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Random Motility of Swimming Bacteria:
Single Cells Compared to Cell Populations
Bret R. Phillips and John A. Quinn
Dept. of Chemical Engineering, University of Pennsylvania, Philadelphia, PA 19104
Howard Goldfine
Dept. of Microbiology, School of Medicine, University of Pennsylvania, Philadelphia, PA 19104
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The motility of a population of swimming bacteria can be characterized by a
random motility coefficient, p, the operational equivalent of a diffusion coefficient
at the macroscopic level and in the absence of interacting chemical gradients. A t
the microscopic level, random motility is related to the singlecell parameters: speed,
tumbling probability, and index of directional persistence (related to the angle a
cell’s path assumes following a change in direction). Various mathematical models
have been proposed f o r relating the macroscopic random motility coefficient to
these microscopic singlecell parameters. In separate experiments, we have measured
motility at both the cellpopulation and singlecell levels for Escherichia coli. The
agreement of these results shows that the macroscopic transport behavior of a
population of motile bacteria can be predicted from straightforward microscopic
observations on single cells.
Introduction
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A characteristic feature of many motile microorganisms is
their zigzag movement which may be represented approximately as a sequence of straight line steps interrupted by turns
(Hall, 1977). The swimming motion of the bacterium Escherichiu coli illustrates this behavior. E. coli swim in a series of
smooth straight paths called “runs” which are terminated by
a rapid turning maneuver called a “tumble.” This gives the
cell a nearly random reorientation from which to begin the
next run (Berg and Brown, 1972). This run and tumble random
motility behavior can best be approximated as a threedimensional random walk. In the presence of a spatial or temporal
gradient of an attractant (sugars and amino acids), a bacterium
swimming up the gradient will suppress its tumbling response.
This results in a longer run in the direction of the higher
attractant concentration (Brown and Berg, 1974; Macnab and
Koshland, 1972; Lovely et al., 1974; Tsang et al., 1973); the
opposite response, that is, a longer run in the direction of
lower concentration is displayed in the presence of a repellent
(pH extremes and aliphatic alcohols). This behavior, termed
chemoklinokinesis, enables an individual bacterium to control
its tumbling frequency, and results in a biased or directed
random walk towards attractants and away from repellents.
At the macroscopic cellpopulation level, this net migration is
termed chemotaxis. This behavior is illustrated in Figure 1.
The bacterium E. coli is a peritrichously (uniformly) flagellated, rodshaped organism approximately 2 pm long and I
pm in diameter. E. coli has on average eight flagellar filaments,
each 2510 pm long and 20 nm in diameter (O’Brien and
Bennett, 1972). Motility is generated by a reversible rotary
motor located at the base of each flagellar filament (Berg and
Anderson, 1973; Silverman and Simon, 1974). When rotated
in the counterclockwise direction (as viewed from the distal
end), the flagella form a coordinated bundle which propels the
cell, resulting in the run type behavior discussed above. A
clockwise rotation of the flagellum results in a disruption of
the coordinated bundle which causes the cell to move chaotically or tumble (Larsen et al., 1974; Macnab and Ornston,
1977). Due to the randomwalk behavior of individual cells,
the dispersion of a population of cells in an isotropic medium
can be described in terms of a random motility coefficient, p,
the operational equivalent of a diffusion coefficient. For a
single cell, motility can be further interpreted in terms of its
speed, s, tumbling frequency (or the reciprocal mean run length
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Correspondence concerning this article should be addressed to J . A. Quinn. Current addresi
of B. R. Phillips: Merck & Co.. Inc.. Danville. PA.
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February 1994 Vol. 40, No. 2
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'andom motility
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chemotaxis
Figure 1. Bacterial motility in absence and presence 01
a chemical gradient.
L.elt: random motility in which the cell alternately swims and
iumbles. Tumbles occur with constant probability (Poisson statistics) and give the cell a nearly random reorientation to begin
i h r next run (see Results and Discussion). Right: biased motility
or ,:hemotaxis, in which the tumbling probability is decreased
*hen the cell is swimming toward higher attractantAower repellent concentration (Macnab, 1979).
time), A,, and the index of directional persistence, (Othmer
et al., 1988). Speed is defined as the cell's linear velocity during
a run. Run length time (or run displacement divided by velocity)
can be defined as the time between tumbles; due to the Poisson
nature of the underlying random processes, the inverse mean
run length time is equivalent to the tumbling frequency or the
tumbling probability. The index of directional persistence is
related to the turn angle, thai. is, the angle between consecutive
runs.
Bacterial motility is a dominant factor in a number of microbial processes. It has been shown to play important rolls
in both nitrogen fixationthe incorporation of atmospheric
nitrogen in plant culture, and denitrificationthe reduction
of nitrate or nitrite to nitrous oxide or elemental nitrogen
(Armitage et al., 1988; Gulash et al., 1984; Kennedy and Lawless, 1985). Evidence exists suggesting that motility may be one
of the mechanisms that controls bacterial interactions with the
mammalian gastrointestinal 'tract (Aldweiss et al., 1977; Freter
et al., 1981; Stanton and Savage, 1983). Chet and coworkers
(1975) found that motility can be used to control biofilm formation. Bacterial motility is also a vital aspect of many bioremediation technologies. Successful bioremediation may
require that biodegradable bacteria move through soil, aquifer
solids or groundwater (Gannon et al., 1991). It is clear that a
quantitative characterization of bacterial motility is necessary
in order to fully understand both naturally occurring microbial
transport and current biotechnological applications.
We have found few reports of quantitative measurements
of motility parameters at either the macroscopic cellpopulation or microscopic singlecell levels. At the cellpopulation
level, less than a dozen measurements of a random motility
coefficient have been reported. At the microscopic level, the
paths of individual swimming cells have been followed by the
ingenious threedimensional tracking microscope first develped
by Berg and coworkers (Berg, 1971; Berg and Brown, 1972;
Brown and Berg, 1974; Lovely et al., 1974; Lowe et al., 1987).
Other researchers (Maeda et al., 1976; Macnab and Koshland,
AlChE Journal
1972; Schneider and Doetsch, 1974; Poole et al., 1988; Taylor
and Koshland, 1975) have used photography and cinematography to investigate singlecell motility. The results of these
investigations in relation to those presented here are discussed
in the Results and Discussion section of this article. For a more
detailed discussion of motility assays see Ford et al. (1991).
Because the random motility of the cell population as a whole
is the result of the swimming behavior of each noninteracting
individual cell, the two are fundamentally related. Furthermore, because the net motion of single cells can be modeled
with three independent parameters, they provide a basis for
describing the transport of the cell population. The theoretical
models of Othmer et al. (1988) and Rivero et al. (1989) relate
microscopic singlecell parameters to the macroscopic random
motility coefficient. These models were tested by Farrell et al.
(1990) who investigated the twodimensional migration of alveolar macrophages, but they have not been rigorously applied
to bacteria swimming in three dimensions.
In this article we present the results of experiments in which
motility parameters for E. coli AW405 have been measured at
both the macroscopic cellpopulation and microscopic singlecell levels. Motility measurements were also gathered for cells
swimming in the presence of two different uniform concentrations of serine, an amino acid known to effect the motility
of this strain (Berg and Brown, 1972). For the macroscopic
measurements, a stoppedflow diffusion cell was used to establish an initial step gradient in cell concentration which decayed over time. At distinct time intervals, bacterial spatial
distributions were recorded photographically. Computeraided
image analysis of the photographic negatives yielded the actual
bacterial concentration profiles. From concentration profiles
a random motility coefficient could be calculated. Ford et al.
(1991) reported preliminary investigations using this same procedure. This approach has proven to be a reliable and precise
method for measuring the random motility of a cell population.
In comparison to the commonly used capillary assay, the
method presented here can be shown to be a more precise and
versatile approach. For the microscopic measurements, computeraided image analysis was used to measure singlecell
parameters (speed and run length time) from a video recording
of swimming cells. Since this method did not easily allow direct
measurement of the turn angle, we were unable to directly
measure the index of directional persistence, $ d . However, by
comparing these measured values of motility parameters at
both the cell population and singlecell levels with the theoretical models relating the two, the index of directional persistence, $ d , was determined. Prior to this investigation, the
only reported measurements of $d for bacteria were obtained
using a tracking microscope; the values measured here compare
well with those measurements.
The following section contains descriptions of two novel
experimental procedures which were used in the measurement
of motility parameters at the cellpopulation and singlecell
levels. The data obtained are then compared with a mathematical model relating the two. Finally, these results are discussed in relation to those reported in the literature.
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Experimental Studies
Cell culture
The bacteria used in all experiments reported here were E.
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coli AW405 (K12, galI, gal2, thr, leu, his4, lac, xyl, ara,
strA, tonA, tsx, and a wild type for chemotaxis) (Armstrong
et al., 1967). Motile cells were selected from the outer edge of
spreading colonies on swarm plates made with Luria broth
and 0.2% agar (Adler, 1966). These cells were introduced to
the swarm assay a second time to ensure a motile culture.
Stocks were made by adding 0.15 mL of glycerol to 0.85 mL
of an overnight culture, and were stored at 55°C. For the
experiments, an inoculating loop was used to transfer a small
amount of the frozen stock into 5 mL of minimal media. The
inorganic portion of the minimal media consisted of 11.2g
K2HP04, 4.8g KH2P04, 2.0g (NH,)2SOJ, 0.25g
MgS047H20,and 0.0005g Fe2(S04)3.5H20per liter distilled
water neutralized to pH = 7.0 (Adler and Dahl, 1967). Separately autoclaved, glucose ( 5 g/L) served as the carbon and
energy source. Growth of this strain required the media to be
supplemented with 0.25 g/L of Lthreonine, Lleucine, and Lhistidine. The inoculated culture, contained in a 25 ml Erlenmeyer flask, was grown in a rotary shaker (LabLine, Model
3527) at 150 rpm and 30°C. Cultures were harvested at an
absorbance of 0.45 to 0.65 at 590 nm (Unicam SP1800 Ultraviolet Spectrophotometer). Cells were observed under phase
contrast microscopy to qualitatively check for motility and
contamination. The cell suspension was diluted (100fold for
population studies and 200fold for singlecell measurements)
in motility buffer (1 1.2g K2HP04, 4.8g KH2P04and 0.029g EDTA per liter distilled water) (Adler and Templeton, 1967),
which had first been passed through sterile 0.2pm filters to
remove any debris or contaminant. This dilution was done
directly, without intermediate washing of the cells, and resulted
in an experimental cell density of approximately 2.4 x lo7cells/
mL for the cellpopulation assay and 1.2 x lo7 cells/mL for
the singlecell assay. All experiments were conducted at room
temperature (25 +2"C).
CellPopulation Assay
Method and procedure
Random motility measurements were carried out using the
stoppedflow diffusion cell (SFDC) assay. The SFDC, originally developed by Staffeld and Quinn (1989) for the study of
diffusiophoresis, is similar in principle to the MachZehnder
interferometry cell and utilizes a flow junction method to bring
in contact two fluids which differ in solute concentration
(Longsworth, 1950; Caldwell et al., 1957).
The SFDC, shown in Figures 2 and 3, operates by allowing
fluid to enter through the top and bottom ports and exit through
the two side ports, thus establishing a flow junction. To ensure
that fluid enters the upper and lower halves of the flow cell
at equal flow rates, a doublesyringe flow pump (Sage Instruments, Model 355) was used to feed the two inlet ports. With
equal flow rates no mixing occurs between the two suspensions.
It is therefore possible to establish a step gradient in solute (in
this case bacteria) concentration. When flow is stopped the
inlet and outlet valves are closed, producing a decaying step
gradient in bacterial concentration. For a more detailed description of the SFDC, see Staffeld and Quinn (1989) and Ford
et al. (1991).
Initially, all air was forced out of the SFDC and connecting
tubes and replaced with motility buffer. Two reservoir flasks
were used to feed the syringes which fed the inlet ports. The
,+
b\ fiirr3rion
I .T
!
Packed
/
Bed
Waste
+
Lower
Reservoir
Figure 2. Stoppedflow diffusion cell (SFDC).
Fluid was fed into the upper and lower inlet ports at equal flow
rates by a doubleflow syringe pump. The fluid which entered the
upper half of the flow cell contained bacteria at concentration
6,. Bacteria present in [he reservoir leading to the lower half of
the SFDC were removed by filtration before reaching the syringe
pump. This prevented bacteria from entering the lower half of
the flow cell. The packed bed regions (I.6mmdia. nylon spheres)
helped disperse the flow across the width of the chamber and
prevented channeling. Microscope slides used for the front and
back walls of the flow cell. The SFDC was designed with a large
depth to width ratio such that the fluid dynamics can be modeled
as HeleShaw flow in two dimensions (Schlichting, 1979). The
dimensions of the flow chamber, excluding the packed bed re5 ( 3 8 . 1 ~ 1 9 . 1 ~ 1 .mm).
9
Dashed
gions, are 1 . 5 ~ 0 . 7 5 ~ 0 . 0 7in.
line indicates crosssectional view in Figure 3 .
reservoir flask which fed the top of the SFDC contained 50
mL of the diluted cell suspension. The reservoir flask which
fed the bottom of the SFDC contained 50 mL of the diluted
cell suspension and 250pL Percoll (Sigma P1644). The Percoll
was added to make the fluid in the lower half of the flow cell
slightly more dense than the fluid in the upper half. This helped
to stabilize the fluid within the SFDC. Inline between the
bottom reservoir and the syringe pump were two membranes
(Standard Nuclepore Polycarbonate Membranes) in series containing 1 .O and 0.8pmdia. pores, respectively. These membranes filtered off the cells before they entered the syringe and
therefore prevented any cells from entering the lower half of
the flow cell (see Figure 2). Placing cells in both reservoirs
ensured that, with the exception of the bacteria present in the
upper half of the SFDC (and Percoll in the lower half), both
solutions were identical. This eliminated any potential chemical
gradients to which the cells might have responded chemotactically (see Results and Discussion).
The SFDC was positioned so that a stereomicroscope (ZeissModel SV8) could be focused on the flow junction region. The
flow cell was illuminated from behind by a fiberoptic ring
light (Schott #KL 1500) emitted at a 45 deg angle producing
darkfield illumination. The bacteria scatter the light, and,
through a port on the microscope, a 35mm camera (Contax
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February 1994 Vol. 40, No. 2
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Figure 4. Cellpopulationdata analysis procedure.
Black and white 35mm negatives were mounted in slide holders
and projected onto a video transfer screen. Video camera focused
on the image. Image digitized and stored in an image processor.
Bacterial profiles determined from the gray levels of the digitized
images. Monitor was used to display the digitized image and
concentration profile.
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Figure 3. Partially exploded crosssectionalview of the
SFDC apparatus.
F ibei optic ring light provides darkfield illumination. Stereoiiiicroscope had possible magnifications of 5 x to 205 x . Stereonncroscope adapted with a camera port parafocal with the oculars.
< amera port contained a reticle which enabled pixel size to be
cdlibrated to real dimensions during image analysis. Black and
M hite photographs were taken with a 35mm camera (2sexposure)
a! a magnification of 12.8 x .
SingleCell Assay
Method and procedure
A cell suspension (1.5 x lo7 cells/mL) was placed on a glass
microscope slide and covered with a cover slip supported by
two other cover slips to provide sufficient volume for the cells
to swim (see Figure 5). The slide was placed on a microscope
stage and observed under phase contrast microscopy (Zeiss,
400 x). The focal plane was established midway between the
top of the microscope slide and the bottom of the cover slip
in order to remove any possible wall effects. A green light
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RTS 11) wa\ used to record black and white images of the
bacterial spatial distributions at discreet time points.
Data analysis
The negatives from the photographs of the bacterial spatial
distributions were mounted in slide holders and the images
projected via a slide projector onto a video transfer screen
(Hama, model 3012). A video camera (Dagemti, 67M Series)
fitted with a 50 mm Cmount TV lens (Fujinon) was focused
on the transfer screen. The image was digitized and the data
were stored for analysis within an image processing system
(Imaging Technology, Series 151 lmage Processor). Figure 4
shows the data analysis procedure. A background image of
the SFDC containing only motility buffer was subtracted from
all analyzed data. The software package Mnemonics (TIPS)
aided in analyzing the bacterial density profiles on each negative. Bacterial profiles were obtained by scanning the gray
levels of the digitized image. The image analysis system employed allowed for 256 discreet gray levels (8 bit). The analysis
was typically able to resolve 1520 gray levels in each profile.
The final profile represents the average of approximately 50
adjacent profiles (each profile one pixel in width), which in
terms of the flow cell corresponds to the average concentration
profile across a 3mm front. By operating the flow cell at
:
AIChE Journal
bacterial concentrations which scatter light linearly
(53.75 x lo7 cells/mL), bacterial concentrations could be directly calibrated to gray level (Ford et al., 1991). For a more
detailed description of the image analysis procedure, see Phillips (1992).
\
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Microscope
Cover Slips
Objstive
Lens
(40xlO.65 NA. Phase 2) &

1
I
:

(=
170 mm hick)
Cell Suspcnsion
Figure 5. Singlecell assay.
Cell suspension (1.2x 10’ cells/mL) placed on a glass microscope
slide. Suspension covered with cover slip supported by two other
cover slips. Focal plane was established midway between the
microscope slide and the cover slip. Cells video recorded in real
time using phase contrast microscopy (400 x ). This provided a
field of view (as seen on the monitor during video playback) of
283x215 Fm.
February 1994 Vot. 40, No. 2
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was monitored until such an event was observed. The recording
was then rewound to the time when the cell had just finished
the first tumble and began a run. Via image analysis, the x
and y pixel (512x480) position of this point was recorded.
The video recording was then advanced frame by frame (30
framesls) until the run was terminated and the second tumble
began. The x and y pixel position of this point was also recorded. The run length time was obtained from the number
of frames which were recorded between the two tumbles. A
micrometer was used to calibrate x and y pixel displacements
to real dimensions to obtain a run displacement. Finally, a run
velocity was calculated by dividing the run displacement (pm)
by the run length time (s).
The insertion of a green light filter between the light source
and cell suspension limited the wavelength of light which illuminated the cell suspension to less than 700 nm. The focal
plane can be defined as the depth of field or “setting accuracy,”
2 i , and is given by (Inouk, 1986):
AnExampleofWhenDam
Would W beTaken
where n, is the refractive index of the immersion medium, X
the wavelength of light and u the half angle of the cone of
light that is captured by the objective lens. The numerical
aperture ( N A ) of the objective lens and Eq. 2 can be used to
calculate u:
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Figure 6. When singlecellmotility data would and would
not be recorded.
Area between the dashed lines represents the focal plane. With
the microscopic and optical systems employed ( X s 7 0 0 nm,
N A = 0.65). focal plane or depth of field is approximately 1.5 pm.
(a) Example of when singlecell motility data would be taken. Cell
initially outside the plane of focus and thus appears white. Cell
appears dark upon entering the focal plane. Cell then tumbles,
runs and tumbles again before exiting the focal plane. Data recorded representative of 3D motility; (b) Example of when singlecell motility data wou/d not be taken. Cell appears dark upon
entering the focal plane and tumbles. Cell begins a run and becomes displaced outside of the plane of focus where it tumbles
again. If data had been recorded, data would represent a 2D
projection of 3D motility.
filter (Zeiss VG9) was mounted between the light source and
the stage to limit the wavelength of light illuminating the cell
suspension (see below). The microscope was adapted with a
video camera (DageMTI, 67M Series) and the images recorded
(JVC BR9000U Time Lapse Video Cassette Recorder) in real
time.
Data analysis
Run length time (the time between tumbles) and run displacement (the distance between tumbles) were determined directly from the video recording with the aid of image analysis
technology (Imaging Technology, Series 151 Image Processor).
Under phase contrast microscopy a cell within the focal plane
will appear dark while a cell immediately above or below the
focal plane will appear to be white. Measurements were taken
only when a cell could be observed to tumble, run and then
tumble again, consecutively, and when all three events took
place within the focal plane (see Figure 6). The video recording
338
u = sin‘
F)
For the microscopic and optical systems employed here, Eq.
2 yields a depth of field of approximately 1.5 pm. As stated
above, data were recorded only when a cell was observed to
tumble, run and then tumble again, all within the focal plane.
This sequence ensues when in the course of its ordinary 3D
trajectory, a bacterium executes these three steps while moving
within the space defined by the focal “volume,” that is, the
focal plane times the depth of focus. Thus, by having a relatively thin focal “plane,” the data gathered are not a twodimensional projection of threedimensional movement, but
are instead a true representation of threedimensional motility.
Mathematical Analysis
Cellpopulation assay
Due to the random walk swimming behavior of an individual
bacterium, the random motility of the population can be expressed in a form analogous to that used for diffusion of inert
particles (Keller and Segel, 1971):
(3)
where b and p represent the bacterial density and random
motility coefficient, respectively. Onespatial dimension, semiinfinite in length, is appropriate for these experimental conditions. Initially, the bacterial concentration in the upper half
of the flow cell is b, and zero elsewhere. Allowing the flow
February 1994 Vol. 40, No. 2
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junction in the SFDC to be represented by x=O (see Figure
2 ) , and applying these boundary and initial conditions, the
solution of E<q. 3 is well known (Crank, 1975):
b 1 erfc(+)
(4)
For experimental times, t , <<L2/4p, where L is the length of
the lower half of the flow cell, the number of bacteria, N ( t ) ,
traveling across x = O as a function of time can, from Eq. 4,
be shown to be:
(5)
where A is the crosssectional area of the flow cell. From this
relationship the number of bacteria entering the lower half of
the flow cell is proportional to the square root of time. By
operating the flow cell at bacterial densities which scatter light
linearly, the number of bacteria below the junction is proportional to the area under the concentration profile, A,, where
A,, is generated directly from image analysis of the photographic negatives. Therefore, the expression used to calculate
a motility coefficient given the methods outlined above is:
r
Theoretical Analysis
Relationship between population and singlecell parameters
The stochastic models of Othmer et al. (I988j and Rivero
et al. (1989) can be used to relate the cellpopulation random
motility coefficient to singlecell parameters as follows:
(7)
where p is the random motility coefficient, s the mean speed
of the swimming cells, nd the dimensionality of the system, AT
the tumbling frequency or reciprocal run length time and
the index of directional persistence. The model assumes that
the probability of tumbling, X T , is the rate intensity of a Poisson
process. From probability theory (see Appendix), the mean
time between tumbles is given by A;'. The index of directional
persistence, $d, accounts for the angle a cell's path takes between adjacent runs. In three dimensions, lc/d is equivalent to
the mean of the cosine of the runtorun angle. I f the angle
between runs is random, the mean runtorun angle will be 90
degrees and $d will be equal to zero (see Figure 7 ) . If the mean
turn angle approaches the extreme value of 180 degrees, then
on average a cell would swim in the opposite direction of its
previous run and $d would be equal to  1; this results in a
value for the motility coefficient of one half of that which
where the calibration constant Crelates gray levels from image
analysis to real bacterial densities. Again, due to the fact that
these experiments are conducted with bacterial concentrations
which scatter light linearly, C 'can be determined directly from
the difference in gray level corresponding to the initial and
zero bacterial densities. The gray level resulting from the initial
bacterial concentration is obtained far above x = 0 where no
net flux of bacteria occurs. Similarly, the gray level corresponding to a zero concentration of bacteria can be obtained
far below x = O where bacteria have not yet migrated. From
Eq. 6, a plot of the calibrated area vs. the square root of time
should produce a linear relationship whose slope is equal to
the square root of p / r .
0
0
Tumble
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Singlecell assay
The theoretical models of Othmer et al. (1988) and Rivero
et al. (1989) assume that the tumbling events are the result of
a Poisson process with rate intensity hT(see Theoretical Analysis). Experimentally, the time intervals between tumbles (run
length times) were measured. The derivation in the Appendix
shows that if the tumbles are the result of a Poisson process,
then the time intervals between tumbles are exponentially distributed with a mean equal to the inverse of the tumbling
frequency or tumbling probability. Because an arithmetic mean
gives no information concerning the distribution about that
mean, a more accurate approach for obtaining the tumbling
probability is to rearrange the data in the form of a cumulative
distribution, as defined by Eq. A6. The tumbling probability
can then be obtained from the slope by plotting the natural
logarithm of the fraction of run length times greater than a
given time vs. time.
AIChE Journal
15
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wd
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= zero: Indicates purely
random orientation
from previous run
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Figure 7. Runtorun angle (6) and characterization of
index of directional persistence, $&
In three dimensions, index of directional persistence is defined as
mean of the cosine of the runtorun angle. I f runtorun angle
was purely random, 6 would have an expected value of 90".
resulting in a value of $, equal to zero.
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would be expected had the turn angle been random. A cell
which only swims and never tumbles, or tumbles and then
continues on in the same direction, would appear from Eq. 7
to have an infinite random motility coefficient. In this case,
however, the assumption that the runs are straight (that is,
noncurved) breaks down and random motility is limited by
rotational Brownian motion (Berg, 1983).
Results and Discussion
Cellpopulation assay
Figure 8 shows the time sequence of the bacterial concentration profiles for one motility experiment. The dimensionless
bacterial concentration (bacterial concentration scaled by the
initial cell density, b,) is plotted vs. position. The vertical
dashed line, x = O , represents the flow junction of the SFDC.
The baselines (horizontal dashed lines) represent the initial
bacterial densities present in the flow cell. The upper baseline
corresponds to the initial bacterial density, b,, present in the
upper half of the flow cell. This line was determined from the
region far left of x = 0 where the gray levels from image analysis
remained constant. This corresponds to the region in the flow
cell above the flow junction where there was no net flux of
cells. Similarly, the lower baseline corresponds to the initial
conditions in the lower half of the flow cell where there were
no bacteria present and was determined from the region far
right of x=O where the gray levels remained constant. This
corresponds to the region in the flow cell where bacteria have
not yet migrated. As discussed above, when operating the flow
cell at bacterial concentrations which scatter light linearly, the
number of bacteria present in the lower half of the flow cell
is proportional to the area of the concentration profile defined
by the lower baseline, x = O , and the bacterial density. This
will be termed the “enhanced” area. From conservation laws,
the number of bacteria entering the lower half of the flow cell
is equal to the number of bacteria which have left the upper
half. From this and linear light scattering, the “depleted” area
(defined by the upper baseline, x = 0, and the bacterial density)
is expected to be equal to the enhanced area. The average of
these two areas was used in determining the random motility
coefficient.
The experimental times indicated include an “offset time”
of 0.25 min to account for the “actual” initial conditions
present in the SFDC when flow was stopped. The offset time
is derived by comparing steady and unsteadystate continuity
equations applied to a solute (bacteria) in the flow cell under
conditions of steady flow. At time equal to zero, the unsteadystate result should be the same as the steadystate result. Using
this approach, the offset time, t o , was found to be given by:
t =I)
0.2
0.1
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where w is the width of the chamber and u the average fluid
velocity. From Eq. 8, it can be seen that an offset time can
be eliminated only by operating the flow cell at an infinite flow
velocity. For the experimental conditions used here, the offset
time was calculated to be 0.25 min. That is, the “ideal” step
function concentration profile had already decayed for 0.25
min before flow was stopped. Stated differently, a “perfect
step function profile” theoretically existed 0.25 min prior to
stopped flow conditions. For a rigorous analysis and further
discussion of the offset time, see Staffeld and Quinn (1989).
The areas measured from concentration profiles presented
in Figure 8 were plotted as a function of the square root of
time and are displayed in Figure 9. The straight lines represent
leastsquares fits of the data. Using Eq. 6 , a random motility
coefficient can be obtained from the slope of this line. In the
example shown here, the random motility coefficient was found
to be 2.1 x
cm2/s. It is important to note that the intercept
of this line is equal to zero. This indicates that no cells have
crossed from the upper to the lower half of the flow cell at
time equal to zero, an ideal step function in bacterial concentration. This would not have been the case had not the offset
time been included in this analysis. Instead, the yintercept
would have been equal to the area expected for a step function
already having decayed for a time period equal to the offset
time. In four experiments, the random motility coefficient was
determined to be 1.9 ( k 0 . 3 ) lo‘
~ cm2/s.
zyxwvutsrqpo
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0.0
0.I
0.2
I (mi
Figure 8. Time sequence of bacterial concentration profiles obtained from image analysis.
Bacterial density, scaled with the initial bacterial concentration
in the upper half of the flow cell, is plotted as a function of
position relative to the flow junction, x=O. “Enhanced” area
defined by zero bacterial density, x = O , and bacterial concentration profile should be equal to the “depleted” area defined by
the initial bacterial density, x = 0, and bacterial concentration
profile. The enhanced, depleted and average of these two areas
plotted as a function of the square root of time in Figure 9.
340
W
2TU
February 1994 Vol. 40, No. 2
AIChE Journal
The first measurements of a random motility coefficient were
reported by Adler and Dahl (1967). Using a slight variation
of the capillary assay, they measurer' the random motility
coefficient of E. coli B275 to be approximately 7 . 0 lo'
~
cm2/s. Using data presented by Adler (1969) obtained from
the capillary assay while investigating the motility of E. coli
W3110, Segel et al. (1977) calculated a random motility coefficient of 8.3 x
cm2/s. In the same investigation and conducting their own capillary experiments, they measured the
motility coefficient of Pseudomonas fluorescens to be
cm2/s. RiveroHudec and Lauffenburger (1986), us5.6 x
ing capillary assay data presented by Adler (1973), calculated
the random motility coefficient of E. coli B14 to be 5 . 4 lo'
~
cm2/s. From capillary assay data presented by Mesibov et al.
(1973) and Eq. 9, we were able to determine the random motility coefficient for E. coliAW518 and E. coli 20SOK. From
13 experiments conducted on five different cultures, values for
the random motility coefficient of AW518 ranged from
7.5 x
to 3.5 x lo' cm2/s, with a mean and standard de~
and 0 . 5 lo'
~ cm2/s, respectively. For
viation of 1 . 4 lo'
20 experiments conducted on the same culture, values for the
random motility coefficient of 20SOK ranged from 4.9 x lo'
to 3.7 x
cm2/s, with a mean and standard deviation of
1.9 x
and 0.9 x
cm2/s, respectively.
Using a light scattering densitometer, Holz and Chen (1979)
concluded that the random motility coefficient of E. coli K12
lies in the range 1  l o x lo' cm2/s. Dahlquist et al. (1972) used
a light scattering apparatus to investigate the motility of Salmonella typhimurium. From the data, Segel and Jackson (1973)
calculated the random motility coefficient to be 6 x lo'
cm2/s. In this laboratory, using the stoppedflow diffusion
cell, Ford et al. (1991) measured the random motility coefficient of E. coli NR5O to be 1 .I (&0.4) x 10 cm2/s. Berg and
Turner (1990), using a fused array of 50 pm diameter capillary
tubes, measured the random motility coefficient of E. coli
AW405 to be 2.6 (&0.4)x
cm2/s. A summary of these
results is presented in Table 1.
Our value for the random motility coefficient of E. coli
AW405 compares well with that of Berg and Turner (1990).
Discrepancies between the two values may possibly be attributed to differences in cell culturing and experimental condi
zyxwvutsrqp
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zyxw
zyxwvuts
Figure 9. Enhanced ( o ) , depleted ( 0 )and average of
these two areas ( A ) displayed in Figure 8 as
a function of the square root of time.
Kanoom motility coefficient can be obtained from the slope. In
this example, using the average area, a random motility coefficient
0:'2 I x
cm'/s was determined.
Due to its experimental simplicity, the capillary assay is the
most commonly used technique in measuring bacterial motility
(Adler, 1973). For random motility measurements, a capillary
tube, approximately 1 pL in volume, is sealed at one end and
filled with motility buffer. The tip of the open end is then
placed into a chamber containing a known bacterial concentration suspended in motility buffer. As time passes, bacteria
will begin to accumulate in the tube. After a given time interval,
usually one hour, the number of bacteria in the tube, N , ( t ) ,
are counted and a random motility coefficient is determined
from Eq. 9 (Segel et al., 1977):
(9)
where I is time, R the inner radius of the capillary tube and
C,, the initial bacterial density in the chamber. Equation 9 is
obtained from the same formalism used in deriving Eq. 5 with
the assumption that the bacterial concentration at the open
end of the tube is constant and equal to C,.
____
Table 1. Measured Values of the Random Motility Coefficient
Species
1 (cm'/s)
Reference
~~
Pseudomonas Fluorescens
Salmonella typhimurium
Escherichia coii
Escherichia coii
Escherichia coii
Escherichia coii
K 12
20SOK
AW518
€314
6 ( * 2 ) x lo'
6 x lo'
1IOX l o  '
1.9 ( k O . 9 ) ~
1.4 (*0.5)x 10"
5.4x 105
Escherichia cor'i NRSO
Escherichia coii W3 1 10
1.1 (=t0.4)xIO'
8.3 x lo'
Escherichia coii B275
Escherichia coli B275
Escherichia coii AW405
Escherichia coii AW405
6 . 9 lo'
~
7 . 2 lo'
~
2.6 (*0.4)x
1.9 ( k 0 . 3 ) ~
____
Segel et al. (1977)"'
Dahlquist et al. (1972)'
Segel and Jackson (1973)**
Holz and Chen (1979)"'
Mesibov et al. (1973);'
Mesibov et al. (1973)*+
Adler (1973)'
RiveroHudec and
Lauffenburger (1986)**
Ford et a]. (1991)"
Adler (1969)*
Segel et al. (1977)**
Adler and Dahl (1967)**'
Adler and Dahl (1967)'* *
Berg and Turner (1990)***
This investigation
'Source of e~perimentaldata.
Source where the random motilil y coefficient was calculated based on the experimental data.
'Random motility coefficient calculated from Eq. 9.
.I
AIChE Journal
February 1994 Vol. 40, No. 2
341
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tions. The cells in this investigation were grown on glucose in
a minimal salts medium whereas theirs were grown on a richer
tryptone broth medium. Lowe et al. (1987) found that for E.
coli HCB437, the rotational frequency of the coordinated flagellar bundle was much higher for cells grown on tryptone
broth than for cells grown on glycerol in a minimal salts medium. This translates to a higher swimming speed for cells
grown in the richer medium. Also, the experiments in this
investigation were performed at room temperature (25 &2”C)
whereas those of Berg and Turner were conducted at 30°C. A
linear dependence of swimming speed on temperature has been
observed for both E. coli and Salmonella typhimurium (Lowe
et al., 1987; Banks et al., 1975; Maeda et al., 1976; Miller and
Koshland, 1977).
As stated above, Berg and Turner (1990) measured their
random motility coefficient for cells swimming in capillary
tubes of diameter 50 pm. In their investigation, they also found
that the value of the random motility coefficient increased by
a factor of 2.3 when measurements were made using 10pm
capillaries. It appears that in the smaller capillary tubes motility
occurs predominantly along the axial direction; this observation is consistent with realistic simulations of motility in small
capillaries (Phillips and Quinn). Although the motility values
obtained with the 10 and 50 pm capillaries are different, it is
not evident that the 50 pm capillary is equivalent to that of a
bulk phase. The macroscopic dimensions of the SFDC provide
unhindered, bulk phase and threedimensional motility.
It is difficult to make other comparisons of the data presented in Table 1 since these measurements have been carried
out on a variety of species and strains. The culturing and
experimental conditions in many cases vary from laboratory
to laboratory, and, as discussed above, these factors also effect
motility. In general, it appears that the random motility coefficient may vary over two orders of magnitude among different
species and strains. If these bacteria were immotile, they would
still “diffuse” as a result of Brownian motion. Using Stokes
law, this yields a motility or diffusion coefficient of approximately 2 x
cm2/s (Berg, 1983).
The random motility coefficient was also measured at two
different uniform concentrations of serine, an amino acid
known to effect the motility of this strain (Berg and Brown,
1972). Conducting four experiments at a serine concentration
of 6.7 x lo’ molar, the random motility coefficient was found
to be 3.8 (*0.4)x
molar serine, a
cm2/s. At 1 x
random motility coefficient of 8.9 (+0.5) x
cm2/s was
measured. These data are presented in Table 2. Holz and Chen
(1979) used their lightscattering densitometer and determined
the random motility coefficient of E. coli K12 as a function
of serine concentration. At a concentration of 1 x
molar
serine they observed approximately a 2fold increase in their
calculated value of the random motility coefficient. At this
same serine concentration and from the data presented in Table
2, a 4.7fold increase was observed in this investigation. In the
absence of serine the measured value of the random motility
coefficient determined here was more than twice their reported
value.
As stated above, it was necessary to place bacteria in the
reservoirs leading to both the upper and lower halves of the
flow cell. The cells in the reservoir leading to the lower half
of the SFDC were then removed by filtration. This was to
ensure that except for the presence of cells in the upper half
of the flow cell (and Percoll in the lower half), both solutions
were identical. When cells were not placed in the lower reservoir, a thin band of migrating cells was seen to form just
below the flow junction. In some instances this band was
observed to form immediately after flow was stopped, and at
other times it became visible over time. The formation of this
band is indicative of a chemotactic response. It is well known
that both E. coli and Salmonella typhimurium respond chemotactically towards oxygen (Baracchini and Sherris, 1959; Adler, 1966; Laszlo and Taylor, 1981; Shioi et al., 1987). We
believe that due to the endogenous metabolism of the cells,
the oxygen concentration in the upper reservoir was reduced
while the oxygen concentration in the lower reservoir, containing only motility buffer, remained saturated. Upon introducing these two fluids into the flow cell, a gradient in oxygen
concentration formed at the flow junction. Cells swimming
near the flow junction region chemotactically responded to the
oxygen gradient. The band was, at most times, easily detectable
by the naked eye. Band formation was observed when the
reservoir containing cells was fed to either the upper or lower
half of the flow cell. Band formation also occurred when
Percoll was absent from the lower reservoir (cells were not
responding chemotactically to Percoll). In all cases, the band
did not move far from the junction. i n some experiments, a
second, and possibly a third band could be seen to form above
the first band. Band formation did not occur when cells were
placed in the reservoir leading to the lower half of the flow
cell and then removed by filtration.
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LSerine Concentration
(lo’ Molar)
p
cm2/s)
1.9*0.3
3.8&0.4
8.9+0.5
‘Dare presented as mean *one standard deviation.
342
The velocity distribution for bacteria swimming in the absence of serine is presented in Figure 10. The data were distributed normally or slightly skewed from normal with a mean
(*standard deviation) of 24.1 &6.8 pm/s. A total of 260 velocity measurements were gathered from three different cell
cultures. The measurements from each culture gave approximately the same distribution. Since the data were gathered
only when a bacterium was observed to tumble, run, and then
tumble again, and when all three events occurred within the
focal plane, it was rare that motility parameters for a single
bacterium were measured more than once. In the rare instance
when an individual bacterium was observed to complete two
consecutive runs within the focal plane the velocity was found
to be relatively constant.
The following is a discussion of previously measured values
of cell velocity which have been reported in the literature. Using
cinematography techniques, Maeda et al. (1976) measured the
swimming speed of E. coli W3110 to be approximately 28
pm/s at 25”C, approximately the same experimental temper
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Table 2. Measured Values of the Random Motility Coefficient
as a Function of Serine Concentration*
0
6.7
100
SingleCell Assay
Cell velocity
Februarv 1994
.. .
Vol. 40, No. 2
I
AIChE Journal
0.15
0.10
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71
a
B
e
.
u'
0.05
zyxwvutsrqponm
zyxwvutsrq
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0.00
0
1977). The tracking microscope provided for a detailed analysis
of swimming velocity, run length time, tumbling time, and
turn angle (index of directional persistence). Using E. coli
AW405, they measured these paramettls for a total of 35
bacteria from three different cell cultures. Mean swimming
speed was found to be 14.2 & 3.4 pm/s and not strongly skewed.
They also found that speed is nearly uniform during a run.
Our value of the velocity determined for the same strain was
significantly higher at 24.1 k 6 . 8 pm/s. It is not clear whether
or not this difference can be attributed to experimental methods. In both investigations cells were grown in a minimal salts
medium, the only difference being that they used glycerol as
the carbon source whereas we used glucose. Their experiments
were conducted at a higher temperature than those reported
here. Lowe et al. (1987) used the tracking microscope and
determined a linear relationship between swimming speed and
temperature. At 32"C, they determined the velocity of E. coli
AW405 grown in a rich T broth medium to be 36.4 1.O
pm/s.
Cell velocity measurements were also determined at the same
two uniform concentrations of serine at which the population
studies were conducted. At a uniform concentration of
molar serine, cell speed was found to
6.7 x lo' and 1 x
be 30.7~t7.1and 30.2*5.2 pm/s, respectively (see Table 3).
This corresponds to approximately a 25 percent increase compared to cell speeds measured in the absence of serine. These
distributions also appeared to be slightly skewed of normal.
Macnab and Koshland (1972) found that the velocity of Salmonella fyphimurium, a related species, remained essentially
unchanged at serine concentrations up to 1 x lo' molar. Nossal and Chen (1973) determined the root mean square velocity
of E. coliK12 over a range of serine concentrations. Cell speed
was found to increase with increasing concentrations of serine
molar where it peaked. At this concentration,
up to 1 X
cell speed was over 30 pm/s, more than double that in the
absence of serine. Berg and Brown, using the same strain, also
studied the effect of serine on motility and found that cell
speed increased 40 percent as serine concentration was increased to 1 x lo) molar.
In brief, the value of cell speed determined in this investigation is consistent with those reported in the literature. The
fact that the value presented here is higher than that reported
by Berg and Brown for the same bacterial strain may be due
to differences in experimental conditions. In addition, the trend
10
20
30
40
50
Cell Velocity (pdsec)
Figure 10. Distribution of cell velocities.
*
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Cell velocities were grouped in 2 pm/swide bins. The number
of runs of a given cell velocity displayed on lefthand side. The
righthand side shows the fraction of runs of a given cell velocity
expressed as probabilit y. A total of 260 measurements were taken
from three different cell cultures. (Not shown is one cell velocity
measured to be 61.3 pm/s.) The mean (*standard deviation)
was found to be 24.1 * 6.8 p m / s . Dashed line represents a normal
divtribution based on this mean and standard deviation.
ature used in this investigation. They also determined that the
swimming velocity increased with increasing temperature. Using number fluctuation spectroscopy, Banks et al. (1975) measured the swimming speed of a smooth swimming strain of E.
coli at 25°C to be 16 pm per second. They also found a linear
relationship between cell speed and temperature between 25
and 35°C. Nossal and Chen (1973) used the technique of laser
intensity correlation spectroscopy and determined the root
mean square speed of a wild type E. coli K12. At 25&2"C,
they measured a value of approximately 1015 pm/s. Using
stroboscopic photography, Macnab and Koshland (1972)
measured the velocity of Salmonella typhimurium to be
28.8 f 5.1 pmis. Schneider and Doetsch (1974) measured cell
velocity directly from videotape. Conducting experiments at
21 " C on two different cultures of Serratia marcexens, they
determined the velocity to be approximately 43.4 pm/s. They
found no statistical difference of velocity measurements between the two cultures. Taylor and Koshland (1975) also made
measurements directly from videotape while investigating motility responses to light. They determined the swimming velocity
of Salmonella typhimurium to be 34.6 pm/s. Poole et al. (1988)
used realtime computer tracking to measure the velocity of
Salmonella typhimurium. From 21 1 cell tracks they determined
a velocity of 18.4*8.8 pm/s. The distribution of these velocities appear to be normal to skewednormal.
The most thorough investigation of singlecell bacterial motility was carried out by Berg and Brown (1972) using a threedimensional tracking microscope (Berg, 1971). The tracking
microscope automatically follows individual cells as they swim
in three dimensions and digitally records their position at a
rate of 12.6 data points per second. Their experiments were
conducted at 32°C in a motility medium with 0.18 percent
(w/v) hydroxypropyl methylcellulose which increased the viscosity of the medium t o 2.7 cp. It has been shown that cell
velocity is nearly optimum under these viscous conditions
(Schneider and Doetsch, 1974; Greenberg and CanaleParola,
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AIChE Journal
Table 3. Measured Values of SingleCell Motility Parameters
as a Function of Serine Concentration*
LSerine
Concentration
Molar)
0
6.7
100
Cell Speed
(pm/s)
24.1 *6.8
30.7 *7.1
30.2*5.2
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Mean Run
Length Time'*
Tumbling
Probabilityt
(S)
(11s)
0.84rt0.71
1.OO*0.87
1.35*1.02
1.37
1.15
0.67
'Data presented as mean +one standard deviation. With no serine present, a
total of 260 velocity and run length measurements were gathered. A total of
280 and 308 measurements were obtained for the intermediate and high serine
concentrations, respectively. Each data set was gathered from three different
cell ,cultures.
For ideal exponential distribution the mean is equal to the standard deviation.
'Obtained from Eq. A6. For ideal Poisson statistics should be equal to the
inverse of the mean run length time in column 3.
February 1994 Vol. 40, No. 2
343
1
zyxw
0.15
zyxwvutsrq
zyxwvutsrqponmlkjihg
zy
I
0
3
2
4
Time (seconds)
0
1
2
3
Run Length Time (seconds)
4
Figure 11. Distribution of cell run length times.
Cell run length times were grouped in 0.2 swide bins. The
number of runs of a given time length are displayed on the lefthand side. Righthand side shows the fraction of runs of a given
time length expressed as a probability. A total of 260 measurements were taken from three different cell cultures. The mean
and standard deviation were found to be 0.84 and 0.71 s, respectively. For an ideal exponential distribution the mean and
standard deviation have the same value. Dashed line represents
an exponential distribution based o n the mean and i s given by
Eq. A5.
Figure 12. Distribution of the cell run length times presented in Figure 11, as defined by Eq. A6.
zyxwvu
Natural logarithm of the fraction of run length times greater
than a given time are plotted as a function of time (solid squares).
Solid line represents the leastsquares fit of the data. From Eq.
A6 the tumbling probability is obtained from the slope. I n this
case, the tumbling probability was found to be 1.37/s. This
corresponds to a mean run length time of 0.73 s. Dashed lines
represent the theoretical standard deviation from the leastsquares
fit and are obtained from Eq. A7.
zyxwvutsrqpo
of serine enhanced velocity reported by other investigators was
also observed here.
Cell run length time
The distribution of run length times for bacteria swimming
in the absence of serine is shown in Figure 11. The data were
distributed exponentially with a mean ( standard deviation)
of 0.84*0.71 s. As noted in the Appendix, the standard deviation of an exponential distribution has the same value as
the mean. In Figure 12, the same data are plotted as defined
by Eq. A6. The tumbling probability can be obtained from
the slope. In this case the tumbling probability was found to
be 1.37 per second. This corresponds to a mean run length
time of 0.73 s, comparable to the value determined from the
arithmetic mean of the data presented in Figure 11. The y intercept of a true exponential distribution logarithmically
plotted in accordance with Eq. A6 would be expected to be
equal to zero. That is, all run length times are greater than
zero. In Figure 12, the intercept was found to be 0.13. This is
due to the fact that the exponential distribution displayed in
Figure 11 appears to be incomplete at short run length times.
The reason for this can be partially attributed to an uncertainty
in the experimental technique when measuring short run length
times. When a run appeared to occur over only a few frames,
it was sometimes very difficult to locate exactly when the first
tumble ended and the run began and also when the run ended
and the second tumble began. For this reason, many of these
runs were not recorded. Thus, the distribution at very short
run length times is somewhat incomplete. This could be accounted for by rewriting Eq. A6:
where t* is the shortest measurable run length time. An ap344
proximation o f t ' can be obtained from the intercept and slope
of a leastsquares fit line. Setting y = 0, t* is approximated to
be 0.095 s. At 30 frames per s, this corresponds to runs of
length less than or equal to three frames. As described above,
it was the run times of this length and less which were very
often not measured due to the uncertainty of the runtumble
distinction. This variation of the analysis, however, did not
effect the tumbling probability.
Maeda et al. (1976) measured the tumbling frequency as a
function of temperature for E. coli W3110. The tumbling frequency was found to have a sharp peak at 34°C and was very
low at both 39 and 20°C. At 25°C the tumbling frequency was
approximately 0.3 per s corresponding to a mean run length
time of 3.33 s. Lowe et al. (1987) used the tracking microscope
and measured the mean run length time of Streptococcus strain
V4051 to be 1.71 *0.90 s. They also found the run statistics
to be Poisson. Berg and Brown (1972) used the tracking microscope and measured the mean ( standard deviation) run
length time of E. coli AW405 to be 0.86 f 1.18 s. This compares
well with the value measured in this investigation of 0.84
*0.71 s.
Run length times were also measured at two uniform concentrations of serine. At a concentration of 6.7 x 10 molar
serine, the run length time distribution was exponential with
a mean (*standard deviation) of 1.00*0.87 s. From Eq. A6
the tumbling probability was determined to be 1.15 per s. At
a serine concentration of 1 x lo' molar, the run length time
distribution was again exponential with a mean (*standard
deviation) of 1.35 f 1.02 s. From Eq. A6 the tumbling probability was determined to be 0.67 per s. These data are presented
in Table 3. We believe that at this high serine concentration
the value for the mean run length time may be underestimated
(tumbling probability overestimated). This is due to the limited
field of view as displayed on the video monitor (see SingleCell Assay Method a n d Procedure). The field of view on the
monitor is 283 x 215 pm (349 pm diagonally). A cell swimming
*
'
zyxwvuts
February 1994 Vol. 40, No. 2
AIChE Journal
zyxwvutsrqpo
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at 30 pmls along the short axis will traverse the field of view
in 7 . 2 s. .4 cell which began its run in the exact middle of the
field of view could at most remain in the field for 5.8 s at this
same velocity. At the high serine concentration, cells were
frequently observed to swim from 4 to 7 s, becoming displaced
outside of the field of view before completing the run (tumbling). Thus, the limited field of view may bias run length time
distributions away from longer runs. The field of view can be
increased bv lowering the magnification, however, visualization of the cells then becomes difficult. The depth of the field
(Figure 6) also biases run length times away from longer runs.
A cell swimming within the focal plane, not exactly parallel
to the field of view will, over time, become displaced above
or below the plane of focus. The longer the run displacement
the more likely that this will occur. Furthermore, small influences due to rotational diffusion would amplify this effect. It
is obvious that the limited field of view and narrow focal plane
would also affect measurements taken on cells swimming in
the absence of or at the intermediate concentration of serine.
Due to shorter run length tim.es, and, in the case of no serine,
a slower velocity, these influences would not be as crucial a
factor in these cases. However, the general effect of the limited
field of view and the narrow local plane biases run length time
distributions away from longer runs. This results in possibly
underestimating both the mean and standard deviation of the
distributions.
Berg and Brown have also measured the effect of serine on
the run length times for E. coli AW405 and found that as the
serine concentration is increased the distributions remain exponential but are shifted towards longer runs. At a concenmolar serine, they found the mean run
tration of 6.7 x
length time to increase approximately 2.5fold over that
observed in the absence of serine. This corresponds to a mean
run length time of 2.15 s. At 1 x W 3 molar serine, the
mean run length time increased 3.5fold corresponding to a
mean run length time of 3.0 s. Although we also observed an
increasing mean run length time as a function of serine concentration, the effect was not as pronounced.
We found no correlation between cell velocity and run length
time. Linear correlation coelficients both in the absence and
presence of serine were less than 0.1. Brown (1974) found cell
speed to be correlated with run length time. From 73 measurements on an individual bacterium, he determined a linear
correlation coefficient of 0.72.
At the intermediate and high serine concentrations, $d was
determined to be 0.28 and 0.49, respectively. Berg and Brown
determined that the mean change in direction from runtorun
decreased 40Vo at higher serine conce .trations resulting in a
slightly higher value of $+ This is the same trend observed in
this investigation. We believe, however, that our value of 0.49
at the high serine concentration may be overestimated due to
an overestimated tumbling probability as discussed above.
Brown (1974) found no correlation between change in direction
and cell speed or run length time.
Conclusions
This is the first investigation in which random motility parameters have been measured at both the cellpopulation and
singlecell levels. The SFDC has proven to be a reliable and
precise method for measuring the random motility coefficient
of a cell population. It has produced highly reproducible results
which are consistent with the few published results to which
direct comparison can be made. Random motility coefficients
were also measured in the presence of two different uniform
concentrations of serine; values were observed to increase with
increasing concentrations of serine, as expected.
We have also developed a very simple optical technique to
measure the singlecell motility parameters, speed and run
length time. The distribution of cell velocities was found to be
slightly skewed of normal. Run length time distributions were
determined to be exponential. From the run length time distribution a tumbling probability was calculated. The values
and distributions of these singlecell parameters are consistent
with those reported in the literature. Singlecell parameters
were also measured at the same two uniform concentrations
of serine used in the population studies. The mean speed value
was found to increase slightly in the presence of serine. As the
concentration of serine increases, the run length time distribution remains exponential but is shifted towards longer times.
These trends are also consistent with reports in the literature.
The limitation of this technique is that it does not readily permit
for measurement of the turn angle between two consecutive
runs. The event in which a cell makes two consecutive runs
while remaining in the focal plane is relatively rare.
The theoretical expressions of Othmer (1988) and Rivero et
al. (1989) were used to estimate the index of directional persistence from the random motility coefficient measured using
the SFDC, and the cell speed and tumbling probability determined microscopically. The values obtained also appear to be
consistent with the limited values reported in the literature.
From this comparison it appears that the theoretical expression
relating the cellpopulation random motility coefficient to singlecell parameters is valid for a bacterium swimming in threedimensions. The agreement of these two experimental methods
demonstrates that the macroscopic transport behavior of a
population of motile bacteria can be predicted from microscopic observations on single cells.
The singlecell observation technique presented here is a
simple and straightforward method for obtaining detailed motility measurements. This approach, however, does not readily
permit for measurement of the index of directional persistence.
This limitation is due to the relative rarity of a cell executing
two consecutive runs while remaining in the focal plane. With
advanced image analysis however, this technique could be au
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Theoretical comparison of cellpopulation and singlecell parameters
Due to the relative infrequency of the event in which an
individual bacterium remains in the focal plane for two consecutive runs, the index of directional persistence, $, could
not be measured directly. However, Eq. 7 and the data presented in Tables 2 and 3 can be used t o estimate the index of
directional persistence. In the absence of serine and using the
tumbling probability obtained from Figure 12, $d was determined to be 0.26. This indicates a positive directional persistence in the random motility behavior. Berg and Brown, using
the tracking microscope, directly measured the runtorun angle and calculated $, to be 0.3 (Brown, 1974). Lowe et al.
(1987), also using the tracking microscope, measured the runtorun angle of a Streptococcus strain and determined approximatelv the same value for $,.
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tomated, thereby overcoming this limitation. This method
would then provide a means for complete and detailed characterization of bacterial random motility.
Acknowledgment
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The authors would like to thank Dr. Paul DiMilla for valuable
discussions concerning the interpretation of results presented here.
This investigation was supported in part by National Science Foundation Grant BCS8612987.
Notation
a
A
A,
b
b,
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in Chemical Gradients,” Rev. Sci. Instrum., 45, 683 (1974).
= time
= crosssectional area of flow cell
= area defined by the bacterial concentration profile
= bacterial density
= initial bacterial density
c = calibration constant relating bacterial density to gray level
c, = initial bacterial density in the capillary assay chamber
g = fraction of run length times greater than some time t
L = length of the lower half of the flow cell
n = number of tumbles
nd = dimensionality of system
n, = refractive index
N = number of bacteria moving from the upper to the lower half
of the flow cell
N, = number of bacteria accumulated in the capillary tube
Np = total number of tumbles
NR = total number of runs
N A = numerical aperture of objective lens
P = probability
R = inner radius of capillary tube
s = cell speed
t = time
t = offset time
t g = shortest run length time which can be measured
T, = time between the (n  1)st tumble and the nth tumble
u = half angle of the cone of light gathered by the objective lens
v = average fluid velocity inside the flow cell
w = flow cell width
x = position in flow cell relative to flow junction
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Greek letters
AT
X
= tumbling probability
=
wavelength of light
p = random motility coefficient
{ = one half of the focal plane thickness
u = standard deviation
Gd = index of directional persistence
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Literature Cited
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Appendix
The models of Othmer et al. (1988) and Rivero et al. (1989)
assume that the tumbling response is the result of a Poisson
process. We measure the time between tumblesthe run length
times. The following analysis shows how the tumbling probability can be determined from these measurements of run
length times (Ross, 1985).
Begin by assuming that we are following one cell starting at
time zero. If N p ( t ) represents the total number of tumbles
which have occurred up to time t , then N p ( t ) is a Poisson
process if the following requirements are met:
(i) Np(0)= 0
(ii) The process has stationary and independent increments
(iii) The number of tumbles in any interval of length t is
Poisson distributed with mean Art (standard deviation JArt).
For all times, a, t r O ,
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P ( N p ( t + a ) N , ( a ) = n ) =
( X,t )“exp’Tr
n!
n = O , 1 , 2, . . . (Al)
Definition (i) simply means that only those tumbles are counted
which occur at time t > 0. From definition (ii), a process is said
to have independent increments if the numbers of events which
occur in disjoint time intervals are independent. The definition
of stationary increments is satisfied if the distribution of the
number of tumbles which occur in any interval of time is
dependent only on the length of the time interval. The process
rate intensity, or in this case the tumbling probability, AT, is
defined by Eq. A2:
Limit
Ar0
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Probability (exactly one event in [f, t + A t ] )
At
(A21
Equation A1 therefore determines the probability that n tumbles occur in the time interval (a, t ) .
The probability that zero tumbles occur in time (0, t ) is equal
to the probability that the first run length time, Ti, is greater
than t. From Eq. A l , this probability is given by:
Since the probability that Ti occurs between times (0, + 0 0) is
one, then the cumulative probability, or the probability that
TI is less than or equal to t is given by Eq. A4:
Equation A4 is the cumulative probability of an exponential
process. The probability density function of a random variable
is the derivative of the cumulative distribution. Taking the
derivative of Eq. A4, the probability that the run length time
February 1994 Vol. 40, No. 2
347
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k!
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zy
TI is equal to some time t is defined by the exponential probability function:
P ( TI= t ) = Xr exp"'
(A5)
The mean and standard deviation of an exponential process
are both given by A;'. Therefore, the run length time, T I ,is
exponentially distributed with mean X; I. This same approach
can be used to show that the entire sequence ( T , , n = 1, 2,
. . . 1 is exponentially distributed with mean A;'. Thus, if the
tumbling rate is the result of a Poisson process with rate intensity AT, then the time between tumbles or the run length
time is exponentially distributed with mean and standard deviation A;'.
The analysis above is for an individual cell. In the experiments conducted here, we are measuring the run length times
for many different cells. Therefore, an underlying assumption
of this approach is that all cells in the population exhibit the
same random motility. This assumption is also present in the
population studies where the dispersion of a cellpopulation
is modeled with a single random motility coefficient, p . Brown
(1974) found both cell speed and turn angle (index of directional persistence) to be uniform from celltocell within a given
population. He also determined that the mean run length times
of individual cells within a cellpopulation were lognormally
distributed. The mean (&standard deviation of the mean) of
the mean run length times of 34 cells was determined to be
1.2&0.3 s. Due to the relatively small standard deviation of
this mean run length time distribution as compared to about
1.2 s standard deviation for each individual cell, the assumption
348
that all cells exhibit the same random motility can be considered
valid. Randomly measuring a single run length time from many
cells is then equivalent to measuring many run length times
from an individual cell. Because a cell's run length time is
independent of whether or not the cell is swimming within the
plane of focus, the cell run length times measured represent a
random sampling.
As stated above, the measured run length times are expected
to be exponentially distributed. From these run length times
the tumbling probability, AT, is determined by two methods.
The first is simply from the arithmetic mean. The second
method is obtained by rearranging Eq. A3. Taking the natural
logarithm of both sides of Eq. A3 gives:
where g represents the fraction of runs of length greater than
t . From Eq. A6, the tumbling probability or reciprocal persistence, X,, can be obtained from the slope of the plot of ln(g)
vs. t . The standard deviation of the lefthand side of Eq. A6
is given by (BlancLapierre and Fortet, 1965):
%(g)
=

where N R is the total number of runs measured.
Manuscripi received Ju1.v 31, 1992. and revision receitied June 30, 1993
February 1994 Vol. 40, No. 2
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