Directional persistence & the optimality of run-and-tumble chemotaxis

E. coli does chemotaxis by performing a biased random walk composed of alternating periods of swimming (runs) and reorientations (tumbles). Tumbles are typically modelled as complete directional randomisations but it is known that in wild type E. coli, successive run directions are actually weakly correlated, with a mean directional difference of not, vert, similar63°. We recently presented a model of the evolution of chemotactic swimming strategies in bacteria which is able to quantitatively reproduce the emergence of this correlation. The agreement between model and experiments suggests that directional persistence may serve some function, a hypothesis supported by the results of an earlier model. Here we investigate the effect of persistence on chemotactic efficiency, using a spatial Monte Carlo model of bacterial swimming in a gradient, combined with simulations of natural selection based on chemotactic efficiency. A direct search of the parameter space reveals two attractant gradient regimes, (a) a low-gradient regime, in which efficiency is unaffected by directional persistence and (b) a high-gradient regime, in which persistence can improve chemotactic efficiency. The value of the persistence parameter that maximises this effect corresponds very closely with the value observed experimentally. This result is matched by independent simulations of the evolution of directional memory in a population of model bacteria, which also predict the emergence of persistence in high-gradient conditions. The relationship between optimality and persistence in different environments may reflect a universal property of random-walk foraging algorithms, which must strike a compromise between two competing aims: exploration and exploitation. We also present a new graphical way to generally illustrate the evolution of a particular trait in a population, in terms of variations in an evolvable parameter

Limits of feedback control in bacterial chemotaxis

Inputs to signaling pathways can have complex statistics that depend on the environment and on the behavioral response to previous stimuli. Such behavioral feedback is particularly important in navigation. Successful navigation relies on proper coupling between sensors, which gather information during motion, and actuators, which control behavior. Because reorientation conditions future inputs, behavioral feedback can place sensors and actuators in an operational regime different from the resting state. How then can organisms maintain proper information transfer through the pathway while navigating diverse environments? In bacterial chemotaxis, robust performance is often attributed to the zero integral feedback control of the sensor, which guarantees that activity returns to resting state when the input remains constant. While this property provides sensitivity over a wide range of signal intensities, it remains unclear how other parameters affect chemotactic performance, especially when considering that the swimming behavior of the cell determines the input signal. Using analytical models and simulations that incorporate recent experimental evidences about behavioral feedback and flagellar motor adaptation we identify an operational regime of the pathway that maximizes drift velocity for various environments and sensor adaptation rates. This optimal regime is outside the dynamic range of the motor response, but maximizes the contrast between run duration up and down gradients. In steep gradients, the feedback from chemotactic drift can push the system through a bifurcation. This creates a non-chemotactic state that traps cells unless the motor is allowed to adapt. Although motor adaptation helps, we find that as the strength of the feedback increases individual phenotypes cannot maintain the optimal operational regime in all environments, suggesting that diversity could be beneficial.Comment: Corrected one typo. First two authors contributed equally. Notably, there were various typos in the values of the parameters in the model of motor adaptation. The results remain unchange

Confinement by biased velocity jumps: aggregation of Escherichia coli

We investigate a linear kinetic equation derived from a velocity jump process modelling bacterial chemotaxis in the presence of an external chemical signal centered at the origin. We prove the existence of a positive equilibrium distribution with an exponential decay at infinity. We deduce a hypocoercivity result, namely: the solution of the Cauchy problem converges exponentially fast towards the stationary state. The strategy follows [J. Dolbeault, C. Mouhot, and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. AMS 2014]. The novelty here is that the equilibrium does not belong to the null spaces of the collision operator and of the transport operator. From a modelling viewpoint it is related to the observation that exponential confinement is generated by a spatially inhomogeneous bias in the velocity jump process.Comment: 15 page

Hook length of the bacterial flagellum is optimized for maximal stability of the flagellar bundle

Most bacteria swim in liquid environments by rotating one or several flagella. The long external filament of the flagellum is connected to a membrane-embedded basal body by a flexible universal joint, the hook, which allows the transmission of motor torque to the filament. The length of the hook is controlled on a nanometer scale by a sophisticated molecular ruler mechanism. However, why its length is stringently controlled has remained elusive. We engineered and studied a diverse set of hook- length variants of Salmonella enterica. Measurements of plate-assay motility, single- cell swimming speed, and directional persistence in quasi-2D and population- averaged swimming speed and body angular velocity in 3D revealed that the motility performance is optimal around the wild-type hook length. We conclude that too-short hooks may be too stiff to function as a junction and too-long hooks may buckle and create instability in the flagellar bundle. Accordingly, peritrichously flagellated bacteria move most efficiently as the distance travelled per body rotation is maximal and body wobbling is minimized. Thus, our results suggest that the molecular ruler mechanism evolved to control flagellar hook growth to the optimal length consistent with efficient bundle formation. The hook-length control mechanism is therefore a prime example of how bacteria evolved elegant but robust mechanisms to maximize their fitness under specific environmental constraints

Cellular memory enhances bacterial chemotactic navigation in rugged environments

The response of microbes to external signals is mediated by biochemical networks with intrinsic time scales. These time scales give rise to a memory that impacts cellular behaviour. Here we study theoretically the role of cellular memory in Escherichia coli chemotaxis. Using an agent-based model, we show that cells with memory navigating rugged chemoattractant landscapes can enhance their drift speed by extracting information from environmental correlations. Maximal advantage is achieved when the memory is comparable to the time scale of fluctuations as perceived during swimming. We derive an analytical approximation for the drift velocity in rugged landscapes that explains the enhanced velocity, and recovers standard Keller–Segel gradient-sensing results in the limits when memory and fluctuation time scales are well separated. Our numerics also show that cellular memory can induce bet-hedging at the population level resulting in long-lived, multi-modal distributions in heterogeneous landscapes

Cell morphology governs directional control in swimming bacteria

The ability to rapidly detect and track nutrient gradients is key to the ecological success of motile bacteria in aquatic systems. Consequently, bacteria have evolved a number of chemotactic strategies that consist of sequences of straight runs and reorientations. Theoretically, both phases are affected by fluid drag and Brownian motion, which are themselves governed by cell geometry. Here, we experimentally explore the effect of cell length on control of swimming direction. We subjected Escherichia coli to an antibiotic to obtain motile cells of different lengths, and characterized their swimming patterns in a homogeneous medium. As cells elongated, angles between runs became smaller, forcing a change from a run-and-tumble to a run-and-stop/reverse pattern. Our results show that changes in the motility pattern of microorganisms can be induced by simple morphological variation, and raise the possibility that changes in swimming pattern may be triggered by both morphological plasticity and selection on morphology

Multi-flagellated bacteria : stochastic model for run-and-tumble chemotaxis

Thesis (MSc)--Stellenbosch University, 2012.ENGLISH ABSTRACT: Bacterial chemotaxis, as observed for Escherichia coli, in a field of chemoattractant molecules is characterised by a run-and-tumble motion. The motion is effected by the clockwise (CW) or counter-clockwise (CCW) rotation of flagella; filamentous appendages attached to molecular motors on the cell body. Runs appear when all flagella turn in the CCW-direction and are used to maintain a favourable direction. Tumbles emerge as soon as one flagellum starts to turn CW and are used for reorientation. Because of the variation observed between individual bacteria displaying run-and-tumble motion, we choose to model this behaviour within a probabilistic framework. An important feature of the chemotactic ability of E.coli is that the cell increases run while moving in the right direction and shortens it in the opposite case. This underlines that tumbles are used for reorientation. It has been found from experiments that there can be significant variation in the tumble fashion depending on the fraction of CW-rotating motors (Turner et al., 2000). The change in angle produced when fewer flagella are rotating CW was found to be smaller when compared to the case for many CW-rotating flagella. In addition, the change of direction contributed by a small portion of CW-rotating flagella is rarely significant for bacteria with many flagella. Based on these observations, we have distinguished between models for the one-flagellated and the multi-flagellated cases. Furthermore, since the tumbling angle change increases with the fraction of CW-rotating motors, it would not be impossible to have some cases where the amount of turn produced by the CW-rotating motors induces the bacterium to have a change of direction greater than 2π. But, this feature could not have been observed because when the bacterium tumbles it can effectuate several revolutions before resuming to a new direction. Therefore, we do not restrict our change of direction to (0,2π) to allow the bacteria to have the possibility to effectuate change of directions of magnitude greater than 2π. To this end, we differentiate between the probability of having directional change of magnitude α and α +2π . Thus we do not use angle change distributions that are defined modulo 2π such as the von Mises distribution or the wrapped normal distribution. The chemotactic ability of the bacterium is modelled by representing the CCW-bias of a single flagellum as a function of the chemoattractant concentration. The model includes the temporal memory of chemoattractant concentration that the bacterium has, which usually spans about 4s. The information about the quality of the current direction of the bacterium is transmitted to the flagellar motor by assuming that this one varies with the chemoattractant concentration level. In addition, the saturation of the bias is incorporated by assuming that the bacterium performs a temporal comparison of the receptor occupancy. The present CCW-bias-Model accounts for the chemotactic ability of the bacterium as well as its adaptation to uniform chemoattractant environment. The models of one-flagellated and multi-flagellated bacterial motion, are used to investigate two main problems. The first one consists of determining the optimal tumbling angle strategy of the bacteria. The second one consists of looking at the effects of the tumble variation on the chemotactic efficiency of the bacteria. In order to address these questions, the chemotactic efficiency measure is defined in such a way that it reflects the ability of the bacteria to converge and to stay in a near neighbourhood of the source so that they gain more nutrients. Since its movement is entirely governed by its single flagellum, the one flagellated bacterium is more able to effectuate a run motion. Tumbling events are modelled to be all equivalent because there is not any fraction of flagella to consider. On the other hand, the tumble variation of the multi-flagellated bacteria is modelled by assuming that the directional change during a tumble is a function of the fraction of CW-rotating motors. By assuming that the number of CW-rotating flagella follows a binomial distribution, we suppose that the multi-flagellated bacteria are less able to effectuate a run motion. This also implies that the change of direction produced by fewer CW-rotating flagella are more likely to happen, and this compensates the lack of run. The models show that the optimal tumbling angle change for the bacteria is less than 2π and that higher flagellated bacteria have higher chemotacitc efficiency. As the number of flagella of the bacteria increases, there can be more tumble variation, in this case the bacteria are more capable of adjusting their direction. There could be some situation were the bacteria are not moving to the right direction, but do not require a large change of direction. This ability to adjust their direction accordingly allows them to converge nearer to the source and to gain more nutrients. In addition, the dependence of the tumbling angle on the fraction of CW-rotating flagella of the mutli-flagellated bacteria, implies that there is a correlation between the tumbling angle deviation and the external environment, because the rotational states CCW-CW of the flagella depends on the external cue. Consequently, it would not be impossible that the average magnitude of tumbling angle change depends on the external environment. To investigate this possibility we analyse the distribution of the tumbling tendency of a single bacterium over time, which is the distribution over time of the average positive tumbling change of the bacterium, within zerogradient environment and within non-zero-gradient environment. We defined the average of these tumbling tendency over time as the directional persistence. We observe that the directional persistence within these different nonzero- gradient environment remains the same. However, the difference between the directional persistence within zero-gradient and non-zeros gradient environment gets larger as the number of flagella of the cell increases. There is more correlation between the external environment and the tumbling tendency of the bacterium. Which is the reason why the higher flagellated bacteria responds the best to the external environment by having the higher chemotactic performance. Finally, the total directional persistence generated by the optimal tumbling angle change of the bacteria is the average directional persistence of the bacteria regardless of their number of flagella. Its value, predicted by the model is 1.54 rad within a non-zero-gradient environment and 1.63 rad within a zero-gradient environment.AFRIKAANSE OPSOMMING: Bakteriese chemotakse, soos waargeneem word vir Escherichia coli, in ’n veld van chemiese lokmiddel molekules word gekenmerk deur ’n hardloopen- tuimel beweging. Die beweging word bewerkstellig deur die regsom of linksom rotasie van flagella; filamentagtige aanhangsels geheg aan molekulêre motors op die selliggaam. ’n Hardloop aksie kom voor as al die flagella linksom roteer en word gebruik om ’m voordelige koers te handhaaf. Tuimels kom voor sodra een van die flagella regsom draai en word gebruik vir heroriënteering. Van wee die variasie wat waargeneem word tussen individuele bakterieë wat hardloop-en-tuimel bewegiging vertoon, verkies ons ’n probabilistiese raamwerk om in te werk. ’n Belangrike eienskap van die chemotakse vermoë van E. coli is dat die sel meer gereeld hardloop terwyl dit in die regte rigting beweeg en minder gereeld in die teenoorgestelde geval. Dit beklemtoon dat tuimels gebruik word vir heroriënteering. Dit is al eksperimenteel vasgestel dat daar betekenisvolle variasie kan wees in die tuimel wyse, wat afhang van die breukdeel regsom roterende motors (Turner et al., 2000). Die hoekverskil afkomstig van minder regsom roterende flagella was vasgestel om kleiner te wees in vergelyking met die menig regsom roterende geval. Verder word die bydrae tot die hoekverskil van ’n klein breukdeel regsom roterende flagella selde beduidend vir bakterieë met baie flagella. As gevolg van hierdie waarnemings, tref ons onderskeid tussen modelle vir een-flagella en multiflagella gevalle. Aangesien die tuimel hoeksverskil vergroot saam met die breukdeel regsom roterende motore, is dit nie onmoontlik om gevalle te hê waar die hoeveelheid draaiaksie gegenereer deur die regsom roterende motore ’n rigtingsverskil groter as 2π kan bewerkstellig nie. Dit was nie moontlik om hierdie eienskap waar te neem nie aangesien die bakterieë ’n paar keer kan tuimel voordat ’n nuwe rigting vasgestel word. Vir hierdie rede beperk ons nie die hoeksverskil tot (0,2π) nie om die bakterieë toe te laat om rigtings veranderinge groter as 2π te ondergaan. Vir hierdie doel, onderskei ons tussen die waarskynlikheid van ’n rigtinsverskil met grootte α en α + 2π. Dus, gebruik ons nie hoekverskil verspreidings wat modulo 2 gedefinieer is nie, soos die von Mises verspreiding of omwinde normaalverdeling. Die chemotakse vermoë van die bakterium word gemodelleer deur die linksom sydigheid van ’n enkele flagellum as ’n funksie van die chemotakse lokmiddel konsentrasie voor te stel. Die model sluit in die tydelike geheue wat die bakterium besit oor chemotakse lokmiddel konsentrasie, wat gewoonlik oor 4s strek. Die informasie oor die kwaliteit van die huidige rigting van die bakterium word deur gegee na die flagella motor toe deur die aanname te maak dat dit wissel met die chemotakse lokmiddel konsentrasie vlak. Die versadiging van die sydigheid word geinkorporeer deur aan te neem dat die bakterium ’n temporale vergelyking maak tussen reseptor okkupasie. Die huidige linksom sydige model neem die bakterium chemotakse vermoë in ag, as ook aanpassing tot ’n uniforme chemotakse lokmiddel omgewing. Die modelle van een-flagella en multi-flagella bakteriële beweging word gebruik om twee hoof probleme te bestudeer. Die eerste, bestaan daaruit om vas te stel wat die optimale tuimel hoek strategie van die bakterieë is. Die tweede kyk na die uitwerking van tuimel variasie op chemotakse effektiwiteit. In orde om hierdie vra te adreseer word die chemotakse effektiwiteit op so mannier gedefinieer dat dit die bakteriese vermoë om die buurt om die oorsprong te nader en daar te bly. Aangesien die beweging heeltemal vasgestel word deur een flagella, in die een-flagella geval, is ’n bakterium meer in staat daartoe om ’n hardloop aksie te bewerkstellig. Tuimel voorvalle word as ekwivalent gemodeleer omdat daar geen breukdeel roterende flagella is om in ag te neem nie. In teenstelling, word die tuimel variasie van multi-flagella bakterieë gemodeleer deur die aanname te maak dat rigtingsverandering gedurende ’n tuimel ’n funksie is van die breukdeel regsom roterende motore. Deur die aanname te maak dat die getal regsom roterende flagella ’n binomiese verspreiding volg, veronderstel ons dat multi-flagella bakterieë minder in staat daartoe is om ’n hardloop aksie te onderneem. Hierdie impliseer ook dat rigtingverandering wat geproduseer word deur minder regsom roterende flagella meer geneig is om voor te kom en dan kompenseer vir ’n tekortkoming aan hardloop gebeure. Die modelle wys dat die optimale tuimelhoek verandering minder as 2 is en dat bakterieë met meer flagella meer chemotaksies effektief is. Soos die getal flagella vermeder, kan daar meer tuimel variasie wees, en in die geval is die bakterieë meer in staat om hul rigting te verander. Daar kan omstandighede wees waar die bakterieë nie in die regtige rigting beweeg nie, maar nie ’n groot rigtingsverskil nodig het nie. Hierdie vermoë om hul rigting byvolglik te verander stel hul in staat om nader aan die oorsprong te konvergeer en dus meer voedingstowwe op te neem. Die afhanklikheid van die tuimel hoek op die breukdeel regsom roterende flagella van multi-flagella bakterieë dui daarop dat daar ’n korrelasie is tussen die tuimel hoek afwyking en die eksterne omgewing, omdat die roterings toestand, regs- of linksom, van die flagella afhanklik is van die eksterne sein. As ’n gevolg, is dit nie onmoontlik dat die gemiddelde grootte van die tuimel hoek verandering van die eksterne omgewing afhang nie. Om hierdie moontlikheid te bestudeer, analiseer ons die verspreiding van die tuimel neiging van ’n enkele bakterium oor tyd, wat die verspreiding oor tyd van die gemiddelde positiewe tuimel verandering is, in ’n nulgradient en nie-nul-gradient omgewing. Ons het hierdie gemiddelde tuimel neigings oor tyd gedefinieer as die rigtings volharding. Ons het waargeneem dat die rigtings volharding binne verskillende nienul- gradient omgewings dieselfde bly. Nogtans is die verskil tussen die rigtings volharding binne nul-gradient en nie-nul-gradient omgewings groter soos die getal flagella vermeder. Daar is meer korrelasie tussen die eksterne omgewing en tuimel neiging van die bakterium. Dit is die rede hoekom bakterieë met meer flagella die beste reageer op die eksterne omgewing deur beter chemotakse effektiwiteit. Ten slotte, die totale rigtings volharding gegenereer deur die optimale tuimel hoek verandering is die gemiddelde rigtings volharding ongeag van die getal flagella. Die waarde wat deur die model voorspel word is 1.54 rad binne ’n nie-nul-gradient omgewing en 1.63 rad binne ’n nul-gradient omgewing

Statistics of C. elegans turning behavior reveals optimality under biasing constraints

Animal locomotion is often subject to constraints arising from anatomical/physiological asymmetries. We use the nematode C.~elegans as a minimal model system to ask whether such constraints might shape locomotion patterns optimized during evolution. We focus on turning behaviours in two contrasting environmental contexts: (i) random exploration in the absence of strong stimuli and (ii) acute avoidance (escape) navigation upon encountering a strong aversive stimulus. We characterise the full repertoire of reorientation behaviours, including gradual reorientations and various posturally distinct sharp turns. During exploration, our measurements and theoretical modelling indicate that orientation fluctuations on short timescales are, on average, optimized to compensate the constraining gradual turn bias on long timescales. During escape, our data suggests that the reorientation is controlled not by an analog logic of continuous turn-amplitude modulations, but rather through the digital logic of selecting discrete turn types, leading to a symmetric escape performance despite strongly asymmetric turning biases.Comment: 32 pages, 17 figure