Hydrodynamic tracer diffusion in suspensions of swimming bacteria

Kasyap, T. V.; Koch, Donald L.; Wu, Mingming

doi:10.1063/1.4891570

We present theoretical predictions, simulations, and experimental measurements of the diffusion of passive, Brownian tracer particles in the bulk of three-dimensional suspensions of swimming bacteria performing run-tumble random walks. In the theory, we derive an explicit expression for the “hydrodynamic” tracer diffusivity that results from the fluid disturbances created by a slender-body model of bacteria by ensemble averaging the mass conservation equation of the tracer over the space of tracer-bacterium interactions which are assumed to be binary. The theory assumes that the orientations of the bacterium before and after a tumble are uncorrelated and the fluid velocity disturbance created by the bacterium is small compared to its swimming speed. The dependence of the non-dimensional hydrodynamic diffusivity

$\widetilde{D_h}$

\tilde{D_{h}}

obtained by scaling the dimensional hydrodynamic diffusivity by nL³U_sL on the persistence in bacterial swimming and the Brownian diffusivity of the tracer are studied in detail through two nondimensional parameters—a Peclet number Pe = U_sL/D which is the ratio of the time scale of bacterial swimming to the tracer diffusion time scale and a non-dimensional persistence time τ^* = U_sτ/L obtained by scaling the dimensional bacterial persistence time by the time that a bacterium takes to swim over a distance equal to its length. Here, n, U_s, τ, and L are the concentration, swimming speed, tumbling time, and the overall length of the bacteria, respectively, and D is the Brownian diffusivity of the tracer.

$\widetilde{D_h}$

\tilde{D_{h}}

is found to be a monotonically increasing function of τ^* and a non-monotonic function of Pe with a Pe^1/2 scaling in the Pe ≪ 1 limit, an intermediate peak and a constant value in the Pe ≫ 1 limit for the typical case of wild-type bacteria with τ^* = O(1). In the simulation study we compute the bacterial contribution to the tracer diffusivity from explicit numerical simulations of binary tracer-bacterium interactions to examine the validity of the weak disturbance assumption made in the theory, and to investigate the effects of correlations in the pre- and post-tumble bacterium orientations and the excluded volume (steric) interactions between the bacterium and the tracer. It is found that the weak disturbance assumption does not have a statistically significant effect on

$\widetilde{D_h}$

\tilde{D_{h}}

and correlations among pre- and post-tumble bacterium orientations and bacterium-tracer excluded volume interactions are found to enhance the tracer diffusivity by modest but statistically significant factors. Finally, we measure the effective diffusion coefficient of 1.01 μm diameter colloidal tracer particles in the bulk of a suspension of wild-type E. Coli cells and compare the experimental measurements with the predictions made by the theory and simulations.

I. INTRODUCTION

Mass transport in bacterial suspensions is an important problem since the survival of bacteria depends crucially on the dispersal of nutrients and other chemical species such as signaling molecules. Recent experiments on suspensions of swimming bacteria^1–5 and simulations of hydrodynamically interacting self-propelled particles^6–8 have shown an enhanced diffusion of passive tracers such as colloidal beads and macromolecules with increasing bacterial concentration. While swimming, bacteria produce Stokesian hydrodynamic disturbances. A tracer particle experiencing the fluid motion created by randomly swimming bacteria would perform a diffusive random walk at long times.^1,6,8 Displacements of colloidal tracers could be influenced by the finite size of the colloidal beads. For a Brownian tracer particle, the effective diffusivity is then the sum of the intrinsic Brownian diffusivity, the “hydrodynamic” diffusivity contributed by bacterial fluid disturbances, and a contribution from possible excluded volume interactions with bacteria. Determination of the effective tracer diffusion coefficient is a problem of significant practical interest and a crucial issue here is that the hydrodynamic component of the effective diffusivity of the tracer in general could be a function of its Brownian diffusivity in addition to the parameters associated with bacteria such as their concentration, geometry, and motility. This is due to the fact that the velocity disturbance of a bacterium sampled by a tracer particle is a function of the position of the tracer relative to the bacterium and this relative position evolves through the Brownian motion of the tracer in addition to bacterial swimming and bacterial velocity disturbances. A similar phenomenon occurs in the well known Taylor-Aris dispersion^9,10 of a solute in a tube in which the Brownian motion of solute particles enables them to sample the spatially varying Poiseuille flow field in the tube resulting in longitudinal dispersion with a coefficient that depends upon the molecular (Brownian) diffusivity of the solute in addition to the tube geometry and flow velocity. In this paper, we calculate the hydrodynamic component of the effective tracer diffusivity in a suspension of swimming bacteria performing run-tumble random walks as a function of the tracer Brownian diffusivity, bacterial concentration, geometry, swimming speed, and persistence time through a theory based on binary interactions between bacteria and tracer particles. We validate the theoretical predictions and study the effect of correlations of bacterial orientations before and after tumbling through explicit simulations of pairwise bacterium-tracer interactions. The simulations also address the effect of steric interactions between bacteria and tracer particles. We perform complementary experiments and compare the experimentally measured tracer diffusivity with predictions from the theory and simulations.

Typical bacteria such as E. Coli are comprised of a spheroidal cell-body propelled by a helical bundle of flagellar filaments. The flagellar filaments are driven by motors within the cell-body and when all the motors rotate in the counter-clockwise direction viewed from behind the cell-body, individual filaments merge to form a helical bundle and propel the cell forward. This leads to a swimming “run,” which persists for around a second. The runs are punctuated by tumbles induced when one or more motors change their direction of rotation.¹¹ Each tumble leads to a new direction for the next swimming run. The duration of the run or the persistence time of bacteria is not a constant but follows a Poissonian distribution and the run-tumble motion is unbiased in the absence of any chemical gradients.¹² While modes of locomotion other than the run-tumble motility have been discovered among bacteria (see Ref. 13), run-tumble motion remains the most extensively characterized one and hence this paper is focused on suspensions of run-tumble bacteria. Owing to the balance of the propulsive force generated by the flagellar bundle and the drag force on the cell-body, a bacterium as a whole is force-free and as a result its hydrodynamic disturbance at large distances r from the bacterium compared with its size is that of a force dipole^6,14 which decays like 1/r².

The phenomenon of enhanced tracer diffusion due to swimming bacteria was first observed by Wu and Libchaber¹ in their experiment with micron-sized particles suspended in a thin liquid film containing swimming E. Coli cells. The motion of particles was found to be correlated at short times and diffusive at long times with the measured diffusivity increasing linearly with the bacteria concentration. While the colloidal particles in the experiment of Wu and Libchaber¹ were too large (4.5 μm and 10 μm) to have significant Brownian motion, macromolecules with relatively large Brownian diffusion also show enhanced diffusion in bacterial suspensions. For instance, Kim and Breuer² observed an effective tracer diffusion coefficient of a high molecular weight fluorescent dye dextran in bacterial suspensions flowing through microfluidic channels that increased linearly with bacterial concentration. While the aforementioned experiments are with liquid cultures, the enhanced transport of passive particles by bacteria-driven fluid flows has been observed in swarming bacterial colonies situated over agar surfaces¹⁵ and even in porous microfluidic devices containing swimming bacteria built to study bacteria-mediated contaminant transport in groundwater.¹⁶

From a recent experimental study on bacterial suspensions close to a solid surface, Miño et al.³ have concluded that the enhancement in tracer diffusivity scales with the product of the concentration and the average velocity of actively swimming bacteria so that the effective tracer diffusivity should be given by D_eff = D + nU_sβ where D is the Brownian diffusivity of the tracer, n is the number of bacteria per unit volume of the suspension, U_s is the bacterial swimming speed, and the factor β arises from averaging tracer displacements from interactions with bacteria and should scale like the fourth power of bacterial size based on dimensional considerations.³ In a recent study, Jepson et al.⁵ have experimentally confirmed the aforementioned linear relationship in the bulk of three dimensional bacterial suspensions with a β value of around 7 μm⁴. The linear dependence signifies single bacterium effects and might be expected to hold only for dilute suspensions for which tracer interactions with more than one bacterium are negligible. However, in experiments with sheared bacterial suspensions² and bacteria suspensions that lie close to no-slip boundaries³ the linear relation has been observed to hold even at relatively large bacteria concentrations—of the order of 10⁹ to 10¹⁰ bacteria per milliliter. For typical E. Coli cells with a cell-body length of 2 μm¹⁷ and flagella bundle length of 5 μm,^18,19 the aforementioned concentration range corresponds to the range of non-dimensional bacterial concentration nL³ = 0.3 − 3 where L is the overall length of the bacterial cell and flagellar bundle. nL³ is a non-dimensional measure of the bacteria concentration that is on the order of the volume fraction of spheres that circumscribe the bacterium cell and its flagella bundle. A bacterial suspension is not truly dilute at these concentrations since the dilute regime corresponds to nL³ ≪ 1. The linear variation of tracer diffusivity with swimmer concentration has also been observed in simulations of hydrodynamically interacting self-propelled particles mimicking bacteria^6,8 and in particular Hernandez-Ortiz, Underhill, and Graham⁸ report a linear scaling even for moderately dense suspensions. Their simulations however do not account for bacterial tumbling and Brownian motion of tracer particles. The importance of bacterial tumbling becomes readily apparent if one thinks of the hydrodynamic tracer diffusivity as the time-integral of the autocorrelation of the bacteria-induced fluid velocity disturbance sampled by the tracer particle. Since the bacterial disturbance velocity field is determined by the bacterial swimming direction, tumbling of bacteria to a new random orientation limits the time for which the disturbance velocity field remains correlated. Thus one may in general expect a reduction in tracer hydrodynamic diffusion with increasing bacterial tumbling frequency. Indeed, in the experiment of Kim and Breuer,² the effective tracer diffusivity in suspensions containing tumbly cells of E. Coli is significantly smaller than those containing wild-type cells.

Motivated by the aforementioned observations, we develop a theory for binary interactions between bacteria performing run-tumble random walks and Brownian tracer particles based on averaging the tracer transport equation over all possible pair-interactions for slowly varying tracer concentration fields where a Fickian constitutive equation relates the average tracer flux to the average tracer concentration gradient. The theory assumes that the runs of the bacterium are independent, random events such that there is no correlation between bacterial orientations before and after a tumble and yields an explicit expression for the hydrodynamic diffusivity of the tracer particle. The bacterium is modelled as a slender-body through a line distribution of stokeslets along its axis spanning the entire length of the bacterium with their strength adjusted in such a way that the net force on the bacterium is zero. While the far-field nature of this slender-body model is the same (dipolar) as that of the conventional hydrodynamic models of the bacterium such as the point force-dipole^4,5 or two force-monopoles,⁴ we shall show that the slender-body model is a more accurate representation of the bacterium since it accounts for the distribution of hydrodynamic forces over the bacterial length instead of lumping them into a single dipole or two stokeslets as in the other two models. In the theoretical derivation the magnitude of the velocity disturbance of the bacterium is assumed to be small compared to its swimming speed and the tracer particle is considered to be a point object which amounts to neglecting the excluded volume interactions between the tracer particle and the bacterium. We also calculate the tracer diffusivity from numerical simulations of pairwise bacterium-tracer interactions to investigate the effects of velocity disturbances with magnitudes comparable to the bacterial swimming speed. Another important purpose served by the simulation is the investigation of the effect of correlations between the bacterium orientations before and after a tumble which in the theory is assumed to be absent. This study is motivated by the experimental observation that cells of E. Coli perform tumbles with a mean turning angle of 68.5° while perfectly uncorrelated run-tumble motion of a bacterium would yield a mean turn angle of 90°. Our simulations also address the effect of excluded volume (steric) interactions between the tracer and the bacterium to see if tracer-bacterium collisions have any significant influence on the tracer diffusivity.

We note here that the idea of calculating the hydrodynamic tracer diffusivity from binary interactions between swimmers and tracers is not new. For example, Thiffeault and Childress²⁰ and Lin, Thiffeault, and Childress²¹ have calculated the effective tracer diffusivity in suspensions of swimmers creating potential flow disturbances and suspensions of Stokesian squirmers, respectively. The methodology followed in these investigations is to first calculate the net displacement that the tracer undergoes in interactions with the swimmer moving along rectilinear trajectories and then to obtain the tracer diffusivity from the rate of change of the mean square displacement of the tracer. The mean square displacement of the tracer is obtained by averaging the net tracer displacements over the ensemble of interactions with the swimmer with each interaction being specified by an impact parameter which is the shortest distance that the swimmer approaches the tracer measured relative to the tracer's initial position. The rate of change of the mean square displacement is obtained using the frequency of swimmer-tracer interactions. Recently, Miño et al.⁴ and Jepson et al.⁵ have used the same strategy to calculate the hydrodynamic tracer diffusivity in three dimensional bacterial suspensions^4,5 and bacterial suspensions that lie close to no-slip boundaries.⁴

A major limitation of the aforementioned investigations is that none of them have addressed the effect of the Brownian motion of the tracer particle on the swimmer-induced diffusivity and none of them have conducted a systematic study on the effect of the persistence length of the swimmer's motion on the induced diffusivity. As we illustrated through the example of Taylor's dispersion in the first paragraph, Brownian motion of the tracer is a determining factor of the hydrodynamic tracer diffusivity especially when one considers practically relevant situations such as mixing of chemical species in bacteria suspensions. While the case of random reorientations of the swimmer and the tumbling of the bacteria have been considered briefly by Lin, Thiffeault, and Childress²¹ and Jepson et al.,⁵ none of them have addressed how the hydrodynamic tracer diffusivity varies with the swimmer's persistence length. The present study includes both of the above-mentioned effects in a consistent fashion and in addition, our simulations address the effects of having a non-zero correlation between bacterial swimming directions before and after tumbling and the effects of bacterium-tracer collisions (excluded volume interactions), both of which have been ignored in the previous studies.^4,5,20,21 Thus the present paper gives a more detailed picture of the tracer transport process in bacteria suspensions and we shall compare our theoretical results with those of Lin, Thiffeault, and Childress,²¹ Miño et al.,⁴ and Jepson et al.⁵ in the appropriate parameter limits.

A major difference between the theoretical formulation employed in the aforementioned investigations and in the present paper is that the former is based on the Einsteinian perspective on diffusion in which the tracer diffusivity is defined as the rate of change of the tracer mean square displacement. We, on the other hand, approach the problem through the Fickian perspective in which the diffusivity of the tracer is obtained by relating the average tracer flux to the average tracer concentration gradient. As we shall see in Sec. II, adopting the Fickian perspective has allowed us to reduce the problem of computing the hydrodynamic tracer diffusivity of Brownian tracers to the evaluation of a one-dimensional integral when the fluid disturbance created by the bacterium is weak. The Einstenian perspective offers no such simplification and instead one would need to explicitly simulate pair-interactions to calculate the tracer mean square displacement when the Brownian motion of the tracer is considered.

We also carry out our own tracer diffusion experiments in the bulk of three dimensional bacterial suspensions to compare with the theoretical and simulation predictions. Most of the previous experiments are either on quasi-two-dimensional bacterial suspensions,¹ bacterial suspensions lying close to a solid surface,^3,4 or strongly sheared bacterial suspensions.² One may expect the tracer hydrodynamic diffusivity in these situations to be different from that in the bulk of a three-dimensional, isotropic, and unbounded suspension under consideration in this paper owing to the boundary conditions imposed by the film or the wall or due to the shear-induced rotation of bacteria. It has been observed that bacteria swimming near a solid wall follow circular trajectories instead of straight ones owing to the hydrodynamic interactions with the wall²² and that the near-wall nature of the fluid disturbance is in general different from that in the bulk owing to the wall-induced hydrodynamic image of the bacteria.⁴ On the other hand, in the case of sheared bacterial suspensions the shear induced rotation of bacteria makes the suspension anisotropic if the shear rate is comparable to or larger than the bacterial tumbling frequency as was the case in the experiment of Kim and Breuer.²

To the best of our knowledge, the recent work of Jepson et al.⁵ is the only experiment available in the literature in which the effective diffusivity of passive particles in the bulk of bacterial suspensions has been measured. In that experiment, the authors used non-motile cells as tracer particles and obtained their diffusivity through a method that involves measuring the intermediate scattering function of the bacteria population. The maximum bacterial concentration reached in that experiment was around 6 × 10⁸ bacteria per milliliter. We, on the other hand, reach much higher bacteria concentrations (of the order of 10¹⁰ bacteria per milliliter) in the present experiment and we employ particle tracking to obtain the tracer diffusivity. Thus the present experiment tests whether the linear relation between the effective tracer diffusivity in the bulk of the bacteria suspension and the bacteria concentration holds at large cell concentrations. The case of dense bacteria suspensions is fundamentally important owing to the possibility of multi-bacterial effects on tracer diffusion resulting in a nonlinear relationship between the tracer diffusivity and the bacteria concentration.

This paper is organized as follows: in Sec. II we present the theory for hydrodynamic tracer diffusion based on the averaged tracer transport equation and present the results in Sec. II A. Next in Sec. III we derive the hydrodynamic tracer diffusivity through pairwise simulations of bacterium-tracer interactions with a detailed description of the simulation method given in Sec. III A and the simulation results in Sec. III B. We then proceed to describe the experimental procedure used to measure the tracer diffusivity in three dimensional bacteria suspensions and give the results in Sec. IV. In Sec. V we draw comparisons between theoretical predictions and experimental data on two aspects of the present investigation. First, we compare the experimentally obtained bacteria-induced fluid velocity field of Drescher et al.²³ with our slender-body model. We then compare our experimental measurements on tracer diffusivity with our theoretical and simulation predictions. Finally, we close the paper with a concluding discussion in Sec. VI.

II. AVERAGED EQUATION THEORY FOR HYDRODYNAMIC DIFFUSION

The method of ensemble averaged equations²⁴ was originally developed to describe the effects of hydrodynamic particle interactions on Stokes flow properties such as effective viscosity, sedimentation rate and permeability of dilute suspensions and fixed beds. It was adapted by Koch and Brady²⁵ to analyze the macro-transport of chemical tracers in random porous media on a length scale much larger than the typical grain size. Specifically, one obtains a macroscopic convection-diffusion equation for the average tracer concentration field in the porous medium with the convection being driven by the bulk fluid velocity, diffusion constituted by the Brownian motion, and the hydrodynamic diffusion driven by fluid disturbances created by grains in the porous medium. The average tracer concentration is defined by averaging over the ensemble of all possible realizations of the porous medium with different arrangements of the microstructural elements.²⁵ In the present study, we apply this method to a bacterial suspension with the ensemble average consisting of different geometries of the bacterium-tracer interaction.

The formulation begins with the transport equation for the local tracer concentration

$c({\bm x},t)$

c (x, t)

²⁵

\begin{equation}\frac{\partial c}{\partial t} + \nabla \cdot \mathbf {q} = 0,\end{equation}

\frac{\partial c}{\partial t} + \nabla \cdot q = 0,

(1)

in which the flux at any given position and time is

\begin{equation}\mathbf {q}(\mathbf {x},t) = \mathbf {u}(\mathbf {x},t) c(\mathbf {x},t) - D \nabla c(\mathbf {x},t),\end{equation}

q (x, t) = u (x, t) c (x, t) - D \nabla c (x, t),

(2)

where uc is the convective flux driven by the local fluid velocity u(x, t) that the tracer experiences and −D∇c is the diffusive flux driven by the local tracer concentration gradient.²⁵ The random medium under consideration in this paper is a suspension of wild-type bacteria executing run-tumble random walks with a swimming speed of U_s and a mean tumbling frequency of τ⁻¹. We assume that the suspension is unbounded, homogeneous, and isotropic with no imposed fluid flow and that the bacteria swim along straight lines between tumbles with the persistence time T (inverse tumbling frequency) following a Poisson distribution as¹²

\begin{equation}P(T) = \frac{\exp \left(-\frac{T}{\tau }\right)}{\tau }.\end{equation}

P (T) = \frac{\exp (- \frac{T}{τ})}{τ} .

(3)

It is also assumed that the swimming directions before and after a tumble are uncorrelated.

We now express the local velocity and tracer concentration fields in the bacterial suspension under consideration as the sum of corresponding bulk quantities (with an ensemble average indicated by ⟨⟩) and fluctuations due to bacteria indicated by primed variables:

\begin{equation}\mathbf {u} = \langle {\mathbf {u}}\rangle +\mathbf {u}^{\prime }, \qquad c = \langle {c}\rangle +c^{\prime }.\end{equation}

u = ⟨ u ⟩ + u^{'}, c = ⟨ c ⟩ + c^{'} .

(4)

The bulk quantities in Eq. (4) are defined as averages over all possible realizations of the bacterial suspension with a particular realization being specified by the trajectories of all bacteria in the suspension. However, in a dilute suspension the tracer particle can be considered at any given time to be sampling the fluid disturbance created by a single bacterium so that the tracer interacts with each bacterium independently. In other words, all interactions between bacteria can be neglected and u′ is the velocity disturbance created by an isolated bacterium. We have provided a detailed discussion of this assumption later in Sec. VI.

A consequence of the above assumption is that any bulk property of the suspension can now be defined as an average over the ensemble of all possible realizations of the trajectory of a single bacterium. Since the orientations p of a bacterium before and after a tumble are assumed to be uncorrelated, the fluid velocity produced by the bacterium in successive runs is also uncorrelated. As a result, one may define the operator ⟨⟩ as the average over the portion of the trajectory that lies between tumbles. The latter trajectory is

\begin{equation}\mathbf {x}_b = \mathbf {x}_{b,0}+U_s{\bm p} \eta ,\end{equation}

x_{b} = x_{b, 0} + U_{s} p η,

(5)

where p is the swimming direction, η is the time variable that parametrizes the trajectory, and x_{b, 0} is the initial position of the center of the cell-body of the bacterium. The trajectory of the bacterium in Eq. (5) evolves until η = T with T following the distribution in Eq. (3) and when η = T, the bacterium tumbles to a random orientation. The averaging over the trajectory in Eq. (5) now requires averaging over the time η, persistence time T, starting point x_{b, 0}, and swimming directions p. For a homogeneous, isotropic suspension the probability densities of the latter two quantities are equal to the number of bacteria per unit volume of the suspension n and 1/(4π), respectively. This leads to the following expression as the definition of the average of any property A(x, t) of the suspension

\begin{equation}\langle {A}\rangle (\mathbf {x},t) = \frac{n}{4\pi \tau }\int \mathbf {d} \mathbf {p} \int \mathbf {d} \mathbf {x}_{b,0} \int _0^\infty dT P(T) \int _0^T d\eta A.\end{equation}

⟨ A ⟩ (x, t) = \frac{n}{4 π τ} \int d p \int d x_{b, 0} \int_{0}^{\infty} d T P (T) \int_{0}^{T} d η A .

(6)

The above averaging process when applied to the tracer transport equation, Eq. (1), yields the following evolution equation for ⟨c⟩

\begin{equation}\frac{\partial \langle {c}\rangle }{\partial t} + \nabla \cdot \langle {\mathbf {q}}\rangle = 0,\end{equation}

\frac{\partial ⟨ c ⟩}{\partial t} + \nabla \cdot ⟨ q ⟩ = 0,

(7)

where the average flux

\begin{equation}\langle {\mathbf {q}}\rangle = \langle {\mathbf {u}^{\prime } c^{\prime }}\rangle -D \nabla \langle {c}\rangle .\end{equation}

⟨ q ⟩ = ⟨ u^{'} c^{'} ⟩ - D \nabla ⟨ c ⟩ .

(8)

This relationship makes use of the fact that there is no macroscopic fluid flow (quiescent suspension) (⟨u⟩ = 0). When the average concentration gradient varies on length scales larger than the bacterial persistence length and time scales larger than the persistence time, the average flux in Eq. (8) can be related to the average concentration gradient through a Fickian diffusion relationship with an effective diffusivity tensor D_eff

\begin{equation}\langle {\mathbf {q}}\rangle = -\mathbf {D_{eff}}\cdot \nabla \langle {c}\rangle .\end{equation}

⟨ q ⟩ = - D_{eff} \cdot \nabla ⟨ c ⟩ .

(9)

The steady solution of Eq. (7) at long times in then

\begin{equation}\langle {c}\rangle = c_0 + \mathbf {G} \cdot \mathbf {x},\end{equation}

⟨ c ⟩ = c_{0} + G \cdot x,

(10)

where G = ∇⟨c⟩ is the average concentration gradient and c₀ is an arbitrary constant. Equations (8) and (9) lead to the following expression for the bacterial contribution to the total flux:

\begin{equation}\langle {\mathbf {u}^{\prime } c^{\prime }}\rangle = -(\mathbf {D_{eff}} - D \mathbf {I}) \cdot \nabla \langle {c}\rangle = -\mathbf {D_h}\cdot \nabla \langle {c}\rangle ,\end{equation}

⟨ u^{'} c^{'} ⟩ = - (D_{eff} - D I) \cdot \nabla ⟨ c ⟩ = - D_{h} \cdot \nabla ⟨ c ⟩,

(11)

where D_h is the hydrodynamic diffusivity tensor and I is the identity tensor.

The governing equation for the fluctuation in the tracer concentration field due to bacteria obtained by subtracting Eq. (7) from Eq. (1) is

\begin{equation}\frac{\partial c^{\prime }}{\partial t} + \nabla \cdot (\mathbf {u}^{\prime } c^{\prime } -\langle {\mathbf {u}^{\prime } c^{\prime }}\rangle +\mathbf {u}^{\prime } \langle {c}\rangle - D \nabla c^{\prime }) = 0,\end{equation}

\frac{\partial c^{'}}{\partial t} + \nabla \cdot (u^{'} c^{'} - ⟨ u^{'} c^{'} ⟩ + u^{'} ⟨ c ⟩ - D \nabla c^{'}) = 0,

(12)

in which the contribution from ∇ · ⟨u′ c′⟩ vanishes owing to the relations in Eqs. (11) and (10). Since the velocity disturbance u′ is from an isolated bacterium whose trajectory is described by Eq. (5), it is useful to rewrite Eq. (12) in a frame fixed on the bacterium as

\begin{equation}\frac{\partial c^{\prime }}{\partial \eta } + \nabla _{\mathbf {r}} \cdot [(\mathbf {u}^{\prime } - U_s\mathbf {p})c^{\prime } - D\nabla _{\mathbf {r}} c^{\prime }] = -\mathbf {u}^{\prime } \cdot \nabla _{\mathbf {r}} \langle {c}\rangle ,\end{equation}

\frac{\partial c^{'}}{\partial η} + \nabla_{r} \cdot [(u^{'} - U_{s} p) c^{'} - D \nabla_{r} c^{'}] = - u^{'} \cdot \nabla_{r} ⟨ c ⟩,

(13)

in which the symbol ∇_r stands for the vector differentiation in the relative coordinate r = x − x_{b, 0} − U_spη. Thus the source of the concentration fluctuation is the convection of the mean tracer concentration field by the fluid disturbance of the bacteria and it evolves due to convection by the swimming of the bacterium (the −U_sp term) and the bacterial velocity disturbance (the u′c′ term), as well as the Brownian diffusion. We now assume that the magnitude of the velocity disturbance created by the bacterium is much smaller than the swimming velocity (|u′| ≪ U_s) so that the u′c′ term in Eq. (13) can be neglected. With this approximation and making use of Eq. (10), we obtain

\begin{equation}\frac{\partial c^{\prime }}{\partial \eta } - U_s \mathbf {p} \cdot \nabla _{\mathbf {r}}c^{\prime } - D \nabla _{\mathbf {r}}^2 c^{\prime } = -\mathbf {u}^{\prime } \cdot {\mathbf {G}}.\end{equation}

\frac{\partial c^{'}}{\partial η} - U_{s} p \cdot \nabla_{r} c^{'} - D \nabla_{r}^{2} c^{'} = - u^{'} \cdot G .

(14)

The assumption |u′| ≪ U_s is valid for a slender-body model of a bacterium in which it is assumed that the overall length of the bacterium including the cell and flagella (L) is large compared with the diameter d of the cell. For a slender-body, the fluid velocity disturbance at distances of O(L) is smaller than the swimming speed by a factor of the logarithm of the aspect ratio. However, in practice we will find this to be a useful approximation even for moderate aspect ratios by comparing the theory with simulations that do not rely upon this approximation. The assumption is not valid close to the bacterium at distances comparable to the diameter of the bacterium since the fluid velocity disturbance has to satisfy the no-slip boundary condition on the surface of the bacterium which means that |u′| = O(U_s) close to the bacterium. Thus, the present treatment of the problem neglects the near-field hydrodynamic effects of the bacterium. The fluid velocity disturbance produced by a bacterium u′ can in general have a complicated spatial dependence relative to the bacterium and hence it is easier to solve Eq. (14) in Fourier space rather than in real space. The Fourier transform of Eq. (14) with respect to the variable r is

\begin{equation}\frac{\partial \widehat{c}}{\partial \eta } + (2\pi k)^2 D \widehat{c} - 2\pi i \mathbf {k} \cdot \mathbf {p} U_s \widehat{c} = -\widehat{\mathbf {u}} \cdot \mathbf {G},\end{equation}

\frac{\partial \hat{c}}{\partial η} + {(2 π k)}^{2} D \hat{c} - 2 π i k \cdot p U_{s} \hat{c} = - \hat{u} \cdot G,

(15)

where k = |k| and

$\widehat{c}(\mathbf {k},\eta )$

\hat{c} (k, η)

and

$\widehat{\mathbf {u}}(\mathbf {k})$

\hat{u} (k)

are the Fourier amplitudes of the concentration and velocity disturbances defined as

\begin{eqnarray}\widehat{c}({\mathbf {k},\eta })&=& \int \mathbf {dr} c^{\prime }(\mathbf {r},\eta )\exp (-2\pi i \mathbf {k} \cdot \mathbf {r}),\end{eqnarray}

\begin{matrix} \hat{c} (k, η) & = & \int dr c^{'} (r, η) \exp (- 2 π i k \cdot r), \end{matrix}

(16)

\begin{eqnarray}\widehat{\mathbf {u}}(\mathbf {k}) &=& \int \mathbf {dr} \mathbf {u}^{\prime }(\mathbf {r})\exp (-2\pi i \mathbf {k} \cdot \mathbf {r}).\end{eqnarray}

\begin{matrix} \hat{u} (k) & = & \int dr u^{'} (r) \exp (- 2 π i k \cdot r) . \end{matrix}

(17)

The solution of Eq. (15) satisfying the initial condition

$\widehat{c}(\mathbf {k}, 0) = 0$

\hat{c} (k, 0) = 0

is

\begin{equation}\widehat{c}(\mathbf {k}, \eta ) = \frac{\exp \lbrace -[(2\pi k)^2 D - 2\pi i \mathbf {k} \cdot \mathbf {p} U_s]\eta \rbrace -1}{(2\pi k)^2 D - 2\pi i \mathbf {k} \cdot \mathbf {p} U_s}\widehat{\mathbf {u}}(\mathbf {k}) \cdot \mathbf {G}.\end{equation}

\hat{c} (k, η) = \frac{\exp {- [{(2 π k)}^{2} D - 2 π i k \cdot p U_{s}] η} - 1}{{(2 π k)}^{2} D - 2 π i k \cdot p U_{s}} \hat{u} (k) \cdot G .

(18)

We have applied an initial condition corresponding to zero concentration disturbance at the beginning of a bacterium's run because any concentration disturbance resulting from previous runs will be uncorrelated with the fluid velocity disturbance due to the present run and will result in no contribution to the mean flux of the tracer. Using the product theorem on Eq. (6), Eq. (11) takes the form

\begin{equation}-\mathbf {D_h} \cdot \mathbf {G} = \frac{n}{4\pi \tau } \int \mathbf {d}\mathbf {p} \int \mathbf {d} \mathbf {k} \int dT P(T) \int _0^T d\eta \widehat{\mathbf {u}}(\mathbf {k})\widehat{c}(-\mathbf {k},\eta ).\end{equation}

- D_{h} \cdot G = \frac{n}{4 π τ} \int d p \int d k \int d T P (T) \int_{0}^{T} d η \hat{u} (k) \hat{c} (- k, η) .

(19)

Using Eq. (18) in Eq. (19) and performing the integration over η and T explicitly yields an expression for the hydrodynamic diffusivity as

\begin{equation}-\mathbf {D_h} \cdot \mathbf {G} = -\frac{n}{4\pi }\left[\int \mathbf {d} \mathbf {p} \int \mathbf {d} \mathbf {k} \frac{\widehat{\mathbf {u}}(\mathbf {k})\widehat{\mathbf {u}}(-\mathbf {k})}{(2\pi k)^2 D + 2 \pi i \mathbf {k} \cdot \mathbf {p} U_s + \frac{1}{\tau }}\right] \cdot \mathbf {G},\end{equation}

- D_{h} \cdot G = - \frac{n}{4 π} [\int d p \int d k \frac{\hat{u} (k) \hat{u} (- k)}{{(2 π k)}^{2} D + 2 π i k \cdot p U_{s} + \frac{1}{τ}}] \cdot G,

(20)

where the integral in brackets is a second order tensor and is independent of k and p. Since the bacteria suspension is isotropic and there is no bulk fluid flow, the term in brackets must be a multiple of the second order identity tensor. This leads to the following expression for the scalar hydrodynamic diffusivity D_h defined by D_h = D_hI

\begin{equation}D_h = \frac{n}{12\pi }\int \mathbf {d} \mathbf {p} \int \mathbf {d} \mathbf {k} \frac{\widehat{\mathbf {u}}(\mathbf {k}) \cdot \widehat{\mathbf {u}}(-\mathbf {k})}{(2\pi k)^2 D + 2 \pi i \mathbf {k} \cdot \mathbf {p} U_s + \frac{1}{\tau }}.\end{equation}

D_{h} = \frac{n}{12 π} \int d p \int d k \frac{\hat{u} (k) \cdot \hat{u} (- k)}{{(2 π k)}^{2} D + 2 π i k \cdot p U_{s} + \frac{1}{τ}} .

(21)

One may view Eq. (21) as the ensemble average of the integral of the disturbance velocity autocorrelation function of the tracer over the time variable η. We only need to perform the integral for the persistence time of the bacterium since the bacterial velocity disturbance autocorrelation becomes zero once the bacterium makes a tumble owing to the uncorrelated pre- and post-tumble bacterial orientations. Thus tumbling of bacteria has a strong influence on the hydrodynamic diffusivity since it limits the correlation time of tracer particles. Physically, one may expect that increasing the mean tumbling frequency of the bacterium τ⁻¹ would reduce the hydrodynamic diffusivity of the tracer.

To proceed further, we need to know the Fourier amplitude of the fluid velocity disturbance produced by the bacterium

$\widehat{\mathbf {u}}(\mathbf {k})$

\hat{u} (k)

in Eq. . While the simplest fluid-dynamical model of a bacterium is the force dipole, it is accurate only at distances from the bacterium much larger than the bacterial length. This is because the propulsive force generated by the bacterial flagella is distributed throughout the flagella and hence at distances of the order of the bacterial length, the velocity disturbance is influenced by the finite length of the bacterium cell and flagellar bundle. The two-monopole model of the bacterium which is an improvement over the point dipole model by representing the forces on the cell-body and flagella bundle by two equal and opposite point forces separated by a finite distance^4,23 still suffers from the approximation of a long flagellar bundle as a point. We therefore model the bacterium as a line distribution of Stokeslets along its axis with the net strength of the Stokeslets on the cell-body being equal to the viscous drag on the cell-body. The force per unit length on the flagella bundle is then adjusted based on the relative lengths of the cell-body and the flagella bundle to make the bacterium as a whole force-free. Since the bacterium is a rod-like object of overall aspect ratio of the order of 10, one may expect a slender-body model of the bacterium to give a reasonably accurate description of the bacterial disturbance field at distances of the order of the bacterial length. We have used this model previously to calculate the two-bacteria velocity correlations²⁶ and to analyze the linear stability of bacteria suspensions.²⁷ A similar slender body formulation was used in the simulations of Saintillan and Shelley.²⁸ The slender-body model leads to the forced Stokes' equations

\begin{equation}-\nabla _{\mathbf {r}}p^{\prime } + \mu \nabla _{\mathbf {r}}^2 \mathbf {u}^{\prime } = -f \mathbf {p} \left[\int _{-\alpha L/2}^{\alpha L/2} \delta (\mathbf {r} - s\mathbf {p})ds - \frac{\alpha }{1-\alpha } \int _{-(1-\alpha /2)L}^{-\alpha L/2} \delta (\mathbf {r} - s\mathbf {p})ds\right],\end{equation}

- \nabla_{r} p^{'} + μ \nabla_{r}^{2} u^{'} = - f p [\int_{- α L / 2}^{α L / 2} δ (r - s p) d s - \frac{α}{1 - α} \int_{- (1 - α / 2) L}^{- α L / 2} δ (r - s p) d s],

(22a)

\begin{equation}\nabla _{\mathbf {r}} \cdot \mathbf {u}^{\prime } = 0\end{equation}

\nabla_{r} \cdot u^{'} = 0

(22b)

in which L is the overall length of the bacterium including the flagella bundle and α is the ratio of the length of the cell-body to the overall length of the bacterium. In Eq. the first integral on the right-hand side is the forcing due to the cell-body with a force per unit length of

\begin{equation}f = \frac{\mu U_s}{M} ,\end{equation}

f = \frac{μ U_{s}}{M},

(23)

where

\begin{equation}M = (8\pi )^{-1}\lbrace -2\gamma ^2(\gamma ^2-1)^{-1}+ (2\gamma ^3-\gamma )(\gamma ^2-1)^{-3/2}\ln [(\gamma +\sqrt{\gamma ^2-1})/(\gamma -\sqrt{\gamma ^2-1})]\rbrace\end{equation}

M = {(8 π)}^{- 1} {- 2 γ^{2} {(γ^{2} - 1)}^{- 1} + (2 γ^{3} - γ) {(γ^{2} - 1)}^{- 3 / 2} \ln [(γ + \sqrt{γ^{2} - 1}) / (γ - \sqrt{γ^{2} - 1})]}

(24)

is the mobility of the cell-body considering the latter as a prolate spheroid of aspect ratio γ.²⁹ The second integral on the right-hand side of Eq. (22a) is the contribution from the force distribution of magnitude αf/(1 − α) on the flagella bundle and is in the opposite direction to the cell-body force distribution so that the net force on the bacterium is zero. For typical E. Coli cells, the length and width of the cell-body are around 2 μm and 1 μm, respectively,¹⁷ with the flagella length being around 5 μm^18,19 which yield α = 2/7 and γ = 2. The value of the cell-body mobility for the aspect ratio γ = 2 calculated from Eq. (24) is M = 0.18. The solution of Eq. (22) is

\begin{equation}\mathbf {u}^{\prime }(\mathbf {r}) = f \mathbf {p} \cdot \left[\int _{-\alpha L/2}^{\alpha L/2} \mathbf {J}(\mathbf {r} - s\mathbf {p})ds - \frac{\alpha }{1-\alpha }\int _{-(1-\alpha /2)L}^{-\alpha L/2}\mathbf {J}(\mathbf {r} - s\mathbf {p})ds\right],\end{equation}

u^{'} (r) = f p \cdot [\int_{- α L / 2}^{α L / 2} J (r - s p) d s - \frac{α}{1 - α} \int_{- (1 - α / 2) L}^{- α L / 2} J (r - s p) d s],

(25)

where J(r) is the Oseen's tensor defined as

\begin{equation}\mathbf {J}(r) = \frac{1}{8 \pi \mu } \left(\frac{\mathbf {I}}{r} + \frac{\mathbf {r} \mathbf {r}}{r^3} \right)\end{equation}

J (r) = \frac{1}{8 π μ} (\frac{I}{r} + \frac{r r}{r^{3}})

(26)

and r = |r|. For r ≫ L the velocity field in Eq. (25) reduces to

\begin{equation}\mathbf {u}^{\prime }(\mathbf {r})=-(\alpha f L^2)/2 [\mathbf {p} \mathbf {p} : \nabla \mathbf {J}(\mathbf {r})]\end{equation}

u^{'} (r) = - (α f L^{2}) / 2 [p p : \nabla J (r)]

(27)

which corresponds to the disturbance caused by a force-dipole of strength −(αfL²)/2.

In Sec. V we compare the results from our slender-body model with the experimental data of Drescher et al.²³ in which the authors have measured the fluid velocity field around a bacterium using particle image velocimetry. In that section, we also provide a comparison with the dipole model and show that the slender-body model described here is indeed a better approximation to the bacterial disturbance field than the dipole model. In addition, we show in Sec. II A that the dipole model will not give the correct dependence of the hydrodynamic diffusivity on the bacterial tumbling frequency in the limit D → 0 owing to the inappropriateness of the dipole model at O(L) distance from the bacterium. An important advantage of the slender-body over the dipole model is that it provides a means to relate the velocity disturbance field of a bacterium to the geometrical parameters of the bacterium (cell length and diameter and flagellar bundle length) and the bacterial swimming speed. On the contrary, a dipole model would need the dipole strength as an input parameter. There is no direct way to determine the force and distance associated with the dipole from the geometrical parameters and the swimming speed of the bacterium and so they must be fit to experimental data related to the fluid velocity disturbance. This remains true for the two-monopole model of the bacterium (see Drescher et al.²³) since there is no rational way to fix the distance between the force monopoles. We would like to point out that the slender-body model described here will not be accurate at distances of the order of the bacterial diameter (≈1μm) from the bacterium since at those length scales short-range hydrodynamic effects of the bacterium would also be important. In short, the slender-body model is accurate at distances from the bacterium r ≫ L and r = O(L) and not accurate when r = O(d) with the bacterial diameter d ≪ L, while the dipole model is only accurate for r ≫ L.

Solving Eq. (22) in Fourier space and using Eq. (23) yields

\begin{equation}\widehat{\mathbf {u}}(\mathbf {k}) = \frac{U_s L^3}{M k^2}\left[\mathbf {p} - \frac{(\mathbf {k} \cdot \mathbf {p}) \mathbf {k}}{k^2}\right]F(\mathbf {k} \cdot \mathbf {p})\end{equation}

\hat{u} (k) = \frac{U_{s} L^{3}}{M k^{2}} [p - \frac{(k \cdot p) k}{k^{2}}] F (k \cdot p)

(28)

in which the wavenumber is non-dimensionalized by (2πL)⁻¹ and the quantity F(k · p) is defined as

\begin{eqnarray}F(\mathbf {k} \cdot \mathbf {p}) = -\frac{1}{i \mathbf {k} \cdot \mathbf {p}}\bigg \lbrace \exp (-i\alpha \mathbf {k} \cdot \mathbf {p}/2)-\frac{1}{1-\alpha }\exp (i\alpha \mathbf {k} \cdot \mathbf {p}/2)\nonumber \\+ \frac{\alpha }{1-\alpha }\exp \left(i[1-\alpha /2]\mathbf {k} \cdot \mathbf {p} \right)\bigg \rbrace .\end{eqnarray}

\begin{matrix} F (k \cdot p) = - \frac{1}{i k \cdot p} {\exp (- i α k \cdot p / 2) - \frac{1}{1 - α} \exp (i α k \cdot p / 2) \\ + \frac{α}{1 - α} \exp (i [1 - α / 2] k \cdot p)} . \end{matrix}

(29)

At long length scales compared to bacterial length for which k ≪ 1,

$\widehat{\mathbf {u}}(\mathbf {k})$

\hat{u} (k)

becomes asymptotically close to the Fourier amplitude of the velocity disturbance of a force-dipole. Substituting Eq. in Eq. and non-dimensionalizing the mean run time τ with the characteristic time scale L/U_s gives an expression for the hydrodynamic diffusivity as

\begin{equation}D_h = n L^3 U_s L \widetilde{D_h},\end{equation}

D_{h} = n L^{3} U_{s} L \tilde{D_{h}},

(30)

in which the non-dimensional hydrodynamic diffusivity

$\widetilde{D_h}$

\tilde{D_{h}}

is given by

\begin{equation}\widetilde{D_h} = \frac{1}{96\pi ^4 M^2}\int d\mathbf {p} \int d \mathbf {k} \frac{\left[k^2-(\mathbf {k} \cdot \mathbf {p})^2\right]F(\mathbf {k} \cdot \mathbf {p})F(-\mathbf {k} \cdot \mathbf {p})}{k^6\left(\frac{1}{\tau ^*}+i \mathbf {k} \cdot \mathbf {p} + \frac{k^2}{Pe}\right)}.\end{equation}

\tilde{D_{h}} = \frac{1}{96 π^{4} M^{2}} \int d p \int d k \frac{[k^{2} - {(k \cdot p)}^{2}] F (k \cdot p) F (- k \cdot p)}{k^{6} (\frac{1}{τ^{*}} + i k \cdot p + \frac{k^{2}}{P e})} .

(31)

In Eq. (31), Pe = U_sL/D measures the strength of convective sampling by bacterial swimming relative to sampling by tracer diffusion and τ^* = U_sτ/L measures the length that a bacterium swims before it tumbles relative to the length of the bacterium itself. Typical wild-type E. Coli has τ^* = O(1) and the typical Pe that one may encounter is primarily determined by the molecular diffusivity of the tracer. For instance, while the colloidal tracer particles used in the experiment of Wu and Libchaber¹ are nearly non-Brownian with Pe = O(10³) corresponding to their thermal diffusivity (around 0.01 μm²/s for 4.5 μm diameter particles and around 0.005 μm²/s for 10 μm particles) typical bacterial length and speed of L = 7 μm and U_s = 15 μm/s, respectively, the fluorescent dye used in the experiment of Kim and Breuer² has Pe ≈ 5 based on its molecular diffusivity of 20 μm²/s (see Fig. 4 in Kim and Breuer²). We also consider the asymptotic limit of τ^* ≫ 1 with Pe ≫ 1 which corresponds to a non-Brownian tracer in a suspension of smooth-swimming bacteria such as the E. Coli strain RP9535.³⁰ We shall derive an analytical expression for the hydrodynamic diffusivity in this limit. It turns out that the product F(k · p)F( − k · p) in Eq. (31) is both real and an even function of k · p. As a result, the imaginary part of the integrand in Eq. (31) becomes an odd function of k · p and vanishes when integrated over k and p space. While the integral in Eq. (31) can be evaluated analytically using asymptotic techniques for the special cases of Pe ≪ 1 with τ^* ⩾ O(1) and Pe ≫ 1 with τ^* ≫ 1, numerical integration is required for general cases. Equation (31) can be reduced to a one-dimensional integral by first writing it as an integral over a cylindrical domain with axis along p for which dk = 2πk_⊥dk_⊥dk_∥ where k_∥ = k · p and k_⊥ = |k − (k · p)p| and evaluating the integral over k_⊥ space analytically. The remaining integration over k_∥ has been performed numerically using the adaptive Gauss-Kronrod quadrature routine.

A. Theoretical results

Fig. 1 shows the results of our calculation of

$\widetilde{D_h}$

\tilde{D_{h}}

as a function of Pe and τ^* with representative values of bacterial geometrical parameters as γ = 2 and α = 2/7. The primary observation from Fig. 1 is that while the non-dimensional hydrodynamic diffusivity increases monotonically with increasing τ^* as expected, the variation is non-monotonic with respect to Pe. In the Pe ≪ 1 regime, the Brownian diffusion of the tracer is much faster compared to bacterial swimming and as a result, the tracer-bacterium interactions are less effective in transporting the tracer. The asymptotic evaluation of the integral in Eq. for Pe ≪ 1 and τ^* ⩾ O(1) (see Subsection A 1 of the Appendix) reveals that

$\widetilde{D_h} \sim \alpha ^2 \sqrt{Pe \tau ^*}/(360 \pi M^2)$

\tilde{D_{h}} \sim α^{2} \sqrt{P e τ^{*}} / (360 π M^{2})

and this asymptotic result shown in Fig. 1(a) as a dashed line agrees well with the numerical calculations shown by the solid line. The origin of the

$\sqrt{Pe \tau ^*}$

\sqrt{P e τ^{*}}

scaling in the limit of Pe ≪ 1 and τ^* ⩾ O(1) can be understood physically by first writing down a generic scaling expression for the hydrodynamic diffusivity as

\begin{equation}D_h \sim n l_s^3 u^{\prime }(l_s)^2 \tau ,\end{equation}

D_{h} \sim n l_{s}^{3} u^{'} {(l_{s})}^{2} τ,

(32)

where l_s is the typical length scale of separation between the bacterium and the tracer particle. This expression arises from the fact that the tracer displacement due to a bacterium's disturbance field stays correlated for a time scale of the order of the bacterium's persistence time τ and the typical velocity with which the tracer moves u′(l_s) is a function of the tracer-bacterium separation distance l_s. Averaging over all possible tracer-bacteria interactions brings the factor

$n l_s^3$

n l_{s}^{3}

⁠. For the case of Pe ≪ 1, the characteristic length scale of separation is the diffusion length of the tracer

$l_s = O(\sqrt{D\tau })$

l_{s} = O (\sqrt{D τ})

since bacteria translate only by a distance of

$U_s \tau \ll \sqrt{D\tau }$

U_{s} τ ≪ \sqrt{D τ}

⁠. Indeed, as given in Subsection A 1 of the Appendix, the dominant contribution to the integral comes from small wavenumbers with

$k \le O(\sqrt{Pe/\tau ^*})$

k \leq O (\sqrt{P e / τ^{*}})

which corresponds to bacteria-tracer separation length scales of order

$\sqrt{D\tau }$

\sqrt{D τ}

or larger. Now if τ^* ⩾ 1,

$l_s/L = \sqrt{\tau ^*/Pe} \gg 1$

l_{s} / L = \sqrt{τ^{*} / P e} ≫ 1

and one may assume that u′(l_s) is the fluid velocity disturbance of a force dipole with a characteristic dipole strength of μU_sL²,

$u^{\prime }(l_s) = O(U_s L^2/l_s^2)$

u^{'} (l_{s}) = O (U_{s} L^{2} / l_{s}^{2})

⁠. Now using these scalings in Eq. gives

$D_h \sim nL^3 U_s L \sqrt{U_s^2 \tau /D}$

D_{h} \sim n L^{3} U_{s} L \sqrt{U_{s}^{2} τ / D}

which in turn yields

$\widetilde{D_h} \sim \sqrt{Pe\tau ^*}$

\tilde{D_{h}} \sim \sqrt{P e τ^{*}}

⁠.

FIG. 1.

$FIG. 1. \documentclass[12pt]{minimal}\begin{document}$\widetilde{D_h}$\end{document}Dh̃ as a function of (a) Pe for various τ* (solid lines) and (b) τ* for various Pe (symbols). The dashed lines in Pe < 1 region of Figure 1(a) correspond to the asymptotic result \documentclass[12pt]{minimal}\begin{document}$\widetilde{D_h} = \alpha ^2 \sqrt{Pe \tau ^*}/(360 \pi M^2)$\end{document}Dh̃=α2Peτ*/(360πM2) for Pe ≪ 1 and τ* ⩾ O(1). The dashed-dotted line in Pe ≫ 1 region of Figure 1(a) corresponds to the limit Pe → ∞. In Figure 1(b) squares are for Pe = 1, up-triangles are for Pe = 10, down-triangles are for Pe = 100, circles are for Pe = 104, and asterisks as for Pe = ∞. The dashed line corresponds to the asymptotic result \documentclass[12pt]{minimal}\begin{document}$\widetilde{D_h} = \alpha ^2/(192\pi M^2)$\end{document}Dh̃=α2/(192πM2) for the case Pe → ∞ and τ* → ∞.$

View large Download slide

$\widetilde{D_h}$

\tilde{D_{h}}

as a function of (a) Pe for various τ^* (solid lines) and (b) τ^* for various Pe (symbols). The dashed lines in Pe < 1 region of Figure 1(a) correspond to the asymptotic result

$\widetilde{D_h} = \alpha ^2 \sqrt{Pe \tau ^*}/(360 \pi M^2)$

\tilde{D_{h}} = α^{2} \sqrt{P e τ^{*}} / (360 π M^{2})

for Pe ≪ 1 and τ^* ⩾ O(1). The dashed-dotted line in Pe ≫ 1 region of Figure 1(a) corresponds to the limit Pe → ∞. In Figure 1(b) squares are for Pe = 1, up-triangles are for Pe = 10, down-triangles are for Pe = 100, circles are for Pe = 10⁴, and asterisks as for Pe = ∞. The dashed line corresponds to the asymptotic result

$\widetilde{D_h} = \alpha ^2/(192\pi M^2)$

\tilde{D_{h}} = α^{2} / (192 π M^{2})

for the case Pe → ∞ and τ^* → ∞.

When Pe ≫ 1, the non-dimensional hydrodynamic diffusivity is independent of Pe as seen in Fig. 1(a) indicating a purely convective transport of the tracer. This is further demonstrated by the coincidence of Pe = 10⁴ data (circles) and Pe = ∞ data (asterisks) in Fig. 1(b). The figures however show that the hydrodynamic diffusivity is still dependent on τ^* thus yielding a scaling expression D_h ∼ nL³U_sLϕ(τ^*) in which the non-dimensional ϕ(τ^*) has to be obtained numerically. Fig. 1(b) indicates that the hydrodynamic diffusivity becomes independent of τ^* in the limit Pe → ∞ and τ^* → ∞. Fig. 1(b) shows that the large τ^* limit can be attained for τ^* ⩾ 5. Experimentally, it is possible to attain the aforementioned limit by controlling the tumbling frequency of bacterium through the control of the level of CheY protein in the bacterium as demonstrated by Alon et al.³¹ The dependence of

$\widetilde{D_h}$

\tilde{D_{h}}

on τ^* when τ^* = O(1) and the independence at large τ^* when Pe is large can be explained again through the scaling expression given in Eq. . First, the characteristic length scale of separation between a bacterium and a tracer particle is of the order of the bacterial persistence length, l_s = O(U_sτ) since the Brownian diffusion of the tracer is negligible in this high Pe regime. Now if τ^* ≫ 1 as in the case of smooth swimming bacteria, l_s ≫ L and the typical velocity disturbance experienced by the tracer follows the dipolar scaling

$u^{\prime }(l_s) = O(U_s L^2/l_s^2)$

u^{'} (l_{s}) = O (U_{s} L^{2} / l_{s}^{2})

⁠. For l_s = O(U_sτ), Eq. yields the scaling D_h ∼ nL³U_sL making

$\widetilde{D_h}$

\tilde{D_{h}}

independent of both Pe and τ^*. Mathematically, the dominant contribution to the integral in Eq. comes from wavenumbers of order 1/τ^* which gives the leading order non-dimensional hydrodynamic diffusivity as

$\widetilde{D_h} = \alpha ^2 /(192 \pi M^2)$

\tilde{D_{h}} = α^{2} / (192 π M^{2})

through the asymptotic evaluation of the integral in Eq. in Subsection A 2 of the Appendix. This expression yields

$\widetilde{D_h} \approx 0.0044$

\tilde{D_{h}} \approx 0.0044

for a typical bacterium with α = 2/7 and M = 0.18 (corresponding to the cell-body aspect ratio γ = 2) in the infrequent-tumbling limit so that the dimensional hydrodynamic diffusivity is given by D_h = 0.0044nL³U_sL. It is clear from Fig. 1(b) that the latter prediction (shown by a dashed line) agrees well with the full numerical calculation for the case with Pe = ∞ (shown by asterisks) when τ^* ≫ 1. Now if τ^* = O(1), l_s = O(L) for which the velocity field experienced by the tracer is not dipolar. This results in a non-trivial relationship

$\widetilde{D_h}$

\tilde{D_{h}}

with τ^* which can be obtained only through the numerical integration of Eq. and hence the scaling D_h ∼ nL³U_sLϕ(τ^*). For a typical bacterium swimming at a speed of 15 μm/s and tumbling at a rate of 1s⁻¹, τ^* = 2.1 for L = 7 μm and for this value of τ^* yields

$\widetilde{D_h} = 0.0039$

\tilde{D_{h}} = 0.0039

⁠.

A surprising feature in Fig. 1(a) is the non-monotonic dependence of the hydrodynamic diffusivity on the Peclet number. This is also evident in the downward shift of the

$\widetilde{D_h}$

\tilde{D_{h}}

versus τ^* curves with increasing Pe (beyond Pe = 1) in Fig. 1(b). The hydrodynamic diffusivity is largest when Pe = O(10) which indicates that weak Brownian diffusion in fact makes bacteria more efficient in transporting the tracer. This is in sharp contrast with the case of fixed beds²⁵ in which the hydrodynamic diffusivity decreases monotonically with a decrease in Pe. In general, the hydrodynamic diffusivity is expected to decrease monotonically with decreasing Pe since the increased Brownian displacement of the tracer relative to its convective displacement makes the tracer motion less correlated. A bacterial suspension differs from this picture owing to the fact that the bacterial disturbance velocity field points in opposite directions at the front and back of the bacterium. The odd nature of the dipolar fluid disturbance produced by the bacterium limits the integral of the tracer velocity autocorrelation and hence its hydrodynamic diffusion. We elucidate this further in Fig. 2 in which we show the interaction between a bacterium and a tracer particle happening over a typical bacterial run with and without Brownian motion for the tracer. Since the hydrodynamic diffusivity of the tracer is given by the integral of the autocorrelation of the bacterial disturbance velocity sampled by the tracer, the contribution to the hydrodynamic diffusivity from a given tracer trajectory can be written as an integral along the tracer trajectory (1/3)∫u′(x) · u′(x + x′(s))ds where x′(s) is the tracer trajectory parametrized by the variable s. In the absence of any tracer Brownian diffusion there exists only a single tracer-bacterium relative trajectory for a given pair of initial locations of the bacterium and the tracer particle and the run duration of the bacterium. For weak bacterial disturbances, the aforementioned trajectory is a straight line as shown in Fig. 2 by the solid line. Depending upon the run duration of the bacterium, the tracer trajectory could span regions where the component of the bacterial disturbance velocity parallel to the tracer trajectory has opposite signs. This decreases the contribution to the integral of the velocity autocorrelation from that trajectory and hence decreases the hydrodynamic diffusivity especially if the trajectory has equal lengths in those two regions with different signs for the velocity disturbance parallel to the trajectory. On the other hand, a non-zero tracer Brownian diffusivity helps the tracer sample an infinite number of possible trajectories like the one shown in Fig. 2 by a dashed-dotted line for any given initial locations of the bacterium and the tracer. The average contribution to the tracer velocity autocorrelation due to these weakly Brownian trajectories will in general be larger than the straight trajectory that one would have in the absence of tracer Brownian motion even when equal lengths of the trajectory lie in regions with opposite signs of velocity disturbance parallel to the swimming direction and this increases the hydrodynamic diffusivity. In other words, having weak Brownian motion helps the tracer in sampling the velocity field around the bacterium in a manner more likely to produce a net displacement for each interaction and a larger hydrodynamic diffusivity averaged over all interactions. Nevertheless, a large tracer Brownian diffusivity would reduce the tracer velocity autocorrelation and hence would reduce the hydrodynamic diffusivity. An important point to be noted here is that our explanation for the non-monotonic nature of

$\widetilde{D_h}$

\tilde{D_{h}}

versus Pe curve relies on the weak disturbance assumption. If the fluid disturbance of the bacterium is comparable to the swimming speed, the tracer trajectory relative to the bacterium will not be rectilinear as in Fig. 2 and as a result the net displacement of the tracer need not be zero. Thus, the non-monotonic nature of the

$\widetilde{D_h}$

\tilde{D_{h}}

versus Pe relationship may change in the presence of finite bacterial disturbances.

FIG. 2.

View large Download slide

A typical tracer-bacterium interaction. The bacterium's velocity disturbance streamlines are shown by dashed lines and the solid line shows the trajectory of the tracer relative to the bacterium in the absence of any Brownian motion. The tracer's trajectory relative to the bacterium in the presence of Brownian motion is shown by a dashed-dotted line.

Now we will explicitly demonstrate that the dependence of the hydrodynamic diffusivity on the tumbling frequency of the bacterium in the limit of vanishing Brownian diffusivity cannot be captured if one models the bacterium as a point force dipole. The dipole velocity field is given in Eq. (27) which in non-dimensional form is

\begin{equation}\mathbf {u}^{\prime }(\mathbf {r}) = -\frac{\alpha }{2M}\mathbf {p} \mathbf {p} : \nabla \mathbf {J}(\mathbf {r})\end{equation}

u^{'} (r) = - \frac{α}{2 M} p p : \nabla J (r)

(33)

in which the Oseen's tensor J is scaled by 1/L and α/(2M) is the non-dimensional dipole strength. Fourier transforming this velocity field yields the Fourier amplitude of the dipole field as

\begin{equation}\widehat{\mathbf {u}}_d(\mathbf {k}) = -\frac{i \alpha U_s L^3}{2M k^2}\mathbf {p} \mathbf {p} : \mathbf {k} \left(\mathbf {I} - \frac{\mathbf {k} \mathbf {k}}{k^2}\right),\end{equation}

{\hat{u}}_{d} (k) = - \frac{i α U_{s} L^{3}}{2 M k^{2}} p p : k (I - \frac{k k}{k^{2}}),

(34)

in which the subscript “d” stands for dipole field. Equation (34) can also be obtained by expanding Eq. (28) in the limit k ≪ 1. Substituting

$\widehat{\mathbf {u}}_d(\mathbf {k})$

{\hat{u}}_{d} (k)

in place of

$\widehat{\mathbf {u}}(\mathbf {k})$

\hat{u} (k)

in Eq. and non-dimensionalizing time using the characteristic time scale L/U_s, we obtain the following expression for the non-dimensional hydrodynamic diffusivity in the limit of vanishing tracer Brownian diffusivity (Pe → ∞)

\begin{equation}\widetilde{D_{hd}} = \frac{\alpha ^2}{384 \pi ^4 M^2}\int \mathbf {dp} \int \mathbf {dk} \frac{(\mathbf {k} \cdot \mathbf {p})^2 \left[k^2 - (\mathbf {k} \cdot \mathbf {p})^2\right]}{k^6 \left( \frac{1}{\tau ^*}-i \mathbf {k} \cdot \mathbf {p} \right)}.\end{equation}

\tilde{D_{h d}} = \frac{α^{2}}{384 π^{4} M^{2}} \int dp \int dk \frac{{(k \cdot p)}^{2} [k^{2} - {(k \cdot p)}^{2}]}{k^{6} (\frac{1}{τ^{*}} - i k \cdot p)} .

(35)

The integral in Eq. (35) can be explicitly evaluated easily by splitting it over k_∥ and k_⊥ spaces and the resulting expression for non-dimensional hydrodynamic diffusivity

\begin{equation}\widetilde{D_{hd}} = \frac{\alpha ^2}{192 \pi M^2}\end{equation}

\tilde{D_{h d}} = \frac{α^{2}}{192 π M^{2}}

(36)

is independent of τ^* no matter what τ^* is and is the same as the Pe ≫ 1, τ^* ≫ 1 asymptote of

$\widetilde{D_h}$

\tilde{D_{h}}

calculated earlier using the slender-body model for the bacterium. The fact the hydrodynamic diffusivity predicted by the dipole model is constant in the limit of Pe ≫ 1 can be rationalized through the generic scaling expression for the hydrodynamic diffusivity given in Eq. . In the absence of any Brownian motion l_s = U_sτ in Eq. and since the velocity disturbance scales like

$u^{\prime }(l_s) = U L^2/l_s^2$

u^{'} (l_{s}) = U L^{2} / l_{s}^{2}

for the dipole model at all τ in Eq. . This gives the scaling D_h ∼ nL³U_sL which means that

$\widetilde{D_h}$

\tilde{D_{h}}

is a constant. The slender-body model, on the other hand, gives a hydrodynamic diffusivity that is dependent on τ^* as seen in Fig. 1. The lack of dependence of the hydrodynamic diffusivity on the bacterial persistence length is an artifact of the dipolar model of the bacterium. Physically, the diffusivity depends on the nature of the hydrodynamic disturbance at separations characteristic of a typical length of a bacterium run. In the point dipole model, the fluid velocity disturbance is similar at all length scales while the slender-body model captures the changes in the velocity disturbance when r = O(L). Experimental results of Kim and Breuer² highlight the physical importance of the persistence length, showing that the tracer diffusivity is significantly smaller in suspensions of bacteria that tumble frequently.

We now compare the results from our analysis with the recent theoretical investigations of Lin, Thiffeault, and Childress²¹ and Miño et al.⁴ In Lin, Thiffeault, and Childress²¹ the authors calculate the hydrodynamic tracer diffusivity in a suspension of swimming microorganisms modeled as spherical squirmers³² swimming along rectilinear trajectories and subjected to random reorientations by calculating the rate of change of the mean square displacement of the tracer resulting from pair-interactions with the squirmer. A squirmer is a Stokesian swimmer model with a prescribed tangential velocity field at its surface. This model sometimes is used for modeling ciliated swimming microorganisms^21,32 and while the near-field fluid disturbance of a squirmer is in general different from that of a bacterium, the far-field disturbance is a force dipole whose strength is determined by a non-dimensional number β in Lin, Thiffeault, and Childress²¹ (not the same as the one mentioned in the fourth paragraph of Sec. I) which is a free model parameter. Comparing the dipolar velocity fields of the squirmer model used in Lin, Thiffeault, and Childress²¹ and that of the bacterium given in Eq. (33) we obtain the relationship between the β parameter used in Lin, Thiffeault, and Childress²¹ and the non-dimensional dipole bacterial strength α/(2M) as αL²/(2M) = 6πβℓ² in which ℓ is the radius of the squirmer. Lin, Thiffeault, and Childress²¹ have not considered the Brownian motion of the tracer particle and give their results only in the limit of large persistence length of the swimmer compared to its size such that their analysis corresponds to the Pe ≫ 1 and τ^* ≫ 1 limit of the present calculation. In that limit we have shown in Subsection A 2 of the Appendix that

$\widetilde{D_h} = \alpha ^2/(192 \pi M^2)$

\tilde{D_{h}} = α^{2} / (192 π M^{2})

which when written in terms of β takes the form

$\widetilde{D_h} = (144 \pi /196)\beta ^2 (\ell /L)^4$

\tilde{D_{h}} = (144 π / 196) β^{2} {(ℓ / L)}^{4}

which reduces to

$\widetilde{D_h} \approx 2.31 \beta ^2$

\tilde{D_{h}} \approx 2.31 β^{2}

when ℓ = L. The dipole contribution to tracer diffusivity calculated by Lin, Thiffeault, and Childress²¹ assuming small tracer displacements (which is equivalent to our small disturbance assumption) is

$\widetilde{D_h} \approx 2.10 \beta ^2$

\tilde{D_{h}} \approx 2.10 β^{2}

⁠. Thus our results scale in the same way with the dipole strength as predicted by Lin, Thiffeault, and Childress²¹ and the small difference between numerical prefactors is most likely due to the fact that Lin, Thiffeault, and Childress²¹ obtain the prefactor from a fitting procedure.

The calculation of diffusivity of non-Brownian tracers of Miño et al.⁴ in bulk bacterial suspensions is very similar to that of Lin, Thiffeault, and Childress²¹ except that Miño et al.⁴ neglect the tumbling of bacteria altogether and instead assume that bacteria swim along infinitely long, straight trajectories. Miño et al.⁴ model the bacterial disturbance field either as two-force-monopoles separated by a distance or as a point-force-dipole with the magnitude of the bacterial velocity disturbance being determined by a non-dimensional parameter γ⁻¹ (not same as the aspect ratio γ that appear in Eq. (24)) which corresponds to our non-dimensional dipole strength through the relationship γ⁻¹a² = α/(16πM)L² where a is a swimmer size used by Miño et al.⁴ Assuming that the tracer displacements are small by expanding in powers of γ⁻¹, Miño et al.⁴ give the scaling for the tracer diffusivity as the fourth power of the dipole strength as

$\widetilde{D_h} \sim \gamma ^{-4}$

\tilde{D_{h}} \sim γ^{- 4}

which is in contrast to the quadratic scaling that we and Lin, Thiffeault, and Childress²¹ predict in the absence of tracer Brownian motion and when the bacterial persistence length is large. The difference between these scalings is due to the fact that Miño et al.⁴ have neglected the reorientation of bacterium entirely while the present work and Lin, Thiffeault, and Childress²¹ have assumed a large but finite persistence length. In fact, neglecting the reorientation of the swimmer gives rise to a non-convergent integral in the calculation of the tracer diffusivity of Miño et al.⁴ at O(γ⁻²) (which corresponds to the O(γ⁻¹) tracer displacement in Eq. (2.12) of Miño et al.) and Miño et al.⁴ wrongly concluded that integral to be zero. The non-convergent nature of the integral in Miño et al.⁴ arises from the assumption of infinitely long, straight bacterial trajectories and one instead needs to have a finite persistence length (though it can be large compared to the bacterial size) for the bacterium to have a well defined tracer diffusivity. In fact, Lin, Thiffeault, and Childress²¹ have already made this observation for the case of squirmers for which the far-field velocity disturbance is dipolar just like bacteria. Apart from the issue of the divergent integral, the approach of Miño et al.⁴ requires a cut-off value for their impact parameter (b_min) which has been chosen somewhat arbitrarily even though they note that their diffusivity predictions are significantly influenced by the choice of b_min. In contrast, the slender-body fluid velocity field used in the present analysis varies more slowly at small bacterium-tracer separations and does not require a cut-off parameter.

III. CALCULATION OF |$\widetilde{D_h}$| $\tilde{D_{h}}$ FROM NUMERICAL SIMULATIONS OF PAIRWISE TRACER-BACTERIUM INTERACTIONS

In deriving the hydrodynamic diffusivity in Sec. II, we have assumed that the disturbance fluid velocity produced by the bacterium is small compared to the swimming velocity, that the tracer is a point and the bacterium a line of forces, and that the pre- and post-tumble orientations of the bacterium are uncorrelated. While the slender-body model of a high aspect ratio bacterium gives a logarithmically small fluid disturbance in the outer region corresponding to separations from the bacterium axis of order L which contribute most to the integrals leading to the diffusivity, the fluid velocity does become of O(U_s) at separations from the axis of order of the bacterium diameter and even diverges logarithmically if (as we have done in the calculations) we extend the fluid velocity description to the axis of the bacterium itself.

In addition, the assumption that the runs of the bacteria are uncorrelated need not be true for the case of realistic bacteria. For instance, the pre- and post-tumble orientations of wild-type E. Coli cells have been observed to be somewhat correlated since their mean turning angle is around 68.5° and not 90° as would be the case for perfectly random tumbles.^27,33 Thus to test for possible effects of the finite ratio of fluid velocity to bacterium swimming speed and the correlation in bacterial runs, we perform numerical simulations of pairwise interaction between a bacterium and a colloidal tracer particle.

A third purpose served by the simulations is the investigation of the effects of excluded volume (steric) interactions between bacteria and tracer particles as most of the experiments on enhanced diffusion in bacteria suspensions are with tracer particles of size comparable to the size of the bacterium.^1,3 A limitation of the present simulation study is that we neglect the detailed hydrodynamic effects involved in the tracer-bacterium interactions at O(d) separations such as the near-field interactions arising from lubrication and the rotation of the bacterial cell-body and the flagella bundle although we do include the modification of the dipolar velocity field due to the distribution of forces over the length scale L. Inclusion of the near-field interactions between the bacterium and the tracer particle would not only require a detailed modeling of the bacterial hydrodynamics, but also colloidal interactions such as electrostatic forces (see Hong and Brown³⁴). We instead focus on the contributions to the tracer diffusion from the slender-body velocity field and the steric repulsion provided by the bacterium since these effects can be accounted for without a detailed model of the near-field physical interactions.

A. Simulation methodology

The fundamental idea behind the simulation method is to first compute the bacteria induced tracer diffusivity as the rate of change of the mean square displacement of the tracer resulting from interactions with bacteria. In a dilute suspension it is rare that two bacteria will affect a tracer's motion simultaneously and so we consider the tracer to be influenced by one bacterium at a time. The mean square displacement of the tracer is then computed by averaging the net quadratic displacement that the tracer undergoes over an ensemble of bacterium-tracer interactions. Bacteria are assumed to be performing the run-tumble random walks with runs along straight trajectories lasting for durations that are Poisson distributed as given in Eq. (3). For the case of uncorrelated bacterial runs, each tracer-bacterium interaction lasts only for the duration of a single bacterial run since the displacement that the tracer particle undergoes during a given bacteria run is uncorrelated with that of the previous run of the same bacterium. Thus, in this case, the ensemble of tracer-bacterium interactions is constituted by independent runs of the bacterium along random directions sampled out of the isotropic distribution. When the bacterial runs are correlated, the induced tracer displacement in a given bacterial run is not independent of that from the previous run. Each time the bacterium performs a tumble, the correlation in the bacterial orientation (and consequently the correlation in the tracer's motion) decreases and eventually becomes negligible after many tumbles of the bacterium. Thus in contrast to the case of uncorrelated bacterial runs, each tracer-bacterium interaction constitutes multiple runs of the bacterium when the bacterial runs are correlated. The number of bacterial runs spanned until the bacterial run direction loses its correlation is dependent upon the mean angle that the bacterium turns in a tumble. We calculate the number of runs using the model of correlated bacterial runs proposed by Subramanian and Koch²⁷ based on a transition probability density function that describes the correlation between bacterial orientations in two consecutive runs in a manner that reproduces the mean turning angle. Since we are calculating the tracer diffusivity in an unbounded bacterial suspension, we first calculate the tracer diffusivity arising from interactions with bacteria that pass through a fixed spherical control volume of radius R centered at the tracer's initial position during an interaction and then extrapolate the result to the limit R → ∞.

We first give the details of the simulation method for the case of uncorrelated bacterial runs. A schematic of the simulation of a typical tracer bacterium interaction with uncorrelated bacterial runs is shown in Fig. 3. We begin the simulation by choosing the bacterial swimming direction p and the run duration T. The former is chosen isotropically on the unit sphere and the latter is chosen from the Poisson distribution given in Eq. (3). The starting point of the tracer trajectory is initialized at the origin x = 0 which is also the center of the spherical control volume. We then stipulate that the bacterium should pass through a random location

$\mathbf {x}_b^{\prime }$

x_{b}^{'}

inside the spherical control volume at some randomly chosen time t′ sampled from a uniform distribution in the interval [0, T] which gives the starting point of the bacterial trajectory as

$\mathbf {x}_b(0) = \mathbf {x}_b^{\prime } - \mathbf {p} t^{\prime }$

x_{b} (0) = x_{b}^{'} - p t^{'}

⁠. Note that the starting point x_b(0) and ending point x_b(T) are not required to lie inside the spherical control volume. During each bacterium run, the positions x and x_b of the tracer and the bacterium, respectively, evolve as

\begin{eqnarray}\mathbf {x}(t+\Delta t) &=& \mathbf {x}(t)+ \mathbf {u}^{\prime }(\mathbf {x}(t) - \mathbf {x}_{b}(t))\Delta t + \sqrt{\frac{6\Delta t}{Pe}}\bm{\xi } + \bm{\delta }_{ex},\end{eqnarray}

\begin{matrix} x (t + Δ t) & = & x (t) + u^{'} (x (t) - x_{b} (t)) Δ t + \sqrt{\frac{6 Δ t}{P e}} ξ + δ_{e x}, \end{matrix}

(37a)

\begin{eqnarray}\hspace*{-65pt}\mathbf {x}_b(t+\Delta t) &=& \mathbf {x}_b(t) + \mathbf {p} \Delta t,\hspace*{65pt}\end{eqnarray}

\begin{matrix} x_{b} (t + Δ t) & = & x_{b} (t) + p Δ t, \end{matrix}

(37b)

in which ξ is a random vector of unit magnitude so that ⟨ξ⟩ = 0 and ⟨ξξ⟩ = (I/3) where I is the identity tensor and δ_ex is the displacement of the tracer due to excluded volume interaction with the bacterium. In Eq. lengths are scaled by the bacterium length L and velocities by the swimming speed U_s. The disturbance velocity field of the bacterium u′(x(t) − x_b(t)) is obtained by the numerical integration of Eq. . To obtain δ_ex, we first model the bacterium as a hard-sphero-cylinder as seen Fig. 3 with an overall length equal to the bacterial length L and radius equal to the radius of the cell-body d/2. The tracer particle is then assumed to be a hard-sphere with a radius R_t so that tracer-bacterium excluded volume is a hard-sphero-cylinder with a non-dimensional radius R_ex = (d/2 + R_t)/L and length L_ex = 1 + 2(R_t/L). Whenever the center of the tracer particle comes within the excluded volume zone, the tracer is pulled back along the normal direction of the surface of the excluded volume and this displacement is added to the tracer trajectory as the quantity δ_ex. We set δ_ex to be zero if the center of the tracer particle is outside the excluded volume.

FIG. 3.

$FIG. 3. Schematic of the pairwise bacterium-tracer interaction simulation. The bacterium-tracer excluded volume is modeled as a hard-sphero-cylinder with non-dimensional radius Rex = (d/2 + Rt)/L and non-dimensional length Lex = 1 + 2(Rt/L) where Rt is the radius of the tracer particle modeled as a hard-sphere. The bacterium swims along p (shown as the dashed-dotted line) chosen from an isotropic distribution at unit speed with a persistence time T chosen from the Poisson distribution given in Eq. (3) and the bacterium is stipulated to pass through a random point \documentclass[12pt]{minimal}\begin{document}$\mathbf {x}_b^{\prime }$\end{document}xb′ inside the control volume at a time t′ randomly chosen from the uniform distribution in [0, T]. This gives the starting point of the bacterial trajectory as \documentclass[12pt]{minimal}\begin{document}$\mathbf {x}_b(0) = \mathbf {x}_b^{\prime } - \mathbf {p} t^{\prime }$\end{document}xb(0)=xb′−pt′ and in the beginning (t = 0), the tracer is located at the center of the spherical control volume (of radius R) which is also the origin. At subsequent times, the tracer moves along a trajectory x(t) (shown by the broken line) that evolves by convective displacements due to bacterial disturbance field, Brownian displacements, and displacements arising from bacterium-tracer excluded volume interactions (see Eq. (37))$

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Schematic of the pairwise bacterium-tracer interaction simulation. The bacterium-tracer excluded volume is modeled as a hard-sphero-cylinder with non-dimensional radius R_ex = (d/2 + R_t)/L and non-dimensional length L_ex = 1 + 2(R_t/L) where R_t is the radius of the tracer particle modeled as a hard-sphere. The bacterium swims along p (shown as the dashed-dotted line) chosen from an isotropic distribution at unit speed with a persistence time T chosen from the Poisson distribution given in Eq. (3) and the bacterium is stipulated to pass through a random point

$\mathbf {x}_b^{\prime }$

x_{b}^{'}

inside the control volume at a time t′ randomly chosen from the uniform distribution in [0, T]. This gives the starting point of the bacterial trajectory as

$\mathbf {x}_b(0) = \mathbf {x}_b^{\prime } - \mathbf {p} t^{\prime }$

x_{b} (0) = x_{b}^{'} - p t^{'}

and in the beginning (t = 0), the tracer is located at the center of the spherical control volume (of radius R) which is also the origin. At subsequent times, the tracer moves along a trajectory x(t) (shown by the broken line) that evolves by convective displacements due to bacterial disturbance field, Brownian displacements, and displacements arising from bacterium-tracer excluded volume interactions (see Eq. )

To calculate the hydrodynamic and excluded volume contributions to the tracer diffusivity, we need to determine the influence of the bacterium on the tracer motion. Thus, in addition to the tracer position x which is influenced by both Brownian motion and the bacterium interaction, we define a displacement x_tb that results from the fluid velocity disturbance due to the bacterium and the excluded volume interaction at the position x and therefore evolves as

\begin{equation}\mathbf {x}_{tb}(t+\Delta t) = \mathbf {x}_{tb}(t)+ \mathbf {u}^{\prime }(\mathbf {x}(t) - \mathbf {x}_{b}(t))\Delta t + \bm{\delta }_{ex},\end{equation}

x_{t b} (t + Δ t) = x_{t b} (t) + u^{'} (x (t) - x_{b} (t)) Δ t + δ_{e x},

(38)

with the initial condition x_tb(0) = 0. Here the subscript “tb” stands for tracer displacement caused by the bacterium. At the end of each bacterial run occurring when t = T, the displacement of the tracer particle x_tb(T) is recorded and the evolution of all trajectories are stopped. The pair-interaction simulation is repeated to accumulate the statistics on tracer displacements and the mean-square tracer displacement induced by a single interaction ⟨|x_tb(T)|²⟩ is calculated in which the angle brackets indicate the averaging over the ensemble of interactions. We first performed simulations for R = 1 and with varying number of bacterial runs to determine the number of tracer-bacterium interactions required for a statistically accurate ⟨|x_tb(T)|²⟩ and then scale the number of bacterial runs proportional to R³ for R > 1. We found that 10⁴ interactions are sufficient to produce a standard relative error of ⟨|x_tb(T)|²⟩ below 10% for R = 1 and Pe ⩽ 250.

Now to obtain the diffusivity of the tracer particle, we make use of the definition of the tracer diffusivity in three dimensions as one-sixth of the rate of change of the mean square tracer displacement. Since there are (4/3)πR³nL³ bacteria inside the control volume of non-dimensional radius R interacting with the particle and since each bacterium tumbles at a rate of τ⁻¹, the non-dimensional tracer diffusivity due to bacterial fluid disturbances and excluded volume interactions is given by

\begin{equation}\widetilde{D_{tb,R}} = \frac{1}{6}\frac{d\langle {|\mathbf {x}_{tb}(T)|^2}\rangle }{dt} = \frac{4 \pi R^3}{3 \tau ^*} \frac{\langle {|\mathbf {x}_{tb}(T)|^2}\rangle }{6}.\end{equation}

\tilde{D_{t b, R}} = \frac{1}{6} \frac{d ⟨ | x_{t b} {(T) |}^{2} ⟩}{d t} = \frac{4 π R^{3}}{3 τ^{*}} \frac{⟨ | x_{t b} {(T) |}^{2} ⟩}{6} .

(39)

Since we are simulating only those bacterial runs which pass through the spherical control volume, the mean square displacement and the diffusivity in Eq. (39) are dependent upon the control volume radius R as indicated by the subscript R in Eq. (39). In order to find the tracer diffusivity corresponding to an unbounded suspension, we extrapolate the diffusivity calculated in Eq. (39) to the limit R → ∞ by fitting the diffusivity computed for various R with the relationship

\begin{equation}\widetilde{D_{tb,R}}=\widetilde{D_{tb}}-A \frac{\tau ^*}{R}\end{equation}

\tilde{D_{t b, R}} = \tilde{D_{t b}} - A \frac{τ^{*}}{R}

(40)

in which

$\widetilde{D_{tb}}$

\tilde{D_{t b}}

is the tracer diffusivity corresponding to an unbounded suspension and A is a constant. This expression arises from the scaling that |x_tb(T)| ∼ τ*/R² at large R for those bacteria trajectories that lie outside the control volume owing to the dipolar nature of the bacterial fluid disturbance at those tracer-bacterium separation length scales. We found that Eq. provides a good fit to simulations with R = 2.5, 3, 3.5, and 4 as shown in Fig. 4(a) for a typical case. We would like to emphasize that the non-dimensional bacteria-induced diffusivity

$\widetilde{D_{tb}}$

\tilde{D_{t b}}

obtained from the above extrapolation procedure is not purely due to the fluid disturbances created by the bacterium but also includes the effect of bacteria-tracer excluded volume interactions, reflected by the δ_ex term in Eq. . When the excluded volume radius R_ex = 0, δ_ex = 0 and the bacteria induced diffusivity obtained from simulations becomes of purely hydrodynamic origin.

FIG. 4.

$FIG. 4. (a) Extrapolation of \documentclass[12pt]{minimal}\begin{document}$\widetilde{D_{tb,R}}$\end{document}Dtb,R̃ to limit of R → ∞ to obtain \documentclass[12pt]{minimal}\begin{document}$\widetilde{D_{tb}}$\end{document}Dtb̃. Symbols are the data from simulation for a typical case and line is the linear fit given in Eq. (40). (b) Variation of autocorrelation of bacteria orientation \documentclass[12pt]{minimal}\begin{document}$\langle {\mathbf {p}_1 \cdot \mathbf {p}_N}\rangle$\end{document}⟨p1·pN⟩ with number of runs N for χ = 1.$

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(a) Extrapolation of

$\widetilde{D_{tb,R}}$

\tilde{D_{t b, R}}

to limit of R → ∞ to obtain

$\widetilde{D_{tb}}$

\tilde{D_{t b}}

⁠. Symbols are the data from simulation for a typical case and line is the linear fit given in Eq. . (b) Variation of autocorrelation of bacteria orientation

$\langle {\mathbf {p}_1 \cdot \mathbf {p}_N}\rangle$

⟨ p_{1} \cdot p_{N} ⟩

with number of runs N for χ = 1.

The simulation method for correlated bacterial runs is essentially the same as that for the case of uncorrelated bacterial runs except that the tracer-bacterium interaction is not terminated at the end of a bacterial run but allowed to continue for multiple runs until the swimming direction of the bacterium becomes nearly uncorrelated with its initial swimming direction. To determine the number of bacterial runs elapsed until the swimming direction of the bacterium becomes uncorrelated, we make use of the transition probability model of Subramanian and Koch²⁷ P_tr(p|p′)

\begin{equation}P_{tr}(\mathbf {p}|\mathbf {p}^{\prime }) = \chi /(4\pi \sinh \chi )\exp (\chi \mathbf {p} \cdot \mathbf {p}^{\prime })\end{equation}

P_{t r} (p | p^{'}) = χ / (4 π \sinh χ) \exp (χ p \cdot p^{'})

(41)

which is the conditional probability density of the bacterium having a post-tumble swimming direction near p when the pre-tumble swimming direction was near p′. P_tr satisfies the normalization condition ∫P_tr(p|p′)dp = ∫P_tr(p|p′)dp′ = 1. The degree of correlation between the pre- and post-tumble swimming directions is determined by the χ parameter. In the limit χ → 0, the run directions become uncorrelated (perfectly random runs) such that P_tr → 1/(4π) and the mean angle that the bacterium turns in a tumble ⟨θ_t⟩ → π/2 with θ_t being defined as cos (θ_t) = p · p′. On the other hand, when χ → ∞, bacterial runs become fully correlated with a negligible difference in pre- and post-tumble orientations of the bacterium such that ⟨θ_t⟩ → 0. χ ≈ 1 for a typical E. Coli cell which corresponds to a mean turn angle of ⟨θ_t⟩ ≈ 68.5° in a tumble.²⁷ We fix the number of runs as N = 8 using the criterion that the autocorrelation in bacterial orientation at the Nth run

$\langle {\mathbf {p}_1 \cdot \mathbf {p}_N}\rangle$

⟨ p_{1} \cdot p_{N} ⟩

is small. Here the orientation of the bacterium in the first run p₁ is chosen from an isotropic distribution and the orientations of the bacterium in the succeeding runs p₂ to p_N is chosen according to the distribution in Eq. . The variation of

$\langle {\mathbf {p}_1 \cdot \mathbf {p}_N}\rangle$

⟨ p_{1} \cdot p_{N} ⟩

for various N is obtained from Monte Carlo realizations of ensembles of bacterial runs each containing N consecutive runs and Fig. 4(b) shows the obtained variation for the typical E. Coli cell with χ = 1. In Fig. 4(b) we find that the correlation has indeed become negligible when N = 8. Once the N bacterial runs are sampled, we proceed to sample the corresponding run times T₁ to T_N from the Poisson distribution in Eq. . As in the case of uncorrelated tumbling, we stipulate that the bacterium should pass through a randomly chosen point

$\mathbf {x}_b^{\prime }$

x_{b}^{'}

inside the spherical control volume of radius R at some point of time during the interaction. We choose this to happen at a time t′ elapsed from the beginning of the interaction and sampled from the uniform distribution [0, T_total] where T_total = T₁ + T₂ + … + T_N is the overall duration of the interaction. As in the case of uncorrelated runs, we record the final position of the tracer at the end of the Nth run x_tb(N) and repeat the interactions and calculate the mean square displacement ⟨|x_tb(N)|²⟩. The R dependent tracer diffusivity resulting from interactions is then calculated from the following expression

\begin{equation}\widetilde{D_{tb,R}} = \frac{1}{6}\frac{d\langle {|\mathbf {x}_{tb}(N)|^2}\rangle }{dt} = \frac{4 \pi R^3}{3 N \tau ^*} \frac{\langle {|\mathbf {x}_{tb}(N)|^2}\rangle }{6} ,\end{equation}

\tilde{D_{t b, R}} = \frac{1}{6} \frac{d ⟨ | x_{t b} {(N) |}^{2} ⟩}{d t} = \frac{4 π R^{3}}{3 N τ^{*}} \frac{⟨ | x_{t b} {(N) |}^{2} ⟩}{6},

(42)

which is analogous to Eq. (39). The factor of N in the denominator in Eq. (42) is due to the fact that the frequency with which a single bacterium interacts with the tracer is τ⁻¹/N owing to the fact that each interaction lasts on an average for a duration of Nτ. Finally, we extrapolate

$\widetilde{D_{tb,R}}$

\tilde{D_{t b, R}}

to the limit of R → ∞ through the same scheme as given in Eq. .

B. Results of the simulations

All results presented in this section are with α = 2/7 and γ = 2 as representative values of the ratio of bacterial cell-body length to its overall length and the aspect ratio of the cell-body, respectively. The results from simulations are shown in Fig. 5 in which Figures 5(a) and 5(b) show the comparison of the tracer diffusivities obtained from simulations without any tracer-bacterium excluded volume interactions (R_ex = 0) and the theoretical prediction of

$\widetilde{D_h}$

\tilde{D_{h}}

made in Sec. II for varying Pe and τ^*, respectively. The comparison of diffusivities obtained from theory and these simulations gives insights about the effect of having bacterial fluid disturbances of magnitude comparable to the swimming speed of the bacterium. Symbols in both figures show data from simulations and lines show the theoretical predictions from Eq. . Circles and squares in Fig. 5(a) are for time steps Δt = 0.01 and Δt = 0.005, respectively, and the agreement between these two sets of data ensures that Δt = 0.01 is adequate for converged results. The latter time step has been used for all of the simulations results discussed. It is clear from Figs. 5(a) and 5(b) that the simulation results agree in general well with the theoretical prediction of hydrodynamic diffusivity. Since numerical simulations have a finite ratio of the fluid velocity disturbance near the bacterium to the swimming speed, the agreement between the simulations and the theoretical results from Eq. shows that the finiteness of the velocity disturbance does not have a statistically significant effect on the hydrodynamic diffusivity.

FIG. 5.

$FIG. 5. (a) Comparison of the theoretical prediction of \documentclass[12pt]{minimal}\begin{document}$\widetilde{D_h}$\end{document}Dh̃ given in Eq. (31) and simulation results \documentclass[12pt]{minimal}\begin{document}$\widetilde{D_{tb}}$\end{document}Dtb̃ (with Rex = 0) for varying Pe and time step Δt at τ* = 1. (b) Comparison the simulation prediction (at Rex = 0) of \documentclass[12pt]{minimal}\begin{document}$\widetilde{D_{tb}}$\end{document}Dtb̃ and theoretical prediction of \documentclass[12pt]{minimal}\begin{document}$\widetilde{D_h}$\end{document}Dh̃ with varying τ* for Pe = 80 . (c) Effect of correlation in bacterial runs—variation of simulation predictions of \documentclass[12pt]{minimal}\begin{document}$\widetilde{D_{tb}}$\end{document}Dtb̃ with varying χ parameter at Pe = 80 and τ* = 0.2. The dashed line is the prediction from theory corresponding to the values of Pe and τ* used. (d) Variation of overall diffusivity \documentclass[12pt]{minimal}\begin{document}$\widetilde{D_{tb}}$\end{document}Dtb̃ and hydrodynamic tracer diffusivity \documentclass[12pt]{minimal}\begin{document}$\widetilde{D_h}$\end{document}Dh̃ obtained from simulations with the non-dimensional excluded volume radius Rex. Here Rex = 0.07 (third data point from left) corresponds to a point tracer interacting with a bacterium with a finite size while the first two points correspond to tracers that can penetrate the interior of the sphero-cylinder representing the bacterium. Error bars in all plots correspond to 95% confidence interval.$

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(a) Comparison of the theoretical prediction of

$\widetilde{D_h}$

\tilde{D_{h}}

given in Eq. and simulation results

$\widetilde{D_{tb}}$

\tilde{D_{t b}}

(with R_ex = 0) for varying Pe and time step Δt at τ^* = 1. (b) Comparison the simulation prediction (at R_ex = 0) of

$\widetilde{D_{tb}}$

\tilde{D_{t b}}

and theoretical prediction of

$\widetilde{D_h}$

\tilde{D_{h}}

with varying τ^* for Pe = 80 . (c) Effect of correlation in bacterial runs—variation of simulation predictions of

$\widetilde{D_{tb}}$

\tilde{D_{t b}}

with varying χ parameter at Pe = 80 and τ^* = 0.2. The dashed line is the prediction from theory corresponding to the values of Pe and τ^* used. (d) Variation of overall diffusivity

$\widetilde{D_{tb}}$

\tilde{D_{t b}}

and hydrodynamic tracer diffusivity

$\widetilde{D_h}$

\tilde{D_{h}}

obtained from simulations with the non-dimensional excluded volume radius R_ex. Here R_ex = 0.07 (third data point from left) corresponds to a point tracer interacting with a bacterium with a finite size while the first two points correspond to tracers that can penetrate the interior of the sphero-cylinder representing the bacterium. Error bars in all plots correspond to 95% confidence interval.

The effect of the correlation in the pre- and post-tumble orientations of the bacterium is shown in Fig. 5(c) for a typical case of Pe = 80 and τ^* = 0.2, where the circles give the values of

$\widetilde{D_{tb}}$

\tilde{D_{t b}}

obtained from simulations with varying χ parameter. Increasing the value of χ results in increased correlation between the pre- and post-tumble bacterial orientations with χ = 0 corresponding to the case of uncorrelated runs (⟨θ_t⟩ = 90°) and χ = 1 corresponding to the case of typical wild-type E. Coli ⟨θ_t⟩ = 68.5°. The dashed line in Fig. 5(c) shows the prediction of tracer diffusivity τ^* = 0.2 by the theory assuming uncorrelated bacterial runs and the agreement between this theoretical prediction and the χ = 0 data point validates the multiple-run simulation method. It is clear from the figure that correlation among the pre- and post-tumble orientations of the bacterium results in a marginally larger tracer diffusivity compared to the case of uncorrelated runs.

We now proceed to investigate the effect of tracer-bacterium excluded volume interactions on the bacteria-induced diffusivity of the tracer by performing simulations with varying excluded volume radius R_ex. When R_ex > 0, the tracer cannot pass through the bacterium. When the bacterium contacts the tracer, it experiences a displacement δ_ex due to a normal force preventing bacterium tracer overlap in addition to the displacements the hydrodynamic flow caused by the bacterium and both these displacements contribute to its bacteria-induced diffusivity

$\widetilde{D_{tb}}$

\tilde{D_{t b}}

⁠. The results of the simulations with excluded volume interactions are presented in Fig. 5(d) in which circles show the variation of the non-dimensional tracer diffusivity

$\widetilde{D_{tb}}$

\tilde{D_{t b}}

with the excluded volume radius R_ex. The purely hydrodynamic contribution to the tracer diffusivity obtained from simulations with R_ex in Fig. 5(d) is given by the squares. The latter is calculated by computing the net hydrodynamic displacement of the tracer particle following a trajectory that includes the effects of the fluid disturbance, the excluded volume interaction and the Brownian motion

\begin{equation}\mathbf {x}_h(t+\Delta t) = \mathbf {x}_h(t)+ \mathbf {u}^{\prime }(\mathbf {x}(t) - \mathbf {x}_{b}(t))\Delta t,\end{equation}

x_{h} (t + Δ t) = x_{h} (t) + u^{'} (x (t) - x_{b} (t)) Δ t,

(43)

and using the definition

\begin{equation}\widetilde{D_{h,R}} = \frac{1}{6}\frac{d\langle {|\mathbf {x}_h(T)|^2}\rangle }{dt} = \frac{4 \pi R^3}{3 \tau ^*} \frac{\langle {|\mathbf {x}_h(T)|^2}\rangle }{6}.\end{equation}

\tilde{D_{h, R}} = \frac{1}{6} \frac{d ⟨ | x_{h} {(T) |}^{2} ⟩}{d t} = \frac{4 π R^{3}}{3 τ^{*}} \frac{⟨ | x_{h} {(T) |}^{2} ⟩}{6} .

(44)

The R dependent hydrodynamic diffusivity in Eq. (44) is then extrapolated to the limit R → ∞ by the same procedure as followed for determining

$\widetilde{D_{tb}}$

\tilde{D_{t b}}

⁠. In Fig. 5(d) the overall bacteria-induced tracer diffusivity

$\widetilde{D_{tb}}$

\tilde{D_{t b}}

is found to first decrease from its value at R_ex = 0 with increasing R_ex, then pass through a minimum and increase. Thus the bacteria-induced diffusivity of a point tracer (R_ex = 0.07) is smaller than the line-of-forces theoretical value owing to the excluded volume of the bacterium preventing the tracer from sampling the velocity disturbances close to the bacterial axis. The hydrodynamic component of the tracer diffusivity shown in Fig. 5(d) by squares keeps decreasing with increasing in R_ex. However, increasing R_ex further causes the tracer displacements arising from excluded volume interactions with bacteria to compensate for the reduced convective transport of the tracer and even enhance the tracer diffusivity beyond the case without any tracer-bacterium excluded volume. For instance, when R_ex = 0.3, the excluded volume contribution is as large as the hydrodynamic contribution to the diffusivity and the tracer diffusivity is larger than the case without any excluded volume.

The excluded volume contribution to the tracer diffusivity can be qualitatively understood by observing that the tracer undergoes the largest excluded volume displacement (|δ_ex|) when it is subjected to a nearly head-on collision with the bacterium. Such an interaction is illustrated in Fig. 6 in which we show the trajectory of a tracer particle in the frame of reference of the bacterium in the absence of any tracer Brownian motion. Once the tracer hits the spherical cap of the model bacterium, it slides along the periphery to the cylindrical portion of the bacterium after which the tracer does not undergo any displacement normal to the axis of the bacterium. In the laboratory frame of reference, the tracer particle would undergo a displacement along the axis of the bacterium since the bacterium pushes the tracer forward. This displacement would be comparable to R_ex since the tracer is pushed by the bacterium for a duration of the order of R_ex/U_s and with a velocity comparable to U_s. Increasing R_ex thus increases the tracer displacement normal and tangential to the bacterium resulting in an increase in the excluded volume contribution to the tracer diffusivity. In the regime when the excluded volume contribution to the tracer diffusivity is significant (R_ex > 0.1 in Fig. 5(d)) the long-range hydrodynamic disturbance of the bacterium has been found to cause a modest alteration of the interaction shown in Fig. 6. However, the overall contribution of the excluded volume interactions to the tracer diffusivity remains comparable to the hydrodynamic contribution on account of the fact that head-on collisions between bacteria and tracers are much rarer than hydrodynamic interactions. This, however, may not be true in the case of tracer diffusion in nearly-two-dimensional bacterial suspensions such as the ones used by Wu and Libchaber¹ since the confinement of the suspension leads to an increased frequency of tracer-bacterium collisions. It is thus possible in the experiment of Wu and Libchaber¹ that a significant portion of the enhancement in tracer diffusivity came from direct collisions between colloidal tracer particles and bacteria.

FIG. 6.

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A head-on collision between the tracer (sphere) and the bacterium in a frame of reference fixed on the bacterium (sphero-cylinder). The bacterium is swimming from left to right such that the tracer moves from right to left in the frame of reference of the bacterium. Brownian motion of the tracer is neglected in this picture and the hydrodynamic disturbance of the bacterium was found to have little influence on the tracer trajectory when R_ex > 0.1.

IV. EXPERIMENTAL PROCEDURE

In this section, we describe our experimental measurements of the effective tracer diffusion coefficient in the bulk of three dimensional bacteria suspensions at relatively large bacteria concentrations, nL³ in the range from 0.24 to 5.61 for a typical bacterial length L = 7 μm. The bacteria used in the experiment are from the wild-type E. Coli strain RP437 and the colloidal tracer particles are fluorescent (red) polystyrene beads of diameter 1.01 μm (Bangs Laboratories, Fishers, IN). To prepare the bacterial suspension, the cells were first grown in tryptone broth (10 g bacto-tryptone powder and 5 g sodium chloride per liter of 10 mM phosphate buffer) overnight in a rotary water bath kept at 30 °C and rotated at 200 rpm. This culture was then diluted into fresh tryptone broth by a factor of 25 and rotated again in the water bath until the bacteria entered the mid-exponential growth phase which corresponds to an optical density of the suspension OD₆₀₀ around 0.5. The cells were then centrifuged, washed, and resuspended in a motility medium consisting of 10 mM lactic acid, 0.1 mM EDTA, 1 μM L - methionine, and 50 mM glucose. Glucose was added to sustain the motility in the event that the oxygen might be depleted due to bacterial respiration especially at higher bacteria concentrations.³⁵ The final concentration of bacteria in the suspension was determined by measuring the OD₆₀₀ optical density of the suspension using a spectrophotometer calibrated by means of the Helber bacteria counting chamber (Hawksley, Sussex, UK). Fluorescent polystyrene tracer particles were added to the bacterial suspension to a final volume fraction of around 0.016%. Experiments were performed with bacterial concentrations ranging from 6.9 × 10⁸ bacteria per milliliter to 1.64 × 10¹⁰ bacteria per milliliter corresponding to non-dimensional bacterial concentration nL³ in the range from 0.24 to 5.61.

The bacteria - tracer suspension was then loaded into the three-channel microfluidic device developed by Cheng et al.³⁶ The device mainly consists of a set of three channels each of depth 170 μm and width 400 μm patterned on a 1 mm thick agarose gel membrane. Only the middle channel was loaded with the bacteria-tracer suspension and channels on either side were filled with the motility medium which helped keep the bacterial suspension oxygenated. Movies³⁷ of tracer particles were taken at the mid-plane of the channel using a fluorescent microscope (Olympus IX-51 microscope with 20× objective lens and EXFO X-Cite 120 fluorescent illumination system) at a rate of 30 frames per second using a CCD camera (Cascade 512B from Photometrics, Tucson, AZ). The positions of the tracer particles were tracked using an in-house MATLAB code²⁶ and the effective diffusivity of tracer particles was obtained from the slope of their mean square displacement versus time curve. While the inlets and outlets of all three channels were sealed by using PDMS filled pipette tips to reduce any evaporation induced drift flow, the sealing was occasionally imperfect resulting in slow, bulk flow of the suspension along the length of the channel with velocities of the order of 3 μm/s. To avoid error due to the bulk flow induced particle motion in calculating the effective diffusivity of tracer particles, we compute the mean square tracer displacement only in the transverse direction of the channel and use the one dimensional diffusion relation D_eff = lim_{t → ∞}(1/2)(d/dt)⟨y²⟩ to calculate the effective diffusivity of the tracer with y being its displacement along the transverse channel direction. The shear rate based on the drift velocity and the smallest channel dimension is of the order of 10⁻² s⁻¹ and much smaller than the bacterial tumbling frequency (≈1 s⁻¹), so that the shear re-orientation of bacteria is negligible and the suspension is nearly isotropic.

A control experiment was carried out using a suspension of tracer particles in the same suspending fluid in the absence of bacteria to measure the Brownian diffusivity of tracer particles in the motility medium. The measured Brownian diffusivity is D = 0.48 ± 0.02 μm²/s with 95% confidence level which is close to the diffusivity of the tracers D ≈ 0.45 μm²/s in water obtained from the Stokes-Einstein relationship at an ambient temperature of 22 °C. In Figs. 7(a) and 7(b) we show a comparison between typical tracks obtained for the control experiment and the case with the highest bacteria concentration (1.64 × 10¹⁰ per milliliter) and in Fig. 7(c) we show the comparison between typical mean square displacement versus time curves for those cases. Typical movies captured by the camera for the control experiment and the 1.64 × 10¹⁰ per milliliter case are available in the supplementary material.³⁷ To obtain the swimming speed U_s and the mean tumbling frequency τ⁻¹, the bacteria were induced to express green fluorescent protein using 0.5 mM of isopropyl β-D-1-thiogalactopyranoside (see Ref. 38) and the swimming cells were imaged at low bacterial concentration of 6 × 10⁸ cells per milliliter (see Ref. 37). Again to avoid errors due to drift motion in the channel, only the component of bacterial velocity in the transverse channel direction was used to compute the swimming speed and the mean run time τ. The three dimensional bacterial swimming speed is obtained from the one-dimensional measurements through the relationship that the average magnitude of the one-dimensional projection of a vector in three dimensional space is half of the magnitude of the vector if the latter is distributed isotropically. The mean run time τ has been obtained by fitting the normalized autocorrelation of the bacterial swimming velocity along the transverse direction defined as VACF = ⟨U_sy(0)U_sy(t)⟩/⟨U_sy(0)²⟩ with an exponentially decaying curve of the form A exp (−t/τ) where A is a fitting parameter as seen in Fig. 7(d). The swimming speed and the run time of bacteria measured from the experiment are U_s = 12.54 μm/s and τ = 0.78 s. These values of U_s and τ are used in Sec. V to compare the measured values of enhancement in tracer diffusivity with predictions from theory and simulations.

FIG. 7.

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Typical tracer particle tracks obtained for (a) control experiment with no bacteria and (b) a bacterial suspension with a cell concentration of 1.64 × 10¹⁰ bacteria per milliliter (c) typical mean square displacement of the tracer versus time curve for the control case (dashed line) and for the case with 1.64 × 10¹⁰ (solid line), and (d) normalized bacteria velocity autocorrelation versus time obtained from the experiment (solid line) and the exponential fit to the experimental data (dashed line).

V. COMPARISON OF THEORY AND SIMULATION WITH EXPERIMENTAL RESULTS

In this section, we first compare the slender-body description of the fluid velocity field induced by a bacterium (Eq. (25)) with the particle image velocimetry measurements of Drescher et al.²³ on swimming E. Coli cells. We then proceed to compare our measurements of tracer diffusivity obtained from the experiments described in Sec. IV with the experimental data of Jepson et al.⁵ and the predictions from our theory and simulations.

A comparison of the slender-body bacterial velocity field with direct experimental measurements is instructive since the theoretical and simulation predictions of the effective tracer diffusivity are dependent upon the description of the fluid velocity disturbance. The results of this comparison are shown in Fig. 8 where the dimensionless magnitude of the bacterial disturbance velocity |u′|/U_s obtained from the slender-body model (Eq. (25)) is plotted as a function of the non-dimensional distance from the center of the bacterium cell, r/L along the axis of the bacterium toward its front and back and along a transverse axis that passes through the center of the cell-body. The fluid velocity disturbance in the slender-body model (Eq. (25)) depends on the aspect ratio of the cell-body γ and ratio of the length of the cell-body to the overall length α. Since this information is not directly available from the observations of Drescher et al.,²³ we assume the cell-body dimensions as 2 μm × 1 μm which correspond to typical E. Coli cells and fix the overall length of the bacterium as L = 6.4 μm so that the theoretical dipole strength −αfL²/2 of the bacterium (see Eq. (27)) provides results that match the far-field fluid velocity from the experiment of Drescher et al. The overall length L = 6.4 μm is comparable with but smaller than the typical overall length of L = 7 μm for E. Coli cells. We also show the velocity field obtained from the dipole model of the bacterium with the same dipole strength in Fig. 8. The matching of the dipole strength ensures the convergence of the slender-body velocity field to that of the dipole as r/L → ∞ so that the comparison between the experimental, slender-body, and the dipole velocity fields allows us to evaluate the accuracy of the slender-body model at distances of O(L) and smaller from the bacterium. Since the velocity field has been spatially averaged over square domains of size 0.63 μm in the experimental data of Drescher et al.,²³ we also average our calculated slender-body and dipole velocity fields over the same length scale.

FIG. 8.

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Comparison of the disturbance velocity profiles—the variation of the magnitude of the disturbance velocity with the distance from the center of the bacteria cell, obtained in the experiment of Drescher et al.²³ (symbols), the slender-body model given in Eq. (25) (solid lines), and the dipole model (Eq. (27)) (broken lines) along (a) the longitudinal axis of the bacterium ahead of the cell-body (b) the longitudinal axis of the bacterium behind the back of the bacterium and (c) transverse axis of the bacterium passing through the center of its cell-body. As in Drescher et al.,²³ velocity magnitudes at each point are averages over a square domain of size 0.63 μm.

We first observe in Fig. 8(a) that the slender-body velocity profile along the axis of the bacterium is in excellent agreement with the experimental data at the front of the bacterium when r/L > 1 and it somewhat under-estimates the magnitude of disturbances at smaller distances from the bacterium. The dipole model predictions in Fig. 8(a) are very close to the slender-body model predictions everywhere. Behind the bacterium as seen in Fig. 8(b), the slender-body calculation has reasonable agreement with experimental data for r/L > 1 and, at smaller distances, the slender-body model significantly overestimates the magnitude of the fluid disturbance. Nevertheless, the slender-body model reproduces the qualitative nature of the disturbance profile when r/L < 1 in the sense that it reaches a maximum and then decreases with decreasing r/L. Note that a portion of the axis of the bacterium in the region r/L < 0.9 in Fig. 8(b) lies within the flagella bundle and since we model the bacterium as a line distribution of stokeslets along the bacterial axis, the slender-body velocity field given by Eq. (25) is actually singular (log divergent) in those regions. It is the spatial averaging of the velocity field that resulted in a non-divergent slender-body velocity profile in the region r/L < 0.9 in Fig. 8(b) and the dipole model with the same spatial averaging fails to reproduce the non-monotonic nature of the experimental profile. The increased accuracy of the slender-body model compared to the dipole model in representing the velocity disturbance of the bacterium at distances of r/L < 1 is apparent in Fig. 8(c) also in which we show the velocity profiles along a transverse axis that passes through the center of the bacterial cell-body. The difference between the slender-body, dipole, and experimental results in region r/L ⩾ 1 in Fig. 8(c) is only marginal but if we consider the full range of r/L in Figures 8(a)–8(c) the slender-body model fares better than the dipole model. This is not surprising since the slender-body model takes account of the finite length of the bacterium over which the propulsive and drag forces are distributed while the dipole model does not. This difference between these two models is also reflected in the fact that the models give different dependence of

$\widetilde{D_h}$

\tilde{D_{h}}

on τ^* in the limit of Pe ≫ 1 as we have shown in Sec. II A.

We now present our measurements of the diffusivity of colloidal tracer particles and compare the results with the data of Jepson et al.⁵ and predictions from our theory and simulations. In Fig. 9 we show the variation of the effective tracer diffusivity measured in the experiment with the bacterial concentration. The experimental data follow a linear trend except at the highest concentration and the linear fit to the data gives a value of the bacteria-induced tracer diffusivity per unit bacterial flux β = (D_eff − D)/(nU_s) = 4.81 ± 0.28 μm⁴ which compares well with the measurement β ≈ 7.1 ± 0.4 μm⁴ made by Jepson et al.⁵ in the bulk of three dimensional bacterial suspensions. The closeness of the values of the β measured in these two experimental studies is remarkable when one considers the fact that the experimental protocols and bacteria concentrations used in these two experiments differ greatly. For instance, while the tracer particles in the present experiment are polystyrene beads of diameter 1.01 μm, non-motile cells act as tracer particles in the experiment of Jepson et al.⁵ While we obtain the diffusivity through particle tracking, Jepson et al.⁵ obtain the tracer diffusivity from the measurement of the intermediate scattering function of the non-motile bacteria. The bacteria concentrations involved in the experiment of Jepson et al.⁵ are of the order of 10⁸ cells per milliliter while in the present experiment, they are as high as 10¹⁰ cells per milliliter. It must also be noted that the quantity β is sensitive to the bacterial geometrical parameters since

$\beta = L^4 \widetilde{D_h}$

β = L^{4} \tilde{D_{h}}

according to Eq. with

$\widetilde{D_h}$

\tilde{D_{h}}

itself being a function of the bacterial geometry. The bacterial strains used in our experiment and in the experiment of Jepson et al.⁵ are different and hence the possible geometrical differences between them may have contributed to the difference in the experimentally measured values of β. An important observation that we make in Fig. 9 is that the diffusivity corresponding to the highest bacteria concentration (1.6 × 10¹⁰ cells per milliliter) does not fall onto the linear trend but is much larger. While the exact reason for this deviation is not clear from our experiments, one possibility is the occurrence of multi-bacterial effects on the tracer diffusivity.

FIG. 9.

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Variation of the effective tracer diffusivity measured in the experiment with bacterial concentration.

The present experimental result therefore reveals the fact that the linear dependence of the effective tracer diffusivity measured in the bulk of the bacteria suspension on the bacteria concentration continues to much higher bacteria concentrations and that the enhancement in tracer diffusivity per bacterium compares well with the that in dilute suspensions.⁵ The question remaining is whether the hydrodynamic tracer diffusivity predicted by our theory and simulations is comparable to the experimental data. To calculate the hydrodynamic diffusivity from theory and simulations we need the geometrical parameters of the bacterium namely the overall bacterial length (L), the ratio of the length of the cell-body to the overall bacterial length (α), and the aspect ratio of the cell-body (γ) along with values of Pe and τ^* which in turn are dependent on the overall bacterial length. We obtain the geometrical parameters of the bacterium as L ≈ 7.6 μm, α ≈ 0.33, and γ ≈ 2.5 from a recent work of Min et al.³⁹ in which the authors have provided high resolution images of single cells of E. Coli RP437, the same strain that we have used in our experiments. With L = 7.6 μm and with the swimming speed and persistence time of the bacteria and the Brownian diffusivity of the tracer measured in our experiments (U_s = 12.54 μm/s, τ = 0.78 s, and D = 0.48 μm²/s), we obtain Pe ≈ 198.6 and τ^* = 1.29. For these values of Pe and τ^*, our theory predicts a hydrodynamic diffusivity of

$\widetilde{D_h} = 0.0038$

\tilde{D_{h}} = 0.0038

corresponding to our experiment and our experimental data give

$\widetilde{D_h} = 0.0014$

\tilde{D_{h}} = 0.0014

⁠. Our prediction of the value of

$\widetilde{D_h}$

\tilde{D_{h}}

corresponding to the experiment of Jepson et al.⁵ (U_s ≈ 15–25 μm/s, D ≈ 0.3 μm²/s, τ = 1s as given in their paper) is also close to

$\widetilde{D_h} = 0.004$

\tilde{D_{h}} = 0.004

if we assume that the geometry and persistence length of the bacteria used in their experiment (which are not repeated in their paper) are the same as ours. Their experimental measurements give the value of

$\widetilde{D_h}$

\tilde{D_{h}}

as

$\widetilde{D_h} = \beta /L^4 = 0.0021$

\tilde{D_{h}} = β / L^{4} = 0.0021

for the assumed value of L = 7.6 μm. The predictions from simulations with excluded volume effects and correlated bacterial runs corresponding to our experiment is

$\widetilde{D_h} = 0.0031$

\tilde{D_{h}} = 0.0031

⁠. Thus the comparison of theory, simulations, and experimental data leads to the conclusion that theory and simulations overestimate the bacterial contribution to tracer diffusivity even though the predicted order of magnitude of the tracer diffusivity matches with the experimental data.

The agreement of the order of magnitude between the predictions and the experimental measurements is encouraging when one considers the fact that our theory and simulations contain no adjustable parameters and are only approximate particularly in describing the bacterial hydrodynamic disturbance. A number of factors may influence the accuracy of the comparison. In our experiments we only measure the bulk concentration of the bacteria in the suspension through the measurement of the optical density while the examination of the vertical bacteria concentration profile in the channel revealed that the bacterial concentration at the mid-plane of the microfluidic channel where the tracer diffusivity is measured is lower than the bulk value by around 30% due to the well-known hydrodynamic-attraction-mediated accumulation of bacteria near the walls of the channel.^3,4,40 The persistence time computed from the autocorrelation of the swimming velocity of the bacterium is only approximate since the bacteria stay in the focus of the microscope only for a time duration that is comparable to persistence time. We do not explicitly identify bacterial tumbles while tracking bacteria in movies.³⁷ We have not conducted a measurement of bacterial geometrical parameters in our experiments and instead we obtained those corresponding to the bacteria that we used in our experiments from Min et al.³⁹ Thus deviations of the actual bacterial length from the assumed value of L = 7.6 μm could also have contributed significantly to the difference between the theoretical prediction and the experimental measurement. Finally, our theory and simulations rely on a simplified hydrodynamic model of the bacterium that ignores the near-field hydrodynamic effects of the bacterium.

VI. CONCLUDING DISCUSSION

In Sec. II, we have provided a theory for the transport of a Brownian tracer particle in an isotropic, homogeneous, and unbounded suspension of swimming bacteria performing run-tumble random walks. The theory is based on a Fickian constitutive law that relates the average convective flux arising from bacterial velocity disturbances with the average tracer concentration gradient through a hydrodynamic diffusion coefficient with the average being defined over all possible pair-interactions between a tracer particle and a bacterium whose fluid velocity disturbance is modeled as that due to a line distribution of forces along its axis. The primary result of the theory is an integral expression for the value of the hydrodynamic diffusion coefficient as a function of a Peclet number Pe = U_sL/D which measures the relative importance of the convective motion due to the bacterial swimming and the diffusive motion of the tracer and the non-dimensional persistence length of the bacterium τ^* = U_sτ/L. For Pe ≪ 1 and τ^* ⩾ O(1)

$D_h \sim nL^3 U_sL \sqrt{U_s^2 \tau /D}$

D_{h} \sim n L^{3} U_{s} L \sqrt{U_{s}^{2} τ / D}

and when Pe ≫ 1 and τ^* = O(1), the hydrodynamic diffusivity follows a purely convective scaling of D_h ∼ nL³U_sLϕ(τ^*) but with a dependence on τ^*. When both Pe and τ^* are much larger than one, the hydrodynamic diffusivity is independent of τ^* so that D_h ∼ nL³U_sL. While the hydrodynamic tracer diffusivity increases monotonically with the bacterial persistence length, it has a non-monotonic dependence on Pe with a peak at Pe of order 10. This peak may be attributed to the fact that the velocity disturbance of the bacterium is an odd function of relative positions and the combined effects of diffusion and convection allow the tracer to sample a less symmetric range of relative position leading to a larger net displacement. The theoretical predictions for the most relevant parametric regime of Pe ⩾ 1 and τ^* = O(1) has been verified through explicit simulations of pair interactions between bacteria and tracer particles and the simulations also revealed the weakening effect of excluded volume of the bacteria and tracer particles on hydrodynamic diffusivity. Additionally, the simulations have revealed that the correlation between the pre- and post-tumble orientations of the bacterium results in a slight increase in the tracer diffusivity compared to the uncorrelated case. Finally, our experimental measurements with suspensions of swimming bacteria and colloidal particles as tracers confirm the predicted linear dependence of the diffusivity on the bacteria concentration with a slope that is within a factor of three of the theoretical and simulation predictions. This constitutes the first attempt at comparing theoretical bacteria-induced tracer diffusivities based solely on the geometry and swimming speed of the bacterium with experimental measurements.

An important insight from this investigation is the dependence of hydrodynamic diffusivity of the tracer on its Brownian diffusivity. It is clear from Fig. 1 that the bacteria induced mixing is in general smaller for smaller tracers with relatively large Brownian diffusivity. For instance, oxygen has Pe ≈ 0.05 based on its molecular diffusivity⁴¹ D = 2000 μm²/s, the bacterial swimming speed U_s = 15 μm/s, and bacterial length L = 7 μm. For a typical tumbling frequency of τ⁻¹ = 1 s⁻¹, the dimensional hydrodynamic diffusivity of oxygen would be around 0.07nL³ μm²/s which would require physically impossible bacterial concentrations of nL³ = O(10³) in order to be comparable to the molecular diffusivity. Thus bacteria induced mixing is irrelevant for small molecules. On the other hand, hydrodynamic diffusion can be significant in the case of chemical signaling between bacteria mediated through membrane vesicles. It is known that bacteria such as E. Coli and P. aeruginosa produce membrane vesicles with diameters of the order of 100 nm which help transport inter-bacterial signaling molecules.^15,42,43 The typical Brownian diffusivity of these vesicles is of the order of 5 μm²/s based on the Stokes-Einstein relationship which yields Pe of the order of 10 for typical bacterial speed and length. The hydrodynamic diffusivity of these vesicles is then of the order of 0.5nL³ μm²/s which is comparable to their Brownian diffusivity even at moderate bacteria concentrations with nL³ = 5. Enhanced transport of chemical signals between bacteria could play an important role in signaling-dependent bacterial behavior such as quorum sensing.⁴²

We now reflect upon the assumptions that we have made to simplify the formulation of our theory and simulations and discuss how they may possibly affect the accuracy of our predictions. The most important assumptions that we have made in this paper is that the motions of bacteria are uncorrelated (no collective motion) and bacterium-tracer interactions are pairwise. We further simplified the theoretical formulation by assuming that the single-bacterial velocity disturbance that the tracer samples is small compared to the bacterial swimming speed. While the weak disturbance assumption in general holds in the far-field, at distances of the order of the bacterial diameter from the bacterium the magnitude of the fluid disturbance is comparable to the swimming speed since the disturbance velocity has to satisfy the no-slip boundary condition on the surface of the bacterium. Even though the outer slender-body velocity field employed here does not satisfy the boundary condition at the bacterial surface, the magnitude of the fluid disturbance at distances of the order of bacterial diameter is still comparable to the swimming speed and the disturbance field even becomes singular at the axis of the bacterium. Thus the weak disturbance assumption made in the theory amounts to neglecting both the near-field hydrodynamic effects of the bacterium and the finite magnitude of the slender-body velocity disturbance compared to the swimming speed near the bacterium. Our simulation study addresses the second of these two assumptions and shows that the finite magnitude of the slender-body disturbance near the bacterium has no significant effect on tracer diffusivity. However, the question of near-field hydrodynamic effects of the bacterium such as the rotational velocity field arising from the counter rotation of the cell-body and the flagella bundle and the lubrication interactions with the tracer particle are not addressed either in theory or in simulations. Thus two issues that remain to be addressed are the neglect of the hydrodynamic interactions (HI) between bacteria and the neglect of the short-range hydrodynamic effects of the bacterium and in the following we discuss these two issues.

We start with the assumption of neglecting the HI between bacteria. This is particularly important since experiments on suspensions of motile bacteria^1,44–48 have revealed the existence of organized fluid and bacterial motion in the suspension with long-ranged correlations and with speeds comparable to the bacterial swimming speed. The aforementioned dynamics termed as collective motion⁴⁹ have also been observed in numerical simulations of suspensions of hydrodynamically interacting self-propelled particles mimicking bacteria^6,7,28 and those studies along with continuum theories of self-propelled particulate suspensions^27,50–53 suggest that collective motion arise from instabilities driven by the mean-field HI between bacteria (see Subramanian and Koch²⁷). The conclusion we made in the previous paragraph that the bacteria induced mixing is irrelevant for small molecules may not hold in the presence of collective motion due to the increased magnitude of fluid disturbances and correlation lengths and times associated with the collective motion. The present investigation does not extend to those cases in which collective motion is present. However, the stability analysis of the mean-field (continuum) equations of bacterial suspensions performed by Subramanian and Koch²⁷ has shown that suspensions of wild-type bacteria would be unstable only if the bacterial concentration exceeds a critical value and as a result, one may expect the present analysis to be applicable to those suspensions with sub-critical bacteria concentrations. For the typical geometry of the bacterium (E. Coli RP437) used in our experiments (L ≈ 7.6 μm, α ≈ 0.33, and γ ≈ 2.5³⁹), the critical concentration calculated using the theory of Subramanian and Koch²⁷ is 1.1 × 10¹⁰ cells per milliliter which is near the maximum bacteria concentration reached in our experiment (see Fig. 9). Thus our theory is appropriate for the lower range of the bacterial concentrations considered in our experiment but it is possible that the deviation of the maximum concentration data point away from the linear trend in Fig. 9 is due to the onset of collective motion.

Apart from the mean-field HI between bacteria that result in hydrodynamic instabilities and collective motion, there can be direct HI between bacteria in which the disturbance field of a given bacterium alters the swimming direction and magnitude of the force per unit length ( f ) of bacteria that pass nearby. Subramanian and Koch²⁷ have shown that the induced rotation of bacteria due to pairwise HI between each other imparts a rotational diffusion to the bacteria at long times with a diffusion coefficient of D_r ≈ 10⁻⁴nL²U_s. Since our theory assumes that bacterial runs are along straight lines with a tumbling frequency of 1/τ, one may expect the HI-induced rotational diffusion of bacteria to significantly alter our theoretical and simulation predictions only if D_rτ = O(1) or larger. The typical maximum bacteria concentration reached in experiments^1,3 including ours is of the order of 10¹⁰ cells per milliliter for which D_rτ is too small (at most O(10⁻³)) to have any significant effect on tracer diffusivity. Now an estimate of the effect of direct HI on the magnitude of the force per unit length f on bacteria can be made by reference to literature on passive fibers. Mackaplow and Shaqfeh⁵⁴ derived a theory for the effective viscosity of passive fibers resulting from two fiber hydrodynamic interactions. For the fiber concentration equal to the typical maximum bacteria concentration of 10¹⁰ cells per milliliter and for the fiber aspect ratio equal to the overall bacterial aspect ratio (defined as L/d) of order 10, the maximum two-particle correction to the effective viscosity is around 19% of the single-particle contribution. It is thus clear from the above discussion that HI between bacteria can only have a modest effect on tracer diffusivity if the bacterial concentration is smaller than the critical concentration required for collective swimming.

We now consider the assumption of neglecting the near-field hydrodynamic effects of the bacterium such as lubrication interactions with the tracer particle in our theory and simulations. Based on the literature on the mechanics of Stokesian particulate suspensions, we do not expect the near-field hydrodynamic effects between the tracer and the bacterium to result in order-of-magnitude changes in our predictions of tracer diffusivity. Such studies have typically found the effects of near-field hydrodynamic interactions to be comparable with those of excluded volume interactions which are retained in our simulations. For example, Foss and Brady⁵⁵ have calculated the long-time self-diffusivity of hard-spherical particles in equilibrium and sheared suspensions using Brownian dynamics simulations that neglect HI between particles and using Stokesian dynamics simulations that accounts for full HI between particles including the lubrication interactions and the predicted values of diffusivities with and without HI differ in that study by less than a factor of two. Another example is Swan and Zia's⁵⁶ calculation of the effective viscosity of hard-sphere suspensions using constant velocity active microrheology in which the mean drag force acting on a probe particle dragged within the suspension at a constant velocity is used to obtain the effective suspension viscosity. This is somewhat analogous to the present problem in the sense that it involves a translating particle creating hydrodynamic disturbances interacts with passive tracer particles. Calculations of Swan and Zia⁵⁶ show that the effective suspension viscosity with and without near-field HI between the translating particle and passive particles differ by less than a factor of four. Thus in the light of these investigations, we expect our theory and simulations to predict the correct order of magnitude of the tracer diffusivity and this is supported by our experimental data.

ACKNOWLEDGMENTS

This work is supported by NSF Grant CBET-1066193.

APPENDIX: ASYMPTOTIC ANALYSIS OF THE HYDRODYNAMIC DIFFUSIVITY FOR LOW AND HIGH PECLET NUMBERS

1. Asymptotic evaluation of |$\widetilde{D_h}$| $\tilde{D_{h}}$ for Pe ≪ 1 and τ^* ⩾ O(1)

We first rewrite Eq. (31) as an integral over a spherical domain defined by k · p = k cos θ as

\begin{equation}\widetilde{D_h} = \frac{\tau ^*}{96\pi ^4 M^2}\int d\mathbf {p} \int _0^{2\pi } d\phi \int _0^{\pi } d\theta \int _0^{\infty } dk \frac{\sin ^3 \theta F(k\cos \theta )F(-k\cos \theta )}{k^2 \left(1+ik \tau ^* \cos \theta + \frac{\tau ^*}{Pe}k^2\right)}.\end{equation}

\tilde{D_{h}} = \frac{τ^{*}}{96 π^{4} M^{2}} \int d p \int_{0}^{2 π} d ϕ \int_{0}^{π} d θ \int_{0}^{\infty} d k \frac{\sin^{3} θ F (k \cos θ) F (- k \cos θ)}{k^{2} (1 + i k τ^{*} \cos θ + \frac{τ^{*}}{P e} k^{2})} .

(A1)

The integral over p and ϕ together yields 8π² and since the product F(k cos θ)F(−k cos θ) = α²k² cos ² θ/4 + O(k⁴) for k ≪ 1, the dominant contribution to the integral for the case with Pe ≪ 1 and τ^* ⩾ O(1) comes from

$k = O(\sqrt{Pe/\tau ^*})$

k = O (\sqrt{P e / τ^{*}})

⁠. Thus at the leading order we simply have

\begin{equation}\widetilde{D_h} = \frac{\alpha ^2 \tau ^*}{48\pi ^2 M^2} \int _0^{\pi } d\theta \sin ^3 \theta \cos ^2 \theta \int _0^{\infty } \frac{dk}{1+\frac{\tau ^*}{Pe}k^2} = \frac{\alpha ^2\sqrt{Pe\tau ^*}}{360\pi M^2}.\end{equation}

\tilde{D_{h}} = \frac{α^{2} τ^{*}}{48 π^{2} M^{2}} \int_{0}^{π} d θ \sin^{3} θ \cos^{2} θ \int_{0}^{\infty} \frac{d k}{1 + \frac{τ^{*}}{P e} k^{2}} = \frac{α^{2} \sqrt{P e τ^{*}}}{360 π M^{2}} .

(A2)

2. Asymptotic evaluation of |$\widetilde{D_h}$| $\tilde{D_{h}}$ for Pe ≫ 1 and τ^* ≫ 1

The leading contribution to the integral in Eq. (31) can be calculated by simply dropping the k²/Pe term in Eq. (31) and taking the limit of the integral for τ^* → ∞. The resulting expression when written as an integral over cylindrical coordinates (k_∥, k_⊥) with the k_∥ axis along p takes the form

\begin{equation}\widetilde{D_h} = \lim _{\tau ^* \rightarrow \infty } \frac{1}{12 \pi ^2 M^2}\int _{-\infty }^{\infty } dk_{\parallel } \int _{0}^{\infty }d k_{\perp } \frac{k_{\perp }^3 F(k_{\parallel })F(-k_{\parallel })}{\left(k_{\parallel }^2+k_{\perp }^2\right)^3\left[\frac{1}{\tau ^*}+ik_{\parallel }\right]} .\end{equation}

\tilde{D_{h}} = \lim_{τ^{*} \to \infty} \frac{1}{12 π^{2} M^{2}} \int_{- \infty}^{\infty} d k_{∥} \int_{0}^{\infty} d k_{⊥} \frac{k_{⊥}^{3} F (k_{∥}) F (- k_{∥})}{{(k_{∥}^{2} + k_{⊥}^{2})}^{3} [\frac{1}{τ^{*}} + i k_{∥}]} .

(A3)

The integral over k_⊥ space in Eq. (A3) is equal to

$1/(4k_{\parallel }^2)$

1 / (4 k_{∥}^{2})

so that the integral in Eq. can be written in the following form by multiplying both the numerator and the denominator by the complex conjugate of the denominator 1/τ^* − ik_∥

\begin{equation}\widetilde{D_h} = \lim _{\tau ^* \rightarrow \infty } \frac{1}{48 \pi ^2 M^2}\int _{-\infty }^{\infty } dk_{\parallel }\frac{\left[\frac{1}{\tau ^*}-ik_{\parallel }\right]F(k_{\parallel })F(-k_{\parallel })}{k_{\parallel }^2 \left[\frac{1}{{\tau ^*}^2}+k_{\parallel }^2\right]}.\end{equation}

\tilde{D_{h}} = \lim_{τ^{*} \to \infty} \frac{1}{48 π^{2} M^{2}} \int_{- \infty}^{\infty} d k_{∥} \frac{[\frac{1}{τ^{*}} - i k_{∥}] F (k_{∥}) F (- k_{∥})}{k_{∥}^{2} [\frac{1}{{τ^{*}}^{2}} + k_{∥}^{2}]} .

(A4)

Since the product F(k_∥)F(−k_∥) is even in k_∥, the imaginary part of the integral in Eq. (A4) vanishes and it is clear from the denominator of the integrand in Eq. (A4) that in the limit of τ^* → ∞ the dominant contribution to the integral comes from the region of k_∥ = O(1/τ^*). We now rescale k_∥ by 1/τ^* to yield the variable

$\bar{k}_{\parallel } = k_{\parallel }\tau ^*$

{\bar{k}}_{∥} = k_{∥} τ^{*}

and rewrite Eq. in terms of

$\bar{k}_{\parallel }$

{\bar{k}}_{∥}

as

\begin{equation}\widetilde{D_h} = \lim _{\tau ^* \rightarrow \infty } \frac{{\tau ^*}^2}{48 \pi ^2 M^2}\int _{-\infty }^{\infty } d\bar{k}_{\parallel }\frac{F(\bar{k}_{\parallel }/\tau ^*)F(-\bar{k}_{\parallel }/\tau ^*)}{\bar{k}_{\parallel }^2\left[1+\bar{k}_{\parallel }^2\right]}.\end{equation}

\tilde{D_{h}} = \lim_{τ^{*} \to \infty} \frac{{τ^{*}}^{2}}{48 π^{2} M^{2}} \int_{- \infty}^{\infty} d {\bar{k}}_{∥} \frac{F ({\bar{k}}_{∥} / τ^{*}) F (- {\bar{k}}_{∥} / τ^{*})}{{\bar{k}}_{∥}^{2} [1 + {\bar{k}}_{∥}^{2}]} .

(A5)

The dominant contribution to the integral in Eq. (A5) comes from

$\bar{k}_{\parallel } = O(1)$

{\bar{k}}_{∥} = O (1)

and hence in the limit τ^* → ∞, one may replace the product

$F(\bar{k}_{\parallel }/\tau ^*)F(-\bar{k}_{\parallel }/\tau ^*)$

F ({\bar{k}}_{∥} / τ^{*}) F (- {\bar{k}}_{∥} / τ^{*})

by its Taylor series near

$\bar{k}_{\parallel } = 0$

{\bar{k}}_{∥} = 0

⁠,

$F(\bar{k}_{\parallel })F(-\bar{k}_{\parallel }) = \alpha ^2 \bar{k}_{\parallel }^2/(4{\tau ^*}^2)$

F ({\bar{k}}_{∥}) F (- {\bar{k}}_{∥}) = α^{2} {\bar{k}}_{∥}^{2} / (4 {τ^{*}}^{2})

to yield

\begin{equation}\widetilde{D_h} = \lim _{\tau ^* \rightarrow \infty } \frac{\alpha ^2}{192 \pi ^2 M^2}\int _{-\infty }^{\infty } d\bar{k}_{\parallel }\frac{1}{1+\bar{k}_{\parallel }^2} =\frac{\alpha ^2}{192 \pi M^2}.\end{equation}

\tilde{D_{h}} = \lim_{τ^{*} \to \infty} \frac{α^{2}}{192 π^{2} M^{2}} \int_{- \infty}^{\infty} d {\bar{k}}_{∥} \frac{1}{1 + {\bar{k}}_{∥}^{2}} = \frac{α^{2}}{192 π M^{2}} .

(A6)

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AIP Publishing LLC

Hydrodynamic tracer diffusion in suspensions of swimming bacteria

I. INTRODUCTION

II. AVERAGED EQUATION THEORY FOR HYDRODYNAMIC DIFFUSION

A. Theoretical results

III. CALCULATION OF |$\widetilde{D_h}$| $\tilde{D_{h}}$ FROM NUMERICAL SIMULATIONS OF PAIRWISE TRACER-BACTERIUM INTERACTIONS

A. Simulation methodology

B. Results of the simulations

IV. EXPERIMENTAL PROCEDURE

V. COMPARISON OF THEORY AND SIMULATION WITH EXPERIMENTAL RESULTS

VI. CONCLUDING DISCUSSION

ACKNOWLEDGMENTS

APPENDIX: ASYMPTOTIC ANALYSIS OF THE HYDRODYNAMIC DIFFUSIVITY FOR LOW AND HIGH PECLET NUMBERS

1. Asymptotic evaluation of |$\widetilde{D_h}$| $\tilde{D_{h}}$ for Pe ≪ 1 and τ^* ⩾ O(1)

2. Asymptotic evaluation of |$\widetilde{D_h}$| $\tilde{D_{h}}$ for Pe ≫ 1 and τ^* ≫ 1

REFERENCES

Supplementary Material

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Hydrodynamic tracer diffusion in suspensions of swimming bacteria

I. INTRODUCTION

II. AVERAGED EQUATION THEORY FOR HYDRODYNAMIC DIFFUSION

A. Theoretical results

III. CALCULATION OF |$\widetilde{D_h}$|Dh̃ FROM NUMERICAL SIMULATIONS OF PAIRWISE TRACER-BACTERIUM INTERACTIONS

A. Simulation methodology

B. Results of the simulations

IV. EXPERIMENTAL PROCEDURE

V. COMPARISON OF THEORY AND SIMULATION WITH EXPERIMENTAL RESULTS

VI. CONCLUDING DISCUSSION

ACKNOWLEDGMENTS

APPENDIX: ASYMPTOTIC ANALYSIS OF THE HYDRODYNAMIC DIFFUSIVITY FOR LOW AND HIGH PECLET NUMBERS

1. Asymptotic evaluation of |$\widetilde{D_h}$|Dh̃ for Pe ≪ 1 and τ* ⩾ O(1)

2. Asymptotic evaluation of |$\widetilde{D_h}$|Dh̃ for Pe ≫ 1 and τ* ≫ 1

REFERENCES

Supplementary Material

Citing articles via

Submit your article

Sign up for alerts

Related Content

Resources

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Connect with AIP Publishing

This Feature Is Available To Subscribers Only

III. CALCULATION OF |$\widetilde{D_h}$| $\tilde{D_{h}}$ FROM NUMERICAL SIMULATIONS OF PAIRWISE TRACER-BACTERIUM INTERACTIONS

1. Asymptotic evaluation of |$\widetilde{D_h}$| $\tilde{D_{h}}$ for Pe ≪ 1 and τ^* ⩾ O(1)

2. Asymptotic evaluation of |$\widetilde{D_h}$| $\tilde{D_{h}}$ for Pe ≫ 1 and τ^* ≫ 1