Steady state running rate sets the speed and accuracy of accumulation of swimming bacterial populations
Margaritis Voliotis, Jerko Rosko, Teuta Pilizota, Tanniemola Liverpool
SSteady state running rate sets the speed and accuracy of accumulation of swimmingbacterial populations
Margaritis Voliotis, Jerko Rosko, Teuta Pilizota, and Tanniemola Liverpool
4, 5 College of Engineering , Mathematics and Physical Sciences , University of Exeter , Exeter EX4 4QF , UK ∗ Laboratoire Jean Perrin , Sorbonne Universit´e , Paris 75005 , France Centre for Synthetic and Systems Biology , University of Edinburgh , Edinburgh , UK School of Mathematics , University of Bristol , Fry Building , Bristol BS8 1UG , UK BrisSynBio , Life Sciences Building , University of Bristol , Bristol BS8 1TQ , UK † (Dated: July 17, 2020)We study the chemotaxis of a population of genetically identical swimming bacteria undergoingrun and tumble dynamics driven by stochastic switching between clockwise and counterclockwiserotation of the flagellar rotary system. Understanding chemotaxis quantitatively requires that onelinks the switching rate of the rotary system in a gradient of chemoattractant/repellant to experi-mental measures of the efficiency of a population of bacteria in moving up/down the gradient. Herewe achieve this by using a probabilistic model and show that the response of a population to thegradient is complex. We find the changes to a phenotype (the steady state switching rate in theabsence of gradients) affects the average speed of the response as well as the width of the distributionand both must be taken into account to optimise the overall response of the population in complexenvironments. This is due to the behaviour of individuals in the ’tails’ of the distribution. Hencewe show that for chemotaxis, the behaviour of atypical individuals can have a significant impact onthe fitness of a population. Bacterial self propulsion, in particular flagellatedmotility, is a phenomenon which captures interest from avariety of disciplines, ranging from physics [1] and biol-ogy [2] to bio-inspired design in engineering [3]. Interestin motility is often in the context of chemotaxis, a be-haviour where membrane bound proteins which act aschemo-receptors sense the presence of certain chemicalsin the environment and affect the flagellar rotation in or-der to move towards or away from the source [4]. Bacteriain nature have evolved, and typically live, in complex en-vironments like the mammalian gastrointestinal tract orthe soil. However, the majority of studies have focusedon the case of dilute aqueous media with a single chemi-cal gradient [4, 5]. While this reductionist approach hasbeen invaluable and generated a large body of knowledgeabout underlying mechanisms of bacterial chemotaxis,our interest is now shifting towards understanding howrobust is bacterial navigation when multiple competingstimuli are present [4–6]. Briefly, and taking the exam-ple of the model organism
Escherichia coli , the bacteriumswims by rotating a flagellar filament bundle that propelsits body through the environment [7, 8]. Each flagellumconsists of a long thin helical filament attached to a bac-terial flagellar motor (BFM), which drives its rotationat rates exceeding 100 Hz. It spins predominantly inthe counter-clockwise (CCW) direction with occasionalswitches to clock-wise (CW) [9]. As long as all the fil-aments are spun CCW they form a stable bundle andwhen one or more participating flagella switches to CWrotation, unbundling occurs resulting in a so-called ”tum-ble” event that brings a change in swimming directiononce all the flagella resume CCW rotation [8]. In a ho-mogeneous environment tumbles happen stochastically, whenever enough copies of the phospho-CheY protein [4](CheY-P) diffuse to the motor and increase the chanceof a CCW-CW switch through their interaction with theBFM [10, 11]. As a result, a single bacterium movesin the pattern of a random walk [13]. The intracellularfraction of CheY that is phosphorylated is controlled bytransmembrane proteins which act as chemosensors [4].They are able to bind very specific chemicals in the cellexterior and, in response, transiently increase or decreasethe concentration of CheY-P inside the cell. This pro-vides a mechanism for biasing the random walk by mak-ing tumbles more or less probable and ensuring, for ex-ample, that tumbles are less probable if the cell is movingtowards a source of food. Only transiently modifying itsreorientation probability allows the bacterium to quicklyrespond to new changes in its surroundings and navigategradients rather than just have a binary response to pres-ence or absence of a chemical [14]. This behaviour, wherethe sensors modulate their own sensitivity to bring theCheY-P concentration to baseline levels only seconds af-ter responding to a stimulus, is called perfect adaptation[15]. It, however, works only if the successive stimuli arein approximately the µ M range and thus are not oversaturating the sensors [16].Biased random walk is not restricted to bacteria butis ubiquitous in the biosphere. Variations of it de-scribe the movement patterns that arise when large her-bivores search for new grazing patches [17] and the waydrosophila larvae search for optimal environmental tem-peratures [18]. Additionally, it also has promising usein bio-inspired swarm robotics as a target search strat-egy [3] and means of controlling the spatial extent of theswarm [19]. Since the changes in the rotational direction a r X i v : . [ q - b i o . CB ] J u l of single motors are at the basis of this kind of directedmotility in bacteria, the quantity that is commonly usedto express how often cells change direction is the CWBias, the total amount of time the motor spent rotat-ing CW in a given interval, divided by the duration ofthe interval[10, 15, 20]. Since most studies on E. coli ’s bi-ased random walk have been done in dilute environments,where the steady state CheY phosphorylation activity isgenerally low, and consequently the CW Bias is low, theefficiency of biased random walked has not been exploredfrom the point of view of steady-state CW Bias changes.Recently, there has been interest in scenarios withoutperfect adaptation, where the CW Bias does not returnto its pre-stimulus levels [21] . Furthermore, experimentsat higher osmolarities, at values typical for the gastroin-testinal tract, have found long term changes in CW Biasfollowing a shift in solute concentration [20]. Becausethe biased random walk arises from transient changes inthe CW Bias due to the concentration gradient, and onthe level of individual bacteria, changing the steady stateCW Bias can affect motion of the bacterial population ina complex manner.Here, therefore, we explore the effect of different steadystate CW biases on the ability of bacteria, or roboticswimmers, to quickly and accurately find their target.To answer such questions requires us to look in detailat the behaviour of whole bacterial populations. Forthe purpose, we introduce a new path integral model ofchemotaxis that sums over whole trajectories of individ-ual bacteria. The model can easily be modified to effi-ciently take account of interactions between the bacteriaand non-local time delay effects in the chemotactic re-sponse. We find that a change to a single cell phenotype(steady-state CW Bias) leads to changes in multiple as-pects of the population. Hence finding an optimal valuefor it is non-trivial and context dependent.We begin by extending our data set from [20]; we mea-sure the CW Bias in 3 different media (see
SupplementaryInformation ), showing a clear change in the CW Biasdistribution and mean value, Fig. 1. Next, to study theimpact of such profound changes in the steady-state CWBias on the speed and accuracy of finding the target withbias random walk, we propose a statistical mechanicaldescription of chemotaxis of individual bacteria, Fig. 2,which can be readily used as tool to quantify chemotacticbehaviour in time-lapse movies of swimming bacteria.We develop a parsimonious model of bacterial chemo-taxis, composed of trajectories of single cells that arestochastic sequences of runs and tumbles. It has somemathematical similarities to wormlike chain (WLC) mod-els describing semi-flexible biopolymers [22, 23]. Its opti-mal length-scales are intermediate between those of PDEmodels [24], which capture average properties of pop-ulations but are insensitive to microscopic details, andagent-based models that link behaviour of individual bac-teria to intracellular biochemistry [25–27] but are difficult
FIG. 1. Variation in flagellar motor’s CW Bias (A) CW biashistogram for cells in VRB Buffer (see
Supplementary Infor-mation for experimental setup and buffer composition). Thedata set includes a total of 118 single motor recordings, each1 min long. (B) Histogram for cells in VRB Buffer with addi-tion of 200 mM sucrose, containing 95 1 min recordings. (C)Histogram for cells in VRB Buffer with addition of 400 mMsucrose, containing 142 1 min recordings. Mean values of CWBias are shown in each panel. to scale to experimentally realistic large populations. In-teractions in space and time between the bacteria are alsoeasy to implement in our framework. We explicitly cal-culate probability distributions of any function of chemo-tactic trajectories as a path integral. The path integralis defined as the weighted sum over all possible individ-ual trajectories of the bacteria, thus the model naturallylinks the CW Bias of individual bacteria with the be-haviour of the population. In this letter, the underlyingbiochemistry is included in a minimal way but it is easyto generalise the model to deal with more detail.We describe a chemotactic trajectory within a con-centration field of chemoattractant/repellant, c ( x ), as achain of random steps (indexed by i ). Each step i rep-resents the state of the bacterium over a time window∆ t i = t i +1 − t i in terms of (i) its chemotactic behaviour s i , either ‘run’ ( s i = +1) or ‘tumble’ ( s i = − τ i . Furthermore, bacteriamove only when in the ‘run’ state at speed v . Hence, thebacterial position, x i , follows from the chain description Steady-state Bias P r obab ili t y D en s i t y Basal running rates
B A behaviourorientationA worm-like chain model of bacterial chemotaxis run (1) run (1) run (1) tumble (-1)
Position ( μ m) P r obab ili t y den s i t y chemical gradientperfectly adaptingcellsnon-adaptingcells C starting position h =0.82 h =0.9 h =1.17 A tumblerunrun run run Orientation s i
0) start to accumulate up achemical gradient, whereas non-adapting cells ( B = 0) are in-capable of performing chemotaxis. Population size N = 5000cells; chemical gradient 0.01 AU · µ m − . of the chemotactic trajectory and the initial position x : x i = x + v i − (cid:88) j =1 ˆ τ j (1 + s j )2 ∆ t . State transitions, i → i + 1, from s i , ˆ τ i to s i +1 , ˆ τ i +1 de-pend only the states, i, i + 1. Hence, starting instate x , the probability of observing the sequence( s , ˆ τ , s , ˆ τ , . . . , s T , ˆ τ T ) ≡ { s T , ˆ τ T } , where T (cid:29) P ( { s T , ˆ τ T }| x ) = p T (cid:89) i =1 p i → i +1 ∝ exp ( − H ( { s T , ˆ τ T } ))with weight defined by: H ( { s T , ˆ t T } ) = (cid:15) T (cid:88) i =2 (1 − s i s i − ) − T (cid:88) i =1 h i s i )+ T (cid:88) i =2 κ ( s i − )(1 − ˆ τ i · ˆ τ i − )Increasing (cid:15) ( >
0) in the first term of H penalises tran-sitions between ‘run’ and ‘tumble’ states, noting that atypical run could extend for several steps. In the secondterm, h i controls the preference for ‘running’ over the‘tumbling’, which in general will depend on the exposureof the bacterium to the chemoattractant/repellent as itmoves through the concentration profile, c ( x ). Here, weenforce perfect adaptation by making h i depend linearlyon the concentration gradient, i.e., h i = h + B ˆ τ i ·∇ c ( x i ).Parameter B controls the strength of the chemotactic re-sponse to the chemical gradient, with B >
B < h , hereafter referred to as the basal running rate,controls the distribution of steady-state tumbling bias(fraction of time a bacterium spends tumbling) (Fig. 2B).For simplicity, and without the loss of generalization, weassume every change in rotational direction of a motorresults in a tumble, hence hereafter we use the termsCW Bias and tumbling bias interchangeably. The modelcould be extended to include any mathematical relationbetween cell’s run/tumble bias and the number and CWBias of the motors, such as the one experimentally ob-served previously [28]. Finally, the third term of H con-trols the change of orientation between steps, which de-pends on the chemotactic state. Since reorientation issignificantly larger during tumbling, κ ( s i ) = (cid:26) κ > s i = − κ < s i = +1with κ > (cid:29) κ < . The final bacterial position is given by x ( t T ) = x j , j = T .Setting (cid:15) = h = 0 and κ constant reduces this modelto the classic wormlike chain of polymer physics. Weevaluate the path integral numerically, using a constanttime step equal to the duration of a typical tumble event,i.e., ∆ t = 0 .
1s [13, 29]; constant speed corresponding tothe average running speed on glucose, i.e., v = 20 µ m · s − [1, 13]; and (cid:15) = 1, B = 1, k > = 1, k < = 0 . h ), initially positioned atthe tip of the base of the triangular profile. Fig. 3B illus-trated how the basal running rate modulates the speedand accuracy with which cells find the target. Lower val-ues of h (Fig. 3B; blue population) achieve consistentexploration of the chemical profile and hence less cell-to-cell variability. However, this comes at a cost of a sloweraverage movement of the bacterial population toward thetarget. As h is increased a portion of the cells approachcloser to the target, but the dispersion of the populationincreases, with a portion of cells completely missing thechemoattractant-rich area (left tail of the red populationin Fig. 3B). Higher values of h give rise to higher levels ofheterogeneity, as prolonged running periods enables cellsto disperse faster and miss the target (Fig. 3B, yellowpopulation).To quantify these observations we introduce the meansquared single-cell distortion (MS-SC distortion), whichis the mean squared distance of a single cell position ( x i )from the optimal position, i.e. the peak of the triangularprofile ( x ∗ ): (cid:88) i ( x i − x ∗ ) = (cid:88) i ( x i − (cid:104) x (cid:105) ) + ( (cid:104) x (cid:105) − x ∗ ) MS-SC distortion can be written as a sum of populationvariance (PV) and squared population distortion (SP dis-tortion), where (cid:104) x (cid:105) = (cid:80) i x i is the mean bacterial posi-tion. We note that the MS-SC distortion is always greaterthan the SP distortion as indicated by the variance de-composition formula.Fig. 3C-E shows the three terms in the equation (MS-SC distortion, PV and SP distortion) as a function ofthe basal running rate ( h ), quantifying the connectionbetween response accuracy and population heterogene-ity. For example, when faced with shallow gradients(Fig. 3C-E, black circles), single cells suffer on averagefrom higher distortion as we increase the basal runningrate, Fig. 3C. The effect could go unobserved, if we fix-ate only on the mean of the population coming closerto the target (Fig. 3E) disregarding the fact that at thesame time the variability in the population is increas-ing rapidly, Fig. 3D. Furthermore, for the range of gra-dients we examined, the SP distortion demonstrates anon-monotonic behaviour as a function of h . This sug-gests that the basal running rate could be regulated toallow bacterial population to optimally adapt to different environments conditions. The values of h inferred fromthe CW Bias data (0.82-1.17; see Fig. 2) are close to thevalues achieving minimum SP distortion in Fig. 3.Fig. 4A illustrates that basal running rate can becontrolled to maximise chemotactic speed and accuracy.Low h give rise to low population velocity due to the in-creased times spent in the tumble state. High h , on theother hand, enables cells to run for longer, but obstructsthem from integrating adequate information about thechemoattractant concentration. The latter gives rise tohigher MS-SC distortion as well as lower average velocity.Hence, for intermediate values of h the system is ableto demonstrate optimal levels of velocity and accuracy.Similarly, Fig. 4B illustrates that successful chemotacticstrategies for the entire bacterial population involve in-termediate values of h , where the population varianceremains low as the population mean comes close to thetarget (low population distrortion). We note that simi-larly to the basal running rate, bacterial running speedalso affects the speed and accuracy of bacterial accumula-tion and must be taken in account to optimise the chemo-tactic response in complex environments (see Supplemen-tary Information ).Our model provides a novel, parsimonious descriptionof bacterial chemotaxis at the single cell level, capturingall the key features of its phenomenology. The statisticalcharacter of the model provides access not only to singlecell chemotactic dynamics as other agent-based chemo-taxis models do [27], but also allows computationally ef-ficient estimation of population measures, bridging thegap between the two scales of description. Despite itssimplicity the model can be straightforwardly extendedto capture more realistic modes of chemtotaxis–involving,for example, changes in the running speed of cells or notperfectly-adapting responses–and study how such modesaffect bacterial accumulation. With advancements in ob-servational techniques and manipulation methods used toprobe bacterial chemotaxis in complex environments, ourmodel could provide a useful inference tool for identifyingtumble/run events and characterising single-cell chemo-tactic responses in more realistic scenarios. Finally, theinfluence of the reorientation frequency of individualswithin a population, on the population level speed andaccuracy of reaching a target could inspire search algo-rithms used in unmanned aerial vehicles [30, 31].
ACKNOWLEDGMENTS
We thank members of Pilizota lab and Richard Berry, Fil-ippo Menolascina and Marco Poiln for useful discussion.TP and JR acknowledge the support from the Office ofNaval Research Global and Defense Advanced ResearchProjects Agency (GRANT12420502). TBL acknowl-edges support of BrisSynBio, a BBSRC/EPSRC Ad-vanced Synthetic Biology Research Centre (grant number basal running rateStarting position Chemical Profile
Single-Cell Distortion
Cell Density (Optimal Position)
A B
Population Average Distortion C Basal Running Rate (h ) M S - S C D i s t o r t i on PV SP D i s t o r t i on Basal Running Rate (h ) Basal Running Rate (h ) Position ( μ m) P r obab ili t y den s i t y ⟨ x ⟩ x cell , i x * ∑ i ( x i − x * ) = ∑ i ( x i − ⟨ x ⟩ ) + ( ⟨ x ⟩ − x * ) Mean Squared Single-Cell (MS-SC) Distortion
Population Variance (PV) Squared Population (SP) Distortion
D E FIG. 3. Bacterial chemotaxis speed and accuracy is influ-enced by the basal running rate. (A) Schematic illustrationof a triangular chemotactic profile, in which chemotactic bac-teria will seek to move towards the optimal position ( x ∗ ).We use the term distortion to denote the distance of singlebacteria to x ∗ , and population-average distortion for the dis-tance between the population average position and x ∗ . MS-SC distortion, PV and SP distortion stand for mean squaredsingle-cell distortion, population variance and squared popu-lation distortion as defined in the main text. (B) Accumula-tion of bacterial populations with different basal running ratesin a triangular chemical profile. Each population consists of N = 10 cells, initialised at the left base point of the triangu-lar profile and followed over 10s. (C) Mean squared single-celldistortion, (D) population variance and (E) squared popula-tion distortion as a function of the basal running rate for dif-ferent heights of the triangular profile. Markers correspondto different gradients of the triangular profile, i.e., 0 .
005 ( (cid:13) ),0 .
01 ( (cid:3) ), and 0 .
02 ( ♦ ) AU · µ m − BB/L01386X/1). And TP, JR, TBL and MV acknowl-edge the support by the Grand Challenge NetworkPlusin Emergence and Physics Far From Equilibrium 2016-19, Engineering and Physical Sciences Research Council (EPSRC), reference EP/P007198/1. Lastly, we thankthe organizers of workshop Physics and Biology of Ac-tive Systems held in June 2015, University of Aberdeen,where this collaboration started. ∗ [email protected] † [email protected][1] Schwarz-Linek J, Arlt J, Jepson A, Dawson A, Vissers T,Miroli D, Pilizota T, Martinez VA, Poon WCK (2016) Escherichia coli as a model active colloid: A practicalintroduction. Coll. and Surf. B: Biointerfaces 137:2-16.[2] Chaban B, Hughes HV, Beeby M. (2015) The flagellum inbacterial pathogens: For motility and a whole lot more.Semin Cell Dev Biol.46:91-103.[3] Bin Yang, Yongsheng Ding, Yaochu Jin and KuangrongHao (2015) Self-organized swarm robot for target search Mean Velocity M S - S C D i s t o r t i on A Population Variance SP D i s t o r t i on B a s a l R unn i ng R a t e ( h ) B FIG. 4. Trade off on chemotactic speed and accuracy imposed by different running rates. (A) Mean squared single-cell distortionversus mean velocity of a bacterial population in a triangular profile as the basal running rate (colour coded) is varied. Optimalbasal running rate achieves the highest mean velocity and lowest distortion. Markers correspond to different gradients of thetriangular profile (0 .
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