Mathematical modeling and experimental validation of chemotaxis ...

Mathematical modeling and experimental validation of chemotaxis under controlle..... (DOI: 10.1039/b924368b)

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DOI: 10.1039/b924368b

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DOI: 10.1039/b924368b

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Mathematical modeling and experimental validation of chemotaxis under controlled gradients of methylaspartate in Escherichia coli Rajitha R. Vuppula a, Mahesh S. Tirumkudulu *b and Kareenhalli V. Venkatesh *c a Department of Chemical Engineering, Indian Institute of Technology Bombay, Mumbai 400076. E-mail: [email protected] b Department of Chemical Engineering, Indian Institute of Technology Bombay, Mumbai 400076. E-mail: [email protected] c Department of Chemical Engineering, Indian Institute of Technology Bombay, Mumbai 400076. E-mail: [email protected] Received 23rd November 2009 , Accepted 12th February 2010 First published on the web 18th March 2010 Escherichia coli has evolved an intracellular pathway to regulate its motion termed as chemotaxis so as to move towards a favorable environment such as regions with higher concentration of nutrients. Chemotaxis is a response to temporal and spatial variation of extracellular ligand concentration and randomness in motion induced by collisions with solvent molecules. Previous studies have reported average drift velocities for a given gradient and do not measure drift velocities as a function of time and space. To address this issue, a novel experimental technique was developed to quantify the motion of E. coli cells to varying concentrations and gradients of methyl-aspartate so as to capture the spatial and temporal variation of the drift velocity. A two-state receptor model accounting for the intracellular signaling pathway predicted the experimentally observed increase in drift velocity with gradient and the subsequent adaptation. Our study revealed that the rotational diffusivity induced by the extracellular environment is crucial in determining the drift velocity of E. coli. The model predictions matched with experimental observations only when the response of the intracellular pathway was highly ultra-sensitive to overcome the extracellular randomness. The parametric sensitivity of the pathway indicated that the dissociation constant for the binding of the ligand and the rate constants of the methylation/demethylation of the receptor are key to predict the performance of the chemotactic behavior. The study also indicates a possible role of oxygen in the chemotaxis response and that the response to a ligand may have to account for effects of oxygen.

1. Introduction Chemotaxis is a phenomenon by which a microorganism moves towards a region with a higher concentration of an attractant or moves away from repellent molecules. This phenomena plays an indispensable role in many biological processes including immune response, embryo-genesis and wound healing.1,2 It is also important in bio-film formation and bio-remediation of subsurface contaminants.3 Of the many organisms that exhibit chemotaxis, the bacterial chemotaxis is the simplest and well characterized. An understanding of such processes in simple systems would help in unraveling similar processes in more complex organisms. There exist a number of different bacterial systems of which the most widely studied is Escherichia coli.4 E. coli responds to the environmental conditions by alternating the rotational direction of their flagella. Counterclockwise (CCW) rotation results in a motion called run, whereas clockwise (CW) rotation leads to tumbling of the bacteria. By modulating the duration of runs and tumbles, the bacteria achieves a net motion towards chemoattractants or away from chemorepellents. The chemotaxis phenomena involves sensing of the ligand, activation of the receptor, response through a signaling pathway and eventually adaptation to the new environment.

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The regulation of chemotaxis in the bacteria is achieved by a network of interacting proteins (see Fig. 1). There are essentially six cytoplasmic proteins,4 namely CheA, CheB, CheR, CheW, CheY and CheZ, that are needed to process the signal and transmit control signals to the flagellar motor. The CheW protein couples CheA to methyl-accepting chemotaxis proteins (MCPs) to form a complex. The CheA autophosphorylates which in turn phosphorylates two other regulator proteins, CheY and CheB. The phosphorylated CheY (CheY-P) activates the flagellar motor switch protein FliM causing the motor to run clockwise thereby producing a tumble. The CheZ promotes the CheY dephosphorylation. The phosphorylated CheB removes the methyl groups from MCPs, whereas CheR continuously adds methyl groups to them. The level of methylation of MCPs decides the tumbling frequency of the organism. The above processes describe the state of the intracellular pathway under steady state conditions resulting in a constant tumbling frequency and therefore a net zero drift velocity (see Fig. 1A). When a ligand (attractant) binds to the receptor, the rate of autophosphorylation of CheA decreases causing the dephosphorylation of CheY-P (see Fig. 1B). This in turn reduces the tumbling frequency allowing the cell to run smoothly in the favorable environment. However, the reduced phosphorylated CheA decreases the phosphorylation of CheB which in turns reduces the demethylation rate thereby increasing the methylation level of the receptors. The latter increases the autophosphorylation of CheA thereby returning the demethylation rate and the tumbling frequency to their original steady state value. E. coli is thus able to adapt exactly to a new environment. Recent studies have shown that oxygen also influences the concentration of CheY-P. Studies indicate that oxygen is sensed indirectly through two receptor proteins, Aer and Tsr, which monitor the cell energy level proportional to the oxygen concentration.5 This aspect is not shown in Fig. 1, since we focus only on the response to a single ligand, such as aspartate.

Fig. 1 Chemotaxis signaling pathways in E. coli (A) in the absence of chemoattractant, and (B) in the presence of chemoattractant. The dotted line represents the reactions with a reduced rate in response to ligand binding to the receptor. It can be noted that on adaptation the pathway returns to steady state as shown in (A). A wide range of experimental methods have been developed to analyze the phenomena of chemotaxis. Four different parameters are typically measured to characterize the motion. First is the run speed that gives the average speed of the cells in between consequent tumbles. The second is the clockwise bias (referred henceforth as CW) which gives the fraction of time spent by the cells in tumble mode. The next two parameters, namely, the drift velocity and the diffusivity give the average behavior of cells over time scales longer than that for a typical run or tumble event. The drift velocity measures the mean speed of cells while the diffusivity characterizes the random motion induced by the collisions with the solvent molecules and due to its own tumble motion. The latter is calculated as the mean square distance traversed by the cells over time. The earliest techniques to characterize chemotaxis were the agarose gel assay6 and the capillary assay7,8 due to their simplicity. In both these methods, the E. coli moves up a gradient set by consumption of the attractant. The drift velocity is used to quantify response to varying concentrations of chemicals. There is no control over the induced gradient 2 of 15

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and the final response depends on the cells metabolism and growth. Later, Berg and Turner9 constructed an assay consisting of two stirred chambers separated by a micro-channel plate comprising a fused array of capillary tubes. Here, the cells were added to one chamber and the attractant to the other and a linear gradient was assumed along the microchannel. The cells migrate to the second chamber via the micro-channel plate and the density of the cells was determined from the intensity of the scattered light in the second chamber. It was found that the flux of bacteria increased on addition of an attractant into the second chamber compared to when there was no attractant. The diffusivity was computed from the flux and found to be independent of the attractant gradients. The drift velocity was calculated from the knowledge of total flux for varying attractant gradients. A similar setup of Liu and Papadopoulos8 involved a single capillary between two tiny reservoirs. Here the single cell parameters such as run speed, run length, turn angles, etc., were measured in the capillary with the help of a microscope. Again, a linear concentration profile of the chemoattractant was assumed between the reservoirs. Ford et al.10 measured the chemotactic behavior of a bacterial population to gradients using a stopped flow diffusion chamber. In this assay, two bacterial suspensions differing in stimulant concentrations were contacted by impinging the two flows. Once the flow was stopped, a transient attractant concentration gradient was created by diffusion and the variations in bacterial density were measured by light scattering. Widman et al.11 used a variant of the diffusion gradient chamber assay that consisted of a square arena bounded by a reservoir on each side. The system was operated for a given time before inoculation to allow the aspartate gradient to form partially. Glucose and oxygen concentrations in the chamber were measured using a micro bio-sensor, and the cell growth and migration patterns were observed with time. In this study, however, the concentration of the aspartate which was used as one of the chemoattractant was not measured. More recent studies have utilized the advantages of the microfluidic technology to establish well controlled spatial and temporal concentration profiles.12,13 Ahmed and Stocker12 injected a solution of -methyl aspartate and fluorescein into a main micro-channel connected at right angles to a side branch. After the main channel was completely filled with the solution, the motility buffer containing cells was injected into the side channel at a constant flow rate so that the cells migrated to a constant gradient into the main channel established by diffusion. Subsequently the trajectories of the bacteria along with the concentration gradient were recorded. The chemotactic drift velocity at a specific position was then calculated from average run speed and swimming direction asymmetry, defined as the sums of travel times for all trajectories up and down the gradient. Using these parameters, the diffusivity was computed from the motion of a single cell as well as that of a population. Kalinin et al.13 used three parallel horizontal micro-channels patterned in an agarose gel for quantification of chemotaxis. A chemoattractant ( -methyl aspartate or L-serine) solution was passed through the source channel (lowest) while a blank buffer flowed through the sink channel (topmost). This established a linear chemical gradient in the central channel because of diffusion across the agarose gel from the source to the sink channel. The cells were then introduced into the central channel and their trajectories were recorded to determine the diffusivity and the chemotactic migration coefficient (CMC), defined as the average vertical position of cells with respect to the central position. The experiments suggest that the cells respond to the spatial gradient of the logarithmic attractant concentration. It is important to note that the motility buffer used in this study contained a carbon source, lactic acid, whose influence on chemotaxis was ignored. In parallel with the experimental work, a number of mathematical models have been developed to describe different aspects of chemotaxis at the level of a single cell,14–17 in addition to descriptions of entire bacterial populations.18–22 Keller and Segel18 were the first to develop a comprehensive mathematical model to describe the behavior of chemotactic bacterial population where the bacterial flux comprises two parts, namely, diffusion (or random motility) and chemotactic motion characterized by drift velocity. This model has been extended to express these parameters in terms of single cell parameters12 such as average run speed, turn angle, tumble time duration, etc. However, the population based models do not consider the intracellular signaling pathway which is necessary to determine the system properties like sensitivity, adaptation, etc. Recent models for chemotaxis incorporate the signaling pathway at the molecular level and integrate it with motor response to predict bacterial motion. Examples include AgentCell,23 E. solo24 and RapidCell.25 AgentCell simulates the complete signaling pathway stochastically in a single cell and gives the motion in a three-dimensional environment. E. solo is a deterministic model that solves ordinary differential equations representing the signaling reactions in the pathway and predicts bacterial movement in two-dimensional environment. RapidCell model utilizes the Monod–Wyman–Changeux (MWC)26,27 two-state model for mixed receptor clusters, incorporates the adaptation dynamics and connects the CheY-P values to the cells tumble/run to predict the motion of E. coli in a two-dimensional environment. Using Systems theory, Yi et al.28 have shown that the robustness of perfect adaptation in chemotaxis is the result of the system possessing the property of integral feedback control. The analysis has demonstrated that the integral control in some form is necessary for a robust implementation of perfect adaptation. Despite the presence of such extensive models, it is only recently that Kalinin et al.13 using the MWC model, predicted bacterial motion to varying gradients of -methyl-DL-aspartate and L-serine, and compared them with experimental results. The study used the Keller–Segel model19 to determine E. coli density distribution functions to varying gradients and varying attractant concentrations, both from experiments and model simulations. They demonstrated via both experiment and model simulations that the cells sense the spatial gradient of the logarithmic ligand concentration. However, this study could not explain E. coli adaptation to varying attractant gradients in both space and time.

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In the current investigation, we have developed a novel technique to measure the chemoattractant gradients with respect to space and time in the absence of fluid flow. Unlike the previous studies, we track the motion of each cell and obtain the local drift velocity as a function of time and space for a broad range of concentrations and gradients of -methylDL-aspartate (MeAsp, a non-metabolized chemoattractant). We used an existing two-state model14 for the intracellular pathway and incorporate the extra-cellular influence such as ligand concentration and Brownian motion to predict the response for the experimental conditions. The statistics of single cell motions such as cell velocity, run angle, turn angle, tumbling frequency, rotational diffusion, drift velocity and diffusivity are obtained independently. The predicted motion matched with the observation only when the response of the intracellular pathway was highly ultra-sensitive. The detailed comparison of the predictions with the observed behavior revealed bounds on the parameters describing the intracellular pathways. Further, our studies also show that oxygen plays a key role in the chemotaxis response and the response to a ligand cannot be analyzed in isolation to oxygen.

2. Materials and methods The motion of E. coli K-12 (MTCC 1302) lacking Trg receptor was observed against aspartate gradients in rectangular micro-capillaries (5 cm (L) × 1000 m (W) × 100 m (H)). The medium preparation for the bacterial growth and subsequent experiments are given in the ESI. Chemotaxis towards MeAsp gradients To observe the movement of E. coli in the absence of gradient, a plug of 5 cm length of chemotaxis medium containing MeAsp at a fixed concentration was drawn into the capillary. Cells were taken into the capillary by contacting the pellet (see ESI for details) with the mouth of the capillary. The capillary ends were sealed with wax and the E. coli movements were recorded. Experiments were performed for 0, 100 and 1000 M MeAsp concentrations. The duration of the experiments varied from 15 to 20 min. Further, to observe the response of E. coli to different MeAsp gradients, 4.5 cm long liquid plugs of different concentrations of chemotaxis medium were drawn into the capillary followed by 0.5 cm of motility buffer. A schematic diagram of the micro-capillary experimental setup is shown in the ESI, Fig. S2. E. coli movements were recorded at distances 500, 1000 and 1500 m from the pellet, where the linear concentration profile is established (Fig. S1 ). Each experiment was repeated six times on different days to capture the variability. Details of the image analysis are reported in the ESI. Chemotaxis signaling model A two-state receptor model proposed by Barkai and Leibler14 was used to simulate the intracellular signaling pathway. The model considers the methyl accepting chemotaxis proteins (MCPs), CheA and CheW, as a single entity (receptor complex) and assumes that these receptor complexes, whose concentration is denoted by T, exist in either an active (TA) or an inactive (TI) state. Further, a receptor is assumed to exist in one of the five methylation states. For a fixed value of the ligand concentration, the model solution yields the active and inactive receptor concentrations for each of the methylation states along with the concentrations of chemotaxis proteins in the phosphorylated and dephosphorylated states. Finally, the concentration of CheY-P was used to determine the tumble frequency. The details of the model are given in the ESI. E. coli motion model The ligand concentration decides the protein CheY-P concentration which is the output of the signaling pathway. In vivo experimental studies using fluorescence correlation spectroscopy29 have reported CW and switching frequency (F) as a function of CheY-P concentration (Yp). Recall that the CW is the fraction of time spent in clockwise rotation of the flagella while the switching frequency is the number of times the motor switched its direction of rotation per unit time. Further, Cluzel et al.29 used a Hill equation to describe the CW bias as a function of Yp,

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(1) where n is the Hill coefficient and K is the half saturation constant. Cluzel et al.29 found n = 10 and K = 3.1 M for their single cell experiments indicating a highly ultra-sensitive response. In our experiments, the experimentally measured CW bias was used to set the value of K so as to yield a steady state value of Yp. It was also observed that the switching frequency F qualitatively behaves as F dCW/dYp.29 So, the inverse of the switching frequency gives the time period containing one change of direction of rotation. So, the time spent in a single tumble mode is given by, (2) while the time spent in a single run mode, also referred to as the run time is, (3) Assuming a Poisson process,25 we can now obtain the probability that E. coli will switch from tumble to run mode and vice versa. If the E. coli is in run mode, then the probability that it will switch to the tumble mode in time dt is Prun tum = dt/trun while 1 − (dt/trun) is the probability that it will continue in the run mode after each time step dt.25 Similarly, Ptum = dt/ttum. run In the simulations, the E. coli starts from its initial position, x = 0, y = 0 with the ligand concentration varying linearly with x, L(x) = L0 + Gx, where G is the gradient. At t = 0, the E. coli is made to run for time duration dt along x, after which Prun tum is determined. A number between 0 and 1 is randomly generated using a Matlab function that has a uniform probability in that range. If the number obtained is less than Prun tum, then the E. coli is made to tumble otherwise it continues to run. In the case of tumble, the position of the cell is held constant and a new direction of the motion is chosen from a gamma distribution of turn angles that is obtained independently from our experiments. The distribution yielded a mean turn angle of 71° ± 1.1 that was close to that observed by Berg and Brown.30 In case of run, the cell is made to move at a constant run speed but at an angle chosen from a normal distribution with mean zero angle (about its previous angle) and a variance , where Dr is the rotational diffusivity and was determined independently from our experiments. Once the E. coli has tumbled or run, the above procedure is repeated at the new location for t = t + dt using the new local ligand concentration. The simulation was obtained for 1000 cells and the mean properties were calculated for comparison with experiments. The parameters used in the simulation are reported in the ESI.

3. Results Gradients of fluorescent glucose (2-(N-(7-nitro-benz-2-oxa-1,3-diazol-4-yl)amino)-2-deoxy-glucose, 2-NBDG) were established in the capillary and their variation with time were recorded. It was found that the gradient was established in the first 1.5 min after which the variation was negligible for almost 30 min (see ESI ). At x = 0, the concentration reaches a constant finite value within this short time beyond which (x > 0) a stable linear gradient was achieved. A wide range of gradients could be established by varying the concentration of the ligand in the liquid plug. It can be noted that the gradient experiments with E. coli were conducted for about 15 min after establishing the gradient. This implies that the stable gradients could be maintained during the period of the experiment (see Fig. S1 ). The experiments were conducted with various gradients of MeAsp and with uniform concentrations in the absence of gradients, and the mean square displacement (MSD) of over

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1000 cells from these experiments are presented as a function of time in Fig. 2. Interestingly, all the measured points from various experiments collapse to give a unique trend, wherein the MSD increases quadratically up to t 3.8 s after which the increase is linear. The MSD was further used to determine the diffusivity, ((x − )2/2t) at large times (t > 3.8 s) yielding a value of 110 ± 26.22 m2 s−1, irrespective of the external conditions. Here, is the mean displacement in time t obtained using a Gaussian distribution. The measured angular displacement data was fit to a Gaussian distribution and was used to calculate the rotational diffusivity, given by Dr = 2 /2t where is the angular displacement and the angular brackets represent an ensemble average. Table 1 presents the various parameters related to E. coli motion obtained from our experiments and the measured values are close to those reported in previous studies.

Fig. 2 Mean square displacements (MSD ± standard error) about the mean value (from Gaussian fit) as a function of time. MSDs were obtained using data from all experiments including those with and without gradients. The solid line represents the linearity obtained for a diffusive regime. The diffusive regime occurs beyond 3.8 s and is shown by a dotted arrow.

Table 1 Comparison of chemotactic parameters in E. coli with those reported in literature. These values were obtained by averaging (± standard deviation) over all experiments Description

Present study

Reported

Average velocity/ m s−1 Mean tumble angle (degrees) Rotational diffusivity (Dr, rad2 s−1)

18.6 ± 2.2 71.80 ± 1.1 0.32 ± 0.07

14.2 ± 3.430 62 ± 2630 0.06–0.2813,30

Steady state CW bias (CWss)

0.16 ± 0.05

—

110 ± 26.22

125–3608,9,12,13

2

−1

Cell diffusivity/ m s

Next, the measured drift velocity, in the presence and the absence of MeAsp, was recorded at three different locations along the capillary and is presented in Fig. 3. In the absence of MeAsp, independent experiments were conducted with only motility buffer that was saturated with air and with pure oxygen. Fig. 3 shows that in the absence of MeAsp the cells show significant drift velocities (u0) initially and the velocity drops steadily beyond 1000 m from the entrance. The high concentration of the cells in the pellet would have rendered the interstitial fluid devoid of oxygen. Consequently, the cells respond to an oxygen gradient established when the cells are brought in contact with the motility buffer in the capillary. However, a significant decrease in drift velocity with increasing distance was observed beyond 5000 m and was probably due to a lack of endogenous energy source since 6 of 15

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the cells become non-motile. Further, the run speeds did not vary up to 2500 m, suggesting that the endogenous energy source was not limiting for x < 2500 m. Fig. 3 also includes the drift velocity when the buffer is saturated with oxygen. The velocities were higher by about 1.5 times everywhere than that of the normal case although the cells again became non-motile for distances beyond 5000 m. The higher velocities could be attributed to the higher gradients of oxygen sensed by the cells.

Fig. 3 Measured drift velocity (u0 ± standard error) as a function of distance in the absence of aspartate using normal buffer ( ), 100 M aspartate in normal buffer ( ), and 10 000 M aspartate in normal buffer ( ). The solid line represents the average trend for the three experiments. The empty circle ( ) represents the drift velocity when the buffer is saturated with oxygen in the absence of aspartate. All experiments in the presence of MeAsp were performed at normal levels of oxygen (saturated with air). Experiments were conducted to first quantify the chemotactic response in the absence of gradient to uniform MeAsp concentrations of 100 ( ) and 10 000 M ( ) and the results are also presented in Fig. 3. The measured drift velocities for both the concentrations were similar to that observed in the absence of MeAsp. This clearly indicates that cells respond solely to oxygen in the absence of MeAsp gradients. To study the chemotactic behavior of E. coli, various gradients of MeAsp were established in the capillary. Fig. 4 presents the measured average run speed as a function of the gradient of MeAsp at three different locations in the capillary. The plot also includes measurements in the absence of MeAsp (indicated by the arrow). It can be noted that the average run speeds are in the range of 16–21 m s−1 irrespective of the gradient or the location. An average value of 18 m s−1 was used in the model simulations.

Fig. 4 Average run speeds (± standard error) as a function of aspartate gradients at: 500 m (+), 1000 m ( ), and 1500 m ( ). The solid line represents the average value for the entire data set. The arrow indicates the average run speed of 18 m s−1 obtained in the absence of MeAsp.

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The measured drift velocities contain the influence of both oxygen and MeAsp gradients. In order to separate the influence of oxygen gradient on the total drift velocity, we subtracted (u0) from measured drift velocity values assuming algebraic addition of the responses.31 Thus, the drift velocity due to MeAsp alone is given by, uasp = u − u0. Note that this procedure not only eliminates the effect of oxygen but also the non-chemotactic drift induced by the no-flux condition on the plug side. The measured drift velocities at fixed locations in the capillary for four different gradients with their corresponding initial concentrations are shown in Fig. 5. For the lowest gradient (G = 0.016 M m−1), the response was minimal indicating that the E. coli had completely adapted even at x = 500 m. However, on increasing the gradient to G = 0.08 M m−1, a drift velocity of 1.4 m s−1 was measured at 500 m which monotonically decreased to 0.8 m s−1 at 1500 m demonstrating adaptation. On increasing the gradient further, a similar profile but with higher initial drift velocities was observed. However, at the highest gradient tested (G = 1.6 M m−1), the measured velocities at 500 and 1000 m matched with that for G = 0.08 M m−1, but was significantly lower at 1500 m indicating early adaptation. The model was simulated for the above mentioned gradients and the predictions are compared with experiments in Fig. 5. Note that the simulations do not incorporate the effects of oxygen. Further, simulations were run in the absence of attractants to determine the non-chemotactic drift velocity as a function of distance. The non-chemotactic drift velocity decreases gradually from 0.35 m s−1 at x = 0 to a negligible value at x = 1500 m (results not shown). These values were subtracted from the values obtained for gradients to compare with experimental data, where the effects of both oxygen and non-chemotactic drift have been eliminated. The spatial variation of the predicted drift velocities matched reasonably well with the measurements for all four gradients. Specifically, the model captures the increase in drift velocities with gradient close to the plug and the subsequent monotonic decrease with distance. Further, the measured drift velocity indicates that the cells adapt faster at higher gradients, again in agreement with experiments.

Fig. 5 Drift velocity (uasp ± standard error) as a function of distance for various gradients of aspartate (G) and initial ligand concentration (L0): (A) G = 0.016 M m−1 and L0 = 16 M; (B) G = 0.08 M m−1 and L0 = 80 M; (C) G = 0.16 M m−1 and L0 = 160 M; (D) G = 1.6 M m−1 and L0 = 1600 M. represents data from experiments and solid line

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represents model prediction. The model was used to obtain the drift velocities at different spatial locations along the capillary as a function of varying gradients (see Fig. 6A). Below a gradient of 0.01 M m−1, the drift velocity is negligible and does not vary with distance indicating no response. With increasing gradients, the drift velocity initially increases at all four locations reaching a maximum after which the velocity drops. A maximum drift velocity of 1.8 m s−1 at 100 m was observed close to a gradient of 0.1 M m−1. Note that the peak value of drift velocity also decreases with spatial distance. A negligible drift velocity is obtained for gradients greater than 10 M m−1 but for locations greater than 3000 m due to adaptation.

Fig. 6 (A) Drift velocity obtained through simulation as a function of aspartate gradients at four different spatial locations: dotted line in gray at 100 m, dotted line in black at 500 m, dashed line at 1500 m and solid line at 3000 m, respectively. (B) Normalized CW bias with time for four different gradients: solid line for G = 0.016 M m−1 and L0 = 16 M, dashed line for G = 0.08 M m−1 and L0 = 80 M, dotted line in black for G = 0.16 M m−1 and L0 = 160 M and dotted line in gray for G = 1.6 M m−1 and L0 = 1600 M. (C) Normalized active receptor concentration with time for four different gradients: solid line for G = 0.016 M m−1 and L0 = 16 M, dashed line for G = 0.08 M m−1 and L0 = 80 M, dotted line in black for G = 0.16 M m−1 and L0 = 160 M and

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dotted line in gray for G = 1.6 M m−1 and L0 = 1600 M. The model also predicted the normalized CW bias with time for measured gradients and the corresponding normalized value (with respect to the steady state value) of the fraction of active receptors, TA/TAss (Fig. 6B and C, respectively). Note that TA decides the concentration of CheY-P and therefore the CW. Perfect adaptation to a new environment requires that the TA and therefore the CW return to their steady state values after the initial transients due to change in the environment. At the lowest gradient tested in our experiments, TA drops by a small value after which it regains the steady state value suggesting quick adaptation. Consequently, this small change in TA leads to a step drop of CW bias drops at x = 0 after which it attains the steady state value within a short time (200 s). This indicates that the run length is higher than the steady state value only for a short distance beyond which there is negligible response. At the higher gradient of 0.08 M m−1, the fraction of active receptors decrease further leading to a slower recovery of the normalized CW bias to its steady state value. The plot indicates that the cells do not completely adapt even after 2000 s. On increasing the gradient further, a similar initial response is observed although the normalized CW bias recovers faster to the steady state value. For very high gradients (G = 1.6 M m−1), the bias takes a negligible value up to 500 s indicating very large run lengths. However, the recovery to the steady state value is achieved within 1500 s, indicating faster adaptation at higher gradients. Note that we were unable to measure the CW bias in our gradient experiments as we acquired images at a rate of 9 frames s−1. This rate was not high enough to capture the low values of CW bias (