DeSantis 1.
The Bacterial Flagellar Motor and E. coli Motion
It is hard to envision a world where we are principally subject to viscous forces as
opposed to inertial ones. One of the closest analogies would be to dive into a pool filled
with molasses restricting the speed at which we move our limbs to a few centimeters per
second, at which point our movements might be akin to microscopic organisms including
bacteria. However, Escherichia coli, otherwise referred to as E. coli, have the capacity to
swim efficiently in low Reynolds environments due to their adapted rotary motors with
attached flagella. The motor itself, which will be discussed at a later point in this paper, is
quite an engineering marvel since it is comprised of several dozen components that all
must work together in order to drive the bacterium along. Furthermore, it is my intention
to incorporate an amount of material discussed in lecture such as basic fluid dynamics
which govern animal motion and elucidate the bacterium‟s need for a means of
propulsion from an analysis of these equations. While viscous forces inhibit E. coli‟s
movements, diffusion and „tumbling‟ effects contribute considerably. Although little is
known about how the bacterial flagellar motor works, its structure and function will be
explored as well as its subsequent motion.
In order to convey the motion of bacterium, specifically in low Reynolds
environments, it is requisite introduce many concepts that are inherent to the field of fluid
dynamics including viscosity and the Reynolds number. Viscosity is typically defined as
a measure of a fluid‟s resistance to flow when shear stress is applied. To better
understand the notion of shear stress, one can consider the wind blowing over the surface
of a body of water or a moving boundary plate that physically induces a velocity gradient,
∂v/∂x, where v is the velocity of the plate and x is the vertical distance from another
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boundary plate stationary at the bottom; hence, fluid layers will move at varying
velocities depending upon the shear stress, Γ, between them:
Γ = η(∂v/∂x)
(1)
such that η is deemed the viscosity, which is characteristic to the particular substance and
has SI units of [Pa*s] or [N*s/m2]; „thicker‟ fluids such as olive oil are considered highly
viscous since greater stresses are required to obtain similar velocity gradients whereas
water would have a comparatively low value for η. The magnitude of the layer‟s
velocities will decrease as the distance from the moving boundary plate increases and
will eventually decay to zero once the stress is removed. It follows that for a plate of
surface area, A, the frictional force acting parallel to the object‟s direction of motion will
be inversely proportional to the depth and is given by equation (2), below.
F = Aηv/x
(2)
It is more convenient, however, to express the net force acting on an object, explicitly a
sphere of radius, a, being dragged through a viscous fluid, by the following equation
which is generally regarded as Stokes‟ law:
F = 6πηav
(3)
For purposes of discussion, E. coli and all animals considered will be modeled
after spheres since these estimates are within an acceptable order of magnitude, and
secondly, equations become more simplified; this last point is demonstrated by equation
(3) which is a solution to the set of Navier-Stokes differential equations as well as a
commonly referenced physics anecdote.1 It is important to understand that equation (3)
1
A farmer has a problem with some of his dairy cows which are all becoming very sick. He calls the
veterinarian who takes a look at the cows but can‟t figure out why they aren‟t producing any milk. Soon
after the farmer asks his friend, the town‟s molecular biologist, to see if he can do anything to
help. Unfortunately, 2 months of testing yields inconclusive results. Desperate and having tried all
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dictates the motion of small bodies through fluids at low velocities, where viscous forces
dominate; this is the low Reynolds number regime. This number is a dimensionless
quantity that signifies the ratio between the inertial and viscous forces and is
approximated as:
R ≈ avρ/η
(4)
Viscosity, η, and density, ρ, are 1x103Pa*s and 1x103kg/m3 respectively for water, and
are likewise very close for cytosol. To illustrate the range of regimes, we can look at the
following examples:
Human (0.25m/s, 0.25m) → R ≈ (0.25)(0.25)(1x103)/(1x103) = 6.25x104
Fish (0.5m/s, 1x10-3m) → R ≈ (1x10-3)(1)(1x103)/(1x103) = 5x102
E. coli (2x10-5m/s, 1x10-6m) → R ≈ (1x10-6)(2x10-5)(1x103)/(1x103) = 2x10-5
While we have a fairly good understanding of our own environment, in terms of
the inertial forces at work, a bacterium will not care since its motion is exclusively
determined by the forces acting on it at the moment, akin to Aristotelian mechanics; a
fact that will be made clear when we look at the total forces acting on the cell. E.coli is
generally regarded as the „workhorse‟ for molecular biology principally because it can be
easily grown, studied and is characteristic of many other microscopic life forms.
Although the above calculations are only estimates, a single rod-shaped E.coli cell will
swim around 30μm/s and is 1μm in diameter by 2μm long. In order for it to move,
however, it must not undergo what Purcell defines as „reciprocal motion,‟ in which a
body returns to its original state by a reversal of swimming steps. There is no net
conventional means he decides to call the local theoretical physicist. He stands there and looks at the cows
for a long time without touching them or anything. Then all of a sudden he starts scribbling away in his
notebook and after several gruesome calculations, he exclaims, "There is a solution! First, we must assume
a spherical cow…”
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movement in any direction even though time is not a factor when R<<1. This is
elucidated when we consider the Navier-Stokes equations for incompressible flow, F = 0:
(5)
For low Reynolds numbers, the inertia term can be disregarded:
(6)
Additionally, steady flow reduces (∂v/(∂t) to 0 and equation (6) can be simplified to:
→
(7)
and is further exemplified by the scallop theorem in which angle approximations for the
force on the scallop can be used to show the net displacement, Δx, is 0. Setting the forces
to be proportionally equal: dx/dt α dθ/dt; we can assume reciprocal motion such that the
initial (0) and final (T) angles are the same over one period, hence:
Δx α ∫(dθ/dt)*dθ = θ(T) – θ(0) = 0
Since the single hinge permits only reciprocal motion, Purcell reasonably invents
the „three-link swimmer,‟ in figure 1 such that its motion is cyclic, beginning with the
configuration at point A and following a sequence of related movements that returns to
A:
Figure 1
The number of possibilities ranging from N hinges to variations on form is
endless, but one of the most intriguing means of propulsion belongs to the previously
discussed organism, E. coli, which utilizes complex rotary motors to drive long helical
filaments, four on average, protruding from random sites of the cell (see figure 2 below).
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Figure 2
Each flagellum is capable of rotating counter-clockwise (CCW), looking towards
the cell body, and clockwise (CW). Collectively, the flagella will act in unison and when
turning CCW, from simple experimentation, will produce motion predominantly in the
direction parallel to its long axis; thus the filaments form a „bundle‟ pushing the cell
forward. However, this motion typically lasts for only a second and is termed as a „run‟
since, for unknown reasons, one or more motors will rotate CW sporadically causing the
bacterium to remain at rest before continuing on in a new direction. This erratic behavior
has been described as a „tumble‟ and will be reviewed more thoroughly towards the end
of the paper. Howard Berg, a professor of biology at Harvard University, has done
considerable research as a pioneer in this field, so understandably most of the references
cited are co-authored by him. He has devoted himself to the study of E. coli‟s motion
which is largely dictated by the various mechanisms of the bacterial motor. As I
previously indicated, the motor is very intricate; it is so advanced, that in an interview on
MSNBC news I recently watched, a researcher at Lehigh University2, has speculated it
may be the single most defining example of a higher power‟s work rather than the result
2
Michael Behe, controversial biochemist and advocate of „intelligent design‟ theory. He has labeled what
he considers vastly complicated systems as “irreducible complexities.” Subsequently, these concepts have
been widely rejected by the scientific community, particularly by the biology department at Lehigh
University which derides him, too.
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of evolutionary adaptations. The motor, constructed from the inside out (based on the
observation of simple structures during genetic mutant testing) is primarily embedded
within the layers comprising the gram-negative bacterium‟s membrane with more than 20
components each identified by the genes that encode them:
Figure 3
Like any rotary motor, it can be viewed as the combination of two distinct parts:
the rotor and the stator; as the name implies, the rotor rotates whereas the stator remains
stationary. Approximately 50nm in diameter, the C-ring (FliG, M and N), otherwise
known as the „switch complex‟ due to rotational changes produced by various gene
mutations, is the largest component and is found in the cytoplasm. Above this ring, in the
cytoplasmic membrane, is the MS-ring (FliF) followed by the rod or drive shaft (FlgB, C,
F and G) which altogether constitute the basal body of the rotor. Connecting the filament
(Fli C) to the rings is the hook (Flg E), a flexible joint allowing the flagellum to rotate
freely. Both structures are located outside of the cell and are formed from crystallized
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polymers which are in turn made up of subunits aligned in parallel on the surface of the
flagellum. In essence the filament acts as a rigid microtubule such that alpha and beta
tubulin subunits are replaced by “short” packing protofilaments (R-type) and “long”
packing protofilaments (L-type), the former being able to pack more tightly. If both types
should be present, there is understandably greater strain inducing a helical conformation
(10 possibilities) whereas like-like packing minimizes strain energy and creates straight
flagella, seen below. To better illustrate the differences in packing, figure 4 compares Ltype with R-type filaments in which the direction of twist is reversed:
Figure 4
Their actual shape is determined by the “amino acid sequence of the flagellin, the
pH, ionic strength, and torsional load.” When motors switch from CCW to CW, the
flagella undergo several transformations changing from “normal to semicoiled to curly”
before reverting to their original state as depicted in figure 5 which shows the entire
sequence for one filament taken by fluorescence microscopy. During this process, the
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proteins found along the hook (FlgK and L) can switch R with L-type hook
protofilaments and visa versa, so long as the flagellar protofilaments are not altered.
Figure 5
Situated outside the rotor rings and attached to the cell wall via the peptidoglycan
network, the stator is a circular group of 12-14 particles chiefly composed of
transmembrane proteins: MotA and MotB (see figure 3). It has been proven that these
proteins serve a very important purpose such that if the genes encoding them, motA and
motB, were to be deleted, the rings are no longer observable and the flagellum is
motionless; likewise, the reintroduction of these particles initiates rotation. Testing has
revealed that MotA has “four membrane-spanning α-helical segments” with a majority of
the protein existing in the cytoplasm; MotB, on the other hand, has only one α-helix
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which helps stabilize MotA in place. Furthermore, this set of particles acts as an
independent torque-generating complex whereby the mutual interaction between “eight
transmembrane segments from two copies of MotA and one transmembrane segment
from one copy of MotB” form two proton channels permitting the movement of protons
(or Na+ ions) from outside to inside the cell. This is demonstrated in the following
diagrams (a) simplified drawing of figure 3 and (b) two possible models for the motor,
each akin to the rotating ATP synthase molecule in the mitochondrial matrix except for
the fact that power is not produced from the hydrolysis of ATP or any other anion:
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Figure 6
Torque, Γ, will be perpendicular to the force exerted on the rings‟ edges by the MotA/B
complexes times the radius of the rotor, and of the usual vector equation:
Γ=Fxr
(8)
To study and quantify the amount of torque generated as protons pass down this
electrochemical gradient, scientists would ideally mount one motor-filament to a
microscope slide and analyze the work done over time (power); each proton carries a
„protonmotive force‟ (PMF), Δp, or work per unit charge that is evaluated in terms of the
transmembrane electrical potential difference, Δψ, and the transmembrane pH difference,
(-2.3kT/e)ΔpH, and is supported by several metabolic processes utilizing chemical energy
to pump protons back out of the cell in order for them to flow down the gradient once
again. Since we do not yet know how the motor produces torque and as this encroaches
on electrochemical subjects that we have yet to cover in lecture, I would prefer to return
to the overall motion of E. coli; however, for future reference, the following are accepted
values for E.coli cells (T = 24°C), whose internal pH range is from 7.6 to 7.8;
Δψ ≈ -120mV and Δp ≈ -140mV. Thus each proton reentering the cell “supplies –eVm =
2.4x10-20J of free energy to drive the motor.”
When the cell rotates at ~10Hz, the motor torque, defined by equation (8), is
approximately equal to the viscous torque, the torque induced on the cell by the
surrounding water as it swims, 3x10 -18Nm; the work done after a single revolution of 2π
radians is subsequently 2.0x10 -17J and the power, P, is equal to 1.88x10-16 W from the
following relation:
P = Γ*w = Γ*2πf
(9)
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According to experimental validation, nearly 1000 protons are involved with
every rotation. Assuming all protons deliver –eVm of energy, the motor requires
2.4x10-17J per revolution; therefore the efficiency of the system is consistently high
(~83%) in a „tethered cell.‟
We finally look at E.coli‟s swimming but must still keep in mind that bacterial
swimming is constantly influenced by viscous forces. Following a run, E.coli‟s motion is
haltered according to the following example:
Applying equation (3) and the velocity of E.coli taken to be roughly 20μm/s, the
characteristic dissociation time, τ, that a bacterium will coast for after its flagella stop
rotating, and is subsequently moving only due to its own acceleration, is given by:
τ = m/6πηa = 2a2ρ/9η
(10)
= 2(1x10-6)2(1x103)/9(1x10-3) = 2x10-7s
d = ∫v(t)dt = v(0)*τ
(11)
= (2x10-5)(2x10-7) = 4x10-12 m = 0.04Å
As this is on the same order of a hydrogen atom, it would appear that E. coli stops
almost instantaneously, however, it is still subject to Brownian motion which can be
modeled more accurately by the diffusion process. A cell, like any other particle in a
fluid, will perform „random walks‟ about its center. As the bacterium is not restricted to
any single axis, the following equation governs simple three-dimensional diffusion:
<r2> = 6Dt
(12)
where r2 = x2 + y2 + z2 and D, the diffusion coefficient, is equal to δ 2/2τ.
While a run requires all filaments acting in unison to produce uniform motion in
one direction, the result is hardly a straight line; rather the cell will wander off course due
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to rotational diffusion. It is worthwhile to note that while particles have an average
kinetic energy of 3kT/2, with each degree of freedom contributing ½kT, they have similar
energies coupled to the rotation about each axis. The equation describing rotational
diffusion is analogous to (12):
<θ2> = 4Drt
(13)
such that a microorganism moving in one direction will meander an angle, θ, by diffusing
along the other two coordinate axes. For E. coli, we are capable of computing the time a
cell significantly deviates from its course (27° on average during a run) by redefining the
rotational diffusion constant, Dr = kT/fr, for a sphere of radius, a, and an associated
rotational frictional drag coefficient, fr = 8πηa3:
Dr = kT/8πηa3
(14)
= (1.38x10-23J/K)(298°K)/8π(1x10-3Pa*s)(1x10-6m)3 = 0.164rad2/s
→ (13) → <θ2> = 4(0.164)t
θ = 0.81t1/2
(15)
→ (θ = 27° = 0.471rad) → t = 0.338s; tracking experiments may use a denser fluid, η =
2.7x10-3, in which case equation (15) has a root-mean-square (RMS) angular deviation of
θ = 0.5t1/2 implying that the cell would stray 27° in almost one second, the time for a
typical run. Although, rotational diffusion is responsible for a certain amount of
observable drift, the primary factor that impedes unidirectional motion is the „tumbling‟
effect.
Test trials have reported that runs and tumbles usually last for 1 and 0.1 seconds
respectively; however, these values can be augmented by directly influencing receptor
occupancy. Experimentally, they are determined by varying the concentration of
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chemical attractant which demonstrates bacterial chemotaxis. While E. coli would prefer
to remain idle due to its diffusion to capture ratio and permit particles to randomly diffuse
through its surface at a rate, where Φ is the flux, given by equation (16), it must ideally
be located in a food-enriched environment.
Φ = 4πDac0
(16)
An attractant, introduced as an exponentially increasing temporal gradient of a
particular chemical, may promote the run (CCW) bias by increasing the rate constant for
the CW → CCW transition, k+, subsequently decreasing the rate constant for the
opposing shift, k-; a reduction in the temporal gradient, though, produces no effect. The
biases are therefore positive indicating that runs are never shortened and can be extended
several additional seconds. In order for biases to change, the bacterium must be sensitive
and be able to detect the attractant as a function of chemoreceptor occupancy with respect
to time. Specifically, when the concentration, C, is relative to the dissociation constant of
the chemoreceptor, Kd, the cell‟s maximal response will be proportional to d(logC)/dt.
Accordingly, the number of receptors that are bound:
P = C/(Kd + C)
(17)
dP/dt = [KdC/(Kd + C)2]*[(dC/dt)/C]
(18)
Equation (18) defines a Gaussian, positioning Kd as the group „average,‟ and will be
minutely affected by altering the concentration in the second term. If C is changed
logarithmically between two extremes, low and high, it can be shown that equation (18)
reduces to:
dP/dt = ¼[d(logC)/dt]
(19)
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Evidence for bias control as a function of the chemoreceptor occupancy‟s time
rate of change is demonstrated by the cell‟s behavior in response to an exponentially
increasing concentration gradient. Letting C = C0e(βt), the cell should therefore exhibit a
shift in bias proportional to β, as seen in figure 7, below, which depicts in graph A the
response due to an exponential ramp from low to high initial concentrations and the
converse in B.
Figure 7
Figure 8
The probability for the CCW bias is understandably greater during the indicated
interval for A and lower in B. These changes in chemoreceptor activity affecting bias, is
best explained by the adopted chemotaxis pathway for E. coli which involves a network
of signaling proteins, namely CheY, in figure 8, above.
Methyl-accepting chemotaxis proteins (MCPs) work in conjunction with CheY, a
response regulator, to affect bias rotation. When less attractant is bound, corresponding to
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an increase in receptor activity, phosphate groups are donated such that CheY is
phosphorylated: CheY → CheY-P; this event prompts CW rotation whilst increased
receptor occupancy improves CCW bias. Additionally, in the absence of CheY and
proteins associated with chemotaxis, the motor rotates solely in the CCW state. Thus, the
motor‟s tendency to switch between the two states can be defined in terms of a ratio of
the bias probabilities:
(CW bias)/(1 – CW bias) = k+/k- = e(-ΔG/kT)
(20)
where ΔG = GCW – GCCW and is considered the standard free energy difference between
the two states. When CheY is deleted, “ΔG changes linearly with temperature” and it has
been found that low temperatures favor CW biases (extreme pH levels, too, above 9 and
below 6, influence efficiency by slowing the rotation of the motor).
Equation (20) reveals that a Boltzmann factor plays the predominant role dictating
the switch in bias. The classification of runs or tumbles can be fitted to an exponential.
Since the number of events, n, occurring is not causally related to the events preceding it,
the process can be modeled after a Poisson interval distribution in which the probability
per unit time, λ, is fixed; while runs are extended in the presence of concentration
gradients, the CCW intervals are still exponential and decay at constant values. The
distribution is defined, in the limit n>>1, as:
P(t,λ)dt = λe-λtdt
(21)
The expectation value of t, <t>, or mean interval, is consequently 1/λ. Computing the
standard deviation:
σ = (<t2> - <t>2)1/2
= ((2/λ2) – (1/λ)2)1/2 = <t>
(22)
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hence, the standard deviation, both mathematically and experimentally verified, equals
the mean. Another general conclusion that can be ascertained from a plot conveys that
shorter intervals are more likely to occur. Likewise, the bias can be transposed to fit a
normal Poisson distribution given by equation (23) with the mean number of events
occurring over an interval of time, t, being equal to λt; the standard deviation is (λt) 1/2.
P(n;λt) = (λt)ne-(λt)/n!
(23)
It proves not uncomplicated to fit the seemingly erratic behavior of a bacterium
such as E. coli to a model that more than adequately describes the radioactive decay of
samples in lab, also. Scientists are continuously trying to understand the organization and
behavior of complex systems and more research will need to be conducted if progress is
to be made for explaining how the motor works as it does or why the CheY-P complex
stabilizes the CW state. It becomes increasingly important to elucidate the structure and
mechanisms governing the bacterial motor as well as its biological analogues including
the ATP synthase molecule and the DNA packaging motor in viruses for numerous
reasons; in principal, they can serve as excellent blueprints for fabricating
micromachines. Additionally, the man-made devices to be built within the field of
nanotechnology will probably not simply be scaled down versions of their macroscopic
counterparts, but resemble, rather, these rotary devices found in nature in order to achieve
optimal efficiency. The bacterial flagellar motor is thus an extraordinary product of
nature and will doubtlessly be studied for years to come.
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Works Cited
Berg, Howard C., Random Walks in Biology. Princeton University Press, Princeton, NJ
1993.
Berg, Howard C., The rotary motor of bacterial flagella. Annu Rev Biochem.
2003;72:19-54. Epub 2002 Dec 11. (October 2005)
http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=pubmed&dopt
=Abstract&list_uids=12500982&itool=iconabstr&query_hl=17 [Online]
Berg, Howard C., Symmetries in bacterial motility. Proc Natl Acad Sci U S A. 1996 Dec.
10;93(25):14225-8. (October 2005) http://www.pnas.org/cgi/reprint/93/25/14225
[Online]
Berry, Richard M., Bacterial Flagella: Flagellar Motor. Encyclopedia of Life Sciences.
Nature Publishing Group, 2001. (October 2005)
http://www.cellcycle.bme.hu/oktatas/mikrofiz/extra/flagella.pdf [Online]
Block, Steven M., Jeffrey E. Segall, and Howard C. Berg. Adaptation Kinetics in
Bacterial Chemotaxis. Journal of Bacteriology Vol. 154, No. 1, Apr. 1983, p. 312323.
Purcell, E.M., Life at low Reynolds number. American Journal of Physics Vol. 45, No. 1,
Jan. 1977, p. 3-11. (October 2005)
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0045000001000003000001&idtype=cvips [Online]
Turner et. al. Temperature Dependence of Switching of the Bacterial Flagellar Motor by
the Protein CheY13DK106YW. Biophysical Journal Vol. 77 July 1999, p. 597-603.
(October 2005) http://www.biophysj.org/cgi/reprint/77/1/597 [Online]