Testing an agent-based model of bacterial cell motility: How nutrient concentration affects speed distribution
VVictor Garcia, Mirko Birbaumer, Frank Schweitzer:Testing an agent-based model of bacterial cell motility
European Physical Journal B vol. 82, no. 3-4 (2011), pp. 235-244
Testing an agent-based model of bacterial cell motility:How nutrient concentration affects speed distribution
Victor Garcia, Mirko Birbaumer, Frank Schweitzer
Chair of Systems Design, ETH Zurich, Kreuzplatz 5, 8032 Zurich, Switzerland
Dedicated to Werner Ebeling on the occasion of his 75th birthday
Abstract
We revisit a recently proposed agent-based model of active biological motion and compareits predictions with own experimental findings for the speed distribution of bacterial cells,
Salmonella typhimurium . Agents move according to a stochastic dynamics and use energystored in an internal depot for metabolism and active motion. We discuss different assump-tions of how the conversion from internal to kinetic energy d ( v ) may depend on the actualspeed, to conclude that d v ξ with either ξ = 2 or < ξ < are promising hypotheses. Totest these, we compare the model’s prediction with the speed distribution of bacteria whichwere obtained in media of different nutrient concentration and at different times. We findthat both hypotheses are in line with the experimental observations, with ξ between 1.67 and2.0. Regarding the influence of a higher nutrient concentration, we conclude that the take-upof energy by bacterial cells is indeed increased. But this energy is not used to increase thespeed, with 40 µ m/s as the most probable value of the speed distribution, but is rather spendon metabolism and growth. Among the many contributions Werner Ebeling made to the interdisciplinary applications ofstatistical physics, his concept of active motion stands out as the most proliferate. More than 35of his own publications deal, directly or indirecly, with such dynamic phenomena that rely on theinflux of energy. A citation analysis by now mentions 415 citations, lead by a paper publishedin
Biosystems in 1999 [10] which also forms the basis of the current publication. But alreadyan earlier publication in 1994 [30] contained in a nutshell the main idea of negative friction toaccelerate the motion of a Brownian particle.The concept of active motion, as proposed by Ebeling, relies on very few, but reasonable assump-tions: particles, which we call agents in the following, have the ability (i) to take up energy fromthe environment, (ii) to store it in an internal energy depot, and (iii) to use this internal energyto accelerate their motion. Without additional energy take-up, the agent’s motion is describedby a stochastic dynamics in terms of a Langevin equation, which denotes the limit case of Brow-nian motion. A Brownian particle moves passively because the friction which would lead to rest,1/20 a r X i v : . [ q - b i o . CB ] S e p ictor Garcia, Mirko Birbaumer, Frank Schweitzer:Testing an agent-based model of bacterial cell motility European Physical Journal B vol. 82, no. 3-4 (2011), pp. 235-244 asymptotically, is compensated by a stochastic force. The fluctuation-dissipation theorem in thespecial form given by Einstein tells us how the mean squared displacement of such a Brownianparticle is related to fundamental properties of the medium it is placed in, such as viscosity ortemperature. This well known scenario is changed if such particles are turned into “agents” bygetting additional internal degrees of freedom, such as the internal energy depot discussed in thefollowing. Then the passive and random motion can, under certain conditions, be turned into an active and directed motion, which is already found on the level of micro organisms and cells [4].The theory developed from the above assumptions makes a number of predictions about theactive motion of agents with an internal energy depot which have, however, not been testedexperimentally. So, it is worth to find out to what extent living organisms, such as cells orbacteria, can be described by “active Brownian particles”. The current paper wants to contributeto this discussion. In addition to the theoretical framework already developed, it can build on aparallel strand of investigations about the motility of cells [21].The paper is organized as follows: In Sect. 2, we recall the analytical framework of active Brownianparticles, by deriving the equation of motion in the presence of an internal energy depot. Then,different assumptions for the conversion of internal into kinetic energy are developed, which leadto three hypotheses to be tested experimentally. Sect 3 describes the experimental observationsin detail. A comparison between theory and experiment is carried out in Sect. 4 at the level of thespeed distribution, which is derived from a Fokker-Planck equation and compared with empiricaldensities. Details of the results are presented in the Appendix. A conclusion summarizes ourfindings and points out the limitations of their interpretation.
Our approach to model the biological motion of bacteria is based on active Brownian particlesor
Brownian agents ([22]). Each of these agents i is described by three continuous variables:spatial position r i , velocity v i and internal energy depot e i . Whereas the spatial position andthe velocity of an agent describe its movement and can be observed by an external observer,the agent’s energy depot, however, represents an internal variable that can only be deducedindirectly from the agent’s motion.For the internal energy depot we assume, in most general terms, the following balance equation: de i dt = q ( r i , t ) − w ( r i , t ) (1)2/20 ictor Garcia, Mirko Birbaumer, Frank Schweitzer:Testing an agent-based model of bacterial cell motility European Physical Journal B vol. 82, no. 3-4 (2011), pp. 235-244 q ( r i , t ) describes the “influx” of energy into the depot, for example through the take-up of nutri-ents, which therefore may depend on the agent’s position and on time. The spatially inhomoge-neous distribution of energy was modelled e.g. in [10, 23]. In the following, we assume both forsimplicity and in accordance with the experiments described below that nutrients are abundant,hence the take-up of energy is homogeneous, i.e. constant in time and space, q = q ( k ) . But, de-pendent on the experimental condition, q varies dependent on the concentration k of nutrients,but not across agents.The “outflux” of energy from the depot w ( r i , t ) depends on those activities of an agent whichrequire additional energy. In [9, 25] we have modeled the case that agents produce chemicalinformation used for communication, e.g. in chemotaxis. Applying our model to the motionof bacteria, we simply assume that energy is spent on two “activities”: (i) Metabolism, whichis assumed to be proportional to the level of internal energy, with the metabolism rate c beingconstant in time and equal across agents. Alternatively, one could assume that metabolism furtherdepends on the size of the bacteria. (ii) Active motion, i.e. internal energy is converted intokinetic energy for the bacteria to move at a velocity much higher than the thermal velocity ofBrownian motion. For this conversion we assume that it proportional to the internal energy andfurther depends monotonously, but nonlinearly on the speed v of the agent. v is a scalar quantity,describing how fast the agent is moving, regardless of direction. Velocity v , on the other hand,describes the direction as well as the speed at which the agent is moving. Our assumption is thatthe conversion rate d ( v ) does not further depend on the position of the agent or on the directionof motion. w ( v i , t ) = e i [ c + d ( v i )] (2)This ansatz satisfies the idea that without internal energy e no metabolism or active motion ispossible. We note that previously the particular ansatz d ( v ) = d v was discussed in detail [7, 10], but not yet confirmed by experiments. Therefore, in this paper we want to find out whetherthis or other possible assumptions are supported by experiments, so we leave d ( v ) unspecifiedfor the moment. But it is important to note the proportion between the two different terms:bacteria spent the vast amount of their internal energy for metabolism, not for active motion.Consequently the approximation c → , which was discussed in previous investigations, does nothold for bacteria.Assuming that the internal energy depot relaxes fast into a quasi-stationary equilibrium allowsto approximate the internal energy depot as e st i ( v i ) = q c + d ( v i ) (3)I.e., the level of the internal energy depot follows instantaneously adjustments of the speed.3/20 ictor Garcia, Mirko Birbaumer, Frank Schweitzer:Testing an agent-based model of bacterial cell motility European Physical Journal B vol. 82, no. 3-4 (2011), pp. 235-244
The equation of motion for a Brownian agent is given by a Langevin equation for the velocity v i .However, because of the conversion of internal into kinetic energy with a rate d ( v ) , we need toconsider an additional driving force the structure of which can be obtained from a total energybalance. In the absense of an external potential, the mechanical energy of the agent is given bythe kinetic energy, E i = mv i / , which can be changed by two different processes: (i) it decreasesbecause of friction, with γ being the friction coefficient (equal for all agents), and (ii) it increasesbecause of conversion of internal energy into energy of motion. Hence, with v i = ˙ r i and ˙ v i = ¨ r i the total balance balance equation gives: ddt E i = m ˙ r i ¨ r i = ed ( v i ) − mγv i (4)Deviding this equation by m ˙ r i results in the deterministic equation of motion: ˙ v i = − v i (cid:20) γ − em d ( v i ) v i (cid:21) (5)Based on this, we propose the Langevin equation for the Brownian agent by adding to the right-hand side of eqn. (5) a stochastic force F i ( t ) with the usual properties, namely that no driftis exerted on average, (cid:104) F i ( t ) (cid:105) = 0 , and that no correlations exist in time or between agents, (cid:104) F i ( t ) F j ( t (cid:48) ) (cid:105) = 2 S δ ij δ ( t − t (cid:48) ) . For physical systems the strength of the stochastic force S isdefined by the fluctuation-dissipation theorem which itself builds on the equipartition law, but formicrobiological objects such as cells and bacteria the situation has proven to be more intricate,as we will outline later.Assuming a quasistationary internal energy depot, eqn. (3) and defining γ = γm , we arrive atthe modified Langevin equation for the Brownian agent: ˙ v i = − γ v i (cid:20) − q γ d ( v i )[ c + d ( v i )] v i (cid:21) + √ S ξ i ( t ) (6)where ξ i denotes Gaussian white noise. Eqn. (6) shall be used as the starting point for the furtherdiscussion. d ( v ) We now specify the function d ( v ) for which we assume a nonlinear dependence on the speed interms of the following power series: d ( v ) = n (cid:88) k =0 d k v k . (7)4/20 ictor Garcia, Mirko Birbaumer, Frank Schweitzer:Testing an agent-based model of bacterial cell motility European Physical Journal B vol. 82, no. 3-4 (2011), pp. 235-244
Using different orders of this power series, we first evaluate the stationary velocity estimatedfrom the deterministic part of eqn. (6) (omitting the index i for the moment) and then comparethe outcome against data from experiments with bacteria.Neglecting the stochastic term of eqn. (6), we first notice that v = 0 is always a solution. Causedby friction, the motion of an agent shall come to rest – but in a stochastic system agents stillpassively move thanks to the impact of the random force F .Secondly, taking into account the influence of the internal energy depot, we notice the importanceof the rate of metabolism, c . For c = 0 , we always find for the nontrivial speed v s = (cid:114) q γ ; ( c = 0 , γ = γm ) (8)independent of further assumptions for d ( v ) . I.e., dependent on the take-up of energy q agentsmove with a constant speed. For c (cid:54) = 0 and n = 0 , i.e. for d ( v ) = d , this nontrivial speed iscorrected leading to v s = (cid:18) q γ (cid:19) / (cid:18)
11 + ( c/d ) (cid:19) / (9)That means dependent on the proportion of metabolism vs active motion, the stationary velocitycan be consirably lowered. For further comparison it is convenient to rewrite eqn. (9) in a differentway v s = (cid:18) cd (cid:19) / (cid:18) Q c/d ) (cid:19) / ; Q n = q d n γ c ; ( n = 0) (10)where the reduced parameter Q n combines the relevant parameters of the model describingtake-up energy and active motion ( q , d n ) and external and internal dissipation ( γ , c ) .Considering the next higher order of the polynom, d ( v ) = d v for n = 1 , the stationary speedfollows from the quadratic equation: v s + (cid:18) cd (cid:19) v s − q γ = 0 (11)One can verify that these solutions are not consistent with other physical considerations, inparticular the speed v s , for large c , is biased toward negative values. Testing another first orderassumption, d ( v ) = d + d v , does not improve the situation, because d is additive with c andthus just rescales the metabolism rate.Consequently, we have to restrict ourselves to the case n = 2 , i.e. we arrive at d ( v ) = d v whichis the known SET model [7, 23] with the stationary solution for the speed v : v s = (cid:114) q γ − cd = (cid:18) q γ (cid:19) / (cid:18) − Q (cid:19) / = (cid:18) cd (cid:19) / ( Q − / (12)5/20 ictor Garcia, Mirko Birbaumer, Frank Schweitzer:Testing an agent-based model of bacterial cell motility European Physical Journal B vol. 82, no. 3-4 (2011), pp. 235-244
Different from the previous cases, for n = 2 we find a bifurcation dependent on the controlparameter Q . For Q ≤ , v = 0 is the only real stationary solution, whereas for Q > anontrivial solution for the speed exist. The possible consequences are already discussed in theliterature. In [11, 24] the supercritical case, Q > was investigated, while in [7, 8, 28] thesubcritical case Q ≤ was considered. Because metabolism consumes the lion share of theinternal energy provided, one would assume that Q (cid:28) is the most realistic case for the motionof bacterial cells – which remains to be tested.For the stochastic motion, eqn. (6), in the supercritical case the contribution of the stochasticterm is small compared to the kinetic energy provided by the energy depot. Hence, the agentshould move forward with a non-trivial velocity (i.e. much above the thermal fluctuations), whichhas a rather constant speed, but can change its direction occasionally. In the subcritical case,on the other hand, the stochastic fluctuations dominate the motion, but the energy depot stillcontributes, this way resulting in the first order approximation of the stationary velocity (in onedimension): v = k B T /m (1 − Q ) [8]. In fact, the authors of [8] put forward a nice argumentthat in the high dissipation regime – or in environments with low nutrition concentration – astrong coupling between the two energy sources (depot and noise) appears that should help microorganisms to search more efficiently for a more favorable environment.We are not going to repeat these theoretical discussions. Instead, we ask a different questionnot investigated so far: which of the above cases is consistent with experimental findings ? Asoutlined above, the SET model with its two regimes, (i) Q ≤ , i.e. subcritical energy supply,and (ii) Q > , i.e. supercritical energy supply, is the most promising ansatz to be tested for d ( v ) . To compare this with a more general setting, instead of integers n = 0 , we may alsoconsider fractional numbers n = ξ with < ξ < , i.e. d ( v ) = d v ξ , which results in the followingequation for the stationary solutions: v ξ − (cid:18) q γ (cid:19) v ξ − + cd = 0 (13)Reasonable values of ξ should be in the interval between 1 and 2 – for which we expect twonontrivial solutions for the stationary velocity, but no bifurcation with respect to the parameter Q . In conclusion, our theoretical investigations provide us with three different hypotheses forthe active motion of biological agents:1. d ( v ) = d v with subcritical take-up of energy, i.e. Q < d ( v ) = d v with supercritical take-up of energy, i.e. Q > d ( v ) = d v ξ with ≤ ξ ≤ and Q > ictor Garcia, Mirko Birbaumer, Frank Schweitzer:Testing an agent-based model of bacterial cell motility European Physical Journal B vol. 82, no. 3-4 (2011), pp. 235-244
In order to test which of the above outlined hypotheses regarding active biological motion iscompatible with real biological motion of bacteria, we proceed as follows: Bacterial cells areplaced in a shallow medium that can be approximated as a two-dimensional system. Keepingall other conditions constant, we vary the nutrition concentration in that medium so that threedifferent nutrition levels are maintained: low, medium, and high. We then measure the velocitydistribution of the bacteria (as described below) in response to these nutrition levels. Eventually,we do a maximum likelihood estimation of the parameters describing the velocity distributionsand compare these with the hypotheses made. Ideally, we would expect that the experimentalvelocity distributions could be better fitted by one of the hypotheses, while the others could berejected.When it comes to the specific setup of the experiments, we soon realize that the devil is in thedetails. In order to be conform with our hypotheses, we would need to test bacteria that movelike Brownian particles in the limit of low nutrition, while performing a rather directed motionfor high nutrition concentration, with arbitrary changes in the direction. Instead, most bacteria,
Escherichia coli being a prominent example, move quite differently, i.e. their movement switchesbetween tumbling and nontumbling phases [6]. Tumbles denote temporary erratic movements,whereas during the nontumbling phases, called runs, bacteria execute a highly directed, ballistic-like motion. Both of these phases describe a different form of active motion, but do not differin the mechanism or level of energy supply. Precisely, the flagellar propellers responsible forthe forward motion [2, 3, 16, 17] rotate with the same efficiency during the tumbling and non-tumbling phases [5].In order to avoid an abritary averaging over these different forms of active motion, we havechosen to study bacteria that do not tumble at all, specifically the non-tumbling strain M935of
Salmonella typhimurium [29, 31]. This type of bacterial cells has another advantage in thatit does not perform chemotaxis, i.e. it does not follow chemical gradients or gradients in thenutrient concentration, which would bias the motility towards directed motion. But mutantstrain is capable to take up the nutrients at the same rate as normal
S. typhimurium .For the medium, we have realized an almost two-dimensional setup, keeping in mind that three-dimensional motion results in a projection error of the trajectories. Further, we need to ensurethat both the temperature T and the viscosity η of the medium is kept constant over time andacross setups with different nutrient concentrations. These were prepared as follows: Medium 0
Used as a reference case where no additional nutrients are available for the bacteria.It consists of a phosphate-buffered saline working solution ( PBS ), with 5 protein.7/20 ictor Garcia, Mirko Birbaumer, Frank Schweitzer:Testing an agent-based model of bacterial cell motility
European Physical Journal B vol. 82, no. 3-4 (2011), pp. 235-244
Medium 2
A nutritionally rich medium which contains of lysogeny broth ( LB ), a substancealso primarily used for the growth of bacteria. Medium 1
A medium with an intermediate concentration of nutrients. It contains an mixtureof medium 0 and 2, equally.We assume that the nutrients are equally dissolved in the whole medium and that nutrientconcentration differences can be neglected. Further, the depletion of nutrients due to consumptionof the bacterial cells can be neglected. Hence, we assume that the take-up of energy per timeunit, q is constant and equal for all bacterial cells, but may change dependent on the nutientconcentration, i.e. q needs to be determined for each of the three different settings as describedbelow. In order to observe trajectories, bacteria of the same
Salmonella strain were grown (detailssee Appendix A) and were put into the three different media. For each medium, we recordedtwo movies at different times: (i) after the bacteria were put into the different media (initialcondition), and (ii) after about one hour, i.e. after a sufficiently long time of relaxation, whichensures a stationary velocity distribution. We had to assume that the cells would adapt ratherslowly to the new environment. Appendix B presents the details of how the trajectories wererecorded.As the trajectories of Fig. 1 verify for both the initial and the stationary conditions,
Salmonellatyphimurium swim in quasi-ballistic manner through the fluid. Their trajectories are mostlyslightly curved. Initially, at least two trajectories are very strongly curved where bacteria seemto swim in narrow circles. The videos make clear that bacteria swim in an isotropic manner andwith a mean velocity. It is remarkable that the bacteria maintain their quasi-ballistic movementeven after more than one hour, despite not being able to take up energy from the medium.Given the trajectories measured at a time resolution of 0.11 sec, we are able to calculate thevelocity vector in the two-dimensional space as v kt j = r kt j − r kt j − t j − t j − (14)where k denotes the bacterial cell and j refers to the time step. The velocity distributions areshown in Fig. 2 both for the initial situation (blue dots at t ) and after a sufficient long time ofrelaxation (red dots at t end ) and for the three different media used. Each of the samples containsabout 1.500 data points. 8/20 ictor Garcia, Mirko Birbaumer, Frank Schweitzer:Testing an agent-based model of bacterial cell motility European Physical Journal B vol. 82, no. 3-4 (2011), pp. 235-244 x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] x [ m m] y [ m m ] Figure 1: All bacterial trajectories recorded in medium 0 (no nutrients) at times t =0min in(top, 54 trajectories) and t end =84min (bottom, 61 trajectories). At both times, the movielength evaluated was about 17 sec with a time resolution of 0.11 sec, i.e. about 700 frames permovie. The points of the trajectories show 26 of these frames. The rectangular boundaries area consequence of the camera calibration.In all cases, we observe the formation of rings of different radii, indicating that bacteria swimat comparable velocities in the time intervals of observation and that their motion is isotropic,i.e. that they do not have a preferrred direction of motion. Most interesting, compared to theinitial distribution the rings either contract (medium 0) or expand (medium 1, 2) in diameter,which means that the bacteria have adjusted their individual velocities according to the nutrientconcentration available. This will be systematically investigated in the next section. The availabledata do not allow us to predict if the rings completely contract (medium 0) or further expand(medium 1,2 ) their extension. The experiments described above have clearly shown that bacteria adjust their velocity dependenton the nutrient available in the medium. It remains (i) to quantify this influence, and (ii) tocompare the outcome with the hypothesis made on the velocity dependent transfer of internalenergy. Such a comparison cannot be made on the level of individual trajectories, but only onthe level of the ensemble average. 9/20 ictor Garcia, Mirko Birbaumer, Frank Schweitzer:Testing an agent-based model of bacterial cell motility
European Physical Journal B vol. 82, no. 3-4 (2011), pp. 235-244
Figure 2: Snapshots of the two-dimensional velocity distributions around times t (blue dots)and t end (red dots), shown for the three different media: (left) medium 0, (middle) medium 1,(right) medium 2. Each of the samples contains about 1.500 data points.Hence, in the following we use the two-dimensional velocity distribution p ( v , t ) , which follows aFokker-Planck equation [11, 24] that corresponds to the Langevin eqn. 6): ∂p ( v , t ) ∂t = ∇ v (cid:20) γ v (cid:18) − q γ d ( v )[ c + d ( v )] v (cid:19) v p ( v , t ) + S ∇ v p ( v , t ) (cid:21) (15)This Fokker-Planck equation is based on the assumption that the internal energy depot hasreached a quasistationary equilibrium fast enough. If we assume the general hypothesis d ( v ) = d v ξ and ˙ p = 0 , we find for the stationary velocity distribution p ( v ) = C (cid:18) d v ξ c (cid:19) q ξmS exp (cid:16) − γ S v (cid:17) , (16)where the normalization condition C is defined through the condition (cid:82)(cid:82) p ( v ) d v . For ξ = 2 the SET model results and the stationary solution, eqn. (16), can be written in first-orderexpansion as: p ( v ) ∼ exp (cid:16) − γ S [1 − Q ] v + · · · (cid:17) (17)As has been discussed in detail [11], for Q < we find a unimodal Maxwell-like velocity distri-bution, whereas for Q > a crater-like velocity distribution results in two-dimensional systems.The amount of data measured does not allow us to reasonably reconstruct the two-dimensionalvelocity distribution by means of density approximations. Therefore, in the following, we restrictourselves to the distribution of the speed which contains sufficient information to test our hy-potheses given. To find the speed distribution p a ( v ) in two dimensions we integrate over a disk B ( v (cid:48) ) : P ( | v | < v (cid:48) ) = (cid:90) (cid:90) B ( v (cid:48) ) p ( v ) dv x dv y = 2 π (cid:90) v (cid:48) p ( v ) dv = (cid:90) v (cid:48) p a ( v ) dv (18)10/20 ictor Garcia, Mirko Birbaumer, Frank Schweitzer:Testing an agent-based model of bacterial cell motility European Physical Journal B vol. 82, no. 3-4 (2011), pp. 235-244 and find p a ( v ) = C a πv (cid:18) d v ξ c (cid:19) q ξmS exp (cid:16) − γ S v (cid:17) (19)When comparing this equation with eqn. 16, one should note the additional prefactor v . So, forthe maximum of p a ( v ) , instead of the compact expression (12) resulting from p ( v ) , we find arather intricate expression which is not reprinted here. Instead, the stationary speed distributionis plotted in Fig. 3 for the SET model and different values of Q . Different from the noticablechange between unimodal and crater-like shape of the stationary velocity distribution p ( v ) , weobserve only a shift in the maximum of p a ( v ) when Q changes from subcritical to supercriticalvalues. At second view, one notices that the ascent of p a ( v ) changes from a linear increase for Q < to an nonlinear increase v n for Q > , where n is an integer. p ( v ) v Figure 3: p a ( v ) , eqn. (19), for ξ = 2 and different values of Q : (blue) Q = 0 . , (green) Q =1 , (red) Q = 3 . . The speed v is given in abitrary units.Specifically, we do not need to explicitely derive the maximum of the speed distribution (whichis also known as the most probable value and different from, e.g., the expecation value or theaverage value), because we want to compare the theoretical and the experimental distributions p a ( v ) rather than their extreme values. These distributions, in addition to their mean value,are further characterized by their width, given by the variance σ = S/γ . Hence, we need todetermine the strength S of the stochastic force in relation to the friction coefficient γ . In statistical physics, the strength of the stochastic force is related to the thermal velocityof microscopic particles, e.g. molecules or Brownian particles, via the fluctuation-dissipationtheorem, which yields for ideal gases
S/γ = k B T /m , where k B is the Boltzmann constant. If onewishes to apply the same relation also to bacteria like Escherichia coli or Salmonella typhimurium ictor Garcia, Mirko Birbaumer, Frank Schweitzer:Testing an agent-based model of bacterial cell motility
European Physical Journal B vol. 82, no. 3-4 (2011), pp. 235-244 at about T = 300 , with a bacterial mass of m ≈ − , one arrives at σ = 2000 µ m/s, whichabout two orders of magnitude larger than the average velocity for such bacteria. Given the rathercomplex nature of bacterial motion described above, this discrepancy is not really surprising.Therefore, in [21] a different method to determine S/γ was proposed, which is based on the speedautocorrelation function g v ( τ ) = (cid:104) v ( t ) v ( t ) (cid:105)(cid:104) v ( t ) v ( t ) (cid:105) ; τ = | t − t | (20)The calculation of g v ( τ ) needs a formal solution of the Langevin equation for the speed v ( t ) .This was provided in [21] for a different model applied to the migration of human granulocytes.It postulates a stationary speed v s and assumes, different from our ansatz of eqn. (6), ˙ v i = − γv i (cid:104) − v s v (cid:105) + √ S ξ i ( t ) (21)It was noted that in this equation an additive Gaussian random noise is physically and mathe-matically problematic as it allows in principle negative speed values. As possible solution, onecan consider appriopriate boundary conditions or a different definition of the velocity-dependentfriction term as suggested recently [19]In the following, we still use eqn. 21, but point out that this is only an approximation in the limitof small noise with respect to the finite stationary speed. The stationary solution for the speedautocorrelation function is reached when the times t and t are larger than the characteristictime γ − , which results into (cid:104) v ( t ) v ( t ) (cid:105) = v s + Sγ exp {− γ | t − t |} (22)In the limit τ → ∞ , the speed autocorrelation function g v ( τ ) becomes a constant, g v = v s / [ v s + S/γ ] , which can be measured experimentally, to obtain: σ = Sγ = v s (cid:18) − g v g v (cid:19) (23)Schienbein and Gruler [21] found for human granulocytes, which are much larger than the bac-terial cells investigated here, g v = 0 . and from the measured speed distribution the maximumvalue v s = 17 µ m/min.As mentioned, our Langevin equation uses a different ansatz for the velocity-dependent frictionfunction, which does not allow us to obtain a simple closed form for g v . However, we can stillapply the results of [21] arguing that the speed of bacteria in the stationary limit reaches valuesaround v s . Hence, in the vicinity of v s , we linearize the dependence on v , which is /v , to /v and use for v s the expression given by eqn. (12) for the SET model, or by eqn. (13) for < ξ < .This approximation allows us to use eqn. (23) to determine S/γ provided we can obtain v s and g v from our own experiments. 12/20 ictor Garcia, Mirko Birbaumer, Frank Schweitzer:Testing an agent-based model of bacterial cell motility European Physical Journal B vol. 82, no. 3-4 (2011), pp. 235-244
With these considerations, we have all ingredients together to compare our experimental datawith the theoretical predictions. In detail. we proceed as follows:1. For the three media described above (Sect. 3.1), we calculate the absolute velocities v kaj ofeach cell k tracked during the time step j . For each medium, two measurements were taken:(a) initially, time t , (b) after sufficiently long time, about one hour, time t end (see alsoFig. 2 for the two-dimensional velocity distributions). From these samples, which containabout 1.500 data points each, we calculate the densities p exp a ( v ) using the Sheather-Jonesmethod of selecting a smoothing parameter for density estimation [26] in R. The resultsare shown by the blue curves in Fig. 4 for t end .2. From the 6 different density plots, we calculate the maximum v s of the experimental speeddistribution p exp a ( v ) . The results are given in Appendix C: Table 1. For t end , we also calculatethe speed autocorrelation function g v , which is shown in Table 1 as well.3. Eventually, we apply the maximum-likelihood estimation (MLE) to find out, which of thestill undertermined parameters fit best the experimental densities p exp a ( v ) . The parameterdetails are presented in Appendix C: Table 2, while the resulting density plots for thehypotheses ξ = 2 and < ξ < are shown by the red curves in Fig. 4, which are to becompared with the experimental findings (blue curves).In the following, we further discuss these findings. In Table 1, one notes slight differences in v s ( t ) for the different media. This indicates that at time t , right after being put into the medium, thebacterial cells already started to adjust to the nutrient concentration in the media. The higherthe concentration, the higher v s . This can be also confirmed at t end , after about 80 min. Thedifferences between medium 1 and 2 (middle and high concentration) is rather small both forat t and t end , indicating that there seems to be a saturation in converting internal into kineticenergy. This saturation can be caused by intracellular processes (e.g. number of receptor proteinsavailable), but is not further discussed here. Noticable, in medium 0 (no nutrients), v s drops down considerably compared to the initial value, which also indicates that the bacterial cells respondto the available energy by adjusting their speed.We further find that the speed autocorrelation function g v returns comparable values for allthree media ( . − . ) which are much larger than for granulocytes, because we have muchsmaller and more motile cells. The estimated standard deviations σ = (cid:112) S/γ range between . µ m/s (medium 0) and . µ m/s (medium 2) and are quite similar for all media, because thetemperature and the viscosity of the media are kept as constant as possible.13/20 ictor Garcia, Mirko Birbaumer, Frank Schweitzer:Testing an agent-based model of bacterial cell motility European Physical Journal B vol. 82, no. 3-4 (2011), pp. 235-244 . . . . . v [ � m/s] den s i t y . . . . . v [ � m/s] den s i t y . . . . . v [ � m/s] den s i t y . . . . . v [ � m/s] den s i t y . . . . . v [ � m/s] den s i t y . . . . . v [ � m/s] den s i t y . . . . . v [ � m/s] den s i t y . . . . . v [ � m/s] den s i t y . . . . . v [ � m/s] den s i t y . . . . . v [ � m/s] den s i t y . . . . . v [ � m/s] den s i t y . . . . . v [ � m/s] den s i t y Figure 4: (blue curves) Estimated densities p exp a ( v ) obtained from experimental measurements ofthe speed (absolute velocity) v [ µ m/s] at time t end . (red curves) Calculated densities p a ( v ) , eqn.(19) using the parameters of the MLE, Table 2. (top row) SET model with ξ = 2 , (bottom row) < ξ < . (left figures) medium 0, (middle figures) medium 1, (right figures) medium 2.The actual width of the speed distribution, however, does not just depend on S/γ but also onthe parameters of the internal energy depot and is therefore larger than σ . In order to calculatethe parameter values that maximize the likelihood, we restrict ourselves to reduced parameters.Keeping in mind that σ = S/γ is given by eqn. (23) and the control parameter is defined as Q = ( q d ) / ( γmc ) , we can rewrite the leading terms in eqn. (19) as: (cid:18) d v ξ c (cid:19) q ξmS exp (cid:16) − γ S v (cid:17) = (cid:18) d c v ξ (cid:19) Q ξσ cd exp (cid:18) − v σ (cid:19) (24)which reduces the number of parameters to be determined to ( d /c ) and Q , while σ is givenby the experiments. ξ on the other hand is either set to 2, in case of the SET model, or used asa free parameter. Given the observations v , v , ..., v n the MLE then determines for which values Θ of these parameters the likelihood function L ( v , v , ..., v n , Θ) is maximised, i.e. what are themost likely model parameters that fit the experimental data best, conditional on the model used.14/20 ictor Garcia, Mirko Birbaumer, Frank Schweitzer:Testing an agent-based model of bacterial cell motility European Physical Journal B vol. 82, no. 3-4 (2011), pp. 235-244
In order to fully appreciate the MLE, we have to notice that no further “logical” assumption aremade, i.e., each for experimental distribution the MLE returns that set of parameter values thatfits this particular distribution best. Precisely, we receive most likely a different set of values foreach of the given distributions. To put it the other way round, from all the observed distributionswe can obtain a range of parameter values that is compatible with the experimental findings,rather than a precise value that is met by all observations.With this in mind, we can interpret Table 2 as follows: For both hypotheses, the SET modelwith ξ = 2 and the general model with < ξ < , we find from the observations a “reasonable”(but different) set of parameters that supports these hypotheses. This is also confirmed by thefits shows in Figure 4. I.e., we cannot reject one of these hypotheses as they both match theexperimental findings. However, we did not observe a subciritical take-up of energy for the SETmodel, as we did not find values Q < for the given observations. The latter conclusion needsa further explanation: In medium 0, we made no nutrients available, so q and Q should bothbe zero. However, the bacteria were grown in a medium that contained nutrients and thus, attime t , started their motion with a filled energy depot e st , eqn. (3), that changes over time onlyrather slowly as v s is adjusted. Hence, the value Q for medium 0 at time t end reflects the valueof the internal energy depot at time t end . As the observations in Figures 2, 4 and Table 1 show,even after a long time the bacteria still have energy enough to move with a non-trivial speed v s despite the fact that no nutrients are provided. But there is a clear trend toward slowing downas the values indicate.As second interesting observation regards the decrease of the d /c values with increasing nutrientconcentration (comparing medium 1 and 2), for both the SET and the ξ model. The ratio M = d v s /c reflects the proportion of energy bacteria spend on the two different processes,active motion and metabolism. If more energy becomes available (from medium 1 to 2) , thisdoes not necessarily lead to a speed-up – the speed was kept almost constant, but the additionalenergy is likely spent on metabolism (and growth). Hence M decreases from 1.50 to 0.62 forthe SET model, and from 0.74 to 0.23 for the ξ model, while the take-up of energy q /γ has increased from 2500 to 4140 for the SET model, and from 5230 to 6800 for the ξ model. So, inconclusion, our model suggests that bacteria indeed take up more energy from the environmentif more nutrients are provided, but the ratio spent on active motion is decreased, while the ratiospent on metabolism is increased. The aim of this paper is twofold: (i) we investigate to what extent a theoretical model of activemotion, namely that of “active Brownian particles”, is compatible with experimental findings15/20 ictor Garcia, Mirko Birbaumer, Frank Schweitzer:Testing an agent-based model of bacterial cell motility
European Physical Journal B vol. 82, no. 3-4 (2011), pp. 235-244 from bacterial cell motility, (ii) we test the impact of available energy in the environment, variedby the nutrient concentration in the medium, on the speed distribution of bacterial cells.For the discussion one has to keep in mind that our results have been obtained for a particularstrain of
Salmonella typhimurium (see Sect. 3.1. and Appendix A) and cannot easily generalizedto other bacterial cells. This is because of the rather complex cellular motion of bacteria which,in many cases, is comprised of tumbling and nontumbling phases (see Sect. 3.1). Hence, for themotion of other types of bacteria we refer to the extensive literature [2, 3, 12–16, 18, 27].Comparing the experiments with moving
S. typhimurium and the theoretical model, we demon-strated that the measured speed distribution can indeed be matched by the analytical prediction.This holds for both tested hypothesis, (a) the SET model with ξ = 2 and (b) the ξ model, whichallows an adjustment of the exponent, < ξ < . We further confirmed that, under the giveexperimental conditions, bacteria move in the supercritical regime, Q > . However, we werenot able to observe as subcritical behavior with Q < as predicted for the SET model [8, 10].This can be probably explained by the fact that even at the end of our experiments bacterial cellshad still enough internal energy available from their growing period, to move with a nontrivialspeed. But we could notice a considerable slowing down of 25 with no additional nutrients.Regarding the impact of available nutrients, we found that the speed of bacteria did not increasein proportion to it. Instead, we observed a more or less constant speed, even if the nutrientconcentration was doubled. From calculating the model parameters for both cases, we conjecturethat indeed more energy was taken up by the cells, but this was used for other internal processessuch as metabolism and growth.In conclusion, the experiments carried out with bacterial cells moving in media of three differentnutrient concentrations could confirm the theoretical predictions and thus indirectly also supportthe assumptions made for our model of active Brownian particles. However, particular detailsof the choice of parameters cannot be fully resolved by our experiments – which is not verysurprising. This regards for example the “correct” value of the exponent ξ for the speed, v ξ .Our findings support values between 1.67 and 2.0 if a take-up of energy from the medium waspossible. The differences between the two assumptions are not so much in the values of ξ but in thetheoretical consequences. In the case of the SET model, there is a clear bifurcation which allowsto distinguish between subcritical (Brownian motion like) behavior, and supercritical behaviorcharacterized by a directed motion. It would still be interesting to find microorganisms for whichthese regimes could be determined. Our experiments had to restrict to the conditions explainedabove and therefore do not support this distinction.16/20 ictor Garcia, Mirko Birbaumer, Frank Schweitzer:Testing an agent-based model of bacterial cell motility European Physical Journal B vol. 82, no. 3-4 (2011), pp. 235-244
Appendix A: Details of Bacterial Probes
The chemotaxis-deficient Salmonella strain M935 (SL1344; cheY::Tn10, [29, 31]) was grown undermild aeration for 12 h/37 ◦ C in LB medium containing 0.3 M NaCL. The culture was diluted 1:20into fresh medium and grown for another 4h / 37 ◦ C. Bacteria were pelleted by centrifugation(8500 rpm, 5 min, 4 ◦ C) and resuspended in phosphate-buffered saline (PBS). The sample wascentrifuged again and the bacteria were resuspended either in PBS (probe A) or LB medium(probe B). From these two probes, samples for live imaging of the bacteria were prepared asfollows:1. probe A was diluted 1:50 in PBS containing 5 (BSA) and transferred to a glass-bottomdish for imaging2. a glass-bottom dish was rinsed with PBS/5 of bacteria to the glass surface. PBS/BSA wasremoved and a 1:50 dilution of probe B in LB was added to the dish for imaging3. probe B was diluted 1:50 in a 1:1 mixture of PBS/5 (final BSA concentration: 2.5All solutions for dilution of probes A and B were prewarmed to 37 ◦ C. Appendix B: Details of Data Evaluation
For time lapse microscopy the different samples were mounted onto a heated specimen holder(37 ◦ C) on a Zeiss Asiovert 200m inverted microscope. Time series of phase contrast images wererecorded using a Plan Neofluoar 20x (NA 0.5) objective at a rate of ca. 20 images per second.Motile bacteria were tracked using Particle tracker software [20] as plugin on the pure Java imageprocessing program ImageJ [1]. Only trajectories appearing over more than 30 time frames wereconsidered. Selected trajectories were manually verified for correct tracking (Wrong trackingoccurred in cases of crossings between identified bacteria and was removed from the trajectories).These criteria for track evaluation were equally applied on all detected trajectories. About 3 hoursof eye selection are necessary to obtain about fifteen to twenty tracks. The evaluation of the datawas carried out by an R-script written by the authors.17/20 ictor Garcia, Mirko Birbaumer, Frank Schweitzer:Testing an agent-based model of bacterial cell motility
European Physical Journal B vol. 82, no. 3-4 (2011), pp. 235-244
Appendix C: Details of Parameters Obtained
Medium v s ( t ) [ µ m/s] v s ( t end ) [ µ m/s] g v ( t end ) v s (10 − m/s) of the experimental speed distribution p exp a ( v ) taken initially( t ) and after long time ( t end ) for 3 different media (see Sect. 3.1). g v ( t end ) gives the value ofthe speed autocorrelation function measured experimentally at time t end .Medium d /c [(s / ( µ m ) ] Q (cid:63) d /c (cid:63) [(s / ( µ m ) ] (cid:63) Q (cid:63) ξ . · − . · − . · − . · − . · − . · − p a ( v ) , eqn. (19), as estimated by MLE for the SET model, ξ = 2 (no-starred values) and for the general model, < ξ < (starred values) at time t end . Acknowledgment
The authors are deeply indebted to Wolf-Dietrich Hardt for providing access to, and use of,his laboratory at the Institute of Microbiology of ETH Zurich, where V.G. could carry outthe experiments. We further gratefully acknowledge scientific discussions with Howard C. Berg,Wolf-Dietrich Hardt and Markus C. Schlumberger.
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