor can sense 7 µM , 18 µM [2], [3] K max Maximum concentration receptor can sense 3000 µM [3, 13] w a Flipping rate of activity 0 . s − Present study ω Switching frequency of motor 1 . s − [14]∆ Nondimensional constant regulating motor switching 10 [14]∆ Nondimensional constant regulating motor switching 20 [14] Y Adopted value of the fraction of CheY-P protein 0 .
34 [14] K Y Phosphorylation rate of CheY molecule 1 . s − [3, 15] K Z Dephosphorylation rate of CheY molecule 2 s − [3, 15] w r Binding rate of bulk CheR to tether site of an unoccupied dimer 0 . s − [5, 8] w b Binding rate of bulk CheB-P to tether site of an unoccupied dimer 0 . s − [5, 8] w u Unbinding rate of bound CheR and CheB-P 5 s − [5, 8] k r Methylation rate of bound CheR 2 . s − [5, 8] k b Demethylation rate of bound CheB-P 3 s − [5, 8] w p CheB phosphorylation rate 3 s − [5, 16] w dp CheB-P dephosphorylation rate 0 . s − [5] L Box length in 1D 2000 µm Present study v Speed of the cell 20 µm/s [17] dt Time step 0 . s Present study L x × L y Box dimension in 2D 2000 × µm Present study D Θ Rotational Diffusivity 0 . µm /s [18–20] c Background nutrient concentration 200 µm Present study1 /x Linear concentration gradient of nutrient 0 . µm − Present study
ACTIVITY FLUCTUATIONS INCREASE WITH RECEPTOR CLUSTERING P r ob a b ilit y d i s t r i bu ti on Activity n=10n=28n=70n=100
FIG. S1. Probability density of activity (defined as the fraction of active clusters) for few representative values of receptorcluster size n . As n increases, distribution gets wider, which is consistent with the experimental observation in [2, 21]. RESULTS IN ONE DIMENSION
In this section, we present our results for the cell motion in one spatial dimension, in presence of a linear c ( x ) = c (1 + x/x ), as in the main paper. All our conclusions presented in the main paper for the two dimensional caseremain valid here. Our simulation data are shown in Figs. S2, S3 and S4. × -4 P ( x ) ( µ m - ) x ( µ m) n=10n=75n=300n=400 × -1 U ( µ m / s ) n (B) V ( µ m / s ) n (A) × -5 S l op e o f P ( x ) ( µ m - ) n FIG. S2. (A) In presence of a linear nutrient concentration profile with weak gradient, the steady state position distribution P ( x ) of the cell is almost linear. The steepness shows a peak with n . Inset shows some representative P ( x ) plots for few chosen n values. (B) Chemotactic drift velocity measured from net displacement in a run (main plot) and from net displacement in afixed time-interval T = 10 s (inset) show peak with n . The data points are averaged over at least 10 histories. The simulationparameters are listed in Table S1. τ R - τ L ( s ) n FIG. S3. During a rightward (leftward) run of the cell when it is headed towards regions with more (less) nutrient, τ R ( τ L )denote the average time till the next tumble. The difference ( τ R − τ L ) shows a peak as a function of n . Each data point isaveraged over at least 10 histories. All simulation parameters are as in Fig. S2. (A) n=10 ∆ F m ( k B T ) t (s) (B) n=30 ∆ F m ( k B T ) t (s) (C) n=50 ∆ F m ( k B T ) t (s) (D) n=100 ∆ F m ( k B T ) t (s) FIG. S4. Average change ∆ F m in methylation free energy (discrete points) of a cluster for first 0 . s during an uphill run for4 different cluster sizes. For comparison, the corresponding change in ligand free energy has been shown by continuous lines.For small n the change in ligand free energy dominates but as n increases, ∆ F m takes over. These data have been averagedover at least 2 × histories. [1] Jacque Monod, Jeffries Wyman, and Jean-Pierre Changeux. On the nature of allosteric transitions: a plausible model. JMol Biol , 12(1):88–118, 1965.[2] Remy Colin, Christelle Rosazza, Ady Vaknin, and Victor Sourjik. Multiple sources of slow activity fluctuations in abacterial chemosensory network.
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