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Journal of Bacteriology, November 1998, p. 5978-5983, Vol. 180, No. 22
0021-9193/98/$04.00+0
Copyright © 1998, American Society for Microbiology. All rights reserved.
Fitness Landscapes for Effects of Shape on
Chemotaxis and Other Behaviors of Bacteria
David B.
Dusenbery*
School of Biology, Georgia Institute of
Technology, Atlanta, Georgia 30332-0230
Received 5 June 1998/Accepted 10 September 1998
 |
ABSTRACT |
Data on the shapes of 218 genera of free-floating or free-swimming
bacteria reveal groupings around spherical shapes and around rod-like
shapes of axial ratio about 3. Motile genera are less likely to be
spherical and have larger axial ratios than nonmotile genera. The
effects of shape on seven possible components of biological fitness
were determined, and actual fitness landscapes in phenotype space are
presented. Ellipsoidal shapes were used as models, since their
hydrodynamic drag coefficients can be rigorously calculated in the
world of low Reynolds number, where bacteria live. Comparing various
shapes of the same volume, and assuming that departures from spherical
have a cost that varies with the minimum radius of curvature, led to
the following conclusions. Spherical shapes have the largest random
dispersal by Brownian motion. Increased surface area occurs in oblate
ellipsoids (disk-like), which rarely occur. Elongation into prolate
ellipsoids (rod-like) reduces sinking speed, and this may explain why
some nonmotile genera are rod-like. Elongation also favors swimming
efficiency (to a limited extent) and the ability to detect stimulus
gradients by any of three mechanisms. By far the largest effect
(several hundred-fold) is on temporal detection of stimulus gradients,
and this explains why rod-like shapes and this mechanism of chemotaxis
are common.
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INTRODUCTION |
The first bacteria to be observed
were classified by their shapes (23), and in the three
centuries since, shape has continued to be used for classification
(e.g., coccus or bacillus). However, only modest efforts have been
devoted to the mechanistic question of how the shapes of single-cell
organisms are determined (3, 9), and the functional question
of why bacteria have particular shapes has been even more neglected.
The latter question is addressed here.
It seems reasonable to assume that early organisms were spherical as a
result of surface tension (9), but the advent of cell walls
opened up the option of different shapes. I consider an organism that
is initially spherical and ask how its shape might evolve, if certain
properties were important components of its fitness.
Shape can potentially affect many of the processes necessary for cell
survival, growth, and reproduction. Some of these, like the transport
of metabolites or chromosome segregation, are internal to the cell; the
impact of shape on such processes depends on the details of the
mechanisms and will not be considered here. Rather, my focus is on
fundamental interactions between the organism and its environment that
are independent of specific mechanisms employed by the organism.
Although the concept of the fitness landscape has been much discussed
since its introduction by Wright (25) in 1932, examples of
fitness landscapes in nature have been hard to come by. However, for
small microbes that live free of any but hydrodynamic constraint on
movement, it is possible to rigorously calculate portions of actual
fitness landscapes in phenotype coordinates and gain insight into why
bacteria have certain shapes and not others. In particular, I asked why
rod-like shapes are common but disk-like shapes are not found.
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MATERIALS AND METHODS |
Data.
I checked each genus of bacteria described in all four
volumes of Bergey's Manual of Systematic Bacteriology
(10). The prime objective was to obtain data for an unbiased
sample of actual microbes, and objective criteria were established for
the selection of data. Only one set of size range and motility status
was recorded for each genus in the hope of reducing the
overrepresentation of human pathogens that would have occurred if each
described species contributed a data set.
Only genera that appeared to consist of unattached, free-swimming, or
free-floating types were included; genera described as having mycelial
growth forms, gliding motility, or magnetic particles or engaging in
intracellular parasitism were excluded. In most cases, a numerical
range for both width and length was given, and these four numbers were
entered into a database. The motility status for each genus was also
recorded in a parameter distinguishing between (i) nearly all strains
(or cells) motile, (ii) nearly all strains (or cells) nonmotile, and
(iii) mixed (some strains [or cells] motile and others not).
If a numerical size range and motility status were not described for
the genus, the type species for the genus was examined;
if its
description was deficient, the first species description
in the genus
that contained the required information was used.
When a genus was
described in more than one location, only the
first description was
used.
All descriptions indicated that the cells were either spherical
(diameter = length) or elongated along one dimension into
a
cylindrical or rod shape (diameter < length). Consequently,
I
characterized all of the different shapes by their axial ratio,
(=length/diameter). (The axial ratio appears to be characteristic
of a
given strain; when bacteria grow under different conditions,
cell
volume increases with growth rate, but the axial ratio remains
constant
(
3 [p.217-218]). Where a range of diameters
(
D) or
length (
L) was given, the geometric mean,
which equals the arithmetic
mean on a log scale, was used. Thus, log
= [log(
Lmin) + log(
Lmax)
log(
Dmin)
log(
Dmax)]/2.
Calculations.
The shapes considered were all ellipsoids,
defined by semiaxes a, b, and c. Included among
them is the special case of the sphere, where a = b = c = r, the radius of the sphere. In comparisons of different
shapes, the volume was kept constant. Since the volume of an ellipsoid
is 4
abc/3 and a sphere is
4r3/3, the two volumes are equal if
r = (abc)1/3.
In general, the surface area of ellipsoids can be calculated only by
evaluation of elliptic integrals. This was done by using
Mathematica to
evaluate equation 3 in reference
15.
The hydrodynamic resistance of rigid ellipsoids can be worked out
exactly in the limit of low Reynolds number (references
in reference
5), which easily applies to bacteria
(
1 [p.
76, 6, 18), and I assume that the organisms
of interest can be
approximated by such ellipsoids. Unfortunately, the
general formulas
involve the values of definite integrals that have no
solution
in common functions except in special cases. General solutions
are as follows:
![S(<IT>a</IT><UP>,</UP>b<UP>,</UP>c) ≡ <LIM><OP>∫</OP><LL>0</LL><UL>∞</UL></LIM> [(a<SUP>2</SUP>+x)(b<SUP>2</SUP>+x)(c<SUP>2</SUP>+x)]<SUP>−1/2</SUP>dx](/content/vol180/issue22/fulltext/5978/img001.gif)
|
(1)
|
(
16 [equation 3]), and for convenience, the
following variations:
![G<IT>i</IT>(a<UP>,</UP>b<UP>,</UP>c) ≡ r<SUB>i</SUB><SUP>2</SUP> <LIM><OP>∫</OP><LL>0</LL><UL>∞</UL></LIM> [(a<SUP>2</SUP>+x)(b<SUP>2</SUP>+x)(c<SUP>2</SUP>+x)<SUP>−1/2</SUP>[r<SUB>i</SUB><SUP>2</SUP>+x]<SUP>−1</SUP>dx](/content/vol180/issue22/fulltext/5978/img002.gif)
|
(2)
|
where
x is a dummy variable,
i refers to
any of the three axes,
ra =
a,
rb =
b, and
rc =
c. A useful relation is
Ga +
Gb +
Gc =
S
(
16 [equation 4]). It is also useful to define
|
(3)
|
where
i,
j, and
k refer to any permutation
of axes
a,
b, and
c.
Motility is most effective when an organism swims along one of its axes
of symmetry, which I call axis
a. (If the organism
is not
motile, it doesn't matter how axes are assigned.) I am
primarily
interested in two types of motion of the organism: translation
parallel
to the
a axis, and rotation of this axis about any
perpendicular
axis (some combination of axes
b and
c.)
For translation parallel to axis
i, the frictional drag
coefficient is
fi(
a,
b,
c) = 16
/(
S +
Gi), which is equivalent to
the formulas of Lamb
(
12 [p. 605, equation 15]) and Perrin
(
16 [equation 5]). A sphere has a translational
frictional coefficient
of
fS(
r) = 6
r
(
16 [equation 11]). Taking the ratio, with
r = (
abc)
1/3
|
(4)
|
At low Reynolds numbers, speed (
v) is equal to the
force (
F) applied to a particle divided by the appropriate
frictional
drag coefficient,
v =
F/
f. Since power
(
P) is work per unit time
and work is force times distance,
P =
Fv. Combining these relations,
|
(5)
|
and speed is inversely proportional to the square root of the
frictional drag
coefficient.
For diffusion in any one direction or rotation about any one axis, the
diffusion coefficient is
Di =
kT/
fi,
where
k is Boltzmann's
constant (1.38 × 10
16 erg K
1) and
T is absolute
temperature (20°C = 293 K). For diffusion
averaged over all
directions, the effective frictional coefficient
is the harmonic mean
of the coefficients for each of the three
orthogonal axes
(
17 [equation 96), and
D = (
kT/3)
(
fa1 +
fb1 +
fc1). Taking the ratio of the diffusion
coefficient of an ellipsoid
(
D) to that of an equal-volume
sphere (
Ds),
|
(6)
|
Brownian motion also causes rotation of particles, and the
rotation of any rigid shape can be described in terms of a time
constant (
) during which time the average of the squares of the
angles rotated increases to one radian (
1 [p. 82])
and the
average of the cosines of the angles of orientation decays from
1 to e
1 (
20 [p. 437]). For rotation
about a single axis or rotation
of a sphere,
=
fR
/ 2
kT, where
fR is the frictional drag
coefficient
for rotation (
20 [equation 25-12]).
For a sphere,
fRS = 8
r3, and
the rotational time constant is
S = 4
r3/
kT (
16 [equation 94]).
For ellipsoids, there are generally distinct frictional constants for
rotation about each of the three axes (
fRa,
fRb, and
fRc), and their values
are
fRi = 16
/3
Hi
(
16 [equation 6]).
For rotation about more than
one axis, the effective frictional
coefficient is the harmonic mean of
the coefficients for each
axis involved. Thus, for rotation of one axis
(
i) about any perpendicular
axis, the time constant for
rotation of the axis is
i = (
kT)
1
(
fRj1 +
fRk1)
1
(
16 [equations 90 and 91). Using the general
formulas (
16 [equation 6]) and dividing by the
time constant for rotation of
an equal-volume sphere,
|
(7)
|
where the subscripts
i,
j, and
k refer to
any permutation of axes
a,
b, and
c.
Plots.
To plot parameter values for all possible ellipsoids
without favoring a particular axis, I take advantage of the
trigonometric identity sin(
) + sin( + 2/3) + sin( + 4/3) = 0 and choose the semiaxes from
|
(8)
|
where
d is the degree of distortion from sphericity
(0 to 1) and
varies the proportion of the three axes. The identity
ensures that
abc = 1 and the volume of all the
ellipsoids is 4
/3,
for any value of
d or
.
The results are presented in polar plots, where the plot radius
represents the minimum radius of curvature of the ellipsoid.
The
minimum radius of curvature (
R) for an ellipsoid is the
minimum
of
ri2/
rj
when
ri and
rj take on
all six combinations of
a,
b, and
c
(
13 [p. 78]).
R' is the minimum radius
of curvature
relative to the radius of the equal-volume sphere
(
R' =
R/
r).
In the plots presented, the horizontal and
vertical positions
are defined by
x =
log(
R')
sin(
) and
y =
log(
R') cos(
),
respectively. The maximum radius in the plots corresponds to a
minimum
radius of curvature of 1% of the equal-volume sphere (
R'
= 0.01), which occurs in prolate ellipsoids of revolution with
semiaxes (10, 10
1/2, 10
1/2) and in oblate
ellipsoids of revolution with semiaxes (10
4/5,
10
2/5, 10
2/5).
The equations were evaluated with Mathematica 2.0, the contours were
plotted with DeltaGraph Professional 2.0, and the plots
were finished
with Adobe Illustrator 5.5 and 6, all on Apple Macintosh
computers.
| |
RESULTS |
Observed shapes.
The selection of data on bacterial shapes and
motility produced usable data for 218 genera, of which 97 were
characterized as motile, 94 were characterized as nonmotile, and 27 were characterized as a mixture of motile and nonmotile types. All
descriptions indicated that the cells were either spherical or
elongated along one dimension into a cylindrical or rod-like shape.
Figure 1A shows that there are two
clusters of axial ratios. Twenty-one percent of the genera are
described as spherical (log = 0), and another peak occurs near an
axial ratio of 3 (log = 0.48).
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FIG. 1.
Distribution of axial ratios. (A) Distribution of axial
ratios for 218 genera of unconstrained bacteria. The median axial ratio
is 2.82. (B) Cumulative distributions for the 97 motile and 94 nonmotile genera. On the cumulative probability scale
(19 [p. 118]), a normal distribution falls on a
straight line.
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|
Since some of the possible adaptations apply only to motile bacteria,
the data were separated according to motility. Nonparametric
statistical testing indicated that the probability that the motile
and
nonmotile groups are drawn from the same distribution is only
0.0012 by
the Mann-Whitney
U test and 0.0001 by the Kolmogorov-Smirnov
test. Cumulative distributions for log
of motile and nonmotile
genera are presented separately in Fig.
1B. A larger fraction
of the
nonmotile genera are spherical (34% versus 10%). The nonmotile
genera
that are not spherical have nearly a normal distribution
of log
,
but among the motile genera, low axial ratios are less
common and large
axial ratios are more common than expected for
a normal distribution.
Clearly, motile genera tend to be more
elongated than nonmotile genera.
Why?
The ellipsoidal model.
To assess the effects of changes in
shape systematically, I model the organisms as rigid ellipsoids in
which all three axes are free to change size independently of one
another. This model has the advantages that it does not bias the study
toward certain shapes and rigorous formulas are available for
ellipsoids but not for some properties of other shapes. Although
bacteria rarely have ellipsoidal shapes, the model can approximate most
simple shapes and be used to rigorously explore the effects of shape on
an organism. Similarly, chemists use the ellipsoidal model to study the
shapes of molecules (20, 22), although molecules are even
less like ellipsoids than are bacteria.
In general, ellipsoids are characterized by three perpendicular
semiaxes (
a,
b, and
c), which are analogous to
radii of a
sphere. Prolate ellipsoids of revolution (e.g.,
a >
b =
c) are
representative of cylindrical shapes; oblate
ellipsoids of revolution
(e.g.,
a <
b =
c)
represent disk-like shapes. An ellipsoid with
all three axes equal
(
a =
b =
c) is a
sphere.
To separate the effects of shape from those of size, different
ellipsoids with the same volume are compared, and the properties
of the
ellipsoids are presented relative to the equal-volume sphere.
Consequently, all of the properties equal unity for the spherical
shape, and it is immediately evident how the property differs
from that
of the spherical
shape.
The results are presented in polar plots, where the proportion of the
three ellipsoid axes varies with direction from the
center of the plot
(azimuth) through all possible combinations,
and the plot radius is a
measure of the degree of distortion from
sphericity. For this measure,
I chose the minimum radius of curvature
of the ellipsoid, which is
thought to provide a good indication
of the difficulty an organism
faces in producing the shape. The
range of axial ratios (1 to 32)
included in the plots includes
all but 4 (motile) genera of the 218 in
Fig.
1.
Surface area.
Organisms must take up nutrients and dispose of
waste products across their surface, and bacteria might find it
advantageous to change shape so as to increase surface area. Figure
2 (Surface area) shows a contour plot of
how surface area changes with shape for equal-volume
ellipsoids. The sphere has the minimum surface area, as is well known.
In addition, the figure reveals that oblate (disk-shaped) ellipsoids
have a larger surface for a given minimum radius of curvature. Thus, an
organism experiencing enhanced fitness from increased surface area is
expected to evolve by spreading in two axes forming a disk-like shape,
other things being equal. Within the limit of radius of curvature equal
to 1% of the equal-volume sphere, the optimal shape has semiaxes
(0.158, 2.51, 2.51) and an axial ratio of 0.063, and the surface is
increased to 3.198 times that of the equal-volume sphere. Prolate
ellipsoids have a maximal surface area 2.48 times that of the sphere.
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FIG. 2.
Contour plots of possible fitness components for
equal-volume ellipsoids of all shapes. The values of the contours are
relative to an equal-volume sphere. The sphere (a = b = c) is at the center, and the distance from the center is
proportional to the negative log of the minimum radius of curvature
occurring in each ellipsoid compared to the radius of the equal-volume
sphere. At the outer edge of the plots, ellipsoids have a minimum
radius of curvature of 1% that of the sphere. The three axes of the
plots encompass the shapes in which two axes of the ellipsoid are
identical (ellipsoids of revolution). At one end of each axis, prolate
ellipsoids, with semiaxes equal to some permutation of (10, 101/2, 101/2), resemble rods, with axial
ratios of 32; at the opposite end, oblate ellipsoids with semiaxes
(104/5, 102/5, 102/5) resemble
disks with axial ratios of 0.063. (Left) in Surface area, the contour
values are the surface area of equal-volume ellipsoids; in Diffusion,
the contours are the diffusion coefficients of equal-volume ellipsoids
(equation 6); in Drag, the contour values are the frictional drag
coefficient (fa') for translation along axis
a (equation 4). For the latter, the values within the
innermost contour are less than 1, indicating that these shapes have a
lower drag than an equal-volume sphere. (Right) Contour values are
measures of the sensitivity a free-swimming organism can have in
detecting the direction of a stimulus gradient by one of three
mechanisms. In all three plots, the contour value is the log of a
parameter proportional to the signal-to-noise ratio and sensitive to
the shape of the organism. In Temporal, the parameter is
(a'3/2
fa'1/2), which is proportional to the
maximum signal-to-noise ratio for organisms employing temporal
mechanisms and swimming in the direction of axis a. In
Fore/aft, the parameter is (a
a'1/2), which is proportional to the
maximum signal-to-noise ratio for organisms employing fore-and-aft
comparisons. In Lateral, the parameter is
(b'1/2), which is proportional to the
signal-to-noise ratio for organisms employing lateral comparisons, and
' is the smaller of a' or
b'.
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|
Hydrodynamic constraints.
The resistance to movement of a
rigid particle in a fluid is summarized by the frictional drag
coefficient, f. This coefficient generally varies for
movement along different axes or rotation about different axes, and
coefficients for different types of motion are distinguished by
subscripts to f. At low Reynolds numbers, which easily
applies to bacteria (6), speed varies in proportion to the
square root of f (equation 5).
Approximate formulas for the frictional drag coefficients are commonly
available for prolate ellipsoids of revolution with
the major axis more
than five times the two equal minor axes (
1 [p. 57, 84],
20 [p. 436],
22 [p.
95]). However, exact equations
for any ellipsoidal shape are available
(
16) and used here,
although they require evaluation of
definite integrals. The ratio
of the frictional coefficient for any
shape to that of the equal-volume
sphere has been called the
coefficient of form resistance (
11).
Random dispersal.
All organisms face the problem of dispersing
progeny away from one another to avoid competition and to new
environments to avoid extinction when the local environment changes.
Bacteria are small enough that Brownian motion may play an important
role in dispersal. The rate of dispersal is then measured by a
diffusion coefficient, and I asked whether an organism can increase its diffusion coefficient by changing its shape.
From knowledge of the frictional drag coefficients for the three
orthogonal axes, it is possible to rigorously calculate the
diffusion
coefficient for any ellipsoidal particle (equation 6).
The value of the
diffusion coefficient relative to that of the
equal-volume sphere is
plotted in Fig.
2 (Diffusion).
All values are less than 1, except for the central point representing a
sphere. Thus, a spherical shape has the largest diffusion
coefficient
for any given volume. If an organism were originally
spherical because
of surface tension, selection for more rapid
dispersal would not lead
to changes in its
shape.
Reduction of sinking.
Most organisms are denser than water,
and those that do not attach to a substrate face the potential problem
that they will sink away from resources such as oxygen or sunlight.
Shape influences the rate of sinking, and some bacteria may adopt
shapes that minimize sinking rate.
The theory for sinking rate has been well documented and exploited to
estimate the size and shape of macromolecules by measuring
sedimentation rates in centrifuges (
1 [p. 59],
20 [p. 365]
22 [p. 116]).
Assuming that the volume and total mass of an
organism are constant and
that the organism is randomly oriented,
its sedimentation velocity is
proportional to its diffusion coefficient
(
1 [p.
59],
20 [p. 380]). Thus, sedimentation rate is
maximum
for a spherical shape and can be reduced by changing to any
other
equal-volume shape. Examination of Fig.
2 (Diffusion) indicates
that the minimum sinking rate with minimum distortion of shape
is
obtained by elongating along one axis into a prolate ellipsoid
or
toward a rod-like shape. Within the curvature limit, the sinking
rate
would be reduced to 0.415 that of the sphere. The smallest
value for
oblate ellipsoids is 0.601.
Swimming efficiency.
Many bacteria expend precious energy on
swimming, and shape influences the efficiency of locomotion. Bacteria
may adopt a shape that allows faster swimming with less energy, because
of streamlining. If we assume that a particular organism requires a
certain volume and devotes a certain amount of power to locomotion, we
can obtain specific predictions about the speeds it can obtain if it
adopts different shapes. In particular, what shape minimizes the
frictional drag coefficient and thus maximizes the efficiency obtained?
Figure
2 (Drag) demonstrates that an organism benefiting from increased
swimming efficiency should evolve from the spherical
shape at the
center to the minimum drag shape above the center.
Although it is often
stated that the sphere is the shape with
minimum drag (24 [p. 247])
(e.g., see reference
22 [p. 95]),
in fact the minimum drag occurs for
a prolate ellipsoid of revolution
with semiaxes in the ratio (1.562, 0.800, 0.800). This ellipsoid
has an axial ratio of 1.952, and its drag
is 0.9555 that of an
equal-volume sphere. For more elongated shapes,
the frictional
coefficient increases because the increase in surface
area increases
drag more than the reduction in cross-sectional area
reduces it
(
24 [p. 247]).
Thus, if swimming efficiency were commonly the major component of
fitness, we would expect motile bacteria to have shapes
similar to
prolate ellipsoids of axial ratio approximately 2 and
swim parallel to
their long axis. In fact, most bacteria are like
this except that 69%
have axial ratios greater than 1.952 and
the median axial ratio is
2.83. Efficiency of swimming provides
no explanation for why such long
rods might
evolve.
Following stimuli.
Swimming is most useful when it is directed
in a favorable direction, and most motile bacteria are probably capable
of moving up or down chemical gradients. To determine the direction of
a gradient, its concentration must be determined at two points
separated by some distance, 
, and for the small distances of
interest here, the difference in concentration is proportional to (6). Detection will also be influenced by the noise in
measuring the two intensities, and the effect of noise is generally
reduced in proportion to the square root of the time (t)
over which the measurement is integrated (6). However, for
free-swimming organisms, this time is limited by the rate at which
orientation is lost as a result of rotational diffusion caused by
Brownian motion (2, 6), which is influenced by shape. These
general considerations lead to the expectation that the signal-to-noise
ratio (S/N) is influenced as S/N 
Gt1/2,
where G is the stimulus gradient (6). Since
S/N 
1 at the limit of detection, the shallowest detectable
gradient is proportional to 1/(t1/2), and
t1/2 is a measure of the relative sensitivity
for gradient detection.
There are several ways in which an organism can separate the two
positions at which intensity is determined. With temporal
(sequential)
comparisons the organism moves between the two positions,
and
vt, where
v is the speed of swimming and
t is the time
between measurements. This is the only
mechanism currently known
to be employed by free-swimming bacteria
(
14). As an estimate
of the maximum time useful to the
organism, I use the relaxation
time (
) for decay of an initial
orientation (equation 7). Since
at any particular specific power
consumption,
v is inversely proportional
to the square root
of the frictional drag coefficient (equation
5), for movement along
axis
a, S/N
a1/2 =
v a3/2 a3/2
fa1/2, where
fa
is the frictional drag coefficient for translation
along axis a
(equation 4) and
a is the relaxation time for loss
of
orientation of the axis (equation
7).
This relationship is plotted in Fig.
2 (Temporal), which indicates that
spherical organisms could improve gradient detection
by elongating
along the axis parallel to the direction of swimming.
Within the
curvature limit, gradient detection is improved by
a factor of
10
2.81 = 647 over the equal-volume
sphere.
An alternative mechanism for gradient direction is to employ spatial
(simultaneous) comparisons (
7 [p. 415]), in which
receptors on different parts of the organism are compared. In
this
case,
represents the distance between these body parts.
If the
comparison is fore-and-aft,
2
a, and at best S/N
a a1/2. (I ignore the
complication that swimming may cause a difference
in stimulation
between the front and back [
2] because the organism
could compensate for this effect.) This relationship is plotted
in Fig.
2 (Fore/aft), which demonstrates that the greatest gain
with minimal
departure from a spherical shape is again obtained
by elongating along
the axis parallel to the direction of locomotion,
toward rod-like
shapes. The same shape is optimal, but performance
is superior to a
sphere by the smaller factor of 10
1.98 =
96.
The other mechanism for spatial comparison is with receptors lateral to
the direction of swimming. Taking the
b axis in the
direction between receptors,
2
b. To detect the
gradient, axis
b must maintain its orientation for a
sufficiently long period.
The organism can then turn in the appropriate
direction. However,
the orientation of this axis might be maintained
while more rapid
rotations occurred around the axis. These latter
rotations would
not interfere with detecting the gradient, but once the
organism
had turned to the gradient, such rotations would randomly
point
the organism up, down, and across the gradient. Thus, this
mechanism
requires that rotations around all three axes be minimized,
in
contrast to previous mechanisms where symmetry made rotation around
one axis inconsequential. For calculation, I use the smaller of
the
time constants for rotation around the
a or
b axis.
Consequently,
S/
N b 1/2, where
is the
smaller of
a and
b.
This relationship is plotted in Fig.
2 (Lateral). A spherical
organism
maximizing this component of fitness with minimal change
in shape
should evolve along a path toward a prolate ellipsoid
of revolution
around axis
b. Within the curvature limit, the optimal
shape
has semiaxes (0.32, 10, 0.32), swims perpendicular to its
long axis,
and is superior to the sphere by the still smaller
factor of
10
1.06 = 11.5. The relatively small improvement explains
why this behavior
is not
observed.
| |
DISCUSSION |
The results for all of the different types of adaptation are
summarized in Table 1. We see that
evolution toward increased surface area should lead to disk-like
shapes, and the fact that these are rarely observed suggests that
increasing surface area is not a major component of fitness for
bacteria.
Adaptation to increase rates of dispersal via Brownian motion does not
explain nonspherical organisms because the spherical shape has the
largest diffusion coefficient, but some bacteria may have retained
spherical shapes for this reason.
Reduction of sinking rate occurs with elongation, and this could be the
reason that some nonmotile bacteria are elongated. For random
orientations, the minimum sinking rate is 0.415 that of the
equal-volume sphere (Fig. 2, Diffusion). If horizontally oriented, the
sinking rate would be 0.349 (=1/2.869 from Fig. 2, Drag). If vertically
oriented, the value is 0.548 (=1/1.826).
Reduction of translational drag for increased swimming efficiency can
occur by elongation in the direction of swimming, but the effect is
small, and an optimum is reached at an axial ratio of only 1.95, while
most motile bacteria have axial ratios greater than 2.8. It might be
noted here that a recent report (4) suggesting much larger
effects of axial ratio on frictional drag, with an optimum at larger
ratios, is based on approximate formulas that are accurate only for
slender shapes (21) and are thus inappropriate for this application.
In contrast to these relatively small effects, shape has a large impact
on a bacterium's ability to detect stimulus gradients, no matter which
sensory mechanism is employed. This results primarily from decreases in
rotational diffusion that allow the organism to measure stimulus
intensities over longer time periods. The effect is largest for
temporal mechanisms with the potential to increase the signal-to-noise
ratio several hundred-fold (Table 1). A recent analysis suggests that
contrary to common notions, spatial gradient detection mechanisms are
not inherently less sensitive than temporal mechanisms for spherical
free-swimming organisms (8). However, the finding here that
temporal mechanisms can be enhanced more effectively by changes in
shape provides a clear explanation for why rod-like shapes and temporal
detection mechanisms are commonly observed.
| |
ACKNOWLEDGMENTS |
I thank Terry Snell for asking the question about rotifer mating
that led to this line of thinking. He, Patricia A. Sobecky, and Marc J. Weissburg also provided valuable suggestions on previous drafts.
| |
FOOTNOTES |
*
Mailing address: School of Biology, Georgia Institute
of Technology, Atlanta, GA 30332-0230. Phone: (404) 894-8426. Fax:
(404) 894-0519. E-mail:
david.dusenbery{at}biology.gatech.edu.
| |
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Top
Abstract
Introduction
