Prospects for Studying Negative Gravitaxis in Paramecium
using Magnetic Field Gradient Levitation
Samuel E. Wurzel
A Senior Thesis
with Professor James Valles, Jr.
Magnetic Levitation for Low Gravity Simulation Laboratory
Department of Physics
Brown University
Providence, RI 02912
email: Samuel [email protected]
April 30, 2003
1
Introduction
Here we provide a brief background on Paramecium with particular emphasis on the phenomenon of
negative gravitaxis. A discussion of low Reynolds number hydrodynamics follows in order to provide
a physics framework to examine physical and physiological hypothesis which have been postulated
and tested (often with contradictory results) to pinpoint a mechanism for negative gravitaxis. The
Magnetic Field Gradient Levitation (MFGL) system is introduced as a tool to probe gravitational
effects on biological systems and ultimately to study negative gravitaxis in Paramecium. Finally we
report on progress in measuring parameters needed to utilize MFGL to achieve these goals.
2
Paramecium
Paramecium are single cell swimming organisms that live in water. They are shaped like slightly
asymmetrical ellipsoids, with the rear of the cell slightly wider than the front, Fig.1. Just barely
visible to the naked eye, they have lengths of approximately 200µm and widths of approximately
50µm. They swim in a helical fashion by beating their cilia, the tiny hair like structures which cover
their bodies.
Figure 1: Paramecium viewed under magnification.
2.1
Negative Gravitaxis
Negative gravitaxis is the tendency for cells to preferentially swim upward, in the direction defined
as opposite to the gravity vector, and collect at the top of their container. This phenomenon is
1
Figure 2: The photograph on the left shows Paramecium immediately after being introduced into a
cuvette. The cells (white dots) are evenly distributed. The photograph on the right shows Paramecium 60 minutes after introduction into the cuvette. Due to negative gravitaxis, most cells have
collected at the top.
2
indeed stimulated by gravity, as the effect is observed in containers without air bubbles or exposure
to light. Paramecium are denser than water with a cell density of 1.04g/cm 3 [1]; thus the preferential
upward motion is not due to buoyancy. In view of their heavier-than-water weight, gravitaxis is a
most curious phenomenon and has attracted much attention since it was first observed in 1889 by
Verworn[2][3]. Although heavily studied, evidence for one definite mechanism has been elusive. Much
of this difficulty arises because of the challenges involved in separating physical and physiological
mechanisms to explain this effect[4]. Physiological explanations assume a biological mechanism in
which the cells sense a gravity induced stimulation along their cell wall or through some internal
sensor. Then, through a biochemical pathway the cell adjusts its cilliary beat and actively changes
its velocity accordingly. Physical explanations assume the mechanical structure of the cell passively
gives rise to a preferred swimming orientation, and derives from properties of fluid flow and drag
appropriate to swimming cells like Paramecium.
Physical effects can be isolated by immobilizing the cells using an aqueous solution of NiCl. A
1mM concentration can immobilize a population of cells in minutes. One caveat of this technique
is that the NiCl solution causes cell deformation over time. Fig.3 shows Paramecium exposed to
1mM NiCl for 5 hours. Such deformation of the cells radically alters their hydrodynamic properties.
Figure 3: Paramecium after 5 hours of 1mM NiCl exposure.
However, if experiments are performed soon after immobilization, this effect can be mitigated.
3
Low Reynolds Number Hydrodynamics
We know Paramecium swim in water. An important quantity to examine when dealing with any type
of fluid flow problem is the Reynolds number, denoted Re. A dimensionless quantity, the Reynolds
3
Fd
ρ0
r
ρ1
Fg
Figure 4: A sphere at terminal velocity in a viscous fluid under the influence of gravity. Re ¿ 1.
number is the ratio of the inertial forces to the viscous forces acting on a body moving in a fluid[5].
For Paramecium, inertial forces arise from having to push away the liquid as it swims. The viscous
forces arise from the stickiness, or viscosity of water. Letting ρ0 be the density of the water, v be
the swimming speed of the Paramecium, r be a length associated with the Paramecium, and η be
the dynamic viscosity of water we have,
Re =
ρ0 v 2 r2
ρ0 vr
Finertial
=
=
.
Fviscous
ηvr
η
For a Paramecium of length 200 µm swimming at 0.4 mm/s in water (η = 0.001
(1)
kg
m·s )
we have
Re = 0.08. Since this is much smaller than unity, viscous forces dominate the lives of Paramecium
swimming in water[6]. The regime of Re ¿ 1 allows us to make useful “creeping flow” simplifications to the Navier-Stokes equations which govern viscous fluid flows[5]. One especially important
consequence of the creeping flow approximation is Stokes’ law,
Fd
=
−6πηrv,
(2)
which describes the drag force that a sphere of radius r feels moving at velocity v through a viscous
medium. As a quick application of this result, we can predict the terminal velocity of a sphere of
density ρ1 under the influence of gravity in a viscous medium of density ρ0 . We restrict the motion
to the vertical direction, with coordinate z.
From Newton’s Law we obtain the differential equation,
mz̈
mz̈
=
Fd + Fg
(3)
=
4
−6πηrż + πr3 (ρ1 − ρ0 )g.
3
(4)
At terminal velocity z̈ = 0 and ż = vs and thus
vs
=
2 (ρ1 − ρ0 )g 2
r .
9
η
(5)
We can apply this result to the sedimentation of a Paramecium by making the spherical Paramecium approximation. We assign the “spherical Paramecium” to have radius r = 25µm. We know
kg
(ρ1 − ρ0 ) = 0.04g/ml, g = 9.8m/s2 , and for water, η = 0.001 m·s
. Plugging this all into Eq. 5 and
4
making the appropriate unit conversions we find vsStokes = 0.056mm/s. This calculation is in reasonable agreement with the median measurement made by Machemer[7] of 84µm/s for sedimenting
Paramecium, as well as our measurements to be discussed in the section 8. Since this simplistic
model does not take into account the fact that Paramecium are not spheres but ellipsoids and can
sediment in any orientation, and our choice of r was somewhat arbitrary, a perfect agreement with
experiment cannot be expected from this first order approximation. Better analytical formulas for
the drag on non spherical objects are available. Happel and Brenner[5] derive the creeping flow
drag on ellipsoids in both the vertical and horizontal orientation and we use their results here. The
simplest way to express the results for this shape is to assign it an effective radius, r ef f , where it and
a sphere of that radius moving at the same speed would feel the same drag force. For an ellipsoid
we denote the length of the semi-major axis a and the length of the semi-minor axes b and c, where
b = c. We also define the ratio of the semi-major axis to the semi minor axis to be (φ ≡ a/b). For
an ellipsoid oriented with its major axis parallel to the direction of motion,
8c
3
ref f k =
1
− φ22φ
−1
+
2φ2 −1
(φ2 −1)3/2
ln[
φ+
√ 2
,
√φ2 −1 ]
φ−
(6)
φ −1
and for an ellipsoid oriented with its major axis perpendicular to the direction of motion,
ref f ⊥ =
8c
3
1
2φ
φ2 −1
+
2φ2 −3
(φ2 −1)3/2
ln[φ +
p
φ2 − 1]
.
(7)
As one might expect intuitively, an ellipsoid sedimenting with its major axis parallel to the direction
of motion will sediment faster than an identical one whose major axis is perpendicular to the direction
of motion. Indeed this is reflected in the growth of the effective radii as a function of a (or φ) in
the parallel and perpendicular case. The effective radius of the perpendicular case is always greater
than that of the parallel, except for the case when φ = 1 (a sphere), when they are equal. This is
illustrated in Fig.5 by plotting the ratio,
p
p
ref f k
φ φ2 − 1 + (2φ2 − 3) ln[φ + φ2 − 1]
√
=
.
p
φ+ φ2 −1
ref f ⊥
−2φ φ2 − 1 + (2φ2 − 1) ln[ √ 2 ]
φ−
(8)
φ −1
From examination of Fig.1 we approximate φ = 5. Then, from an numerical evaluation of the
function plotted in Fig.5 we expect the effective radius of the ellipsoid falling parallel to be 74 %
of the perpendicular case. To obtain the terminal velocity of an ellipsoid we must appeal back to
Newton’s second law; plugging the effective radius back into the formula for the sedimentation rate
for a sphere is incorrect because that formula uses the same radius for the volume of the sphere. In
the ellipsoid case, the effective radius is not related to the volume; it is only applicable for calculating
the drag force. We recall that the volume of an ellipsoid is 43 πabc. Thus for the perpendicular case,
mz̈
mz̈
=
Fd + Fg
=
4
−6πηref f ⊥ ż + πabc(ρ1 − ρ0 )g.
3
(9)
5
(10)
Figure 5: Ratio of
ref f k
ref f ⊥
plotted as a function of φ. Note that its value is unity for φ = 1 (a sphere)
and is less than 1 for all φ > 1, as the effective radius of an ellipsoid is always greater in the horizontal
configuration than in the vertical configuration.
At terminal velocity z̈ = 0 and ż = vs⊥ and thus
vs⊥
=
2 (ρ1 − ρ0 )g abc
.
9
η
ref f ⊥
(11)
vsk
=
2 (ρ1 − ρ0 )g abc
.
9
η
ref f k
(12)
Similarly for the parallel case,
In stark contrast to the sedimentation rate of the sphere, the sedimentation rate of an ellipsoid
falling horizontally or perpendicularly goes as
1
ref f
. This should not be too surprising though, as
the effective radius is not related to the volume but to φ. Finally, it is interesting to write the ratio
of the sedimentation rates:
Since we saw for φ = 5 that
ref f k
vs⊥
=
.
vsk
ref f ⊥
rsk
rs⊥
(13)
= 0.74, Eq.13 tells us that this ellipsoid in the parallel orientation
will sediment 1.35 times faster than it will in the perpendicular orientation. We will apply this fact
in section 8 when discussing Paramecium sedimenting in different orientations.
Another extremely useful application of Stokes law is constructing barbell models to model the
angular reorientation of cells due to front-rear asymmetries. This approach will be investigated
concurrently with the discussion of the Paramecium models of gravitaxis to which it applies.
4
Theories of Negative Gravitaxis
Here we delineate mechanisms which have been proposed and tested to explain negative gravitaxis
in Paramecium.
6
4.1
Physiological: Electromotor Coupling
In order for Paramecium to respond to a stimulus physiologically, the stimulus must be large enough
to sense. In the case of the direction of gravity, one stimulus is the pressure of the cell cytoplasm on
the cell membrane. Machemer[4] approximates this pressure by appealing to the buoyant force in
Eq.4. Since we want to know the volume and the difference in densities, we can more accurately approximate the volume of a Paramecium as an ellipsoid. Using the same definition for the parameters
of an ellipsoid as we did in section 3, we set a = 100µm, b = c = 25µm, and find that the volume
of this ellipsoid is, V = 2.62 × 105 µm3 . The difference in density between Paramecium and water,
(ρ1 − ρ0 ) = 0.04g/ml. The buoyancy equation where F is the downward force on the cytoplasm
which is transmitted to the cytoplasm-membrane interface is
F = V (ρ1 − ρ0 )g.
(14)
Evaluating this we find the downward force the cytoplasm is 1.0 × 10−10 N . If we approximate this
force to be distributed over a surface of a circle of radius 25µm, the pressure is 0.053P a, which
Machemer rounds up to 0.1P a and reports to be of the same order of magnitude which plant cells
respond to. Their experiments showed this pressure was sufficient to be communicated through
the cell wall by way of depolarizing sensors on the front of the cell and hyperpolarizing sensors on
the rear of the cell. When the cell is oriented downward, the pressure on front increases and the
membrane becomes depolarized, reducing the swimming speed. When the cell swims upwards the
pressure increases on the rear, polarizing the membrane and increasing the swimming speed. This
orientation dependent velocity is known as gravikinesis and is purported to explain the phenomenon
of gravitaxis from a physiological point of view. However, an orientation dependent velocity does
not necessarily imply that cells will collect on the top of the container. Regardless of swimming
speed, cells tumble and change direction when they reach a barrier. Thus when a fast swimming cell
reaches the top, it will tumble and move off in another direction more slowly with some downward
component, negating gravitaxis.
An upward curvature in the motion of the cells will result in gravitaxis because non vertically
swimming cells are compelled to reorient, while upward swimming cells continue to swim upward.
This upward curvature can be predicted by physical models of the cells.
4.2
Physical: Back Heavy Hypothesis
The back heavy hypothesis[3][2] assumes that the density of the cell is not constant, but a density
gradient exists across the major axis of the cell. This can be modeled by using a barbell approximation; we apply Stokes’ law to two spheres connected by a massless rigid rod of length 2L, as shown
in Fig.6. We assume no hydrodynamic interaction between the spheres. To take into account the
density gradient, we make the densities of the spheres unequal (ρ1 > ρ2 ). Thus from Eq.5, their
respective sedimentation velocities will be,
vs 1
=
−
2 (ρ1 − ρ0 )g 2
r
9
η
7
(15)
r
θ
r
ρ2
2L
vs2
ρ1
η
vs1
Figure 6: The barbell model of a longitudinal density gradient consists of two spheres of equal radii
and different densities.
vs 2
=
−
2 (ρ2 − ρ0 )g 2
r
9
η
(16)
If both ends of the barbell fall with different velocities, from vector analysis we can write θ̇ as,
θ̇
=
vs 1 − v s 2
sin θ
L
(17)
and substituting in the individual sedimentation rates, we obtain the interesting fluid density independent result,
θ̇
=
2 gr2
(ρ1 − ρ2 ) sin θ.
9 ηL
(18)
This result implies that if this is indeed the mechanism which gives rise to gravitaxis, then changing
the density of the medium should have no effect on the reorientation rate, and thus on the gravitactic
behavior. Experiments have shown[8] the orientation rate of Paramecium swimming in heavy water
to decrease and become zero at a density of 1.10 g/ml, contradicting the expectation in this model
that the orientation rate would be density independent. However using heavy water has other
consequences besides increasing the density of the fluid. It readily diffuses across the cell membrane,
thereby increasing the cell density. Heavy water can also alter biological processes, calling into
question possible physiological effects of heavy water exposure. Ficoll 400, a polymer of sucrose is
better suited for this purpose because it is large molecule with a molecular weight of 400, 000 ±
100, 000 and does not penetrate the cell membrane. However, addition of Ficoll drastically increases
the viscosity of the solution which confounds our intent to change only the density of the fluid to
test this hypothesis.
4.3
Physical: Front-Rear Body Asymmetry
This hypothesis arises from a visual inspection of a Paramecium. The rear of the cell is observed
to be slightly wider than the front, giving the cell an asymmetrical shape. From a different barbell
8
r2
θ
r1
2L
ρ1
vs2
ρ1
vs1
Figure 7: The asymmetrical barbell model consists of two spheres of unequal radii and equal density.
model, originally proposed by Roberts[9], and then further developed[10], we can model reorientation
arising from front-rear asymmetries. Fig.7 shows a barbell consisting of two spheres of equal density
ρ1 with radii r1 , r2 , (r1 > r2 ) connected by a massless rigid rod of length 2L. We ignore the
hydrodynamic interaction between the two spheres. The angle of the rod from vertical is θ. We
begin our analysis by calculating the sedimentation rate of the center of mass of the system. From
a simple free body diagram we write out all the forces in the z direction longhand,
4
4
Fnetz = −6πηr1 ż − 6πηr2 ż + πr13 (ρ1 − ρ0 )g + πr23 (ρ1 − ρ0 )g
3
3
(19)
and at terminal velocity Fnet = 0 and ż = vs we obtain,
vs =
2 (ρ1 − ρ0 )g r13 + r23
.
9
η
r1 + r 2
Next we consider the same barbell falling horizontally at terminal velocity (θ =
(20)
π
2 ).
If we take
both spheres to fall at their respective vs1 , vs2 , from vector analysis, the orientation rate θ̇, in rad/s
is,
θ̇
=
θ̇
=
vs 1 − v s 2
sin θ,
L
2 (ρ1 − ρ0 )g 2
(r1 − r22 ) sin θ.
9
ηL
(21)
(22)
If we pick the following reasonable parameters to approximate a Paramecium barbell in water,
r1 = 20µm, r2 = 18µm and L = 150µm and taking the density of Parameicum to be 1.04 g/ml, we
obtain θ̇ = −2.53o /s. Thus with a front-rear asymmetry of 4µm, an orientation rate on the low end
of the same order of magnitude of that measured by Taneda and Miyata can be obtained.
4.4
Physical: Gravity Propulsion Hypothesis
This hypothesis[11] assumes that the center of gravity and center of propulsion of the cell do not
coincide. Rather, the center of propulsion is located closer to the front of the cell. Since the cell
9
gyrates as it swims, for a horizontally swimming cell, the front of the cell alternately points at a
small angle below the horizontal and then a small angle above as it completes one gyration. This
model states that a torque acts to reorient the cell upwards when the cell is in the upward phase of
gyration. However the origin of this torque is difficult to interpret.
5
Magnetic Field Gradient Levitation
5.1
Theory[12]
With the eventual goal of developing a new tool to further probe the gravitational response of
biological systems, we introduce here the capabilities and limitations of MFGL. MFGL is based on
our ability to generate a large enough magnetic field and field gradient to exert forces which are
close in magnitude to the gravitational forces on common diamagnetic materials. Diamagnetic forces
arise on the atomic scale from the repulsion of orbital electron currents from an external magnetic
field[13]. All materials are diamagnetic, however it is a very weak property and is only evident in
atoms or molecules when not overshadowed by ferromagnetic or paramagnetic properties. We denote
the magnetic susceptibility of a material as χ. For diamagnetic materials, χ < 0, for paramagnetic
materials, χ > 0. Common diamagnetic substances such as water and protein have χ on the order
of −10−6 . The potential energy of a diamagnetic substance in a gravitational field is[14],
U (r, z)
1
−mχρ B2 (r, z) + mgz,
2
=
where
χρ =
χ
.
ρ
(23)
(24)
Since
F = −∇U,
(25)
the force exerted on a diamagnetic substance in an external magnetic field in the presence of gravity
in the z direction is
F
Setting
dB
dz
=
mχρ B
dB
− mg.
dz
(26)
≡ B 0 , the forces on the object cancel in the z direction when,
BB 0
=
g
,
χρ
(27)
and it is stably levitated if the radial diamagnetic forces are restoring (see Appendix).
5.2
Apparatus
To generate sufficient BB 0 necessary for MFGL, we employ a superconducting solenoid cooled by
liquid helium, sketched in Fig.8. The room temperature bore of the solenoid has a radius of 12mm.
Visualization in the bore is achieved by way of a boroscope which moves vertically with the sample
10
+
-
room temperature
bore
z
cooled
region
g
x
y
superconducting
solenoid
sample holder
boroscope
CCD camera
Figure 8: A schematic of the solenoid apparatus and its orientation is shown. The boroscope can
be moved in the z direction into and out of the bore.
Magnetic Field vs Axial Position
9
8
7
B (T)
6
5
4
3
2
1
0
-6
-4
-2
0
2
Axial position (cm)
4
6
Figure 9: This plot shows the magnetic field in the bore as a function of axial position.
11
inside the bore. The boroscope is connected to a CCD camera. Movies and still images can be
recorded. Fig.9 shows the magnetic field as a function of axial position within the bore. Position
zero is the center of the solenoid. Negative position indicates the position below the center. Fig.10
BB’ vs Axial Position
15
BB’ (T^2/cm)
10
5
0
-5
-10
-15
-6
-4
-2
0
2
Axial position (cm)
4
6
Figure 10: This plot shows BB 0 in the bore as a function of axial position.
shows the value of BB 0 as a function of axial position within the bore. The maximum values of BB 0
occur at ±1.5cm. Fig.11 shows both radial and vertical stability parameters K r , and Kz , both of
which are derived in the Appendix (Eqs. 59, 60). Stability exists in the region where both K r and
Kz are positive. From examination of the plotted data we see this condition occurs below the center
in the range [−2.1, −1.5]cm and above the center in the range [1.5, 2.1]cm. It is in this region that
BB 0 is maximum, giving the greatest force in the stable region.
6
Inhomogeneous objects
For inhomogeneous objects like Paramecium, made up of protein, lipids and water, their effective
magnetic susceptibility is non uniform within their volume. However, an overall susceptibility can be
prescribed to a Paramecium by forming a linear combination of the susceptibilities of the constituent
components. Letting α, β and γ be scalars representing the mass fraction of each component then,
χParamecium
=
αχwater + βχprotein + γχlipid ,
(28)
with
α+β+γ
12
=
1.
(29)
Stability Parameters, K_z, K_r
35
K_z
K_r
30
25
20
(T/cm)^2
15
10
5
0
-5
-10
-15
-20
-6
-4
-2
0
Position (cm)
2
4
6
Figure 11: Both radial and vertical stability functions are plotted as derived in the Appendix.
Stability exists in the region where both functions are positive.
Difficulties arise when measuring this parameter for individual cells because isolating them outside
of their environment causes rapid evaporation of water from within the cell, altering their susceptibility. Since the eventual goal of measuring their susceptibility is to manipulate their swimming,
determination of their susceptibility in water is desired. This consideration motivates the description
of the behavior of a Paramecium with specific susceptibility χρP , in a fluid of specific susceptibility
χ ρf .
7
Magnetic susceptibility of a Paramecium immersed in a
fluid[12]
In the absence of a magnetic field, the net force that a Paramecium of volume V feels in a fluid due
to buoyancy is,
Fbuoyancy = (ρP − ρf )V g.
(30)
If we place this system into a solenoid, both the fluid and the Paramecium experience diamagnetic
forces. We can treat the diamagnetic forces on the fluid and the Paramecium together by replacing
the susceptibility in Eq. (26) with the difference between the susceptibilities, 4χ = χ P − χf . Also
replacing the gravitational force with the buoyant force writing the susceptibility in terms of χ ρ ,
13
and taking into account the velocity dependent drag force, we arrive at
Fnet
=
[ρP (χρP BB 0 − g) − ρf (χρf BB 0 − g)]V − 6πηrż.
(31)
Thus, BB 0 gives us a convenient knob to adjust the net force on any given Paramecium swimming in
a fluid. However, the efficacy of this knob depends not only on our range of achievable BB 0 but also
on the difference of the densities and susceptibilities of the Paramecium and their liquid medium.
If Paramecium has a susceptibility of magnitude less than that of the liquid they swim in, their
effective susceptibility, ∆χ becomes positive and the cells become effectively paramagnetic! Since
paramagnetic materials cannot be stably levitated, this radial instability may cause problems when
attempting to manipulate them inside the solenoid. We have observed this effect in our magnet.
Since our goal is to measure χρP and we can readily measure the terminal velocity of immobilized
cells, from Eq.31 we see that we need to simply fill in the parameters for the densities of the fluid
and Paramecium as well as the susceptibility of the fluid and the coefficient of the drag force. It is
these requirements that motivate the measurements of the next section.
8
Measured Parameters
We have discussed in the previous section the parameters we wish to measure. Some of these
parameters can be measured by examining the sedimentation rate of immobilized specimens. This
parameter is an important characteristic of Paramecium, as it gives an indication of the sedimenting
velocity which is always superimposed on the velocity of a swimming cell. Studying the dynamics
of sedimenting cells allows us to measure their density as well and the effect of viscosity and density
on the terminal velocity.
Kuroda and Kamiya[1] have measured the density of Paramecium to be 1.04g/ml by placing
the cells in graded density solutions and noting the region in which they collect. We performed an
independent measurement of this parameter by introducing Paramecium into water and four Ficoll
solutions with densities above and below 1.04g/ml.
We rewrite equation 5 here because of it’s central importance in this experiment.
vs
=
2 (ρ1 − ρ0 )g 2
r .
9
η
(32)
As we increase the density of Ficoll, ρ0 , the viscosity, η changes. Thus there exists a function
η(ρ0 ). This function is known and is reported by Dryer and Late[15] in plot form. We fitted a
third order polynomial to the published curve and it is shown in Fig.12. By measuring the terminal
velocity of the cells as a function of Ficoll solution density we hope to measure the density of the
cells as well as fit a curve from the Stokes’ law prediction.
Four samples of Ficoll solution of different densities, and one water solution were prepared to
observe sedimentation. The density of the solutions were determined by measuring the change in
mass by removing of 200µL of fluid. This measurement was repeated 5 times and the mean of these
measurements is the reported value. The biggest challenge in measuring the sedimentation rate of
14
Viscosity of Ficoll Solution vs Density
0.035
Viscosity (kg/m*s)
0.03
0.025
0.02
0.015
0.01
0.005
0
1
1.02
1.04
Density (g/ml)
1.06
1.08
Figure 12: Ficoll viscosity as a function of density, adapted from plot in Dryer and Late[15]
.
the cells was circulating currents within the fluid samples. The curl of these circulating currents was
always perpendicular to gravity, seriously affecting sedimentation rate measurements. The curl of
the currents was never observed parallel to gravity. The currents were observable because bacteria
(observable as the tiny dots in Fig.1) with diameters < 20µm flowed along with the currents. To
minimize these currents a number of techniques were employed. A long sedimentation chamber
was fabricated with dimensions 13.8cm × 1.3cm × 0.48cm, with the thought that the circulating
currents would dissipate by the time the cells reached the bottom of the chamber where they could
be observed. The currents abated slightly but not completely. From observation of the motion of
the bacteria, we observed that the currents were minimized in the centers of the containers ? in the
“eye of the storm”. Data was only taken in this region. The trade off to restricting our field of view
to this region was a high rejection rate of cells caught up in the circulating currents and a lower
number of velocities measured than initially anticipated.
We hoped that with increasing density, the increased viscosity would reduce the magnitude of
circulating currents. Observing a Ficoll solution with the naked eye at a density of about 1.05g/ml
or higher one notices that the movement immediately after pouring a sample is non existent when
compared with that of water. However, under magnification these currents simply persist at much
lower velocities. Since the velocities of sedimenting Paramecium are also greatly reduced as such
viscosities, the circulating currents still interfere with measurements of terminal velocity.
The terminal velocities plotted in Fig.13 were observed both under a stereoscope using a 45 o angle
15
mirror to view the sedimenting Paramecium horizontally, and also with a CCD camera attached to a
VCR. Length scales were calibrated by scribe lines 1mm apart etched onto the sample holders. Most
velocities were measured by measuring the time for a cell to traverse 1mm. A few were observed
for longer distances. For two of the Ficoll samples, 1.038g/ml and 1.069g/ml no observable velocity
could be seen in regions were circulation was absent. For the 1.038g/ml This could be attributed to
the densities of the Paramecium and Ficoll becoming very close and thus achieving neutral buoyancy.
The lack of observable velocities at 1.069g/ml is not as easily explained; as we shall see Stokes’ law
predicts the velocity to be near that of the cells observed at a Ficoll density of 1.058g/ml. Perhaps
the increased viscosity caused incomplete mixing of the injected Paramecium-filled water drops,
leading to a situation where the cells were caught in some viscosity interface.
Terminal Velocity vs Ficoll Density
0.02
Terminal Velocity (mm/s)
0
-0.02
-0.04
-0.06
-0.08
-0.1
Experiement
1.15*(Stokes’ Law, r=0.040mm)
Stokes’ Law, r=0.040mm
0.85*(Stokes’ Law, r=0.040mm)
-0.12
-0.14
-0.16
1
1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09
Ficoll Density (g/ml)
Figure 13: Plotted are the experimental sedimentation rate data and Stokes’ law prediction as a
function of Ficoll density. The Stokes’ law prediction is not linear in density because the viscosity
increases as density increases.
The terminal velocities of 25 Paramecium were observed in water with a measured density of
0.998±0.003g/ml, 5 were observed in Ficoll solutions of density 1.018±0.002g/ml, 13 were observed
in Ficoll solutions of 1.058 ± 0.008g/ml. For the 1.038 ± 0.002g/ml and 1.069 ± 0.008g/ml samples
at least 10 cells were observed moving with the currents.
The possibility for systematic error in the density measurements arose when a slight residue was
observed on the outside of the pipette after it was withdrawn from the sample. By lightly brushing
16
the pipette against the container, the residue was transferred back to the container, mitigating this
source of systematic error. Because this was possible, the systematic error was neglected. The
reported error in the density measurement is determined by the sample standard deviation of the
mean of five density measurements.
The error associated with the measurements of the terminal velocity was calculated by the sample
standard deviation of the mean of the velocity measurements for each Ficoll density. The same error
in terminal velocity was assigned to the two cases where no terminal velocity was measured as the
1.058g/ml case, since the conditions and therefore random error was most similar.
We have plotted 3 curves on top of the data in Fig.13. These curves are the Stokes’ law predictions taking into account the viscosity increasing with Ficoll concentration. From these plots we
see how the increased viscosity and increased buoyancy work together or compete in affecting the
sedimentation rate as the density is increased. For Ficoll densities less than that of the Paramcium,
increasing the density serves to reduce the cell speed on two fronts:
• The buoyant force increases which acts in the opposite direction of the velocity of the cell.
• The increased viscous drag also acts in the opposite direction of the velocity.
For Ficoll densities greater than that of water, the two forces compete.
• The buoyant force upward continues to increase but is now in the same direction of the velocity.
• The viscous force still acts in the opposite direction of motion which has reversed.
The curve representing Stokes’ Law with radius 0.040mm was chosen to best fit the data. It
is a reasonable choice, as the diameter, 0.080mm, is less than the 0.2mm length of a cell and
greater than the 0.05mm width of a cell. The ratio of the terminal velocities of the two curves
above and below is 1.35, representing the ratio of the terminal velocity of an ellipsoid moving in
the parallel orientation with that of an ellipsoid moving in the perpendicular orientation. They are
centered around the Stokes’ Law prediction in order to show the spread in terminal velocities due
to orientational differences. The experimental data show reasonable agreement with the spread of
terminal velocity differences due to non uniform orientation of the cells.
From examination of the data, the previously measured value of ρP = 1.04g/ml seems reasonable. While this study is useful to confirm previous measurements of ρP , its main utility is in
understanding how the terminal velocities of Paramecium are affected by changing the density of
fluids whose viscosity is highly density dependent. It is this technique, measuring the terminal velocities of Paramecium in Ficoll which are affected by BB 0 , that has been used to make preliminary
measurements of χρP . Therefore, understanding the response of terminal velocity to variations in
density and viscosity is important in order to isolate diamagnetic forces.
8.1
Orientation Rate of an Immobilized Specimen
From video observation of one immobilized sedimenting specimen in water, the angle from vertical,
θ was recorded as a function of time in 0.5 second intervals. An error of 2o was the estimated
uncertainty of the angle measurements made from the video. This orientation angle is plotted as
a function of time in Fig. 14. We measured a mean orientation rate of −3.1o /s over a 17 second
17
Orientation of Sedimenting Immobilized Paramecium vs Time
100
theta (degrees)
90
80
70
60
50
40
30
0
2
4
6
8
10
time (s)
12
14
16
18
Figure 14: Orientation rate of a single immobilized cell as a function of time. The mean value of θ̇
is −3.1o /second.
18
observation period. This individual cell cannot be taken as representative of the orientation rate
of all cells. However, the cells we observed which oriented, including this specimen, did show a
front-rear shape asymmetry. Because this specimen was immobilized, the reorientation could only
be due to the shape and density distribution of the cell. From our barbell models, Eqs.18 and 22,
we expect the reorientation rate, θ̇ to depend on the orientation angle θ like sin θ. The effect of this
prediction on a plot of θ vs. time is that the slope of this curve approaches zero as time goes to
infinity. Since our window of observation was limited to 17 seconds, this effect is difficult to see in
its entirety. However, one can observe the slope of the data decreasing near the final two seconds of
observation.
Taneda and Miyata[16] measured the orientation rate of swimming cells as their paths curved
towards the top of the container. They observed orientation rates from near zero to about −10 o /s.
Our measurement of an immobilized specimen confirms that there exists a purely physical component
to the orientation of the cells, and to negative gravitaxis. Indeed, comparing this with our result
from the barbell model, Fig.6, of 2.53o /s indicates that shape asymmetry can contribute significantly
to reorientation.
8.2
Magnetic Susceptibility of Paramecium
Individual Paramecium immersed in Ficoll solutions have recently been levitated in our magnet with
the goal of measuring their magnetic susceptibility. By observing the vertical terminal velocities of
immobilized cells in Ficoll solutions, the direction and magnitude of the net force on the cells minus
the viscous drag can be approximated from Stokes’ Law. If we know the terminal velocites as
function of BB 0 (the axial position in the solenoid) we can recover χρP by fitting a curve dictated
by Eq. 31.
One challenge in making this measurement is thatParamecium in Ficoll act like paramagnetic
substances, preventing stable levitation. Our experiments have indeed been complicated by the radial
forces which the Paramecium feel, pushing them to the sides of the sample cuvette. Nonetheless, we
have succeeded in observing cells moving up and down in the magnet by adjusting the axial position
within the bore and thus the BB 0 experienced by the cells. A complete analysis of this data is
underway.
9
Conclusion
From our recent success observing the changing terminal velocities arising from variable buoyancy in
Paramecium due to MFGL, we’ve seen the applicability of the topics discussed in this work. Applying
the consequences of low Reynolds number hydrodynamics has proved useful in understanding the
drag forces. Studying the consequences of the increasing viscosity of Ficoll with concentration as
well as confirming the density of Paramecium has increased our familiarity with the methods and
materials related to the experiment, as well as bolstered our confidence about reported values.
Observing reorientation of immobilized cells has indicated to us a physical component to gravitaxis.
19
Finally, studying the theories which explain negative gravitaxis has familiarized us with the current
state of research, helping us to plan for future experiments to finally settle this question.
10
Appendix A: Stable levitation
What follows is a discussion of the requirements for stable diamagnetic levitation in a magnetic field,
following that of Berry and Geim[17].
Any particle in stable equilibrium will experience a restoring force that opposes a slight displacement. A spring is the simplest example of this situation; any displacement x from equilibrium
results in a force,
F (x) = −kx
(33)
where the minus sign indicates that the direction of the force is opposite to the displacement, x.
Writing the potential energy of the spring,
U (x) =
we see that the shape is a parabola with
d2 U
dx2
1 2
kx ,
2
(34)
> 0. This parabola is the prototypical potential well.
Extending this feature to three dimensions we guarantee stability by requiring the Laplacian of the
potential energy to be positive,
∇2 U > 0,
(35)
and plugging in Eq.23 we find that the gravitational term disappears and,
1
(36)
∇2 U = −mχρ ∇B2 (r, z),
2
Let’s for the moment assume that we are trying to levitate a paramagnet, so χ ρ > 0. Therefore, to
ensure the left hand side of this equation is positive we require,
∇2 B2 (r, z) < 0,
(37)
to cancel the negative sign on the right hand side. Before proceeding, let’s think about what the
magnetic field in the bore of a solenoid looks like. Of course, it must satisfy Maxwell’s equation
forbidding magnetic monopoles,
∇ · B = 0,
(38)
and since there are no currents inside the bore, Ampere’s law reduces to
∇ × B = 0.
(39)
These two conditions will soon be of great utility. It is instructive to initially consider the magnetic
field in Cartesian coordinates, x, y and z. Breaking up B2 into these three components and then
applying the chain rule we see,
∇2 B2 (x, y, z)
=
=
∇2 (Bx 2 + By 2 + Bz 2 )
2
2
(40)
2
2[Bx ∇ Bx + By ∇ By + Bz ∇ Bz
+(∇Bx )2 + (∇By )2 + (∇Bz )2 ].
20
(41)
(42)
Since B is divergenceless and without curl, all the terms which have a factor ∇ 2 Bi go to zero.
Therefore, we obtain,
∇2 B2 (x, y, z)
=
2[(∇Bx )2 + (∇By )2 + (∇Bz )2 ].
(43)
From this result we see that that because all the terms on the right hand side are squared,
∇2 B2 (x, y, z) > 0,
(44)
contradicting Eq. 37. Therefore, for stable levitation of paramagnets is impossible. However, if χ ρ
is negative (the substance is diamagnetic), stable levitation becomes possible.
We now turn our attention to deriving the conditions for stable levitation of diamagnetic substances. We can express the horizontal and radial stability conditions separately. The condition for
vertical stability is,
∂2 2
B (x, y, z) > 0,
(45)
∂z 2
and the horizontal stability conditions (which are equivalent, as B is symmetric about the z axis)
are
∂2 2
B (x, y, z) > 0
∂x2
2
∂
B2 (x, y, z) > 0.
∂y 2
(46)
(47)
Since we are only concerning ourselves with the magnetic field inside the bore, we recall Eq.39.
Thus we can define a magnetic scalar potential, Φ(x, y, z) with the usual definition, yet unusual
appearance of B,
B(x, y, z) = ∇Φ(x, y, z).
(48)
We can then expand Φ(x, y, z) in a two dimensional Taylor series to second order about any point
on the vertical axis, (0, 0, z). Letting Φ(0, 0, z) = φ
Φ(x, y, z0 + z)
=
φ + [φx x + φy y + φz z]
1
+ [φxx x2 + φyy y 2 + 2φxy xy] + . . .
2!
(49)
(50)
Happily, we can simplify this expression considerably. All three first order terms disappear because
the magnetic field in the x or y direction is zero at the center. The second order cross term also
disappears from symmetry considerations. Simplifying,
φ(x, y, z) = φ(0, 0, z) +
1
[φxx (0, 0, z0 )x2 + φyy (0, 0, z0 )y 2 ] + . . .
2!
(51)
The second order term can be expressed in terms of z. Since Φ is a scalar potential (it satisfies the
Laplace equation)
∇2 Φ = 0
21
(52)
writing out the components,
Φxx + Φyy + Φzz = 0
(53)
Φzz = −Φxx − Φyy ,
(54)
and since we have symmetry in the x and y directions,
Φxx = Φyy
(55)
Φzz = −2Φxx
(56)
so,
Plugging this into the second order term of the expansion we obtain the expansion only in terms of
the potential on the axis, along with the radial displacement squared x 2 + y 2 ,
1
φ(x, y, z) = φ(0, 0, z) + (x2 + y 2 )φzz (0, 0, z) + . . .
4
(57)
Taking the gradient, and squaring it we obtain B 2 (r, z) For the sake of clarity we write B(0, z) = B.
Thus,
1
(58)
B2 (r, z) = B 2 + 2BB 0 z + z 2 [BB 00 + B 02 ] + r2 [B 02 − 2BB 00 ]
4
At this point we need only ensure that the coefficients of the z 2 and r 2 of B 2 (r, z) are positive,
enforcing the stability conditions, Eqs. 45, 47. We denote these parameters which carry units of
(Gauss/cm)2 as Kz and Kr , respectively. For radial stability,
Kr = B(0, z)B 00 + B 02 > 0,
(59)
Kz = B 02 − 2B(0, z)B 00 > 0.
(60)
and for vertical stability,
By plugging in our measured values of B, B 0 and B 00 along the axis of our magnet, we can plot Kz
and Kr as functions of position and find stable levitation region(s), Fig.11.
References
[1] N Kamiya K. Kuroda. Propulsive force of paramecium as revealed by the video centrifuge
microscope. Experimental Cell Research, 184:268–272, 1989.
[2] M. Verworn. Psychophysiologische protistenstudien. Gustav Ficher, Jena, pages 1–219, 1889.
[3] H. Machemer et al. Graviperception in unicellular organisms: a comparative behavioral study
under short-term microgravity. Microgravity Science and Technology, 1992.
[4] H. Machemer. A theory of gravikinesis in paramecium. Advanced Space Research, 17(6/7):11–20,
1996.
22
[5] H. Brenner J. Happel. Low Reynolds Number Hydrodynamics. Noordhoff International Publishing, 1973.
[6] E. M. Purcell. Life at low reynolds number. Americal Journal of Physics, 45:3–11, 1977.
[7] H. Machemer et al. Gravikinesis in paramecium: Theory and isolation of a physiological response
to the natural gravity vectory. Journal of Comparative Physiology A, 168:1–12, 1991.
[8] S. Miyata K. Taneda. Analysis of motile tracks of paramecium under gravity field. Comparative
Biochemical Physiology, 111A(4):676–680, 1995.
[9] A.M. Roberts. Geotaxis in motile micro-organisms. Journal of Experimental Biology, 53:687–
699, 1970.
[10] F. M. Deacon A.M. Roberts. Gravitaxis in motile micro-organisms: the role of fore-aft body
asymmetry. Journal of Fluid Mechanics, 452:405–423, 2002.
[11] T. L. Jahn H. Winet. Geotaxis in protozoa i. a propulsion-gravity model for tetrahymena
(cilliata). Journal of Theoretical Biology, 46:449–465, 1974.
[12] J.M Valles K. Guevorkian. Low gravity on earth by magnetic levitation of biological material.
Not yet published.
[13] M.D Simon et al. Diamagnetically stabilized magnet levitation. American Journal of Physics,
69(6), June 2001.
[14] M.A Weilert et al. Magnetic levitation and noncoalescence of liquid helium. Physical Review
Letters, 77(23), December 1996.
[15] G.F Late R.L Dryer. Experimental Biochemistry. Oxford University Press, 1989.
[16] S. Miyata K. Taneda. Analysis of motile tracks of paramecium under gravity field. Comparative
Biochemical Physiology, 111A(4):673–680, 1995.
[17] A.K. Geim M.V. Berry. Of flying frogs and levitrons. European Journal of Physics, 18:307–313,
1997.
11
Acknowledgements
Many thanks to my thesis advisor Prof. Jim Valles for his contagious enthusiasm about this project
in particular and science in general. In addition to being a great advisor, he has been a terrific
professor. Also, many thanks to Karine Guevorkian for her patience in answering all of my questions
and for her incredibly hard work designing and maintaining so much of the apparatus we use in the
lab. Thanks to Andres Morey for his camaraderie and always enjoyable lunch conversations at Luis’
Restaurant, South Station, and Chinatown.
And of course, thanks to my family for supporting me in my academic endeavors.
23