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Impact of external flow on the dynamics of swimming microorganisms near surfaces
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 115101 (http://iopscience.iop.org/0953-8984/26/11/115101) View the table of contents for this issue, or go to the journal homepage for more
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 115101 (8pp)
doi:10.1088/0953-8984/26/11/115101
Impact of external flow on the dynamics of swimming microorganisms near surfaces Sandeep Chilukuri, Cynthia H Collins and Patrick T Underhill Howard P Isermann Department of Chemical and Biological Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180, USA Center for Biotechnology and Interdisciplinary Studies, Rensselaer Polytechnic Institute, Troy, NY 12180, USA E-mail: [email protected] Received 8 September 2013, revised 12 December 2013 Accepted for publication 16 January 2014 Published 3 March 2014
Abstract
Swimming microorganisms have been previously observed to accumulate along walls in confined systems both experimentally and in computer simulations. Here, we use computer simulations of dilute populations for a simplified model of an organism to calculate the dynamics of swimmers between two walls with an external fluid flow. Simulations with and without hydrodynamic interactions (HIs) are used to quantify their influence on surface accumulation. We found that the accumulation of organisms at the wall is larger when HIs are included. An external fluid flow orients the organisms parallel to the fluid flow, which reduces the accumulation at the walls. The effect of the flow on the orientations is quantified and compared to previous work on upstream swimming of organisms and alignment of passive rods in flow. In pressure-driven flow, the zero shear rate at the channel center leads to a dip in the concentration of organisms in the center. The curvature of this dip is quantified as a function of the flow rate. The fluid flow also affects the transport of organisms across the channel from one wall to the other. Keywords: swimming microorganisms, accumulation, hydrodynamic interactions, upstream swimming (Some figures may appear in colour only in the online journal)
1. Introduction
Swimming microorganisms are often found in confined spaces or on surfaces, where attachment to surfaces and biofilm formation can have detrimental effects, ranging from infection to biofouling and corrosion [13]. An early observation that swimming organisms tend to concentrate near surfaces was made by Rothschild, who observed that the density distribution of bull spermatozoa between two glass surfaces was non-uniform [14]. Subsequent studies have shown similar observations in dilute populations of swimming microbial cells such as Escherichia coli and Caulobacter crescentus. Simulation studies have shown an accumulation near the walls without external flow even for spherical particles [15]. This paper will focus on the dynamics of swimming organisms in an external flow. A number of researchers have previously examined how external flows can alter the distribution and dynamics of self-propelled particles. For example, Zilman
Motility is required for a range of important biological processes, including chemotaxis [1], reproduction [2] and infection [3]. While these processes have been extensively studied from the biological perspective, the dependence of these processes on the mechanical principles of locomotion at the micron scale is not fully understood. Many experimental and theoretical studies on swimming microorganisms have reported collective behavior [4, 5], enhanced transport [6–8], interesting rheological properties [9], and migration towards surfaces [10–12]. A key challenge in this area is to tie our physical understanding of these observations across different scales (from the levels of individual cells to large cell populations) to biological processes. 0953-8984/14/115101+08$33.00
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et al [16] experimentally observed the dependence of an external flow on the trajectories of spherical larvae, Bugula neritina, in the vicinity of a collector surface. ‘Cellular flows’ have also been used to examine the dynamics and ‘aggregation’ in Newtonian [17] and non-Newtonian [18] suspending fluids. The focus of this paper is the dynamics of self-propelled particles in Poiseuille and Couette flow between two parallel walls. Z¨ottl and Stark have previously examined the dynamics of a microswimmers, both spherical [19] and elongated [20], in Poiseuille flow and report that organisms exhibited periodic motions of their positions and orientations when there was no translational or rotational Brownian motion. We will use some of these results in this paper, as described later. The key contribution of our work, which builds on theirs by the addition of Brownian motion, is we examine the probability density distribution of the organisms, which cannot be calculated if Brownian motion is not considered. Two key models and theories have been developed to explain the surface accumulation in dilute suspensions of rodlike microorganisms. In the mechanism suggested by Berke et al [10], the microorganisms are assumed to be oriented parallel to the surface and hydrodynamic interactions (HIs) between the cells and the surface cause the accumulation. From the mechanism suggested by Li et al [11, 12], the accumulation results from a relatively larger flux of organisms swimming into and colliding with a wall than the flux of organisms that leave the wall by rotational Brownian motion. Neither of these models consider the role of an external fluid flow. Since swimming microbes are found in environments with a wide range of flow types and velocities, it is essential that we examine how flow affects motility and surface accumulation of swimming organisms. For microorganisms that can be represented as slender objects (including both the body and flagellar bundle), it is expected that the flow will align the organisms parallel to the flow. In the mechanism suggested by Berke et al, the microorganisms are assumed to be oriented parallel to the surface, so we would expect qualitatively similar observations under conditions with and without an external flow. In the mechanism suggested by Li et al, flow should reduce both the collisions with the wall and the rate of leaving the wall. Therefore, the external flow could significantly change the accumulation of swimming organisms near the surface. Within the context of these two mechanisms, it is difficult to predict beforehand how the accumulation of selfpropelled particles (such as microorganisms) will be altered by an external flow, which is one of the goals of this study. In this paper, changes in the dynamics of swimming microorganisms in confined geometries in the presence of external flow are addressed. Using two systems of modeled freely swimming microorganisms, with and without HI between the cells and surface, we look at the surface accumulation of swimming microorganisms in two simple flows, uniform shear and a parabolic flow. In addition to location, the orientations of the organisms are quantified. Experimental observations [21, 22] and computer simulations [23] of microorganisms under flow have shown that they swim along surfaces, causing upstream migration. How the flow affects the orientations of the swimmers is important for our overall understanding of the accumulation of organisms at the walls and the movement of organisms from one wall to the other.
Figure 1. Representation of the swimmer, the different regions
between the surfaces and the calculation of hydrodynamic interactions (HIs) between the swimmer and walls. (Left) The regions I, II and III represent the regions near the wall, an intermediate region between the wall and the center, and region at the center, respectively. (Center) e is a unit vector in the direction of swimming. (Right) The body bead (b) and flagellum bead (f) of a swimmer and the images of the body bead (b1, b2) and flagellum bead (f1, f2) in the two walls.
2. Swimmer model
Our simplified model of a swimmer consists of two beads connected by a very stiff spring with equilibrium length l. The size of the swimmer changes by less than 8% even in the strongest flow rate considered here. One bead (body bead) represents the center of the body and the other (flagellum bead) the center of the flagellar bundle, as shown in figure 1. The orientation of each swimmer is denoted by a unit vector e. We integrate forward in time using the Brownian dynamics method [24]. A force balance for each bead is written as Fh + Fbr + Fnh = 0, neglecting inertia due to their small size [25]. The forces are the hydrodynamic force Fh , the Brownian force Fbr , and the non-Brownian and non-hydrodynamic forces Fnh , which includes the spring force and the excluded volume forces between swimmers and between a swimmer and a wall. The drag force on the body bead is given by the Stokes’ law Fh,b = −6π ηa[˙r − u(r)], where a is the bead radius, η is the fluid viscosity, r is the position of the bead and u(r) is the fluid velocity at r. For the flagellum bead, the drag force is given by Fh, f = −6π ηa[˙r − u(r) − 2vsw e], where vsw is the isolated swimming speed of an organism. This additional term is equivalent to the flagellum force used by Hernandez-Ortiz et al [26] and arises from the torque on the flagellar bundle. The force balance equations are rewritten as stochastic differential equations, where the Brownian forces are determined such that the system satisfies the fluctuation–dissipation theorem [27] using Fixman’s mid-point algorithm [28]. The Brownian forces must be included for the swimmers to reach a steady state distribution across the channel. Both tumbling and rotational diffusion lead to changes in the orientation of an organism over a particular timescale. Previous work by Saintillan [29, 30] that included both mechanisms showed that an external flow orients the organisms in qualitatively the same way for each mechanism. Therefore, only rotational Brownian motion is included in our simulations. We consider two different values for the distance between the walls (confinement), 2L = 10l and 16l. The domain is periodic in the x and y directions, which are parallel to the walls with side 2L. These separation of the walls, 2
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which correspond to 10l ≈ 100 µm and 16l ≈ 160 µm for E. coli, are consistent with experimental and theoretical studies without flow. In the model of Li and Tang [12], which did not include HI or external flow, a key parameter is the persistence length, which is the characteristic length over which the organism trajectories are straight and equals L p = vsw /Dr , where Dr is the rotational diffusivity. If (2L)/ L p < 1, significant accumulation is expected without flow. For the two confinements here, (2L)/L p = 0.25, 0.40. A potential-free algorithm, which is a modified version of the Heyes–Melrose hard-sphere algorithm [31], is used for bead–wall interactions, where a bead is positioned at a distance a 0 = 7a/6 when it tries to penetrate through the wall during a timestep. The body beads of the swimmers interact by a Weeks–Chandler–Andersen potential: ( σ 12 σ 6 − 4 + 4 UWCA = r r 0
for r < 21/6 σ for r ≥ 21/6 σ
small in comparison to the contribution from the first images (the image flow field decays as 1/r 2 ). Torques on each bead that lead to circular motion of organisms near a wall [34] are not included since they should not have a large impact on the quantities calculated here. 3. Computational results
We have calculated the impact of Couette and Poiseuille flow on the dynamics of the organisms. Using the coordinate system defined in figure 1, the two flows are vx (z) = γ˙w z for a uniform shear and vx (z) = γ˙w L(1 − (z/L)2 )/2 for a parabolic flow, where γ˙w is the shear rate at the wall. If the rate of shear is faster than the organisms change their directions by rotational diffusion, the organisms will be aligned parallel to the flow. In suspensions of passive particles, this competition leads to a key dimensionless group that quantifies the strength of the external flow, called the rotational Peclet number. Because of the importance of the orientational distribution of the swimming microorganisms, we quantify the external flow as Per = γ˙w /Dr , where the wall shear rate is used and the rotational diffusivity Dr of a swimmer is calculated from the orientational correlation function of a swimmer in a simulation in an unbounded domain. Even though the fluctuating forces are small at high Per , the singular nature of the limit causes Brownian motion of the swimmers to still be important. A system without Brownian motion will not produce a steady state angular distribution, and it is Brownian motion (in the flow gradient direction) that allows the swimmers to move between the walls when they are aligned by the flow. For the two confinements 2L = 10l and 16l and both Couette and Poiseuille flow types, the distribution of the swimmers between the walls (section 3.1), the orientations of the swimmers (section 3.2) and the dynamics of the swimmers across the channel (section 3.3) are discussed for a wide range of flow rates. Quantification of the swimmer distributions and orientations of the swimmers was done to understand the importance of HI and the influence of flow on the alignment of the organisms.
(1)
where r is the distance between two body beads, σ = 2a is the diameter of a bead, and = 2a f /48 is the potential well depth. The flagellum beads are assumed to have no excluded volume interactions with beads of any other swimmer. The volume N 4 πa 03
3 . The values here are fraction is defined by φe = (2L)(2L)(2L) N = 1500 swimmers and φe = 0.03 (for confinement 10l) and φe = 0.007 (for confinement 16l). The ratio of bead radius (excluded volume) to the swimmer size is a 0 /l = 1/6, which is a close representation of most flagellated microorganisms. HI between a swimmer and the surface is one mechanism for accumulation at the surface. These are calculated using Blake’s solution for the flow caused by a point force near a no-slip boundary [32]. This is obtained by the superposition of the flow due to the point force and the flow due to an image system, located on the other side of the boundary, to satisfy the no-slip condition at the boundary. From previous work including HI between organisms [33], we can safely ignore HI between organisms for the relatively dilute concentrations considered here and include only the flow due to a swimmer’s own images. Specifically, the impact of those interactions would be to affect the translational and rotational motion of the organisms. For the concentrations used here, those contributions are smaller than the intrinsic motions of the individuals. Note that there are equal and opposite point forces exerted by a swimmer on the fluid at the positions of the body bead, b, and flagellum bead, f (the net force on the fluid is zero). As shown in figure 1, the net flow at the location of the body bead (b) of a swimmer is the sum of the flow due to the flagellum bead (f), images of the body bead and flagellum bead from the two surfaces (b1, b2, f1, f2), and any external flow. This can be written as
3.1. Distribution across the channel
Figure 2 shows the probability density of swimmers between the two walls for 2L = 10l at a relatively high shear rate of Per = 400 for both Couette and Poiseuille flows compared with the no flow case. We observed qualitatively similar results for the simulations performed without excluded volume interactions between the body beads (which are essentially multiple single swimmer simulations). Therefore, the data and mechanisms described here represent the response seen in the dilute limit. A key feature of the data is that the accumulation at the wall is smaller with flow than without. We also examined the role of HI and found that in the presence of these interactions (figure 2(a)), there is an increased accumulation of the swimmers at the walls compared to the system without them (figure 2(b)). The other key feature of the data is that, in parabolic flow, there is a dip in the probability density in the center of the channel. In order to understand the influence of
u(b) = uf (b) + ub1 (b) + ub2 (b) + uf1 (b) + uf2 (b) + uext , (2) where uf (b) is the flow at b due to the flagellum bead f, uext is the external flow, ub1 (b) is the flow at b due to the image b1, etc. The flagellum bead (f) only sees the net flow from b, b1, b2, f1 and f2. Multiple reflections (reflections of images in the two walls) are not considered here as their contribution is very 3
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Figure 2. The distribution of swimmers between the walls with (a) and without (b) HI in a confinement 2L = 10l. The density distribution of swimmers for different flow conditions is shown—no external flow (black-circles), Poiseuille flow (red-dashed line), Couette flow (blue-solid line). The flow rate corresponds to Per = 400. The dotted-green line represents the uniform distribution of swimmers between the walls.
Figure 3. Quantifying the excess number of swimmers at the wall
using a normalized area under the peaks from the swimmer density distribution at different flow rates with HI (a) and without HI (b) between the swimmers and the walls. The flow types and confinements are: Couette at 2L = 10l (blue circles), Couette at 2L = 16l (blue triangles), Poiseuille at 2L = 10l (red stars), and Poiseuille at 2L = 16l (red squares).
shear rate, separation of the walls, flow type, and HI, we quantify the peaks in the distribution at the walls. To do this, the area under the probability density is calculated for positions within a distance of l of the wall. The accumulation of swimmers when there is no external flow is approximately of width l (figure 2). This area is normalized by the value of the area if the organisms were uniformly distributed within the channel. Figure 3 shows the normalized area of the peaks as a function of flow rate for two confinements 2L = 10l and 2L = 16l, for both Couette and Poiseuille flow, and for both with (figure 3(a)) and without HI (figure 3(b)). Note that the normalized peak area can drop below 1. This results from the excluded volume interactions between the organism and the wall, which gives a zero probability for the cell to be at the wall. This zero probability reduces the area under the curve. The area under the peak, which quantifies the accumulation at the wall, is always larger with HI than without. In all cases, the size of the peak at the wall is maximum for no flow, begins to decrease when Per is between 1 and 10, and asymptotes for Per above 100. The results in figure 2 (at Per = 400) are in this high flow region. The fact that the change in accumulation occurs when Per rises above 1 suggests that the mechanism for the change is the alignment of the organisms parallel to the flow direction. The similarity of the curves for different confinements and flow types when plotted versus Per also suggests that the rotational Peclet number is the key dimensionless group. The alignment of the organisms reduces their collisions with the wall and their swimming away from the wall. The only mechanism for
transport across the channel is passive Brownian motion or HI with the wall, which produces a smaller accumulation at the wall at high flow rates. For Poiseuille flow the probability densities show a dip in the center of the channel where the shear rate is zero. Near the channel center, the probability is parabolic in shape, and we can use the curvature of that parabola to quantify the dip in the center. The probability distribution is fit with m 1 (1 + m 2 (z/2L)2 ) over the region z/2L = [−0.15, 0.15]. The parameter m 1 is affected by the normalization of the probability but the parameter m 2 quantifies the shape. Figure 4 shows how this curvature varies with flow rate for two confinements 2L = 10l and 2L = 16l and with and without HI. For small flow rates, there is a small slight positive curvature. As the flow rate increases, the curvature drops just below zero before increasing dramatically at large flow rates. The quantification of the curvature (dip) at high flow rates is the key feature of this data. In this region, HI with the walls does not influence the curvature, but the confinement does. Figure 4 shows the data with the flow rate scaled such that the different confinements collapse onto a single curve. The important measure of the flow rate is not the rotational Peclet number, but v f = γ˙w L/(2vsw ), which is the ratio of the fluid velocity at the channel center (γ˙w L/2) to the swimming speed of an organism (vsw ). The inset to the figure shows the data plotted versus rotational Peclet number, which confirms explicitly that the data do not collapse when plotted in that way. 4
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the distribution of values of C0 ; the swimmer density is then an average over these orbits. However, it is not necessary to determine the distribution of C0 and perform the average in order to understand the dependence of the probability on position or flow rate. Since each orbit produces a P(z) with the same functional form, we expect the simulations to produce that functional form. This is confirmed in figure 4, in which the curvature is proportional to v f for large flow rates. 3.2. Swimmer orientations
Because an organism swims in the direction of its orientation, there is a coupling between the orientation and position of a swimmer. In figure 3, the accumulation at the walls was shown to be determined by the rotational Peclet number, since the flow aligns the organisms with the flow, thereby affecting the positions (accumulation). Recent experiments and simulations showing upstream swimming of E. coli in an external flow have highlighted the importance of examining how orientation is affected by flow [21–23]. While some mechanisms to explain this phenomenon rely on a hydrodynamic difference between the cell body and flagellar bundle, simulations have indicated that upstream motility can occur even for symmetric organisms. This is particularly interesting, since it shows that swimming of a symmetric object can lead to an asymmetric distribution of orientations. Motivated by the density distribution in figure 2, we can identify three regions across the channel as shown in figure 1: region I of width l near the walls, corresponding to the accumulated organisms at the wall, region III of width 2l in the channel center, which consists of organisms within the dip in Poiseuille flow, and region II, which corresponds to the organisms between the wall and the center. In order to quantify how much the swimmers are oriented parallel to the flow, we compute the dot product of the orientation of the organism e with a vector that is parallel to the flow direction, defined here as p. As shown in figure 5, we define p such that it retains the symmetry of the flow; Couette flow has a rotational symmetry between the two sides, while Poiseuille flow has a mirror symmetry. Specifically, we define the unit vector p as p = n × (∇ × v)/k∇ × vk, where ∇ × v is the flow vorticity. The unit vector n is in the negative z-direction for z > 0 and in the positive z-direction for z < 0. Figure 5 shows the distribution of swimmer orientations, quantified as p · e, for a confinement of 2L = 10l at Per = 400 for the two types of flow and the three regions within the channel. At the walls (region I), swimmers are found to be in the direction of p, which is in the direction of the flow in uniform shear and against in parabolic flow. This shows the same upstream motility seen in experiments and simulations in parabolic flow. The fact that organisms point in the direction of p at the walls shows that the flow vorticity (and not the flow direction) determines the orientation of the swimmers. Away from the walls (regions II and III), the organisms in Couette flow are equally likely to be oriented in the positive or negative p direction. This is not surprising for our model, in which the body and flagellum beads have the same hydrodynamic drag coefficient. However, in Poiseuille flow we find that the swimmers in the center are directed preferentially in the
Figure 4. Quantifying the dip in the swimmer distribution at the
channel center in Poiseuille flow using the curvature of the distribution m 2 . The value of m 2 is shown with HI at 2L = 10l (black squares), with HI at 2L = 16l (black stars), without HI at 2L = 10l (red circles), and without HI at 2L = 16l (red triangles). The dashed line is a power law of +1. Inset: the same data shown as m 2 versus rotational Peclet number for the two confinements 2L = 10l and 2L = 16l, showing that the data do not collapse when using this measure of flow strength.
Recently, Z¨ottl and Stark [20] examined the motion of an elongated microswimmer in Poiseuille flow. They found that organisms will undergo periodic orbits of their position and orientation within the channel when rotational and translational Brownian motion are not included. At large enough flow rates (large Per ), Brownian motion will have a relatively small effect on the motion within these orbits. However, without Brownian motion, the organisms stay on the orbits and will not approach a steady distribution. In order to understand the steady state dip in swimmer concentration in the channel center, we must go beyond their analysis. Two possible approaches to derive the probability distribution in the strong flow limit are using singular perturbation and boundary layer methods or by finding the probability density on an orbit and then calculating the distribution of swimmers across different orbits, similar to the analysis of Leal and Hinch for Jeffery orbits [35]. We will use the second approach here, in which we use the results of Z¨ottl and Stark for the dynamics within an orbit. As an organism moves along its periodic orbit, the position z changes in time and the effective number of organisms measured at a value of z is inversely proportional to the rate at which z changes in time. Therefore, P(z) ∝ 1/|˙z |, where the dot denotes a time derivative. Using the results of Z¨ottl and Stark, we can express the rate of change z˙ as a function of the flow rate, the position z, a parameter denoting the orbit the swimmer is on, and a geometric factor G = (γ 2 − 1)/(γ 2 + 1), where γ is the aspect ratio of the organism. If the orientation of the organism is restricted to the x–z plane, the probability along an orbit is calculated to be 1 P(z) ∝ q √ 2 1+G 1 − 2G tanh 2G(1 + G)
1 v (z/L)2 2 f
+ 1 − C0
(3)
where C0 is a parameter which determines the periodic orbit. Expanding this expression for small z shows that, within an orbit, the probability is quadratic in z/L, with the curvature proportional to v f . Brownian motion will determine 5
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that are oriented with the flow in the channel center. The observations for systems with and without HI between the swimmer and the walls are the same. This shows that the external flow (at high shear rates) plays a dominant role in determining the orientations of the microorganisms. As the flow aligns the organisms, the component of the swimmer orientation vector in the gradient direction (z) will reduce while the component in the flow direction will approach ±1. We can quantify the alignment by calculating he2z i and 1 − he2x i, where the angle brackets denote an average. Figure 6 shows he2z i as a function of Per for the three regions of the channel, for the two flow types, and for two confinements. For Per > 1 the organisms become aligned with the flow, reducing the value of he2z i. The different confinements collapse when plotted versus Per , which again shows that the rotational Peclet number is the correct dimensionless group to understand how the flow aligns the swimmers. In all cases, the organisms at the walls are more aligned with the flow even when no external flow is applied. Many previous studies have examined the alignment of passive slender bodies in an unbounded Couette flow [35–39], −1/3 where it has been shown that the alignment scales as Per for strong flows. Figure 6 compares our numerical results with this power-law scaling. The numerical results for Couette flow match exactly with this power law for organisms that are not located at the walls. For Poiseuille flow, the organisms in the intermediate region (not at the walls or in the center) match qualitatively with the theory for Couette flow. Organisms in the center, where the shear rate is zero, show less alignment with the flow. In figure 7, we show 1 − he2x i, which is a different measurement of the alignment parallel to the flow. It shows the same qualitative features: Couette flow away from the walls matches exactly with the power-law scaling while Poiseuille flow shows less alignment in the channel center.
Figure 5. The distribution of swimmer orientations e compared to vector p in different regions of the channel at Per = 400 with HI for
confinement 2L = 10l in Couette (a) and Poiseuille (b) flow. The directions of vectors p and n are shown in the inset. The regions shown are near the wall (red-solid line), the intermediate region (blue-circles) and the center (black-dashed line). At the walls the swimmers are more likely to point against the direction of flow in parabolic flow, and are more likely to point in the direction of flow in a uniform shear flow.
direction of the flow. We are not aware of previous experiments or simulations that explicitly show this, and it is surprising based on the recent work of Z¨ottl and Stark [20, 19]. They showed that, if Brownian motion is ignored, the periodic orbits which cross the channel center and stay near the center are more likely to be oriented against the flow. Our simulation data suggests that Brownian motion preferentially selects orbits
3.3. Dynamics in the gradient direction
An important consequence of the alignment of organisms by the flow is that it changes their ability to move from one wall to
Figure 6. Orientation moments of swimmers in the velocity gradient direction (he2z i) for Couette (a) and Poiseuille (b) flow including HI
with the walls. The regions of the channel and confinements correspond to wall region at 2L = 10l (red circles), wall region at 2L = 16l (red squares), intermediate region at 2L = 10l (blue upward triangles), intermediate region at 2L = 16l (blue diamonds), center region at 2L = 10l (black downward triangles), and center region at 2L = 16l (black stars). The black dash-dot line represents a power law of −1/3. 6
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Figure 7. Orientation moments of swimmers in the direction of flow (1 − he2x i) for Couette (a) and Poiseuille (b) flow including HI with the walls. The regions of the channel and confinements correspond to wall region at 2L = 10l (red circles), wall region at 2L = 16l (red squares), intermediate region at 2L = 10l (blue upward triangles), intermediate region at 2L = 16l (blue diamonds), center region at 2L = 10l (black downward triangles), and center region at 2L = 16l (black stars). The black dash-dot line represents a power law of −1/3.
the other. The movement to and from the walls could be important for the adhesion of cells on a surface, which is the initial step in the formation of biofilms [40]. We assessed the impact of external flow on movement between the walls by looking at the mean-squared displacement of the swimmers in the z-direction (MSDz = h(z(t + 1t) − z(t))2 i). This meansquared displacement is shown as a function of 1t in figure 8 for both confinements 2L = 10l and 2L = 16l and a range of flow rates. When no external flow is applied, the MSD is ballistic at short times (scales as 1t 2 ) until the MSD levels off. This saturation is because the displacement in the z-direction cannot be larger than 2L. As the flow rate is increased, the alignment of the organisms reduces the component of their swimming motion in the z-direction, which leads to a smaller MSD. At high enough flow rate, the alignment is sufficient that the dominant mode for transport in the z-direction is Brownian motion. This leads to a diffusive scaling of the MSD proportional to 1t. The change in transport to and from the surface could impact other processes such as adhesion and biofilm formation. 4. Conclusions
In conclusion, we have used computer simulations to examine how simple flows, such as Couette and Poiseuille flow, change the properties of suspensions of swimming microorganisms in confined environments. The external flow acts to reduce the accumulation of organisms at the walls. This occurs in qualitatively the same way whether HIs with the walls are included or not (figure 2). However, more accumulation occurs when HIs are included (figure 3). The flow aligns the organisms, which leads to fewer collisions with the walls and less accumulation. A quantification of the swimmer orientations shows that in the intermediate region II (not right at the wall or the center of the channel) the alignment follows the scaling expected from an analysis of passive slender bodies in a shear flow, which is governed by a rotational Peclet number (figures 6 and 7). In addition to its impact on the density of organisms within the channel, the alignment also impacts the movement of organisms from one wall to the other. At low
Figure 8. The mean-squared displacement of swimmers between the
walls (in z-direction) for confinement 2L = 10l (a) and 2L = 16l (b). The flow conditions are no external flow (black solid line), Poiseuille flow (red) and Couette flow (blue). The flow rates are Per = 40 (dashed lines) and Per = 400 (solid lines). The ability of swimmers to move between the walls is reduced as the strength of flow increases.
shear rates, the organisms can swim ballistically from one wall to the other, while at high shear rates they can only move across the channel by Brownian motion (figure 8). Finally, the coupling of swimmer positions and orientations led to a dip in the concentrations of the organisms in the channel center in Poiseuille flow. It was determined that at high shear rates the governing parameter that determines this dip is the ratio of the velocity of the fluid at the channel center to the swimming speed of the organism (figure 4). This dependence is consistent 7
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with a theoretical prediction. These findings may provide insight into the formation of biofilms in various industrial and medical devices under different flow conditions, where surface accumulation and subsequent adhesion are critical steps in these processes. Further, this model may be used to explore how other flows and geometries could enable more efficient sorting of cells based on size, shape and motility for a range of applications, including diagnostics or quantifying components of complex microbial communities.
[18] Ardekani A M and Gore E 2012 Emergence of a limit cycle for swimming microorganisms in a vortical flow of a viscoelastic fluid Phys. Rev. E 85 056309 [19] Z¨ottl A and Stark H 2012 Nonlinear dynamics of a microswimmer in Poiseuille flow Phys. Rev. Lett. 108 218104 [20] Z¨ottl A and Stark H 2013 Periodic and quasiperiodic motion of an elongated microswimmer in Poiseuille flow Eur. Phys. J. E 36 1–10 [21] Hill J, Kalkanci O, McMurry J L and Koser H 2007 Hydrodynamic surface interactions enable Escherichia coli to seek efficient routes to swim upstream Phys. Rev. Lett. 98 068101 [22] Kaya T and Koser H 2012 Direct upstream motility in Escherichia coli Biophys. J. 102 1514–23 [23] Costanzo A, Di Leonardo R, Ruocco G and Angelani L 2102 Transport of self-propelling bacteria in micro-channel flow J. Phys.: Condens. Matter 24 065101 ¨ [24] Ottinger H C 1996 Stochastic Processes in Polymeric Fluids: Tools and Examples for Developing Simulation Algorithms (Berlin: Springer) [25] Purcell E M 1977 Life at low Reynolds number Am. J. Phys. 45 3–11 [26] Hern´andez-Ortiz J P, Stoltz C G and Graham M D 2005 Transport and collective dynamics in suspensions of confined swimming particles Phys. Rev. Lett. 95 204501 [27] Grassia P S, Hinch E J and Nitsche L C 1995 Computer simulations of Brownian motion of complex systems J. Fluid Mech. 282 373–403 [28] Fixman M 1978 Simulation of polymer dynamics. II. Relaxation rates and dynamic viscosity J. Chem. Phys. 69 1538 [29] Saintillan D 2010 The dilute rheology of swimming suspensions: a simple kinetic model Exp. Mech. 50 1275–81 [30] Saintillan D 2010 Extensional rheology of active suspensions Phys. Rev. E 81 056307 [31] Heyes D M and Melrose J R 1993 Brownian dynamics simulations of model hard-sphere suspensions J. Non-Newton. Fluid Mech. 46 1–28 [32] Blake J R 1971 A note on the image system for a stokeslet in a no-slip boundary Proc. Camb. Phil. Soc. 70 303–10 [33] Hern´andez-Ortiz J P, Underhill P T and Graham M D 2009 Dynamics of confined suspensions of swimming particles J. Phys.: Condens. Matter 21 204107 [34] Lauga E, DiLuzio W R, Whitesides G M and Stone H A 2006 Swimming in circles: motion of bacteria near solid boundaries Biophys. J. 90 400–12 [35] Leal L G and Hinch E J 1971 The effect of weak Brownian rotations on particles in shear flow J. Fluid Mech. 46 685–703 [36] Hinch E J and Leal L G 1972 The effect of Brownian motion on the rheological properties of a suspension of non-spherical particles J. Fluid Mech. 52 683–712 [37] Chen S B and Jiang L 1999 Orientation distribution in a dilute suspension of fibers subject to simple shear flow Phys. Fluids 11 2878 [38] Chen S B and Koch D L 1996 Rheology of dilute suspensions of charged fibers Phys. Fluids 8 2792 [39] Park J and Butler J E 2009 Inhomogeneous distribution of a rigid fibre undergoing rectilinear flow between parallel walls at high P´eclet numbers J. Fluid Mech. 630 267 [40] Hall-Stoodley L, Costerton J W and Stoodley P 2004 Bacterial biofilms: from the natural environment to infectious diseases Nature Rev. Microbiol. 2 95–108
References [1] Adler J and Tso W-W 1974 ‘Decision’-making in bacteria: chemotactic response of Escherichia coli to conflicting stimuli Science 184 1292–4 [2] Suarez S S and Pacey A A 2006 Sperm transport in the female reproductive tract Hum. Reprod. Update 12 23–37 [3] Josenhans C and Suerbaum S 2002 The role of motility as a virulence factor in bacteria Int. J. Med. Microbiol. 291 605–14 [4] Dombrowski C, Cisneros L H, Chatkaew S, Goldstein R E and Kessler J O 2004 Self-concentration and large-scale coherence in bacterial dynamics Phys. Rev. Lett. 93 098103 [5] Cisneros L H, Cortez R, Dombrowski C, Goldstein R E and Kessler J O 2007 Fluid dynamics of self-propelled microorganisms, from individuals to concentrated populations Exp. Fluids 43 737–53 [6] Underhill P T, Hern´andez-Ortiz J P and Graham M D 2008 Diffusion and spatial correlations in suspensions of swimming particles Phys. Rev. Lett. 100 248101 [7] Leptos K C, Guasto J S, Gollub J P, Pesci A I and Goldstein R E 2009 Dynamics of enhanced tracer diffusion in suspensions of swimming eukaryotic microorganisms Phys. Rev. Lett. 103 198103 [8] Wu X-L and Libchaber A 2000 Particle diffusion in a quasi-two-dimensional bacterial bath Phys. Rev. Lett. 84 3017 [9] Sokolov A and Aranson I S 2009 Reduction of viscosity in suspension of swimming bacteria Phys. Rev. Lett. 103 148101 [10] Berke A P, Turner L, Berg H C and Lauga E 2008 Hydrodynamic attraction of swimming microorganisms by surfaces Phys. Rev. Lett. 101 038102 [11] Li G, Tam L-K and Tang J X 2008 Amplified effect of Brownian motion in bacterial near-surface swimming Proc. Natl Acad. Sci. USA 105 18355–9 [12] Li G and Tang J X 2009 Accumulation of microswimmers near a surface mediated by collision and rotational Brownian motion Phys. Rev. Lett. 103 078101 [13] Watnick P and Kolter R 2000 Biofilm, city of microbes J. Bacteriol. 182 2675–9 [14] Rothschild L 1963 Non-random distribution of bull spermatozoa in a drop of sperm suspension Nature 198 1221–2 [15] Elgeti J and Gompper G 2013 Wall accumulation of self-propelled spheres Europhys. Lett. 101 48003 [16] Zilman G, Novak J, Liberzon A, Perkol-Finkel S and Benayahu Y 2013 The hydrodynamics of contact of a marine larva, Bugula neritina, with a cylinder J. Exp. Biol. 216 2789–97 [17] Torney C and Neufeld Z 2007 Transport and aggregation of self-propelled particles in fluid flows Phys. Rev. Lett. 99 078101 8
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Impact of external flow on the dynamics of swimming microorganisms near surfaces
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 115101 (8pp)
doi:10.1088/0953-8984/26/11/115101
Impact of external flow on the dynamics of swimming microorganisms near surfaces Sandeep Chilukuri, Cynthia H Collins and Patrick T Underhill Howard P Isermann Department of Chemical and Biological Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180, USA Center for Biotechnology and Interdisciplinary Studies, Rensselaer Polytechnic Institute, Troy, NY 12180, USA E-mail: [email protected] Received 8 September 2013, revised 12 December 2013 Accepted for publication 16 January 2014 Published 3 March 2014
Abstract
Swimming microorganisms have been previously observed to accumulate along walls in confined systems both experimentally and in computer simulations. Here, we use computer simulations of dilute populations for a simplified model of an organism to calculate the dynamics of swimmers between two walls with an external fluid flow. Simulations with and without hydrodynamic interactions (HIs) are used to quantify their influence on surface accumulation. We found that the accumulation of organisms at the wall is larger when HIs are included. An external fluid flow orients the organisms parallel to the fluid flow, which reduces the accumulation at the walls. The effect of the flow on the orientations is quantified and compared to previous work on upstream swimming of organisms and alignment of passive rods in flow. In pressure-driven flow, the zero shear rate at the channel center leads to a dip in the concentration of organisms in the center. The curvature of this dip is quantified as a function of the flow rate. The fluid flow also affects the transport of organisms across the channel from one wall to the other. Keywords: swimming microorganisms, accumulation, hydrodynamic interactions, upstream swimming (Some figures may appear in colour only in the online journal)
1. Introduction
Swimming microorganisms are often found in confined spaces or on surfaces, where attachment to surfaces and biofilm formation can have detrimental effects, ranging from infection to biofouling and corrosion [13]. An early observation that swimming organisms tend to concentrate near surfaces was made by Rothschild, who observed that the density distribution of bull spermatozoa between two glass surfaces was non-uniform [14]. Subsequent studies have shown similar observations in dilute populations of swimming microbial cells such as Escherichia coli and Caulobacter crescentus. Simulation studies have shown an accumulation near the walls without external flow even for spherical particles [15]. This paper will focus on the dynamics of swimming organisms in an external flow. A number of researchers have previously examined how external flows can alter the distribution and dynamics of self-propelled particles. For example, Zilman
Motility is required for a range of important biological processes, including chemotaxis [1], reproduction [2] and infection [3]. While these processes have been extensively studied from the biological perspective, the dependence of these processes on the mechanical principles of locomotion at the micron scale is not fully understood. Many experimental and theoretical studies on swimming microorganisms have reported collective behavior [4, 5], enhanced transport [6–8], interesting rheological properties [9], and migration towards surfaces [10–12]. A key challenge in this area is to tie our physical understanding of these observations across different scales (from the levels of individual cells to large cell populations) to biological processes. 0953-8984/14/115101+08$33.00
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et al [16] experimentally observed the dependence of an external flow on the trajectories of spherical larvae, Bugula neritina, in the vicinity of a collector surface. ‘Cellular flows’ have also been used to examine the dynamics and ‘aggregation’ in Newtonian [17] and non-Newtonian [18] suspending fluids. The focus of this paper is the dynamics of self-propelled particles in Poiseuille and Couette flow between two parallel walls. Z¨ottl and Stark have previously examined the dynamics of a microswimmers, both spherical [19] and elongated [20], in Poiseuille flow and report that organisms exhibited periodic motions of their positions and orientations when there was no translational or rotational Brownian motion. We will use some of these results in this paper, as described later. The key contribution of our work, which builds on theirs by the addition of Brownian motion, is we examine the probability density distribution of the organisms, which cannot be calculated if Brownian motion is not considered. Two key models and theories have been developed to explain the surface accumulation in dilute suspensions of rodlike microorganisms. In the mechanism suggested by Berke et al [10], the microorganisms are assumed to be oriented parallel to the surface and hydrodynamic interactions (HIs) between the cells and the surface cause the accumulation. From the mechanism suggested by Li et al [11, 12], the accumulation results from a relatively larger flux of organisms swimming into and colliding with a wall than the flux of organisms that leave the wall by rotational Brownian motion. Neither of these models consider the role of an external fluid flow. Since swimming microbes are found in environments with a wide range of flow types and velocities, it is essential that we examine how flow affects motility and surface accumulation of swimming organisms. For microorganisms that can be represented as slender objects (including both the body and flagellar bundle), it is expected that the flow will align the organisms parallel to the flow. In the mechanism suggested by Berke et al, the microorganisms are assumed to be oriented parallel to the surface, so we would expect qualitatively similar observations under conditions with and without an external flow. In the mechanism suggested by Li et al, flow should reduce both the collisions with the wall and the rate of leaving the wall. Therefore, the external flow could significantly change the accumulation of swimming organisms near the surface. Within the context of these two mechanisms, it is difficult to predict beforehand how the accumulation of selfpropelled particles (such as microorganisms) will be altered by an external flow, which is one of the goals of this study. In this paper, changes in the dynamics of swimming microorganisms in confined geometries in the presence of external flow are addressed. Using two systems of modeled freely swimming microorganisms, with and without HI between the cells and surface, we look at the surface accumulation of swimming microorganisms in two simple flows, uniform shear and a parabolic flow. In addition to location, the orientations of the organisms are quantified. Experimental observations [21, 22] and computer simulations [23] of microorganisms under flow have shown that they swim along surfaces, causing upstream migration. How the flow affects the orientations of the swimmers is important for our overall understanding of the accumulation of organisms at the walls and the movement of organisms from one wall to the other.
Figure 1. Representation of the swimmer, the different regions
between the surfaces and the calculation of hydrodynamic interactions (HIs) between the swimmer and walls. (Left) The regions I, II and III represent the regions near the wall, an intermediate region between the wall and the center, and region at the center, respectively. (Center) e is a unit vector in the direction of swimming. (Right) The body bead (b) and flagellum bead (f) of a swimmer and the images of the body bead (b1, b2) and flagellum bead (f1, f2) in the two walls.
2. Swimmer model
Our simplified model of a swimmer consists of two beads connected by a very stiff spring with equilibrium length l. The size of the swimmer changes by less than 8% even in the strongest flow rate considered here. One bead (body bead) represents the center of the body and the other (flagellum bead) the center of the flagellar bundle, as shown in figure 1. The orientation of each swimmer is denoted by a unit vector e. We integrate forward in time using the Brownian dynamics method [24]. A force balance for each bead is written as Fh + Fbr + Fnh = 0, neglecting inertia due to their small size [25]. The forces are the hydrodynamic force Fh , the Brownian force Fbr , and the non-Brownian and non-hydrodynamic forces Fnh , which includes the spring force and the excluded volume forces between swimmers and between a swimmer and a wall. The drag force on the body bead is given by the Stokes’ law Fh,b = −6π ηa[˙r − u(r)], where a is the bead radius, η is the fluid viscosity, r is the position of the bead and u(r) is the fluid velocity at r. For the flagellum bead, the drag force is given by Fh, f = −6π ηa[˙r − u(r) − 2vsw e], where vsw is the isolated swimming speed of an organism. This additional term is equivalent to the flagellum force used by Hernandez-Ortiz et al [26] and arises from the torque on the flagellar bundle. The force balance equations are rewritten as stochastic differential equations, where the Brownian forces are determined such that the system satisfies the fluctuation–dissipation theorem [27] using Fixman’s mid-point algorithm [28]. The Brownian forces must be included for the swimmers to reach a steady state distribution across the channel. Both tumbling and rotational diffusion lead to changes in the orientation of an organism over a particular timescale. Previous work by Saintillan [29, 30] that included both mechanisms showed that an external flow orients the organisms in qualitatively the same way for each mechanism. Therefore, only rotational Brownian motion is included in our simulations. We consider two different values for the distance between the walls (confinement), 2L = 10l and 16l. The domain is periodic in the x and y directions, which are parallel to the walls with side 2L. These separation of the walls, 2
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which correspond to 10l ≈ 100 µm and 16l ≈ 160 µm for E. coli, are consistent with experimental and theoretical studies without flow. In the model of Li and Tang [12], which did not include HI or external flow, a key parameter is the persistence length, which is the characteristic length over which the organism trajectories are straight and equals L p = vsw /Dr , where Dr is the rotational diffusivity. If (2L)/ L p < 1, significant accumulation is expected without flow. For the two confinements here, (2L)/L p = 0.25, 0.40. A potential-free algorithm, which is a modified version of the Heyes–Melrose hard-sphere algorithm [31], is used for bead–wall interactions, where a bead is positioned at a distance a 0 = 7a/6 when it tries to penetrate through the wall during a timestep. The body beads of the swimmers interact by a Weeks–Chandler–Andersen potential: ( σ 12 σ 6 − 4 + 4 UWCA = r r 0
for r < 21/6 σ for r ≥ 21/6 σ
small in comparison to the contribution from the first images (the image flow field decays as 1/r 2 ). Torques on each bead that lead to circular motion of organisms near a wall [34] are not included since they should not have a large impact on the quantities calculated here. 3. Computational results
We have calculated the impact of Couette and Poiseuille flow on the dynamics of the organisms. Using the coordinate system defined in figure 1, the two flows are vx (z) = γ˙w z for a uniform shear and vx (z) = γ˙w L(1 − (z/L)2 )/2 for a parabolic flow, where γ˙w is the shear rate at the wall. If the rate of shear is faster than the organisms change their directions by rotational diffusion, the organisms will be aligned parallel to the flow. In suspensions of passive particles, this competition leads to a key dimensionless group that quantifies the strength of the external flow, called the rotational Peclet number. Because of the importance of the orientational distribution of the swimming microorganisms, we quantify the external flow as Per = γ˙w /Dr , where the wall shear rate is used and the rotational diffusivity Dr of a swimmer is calculated from the orientational correlation function of a swimmer in a simulation in an unbounded domain. Even though the fluctuating forces are small at high Per , the singular nature of the limit causes Brownian motion of the swimmers to still be important. A system without Brownian motion will not produce a steady state angular distribution, and it is Brownian motion (in the flow gradient direction) that allows the swimmers to move between the walls when they are aligned by the flow. For the two confinements 2L = 10l and 16l and both Couette and Poiseuille flow types, the distribution of the swimmers between the walls (section 3.1), the orientations of the swimmers (section 3.2) and the dynamics of the swimmers across the channel (section 3.3) are discussed for a wide range of flow rates. Quantification of the swimmer distributions and orientations of the swimmers was done to understand the importance of HI and the influence of flow on the alignment of the organisms.
(1)
where r is the distance between two body beads, σ = 2a is the diameter of a bead, and = 2a f /48 is the potential well depth. The flagellum beads are assumed to have no excluded volume interactions with beads of any other swimmer. The volume N 4 πa 03
3 . The values here are fraction is defined by φe = (2L)(2L)(2L) N = 1500 swimmers and φe = 0.03 (for confinement 10l) and φe = 0.007 (for confinement 16l). The ratio of bead radius (excluded volume) to the swimmer size is a 0 /l = 1/6, which is a close representation of most flagellated microorganisms. HI between a swimmer and the surface is one mechanism for accumulation at the surface. These are calculated using Blake’s solution for the flow caused by a point force near a no-slip boundary [32]. This is obtained by the superposition of the flow due to the point force and the flow due to an image system, located on the other side of the boundary, to satisfy the no-slip condition at the boundary. From previous work including HI between organisms [33], we can safely ignore HI between organisms for the relatively dilute concentrations considered here and include only the flow due to a swimmer’s own images. Specifically, the impact of those interactions would be to affect the translational and rotational motion of the organisms. For the concentrations used here, those contributions are smaller than the intrinsic motions of the individuals. Note that there are equal and opposite point forces exerted by a swimmer on the fluid at the positions of the body bead, b, and flagellum bead, f (the net force on the fluid is zero). As shown in figure 1, the net flow at the location of the body bead (b) of a swimmer is the sum of the flow due to the flagellum bead (f), images of the body bead and flagellum bead from the two surfaces (b1, b2, f1, f2), and any external flow. This can be written as
3.1. Distribution across the channel
Figure 2 shows the probability density of swimmers between the two walls for 2L = 10l at a relatively high shear rate of Per = 400 for both Couette and Poiseuille flows compared with the no flow case. We observed qualitatively similar results for the simulations performed without excluded volume interactions between the body beads (which are essentially multiple single swimmer simulations). Therefore, the data and mechanisms described here represent the response seen in the dilute limit. A key feature of the data is that the accumulation at the wall is smaller with flow than without. We also examined the role of HI and found that in the presence of these interactions (figure 2(a)), there is an increased accumulation of the swimmers at the walls compared to the system without them (figure 2(b)). The other key feature of the data is that, in parabolic flow, there is a dip in the probability density in the center of the channel. In order to understand the influence of
u(b) = uf (b) + ub1 (b) + ub2 (b) + uf1 (b) + uf2 (b) + uext , (2) where uf (b) is the flow at b due to the flagellum bead f, uext is the external flow, ub1 (b) is the flow at b due to the image b1, etc. The flagellum bead (f) only sees the net flow from b, b1, b2, f1 and f2. Multiple reflections (reflections of images in the two walls) are not considered here as their contribution is very 3
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Figure 2. The distribution of swimmers between the walls with (a) and without (b) HI in a confinement 2L = 10l. The density distribution of swimmers for different flow conditions is shown—no external flow (black-circles), Poiseuille flow (red-dashed line), Couette flow (blue-solid line). The flow rate corresponds to Per = 400. The dotted-green line represents the uniform distribution of swimmers between the walls.
Figure 3. Quantifying the excess number of swimmers at the wall
using a normalized area under the peaks from the swimmer density distribution at different flow rates with HI (a) and without HI (b) between the swimmers and the walls. The flow types and confinements are: Couette at 2L = 10l (blue circles), Couette at 2L = 16l (blue triangles), Poiseuille at 2L = 10l (red stars), and Poiseuille at 2L = 16l (red squares).
shear rate, separation of the walls, flow type, and HI, we quantify the peaks in the distribution at the walls. To do this, the area under the probability density is calculated for positions within a distance of l of the wall. The accumulation of swimmers when there is no external flow is approximately of width l (figure 2). This area is normalized by the value of the area if the organisms were uniformly distributed within the channel. Figure 3 shows the normalized area of the peaks as a function of flow rate for two confinements 2L = 10l and 2L = 16l, for both Couette and Poiseuille flow, and for both with (figure 3(a)) and without HI (figure 3(b)). Note that the normalized peak area can drop below 1. This results from the excluded volume interactions between the organism and the wall, which gives a zero probability for the cell to be at the wall. This zero probability reduces the area under the curve. The area under the peak, which quantifies the accumulation at the wall, is always larger with HI than without. In all cases, the size of the peak at the wall is maximum for no flow, begins to decrease when Per is between 1 and 10, and asymptotes for Per above 100. The results in figure 2 (at Per = 400) are in this high flow region. The fact that the change in accumulation occurs when Per rises above 1 suggests that the mechanism for the change is the alignment of the organisms parallel to the flow direction. The similarity of the curves for different confinements and flow types when plotted versus Per also suggests that the rotational Peclet number is the key dimensionless group. The alignment of the organisms reduces their collisions with the wall and their swimming away from the wall. The only mechanism for
transport across the channel is passive Brownian motion or HI with the wall, which produces a smaller accumulation at the wall at high flow rates. For Poiseuille flow the probability densities show a dip in the center of the channel where the shear rate is zero. Near the channel center, the probability is parabolic in shape, and we can use the curvature of that parabola to quantify the dip in the center. The probability distribution is fit with m 1 (1 + m 2 (z/2L)2 ) over the region z/2L = [−0.15, 0.15]. The parameter m 1 is affected by the normalization of the probability but the parameter m 2 quantifies the shape. Figure 4 shows how this curvature varies with flow rate for two confinements 2L = 10l and 2L = 16l and with and without HI. For small flow rates, there is a small slight positive curvature. As the flow rate increases, the curvature drops just below zero before increasing dramatically at large flow rates. The quantification of the curvature (dip) at high flow rates is the key feature of this data. In this region, HI with the walls does not influence the curvature, but the confinement does. Figure 4 shows the data with the flow rate scaled such that the different confinements collapse onto a single curve. The important measure of the flow rate is not the rotational Peclet number, but v f = γ˙w L/(2vsw ), which is the ratio of the fluid velocity at the channel center (γ˙w L/2) to the swimming speed of an organism (vsw ). The inset to the figure shows the data plotted versus rotational Peclet number, which confirms explicitly that the data do not collapse when plotted in that way. 4
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the distribution of values of C0 ; the swimmer density is then an average over these orbits. However, it is not necessary to determine the distribution of C0 and perform the average in order to understand the dependence of the probability on position or flow rate. Since each orbit produces a P(z) with the same functional form, we expect the simulations to produce that functional form. This is confirmed in figure 4, in which the curvature is proportional to v f for large flow rates. 3.2. Swimmer orientations
Because an organism swims in the direction of its orientation, there is a coupling between the orientation and position of a swimmer. In figure 3, the accumulation at the walls was shown to be determined by the rotational Peclet number, since the flow aligns the organisms with the flow, thereby affecting the positions (accumulation). Recent experiments and simulations showing upstream swimming of E. coli in an external flow have highlighted the importance of examining how orientation is affected by flow [21–23]. While some mechanisms to explain this phenomenon rely on a hydrodynamic difference between the cell body and flagellar bundle, simulations have indicated that upstream motility can occur even for symmetric organisms. This is particularly interesting, since it shows that swimming of a symmetric object can lead to an asymmetric distribution of orientations. Motivated by the density distribution in figure 2, we can identify three regions across the channel as shown in figure 1: region I of width l near the walls, corresponding to the accumulated organisms at the wall, region III of width 2l in the channel center, which consists of organisms within the dip in Poiseuille flow, and region II, which corresponds to the organisms between the wall and the center. In order to quantify how much the swimmers are oriented parallel to the flow, we compute the dot product of the orientation of the organism e with a vector that is parallel to the flow direction, defined here as p. As shown in figure 5, we define p such that it retains the symmetry of the flow; Couette flow has a rotational symmetry between the two sides, while Poiseuille flow has a mirror symmetry. Specifically, we define the unit vector p as p = n × (∇ × v)/k∇ × vk, where ∇ × v is the flow vorticity. The unit vector n is in the negative z-direction for z > 0 and in the positive z-direction for z < 0. Figure 5 shows the distribution of swimmer orientations, quantified as p · e, for a confinement of 2L = 10l at Per = 400 for the two types of flow and the three regions within the channel. At the walls (region I), swimmers are found to be in the direction of p, which is in the direction of the flow in uniform shear and against in parabolic flow. This shows the same upstream motility seen in experiments and simulations in parabolic flow. The fact that organisms point in the direction of p at the walls shows that the flow vorticity (and not the flow direction) determines the orientation of the swimmers. Away from the walls (regions II and III), the organisms in Couette flow are equally likely to be oriented in the positive or negative p direction. This is not surprising for our model, in which the body and flagellum beads have the same hydrodynamic drag coefficient. However, in Poiseuille flow we find that the swimmers in the center are directed preferentially in the
Figure 4. Quantifying the dip in the swimmer distribution at the
channel center in Poiseuille flow using the curvature of the distribution m 2 . The value of m 2 is shown with HI at 2L = 10l (black squares), with HI at 2L = 16l (black stars), without HI at 2L = 10l (red circles), and without HI at 2L = 16l (red triangles). The dashed line is a power law of +1. Inset: the same data shown as m 2 versus rotational Peclet number for the two confinements 2L = 10l and 2L = 16l, showing that the data do not collapse when using this measure of flow strength.
Recently, Z¨ottl and Stark [20] examined the motion of an elongated microswimmer in Poiseuille flow. They found that organisms will undergo periodic orbits of their position and orientation within the channel when rotational and translational Brownian motion are not included. At large enough flow rates (large Per ), Brownian motion will have a relatively small effect on the motion within these orbits. However, without Brownian motion, the organisms stay on the orbits and will not approach a steady distribution. In order to understand the steady state dip in swimmer concentration in the channel center, we must go beyond their analysis. Two possible approaches to derive the probability distribution in the strong flow limit are using singular perturbation and boundary layer methods or by finding the probability density on an orbit and then calculating the distribution of swimmers across different orbits, similar to the analysis of Leal and Hinch for Jeffery orbits [35]. We will use the second approach here, in which we use the results of Z¨ottl and Stark for the dynamics within an orbit. As an organism moves along its periodic orbit, the position z changes in time and the effective number of organisms measured at a value of z is inversely proportional to the rate at which z changes in time. Therefore, P(z) ∝ 1/|˙z |, where the dot denotes a time derivative. Using the results of Z¨ottl and Stark, we can express the rate of change z˙ as a function of the flow rate, the position z, a parameter denoting the orbit the swimmer is on, and a geometric factor G = (γ 2 − 1)/(γ 2 + 1), where γ is the aspect ratio of the organism. If the orientation of the organism is restricted to the x–z plane, the probability along an orbit is calculated to be 1 P(z) ∝ q √ 2 1+G 1 − 2G tanh 2G(1 + G)
1 v (z/L)2 2 f
+ 1 − C0
(3)
where C0 is a parameter which determines the periodic orbit. Expanding this expression for small z shows that, within an orbit, the probability is quadratic in z/L, with the curvature proportional to v f . Brownian motion will determine 5
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that are oriented with the flow in the channel center. The observations for systems with and without HI between the swimmer and the walls are the same. This shows that the external flow (at high shear rates) plays a dominant role in determining the orientations of the microorganisms. As the flow aligns the organisms, the component of the swimmer orientation vector in the gradient direction (z) will reduce while the component in the flow direction will approach ±1. We can quantify the alignment by calculating he2z i and 1 − he2x i, where the angle brackets denote an average. Figure 6 shows he2z i as a function of Per for the three regions of the channel, for the two flow types, and for two confinements. For Per > 1 the organisms become aligned with the flow, reducing the value of he2z i. The different confinements collapse when plotted versus Per , which again shows that the rotational Peclet number is the correct dimensionless group to understand how the flow aligns the swimmers. In all cases, the organisms at the walls are more aligned with the flow even when no external flow is applied. Many previous studies have examined the alignment of passive slender bodies in an unbounded Couette flow [35–39], −1/3 where it has been shown that the alignment scales as Per for strong flows. Figure 6 compares our numerical results with this power-law scaling. The numerical results for Couette flow match exactly with this power law for organisms that are not located at the walls. For Poiseuille flow, the organisms in the intermediate region (not at the walls or in the center) match qualitatively with the theory for Couette flow. Organisms in the center, where the shear rate is zero, show less alignment with the flow. In figure 7, we show 1 − he2x i, which is a different measurement of the alignment parallel to the flow. It shows the same qualitative features: Couette flow away from the walls matches exactly with the power-law scaling while Poiseuille flow shows less alignment in the channel center.
Figure 5. The distribution of swimmer orientations e compared to vector p in different regions of the channel at Per = 400 with HI for
confinement 2L = 10l in Couette (a) and Poiseuille (b) flow. The directions of vectors p and n are shown in the inset. The regions shown are near the wall (red-solid line), the intermediate region (blue-circles) and the center (black-dashed line). At the walls the swimmers are more likely to point against the direction of flow in parabolic flow, and are more likely to point in the direction of flow in a uniform shear flow.
direction of the flow. We are not aware of previous experiments or simulations that explicitly show this, and it is surprising based on the recent work of Z¨ottl and Stark [20, 19]. They showed that, if Brownian motion is ignored, the periodic orbits which cross the channel center and stay near the center are more likely to be oriented against the flow. Our simulation data suggests that Brownian motion preferentially selects orbits
3.3. Dynamics in the gradient direction
An important consequence of the alignment of organisms by the flow is that it changes their ability to move from one wall to
Figure 6. Orientation moments of swimmers in the velocity gradient direction (he2z i) for Couette (a) and Poiseuille (b) flow including HI
with the walls. The regions of the channel and confinements correspond to wall region at 2L = 10l (red circles), wall region at 2L = 16l (red squares), intermediate region at 2L = 10l (blue upward triangles), intermediate region at 2L = 16l (blue diamonds), center region at 2L = 10l (black downward triangles), and center region at 2L = 16l (black stars). The black dash-dot line represents a power law of −1/3. 6
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Figure 7. Orientation moments of swimmers in the direction of flow (1 − he2x i) for Couette (a) and Poiseuille (b) flow including HI with the walls. The regions of the channel and confinements correspond to wall region at 2L = 10l (red circles), wall region at 2L = 16l (red squares), intermediate region at 2L = 10l (blue upward triangles), intermediate region at 2L = 16l (blue diamonds), center region at 2L = 10l (black downward triangles), and center region at 2L = 16l (black stars). The black dash-dot line represents a power law of −1/3.
the other. The movement to and from the walls could be important for the adhesion of cells on a surface, which is the initial step in the formation of biofilms [40]. We assessed the impact of external flow on movement between the walls by looking at the mean-squared displacement of the swimmers in the z-direction (MSDz = h(z(t + 1t) − z(t))2 i). This meansquared displacement is shown as a function of 1t in figure 8 for both confinements 2L = 10l and 2L = 16l and a range of flow rates. When no external flow is applied, the MSD is ballistic at short times (scales as 1t 2 ) until the MSD levels off. This saturation is because the displacement in the z-direction cannot be larger than 2L. As the flow rate is increased, the alignment of the organisms reduces the component of their swimming motion in the z-direction, which leads to a smaller MSD. At high enough flow rate, the alignment is sufficient that the dominant mode for transport in the z-direction is Brownian motion. This leads to a diffusive scaling of the MSD proportional to 1t. The change in transport to and from the surface could impact other processes such as adhesion and biofilm formation. 4. Conclusions
In conclusion, we have used computer simulations to examine how simple flows, such as Couette and Poiseuille flow, change the properties of suspensions of swimming microorganisms in confined environments. The external flow acts to reduce the accumulation of organisms at the walls. This occurs in qualitatively the same way whether HIs with the walls are included or not (figure 2). However, more accumulation occurs when HIs are included (figure 3). The flow aligns the organisms, which leads to fewer collisions with the walls and less accumulation. A quantification of the swimmer orientations shows that in the intermediate region II (not right at the wall or the center of the channel) the alignment follows the scaling expected from an analysis of passive slender bodies in a shear flow, which is governed by a rotational Peclet number (figures 6 and 7). In addition to its impact on the density of organisms within the channel, the alignment also impacts the movement of organisms from one wall to the other. At low
Figure 8. The mean-squared displacement of swimmers between the
walls (in z-direction) for confinement 2L = 10l (a) and 2L = 16l (b). The flow conditions are no external flow (black solid line), Poiseuille flow (red) and Couette flow (blue). The flow rates are Per = 40 (dashed lines) and Per = 400 (solid lines). The ability of swimmers to move between the walls is reduced as the strength of flow increases.
shear rates, the organisms can swim ballistically from one wall to the other, while at high shear rates they can only move across the channel by Brownian motion (figure 8). Finally, the coupling of swimmer positions and orientations led to a dip in the concentrations of the organisms in the channel center in Poiseuille flow. It was determined that at high shear rates the governing parameter that determines this dip is the ratio of the velocity of the fluid at the channel center to the swimming speed of the organism (figure 4). This dependence is consistent 7
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with a theoretical prediction. These findings may provide insight into the formation of biofilms in various industrial and medical devices under different flow conditions, where surface accumulation and subsequent adhesion are critical steps in these processes. Further, this model may be used to explore how other flows and geometries could enable more efficient sorting of cells based on size, shape and motility for a range of applications, including diagnostics or quantifying components of complex microbial communities.
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