Evolution of chemotactic hitchhiking
EEvolution of chemotactic hitchhiking
Gurdip Uppal , Weiyi Hu , , Dervis Can Vural ∗ Department of Physics, University of Notre Dame, USA Department of Mathematics, Sichuan University, China ( Dated: May 5, 2020)Bacteria typically reside in heterogeneous environments with various chemogradients where motile cells cangain an advantage over non-motile cells. Since motility is energetically costly, cells must optimize their swim-ming speed and behavior to maximize their fitness. Here we investigate how cheating strategies might evolvewhere slow or non-motile microbes exploit faster ones by sticking together and hitching a ride. Starting withphysical and biological first-principles we computationally study the effects of sticking on the evolution ofmotility in a controlled chemostat environment. We find stickiness allows slow cheaters to dominate when nu-trients are dispersed at intermediate distances. Here, slow microbes exploit faster ones until they consume thepopulation, leading to a tragedy of commons. For long races, slow microbes do gain an initial advantage fromsticking, but eventually fall behind. Here, fast microbes are more likely to stick to other fast microbes, andcooperate to increase their own population. We therefore find the nature of the hitchhiking interaction, parasiticor mutualistic, depends on the nutrient distribution.
INTRODUCTION
Microbial motility plays an essential role in biofilm forma-tion [1], dispersal [2], virulence [3], and biogeochemical pro-cesses [4]. However, motility also comes with metabolic [5, 6]and ecological [4] costs and can also increase the rate of pre-dation [7] and viral infection [8]. Between such costs andbenefits, evolution optimizes how to, when to, where to, andhow fast to swim.Interestingly, motility also promotes aggregation and adhe-sion of microbes [9]. There appears to be a connection be-tween the metabolic pathways that regulate chemotaxis andthose that regulate clumping behavior [10, 11]. Furthermore,non-motile bacteria can stick to motile ones, thereby dispers-ing without paying the energetic cost. This phenomenon isknown as microbial hitchhiking. Hitchhiking behavior hasbeen observed in a variety of microbial species [12–14], wherethe co-dynamics of motile and non-motile species can lead topattern formation [15]. In the oral microbiome, multiple non-motile species of bacteria hitchhike gliding bacteria
C. gin-givalis to disperse and shape the spatial diversity of the mi-crobiome [13, 14]. In ocean environments, zooplankton cantransport microbes across otherwise untraversable strata [16].Co-swarming of motile species can also allow species to com-bine their skills [17].However, the ecology of microbial hitchhiking is not en-tirely clear. For example, some experimental studies of theswarming bacteria
P. vortex aiding the dispersal of non-motilemicrobes seem to suggest a possible mutualistic relationship.In experiments with the non-motile
X. perforans , X. perforans attracted and directed the motility of
P. vortex to facilitate itsown dispersal [18]. In another case,
P. vortex helped transportconidia of the filamentous fungus
A. fumigatus [19] and wereable to rescue fungal spores from areas harmful to the fungus.Here, the bacteria also gain an advantage by utilizing germi-nating mycelia as bridges to cross air gaps [20, 21]. In another ∗ [email protected] experiment, P. vortex carried non-motile
E. coli strain as cargoto help degrade antibiotics [22]. Here
P. vortex used a bethedging strategy where it only carried the cargo bacteria whenneeded. In this case, the motile
P. vortex actually seemed togain more from the relationship. Both species had an increasein population when co-inoculated but
P. vortex grew -foldcompared to E. coli which increased 100-fold [22].Though hitchhiking behavior has been observed in a vari-ety of microbial species, a theoretical understanding incorpo-rating the effects of cell and nutrition density, propensity tostick, and the hydrodynamic interactions between microbes islacking.Here we fill this gap by studying the evolution of swim-ming strategies of chemotactic microbes that interact witheach other and the habitat fluid through contact and hydrody-namic forces. Studies have shown hydrodynamic effects de-pending on microbial shape, swimming mechanism, and in-teractions with boundaries can strongly influence swimmingpatterns [23]. Hydrodynamic interactions between microbeshave been shown to promote aggregation in spherical [24] androd shaped bacteria [25]. Sperm cells can aggregate to betteralign and increase their overall velocity [26]. The shape ofcells will also determine the convective [27] and drag [28, 29]forces. It has also been shown that pair-wise swimming isnot stable without extra aggregation mechanisms [30]. Inter-actions with self-generated flows can also drastically effectmotility. Fluid flows created by bacteria can drive self orga-nization [31] and influence chemotactic motion [32]. Despitethese developments, the evolutionary and ecological conse-quences of hydrodynamic and contact forces between motilemicroorganisms have not been explored.Our goal is to start with the physics of flow, drag and aggre-gation, and from here draw ecological and evolutionary impli-cations on the emergence of social and anti-social behavioralstrategies in microbial swarms. Specifically, in our evolution-ary simulations, we account for the possibility that two mi-crobes might temporarily stick upon colliding; for their indi-rect pushes, pulls, and torques on each other from a distance(as their swimming alters the fluid flow surrounding them);and for the difference in frictional (drag) forces exerted by a r X i v : . [ q - b i o . P E ] M a y the fluid when they are swimming in solitude, versus stucktogether.One common mode of motility in the bacterial world isrun-and-tumble chemotaxis, where bacteria perform a randomwalk with step lengths that depend on the local concentrationgradient along the swim direction [33, 34]. The bacteria willrun straight for a longer duration if the nutrients or toxins arechanging favorably; if not, it will keep the run short and “tum-ble”, picking a new random direction. In the present work too,we model the evolution of microbes carrying out this mode ofchemotaxis.In summary, our model assumptions, stated qualitatively,are as follows: (1) Microbes perform run-and-tumble chemo-taxis [33, 34], for which we use precise chemotaxis responsefunctions derived from empiric data [35, 36] (2) Microbes areplaced in the low end of a nutrition gradient every τ hours,as if in an evolution experiment, or as if in a still fluid bodyin which resources appear at a certain distance every τ hours.We call each such time interval, a “race”. (3) Microbes paya metabolic cost directly proportional to their run speed, andrun speed is heritable (4) the microbial growth rate at a givenlocation depends on the nutrient concentration at that loca-tion, in accordance with empiric data [37, 38] (5) when twomicrobes collide, they either stick, or not stick depending onthe stickiness trait of the microbes. We study three separatecases: (a) all microbes are sticky, (b) no microbe is sticky,(c) only some microbes are sticky, and stickiness is herita-ble except for random mutations. (6) swimming microbes al-ter the flows surrounding them, which causes them to push,pull, and reorient each other. The microbes also experiencedrag (friction) force from the fluid, which is different forstuck and unstuck microbes. The spatial and angular formof physical forces used in our simulations are gathered fromfirst-principles experimental and computational fluid dynam-ics studies [23, 28, 29, 39].Operating under these assumptions, we find that (1) Forshort races, the best strategy for everyone is to not swim. (2)For intermediate-length races, sticking allows slow runnersto make it to large nutrition concentrations without exertingmuch effort themselves and exploiting the motility of fasterswimmers. (3) For long races, fast runners are ultimately ableto leave behind the slow hitchhikers regardless of whether theythemselves are sticky, and ultimately dominate the population.In summary, we find that stickiness selects for hitchhiking be-havior only for races that are intermediately long.We also find that (4) the evolutionarily stable strategy willsensitively depend on the initial distribution of run speed andstickiness as well as the difference between drag forces ex-perienced by stuck versus unstuck microbes. We find thatwhen the drag experienced by a stuck pair is sufficiently lowerthan the drag on two individuals (which is a matter of mi-crobial shape), fast microbes develop cooperation by stickingtogether and further increasing their net speed. In this case,sticking goes from mediating a parasitic interaction (leadingto tragedy of commons, i.e. all microbes slowing down) forintermediate-length races, to an evolutionarily stable mutual-istic interaction amongst fast microbes for long races.This paper is outlined as follows, we first study the effects FIG. 1.
Model schematics.
A band of microbes perform run-and-tumble chemotaxis in a channel with linear chemoattractant gradient(yellow). Colliding microbes stick together until the next tumble.Sticking can be beneficial, since pairs experience less drag force.However on the flip side, slow microbes (red) can also exploit fastones (green). We simulate the evolution of run speed distribution inthe presence and absence of sticking, for different channel lengths. of sticking when all microbes are equally sticky, and sticki-ness does not mutate. We then study the effects of varyingmicrobial density and steepness of the chemogradient. Fi-nally, we study how sticking strategies evolve depending onmicrobe shape (i.e. depending on how drag forces change asmicrobes pair up) as well as when sticking is a evolvable trait.
METHODS
We study an evolving system of actively swimming bacte-ria in a two dimensional chemostat where a chemoattractantis held at a concentration of at the left end and increases lin-early with constant slope m to reach a maximum value at theright end. The chemostat is a rectangular domain, with Neu-mann (reflecting) boundary conditions at the walls. In the be-ginning of each race, bacteria are initialized at the low concen-tration end of the chemostat, at a fixed distance away from thezero point, and perform run-and-tumble chemotaxis towardsthe high concentration end, while subject to inter-microbialhydrodynamic interactions, sticking and drag (Fig. 1). Belowwe describe how we account for these factors that govern themotion of microbes, and how these factors select for stickinessand run speed over repeated races.The chemotaxis of microbes is implemented as follows.Every microbe stores a history of chemical concentration ofthe last 20 time steps, which is then convolved with a responsefunction K ( t ) to determine the tumble probability ˜ ω ( t ) at agiven time t , ˜ ω ( t ) = ω (cid:20) − (cid:90) t K ( t − s ) c ( s ) ds (cid:21) . This equation, together with the chemotaxis response func-tion, is taken from the rigorous experiments of [33]. Here c ( t ) is the surrounding nutrient concentration of the bacteriaat time t , and the chemotactic response K ( t ) is K ( t ) = κλe − λt (cid:2) β ( λt ) + β ( λt ) (cid:3) , The response kernel takes the shape of a positive lobe fol-lowed by a negative lobe so that a swimming microbe essen-tially takes a difference between more recent and less recentconcentrations to determine whether it is swimming towardsa better place.
Hydrodynamic and adhesive interactions between mi-crobes.
We assign each microbe a hydrodynamic radius R h and a sticking radius R s . If two bacteria are within a distance R h from each other, we add the flow velocity generated byeach bacteria to the other’s motion. Each bacteria generates adipole flow around itself given by [23] u ( r ) = p πηr (cid:2) θ − (cid:3) r , (1)where p gives the strength of the dipole flow and η corre-sponds to the viscosity of the surrounding fluid. The flowstrength p ∼ v scales linearly with the run speed of the mi-crobe. We therefore simulate the dipole flow generated by abacteria moving at speed v i as (˜ pv i /r )[3 cos θ − r , where ˜ p gives the rescaled strength of the dipole flow for all microbes.Here we only consider pusher microbes, and therefore take ˜ p > . Equation (1) will lead to microbes swimming in thesame direction to attract, whereas those swimming in oppositedirections will repel.In addition to attracting and repelling, microbes also exerta torque on each other from a distance. This torque scales as /r , and generally results in microbes aligning parallel witheach other if they are pushing swimmers and aligning anti-parallel if they are pulling swimmers. Since we only considerpushers here, and since /r falls of rapidly, we take into ac-count torques pragmatically, by simply aligning the velocityof two microbes that come within distance R s of each other. Aggregation.
Bacteria carry a stickiness trait s i , whichtakes values of either (non-sticky) or (sticky). If two mi-crobes come within a distance R s they stick together if bothof them are sticky; they do not stick if both of them are non-sticky; and they do not stick if one is sticky and the other isnon-sticky. This assumption is a special case of that exploredin [40] (in our case, stickiness is a binary trait rather than acontinuous one).We have also explored what happens if the latter of theseassumptions is modified (i.e. when a sticky microbe collideswith a non-sticky one, they stick) and found that this does notmake a qualitative difference in any of our results (cf. Ap-pendix B).Once two microbes stick, we assume that they swim to-gether until one of them tumbles. Stuck pairs move at a mod-ified speed determined from the motile forces exerted by eachmicrobe and the drag forces experienced by the pair, whichwe discuss below. Drag force.
For low Reynolds number, which is the typicalenvironment for bacteria [23], the drag force on a sphere isgiven by the Stokes law, D = 6 πηRv , where η is the fluidviscosity, R the radius of the sphere, and v the velocity of thesphere relative to the fluid. The drag experienced by a pairof spheres is less than twice the drag force experienced by asingle sphere, since the stuck pair has less total contact areawith the liquid [28]. We model the effect of a reduced drag bythe factor γ and take γ from earlier theoretical and empiricalstudies. Since the microbes are still exerting the same forcewhen stuck, the pair will accelerate to reach a new terminal velocity given by v pair = v + v γ . In general γ will depend on the shape and orientation of themicrobes, and can be viewed as a general ‘cooperative fac-tor’ for sticking. Most of our figures are generated setting γ = 0 . , taken from [29] with the assumption that thetwo microbes are spheres stuck along an axis perpendicular tothe direction of motion. However, we also briefly explore theeffects of varying this parameter. Evolutionary dynamics.
Bacteria reproduce at a rate de-termined by their local nutrient concentration, cost of moving,and cost of sticking. Specifically, the reproduction rate of bac-teria i at position x i is given by f i = a c ( x i ) c ( x i ) + d − bv − cs i where a is the benefit received by the nutrient c ( x ) , b the costof moving, and c is the cost of being sticky. If f i ∆ t is nega-tive, the bacteria dies with probability f i ∆ t , if f i ∆ t is posi-tive, bacteria will reproduce with a probability given by f i ∆ t .The first term in fitness is a Monod growth function that isempirically verified and commonly used in ecological mod-elling [37, 38]. If the chemoattractant (e.g. nutrition) con-centration is much above d (which we set to 1 throughout)the microbe receives diminishing returns. The second term infitness assumes that the energetic cost of swimming is propor-tional to velocity. Since the microbe is working against fluiddrag and since fluid drag is proportional to velocity in thisphysical regime, this assumption is reasonable. The last termis essentially an added constant c for sticky microbes and fornon-sticky ones. Here c would be the amount of slowdown ingrowth rate due to assembling sticky surface glycoproteins orpili, or secreting extracellular polymer substances.To eliminate discrete-time artefacts, fitness constants andtime steps are chosen such that | f i ∆ t | (cid:28) . When a celldivides, a new microbe is placed a distance R s in a randomdirection away, with a random swim direction, and zero his-tory of past chemical concentrations. The run speed v andstickiness s is inherited. However a random mutations can al-ter either. Mutations occur at a rate µ v for velocities and µ s for stickiness. A mutation updates the current velocity by anamount δ picked from a normal distribution with mean 0 andvariance σ v . A mutation on stickiness toggles s i from 0 to 1or vice versa.We simulate multiple races. After a pre-specified race du-ration τ , a fixed number n of randomly chosen bacteria arereset to their original position in the chemostat, as would beduring the dilution step of an evolution experiment. Bacteriaare placed at the location corresponding to chemical concen-tration c along the horizontal x-axis and uniformly along thevertical y-axis. The repeated races take place up until a totalrun time T .A summary of physical parameters is given in Table I. Pa-rameter values were chosen to fit typical values observed forrun lengths [36], bacteria sizes [41], and growth kinetics [38]for bacteria populations. Parameter Definition Value a Sugar benefit constant × − b Cost of moving . × − c Cost of being sticky (0 to 0.36) × − m Slope of nutrient concentration (4.5 to 9) × − c Minimum nutrient concentration 0.1 ω Tumble rate 0.1 λ Response time scale 0.5 β Response shape parameter 2 β Response shape parameter -1 κ Response scaling factor 50 R s Microbe sticking radius 20 R h Hydrodynamic radius 50 γ Hydrodynamic drag factor 0.5 to 1.0 ˜ p Hydrodynamic dipole factor 50 s Microbe stickiness 0, 1.0 µ v Velocity mutation rate × − σ v Velocity mutation strength 1.0 µ s Stickiness mutation rate × − n Number microbes reset 1000 τ Race duration 10 to 300 T Total evolutionary duration 50000 ∆ t Time step 1 H Domain height × to × W Domain width × TABLE I. A summary of the model parameters and the default valuesused in simulations.
Before we move on to describing our results, we shouldwarn that in all of our simulations, we consider only pairwiseinteractions between microbes. This means that our resultsare valid only when the microbial swarm is moderately sparse.More specifically, our model will hold true if the number of in-stances where 3 or more sticky microbes happens to be within R h (and thus R s ) is negligible compared to the number ofinstances where a radius of R h contains one or two sticky mi-crobes. RESULTSOptimal velocity in the absence of hitchhiking
We first determine the evolutionarily optimal swimmingspeeds when there is no cell-cell sticking (and no stickinesscost). We ran simulations varying the race duration τ andnutrient slope m . Overall we observe a uni-modal distribu-tion with the mean velocity increasing to a maximum optimalvalue for longer races (Fig. 2). This optimal value is indepen-dent of the initial velocity distributions and is evolutionarilystable.For very long races, one might guess that faster is alwaysbetter, since those that reach the high end of the chemogra-dient early on will have the most offspring. However this is FIG. 2.
Optimal velocity versus race duration without hitchhik-ing.
Mean optimal velocity versus race duration for different nutrientslopes. In shorter races, faster moving microbes pay a larger cost anddo not gain as much of an advantage from moving. In longer races,the optimal mean velocity increases to a saturating value given bythe maximum viable velocity which can be sustained at the mini-mum nutrient concentration c (equation (2)). As we vary the nutri-ent slope, we effectively rescale space. For larger slopes, the benefitsof moving are realized at shorter races. We can determine the optimalmean velocity analytically (equation (A2)). Points are from simula-tion data ran for a total duration of T = 50000 and dashed linesfrom equation (A2). Numerical results are independent of the initialvelocity distribution. The mean speeds are therefore evolutionarilystable strategies. The chemostat width is H = 2000 , and the rest ofthe parameters are as given in Table I. not the case. Microbes that swim too fast cannot recover theenergy they expend while they are at the low end of the gra-dient, thus, the optimal velocity for large race durations is de-termined by the maximum viable run speed at the beginningof each race. That is, v max = ab c c + d . (2)Mutations may allow larger velocities to emerge once slowmicrobes reach higher nutrition values, but these faster swim-ming microbes will die out in the beginning of the next race.Therefore, in short races, microbes do better by not swim-ming. Beyond a critical race duration τ > τ c , it becomes bestfor microbes to swim at their maximum viable run speed givenby equation (2). We also obtain this critical transition time τ c analytically as given by dashed lines in Fig. 2, and derived inAppendix A and get good agreement with simulations. Effects of hitchhiking for fixed stickiness
We now study the effect of sticking on the optimal swim-ming speed of bacteria. We first investigate the case wheresticking has no fitness cost, and where everyone has the samestickiness.We find, starting from an initially uniform velocity distribu-tion, sticking mostly benefits slower moving microbes, giving
FIG. 3.
Effects of sticking. a,
Effect of stickiness on drift velocity for various drag factors γ . The solid blue line gives the drift velocity inthe x direction versus run speed in the absence of sticking. The purple line corresponds to sticking with γ = 1 . Here stickiness gives a boostto the slowest microbes allowing them to move faster than would be possible on their own, and slows down fast moving microbes. As wedecrease the factor γ , microbes with a larger range of velocities benefit from sticking. For γ = 0 . , we see that everyone moves faster than thenon-sticking case. The drift velocity was found by taking the final displacement of a population after a time T = 300 , starting with a uniforminitial distribution of run velocities. b, Optimal mean velocity versus race duration for non-sticking (blue) and sticking populations. Forshort races, stickiness does not affect the optimal mean velocity. At intermediate race durations, stickiness allows slow microbes to dominatewhere they otherwise would not. For long races, fast microbes can cooperate through sticking and lower their mean velocity compared tothe case without sticking. As we lower the drag factor γ , sticking becomes advantageous to fast microbes at shorter race durations. c, Finalpopulation after T = 50000 for non-sticking (blue) and sticking populations for different drag factors. For short races, stickiness does noteffect the final population. For intermediate-length races, stickiness allows slow microbes to reach larger nutrient concentrations and dominatethe population. This leads to a tragedy of the commons and the final population is lower than without stickiness. For long races, fast microbescooperate through sticking and the final population increases. As we lower γ , the region where the tragedy of commons shrinks and thecooperative region where a sticky fast population outperforms a non-sticky one comes at an earlier race duration. d, Evolution of velocitydistribution for intermediate-length race with τ = 160 and drag constant γ = 0 . . An initially uniform velocity distribution becomestransiently bimodal as slow microbes exploit fast ones to move to larger nutrient regions. Finally, the slow microbes dominate, leading to atragedy of the commons where there are no longer fast microbes for slow ones to exploit. them the largest velocity boost, and the fastest microbes areharmed by being slowed down from sticking to slower mi-crobes (Fig. 3a). As we lower the drag factor γ , a largerproportion of run velocities is benefited by sticking and at γ = 0 . we see that everyone moves faster through stickingthan without. This effect then benefits the slower microbesbest at intermediate-length races, τ (Fig. 3b). For shorterraces, slow microbes already do the best. For long races, slowmicrobes gain an initial advantage, but eventually fall behind.Fast microbes on their own move faster than pairs of slow andfast microbes, and thus still dominate in long races. For inter-mediate races however, slow microbes are able to make it toregions of larger nutrient concentration without expending asmuch energy as fast swimmers. Over the course of many re-peated races, the population distribution transiently becomes bimodal and slow microbes benefit from hitchhiking on fastones. Eventually the population becomes dominated by slowmicrobes (Fig. 3d). This is a typical “tragedy of the com-mons” scenario, where the cheating strategy takes over andfast microbes no longer exist to help disperse slow microbes.To see this clearly, we plot the population for sticking and non-sticking populations versus race duration in Fig. 3c. For inter-mediate races, the population decreases from sticking, sinceslow microbes cause a tragedy of the commons. In long races,fast microbes are able to cooperate with each other via stick-ing and the overall population increases compared to a non-sticking population. As the drag factor γ decreases, this re-gion of tragedy of the commons shrinks and the region wherefast microbes benefit by cooperating and sticking comes at anearlier race duration. FIG. 4.
Coupled effects of sticking, density, and nutrient slope.a,
Mean velocity difference versus race duration for various popula-tion densities. The mean velocity difference is given as the optimalmean velocity without stickiness minus the optimal mean velocitywith stickiness set to one. The velocity difference peaks at intermedi-ate race durations. For long races there is a small velocity differencefrom fast microbes cooperating to lower their mean velocity to a newoptimum. As the population density decreases, the effect of stickingdiminishes, and the race duration region where stickiness benefitsslow microbes shrinks. b, As we vary the slope, the position of thepeak shifts. A larger slope shifts the position of the peak to shorterrace durations. The width of race durations where sticking makes adifference also shrinks with larger slope. The effect of varying slopecan essentially be understood from a rescaling of space.
We next study the effects of stickiness for varying micro-bial density and nutrient gradients. To tune the microbial den-sity, we varied the height H of the simulation domain, keepingthe number of bacteria at the beginning of each reset n con-stant. We plot the mean velocity difference – the mean veloc-ity when there is no sticking minus the mean velocity whenstickiness is one – versus race duration, for varying densityand nutrient slope in Fig. 4. The plots show a maximal differ-ence at an intermediate race duration as discussed above.When we vary the microbial density, we find the peak be-comes less wide for sparser populations. For denser popula-tions, sticking events are more frequent and the effect of stick-iness is more pronounced.When we vary the slope we see the peaks shift to differentrace durations. For higher nutrient slopes, the peaks occur atshorter race durations. This is due to the shift in the transitionrace duration seen in the case without sticking (Fig. 2). Thiscan also be seen as a rescaling of space. A larger slope bringsthe high nutrient concentration region closer, and so at largernutrient slopes the benefit of swimming is realized at shorterraces. The advantage from sticking is therefore also realized atshorter races, and the peaks shift towards lower race durationsat larger slopes.Thus, we see that the optimal conditions for employing asticking strategy vary with population density and nutrientslope. The slow and sticky cheaters are better off always atintermediate-length races, which can be interpreted as sparsenutrient concentrations and/or large consumption and decayrates of nutrients. Next we show how sticking strategies mayevolve naturally for microbes for varying costs and hydrody-namic drag factors associated with sticking. FIG. 5.
Evolution of stickiness. a,
For short races, microbes evolveto swim at slower velocities. Here, stickiness is neutral and evolves tobe around (cid:104) s (cid:105) = 0 . on average due to genetic drift. For long races,microbes are narrowly distributed around a faster velocity distribu-tion. Here sticking allows microbes to cooperate and move fasterthan alone. Stickiness therefore evolves to be around (cid:104) s (cid:105) = 1 . . b, Evolution of run speed and stickiness for long race durations( τ = 200 ). Here, an initially uniform population quickly evolvesto have a large mean velocity. It then becomes advantageous to stick.At this point, microbes evolve to become sticky and lower their runspeed to out-compete non-sticky, faster microbes. c, Mean veloc-ity and stickiness versus drag factor γ for τ = 200 and zero cost.For long race durations, the population predominately consists offast microbes. Since nearby microbes are close to the same veloc-ity, for low drag factors γ stickiness offers an advantage to microbes.They therefore evolve to be sticky and can lower their velocity andout-compete faster microbes. The mean velocity therefore decreasesslightly. For a larger drag factor, stickiness actually harms microbessince sticking to a randomly moving microbe slows it down on aver-age. Around γ = 0 . , microbes evolve to not be sticky and moveon their own at a larger run speed. d, Mean velocity and stickinessversus sticking cost for τ = 200 , and γ = 0 . . As the cost of sticki-ness increases, there is a trade-off between sticking to boost the driftvelocity and moving on one’s own without expending resources tostick. Once the cost of sticking is too large, microbes evolve to notstick and swim at a larger velocity instead. Coevolution of run speed and stickiness
Finally, we explore how microbes may adapt their stick-ing strategies by allowing stickiness to mutate. We determinehow sticking strategies may evolve over time and the effects ofreduced drag γ and sticking cost. One method of sticking to-gether is through the use of secreted extracellular substances.These substances may be costly to produce, but advantageousto slow cheaters or mutually cooperating fast microbes. Wetherefore add an associated cost cs i to sticking. The probabil-ity for two bacteria to stick p is given by the product of the twobacteria’s stickiness constants p = s s ∈ { , } . We also ex-plore the alternative case where p = max( s , s ) ∈ { , } inAppendix B and find no qualitative differences.We first study stickiness evolution for varying race dura-tions and plot the mean stickiness and mean velocity (Fig. 5a).We find that when there is no cost, slow microbes evolve to anaverage stickiness close to . for short races. This is the casewhether we start with an initial population of all stickiness ornear zero stickiness. Here, since the population is composedof essentially non-motile microbes, stickiness does not have asignificant effect. Hence, the stickiness of microbes evolvesprimarily due to genetic drift. For long races, the populationis composed of faster microbes. Since the velocity distribu-tion is concentrated around fast microbes, sticking helps fastmicrobes as they stick to other fast microbes and reduce theirdrag force. We see in Fig. 5b, that microbes evolve to besticky after they have evolved to have fast velocities. Here,the stickiness of fast microbes evolves to near one and themean velocity slightly drops to where slower microbes out-compete the very fast ones. Hence, what is seen as a parasiticinteraction between slow and fast microbes, becomes a coop-erative interaction between fast microbes themselves. Sincethe population consists of predominately fast microbes, stick-ing is mutually beneficial in long races.The amount by which sticking helps microbes will in gen-eral depend on their shape and hydrodynamic properties. InFig. 5c, we plot the mean velocity and stickiness as a functionof the drag factor γ . For lower values of γ sticking allowspairs of microbes to reduce their hydrodynamic drag and in-crease their drift velocity. Fast microbes therefore evolve tobecome sticky. At larger values of γ , sticking no longer be-comes beneficial, and in fact begins to slow microbes downas they stick to other microbes moving in random directions.Therefore, around γ = 0 . (Fig. 5c), microbes evolve to losetheir stickiness. Finally, we study the effects of having a stick-ing cost, with the drag factor fixed at γ = 0 . . Even with somecost, sticking offers a larger advantage to microbes. Once thecost becomes too large however, around c = 0 . , sticking nolonger becomes beneficial and microbes evolve to lose sticki-ness, and increase their mean velocity instead (Fig. 5d). Cheating and tragedy of commons:
At intermedi-ate distances between nutrient patches, slow movingcheaters gain the most benefit by sticking to faster mi-crobes to move to “greener pastures” without expend-ing effort on their own. Over many evolutionary runs,parasitic slow microbes out-compete fast ones, lead-ing to a tragedy of the commons where there are nolonger fast microbes left to exploit. The final popula-tion of microbes is then lower compared to the non-sticking case (Fig. 3c).
Hydrodynamic cooperation:
For long races, fast mi-crobes leave slow ones behind. They can then coop-erate with each other by sticking and reducing theirhydrodynamic drag. Sticking fast microbes then dobetter compared to non-sticking (Fig. 3c). When al-lowing sticking to be a mutable trait, we see fast mi-crobes naturally evolve to stick at long race durationsand sufficiently low drag and cost to sticking (Fig. 5).
DISCUSSION
The phenomena of hitchhiking has been observed experi-mentally, but a theoretical understanding of its evolution andecological function has been lacking. Here we study a simplemodel in which slow microbes can stick to faster ones to hitcha ride for free, as well as faster microbes sticking together tomutually benefit from reduced drag.In addition to aggregation, we also accounted for hydro-dynamic forces in the evolution of microbial motility. Specifi-cally we investigated the effects of self generated flows and re-duced drag forces experienced by pairs of microbes. We alsoaccounted for the drag force modification factor γ (which willdepend on the shape and orientation of microbes) and studiedits role in the evolution of different motility strategies.We ran “evolutionary experiments” where microbes ac-tively swim up a nutrient gradient for a predetermined raceduration. The race duration can be interpreted as the aver-age distance between nutrient patches or the decay time oftransient nutrition concentrations. After this decay time thechemical concentration is reset and microbes swim to the nextpatch.Through our first-principle simulations, we find that whennutrients are distributed at short distances, the best strategyis for no one to swim. At intermediate nutrient distributions,slow microbes evolve to hitchhike on faster ones, leading to atragedy of commons where there are no longer fast microbesleft to exploit. When nutrient sources are distributed far apart,fast microbes evolve to adhere to each other to cooperate andreduce their hydrodynamic drag, benefitting the whole popu-lation.Sticking therefore goes from meditating a parasitic interac-tion, leading to a tragedy of commons, at intermediate nutrientdistributions, to an evolutionarily stable mutualistic interac-tion amongst fast microbes when nutrients are scarcely dis-tributed. We therefore find the ecological nature of hitchhik-ing will depend on the nutrient landscape and on the hydrody-namic drag forces on microbes, which are related to microbialshape and orientation.Throughout, we paid close attention to physical realism,however we also made important simplifying assumptions.To simplify our analysis and to capture the relevant phenom-ena, we implemented evolutionary simulations in a controlledchemostat environment with a linearly increasing chemicalprofile. We also focused on pair-wise interactions betweenmicrobes. Other mechanisms may also contribute to the ag-gregation of microbes and would be interesting to investigatein this context. For example, turbulent forces can cause ac-cumulation of cells. This effect also depends on the shape ofmicrobes [27]. [1] Sarah B Guttenplan and Daniel B Kearns. Regulation of flag-ellar motility during biofilm formation. FEMS microbiologyreviews , 37(6):849–871, 2013.[2] J ´aB Kaplan. Biofilm dispersal: mechanisms, clinical implica-tions, and potential therapeutic uses.
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Appendix A: Semi-analytical results for optimal velocity in theabsence of hitchhiking
Here we derive semi-analytical results for determining thecritical time τ c where swimming becomes advantageous tomicrobes.We can describe the run-and-tumble motion of a populationat scales larger than the run length and time scales longer thanthe tumble time, via an effective diffusion-advection equa-tion. Adding in mutations and reproduction terms, we caneffectively describe the model with a continuous system ofpartial differential equations for non-sticky microbial density n = n ( x, y, v ) as, ˙ n = (cid:18) D ∇ − εv∂ x + µ v σ v ∂ v + a mx + c mx + c +1 − bv (cid:19) n where the effective diffusion D and chemotactic efficiency ε will depend on the response kernel K ( t ) . The effective diffu-sion is simply given as D = v / ω . Following the proceduregiven in [33], we also obtain an expression for the chemotacticefficiency ε , ε = mκv ω (cid:20) β λ ( λ + ω ) + β λ ( λ + ω ) (cid:21) . To determine the mean velocity versus race duration the-oretically, we first simplify our system by ignoring diffusionand mutations, and assume everyone moves at a velocity ˜ εv .Here, due to additional hydrodynamic interactions as well asthe effects of diffusion and reproduction, we have ˜ ε > ε asgiven in equation (A), since alignment generally helps orientvelocities towards nutrients and the growth rate of microbesthat diffuse ahead of the mean is larger than those that fall be-hind. This value is not straightforward to obtain theoreticallybecause of the saturated growth. We therefore measure thisquantity from simulations.We then obtain an ordinary differential equation describingthe growth of the population n ( v, t ) , ˙ n = n (cid:20) a m ˜ εvt + c m ˜ εvt + c + 1 − bv (cid:21) . We can solve this analytically to get, n = n ( v ) e t ( a − bv ) (cid:20) c + 1 c + 1 + ˜ εmtv (cid:21) a/ ˜ εmv . Where n ( v ) is the initial velocity distribution. We candescribe the result of restarting the run N = T /τ times, bytaking the distribution at the end of a race as the initial distri-bution and repeating the process, times a normalization factor.Therefore, after N iterations, the distribution asyptotically ap-proaches, n N ( v, τ ) ∼ e Nτ ( a − bv ) (cid:20) c + 1 c + 1 + ˜ εmτ v (cid:21) aN/ ˜ εmv . (A1)We can then get the mean velocity after N resets and raceduration τ by taking the average, (cid:104) v (cid:105) N ( τ ) = (cid:82) v max vn N ( v, τ ) d v (cid:82) v max n N ( v, τ ) d v . (A2)We compare this to simulation results in Fig. 2 and get goodagreement.We note in equation (A1), as race duration τ goes to infin-ity, the optimal velocity goes to zero, since any small posi-tive velocity will reach high enough saturating goods and out-compete faster microbes. However, in a more natural setting,microbes will consume the resources and slow ones may notactually make it to the resource in time. For shorter races then,there is an advantage to swimming, and the optimal run speedbehaves as in Fig. 2. Appendix B: Significance of sticking assumptions
Here we determine the significance of the sticking assump-tions made in the paper. Specifically, we explore what hap-pens if we modify our assumption that a sticky and non-stickymicrobe do not stick and instead have them stick. We find thismodification does not make a qualitative difference in any ofour results.For results where sticking is not subject to mutation, theassumption makes no difference at all since all microbes aretaken to be either fully sticking or non-sticking. Here the caseof interest where a non-sticky microbe encounters a stickingone does not occur.In the case where we do allow stickiness to mutate, we findno change in our results when varying race duration and dragfactor (Fig. 1a-c). We do see a quantitative change whenvarying sticking cost (Fig. 1d), but observe the same quali-tative behavior. Here, as we increase sticking cost, a fractionof the population evolves to not be sticky but can still hitch-hike due to other sticking microbes. As the cost increases, alarger fraction of the population evolves to be non-sticky untila critical cost where the cost of sticking outweighs the ben-efit and microbes evolve to be non-sticky and swim alone ata faster speed. Compared to Fig. 5d, we see the critical costwhere it is no longer advantageous to stick is now at a largervalue and the transition from non-sticking to sticking is moregradual.0
Supplementary Figure 1.
Evolution of stickiness with modifiedsticking scheme.
Here we reproduce figure 5 in the text with themodified assumption that a sticky and non-sticky microbe stick to-gether when coming into contact. For mean velocity and stickinessversus race duration a, over reset cycles b, and versus drag factor γ c, we see no significant quantitative difference. For the mean velocityand stickiness versus sticking cost d,d,