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BioSystems 91 (2008) 171–182

Available online at www.sciencedirect.com

Investigations into the design principles in thechemotactic behavior of Escherichia coli

Tae-Hwan Kim a, Sung Hoon Jung b, Kwang-Hyun Cho c,∗a Interdisciplinary Graduate Program in Genetic Engineering, Seoul National University, Gwanak-gu, Seoul 151-747, Republic of Korea

b Department of Information and Communication Engineering, Hansung University, Seoul 136-792, Republic of Koreac Department of Bio and Brain Engineering and KI for the BioCentury, Korea Advanced Institute of Science and Technology,

335 Gwahangno, Yuseong-gu, Daejeon 305-701, Republic of Korea

Received 7 August 2006; received in revised form 21 March 2007; accepted 28 August 2007

bstract

Inspired by the recent studies on the analysis of biased random walk behavior of Escherichia coli[Passino, K.M., 2002. Biomimicryf bacterial foraging for distributed optimization and control. IEEE Control Syst. Mag. 22 (3), 52–67; Passino, K.M., 2005.iomimicry for Optimization, Control and Automation. Springer-Verlag, pp. 768–798; Liu, Y., Passino, K.M., 2002. Biomimicryf social foraging bacteria for distributed optimization: models, principles, and emergent behaviors. J. Optim. Theory Appl. 115 (3),03–628], we have developed a model describing the motile behavior of E. coli by specifying some simple rules on the chemotaxis.ased on this model, we have analyzed the role of some key parameters involved in the chemotactic behavior to unravel thenderlying design principles. By investigating the target tracking capability of E. coli in a maze through computer simulations, we

ound that E. coli clusters can be controlled as target trackers in a complex micro-scale-environment. In addition, we have exploredhe dynamical characteristics of this target tracking mechanism through perturbation of parameters under noisy environments. Iturns out that the E. coli chemotaxis mechanism might be designed such that it is sensitive enough to efficiently track the target andlso robust enough to overcome environmental noises.

2007 Elsevier Ireland Ltd. All rights reserved.

namics;

eywords: Bacterial foraging; Biased random walk; Chemotaxis; Dy

. Introduction

Escherichia coli, often abbreviated to E. coli, arerobably one of the best understood microorganisms.heir cylindrically shaped cell bodies are about 1 �m iniameter and 2 �m long, and they use flagella attached

o a set of biological rotary motors, running eitherlockwise (CW) or counterclockwise (CCW), for theirocomotion (Passino, 2002, 2005; Berg, 2000). When

∗ Corresponding author. Tel.: +82 42 869 4325;ax: +82 42 869 4310.

E-mail address: ckh@kaist.edu (K.-H. Cho).URL: http://sbie.kaist.ac.kr (K.-H. Cho).

303-2647/$ – see front matter © 2007 Elsevier Ireland Ltd. All rights reservdoi:10.1016/j.biosystems.2007.08.009

Escherichia coli; Mathematical modeling; Simulation

all motors rotate CCW, E. coli“run” steadily forward,otherwise they “tumble” by randomly changing theirdirections (Berg, 2000). To decide whether to “run” or“tumble”, E. coli have the control mechanism calledchemotaxis which enables them to forage or run awayfrom noxious chemicals (Passino, 2005). E. coli detectthe change in the chemical concentration gradient anddecide more often to “run” if they are moving towardsthe attractants, or to “tumble” if they are moving towardsthe repellents. Overall, their motile behavior resembles

biased random walks, having a tendency to swim towardsspecific attractants and away from repellents (Falke et al.,1997; Berg and Brown, 1972; Adler, 1966; Muller et al.,2002).

ed.

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The chemotaxis system of E. coli has been widelystudied and its underlying molecular mechanism is nowwell-known (Adler, 1966; Baker et al., 2006a,b). Inthe signaling pathway of chemotaxis, the binding ofchemo-attractants to chemo-receptors reduces the phos-phorylation rate of CheY by CheA. The phosphorylatedCheY binds to the base of the flagellar motor andinduces tumbles by increasing the probability of CWrotation (Berg, 2000; Alon et al., 1999). Thus, it maybe possible to control the motile behavior of E. coli inchemotaxis by changing the tumbling frequency throughover-expression or down-regulation of cheY. Assumingthat the tumbling frequency of E. coli can be controlled inthis way, the goal is to unravel the underlying design prin-ciples by analyzing the role of key parameters involved inthe chemotactic behavior. By controlling the chemotacticbehavior of E. coli based on the understanding about therole of key parameters, the quorum sensing mechanismcan be disrupted such that undesirable biofilm formationis prevented. This might also be applicable to synthe-sizing bacteria that lack virulence against their host forvarious medical applications. On the other hand, the bac-terial chemotaxis system has been known robust in thesense of exact adaptation (Alon et al., 1999), but therobustness with respect to environmental noises has notyet been studied. Hence, we further address this issuein this paper. To this end, we first mimic the motilebehavior of E. coli by specifying two simple rules thatdescribe the chemotaxis mechanism. By investigatingthe target tracking capability of E. coli in a maze, interms of the time taken for E. coli to detect the tar-get, we show that E. coli clusters can be controlled astarget trackers in a complex micro-scale-environment.Then, we analyze the influence of changing tumblingfrequencies on the target tracking capability of E. colithrough perturbation of some key parameters involvedin the chemotactic behavior. In addition, to explorenoise robustness of the chemotactic mechanism of E.coli, we investigate the dynamical characteristics of thistarget tracking mechanism under noisy environments.Throughout these investigations, we aim to characterizeE. coli chemotaxis mechanism at a system-level.

2. Materials and methods

2.1. The Mathematical Modeling of E. coli Chemotaxis

E. coli sense the gradient of attractant or repellent moleculesthrough their specialized transmembrane receptors calledchemoreceptors on the cell surface. Attractant molecules arebound to these chemoreceptors and the increased fraction ofoccupied receptors transiently raises the probability of CCW

91 (2008) 171–182

rotation, resulting in extended runs, while a decrease of theoccupied fraction causes tumbling (Mittal et al., 2003; Berg,2000). It was revealed that, under certain experimental condi-tions, the cell compares its average chemoreceptor occupancyover the past 1 s to that over the previous 3 s and responds tothe difference (Segall et al., 1986). Based on this result, Mittalet al. (2003) made a criterion on the tumbling rate of E. colicells for their simulation by comparing the average receptoroccupancy over the last second, 〈c〉0−1, to that between 4 and1 s ago, 〈c〉1−4. If this difference 〈c〉0−1 − 〈c〉1−4 is greater than0, the E. coli cell is climbing up the attractant gradient andtherefore reduces the tumbling rate resulting in extended runs.

2.1.1. The Decision RuleIn this paper, we assume a similar decision rule that

describes the chemotactic behavior of E. coli climbing upthe attractant concentration gradient and moving towards theattractant. For simplicity, we employ a discrete simulationframework and assume 3 simulation time steps to be 1 s. Hence,it is assumed that each E. coli cell compares its average recep-tor occupancy of attractant molecules during the recent 3 steps,〈c〉1−3, to that between 12 and 4 steps ago, 〈c〉4−12. If 〈c〉1−3 isless than or equal to 〈c〉4−12, implying that it is not climbingup the increasing gradient of attractant molecules, it decidesto tumble in a random direction. Otherwise, it decides to runmore unless it reaches the maximal running steps that will bedescribed later. The average receptor occupancy between a andb steps before the current time step t is as follows:

〈c〉a−b = 1

b − a + 1

b∑

i=a

[ATTR]t−i (1)

where [ATTR]k represents the amount of attractant moleculesin a cell at time step k. According to the decision rule, thecondition for an E. coli cell to run more is

〈c〉1−3 > 〈c〉4−12. (2)

In representing this decision rule, we introduce two parameters,DRnew and DRold where DRnew is the number of recent simula-tion time steps and DRold is the number of previous time stepsexcluding DRnew steps. For instance, DRnew = 3 and DRold = 9for (2).

2.1.2. The Behavior RuleAccording to the decision rule, E. coli cells will never

change their swimming direction as long as there is an increasein the fraction of occupied receptors through binding of attrac-tant molecules. However, this does not actually happen asE. coli cells still have the possibility of tumbling even ifthey are swimming up the concentration gradient of attrac-tant molecules (Mittal et al., 2003; Berg, 2000). To describe

this real feature, we introduce a limit on the maximal and min-imal consecutive running steps in the behavior rule. Hence,each E. coli cell must run straight forward at least for BRmin

steps unless it confronts with an obstacle, and it must tumbleat most after BRmax steps running straight forward regardless

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ig. 1. Illustration of the behavior and decision rule. The number wrischerichia coli cell has been running steadily forward. Since BRmin

umble at every three steps and the maximum number of time steps fo

f the right direction. An example of the behavior rule withRmin = 3 and BRmax = 7 is illustrated in Fig. 1.

.2. Simulation Methods

With the E. coli model described above, we simulate thehemotactic motile behavior of E. coli by using two automataodels that are synchronously working together: one is cellular

utomata for E. coli cells and the other is continuous automataor the environment. Cellular automata are discrete modelsomposed of regular grid of cells (each corresponds to one ofnite number of states) acting together to produce complicatedatterns of behavior (Packard and Wolfram, 1985; Wolfram,984). Continuous automata are extended frameworks of cel-ular automata where the state of a cell is continuous. Suchutomata can represent various physical reactions such as dif-usion in a more realistic way. In both automata, every cellpdates its state for the next time step based on both its presenttate and its neighboring cells by using the same transitionule. We divide the grid of cellular and continuous automatanto equally sized squares such that each cell is surrounded byneighboring cells.

.2.1. Continuous Automata for the EnvironmentWe employed continuous automata to represent the dif-

usion and decay of attractant molecules. On the grid ofontinuous automata for the environment, it is assumed that

ig. 2. A simple example illustrating the diffusion and decay of attractant m.02(2%) and the decay coefficient cdecay is 0.002(0.2%). It is assumed that theighboring cells are 0 initially (left figure). After one time step, 2% of attraells (center figure) and 0.2% of attractant molecules at the center cell decay

ow each time step indicates the number of time steps over which thed BRmax = 7 in the behavior rule, the E. coli cell decides whether tocutive forward running is limited to seven.

chemical molecules diffuse and decay along with time. In oursimulation, attractant molecules diffuse from the current cell tothe four neighboring cells located at N, S, E and W positions,and decay at each cell along with time steps. To reflect thediffusion and decay of attractant molecules in a quantitativeway, we introduce the diffusion and decay coefficients. Thediffusion coefficient describes how fast the chemo-attractantmolecules generated from the target diffuse across the wholegrid whereas the decay coefficient represents the life-time ofchemo-attractant molecules (see Fig. 2).

2.2.2. Cellular Automata for E. coli CellsIn the cellular automata for E. coli cells, each E. coli cell

decides to run or tumble based on the behavior and decisionrules at each time step (see Step 3 of the simulation algorithmin Section 2.2.3). The concentration of attractant molecules atthe corresponding cell of the continuous automata is requiredto calculate the gradient of attractant molecules for the decisionrule (see (1)).

2.2.3. Simulation AlgorithmWe summarize the simulation algorithm as follows:

Step 1. Initialization (see Figs. 3A, 4A and 6A):The initial concentration of attractant molecules at

the target is set to 100. Then the initial concentration ofattractant molecules at the other cells is determined by

olecules after one time step where the diffusion coefficient cdiff ise concentration of attractant molecules at the center cell is 1 and otherctant molecules at the center cell diffuse to each of four neighboring(right figure).

174 T.-H. Kim et al. / BioSystems 91 (2008) 171–182

Fig. 3. The trail of one E. coli cell shows a biased random walk. (A) The initial position of the E. coli (the red automaton cell at the lower leftrner). Gn cell isFor inte

corner) and the target (the yellow automaton cell at the upper right coof chemo-attractants diffused from the target. The darker the automatotrajectory (blue lines) of the E. coli, showing a biased random walk. (is referred to the web version of the article.)

the diffusion coefficient (cdiff) and the decay coefficient(cdecay) of attractant molecules through pre-running

20,000 steps before E. coli cells begin to move. This isrequired for attractant molecules at the target to diffuseacross the whole map so that E. coli cells at the initialstarting position (lower left or right corner of the map)

Fig. 4. 100 E. coli cells move toward a fixed target in the maze. The time stepBlue automata cells show E. coli cells and the black automaton cell in the centhe concentration gradient of chemo-attractant molecules. The darker the cellof the references to colour in this figure legend, the reader is referred to the w

ray-scaled automata cells around the target indicate the concentration, the more chemo-attractant molecules in the automaton cell. (B) Therpretation of the references to colour in this figure legend, the reader

can sense and climb up the concentration gradient ofattractant molecules when a simulation begins.

Step 2. Continuous automata for the environment:The diffusion and decay of attractant molecules

occur at every automaton cell except the target as itis assumed to be a constant source or a source having

is 0, 2500, 5000, and 10,000 for (A), (B), (C), and (D), respectively.ter of the maze denotes the target. Gray-scaled automata cells indicateis, the more chemo-attractant molecules in the cell. (For interpretationeb version of the article.)

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a high enough concentration that can be consideredinvariant over the simulations.

tep 3. Each E. coli cell decides its direction to move in thenext time step according to the decision and behaviorrule as follows:

tep 4. Cellular automata for E. coli cells:Each E. coli cell moves one automaton cell to the

direction decided in Step 3 and we update the entireautomata cells by integrating these information. Forthe next time step simulation, go to Step 2 and con-tinue.

.3. Simulation Schemes

In consideration of the chemotactic behavior of E. coli fol-owing a biased random walk, we have described the motileehavior of E. coli by specifying the two rules and applyhis to simulate the tracking of a target in a maze. A maze

s employed here to represent the complex micro-environmentn vivo. We first examine the target tracking capability of E.oli with the normal parameter set summarized in Table 1.hen, we change each parameter one at a time to analyze theffect of each parameter on the time required for E. coli cells to

able 1he parameter set under a normal condition

arameter Value

diff 0.2

decay 0.002Rmin 3Rmax 9Rnew 3Rold 9

diff is the diffusion coefficient of attractant molecules; cdecay is theecay coefficient of attractant molecules; BRmin is the minimum con-ecutive running steps in the behavior rule; BRmax is the maximumonsecutive running steps in the behavior rule; DRnew is the numberf recent time steps in the decision rule; DRold is the number of pastime steps in the decision rule.

91 (2008) 171–182 175

detect the target. Moreover, we analyze the noise robustness ofthe chemotactic mechanism of E. coli by introducing an envi-ronmental noise to the chemo-attractant concentration in eachautomaton cell.

2.3.1. Simulation Scheme for Parameter PerturbationAnalysis

For a quantitative parameter perturbation analysis, we setthe simulation environment as shown in Fig. 6A. Initially, 25E. coli cells are located at the lower left corner of the maze. Atsimulation time t, we have counted the number of E. coli cellsinside the target boundary (the dotted box around the target inFig. 6A), n. The dotted line in Fig. 6B shows the result from arandom seed for the normal parameter set in Table 1. We havefitted the original data with the hill function in (3) by usingthe Curve Fitting Toolbox in MATLAB to estimate the steadystate value n as follows:

nf(t) = atm

bm + tm(3)

where nf(t) is the fitted data of n at time t, m the hillcoefficient, a the maximum of n that nf(t) can reach, andb is the time taken for nf(t) to reach a/2. From the fitteddata (solid line in Fig. 6B), we determine nss, the num-ber of E. coli cells inside the target boundary at the steadystate, and tss, the time taken for 0.9nss E. coli cells to getinside the target boundary. Mathematically, nss = a and tss =mint(nf(t) ≥ 0.9nss).

2.3.2. Noise Model for the Robustness Analysis of E. coliChemotaxis

In a real environment, there might be fluctuations of thechemo-attractant gradients caused by non-ideal diffusion due

to environmental noises. As the diffusion is related to the attrac-tant molecule itself, we assume the noise varies dependingon the concentration of the attractant molecules. To analyzerobustness of the chemotaxis mechanism of E. coli, we haveadded this environmental noise to the concentration profile of

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chemo-attractant molecules at each automaton cell and evalu-ated their effect on nss and tss. The noise-added concentrationof each automaton cell, [ATTR]update, is

[ATTR]update = [ATTR](1 + N(0, σ)) (4)

where [ATTR] is the concentration of attractant molecules ateach automaton cell and N(0, σ) is the normal distributionwith a zero mean and a standard deviation of σ. Increasingσ adds more environmental noises to the concentration profileof chemo-attractants.

3. Results

The E. coli model described by the decision andbehavior rules shows a biased random walk behavior.Fig. 3 illustrates the trajectory of one E. coli cell mov-ing toward the target. Initially, the E. coli cell lies at thelower left corner and a fixed target is located at the upperright corner of the simulation grid as in Fig. 3A. FromStep 1 of the algorithm, every automaton cell has a cer-tain level of chemo-attractant molecules generated by the

target so that the E. coli can sense the gradient of attrac-tant molecules. To the direction decided at Step 3 of thealgorithm, the E. coli moves one automaton cell at eachtime step. As time passes by, the E. coli swims toward

Fig. 5. Distribution of 100 E. coli cells toward the two fixed targets. (A) Theretwo targets. Each figure right next to the maze map of A and B shows the attrin log10-scale, and the magnification of the black-boxed area is shown right noverlapping of an E. coli cell with the target. (For interpretation of the refereversion of the article.)

91 (2008) 171–182

the target and its trajectory shows a biased random walk(see Fig. 3B).

3.1. Target Tracking Based on E. coli ChemotaxisMechanism

In a real environment, the paths to the target may bemore complicated. To see the target tracking capabil-ity of E. coli in a complex micro-scale-environment, wehave simulated the tracking of one fixed target in a maze(see Fig. 4). The maze map has narrow passages of 5cells width and there is only one path from the start-ing position of E. coli to the target. Initially, 100 E. colicells are gathered at the lower right corner of the maze,then as the simulation time step increases, E. coli cellsmove and gather around the target at the center of themaze. If there is more than one path to the target, weobserve that E. coli cells move along the shortest pathof all. When there are two fixed targets, E. coli cells areproportionally distributed to the two targets, dependingon the concentration of chemo-attractant molecules at

each target. This is shown in Fig. 5 where the distancefrom the decision point, a section where the branchingof the path to each target occurs (black-boxed area of themap in the middle figure of A and B), to each target is

is no short path between two targets. (B) There is a short path betweenactant molecule concentration profile of its corresponding maze mapext to this figure. The yellow automaton cell in the maze indicates thences to colour in this figure legend, the reader is referred to the web

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lmost equal and the left target has twice higher concen-ration of chemo-attractant molecules than the one onhe right. Note that E. coli cells can also track a movingarget.

.2. Parameter Perturbation Analysis

With the target tracking capability of E. coli in a maze,e have analyzed the effect of parameter perturbations

see Section 2.3.1 for simulation details).

.2.1. The Role of Key Parameters in the Decisionule

In the decision rule, there are two parameters, DRnewnd DRold. DRnew is the number of recent simulationime steps, which is to be compared with DRold, the num-er of previous time steps excluding DRnew steps. If theverage receptor occupancy by attractant molecules overecent DRnew steps is greater than that of previous DRoldteps, the E. coli cell decides to keep running, otherwiset decides to tumble. Increasing DRnew desensitizes theecision rule, i.e. E. coli cells respond less sensitively tohe environmental change during the DRnew steps, sinceveraging more steps smooths out the effect of abrupthanges in the concentration of chemo-attractants. Fromig. 7A and B, it turns out that the optimal sensitivityor tracking the target in the maze shown in Fig. 6A ischieved when DRnew is about 3 for any value of DRoldince the maximum nss occurs around the minimum tss.ncreasing DRold extends the history of each E. coli cell,.e. each E. coli cell uses more time steps in evaluating

he environmental changes to decide its tumbling. Thus,f the E. coli cell is swimming towards the target, increas-ng DRold gives more chances to extend runs due to theorse evaluation of previous tendency. This results in an

ig. 6. The environment for parameter perturbation experiments. (A) The maaze. The dotted box around the target is the region where E. coli cells are co

otted box, n, vs. simulation time t for the normal parameter set in Table 1.

91 (2008) 171–182 177

increased nss achieved at reduced tss as DRold increases(see Fig. 7C and D).

3.2.2. The Role of Key Parameters in the BehaviorRule

The behavior rule limits the motion of E. coli byaffecting the decision made by the decision rule. TheE. coli cannot change its direction for at least BRminsteps even though its environment gets worse, and it mustchange its direction after running straight for BRmaxsteps even though its environment gets better. BRmin inthe behavior rule determines the decision point wherethe decision for tumbling or extended run must be made.Increasing BRmin prevents elaborate movement of E. colisince the minimum time steps over which E. coli cannotmake any decision increases, taking more time to find thetarget (see Fig. 8B). Moreover, although the E. coli cellfound the target, it can also easily leave the target, result-ing in a decreased nss since it has to swim for BRmin stepseven if its environment gets worse (see Fig. 8A). Increas-ing BRmax relaxes the maximum limit of time steps thatE. coli can run straight. This makes E. coli swim moredirectly to the target resulting in an increased nss anddecreased tss (see Fig. 8C and D).

3.3. Noise Robustness of E. coli ChemotaxisMechanism

If we increase the level of environmental noises inthe concentration of chemo-attractants, then it becomesmore difficult for E. coli to find the target (see Section

2.3.2 for the noise model). This results in decreasednss and increased tss as shown in Fig. 9 for the nor-mal parameter set in Table 1. Compared to the casewithout any environmental noise, it does not make any

ze map for a parameter perturbation. There are 25 E. coli cells in thensidered to be within the target. (B) The number of E. coli cells in the

178 T.-H. Kim et al. / BioSystems 91 (2008) 171–182

endenc0 rando

Fig. 7. The effects of parameter changes in the decision rule. (A) Depof nss on DRold. (D) Dependence of tss on DRold. The results from 1standard deviations.

noticeable difference till σ increases up to 0.05— nssstays at the same level with only slightly increased tss.With more increased σ, nss gets decreased and tss getsincreased. Overall, we found that the chemotactic motilebehavior of E. coli cell is quite robust to environmentalnoises. Some of E. coli cells can still track the target ina maze for σ equal to 0.45 which is rather a strong noiselevel.

Let us examine the effects of parameter changes in thisnoisy environment. We setσ to 0.40 assuming an extremenoise condition. From Fig. 10A and B, we found thatchanging DRnew causes a shift of the optimal sensitivityto DRnew = 4. From Fig. 10C and D, it turns out thatchanging DRold shows similar tendencies as in the casewithout noises in Fig. 7C and D. The same also holds forthe parameter changes in the behavior rule (see Fig. 11).

4. Discussion

Motivated by the motile behavior of E. coli showinga biased random walk, we have investigated the chemo-tactic motile behavior of E. coli by simulating the E.coli model based on the decision and behavior rules.

e of nss on DRnew. (B) Dependence of tss on DRnew. (C) Dependencem seeds are averaged for each data point and error bars indicate the

By showing the target tracking capability of E. coli ina maze, we presented a possible usage of E. coli as tar-get trackers in a complicated micro-scale-environment.Throughout perturbation of the key parameter values inthe decision and behavior rules, we have unraveled thedynamical role of each parameter.

The decision rule drives E. coli cells to climb up theconcentration gradient of chemo-attractant moleculesgenerated from the target by making proper decisionson whether to tumble or run. Parameters in the decisionrule are related to the sensitivity of E. coli chemotaxis toenvironmental changes. The increase of DRnew desensi-tizes the decision rule by averaging more time steps andsmoothing out the effect of abrupt changes in the environ-ment. For survival of E. coli, its chemotaxis mechanismshould not be too sensitive nor too insensitive to envi-ronmental changes. If it is too sensitive, it may drive E.coli cells more directly to the target since E. coli cellscan respond more rapidly to the changes in their environ-

ment. However, in a noisy environment, this may lead E.coli cells to wander off on their way to the target by forc-ing them to climb up every nearby places having higherchemo-attractant concentrations. Too insensitive chemo-

T.-H. Kim et al. / BioSystems 91 (2008) 171–182 179

F endenco ) and (Ba tions.

tcoNaevmttct

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ig. 8. The effects of parameter changes in the behavior rule. (A) Depf nss on BRmax. (D) Dependence of tss on BRmax. BRmax is 9 in (Averaged for each data point and error bars indicate the standard devia

axis may lead to too slow responses to the environmentalhanges by lowering the threshold for a decision to runr tumble. Hence, a balanced chemotaxis is required.ote that the optimal sensitivity for the target tracking is

chieved when DRnew is about 3. The increase of DRoldxtends the history of each E. coli in evaluating the pre-ious tendency of the environment and thereby providesore chances for extended runs by making the previous

endency worse when the E. coli cells swim towards thearget. This results in getting a larger number of E. coliells inside the target boundary in a shorter period ofime (see Fig. 7C and D).

ig. 9. The effects of environmental noises. (A) Dependence of nss on σ. (B)rom 10 random seeds are averaged for each data point and error bars indicate

e of nss on BRmin. (B) Dependence of tss on BRmin. (C) Dependence). BRmin is 3 in (C) and (D). The results from 10 random seeds are

The behavior rule puts limitations to the decision ruleand thereby drives E. coli cells to swim in a biasedrandom walk. Thus it is related to the basal tumblingfrequency. E. coli might have adopted the biased ran-dom walk behavior to avoid falling into a local maximumsource and to eventually find the global maximum sourceby being properly distributed to each source as shown inFig. 5. BRmin in the behavior rule determines the decision

point where the decision for tumbling or extended runto be made. As shown in Fig. 8A and B, BRmin is nega-tively correlated to nss, the number of E. coli cells insidethe target boundary, and positively correlated to tss, the

Dependence of tss on σ. Parameters in Table 1 are used. The resultsthe standard deviations.

180 T.-H. Kim et al. / BioSystems 91 (2008) 171–182

Fig. 10. The effects of parameter changes in the decision rule for σ = 0.40. (A) Dependence of nss on DRnew. (B) Dependence of tss on DRnew. (C)Dependence of nss on DRold. (D) Dependence of tss on DRold. DRold is 9 in (A) and (B). DRnew is 3 in (C) and (D). The results from 10 randomseeds are averaged for each data point and error bars indicate the standard deviations.

Fig. 11. The effects of parameter changes in the behavior rule for σ = 0.40. (A) Dependence of nss on BRmin. (B) Dependence of tss on BRmin. (C)Dependence of nss on BRmax. (D) Dependence of tss on BRmax. BRmax is 9 in (A) and (B). BRmin is 3 in (C) and (D). The results from 10 randomseeds are averaged for each data point and error bars indicate the standard deviations.

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ime taken for 90% E. coli cells to reach their steadytates. On the contrary, the increase of BRmax relaxeshe maximum limit of time steps over which E. coli canun straight resulting in an increased nss and decreasedss (see Fig. 8C and D). This is because decreasing BRminr increasing BRmax increases run intervals and therebyeduces the basal tumbling frequency.

The chemotaxis system of E. coli described by theecision and behavior rules has been shown to be robusto environmental noises. By analyzing the capabilityf E. coli in target tracking as the environmental noisencreases, we found that some of E. coli cells still couldrack the target even when σ is 0.45 (see Fig. 9). Chang-ng parameters in the decision and behavior rules undern extremely noisy environment (σ = 0.40) showed sim-lar results as in the case without any noise except for thearameter DRnew. By increasing DRnew, we observedshifted optimal sensitivity of DRnew = 4 which may

ndicate a property change in the chemotaxis systemsee Fig. 10A and B). This property change might beaused since more steps are required to reflect the globalendency of gradient profile under a noisy environment.

In the simulation of our E. coli model, we have notonsidered any external force acting on E. coli move-ents. However, since the E. coli cells live in fluids,

hey are subject to rotational Brownian movements thatan cause them to wander off course by ca. 30◦ in 1 sPassino, 2005; Ermak and McCammon, 1978; Berg,000). Analysis with the inclusion of this rotationalrownian movement among the E. coli cells remainss further studies. In addition, we note that E. coli havemechanism called quorum sensing which realizes the

ommunication among the E. coli (Miller and Bassler,001; Park et al., 2003a,b). Though it may not be veryophisticated to have sterile worker castes as in socialnsects such as bees and ants, quorum sensing allowsacteria to behave in a quorum-dependent synchronizedanner which might be similar to the behavior of social

nsects (Waters and Bassler, 2005). This quorum sensingodule might further provide E. coli with the capability

f moving in a group by communicating with each othernd thereby result in more efficient target tracking. Onhe other hand, there has been a growing interest in thepplication of the chemotactic motile behavior of bacte-ia to control micro-robots (Edd et al., 2003; Dhariwal etl., 2004). The present work can be applied for designingarameters to acquire the optimal collective behavior ofultiple micro-robots.

In this paper, we have developed a model describing

he motile behavior of E. coli by specifying two sim-le rules on the chemotaxis. Then, we have explored theole of some key parameters in the E. coli chemotaxis

91 (2008) 171–182 181

through parameter perturbation analysis to characterizeE. coli chemotaxis mechanism at a system-level. Fromthe simulation results, we conclude that the chemotaxismechanism of E. coli might be designed such that itis well balanced between sensitivity and robustness. Inother words, it might be designed such that it is sensitiveenough to efficiently track the target while robust enoughto overcome environmental noises, which however needsto be further investigated experimentally.

Acknowledgments

This work was supported from the Korea Min-istry of Science and Technology through the KoreanSystems Biology Research Grant (M10503010001-07N030100112), the Nuclear Research Grant(M20708000001-07B0800-00110), the 21C Fron-tier Microbial Genomics and Application CenterProgram (Grant MG05-0204-3-0), and in part fromthe Korea Ministry of Commerce, Industry & Energythrough the Korea Bio-Hub Program (2005-B0000002).S.H. Jung acknowledges the financial support receivedfrom Hansung University in the year of 2007.

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