Traveling concentration pulses of bacteria in a generalized Keller-Segel model

We formulate the Smoluchowski equation for a run-and-tumble particle. It includes the mean tumble rate in a chemical field, for which we derive a Markovian response theory. Using a multipole expansion and a reaction-diffusion equation for the chemoattractant field, we derive a polarization extended model, which also includes the recently discovered angle bias. In the adiabatic limit we recover generalized Keller-Segel equations with diffusion and chemotactic coefficients that depend on the microscopic swimming parameters. By requiring the tumble rate to be positive, our model possesses an upper bound of the chemotactic drift velocity, which is no longer singular as in the original Keller-Segel equations. Using the Keller-Segel model, we present an extensive study of traveling bacterial concentration pulses demonstrating how speed, width, and height of the pulse depend on the microscopic parameters. Most importantly, we discover a maximum number of bacteria that the pulse can sustain - the maximum carrying capacity. Finally, we obtain a remarkably good match to experimental results on the traveling bacterial pulse. It does not require a second, signaling chemical field nor a singular chemotactic drift velocity.

The motility mechanism of the run-and-tumble bacterium E.coli has been extensively studied [33][34][35][36][37][38][39][40]. Bacteria perform chemotaxis, the ability to sense and respond to chemical gradients in order to find better living conditions. They realize the chemotactic drift motion along a chemical gradient by elongated run phases if the environment becomes more favorable while runs are shortened in the opposite case [39][40][41]. The internal chemotaxis machinery of the bacterium senses and compares the nutrient concentration in time, which is rationalized in a linear response theory for the tumble rate [34,[42][43][44]. More recently, a second chemotaxis strategy, called angle bias, has been reported [36,39,40]. The mean reorientation angle during tumbling is reduced if the bacterium swims along a chemical gradient and increased in the opposite case. This also generates a net drift motion in the favorable direction. Finally, using logarithmic sensing, E.coli is able to perform chemotaxis in concentration fields varying by many orders of magnitude [45][46][47]. Such an ability is commonly described by Weber's law in different physical areas [48,49].
A very interesting collective phenomenon in a bacterial population is a concentration pulse that travels along a capillary tube with almost no dispersion nearly like a soliton [6,36], most recently also observed in a population with non-genetic variations [50]. The pulse is initiated in an initially uniform environment of a chemoattractant. A bacterial population concentrated in space eats the nutrient and thereby creates a chemical gradient along which it drifts towards untouched regions. Moreover, Adler in his experiments also observed that not all bacteria travel with the pulse but are left behind at the initial location [6], which indicates a finite carrying capacity of the traveling pulse. Further chemoattractants present in Adler's experiments then initiated further pulses emerging from the bacteria left behind.
A very prominent theoretical approach to describe the traveling bacterial pulse is the celebrated Keller-Segel model [51], originally introduced for the aggregation of slime molds [52]. It couples a diffusion-drift equation for the bacterial density to a reaction equation for the nutrient. However, the Keller-Segel model has two drawbacks. First, a soliton solution (classified as unstable [53]) only occurs if the chemotactic drift velocity diverges for vanishing nutrient concentration. Second, nutrient diffusion was neglected. Later, based on analytic arguments, [53] demonstrated that traveling pulses also exist in the presence of nutrient diffusion. More importantly, Brenner et al. showed that the singularity in the chemotactic drift velocity is not necessary if one introduces a second chemoattractant, which the bacteria excrete themselves [54]. Reference [36] followed this approach to formulate a kinetic model (inspired by [55]), which describes traveling pulses in their experiments. Finally, a modification of this kinetic model has recently been used to investigate pulse propagation in the presence of two E.coli populations [56]. The Keller-Segel equations find wide applications in modeling bacterial chemotaxis as reviewed in [57]. They have also been derived for active Brownian particles, which propel by selfdiffusiophoresis, and for quorum-sensing run-and-tumble particles [58].
Multipole expansions have frequently been applied to microswimmers in order to approximate the Smoluchowski equation for the full distribution function in the microswimmer's position and orientation [13,27,[58][59][60]. Besides for density such expansions also provide an additional dynamic equation for the polarization, which unraveled interesting collective behavior of Janus particles [29] and which also allowed to investigate steady-state distributions of run-and tumble particles [61]. Our derivation is inspired by the approach of the latter reference but extends it by introducing the concentration field of a chemoattractant.
In this article we formulate a Markovian response theory for the tumble rate. It includes logarithmic sensing for which we introduce an upper threshold. We use the tumble rate in the Smoluchowski equation and derive a polarization extended model (PE) to treat chemotaxis of non-interacting E.coli bacteria. The PE model contains equations for the bacterial density, the bacterial polarization, and the chemical concentration field. In a second step, we also include the recently discovered angle bias. In the adiabatic limit the PE model simplifies to a generalized Keller-Segel model (KS) where the coefficients for diffusion and chemotactic drift velocity depend on the microscopic swimming parameters of the bacterium. In particular, the chemotactic coefficient is not singular in the chemical concentration. We numerically solve both models for an initially uniform chemoattractant and a bacterial population concentrated in space using parameters that are realistic for the E.coli bacterium. The traveling bacterial pulse generated by both the PE and KS model are identical thus the KS model is a valid approximation of the full kinetic formalism. We present a detailed parameter study of the traveling pulse and identify a maximum carrying capacity as a consequence of the bounded chemotactic drift velocity, which has not been mentioned so far. It means that the pulse can only sustain a finite number of bacteria. Finally, we tune our parameters to match the experimental realization of the bacterial pulse in [36]. Hence, our generalized Keller-Segel model is able to describe traveling bacterial pulses without the need neither for a singular chemotactic drift velocity nor for a second chemoattractant.
The remainder of the article is organized as follows. We present the Markovian response theory for the tumble rate in section 2.1. We use it to derive the polarization extended model (PE) and the generalized Keller-Segel model (KS) in sections 2.2 and 2.3. We also incorporate the angle bias and formulate a non-dimensional version of the KS model in sections 2.4 and 2.5. Details of the numerical solution scheme are given in sections 3 and 4 presents our detailed numerical study. We close with conclusions and an outlook in section 5.

Model
2.1. Markovian response theory for tumble rate Bacteria tumble less when moving up a chemical gradient. Based on the established linear-response theory, we formulate an equation for the tumble rate r e , l ( ) as a function of the swimming direction e. Below, we will relate it to the angle θ relative to the local gradient c  of a chemoattractant with density c.
We start with the linear-response theory [43]. It gives the tumble rate λ(t) as a function of time and depends on the bacterium's past trajectory t where we have introduced the response kernel R(t) and equ l is the tumble rate without any chemical gradient. Note that equation (1) describes a non-Markovian process. In appendix A we convert it to a Markovian process with λ depending on location r and swimming direction e by averaging over all possible bacterial trajectories in order to obtain the mean history for a given swimming direction e. Thus, the tumble rate we give below is an averaged quantity suitable to be used in a continuum theory. To derive it, we split the integral on the rhs of equation (1) into contributions from individual runs, during which the according swimming directions e i are assumed to be constant. Averaging over the history of all possible paths, we can show that each of these contributions gives a term proportional to the scalar product e c c  · . The factor c 1 results from the fact that the response kernel is proportional to the inverse background concentration, R ∝ 1/c , which was indeed measured in experiments for the chemoattractant alpha-methyl-aspartate in [38]. This chemoattractant is widely used, e.g. in the seminal experiments of [33,34,38,62] and also in experiments of [39,40]. For the detailed derivation we refer to appendix A and present the final result Here, v 0 is the swimming velocity of the bacterium and χ 0 is a unitless measure of the chemotactic strength. It depends on integrals over the response function R and moments of the tumble angle distribution P(β). Note that we obtain here c ln l  µ ( )commonly known as logarithmic sensing and Weber's law. It was measured, for example, in [47]. The linear dependence of the tumble rate λ on c r e  ( ) · was already introduced by Schnitzer as the leading order for the angular variation of λ [61]. It follows directly from equation (1) by choosing R t t -¢ ( ) proportional to the time derivative of the δ function as demonstrated by Locsei in [63]. This article also calculates the chemotactic drift speed (see section 2.3) by performing an average over all possible bacterial trajectories similar to our approach. Indeed, in appendix A we show that we arrive at the same chemotactic drift speed for the specific response function chosen in [63].
When we define the orientation angle relative to the negative chemical gradient, e cos In figure 1 adapted from [39], the red points show experimental data for the mean tumble rate r, l q ( ). It was obtained by averaging over a population of around 1000 individual bacteria in a linear gradient. The appropriate cosine fit (red line) confirms our theoretically derived result of equation (2) and was the motivation to derive it from the linear-response theory.
A biologically relevant tumble rate should have both a sensing threshold [64] as well as a saturation of the response to the chemoattractant [65], which is not present in equation (2). First, it is known that the bacterium needs a small threshold concentration c t to perform chemotaxis since it senses the chemical field by chemoreceptors [64]. Second, equation (2) produces a negative tumble rate for a sufficiently large gradient of c log , which is even singular at c=0. In our derivation the singularity arises from the relation R ∝ 1/c for the response function mentioned earlier. While R ∝ 1/c was measured for a wide range of background concentrations [38], clearly the second term on the right-hand side of equation (2) has to saturate to a value smaller than equ l to keep the tumble rate positive. There a several approaches to remedy these shortcomings. The threshold concentration was implemented, e.g. in [51,66,67] by shifting the singularity in equation (2) from coli tumbling from its previous direction e¢ to e with tumble angle β. Right: mean tumble rate λ(θ) (red) as a function of the swimming angle θ measured against the negative chemical gradient. The tumble rate was averaged over a population of around 1.000 E.coli in a linear gradient of α-methyl-aspartate. The fit with a cosine function, y x a a x cos 1 2 = + ( ) ( ) (red line) according to equation (2), yields regression parameters a 1 =0.278 and a 2 =0.115. The blue line represents the mean tumble angle cos b q á ñ( ) as a function of θ of the same population. It indicates an angle bias for tumbling in a chemical gradient. Adapted from [39]. (saturation in the chemotactic response) or it tends to equ l for c<c t (sensing threshold). The chemotactic length δ quantifies the strength of the logarithmic derivative of c r ( ). Appendix B presents a parameter study for the tumble rate of equation (4). Finally, we note that in this article the swimming speed v 0 is constant. If it varies, one always has to make sure that v 0 0 e q u c d l < to keep the tumble rate positive.

Smoluchowski equation
We first construct dynamic equations for the evolution of the one-particle distribution function t r e , , y ( )of position r and orientation e at time t and the concentration of chemoattractant, c t r, ( ). We begin with a generalized Smoluchowski equation for ψ [55,61,[70][71][72], which contains the usual contributions from translational and rotational currents, J trans and J rot , but also contributions from tumble events represented by  y { }and from cell division and death, t r e e , , d Here, , where D rot is the rotational diffusion coefficient. According to [33] we take a Poisson distribution for the run times and write the term for the tumble events as We introduced the tumble rate r e , l ( ) and P e e e , -¢ ¢ ( ) is the probability of a bacterium to reorient from orientation e¢ to e. In equation (7) the first term on the rhs represents events, which cause bacteria with orientation e to tumble into any orientation, and the second term represents all events, which cause bacteria with other orientations to tumble into orientation e. The

Multipole expansion
In order to proceed with equations (8) and (9), we assume that the probability distribution for a specific tumble event does not depend on the initial orientation of the bacterium, P P e e e e e , -¢ ¢ = -¢ ( ) ( ) . Therefore, we can write for the zeroth and first moment where cos b á ñ is the mean of the cosine of the reorientation angle β [61]. We take the tumble rate r e , l ( ) to vary as in equation (4). Now, we integrate equation (8) We have also used e d 0 2  ò y = and the normalization condition equation (10) to show that tumbling does not contribute to equation (12).
In order to derive a dynamic equation for the polarization, we compute e ò equation ( ) and equation (11). To truncate the multipole expansion, we neglect the quadrupole moment Q, which strictly means that the orientational distribution hardly deviates from the isotropic distribution and the deviation can be well described by the polarization. We also define the relaxation rate with which polar order relaxes or decorrelates in time. Thus, we ultimately obtain applies, the polarization from equation (15) becomes Substituting equation (17) into (12), we obtain the generalized Keller-Segel model is the typical translational diffusion coefficient of an active particle, the orientation of which decorrelates on the characteristic time ω −1 . The approximate expression for D eff is valid at high Peclet numbers.
From the third term on the rhs of equation ( This shows that an appropriately bounded tumble rate is closely linked to a physically bounded chemotactic drift speed. Furthermore, in appendix C we will show that this upper bound implies an upper bound for the speed of the traveling bacterial pulse and thereby for the maximum number of bacteria it can carry. We will address this point in section 4.

Bias of tumble angles
Up to this point we have ignored the effect of a bias in the tumble angle towards smaller mean values when swimming up the chemical gradient. This has recently been observed in experiments [36,39]. We now introduce it by allowing the probability distribution for a specific tumble event to explicitly depend on the initial orientation of the bacterium, e¢. Hence equations (10) and (11) Equation (22) states that the distribution is always normalized irrespective of the initial orientation of the bacterium. In equation (23) the value of the mean cosine of the tumble angle e cos b á ñ ¢ ( ) now explicitly depends on the initial orientation e¢ before the tumble event.
The effect of an angle bias is to lower the mean tumble angle when the bacterium aligns with the chemoattractant gradient, hence the value of e cos b á ñ ¢ ( )will increase for stronger alignment. Expanding e cos b á ñ ¢ ( )up to the first Legendre polynomial, thus taking into account the leading polar correction, yields c c e es cos cos , 24 where σ is a positive and monotonically increasing function. It is bounded such that its maximum value 1 cos ñ being the mean cosine of the tumble angle, when the angle bias is not taken into account. Using equations (2), (22)-(24), we can retrace the steps of the multipole expansion (see appendix D for details) to obtain an extended form for equation (15) with equations (12) and (16) Here, Ä means dyadic product and 1 s s P P ss P 2 2 + Ä = + (ˆˆ)ˆˆ· , where · means scalar product. One immediately recognizes that the angle bias renormalizes the relaxation rate of the polarization and makes it anisotropic. Thus polarizations along and perpendicular to the chemical gradient relax with different rates. In the adiabatic limit 0 t P » ¶ ¶ and for large Pe we can again solve for the polarization by inverting the matrix in front of P in equation (25). Substituting the resulting equation into equation (12), we again obtain a generalized Keller-Segel equation and a chemotactic velocity v ch along the chemical gradient. It now also depends on the angle bias quantified by σ. We refrain from giving the lengthy expression.

Rescaling the Keller-Segel equations
In order to identify essential parameters especially in the generalized Keller-Segel equations (18) and (19), we introduce unitless quantities. First, we rescale time with the chemical consumption rate, t kt = , lengths by the distance l v dk = . This allows us to write the generalized Keller-Segel equations (18) and (19), where chemotactic response is bounded by the hyperbolic tangents, in rescaled form: , where we neglected the thermal contribution to D eff . We recognize that the rescaled Keller-Segel equations are described by a set of six relevant parameters:

Details of numerical solution scheme
In the following we study in detail traveling bacterial pulses in an initally uniform density field of a chemoattractant by numerically solving both the polarization extended model (PE) of equations (12), (15) and (16) and the generalized Keller-Segel model (KS) of equations (18) and (19). The experiments in [36] are performed in microchannels with cross section A. Neglecting any influence from the channel walls, we take the three-dimensional system (d=3) to be quasi infinitely extended perpendicular to the channel axis. Thus all densities just depend on the x coordinate along the channel and by symmetry only the x component of the polarization is non-zero.
To solve the respective system of equations, we apply a predictor-corrector method at any given time step to efficiently propagate the field variables in time [74]. As initial field values we choose an exponentially distributed bacterial density, ), a uniform density of the chemoattractant, c(x, t=0)=c 0 , and zero polarization P x (x, t=0)=0. During time integration no-flux boundary conditions are employed at x=0 and at sufficiently large x ∞ , such that c 0 and 0 28 while we assume that polarization stays zero, P 0 [58]. Note, this assumption neglects accumulation of bacteria at boundaries due to their persistent motion. However, since the accumulation only occurs within a thin region with thickness given by the persistence length v 0 /ω, it is not relevant for the pulse propagation in the bulk, which we study here. To arrive at the conditions of equation (28), we consider the governing equations of the KS and PE model, separately. The second condition for c in equation (28) means that the chemical flux at the boundaries vanishes. For the KS model, the bacterial flux is D v eff ch r r - + , which we identify from equations (18) and (20). The flux vanishes at the boundaries due to the first condition of equation (28) and since v 0 ch = for zero chemical gradient. For the PE model, equation (12) gives the bacterial flux v D P 0 r - , which also vanishes at the boundary due to the first condition of equation (28) and the vanishing polarization 5 . Finally, when integrating equation (16) the sink term can produce negative concentrations of the chemoattractant [75,76]. To avoid this, we set the concentration c to zero whenever it would become negative. This allows the bacteria to fully degrade the chemoattractant without producing negative values for c.
When we solve our equations with real parameters, we rely on [36] and take the channel length This ensures that at t=0 99% of the bacteria can be found within 200 m m at the channel end at x 0 =0. To be concrete, we also assume a channel cross section A 500 m 100 m m m =´to calculate the initial number of bacteria N 0 =ρ 0 x 0 A, which we use as a parameter instead of ρ 0 in the following. We divide the channel length into 5 10 4 grid points so that the grid length is 2 m m and use the time step 0.01 s for integrating our equations in time. All the relevant parameters are given in table 1. Finally, we will also numerically solve the rescaled Keller-Segel equations (26) and (27) in order to explore the dependence on some of the relevant dimensionless parameters.

Traveling concentration pulses of bacteria
We first introduce the traveling bacterial pulse for a reference system using two values for the initial number of bacteria, then perform a systematic parameter study, and finally demonstrate a perfect match with the experimentally observed bacterial pulse reported in [36]. , where n is the unit vector normal to the bounding surface. In our geometry this implies v 0 P x −D∂ρ/∂x=0 and −v 0 /d ρ+D∂P x /∂x=0, where in the second condition we neglected the contribution from the quadrupole moment Q.  (20) has an upper bound. In contrast, in the original KS model v ch diverges at small chemoattractant concentrations [51], which allows all bacteria to stay in the traveling pulse. Thus, we demonstrate when one allows dispersion a singular chemotactic drift velocity is no longer necessary for Table 1. List of parameters used for the reference pulse in figure 2 and for the match to the experimental system shown in figure 6.  traveling pulses to occur. Note, already [66] used the KS model with a bounded chemotactic drift and observed traveling pulses which was ignored in more recent works [36,54]. Instead, a second chemoattractant was proposed as explained in the introduction. One realizes that the profiles generated from the KS and PE model are identical except in the beginning. We start with P x =0 in the PE model whereas in the KS model a non-zero polarization is calculated from equation (17). It is due to the initial gradient in the bacterial density. Thus, we conclude that the adiabatic limit 0 P t » ¶ ¶ as a condition for deriving the generalized KS model is fulfilled. Indeed, the decorrelation or decay time 0.49 s 1 w = is much smaller than the characteristic time for the pulse propagation. For the latter, we approximately find 200 s for the pulse to travel its own width and the polarization always assumes its stationary value. Our finding also means that the kinetic models of [36,41,56], which work with the full one-particle distribution function t r e , , y ( ), are not necessary to describe pulse propagation. They can be reduced to the Keller-Segel equations.

Reference system
In figure 2(a) not all bacteria travel with the pulse but some remain at the initial location. This also occurs in the experiments of [6]. However, there the remaining bacteria perform chemotaxis in oxygen as a second chemoattractant and thereby initiate a secondary pulse. Since we do not incorporate another chemoattractant, the bacterial distribution at the initial location only broadens by diffusion. Finally, we mention previous numerical work on the KS model that also showed the bacteria left behind [66].
In their original work Keller and Segel derived an analytic expression for the speed of the bacterial soliton. It is a function of the number of bacteria in the soliton N p , the consumption rate k, the cross section A, and the initial chemoattractant concentration c 0 [51] v N k Ac .  In figure 2(b) we lower the number of bacteria N 0 by a factor of three. Now, all bacteria travel in the pulse and none are left behind. This suggests that the traveling pulse can only carry a certain number of bacteria and thus has a maximum carrying capacity. Indeed, in appendix C we show that the pulse speed v p is bounded from above by the chemotactic drift speed of equation (20), which itself cannot grow to infinity since we bound the chemotactic response through the tumble rate. Thus, the number N p of bacteria in the pulse, given in equation (29), cannot become arbitrarily large. We will investigate the maximum carrying capacity in more detail in the following parameter studies, where we use the two traveling pulses from figure 2 as a reference. For the traveling pulse in

Parameter studies 4.2.1. Influence of bounded chemotactic drift
To keep the tumble rate positive, we introduced the chemotactic length δ in equation (4), which prevents the chemotactic drift velocity in equation (20) to become arbitrarily large. Furthermore, for the chemotactic response the lower chemical threshold c t was introduced. In figure 3 we explore the influence of both parameters on the traveling bacterial pulse. The chemotactic length δ increases from left to right and the threshold concentration c t from top to bottom. The reference system of figure 2(a) is in the center.
A smaller chemotactic length δ means that the bacterium can sense larger chemical gradients and that the drift velocity v ch saturates at a larger value proportional to δ −1 . However, δ cannot be chosen arbitrarily small since then the tumble rate in equation (4) becomes negative. In this case our numerical solution scheme is unstable and the bacterial density becomes negative.
The length 384 m d m = is close to the minimal value. In particular for small c t nearly all bacteria travel in the pulse, none are left at the origin. As a result, the pulse travels the fastest. Increasing δ to 600 m m , it is clearly visible that some bacteria are left behind. Thus, the pulse contains less bacteria and, therefore, is slower. This is also in agreement with the smaller chemotactic drift velocity. Interestingly, for the smallest c t we observe a second propagating pulse strongly decreasing in height. Finally, if we increase δ by a factor of 10-6000 m m , the majority of bacteria stay close to the initial location while only a smaller number of them travels in the pulse (note the 10 times smaller range of the vertical axis). Thus, the pulse speed is small and the pulse has not yet separated from the non-propagating bacteria. In conclusion, increasing δ decreases the maximum carrying capacity significantly and makes the pulse slower.
Increasing the threshold concentration c t from nearly zero to c t =0.1c 0 has three effects. First, the dispersion of the pulse increases which slows down the pulse. This is most notable when comparing the second to the third row. Second, the shape of the pulse becomes more asymmetric as bacteria at the rear flank cannot follow the pulse. Third, the number of bacteria left behind at the initial location increases slightly. In the upper middle plot we recognize that the threshold c t is so low that the remaining bacteria can still travel by chemotaxis, although with a stronger dispersion as the first pulse. Finally, even for the vanishing threshold of the upper left plot a slight dispersion is visible. This again suggests that a true propagating soliton, for which the pulse shape does not vary in time, is not possible as long as the chemotactic drift velocity is bounded.

Quantitative study of the rescaled Keller-Segel equations
We now consider the rescaled Keller-Segel equations and study the propagating bacterial pulse in detail. Hence, we plot the rescaled pulse speed v p , the pulse full width at half maximum Δx h , and the pulse amplitude ρ max /ρ 0 as a function of the remaining parameters ρ 0 /c 0 , D c /D eff and 1 cos ) . Again, we neglect bacterial growth by setting 0 a = . Figure 4 shows all results and the relevant parameters are given in the figure caption. In figure 4(a) we see that the pulse speed depends linearly on ρ 0 /c 0 in agreement with the Keller-Segel prediction of equation (29) but then saturates at a constant value. The reference pulse from figure 2(b) (purple disc), where all bacteria travel with the pulse, is located in the linear regime, while the reference pulse from figure 2(a) (orange star), where some bacteria remain close to the origin, propagates in the saturated regime. Thus, in the first case adding more bacteria to the system increases the number of bacteria in the traveling pulse and speeds it up. In contrast, in the second case additional bacteria remain close to the initial location. Thus, the traveling concentration pulse has a maximum carrying capacity N * with respect to the amount of bacteria it can carry while further bacteria exceeding the carrying capacity are left behind. This is in agreement with appendix C, where we derive an upper bound for N * . The transition between both regimes occurs at the critical ratio (ρ 0 /c 0 ) * . To illustrate the transition, we discuss the following scenario. Lowering c 0 at constant ρ 0 speeds up the pulse in the linear regime since bacteria degrade the chemoattractant faster and in agreement with equation (29). However, once c 0 0 * r ( ) is reached, the pulse looses bacteria to keep the pulse velocity constant, again according to equation (29). Thus the carrying capacity of the pulse decreases when c 0 decreases below a treshold.
The ratio ρ 0 /c 0 also influences the pulse shape. In the linear regime of increasing ρ 0 /c 0 the pulse becomes narrower while its absolute height ρ max roughly increases with ρ 0 2 . When reaching the saturation regime, the pulse width stays constant as should ρ max . Thus for the relative height we find c ) . In figure 4(b) we show the pulse speed does not significantly depend on the ratio of diffusion constants, D c /D eff , for both study cases I (blue stars) and II (green circles). This is in contrast to [77] where the authors proposed a correction term to equation (29), which predicts a decrease of the pulse speed with increasing D c . However, when examining the bacterial pulse profile, we observe that for larger D c /D eff the pulse needs more time to form. It needs more time to consume all the chemoattractant at the origin due to the larger diffusive flux ) is varied. For values larger than 50 the numerical scheme becomes unstable similar to the instability in the chemotactic length δ already discussed in section 4.2.1. The pulse speed increases linearly in the chemotactic parameter for the study case I (blue stars) and also for the study case II (green circles) in the range 1 cos 20 To understand this finding, we looked in detail at the bacterial profiles. In study case II we find that with increasing 1 cos 0 c b -á ñ ( ) more and more bacteria from the vicinity of the initial location enter the pulse, which according to equation (29) then speeds up. Thus we conclude for the maximum carrying capacity of the pulse, N 1 cos . A similar observation in connection with the scenario of figure 4(b) gives N D eff * µ . Note that our results are in contrast to [78] which found v p 0 c µ . In study case II (green circles) we start with a smaller number of bacteria. Thus, at 1 cos 20 all bacteria have entered the pulse, which then travels with constant speed when 1 cos ) is further increased. Finally, with growing chemotactic parameter the width of the traveling pulse decreases in the study case I (blue stars). The curve of study case II (green circles) follows this trend until the pulse speed becomes constant and then steadily increases.
For the pulse amplitude of the two study cases the behavior of the curves are inverted compared to the pulse width. The curves are well fitted by y(x)=Ax B , where the constants A differ approximately by a factor of three. This is the factor by which the density ratios ρ 0 /c 0 of study cases I and II differ. The factor of three appears since we plot the reduced amplitude ρ max /ρ 0 . The exponents are nearly the same. The amplitude of the study case II (green circles) decreases for 1 cos 20 and thereby compensates the increasing pulse width as the number of bacteria in the pulse is constant.

Influence of growth rate α
Finally, we investigate the influence of the growth term in the Keller-Segel equation (18).  =´It is below the maximum carrying capacity of the pulse and was used in figure 2(b). Consequently, in the upper panel the pulse grows due to the non-zero growth rate and speeds up in time until the maximum carrying capacity is reached at around 2600 s. Then, the pulse propagates with constant shape like a perfect soliton. However, in our case the pulse leaves a trail of bacteria behind, which originates from the continuous bacterial growth.
In the middle and lower panel, the maximum carrying capacity is reached after 1000 s and 50 s, respectively. Interestingly, the pulse does no longer separate from the broad distribution of bacteria, which spreads from the initial location, but rather sits on top of the distribution's right flank. In the lower panel the pulse is fastest and its amplitude is highest. This comes from the fact that the broad distribution around the origin contains much more bacteria compared to the middle panel and, thus, more bacteria actively take part in the degradation of the chemoattractant. As a consequence, the pulse propagates faster.
Last, we observe that with increasing growth rate the pulse becomes more peaked. This is reminiscent to figures 4(a) and (c), where a faster pulse has a smaller pulse width. m =´=was chosen [64]. This was necessary to match the asymmetry and dispersion of the pulse. The full parameter set is given in table 1.

Discussion and conclusions
In the first part of this article we demonstrated how to derive the celebrated Keller-Segel equation using a generalized Smoluchowski equation for the full distribution function in position and orientation and a multipole expansion. An important ingredient is the bacterial tumble rate in a chemical field. We derived the known Markovian response theory for the mean tumble rate starting from the classical chemotaxis strategy based on temporal sensing in order to explain experimental results from [39]. Our expression for the tumble rate includes logarithmic sensing and we added a lower chemical threshold and an upper bound for the chemotactic response to keep the tumble rate positive. The multipole expansion provides a polarization extended model (PE) from which we derived the Keller-Segel equation in the adiabatic limit, where the bacterial polarization instantly follows variations in the density. We thereby obtain microscopic expressions for the diffusion coefficient and the chemotactic drift velocity. Due to the bounded chemotactic response the inherent and unrealistic singularity in the drift velocity is removed. Our detailed study of the traveling bacterial pulse shows that its characteristic time is much larger than the relaxation time of the bacterial polarization. Thus, PE and KS model provide identical results except for the initial fields and we conclude that the full Smoluchowski equation as used for example in [36,56] is not necessary. This drastically reduces the computational effort and allowed us to perform extensive numerical studies of the bacterial pulse propagation.
We find that due to the upper bound of the chemotactic velocity the traveling pulse can only carry a limited number of bacteria. To the best of our knowledge such a maximum carrying capacity has not yet been reported in the context of traveling bacterial pulses. In particular, it is not predicted by the analytic soliton solution of the original Keller-Segel model [51]. Another consequence of the upper bound of the chemotactic velocity is an effective dispersion of the pulse. While propagating, the pulse leaves a trail of bacteria behind and hence the pulse height decreases and the width expands. This is consistent with results from [66]. The loss of bacteria can be compensated by a non-zero growth rate and we have seen that soliton-like pulses, which propagate with constant shape, are possible.
Exploiting a rescaled version of our KS model, we quantify how pulse speed, pulse width, and pulse amplitude depend on the different unitless parameters. We mention some key results. First, throughout our parameter study we find that the analytic soliton solution of the original KS model still provides a correct estimate for the pulse speed as a function of the number of bacteria in the pulse. Second, we find the maximum carrying capacity to be proportional to the chemotactic strength 1 cos As a consequence these parameters affect the pulse speed as long as there are sufficient bacteria in the system so that the maximum carrying capacity is reached. Third, the diffusion coefficient of the chemoattractant does not influence the pulse speed as predicted in a theoretic model in [77]. The pulse only takes longer to eat up all the chemoattractant at the origin due to the larger diffusive flux of chemoattractant into depleted areas.
Finally, we show that our simulated pulse propagation is able to match quantitatively the traveling bacterial pulse in the exeriments of [36] in speed and shape. In contrast to the models used in [54] and [36], we do not need a second chemoattractant to generate a traveling concentration pulse as a solution of our generalized KS model.
We mention four directions into which our approach can be extended. First, so far we did not explore the full PE model, which will be relevant for dynamic processes with typical time scales of the order of 1/ω. In future works it would be interesting to explore the possibility of having the bacterial polarization as an independent field and its potential to induce complex dynamics. For example, the alignment or polarization of magnetotatic bacteria can be controlled by an external magnetic field [79,80], which offers the possibility to address polarization as an independent field variable, e.g. by a time-varying external stimulus. The dependence of cell characteristics on polarization could also evoke a feedback loop in highly nonlinear equations. For example, it was shown that the nutrient uptake of bacteria depends on cell shape [81], meaning that the consumption rate may depend on the polarization, which then influences chemotaxis [82]. For Janus colloids with effective phoretic repulsion this can generate interesting collective dynamics on times much smaller than the characteristic time scale of the bacterial pulse [29]. Finally, in complex geometries with characteristic lengths similar to the persistence length of the bacterium, we expect the polarization equation also to become important.
Second, to describe the multiple pulses that have been observed in experiments with several nutrients [6,83], one can extend our model by coupling the bacterial density to several nutrient fields. Bacteria that are left behind by the first pulse can then perform chemotaxis in a second nutrient field and thereby create a second pulse.
Third, in our generalized Smoluchowski equation the swimming speed is a constant as it is commonly done for the E coli . bacterium also during chemotaxis [33,35,40]. However, some bacteria are known to couple their swimming speed to the concentration of a chemical field [84][85][86], a strategy which is called chemokinesis. Reference [58] derived coupled equations for bacterial polarization and density from a Smoluchowski equation where the swimming velocity depends on the chemical concentration. It is certainly interesting to extend our theory in order to investigate the combined effect of chemokinesis and chemotaxis. A cell that performs both strategies is the sperm cell [44].
Finally, it would be interesting to extend our approach to chemoattractants to which E.coli is not perfectly adapted such as serine [33,40,87]. For this chemoattractant the mean tumble rate drops as the concentration of serine increases. Thus, when swimming up a chemical gradient, the chemotactic velocity increases [87]. In our generalized KS model, the effective diffusion coefficient of bacteria D eff , which directly depends on the tumble rate λ equ , now is enhanced in front of the pulse as runs are longer, while it is smaller in the back of the pulse where runs are shorter. This should affect the pulse propagation and indeed, experiments with serine showed that below a certain strength of the chemical gradient traveling pulses do not form [88].
where we applied the property of perfect adaption to set R t t c c t t r rd 0 and that c  is constant within the memory time τ m .
To proceed, we write the trajectory of a bacterium that swims with constant velocity v 0 along the direction given by unit vector t e( ) as t where the bacterium has been at location t r ¢ ( ) before reaching t r( ), thus t t  ¢ . In the following, we use it in the form Changes in the swimming direction due to rotational diffusion are much smaller than due to tumbling. Consequently, the trajectory of the bacterium becomes a sequence of straight runs and instantaneous tumble events. Denoting tumble events by index i, the tumble time t i , and e i the direction prior to tumble event i, we can write the orientation vector in equation (A4) at any time t t ¢¢ < with the help of a telescope sum: where the only remaining orientation vector is the current one, t e( ). In general, the probability distribution P n t t , -¢¢ ( )is a complex quantity as it depends on the tumble rates at previous times. Only for weak chemotaxis we can approximate the tumble rate by the constant value equ l and P becomes the Poisson distribution The weak-chemotaxis approximation was used, e.g. by Locsei in [63]. However, knowing the exact form of P n t t , -¢¢ ( )is not relevant for our further argumentation. Now, with equations (A8) and (A4) we can formulate the average location Setting c c r a = ( ) we then recover equation (2) from the main text. It is clear that χ 0 is a complicated quantity, which depends on the history of all trajectories, we average over. In the article we take a more pragmatic approach. We take χ 0 as a constant and concentrate on how the tumble rate depends on t c c e cos q   = -( ) · | | . This dependence was measured in experiments as we show in the plot of figure 1, right reproduced from [39].
Locsei in [63] calculated the chemotactic drift speed v ch in the weak-chemotaxis approximation using an explicit form for the reponse function. We can reproduce his formula for v ch , which demonstrates that our approach agrees with his work. For weak chemotaxis equation (A9) is valid. Following [89] we can also include rotational diffusion and obtain for the mean orientation  (14) in the main text. We will further use it. Now the exponential function of the last equation becomes the integrand in equation (A12). Integrating over t¢¢, we arrive at In figure E1(b) we investigate the effect of the initial bacterial configuration on the pulse propagation. Changing the initial distribution from an exponential to a step profile or a linear profile has no effect on the pulse speed, indeed the curves are very close. The same is true when we change the decay length to , which corresponds to 5% of the system length and which is larger than the pulse width, the curve starts to look different. Probably, when one simulates for much larger time, the pulse assumes the speed of the previous cases. As the pulse speed is directly linked to the number of bacteria in the pulse, we conclude that the maximum carrying capacity is stable for reasonable variations in the initial bacterial configuration.